Periodic twisted cohomology and T-duality

Ulric h Bunk e Thomas Sch ic k Markus Spitzwec k PERIODIC TWISTED COHOMOLOGY AND T-DUALITY Ulrich Bunke Mathematische F akult¨ at, Universit¨ at Regensbur g, 9 3 040 Regensbur g , Germany. E-mail : ulrich .bunk e@mathematik.uni-regensburg.de Url : http:/ /www. mathematik.uni-regensburg.de/Bunke/index.html Thomas Schick Mathematisches Institut, Geo rg-August- Universit¨ at G¨ ottingen, Buns e ns tr. 3-5, 37073 G¨ ottingen, Germany. E-mail : schick @uni- math.gwdg.de Url : http:/ /www. uni-math.gwdg.de/schick Markus Spitzwe c k Mathematische F akult¨ at, Universit¨ at Regensbur g, 9 3 040 Regensbur g , GErmany. E-mail : markus .spit zweck@mathematik.uni-regensburg.de 2000 Mathematics Subj e ct C lassiﬁc ation . — 55N30,4 6M20,14 A20. Key wor ds and phr ases . — t wisted cohomolog y , V er dier dualit y , top ological stacks, sheaf theory , T- duality , orbispa c es, unbounded derived category , twisted de Rham coho mology . PERIODI C TWISTED COHOMOLOG Y AND T-DUALITY Ulric h Bunk e, Thomas Schic k, Markus S pitzwe c k Abs tr act . — Using the diﬀerentiable structure, twisted 2-per io dic de Rham coho- mo ology is well known, and showing up as the ta r get of Chern characters for t wisted K-theory . The main motiv ation of this work is a top olog ical in terpr etation of t wo- per io dic twisted de Rham coho mology whic h is generaliza ble to arbitrary top ologica l spaces and a t the same time to arbitra ry co eﬃcients. T o this end we dev elop a sheaf theory in the context of lo cally compact top ologica l stacks with emphasis on: – the construction of the sheaf theory op er ations in un b o unded derived categories – elemen ts of V erdier duality – and integration. The main result is the co nstruction of a functorial p erio dizatio n asso cia ted to a U (1)- gerb e. As a application we verify the T -duality isomorphis m in p erio dic twisted co homol- ogy and in p erio dic twisted or bispace coho mo logy . R ´ esum ´ e (Cohomolog ie p´ erio dique tordue et T-dualit´ e) La cohomology de de Rham tor due (p erio dique av ec p ´ erio de 2) est une construction bien connue, et elle est imp or tante c o mme co do maine d’un charact` ere de Chern p our la K - theorie tordue. La motiv a tion principale de notre livre est une interpretation top olog ique de la cohomolog y de de Rham tordue, une interpretation avec g´ eneraliza tions ´ a des espaces arbitrair e, et aux c o´ eﬃcients quelconque. ` A ce but, nous developpons une th ´ eorie des fais ceaux s ur des stacks to po logiques lo calement c ompacts. Nous a ppuyons – la constr uctio n des oper ations de la th´ eo rie des faisceaux dans le s cat´ egor ies deriv´ ees no n-b orn´ ees – ´ elemen ts de la dualit´ e de V erdier – e t integration. Le r e s ultat principa l est la cons truction d’une p erio dizatio n fonctor ielle asso ci´ e a une U (1)-gerb e. Une a pplica tion est la veriﬁcation d’un isomo r phisme de T-dualit ´ e p our la coho - mologie p erio dique tordue et la co ho mologie p erio dique tor due des orbi-e spaces. CONTENTS 1. In tro duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Perio dic twisted cohomolo gy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. T -duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3. Duality for sheaves on lo cally compact stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Gerb es and p erio dization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1. Sheav es o n the lo cally compa c t site of a s ta ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2.2. Algebra ic structur es on the cohomo lo gy of a g erb e . . . . . . . . . . . . . . . . . . . . . . . 13 2.3. Identiﬁcation of the transfor mation D G in the s mo oth case . . . . . . . . . . . . . . 17 2.4. Two-per io dization — up to isomor phis m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5. Calculatio ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3. F unctorial p e ri o dization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7 3.1. Flabby re s olutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2. A mo del for the push-forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3. Zig-z ag diagr ams and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4. The functorial p erio diza tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5. Pro p erties of the p erio diza tion functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 3.6. Perio dicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4. T -dualit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 4.1. The universal T -duality diag ram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 4.2. T -duality and p er io dization diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3. Twisted coho mology and the T -duality tra nsformation . . . . . . . . . . . . . . . . . . . 61 5. Orbispaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.1. Twisted p erio dic delo calized co homology of orbispa ces . . . . . . . . . . . . . . . . . . . 6 5 5.2. The T -duality tra nsformation in twisted p er io dic delo ca lized coho mology 67 5.3. The geo metry of T -duality diag rams over or bis paces . . . . . . . . . . . . . . . . . . . . . 69 5.4. The T -duality tra ns formation in twisted p erio dic delo ca lized coho mology is an is o morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0 vi CONTENTS 6. V erdier duality for locall y compact stac ks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 6.1. Elements of the theory of stacks on T op a nd shea f theory . . . . . . . . . . . . . . . . 75 6.2. T ensor pro ducts and the pro jection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3. V erdier duality for lo ca lly compa ct stacks in detail . . . . . . . . . . . . . . . . . . . . . . . 96 6.4. The integration map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 1 6.5. Op era tions with unbounded der ived categ ories . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.6. Extended sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 CHAPTER 1 INTRODUCTION 1.1. P erio di c t wis ted cohomo l ogy 1.1.1. — The twisted de Rham cohomo logy H dR ( M , ω ) of a manifold M equipp ed with a clos ed three form ω ∈ Ω 3 ( M ) is the tw o-p erio dic coho mo logy of the complex (1.1.1) Ω( M , ω ) per : · · · → Ω ev ( M ) d ω → Ω odd ( M ) d ω → Ω ev ( M ) → . . . , where d ω := d dR + ω is the sum of the de Rham diﬀerential and the op eration of taking the wedge pro duct with the form ω . The tw o -p erio dic twisted de Rham cohomolog y is interesting as the targ et of the Chern character from twisted K - theo ry [ AS04 ], [ MS03 ], [ BCM02 ], or as a cohomolo g y theory which admits a T - dua lity isomorphism [ BE M 04 ], [ BS0 5 ]. 1.1.2. — In [ BSS07 ] we developed a sheaf theory for smo oth sta cks. Let f : G → X be a gerb e with band U (1) over a smo o th stack X , and consider a c losed three-for m ω ∈ Ω 3 X ( X ) whic h represents the image of the Dixmier- Douady cla ss o f the g erb e G → X in de Rham coho mology . The main result of [ BSS07 ] states that there ex ists an iso morphism (1.1.2) Rf ∗ f ∗ R X ∼ ← − Ω X [[ z ]] ω in the b ounded b elow derived categ ory D + ( Sh Ab X ) of sheaves of abelian groups on X . Here R X denotes the consta nt s heaf with v alue R on X . F ur thermore, Ω X [[ z ]] ω is the shea f of formal p ower series of smo oth forms on X , where deg( z ) = 2, and its diﬀerential is given by d ω := d dR + ω d dz . The isomor phism is not canonical, but depe nds on the choice of a connection on the gerb e G with characteris tic form ω . 1.1.3. — The co mplex (1.1 .1) can b e deﬁned for a smo oth stack X equipp e d with a three-form ω ∈ Ω 3 X ( X ). It is the c o mplex o f g lobal sectio ns o f a shea f of tw o-p erio dic complexes Ω X,ω ,per on X . The co mplex of sheaves Ω X [[ z ]] ω is no t tw o-p erio dic. The r elation b etw een Ω X [[ z ]] ω and Ω X,ω ,per has b een discussed in [ BSS07 , 1.3.23]. 2 CHAPTER 1. INTRODUCTION Consider the diag ram (1.1.3) D : Ω( X )[[ z ]] ω d dz ← Ω( X )[[ z ]] ω d dz ← Ω( X )[[ z ]] ω d dz ← . . . . Then there exis ts an iso morphism (1.1.4) Ω X,ω ,per ∼ = holim D . 1.1.4. — As mentioned above, the is o morphism (1.1.2) depends on the choice of a connection o n the gerb e G . Mor eov er, the diagram D dep ends on these choices via ω . In or der to c o nstruct a natural t wo-per io dic cohomolog y one m ust ﬁnd a natural replacement o f the op eration d dz which a cts on the left-hand side Rf ∗ f ∗ R X of (1.1.2). It is the ﬁrst g o al of this pap er to carry this out pro p e r ly . 1.1.5. — O ne can do this co ns truction in the fra mework o f smo oth stacks develop e d in [ BSS07 ]. But for the pres ent pa pe r we c ho o se the setting of topo logical stacks. Only in Subsection 2.3 we w ork in smo oth stacks and dis c uss the connection with [ BSS07 ]. In Section 6 we develop some asp ects of the theory of lo cally compac t stacks and the sheaf theory in this context. F or the purp os e of this introductio n we freely use no tions a nd co nstructions from this theory . W e ho pe that the ideas ar e understandable b y ana logy with the usual case of sheaf theo ry o n lo cally compact spaces. 1.1.6. — Let G → X b e a U (1)-banded gerb e ov er a lo ca lly compact stack. The main ob ject o f the present pap er is a p erio dization functor P G : D + ( Sh Ab X ) → D ( Sh Ab X ) which is functoria l in G → X , and where D + ( Sh Ab X ) and D ( Sh Ab X ) denote the bo unded b elow a nd un b o unded derived categories of sheav es o f ab elian gr oups on the site X o f the stack X . A simple co nstruction of the isomorphism class o f P G ( F ) is given in Deﬁnition 2 .4.2. The functoria l version is muc h more complicated. Its construction is co mpleted in Deﬁnition 3 .4.5. 1.1.7. — Let us sketc h the construction of P G . Recall that gerb es with band U (1) ov er a lo cally co mpact stack Y are classiﬁed by H 3 ( Y ; Z ), and a utomorphisms of a given U (1)-gerb e are classiﬁed by H 2 ( Y ; Z ) [ Hei05 ]. W e co ns ider the diagra m T 2 × G p   % % K K K K K K K K K K u / / T 2 × G p   y y s s s s s s s s s s G f % % L L L L L L L L L L L L T 2 × X   G f y y r r r r r r r r r r r r X , where the automo r phism u of gerb es ov er T 2 × X is class iﬁed by or T 2 × 1 ∈ H 2 ( T 2 × X ; Z ), and where or T 2 denotes the orientation cla ss of the t wo-torus. W e deﬁne a 1.1. PERIODIC TWISTED COHOMOLOGY 3 natural transforma tion D : Rf ∗ f ∗ → R f ∗ f ∗ : D + ( Sh Ab X ) → D + ( Sh Ab X ) of degr ee − 2 as the comp os itio n D : Rf ∗ f ∗ units → Rf ∗ Rp ∗ Ru ∗ u ∗ p ∗ f ∗ f pu = f p → Rf ∗ Rp ∗ p ∗ f ∗ R p → R f ∗ f ∗ , where R p : R p ∗ p ∗ → id is the int egra tion ma p of the o riented T 2 -bundle T 2 × G → G . F or F ∈ D + ( Sh Ab X ) we form the diag ram S G ( F ) : Rf ∗ f ∗ ( F ) D ← R f ∗ f ∗ ( F )[2] D ← R f ∗ f ∗ ( F )[4] D ← . . . in D ( Sh Ab X ). Deﬁnition 1.1.5 . — We deﬁne the p erio dization P G ( F ) ∈ D ( Sh Ab X ) of F by P G ( F ) := holim S G ( F ) ∈ D ( Sh Ab X ) . Note that this intro duction is meant as a sketc h. In pa r ticular, one ha s to b e aware of the fact tha t the notion of h olim in a triag ulated categ ory is a mb iguous a nd has to b e used with great care, as will b e explained b elow and in the b o dy of the pa p er . A t present, the ab ov e deﬁnition only ﬁx e s the isomo rphism cla ss o f P G ( F ). 1.1.8. — The same construction can b e applied in the ca se o f smo o th stacks X . It is an immediate consequence of Theor em 2.3 .2 that there exists an isomo rphism of the diagrams S G ( R X ) a nd D (s e e (1.1.3)). Equation (1.1.4) implies the following result. Cor ol lar y 1.1.6 . — If X is a smo oth m anifold, then ther e exists an isomorphism P G ( R X ) ∼ = Ω X,ω ,per in D ( Sh Ab X ) . In p articular we have an isomorphism of two-p erio dic c ohomolo gy gr oups H ∗ dR ( X, ω ) ∼ = H ∗ ( X ; P G ( R X )) . The existence o f this is o morphism play ed the role of a design c r iterion for the construction of the p erio dization functor P G . 1.1.9. — The op era tio n D : R f ∗ f ∗ ( F ) → R f ∗ f ∗ ( F ) is a well-deﬁned mo r phism in the derived catego r y . In particula r, we g et a well-deﬁned diag ram S G ( F ) ∈ D ( Sh Ab X ) N op , where we consider the o rdered set N as a categor y . This determines the isomor phism class o f the ob ject P G ( F ) ∈ D ( Sh Ab X ). W e actually wan t to deﬁne a p erio dization functor P G : D + ( Sh Ab X ) → D ( Sh Ab X ) , which a lso dep ends functorially o n the gerb e G → X . These functor ial pr op erties ar e required in our applications to T -duality , or if one w ants to for mulate a sta temen t ab out the natura lit y of a Chern character from G -t wisted K - theory with v alues in the p er io dic twisted co homology H ∗ ( X ; P G ( R X )). In or der to deﬁne P G ( F ) in a functorial wa y we must re ﬁne the diagr am S G ( F ) ∈ D ( Sh Ab X ) N op to a diagram in D (( Sh Ab X ) N op ). This is the tech nical heart of the pres e n t 4 CHAPTER 1. INTRODUCTION pap er. The details of this construction ar e contained in Sectio n 3 and will b e com- pleted in Deﬁnition 3 .4.5. Alo ng the w ay , we hav e to use the enhancement of the category of sheaves to b ounded b elow complexe s of ﬂasque sheav es. 1.1.10 . — The p erio dizatio n functor P G can be a pplied to arbitrar y ob jects in D + ( Sh Ab X ). In Prop o sition 2.5.1 we calculate examples which indicate some int eresting arithmetic features of this functor. 1.2. T -duality 1.2.1. — T o p ological T -duality is a concept whic h mo dels the underlying topolo gy of mirror symmetry in a lgebraic geometry or T -dua lity in string theory . W e refer to [ BRS ] for a more detailed discussion of the literature. In the present pap er we int ro duce the co ncept o f T -duality for pairs ( E , G ) of a U (1)-principal bundle E → B ov er a top olog ical stack B together with a top olo gical gerb e G → E with band U (1) using the notio n of a T -duality diag ram. 1.2.2. — Consider a dia g ram (1.2.1) p ∗ G q ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ # # H H H H H H H H H u / / ˆ p ∗ ˆ G { { v v v v v v v v v ˆ q @ @ @ @ @ @ @ @ G f A A A A A A A A E × B ˆ E p z z v v v v v v v v v v ˆ p $ $ H H H H H H H H H H ˆ G ˆ f ~ ~ } } } } } } } } E π $ $ I I I I I I I I I I I ˆ E ˆ π z z u u u u u u u u u u u B , where π , ˆ π ar e U (1)-principa l bundles, and f , ˆ f are gerb es with band U (1). In 4 .1.3 we describ e the isomorphis m class of the universal T -duality diagram over the classifying stack B U (1). Deﬁnition 1.2.2 (Deﬁnitio n 4.1. 3) . — The diagr am (1.2.1) is a T - duality dia- gr am, if it is lo c al ly i somorphic to the universal T -duality diagr am. The pair ( ˆ G, ˆ E ) is then called a T -dual o f ( E , G ). 1.2.3. — In L e mma 4.1.5 we will chec k that this gener alizes the co nc e pt of T-duality (for U (1)-bundles) from the c la ssical situatio n of principal bundles in the c a tegory of spaces [ BS06, BRS ] a nd the slightly more general situation of such bundles in orbispaces [ BS06 ] to a rbitrary U (1)-actions. The situation of semi- fr ee actions is discussed (in a completely diﬀerent wa y) in [ P an06 ]. It is an interesting op en problem to rela te his appr oach to the approa ch used here. 1.2. T -DUALITY 5 1.2.4. — O ne of the main themes of top olo gical T -duality is the T -duality trans for- mation in twisted co homology theo ries. In [ BS0 6 ] we observed that if the T -dua lit y transformatio n is an isomorphis m, then the corr esp onding twisted cohomology theo r y m ust b e tw o- p er io dic. This applies e.g. to twisted K -theo ry . In fact, one c an arg ue that t wisted K -theo r y is the universal twisted cohomolog y theory for which the T -dua lit y transformatio n is an iso morphism (1) . 1.2.5. — Our constructio n of P G is designed such that the corr esp onding T -duality transformatio n is an isomor phis m. T o this end we deﬁne the p er io dic G -twisted cohomolog y o f E with co eﬃcients in π ∗ F , F ∈ D + ( Sh Ab B ), by H ∗ per ( E , G ; π ∗ F ) := H ∗ ( E ; P G ( π ∗ F )) . In this c ase the T -duality tra nsformation T : H ∗ per ( E , G ; π ∗ F ) → H ∗ per ( ˆ E , ˆ G ; ˆ π ∗ F ) is induced by the co mp os ition Rπ ∗ P G ( π ∗ F ) unit → Rπ ∗ Rp ∗ p ∗ P G ( π ∗ F ) ∼ = Rπ ∗ Rp ∗ P p ∗ G ( p ∗ π ∗ F ) u ∗ ∼ = Rπ ∗ Rp ∗ P ˆ p ∗ ˆ G ( p ∗ π ∗ F ) π p = ˆ π ˆ p → R ˆ π ∗ R ˆ p ∗ P ˆ p ∗ ˆ G ( ˆ p ∗ ˆ π ∗ F ) ∼ = → R ˆ π ∗ R ˆ p ∗ ˆ p ∗ P ˆ G ( ˆ π ∗ F ) R ˆ p → R ˆ π ∗ P ˆ G ( ˆ π ∗ ( F )) . Note that here we use the functoriality of the p erio dizatio n in an essential wa y . The or em 1.2.3 (Theorem 4.3.7) . — The T -duality tr ansformation in t wiste d p e- rio dic c ohomolo gy is an isomorphism. 1.2.6. — If G → X is a gerb e ov er a nice no n-singular space X , then H ∗ per ( X, G ; R X ) is the correct target o f a Chern character fr om twisted K -theo ry . If X is a top o logical stack with non-trivial automorphisms o f p oints, then this no longer correc t. At the moment we do understand the sp ecial case of orbispaces. In [ BSS08 , Sec. 1 .3] w e give a detailed mo tiv ation for the introductio n of the twisted delo calized co homology . Let G → X be a top o lo gical ger be with band U (1) ov er an orbispace X . In [ BSS08 , Deﬁnition 3.4] we show that it gives rise to a sheaf L ∈ Sh Ab LX , where LX is the lo op o rbispace o f X . (1) W e thank M . Hopkins for p ointing out a pro of of this fact. 6 CHAPTER 1. INTRODUCTION The G -twisted delo calized p erio dic coho mo logy of X (with complex co eﬃcients) is deﬁned a s (see [ BSS08 , Deﬁnition 3 .5 ]) H ∗ deloc,per ( X, G ) := H ∗ ( LX ; P G L ( L )) , where G L → L X is deﬁned by the pull-back G L   / / G   LX / / X . Let us now cons ider a T -duality diag r am (1.2.1) ov er an orbispace B . Then we deﬁne a T -duality transforma tio n T : H ∗ deloc,per ( E , G ) → H ∗ deloc,per ( ˆ E , ˆ G ) by a mo diﬁcation o f the constructio n 1.2.5. The or em 1.2.4 (Theorem 5.4.2) . — The T -duality t r ansformation in twiste d de- lo c alize d p erio dic c ohomolo gy i s an isomorphism. So the situation with twisted delo calized p er io dic cohomolo gy is b etter than with orbispace K -theory . At the moment we do not know a pr o of that the T -duality transformatio n in twisted orbifold K -theory is an isomorphism (see the co r resp onding comments in [ BS06 ]). Using the fact that the Chern c har acter is an isomorphism, our result implies that the T -duality transforma tio n in twisted orbifold and or bispace K - theory is an isomo rphism after complexiﬁcation. 1.3. Duality for shea v es on l o cally com pact s tac ks 1.3.1. — In Section 6 of the present pap er we develop some features of a sheaf theory for lo cally compact sta cks. O ur main results are the co nstruction o f the basic setup, of the functor f ! , and the integration R f for o r iented ﬁber bundles. Section 6 not only provides the technical background for the applica tions of sheaf theory in the previous sec tions, but also co ntains so me additiona l materia l of indep endent int erest (in pa rticular the res ults connected with f ! ). 1.3.2. — A presheaf F of sets on a top olo gical space X ass o ciates to each o pen subset U ⊆ X a set of sections F ( U ), a nd to every inclusion V → U of o pe n s ubsets a functoria l restriction ma p F ( U ) → F ( V ), s 7→ s | V . In short, a pr e s heaf it is contra v ariant a functor from the ca teg ory ( X ) op en subsets o f X to sets. A pr esheaf is a s heaf of it has the fo llowing t wo prop erties: (1) If s, t ∈ F ( U ) are t wo sections and there exists an op en cov ering ( U i ) of U suc h that s | U i = t | U i for a ll i , then s = t . 1.3. DUALITY FOR SHE A VES ON LOCALL Y COMP ACT ST ACKS 7 (2) If ( U i ) is an op en covering of U and ( s i ) is a collection of sec tio ns s i ∈ F ( U i ) such that s i | U i ∩ U j = s j | U i ∩ U j for all pa irs i, j , then ther e exists a sectio n s ∈ F ( U ) s uch that s | U i = s i for a ll i . The notion of a sheaf is thus determined by the Grothendieck to po logy on ( X ) given by the c o llections o f op en coverings of op en subsets. W e will call ( X ) the s mall site asso ciated to X . If X is a top olog ical s tack, then the op en substacks form a tw o- categor y which do es not give the appropr iate setting for sheaf theo ry o n X . F or example, if G is a ﬁnite group, then the quo tient stac k [ ∗ /G ] is quite non-trivial but do es not have proper op en substacks. On the other hand its identit y o ne-morphism has the tw o-a utomorphism group G , and in a non-triv ial theory sheav es should reﬂect the tw o-automor phisms. 1.3.3. — F or applications to twisted co homology a s e tting for sheaf theory on smo o th stacks has b een introduced in [ BSS07 ]. In the prese nt paper we develop a similar theory for top ologica l stacks. There ar e v a rious choices to b e made in order to deﬁne the site o f a stack in top ologica l spaces. The sheaf theor ies asso ciated to these choices will have many features in common, but will diﬀer in o thers. The main goa l of the present pap er is the construction of the p erio diza tion functor P G asso ciated to a U (1)-banded gerbe G → X . One o f the main ing redients of the construction is an int egra tio n R f for orie n ted ﬁber bundles f with a clo sed top olog ical manifold as ﬁb er . In o rder to deﬁne the integration map we nee d a pro jection fo rmula which expresses a compatibilit y of the pull-ba ck and push-forward op era tio ns with tensor products, see Lemma 6.2.11. Alre a dy for the pro jection formula in ordinary sheaf theo ry o ne needs lo ca l co mpactness assumptions . F or this reason we dec ide d to work genera lly with lo cally compact stacks and s paces though muc h of the theor y would go through under mor e gene r al or diﬀerent ass umptions. 1.3.4. — A stack in top olo g ical spaces is topologic a l if it admits an atlas A → X . F rom the a tlas we ca n derive a gro upo id A × X A ⇒ A which represents X in an appropria te sense. The stack is called lo cally co mpact if one ca n ﬁnd an atlas A → X such that the resulting group o id is loca lly compact (i.e. A and A × X A ar e locally compact s paces). The site X ass o ciated to a lo ca lly c o mpact stack is the catego ry o f lo cally compact spaces ( U → X ) o ver X such that the mo rphisms a re morphisms o f spaces o ver X (i.e. pairs of a morphism b etw een the spa ces a nd a tw o-morphism ﬁlling the obvious triangle.) W e require that the structure morphism U → X has local sections. The top ology on X is again given by the collections of co verings b y open subsets of the ob jects ( U → X ). F or man y cons tr uctions and calculations the restriction functors from sheaves on X to sheav e s on ( U ) play a disting uis hed role. They ar e use d to build the connection betw een ope r ations with shea ves on the stack X a nd corresp onding classical o p e rations in shea f theo ry on the space s U . 8 CHAPTER 1. INTRODUCTION 1.3.5. — F or the theo ry of stacks in top ologica l spaces in ge ne r al we refer to [ He i05 ], [ BSS08 ], [ No o ]. Some sp ecial asp ects of lo cally compact stacks are discus s ed in Subsection 6 .1 of the pre sent pap er . In o ur treatment of sheaf theory o n the site X we give a description o f the clo sed monoidal structure on the categorie s of sheav e s and pre sheav e s of ab elian gr oups Sh Ab X a nd Pr Ab X o n X . The interplay b etw een shea ves and pr esheav es will be im- po rtant when we study the compatibility of the monoida l structures with the functors f ∗ : Sh Ab Y ⇆ Sh Ab X : f ∗ asso ciated to a morphism o f lo cally compact s ta cks f : X → Y . In g eneral these functors do not come from a morphisms of sites but ar e constr ucted in an ad-ho c manner. Because of this we must check under which co nditions prop erties exp ected from the cla s sical theory car ry ov er to the present case. The derived versions of these functor s on the bo unded b elow and unbounded de- rived categor ies D + ( Sh Ab X ) and D ( Sh Ab X ) will play an imp ortant role in the present pap er. In order to deal w ith the unbounded der ived category we use a n a pproach via mo del categor ies. 1.3.6. — Bes ides the developmen t o f the basic set up which we will not discuss fur- ther in the introduction let us no w ex pla in the tw o main r esults which ma y be of independent interest. The or em 1.3.1 (Theorem 6.3.2) . — If f : X → Y is a pr op er r epr esent able map b etwe en lo c al ly c omp act stacks such t hat f ∗ has ﬁnite c ohomolo gic al dimension, then the functor Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) has a right-adjoint, i.e. we have an adjoint p air (1.3.2) Rf ∗ : D + ( Sh Ab X ) ⇆ D + ( Sh Ab Y ) : f ! . W e think that one could prove a mor e gener al theor em stating the existence of a right adjoint of a functor R f ! where f ! is the push-forward with prop er supp or t alo ng an arbitrar y ma p b et ween lo c a lly compact stacks such that f ! has ﬁnite cohomo logical dimension, thoug h we have not chec ked all details. This theorem gener alizes a well-known r esult ([ V er95 ], [ KS94 , Ch. 3 ] in or dina ry sheaf theor y . Its impo rtance is due to the cla ssical calc ula tion (1.3.3) f ! ( F ) ∼ = f ∗ ( F )[ n ] (compare [ KS94 , Prop.3.3 .2]) for F ∈ D + ( Sh Ab ( Y )), if f : X → Y is an oriented lo cally trivia l bundle of clos e d connected top olo gical n -dimensional manifo lds on a lo cally compact s pace Y . If we would know s uch an isomor phism in the present case (for sheav es on the sites X , Y a nd stacks X , Y ), then we could deﬁne the integration map as the comp ositio n Z f : R f ∗ f ∗ ( F ) ∼ → R f ∗ f ! ( F )[ − n ] counit → F [ − n ] , 1.3. DUALITY FOR SHE A VES ON LOCALL Y COMP ACT ST ACKS 9 where the la st map is the co-unit of the adjunction (1.3.2). Unfortunately , at the moment we are not a ble to calcula te f ! ( F ) in any interesting example. Howev er, we can construct the int egra tion ma p in a direct manner avoiding the k nowledge of (1.3 .3). Some element s of the theory develop ed here are formally similar to the work [ O ls07 ] on shea ves o n the liss e ´ etale s ite o f an Artin stac k. In this framew ork in [ LO05 ] a functor f ! was introduce d b et ween derived catego ries of constructible sheav es. On the one ha nd the metho ds se e m to be completely diﬀerent. On the o ther ha nd this functor has the ex p ected b ehavior for smo o th maps, i.e. it s a tisﬁes a relation like (1.3.3). At the moment we do not see even a formal r elation b etw een the co nstruction of [ LO05 ] with the constructio n in the prese nt pa p er which could b e exploited for a calculation of f ! ( F ). 1.3.7. — The fo llowing Theorem is the res ult of Subsection 6.4. The or em 1.3.4 . — If the map f : X → Y of lo c al ly c omp act stacks is an oriente d lo c al ly trivial ﬁb er bund le with a close d c onne cte d t op olo gic al n - dimensional manifold as ﬁb er, then ther e exist s an int e gr ation m ap, a natur al tr ansformation of functors Z f : R f ∗ f ∗ → id [ − n ] : D + ( Sh Ab X ) → D + ( Sh Ab X ) which has the exp e cte d c omp atibility with pul l-b ack and c omp ositions. In Subsection 6 .5 we ex tend the push-for ward and pull-back op era tions to the un b ounded derived catego ries and construct the integration map in this setting. CHAPTER 2 GERBES AND PERI ODIZA TION 2.1. Shea v e s on the lo cally compact site of a stac k 2.1.1. — Let Top denote the site of top olog ical spaces. The top o logy is genera ted by covering families cov Top ( A ) of the ob jects A ∈ Top , where cov Top ( A ) is the set of cov er ings by co llections o f o pe n subsets. A stack will be a s ta ck on the site Top . Spaces are consider ed as stacks thro ugh the Y o neda embedding. A map A → X fr om a space A to a sta ck X w hich is s urjective, representable, and has lo cal se ctions is called a n atlas. W e refer to 6.1 .2 for deﬁnitions and mor e details ab out stacks in top o logical spa ces. Deﬁnition 2.1.1 . — A top olo gic al stack is a st ack which admits an atlas. Deﬁnition 2.1.2 . — A top olo gic al sp ac e is lo c al ly c omp act if it is Hausdorﬀ and every p oint admits a c omp act n eighb orho o d. A stack is c al le d lo c al ly c omp act if it admits an atlas A → X such that A and A × X A ar e lo c al ly c omp act. If X is a lo cally compact stack, then the site o f X is the sub c ategory Top lc /X of lo cally compact spaces ov er X such that the structure map A → X has lo cal sections. The top o logy is induced from Top . W e denote this site by X or Site ( X ). See 6.1.6 for mo r e details . 2.1.2. — As will b e explained in 6.1.9, a morphism of lo cally compact stacks f : X → Y gives r ise to an adjoint pair of functor s f ∗ : Sh Y ⇆ Sh X : f ∗ . The functor f ∗ is left-exa ct on the ca tegories of sheav es of ab elian gr oups and admits a r ig ht-deriv ed Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) betw een the b ounded b elow derived ca teg ories, co mpare 6.1 .9. 12 CHAPTER 2. GERBES AND PERIODIZA TION 2.1.3. — Let M be some s pa ce. Deﬁnition 2.1.3 . — A map b etwe en t op olo gic al stacks f : X → Y is a lo c al ly t rivial ﬁb er bund le with ﬁb er M if for every sp ac e U → X the pul l-b ack U × Y X → U is a lo c al ly trivial ﬁb er bund le of sp ac es with ﬁb er M . Assume that M is a clo s ed connected and or ientable n -dimens io nal top olog ical manifold. Deﬁnition 2.1.4 . — L et f : X → Y b e a map of lo c al ly c omp act stacks which is a lo c al ly trivial ﬁb er bund le with ﬁb er M . It is c al le d orientable if ther e exists an iso- morphism R n f ∗ ( Z X ) ∼ = Z Y . An orientation of f is a choic e of su ch an isomorphism. 2.1.4. — Let f : X → Y b e a lo ca lly triv ial o riented ﬁber bundle with n -dimensio nal ﬁber M ov er a lo ca lly compact s tack Y . Under these ass umption we can gener alize the integration map (se e [ KS94 , Sec. 3.3 ]) The or em 2.1.5 (Deﬁniti o n 6.4. 6) . — If f : X → Y b e a lo c al ly trivial oriente d ﬁb er bund le over a lo c al ly c omp act st ack with ﬁb er a close d top olo gic al manifold of dimension n , t hen we have an inte gr ation map, i.e. a natur al tr ansformation of functors Z f : R f ∗ ◦ f ∗ → id : D + ( Sh Ab Y ) → D + ( Sh Ab Y ) of de gr e e − n . 2.1.5. — W e consider a map of lo cally co mpact stacks f : X → Y which is a lo cally trivial or ie nted ﬁb er bundle with ﬁb er a close d top olog ical manifo ld o f dimension n . F urthermore let U → X b e a morphisms of lo cally compact stacks which has lo ca l sections. Then we for m the Car tesian (1) diagram V v / / g   X f   U u / / Y . Note that g : V → U is aga in a lo cally trivial o riented ﬁb er bundle with ﬁb er a clos e d top ological ma nifold o f dimensio n n . The orientation o f f (which gives the mar ked isomorphism b elow) induces an o rientation of g by R n g ∗ ( Z V ) ∼ = R n g ∗ v ∗ ( Z X ) ( 6.1.15 ) ∼ = u ∗ R n f ∗ ( Z X ) ! ∼ = u ∗ ( Z Y ) ∼ = Z U . (1) In the pr esent paper by a Cartesian diagram in the tw o-category of stac ks we mean a 2-Cartesian diagram. In particular, the square commu tes up to a 2-isomorphis m which we often omi t to write in order to si mplify the notation. More generally , when we talk ab out a commutat ive diagram in stac ks, then we mean a diagram of 1-m orphisms together with a collection of 2-isomorphism ﬁll ing all faces in a compatible wa y , and again we will usually not wr ite the 2-i somorphisms explicitly . 2.2. ALGEBRAIC STR UCTURES ON THE COHOMOLOGY OF A GERBE 13 L emma 2.1.6 . — The fol lowing diagr ams c ommu te (2.1.7) u ∗ ◦ Rf ∗ ◦ f ∗ ∼ = / / u ∗ R f   Rg ∗ ◦ v ∗ ◦ f ∗ ∼ =   u ∗ Rg ∗ ◦ g ∗ ◦ u ∗ R g o o Ru ∗ ◦ Rg ∗ ◦ g ∗ ∼ = / / Ru ∗ R g   Rf ∗ ◦ Rv ∗ ◦ g ∗ ∼ =   Ru ∗ Rf ∗ ◦ f ∗ ◦ Ru ∗ R f Ru ∗ o o . Pr o of . — Commut ativity of the ﬁrst diagr am follows immedia tely from the stronge r (beca use v alid in the der ived c ategory of unbounded c omplexes) Lemma 6.5.31. Com- m utativity of the second diagra m is prov ed in Lemma 6 .5.31, but only for the b ounded below der ived ca tegory . 2.2. Algebraic structures on the cohomol ogy of a gerb e 2.2.1. — Le t X b e a locally compa c t stack and f : G → X be a topolo g ical gerbe with band U (1). Then G is a lo cally compac t s ta ck. Indeed, we can choose an atlas A → X such that A and A × X A a re lo cally compact, and there exists a section G   A > > / / X . Then A → G is a n atlas and A × G A → A × X A is a lo cally trivial U (1)-bundle. In particular, A × G A is a lo cally compact space. 2.2.2. — By T 2 we denote the t wo-dimensional torus. W e ﬁx an orientation of T 2 . W e co nsider the pull-back pr ∗ 2 G ∼ = T 2 × G → T 2 × X . The isomorphism classes of automorphisms o f this g erb e a re class iﬁed by H 2 ( T 2 × X ; Z ). Let pr ∗ 2 G φ / / $ $ I I I I I I I I I pr ∗ 2 G z z u u u u u u u u u T 2 × X be an automorphism cla ssiﬁed by or T 2 × 1 X ∈ H 2 ( T 2 × X ; Z ). W e consider the diagram (2.2.1) pr ∗ 2 G $ $ I I I I I I I I I φ / / p   pr ∗ 2 G z z u u u u u u u u u p   G f % % J J J J J J J J J J J T 2 × X   G f y y t t t t t t t t t t t X . 14 CHAPTER 2. GERBES AND PERIODIZA TION Notice that φ is unique up to a non-c a nonical 2-iso morphism. In the prese n t pap e r we pr efer a more ca nonical choice. W e will ﬁx the mo rphism φ once and for all in the sp ecial ca se that X is a p oint and G = B U (1), i.e. we ﬁx a dia gram T 2 × B U (1)   φ univ / / % % J J J J J J J J J J T 2 × B U (1)   y y t t t t t t t t t t B U (1) % % L L L L L L L L L L T 2   B U (1) y y r r r r r r r r r r ∗ . If G → X is a top ologica l g erb e with band U (1), then we obtain the induced diagr a m by taking pro ducts G × T 2 × B U (1)   id G × φ univ / / ' ' O O O O O O O O O O O O G × T 2 × B U (1)   w w o o o o o o o o o o o o G × B U (1) ( ( P P P P P P P P P P P P P X × T 2   G × B U (1) v v n n n n n n n n n n n n n X . W e now replace the pr o ducts B U (1) × G by the tensor pro duct of gerb es as explained in [ BSST , 6.1.9 ] and identify B U (1) ⊗ G with G using the canonica l isomorphism in order to g et pr ∗ 2 G p   φ / / $ $ I I I I I I I I I pr ∗ 2 G p   z z u u u u u u u u u G f % % J J J J J J J J J J J T 2 × X   G f y y t t t t t t t t t t t X . In this way we hav e cons tructed a 2-functor from the 2-ca tegory of U (1)-banded gerb es ov er X to the 2-ca tegory of diagr ams of the form (2.2.1). By taking prefered mo dels for the pr o ducts we can, if we wan t, assume a s trict equality f ◦ p ◦ φ G = f ◦ p . 2.2.3. — Observe that the map of lo cally co mpact stacks p : pr ∗ 2 G → G is a locally trivial oriented ﬁb er bundle with ﬁber T 2 . Therefore we have the integration map (see 2.1 .5) Z p : Rp ∗ ◦ p ∗ → id . 2.2. ALGEBRAIC STR UCTURES ON THE COHOMOLOGY OF A GERBE 15 Deﬁnition 2.2.2 . — We deﬁne a natur al endo-tr ansformation D G of the fun ctor Rf ∗ ◦ f ∗ : D + ( Sh Ab X ) → D + ( Sh Ab X ) of de gr e e − 2 which asso ciates to F ∈ D + ( Sh Ab X ) the morphism Rf ∗ ◦ f ∗ ( F ) units − → R f ∗ ◦ Rp ∗ ◦ Rφ ∗ ◦ φ ∗ ◦ p ∗ ◦ f ∗ ( F ) f ◦ p ◦ φ = f ◦ p − − − − − − − → R f ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F ) R p → R f ∗ ◦ f ∗ ( F ) . 2.2.4. — It follows from L emma 2 .1 .6 that D G is co mpatible with pull- back diagra ms . In fa ct, consider a Cartes ian diagra m G ′ f ′   / / G f   X ′ g / / X . Using the ca no nical co nstruction explained in 2.2.2 we e x tend this to a morphism betw een dia grams o f the for m (2.2.1). Then we hav e the co mmutative diagra m g ∗ ◦ Rf ∗ ◦ f ∗ ∼ / / g ∗ D G   Rf ′ ∗ ◦ ( f ′ ) ∗ ◦ g ∗ D G ′ ◦ g ∗   g ∗ ◦ Rf ∗ ◦ f ∗ ∼ / / Rf ′ ∗ ◦ ( f ′ ) ∗ ◦ g ∗ . 2.2.5. — W e compute the action of D G in the case of the trivial gerb e f : G → ∗ and the shea f F ∈ Sh Ab Site ( ∗ ) r epresented by a discrete ab elian gro up F . Note that Rf ∗ ◦ f ∗ ( F ) is a n ob ject of D + ( Sh Ab Site ( ∗ )). W e get an ob ject Rf ∗ ◦ f ∗ ( F )( ∗ ) ∈ D + ( Ab ) by ev aluation at the ob ject ( ∗ → ∗ ) ∈ S ite ( ∗ ). L emma 2.2.3 . — Ther e exists an isomorphi sm H ∗ ( Rf ∗ ◦ f ∗ ( F )( ∗ )) ∼ = F ⊗ Z [[ z ]] , wher e deg( z ) = 2 . On c ohomolo gy the tr ansformation D G is given by D G = id ⊗ d dz . Pr o of . — W e choose a lift ∗ → G . F orming iterated ﬁb er pro ducts we get a simplicial space · · · ∗ × G ∗ × G ∗ × G ∗ → ∗ × G ∗ × G ∗ → ∗ × G ∗ → ∗ . Note that ∗ × G ∗ ∼ = U (1). One chec ks that the simplicial space is equiv a lent to the simplicial spa ce B U (1) · , the c la ssifying spac e of the gr oup U (1), U (1) × U (1) × U (1) → U (1) × U (1) → U (1) → ∗ . Let ( U → ∗ ) ∈ Si te ( ∗ ). If H ∈ Sh Ab G , then we co nsider an injective resolution 0 → H → I · . The ev a luation I · ( U × B U (1) · ) g ives a c o simplicial complex, and after normaliza tion, a double complex. Its total co mplex represents Rf ∗ ( H )( U → ∗ ) (see [ BSS07 , Lemma 2 .41] for a pro of of the corre sp onding s tatement in the smo oth 16 CHAPTER 2. GERBES AND PERIODIZA TION context). W e c alculate the co homology of Rf ∗ ( H )( U → ∗ ) using the a sso ciated sp ectral sequence. Its second page has the fo rm E p,q 2 ∼ = H p ( U × B U (1) q ; H ) . W e no w s pe c ialize to the sheaf H = f ∗ ( F ) ∼ = F G , where F is a discrete a b elian group, and U = ∗ . In this case the spectral sequenc e is the usual s pec tr al sequence which ca lc ula tes the cohomolo gy of the rea liz ation of the simplicia l space B U (1) · with co eﬃcients in F . Note that H ∗ ( B U (1); Z ) ∼ = Z [[ z ]] a s ring s with deg( z ) = 2 . Since it is tor sion fre e as an ab elian gr oup we g et H ∗ ( R ∗ f ∗ ◦ f ∗ ( F )( ∗ )) ∼ = F ⊗ H ∗ ( B U (1); Z ) ∼ = F ⊗ Z [[ z ]] . In a similar manner we calculate R f ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F )( ∗ ). Its cohomolo g y is H ∗ ( T 2 × B U (1); F ), hence we hav e H ∗ ( Rf ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F )( ∗ )) ∼ = F ⊗ H ∗ ( T 2 × B U (1); Z ) ∼ = F ⊗ Λ( u , v ) ⊗ Z [[ z ]] , where u , v ∈ H 1 ( T 2 , Z ) are the ca nonical generato r s. F or every top o logical gro up Γ we have a natura l map Γ → Ω( B Γ). By a djo intness we get a ma p c : U (1) × Γ → U (1) ∧ Γ → B Γ. W e will need a simplicial mo del c · of this map. W e co nsider the standar d simplicial mo del S · of U (1) with tw o non- degenerate simplices, one in deg ree 0, and one in degr ee 1. Then S · × Γ is a simplicial mo del of U (1) × Γ. It suﬃces to descr ib e the ma p c · on the non-degenera te par t of S · × Γ. The comp onent c 0 maps S 0 × Γ to the base p oint ∗ o f B Γ · . The co mp one nt c 1 is the natural identiﬁ cation of the non-degener ate c o py o f Γ ⊂ S 1 × Γ with Γ ∼ = B Γ 1 . W e now sp ecialize to the ca se Γ = U (1). W e get a map c : T 2 ∼ = U (1) × U (1) → B U (1), or on the simplicial level, a ma p c · : S · × U (1) → B U (1) · . W e hav e H ∗ ( B U (1); Z ) ∼ = Z [[ z ]] with z o dd degree 2, and one chec ks that uv = c ∗ ( z ) ∈ H 2 ( T 2 ; Z ) (after choos ing an appro priate basis u, v ∈ H 1 ( T 2 ; Z )). Note that B U (1) · is a simplicial ab elian gr oup. The discussion ab ove shows that the automor phism φ : G → G in (2.2.1) with X = ∗ and classiﬁed by uv ∈ H 2 ( T 2 ; Z ) can b e ar ranged so that it induces an automorphism of bundles of B U (1) · -torsor s (2.2.4) S · × U (1) × B U (1) · ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q φ · ( t,x ) 7→ ( t,c · ( t ) x ) / / S · × U (1) × B U (1) · v v m m m m m m m m m m m m m S · × U (1) . Under this iso mo rphism the action of (2.2.5) φ ∗ : H ∗ ( Rf ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F )( ∗ )) → H ∗ ( Rf ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F )( ∗ )) is induced b y z 7→ z + uv , u 7→ u , v 7→ v . In o rder to see this no te that m ∗ ( z ) = z 1 + z 2 , where m : B U (1) × B U (1) → B U (1) is the multiplication, and H ∗ ( B U (1) × 2.3. IDENTIFICA TION OF THE TRANSFORMA TION D G IN THE SMOOTH CASE 17 B U (1); Z ) ∼ = Z [[ z 1 , z 2 ]]. After realizatio n the ma p φ · leads to the comp ositio n T 2 × B U (1) ( id T 2 ,c ) × id → T 2 × B U (1) × B U (1) id T 2 × m → T 2 × B U (1) which maps z ( id T 2 × m ) ∗ 7→ z 1 + z 2 (( id T 2 ,c ) × id ) ∗ 7→ uv + z . In c ohomolog y of the ev a luations at the p oint the int egra tion map Z p : Rf ∗ ◦ Rp ∗ ◦ p ∗ ◦ f ∗ ( F ) → Rf ∗ ◦ f ∗ ( F ) induces the map F ⊗ Λ( u, v ) ⊗ Z [[ z ]] → F ⊗ Z [[ z ]] which takes the co eﬃcient a t uv . This implies the as sertions of Lemma 2.2 .3. 2.3. Iden tiﬁcation of the transformation D G in the sm o oth ca se 2.3.1. — In this subsection w e work in the context of [ BSS07 ] of manifolds and smo oth s ta cks. It ca n b e considered as a supplement to [ BSS0 7 ] co ncerning the transformatio n D G int ro duced in Deﬁnition 2 .2.2 which can b e deﬁned in the smo o th context in a par allel manner . If X is a smo o th stack, then Ω X denotes the sheaf of de Rham complexes on X . It asso ciates to ( U → X ) ∈ X the de Rham complex Ω X ( U → X ) := Ω( U ) of the manifold U . Note that in this s ubsection X denotes the site of a smo oth stack int ro duced in [ BSS07 ]. If ω ∈ Ω 3 X ( X ) is a clos ed 3-form, then we form the sheaf of twisted de Rha m complexes Ω X [[ z ]] ω . Its ev a luation at ( U → X ) ∈ X is the complex Ω X [[ z ]] ω ( U → X ) := Ω( U )[[ z ]] ∼ = Ω( U ) ⊗ Z Z [[ z ]] with diﬀerent ial d dR + ω d dz . In this formula the form ω a cts by wedge multiplication with the pull-back of ω to U . Let f : G → X be a gerb e with band U (1) ov er a smo oth manifold X . The choice of a ger b e connection deter mines a closed 3 - form ω ∈ Ω 3 X ( X ) whic h represents the Dixmier-Douady class of the gerb e. By [ BSS07 , Theorem 1 .1] w e hav e an isomor- phism (2.3.1) Rf ∗ f ∗ R X ∼ → Ω X [[ z ]] ω in the der ived ca tegory D + ( Sh Ab X ). 2.3.2. — The or em 2.3.2 . — We have a c ommut ative diagr am Rf ∗ f ∗ R X ∼ = ← − − − − − ( 2.3.1 ) Ω X [[ z ]] ω   y D G   y d dz Rf ∗ f ∗ R X ∼ = ← − − − − − ( 2.3.1 ) Ω X [[ z ]] ω . 18 CHAPTER 2. GERBES AND PERIODIZA TION Pr o of . — The is omorphism (2.3.1) was constructed in [ BSS07 , Section 3] using a particular mo del of R f ∗ f ∗ ( R X ). W e ﬁr st reca ll its co nstruction. Let A → G be an atlas. F or ( U → X ) ∈ X we form the simplicial ob ject ( A · U → G ) ∈ G ∆ op with n th piece A n U := A × G · · · × G A | {z } n +1 factors × X U → G . The b oundar ies and degener ations are given b y the pro jections a nd diagona ls as us ual. If F ∈ C + ( Pr Ab G ) is a b ounded below complex o f pres he aves, then we form the sim- plicial complex of pre s heav es ( U → X ) 7→ F ( A · U → G ). W e let C A ( F ) ∈ C + ( Pr Ab X ) denote the presheaf of asso ciated total complexes. Sometimes we will wr ite C m,n A ( F ) for the summa nd of bidegree ( m, n ), wher e the ﬁrst entry m denotes the cos implicial degree. If F is a complex of ﬂabby sheav es, then by [ BSS07 , Lemma 2.41] w e have a natural isomorphism Rf ∗ ( F ) ∼ = C A ( F ). Here we use in particular that the functor C A preserves sheav es. Note that the r esolution R G → Ω G of the constant s heaf with v alue R by the shea f of de Rham c o mplexes is a ﬂabby resolutio n (see [ BSS07 , Subsection 3.1 ]). Therefor e we have a natura l isomo rphism Rf ∗ ( R G ) ∼ = C A (Ω G ). W e cho ose an atlas A → X g iven by the disjoint union of a collection of op en subsets o f X s uch that there exists a lift in G f   A / / > > X . This lift is an a tlas A → G of G . W e furthermo re choo se a connection datum ( α, β ) ∈ Ω 1 ( A × G A ) × Ω 2 ( A ). The one-form α is a c onnection of the U (1)-princ ipa l bundle A × G A → A × X A . It is re la ted with the tw o- form β by d dR α = δ β . This eq uation implies that δ d dR β = 0 so that d dR β assembles to a uniquely determined closed form ω ∈ Ω 3 X ( X ) (compare [ BSS07 , Sectio n 3 .2]). The 3-for m ω repr esents the Dixmier- Douady clas s of the ger be G → X and will b e us ed for twisting the de Rha m complex. The isomo rphism (2.3 .1) is given by an explicit q ua si-isomor phism (2.3.3) Ω X [[ z ]] ω → C A (Ω G ) . Note that Ω X [[ z ]] ω and C A (Ω G ) are sheav es o f asso c iative D G -algebra s central over the sheaf o f D G -alg ebras Ω X , and that z generates Ω X [[ z ]] ω . The quas i- isomorphism (2.3.3) is the unique morphism o f she aves o f a sso ciative D G -algebras , c ent ral ov er Ω X , with z 7→ ( α, β ) ∈ C 1 , 1 A (Ω G )( X ) ⊕ C 0 , 2 A (Ω G )( X ) . F or more details we refer to [ BSS07 , Subsection 3.2] 2.3. IDENTIFICA TION OF THE TRANSFORMA TION D G IN THE SMOOTH CASE 19 2.3.3. — F or i = 1 , . . . , n there a re U (1)-principal bundle structures p i : A × G · · · × G A | {z } n +1 fac tors → A × G · · · × G A | {z } i fact ors × X A × G · · · × G A | {z } n − i +1 fac tors . F urthermore, we hav e e m b eddings j i : A × G · · · × G A | {z } n factors → A × G · · · × G A | {z } i factors × X A × G · · · × G A | {z } n − i +1 fac tors given b y j i := id A · · · × id A | {z } i − 1 fac tors × ∆ A × id A · · · × id A | {z } n − i fac tors , where ∆ : A → A × X A is the diago nal. If ( U → X ) ∈ X , then the maps p i and j i induce similar maps on the pro duct · · · × X U of these manifolds o ver X with U which we denote b y the same symbols. F or i = 1 , . . . , n we deﬁne the map of degree − 1 v i : Ω( A n U ) → Ω( A n − 1 U ) as the co mpo sition o f the integration over the ﬁb er of p i with the pull-ba ck along j i , i.e. v i := j ∗ i ◦ R p i . Since the co nstruction o f v i is natural with resp ect to U we can view v i as a mor phism of sheav es C n,m A (Ω G ) → C n − 1 ,m − 1 A (Ω G ). W e deﬁne the family of mor phis ms D n := n X i =1 ( − 1) i v i : C n, ∗ A (Ω G ) → C n − 1 , ∗− 1 A (Ω G ) and let D : C A (Ω G ) → C A (Ω G ) b e the endomo r phism of sheaves of degr ee − 2 given by D n in bideg ree ( n, ∗ ). 2.3.4. — L emma 2.3.4 . — The map D : C A (Ω G ) → C A (Ω G ) is a derivation of Ω X -mo dules. Pr o of . — Note that v j commutes with the de Rham diﬀerential. Moreov er, if q k : A × G · · · × G A | {z } n +1 factors → A × G · · · × G A | {z } n factors is the pro jection whic h leav es out the k -th factor ( k = 0 , . . . , n ), then we hav e the relations v j q ∗ k = q ∗ k − 1 v j , j < k v j q ∗ k = q ∗ k v j − 1 , j > k + 1 v j q ∗ k = 0 , j = k , k + 1 . Observe that in the la st case q k factors o ver the bundle whic h is used for the inte- gration in the deﬁnition of v k or v k +1 , and the comp osition o f a pullback alo ng a 20 CHAPTER 2. GERBES AND PERIODIZA TION bundle pro jection follow ed by an integration along the s ame bundle pro jectio n v an- ishes. These rela tions imply by a dir ect calculation that D is a chain map for the ˇ Cech-de Rha m diﬀerential o f C A (Ω G ). Moreov er, it follows immediately from the deﬁnition of D that it is Ω X -linear (even Ω A -linear). It is aga in a straightforward calculation to verify that D is a deriv atio n for that asso ciative pro duct on C A (Ω G ) (compare [ BSS07 , 2 .4 .9] for the pro duct structure). 2.3.5. — L emma 2.3.5 . — We have a c ommut at ive diagr am Ω X [[ z ]] ω ( 2.3.3 ) − − − − − → C A (Ω G )   y d dz   y D Ω X [[ z ]] ω ( 2.3.3 ) − − − − − → C A (Ω G ) . Pr o of . — Since α is the connection one-for m of a U (1)-connection on the total space of the U (1)-principa l bundle p 1 : A × G A → A × X A we hav e R p 1 α = 1. Cons equently , D ( α, β ) = 1 . This implies the assertion, since D and d dz are Ω X -linear deriv ation, and z generates Ω X [[ z ]] ω . In view o f Lemma 2.3.5, in order to ﬁnish the pr o of of Theore m 2.3.2 is s uﬃce s to show tha t the op era tio n D coincides with the op era tion of R p ◦ φ ∗ ◦ p ∗ on C A (Ω G ). 2.3.6. — Let M · be a simplicial manifold and cons ider the bundle U (1) × M · → M · . W e describ e the integration map Z : Ω( U (1) × M · ) → Ω( M · ) in the s implicial picture, i.e. as a map Z : Ω( S · × M · ) → Ω( M · ) . F or n ≥ 1 the manifolds S n × M n consists of n copies σ 1 ( M n ) , . . . , σ n ( M n ) o f M n which co rresp ond to the p oints of S n which a re degener ations of the non-degener ated po int of S 1 (where the index meas ur es which 1-simplex in the b oundary is non- degenerate), and an additional co py o f M n corres p o nding the po int of S n which is the degeneration o f the p oint in S 0 . F or k = 1 , . . . , n + 1 let j k : M n → S n +1 × M n +1 be the ma p M n → σ k ( M n +1 ) ⊂ S n +1 × M n +1 , which cor resp onds the k th deg eneration [ n + 1] → [ n ]. W e now deﬁne a chain map of tota l complexes Z : Ω( S · × M · ) → Ω( M · ) 2.4. TWO-PERIODIZA TION — UP TO IS OM ORPHISM 21 of degr ee − 1 which is given by (2.3.6) n +1 X k =1 ( − 1) k j ∗ k : Ω( S n +1 × M n +1 ) → Ω( M n ) , and is zero on Ω( S 0 × M 0 ). This map realiz es the integration in the simplicial picture. 2.3.7. — F or ( U → X ) ∈ X the automorphism of gerb es φ : T 2 × G → T 2 × G induces an auto morphism o f simplicial se ts φ · : S · × U (1) × A · U → S · × U (1) × A · U which we now describ e explicitly by an extens ion o f the sp ecia l case (2.2 .4) to genera l base s paces. If t ∈ S n × U (1) b elo ng s to U (1) ∼ = σ k ( U (1)) ⊂ S n × U (1), k = 1 , . . . , n , then φ · ( t, a ) = ( t, m k ( t, a )), where m k : U (1) × A n U → A n U is the action of U (1) on the principal ﬁbra tio n p k . If t ∈ S n × U (1) b elongs to the degeneration of S 0 × U (1), then φ · ( t, a ) = ( t, a ). This formula provides a simplicial description of the action o f φ ∗ : C S · × U (1) × A (Ω G ) → C S · × U (1) × A (Ω G ) . Combining the description of the integration map (2 .3 .6) with this for mula for the action o f φ ∗ it is now straig ht forward to show the equality of maps D = Z p ◦ φ ∗ ◦ p ∗ : C A (Ω G ) → C A (Ω G ) . ✷ 2.4. Tw o-p erio dization — up to isomorphism 2.4.1. — Let f : G → X b e a top ologica l ger b e with ba nd U (1) ov er a lo cally compact stack X . In Deﬁnition 2 .2.2 we hav e co nstructed a natur al endo morphism D G ∈ End ( Rf ∗ ◦ f ∗ ) of degre e − 2. T o any ob ject F ∈ D + ( Sh Ab X ) we ass o ciate the inductive system (2.4.1) S G ( F ) : R f ∗ ◦ f ∗ ( F ) D G ← Rf ∗ ◦ f ∗ ( F )[2] D G ← Rf ∗ ◦ f ∗ ( F )[4] D G ← . . . indexed by { 0 , 1 , 2 . . . } . Using the inc lus ion D + ( Sh Ab X ) → D ( Sh Ab X ) of the b ounded b elow into the un- bo unded der ived ca tegory of sheav e s of a belia n gro ups on X we can consider S G ( F ) ∈ D ( Sh Ab X ) N op , where the o rdered set of integers N is considere d a s a categ ory . 2.4.2. — Using the triang ulated structur e of D ( Sh Ab X ) one can deﬁne for each ob ject S ∈ D ( Sh Ab X ) N op an o b ject hol im S ∈ D ( Sh Ab X ) which is unique up to non- canonical isomorphism (see [ Nee0 1 ]). An ex plic it co nstruction of this homo topy limit uses the extension o f maps in D ( S h Ab X ) to exa c t triangles by a mapping cylinder construction. 22 CHAPTER 2. GERBES AND PERIODIZA TION In par ticular, we obtain holi m S G ( F ) b y the extension to a triangle of the map 1 − ˆ D in holim S G ( F ) → Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] 1 − ˆ D − → Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] → holim S G ( F )[1] , where ˆ D : Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] → Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] maps the seq uence ( x i ) i ≥ 0 to the sequence ( D G x i +1 ) i ≥ 0 . 2.4.3. — W e can now deﬁne the p erio diza tion P G ( F ) ∈ D ( S h Ab X ) of a n ob ject F ∈ D + ( Sh Ab X ). Deﬁnition 2.4.2 . — F or F ∈ D + ( Sh Ab X ) we deﬁne P G ( F ) ∈ D ( Sh Ab X ) by P G ( F ) := holim S G ( F ) . Note that P G ( F ) is wel l-deﬁne d up to non-c anonic al isomorphism. 2.4.4. — The o p e rator Y i ≥ 0 D G : Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] → ( Y i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ])[ − 2 ] commutes with ˆ D and there fo re induces a map W : P G ( F ) → P G ( F )[ − 2] via an extension in the dia gram P G ( F ) W − − − − → P G ( F )[ − 2]   y   y Q i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] Q i ≥ 0 D G − − − − − − → Q i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ])[ − 2 ]   y 1 − ˆ D   y 1 − ˆ D Q i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ] Q i ≥ 0 D G − − − − − − → Q i ≥ 0 Rf ∗ ◦ f ∗ ( F )[2 i ])[ − 2 ]   y   y P G ( F )[1] W − − − − → P G ( F )[1][ − 2 ] . Note tha t such an extension exists b y the axioms of a triangulated category , but it might not b e unique. The following pro p o sition asser ts that P G ( F ) is tw o -p erio dic. Pr op osi t i on 2.4.3 . — The map W : P G ( F ) → P G ( F )[ − 2] is an isomorphi sm. Pr o of . — F or notational conv enience, we consider the following gener al situation. Let D ( A ) b e the unbo unded derived catego ry o f a Gr othendieck ab elian ca teg ory . Note that Sh Ab ( X ) is such a categor y (see Section 3 .3.1). W e consider a n ob ject X ∈ D ( A ) 2.4. TWO-PERIODIZA TION — UP TO IS OM ORPHISM 23 together with a mor phism D : X → X [ − 2]. W e ca n assume that D is r e pr esented by a map o f complexes D : X → X [ − 2]. W e obtain the extension 1 − ˆ D to a tr iangle (2.4.4) Y → Y i ≥ 0 X [2 i ] 1 − ˆ D → Y i ≥ 0 X [2 i ] → Y [1 ] where Y := Q i ≥ 0 X [2 i ] ⊕ ( Q i ≥ 0 X [2 i ])[1] with the diﬀerential δ :=  d 1 − ˆ D 0 − d  , where d is the diﬀerential of X . The induced map W : Y → Y [ − 2 ] is given by W := Q i ≥ 0 D 0 0 Q i ≥ 0 D ! . Let E : Y i ≥ 0 X [2 i ] → ( Y i ≥ 0 X [2 i ])[2] be the shift E ( x i ) i ≥ 0 := ( x i +1 ) i ≥ 0 . Note that E co mm utes with 1 − ˆ D , to o. Therefore we obta in the extension S : Y → Y [2] in the diagr am Y / / S   Q i ≥ 0 X [2 i ] E   1 − ˆ D / / Q i ≥ 0 X [2 i ] E   / / Y [1] S   Y [2] / / ( Q i ≥ 0 X [2 i ])[2] 1 − ˆ D / / ( Q i ≥ 0 X [2 i ])[2] / / Y [1][2] . by the matrix S :=  E 0 0 E  . Prop ositio n 2.4.3 is a co ns equence of the following Lemma. L emma 2.4.5 . — We have the e qualities W ◦ S = id = S ◦ W . Pr o of . — First obse rve that Q i ≥ 0 D ◦ E = ˆ D = E ◦ Q i ≥ 0 D . Therefore W ◦ S = S ◦ W =  ˆ D 0 0 ˆ D  . In or der to show that W ◦ S = id we show that the map I := ˆ D 0 0 ˆ D ! . on Y is homotopic to the identit y and therefor e is equal to the identit y in the der ived category . This follows from 1 − I = δ ◦ J + J ◦ δ with J :=  0 0 1 0  . 24 CHAPTER 2. GERBES AND PERIODIZA TION 2.4.5. — W e contin ue with the notation intro duced in the pro of o f Prop os ition 2 .4.3. Applying a homolo gical functor to the triangle (2.4.4) we get the lo ng exact s equence · · · → H ∗ ( Y ) → Y i ≥ 0 H ∗ ( X [2 i ]) → Y i ≥ 0 H ∗ ( X [2 i ]) → H ∗ ( Y [1]) → . If we analyze the middle map and compar e it with the ordina r y deﬁnition of limits in ab elian ca tegories we g e t the following result. Cor ol lar y 2.4.6 . — We have an ex act se quen c e: 0 → lim i 1 H ∗ ( X [2 i ])[ − 1 ] → H ∗ ( Y ) → lim i H ∗ ( X [2 i ]) → 0 . 2.4.6. — Note tha t the co nstruction holim : D ( A ) N op → D ( A ) is not a functor. The constructio n of the homoto py limit hol im ( S ) for S ∈ D ( A ) N op via ma pping cylinders uses explicit repres ent atives of the maps of the system S and depe nds non-trivially on this choice. A homotopy limit functor ho lim : D ( A N op ) → D ( A ) ca n b e deﬁned as the righ t- derived functor of lim : A N op → A . Note that in the domain we take the derived category of the a b elian category of N op -diagra ms in A as oppo s ed to N op -diagra ms in the der ived categ ory of A . In Section 3 we will use this idea a nd reﬁne P G to a per io dization functor P G : D + ( Sh Ab X ) → D ( Sh Ab X ) which is a triangulated functor and natural in G → X . The main idea is the con- struction of a r eﬁnement of the dia gram (2.4.1) to a diagr am in D (( Sh Ab X ) N op ), see 3.4.6 (the details a re in fact more complicated). 2.5. Calculations 2.5.1. — In this subsection we calculate P G ( F ) in the sp ecial case , wher e G → ∗ is the (trivial) U (1)-gerb e ov er the p oint, and F ∈ Sh Ab Site ( ∗ ) is the sheaf repres ent ed by a discrete a b elia n gro up F . W e will calculate the ab elian gr oup H ∗ ( ∗ ; P G ( F )). This co homology is tw o-p erio dic so that we only have to distinguish the even a nd the o dd-degree case. In the table b elow A Q f denotes the gro up of ﬁnite adeles of Q , which contains Q via the diag onal embedding. Pr op osi t i on 2.5.1 . — We have the fol lowing table for the c ohomolo gy H ∗ ( ∗ ; P G ( F )) . F H ev ( ∗ ; P G ( F )) H odd ( ∗ ; P G ( F )) Z 0 A Q f / Q Q Q 0 Z /n Z 0 0 Q / Z A Q f 0 . 2.5. CALCULA TIONS 25 2.5.2. — T o prov e Pro po sition 2.5.1, we use the ex act seq uence 2.4 .6 where H ∗ ( X ) = H ∗ ( ∗ ; R f ∗ ◦ f ∗ ( F )) ∼ = F ⊗ Z [[ z ]] ∼ = F [[ z ]] by Lemma 2 .2.3 with z of degree 2. W e must dis cuss the cohomolo gy of the complex 0 → Y i ≥ 0 F [[ z ]][2 i ] 1 − ˆ D → Y i ≥ 0 F [[ z ]][2 i ] → 0 , where ˆ D ( x i ) i ≥ 0 = ( D G x i +1 ) i ≥ 0 with D G = d dz . This mea ns that w e ha ve to study the s olution theor y for the sys tem (2.5.2) x i − d dz x i +1 = a i , i ≥ 0 , x i ∈ F [[ z ]] . 2.5.3. — L e t us star t with the ca s e F = Q . Since we c a n divide b y ar bitrary integers the op er ator D G is s urjective a nd we can for any ( a i ) i ∈ N solve this system inductively . Therefore the cok ernel lim 1 i Q [ u ] of 1 − ˆ D is trivial. A solution of the homogeneous system is uniquely determined by the choice of x 0 and the consta nt ter ms of the x i , i ≥ 1. Note that the constant term of x i is in degr ee − 2 i . It follows that H ev ( ∗ ; P G ( Q )) ∼ = Q , H odd ( ∗ ; P G ( Q )) ∼ = 0 . 2.5.4. — W e now discuss torsio n co eﬃcients F = Z /n Z . W rite x i = P x i,k z k , a i = P a i,k z k with x i,k , a i,k ∈ Z /n Z . Then we hav e to solve ∞ X k =0 x i,k z k − ( k + 1) x i +1 ,k +1 z k = ∞ X k =0 a i,k z k ∀ i ≥ 0 . Equating co eﬃcie nts this system deco uples into ﬁnite sy stems x i,kn − ( k n + 1) x i +1 ,kn +1 = a i,kn x i,kn +1 − ( k n + 2) x i +1 ,kn +2 = a i,kn +1 . . . x i,kn + n − 2 − ( k n + n − 1) x i +1 ,kn + n − 1 = a i,kn + n +2 x i,kn + n − 1 − r + ( k n + n ) x i +1 ,kn + n | {z } =0 = a ikn + n − 1 , where i , k ≥ 0. W e see tha t we ca n always solve this system uniquely by backw ards induction. W e get H ev ( ∗ ; P G ( Z /n Z )) ∼ = 0 , H odd ( ∗ ; P G ( Z /n Z )) ∼ = 0 . 2.5.5. — Let us now assume that F = Q / Z . Since this g roup is divisible we can solve the s ystem (2.5 .2) for every ( a i ) i ∈ N . It follows that H odd ( ∗ ; P G ( Q / Z )) ∼ = 0 . W e now discuss the solution of the homogeneous system in deg r ee 0 . W e can choose x 0 arbitrar y . If we hav e found x i for i = 0 , . . . , n − 1 , then we must solve x n − 1 = nx n 26 CHAPTER 2. GERBES AND PERIODIZA TION in the next step. W e see that x n is well-deﬁned up to the imag e of Z /n Z ∼ = n − 1 Z / Z ⊂ Q / Z . W e s ee that H ev ( ∗ ; P G ( Q / Z )) a dmits a seque nc e of quotients H ev ( ∗ ; P G ( Q / Z )) → · · · → Q n → Q n − 1 → · · · → Q 0 where Q n ∼ = Q / Z and Q n → Q n − 1 is giv e n b y m ultiplication with n for all n ∈ N . The limit A Q f ∼ = lim ← − n ∈ N ( Q /n ! Z ) is the ring A Q f of ﬁnite a deles o f Q , and Q ⊂ A Q f is a s ubgroup. W e thus g et H ev ( ∗ ; P G ( Q / Z )) ∼ = A Q f . 2.5.6. — Finally assume that F = Z . W e must again consider the system (2.5.2) of eq uations a bove. Let us discuss this system in degr ee 2 r . The n the relev ant co eﬃcients of x i and a i are seq uences of integers, a nd (writing out only these) dx i +1 = ( r + i + 1) x i +1 . W e see that the ho mogeneous equation has only the triv ial solution since other wise the integer x 0 m ust b e divisible by n + i + 1 for all i ≥ 0. Hence H ev ( ∗ ; P G ( Z )) ∼ = 0 . In o rder to calculate H odd ( ∗ ; P G ( Z )) we consider the exa ct sequence 0 → Z → Q → Q / Z → 0 . It g ives rise to an exact sequence of sheav es 0 → Z → Q → Q / Z → 0 . and a lo ng exa ct cohomolog y seq uence. In Sec tion 3.4 we will construct a functorial version of P G which is a triangula ted functor , a nd which coincides with the isomor- phism c lass constr ucted ab ov e. Using this functor , we g e t a tria ng le P G ( Z ) → P G ( Q ) → P G ( Q / Z ) → P G ( Z )[1] and therefo r e a long exact cohomo logy sequence H ∗ ( ∗ ; P G ( Z )) → H ∗ ( ∗ ; P G ( Q )) → H ∗ ( ∗ ; P G ( Q / Z )) → H ∗ ( ∗ ; P G ( Z ))[1] . By the calc ulations fo r Q and Q / Z we get e x act seq uences 0 → Q c → A Q f → H odd ( ∗ ; P G ( Z )) → 0 , where c is the cano nical embedding. Therefo re H odd ( ∗ ; P G ( F )) ∼ = A Q f / Q . ✷ CHAPTER 3 FUNCTORIAL PERIODI ZA TION 3.1. Flabb y resolutions 3.1.1. — L e t X b e a site, e.g. the site of a lo ca lly co mpact stack. F or U ∈ X let τ := ( U i → U ) i ∈ I ∈ c ov X ( U ) b e a covering family . T hen we consider V := F i ∈ I U i → U . F orming iterated ﬁbe r pro ducts we o btain a simplicia l ob ject V · in X with V n = V × U · · · × U V | {z } n +1 f actors . If F ∈ Pr X is a presheaf on X , then we fo r m the cos implicia l set C · ( τ , F ) := F ( V · ). 3.1.2. — If F is a presheaf of a b elia n gro ups, then we form the ˇ Cech complex ˇ C ( τ , F ) which is the chain complex asso ciated to the cosimplicial ab elian gro up C · ( τ , F ). If F is a sheaf, then H 0 ˇ C ( τ , F ) ∼ = F ( U ). W e recall the following deﬁnition (see [ T am94 , Deﬁnition 3.5 .1]). Deﬁnition 3.1.1 (see 3.5.1, [ T am 9 4 ] ) . — A she af F ∈ Sh Ab X is c al le d ﬂabby if for al l U ∈ X and τ ∈ cov X ( U ) we have H i ˇ C ( τ , F ) ∼ = 0 for al l i ≥ 1 . By [ T am94 , Cor. 3.5 .3] a sheaf F ∈ Sh Ab X is ﬂabby if and o nly if R k i ( F ) = 0 fo r all k ≥ 1, where i : Sh Ab X → Pr Ab X is the inc lus ion of sheaves into presheaves. As an immediate consequence of the deﬁnition a shea f F ∈ S h Ab X is ﬂabby if and only if the restriction F U of F to the site ( U ) is ﬂabby for all ( U → X ) ∈ X (see 6.1.14 fo r the nota tion). 3.1.3. — Let now X be a lo cally co mpact stack and X b e the site of X . Occasionally , in the pr esent pa p er we need the strong er notion of a ﬂasque sheaf. Deﬁnition 3.1.2 . — A she af F ∈ Sh Ab X is c al le d ﬂasque if for every ( U → X ) ∈ X and op en su bset V ⊆ U the r estriction F ( U → X ) → F ( V → X ) is surje ctive. In the literature , e.g . in [ KS94 ] or [ B re9 7 ], this is used as the deﬁnition o f ﬂabbiness. 28 CHAPTER 3. FUNCTORIAL P E RIODIZA TION L emma 3.1.3 . — A ﬂasque she af is ﬂabby. Pr o of . — F or U ∈ X let Γ U : Sh Ab X → Ab be the section functor F 7→ Γ U ( F ) := F ( U ). F o r V ⊆ U we hav e Γ V ( F U ) = Γ V ( F ). A sheaf F ∈ S h Ab X is ﬂasque b y deﬁnition if a nd only if F U is ﬂas q ue for all U ∈ X . But a ﬂa sque sheaf is Γ V -acyclic for e very V ⊆ U by [ Bre97 , Ch. 2, Thm. 5.4] (note that in this refer ence our ﬂasque is calle d ﬂa bby). By [ T am94 , Cor . 3.5 .3] it is ﬂabby in the sense o f 3.1.1. This argument shows that F U is ﬂabby for all ( U → X ) ∈ X and implies that F itself is ﬂabby . W e do not know if the conv e r se of Lemma 3.1.3 is true. Therefor e we m us t b e careful whe n us ing re s ults from the literature. 3.1.4. — L emma 3.1.4 . — If f : X → Y is a r epr esentable map of lo c al ly c omp act stacks, then a ﬂabby she af is f ∗ -acyclic. Pr o of . — Let F ∈ Sh Ab X b e a ﬂabby sheaf. W e must show tha t R k f ∗ ( F ) = 0 for all k ≥ 1. W e hav e a mor phism of sites f ♯ : Y → X , see 6 .1.10. The functor p f ∗ : Pr X → Pr Y is given by p f ∗ F := F ◦ f ♯ . It is in particula r exa ct. There fore we hav e Rf ∗ ∼ = i ♯ ◦ p f ∗ ◦ R i . Since a ﬂabby sheaf is i -a cyclic we conclude that R k i ( F ) = 0 for k ≥ 1. This implies R k f ∗ ( F ) = 0 for k ≥ 1. 3.1.5. — L emma 3.1.5 . — If a morphism f : X → Y of lo c al ly c omp act stacks has lo c al se c- tions, t hen the functor f ∗ : Sh Ab Y → Sh Ab X pr eserves ﬂ abby she aves. Pr o of . — Let F ∈ S h Ab Y b e ﬂabby . W e consider an o b ject ( U → X ) ∈ X and a cov er ing family τ ∈ cov X ( U ). Then we must show that the higher cohomo logy groups o f ˇ C ( τ , f ∗ F ) v anish. W e obtain a covering fa mily f ♯ τ ∈ cov Y ( f ♯ U ), see 6.1.11. Let V · be the simplicial ob ject asso ciated to τ as in 3.1 .1. Since f ♯ preserves ﬁb er pro ducts in the sens e of [ T am94 , 1 .2 .2(ii)] w e see that f ♯ V · is the s implicial ob ject in Y asso ciated to f ♯ τ . The rule f ∗ F ( U ) ∼ = F ( f ♯ U ) (see a gain 6.1.11) gives the isomorphism of cosimplicial sets f ∗ F ( V · ) ∼ = F ( f ♯ V · ) and hence an isomor phism of complexes ˇ C ( τ , f ∗ F ) ∼ = ˇ C ( f ♯ τ , F ) . Since F is ﬂabby the hig her co homology gr oups of the right-hand side v anish. 3.1.6. — W e now construct a ca nonical ﬂabby reso lution functor F l : Sh Ab X → C + ( Sh Ab X ) , id → F l . It asso ciates to a F a sort of Go dement re solution which consists in fact of ﬂas que sheav e s. 3.1. FLABBY RESOLUTIONS 29 F or a spac e U let ( U ) denote the s ite of op en subsets of U with the topolog y of op en cov erings. W e will ﬁrs t construct ﬂabby resolutio n functors F l U : Sh Ab ( U ) → C + ( Sh Ab ( U )) , i d → F l U for all ( U → X ) ∈ X whic h are compa tible with the morphis ms V → U in X . F or F ∈ Sh Ab X we obtain a c o llection of ﬂabby resolutions ( F U → F l U ( F U )) U ∈ X , which by 6 .1.14 give rise to a res olution F → F l ( F ). In the following we discuss thes e steps in de ta il. 3.1.7. — L e t p U : ˆ U → U b e the ident ity map, where ˆ U is the set U with the discrete top ology . Let F ∈ Sh Ab ( U ). W e set F l 0 U ( F ) := ( p U ) ∗ ◦ p ∗ U ( F ) a nd let F → F l 0 U ( F ) be g iven by the unit id → ( p U ) ∗ ◦ p ∗ U . L emma 3.1.6 . — The se quenc e 0 → F → ( p U ) ∗ ◦ p ∗ U F is exact. Pr o of . — Let w ∈ U . W e must show that the induced map on stalks F w → (( p U ) ∗ ◦ p ∗ U F ) w is injective. This immediately follows fro m the de s cription (( p U ) ∗ ◦ p ∗ U F ) w = colim w ∈ W ⊆ U Y v ∈ W F v . ✷ 3.1.8. — W e now constr uct F l U ( F ) inductively . Assume that we have a lready con- structed F l 0 U ( F ) → · · · → F l k U ( F ). Then we let F l k +1 U ( F ) := ( p U ) ∗ ◦ p ∗ U ( coker ( F l k − 1 U ( F ) → F l k U ( F )) and F l k U ( F ) → F l k +1 U ( F ) b e aga in given b y F l k U ( F ) → coke r ( F l k − 1 U ( F ) → F l k U )) unit → F l k +1 U ( F ) . In this way we construct an exact complex 0 → F → F l 0 U ( F ) → F l 1 U ( F ) → · · · → F l k U ( F ) → . . . . All pieces of the co nstruction a re functorial. Hence, the as so ciation F 7→ F l U ( F ) is functorial in F . The inclusio n F → F l 0 U ( F ) gives the natural tra nsformation id → F l U . 3.1.9. — L emma 3.1.7 . — F or any she af F ∈ Sh Ab ( U ) t he she af ( p U ) ∗ ◦ p ∗ U ( F ) is ﬂasque and ﬂabby. Pr o of . — F or W ⊆ U w e hav e (3.1.8) ( p U ) ∗ ◦ p ∗ U ( F )( W ) ∼ = Y w ∈ W F w . 30 CHAPTER 3. FUNCTORIAL P E RIODIZA TION It is now obvious that ( p U ) ∗ ◦ p ∗ U ( F )( U ) → ( p U ) ∗ ◦ p ∗ U ( F )( W ) is surjective. A ﬂasque sheaf is ﬂa bby by Lemma 3.1.3. 3.1.10 . — W e now consider a sheaf F ∈ Sh Ab X . F or ( U → X ) let F U ∈ Sh Ab ( U ) denote its restric tion to ( U ). W e apply the previous co nstruction to all ob jects ( U → X ) ∈ X and the sheaves F U . Then we get a collection of complex es o f sheav es F l U ( F U ) for all ( U → X ) ∈ X . Let f : V → U b e a mo rphism in X . W e shall constr uct a functoria l morphism f ∗ F l U ( F U ) → F l V ( F V ). Let G ∈ Sh ( U ), H ∈ Sh ( V ), and f ∗ G → H b e a morphism of s heav es . W e consider the dia gram ˆ V ˆ f / / p V   ˆ U p U   V f / / U . It induces the trans fo rmation, natura l in G and H , f ∗ ◦ ( p U ) ∗ ◦ p ∗ U ( G ) → ( p V ) ∗ ◦ ˆ f ∗ ◦ p ∗ U ( G ) ∼ = ( p V ) ∗ ◦ p ∗ V ◦ f ∗ ( G ) → ( p V ) ∗ ◦ p ∗ V ( H ) W e now co nstruct the map f ∗ F l U ( F U ) → F l V ( F V ) of c omplexes inductively . As- sume that we have alrea dy constructed the morphisms f ∗ ( F l i U ( F U )) → F l i V ( F V ) for all i ≤ k co mpatible with the diﬀerential. Using that f ∗ is right exact (Lemma 6 .1 .9), we have a n induced morphism f ∗ coker ( F l k − 1 U ( F U ) → F l k U ( F U )) → co ker ( F l k − 1 V ( F V ) → F l k V ( F V )) . The construction ab ove gives a morphism f ∗ F l k +1 U ( F U ) → F l k +1 V ( F V ), a gain com- patible with the diﬀer ent ial of the complexes. In this way we g et the morphism f ∗ F l U ( F U ) → F l V ( F V ). By an ins pec tion of the construction w e c heck tha t for a second mor phism g : W → V in X the morphisms g ∗ f ∗ F U ( F U ) → g ∗ F V ( F V ) → F W ( F W ) a nd ( f ◦ g ) ∗ F U ( F U ) → F W ( F W ) co incide. The colle ctions o f resolutions F U → F l U ( F U ), ( U → X ) ∈ X , deter mines a resolu- tion F → F l ( F ) in C + ( Sh Ab X ). 3.1.11 . — L emma 3.1.9 . — The asso ciation F 7→ ( F → F l ( F )) is a fun ctorial ﬂ abby r esolu- tion. Pr o of . — The lo ca l constructions F U 7→ F l U ( F U ) ar e functoria l in F U . The connect- ing maps f ∗ F l U ( F U ) → F l V ( F V ) are co mpatible w ith this functoriality . It follo ws that the c o nstruction F → F l ( F ) is functor ial in F . The res tr ictions S h X → Sh ( U ) detect ﬂa bbiness a nd exact seq uences (see 6 .1.14). Therefore the lo c a l statement s 3.1.6 and 3 .1.7 imply that the sequence 0 → F → 3.2. A MODEL FOR THE PUSH-FOR W ARD 31 F l ( F ) is a quasi-isomo rphism, and that the she aves F l k ( F ) are ﬂabby for all k ≥ 0. Deﬁnition 3.1.10 . — We c al l F → F l ( F ) the functorial ﬂabby r esolution of F . Note that it actually pro duces res olutions by ﬂasque sheav es. 3.1.12 . — Let f : X → Y b e a map of lo c a lly c o mpact stacks which has lo ca l sec tions. Let F l X and F l Y denote the ﬂabby resolution functors for X and Y accor ding to Deﬁnition 3 .1.10. L emma 3.1.11 . — We have a natur al isomorphism of funct ors f ∗ ◦ F l Y ∼ = F l X ◦ f ∗ . Pr o of . — F or ( U → X ) ∈ X we have by 6.1 .11 a na tural isomor phism f ∗ F U ∼ = F f ♯ U . It gives natural is o morphisms F l U (( f ∗ F ) U ) ∼ = F l f ♯ U ( F f ♯ U ) and thus F l X ( f ∗ F ) U ∼ = ( f ∗ F l Y ) U . Finally this collection of isomorphisms gives a na tural is omorphism F l X ( f ∗ F ) ∼ = f ∗ F l Y ( F ) . 3.1.13 . — L emma 3.1.12 . — The fun ct orial ﬂabby r esolution functor pr eserves ﬂatn ess . Pr o of . — Consider a spa ce U , p : ˆ U → U as ab ove and a ﬂa t sheaf F ∈ Sh Ab ( U ). Then cok er ( F → p ∗ p ∗ ( F )) is ﬂat as s hown in the pro o f of [ KS94 , Lemma 3.1.4]. This implies inductiv ely that the sheav es F l k U ( F ) are ﬂat for a ll k ≥ 0. The result for the functorial ﬂabby res olution functor on Sh Ab X now follows fro m the fact that the restriction functor s Sh Ab X → Sh Ab ( U ) detect ﬂatness (see 6.2.6). 3.1.14 . — W e can extend the ﬂabby r esolution functor 3.1.10 to a quasi-iso morphism preserving functor F l : C + ( Sh Ab X ) → C + ( Sh Ab X ) by apply ing F l to a complex term-wise a nd for ming the total complex o f the resulting double co mplex . 3.2. A mo de l for the push-forw ard 3.2.1. — Le t f : G → X b e a morphism of lo cally compact stacks w hich has local sections. F ollowing [ BSS0 7 , Sec. 2.4] we construct a nice mo de l for the functor Rf ∗ ◦ f ∗ : D + ( Sh Ab X ) → D + ( Sh Ab X ). W e choos e an atlas a : A → G . Then b y Prop ositio n 6.1.1 the comp o s ition f ◦ a : A → G → X is r epresentable. Then we can deﬁne the functor p C A : C + ( Pr Ab G ) → C + ( Pr Ab X ) as in [ BSS07 , Sec. 2 .4] (the subscript p indicates that it acts b etw een c ategories of presheav es). 32 CHAPTER 3. FUNCTORIAL P E RIODIZA TION 3.2.2. — W e recall the deﬁnition p C A . F o r ( U → X ) consider the C a rtesian diagr am G U   / / G f   U / / X . Then for F ∈ Pr Ab G we have (3.2.1) p C k A ( F )( U → X ) = F (( A × G · · · × G A | {z } k +1 f ac tor s ) × G G U → G ) . The diﬀerential p C k A ( F )( U → X ) → p C k +1 A ( F )( U → X ) is induced as usua l as an alternating sum by the pro jections ( A × G · · · × G A | {z } k +2 f ac tor s ) → ( A × G · · · × G A | {z } k +1 f ac tor s ) . W e extend the functor p C A to sheav es by the fo r mula C A := i ♯ ◦ p C A ◦ i . 3.2.3. — The functor i ♯ : C + ( Pr Ab X ) → C + ( Sh Ab X ) is exact by 6.1.8. The functor p C A is exa ct, see [ BSS07 , 2.4.8]. Since ﬂabby sheav e s are i -acyclic the functor i ◦ F l : C + ( Sh Ab X ) → C + ( Pr Ab X ) pres erves qua s i- isomorphisms. Therefore the co mpo sition i ♯ ◦ p C A ◦ i ◦ F l = C A ◦ F l : C + ( Sh Ab G ) → C + ( Sh Ab X ) preserves qua si-isomor phisms and des cends to the homo topy catego ries (1) C A ◦ F l : hC + ( Sh Ab G ) → hC + ( Sh Ab X ) . After iden tiﬁca tio n o f the homotopy categor ies with the der ived categor ies we hav e by [ BSS07 , 2.41 ] that C A ◦ F l ∼ = Rf ∗ : D + ( Sh Ab G ) → D + ( Sh Ab X ) . 3.2.4. — Since f : G → X has lo c a l sectio ns the functor f ∗ is exact. It therefore descends to f ∗ : hC + ( Sh Ab X ) → hC + ( Sh Ab G ) . The comp osition C A ◦ F l ◦ f ∗ : hC + ( Sh Ab X ) → hC + ( Sh Ab X ) th us repres ent s Rf ∗ ◦ f ∗ : D + ( Sh Ab X ) → D + ( Sh Ab X ) . (1) By abuse of notation we use the same s ymbol 3.2. A MODEL FOR THE PUSH-FOR W ARD 33 3.2.5. — W e now study the dep endence o f C A on the choice of the atlas A → G . Let us cons ide r a diagr am (3.2.2) A ′ φ / / a ′ A A A A A A A A a          G , where a ′ satisﬁes the same a ssumptions as a (see 3.2.1). The map φ induces maps ( A ′ × G · · · × G A ′ ) × G G U φ k +1 × id G U / / ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q ( A × G · · · × G A ) × G G U v v m m m m m m m m m m m m m m G and therefo r e p C k A ( F )( U → X ) = F (( A × G · · · × G A | {z } k +1 f actor s ) × G G U → G ) → F (( A ′ × G · · · × G A ′ | {z } k +1 f ac tor s ) × G G U → G ) = p C k A ′ ( F )( U → X ) . This map is natural in F a nd preser ves the cosimplicial s tructures. In other words, the dia gram (3.2 .2) induces a natura l tra nsformation p C φ : p C A → p C A ′ . Comp osing with i ♯ and i ◦ F l we get a mor phism of functor s C φ : C A ◦ F l → C A ′ ◦ F l : hC + ( Sh Ab G ) → hC + ( Sh Ab X ) . Both C A ◦ F l and C A ′ ◦ F l repre sent R f ∗ . Using the ex plicit constructio ns and the pro of o f [ BSS07 , Lemma 2.3 6 ] one chec ks that the diagra m H 0 ( C A ◦ F l )( F ) H 0 ( C φ ) / / ' ' O O O O O O O O O O O H 0 ( C A ′ ◦ F l )( F ) w w o o o o o o o o o o o f ∗ ( F ) commutes. Therefore, o n the level of der ived categ ories, C φ : C A ◦ F l → C A ′ ◦ F l is the c a nonical iso mo rphism b etw een tw o r ealizations o f Rf ∗ . 34 CHAPTER 3. FUNCTORIAL P E RIODIZA TION 3.2.6. — Let q : H → G b e a re pr esentable morphism with local sec tions. Consider the pullba ck dia g ram B b / / l   H q   A a / / G f   X . Then b : B → H is an atlas, and w e can form the functor C B : C + ( Pr Ab H ) → C + ( Pr Ab X ). Observe that B × H · · · × H B ∼ = ( A × G · · · × G A ) × G H . F or ( U → X ) consider the diagr am H U / /   H q   G U   / / G f   U / / X . Observe further that ( B × H · · · × H B ) × H H U ∼ = ( A × G · · · × G A ) × G G U × G H . F or a preshe a f F ∈ Pr H and ( V → G ) ∈ G we have p q ∗ ( F )( V ) = F ( V × G H ). W e now have the following identit y p C k A ◦ p q ∗ ( F )( U → X ) ∼ = p q ∗ ( F )(( A × G · · · × G A ) | {z } k +1 f actor s × G G U → G ) ∼ = F ((( A × G · · · × G A | {z } k +1 f actor s ) × G G U ) × G H → H ) ∼ = F (( B × H · · · × H B | {z } k +1 f actor s ) × H H U → H ) ∼ = p C k B ( F )( U → X ) This iso mo rphism is functorial in F and induces a natural isomor phism p C A ◦ p q ∗ ∼ = p C q ∗ A , where we write q ∗ A := B . 3.2. A MODEL FOR THE PUSH-FOR W ARD 35 The functor p q ∗ preserves sheav es [ BSS07 , Lemma 2 .13]. Therefor e we get the ident ity i ◦ i ♯ ◦ p q ∗ ◦ i = p q ∗ ◦ i and thus an isomor phism (3.2.3) C A ◦ q ∗ ∼ = i ♯ ◦ p C A ◦ i ◦ i ♯ ◦ p q ∗ ◦ i ∼ = i ♯ ◦ p C A ◦ p q ∗ ◦ i ∼ = i ♯ ◦ p C q ∗ A ◦ i ∼ = C q ∗ A . 3.2.7. — Consider a Ca rtesian diagra m H g   v / / G   Y u / / X where u has lo cal sections . W e extend the diag ram to B   / / A   H v / / g   G f   Y u / / X . The map B → H is again an a tlas. L emma 3.2.4 . — We have a natur al isomorphism of functors u ∗ ◦ C A ∼ = C B ◦ v ∗ . Pr o of . — W e ﬁr st ﬁnd a na tur al iso morphism p u ∗ ◦ p C A ∼ = p C B ◦ p v ∗ . Let ( U → Y ) ∈ Y a nd F ∈ Pr Ab G . Then we have p u ∗ ◦ p C A ( F )( U ) ∼ = p C A ( F )( u ♯ U ) . W e hav e a diag ram H U ∼ = G u ♯ U / /   H v / / g   G   U / / Y u / / X . W e calculate ( A × G · · · × G A ) × G G u ♯ U ∼ = ( A × G · · · × G A ) × G H × H G u ♯ U ∼ = v ♯ ( B × H · · · × H B ) × H H U 36 CHAPTER 3. FUNCTORIAL P E RIODIZA TION This implies that p u ∗ ◦ C A ( F )( U ) ∼ = p C A ( F )( u ♯ U ) ∼ = F (( A × G · · · × G A ) × G G u ♯ U ) ∼ = F ( v ♯ (( B × H · · · × H B ) × H H U )) ∼ = ( p v ∗ F )(( B × H · · · × H B ) × H H U ) ∼ = p C B ◦ p v ∗ ( F )( U ) Since u and v ha ve lo cal sections, by 6 .1.11 the functors p u ∗ and p v ∗ commute with i ◦ i ♯ , and this iso morphism induces u ∗ ◦ C A ∼ = C B ◦ v ∗ (compare with the c a lculation (3.2.3)). 3.2.8. — The isomor phisms of Lemma 3.2.4 and L e mma 3.1.11 induce a n is o mor- phism (3.2.5) u ∗ ◦ C A ◦ F l ∼ = C B ◦ u ∗ ◦ F l ∼ = C B ◦ F l ◦ v ∗ . On the other hand, by Lemma 6.1.12 we have a n isomo rphism u ∗ ◦ Rf ∗ ∼ = Rg ∗ ◦ v ∗ . L emma 3.2.6 . — The fol lowing diagr am of natura l isomorphisms of functors D + ( Sh Ab G ) → D + ( Sh Ab H ) c ommutes. u ∗ ◦ C A ◦ F l ∼ =   ∼ = / / C B ◦ F l ◦ v ∗ ∼ =   u ∗ ◦ Rf ∗ ∼ = / / Rg ∗ ◦ v ∗ Pr o of . — It is easy to chec k that this commutativit y holds tr ue on the lev el of ze- roth co homology sheav es. Since all functors are the deriv ed versions of their zeroth cohomolog y functor s the req uired co mm utativity fo llows. Cor ol lar y 3.2.7 . — The fol lowing diagr am o f natura l isomorphisms c ommu tes u ∗ ◦ C A ◦ F l ◦ f ∗ ∼ =   ∼ = / / C B ◦ F l ◦ g ∗ ◦ u ∗ ∼ =   u ∗ ◦ Rf ∗ ◦ f ∗ ∼ = / / Rg ∗ ◦ g ∗ ◦ u ∗ 3.3. ZIG-ZAG DIAGRAMS AND LIM ITS 37 3.3. Zig-zag diagrams and limits 3.3.1. — W e deﬁne the unbo unded der ived catego ry D ( A ) o f an ab elian catego ry as the ho motopy categ ory hC ( A ) of complexes (with no restrictio ns ) in A . Deﬁnition 3.3.1 . — An ab elian c ate gory A with the fol lowing pr op erties (1) A is c o c omplete, (2) ﬁ lt er e d c olimits ar e exact, (3) A has a gener ator, i.e. ther e is an obj e ct Z such that for every o bje ct F with pr op er sub obje ct F ′ ⊂ F , Hom ( Z, F ′ ) → Hom ( Z , F ) is not surje ctive. is c al le d a Gr othendie ck ab elian c ate gory. In this section, we will consider a Grothendieck categ ory in which countable pro d- ucts exist, e.g. a complete Grothendieck catego r y . The catego ry Sh Ab X of sheav es of ab elian g roups on a site X is a co mplete Grothendieck ab elian ca tegory [ T am9 4 , Chapter I, Thm. 3.2.1]. L emma 3.3.2 . — If Z is a sm al l c ate gory and A is a Gr othendie ck ab elian c ate- gory in which c ountable pr o ducts exists, then the diagr am c ate gory A Z is again a Gr othendie ck ab elian c ate gory in which c ountable pr o ducts exist . This is pr oved in [ T am94 , 1 .4.3]. 3.3.2. — W e co nsider the catego ry C ( A ) of c omplexes in a Gr o thendieck ab elia n category A . It is known that C ( A ) has a mo del catego ry s tructure (see [ Hov 01 , Theorem 2.2] where this fact is attributed to Joy al, [ Hov99 , Thm. 2 .3.12] for the example o f the catego ry of mo dules over a r ing, a nd [ Bek00 ] for a pr o of in genera l). This mo del str uc tur e is g iven by the following da ta: (1) The weak equiv ale nc e s ar e the quas i-isomorphisms . (2) The co ﬁbrations a re the degree- wise injections. (3) The ﬁbr ations ar e deﬁned by the right lifting prop erty . By hC ( A ) we denote the homotopy categor y of C ( A ). The categ ory hC ( A ) is trian- gulated with the shift functor T : hC ( A ) → hC ( A ) given b y the shift of complexe s T ( X ) = X [1]. The class of distinguished triang le s is genera ted by the mapping cone sequences on C ( A ), · · · → A f → B → C ( f ) → T ( A ) . . . . The extensio n of a mor phism in [ f ] ∈ hC ( A ) with chosen repre s entativ e f ∈ C ( A ) to a tr iangle can thus naturally b e deﬁned using the mapping cone C ( f ). 3.3.3. — Let A b e a Gro thendieck ab elia n ca tegory , and consider a s ma ll ca tegory Z . Then w e hav e an equiv a lence C ( A ) Z ∼ = C ( A Z ). Because A Z is a Gro thendieck category by Lemma 3 .3 .2, we ca n equip the catego ry of Z -diagra ms C ( A ) Z with the injective mo del category s tr ucture. By trans lation of 3.3.2 we get the following description. 38 CHAPTER 3. FUNCTORIAL P E RIODIZA TION (1) The weak equiv ale nc e s ar e the level-wise quasi-is omorphisms. (2) The co ﬁbrations a re the level-wise injections. (3) The ﬁbr ations ar e deﬁned by the right lifting prop erty . 3.3.4. — W e consider the categor y U pictured by • • o o / / • • o o   • O O • . W e let D ( A ) ⊂ C + ( A ) U be the s ubca tegory o f ob jects of the form (3.3.3) Y 0 Y 1 ∼ o o / / Y 2 Y 3 ∼ o o   X O O X [ − 2] with b ounded b elow complex e s Y i , X . A morphism in the category D ( A ) is given by maps Y i → Y ′ i , i = 0 , 1 , 2 , 3, and X → X ′ which a re co mpatible with the struc- ture maps. A qua si-isomor phism in this ca tegory is a morphism which is a quasi- isomorphism level-wise. 3.3.5. — W e let Z be the categ ory pictured by . . .   ; ; ; ; ; ; ; ; . . . • / /   @ @ @ @ @ @ @ • • / /   @ @ @ @ @ @ @ • • / /   @ @ @ @ @ @ @ • • / / • . Let C ( A ) Z be the c ategory of Z -diagra ms of c o mplexes in A . W e deﬁne a functor R 1 : D ( A ) → C ( A ) Z 3.3. ZIG-ZAG DIAGRAMS AND LIM ITS 39 which maps the dia gram (3.3 .3) to the Z -diagra m . . . " " D D D D D D D D D D . . . Y 3 [4] / / " " F F F F F F F F Y 2 [4] Y 1 [2] / / " " F F F F F F F F Y 0 [2] Y 3 [2] / / Y 2 [2] . The maps a re induced by the shifted ma ps of the diagr am (3.3.3), a nd the comp osition Y 3 [2 k + 2 ] → X [2 k ] → Y 0 [2 k ]. The functor R 1 preserves quasi- isomorphisms, since those a re deﬁned level-wise. 3.3.6. — W e now deﬁne a tr iangulated functor lim : h ( C ( A ) Z ) → hC ( A ) by a dir ect co nstruction o n the level of co mplex es. Consider a Z - dia gram X ∈ C ( A ) Z C 3 c 3 / / d 3 B B B B B B B B B 3 C 2 c 2 / / d 2 B B B B B B B B B 2 C 1 c 1 / / d 1 B B B B B B B B B 1 C 0 c 0 / / B 0 . W e deﬁne the mo rphism in C ( A ) φ X : Y i ≥ 0 C i → Y i ≥ 0 B i which maps ( x i ) i ≥ 0 to ( c i ( x i ) − d i +1 ( x i +1 )) i ≥ 0 . Then we deﬁne lim( X ) as a shifted cone o f φ X : lim( X ) := C ( φ X )[ − 1] ∈ C ( A ) . Since qua si-isomor phisms in C ( A ) Z are deﬁned level-wise, the functorial cons truction X → lim X pr eserves q uasi-isomo rphisms and thu s de s cends to a functor lim : h ( C ( A ) Z ) → hC ( A ) . Note that lim commutes with the shift and sum, so that it is a tr iangulated functor. 40 CHAPTER 3. FUNCTORIAL P E RIODIZA TION 3.3.7. — W e now consider the co mpo sition lim ◦ R 1 : D ( A ) → hC ( A ). The comp osi- tion of the maps (or their inverses, resp ectively) in the diag r am (3.3.3) gives r ise to a morphism D [ − 2] : X → X [ − 2] in hC ( A ). W e co ns ider the sequence (3.3.4) X • : X D ← X [2] D [2] ← X [4] ← . . . . in hC ( A ). As alre a dy explained in 2 .4, for suc h a diagr am in the tr iangulated category hC ( A ) the homotopy limit holim ( X • ) ∈ hC ( A ) is a well-deﬁned isomorphis m class of ob jects. It is g iven by the mapping cone shifted by − 1 o f the mo rphism Y i ≥ 0 X [2 i ] → Y i ≥ 0 X [2 i ] which maps ( x i ) i ≥ 0 to ( x i − D [2 i ] x i +1 ) i ≥ 0 (see [ Nee 01 , Sec. 1 .6]). L emma 3.3.5 . — F or a diagr am W ∈ D ( A ) of the form (3.3.3) we have a non- c anonic al isomorphism holim ( X • ) ∼ = lim ◦ R 1 ( W ) . Pr o of . — W e use the dual statement of [ Nee01 , Lemma 1.7.1]. F or i ≥ 1 let C 2 i − 1 = Y 3 [2 i ], C 2 i := Y 1 [2 i ], B 2 i − 1 := Y 2 [2 i ] and B 2 i := Y 0 [2 i ]. No te that we hav e morphisms v i : C i → B i in C ( A ) which b e come is o morphisms in hC ( A ). Mor eov er , we hav e maps w 2 i : C 2 i → B 2 i − 1 coming from the map Y 1 → Y 2 of (3.3.3), and morphisms w 2 i +1 : C 2 i +1 → B 2 i coming from Y 3 [2] → X → Y 0 of (3 .3 .3). W e consider the diagram in hC ( A ), using the inv ertibility o f v i in h C ( A ), Q i ≥ 1 C i Q v i − Q w i − − − − − − − → φ R 1 ( W ) Q i ≥ 1 B i   y id   y Q i ≥ 1 v − 1 i Q i ≥ 1 C i − − − − → Q i ≥ 1 C i , whose vertical maps a re isomor phism. By deﬁnition, the mapping cone o f the upp er horizontal map is lim ◦ R 1 ( W ). Because the vertical maps are iso mo rphisms in h C ( A ), this is isomorphic to the mapping cone of the lower ho rizontal map, which gives the homotopy limit of the sequence Y 3 [2] ← Y 1 [2] ← Y 3 [4] ← Y 1 [4] ← Y 3 [6] . . . . W e can expa nd this sequence to (3.3.6) X ← Y 3 [2] ← Y 2 [2] ← Y 1 [2] ← Y 0 [2] ← X [2] ← Y 3 [4] ← Y 2 [4] ← Y 1 [4] ← Y 0 [4] ← X [4] ← Y 3 [6] . . . , and because the s equence (3.3 .4) is just another contraction o f (3.3 .6), b y [ N ee01 , Lemma 1.7 .1] its homotopy limit hol im ( X • ) is then a lso isomo r phic to lim ◦ R 1 ( W ). 3.4. THE F UNCTORIAL PERIODIZA TION 41 3.4. The func torial p e rio dization 3.4.1. — L e t X be a lo cally compa ct stack. Deﬁne C + ( Sh f lat Ab X ) ⊆ C + ( Sh Ab X ) to b e the full sub ca teg ory of b ounded b elow complexe s of ﬂat sheav es. L emma 3.4.1 . — This inclusion induc es an e quivalenc e of homotopy c ate gories hC + ( Sh f lat Ab X ) ∼ → hC + ( Sh Ab X ) . Pr o of . — W e ﬁr st construct a functor ial ﬂa t r esolution functor R : S h Ab X → C b ( Sh f lat Ab X ) . Note that a torsion free sheaf is ﬂat. If F ∈ Sh Ab X , then let ˆ F ∈ Pr X denote the underlying preshea f o f sets. Let Z ˆ F ∈ Pr Ab X b e the presheaf of free ab elia n g roups generated by ˆ F , and Z F := i ♯ Z ˆ F b e its sheaﬁﬁca tion. Then we have a natural ev aluation Z ˆ F → F , whic h extends uniquely to e : Z F → F since F is a sheaf. W e deﬁne R ( F ) to b e the complex ker( e ) → Z F , where Z F is in degree zero . The natural map R ( F ) → F is a qua si-isomor phism. Moreover, Z F a nd its subsheaf ker( e ) ar e torsion-fr e e , hence ﬂat. W e extend R to a functor R : C + ( Sh Ab X ) → C + ( Sh f lat Ab X ) by applying R ob jectwise and taking the total co mplex of the resulting double complex. The inclusion C + ( Sh f lat Ab X ) → C + ( Sh Ab X ) and R : C + ( Sh Ab X ) → C + ( Sh f lat Ab X ) induce m utually inv ers e functors of the homotopy categor ies. 3.4.2. — Let f : G → X b e a top ologica l ger b e with ba nd U (1) ov er a lo cally compact stack. Reca ll the asso ciated geometry introduced in 2.2 .1. Using the functorial version we ge t the diagr am (3.4.2) T 2 × G p { { w w w w w w w w w m # # G G G G G G G G G G f # # G G G G G G G G G G f { { w w w w w w w w w X which 2-functorially depends o n the g erb e G → X . The map p : T 2 × G → G is the pro jection o nt o the s econd facto r, a nd m := p ◦ φ . 3.4.3. — Obser ve that p is a triv ial oriented ﬁb er bundle with ﬁb er T 2 . Let 0 → Z Site ( T 2 × G ) → F l ( Z Site ( T 2 × G ) ) be the functoria l ﬂat and ﬂabby r esolution o f Z G constructed in 3 .1.10, see also 3 .1.12 for ﬂa tness. By K · : 0 → K 0 → K 1 → K 2 → 0 42 CHAPTER 3. FUNCTORIAL P E RIODIZA TION we denote the tr unca tion of F l ( Z Site ( T 2 × G ) ) a fter the seco nd term, i.e. with K 2 := ker ( F l 2 ( Z Site ( T 2 × G ) ) → F l 3 ( Z Site ( T 2 × G ) )) . The complex K · is still a ﬂat a nd p ∗ -acyclic r esolution of Z Site ( T 2 × G ) (Lemma 6.3 .3). Let T : C + ( Sh Ab Site ( T 2 × G )) → C + ( Sh Ab Site ( T 2 × G )) be the functor given on ob jects by T K · ( F ) := F ⊗ K · . 3.4.4. — W e co nsider the comm uta tive diagr am 3.4.2. Since f ◦ p ∼ = f ◦ m (recall that we a ctually can ass ume eq uality) we have by Lemma 6 .6 .8 and Co rollar y 6.6.9 isomorphisms o f functors m ∗ ◦ f ∗ ∼ = p ∗ ◦ f ∗ and f ∗ ◦ m ∗ ∼ = f ∗ ◦ p ∗ . W e ﬁx an atlas A → G and deﬁne X : C + ( Sh f lat X ) → C + ( Sh X ) by X := C A ◦ f ∗ ◦ F l . Since f has lo cal sections we hav e f ∗ ◦ F l ∼ = F l ◦ f ∗ by Le mma 3.1.11. It now follows from 3.2.4 that X ∼ = C A ◦ F l ◦ f ∗ preserves q ua si-isomor phisms. It therefore descends to the homoto py ca teg ories and induces the functor Rf ∗ ◦ f ∗ D + ( Sh Ab G ) Lemma 3.4 .1 ∼ = hC + ( Sh f lat Ab G ) X → hC + ( Sh Ab X ) ∼ = D + ( Sh Ab X ) . 3.4.5. — W e fur ther form B := m ∗ A × T 2 × G p ∗ A . It comes with a natura l mor phism B → m ∗ A over T 2 × G which induces a transfor mation C m ∗ A → C B . Using the unit id → m ∗ ◦ m ∗ , the inclusion id → T K · , and the iso morphisms m ∗ ◦ f ∗ ∼ = p ∗ ◦ f ∗ , and using that by 3.2.6 C A ◦ m ∗ ∼ = C m ∗ A , we deﬁne a natura l trans fo rmation X = C A ◦ f ∗ ◦ F l → C A ◦ m ∗ ◦ m ∗ ◦ f ∗ ◦ F l → C A ◦ m ∗ ◦ T K · ◦ m ∗ ◦ f ∗ ◦ F l ∼ = C m ∗ A ◦ T K · ◦ m ∗ ◦ f ∗ ◦ F l ∼ = C m ∗ A ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l → C m ∗ A ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l → C B ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l =: Y 0 Using the other pro jection B → p ∗ A we deﬁne Y 1 := C p ∗ A ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ∼ → C B ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ∼ → C B ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l = Y 0 . 3.4. THE F UNCTORIAL PERIODIZA TION 43 Using the identit y C p ∗ A ∼ = C A ◦ p ∗ we deﬁne Y 1 = C p ∗ A ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ∼ = C A ◦ p ∗ ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ → C A ◦ F l ◦ p ∗ ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ =: Y 2 Note that p ∗ ◦ T K is an e xact functor by Lemma 6.3.6 and ca lculates R p ∗ by Co r ollary 6.4.4. Since p ∗ ◦ F l ◦ T K represents the s ame functor the map p ∗ ◦ T K → p ∗ ◦ F l ◦ T K induces a qua si-isomor phism whic h is prese rved b y C A ◦ F l . The natural transformatio n T p ∗ K · ∼ − → p ∗ ◦ T K · ◦ p ∗ is an iso morphism, if applied to complexes of ﬂat sheaves by 6.2 .11. By Lemma 6.1.11 the pull-back f ∗ preserves ﬂatness. These tw o fac ts explain the quasi-is omorphisms in Y 3 := C A ◦ F l ◦ T p ∗ K · ◦ f ∗ ∼ → C A ◦ F l ◦ p ∗ ◦ T K · ◦ p ∗ ◦ f ∗ ∼ → C A ◦ F l ◦ p ∗ ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ = Y 2 . Using the pro jection T p ∗ K [ − 2] → id of (6.5.8) we deﬁne the natur al tra nsformation Y 3 = C A ◦ F l ◦ T p ∗ K · ◦ f ∗ (3.4.3) → C A ◦ F l ◦ f ∗ [ − 2] ∼ = C A ◦ f ∗ ◦ F l [ − 2] = X [ − 2] . Observe that all functors Y i preserve quasi-iso morphisms, using that f ∗ , p ∗ , C A ◦ F l , p ∗ ◦ T K (and by Lemma 6 .2.11 ther efore also T p ∗ K ) do so. 3.4.6. — The co nstruction 3.4.4, 3 .4.5 g ives a quasi- is omorphism pre s erving functor R 0 : C + ( Sh f lat Ab X ) → D ( Sh Ab X ) (see 3.3.4 for the deﬁnition of the targ e t). By comp osition with the functor R 1 (see 3.3.5) we g e t a functor R := R 1 ◦ R 0 : C + ( Sh f lat Ab X ) → C ( Sh Ab X ) Z . It prese r ves quasi-isomo r phisms and ther efore descends to (a gain using Lemma 3.4.1) R : D + ( Sh Ab X ) → h ( C ( Sh Ab X ) Z ) . 3.4.7. — The c o nstruction of the functor R 0 explicitly dep ends on the c hoic e of an atlas A → G . These choices form a sub c ategory Z ⊂ Sta cks /G . The choice o f A → G enters the deﬁnition via the functor C A . F o r the moment let us indicate the depe ndence on A in the notation and write R A 0 for the fun ctor R 0 deﬁned with the choice A . 44 CHAPTER 3. FUNCTORIAL P E RIODIZA TION Observe, that A → m ∗ A , A → p ∗ A and A → m ∗ A × T 2 × G p ∗ A ar e functors Stacks /G → S tacks / ( T 2 × G ). The co nstruction 3.2.5 s hows that for a giv en F ∈ D + ( Sh Ab X ) the asso cia tion A → R A 0 ( F ) extends to a functor R ... 0 ( F ) : Z op → D ( Sh Ab X ) . The co mpo nent s X ∼ = C A ◦ F l ◦ f ∗ and Y i ∼ = C ∗ ◦ F l ◦ . . . (where ∗ ∈ { A, p ∗ A, m ∗ A, m ∗ A × T 2 × G p ∗ A } ) all in volv e a ﬂabby resolutio n functor in front of C ∗ . If A → A ′ is a mor phism in Z , then the transfor mation C A ′ ◦ F l → C A ◦ F l (or the s imila r transfor mations for the o ther subscripts) pro duce qua si-isomor phisms by 3.2 .5. It follows that the functor R ... 0 ( F ) : Z op → D ( Sh Ab X ) maps all mo rphisms to quasi- isomorphisms. W e now consider the comp os ition R ... ( F ) := R 1 ◦ R ... 0 ( F ) : Z op → h ( C ( Sh Ab X ) Z ). F or tw o ob jects A, B ∈ Z w e co nsider the diagr am A × B s | | x x x x x x x x x t # # F F F F F F F F F A B , where the ﬁb er pr o duct is taken in Stack s /G . W e co nsider the isomo r phism R ( A, B ) := R t ◦ ( R s ) − 1 : R A ( F ) → R B ( F ) in h ( C ( Sh Ab ( X )) Z ). Using the commutativit y of the s quares in the diag ram A × B × C x x q q q q q q q q q q & & M M M M M M M M M M   A × B | | x x x x x x x x x & & M M M M M M M M M M M M A × C t t i i i i i i i i i i i i i i i i i i i i * * U U U U U U U U U U U U U U U U U U U U B × C # # F F F F F F F F F x x q q q q q q q q q q q q A B C we chec k tha t R ( A, B ) ◦ R ( B , C ) = R ( A, C ) . This has the following conseq uence:. L emma 3.4.4 . — The functor R : D + ( Sh Ab X ) → hC (( Sh Ab X ) Z ) is indep endent of the choic e of t he atlas A → G up to c anonic al isomorphism. Consider an automorphism φ : A → A in Z and observe tha t it induces the identit y on the level of coho mology , i.e . H ∗ ( R φ ) = id . It is an interesting ques tion whether R φ is the identit y . 3.5. PROPER TIE S OF THE PERIODIZA TION FUNCTOR 45 3.4.8. — Deﬁnition 3.4.5 . — We deﬁne the p erio dization f unctor P G := lim ◦ R : D + ( Sh Ab X ) → h ( C (( Sh Ab X ) Z )) → hC ( Sh Ab X ) . By Lemma 3.4.4 it is well deﬁned up to canonical iso morphism. 3.4.9. — Let F ∈ D + ( Sh Ab X ). By 3.2.4 X ( F ) = C A ◦ f ∗ ◦ F l ( F ) represents R f ∗ ◦ f ∗ ( F ). The c o mpo sition D [ − 2] : X → X [ − 2 ] of the maps (or their in verses, resp ec- tively) in the diagra m R A 0 ( F ) ∈ D ( Sh Ab X ) re pr esents the map D G : R f ∗ ◦ f ∗ ( F ) → Rf ∗ ◦ f ∗ ( F )[ − 2] deﬁned in Deﬁnition 2 .2.2. By Lemma 3.3.5 we see that P G ( F ) (a c- cording to 3.4.5) is isomor phic to our former Deﬁnition 2.4.2 of the iso morphism class P G ( F ). 3.5. Prop erties of the p erio dization functor 3.5.1. — The domain and the target of P G are triangula ted categories . Distinguished triangles in both categories are a ll triangles whic h are isomorphic to mapping cone sequences · · · → C ( f )[ − 1] → X f → Y → C ( f ) → . . . . L emma 3.5.1 . — The funct or P G : D + ( Sh Ab X ) → hC ( Sh Ab X ) is triangulate d. Pr o of . — W e m ust show that it is additive, preser ves the shift, and maps distin- guished tr iangles to distinguished triangles . It follows from the ex plicit constructions that the functors lim a nd R 1 are additive and preserve the shift. The functor ia l ﬂabby r e solution F l on sheav es is additive. On complexes of sheaves it is deﬁned a s the level-wise applicatio n of the ﬂabby reso lutio n functor compo sed with the total complex co nstruction. Ther efore it also c ommut es w ith the shift. All o ther functors inv olved in the constr uction of R 0 (e.g. C A , q ∗ , T K · ) are additive and commute with the s hift, to o. Since the distinguished triang les in D + ( Sh Ab X ), h ( C ( Sh Ab X ) Z ), and hC ( Sh Ab X ) are deﬁned as triangles which are isomorphic to mapping cone sequences, and the latter o nly dep end on the additive structure and the shift, we see that lim and R preserve triangles. 3.5.2. — L emma 3.5.2 . — F or F ∈ D + ( Sh Ab X ) t he obje ct P G ( F ) ∈ h C ( Sh Ab X ) is two- p erio dic. Pr o of . — The isomo r phism P G ( F )[2] → P G ( F ) is given by the is o morphism W in 2.4.3. The tw o p erio dicit y will b e a nalyzed in mo re detail in Subsection 3.6. 46 CHAPTER 3. FUNCTORIAL P E RIODIZA TION 3.5.3. — Let u : Y → X be a map of top ologica l stacks which admits lo cal s e ctions. Then we co nsider a Car tesian diagr am (3.5.3) H v − − − − → G   y g   y f Y u − − − − → X . L emma 3.5.4 . — The diagr am (3.5.3) induc es an isomorphism u ∗ ◦ P G ∼ − → P H ◦ u ∗ . Pr o of . — By taking the pull-back of (3 .4.2) alo ng u we g et the extension of the Cartesian dia gram ab ove to T 2 × H n,q   w / / T 2 × G m,p   H v / / g   G f   Y u / / X . Note that there is no 2-isomor phism b etw een n a nd q or m and p , resp ectively . Since u has loca l sections the functor u ∗ : Sh Ab X → Sh Ab Y is e xact b y Lemma 6.1.11. It therefore extends to functors u ∗ : D ( Sh Ab X ) → D ( Sh Ab Y ) a nd u ∗ : C ( Sh Ab X ) Z → C ( Sh Ab Y ) Z which b o th preserve qua si-isomor phisms. W e therefore also have co rre- sp onding functors on the derived categ ories which will all b e denoted by u ∗ . In the following we a re going to show that there a re na tural is omorphisms (1) u ∗ ◦ R 1 ∼ = R 1 ◦ u ∗ (2) u ∗ ◦ lim ∼ = lim ◦ u ∗ (3) u ∗ ◦ R 0 ∼ = R 0 ◦ u ∗ of functors o n the level of homotopy categor ies. In fact it follows fr om an insp ection of the construction of R 1 that alrea dy u ∗ ◦ R 1 ∼ = R 1 ◦ u ∗ on the level of functors D ( Sh Ab X ) → C ( Sh Ab Y ) Z , i.e. before descending to the ho motopy categ ory . Assertion (1 ) fo llows. Since u ∗ : C ( S h Ab X ) Z → C ( Sh Ab Y ) Z preserves pro ducts and mapping cones we again have u ∗ ◦ lim ∼ = lim ◦ u ∗ befo re going to the ho motopy categ o ries. This implies (2). In or der to see (3), using v we construct a canonica l isomorphism u ∗ ◦ R A 0 ∼ = R C 0 ◦ u ∗ : C + ( Sh f lat Ab X ) → D ( Sh Ab Y ) , 3.5. PROPER TIE S OF THE PERIODIZA TION FUNCTOR 47 where we indicate the dep endence o f the functor R 0 on the choices by a sup erscr ipt as in 3 .4.7. The atlas C → H is given by the diag ram C   / / A   H v / / g   G f   Y u / / X , where the upp er s quare is also Car tes ian. The isomorphism (3) is induced by a collection o f isomor phisms indexed by the ob jects o f the diagra m U (3.3.4) which induce a morphism of dia grams in h D ( Sh Ab Y ). First we hav e u ∗ ◦ X = u ∗ ◦ C A ◦ f ∗ ◦ F l ∼ = C C ◦ v ∗ ◦ f ∗ ◦ F l ∼ = C C ◦ g ∗ ◦ u ∗ ◦ F l ∼ = C C ◦ g ∗ ◦ F l ◦ u ∗ = X ◦ u ∗ (3.5.5) where we use Lemma 3 .2.4, v ∗ ◦ f ∗ ∼ = g ∗ ◦ u ∗ (see Lemma 6.6.9) and the fact that the ﬂabby resolution functor commutes with the pull-back by u , since u has lo cal sections (Lemma 3.1.11). Let D := n ∗ C × T 2 × H q ∗ C . W e write K · T 2 × G for the complex formerly de no ted by K · . Next w e obser ve that there is a canonical is omorphism w ∗ K · T 2 × G ∼ = K · T 2 × H . In fact K · T 2 × G and K · T 2 × H are g iven by truncations of the complexes F l ( Z Site ( T 2 × G ) ) and F l ( Z Site ( T 2 × H ) ). The isomorphism is induced by the fact that w ∗ commutes with the ﬂa bby r e s olution functor, and the isomor phism w ∗ Z Site ( T 2 × G ) ∼ = Z Site ( T 2 × H ) . This implies by Lemma 6.2.5 that w ∗ ◦ T K · T 2 × G ∼ = T K · T 2 × H ◦ w ∗ . In o rder to increase readability of the for mulas we will omit the double subscript from now on and write T K · for bo th functors. Using this observ a tion, Le mma 3.2.4, a nd the other pr eviously 48 CHAPTER 3. FUNCTORIAL P E RIODIZA TION used iso mo rphisms, we get u ∗ ◦ Y 0 ∼ = u ∗ ◦ C B ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l ∼ = C D ◦ w ∗ ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l ∼ = C D ◦ F l ◦ w ∗ ◦ T K · ◦ p ∗ ◦ f ∗ ◦ F l ∼ = C D ◦ F l ◦ T K · ◦ w ∗ ◦ p ∗ ◦ f ∗ ◦ F l ∼ = C D ◦ F l ◦ T K · ◦ q ∗ ◦ v ∗ ◦ f ∗ ◦ F l ∼ = C D ◦ F l ◦ T K · ◦ q ∗ ◦ g ∗ ◦ u ∗ ◦ F l ∼ = C D ◦ F l ◦ T K · ◦ q ∗ ◦ g ∗ ◦ F l ◦ u ∗ ∼ = Y 0 ◦ u ∗ In a simila r manner we get u ∗ ◦ Y 1 ∼ = u ∗ ◦ C p ∗ A ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ ∼ = C q ∗ C ◦ w ∗ ◦ F l ◦ T K · ◦ p ∗ ◦ f ∗ . . . ∼ = Y 1 ◦ u ∗ u ∗ ◦ Y 2 ∼ = Y 2 ◦ u ∗ u ∗ ◦ Y 3 ∼ = Y 3 ◦ u ∗ F or these iso mo rphisms, we use in particular Lemma 6.1.12 to g et v ∗ p ∗ ∼ = q ∗ w ∗ , and moreov er Lemma 6 .2.5 to ge t the chain of isomor phis ms v ∗ ( F ⊗ p ∗ K ) ∼ = v ∗ F ⊗ v ∗ p ∗ K ∼ = v ∗ F ⊗ q ∗ w ∗ K ∼ = v ∗ F ⊗ q ∗ K ∼ = T q ∗ K ( v ∗ F ) , which gives the iso morphism v ∗ ◦ T p ∗ K ∼ = T q ∗ K ◦ v ∗ . By a tedio us chec k of the commutativit y of many little sq ua res we see that these maps indeed deﬁne an iso morphism of functors u ∗ ◦ R A 0 ∼ = R C 0 ◦ v ∗ . As a n example of these chec ks, let us indicate some details of the ar gument for the map Y 3 → X [ − 2]. F or F ∈ D + ( Sh Ab X ) we have the ma ps φ : Y 3 ( F ) → X [ − 2 ]( F ) and ψ : Y 3 ( u ∗ F ) → X [ − 2]( u ∗ F ) given by (3.4.3). W e must show that u ∗ Y 3 ( F ) ∼ = / / u ∗ φ   Y 3 ( u ∗ F ) ψ   u ∗ X [ − 2]( F ) ∼ = / / X [ − 2]( u ∗ F ) 3.5. PROPER TIE S OF THE PERIODIZA TION FUNCTOR 49 commutes. This indeed follows fr om the sequence of co mmutative diag rams (3.5.6) u ∗ Y 3 u ∗ C A F l T p ∗ K f ∗ T p ∗ K [2] − → i d − − − − − − − → u ∗ C A F l f ∗ [ − 2] u ∗ X [ − 2]   y ∼ =   y ∼ = C B v ∗ F l T p ∗ K f ∗ T p ∗ K [2] − → i d − − − − − − − → C B v ∗ F l f ∗ [ − 2]   y ∼ =   y ∼ = C B F l v ∗ T p ∗ K f ∗ T p ∗ K [2] − → i d − − − − − − − → C B F l v ∗ f ∗ [ − 2]   y ∼ =   y ∼ = Y 3 u ∗ C B F l T q ∗ K g ∗ u ∗ T q ∗ K [2] − → id − − − − − − − → C B F l g ∗ u ∗ [ − 2] X [ − 2] u ∗ where for the las t we use that w preserves the orientation of the ﬁb er T 2 . The following statement directly follows from the constructions. L emma 3.5.7 . — The isomorphism of L emma 3.5.4 b ehaves functorial ly under c om- p ositions of diagr ams of the form (3.5.3). 3.5.4. — Let F ∈ D + ( Sh Ab X ). Reca ll that P G ( F ) is the ho mo topy limit of a Z - diagram consisting of sheaves Y 0 [2 i ], Y 1 [2 i ], Y 2 [2 i ], Y 3 [2 i ]. F o r all i ≥ 0 we construct an ev a luation tr ansformation e i : P G ( F ) → R f ∗ ◦ f ∗ ( F )[2 i ] as the c o mpo sition of the cano nical map from the limit to Y 3 [2 i + 2] with the str uc tur e map to X [2 i ] and the ide ntiﬁcation X [2 i ]( F ) ∼ = Rf ∗ ◦ f ∗ [2 i ]( F ). T o b e pr ecise we consider Rf ∗ f ∗ ( F ) ∈ D ( Sh Ab X ) via the inclus ion D + ( Sh Ab X ) → D ( Sh Ab X ). In the situation o f 3.5.3 an insp ection of the pro of of Lemma 3.5 .4 together with Co rollary 3.2.7 shows that we hav e a commutativ e diagr am in D ( Sh Ab X ) (3.5.8) u ∗ P G ( F ) ∼ = − − − − → v ∗ P H ( u ∗ F )   y u ∗ e i   y e i u ∗ Rf ∗ f ∗ ( F )[2 i ] ∼ = − − − − → v ∗ Rg ∗ g ∗ ( u ∗ F )[2 i ] . Note, how ever, that the morphism in the b ottom line is only deﬁned on D + ( Sh Ab X ) (or equiv alently on its image in D ( Sh Ab X )), a nd w e do not know whether we can extend it to the full unbounded der ived ca tegory . F ortunately , we do not hav e to do this for the pur po ses of the prese n t pap er . 50 CHAPTER 3. FUNCTORIAL P E RIODIZA TION 3.5.5. — Consider the sp ecial case of the diagra m (3.5.3) where Y = X , u = id X , H = G , and v is an automorphism o f the gerbe G . Le mma 3 .5.4 provides an a uto - morphism v ∗ : P G → P G of p e r io dization functors. 3.5.6. — Let us illustr a te this a uto morphism b y an ex a mple. W e cons ide r the trivial U (1)-gerb e G → S 2 ov er S 2 and let φ ∈ Au t ( G/S 2 ) b e classiﬁed by 1 ∈ H 2 ( S 2 ; Z ) ∼ = Z . It induces a n automor phism of the co homology H ∗ ( S 2 ; P G ( F S 2 )), where F S 2 is the sheaf r e presented b y a discr ete ab elian gro up F . W e hav e a Ca r tesian diagram G / / g   B U (1)   S 2 f / / ∗ . Since f ∗ F ∗ ∼ = F S 2 we hav e H ∗ ( S 2 ; P G ( F S 2 )) ∼ = H ∗ ( S 2 ; P G ( f ∗ F ∗ )) Lemma 3.5.4 ∼ = H ∗ ( S 2 ; f ∗ P B U (1) ( F ∗ )) Lemma 6.2.13 ∼ = H ∗ ( S 2 ; Z ) ⊗ H ∗ ( ∗ ; P B U (1) ( F ∗ )) ∼ = Z [ w ] / ( w 2 ) ⊗ H ∗ ( ∗ ; P B U (1) ( F ∗ )) , where H ∗ ( ∗ ; P B U (1) ( F ∗ )) has b een calculated in examples in P rop osition 2.5 .1. If F = Q or Q / Z , then H ev ( ∗ ; P B U (1) ( F ∗ )) ∼ = Q or . . . ∼ = A Q f , respec tively . If F = Z , then H odd ( ∗ ; P B U (1) ( Z ∗ )) ∼ = A Q f / Q . L emma 3.5.9 . — In al l these c ases the action of φ ∗ is given by φ ∗ (1 ⊗ λ + w ⊗ µ ) = 1 ⊗ λ + w ⊗ ( λ + µ ) , wher e λ, µ ∈ Q , A Q f , or A Q f / Q , r esp e ctively. Pr o of . — W e will use the description of H ∗ ( S 2 , P G ( F S 2 )) given in Cor o llary 2.4.6. In Le mma 2.2.3 have alr eady calcula ted the auto mo rphism o n H ∗ ( S 2 , Rg ∗ g ∗ F S 2 ) ∼ = F [ w ][[ z ]] / ( w 2 ) induced by the diagr am G φ / / g A A A A A A A G g ~ ~ } } } } } } } S 2 . It is given by z 7→ z + w , w 7→ w . The op era tion induced by D G is d dz , and the per io dized cohomolo gy is giv e n as the kernel (in the cases F = Q and F = Q / Z ) or cokernel (in the case F = Z ) o f Q i ≥ 0 id [2 i ] − Q i ≥ 0 D G [2 i ] on Q i ≥ 0 F [ w ][[ z ]] / ( w 2 )[2 i ]. Recall from 2.5.3 that the cla ss a ∈ H 0 ( S 2 , P G ( Q S 2 )) ∼ = Q [ w ] / ( w 2 ) is represented by ( a, az , az 2 / 2 , . . . , az k /k ! . . . ), which is mapp ed by φ ∗ to ( a, a ( w + z ) , a ( w + z ) 2 / 2 , . . . ). W e must read oﬀ a repr esentativ e o f this class in the form ab ov e. If a = w then 3.6. PERIODICITY 51 w ( w + z ) k /k ! = w z k /k ! a nd therefore φ ∗ w = w . On the other hand, if a = 1 , then a ( w + z ) k /k ! = z k /k ! + wz k − 1 / ( k − 1)!, s o that φ ∗ (1) = 1 + w . Exactly the sa me a rgument applies if F = Q / Z . Finally , the cohomo lo gy with co eﬃcients F = Z is the cokernel (up to shift of degree) of the map induced b y the inclusion Q ֒ → A Q f , which implies the as s ertion also for F = Z . 3.6. P erio di ci t y 3.6.1. — W e consider a top ologic a l U (1)-gerb e f : G → X ov er a lo ca lly co mpa ct stack. Let F ∈ D + ( Sh Ab X ). In Le mma 3.5.2 we have argued that P G ( F ) ∈ D ( Sh Ab X ) is tw o -p erio dic. The per io dicity is implemented by a certain isomor phism W : P G ( F )[2] → P G ( F ) which may dep end on additional choices, see also the discussion in 2.4.4. In the present subsection we s how that there is a ca nonical tw o -p erio dicity isomorphism. 3.6.2. — The ger be G → X giv es r ise in a 2-functoria l way to the diagram (see 2.2.1 for deta ils ) (3.6.1) ˜ G s # # F F F F F F F F F r   φ / / ˜ G s { { x x x x x x x x x r   G f # # G G G G G G G G G X × T 2 p   G f { { w w w w w w w w w X . This diagra m induces the desir ed p erio dization isomor phism as the following comp o- sition of na tural tr a nsformations (3.6.2) W : P G ( F ) unit → R p ∗ p ∗ P G ( F ) Lemma 3.5.4 → Rp ∗ P ˜ G ( p ∗ F ) φ ∗ − → Rp ∗ P ˜ G ( p ∗ F ) ∼ = Rp ∗ p ∗ P G ( F ) R p − → P G ( F )[ − 2] . Pr op osi t i on 3.6.3 . — The t r ansformation (3.6.2) W : P G ( F ) → P G ( F )[ − 2] is a c anonic al choic e fo r the isomorphism in Pr op osition 2.4.3. 3.6.3. — T o start the pro of of Pr op osition 3.6 .3, r ecall the deﬁnition D G : Rf ∗ f ∗ ( F ) → Rf ∗ f ∗ ( F )[ − 2] as the co mpo sition Rf ∗ f ∗ ( F ) unit − − → R f ∗ Rr ∗ Rφ ∗ φ ∗ r ∗ f ∗ ( F ) ! ∼ = Rf ∗ Rr ∗ r ∗ f ∗ ( F ) R r − → R f ∗ f ∗ ( F )[ − 2] , 52 CHAPTER 3. FUNCTORIAL P E RIODIZA TION where at the ma r ked iso morphism ”!” w e use the na tural isomor phisms 6.6.13 and 6.6.9 as so ciated to the identit y f ◦ r = f ◦ r ◦ φ Recall from 3.5 .4 the deﬁnition of the natural e v aluation tra nsformation e i : P G ( F ) → Rf ∗ f ∗ ( F )[2 i ] for all i ≥ 0 . L emma 3.6.4 . — The fol lowing diagr am c ommu tes: P G ( F ) e i +1   W / / P G ( F ) e i   Rf ∗ f ∗ ( F )[2 i + 2] D G / / Rf ∗ f ∗ ( F )[2 i ] . Pr o of . — W e s plit this s quare in par ts. First we obse r ve that in D ( Sh Ab X ) P G ( F ) unit − − − − → Rp ∗ p ∗ P G ( F ) Rp ∗ r ∗ − − − − → ∼ = Rp ∗ P ˜ G ( p ∗ F )   y e i +1   y Rp ∗ p ∗ e i +1   y Rp ∗ e i +1 Rf ∗ f ∗ ( F )[2 i + 2] unit − − − − → R p ∗ p ∗ Rf ∗ f ∗ ( F )[2 i + 2] Rp ∗ r ∗ − − − − → ∼ = Rp ∗ Rs ∗ s ∗ p ∗ ( F )[2 i + 2]   y =   y ∼ = Rf ∗ f ∗ ( F )[2 i + 2] Rf ∗ f ∗ unit − − − − − − − → Rf ∗ f ∗ Rp ∗ p ∗ ( F ) ∼ = − − − − → R f ∗ Rr ∗ r ∗ f ∗ ( F )[2 i + 2] commutes (use Lemma 6.1.1 2 for the upper left and the low er and 3.5.4 for the upp er right r ectangle). In the next step we observe that Rp ∗ P ˜ G ( p ∗ F ) id − − − − − → Rp ∗ P ˜ G ( p ∗ F ) Rp ∗ φ ∗ − − − − − → ∼ = Rp ∗ P ˜ G ( p ∗ F )   y Rp ∗ e i +1   y Rp ∗ e i +1 Rp ∗ Rs ∗ s ∗ p ∗ ( F )[2 i + 2] unit − − − − − → R p ∗ Rs ∗ Rφ ∗ φ ∗ s ∗ p ∗ ( F )[2 i + 2] ∼ = − − − − − → R p ∗ Rs ∗ s ∗ p ∗ ( F )[2 i + 2]   y ∼ =   y ∼ =   y ∼ = Rf ∗ Rr ∗ r ∗ f ∗ ( F )[2 i + 2] unit − − − − − → R f ∗ Rr ∗ Rφ ∗ φ ∗ r ∗ f ∗ ( F )[2 i + 2] ∼ = − − − − − → R f ∗ Rr ∗ r ∗ f ∗ ( F )[2 i + 2] commutes, where we us e for the upper rectangle again 3.5.4, and p ◦ s ◦ φ = p ◦ s , p ◦ s = f ◦ r , f ◦ r ◦ φ = f ◦ r and Lemma 6.1.1 2 for the remaining s quares. 3.6. PERIODICITY 53 In the last step we obser ve the co mm utativity of Rp ∗ P ˜ G ( p ∗ F ) ( r ∗ ) − 1 − − − − → ∼ = Rp ∗ p ∗ P G ( F ) R p − − − − → P G ( F )[ − 2] T − 2 ∼ = P G ( F )   y Rp ∗ e i +1   y Rp ∗ p ∗ e i +1   y T − 2 e i +1 ∼ = e i Rp ∗ Rs ∗ s ∗ p ∗ ( F )[2 i + 2] ( r ∗ ) − 1 − − − − → ∼ = Rp ∗ p ∗ Rf ∗ f ∗ ( F )[2 i + 2] R p − − − − → Rf ∗ f ∗ ( F )[2 i ]   y ∼ =   y = Rf ∗ Rr ∗ r ∗ f ∗ ( F )[2 i + 2] Rf ∗ ( R r ) − − − − − → Rf ∗ f ∗ ( F )[2 i ] . Again, for the c ommut ativity of the upp er left r ectangle we use (3.5 .8) of 3.5.4. F or the upper right co rner w e use the fact that R p is a natur al transformation betw een the functors R p ∗ p ∗ and id o n D ( Sh Ab X ). F or the low er r ectangle we use Lemma 6.5.31. 3.6.4. — W e now ﬁnish the pro o f of Pr op osition 3.6 .3. W e have a n exact triangle · · · → P G ( F ) Q i ≥ 0 e i → Y i ≥ 0 Rf ∗ f ∗ ( F )[2 i ] α → Y i ≥ 0 Rf ∗ f ∗ ( F )[2 i ] [1] → . . . where (using the lang ua ge of elements) the map α is given by α ( x i ) i ≥ 0 = ( x i − D G x i +1 ) i ≥ 0 . By Lemma 3 .6.4 we hav e a morphism of e x act tria ngles P G ( F ) W   Q i ≥ 0 e i / / Q i ≥ 0 Rf ∗ f ∗ ( F )[2 i ] β   α / / Q i ≥ 0 Rf ∗ f ∗ ( F )[2 i ] β   P G ( F )[ − 2] Q i ≥ 0 e i / / Q i ≥ 0 Rf ∗ f ∗ ( F )[2 i − 2] α / / Q i ≥ 0 Rf ∗ f ∗ ( F )[2 i − 2] , where the ma p β is given by β ( x i ) i ≥ 0 := ( D G x i ) i ≥ 0 . In Lemma 2 .4.5 we hav e shown that W is an iso morphism. ✷ CHAPTER 4 T -DUALITY 4.1. The univ ersal T -duality di agram 4.1.1. — T o p ological T -duality int ends to mo del the underlying top olog y of string theoretic T -duality o n the level of targets and quantum ﬁeld theory . In the s pe c ia l case of targe ts mo deled by a gerb e on top of a T n -principal bundle over a spa ce, top ological T -duality is by now a well-deﬁned mathematical concept, see [ BSST ], [ BRS ] and the literature cited therein. In the ca se of T -principa l bundles it was extended to orbifolds in [ BS06 ]. In the present pap er we pro p o se a deﬁnition of T -duality in the case of T -bundles over arbitrary stacks. This fra mework includes arbitrar y T -actions on spaces. The special case of an almost free a ction (i.e. ev er y orbit is either free or a ﬁxed p oint) ha s b een trea ted with co mpletely diﬀerent methods in [ Pan06 ]. 4.1.2. — The notio n of a T -duality diagr am has ﬁrst b een introduced in [ BRS ]. In the presen t paper we ﬁrst pro duce a universal T -duality diag ram o ver the stack B U (1) = [ ∗ /U (1)]. Then we pro ceed to deﬁne a T -duality diag ram ov er a g eneral stack as one which is lo cally isomo rphic to the universal one. 4.1.3. — The universal T -duality diagr am is a diag ram of stacks (4.1.1) p ∗ univ G univ y y s s s s s s s s s s s % % K K K K K K K K K K u univ / / ˆ p ∗ univ ˆ G univ y y s s s s s s s s s s % % K K K K K K K K K K G univ f univ % % K K K K K K K K K K K F univ p univ y y s s s s s s s s s s s ˆ p univ % % K K K K K K K K K K ˆ G univ ˆ f univ y y s s s s s s s s s s E univ π univ % % L L L L L L L L L L ˆ E univ ˆ π univ y y r r r r r r r r r r B univ . 56 CHAPTER 4. T -DUALITY In the fo llowing we explain the stacks and the maps . – B univ := B U (1) – E univ := ∗ and π univ is the ma p which classiﬁes the triv ial U (1)-bundle ov er the p o int ∗ . – G univ := B U (1), and f univ is the unique map. – ˆ E univ := B U (1) × U (1), and ˆ π univ is the pro jection o nt o the ﬁr st factor . – ˆ f univ : ˆ G univ → ˆ E univ is a gerb e with band U (1) c la ssiﬁed by z ⊗ v ∈ H 2 ( B U (1); Z ) ⊗ H 1 ( U (1); Z ) ∼ = H 3 ( B U (1) × U (1); Z ), where z ∈ H 2 ( B U (1); Z ) and v ∈ H 1 ( U (1); Z ) are the sta nda rd g enerators . – F univ := E univ × B univ ˆ E univ ∼ = U (1), and p univ , ˆ p univ are the canonical pro jec- tions. – Since H 2 ( F univ ; Z ) ∼ = 0 ∼ = H 3 ( F univ ; Z ), the pull-back ˆ p ∗ univ ˆ G univ can be iden ti- ﬁed with the trivial ger b e p ∗ univ G univ ∼ = U (1) × B U (1) b y a uniq ue isomorphis m class of maps r epresented by u univ . Let us ﬁx onc e and for all a universal T -dua lity diagram (i.e. a choice of u univ in its isomorphism cla ss a nd 2-is o morphisms ﬁlling the faces). 4.1.4. — Let B be a top olo gical stack and consider a diagr am (4.1.2) p ∗ G ~ ~ } } } } } } } } A A A A A A A A u / / ˆ p ∗ ˆ G ~ ~ } } } } } } } } A A A A A A A A G f A A A A A A A A F p ~ ~ } } } } } } } } ˆ p A A A A A A A A ˆ G ˆ f ~ ~ } } } } } } } } E π ! ! B B B B B B B B ˆ E ˆ π } } | | | | | | | | B of top olo g ical s tacks where the sq ua res ar e Cartesian, f : G → E and ˆ f : ˆ G → ˆ E are top ological U (1)-gerb es, and u is an isomor phis m of gerb es over F . An iso morphism b etw een tw o s uch dia grams over B is ﬁrs t of all a large commu- tative diagra m in s tacks, but we furthermore requir e that the horizontal morphisms are mor phis ms of U (1)-banded gerb es in all places where this condition ma kes sense. Deﬁnition 4.1.3 . — The diagr am (4.1.2) is c al le d a T -duality diagr am if for every obje ct ( U → B ) ∈ B ther e ex ists a c overing ( U i → U ) i ∈ I ∈ co v B ( U ) such that for al l i ∈ I the pul l-b ack of t he diagr am (4.1.2) along the map U i → U → B is isomorphic to t he pul l-b ack of the universal T -duality diagr am (4.1.1) along a map U i → B univ . 4.1. THE UN I VE RSAL T -DUALITY DIAGRAM 57 4.1.5. — In the following we descr ib e the c o ncept of T - duality . Let B b e a top ologic a l stack. A pair ( E , G ) over B consists of a T -principa l bundle π : E → B and a U (1)- gerb e f : G → E . Deﬁnition 4.1.4 . — We say t hat a p air ( E , G ) admits a T -dual, if it app e ars as a p art of a T -duality diag r am 4.1.2 . In this c ase the p air ( ˆ E , ˆ G ) is c al le d a T -dual of ( E , G ) . This is our prop osa l for the mathematical c oncept o f T -duality for pa irs of T - principal bundles and gerb es . Using the T n -bundle v a riant of the universal T -duality diagram o ne can easily gener alize this deﬁnition to the higher -dimensional ca s e. But note that, in contrast to the case of one-dimensio nal ﬁb ers, a unique isomorphism u univ do es not exis t for T n if one us es the exact parallel setup. This explains why suitable mo diﬁcations are necessar y in [ BRS ]. In particular , the universal ba se space is not simply the n - fold pro duct of copies of B univ used in the one-dimensional case. 4.1.6. — In the following we show that the concept of to po logical T -dua lit y as deﬁned ab ov e really coincides with the former deﬁnitions. L emma 4.1.5 . — Deﬁnitions 4.1.3 and 4.1.4 r e duc e t o the notion of T -duality as use d in [ BR S ] , [ BS0 5 ] , if B is a lo c al ly acyclic sp ac e. Pr o of . — By Deﬁnition 4.1.3 a T -dua lity tr iple ov er a space B is given b y the following data: (1) lo c ally tr iv ial U (1)-principal bundles E , ˆ E over B , (2) U (1)-banded gerb es G , ˆ G over E or ˆ E , resp ectively , (3) a n isomorphism u b etw een the pullbacks of G and ˆ G to the co rresp ondence space E × B ˆ E . Every p oint b ∈ B admits an acyclic neighborho o d b ∈ U ⊆ B . The bundles E and ˆ E are trivia l ov er U , i.e. we have E | U ∼ = U × U (1) ∼ = ˆ E | U . Since H 3 ( U × U (1); Z ) ∼ = 0, the r estrictions of the ger b es G | E | U and ˆ G | ˆ E | U are tr ivial, too. The Deﬁnition 4.1.3 requires that the isomor phism of trivial gerb es u | E | U × U ˆ E | U is classiﬁed by the generator of H 2 ( E | U × U ˆ E | U ; Z ) (no te that E | U × U ˆ E | U ∼ = U × U (1) × U (1)). This refor mulation of the deﬁnition o f a T -duality triple ov er a lo c a lly acyclic s pace B is exactly the deﬁnition of a T -dua lit y triple in [ BRS ]. In the a ppr oach of [ BS05 ] to T -dua lit y we start with a pair ( E , G ). W e characterize T -dual pa irs by topo logical conditions . W e then analyze the clas s ifying space of pairs and obse rve that the universal pair has a uniq ue T -dua l pair which gives rise to the T -duality transfor mation. It turns out that the cla ssifying s pace of pa irs in [ BS0 5 ] is equiv alent to the classifying space of T - duality triples in [ BR S ], and that the universal pair and its dual are par ts o f the universal T -duality triple. This sho ws that the approaches of [ BS05 ] a nd [ BR S ] ar e equiv a lent. 58 CHAPTER 4. T -DUALITY 4.2. T -duality and p erio di zation di agrams 4.2.1. — Reca ll that the constructio n of the p erio dizatio n functor P G was based o n the dia grams introduced in 2.2.1. In the present subsection we rela te these diagra ms to T -duality . 4.2.2. — The double o f the universal T -duality diag ram (4.1.1) is (by deﬁnition) the big universal p erio diza tion diag ram (4.2.1) pr ∗ 0 p ∗ univ G univ ˜ pr 0 w w o o o o o o o o o o o o ( ( R R R R R R R R R R R R R R pr ∗ 0 u univ / / pr ∗ ˆ E ˆ G univ pr ∗ 1 u − 1 univ / /   pr ∗ 1 p ∗ univ G univ v v l l l l l l l l l l l l l l ˜ pr 1 ' ' O O O O O O O O O O O O p ∗ univ G univ f ∗ univ p univ   p ∗ univ f univ ' ' P P P P P P P P P P P P F univ × ˆ E univ F univ pr 0 u u l l l l l l l l l l l l l l pr 1 ) ) R R R R R R R R R R R R R R p ∗ univ G univ f ∗ univ p univ   p ∗ univ f univ w w n n n n n n n n n n n n F univ p univ ) ) S S S S S S S S S S S S S S S F univ p univ u u k k k k k k k k k k k k k k k G univ f univ / / E univ G univ f univ o o Note that all sq uares ar e Cartesia n, with the exception of the central s quare F univ × ˆ E univ F univ w w n n n n n n n n n n n n ' ' P P P P P P P P P P P P F univ ( ( P P P P P P P P P P P P F univ v v n n n n n n n n n n n n E univ which do es not co mm ute. The same remar k applies to similar diagra ms we intro duce later. 4.2.3. — W e form the diagra m (1) (4.2.2) pr ∗ 0 p ∗ univ G univ q univ ( ( m univ 6 6 G univ f univ / / E univ , where m univ := f ∗ univ p univ ◦ ˜ pr 1 ◦ pr ∗ 1 u − 1 univ ◦ pr ∗ 0 u univ , q univ := f ∗ univ p univ ◦ ˜ pr 0 . (1) This diagram do es not commute. It i s a short-hand for a square of the form (3.4.2) with a 2- isomorphism b et wee n f univ ◦ q univ and f univ ◦ m univ . W e will adopt a simi lar con ve ntion for other diagrams wr itten i n this shor t- hand for m b elow. 4.2. T -DUALITY AND PERIODIZA TION DIAGRAMS 59 Deﬁnition 4.2.3 . — The diagr am (4.2.2) is c al le d the smal l universal p erio dization diagr am. 4.2.4. — Let f : G → X b e a top olog ical ger b e with ba nd U (1) ov er a sta ck X . Then we consider the pull-back of the s ma ll universal p erio dization diagr a m to X v ia the pro jection r : X → E univ ∼ = ∗ . W e form the tensor pro duct with the gerb e G (see [ BSST , 6 .1.9] for s ome details on such tensor pro ducts) and obtain the diagra m (4.2.4) ˜ H q " " m < < H f / / X , where ˜ H := p r ∗ X G ⊗ pr ∗ F univ × ˆ E univ F univ pr ∗ 0 p ∗ univ G univ , H := G ⊗ r ∗ G univ , pr X : X × F univ × ˆ E univ F univ → X , pr F univ × ˆ E univ F univ : X × F univ × ˆ E univ F univ → F univ × ˆ E univ F univ are the pr o jections, a nd m , q are induced by the corr e sp onding universal maps m univ or q univ , resp ectively . Deﬁnition 4.2.5 . — The diagr am (4.2 .4) is c al le d the smal l p erio dization diagr am of G → X . In fact we hav e deﬁned a 2 -functor from ge rbes /X to a 2-ca tegory of such small per io dization diagrams . Using the fact that G univ = B U (1) we hav e a cano nic a l ident iﬁcation H ∼ = G . F urther mo re, F univ × ˆ E univ F univ ∼ = T 2 , a nd we can identify ˜ H → X × F univ × ˆ E univ F univ with G × T 2 → X × T 2 . L emma 4.2.6 . — With these identiﬁc ations the smal l p erio dization diagr am (4.2.4) is isomorphic to the diagr am (3.4.2) use d in the deﬁnition of P G . Pr o of . — This follows directly from the deﬁnitions o f these maps. 4.2.5. — The T -duality diagra m (4.1.2) gives rise to the big double T -duality diagram (4.2.7) pr ∗ 0 p ∗ G ˜ pr 0 { { w w w w w w w w w $ $ J J J J J J J J J J pr ∗ 0 u / / pr ∗ ˆ E ˆ G pr ∗ 1 u − 1 / /   pr ∗ 1 p ∗ G z z t t t t t t t t t t ˜ pr 1 # # G G G G G G G G G p ∗ G f ∗ p   p ∗ f $ $ I I I I I I I I I I F × ˆ E F pr 0 y y r r r r r r r r r r r pr 1 % % L L L L L L L L L L L p ∗ G f ∗ p   p ∗ f z z u u u u u u u u u u F p & & L L L L L L L L L L L L F p x x r r r r r r r r r r r r G f / / E G f o o 60 CHAPTER 4. T -DUALITY Note that the middle s quare do es not commute. W e have F × ˆ E F ∼ = ( E × B ˆ E ) × ˆ E ( ˆ E × B E ) ∼ = E × B ˆ E × B E ∼ ← E × B ˆ E × U (1) , where the last arrow is g iven by ( e, ˆ e, eu ) ← ( e, ˆ e, u ). Under this identiﬁcation pr 0 ( e, ˆ e, u ) = ( e, ˆ e ) and pr 1 ( e, ˆ e, u ) = ( e u, ˆ e ). W e can cor rect this non-co mm utativity as follows. Let c : F × ˆ E F → F × ˆ E F b e the isomo rphism, whic h under the abov e ident iﬁcation is given by c ( e, ˆ e, u ) := ( eu − 1 , ˆ e, u ). Note that pr 1 ◦ c = pr 0 . F ur ther- more note that pr ˆ E = pr ˆ E ◦ c : F × ˆ E F → ˆ E . Therefore we get a ca nonical morphism ˆ c sa tisfying pr ˆ E = pr ˆ E ◦ ˆ c in the diagra m pr ∗ ˆ E ˆ G   ˆ c / / pr ∗ ˆ E ˆ G   pr ˆ E / / ˆ G   F × ˆ E F c / / F × ˆ E F pr ˆ E / / ˆ E . If we plug this in the big double T -duality dia g ram, then we get the big commutativ e T -duality diagra m diagra m (4.2.8) pr ∗ 0 p ∗ G ˜ pr 0 { { w w w w w w w w w $ $ J J J J J J J J J J pr ∗ 0 u / / pr ∗ ˆ E ˆ G   ˆ c / / pr ∗ ˆ E ˆ G pr ∗ 1 u − 1 / /   pr ∗ 1 p ∗ G z z t t t t t t t t t t ˜ pr 1 # # G G G G G G G G G p ∗ G f ∗ p   p ∗ f $ $ I I I I I I I I I I F × ˆ E F pr 0 y y r r r r r r r r r r r c / / F × ˆ E F pr 1 % % L L L L L L L L L L L p ∗ G f ∗ p   p ∗ f z z u u u u u u u u u u F p & & L L L L L L L L L L L L F p x x r r r r r r r r r r r r G f / / E E G f o o ¿F rom this we derive the diagr am (4.2.9) pr ∗ 0 p ∗ G q T $ $ m T : : G f / / E , where q T := f ∗ p ◦ ˜ pr 0 , m T := f ∗ p ◦ ˜ pr 1 ◦ pr ∗ 1 u − 1 ◦ ˆ c ◦ pr ∗ 0 u . Deﬁnition 4.2.10 . — The diagr am (4.2.9) is c al le d the small double T -duality di- agr am asso ciate d t o (4.1.2). 4.3. TWISTED COHOMOLOGY AND THE T -DUALITY TRANSFORMA TION 61 4.2.6. — The fo llowing fact is an immediate conseq uence of the deﬁnitions. Pr op osi t i on 4.2.11 . — The smal l double T -duality diagr am (4.2.9) is lo c al ly iso- morphic to the smal l p erio dization diagr am (4. 2.4) of G → E . 4.3. Twisted c ohomol ogy and the T -duality transformation 4.3.1. — Let E be a top ologica l s ta ck. In o rder to write out o p e r ations on t wisted cohomolog y eﬀectively we introduce some notation for oper ations on D + ( Sh Ab E ) o r D ( Sh Ab E ). If p : F → E is a map of top olog ic al stacks, then we let p ∗ : id → R p ∗ p ∗ denote the unit. If p is an oriented ﬁb er bundle, then we let p ! : Rp ∗ p ∗ → id denote the integration map. If π : E → B is a second map, then we write π ∗ p ∗ , π ∗ p ! or simply also p ∗ and p ! for the induced transfo rmations R π ∗ π ∗ → Rπ ∗ Rp ∗ p ∗ π ∗ and Rπ ∗ Rp ∗ p ∗ π ∗ → R π ∗ π ∗ . If G v / /   H   E u / / π @ @ @ @ @ @ @ F ˆ π ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ B is a diagram with U (1)-gerb es H → F a nd G → E suc h that the sq uare is Ca rtesian, then we write P ( v ) for the transfor mation u ∗ ◦ P H → P G ◦ u ∗ , and we use the sa me symbol for the induced tra nsformation R π ∗ u ∗ P H ˆ π ∗ → R π ∗ P G u ∗ ˆ π ∗ . In a commutativ e diagr am F p           ˆ p   ? ? ? ? ? ? ? E π   ? ? ? ? ? ? ? ? ˆ E ˆ π          B we will use the symbol I o r , if neces s ary , I π ◦ p = ˆ π ◦ ˆ p in order to denote the transfor ma- tion Rπ ∗ Rp ∗ p ∗ π ∗ ∼ → R ˆ π ∗ R ˆ p ∗ ˆ p ∗ ˆ π ∗ . 4.3.2. — W e consider a top olog ical gerb e f : G → E with band U (1) o ver a loca lly compact stack. In [ BSS07 ] we deﬁne the G -twisted cohomolog y o f E with co eﬃcients in F ∈ D + ( Sh Ab E ) by H ∗ ( E , G ; F ) := H ∗ ( E ; R f ∗ f ∗ ( F )) . 62 CHAPTER 4. T -DUALITY 4.3.3. — Assume now that f : G → E is a par t o f a T -duality diag r am (4.3.1) p ∗ G q ~ ~ } } } } } } } } A A A A A A A A u / / ˆ p ∗ ˆ G ~ ~ } } } } } } } } ˆ q A A A A A A A A G f A A A A A A A A F p ~ ~ } } } } } } } } ˆ p A A A A A A A A ˆ G ˆ f ~ ~ } } } } } } } } E π ! ! B B B B B B B B ˆ E ˆ π } } | | | | | | | | B . Then we deﬁne the tra nsformation (4.3.2) J := ˆ q ! ◦ I ◦ ( u − 1 ) ∗ ◦ q ∗ : R π ∗ Rf ∗ f ∗ π ∗ → R ˆ π ∗ R ˆ f ∗ ˆ f ∗ ˆ π ∗ . Note that here I = I π f qu − 1 = ˆ π f ˆ q . Consider a sheaf F ∈ D + ( Sh Ab B ). Note that, by deﬁnition, H ∗ ( E , G ; π ∗ F ) = H ∗ ( B ; Rπ ∗ Rf ∗ f ∗ π ∗ F ). Deﬁnition 4.3.3 . — F or F ∈ D + ( Sh Ab E ) t he T -duality tr ansformation is deﬁne d as t he map T : H ∗ ( E , G ; π ∗ F ) → H ∗− 1 ( ˆ E , ˆ G ; ˆ π ∗ F ) induc e d by the natur al tr ansformation (4.3.2). 4.3.4. — Let us ca lculate the eﬀect of the T -duality transfor mation in a s imple exa m- ple. The r e is a unique iso morphism clas s o f T -duality diagr ams ov er the p oint B = ∗ . In this case E = U (1) and G = U (1) × B U (1). W e conside r a discrete ab elian g r oup F . Then we hav e H ∗ ( E , G ; π ∗ F B ) ∼ = Z [[ z ]][ v ] / ( v 2 ) ⊗ F , H ∗ ( ˆ E , ˆ G ; ˆ π ∗ F B ) ∼ = Z [[ z ]][ ˆ v ] / ( ˆ v 2 ) ⊗ F , where deg ( v ) = 1 = deg ( ˆ v ) and deg ( z ) = 2. T o ex plicitly calculate the eﬀect o f T in this case, obser ve that the cohomolo gy of Rf ∗ Rq ∗ q ∗ f ∗ F is Z [[ z ]] ⊗ Λ( v , ˆ v ) ⊗ F with v and ˆ v the g enerator s of the t wo S 1 -factors E a nd ˆ E in F . The automor phism u induces in co homology , i.e. on Z [[ z ]] ⊗ Λ( v , ˆ v ) ⊗ F , the a lgebra ho momorphism given by z 7→ z + v ˆ v , v 7→ v , ˆ v 7→ ˆ v . It follows that T ( z n ⊗ f ) = Z F / ˆ E ( z n ⊗ f + nz n − 1 v ˆ v ⊗ f ) = nz n − 1 ˆ v ⊗ f T ( z n v ⊗ f ) = Z F / ˆ E z n v ⊗ f = z n ⊗ f . W e see that the T -duality transformatio n is not a n isomo r phism. 4.3. TWISTED COHOMOLOGY AND THE T -DUALITY TRANSFORMA TION 63 4.3.5. — Our ma in mo tiv a tion for intro ducing the perio dizatio n functor is the con- struction of twisted sheaf coho mology whic h admits a T -duality iso morphism. L et G → E be a top olog ical gerb e with band U (1) ov er a lo cally compac t stack E . Deﬁnition 4.3.4 . — We deﬁne the p erio dic G -twiste d c ohomolo gy of E with c o eﬃ- cients in F ∈ D + ( Sh Ab E ) by H ∗ per ( E , G ; F ) := H ∗ ( E ; P G ( F )) . Note tha t here we use the sheaf theory op eratio ns for the un b ounded derived category , see Subs e ction 6.5 for details. 4.3.6. — Assume a gain that f : G → E is part o f a T -duality diagram (4.3.1). W e deﬁne a natur al tra nsformation (4.3.5) J : R π ∗ ◦ P G ◦ π ∗ → R ˆ π ∗ ◦ P ˆ G ◦ ˆ π ∗ by J := ˆ p ! ◦ I ◦ P ( u ) − 1 ◦ p ∗ . It aga in inv olves shea f theory op era tions in the unbounded derived catego ry . Consider a sheaf F ∈ D + ( Sh Ab B ). Note that by deﬁnition H ∗ per ( E , G ; π ∗ F ) = H ∗ ( B , Rπ ∗ P G ( π ∗ ( F ))). Deﬁnition 4.3.6 . — F or F ∈ D + ( Sh Ab E ) t he T -duality tr ansformation in p erio dic twiste d c ohomolo gy T : H ∗ per ( E , G ; π ∗ F ) → H ∗− 1 per ( ˆ E , ˆ G ; ˆ π ∗ F ) is t he map induc e d by the n atur al t r ansformation (4.3.5). 4.3.7. — As an illustratio n let us calculate the ac tio n o f the T -dua lit y tra nsforma- tion in the example star ted in 4.3.4. The sequence S G ( F ) for F = Z , Q , Q / Z either has triv ia l lim o r trivial lim 1 . Therefor e in this special case the mor phism T calcu- lated in 4.3 .4 deﬁnes uniquely an endomo r phism of H ∗ per ( E , G ; π ∗ F B ) (w e identify E ∼ = ˆ E ). F or exa mple if F = Q , then we read oﬀ dir e ctly from 4.3 .4 that (with H 0 per ( E , G ; π ∗ Q ) ∼ = Q [ v ] /v 2 ) the T -duality morphism is T : Q [ v ] /v 2 → Q [ v ] /v 2 , T ( v ) = 1 , T (1) = v . In pa rticular, we see in this example that now we ge t a n isomo rphism. 4.3.8. — In the r emainder o f the present subsection we show the following theo rem. The or em 4.3.7 . — The T -duality tr ansformation in t wiste d p erio dic c ohomolo gy 4.3.6 is an isomorphism. Pr o of . — The opp osite of the T -duality diagr am (4.3.1) is obtained by reﬂecting it in the middle vertical, and by repla cing u by its inv erse. W e let T ′ : H ∗ per ( ˆ E , ˆ G ; ˆ π ∗ F ) → H ∗− 1 per ( E , G ; π ∗ F ) b e the asso c iated T -duality trans fo rmation. 64 CHAPTER 4. T -DUALITY Both, the T -duality diagra m and its o pp o site can b e recognize d as sub diagrams of the (slig htly extended) big c o mmut ative T -duality diagr am (4.3.8) pr ∗ 0 p ∗ G ˜ pr 0                      7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 pr ∗ 0 u / / pr ∗ ˆ E ˆ G " " D D D D D D D D D   ˆ c / / pr ∗ ˆ E ˆ G | | z z z z z z z z z pr ∗ 1 u − 1 / /   pr ∗ 1 p ∗ G                    ˜ pr 1   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ˆ p ∗ ˆ G   / / ˆ G   ˆ p ∗ ˆ G o o   u − 1 # # G G G G G G G G G p ∗ G u ; ; w w w w w w w w w f ∗ p   p ∗ f # # H H H H H H H H H H F × ˆ E F pr 0 y y s s s s s s s s s s s s # # F F F F F F F F F c / / F × ˆ E F pr 1 % % K K K K K K K K K K K { { x x x x x x x x x p ∗ G f ∗ p   p ∗ f { { v v v v v v v v v v F p % % L L L L L L L L L L L L ˆ p / / ˆ E F ˆ p o o p y y r r r r r r r r r r r r G f / / E E G f o o W e now calculate the comp os ition T ′ ◦ T . The compatibility o f the integration with pull-back in the Cartes ian diag ram F ˆ p   F × ˆ E F pr 1   pr 0 o o ˆ E F ˆ p o o is employ ed in the equality ma rked by ! b elow. The equa lity ˆ p ◦ pr 0 ◦ c − 1 = ˆ p ◦ pr 0 is used in the equa lit y !!. Finally we use pr 0 ◦ c = pr 1 at !!!. W e have J ′ ◦ J = p ! ◦ I ◦ P ( u ) ◦ ˆ p ∗ ◦ ˆ p ! ◦ I ◦ P ( u ) − 1 ◦ p ∗ ! = p ! ◦ I ◦ P ( u ) ◦ pr 1 ! ◦ I ◦ pr 0 ∗ ◦ I ◦ P ( u ) − 1 ◦ p ∗ !! = p ! ◦ I ◦ P ( u ) ◦ pr 1 ! ◦ I ◦ P ( ˆ c − 1 ) ◦ ( c − 1 ) ∗ ◦ pr 0 ∗ ◦ I ◦ P ( u ) − 1 ◦ p ∗ !!! = p ! ◦ pr 1 ! ◦ P ( pr ∗ 1 u ) ◦ P ( ˆ c − 1 ) ◦ P ( pr ∗ 0 u ) − 1 ◦ pr 1 ∗ ◦ p ∗ = p ! ◦ pr 1 ! ◦ P ( pr ∗ 1 u ◦ ˆ c − 1 ◦ ( pr ∗ 0 u ) − 1 ) ◦ pr 1 ∗ ◦ p ∗ This is ex a ctly the tra nsformation asso ciated to the asso ciated small double T - duality diag ram (4.2.9) (actually its mirr or). Since this is lo cally iso morphic to the small p er io dization diagr am we s ee that lo cally J ′ ◦ J coincides with π ∗ W , where W is a s in Pr op osition 3 .6.3. By Pr o p osition 3 .6 .3 this transfor mation is an isomor phism on perio dic sheav es of the form R π ∗ P G ( π ∗ F ). Therefore T ◦ T ′ is an isomor phis m. W e can interc hange the r oles of T and T ′ , hence T ◦ T ′ is an is omorphism, to o. This implies the result. CHAPTER 5 ORBISP ACES 5.1. Twisted perio dic del o calized coho m ology of orbispaces 5.1.1. — Let us r ecall so me notions related to orbis paces (co mpare [ BSS0 8 ]). Or - bispaces a s par ticular kind of to po logical sta cks have previously been in tro duced in [ BS06 , Sec. 2.1] and [ No o , Sec. 1 9.3]). In the pre s ent pa p er w e use the se t-up of [ BS06 ] but add the additional condition that an o rbifold atla s should b e sepa rated. This condition is needed in o rder to show that the lo op sta ck of a n orbifold is aga in an or bifold. (1) A topolo gical group o id A : A 1 ⇒ A 0 is called separated if the iden tit y 1 A : A 0 → A 1 of the gr o up oid is a closed map. (2) A top ologica l group oid A 1 ⇒ A 0 is called prop er if ( s, r ) : A 1 → A 0 × A 0 is a prop er map. (3) A top ologic a l g r oup oid is calle d ´ etale if the sour ce a nd range maps s, r : A 1 → A 0 are ´ etale. (4) A prop er ´ etale top olo gical gro up o id A 1 ⇒ A 0 is called very prop er if there exists a contin uous function χ : A 0 → [0 , 1] such that (a) r : supp ( s ∗ χ ) → A 0 is pro pe r (b) P y ∈ A x χ ( s ( y )) = 1 for all x ∈ A 0 . (5) A topolog ic al stack is calle d (v er y) prop er (or ´ etale, separ ated, resp ectively), if it admits an a tlas A → X such that the top ologic al gr oup oid A × X A ⇒ A is (very) prop er (or ´ etale, separ a ted, resp ectively). (6) An orbispace atlas of a top ologica l stack X is an atlas A → X such that A × X A ⇒ A is a very pr op er ´ etale and separa ted group oid. (7) An or bispace X is a top olo gical stack which admits an orbispace atlas. (8) If X , Y ar e orbispa ces, then a morphism of orbispaces X → Y is a representable morphism o f s tacks. (9) A lo cally compa ct orbispa ce is an or bispace X which admits an orbispace atlas A → X such that A is lo cally compact. 66 CHAPTER 5. ORBISP AC ES 5.1.2. — If X is a stac k , then its iner tia stack (sometimes ca lled lo op stack) LX is deﬁned a s the tw o-categ o rical eq ua lizer of the diag r am X id X id X + 3 X . In [ BSS08 , Sec 2.2] we have intro duced an explicit mo del o f L X a nd studied its prop erties. The lo op s tack L X depends 2-functor ia lly on X . Indeed, since Hom Cat is a strict 2-functor, the lo op functor is a s trict functor b etw een 2-ca tegories . As alrea dy men tioned b efore, later we will suppress the 2-mo rphisms in 2-commutativ e diagrams in 2 -categor ies for b etter legibility . If X is a top ologica l stack (orbispace), then LX is a topolog ical stack (orbispace), too (see [ BSS08 , Lemma 2.25], [ BSS08 , Lemma 2.33]). L emma 5.1.1 . — If X is a lo c al ly c omp act orbisp ac e, then LX is a lo c al ly c omp act orbisp ac e, to o. Pr o of . — Let A → X b e a lo ca lly compact orbispace a tlas o f X . Then we hav e the prop er, separa ted and ´ etale top o logical gro upo id A × X A ⇒ A . Since the sour ce map of this gro upo id is ´ etale, the space of morphisms A × X A of this group oid is lo cally compact, to o . In the pro of of Lemma [ BSS08 , Lemma 2.25 ] w e constructed an or bispace a tlas W → LX of L X , whe r e W was given by the pull- back o f space s W / / w   A × X A ( pr 1 , pr 2 )   A diag / / A × A . This implies that W is lo ca lly compa ct. 5.1.3. — Let G → X b e a top o logical g erb e with band U (1) ov er a lo cally compact o r- bispace. The truly interesting G -twisted cohomolo gy of X (with complex co eﬃcients) is not the cohomolo gy H ∗ per ( X, G ; C ) (see 4 .3.6), but a more complica ted delo calize d version H ∗ deloc,per ( X, G ), which w e will deﬁne b elow (see [ BSS08 , Sec. 1.3] for an explanation). As shown in [ BSS 0 8 , Sec. 2.5] the ger b e gives ris e to a principal bundle ˜ G δ → L X with s tr ucture g roup U (1) δ in a functoria l wa y , where U (1) δ denotes the g r oup U (1) with the discrete top olo gy . By L ∈ Sh Ab LX we denote the she a f of lo c a lly cons tant sections of the ass o ciated vector bundle ˜ G δ × U (1) δ C → L X . W e deﬁne the ger be G L → L X as the pull- ba ck G L f L   / / G f   LX / / X . 5.2. T -DUALITY IN TWIS TED PERIODIC DELOCALIZED COHOMOLOGY 67 Deﬁnition 5.1.2 . — We deﬁne L G := P G L ( L ) ∈ D ( Sh Ab LX ) . The G - t wiste d delo c alize d p erio dic c ohomolo gy of X is deﬁne d as H ∗ deloc,per ( X, G ) := H ∗ ( LX ; L G ) . 5.2. The T -duality transformation in twisted p erio di c delo calized coho- mology 5.2.1. — W e consider a T -duality diagra m (5.2.1) p ∗ G ~ ~ } } } } } } } } A A A A A A A A u / / ˆ p ∗ ˆ G ~ ~ } } } } } } } } A A A A A A A A G f A A A A A A A A F p ~ ~ } } } } } } } } ˆ p A A A A A A A A ˆ G ˆ f ~ ~ } } } } } } } } E π ! ! B B B B B B B B ˆ E ˆ π } } | | | | | | | | B (see Deﬁnition 4.1.3), wher e B is a lo cally compa c t orbis pa ce. W e apply the lo o ps functor L : orbi spaces → or bispace s to the sub diag ram F ˆ p   ? ? ? ? ? ? ? p           E π   ? ? ? ? ? ? ? ? ˆ E ˆ π          B and get LF L ˆ p ! ! D D D D D D D D Lp } } z z z z z z z z LE Lπ ! ! D D D D D D D D L ˆ E L ˆ π } } z z z z z z z z LB . In the ﬁrst diagram the maps p, ˆ p, π , ˆ π are all U (1)-principal bundles. The ma ps Lp, L ˆ p, L π , L ˆ π a re no t ne c essarily s urjective. Thus in gener a l the derived diagra m of 68 CHAPTER 5. ORBISP AC ES lo op stacks is not par t o f a T -duality diagr am. But it is so lo cally in a certain sens e which we will explain in the following. 5.2.2. — W e can extend the second diagra m b y the lo c al sy s tems (see 5 .1 .3) (5.2.2) Lp ∗ L } } | | | | | | | | | " " E E E E E E E E E u / / L ˆ p ∗ ˆ L | | y y y y y y y y ! ! B B B B B B B B L ! ! C C C C C C C C C LF Lp | | y y y y y y y y y L ˆ p " " E E E E E E E E E ˆ L } } { { { { { { { { LE Lπ # # F F F F F F F F F L ˆ E L ˆ π { { x x x x x x x x LB and the pull-backs o f gerb es (5.2.3) Lp ∗ G L | | x x x x x x x x x " " F F F F F F F F F u / / L ˆ p ∗ ˆ G L | | x x x x x x x x x " " F F F F F F F F G L f L # # F F F F F F F F F LF Lp { { w w w w w w w w w L ˆ p # # G G G G G G G G G ˆ G L ˆ f L { { x x x x x x x x x LE Lπ # # H H H H H H H H H L ˆ E L ˆ π { { v v v v v v v v v LB In pa rticular, we hav e a n isomo r phism (5.2.4) u : Lp ∗ L G ∼ → L ˆ p ∗ ˆ L ˆ G . 5.2.3. — Note that ˆ p : F → ˆ E is a U (1)-principal bundle. In [ BSS08 , Lemma 2 .3 4] we ha ve constructed a map h : L ˆ E → U (1) δ which measures the action of the au- tomorphisms of the p oints of ˆ E o n the ﬁber s of ˆ p . W e get a decomp ositio n in to a disjoint union of o pen substacks L ˆ E ∼ = G u ∈ U (1) L ˆ E u , where L ˆ E u := h − 1 ( u ). Here and in the following we use the simpliﬁed notation h − 1 ( u ) for the pullback of h : L ˆ E → U (1) δ along the inclusion i u : ∗ → U (1) with i u ( ∗ ) := u . By [ BSS08 , Lemma 2.3 6], the map L ˆ p : L F → L ˆ E factors ov er the inclusion J : L ˆ E 1 → L ˆ E , a nd the co rresp onding map L ˆ p 1 : L F → L ˆ E 1 is a U (1)- principal bundle. The integration L ˆ p 1 ! : R ( L ˆ p 1 ) ∗ ◦ L ˆ p ∗ 1 → id 5.3. THE GE OMETR Y OF T -DUALITY DIAGRAMS OVER ORBISP ACES 69 is well-deﬁned. The op en inclusion J induces a natural tra nsformation J ! : RJ ∗ ◦ J ∗ → id . W e can thus deﬁne L ˆ p ! := J ! ◦ L ˆ p 1 ! : RL ˆ p ∗ ◦ L ˆ p ∗ → id . 5.2.4. — Deﬁnition 5.2.5 . — The lo c al T -duality tr ansformation asso ciate d to the di agr am (5.2.1) is given by the c omp osition T loc := L ˆ p ! ◦ u ◦ Lp ∗ : RLπ ∗ L G → R L ˆ π ∗ ˆ L ˆ G , wher e u is induc e d by (5.2.4). Note that H ∗ deloc,per ( E , G ) ∼ = H ∗ ( LB ; RL π ∗ L G ). Hence we can make the following deﬁnition. Deﬁnition 5.2.6 . — The T -duality tr ansformation in twist e d p erio dic delo c alize d c ohomolo gy asso ciate d to the T -duality diagr am (5.2.1) is the tr ansformation T : H ∗ deloc,per ( E , G ) → H ∗ deloc,per ( ˆ E , ˆ G ) induc e d by the lo c al T -duality tra nsformation T loc deﬁne d in 5.2.5. 5.3. The geometry of T -duali t y diagrams ov er o rbi spaces 5.3.1. — W e co nsider a T -duality dia g ram (5.2.1) ov er a lo cally compact or bispace. As explained in [ BSS08 , Sec. 2.5] (see a lso 5.1 .3) the ger be G → E natura lly g ives rise to a U (1) δ -principal bundle ˜ G δ → LE . Let g : L B 1 → U (1) δ be the function whic h describ es the holonomy of the bundle ˜ G δ → LE along the ﬁb ers of LE → L B 1 (see [ BSS08 , 2.6.3 ]). In the following we reca ll fro m [ BSS0 8 ] a co homologica l description of the functions g and h (int ro duced in 5.2.3). Let c 1 ∈ H 2 ( B ; Z ) denote the ﬁrst Chern class of the U (1)-principal bundle π : E → B , and let d ∈ H 3 ( E ; Z ) denote the Dixmier- Douady class of the g e rb e f : G → E . By int egra tio n ov er the ﬁb er it gives rise to a clas s R π d ∈ H 2 ( B ; Z ). In [ BSS08 , 2.4 .11] we hav e shown that a class χ ∈ H 2 ( B ; Z ) gives rise to a function ¯ χ : LB → U (1) δ in a natur al way . Pr op osi t i on 5.3.1 (Lemma 2.38 and Prop. 2.49 [ B SS0 8 ] ) We have the e qualities (1) c 1 = h : L B → U (1) δ . (2) Z π d | LB 1 = g : L B 1 → U (1) δ . 70 CHAPTER 5. ORBISP AC ES 5.3.2. — W e now have functions h, ˆ h : LB → U (1) δ asso ciated to the U (1)-principal bundles π : E → B and ˆ π : ˆ E → B . W e deﬁne LB (1 , ∗ ) := h − 1 (1) , L B ( ∗ , 1) := ˆ h − 1 (1) . W e furthermore have functions (se e 5.2 .1) g : L B (1 , ∗ ) → U (1) δ , ˆ g : LB ( ∗ , 1) → U (1) δ measuring the holo nomy of ˜ G δ → L E and ˜ ˆ G δ → L ˆ E along the ﬁber s. Pr op osi t i on 5.3.2 . — We have the e qualities ˆ g = h − 1 | LB ( ∗ , 1) , g = ˆ h − 1 | LB (1 , ∗ ) . Pr o of . — Let d ∈ H 3 ( E ; Z ) , ˆ d ∈ H 3 ( ˆ E ; Z ) be the Dixmier-Doua dy cla sses of the gerb es G L → E and ˆ G L → ˆ E . F urthermor e let c 1 , ˆ c 1 ∈ H 2 ( B ; Z ) denote the ﬁrst Chern class es of the U (1)-principal bundles π : E → B and ˆ π : ˆ E → B . The theory of T -duality for o rbispaces [ BS0 6 ] gives the equalities c 1 = − ˆ π ! ( ˆ d ) , ˆ c 1 = − π ! ( d ) . Hence the a ssertion follows from Pr op osition 5.3.1. 5.4. The T -duality transformation in twisted p erio di c delo calized coho- mology is an i s omorphism 5.4.1. — Let us consider a U (1)-principal bundle π : E → B in lo c a lly compa c t orbispaces with ﬁrst Chern class c 1 ∈ H 2 ( B ; Z ) and a top olo g ical U (1)-banded g erb e f : G → E with Dixmier -Douady c la ss d ∈ H 3 ( E ; Z ). In Deﬁnition 5 .1 .2 we hav e int ro duced the ob ject L G ∈ D ( Sh Ab LE ). F ur ther more we have U (1) δ -v alued functions h = c 1 and g = π ! ( d ) on LB . Let L B 1 := h − 1 (1) and no te that L π : LE → LB factors ov er the U (1)-principal bundle Lπ : LE → LB 1 . W e ﬁx u ∈ U (1) δ \ { 1 } a nd consider the c o mpo nent LB (1 ,u ) := h − 1 (1) ∩ g − 1 ( u ). L emma 5.4.1 . — We have R π ∗ ( L G ) | LB (1 ,u ) ∼ = 0 . 5.4. T -DUALITY IS AN ISOMORP HISM 71 Pr o of . — Let ( T → LB (1 ,u ) ) ∈ LB (1 ,u ) . After re ﬁning T by a covering we can assume that there is a diag ram B U (1) y   U (1) × B U (1) z o o x   s ∗ G L   o o / / G L (1 ,u )   ∗ U (1) q o o q   T × U (1) v o o s / / p   LE (1 ,u ) π   ∗ T w o o t / / LB (1 ,u ) of Car tesian s q uares. W e get t ∗ Rπ ∗ ( L G ) ∼ = Rp ∗ s ∗ ( L G ) = Rp ∗ s ∗ ( P G L ( L )) ∼ = Rp ∗ P s ∗ G L ( s ∗ L ) . Let H ∈ Sh Ab ( Site ( U (1))) b e the lo cally constant shea f ov er U (1) with ﬁber C and holonomy u ∈ U (1) \ { 1 } . Then we hav e s ∗ L ∼ = v ∗ H . W e calcula te further Rp ∗ P s ∗ G L ( s ∗ L ) ∼ = Rp ∗ P s ∗ G L ( v ∗ H ) ∼ = Rp ∗ v ∗ P U (1) ×B U (1) ( H ) ∼ = w ∗ Rq ∗ P U (1) ×B U (1) ( H ) . It r emains to show that Rq ∗ P U (1) ×B U (1) ( H ) ∼ = 0 . Recall from 3.4.9 that the ob ject P U (1) ×B U (1) ( H ) ∈ D ( Sh Ab Site ( U (1))) is giv e n (up to non-c anonical iso morphism) by the hol im of a diagr am 0 ← R x ∗ x ∗ ( H ) D ← R x ∗ x ∗ ( H )[2] D ← R x ∗ x ∗ ( H )[4] D ← R x ∗ x ∗ ( H )[6] . . . . The functor Rq ∗ commutes with this hol im (1) . Therefor e R q ∗ P U (1) ×B U (1) ( H ) is given by the ho lim o f the diag ram 0 ← R q ∗ Rx ∗ x ∗ ( H ) Rq ∗ ( D ) ← Rq ∗ Rx ∗ x ∗ ( H )[2] Rq ∗ ( D ) ← Rq ∗ Rx ∗ x ∗ ( H )[4] Rq ∗ ( D ) ← R q ∗ Rx ∗ x ∗ ( H )[6] . . . . The following calculation uses the pro jection formula twice, ﬁrst by Lemma 6.2.1 0 for the non-repr e s entable map x and a tensor pr o duct with a one- dimensional lo cal system of complex vector spaces H , secondly using Lemma 6.2.13 for the prop er representable (1) Rq ∗ is a right-adjoin t and commute s with pro ducts and mapping cones 72 CHAPTER 5. ORBISP AC ES map q and the tensor pro duct w ith the b ounded b elow ob ject Ry ∗ ( i ♯ Z Site ([ ∗ /U (1)]) ) ∈ D + ( Sh Ab Site ( U (1))) Rq ∗ Rx ∗ x ∗ ( H ) ∼ = Rq ∗ Rx ∗ ( Z Site ( U (1) ×B U (1) ⊗ x ∗ ( H )) ∼ = Rq ∗ ( Rx ∗ ( Z Site ( U (1) × B U (1)) ) ⊗ H ) ∼ = Rq ∗ ( Rx ∗ ( z ∗ Z Site ( B U (1)) ) ⊗ H ) ∼ = Rq ∗ ( q ∗ ( Ry ∗ Z Site ( B U (1)) ) ⊗ H ) ∼ = Ry ∗ Z Site ( B U (1)) ⊗ Rq ∗ ( H ) . Since the holonomy of H along U (1) is non-tr ivial, and the cohomolog y of S 1 with co eﬃcients in a non- tr ivial ﬂa t line bundle is trivia l, we hav e Rq ∗ ( H ) ∼ = 0 . 5.4.2. — W e now consider a T -duality diagra m (5 .2.1) where B is a lo ca lly compact orbispace. The or em 5.4.2 . — The lo c al T - duality tra nsformation (D eﬁ nition 5.2.5) T loc : R Lπ ∗ ( L G ) → RL ˆ π ∗ ( ˆ L ˆ G )[ − 2] is an isomorphism in D ( Sh Ab LB ) . In p articular, t he T -duality t r ansformation T : H ∗ deloc,per ( E , G ) → H ∗ deloc,per ( ˆ E , ˆ G ) is an isomorphism. Pr o of . — W e hav e functions h, ˆ h : L B → U (1) which deﬁne substa cks L B (1 , ∗ ) := h − 1 (1) a nd LB ( ∗ , 1) := ˆ h − 1 (1). By Pro po sition 5.3.2 we have g = ˆ h − 1 | LB (1 , ∗ ) : L B (1 , ∗ ) → U (1) δ . By Le mma 5.4.1 the ob ject R Lπ ∗ ( L G ) ∈ D ( Sh Ab LB ) is supp orted o n g − 1 (1) = LB (1 , ∗ ) ∩ LB ( ∗ , 1) =: LB (1 , 1) . Note that ˆ g = h − 1 | LB ( ∗ , 1) , so that RL ˆ π ∗ ˆ L ˆ G is suppo rted on LB (1 , 1) , to o. Let i : LB (1 , 1) → LB de no te the inclusio n. The following diagr am is the pull-ba ck of 5.4. T -DUALITY IS AN ISOMORP HISM 73 (5.2.1) via the ma p LB (1 , 1) → L B → B (5.4.3) p ∗ L ( G L ) | LE | LB (1 , 1) w w n n n n n n n n n n n n n & & N N N N N N N N N N N N u L / / ˆ p ∗ L ( ˆ G L ) | L ˆ E | LB (1 , 1) x x p p p p p p p p p p p p ' ' P P P P P P P P P P P P ( G L ) | LE | LB (1 , 1) f L ( ( P P P P P P P P P P P P P LF | LB (1 , 1) Lp w w p p p p p p p p p p p p p ˆ Lp ' ' N N N N N N N N N N N N ( ˆ G L ) | L ˆ E | LB (1 , 1) ˆ f L v v n n n n n n n n n n n n LE | LB (1 , 1) Lπ 1 ' ' P P P P P P P P P P P P L ˆ E | LB (1 , 1) L ˆ π 1 w w n n n n n n n n n n n n LB (1 , 1) W e consider L 1 := L | LE | LB (1 , 1) , ˆ L 1 := ˆ L | L ˆ E | LB (1 , 1) . Because we r estrict to the subset LB (1 , 1) of trivia l holo no my we hav e isomor phisms L 1 ∼ = Lπ ∗ 1 C LB (1 , 1) ˆ L 1 ∼ = L ˆ π ∗ 1 C LB (1 , 1) . The lo cal T -duality transfor mation T loc is now lo cally equal to the tr ansformatio n J deﬁned in 4.3.5 a pplied to the T -duality diagra m (5.4.3) a nd the sheaf C LB (1 , 1) . As in the pro of of Theo rem 4.3.7 one shows, us ing the comm utative double T -duality diagram, that T loc is an is o morphism. The glo ba l seco nd a ssertion can be deduced directly from Theorem 4.3.7. By the observ a tio n on the supp or t o f RL π ∗ ( L G ) ∈ D ( Sh Ab LB ) made ab ove we get H ∗ deloc,per ( E , G ) ∼ = H ∗ per ( LB (1 , 1) ; RL ( π 1 ) ∗ P ( G L ) | LE | LB (1 , 1) ( Lπ ∗ 1 C LB (1 , 1) )) , and similar ly H ∗ deloc,per ( ˆ E , ˆ G ) ∼ = H ∗ per ( LB (1 , 1) ; RL ( ˆ π 1 ) ∗ P ( ˆ G L ) | L ˆ E | LB (1 , 1) ( Lπ ∗ 1 C LB (1 , 1) )) . With these iden tiﬁcatio ns the T -dua lity tr ansformation in twisted per io dic deloca l- ized cohomology is then equal to the T -duality transformatio n in t wisted p erio dic cohomolog y for the diag ram (5.4.3) and the sheaf C LB (1 , 1) ∈ D + ( Sh Ab LB 1 , 1 ). CHAPTER 6 VERDIE R DUALITY F OR LOCALL Y COMP A CT ST A CKS 6.1. Elem e n ts of the theory of s tac ks on To p and sheaf theory 6.1.1. — In the present pap er we consider stac ks on the site Top . A pres tack is a lax pr e sheaf X of group oids on To p . The preﬁx ”lax” indica tes that for a pair of comp osable morphisms u : U → V , v : V → W we hav e a natural transfor ma tion of functors φ u,v : X ( u ) ◦ X ( v ) → X ( v ◦ u ) which is not necessarily the iden tity , and which satisﬁes a c ompatibility condition for triples. A prestack is a s tack if it sa tisﬁes the standard descen t c onditions on the level of ob jects and morphisms. A sheaf of sets can b e consider ed as a stack in the canonical w ay . Via the Y oneda em b edding Top → ShTop (note that the top ology of To p is s ub-canonical, i.e . representable presheav es are sheav es) we co nsider top ologica l spaces as stacks in the natural wa y . 6.1.2. — In the following we collect so me deﬁnitions and facts of the theo ry of sta cks in topo logical spaces. Stacks are ob jects of a tw o-categ ory , and ﬁbre pro ducts and more g eneral limits in stacks are under sto o d in the tw o-categ orial se nse. Note that t wo-categorial limits in stacks exists (see [ BSS08 ] for mor e information), and that the inclusion of spaces into stacks pres erves those limits. A useful reference for stacks in to po logical s paces and manifolds is the survey [ Hei05 ]. (1) A morphism of stacks G → H is called r epresentable, if for each spa ce U a nd map U → H the ﬁbre pro duct U × H G is equiv a lent to a space. (2) A representable map G → H b etw een s tacks is calle d prop er if fo r every ma p K → H from a compact space the ﬁbre pro duct K × H G is a co mpact space . (3) A map f : A → B of topo logical spa ces has lo cal sections if for each p oint b ∈ B in the imag e of f there exis ts a neighbourho o d b ∈ U ⊆ B and a map s : U → A such that f ◦ u = id U . (4) A representable morphism G → H has lo cal sections if for every map U → H from a space the induced map U × H G → U of spaces has lo cal sections. (5) A repres entable map G → H is surjective if for every map U → H from a space the induced ma p U × H G → U is a surjective map of spa ces. 76 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS (6) A map A → X from a space A to a stack X is called an atlas of X , if it is surjective, repres ent able and admits lo cal sections. A stac k which admits an atlas is ca lle d a top olo gical stack. (7) A morphism (not nec essarily r epresentable) b etw een to po logical sta cks G → H is sur jective (or has lo c al sections, resp ectively) if for an a tlas A → G the comp osition A → G → H is surjective (or has lo cal s e ctions, resp ectively) (note that this comp osition is r epresentable by Pr o p osition 6.1.1 b elow). (8) A co mpo sition of maps with lo ca l sectio ns has lo cal sections. The corr esp ond- ing as sertion is true fo r the following prop erties of maps: (a) representable (b) representable a nd pro p er (c) surjective. (9) Co nsider a tw o-car tesian dia gram of sta cks H − − − − → v G   y g   y f Y − − − − → u X If u has lo cal sections, then s o has v . If f is representable, then so is g . 6.1.3. — The inclusion of spaces into sheav e s and of sheaves into stacks preserves small limits, where limits in stacks are under sto o d in the tw o-categ orical s ense. This implies that a map of spaces X → Y is re presentable. In fact we have the following more general r esult. Pr op osi t i on 6.1.1 . — L et G b e a top olo gic al st ack and X a sp ac e. Then every morphism f : X → G is r epr esentable. The pro o f will b e given in 6.1.5 and needs s ome pr eparations. 6.1.4. — W e will nee d the no tion of an op en substack. Deﬁnition 6.1.2 . — L et G b e a stack in top olo gic al sp ac es. A morphism H → G of stacks is an emb e dding of an op en substack, if it is re pr esentable and for e ach map T → G fr om a sp ac e T the induc e d map of s p ac es T × G H → T is an op en emb e dding of top olo gic al sp ac es. Note that, via Y oneda , a n op en embedding o f space s is an op en embedding of stacks. Deﬁnition 6.1.3 . — A morphism U → G of top olo gic al stacks is lo c al ly an op en emb e dding if U ∼ = F i ∈ I U i for a c ol le ction ( U i ) i ∈ I of top olo gic al stacks and U i → G is an emb e dding of an op en s ubstack for every i ∈ I . Let us ﬁrst c har acterize spaces as stacks which can be cov er ed by a collec tion o f spaces. 6.1. ELEM E NTS OF THE THEOR Y OF ST ACKS ON To p AND SHE AF THEOR Y 77 L emma 6.1.4 . — L et X b e a stack in top olo gic al sp ac es for which ther e exists a morphism U → X fr om a sp ac e which is surje ctive and lo c al ly an op en emb e dding. Then X is e quivalent t o a sp ac e. Pr o of . — Let U ∼ = ⊔ i U i be s uch that U i → X is an op en embedding for all i . Then we deﬁne the spa c e B as the co equa liz er in spaces (6.1.5) B := co eq( G i,j U i × X U j ⇒ G i U i ) . Since U i → X is an open embedding we see that pr U i : U i × X U j → U i is an op en embedding. W e can now refer to [ No o , Pro p. 16.1 ] and deduce tha t the equalizer in spaces B is also the t wo-categorica l equalizer in stacks of the diagr am (6.1.5), which is of co urse equiv alent to X . Note that the diﬃculty at this p oint is that the embedding of the catego ry of spaces (viewed as a two-category) into the tw o-categ ory of stacks do es no t pr eserve g eneral sma ll colimits, as opp osed to the case of limits. F or completeness we will give an arg umen t. First note tha t pr U i : U i × X U i ∼ → U i is a homeo morphism. It thus follows from the group oid structure o f the co eq ua lizer diagram that U i → B is injective fo r all i . Since F i U i → B is a top ologica l quotient map it is o pe n. The r efore F i U i → B is a o pe n co vering. W e further conclude that the na tural ma p U i × X U j → U i × B U j is in fact a homeomo r phism. The claim is that X is equiv alent to B . W e ﬁrst c o nstruct a morphism X → B . Let ( T → X ) ∈ X ( T ). Then ( T i := T × X U i ) i is an op en co vering of T . Using the ident iﬁcation T i × T T j ∼ = T × X ( U i × X U j ) we get a diagr am F i,j T i × T T j     / / U i × X U j     F i T i / /   F i U i   T / / B , where the horizontal maps a r e induced b y the pro jections T × X U i → U i , a nd the left vertical is the repres ent ation of T as a co equalize r . There fo re we obta in a unique factorization ( T → B ) ∈ B ( T ). The construction is functorial in T and therefore induces a mo rphism X → B . In o r der to see that it has an inv ers e let ( T → B ) ∈ B ( T ) be given. Then we deﬁne the o p e n c overing ( T i := T × B U i ) i of T . The comp ositions φ i : T i ∼ = T × B U i pr U i → U i → X 78 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS can b e co ns idered a s a collectio n of ob jects ( φ i ∈ X ( T i )) i . The induced map T i ∩ T j ∼ = T i × T T j ∼ = ( T × B U i ) × T ( T × B U j ) ∼ = T × B ( U i × B U j ) pr U i × B U j → U i × B U j ∼ = U i × X U j → X × X X can b e co ns idered a s a co llection o f is omorphisms φ ij : ( φ i ) | T i ∩ T j ∼ → ( φ j ) | T i ∩ T j which satisfy the co cycle condition on triple intersections. Since X is a s tack w e can therefo re glue the lo cal maps and get a map ( T → X ) ∈ X ( T ) which is unique up to unique isomorphism. This construction is a gain functor ia l in T and provides the map B → X . It is easy to see that b oth maps X → B and B → X constructed ab ov e a re m utually inv ers e . 6.1.5. — W e now show Pr op osition 6.1 .1 Pr o of . — Consider a map T → G fr om a space T . W e ha ve to prov e that the ﬁb er pro duct T × G X is equiv alent to a space. Using the assumption that G is top olog ical we choose a n a tlas A → G of G . Beca use A → G ha s lo ca l s ections, we can ﬁnd an op en covering F i ∈ I U i =: U → X such that U × G A → U has a section s : U → U × G A . W e ﬁr st want to sho w that T × G U is a spa ce. Since the structure map A → G of an atlas is repr esentable we know that U × G A and T × G A ar e spac e s. Therefo re, T × G U × G A ∼ = ( T × G A ) × A ( U × G A ) is a space, to o . T he section s pulls back to a section ˆ s : T × G U → T × G U × G A which implements T × G U a s a subs pa ce of the space T × G U × G A . T × G U × G A / / w w o o o o o o o o o o o U × G A { { w w w w w w w w w   T × G U ˆ s * * / /   U   s , , T × G X / /   X   A z z v v v v v v v v v v T / / G . Since the map U → X is surjective and lo ca lly a n op en embedding its pull-back T × G U → T × G X is surjective and lo cally an open em b edding, too. Therefor e by Lemma 6.1 .4 the sta ck T × G X is equiv alent to a spa c e . 6.1. ELEM E NTS OF THE THEOR Y OF ST ACKS ON To p AND SHE AF THEOR Y 79 6.1.6. — Recall that a top olog ical stack is called lo ca lly compac t if it admits a lo cally compact atlas A → G such that A × G A is a lo c ally compact spa c e. F urthermo re recall that the s ite X = Si te ( X ) asso cia ted to a lo cally compact stack X is the full sub c ategory o f lo c a lly compa ct spaces U → X over X such that the structure ma p has lo c al sec tio ns. A mor phism in this site X is a diagram (6.1.6) U @ @ @ @ @ @ @ / /   V ~ ~ ~ ~ ~ ~ ~ ~ ~ X consisting of a morphis m of spaces ov e r X a nd a tw o-mor phis m. The top olo gy on X is given by the cov ering fa milies of the ob jects ( U → X ) induced by op en covering of U . Much of the general theory would work without the assumption of lo ca l compact- ness. B ut lo ca l compactness is imp or ta nt in co nnection with the pr o jection formula Lemma 6.2.11 which is a c r ucial ingredient of the theor y of integration. Since the latter is o ur main g oal of the present section w e ge nerally adopt the restriction to lo cally compact s tacks. 6.1.7. — The s heaf theory for topolog ical stac ks can be built in a parallel manner to the sheaf theo ry for smo o th stacks develop ed in [ BSS07 ]. The tra nsition go es v ia the fo llowing repla cements of words: (1) F o r the deﬁnition of stacks the site of s mo oth manifolds Mf ∞ is r eplaced by the site of top o logical s paces Top . In the deﬁnition of the site of a lo cally c ompact stack manifolds a re r eplaced by lo ca lly compa ct spaces . (2) The co ncept of a s m o oth stack is repla ced by the co ncept o f a lo c al ly c omp act stack . (3) The notion of a smo oth map is r eplaced by the notion of a map which admits lo c al se ctions . In the present pa pe r we freely use results in the general shea f theory for top ologica l stacks from [ BSS07 , Sec . 2] in the case of stacks in topo logical spaces which a re prov ed there for manifolds. It should b e noted that with the conven tions just made, all sta tements and pro ofs ca r ry over verbatim 6.1.8. — Let X b e a lo c ally compact stack. By Pr X and S h X we denote the ca teg ories of pres he aves and sheaves on X . They are related by a pa ir o f adjoint functors i ♯ : Pr X ⇆ Sh X : i . The sheaﬁﬁcation functor i ♯ is exa ct. 6.1.9. — Let f : X → Y be a morphism o f lo cally compact stacks. In induces a functor p f ∗ : Pr X → Pr Y by p f ∗ F ( V → Y ) := lim F ( U → X ) , 80 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS where the limit is taken over the catego ry of diagra ms (6.1.7) U   / / X f   V / / : B Y with ( U → X ) ∈ X . F or details we r efer to [ BSS 0 7 , Sections 2.1, 2 .2]. This functor ﬁts int o an adjo int pa ir p f ∗ : Pr Y ⇆ Pr X : p f ∗ . The functor p f ∗ is given by p f ∗ G ( U → X ) = colim G ( V → Y ) , where the co limit is ag ain taken ov er the categor y of diagr a ms with ( V → Y ) ∈ Y . W e extend these functors to sheaves by f ∗ := i ♯ ◦ p f ∗ ◦ i , f ∗ := i ♯ ◦ p f ∗ ◦ i and obta in a n adjoint pair f ∗ : Sh Y ⇆ Sh X : f ∗ . Note that p f ∗ preserves shea ves (see [ BSS0 7 , Lemma 2.13]). The right-adjoin t functor f ∗ : Sh Ab X → Sh Ab Y is le ft exa ct and there fo re a dmits a right-derived functor Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) betw een the b ounded b elow derived ca teg ories. 6.1.10 . — If g : Y → Z is a second morphism of lo cally compact s tacks, then we hav e natural isomorphisms of functors ( g ◦ f ) ∗ ∼ = g ∗ ◦ f ∗ , f ∗ ◦ g ∗ ∼ = ( g ◦ f ) ∗ (see 6.6 .9). F urthermore, we hav e Rg ∗ ◦ Rf ∗ ∼ = R ( g ◦ f ) ∗ on the level of b ounded b elow derived categor ie s by Lemma 6.6.1 3. The relation f ∗ ◦ g ∗ ∼ = ( g ◦ f ) ∗ descends to the der ived categor ie s if the pull-ba ck functors ar e exact, e.g. if f a nd g hav e lo ca l sections (see 6.1.11). These facts g eneralize co rresp onding results s hown in [ BSS0 7 ]. 6.1.11 . — Let f : G → H b e a morphism b etw een top olo gical stacks which has lo cal sections. It induces a mor phism b etw een sites f ♯ : G → H by compo sition. On ob jects it is given by f ♯ ( U → G ) := ( U → G → H ) (we will o ften use the short hand U for ( U → G ) and write f ♯ U ). In fact, since U → G and f hav e lo cal sec tions, the comp osition U → H has lo ca l sections. F urthermore , the ma p U → H from a s pace to a to po logical stack is repres e ntable by Lemma 6.1.1. One chec k s that f ♯ maps cov er ing fa milies to covering families and pr eserves the ﬁber pro ducts as in [ T am94 , 1.2.2]. 6.1. ELEM E NTS OF THE THEOR Y OF ST ACKS ON To p AND SHE AF THEOR Y 81 If f : G → H has lo cal sections, then the functor f ∗ : Sh H → Sh G is the pull-back f ∗ = ( f ♯ ) ∗ asso ciated to a mor phism of sites. Ex plicitly it is given b y f ∗ F ( U ) := F ( f ♯ U ), compar e Lemma [ BSS0 7 , 2.7]. In addition, the functor f ∗ : Sh H → Sh G is exact (se e [ BSS07 , 2 .5.9]) and pres erves ﬂat sheaves of a b elian groups . L emma 6.1.8 . — If f : X → Y is a morphism b etwe en lo c al ly c omp act stacks which has lo c al se ctions, then we have t he derive d adjunction f ∗ : D + ( Sh Ab Y ) ⇆ D + ( Sh Ab X ) : R f ∗ . Pr o of . — Since f ∗ is exact its right a djoint f ∗ preserves injectives. If G ∈ C + ( Sh Ab X ) is a co mplex of injectives and F ∈ C + ( Sh Ab Y ), then we have R Hom Sh Ab Y ( F, Rf ∗ ( G )) ∼ = Hom Sh Ab Y ( F, f ∗ ( G )) ∼ = Hom Sh Ab X ( f ∗ ( F ) , G ) ∼ = R Hom Sh Ab X ( f ∗ ( F ) , G ) . This implies the as sertion. 6.1.12 . — L emma 6.1.9 . — L et X b e a lo c al ly c omp act stack. If C, B → X ar e maps fr om lo c al ly c omp act sp ac es, t hen C × X B is lo c al ly c omp act. Pr o of . — By as sumption X is lo cally compac t s o that we can chose a n atlas A → X such that A and A × X A are lo cally compact. Since A → X is surjective and has lo cal s ections, ther e exists a n op en cov ering ( B i ) o f B suc h that we hav e lifts A   B i 7 7 / / B / / X . Then ( A × X B i ) is a n o p en covering o f A × X B . In order to show that A × X B is lo cally compact it suﬃces to show that the space A × X B i is loca lly compact. By A × X B i ∼ = ( A × X A ) × A B i ⊆ A × X A × B i , this space is a closed (note tha t A is Hausdorﬀ ) subspace of a lo cally compa c t space and hence itself lo cally compact. The same arg umen t shows that C × X A is lo cally compact. W e now write C × X B i ∼ = ( C × X A ) × A B i ⊆ ( C × X A ) × B i in order to se e that C × X B i is lo ca lly compact. Since ( C × X B i ) is an op en covering o f C × X B we conclude that C × X B is lo cally compact. 6.1.13 . — Let f : X → Y b e a mor phism betw een lo cally c o mpact sta cks. L emma 6.1.10 . — If f is r epr esentable, then it induc es a morphism of sites f ♯ : Y → X given by f ♯ ( V → Y ) := ( X × Y V → X ) . 82 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS Pr o of . — Let B → X b e a lo cally compact atlas . W e consider ( V → Y ) ∈ Y and form the diag ram of Car tesian squar es V × Y B / /   B   U / /   X f   V / / Y . In o rder to check that ( U → X ) ∈ X we must show that U is lo cally compact. Since B → X is surjective and has loca l sections w e see that V × Y B → U is surjective and has lo cal sections, too. Since Y is lo cally compact we see by Lemma 6.1.9 that V × Y B is lo ca lly compact. Let u ∈ U and W ⊆ U b e a ne ig hborho o d of u such that there exis ts a se ction V × Y B π   W s : : / / U . Let K ⊆ π − 1 ( W ) b e a compact neigh bo rho o d of s ( u ). Then s − 1 ( K ) is a compact neighborho o d o f u . Indeed, s − 1 ( K ) is a clos ed subset of the compact set π ( K ). It is easy to see that f ♯ maps covering families to co vering families and pr eserves the ﬁb er pro ducts r equired for a mor phism of sites, see [ T am94 , 1.2.2]. If f : X → Y is a representable mor phism betw een loca lly compa c t stacks, then we hav e the relations f ∗ = ( f ♯ ) ∗ : Sh Y → Sh X and f ∗ = ( f ♯ ) ∗ : S h X → Sh Y , see [ BSS07 , Lemma 2.9 ]. 6.1.14 . — Let X b e a top o logical stack a nd ( U → X ) ∈ X . Let ( U ) denote the site whose o b jects and morphisms are the open subsets of U and inclusions, and whose cov er ings ar e cov er ings by families of op en subsets. W e have restriction functors ν U : Sh X → Sh ( U ) and p ν U : Pr X → Pr ( U ). F o r F ∈ Sh X w e a lso write ν U ( F ) =: F U . W e hav e the following a ssertions, most of which are stra ig htf orward to prove. (1) Let i ♯ and i ♯ U denote the sheaﬁﬁcation functor s on the s ites X and ( U ). Then we have a natura l isomo rphism i ♯ U ◦ p ν U ∼ = ν U ◦ i ♯ , see [ BSS0 7 , Lemma 2.4 .7] (2) Let F ∈ S h X . If f : U → V is a mor phism (6.1.6) in X , then we hav e a natural map f ∗ F V → F U . (3) Ther e is a o ne-to one corresp ondence of shea ves F ∈ Sh X on the one hand, and of colle c tio ns ( F U ) ( U → X ) ∈ X of sheav es F U ∈ Sh ( U ) together with functorial maps f ∗ F V → F U for a ll morphis ms f : U → V in X on the other hand. 6.1. ELEM E NTS OF THE THEOR Y OF ST ACKS ON To p AND SHE AF THEOR Y 83 (4) Let F, G ∈ Sh X . There is a o ne-to-one co rresp ondence b etw een compatible collections of morphisms g U : F U → G U for all ( U → X ) ∈ X a nd maps g : F → G . (5) If F , G ∈ Sh X or F , G ∈ D + ( Sh Ab X ), then a ma p F → G is an is omorphism if and only if the induced map F U → G U is an isomorphism for all ( U → X ) ∈ X . (6) Let f : X → Y be a repres ent able map of lo cally compact stacks, ( A → Y ) ∈ Y and ( B := A × Y X → X ) ∈ X . Let g : B → A b e the pro jection onto the ﬁrst factor a nd g ∗ : Sh ( B ) → Sh ( A ). Then we hav e for F ∈ S h X or G ∈ D + ( Sh Ab X ) ( f ∗ F ) A ∼ = g ∗ ( F B ) , ( Rf ∗ G ) A ∼ = Rg ∗ ( G B ) . The second is o morphism follows from the ﬁrst using the fact that the res tr iction ν B preserves ﬂabby or even injective sheaves (see Lemma 6.1.11). (7) If f : X → Y is a map of top ologica l stac ks which ha s loca l sections, ( B → X ) ∈ X , then we hav e ( B → X → Y ) ∈ Y and for F ∈ Sh Y ( f ∗ F ) B ∼ = F B . (8) The collection of r estriction functor s ( ν U ) ( U → X ) ∈ X detects ﬂabby (ﬂasque, ﬂat) sheav e s (see Deﬁnition 3.1.1), i.e. a shea f F ∈ Sh Ab X is ﬂabb y (ﬂasque, ﬂat) if a nd only if F U ∈ S h Ab ( U ) is ﬂa bby for a ll ( U → X ) ∈ X (compare 6.2.6 for the ﬂa t c a se). (9) The collection of restrictio n functor s ( ν U ) ( U → X ) ∈ X detects exact seq uences, i.e. a s equence F → G → H of sheaves of ab elia n g roups on X is exact if and only if F U → G U → H U is exa ct for a ll ( U → X ) ∈ X . L emma 6.1.11 . — L et ( U → X ) ∈ X . The functor ν U : S h Ab X → Sh Ab ( U ) pr e- serves inje ct ive she aves. Pr o of . — W e show that ν U has a n exa ct left adjoint ν U Z : Sh Ab ( U ) → Sh Ab X . W e ﬁr st show tha t the restr ic tio n functor p ν U : Pr Ab X → Pr Ab ( U ) ﬁts into an adjoint pa ir p ν U Z : Pr Ab ( U ) ⇆ P r Ab X : p ν U . The left-adjoint is g iven by p ν U Z ( F )( A → X ) := colim F ( V ) , where the co limit is taken ov er the categ o ry of diagra ms V   A φ ~ ~   o o U / / X , where V → U is the embedding o f an op en subset. As expla ined in [ Mil 80 , I I.3.1 8] we hav e a deco mp o sition of this ca tegory in to a union of categ ories S ( φ ) with φ ∈ Hom X (( A → X ) , ( U → X )). The c ategory S ( φ ) is the catego r y of o pen neighbo rho o ds 84 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS of φ ( A ) and their inclusio ns. It is coﬁltered. Therefore F 7→ colim S ( φ ) F ( V ) preser ves ﬁnite limits a nd is in par ticular left exact. This implies that p ν U Z given b y p ν U Z ( F )( A → X ) ∼ = M φ colim S ( φ ) F ( V ) is left-exa c t, to o. W e now get ν U Z := i ♯ ◦ p ν U Z ◦ i U . As a left-adjoint it is rig h t-exact. Since i U is left exact and i ♯ is exact, this comp osition is a lso left-ex a ct. 6.1.15 . — L emma 6.1.12 . — Consider the fol lowing Cartesian diagr am in lo c al ly c omp act top olo gic al st acks H v − − − − → G   y g   y f Y u − − − − → X In this situation the two c anonic al ways to deﬁne a n atur al tr ansformation u ∗ f ∗ → g ∗ v ∗ : Sh Ab ( G ) → Sh Ab ( Y ) give t he same r esult, i.e. the diagr am (6.1.13) u ∗ f ∗ unit / / g ∗ g ∗ u ∗ f ∗ ug = f v / / g ∗ v ∗ f ∗ f ∗ counit / / g ∗ v ∗ u ∗ f ∗ unit / / u ∗ f ∗ v ∗ v ∗ ug = f v / / u ∗ u ∗ g ∗ v ∗ counit / / g ∗ v ∗ c ommutes. This tr ansformation is functorial with r esp e ct to c omp osition of Cartesian diagr ams. Mor e over, if u has lo c al se ctions, then this tr ansformation induc es isomorphi sms u ∗ f ∗ ∼ = g ∗ v ∗ : Sh Ab ( G ) → Sh Ab ( Y ) , (6.1.14) u ∗ Rf ∗ ∼ = Rg ∗ v ∗ : D + Sh Ab ( G ) → D + Sh Ab ( Y ) . (6.1.15) If u and f have lo c al se ctions, t hen we get c ommutative diagr ams u ∗ unit % % K K K K K K K K K K unit y y s s s s s s s s s s u ∗ g ∗ g ∗ ∼ / / f ∗ v ∗ g ∗ f ∗ f ∗ u ∗ ∼ o o , v ∗ f ∗ f ∗ v ∗ counit 9 9 s s s s s s s s s s ∼ / / f ∗ u ∗ g ∗ ∼ / / v ∗ g ∗ g ∗ counit e e K K K K K K K K K K u ∗ unit % % K K K K K K K K K K unit y y s s s s s s s s s s u ∗ f ∗ f ∗ ∼ / / g ∗ v ∗ f ∗ g ∗ g ∗ u ∗ ∼ o o , v ∗ v ∗ f ∗ f ∗ counit 9 9 s s s s s s s s s s ∼ / / g ∗ u ∗ f ∗ ∼ / / g ∗ g ∗ v ∗ counit e e K K K K K K K K K K and their derive d versions, e.g. 6.1. ELEM E NTS OF THE THEOR Y OF ST ACKS ON To p AND SHE AF THEOR Y 85 (6.1.16) u ∗ unit & & M M M M M M M M M M M unit x x q q q q q q q q q q q u ∗ Rf ∗ f ∗ ∼ / / Rg ∗ v ∗ f ∗ Rg ∗ g ∗ u ∗ ∼ o o , and also (6.1.17) Ru ∗ u ∗ unit + + X X X X X X X X X X X X X X X X X X X X X X X X X unit s s f f f f f f f f f f f f f f f f f f f f f f f f f f Ru ∗ u ∗ Rf ∗ f ∗ ∼ / / Ru ∗ Rg ∗ v ∗ f ∗ ∼ / / Rf ∗ Rv ∗ v ∗ f ∗ ∼ / / Rf ∗ Rv ∗ g ∗ u ∗ Rf ∗ f ∗ Ru ∗ u ∗ ∼ o o Pr o of . — Most of the fol lowing ar gument s and the lar ge diagr ams wer e supplie d by A. Schneider . F or convenience we prese nt a pro of of (6.1.13), see also [ Del73 , Exp os e XVII, P r op osition 2.1 .3]. W e ﬁrst o bserve that (6.1.18) v ∗ f ∗ f ∗ v ∗ counit / / ∼   v ∗ v ∗ counit   ( f v ) ∗ ( f v ) ∗ counit / / id commutes. Using this in addition to standard functorial prop er ties we check that a ll squares in the following diagr am commute: u ∗ f ∗ unit / / g ∗ g ∗ u ∗ f ∗ ∼ / / unit   g ∗ ( ug ) ∗ f ∗ = / / unit   g ∗ ( f v ) ∗ f ∗ ∼ / / unit   g ∗ v ∗ f ∗ f ∗ counit / / unit   g ∗ v ∗ unit   id } } g ∗ g ∗ u ∗ f ∗ v ∗ v ∗ ∼ / / g ∗ ( ug ) ∗ f ∗ v ∗ v ∗ = / / ∼   g ∗ ( f v ) ∗ f ∗ v ∗ v ∗ ∼ / / ∼   g ∗ v ∗ f ∗ f ∗ v ∗ v ∗ counit / / ∼   g ∗ v ∗ v ∗ v ∗ counit   g ∗ g ∗ u ∗ f ∗ v ∗ v ∗ ∼ / / g ∗ ( ug ) ∗ ( f v ) ∗ v ∗ = / / g ∗ ( f v ) ∗ ( f v ) ∗ v ∗ g ∗ ( f v ) ∗ ( f v ) ∗ v ∗ counit / / g ∗ v ∗ g ∗ g ∗ u ∗ f ∗ v ∗ v ∗ ∼ / / g ∗ ( ug ) ∗ ( f v ) ∗ v ∗ = / / g ∗ ( ug ) ∗ ( ug ) ∗ v ∗ = O O g ∗ ( ug ) ∗ ( ug ) ∗ v ∗ counit / / = O O g ∗ v ∗ g ∗ g ∗ u ∗ f ∗ v ∗ v ∗ ∼ / / g ∗ g ∗ u ∗ ( f v ) ∗ v ∗ = / / ∼ O O g ∗ g ∗ u ∗ ( ug ) ∗ v ∗ ∼ / / ∼ O O g ∗ g ∗ u ∗ u ∗ g ∗ v ∗ counit / / ∼ O O g ∗ g ∗ g ∗ v ∗ counit O O u ∗ f ∗ unit / / u ∗ f ∗ v ∗ v ∗ ∼ / / unit O O u ∗ ( f v ) ∗ v ∗ = / / unit O O u ∗ ( ug ) ∗ v ∗ ∼ / / unit O O u ∗ u ∗ g ∗ v ∗ counit / / unit O O g ∗ v ∗ . unit O O id a a The tw o wa ys to go a long the b oundary from the upp er left to lower right co rner give the tw o maps u ∗ f ∗ → g ∗ v ∗ in q uestion. 86 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS The isomorphism (6.1.14) can be shown as in [ BSS07 , Lemma 2.16], wher e the assumption of smo othness of u in [ BS S0 7 ] corre s po nds to the assumption of lo ca l sections in the present setting. The der ived version (6.1 .15) can b e shown using the simplicial mo dels as in [ BSS07 , Lemma 2.43 ]. Alternatively one can use the commutativit y o f the diagra m as serted in L e mma 3.2 .6 a nd the isomor phism (3.2 .5). W e now s how the co mpatibilit y o f the units and counits with Cartesia n diagra ms. The arguments are purely formal and only use that the functors involv ed o ccur as parts of adjoin t pairs. W e will only give the deta ils for the t wo triangles in volving derived functor s. If in addition to u also f has lo cal se c tio ns, then so has g . In this case w e ha ve the adjoint pa irs ( f ∗ , Rf ∗ ) and ( g ∗ , Rg ∗ ). In order to see (6.1.16) w e m ust show that u ∗ unit / / unit 4 4 u ∗ Rf ∗ f ∗ Ψ / / Rg ∗ v ∗ f ∗ ∼ / / Rg ∗ ( f v ) ∗ = / / Rg ∗ ( ug ) ∗ ∼ / / Rg ∗ g ∗ u ∗ , commutes, where Ψ : u ∗ Rf ∗ f ∗ → Rg ∗ v ∗ f ∗ is induced by (6 .1.15). This is a co ns e- quence o f the co mm utativity o f u ∗ unit / / u ∗ Rf ∗ f ∗ unit   Ψ / / Rg ∗ v ∗ f ∗ v v Rg ∗ g ∗ u ∗ f ∗ f ∗ ∼ / / Rg ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rg ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / Rg ∗ v ∗ f ∗ Rf ∗ f ∗ counit O O u ∗ unit / / Rg ∗ g ∗ u ∗ ∼ / / unit O O Rg ∗ ( ug ) ∗ unit O O = / / h h Rg ∗ ( f v ) ∗ unit O O ∼ / / i i Rg ∗ v ∗ f ∗ i i unit O O id a a which follows fro m standar d functoria l prop er ties of units and counits. The same prop erties are used in the pr o of o f (6.1.17) which is repr esented by the bo undary of the following big arr ay of small commutativ e squar es and triangles Rf ∗ f ∗ u ∗ u ∗ unit / / Φ - - unit   Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ ∼ / / unit   Rf ∗ f ∗ R ( ug ) ∗ g ∗ u ∗ = / / unit   Rf ∗ f ∗ R ( f v ) ∗ g ∗ u ∗ ∼ / / unit   Rf ∗ f ∗ Rf ∗ v ∗ g ∗ u ∗ counit / / unit   Rf ∗ v ∗ g ∗ u ∗ Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q unit   Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ g ∗ u ∗ Rf ∗ f ∗ = / / ∼   Rf ∗ f ∗ R ( f v ) ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / ∼   Rf ∗ f ∗ Rf ∗ v ∗ g ∗ u ∗ Rf ∗ f ∗ counit / / ∼   Rf ∗ v ∗ g ∗ u ∗ Rf ∗ f ∗ ∼   Rf ∗ v ∗ g ∗ u ∗ ∼   unit o o Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( f v ) ∗ ( ug ) ∗ Rf ∗ f ∗ ∼ / / =   Rf ∗ f ∗ Rf ∗ v ∗ ( ug ) ∗ Rf ∗ f ∗ counit / / =   Rf ∗ v ∗ ( ug ) ∗ Rf ∗ f ∗ =   Rf ∗ v ∗ ( ug ) ∗ =   unit o o Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( f v ) ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ Rf ∗ v ∗ ( f v ) ∗ Rf ∗ f ∗ counit / / ∼   Rf ∗ v ∗ ( f v ) ∗ Rf ∗ f ∗ ∼   Rf ∗ v ∗ ( f v ) ∗ unit o o ∼   Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( f v ) ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ Rf ∗ v ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / Rf ∗ v ∗ v ∗ f ∗ Rf ∗ f ∗ counit ( ( Q Q Q Q Q Q Q Q Q Q Q Q Rf ∗ v ∗ v ∗ f ∗ unit o o id   u ∗ u ∗ unit A A unit   Rf ∗ v ∗ v ∗ f ∗ Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( f v ) ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ Rf ∗ v ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / Rf ∗ f ∗ Rf ∗ v ∗ v ∗ f ∗ counit 6 6 m m m m m m m m m m m m Rf ∗ v ∗ v ∗ f ∗ unit o o id O O Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( f v ) ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( f v ) ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / ∼ O O Rf ∗ f ∗ R ( f v ) ∗ v ∗ f ∗ ∼ O O R ( f v ) ∗ v ∗ f ∗ ∼ O O unit o o Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ R ( ug ) ∗ ( ug ) ∗ Rf ∗ f ∗ = / / Rf ∗ f ∗ R ( ug ) ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / = O O Rf ∗ f ∗ R ( ug ) ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / = O O Rf ∗ f ∗ R ( ug ) ∗ v ∗ f ∗ = O O R ( ug ) ∗ v ∗ f ∗ = O O unit o o Rf ∗ f ∗ u ∗ u ∗ Rf ∗ f ∗ unit / / Rf ∗ f ∗ u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / Rf ∗ f ∗ u ∗ Rg ∗ ( ug ) ∗ Rf ∗ f ∗ = / / ∼ O O Rf ∗ f ∗ u ∗ Rg ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / ∼ O O Rf ∗ f ∗ u ∗ Rg ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / ∼ O O Rf ∗ f ∗ u ∗ Rg ∗ v ∗ f ∗ ∼ O O u ∗ Rg ∗ v ∗ f ∗ ∼ O O unit o o u ∗ u ∗ Rf ∗ f ∗ unit / / Ψ 1 1 unit O O u ∗ Rg ∗ g ∗ u ∗ Rf ∗ f ∗ ∼ / / unit O O u ∗ Rg ∗ ( ug ) ∗ Rf ∗ f ∗ = / / unit O O u ∗ Rg ∗ ( f v ) ∗ Rf ∗ f ∗ ∼ / / unit O O u ∗ Rg ∗ v ∗ f ∗ Rf ∗ f ∗ counit / / unit O O u ∗ Rg ∗ v ∗ f ∗ m m m m m m m m m m m m m m m m m m m m m m m m unit O O 88 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.2. T ens or pro ducts and the pro jection form ula 6.2.1. — W e consider a Gro thendieck site X a nd a co mmu tative ring R . The goal of the present Subsection is to discuss as pe c ts of the close d monoida l s tructures on the categorie s of presheaves Pr R − Mod X a nd sheaves Sh R − Mod X o f R -mo dules on X . The material is standard, but w e need to unders tand in detail the r elation betw ee n the sheaf and preshea f v er sions in order to show the compatibilit y with the op erations induced by a morphism o f stacks. 6.2.2. — Let F , G ∈ Pr R − Mod X be presheaves of R - mo dules. The tenso r pro duct F ⊗ p G ∈ Pr R − Mod X is deﬁned a s the preshea f which asso ciates to ( U → X ) the R -mo dule F ( U ) ⊗ p R G ( U ). In this wa y Pr R − Mod X b ecomes a symmetric monoidal category . Since colimits of pres heaves are deﬁned ob ject wise we hav e for a dia gram of presheav es of R -mo dules ( F i ) i ∈ I that colim i ∈ I ( F i ⊗ p R G ) ∼ = (colim i ∈ I F i ) ⊗ p R G . 6.2.3. — F or U ∈ X let h U ∈ P r X denote the presheaf repre sented by U and h R U ∈ Pr R − Mod X b e the pr esheaf of R - mo dules g e ne r ated by h U . Let F , G ∈ Pr R − Mod X . W e deﬁne the pr esheaf Hom p ( F, G ) ∈ Pr R − Mod X by Hom p ( F, G )( U ) := Hom Pr R − Mod X ( h R U ⊗ p F, G ) . The top ology of the site of a lo cally co mpa ct stack is sub-ca nonical. Hence , in this case h U is actually a shea f. But even in the case of a s ub-canonical top olo gy h R U is only a pres he a f, in gener al. If U → V is a morphism in X , then Hom p ( F, G )( V ) → Hom p ( F, G )( U ) is induced by the mo rphism h U → h V . If H ∈ Pr R − Mod X , then we have Hom Pr R − Mod X ( H, Hom p ( F, G )) ∼ = Hom Pr R − Mod X (colim h R V → H h R V , Hom p ( F, G )) ∼ = lim h R V → H Hom Pr R − Mod X ( h R V , Hom p ( F, G )) ∼ = lim h R V → H Hom p ( F, G )( V ) = lim h R V → H Hom Pr R − Mod X ( h R V ⊗ p F, G ) ∼ = Hom Pr R − Mod X (colim h R V → H ( h R V ⊗ p F ) , G ) ∼ = Hom Pr R − Mod X ((colim h R V → H h R V ) ⊗ p F, G ) ∼ = Hom Pr R − Mod X ( H ⊗ p F, G ) 6.2. TENSOR PRODUCT S AND THE PROJECTION FORMULA 89 In other words, the pair ( ⊗ p , Hom p ) together with this natural iso morphism deﬁnes a clos ed symmetric monoidal structure on Pr R − Mod X . In particular, if ( F i ) i ∈ I is a diagram o f pr esheav es, then we hav e (6.2.1) Hom p (colim i ∈ I F i , G ) ∼ = lim i ∈ I Hom p ( F i , G ) . 6.2.4. — An element o f Hom ( F, G )( U ) = Hom Pr R − Mod X ( h R U ⊗ p F, G ) is given by a collection of R -linea r maps ( φ V → U : F ( V ) → G ( V )) ( V → U ) ∈ X /U such that for a mor phism ( W → U ) 7→ ( V → U ) in X /U the diagr am F ( V ) / / φ V → U   F ( W ) φ W → U   G ( V ) / / G ( W ) commutes. Therefore Hom ( F, G )( U ) ∼ = Hom Pr R − Mod X /U ( F | U , G | U ) . L emma 6.2.2 . — If G is a she af, then H om ( F, G ) is a she af. Pr o of . — Let U ∈ X and ( U i → U ) i ∈ I be a cov er ing. In order to simplify the no tation we co ns ider V := ⊔ i ∈ I U i . W e must show that the s equence 0 → Ho m ( F, G )( U ) → Ho m ( F, G )( V ) → Ho m ( F, G )( V × U V ) is exa ct. Let ψ ∈ Hom Pr R − Mod X /U ( F | U , G | U ) b e such that its restriction to V v anishes. If ( W → U ) ∈ X /U , then W × U V → W is a cov er ing of W , and pr ∗ W : G ( W ) → G ( W × U V ) is injective since G is a shea f. In view of the commutativ e diag ram F ( W ) pr ∗ W / / ψ W   F ( W × U V ) ( ψ | V ) W × U V   G ( W ) pr ∗ W / / G ( W × U V ) we see that ψ W = 0. Let now φ ∈ H om Pr R − Mod X /V ( F | V , G | V ) b e such that the induced map Φ ∈ H om Pr R − Mod X / ( V × U V ) ( F | V × U V , G | V × U V ) v anishes. W e will co nstruct ψ ∈ Hom Pr R − Mod X /U ( F | U , G | U ) such tha t ψ | V = φ . Let ( W → U ) ∈ X /U and f ∈ F | U ( W → U ) = F ( W ). Then W × U V → W is a covering of W a nd pr ∗ W f ∈ F | V ( W × U V → V ) = F ( W × U V ). W e get an e lement φ W × U V → V ( pr ∗ W ( f )) ∈ G ( W × U V ) = G | V ( W × U V → V ) . 90 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS Note that ( W × U V ) × W ( W × U V ) ∼ = W × U ( V × U V ). The diﬀerence o f the pull-backs of φ W × U V → V ( pr ∗ W ( f )) with res pec t to the t wo pro jections to W × U V induces Φ W × U ( V × U V ) ( pr ∗ W ( f )) = 0 ∈ G (( W × U V ) × W ( W × U V )) . Again, sinc e G is a sheaf there is a unique ele ment ψ W ( f ) ∈ G ( W ) such that ψ W ( f ) | W × U V = φ W × U V → V ( pr ∗ W ( f )) . The morphism ψ is now g iven by the collection ( ψ W ) ( W → U ) ∈ X /U . 6.2.5. — If F, G ∈ Sh R − Mod X , then we deﬁne F ⊗ G ∈ S h R − Mod X to be F ⊗ G := i ♯ ( i ( F ) ⊗ p i ( G )) . W e furthermore deﬁne Hom ( F, G ) := i ♯ Hom p ( i ( F ) , i ( G )) . Using the fact 6.2 .2 that Hom p ( i ( F ) , i ( G )) is a sheaf at the is omorphism marked by ! we ge t for every H ∈ Sh R − Mod X tha t Hom Sh R − Mod X ( H ⊗ F , G ) ∼ = Hom Sh R − Mod X ( i ♯ ( i ( H ) ⊗ p i ( F ) , G ) ∼ = Hom Pr R − Mod X ( i ( H ) ⊗ p i ( F ) , i ( G )) ∼ = Hom Pr R − Mod X ( i ( H ) , Ho m p ( i ( F ) , i ( G )) ! ∼ = Hom Pr R − Mod X ( i ( H ) , i ◦ i ♯ ( Hom p ( i ( F ) , i ( G )))) ∼ = Hom Sh R − Mod X ( i ♯ ◦ i ( H ) , Ho m ( F, G )) ∼ = Hom Sh R − Mod X ( H, Hom ( F, G )) . In o ther words, the pair ( ⊗ , H om ) tog ether with this natural is o morphism ma ke Sh R − Mod X into a closed symmetric mono idal catego r y . 6.2.6. — Let F , G ∈ Sh R − Mod X a nd ( U → X ) ∈ X . Then we hav e ( F ⊗ G ) U ∼ = F U ⊗ G U . Indeed, this follows fr om the fa ct that s hea ﬁﬁcation commut es with the restric- tion from the s ite X to the site ( U ), see 6.1.14. Since the co llection of functors ( ν U ) ( U → X ) ∈ X detects exact se quences it now follows that a sheaf F ∈ Sh R − Mod X is ﬂat if a nd o nly if F U ∈ Sh R − Mod ( U ) is ﬂat for a ll ( U → X ) ∈ X . This fact was cla imed in 6 .1.14. 6.2.7. — L emma 6.2.3 . — F or F, G ∈ P r R − Mod X we have i ♯ ( F ⊗ p G ) ∼ = i ♯ ( F ) ⊗ i ♯ ( G ) . 6.2. TENSOR PRODUCT S AND THE PROJECTION FORMULA 91 Pr o of . — This follows from (we o mit the functor i at v ar ious places in order to sim- plify the for mula) Hom Sh R − Mod X ( i ♯ ( F ⊗ p G ) , H ) ∼ = Hom Pr R − Mod X ( F ⊗ p G, H ) ∼ = Hom Pr R − Mod X ( F, Hom p ( G, H )) ! ∼ = Hom Pr R − Mod X ( i ♯ ( F ) , H om p ( G, H )) ∼ = Hom Pr R − Mod X ( i ♯ ( F ) ⊗ p G, H ) ∼ = Hom Pr R − Mod X ( G, Hom p ( i ♯ F, H )) ! ∼ = Hom Pr R − Mod X ( i ♯ G, Hom p ( i ♯ F, H )) ∼ = Hom Pr R − Mod X ( i ♯ G ⊗ p i ♯ F, H ) ∼ = Hom Sh R − Mod X ( i ♯ G ⊗ i ♯ F, H ) for arbitr ary H ∈ Sh R − Mod X , where we use Lemma 6 .2.2 at the is omorphisms ma rked by !. 6.2.8. — Let f : X → Y be a morphism o f lo cally compact sta cks. Let X a nd Y b e the s ites as s o ciated to X and Y . Conside r the adjoint pair of functor s p f ∗ : Pr R − Mod Y ⇆ Pr R − Mod X : p f ∗ . The pro of of the following Lemma uses the pr o duct in Y describ ed in [ BSS07 , Lemma 3.1] in a sp eciﬁc wa y . L emma 6.2.4 . — F or F, G ∈ P r R − Mod Y we have a natu r al isomorphism p f ∗ ( F ⊗ p G ) ∼ = p f ∗ F ⊗ p p f ∗ G . Pr o of . — W e use the notation introduced in [ BSS07 , 2.1 .4]. F o r ( U → X ) ∈ X we consider the categ ory U / Y o f diagr ams U / /   X   V / / Y . The functor p f ∗ is deﬁned in [ BSS07 , Deﬁnition 2 .3] as a colimit over this categor y . W e consider the diag onal functor U / Y → U / Y × U / Y given on o b jects by U / /   X   V / / Y 7→ ( U / /   X   V / / Y , U / /   X   V / / Y ) . In v iew of the deﬁnition of p f ∗ by co limits it induces a map p f ∗ ( F ⊗ p G ) → p f ∗ F ⊗ p p f ∗ G . 92 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS In the o ther dir e ction we hav e the functor U / Y × U / Y → U / Y g iven by ( U / /   X   V / / Y , U / /   X   V ′ / / Y ) 7→ U / /   X   V × Y V ′ / / Y . This toge ther with the pro jections V × Y V ′ → V and V × Y V ′ → V ′ it induces the inv ers e map p f ∗ F ⊗ p p f ∗ G → p f ∗ ( F ⊗ p G ) . 6.2.9. — Let f : X → Y be a mor phism of lo cally compact stacks. L emma 6.2.5 . — F or F, G ∈ S h R − Mod Y we have a natu r al isomorphism f ∗ ( F ⊗ G ) ∼ = f ∗ F ⊗ f ∗ G . Pr o of . — F or H ∈ Sh R − Mod X , using the fact that p f ∗ preserves s heav es (see 6.1 .9) and Lemma 6 .2.3, we hav e Hom Sh R − Mod X ( f ∗ ( F ⊗ G ) , H ) ∼ = Hom Sh R − Mod X ( F ⊗ G, f ∗ ( H )) ∼ = Hom Sh R − Mod Y ( i ♯ ( i ( F ) ⊗ p i ( G )) , i ♯ ◦ f p ∗ ◦ i ( H )) ∼ = Hom Pr R − Mod Y (( i ( F ) ⊗ p i ( G )) , f p ∗ ◦ i ( H )) ∼ = Hom Pr R − Mod X ( p f ∗ ( i ( F ) ⊗ p i ( G )) , i ( H )) ∼ = Hom Pr R − Mod X ( p f ∗ ◦ i ( F ) ⊗ p p f ∗ ◦ i ( G ) , i ( H )) ∼ = Hom Sh R − Mod X ( i ♯ ( p f ∗ ◦ i ( F ) ⊗ p p f ∗ ◦ i ( G )) , H ) ∼ = Hom Sh R − Mod X ( f ∗ ( F ) ⊗ f ∗ ( G ) , H ) 6.2.10 . — F o r a deriv ed version o f Lemma 6.2.5 w e assume that the morphism f : X → Y of lo cally compac t stacks has lo cal sec tions. F o r simplicity we only c onsider the case R = Z , i.e. sheaves o f ab elian gr oups (ﬁnite cohomologica l dimension of R would suﬃce). Then the exact functor f ∗ = ( f ♯ ) ∗ preserves tors ion-free sheav es of ab elian gro ups. Since the derived tenso r pr o duct can b e ca lculated using tors ion-free resolutions we get the co rollary Cor ol lar y 6.2.6 . — If f : X → Y has lo c al se ctions, t hen for F, G ∈ D + ( Sh Ab Y ) we have a natur al isomorphism f ∗ ( F ⊗ L G ) ∼ = f ∗ F ⊗ L f ∗ G . of Lemma 6.2.5. 6.2. TENSOR PRODUCT S AND THE PROJECTION FORMULA 93 6.2.11 . — Let f : X → Y be a mor phism of lo ca lly compa ct stacks. L emma 6.2.7 . — F or F ∈ Sh R − Mod Y and G ∈ S h R − Mod X we have a natu r al isomor- phism Hom ( F, f ∗ G ) ∼ = f ∗ Hom ( f ∗ F, G ) in S h R − Mod Y Pr o of . — F or a ny T ∈ S h R − Mod Y we calcula te Hom Sh R − Mod Y ( T , f ∗ Hom ( f ∗ F, G )) ∼ = Hom Sh R − Mod X ( f ∗ T , H om ( f ∗ F, G )) ∼ = Hom Sh R − Mod X ( f ∗ T ⊗ f ∗ F, G ) ∼ = Hom Sh R − Mod X ( f ∗ ( T ⊗ F ) , G ) ∼ = Hom Sh R − Mod Y ( T ⊗ F, f ∗ G ) ∼ = Hom Sh R − Mod Y ( T , H om ( F, f ∗ G )) 6.2.12 . — Let f : X → Y be a mor phism of lo ca lly compa ct stacks. L emma 6.2.8 . — F or F ∈ Sh R − Mod Y and G ∈ Sh R − Mod X we have a natur al mor- phism f ∗ G ⊗ F → f ∗ ( G ⊗ f ∗ F ) . Pr o of . — The tr ansformatio n is the image of the identit y under the following chain of maps, where the ﬁrst is induced b y the counit f ∗ ◦ f ∗ → id of the a djo int pair ( f ∗ , f ∗ ), a nd the second isomor phism is given by L emma 6.2 .5. Hom Sh R − Mod X ( G ⊗ f ∗ F, G ⊗ f ∗ F ) → Hom Sh R − Mod X ( f ∗ f ∗ G ⊗ f ∗ F, G ⊗ f ∗ F ) ∼ = Hom Sh R − Mod X ( f ∗ ( f ∗ G ⊗ F ) , G ⊗ f ∗ F ) ∼ = Hom Sh R − Mod Y ( f ∗ G ⊗ F, f ∗ ( G ⊗ f ∗ F )) . L emma 6.2.9 . — If f has lo c al se ctions, then for F ∈ Sh Ab Y and G ∈ Sh Ab X we have a natura l morphism f ∗ G ⊗ L F → f ∗ ( G ⊗ L f ∗ F ) . Pr o of . — W e use the same argumen t as for Lemma 6 .2.8 based on the adjo int pair ( f ∗ , Rf ∗ ) a nd Lemma 6.2 .6. 94 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.2.13 . — Let f : X → Y be a mor phism of lo ca lly compa ct stacks. L emma 6.2.10 . — L et F ∈ Sh R − Mod Y b e she af whic h is lo c al ly isomorphic to R Y , i.e. ther e ex ist an atlas a : U → Y su ch that a ∗ F ∼ = R U . In this c ase we have the pr oje ction formula: F or al l G ∈ Sh R − Mod X or H ∈ D + ( Sh Ab X ) t he natura l morphism f ∗ G ⊗ F → f ∗ ( G ⊗ f ∗ F ) , Rf ∗ H ⊗ L F → Rf ∗ ( H ⊗ L f ∗ F ) ar e isomorphisms. Pr o of . — This can b e check ed lo cally on the atlas U → Y . W e c onsider the pull-back V b / / g   X f   U : B a / / Y . W e m ust chec k that a ∗ ◦ ( f ∗ G ⊗ F ) → a ∗ ◦ f ∗ ( G ⊗ f ∗ F ) is an is o morphism. T his map is eq uiv alent to a ∗ ( f ∗ G ⊗ F ) ∼ = a ∗ f ∗ G ⊗ a ∗ F ∼ = a ∗ f ∗ G ⊗ R U ∼ = a ∗ f ∗ G ∼ = g ∗ b ∗ G ∼ = g ∗ b ∗ ( G ⊗ R X ) ∼ = g ∗ ( b ∗ G ⊗ b ∗ f ∗ R Y ) ∼ = g ∗ ( b ∗ G ⊗ g ∗ a ∗ R Y ) ∼ = g ∗ ( b ∗ G ⊗ g ∗ a ∗ F ) ∼ = g ∗ b ∗ ( G ⊗ f ∗ F ) ∼ = a ∗ f ∗ ( G ⊗ f ∗ F ) . The derived version is shown in similar manner. 6.2.14 . — W e w ill a lso need the pro jectio n fo rmula with diﬀerent assumptions. L e t f : X → Y b e a map of lo cally compact stacks. W e consider F ∈ Sh R − Mod Y and G ∈ Sh R − Mod X . L emma 6.2.11 . — A ssume that f is pr op er and r epr esent able, and that F is ﬂat. Then the natura l tr ansformation f ∗ G ⊗ F → f ∗ ( G ⊗ f ∗ F ) of 6.2.8 is an isomorphism. 6.2. TENSOR PRODUCT S AND THE PROJECTION FORMULA 95 Pr o of . — Using the observ a tions 6.1.1 4 we see that it suﬃces to show that for all ( U → Y ) ∈ Y the induced morphism (6.2.12) g ∗ G V ⊗ F U → g ∗ ( G V ⊗ g ∗ F U ) is an isomo rphism. Here g : V → U is the prop er map of lo cally compact spaces deﬁned by the Car tesian diagr am V g   / / X f   U / / Y . The map (6.2 .12) is an isomor phism by [ KS94 , Prop. 2.5.1 3]. 6.2.15 . — W e also have a deriv ed version of the pro jection for mula in the c ase of sheav e s of ab elian groups. The main p o int is that the ring Z has a ﬁnite c o homologic a l dimension (in fact equa l to 1). Let f : X → Y b e a morphism of lo ca lly compact stacks. L emma 6.2.13 . — A ssume that f is p r op er and re pr esentable. If G ∈ D + ( Sh Ab Y ) and F ∈ D + ( Sh Ab X ) , then we have Rf ∗ G ⊗ L F ∼ → R f ∗ ( G ⊗ L f ∗ F ) in D + ( Sh Ab Y ) . Pr o of . — As in the pro o f o f Lemma 6.2.11 we can r educe to the small sites ( U ) for all o b jects ( U → Y ) ∈ Y . After this reduction we apply [ KS94 , Pr o p. 2.6 .6 ] and the fact that the coho mo logical dimensio n of Z is 1, he nce ﬁnite. 6.2.16 . — The following derived adjunction again uses the ﬁniteness o f the cohomo- logical dimensio n of Z . L emma 6.2.14 . — F or F, G, H ∈ D + ( Sh Ab X ) we have a natur al isomorphism R Hom Sh Ab X ( F ⊗ L G, H ) ∼ = R Hom Sh Ab X ( F, R Ho m ( G, H )) . Pr o of . — If G ∈ Sh Ab X is ﬂat and H ∈ Sh Ab X is injective, then the functor Sh Ab X ∋ F 7→ Hom Sh Ab X ( F, Hom ( G, H )) ∼ = Hom Sh Ab X ( F ⊗ G, H ) ∈ A b is, as a co mpo sition of exact functors, ex a ct. It follows that Hom ( G, H ) is aga in injectiv e. W e now show the Lemma. W e c a n assume that H is a complex of injectives. F urthermore, since the cohomolo gical dimensio n o f Z is one, hence in par ticular ﬁnite, we ca n assume that G is a complex of ﬂat s he aves. Then we have R Hom Sh Ab X ( F ⊗ L G, H ) ∼ = Hom Sh Ab X ( F ⊗ G, H ) ∼ = Hom Sh Ab X ( F, Hom ( G, H )) ∼ = R Hom Sh Ab X ( F, Hom ( G, H )) . 96 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.3. V e rdi er dual i t y for lo cally compact stac ks in detail 6.3.1. — Let f : X → Y be a map of lo cally compac t stacks. Deﬁnition 6.3.1 . — We say that the c ohomolo gic al dimension of f ∗ is not gr e ater than n ∈ N if the derive d functor R i f ∗ : Sh Ab X → Sh Ab Y vanishes for al l i > n . The main theor em of the present subsection is The or em 6.3.2 . — Assume that f : X → Y is a r epr esentable and pr op er map b etwe en lo c al ly c omp act stacks such that f ∗ has ﬁnite c ohomolo gic al dimension. Then the functor Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) admits a right adjoint f ! : D + ( Sh Ab Y ) → D + ( Sh Ab X ) . The pro of of Theorem 6.3 .2 will b e ﬁnished in 6.3.6. The main idea is to tra nsfer the c o nstruction of f ! from [ KS94 , Section 3 .1] to the pr esent situa tio n. 6.3.2. — W e co nsider the functoria l ﬂabby r esolution (see 3.1.10) of the constant sheaf Z X → F l ( Z X ) and form the truncated complex K := τ ≤ n F l ( Z X ) so that in particular K n = ker ( F l n ( Z X ) → F l n +1 ( Z X )). L emma 6.3.3 . — Assume that f is r epr esentable and that f ∗ has c ohomolo gic al di - mension n ot gr e ater than n . Then the c omplex (6.3.4) 0 → Z X → K 0 → K 1 → · · · → K n → 0 is a ﬂat and f ∗ -acyclic r esolution of Z X . Pr o of . — The she a f ker( K n → K n +1 ) is a torsion- fr ee subsheaf of a tors ion-free sheaf and therefore ﬂa t (compar e [ KS94 , Lemma 3.1 .4]). By Lemma 3.1.4 the ﬂabby sheav e s K i for i = 0 , . . . , n − 1 are f ∗ -acyclic. In order to se e tha t K n is f ∗ -acyclic, it suﬃces to show that R i f ∗ (ker( K n → K n +1 )) ∼ = 0 for i ≥ 1. In fact, we hav e R i f ∗ (ker ( K n → K n +1 )) ∼ = R i + n f ∗ Z X ∼ = 0. 6.3.3. — The ﬁb ers of a re pr esentable and prop er morphism of top ologica l stacks a re compact. This is ex plic itly used in the pro o f of the following Lemma. L emma 6.3.5 . — If f : X → Y is a r epr esentable and pr op er morphism of lo c al ly c omp act stacks, then the fun ctor f ∗ : Sh Ab X → Sh Ab Y pr eserves dir e ct su m s. Pr o of . — Let ( G i ) i ∈ I be a family of sheav es in S h Ab X . Then we ha ve a canonical map M i ∈ I ◦ f ∗ ( G i ) → f ∗ ◦ M i ∈ I ( G i ) . 6.3. VERDIER DUALITY FOR LOCALL Y COMP AC T ST ACKS IN DET AIL 97 In o rder to show that this map is an iso morphism we show that the induced map ( M i ∈ I ◦ f ∗ ( G i )) U → ( f ∗ ◦ M i ∈ I ( G i )) U is an isomor phism for all ( U → Y ) ∈ Y . Cho ose such ( U → Y ) and consider the Cartesian dia gram V g   / / X f   U / / Y . It s uﬃces to show that the induced map M i ∈ I ◦ g ∗ ( G i ) U → g ∗ ◦ M i ∈ I ( G i ) U is an isomorphism. W e consider the induced map o n the stalk at x ∈ U . Since the restriction to g − 1 ( x ) commutes with the sum and g − 1 ( x ) is co mpact it is given by M i ∈ I ◦ Γ( g − 1 ( x ) , [( G i ) U ] | g − 1 ( x ) ) → Γ( g − 1 ( x ) , M i ∈ I [( G i ) U ] | g − 1 ( x ) ) (see [ KS94 , P rop osition 2.5.2 ]). But this last map is an isomorphism since the globa l section functor on sheaves on a compact space commutes w ith sums. 6.3.4. — Fix j ∈ { 0 , 1 , 2 . . . , n } a nd set K := K j , see 6 .3.2 L emma 6.3.6 . — L et f : X → Y b e a r epr esentable, pr op er morphism of lo c al ly c omp act stacks su ch that f ∗ has c ohomolo gic al dimension not gr e ater than n . The n the functor G 7→ f ∗ ( G ⊗ K ) is an exact functor Sh Ab X → Sh Ab Y . F urt hermor e, G ⊗ K is f ∗ -acyclic. Pr o of . — In the following pro of we freely use the facts lis ted in 6 .1.14. Let G · be an exact co mplex in Sh Ab X . F or ( U → Y ) ∈ Y cons ider the Car tesian diagr am V g   / / X f   U / / Y . Note that ( V → X ) ∈ X . By co nstruction (see [ KS94 , Lemma 3.1.4]) K V is ﬂat and g -soft. The co mplex G · V is exact. By [ KS9 4 , Lemma 3.1.2 (ii)] the complex g ∗ ( G · V ⊗ K V ) = ( f ∗ ( G · ⊗ K )) U is exact. Since this is true for all ( U → Y ) ∈ Y w e conclude that f ∗ ( G · ⊗ K ) is exact. W e now show tha t G ⊗ K is f ∗ -acyclic. W e must show that R i f ∗ ( G ⊗ K ) ∼ = 0 for all i ≥ 1. F or ( U → Y ) ∈ Y as ab ove we hav e ( R i f ∗ ( G ⊗ K )) U ∼ = R i g ∗ ( G U ⊗ K U ) ∼ = 0, since G U ⊗ K U is g -soft by [ KS94 , Lemma 3.1 .2 (i)] (note that K U is ﬂasque and ﬂat). Since ( U → Y ) was ar bitrary this implies that R i f ∗ ( G ⊗ K ) ∼ = 0 98 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.3.5. — F o r ( V → X ) ∈ X let ˆ h Z V denote the she a ﬁﬁcation of the pr esheaf h Z V , the preshea f of fr ee ab elian gro ups ge nerated by the sheaf h V represented by V . W e consider the functor f ! K : Sh Ab Y → Pr Ab X which asso ciates to a s heaf F ∈ Sh Ab Y the presheaf f ! K ( F ) ∈ Pr Ab X g iven by X ∋ ( V → X ) 7→ f ! K F ( V ) := Hom Sh Ab Y ( f ∗ ( ˆ h Z V ⊗ K ) , F ) ∈ Ab . Note that K → f ! K ( F ) is also a functor in K (for ﬁxed F ). L emma 6.3.7 . — L et K b e as in 6.3.4 and f : X → Y b e a r epr esentable, pr op er morphism of lo c al ly c omp act stacks such that f ∗ has c ohomolo gic al dimension not gr e ater t han n . Assume that F ∈ Sh Ab Y is an inje ctive she af. The n f ! K ( F ) is an inje ctive she af. F u rt hermor e, for G ∈ Sh Ab X t her e is a c anonic al isomorphism (6.3.8) Hom Sh Ab Y ( f ∗ ( G ⊗ K ) , F ) ∼ = Hom Sh Ab X ( G, f ! K ( F )) . Pr o of . — W e show that f ! K F is a sheaf b y copying the cor resp onding argument in the pro o f of [ KS94 , Lemma 3.1.3]. The functor G 7→ Hom Sh Ab Y ( f ∗ ( G ⊗ K ) , F ) is exact by Lemma 6.3.6 a nd injectivity of F . If we es tablish the isomorphism (6.3 .8), then we als o have shown that f ! K ( F ) is injective. F or ( W → X ) ∈ X we hav e a canonica l isomorphism (6.3.9) Hom Sh Ab Y ( f ∗ ( ˆ h Z W ⊗ K ) , F ) = f ! K ( F )( W ) ∼ = Hom Sh Ab X ( ˆ h Z W , f ! K ( F )) . F or a system ( G i ) i ∈ I of sheav es we hav e a natural map co lim i ∈ I ◦ f ∗ ( G i ) → f ∗ ◦ colim i ∈ I ( G i ). F or G ∈ Sh Ab X we get Hom Sh Ab Y ( f ∗ ( G ⊗ K ) , F ) ∼ = Hom Sh Ab Y ( f ∗ ((colim ˆ h Z W → G ˆ h Z W ) ⊗ K ) , F ) ! ∼ = Hom Sh Ab Y ( f ∗ ◦ colim ˆ h Z W → G ( ˆ h Z W ⊗ K ) , F ) → Hom Sh Ab Y (colim ˆ h Z W → G ◦ f ∗ ( ˆ h Z W ⊗ K ) , F ) ∼ = lim ˆ h Z W → G Hom Sh Ab Y ( f ∗ ( ˆ h Z W ⊗ K ) , F ) ∼ = lim ˆ h Z W → G Hom Sh Ab X ( ˆ h Z W , f ! K ( F )) ∼ = Hom Sh Ab X (colim ˆ h Z W → G ˆ h Z W , f ! K ( F )) ∼ = Hom Sh Ab X ( G, f ! K ( F )) . The mar ked isomorphism uses that the tenso r pr o duct of sheaves commutes with colimits, a consequence of the fact 6.2.5 that it is part of a clos ed monoidal structure. It rema ins to show that this c omp osition is an isomor phism. If we write out the deﬁnition of the colimit in G ∼ = colim ˆ h Z W → G ˆ h Z W we obtain an ex act sequence of the form (6.3.10) M j ∈ J ˆ h Z W j → M i ∈ I ˆ h Z V i → G → 0 . 6.3. VERDIER DUALITY FOR LOCALL Y COMP AC T ST ACKS IN DET AIL 99 Now o bs erve that for any collection ( G i ) i ∈ I of sheav es in S h Ab X we have Hom Sh Ab Y ( f ∗ (( M i G i ) ⊗ K ) , F ) ∼ = Y i ∈ I Hom Sh Ab Y ( f ∗ ( G i ⊗ K ) , F ) since f ∗ (Lemma 6.3 .5) and · · · ⊗ K commute with sums . Clearly we a lso hav e Hom Sh Ab X ( M i G i , f ! K ( F )) ∼ = Y i ∈ I Hom Sh Ab X ( G i , f ! K ( F )) . F rom (6 .3.10) we thus get the diagr am 0 0   y   y Hom Sh Ab Y ( f ∗ ( G ⊗ K ) , F ) − − − − → Hom Sh Ab X ( G, f ! K ( F ))   y   y Q i ∈ I Hom Sh Ab Y ( f ∗ ( ˆ h Z V i ⊗ K ) , F ) α − − − − → Q i ∈ I Hom Sh Ab X ( ˆ h Z V i , f ! K ( F ))   y   y Q j ∈ J Hom Sh Ab Y ( f ∗ ( ˆ h Z W j ⊗ K ) , F ) β − − − − → Q j ∈ J Hom Sh Ab X ( ˆ h Z W j , f ! K ( F )) . Because of the isomor phis m (6.3.9) the maps α and β ar e isomorphisms . The left vertical sequence is exact by Lemma 6.3.6. The right vertical sequence is exa ct b y the left-exactness o f the Ho m -functor. It follows fro m the ﬁve L e mma that (6.3.8) is an iso morphism. 6.3.6. — Let I S h Ab X ⊂ Sh Ab X denote the full sub categor y of injective ob jects a nd K + ( I S h Ab X ) b e the categ ory of complexes in I S h Ab X which ar e b ounded b elow, and whose morphisms ar e homotopy cla sses of chain maps. Then we hav e an equiv a lence of tria ng ulated ca tegories K + ( I S h Ab X ) ∼ = D + ( Sh Ab X ) . Let f : X → Y b e a r epresentable, prop er morphism of lo cally co mpact stacks such that f ∗ has cohomological dimension not g reater than n , and le t K · be as in 6.3 .2. W e then deﬁne the functor f ! : K + ( I S h Ab Y ) → K + ( I S h Ab X ) by f ! ( F · ) = ( f ! K · ( F · )) tot , where E · , · tot denotes the total complex of the double complex E · , · . Since f ! K preserves injectiv e sheav es by Lemma 6.3.7 this functor is w ell- deﬁned. F urthermor e, for F ∈ K + ( I S h Ab Y ) and G ∈ K + ( I S h Ab X ) we have by Lemma 6 .3.7 a natur a l isomorphism betw een s paces of chain ma ps Hom C + ( Sh Ab Y ) ( f ∗ ( G · ⊗ K · ) tot , F · ) ∼ = Hom C + ( Sh Ab X ) ( G · , f ! ( F · )) 100 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS which descends to an isomor phism on the level of homotopy cla sses Hom K + ( I Sh Ab Y ) ( f ∗ ( G · ⊗ K · ) tot , F · ) ∼ = Hom K + ( I Sh Ab X ) ( G · , f ! ( F · )) . Since f ! ( F · ) is a co mplex of injective sheav es we hav e Hom K + ( I Sh Ab X ) ( G · , f ! ( F · )) ∼ = Hom D + ( Sh Ab X ) ( G · , f ! ( F · )) . Note that G · ∼ = G · ⊗ Z X → ( G · ⊗ K · ) tot is a quasi-isomo r phism, and the complex G · ⊗ K · consists of f ∗ -acyclic sheaves by Lemma 6.3.6. Ther e fore f ∗ ( G · ⊗ K · ) tot ∼ = Rf ∗ ( G · ). Since F · is injective we have Hom K + ( Sh Ab Y ) ( f ∗ ( G · ⊗ K · ) tot , F · ) ∼ = Hom D + ( Sh Ab Y ) ( Rf ∗ ( G · ) , F · ) . W e conclude that Hom D + ( Sh Ab Y ) ( Rf ∗ ( G · ) , F · ) ∼ = Hom D + ( Sh Ab X ) ( G · , f ! ( F · )) . This ﬁnishes the pr o of of Theorem 6.3.2. ✷ 6.3.7. — W e co nsider morphisms f : X → Y and u : U → Y of loc ally compact stacks and for m the Ca rtesian diag ram V v / / g   X f   U u / / Y . L emma 6.3.11 . — A ssume the f is r epr esent able, pr op er and that f ∗ has ﬁnite c ohomolo gic al dimension. Assume furt hermor e that u has lo c al se ctions. Then we have a natura l tr ansformation v ∗ ◦ f ! → g ! ◦ u ∗ . Pr o of . — First note that g is r e pr esentable, pro pe r a nd that g ∗ has ﬁnite coho- mological dimension. F urthermo r e, v has lo cal sections. W e apply f ! to the unit id → Ru ∗ ◦ u ∗ and obtain a mor phism (6.3.12) f ! → f ! ◦ Ru ∗ ◦ u ∗ . Since f is r epresentable and u has lo ca l sections we have the iso morphism (see Lemma 6.1.12 o r [ BSS07 , Lemma 2.43 ]) u ∗ ◦ Rf ∗ ∼ = Rg ∗ ◦ v ∗ . T aking its rig ht adjoint yie lds the isomo rphism f ! ◦ Ru ∗ ∼ = Rv ∗ ◦ g ! . W e plug this into (6.3 .12) and obtain a transfor mation f ! → R v ∗ ◦ g ! ◦ u ∗ . Its adjoint is the desired transfor mation 6.4. THE I NTE GRA TION MAP 101 6.3.8. — The following adjunction is a conse quence of the der ived pr o jection formula Lemma 6.2 .13 and the derived adjunction Lemma 6.2.14 L emma 6.3.13 . — If f : X → Y is a r epr esentable pr op er morphism of lo c al ly c omp act stacks which has lo c al se ctions and is such that f ∗ has ﬁnite c ohomolo gic al dimension, t hen for G ∈ D + ( Sh Ab X ) and F ∈ D + ( Sh Ab X ) we have a natur al isomor- phism Rf ∗ R Hom ( G, f ! F ) ∼ = R Hom ( Rf ∗ G, F ) . Pr o of . — Let H ∈ D + ( Sh Ab X ) be arbitrary . Then w e calculate using Lemma 6.1 .8 and Lemma 6 .2.13 that R Hom Sh Ab Y ( H, Rf ∗ R Hom ( G, f ! F )) ∼ = R Hom Sh Ab X ( f ∗ H, R H om ( G, f ! F )) ∼ = R Hom Sh Ab X ( f ∗ H ⊗ L G, f ! F ) ∼ = R Hom Sh Ab Y ( Rf ∗ ( f ∗ H ⊗ L G ) , F ) ∼ = R Hom Sh Ab Y ( H ⊗ L Rf ∗ G, F ) ∼ = R Hom Sh Ab Y ( H, R H om ( Rf ∗ G, F )) . 6.3.9. — Deﬁnition 6.3.14 . — If f : X → Y is a pr op er morphism of lo c al ly c omp act stacks such that f ∗ has ﬁnite c ohomolo gic al dimension, then we deﬁne t he r elative dualizing c omplex by ω X/ Y := f ! ( Z Y ) . It would b e interesting to know the structure of ω X/ Y for a top olog ical submersion f in the sens e of [ KS94 , Def. 3.3.1 ]. 6.3.10 . — In a diﬀerent setup o f Artin stacks and the liss e -´ etale s ite in [ LO05 ] a six functor ca lculus was constr ucted. Starting with the observ a tion that dualizing she aves on the small sites are s uﬃcient ly functorial the functors R f ! and f ! are constructed on c onstructible sheaves by duality . In this a pproach o ne ca n relate the global f ! with the lo cal versions without a n y diﬃculty . A simila r approa ch may work in the present top olo gical context a s w ell, but it is not clear how the resulting f ! will relate to the cons truction in the pre s ent pap er. 6.4. The in tegration map 6.4.1. — Let M b e a closed co nnected orientable n -dimensional to po logical manifold. Deﬁnition 6.4.1 . — A map b etwe en lo c al ly c omp act stacks f : X → Y is a lo c al ly trivial ﬁ b er bund le with ﬁ b er M if for every s p ac e U → X t he pul l-b ack U × Y X → U is a lo c al ly trivia l ﬁb er bund le of sp ac es with ﬁb er M . 102 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS Note that a lo ca lly trivial ﬁb er bundle f with ﬁb er M is repr esentable, pro p er and has lo cal sections, and f ∗ has ﬁnite cohomolog ical dimension. In or der to see the last fact and to calculate R n f ∗ ( Z X ) w e consider ( U → Y ) ∈ Y and let V → U b e surjective and lo cally a n op en embedding such that we hav e a diagra m with Ca rtesian squares (6.4.2) M q   V × Y X o o h   / / U × Y X g   / / X f   ∗ V p o o / / U u / / Y . The ma p g is a top o logical submer sion in the s ense of [ KS9 4 , Def. 3.3.1]. As remarked in [ KS94 , Sec. 3.3] the coho mologica l dimension of g ∗ is not greater than n . This implies ( R i f ∗ F ) U ∼ = R i g ∗ ( F U × Y X ) = 0 for all i > n . Since this holds tr ue for all ( U → Y ) ∈ Y we c o nclude that R i f ∗ F = 0 for all i > n . W e use the left part o f the diag ram (6.4.2) in order to see that R n f ∗ ( Z X ) is lo cally isomorphic to Z Y . In fact, we have Rf ∗ ( Z X ) V ∼ = Rh ∗ Z ( V × Y X ) ∼ = p ∗ Rq ∗ Z ( M ) . A choice of an orientation o f M giv es an is omorphism R n q ∗ Z ( M ) ∼ = Z ( ∗ ) and therefore R n f ∗ ( Z X ) V ∼ = p ∗ Z ( ∗ ) ∼ = Z ( V ) . Deﬁnition 6.4.3 . — A lo c al ly trivial ﬁb er bun d le f : X → Y with ﬁb er M is c al le d orientable if ther e exists an isomorphism R n f ∗ ( Z X ) ∼ = Z Y . An orientation of f is a choic e o f such an isomorphism. 6.4.2. — Le t f : X → Y b e a lo ca lly trivial ﬁb er bundle with ﬁb er M , where M is a compact clo sed n -dimensional topolog ical manifold. W e consider the f ∗ -acyclic and ﬂat res olution K deﬁned in (6.3.4). The following was obser ved in 6.3.6 Cor ol lar y 6.4.4 . — The funct or Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) is r epr esente d by f ∗ ◦ T K , wher e T K is tensor pr o duct with the c omplex K . W e now deﬁne a natural transfor ma tion R Hom ( R n f ∗ ( Z X ) , F ) → Rf ∗ ◦ f ! ( F ) . Let F ∈ C + ( I S h Ab Y ) b e a complex o f injectives. W e s ta rt fro m the obser v ation that R n f ∗ ( Z X ) ∼ = f ∗ ( K n ) / im ( f ∗ ( K n − 1 ) → f ∗ ( K n )) . 6.4. THE I NTE GRA TION MAP 103 F or ( U → Y ) ∈ Y w e thus obtain a chain o f ma ps of complexes Hom ( R n f ∗ Z X , F )( U ) ∼ = Hom Sh Ab Y ( ˆ h Z U , Hom ( R n f ∗ Z X , F )) ∼ = Hom Sh Ab Y ( ˆ h Z U ⊗ R n f ∗ ( Z X ) , F ) ∼ = Hom Sh Ab Y ( ˆ h Z U ⊗ f ∗ ( K n ) / im ( f ∗ ( K n − 1 ) → f ∗ ( K n )) , F ) ! → Hom Sh Ab Y ( ˆ h Z U ⊗ f ∗ ( K ) , F ) 6.2.11 ∼ = Hom Sh Ab Y ( f ∗ ( f ∗ ˆ h Z U ⊗ K ) , F ) 6.3.7 ∼ = Hom Sh Ab X ( f ∗ ˆ h Z U , f ! K ( F )) ∼ = Hom Sh Ab X ( ˆ h Z U , f ∗ ◦ f ! K ( F )) ∼ = f ∗ ◦ f ! K ( F )( U ) , where the map marked by ! has deg ree n . The pro jection fo rmula Lemma 6.2.11 can b e applied since f ∗ ˆ h Z U is ﬂat. This transforma tio n preserves homo topy cla sses of morphisms F → F ′ . Since F is injective we have Hom ( R n f ∗ Z X , F ) ∼ = R Hom ( R n f ∗ Z X , F ) . F urther note that f ! K ( F ) is still a complex of injectiv es by Lemma 6.3.7. Therefor e f ∗ ◦ f ! K ( F ) ∼ = Rf ∗ ◦ f ! ( F ). Hence this c hain of ma ps of complexes induces a tra nsformation (6.4.5) R Hom ( R n f ∗ Z X , F ) → Rf ∗ ◦ f ! ( F ) . 6.4.3. — Its a djoint is a natura l trans fo rmation Rf ∗ f ∗ R Hom ( R n f ∗ Z X , F ) → F . Let us now a ssume that f : X → Y is in additio n or ient ed by an iso morphism R n f ∗ Z X ∼ = Z Y . W e precomp ose w ith this iso morphism and g e t the integration map. Deﬁnition 6.4.6 . — The inte gr ation m ap Z f : R f ∗ ◦ f ∗ → id is t he n atur al tr ansformation of functors D + ( Sh Ab Y ) → D + ( Sh Ab Y ) of de gr e e − n deﬁne d as t he c omp osition Rf ∗ f ∗ ( F ) ∼ = Rf ∗ f ∗ ( Hom ( Z Y , F )) ∼ = Rf ∗ f ∗ ( Hom ( R n f ∗ ( Z X ) , F )) → F . In Lemmas 6.5.2 0 and 6.5.31 we will verify in the more g eneral case of unbounded derived catego r ies that the integration map is functor ia l with r esp ect to comp ositio ns and compa tible w ith pull-ba ck dia grams. 104 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.5. Op erations with un b ounded deriv ed categori e s 6.5.1. — The catego ry of sheaves Sh Ab X on a lo ca lly compact stack is a Grothendieck ab elian category (see 3.3 .1). The categ o ry of c omplexes in a Gro thendieck ab elian cat- egory ca rries a mo del ca tegory str uc tur e (see 3.3.2). The unbounded derived categor y is the ass o ciated ho motopy categ ory . The g oal of the present subsection is to extend the sheaf theory op eratio ns ( f ∗ , f ∗ ) and the integration map R f to the unbo unded derived category . Many results of the pr esent subsection would contin ue to hold if one drops the assumption of lo ca l compactness in the deﬁnition o f the site a sso ciated to stacks as well as for the sta cks themselves. But the assumption o f lo ca l compactness is impo rtant for the integration map since it uses versions of the pro jectio n formula. 6.5.2. — Let f : X → Y be a morphism betw een locally compact stac k s. Then w e hav e a n adjoint pa ir of functors f ∗ : C ( Sh Ab Y ) ⇆ C ( Sh Ab X ) : f ∗ . In or der to descend the functor f ∗ to the b ounded b elow derived ca tegory it w as suﬃcient to know that f ∗ is left exa ct. In this cas e the idea is to apply f ∗ to injective resolutions. The desce n t o f the other functor f ∗ is usually only cons idered if it exact, but see e.g. [ Ols07 ] fo r mo re g e ne r al co nstructions. W e know by 6.1 .11 that the functor f ∗ is exa ct if f has lo cal sections . It is no t p ossible to show using the left exa ctness that f ∗ preserves quasi- isomorphisms be t ween unbounded co mplexes of injectives. E ven worse, it is not clear how to res olve an unbounded complex by an injective complex. The metho d to descend f ∗ to the derived catego ry use s abstract homotopy theory and works under the a dditional a ssumption that f has lo cal sections. Recall tha t we use a mo del str ucture o n the ca tegory C ( Sh Ab X ) o f unbounded complexes of sheav es for which the equiv alences are the qua si-isomor phisms, a nd the coﬁbrations ar e the level-wise injections (see 3.3.2). The inclusion C + ( Sh Ab X ) ֒ → C ( Sh Ab X ) of the full sub catego ry of b ounded b elow co mplexes induces an identiﬁca- tion D + ( Sh Ab X ) ∼ = hC + ( Sh Ab X ) ֒ → hC ( Sh Ab X ) =: D ( Sh Ab X ) of the bounded below derived category as a full sub categ ory of the unbounded derived catego r y . The functor Rf ∗ : D + ( Sh Ab X ) → D + ( Sh Ab Y ) is the adjoint o f the restriction of f ∗ to the b ounded b elow der ived categorie s , and it is therefo re the restr iction o f Rf ∗ : D ( Sh Ab X ) → D ( Sh Ab Y ) to b e deﬁned b elow. L emma 6.5.1 . — If the morphism f : X → Y of lo c al ly c omp act stacks has lo c al se ctions, then ( f ∗ , f ∗ ) is a Quil len adjoint p air. Pr o of . — W e use the criterio n [ Hov 99 , Def. 1.3.1 (2 )] in order to show that f ∗ is a left Quillen functor . W e must show that it preserves coﬁbrations and trivial coﬁbrations. In other words, we must show that f ∗ preserves injections and injections 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 105 which induce isomo rphisms on cohomology . B oth prop erties follow from the ex actness of f ∗ : Sh Ab Y → Sh Ab X . 6.5.3. — Let f : X → Y b e a ma p b etw een locally compact stac k s which has local sections. Since ( f ∗ , f ∗ ) is a Quillen adjoint pair it induces a derived adjoint pa ir Lf ∗ : hC ( Sh Ab Y ) ⇆ hC ( Sh Ab X ) : Rf ∗ (see Lemma [ Hov99 , Lemma 1.3.1 0]). Since f ∗ is exact it directly descends to the homotopy ca teg ory . 6.5.4. — Let g : Y → Z b e a second map of lo cally compact stacks which admits lo cal s ections. Then we have a djoint cano nical isomor phisms (6.5.2) ( g ◦ f ) ∗ ∼ = f ∗ ◦ g ∗ , ( g ◦ f ) ∗ ∼ = g ∗ ◦ f ∗ . L emma 6.5.3 . — We have a c anonic al isomorphism R ( g ◦ f ) ∗ ∼ = Rg ∗ ◦ Rf ∗ . Pr o of . — Using [ Hov 99 , T hm. 1 .3.7] we have a natur a l transfor ma tion (6.5.4) R ( g ◦ f ) ∗ ∼ = R ( g ∗ ◦ f ∗ ) → Rg ∗ ◦ Rf ∗ which is a djo int to (6.5.5) Lf ∗ ◦ Lg ∗ → L ( f ∗ ◦ g ∗ ) ∼ = L ( g ◦ f ) ∗ . Since Lf ∗ , Lg ∗ , and L ( g ◦ f ) ∗ are plain descents of f ∗ , g ∗ , a nd ( g ◦ f ) ∗ to the homotopy category it follows from (6.5 .2) that (6.5 .5) is an is omorphism. Therefore its adjoint (6.5.4) is als o an isomo r phism. 6.5.5. — Consider a Ca rtesian diagra m o f lo cally compac t stacks U g   v / / X f   V u / / Y , where all maps hav e lo ca l sections. Using the unit id → v ∗ ◦ v ∗ , the counit u ∗ ◦ u ∗ → id , and (6.5 .2) we deﬁne (see Lemma 6.1.12) a tr ansformation u ∗ ◦ f ∗ → u ∗ ◦ f ∗ ◦ v ∗ ◦ v ∗ ∼ = u ∗ ◦ u ∗ ◦ g ∗ ◦ v ∗ → g ∗ ◦ v ∗ . It is functorial with res p ect to co mpo sitions of s uch Cartesia n diagr ams. By the same metho d we obtain a tra nsformation (6.5.6) Lu ∗ ◦ Rf ∗ → R g ∗ ◦ Lv ∗ . 106 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.5.6. — By Lemma 6 .1.12 we know that the transfor mation u ∗ ◦ f ∗ → g ∗ ◦ v ∗ is in fact an isomorphism. The derived version is more complicated and needs an additional assumption. L emma 6.5.7 . — Assume that g is r epr esentable and g ∗ : Sh Ab U → Sh Ab V has ﬁn ite c ohomolo gic al dimension. Then the tr ansformation (6.5.6 ) is an isomorphism. Pr o of . — W e cho ose ﬁbrant replacement functors I X : C ( Sh Ab X ) → C ( Sh Ab X ) , I U : C ( Sh Ab U ) → C ( Sh Ab U ) . In ter ms of these r eplacement functor s w e can write the comp ositions of der ived functors as descents of quasi-isomor phis m preserving functors on the level of c hain complexes: Lu ∗ ◦ Rf ∗ ∼ = u ∗ ◦ f ∗ ◦ I X , R g ∗ ◦ Lv ∗ ∼ = g ∗ ◦ I U ◦ v ∗ . Let F ∈ C ( Sh Ab X ). W e m ust show that the marked a rrows (induced by id → I U and id → I X ) in the following sequence are quasi- isomorphisms u ∗ f ∗ I X ( F ) ∼ = g ∗ v ∗ I X ( F ) ( ∗ ) → g ∗ I U v ∗ I X ( F ) ( ∗∗ ) ← g ∗ I U v ∗ ( F ) . The arrow marked by ( ∗∗ ) is a quasi-iso morphism sinc e the functors g ∗ I U and v ∗ preserve quasi-iso mo rphisms, and F → I X ( F ) is a quasi-is omorphism. The mor phis m ( ∗ ) is mor e complicated, and it is here where we need the a s sump- tion. It is a prop erty o f the injectiv e mo del structure on the chain complexes of a Grothendieck ab elian c a tegory that a ﬁbrant complex consists o f injective ob jects. An injectiv e sheaf is in particular ﬂabb y . Since v has local sections v ∗ preserves ﬂabby sheav e s (Lemma 3 .1 .5). W e c o nclude that v ∗ I X ( F ) is a complex of ﬂabby sheav es. Let G ∈ C ( Sh Ab U ) b e a complex of ﬂabby sheav es. W e must s how that g ∗ ( G ) → g ∗ I U ( G ) is a quasi-iso morphism. Since g ∗ is an additive functor this asser tio n is equiv alent to the asser tion that g ∗ ( C ) is e xact, where C is the mapping cone of G → I U ( G ). No te that C is an exact complex of ﬂabby sheav es. It deco mpo ses into short exact s equences 0 → Z n → C n → Z n +1 → 0 , where Z n := ker( C n → C n +1 ). Since g is r epresentable we know by Lemma 3.1.4 that ﬂabby s he aves ar e g ∗ -acyclic. Therefor e we o btain the exact s equence 0 → g ∗ ( Z n ) → g ∗ ( C n ) → g ∗ ( Z n +1 ) → R 1 g ∗ ( Z n ) → 0 and the isomo rphisms R k g ∗ ( Z n +1 ) ∼ = R k +1 g ∗ ( Z n ) for a ll k ≥ 1. By induction we show that for k ≥ 1 and all l ∈ N we hav e R k g ∗ ( Z n ) ∼ = R k + l g ∗ ( Z n − l ) . 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 107 Since we assume that g ∗ has b ounded cohomolog ical dimension we co nclude that R k ( Z n ) ∼ = 0 for all n ∈ Z and k ≥ 1. In pa rticular the seque nce s 0 → g ∗ ( Z n ) → g ∗ ( C n ) → g ∗ ( Z n +1 ) → 0 are exa c t for all n ∈ Z . This shows the ex actness of g ∗ ( C ). 6.5.7. — L e t now f : X → Y b e a representable ma p betw een lo cally compa ct stacks which is an oriented lo cally trivial ﬁb er bundle of closed o riented manifolds of dimen- sion n . In particular, f has loca l sections a nd is proper, a nd f ∗ has cohomological dimension ≤ n . W e co nsider the cano nic a l ﬂabby reso lution (see 3.1.1 0) 0 → Z X → F l 0 ( Z X ) → F l 1 ( Z X ) → . . . . Then we know that f ∗ F l ( Z X ) is exact ab ove degree n . W e let K denote the truncation (6.3.4) of this resolution at n . Then the or ientation of the bundle (see 6.4.3) gives the last iso morphism in the following comp os itio n f ∗ K n → f ∗ K n / im ( f ∗ K n − 1 → f ∗ K n ) ∼ = R n f ∗ Z X ∼ = Z Y . W e let T K : C ( Sh Ab X ) → C ( Sh Ab X ) denote the functor which ass o ciates to the complex F the tota l complex T K ( F ) of F ⊗ K . The pr o jection for mula Lemma 6.2.1 1 for the prop er representable map f gives a n isomo r phism f ∗ ◦ T K ◦ f ∗ ( F ) ∼ = T f ∗ K ( F ) for co mplexes o f ﬂa t s heav es F ∈ C ( Sh Ab Y ). The inclusio n Z X → K and the pr o jec- tion f ∗ K → Z Y [ − n ] induces trans fo rmations (6.5.8) id → T K , T f ∗ K · → id [ − n ] . 6.5.8. — W e know by L e mma 6.3 .6 that the functor f ∗ ◦ T K : Sh Ab X → S h Ab Y is ex a ct. W e ch o ose a functoria l ﬁbra nt replacement functor id → I o n C ( Sh Ab X ). Let R : C ( Sh Ab Y ) → C ( Sh Ab Y ) b e the functorial ﬂat re s olution functor o f 3.4.1, extended to unbounded co mplex es. Then we consider seq uence (6.5.9) f ∗ ◦ I ◦ f ∗ → f ∗ ◦ T K ◦ I ◦ f ∗ ! ← f ∗ ◦ T K ◦ f ∗ ! ← f ∗ ◦ T K ◦ f ∗ ◦ R ∼ = T f ∗ K ◦ R → R [ − n ] → i d [ − n ] . All functors in this sequence preserve q ua si-isomor phisms a nd therefore descend plainly to the homotopy categ ory hC ( Sh Ab X ). Since f ∗ ◦ T K is exact the ar rows marked b y ! induce isomo rphisms of functors on the homotopy category . Now observe that the des cent of f ∗ ◦ I ◦ f ∗ to the homotopy categor y is is omorphic to Rf ∗ ◦ Lf ∗ . Therefore (6.5 .9) induces a tra nsformation (6.5.10) Z f : R f ∗ ◦ Lf ∗ → id [ − n ] . Deﬁnition 6.5.11 . — The t ra nsformation (6.5.10 ) is c al le d the int e gr ation map. 108 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS It g eneralizes Deﬁnition 6.4 .6 fro m the bounded b elow to the unbounded der ived category . 6.5.9. — In o rder to ha ve a simple deﬁnition we have deﬁned the integration map using a ca nonical resolution of Z X of length n . In fact, we c an use mor e ge ner al resolutions. This will turn o ut to b e useful for the veriﬁcation of functor ial prop er ties of the integration ma p. 6.5.10 . — Let us ﬁrst reca ll some notation. An ob ject ( U → X ) ∈ X repres ents the presheaf h U ∈ P r X (see also 6 .2.3). W e let h Z U ∈ Pr Ab X b e the free a belia n pres hea f generated by h U and for m ˆ h Z U := i ♯ h Z U ∈ Sh Ab X . Deﬁnition 6.5.12 . — L et f : X → Y b e a map of lo c al ly c omp act stacks. A she af F ∈ Sh Ab X is c al le d lo c al ly f ∗ -acyclic, if for every ( U → X ) ∈ X and k ≥ 1 we hav e R k f ∗ ( ˆ h Z U ⊗ F ) ∼ = 0 . 6.5.11 . — Let f : X → Y b e a map o f lo c a lly compact stacks. L emma 6.5.13 . — A ssume t hat the c ohomolo gic al dimension of f ∗ is b ounde d by n . If L 0 → L 1 → · · · → L n − 1 → L n → 0 is an exact c omplex su ch that the L i ar e f ∗ -acyclic (or lo c al ly f ∗ -acyclic) for i = 0 , . . . , n − 1 , then L n is f ∗ -acyclic (or lo c al ly f ∗ -acyclic, r esp e ctively). This can b e shown by a s imilar induction argument as used in the pro o f of Lemma 6.5.7. ✷ 6.5.12 . — Let f : X → Y b e a map o f lo c a lly compact stacks. L emma 6.5.14 . — L et ( V → X ) ∈ X and F b e lo c al ly f ∗ -acyclic. Then ˆ h Z V ⊗ F is lo c al ly f ∗ -acyclic. Pr o of . — Indeed, let ( U → X ) ∈ X . Then we have ˆ h Z U ⊗ ( ˆ h Z V ⊗ F ) ∼ = ( ˆ h Z U ⊗ ˆ h Z V ) ⊗ F . F urthermore we hav e ˆ h Z U ⊗ ˆ h Z V Lemma 6.2.3 ∼ = i ♯ ( h Z U ⊗ p h Z V ) ∼ = i ♯ ( h U × h V ) Z ∼ = i ♯ h Z U × X V ∼ = ˆ h Z U × X V , where we us e the fact, tha t the absolute pro duct in X is given by the ﬁb er pro duct spaces over X ([ BSS07 , Lemma 2.3 .3]). It follows that R k f ∗ ( ˆ h Z U ⊗ ( ˆ h Z V ⊗ F )) ∼ = R k f ∗ ( ˆ h Z U × X V ⊗ F ) ∼ = 0 for a ll k ≥ 1. 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 109 6.5.13 . — Let f : X → Y b e a map o f lo c a lly compact stacks. L emma 6.5.15 . — A ssume that f is pr op er, r epr esent able, and that the c ohomolo g- ic al dimension of f ∗ is b ounde d. If F ∈ S h Ab X is ﬂat and lo c al ly f ∗ -acyclic, then for any she af G ∈ Sh Ab X t he tens or pr o duct G ⊗ F is f ∗ -acyclic and lo c al ly f ∗ -acyclic). Pr o of . — W e construct a r esolution · · · → G j → G j − 1 → · · · → G 0 → G , where all G j are copr o ducts of sheaves of the form ˆ h Z U . In fact, we hav e a surjection M ˆ h Z U → G ˆ h Z U → G . If we hav e alrea dy constructed G j → · · · → G 0 → G , then we extend this co mplex by M ˆ h Z U → ker( G j → G j − 1 ) ˆ h U Z → G j . Since F is ﬂat, the complex · · · → G j ⊗ F → · · · → G 0 ⊗ F → G ⊗ F is exact. The tensor pro duct co mm utes with dire c t s ums. Therefore G j ⊗ F is a sum of f ∗ -acyclic sheaves, a nd by Lemma 6 .5.14 also of lo cally f ∗ -acyclic sheaves. Since f ∗ commutes with direct sums (Lemma 6.3.5) the sheaves G j ⊗ F a re themselves f ∗ -acyclic a nd lo cally f ∗ -acyclic. With Lemma 6.5.1 3 we conclude that G ⊗ F is f ∗ -acyclic a nd lo cally f ∗ -acyclic. 6.5.14 . — Let f : X → Y b e a map o f lo c a lly compact stack. L emma 6.5.16 . — If f is r epr esentable, then a ﬂasque she af is lo c al ly f ∗ -acyclic. Pr o of . — Let F ∈ S h Ab X b e ﬂasque. W e co ns ider ( U → Y ) ∈ Y and form the Cartesian dia gram V / / g   X f   U / / Y . Then ( V → X ) ∈ X and we hav e Rf ∗ ( F ) U ∼ = Rg ∗ ( F V ). The re striction F V ∈ Sh Ab ( V ) is still ﬂasque. A ﬂasque s heaf on ( V ) is g -soft (see [ KS94 , Deﬁnition 3.1.1]). But this implies that R k g ∗ ( F V ) = 0 for k ≥ 1. Since U → Y was a rbitrary we see that R k f ∗ ( F ) = 0 for k ≥ 1. 110 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.5.15 . — L e t us from now on until the end of this subsection assume that f : X → Y is a prop er repres e n table ma p o f lo cally compact stacks which is a n o riented lo cally trivial ﬁber bundle with ﬁb er a clos ed c o nnected topologica l manifold of dimension n . Since a ﬂat a nd ﬂasque sheaf is lo cally f ∗ -acyclic and K is a truncation of a ﬂat and ﬂas que r esolution of Z X we s e e by L e mma 6.5.13 that K is a complex of ﬂat a nd lo cally f ∗ -acyclic s he aves. These a re the t wo prop erties which make the theor y work. Let L → M b e a q ua si-isomor phism b etw een upper b ounded complexes of lo cally f ∗ -acyclic a nd ﬂa t sheaves. L emma 6.5.17 . — F or every c omplex F ∈ C ( Sh Ab X ) t he induc e d map f ∗ ( F ⊗ L ) → f ∗ ( F ⊗ M ) is a quasi-isomorphism. Pr o of . — W e for m the mapping cone C o f L → M . It is an exa ct complex o f lo- cally f ∗ -acyclic and ﬂat s heav es . Since the tensor pro duct a nd g ∗ commute with the formation of a mapping co ne it suﬃces to show tha t f ∗ ( F ⊗ C ) is exact. W e know by Lemma 6.5.15 tha t F ⊗ C is a complex of f ∗ -acyclic sheaves. W e claim that F ⊗ C is exa ct. T o this end we ﬁrst show that H ⊗ C is exact for an arbitra ry sheaf H ∈ S h Ab X . W e decomp ose the exact co mplex C into short exact sequences S ( k ) : 0 → Z k → C k → Z k +1 → 0 where Z k := ker ( C k → C k +1 ). Using the fact that the sheaves C k are ﬂat we obtain 0 → To r 1 ( H, Z k +1 ) → H ⊗ Z k → H ⊗ C k → H ⊗ Z k +1 → 0 and the isomo rphisms Tor m +1 ( H, Z k +1 ) ∼ = Tor m ( H, Z k ) for all m ≥ 1. Since Z is one-dimensional we know that To r m ∼ = 0 for m ≥ 2. Inductively we conclude that Tor 1 ( H, Z k ) ∼ = 0 for all k ∈ Z . It follows that H ⊗ S ( k ) is exact fo r all k ∈ Z . This implies that H ⊗ C is exact. Let now F b e a complex . Using the previous result and a sp ectral sequence arg u- men t we c o nclude that the total complex asso cia ted to the double co mplex F ⊗ C is exact. Let now C ∈ C ( Sh Ab X ) b e an exa ct complex of f ∗ -acyclic sheaves. W e s how that this implies tha t f ∗ ( C ) is exact. The complex C decomp o ses in to shor t exa ct sequences 0 → Z n → C n → Z n +1 → 0 , where Z n := k er( C n → C n +1 ). Using the fact that C n is f ∗ -acyclic w e obtain the exact s equence 0 → f ∗ ( Z n ) → f ∗ ( C n ) → f ∗ ( Z n +1 ) → R 1 f ∗ ( Z n ) → 0 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 111 and the isomo rphisms R k f ∗ ( Z n +1 ) ∼ = R k +1 f ∗ ( Z n ) for a ll k ≥ 1. By induction we show that for k ≥ 1 and all l ∈ N we hav e R k f ∗ ( Z n ) ∼ = R k + l f ∗ ( Z n − l ) . Since f ∗ has b ounded cohomolo gical dimension we conc lude that R k f ∗ ( Z n ) ∼ = 0 for all n ∈ Z and k ≥ 1. In pa rticular the seq uences 0 → f ∗ ( Z n ) → f ∗ ( C n ) → f ∗ ( Z n +1 ) → 0 are exa c t for all n ∈ Z . This shows the ex actness of f ∗ ( C ). 6.5.16 . — L emma 6.5.18 . — The inte gr ation map is indep endent of the choic e of a ﬂ at lo c al ly f ∗ -acyclic r esolution K of Z X of length n . Pr o of . — Let K, L a re two such r esolutions. Assume that there exists a quasi- isomorphism K → L . The identiﬁcation coker ( f ∗ L n − 1 → f ∗ L n ) ∼ = coker ( f ∗ K n − 1 → f ∗ K n ) ∼ = R n f ∗ ( Z X ) ∼ = Z Y gives a map f ∗ L → Z Y [ − n ] whic h induces the transformation T f ∗ L → id of degree − n . It induces a commutativ e diag ram f ∗ I f ∗ / / f ∗ T K I f ∗   f ∗ T K f ∗   ∼ o o f ∗ T K f ∗ R ∼ o o ∼ = / /   T f ∗ K R / /   R / / id f ∗ I f ∗ / / f ∗ T L I f ∗ f ∗ T L f ∗ ∼ o o f ∗ T L f ∗ R ∼ o o ∼ = / / T f ∗ L R / / R / / id The upper ho rizontal comp osition is the in tegratio n map deﬁned us ing K (see 6.5.9), and the low er horizontal co mpo sition is the integration map deﬁned using L . W e see that b oth maps ar e equal. Let now K, L aga in b e ﬂa t and lo cally f ∗ -acyclic reso lutions of Z X of length n . W e complete the pr o of of the Lemma by showing that there ex is ts a third such resolution M together with q ua si-isomor phisms K ∼ → M ∼ ← L . The maps Z X → K and Z X → L , resp ectively , induce maps K → K ⊗ L and L → K ⊗ L whic h are quasi- isomorphisms . W e further get induced quasi-iso morphisms (6.5.19) K → F l ( K ⊗ L ) , L → F l ( K ⊗ L ) . W e let M := τ ≤ n F l ( K ⊗ L ). The maps (6.5.19) factor ize ov e r M . Note that K ⊗ L is ﬂat. Since F l and truncation pr e serve ﬂatnes s (see Lemma 3.1.12), we see that M is ﬂat. Since F l in fact pro duces ﬂasque and hence lo ca lly f ∗ -acyclic resolutio ns, and the cohomo logical dimension o f f ∗ is b ounded b y n we co nclude by Lemma 6 .5.13 that M is lo cally f ∗ -acyclic. 112 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS 6.5.17 . — In this pa ragr a ph we show that the integration map is functoria l. Let g : Y → Z b e a s econd pr op er and repre s ent able map of lo ca lly compact stacks which is an o riented lo cally trivial ﬁb er bundle of clos ed m -dimensio nal manifolds. L emma 6.5.20 . — We have a c ommut ative diagr am Rg ∗ ◦ Rf ∗ ◦ Lf ∗ ◦ Lg ∗ Rg ∗ ( R f )   ∼ = / / R ( g ◦ f ) ∗ ◦ L ( g ◦ f ) ∗ R g ◦ f   Rg ∗ ◦ Lg ∗ [ − n ] R g / / id [ − n − m ] . Pr o of . — The following s equence o f mo diﬁcations tra nsforms the down-right comp o - sition into the r ight-do wn comp osition. (6.5.21) g ∗ I f ∗ I f ∗ g ∗ → g ∗ I f ∗ T K I f ∗ g ∗ ∼ ← g ∗ I f ∗ T K f ∗ g ∗ R → g ∗ I g ∗ R → g ∗ T L I g ∗ R ∼ ← g ∗ T L g ∗ R → i d (6.5.22) g ∗ I f ∗ I f ∗ g ∗ → g ∗ T L I f ∗ I f ∗ g ∗ → g ∗ T L I f ∗ T K I f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K I f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K f ∗ g ∗ R → g ∗ T L g ∗ R → i d (6.5.23) g ∗ I f ∗ I f ∗ g ∗ → g ∗ T L I f ∗ I f ∗ g ∗ ∼ ← g ∗ T L f ∗ I f ∗ g ∗ → g ∗ T L f ∗ T K I f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K f ∗ g ∗ R → g ∗ T L g ∗ R → i d (6.5.24) g ∗ I f ∗ I f ∗ g ∗ ∼ ← g ∗ f ∗ I f ∗ g ∗ → g ∗ T L f ∗ I f ∗ g ∗ → g ∗ T L f ∗ T K I f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K f ∗ g ∗ R → g ∗ T L g ∗ R → i d (6.5.25) g ∗ f ∗ I f ∗ g ∗ → g ∗ T L f ∗ I f ∗ g ∗ → g ∗ T L f ∗ T K I f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K RI f ∗ g ∗ ∼ ← g ∗ T L f ∗ T K Rf ∗ g ∗ R → g ∗ T L g ∗ R → i d (6.5.26) g ∗ f ∗ I f ∗ g ∗ → g ∗ T L f ∗ T K I f ∗ g ∗ ∼ ← g ∗ f ∗ T f ∗ L ⊗ K RI f ∗ g ∗ ∼ ← g ∗ f ∗ T f ∗ L ⊗ K Rf ∗ g ∗ R → g ∗ T L g ∗ R → i d (6.5.27) ( g ◦ f ) ∗ I ( g ◦ f ) ∗ → ( g ◦ f ) ∗ T M I ( g ◦ f ) ∗ ∼ ← ( g ◦ f ) ∗ T M ( g ◦ f ) ∗ R → id The transitio n from (6.5.21) to (6.5.2 2) uses the fact that tensoring with L and the map id → T L can b e commuted with the intermediate oper ations. In order to g o from (6.5.2 2) to (6.5.23) we use the fact that g ∗ T L preserves quasi-isomo rphisms. The same rea son and the fact tha t f ∗ preserves ﬁbran t ob jects is b ehind the tra ns ition from (6.5.23) to (6.5.24). W e use e.g. the isomorphism g ∗ f ∗ I f ∗ g ∗ ∼ → g ∗ I f ∗ I f ∗ g ∗ . There is a vertical quas i-isomorphis m from (6.5.25) to (6.5.24). The step fr o m (6.5.2 5) to (6.5.26) uses the isomorphism T L f ∗ T K R ∼ → f ∗ T f ∗ L ⊗ K R given b y the pro jection 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 113 formula. The w eak equiv a lence in (6.5.26) is not obvious (since f ∗ L ⊗ K is not obviously g ∗ f ∗ -acyclic), but follows from the fact, that this line is iso morphic to the pre vious (6.5.25). In the last step from (6.5.2 6) to (6.5.27) we use the map f ∗ L ⊗ K → M given by a truncated ﬂabby reso lution of f ∗ L ⊗ K and the fac t that the integration map is indep endent o f the choice of the r esolution. 6.5.18 . — Consider a cartesia n diagram o f lo cally c o mpact sta cks (6.5.28) V g   v / / X f   U u / / Y . W e a ssume that f and u , and hence also g and v ha ve lo ca l s ections. F urther mo re we ass ume that f is representable and a lo cally trivial o r iented ﬁb er bundle with a closed manifold as ﬁb er . Then g has these proper ties, to o. The orientation o f g is induced by R n g ∗ Z V ∼ = R n g ∗ v ∗ Z X ∼ = u ∗ R n f ∗ Z X ∼ = u ∗ Z Y ∼ = Z U W e get diag rams (6.5.29) u ∗ Rf ∗ f ∗ u ∗ R f   ( 6.5.6 ) / / Rg ∗ v ∗ f ∗ ( 6.5.5 )   u ∗ Rg ∗ g ∗ u ∗ R g o o (6.5.30) R u ∗ Rg ∗ g ∗ Ru ∗ R g   / / Rf ∗ Rv ∗ g ∗ Ru ∗ Rf ∗ f ∗ Ru ∗ R f Ru ∗ o o ∼ O O F or the upp er hor izontal trans formation in (6.5.29) we use 6 .5 .3, and for the right vertical one (6.1.1 5) or 6.5.7. Note that o nly in the b ounded b elow derived categ ory the right vertical morphis m is an equiv alence for gener al u (whic h is anyw ay the situation in which we will apply the asser tion). L emma 6.5.31 . — The diagr ams ( 6.5.29 ) and (6.5.30) c ommutes. T o pr ov e Le mma 6.5 .31, we start with the fo llowing t wo technical lemmas. L emma 6.5.32 . — Given a Cartesian diagr am (6.5.28) of lo c al ly c omp act stacks such that f and u have lo c al se ctions, t hen for she aves K ∈ Sh Ab X and F ∈ S h Ab U 114 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS the fol lowing diagr am c ommutes: f ∗ K ⊗ u ∗ F = − − − − → f ∗ K ⊗ u ∗ F   y 6.2.8   y 6.2.8 f ∗ ( K ⊗ f ∗ u ∗ F ) u ∗ ( u ∗ f ∗ K ⊗ F ) ∼   y 6.1.12 ∼   y 6.1.12 f ∗ ( K ⊗ v ∗ g ∗ F ) u ∗ ( g ∗ v ∗ K ⊗ F )   y 6.2.8   y 6.2.8 f ∗ v ∗ ( v ∗ K ⊗ g ∗ F ) u ∗ g ∗ ( v ∗ K ⊗ g ∗ F ) ∼   y 6.6.8 ∼   y 6.6.8 h ∗ ( v ∗ K ⊗ g ∗ F ) = − − − − → h ∗ ( v ∗ K ⊗ g ∗ F ) , wher e h := f ◦ v = u ◦ g . Pr o of . — By Deﬁnition 6.2.8, the left vertical morphism is the image of the identit y under the following seq uence of maps Hom ( v ∗ K ⊗ g ∗ K, v ∗ K ⊗ g ∗ K ) → Hom ( v ∗ f ∗ f ∗ K ⊗ v ∗ v ∗ g ∗ K, v ∗ K ⊗ g ∗ K ) → Ho m ( v ∗ ( f ∗ f ∗ K ⊗ f ∗ u ∗ K ) , v ∗ K ⊗ g ∗ K ) → Hom ( f ∗ ( f ∗ K ⊗ u ∗ K ) , v ∗ ( v ∗ K ⊗ g ∗ K )) → Ho m ( f ∗ K ⊗ u ∗ K, f ∗ v ∗ ( v ∗ K ⊗ g ∗ K )) → Hom ( f ∗ K ⊗ u ∗ K, h ∗ ( v ∗ K ⊗ g ∗ F )) . The right vertical mor phism, on the other hand, is given by Hom ( v ∗ K ⊗ g ∗ K, v ∗ K ⊗ g ∗ K ) → Hom ( g ∗ g ∗ v ∗ K ⊗ g ∗ u ∗ u ∗ K, v ∗ K ⊗ g ∗ K ) → Ho m ( g ∗ ( u ∗ f ∗ K ⊗ u ∗ u ∗ K ) , v ∗ K ⊗ g ∗ K ) → Hom ( u ∗ ( f ∗ K ⊗ u ∗ K ) , g ∗ ( v ∗ K ⊗ g ∗ K )) → Ho m ( f ∗ K ⊗ u ∗ K, u ∗ g ∗ ( v ∗ K ⊗ g ∗ K )) → Hom ( f ∗ K ⊗ u ∗ K, h ∗ ( v ∗ K ⊗ g ∗ F )) . In bo th cases, we ﬁrst use the counit, then “co mm ute” pushdown and pullback using Lemma 6.1.12 and ﬁnally use a djunctions. By Lemma 6.1.12, the tw o wa ys to apply the counit and the push-pull isomorphism commute. This implies commutativit y of the diagr am of homo morphism sets, and there fore the commutativit y of the orig inal diagram. 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 115 L emma 6.5.33 . — In the situation of Le mma 6.5.3 2 for K ∈ Sh Ab X and F ∈ Sh Ab Y the fol lowing diagr am c ommutes: u ∗ ( f ∗ K ⊗ F ) 6.2.8 − − − − → u ∗ f ∗ ( K ⊗ f ∗ F )   y 6.2.5   y 6.1.12 u ∗ f ∗ K ⊗ u ∗ F g ∗ v ∗ ( K ⊗ f ∗ F )   y 6.1.12   y 6.2.5 g ∗ v ∗ K ⊗ u ∗ F g ∗ ( v ∗ K ⊗ v ∗ f ∗ F )   y =   y 6.6.9 g ∗ v ∗ K ⊗ u ∗ F − − − − → 6.2.8 g ∗ ( v ∗ K ⊗ g ∗ u ∗ F ) . Pr o of . — The left vertical and low er comp os ition is b y deﬁnition the image of the ident ity under the sequence o f maps Hom ( K ⊗ f ∗ F, K ⊗ f ∗ F ) unit − − − → Hom ( K ⊗ f ∗ F, v ∗ v ∗ ( K ⊗ f ∗ F )) ad j − − → H om ( v ∗ ( K ⊗ f ∗ F ) , v ∗ ( K ⊗ f ∗ F )) → Hom ( v ∗ K ⊗ g ∗ u ∗ F, v ∗ K ⊗ g ∗ u ∗ F ) counit − − − − → Hom ( g ∗ g ∗ v ∗ K ⊗ g ∗ u ∗ F, v ∗ K ⊗ g ∗ u ∗ F ) ad j − − → Ho m ( g ∗ v ∗ K ⊗ u ∗ F, g ∗ ( v ∗ K ⊗ g ∗ u ∗ F )) → Hom ( u ∗ ( f ∗ K ⊗ F ) , g ∗ ( v ∗ K ⊗ g ∗ u ∗ F )) . The upp er and r ight vertical comp osition is the image of the iden tity under the sequence o f ma ps Hom ( K ⊗ f ∗ F, K ⊗ f ∗ F ) counit − − − − → Ho m ( f ∗ f ∗ K ⊗ f ∗ F, K ⊗ f ∗ F ) ad j − − → Ho m ( f ∗ K ⊗ F, f ∗ ( K ⊗ f ∗ F )) unit − − − → Hom ( f ∗ K ⊗ F , u ∗ u ∗ f ∗ ( K ⊗ f ∗ F )) ad j − − → Hom ( u ∗ ( f ∗ K ⊗ F ) , u ∗ f ∗ ( K ⊗ f ∗ F )) → H om ( u ∗ ( f ∗ K ⊗ F ) , g ∗ v ∗ ( K ⊗ f ∗ F )) → Hom ( u ∗ ( f ∗ K ⊗ F ) , g ∗ ( v ∗ K ⊗ v ∗ f ∗ F )) → Hom ( u ∗ ( f ∗ K ⊗ F ) , g ∗ ( v ∗ K ⊗ g ∗ u ∗ F )) . These tw o maps coincide, a s follows from the fact that units and counits commute (in the a ppropriate se nse) with α ∗ and β ∗ . 6.5.19 . — W e no w show that (6.5.29) c ommut es. W e simplify the deﬁnition o f the int egra tio n map which is repr esented by a ll hor izontal comp ositio ns in the following 116 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS diagram. f ∗ I f ∗ / / ∼   f ∗ T K I f ∗   f ∗ T K f ∗ R ∼ o o / /   id ∼   f ∗ I f ∗ I ∼ / / f ∗ T K I f ∗ I f ∗ T K f ∗ RI ∼ o o / / I f ∗ f ∗ I ∼ O O / / ∼   f ∗ T K f ∗ I ∼   ∼ O O f ∗ T K f ∗ RI ∼ o o ∼   / / I ∼   f ∗ f ∗ I F l / / f ∗ T K f ∗ I F l f ∗ T K f ∗ RI F l ∼ o o / / I F l f ∗ f ∗ F l ∼ O O / / f ∗ T K f ∗ F l ∼ O O f ∗ T K f ∗ R F l ∼ o o ∼ O O / / F l ∼ O O Let us comment ab out the is omorphisms in the ﬁrst column. Let F ∈ C ( Sh Ab X ). Then f ∗ I f ∗ ( F ) → f ∗ I f ∗ I ( F ) is a quasi- is omorphism since f ∗ I f ∗ preserves quasi- isomorphisms a nd F → I ( F ) is a quasi-iso morphism. The ma p f ∗ f ∗ I ( F ) → f ∗ I f ∗ I ( F ) is a quasi- isomorphism sinc e I ( F ) is a complex of injectiv e, hence ﬂabby sheav e s, the functor f ∗ preserves ﬂabby sheaves, and therefore the a cyclic ma pping cone of C := C ( f ∗ I ( F ) → I f ∗ I ( F )) is an exa ct co mplex of ﬂabby sheav es. In particular it is an exact complex of f ∗ -acyclic sheaves. Since f ∗ has b ounded coho- mological dimension this implies tha t f ∗ ( C ) is exact (see the argument in the pro o f of Lemma 6 .5.17), and therefor e f ∗ f ∗ I ( F ) → f ∗ I f ∗ I ( F ) is a quasi-isomo r phism. The ma p f ∗ f ∗ I ( F ) → f ∗ f ∗ I F l ( F ) is a quasi-isomor phism by a similar argument. In fact, f ∗ F l ( F ) → f ∗ I F l ( F ) is a quasi-iso mo rphism of f ∗ -acyclic she aves. This implies ag a in by the mapping cone ar gument, that f ∗ f ∗ F l ( F ) → f ∗ f ∗ I F l ( F ) is a quasi-isomo rphism. The lower line of the diagram (6.5 .29) ex pr esses the int egra tio n map in terms of the ﬂabby res olution functor F l . Since we know tha t F l preserves ﬂat sheav es (we do not know this for I ) we can dro p the ﬂat r esolution functor R from the construction of the integration by ado pting the conv e ntion that the functor s a r e applied to complexes of ﬂat sheaves. 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 117 W e get the following commutativ e diag r am (6.5.34) u ∗ Rf ∗ f ∗ ∼ − − − − → u ∗ Rf ∗ f ∗ u ∗ R f − − − − → u ∗   y ∼   y ∼   y ∼ u ∗ f ∗ T K f ∗ F l ∼ ← − − − − u ∗ T f ∗ K F l − − − − → u ∗ F l   y ∼   y ∼   y ∼ g ∗ v ∗ T K f ∗ F l T u ∗ f ∗ K u ∗ F l − − − − → T u ∗ Z u ∗ F l   y ∼   y ∼   y ∼ g ∗ T v ∗ K v ∗ f ∗ F l T g ∗ v ∗ K u ∗ F l − − − − → u ∗ F l   y ∼   y =   y = g ∗ T v ∗ K g ∗ u ∗ F l ∼ ← − − − − T g ∗ v ∗ K u ∗ F l − − − − → u ∗ F l   y ∼   y ∼   y = g ∗ T v ∗ K g ∗ F l u ∗ ∼ ← − − − − T g ∗ v ∗ K F l u ∗ − − − − → F l u ∗   y ∼   y ∼   y ∼ Rg ∗ g ∗ u ∗ = − − − − → R g ∗ g ∗ u ∗ R g u ∗ − − − − → u ∗ The commutativit y of all the small squa res is evident. The commutativit y o f the larg e rectangle r elies o n the fac t that the pr o jection formula is co mpatible with pullbacks, this is the sta temen t of Lemma 6.5.33. The commut ativity of the bo undary of this diagram g ives (6.5.2 9). 6.5.20 . — In or der to sho w that (6.5.30) co mmut es we start with the fo llowing ob- serv ation. L emma 6.5.35 . — A ssume, i n the situation of L emma 6.5. 32 , that K is a ﬂat lo- c al ly f ∗ -acyclic r esolution of Z X of length n , and that f is a pr oje ction of a lo c al ly triv- ial orientable ﬁb er bund le of n -dimensional close d manifolds. Assu me t hat f ∗ K → Z Y is an orientation. L et g ∗ v ∗ K → Z U b e the induc e d orientation of the pul lb ack bund le g . Then the fol lowing diagr am c ommutes, wher e al l t he horizontal maps ar e given by the orientations. f ∗ K ⊗ u ∗ F − − − − → Z Y ⊗ u ∗ F   y   y u ∗ ( u ∗ f ∗ K ⊗ F ) − − − − → u ∗ ( u ∗ Z Y ⊗ F )   y ∼   y ∼ u ∗ ( g ∗ v ∗ K ⊗ F ) − − − − → u ∗ ( Z U ⊗ F ) 118 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS Pr o of . — The upp er diagram co mm utes b eca us e of the natur ality of the homomor- phism of the pr o jection formula, the low er dia gram commutes by the deﬁnition of the induced o rientation o f g . T o unders tand the relation betw een deriv ed pushdown along a non-representable map a nd in teg ration w e need to use an explicit model of the derived pushdown. If u : U → Y is a morphism b etw een lo cally compact stacks which has local sections, then Ru ∗ is giv e n b y C A ◦ F l , where F l is the functorial ﬂabb y resolution funct or, and C A is deﬁned in Section 3 .2, using an a tlas A → U . Note that C A indeed can b e decomp osed as the co mpo sition o f a functor L A on sheaves on U and u ∗ . Her e L A is the s heaﬁﬁcation of the functor on pres heav es given by p L k A F ( W → U ) := F ( A × U · · · × U A | {z } k + 1 factors × U W → U ) . i.e. p L k A = p k ∗ p ∗ k , with p k : A × U · · · × U A | {z } k + 1 factors → U . L emma 6.5.36 . — In t he situ ation of L emma 6.5.35, we obtain a c ommu t ative di- agr am f ∗ T K f ∗ u ∗ L A F l = − − − − → f ∗ T K f ∗ u ∗ L A F l ∼ ← − − − − T f ∗ K u ∗ L A F l − − − − → u ∗ L A F l   y ∼   y ∼   y   y = f ∗ T K v ∗ L g ∗ A g ∗ F l 3.2.4 − − − − → ∼ f ∗ T K v ∗ g ∗ L A F l u ∗ T u ∗ f ∗ K L A F l − − − − → u ∗ L A F l   y   y ∼   y = f ∗ v ∗ T v ∗ K g ∗ L A F l u ∗ T g ∗ v ∗ K L A F l − − − − → u ∗ L A F l   y ∼   y =   y = u ∗ g ∗ T v ∗ K g ∗ L A F l ← − − − − u ∗ T g ∗ v ∗ K L A F l − − − − → u ∗ L A F l . Her e, the right horizontal maps ar e given by the orientations f ∗ K → Z Y and g ∗ v ∗ K → Z U . Pr o of . — This is the direc t tr anslation of Lemma 6.5.3 2 and Lemma 6.5.3 5. Note that the upp er comp osition is a repres ent ation (when a pplied to ﬂa t sheav es) of Rf ∗ f ∗ Ru ∗ R f − → R u ∗ . The leftmost vertical a rrow repre sents the morphism (6.5.37) Rf ∗ f ∗ Ru ∗ → Rf ∗ Rv ∗ g ∗ , since g ∗ preserves ﬂabby sheaves, and v ∗ L g ∗ A indeed is a mo de l fo r C g ∗ A , which can be us ed to ca lc ulate R v ∗ . 6.5. OPERA TIONS WITH UNBOUNDED DERIVED CA TEGORIES 119 Therefore the diag ram in Lemma 6.5.3 6 contains one part (lower r ight-up) of the diagram (6 .5 .30). 6.5.21 . — T o r e present the other comp osition of the diagra m (6.5.30), we have to commute not only u ∗ but also L A with the other op er a tions. Recall that L A provides some kind of a resolution, i.e. we have a canonical map id → L A , whic h is us ed in the Le mma b elow. L emma 6.5.38 . — In the situation of L emma 6.5. 35, the fol lowing diagr am c om- mutes, wher e the horizontal maps ar e induc e d by the orientation of g . u ∗ T g ∗ v ∗ K L A F l − − − − → u ∗ T Z L A F l   y   y u ∗ T L A g ∗ v ∗ K L A F l − − − − → u ∗ T L A Z L A F l   y   y u ∗ L A T g ∗ v ∗ K F l − − − − → u ∗ L A T Z F l The s e c ond vertic al map in e ach c olumn fol lows fr om a variant of the pr oje ction for- mula, using that L A is given by applic ation of ( p k ) ∗ p ∗ k (or by dir e ctly insp e ct ing the deﬁnitions). Pr o of . — If G → H is a morphism of sheav es, then we get a natur al tra nsformation of functors T G → T H . This naturality implies the commutativit y of the ﬁrst square . The second square is comm utative by the naturality of the morphism in the pro jection formula. Observe that we have a natural isomorphism g ∗ L A ∼ = L g ∗ A g ∗ . L emma 6.5.39 . — In the situ ation of L emma 6.5.35, we obtain the fol lowing c om- mutative diagr am u ∗ g ∗ T v ∗ K g ∗ L A F l ← − − − − u ∗ T g ∗ v ∗ K L A F l   y   y u ∗ g ∗ T L g ∗ A v ∗ K g ∗ L A F l ← − − − − u ∗ T g ∗ L g ∗ A v ∗ K L A F l 3.2.4   y ∼   y ∼ u ∗ g ∗ T L g ∗ A v ∗ K L g ∗ A g ∗ F l u ∗ T L A g ∗ v ∗ K L A F l   y   y u ∗ g ∗ L g ∗ A T v ∗ K g ∗ F l u ∗ L A T g ∗ v ∗ K F l 3.2.4   y ∼   y = u ∗ L A g ∗ T g ∗ v ∗ K g ∗ F l ← − − − − u ∗ L A T g ∗ v ∗ K F l 120 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS Pr o of . — The upp er square is commutativ e b ecause of the naturality of the morphism in the pro jection formula. The co mm utativity o f the low er rectangle follows from Lemma 6.5.32, as we basically hav e to co mmute tw o diﬀerent applicatio ns of the pro jection for mula. W e now pr ov e the c ommut ativity of (6.5.30). Using explicit representatives of the maps in ques tion, we obta in (applied to ﬂat sheav es) Rf ∗ f ∗ Ru ∗ − − − − → = Rf ∗ f ∗ Ru ∗ R f Ru ∗ − − − − → R u ∗   y ∼   y ∼   y ∼ f ∗ T K f ∗ u ∗ L A F l ∼ ← − − − − T f ∗ K u ∗ L A F l − − − − → u ∗ L A F l   y   y   y = u ∗ g ∗ T v ∗ K g ∗ L A F l ← − − − − u ∗ T g ∗ v ∗ K L A F l − − − − → u ∗ L A F l   y   y   y = u ∗ L A g ∗ T g ∗ v ∗ K g ∗ F l ← − − − − u ∗ L A T g ∗ v ∗ K F l − − − − → u ∗ L A T Z F l   y   y   y = u ∗ L A F l g ∗ T g ∗ v ∗ K g ∗ F l ← − − − − u ∗ L A F l T g ∗ v ∗ K F l − − − − → u ∗ L A F l   y ∼   y ∼   y ∼ Ru ∗ Rg ∗ g ∗ − − − − → = Ru ∗ Rg ∗ g ∗ Ru ∗ R g − − − − → Ru ∗ Here, the ﬁrst a nd the last rows are just added as illustration what the next or preceding line, resp ectively , computes in the derived c a tegory . The map from the third-last to the sec o nd-last row is induced by the inclusion into the ﬂa bby resolution. This step is necessary because w e don’t know that the functors in question are u ∗ - acyclic, and explains why one ca n directly deﬁne only the map f ∗ Ru ∗ → R v ∗ g ∗ , and why it is hard to show that this is an equiv a lence. The other vertical maps, a nd the commutativit y of the rema ining four squar es, is given by Lemmas 6 .5.36, 6 .5 .38, 6.5.39. Note that the left vertical comp os itio n is the comp osition Rf ∗ f ∗ Ru ∗ → R f ∗ Rv ∗ g ∗ → R u ∗ Rg ∗ g ∗ , as s hown in the reaso ning for (6.5.37). The ass ertion follows. ✷ 6.5.22 . — Compared w ith the simplicity of its statement the pro of of Lemma 6 .5.31 seems to b e to o long. But let us mention that the pro of of a similar result in the algebraic context is quite in volv ed, to o. The b o o k [ Con00 ] is dev oted to this problem. 6.6. EXTENDED S ITES 121 6.6. Extended sites 6.6.1. — W e consider the low er right Cartesian s quare of the diag ram U × Y B / /   B   A × Y X   / / U × Y X   / / X f   A / / U / / Y in s tacks where U, X , Y are lo c a lly co mpact. L emma 6.6.1 . — If U is a sp ac e or f is r epr esentable, then U × Y X is a lo c al ly c omp act stack. Pr o of . — W e ﬁrst assume that U is a lo ca lly compact space. Let B → X be a lo cally compac t a tlas. Then U × Y B → U × Y X is an atlas. Indee d, surjectivity , representabilit y , and lo ca l sectio ns for this ma p a re implied by the co r resp onding prop erties of the map B → X . The stack U × Y B is a spa ce since U → Y is representable by Prop o s ition 6.1.1. By Lemma 6.1.9 the spa ce U × Y B is lo cally compact. F urthermore, aga in by Lemma 6 .1.9, ( U × Y B ) × ( U × Y X ) ( U × Y B ) ∼ = U × Y ( B × X B ) is lo c a lly co mpact s ince B × X B is lo cally compact. Hence the atlas U × Y B → U × Y X has the properties required in Deﬁnition 2.1.2 so that U × Y X is a lo cally compact stack. W e no w assume that f is representable. Let A → U b e a lo cally compact atlas such that A × U A is lo ca lly compact. Then A × Y X ∼ = A × U ( U × Y X ) → U × Y X is an atla s of U × Y X . W e a gain verify the pr o p erties requir ed in Deﬁnition 2 .1.2. By the sp ecial case of the Lemma already shown this atlas is lo ca lly compact. Mo reov er [ A × U ( U × Y X )] × U × Y X [ A × U ( U × Y X )] ∼ = ( A × U A ) × Y X is lo cally c ompact. 6.6.2. — If f : X → Y is a repr esentable map with lo cal sections b etw een lo ca lly compact stacks, then for ( U → Y ) ∈ Y we hav e p f ∗ h U ∼ = h U × X Y (see the pro of of L e mma 6.6.6 below). If w e drop the assumption that f is representable, then in general p f ∗ h U is not represen table. In or der to o vercome this defect we e nla rge the site X to ˜ X s o that it contains the stacks U × X Y → X over X . W e consider the 2-categ ory Stack s top,lc / ls,r ep X of lo ca lly compa ct stacks U → X ov er X , wher e the structure map is representable and has lo cal sections. A morphism 122 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS in this c ategory is a diagr am U @ @ @ @ @ @ @ / /   V ~ ~ ~ ~ ~ ~ ~ ~ ~ X consisting of a one-mo rphism and a tw o -morphism. The co mpo sition is deﬁned in the obvious way . If there is a t wo-morphism b etw een tw o such one-mo rphisms, then it is unique by the repres ent ability o f the structure maps. Therefo r e Sta cks top,lc / ls,r ep X is equiv alent in t wo-categories to the one-catego r y obtained b y ident ifying all isomorphic one-morphisms. 6.6.3. — Let f : X → Y be a map b etw een lo ca lly co mpact stacks. Deﬁnition 6.6.2 . — We let ˜ X b e the c ate gory obtaine d fr om Stack s top,lc / ls,r ep X by identifying al l isomorphic one-morphisms. W e now deﬁne the top olo gy on ˜ X . A covering family ( U i → U ) o f ( U → X ) ∈ ˜ X is a family of lo ca lly compact stacks ov er U such that U i → U is repres entable, has lo cal sections a nd ⊔ i ∈ I U i → U is sur jective (1) . Using Lemma 6.6 .1 one eas ily chec ks the a xioms listed in [ T am94 , 1.2.1]. Let ˆ X b e the site with the same underlying categ ory as ˜ X , but with the top ol- ogy gener ated b y the cov ering families o f ( U → X ) g iven b y families ( U i → U ) ∈ Stacks top,lc /X such that U i → U is a map fro m a locally co mpact space w ith local sections and ⊔ i U i → U is sur jective. L emma 6.6.3 . — We have a c anonic al isomorphism Sh ˜ X ∼ = Sh ˆ X . Pr o of . — The cov er ing families of ˆ X a re cov ering families in ˜ X . Here we use Prop o - sition 6.1.1 in or der to s e e that the maps U i → U from spa ces U i are representable. On the other hand, every cov ering family ( U i → U ) of ( U → X ) in ˜ X c a n b e reﬁned to a covering family in ˆ X by choos ing a lo cally co mpa ct a tlas A i → U i for each U i . This implies the lemma. 6.6.4. — The natural functor Top lc /X → Sta cks top,lc /X fr om lo c ally compac t spaces ov er X to lo cally c o mpact sta cks ov er X induces a map of sites j : X → ˜ X . L emma 6.6.4 . — The r estriction functor j ∗ : Sh ˜ X → Sh X is an e quivalenc e of c ate gories. (1) These maps are actually equiv alence classes, bu t in order to sim plify the l anguage we will not men tion this explicitly in the following 6.6. EXTENDED S ITES 123 Pr o of . — The inv ers e of j ∗ is the functor j ∗ given by j ∗ F ( U ) := lim ( V → U ) ∈ X //U F ( V ) for a ll ( U → X ) ∈ ˜ X , wher e X // U is the categor y of all pa irs ( V ∈ X , j ( V ) → U ∈ Mor( ˜ X )) such tha t the ma p j ( V ) → U has lo cal se ctions. If U ∈ j ( X ), then ( U, id j ( U ) : j ( U ) → j ( U )) it is the ﬁnal ob ject of X //U . This gives a natura l isomo rphism j ∗ j ∗ ( F )( U ) ∼ = F ( U ). W e now deﬁne a natural isomorphism j ∗ j ∗ ( F ) → F for all F ∈ Sh ˜ X . Let ( U → X ) ∈ ˜ X . The family ( V → U ) X //U is a covering family of U → X in ˆ X . Since F is also a s he a f on ˆ X by Lemma 6 .6 .3 we get an isomorphism j ∗ j ∗ ( F )( U ) ∼ = lim ( V → U ) ∈ X //U j ∗ ( F )( V ) ∼ = F ( U ) . 6.6.5. — L emma 6.6.5 . — A map f : X → Y b etwe en lo c al ly c omp act stacks induc es a map of sites ˜ f ♯ : Y → ˜ X by ˜ f ♯ ( U → Y ) := U × Y X → X . Pr o of . — Indeed, if U → Y is a map from a lo ca lly compact space, then the stack U × Y X is lo cally compact by Lemma 6.6.1. If ( U i → U ) is a covering family of ( U → Y ) ∈ Y by op en subspaces, then ( U i × Y X → U × Y X ) is a covering family in ˜ X by op en substa cks. F urthermore it is easy to see that ˜ f ♯ preserves ﬁb er pr o ducts, i.e. if ( U i → U ) is a cov er ing family a nd V → U is a mor phism in Y , then ˜ f ♯ ( U i × U V ) ∼ = ˜ f ♯ ( U i ) × ˜ f ♯ ( U ) ˜ f ♯ ( V ). 6.6.6. — W e consider a map f : X → Y b etw een locally co mpact stacks. Then we hav e a n adjoint pa ir of functors ˜ f ♯ ∗ : Sh Y ⇆ Sh ˜ X : ( ˜ f ♯ ) ∗ . L emma 6.6.6 . — We have an isomorphism of fu n ctors j ∗ ◦ ˜ f ♯ ∗ ∼ = f ∗ : Sh Y → Sh X Pr o of . — The ma p j : X → ˜ X induces a map p j ∗ : P r ˜ X → Pr X . W e show the relation ﬁrst o n representable presheav es. Le t ( U → Y ) ∈ Y and obser ve that ( U × Y X → X ) ∈ ˜ X by L e mma 6.6 .1. The following chain o f na tural iso morphisms 124 CHAPTER 6. VE RDIER DUALITY FOR LOCALL Y COMP ACT ST ACKS (for a r bitrary F ∈ Pr ˜ X ) shows tha t ˜ f ♯ ∗ h U ∼ = h U × Y X : Hom Pr ˜ X ( ˜ f ♯ ∗ h U , F ) ∼ = Hom Pr Y ( h U , ( ˜ f ♯ ) ∗ F ) ∼ = ( ˜ f ♯ ) ∗ F ( U ) ∼ = F ( ˜ f ♯ ( U )) ∼ = F ( U × Y X ) ∼ = Hom Pr ˜ X ( h U × Y X , F ) . F or ( U → Y ) ∈ Y w e hav e p f ∗ h U ∼ = p j ∗ h U × Y X . Indeed, for ( V → X ) ∈ X we hav e p j ∗ h U × Y X ( V ) ∼ = Hom ˜ X ( j ( V ) , U × Y X ) ! ∼ = p f ∗ h U ( V ) , where the marked is omorphism can b e seen by making the deﬁnition o f p f ∗ explicit. Since p j ∗ ◦ p ˜ f ♯ ∗ and p f ∗ commute with colimits the equation p j ∗ ◦ p ˜ f ♯ ∗ ∼ = p f ∗ holds on a ll presheav es. The res tr iction to sheaves (note that a ll functors preserve sheav es) gives j ∗ ◦ ˜ f ♯ ∗ ∼ = f ∗ . By adjo intness we get (6.6.7) ( ˜ f ♯ ) ∗ ◦ j ∗ ∼ = f ∗ . 6.6.7. — Consider t wo comp oseable ma ps b etw een lo cally compact stacks. X f → Y g → Z . The following lemma generalizes [ BSS07 , Lemma 2 .23] by dropping the unneces sary additional assumptions that f has lo c al sec tio ns or g is repr esentable. L emma 6.6.8 . — We have an isomorphism of fu nctors g ∗ ◦ f ∗ ∼ = ( g ◦ f ) ∗ : Sh X → Sh Z . Pr o of . — W e c onsider the following diag ram: Sh X ( g ◦ f ) ∗ ) ) j X ∗ / / f ∗   Sh ˜ X ( ^ ( g ◦ f ) ♯ ) ∗ u u ( ˜ f ♯ ) ∗   Sh Y j Y ∗ / / g ∗   Sh ˜ Y (˜ g ♯ ) ∗   Sh Z j Z ∗ / / Sh ˜ Z . W e know that the squar es commut e (Equation (6 .6.7)), and that the horizontal arrows are isomo rphisms (Le mma 6.6 .4). It follows fro m the constructio ns that ˜ f ♯ ◦ ˜ g ♯ = ^ ( g ◦ f ) ♯ on the level of sites. Hence the rig ht triangle co mmutes, to o. This implies commuta- tivit y of the le ft tria ngle. 6.6. EXTENDED S ITES 125 T aking adjoints we get: Cor ol lar y 6.6.9 . — We have an isomorphism f ∗ ◦ g ∗ ∼ = ( g ◦ f ) ∗ : Sh Z → Sh X . 6.6.8. — W e consider a topo logical stack X and the inclusio n j : X → ˜ X which induces by Le mma 6.6 .4 an equiv ale nc e of categ ories of sheaves j ∗ : Sh ˜ X ⇆ Sh X : j ∗ . Note that the notion o f ﬂabbiness dep ends on the site. Deﬁnition 6.6.10 . — We c al l a she af F ∈ Sh Ab X str ongly ﬂabby if j ∗ ( F ) is ﬂabby. Since ﬂabbiness is a co ndition to b e chec ked for all covering families and since all cov er ing families in X induce cov ering families in ˜ X it follows that a strongly ﬂabby sheaf is ﬂabby . Since injective sheav es are strongly ﬂabby each sheaf admits a stro ngly ﬂabby r esolution. 6.6.9. — Let f : X → Y be a mor phism of lo cally compact stacks. L emma 6.6.11 . — S tr ongly ﬂabby she aves ar e f ∗ -acyclic. Pr o of . — In view o f Lemma 6.6.6 it suﬃces to show that ﬂa bby sheav es in Sh Ab ˜ X ar e ˜ f ∗ -acyclic. W e now can write ˜ f ∗ = ˜ i ♯ ◦ p ˜ f ∗ ◦ ˜ i , where ˜ i ♯ and ˜ i ar e the sheaﬁﬁcation functor and the inclusion o f shea ves into pr esheav es for the tilded sites, and p ˜ f ∗ = p ( ˜ f ♯ ) ∗ : P r ˜ X → P r ˜ Y . Since p ˜ f ∗ ( F )( V → Y ) = F ( V × Y X → X ) we see that p ˜ f ∗ is exact. Since strongly ﬂabby sheav es a r e ˜ i -acyclic, and ˜ i ♯ is exact, it follo ws that strongly ﬂa bb y s heav es are ˜ f ∗ -acyclic. L emma 6.6.12 . — The fun ct or f ∗ : Sh Ab X → Sh Ab Y pr eserves str ongly ﬂabby she aves. Pr o of . — W e must show that ˜ f ∗ preserves ﬂabby she aves. Let F ∈ S h Ab ˜ X and τ = ( U i → U ) b e a covering family of ( U → Y ) in Y . W e must show that the ˇ Cech complex C ( τ , ˜ f ∗ F ) is acyclic. Note that ˜ f ∗ F ( V ) = F ( V × Y X ). The fa mily f ♯ ( τ ) := ( U i × Y X → U × Y X ) is a covering family o f U × Y X in ˜ X . W e see that C ( τ , ˜ f ∗ F ) ∼ = C ( f ♯ ( τ ) , F ). 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Periodic twisted cohomology and T-duality

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