Sign lemma for dimension shifting

Sign lemma for dimension shifting Nitin Nitsure Abstract There is a surprising o ccurrence of s ome minus signs in the isomorphisms pro du ced in the w ell-kno wn tec hniqu e of d imension shifting in calculating de- riv ed functors in homological algebra. W e explicitly d etermine these s igns. Getting these signs r igh t is imp ortan t in order to a void b asic con tradictions. W e illustrate th e lemma b y some de Rh am cohomology an d Chern class con- siderations for compact Riemann surfaces. Statemen t of the main result Let A b e a n a b elian category with enough injectiv es, and let F : A → B b e an additiv e, left-exact functor fro m A to a nother ab elian category B . F or eac h ob j ect M of A , w e choose an injectiv e resolution 0 → M → I • , and define the v a lue o f the deriv ed functor R i F o n M to b e H i ( F I • ). (T o mak e suc h a c hoice for each ob ject, w e need to assume some foundationa l framew ork, whic h is fairly standard so we w ill omit all referenc e to it.) If 0 → M → J • is an y other resolution of M , w e hav e a homomorphism of complex es f • : J • → I • whic h is unique up to homotop y , and is iden tit y o n M . If J • is F -acyclic, that is, if R i F is zero on J k for eac h i ≥ 1 and k ≥ 0, then for eac h n ≥ 1, f • induces a n isomorphism c n = H n ( F f • ) : H n ( F J • ) → H n ( F I • ) = R n F M . W e will call c n : H n ( F J • ) → R n F M as the canonical isomorphism for the a cyclic resolution J of M . There is ano ther v ery useful isomorphism d n : H n ( F J • ) → R n F M for n ≥ 1 , kno wn as the dimension shifting isomorphism . T o define it, w e b egin b y breaking-up the r esolution J into a sequence E 1 , . . . , E n of short exact sequen ces E 1 = (0 → Z 0 → J 0 → Z 1 → 0) , where Z 0 = M . E 2 = (0 → Z 1 → J 1 → Z 2 → 0) , . . . E n = (0 → Z n − 1 → J n → Z n → 0) . As the J i as F -acyclic, the corresp onding connecting homomorphisms ar e isomor- phisms f or p ≥ 1 and q ≥ 1, whic h w e denote b y ∂ p E q : R p F Z q → R p +1 F Z q − 1 . F or p = 0 and q ≥ 1, the connecting homomorphism ∂ 0 E q : F Z q → R 1 F Z q − 1 is epic, and induces an isomorphism ∂ 0 E q : F Z q im F J q − 1 → R 1 F Z q − 1 . 1 F or q = n , w e th us hav e a sequence of isomorphisms H n ( F J • ) = F Z n im F J n ∂ → R 1 F Z n − 1 ∂ → . . . ∂ → R n F M The comp osite of these is an isomorphism d n : H n ( F J • ) → R n F M whic h is b y definition the dimension shifting isomorphism fo r n ≥ 1. In this note, w e compare the tw o isomorphisms c n : H n ( F J • ) → R n F M and d n : H n ( F J • ) → R n F M for all n ≥ 1, and w e find the following, whic h is our main result. Sign lemma for dimension shifting With notation as ab ove, the c anonic al iso- morphism c n and the d imension shifting i somorphism d n ar e r elate d by d n = ( − 1) ( n 2 + n ) / 2 c n . T o prov e the ab o v e lemma, w e need some preliminaries. Preliminary lemmas An y ob ject X in a category C defines a con trav ariant functor h X = H om C ( − , X ), whic h w e call its functor of el ements , and fo r any other ob ject T , an elemen t x ∈ h X ( T ) will b e called a T -v a lued elemen t o f X . When w e do no t w an t to explicitly men tion T , then suc h an x will b e just called a v alued elemen t of X , and b y abuse of notation we write it as x ∈ X . By the Y oneda lemma, an y morphism f : X → Y in C is determined b y its effect on all v alued elemen ts of X . If g : X ′ → X is epic, then (ev en though not a ll T -v alued p oints of X lift to X ′ ) the morphism f : X → Y is determined b y the effect of g ◦ f on all v alued elemen t s of X ′ . In what follow s, A will b e an ab elian category with enough injective s, and F : A → B will b e a n additiv e, left- exact functor from A to another ab elian category B . Let there be giv en o b j ects A and B in A , together with F - a cyclic resolutions 0 → A → J • and 0 → B → K • . Let c i B : H i ( F J • ) → R i F A and c i B : H i ( F K • ) → R i F B b e t he corresp onding canonical isomorphisms. Let t here b e giv en a short- exact sequence E = (0 → A → C → B → 0) and let ∂ i E : R i F B → R i +1 F A b e the connecting homomorphism. Let t here b e c hosen an F - acyclic resolution 0 → C → L • , together with a short- exact sequenc e of complexes E = (0 → J • → L • → K • → 0) 2 suc h that the following diagram comm utes: 0 → A → C → B → 0 ↓ ↓ ↓ 0 → J • → L • → K • → 0 Note that b y the so called ‘horse-shoe lemma’, suc h a resolution L • alw ay s exists. As J • is F - acyclic, applying F g iv es a short-exact sequence of complex es F E = (0 → F J • → F L • → F K • → 0) Let δ i F E : H i ( F K • ) → H i +1 ( F J • ) b e the corresp onding connecting homomorphism. Lemma A. With notation as a b ove, the f ol lowing diagr am c ommutes. H i ( F K • ) δ i F E → H i +1 ( F J • ) c i ↓ ↓ c i +1 R i F A ∂ i E → R i +1 F B Pro of. W e leav e the pro of of Lemma A as a n exercise to the reader. Remark. In particular, the ho momo r phism δ i F E : H i ( F K • ) → H i +1 ( F J • ) do es not dep end on the choice of the resolution 0 → C → L • , or on the short exact sequence of complexes E , but only dep ends on the giv en short exact sequence E = (0 → A → C → B → 0). Hence in what fo llo ws w e will denote it simply b y δ i E : H i ( F K • ) → H i +1 ( F J • ). W e now return to the situation of our main result, t he Sign Lemma. Recall that w e b egan with an F -acyclic resolution 0 → M → J • of a n o b ject M of A . Let δ i J : J i → J i +1 denote the differen tials. W e brak e up the resolution in to short- exact sequence s E q = (0 → Z q − 1 u q − 1 → J q − 1 v q − 1 → Z q → 0) for 1 ≤ q ≤ n , with δ i = u i +1 v i . F or eac h i ≥ 0, we get an F - a cyclic resolution 0 → Z i → K • i of Z i defined b y K p i = J i + p , δ i K = δ i + p J , and Z i → K 0 i the ‘inclusion’ homomorphism u i : Z i → J i . Lemma B. F or al l n ≥ 1 and p ≥ 1 , the fol lowing diagr ams ar e c ommutative. H n ( F J • ) − 1 → H n ( F J • ) k k H 0 ( F K • n ) im F J n − 1 δ E n → H 1 ( F K • n − 1 ) k ↓ c 1 F Z n im F J n − 1 ∂ E n → R 1 F Z n − 1 and H n ( F J • ) ( − 1) p +1 → H n ( F J • ) k k H p ( F K • n − p ) δ E n − p → H p +1 ( F K • n − p − 1 ) c p ↓ ↓ c p +1 R p F Z n − p ∂ E n − p → R p +1 F Z n − p − 1 3 Pro of By Lemma A, the homomorphism δ E q can b e computed in terms of an y resolution L • i of J q whic h fits in a short-exact sequence E of comm uting resolutions of 0 → Z q → J q → Z q +1 → 0, and the low er sq uares in the ab ov e dia grams comm ute b y Lemma A. T o see that the upp er squares comm ute, w e construct a particular suc h resolution 0 → J i → L • i as follo ws. F or an y p w e put L p i = K p i ⊕ K p i +1 = J i + p ⊕ J i + p +1 . W e write v alued elemen ts of L p i as (2 × 1) -column v ectors. With this notation, the inclusion of J i in to L 0 i is defined in matrix terms b y  1 J i δ i J  : J i → J i ⊕ J i +1 = L 0 i . The differen t ia l δ p L i : L p i → L p +1 i is defined in matrix terms (acting on column vec to rs) b y δ p L i =  δ i + p J ( − 1) p +1 1 J i + p +1 0 δ i + p +1 J  : J i + p ⊕ J i + p +1 → J i + p +1 ⊕ J i + p +2 . With t hese definitions, 0 → J i → L • i is indeed exact. Moreov er, the follo wing is a comm utativ e diag r a m with exact rows, where the second ro w is g iven b y inclusions and pro j ections for the lev el-wise direct sum L p i = K p i ⊕ K p i +1 = J i + p ⊕ J i + p +1 . 0 → Z i → J i → Z i +1 → 0 ↓ ↓ ↓ 0 → K • i → L • i → K • i +1 → 0 No w tak e i = n − p − 1 in the ab ov e. Giv en any v alued elemen t x ∈ F Z n whic h represen ts a v alued elemen t x ∈ F Z n / im F J n − 1 = H p ( F K • n − p ) , the v alued elemen t y =  0 x  ∈ F J n − 1 ⊕ F J n = F K p n − p − 1 ⊕ F K p n − p = F L p n − p − 1 has the prop erty that under the pro jection F L p n − p − 1 → F K p n − p , w e hav e y 7→ x . No w note that as δ n F J x = 0, w e hav e  δ n − 1 F J ( − 1) p +1 1 F J n 0 δ n F J   0 x  =  ( − 1) p +1 x δ n F J x  =  ( − 1) p +1 x 0  ∈ F L p +1 n − p − 1 4 This is the image of ( − 1) p +1 x ∈ F K p +1 n − p − 1 = F J n under the inclusion F K p +1 n − p − 1 → F L p +1 n − p − 1 . This sho ws that under the connecting morphism δ p F E , the image of x is ( − 1) p +1 x . As F Z n → F Z n / im F J n − 1 = H p ( F K • n − p ) is epic, this calculation is enough to sho w that δ p F E acts as ( − 1) p +1 on all v alued elemen ts of H p ( F K • n − p ) = F Z n / im F J n − 1 . This completes the pro o f of Lemma B. Pro of of the sign lemma By Lemma B, the f ollo wing squares comm ute fo r all n ≥ 1 and p ≥ 1. H n ( F J • ) − 1 → H n ( F J • ) k ↓ c 1 F Z n im F J n − 1 ∂ → R 1 F Z n − 1 and H n ( F J • ) ( − 1) p +1 → H n ( F J • ) c p ↓ ↓ c p +1 R p F Z n − p ∂ → R p +1 F Z n − p +1 As P n − 1 p =0 ( p + 1) = ( n 2 + n ) / 2, the horizon ta l comp osition of the ab ov e diagrams giv es a commutativ e square H n ( F J • ) ( − 1) ( n 2 + n ) / 2 → H n ( F J • ) k ↓ c n F Z n im F J n − 1 d n → R n F Z 0 where d n is the compo site H n ( F J • ) ∂ → R 1 F Z n − 1 ∂ → . . . ∂ → R n F M , whic h is by definition t he dimension-shifting isomorphism . Thus d n = ( − 1) ( n 2 + n ) / 2 c n whic h completes the pro o f of the sign lemma. Illustration: Chern class and de Rham’s theorem. Let X b e a diff erential manifold (paracompact), a nd A the category o f shea v es of real v ector spaces (or complex v ector spaces) on X . Let B b e the category of real (resp. complex) v ector spaces, and let F : A → B b e the global section functor Γ( X , − ), with deriv ed f unctors the sheaf coho mo lo gies H i ( X , − ). Let C i b e the sheaf of real (resp. complex) differen tia l i -forms, and let δ i : C i → C i +1 b e the exterior deriv ativ e. As this defines an F -acyclic resolution 0 → R X → C • (resp. 0 → C X → C • ), we get a canonical isomorphism c i : H i dR ( X ) = H i ( F C • ) → R i F ( R X ) = H i ( X , R X ) (a canonical isomorphism c i : H i dR ( X ) = H i ( F C • ) → R i F ( C X ) = H i ( X , C X ) 5 in the complex case), where H i dR ( X ) denotes the real (resp. complex) de Rham cohomology of X . Some authors (for example [G-H]) pro ve de Rham’s theorem b y iden tifying de Rham cohomolog y and sheaf cohomology b y the dimension shifting isomorphism d i : H i dR ( X ) → H i ( X , C X ). By the sign lemma, this is ( − 1) ( i 2 + i ) / 2 -times the canoni- cal isomorphism. Omission of this s ign can lead to s ig n mistak es and confusion later on. One suc h confusion, whic h we describ e next to end this note, o ccurs in the basic calculation of Chern classes of line bundles on Riemann surfaces. Let C ∗ b e t he m ultiplicativ e sheaf non-v anishing complex v alued smo oth functions on X . Let exp : C → C ∗ the ma p defin ed at the lev el of lo cal sections b y f 7→ e 2 π if . This map is surjectiv e at the lev el of germs, and defines a short exact sequence of shea v es 0 → Z X → C exp → C ∗ → 0 called as exp onen tial sequence. An y complex line bundle on X defines an ele ment ( L ) ∈ H 1 ( X , C ∗ ), whose image c 1 ( L ) = ∂ ( L ) ∈ H 2 ( X , Z X ) under the connecting homomorphism ∂ : H 1 ( X , C ∗ ) → H 2 ( X , Z X ) is t he first Chern class of L . F or a compact Riemann surface X , let η X ∈ H 2 ( X , Z X ) denote the p ositiv e gener- ator. F or a complex line bundle L on a compact Riemann surface it can b e directly calculated tha t ∂ ( L ) = deg( L ) η X ∈ H 2 ( X , Z X ) where deg ( L ) is t he degree of L , whic h is p ositiv e for ample line bundles. Hence w e get the relation c 1 ( L ) = deg( L ) η X relating first Chern class and degree. W e ha v e a comm uta tiv e diagra m with exact rows 0 → Z X → C → C ∗ → 0 ↓ k ↓ 0 → C X → C → Z 1 → 0 where Z 1 is the sheaf of closed 1-forms, a nd C ∗ X → Z 1 is defined b y f 7→ d f / 2 π if . This give s a comm uta t iv e diagram H 1 ( X , C ∗ ) ∂ → H 2 ( X , Z X ) ↓ ↓ H 1 ( X , Z 1 ) ∂ → H 2 ( X , C X ) in whic h the b ot t o m ro w is an isomorphism. Th us, any comple x line bundle L defined b y t r a nsition f unctions ( g a,b ) ∈ H 1 ( X , C ∗ ) defines a class c ( L ) = ( dg a,b / 2 π ig a,b ) ∈ 6 H 1 ( X , Z 1 ) whose image ∂ ( c ( L )) ∈ H 2 ( X , C X ) is c 1 ( L ). W e ha ve connec ting isomor- phisms H 2 dR ( X ) = H 0 ( X , Z 2 ) im H 0 ( X , C 1 ) ∂ → H 1 ( X , Z 1 ) ∂ → H 2 ( X , C X ) whose comp osite is the dimension shifting isomorphism d 2 : H 2 dR ( X ) → H 2 ( X , C X ). A simple calculation (see for example [Na]) sho ws that if X is a compact Riemann surface and if α X ∈ H 2 dR ( X ) is the p ositiv e inte g r al generator (means R X α X = 1), then c ( L ) = − deg ( L ) ∂ ( α X ) . This is consisten t with the sign lemma, b y whic h d 2 : H 2 dR ( X ) → H 2 ( X , C X ) is ( − 1)-times the canonical isomorphism, so that ∂ ◦ ∂ ( α X ) = − η X , and so ∂ ( c ( L )) = deg( L ) η X . Not ta king the sign ( − 1) in to accoun t will lead to a parado x at this p oin t. An y attempt to resolv e it b y trying to define the firs t Chern class as − ∂ ( L ) ∈ H 2 ( X , Z X ) will in turn b e contradicted b y a direct calculation of ∂ : H 1 ( X , C ∗ ) → H 2 ( X , Z X ). The author of t his note a ctually go t into t his contradiction, whic h led to this w o rk whic h in particular resolv es it. References [G-H] Griffiths, P . and Harris, J. : A lgebr ai c Ge ometry Wiley-In terscience Pub., 1978. [Na] Nara simhan, M.S. ‘V ector bundles o n compact Riemann surfaces’, in Complex analysis and its applic ations V ol. 3, IAEA Vienna, 197 6. School of Mathematics T ata Institute of F undamental Research Homi Bhabha Road Mum ba i 400 005, India e-mail: nitsure@math.tifr.res.in 15 June 2007 7