A category of kernels for equivariant factorizations and its implications for Hodge theory

A CA TEGOR Y OF KERNELS F OR EQUIV ARIANT F A CTORI ZA TION S AND ITS IMPLICA TI ONS FOR H ODGE TH EOR Y MA TTHEW BALLARD, D A VID F A VERO, AND LUDMIL KA TZ ARKO V Abstract. W e provide a factorization mo del for the c o ntin uous in terna l Hom, in the ho mo- topy ca tegory of k -linear dg -categor ies, b et ween dg-ca tegories of equiv ariant factor izations. This motiv ates a notion, simila r to that o f Kuznetsov, which we call the ex tended Hochsc hild cohomolog y a lgebra o f the category of equiv ariant factorizations . In some cas e s of g eometric int er est, extended Ho chschild cohomolo gy co nt a ins Ho chschild coho mology a s a subalg ebra and Hochschild ho mo logy as a homogeneous comp onent. W e use our factorization mo del for the in ter nal Hom to ca lculate the extended Hochschild coho mology for equiv aria nt fac- torizations o n affine space. Combining the computation of extended Hochsc hild cohomology with the Ho chschild- Kostant-Rosenberg isomorphism and a theorem of Orlov reco vers and extends Griffiths’ classical des cription o f the primitive cohomolo g y of a smo oth, complex pro jective h yp ersur - face in terms of homogeneous pieces of the Jacobia n a lgebra. In the pro cess, the primitive cohomolog y is ident ified with the fixed subspace of the cohomologica l endomorphism asso- ciated to a n interesting endofunctor of the b ounded derived category of co he r ent sheaves on the hypersurface . W e also demonstrate how to under stand the whole Jaco bian alge bra as morphisms b etw een k erne ls of endofunctors of the derived categor y . Finally , we present a b o otstrap metho d for pr o ducing algebr a ic cyc les in categ ories of equiv ariant factoriza tions. As pro of of concept, we show how this repr ov es the Ho dge con- jecture for all self-pr o ducts of a particular K3 surface c losely rela ted to the F ermat cubic fourfold. 1. Introduction The sub ject of matrix factorizations has, in recen t y ears, found itself a t the crossroads b et w een comm utativ e algebra, homological algebra, theoretical phy sics, and algebraic ge- ometry . One of the deepest manifestations of this junction is D. Orlo v’s σ - mo del/Landau- Ginzburg corresp ondence [Orl09] whic h in timately links pro jectiv e v arieties to equiv ariant factorization categories. With Orlov’s w ork as inspiration, this pap er pro vides a thorough in v estigation of equiv ariant factorizations in bro ad generality . The cen tral tec hnical result is a factorization mo del for B. T¨ oen’s in ternal Hom dg- cat ego ry [T o¨ e07] b et w een t hese dg- categories. The no v elty lies in the range of applications, including tho se to classical problems in algebraic geometry and Ho dge theory . In this article, w e w ill e xamine some o f the more immediate conse quences of the main result, suc h as some sp ecial cases o f the Ho dge conjecture a nd a new pro of of Griffith’s clas- sical result [Gri69] r elating the Do lb eault cohomolo g y o f a complex pro jectiv e h yp ersurface to the Jacobian algebra of its defining p olynomial. In the sequel to this article [BFK13], w e will construct categorical cov erings, calculate Rouquier dimension, in v estigate Orlov sp ectra, and connect our work to Homological Mirror Symmetry , all a s applications o f the cen tral theorem presen ted here. No w, b efore w e delv e in to detailed statemen ts, let us t r y to pro vide some con text for the results. 1 2 BALLARD, F A VERO, A N D KA TZAR KO V P erhaps t he simplest class of singular rings is that of hypersurface rings, i.e. rings whic h are the quotient of a regular ring b y a single elemen t (also called h yp ersurface singulari- ties). In the foundational pa p er, [Eis80], D. Eisen bud introduced matrix factorizations and demonstrated their precise relationship with maximal Cohen-Macaulay (MCM) mo dules ov er a hypersurface singularit y . Building o n Eisen bud’s description, R.-O . Buc h we itz in tro duced the proper categorical framew ork in [Buc86]. Buc h w eitz sho w ed that the homotop y c at - egory of matrix factorizations, the stable category of MCM modules ov er the asso ciated h yp ersurface ring , and the stable deriv ed category of the asso ciated hy p ersurface ring are all equiv alen t descriptions of the same triangulated category . Outside of comm utativ e a lgebra, interes t in matrix fa cto r izat io ns grew due to intimate connections with ph ysics; phys ical theories with p oten tials, called Landau-Ginzburg mo dels, are ubiquitous. Building on the large b o dy o f w ork on Landau-Ginzburg mo dels without b oundary , (see, for example, C. V afa’s computation, [V af91], of the closed string top o logical sector as the Jacobia n algebra of the p oten tia l) , M. Kon tsevic h prop osed matrix factoriza- tions as the appropriate category of D - branes for the to p ological B-mo del in the presence of a p otential [KL03a, Section 7.1]. In phys ics, A. Kapustin and Y. Li confirmed K on tsevic h’s prediction and gav e a mathe- matically conjectural description of the Chern c haracter map and the pairing on Ho ch sch ild homology fo r the category of matrix factorizations, [KL03a] [KL03b]. In mathematics, sev eral foundational pap ers b y Orlo v so on follow ed: [Orl04, Orl06, Orl09 ]. In particular, Orlo v g av e a global mo del for the stable b ounded derive d category o f a No e- therian sc heme possessing enough lo cally-free shea ve s. He called this the category of sin- gularities. Orlov also prov ed that the category of B-bra nes for an LG-mo del is equiv alent to the copro duct o f the categories of singularities of the fib ers, and, to reiterate, the main inspiration f o r this w or k w as the tigh t relationship he provided b etw een the b ounded deriv ed categories of coheren t shea ves on a pro jectiv e hy p ersurface and the equiv ariant factorization category of affine space together with the defining p olynomial. In another early dev elopmen t, signaling the fertility of the marr ia ge of ph ysical inspiration to matr ix fa cto r izations, M. Khov anov and L. Rozansky categorified the HOMFL Y p olyno- mial using matrix factorizations, [KR08a, KR08b]. In the pro cess, Kho v ano v and Rozansky also in tro duced sev eral imp ortan t ide as to the study of matrix factorizatio ns. Central to their w ork is a construction whic h asso ciates functors b etw een categories of matrix fa cto r - izations to matrix factorizations of the difference p oten tial. A strong, and precise, analog y exists b etw een Khov a no v and Rozansky’s construction and the calculus o f k ernels of in tegral transforms b etw een deriv ed categor ies of coheren t sheav es on algebraic v arieties. Through this analogy , factorizat io ns o f the difference p o ten tial can b e view ed as categorified corre- sp ondences f o r factorization categor ies. Numerous further ar t icles hav e elucidated the connection b et w een factorization categories and Ho dge theory . In [KKP08], the third author, K on tsevic h, and T. P antev giv e explicit constructions describing the Ho dge theory a sso ciated to the catego r y of matrix f a ctoriza- tions. F or t he case of an isolated lo cal h yp ersurface singularity , T. Dyc k erhoff prov ed, in [Dyc11], that the category of ke rnels introduced in [K R08a] is the correct one from t he p er- sp ectiv e of [T o ¨ e07]. More precisely , the dg-category of k ernels from [KR08a] a nd [Dyc11] is quasi-equiv alent to the inte rnal homomorphism dg-category in the homotopy category of dg-categories. Using this result, Dyc k erhoff rigor o usly es tablished Kapustin and Li’s de - scription of the Ho c hsc hild homology of the dg-category of matrix factorizations. D. Murfet KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 3 ga ve a mat hematical deriv atio n of the Kapustin-Li pairing [Mur09] whic h subsequen tly was expanded in [D M12]. In additio n, E. Segal gav e a description of the Kapustin-Li pac k age in [Seg09]. F ollowin g this lead, sev eral groups of authors extended Dyc k erhoff ’s results. F or a finite group, G , A. Polishc h uk and A. V aintrob ga v e a description of the Chern c hara cter, the bulk-b oundary map, and pro v ed an analog of Hirzebruch - Riemann-Ro ch in the case o f the G -equiv ariant catego r y o f singularities of a lo cal isolated hypersurface ring [PV12]. Orlo v defined a cat ego ry of mat r ix factorizations fo r a non-a ffine sc heme with a global regular function and pro ved it is equiv alen t to the category o f singularities of the asso ciated hyper- surface in the case when the am bient sc heme is regular [Orl12]. K. Lin a nd D. Pomerle ano also tac kled non-affine matrix factorizatio ns [LP11]. Con temp oraneously , A. Preygel, using gen uinely new ideas ro ot ed in deriv ed algebraic geometry , handled matrix fa cto r izations o n deriv ed sche mes, [Pre11]. Extending Dyck erhoff ’s results from the case of a lo cal hypersurface to a global h yp er- surface, i.e. using a section of a line bundle instead of a glo bal regular function, w as also vigorously pursued. The first suc h results w ere obta ined b y A. C˘ a ld˘ araru a nd J. T u. in [CT10]. C˘ ald˘ ara r u and T u defined a c urved A ∞ -algebra asso ciat ed t o a h yp ersurface in pro jectiv e space and computed the Borel-Mo or e homology of the curv ed a lgebra. F urther- more, in [T u10], T u clarified the r elat io nship b etw een Borel-Mo ore homology and Ho c hsc hild homology . In [PV10 ], Polis hch uk and V ain trob gav e a definition of a category of matrix fac- torizations on a stack satisfying appropriat e conditions and prov ed that their category of matrix factorizatio ns coincided with the catego ry of singularities of the underlying h yp er- surface. In [P os09 ], L. Pos itselski, using his w ork on co- and con tra-derived cat ego ries of curv ed dg -mo dules o ve r a curv ed dg- a lgebra, defined an enlargemen t of the categor y o f ma- trix factorizations in the case of a section of line bundle. He also defined in [P os11], a relat ive singularit y category for an em b edding of Y in X and pro v ed that the relative singularity category of t he h yp ersurface defined by a section of a line bundle coincides with his category of factorizations ev en if the am bient sch eme is not regular. Con tin uing in this direction, this pap er completely handles the case of a global h yp ersur- face. Moreov er, it also allo ws for an action of a n affine algebraic g r oup. Th us, in particular, it handles factorizations on a ny smo oth algebraic stac k with enough lo cally-free shea v es [T ot04]. The first main result of our pap er pro vides an inte rnal description o f the functor category b etw een categories of eq uiv arian t factorizations i.e. as another category of equi- v ariant fa ctorizations. T o state it appropria tely , let us recall some w ork of T¨ oen, with the simplifying assumption that k is a field. In [T o¨ e07], T¨ oen studies the structure of the lo calization of the categor y of dg-categories o v er a field, dg-cat k , by the class o f quasi-equiv alences. T¨ oen calls this lo calization, the homotop y category o f dg-categories, and denotes it as Ho(dg-cat k ). F or t w o dg-categories, C and D , T¨ oen t hen defines a dg-category , denoted R Hom( C , D ), whic h is the in ternal Ho m dg-category in Ho(dg-cat k ). T¨ oen defines R Hom c ( C , D ) to b e the full dg-subcat ego ry of R Hom( C , D ) whose o b jects induce copro duct-preserving functors b et w een the homotop y categories. He calls suc h functors contin uous. The category , R Hom c ( C , D ), lies at the heart o f T¨ oen’s deriv ed Morita result of [T o¨ e07]. Indeed, it seems to b e a robust and general prescription fo r pic king out the “g eometrically correct” functor categor y fo r familiar dg/tria ngulated categories. Let us giv e atten tion to an imp ortant example: de rived categories of shea v es o n v a rieties, X and Y . 4 BALLARD, F A VERO, A N D KA TZAR KO V An ob ject, K ∈ D (Qcoh X × Y ), giv es a copro duct-preserving, exact functor, R q ∗ ( K L ⊗ O X × Y L p ∗ • ) : D(Qcoh X ) → D(Qcoh Y ) , where p : X × Y → X and q : X × Y → Y are the pro jections. Ho w eve r, it is w ell-kno wn that the category o f exact, copro duct-preserving f unctors fr o m D(Qcoh X ) to D( Q coh Y ) is not equiv alen t to D(Qcoh X × Y ), see [CS10] for an example. P a ssage fro m the category of c hain complexes to t r iangulated categories is to o brutal, we need to remem b er a bit more information. In [T o¨ e07], T o¨ en pro ves that, in Ho(dg-cat k ), there is an isomorphism, R Hom c ( Inj ( X ) , Inj ( Y )) ∼ = Inj ( X × Y ) where Inj ( Z ) is a particular dg-enhancemen t of D(Qcoh Z ). Similar work for v arieties a nd other higher ob jects w as carr ied out in [BFN10]. Hence, the f a ilure o f a Morita-type result for deriv ed categor ies is remedied b y lifting to dg-categories and w orking in Ho(dg-cat k ). This mak es R Hom c the correct functor category to study . Ho w ev er, in general, if tw o dg-categories, C and D , come fr o m some geometric framew ork, suc h as derived categories o f shea v es, it is not clear a priori from T¨ oen’s definition of the internal Hom how R Hom c ( C , D ) reflects the underlying geometry . One m ust iden tify R Hom c ( C , D ) geometrically . This is the first go a l of the pap er. Let us define our dg-categories of matrix factorizatio ns. Let k b e an algebraically closed field of c haracteristic zero and let G and H b e affine algebraic groups. Let X and Y b e smo oth v arieties. Assume that G acts on X a nd H acts on Y . Let L b e an in vertible G -equiv ariant sheaf on X a nd le t w ∈ H 0 ( X , L ) G . Similarly , let L ′ b e an in ve rtible H - equiv arian t sheaf on X and let v ∈ H 0 ( Y , L ′ ) H . Let I nj ( X , G, w ) and Inj ( Y , H , v ) b e the dg- categories of equiv ar ia n t factorizatio ns with injectiv e comp onen ts. Let U( L ) b e the geometric v ector bundle corresp onding to L with the zero sec tio n remo v ed and denote the regular function induced b y w o n U( L ) b y f w . Similarly , let U( L ′ ) b e the geometric vec tor bundle corresp onding to L ′ with the zero section r emov ed and denote the regular function induced b y v on U( L ′ ) b y f v . Equip U( L ) × U( L ′ ) with the natural G × H -action and allow G m to scale the fib ers of U( L ) × U( L ′ ) diagonally . Let ( − f w ) ⊞ f v := − f w ⊗ k 1 + 1 ⊗ k f v . The follow ing is the main result of Section 5. Theorem 1.1. In the homotopy c ate gory of k -line ar dg-c ate gories, ther e is an e quivalenc e, R Hom c ( Inj ( X , G, w ) , Inj ( Y , H , v )) ∼ = Inj (U( L ) × U( L ′ ) , G × H × G m , ( − f w ) ⊞ f v ) . This result follow s work in the ungraded case b y D yc k erhoff, [Dyc11]. Our metho ds in pro ving Theorem 1.1 are in line with [LP11] as w e rely on generation statemen ts for singularit y categories and use P ositselski’s absolute deriv ed category , [P os09 , Pos11] as the mo del for our “large” triangulated category whose compact ob jects are (up to summands) coheren t factorizations. In con temp o raneous a nd indep enden t w ork, [PV11], Polis hch uk and V ain trob pro ve The- orem 1.1 in the case X and Y are affine, G a nd H are finite extensions of G m , and b o t h w and v hav e an isolated critical lo cus. P olishc huk and V aintrob also giv e a computation o f the Ho chs child homology of the category of equiv ariant matrix factorizatio ns in this case. Despite the o ve rla p in these foundational results, their inspiration and fo cus are ultimately distinct from the w ork here. They pro vide a purely algebraic v ersion of F JR W-theory [FJR07] KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 5 b y w ay of ma t rix factorizations. The authors find this to b e a b eautiful illustration of the range and mag nitude of this sub ject of study . One significant adv a n tage of a geometric description of the in ternal Hom category is greater computational p ow er. As defined b y T¨ o en, Ho chsc hild cohomology of a dg-category is the cohomology of t he dg-algebra o f endomorphisms o f the iden tity , view ed as an ob ject of the in ternal Hom dg-category in Ho(dg-cat k ). In the setting of G -equiv ariant factorizations, there is a natural extension, whic h w e call extended Ho chs child cohomolog y . F or a dg- cat ego ry , C , w e denote its homotopy catego r y b y [ C ]. Let b G b e the group of c haracters o f G . The extended Ho c hsc hild cohomology is defined as HH ( χ,t ) e ( X , G, w ) := M χ ∈ b G,t ∈ Z Hom [ R Hom c ( Inj ( X,G,w ) , Inj ( X,G, w ))] (Id , ( χ )[ t ]) . Under certain assumptions o n X , G , and w , the Ho chsc hild homology of is a homogeneous comp onen t of HH • e ( X , G, w ). W e use Theorem 1.1 to compute the extended Ho chs child cohomology of ( X , G, w ) when X is affine, G is a finite extension of G m , and w is semi-homogeneous regular function of non-torsion degree. The computation is along the lines o f [PV12]. Theorem 1.2. L et G act line arly on A n and let w ∈ Γ( A n , O A n ( χ )) G . A ssume that the kernel of χ , K χ , is fi nite and χ : G → G m is surje c tive. Assume that the s ingular lo cus of the zer o set, Z ( − w ) ⊞ w , is c ontaine d in the pr o duct of the zer o sets, Z w × Z w . Then, HH ( ρ,t ) e ( A n , G, w ) ∼ =     M g ∈ K χ ,l ≥ 0 t − rk W g =2 u H 2 l (d w g )( ρ − κ g + ( u − l ) χ ) ⊕ M g ∈ K χ ,l ≥ 0 t − rk W g =2 u +1 H 2 l +1 (d w g )( ρ − κ g + ( u − l ) χ )     G wher e H • (d w g ) denotes the Koszul c ohomol o gy of the Jac obian id e al of w g := w | ( A n ) g , W g is the c onormal she af of ( A n ) g , and κ g is the char acter of G c orr e sp ondi ng to Λ rk W g W g . If, additional ly, we assume the supp ort of the Jac obian ide al (d w ) is { 0 } , then w e have HH ( ρ,t ) e ( A n , G, w ) ∼ =     M g ∈ K χ t − rk W g =2 u Jac( w g )( ρ − κ g + uχ ) ⊕ M g ∈ K χ t − rk W g =2 u +1 Jac( w g )( ρ − κ g + uχ )     G . wher e Jac( w ) den otes the Jac obian algebr a of w . After building these foundations, w e apply our res ults to Ho dge theory . The primary observ ation is tha t Orlo v’s relationship b et w een graded categor ies of singularities and deriv ed categories of coheren t shea v es [Orl09] has some v ery in teresting geometric consequen ces when com bined with Theorem 1.1. Let C b e a saturated dg-category ov er k . The Ho c hsc hild homo lo gy of C , HH ∗ ( C ), is an in v ar ia n t that play s an imp orta nt ro le in the noncomm utative Ho dge theory of C , [KKP08]. When X is a smo oth prop er algebraic v ariet y ov er k , one can let C = Inj coh ( X ) b e the dg- category of b ounded b elo w complexes of injectiv e O X -mo dules with b ounded and coheren t 6 BALLARD, F A VERO, A N D KA TZAR KO V cohomology . Th ere is a Ho c hsc hild-Kostant-Rosen b erg is omorphism, see [HKR6 2, Swa96, Kon03] φ HKR : HH i ( Inj coh ( X )) = : HH i ( X ) → M q − p = i H p ( X , Ω q X ) . The HKR isomor phism allo ws one to study questions o f Ho dge theory b y means of cate- gory theory . In Section 6.1 , w e com bine Orlov’s theorem, the HKR isomorphism, and the computations of Theorem 1.2 to repro duce a classic result of Griffiths [Gri69] describing t he primitiv e cohomology o f a pro jectiv e hypersurface. Theorem 1.3. L et Z b e a smo oth, c om plex pr oje c tive hyp e rsurfac e define d by a homo gene ous p olynom i a l w ∈ C [ x 1 , . . . , x n ] of d e gr e e d . F or e ach 0 ≤ p ≤ n/ 2 − 1 , Orlov’s the or em and the HKR i s o morphism induc e an isomorp h ism, H p,n − 2 − p prim ( Z ) ∼ = Jac( w ) d ( n − 1 − p ) − n . In the pro cess, w e sho w that the primitive cohomology of Z is exactly the fixed lo cus of the action of the endofunctor { 1 } := L O Z ◦ T O Z (1) : D b (coh Z ) → D b (coh Z ) E 7→ Cone  ⊕ i ∈ Z Hom D b (coh Z ) ( O Z , E ( i )[ i ]) ⊗ k O Z [ − i ] ev → E (1)  on Ho c hsc hild homolog y , HH • ( Z ). F urthermore, when Z is Calabi-Y a u, for the ke rnel, K ∈ D b (coh Z × Z ), of { 1 } , we ha ve an injectiv e homomorphism of g r a ded rings, Jac( w ) → M i ≥ 0 Hom D b (coh Z × Z ) (∆ ∗ O Z , K ∗ i ) whose appropriate gra ded pieces are t he isomorphisms of Theorem 1.3, at least after ap- plication of the HKR is omo r phism. Thus , w e hav e a categorical realization o f Griffiths’ fundamen tal result that sees the en tire Jacobian alg ebra. F ollowin g this categorical path further, w e study algebraic cycles b y understanding the image of the Chern c haracter map in Ho chsc hild homology . In Section 6.2, w e prov e a result that allo ws one to b o o t strap, via group homomor phisms, the Ho dge conjecture for categories of equiv arian t matrix factorizations. W e give one application of this pro cedure to the Ho dge conjecture for v arieties: w e apply the results of Orlov in [Orl09] a nd w or k of Kuznets ov [Kuz09, Kuz10] to reprov e t he Ho dge conjecture for n -fold pro ducts a certain K 3 surface asso ciated to a F ermat cubic fourfold. This case of the Ho dge conjecture was originally handled in [R M08]. W e thank P . Stellari for p ointing out the reference, [RM08]. Ac knowled gments: The a uthors are greatly appreciativ e of the v aluable insigh t gained from con v ersations and corresp ondence with Mohammed Ab ouzaid, Denis Auroux, Andrei C˘ ald˘ ara r u, Drag os Deliu, Colin Diemer, T obias D yc k erhoff, Manfred Herbst, M. Umut Isik, Gabriel Kerr, Maxim Kontse vic h, Alexander Kuznetso v, Jacob Lewis, Dmitri Orlov, Pranav P andit, T ony P ante v, Anato ly Preygel, Victor Przy ja lk o wski, Ed Segal, P aul Seid el, and P aolo Stellari and would lik e to thank them for their time and pat ience. F urthermore, the authors are deeply gra t eful to Alexander P olishc h uk and Ark ady V a introb for providing us with a preliminary v ersion o f their work [PV11] and for a llowing us time to prepare the original v ersion of this pap er in o rder to sync hronize p osting of the articles due to the o ve rla p. The first named author was funded by NSF D MS 06 36606 R TG, NSF DMS 08382 10 R TG, and NSF DMS 0854 977 FR G. The second and third named autho rs w ere funded by NSF KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 7 DMS 0 8 54977 FR G, NSF DMS 06 0 0800, NSF D MS 0652633 F R G, NSF DMS 0854 9 77, NSF DMS 090 1330, FWF P 2 4572 N25, by FWF P20778 a nd b y an ERC Gran t. 2. Back gro und on equiv ariant shea ves F or the en tiret y of this pap er, k will denote an algebraically-closed field of characteris tic zero. In this section, w e recall some facts ab out quasi-coheren t equiv a rian t shea ve s on separated, sc hemes/algebraic spaces of finite type f o llo wing [MFK94]. A nice reference for basic f a cts, with a full set of details, is [Blu07, Chapter 3]. Th e results here will b e used in later sections. Let X b e a separated sc heme of finite type ov er k and G b e an affine alg ebraic group o v er k acting on X . Denote by m : G × G → G , i : G → G , and e : Sp ec k → G , the group actio n, the inv ersion and the iden tit y , resp ectiv ely . Let σ : G × X → X denote the G -actio n and π : G × X → X the pro jection o nto X . Definition 2.1. A quasi-coheren t G -equivariant sheaf on X is a quasi-coheren t sheaf, F , on X together with an isomorphism, θ : σ ∗ F → π ∗ F , satisfying, ((1 G × σ ) ◦ ( τ × 1 X )) ∗ θ ◦ (1 G × π ) ∗ θ = ( m × 1 X ) ∗ θ , on G × G × X where τ : G × G × X → G × G × X switc hes the tw o factors of G , and, s ∗ θ = 1 F , where s : X → G × X is induced by e . If F is a coheren t, resp ectiv ely lo cally-free, sheaf on X , then w e sa y the equiv ariant sheaf, ( F , θ ), is coheren t, resp ectiv ely lo cally-free. The isomorphism, θ , is called the equivariant structure . W e often refer to a quasi-coheren t G -equiv ariant sheaf simply a s E . If the con text is am biguous, w e denote the equiv aria nt structure of E b y θ E . Remark 2.2. F or eac h closed p oin t g ∈ G , w e get an automorphism σ g := σ ( g , • ) : X → X . These satisfy σ g 1 ◦ σ g 2 = σ g 1 g 2 . If E is a quasi-coheren t G -equ iv arian t sheaf, then θ g iv es isomorphisms θ g := θ | { g }× X : σ ∗ g E → E . for each g ∈ G with θ g 2 g 1 = θ g 1 ◦ σ ∗ g 1 θ g 2 . Chec king a subsheaf F of E inherits t he equiv ariant structure, i.e. θ ( σ ∗ F ) ⊆ π ∗ F , b oils do wn to chec king t ha t it is preserv ed b y eac h θ g . Definition 2.3. Let Qcoh G X b e the Ab elian cat ego ry of quasi-coheren t G -equiv arian t shea v es on X . Analogously , w e let coh G X b e the Ab elian category of coheren t G -equiv ar ia n t shea v es. Definition 2.4. Let E and F b e quasi-coheren t G -equiv ariant shea ve s on X . The tenso r p ro duct of E and F is the quasi-coheren t sheaf E ⊗ O X F together with the equiv arian t structure, θ E ⊗ O G × X θ F . The sheaf of homomorphisms from E to F is the quasi-coheren t sheaf H om X ( E , F ) together with the equiv a rian t structure θ F ◦ ( • ) ◦ ( θ E ) − 1 . Definition 2.5. Let X and Y b e separated, finite-ty p e sch emes equipped with actions, σ X and σ Y , of G a nd pro j ections π X , π Y . A morphism of sc hemes, f : X → Y , is G - equiv a riant if the diagr a m 8 BALLARD, F A VERO, A N D KA TZAR KO V G × X G × Y X Y 1 × f σ X σ Y f comm utes. Giv en such an f , w e get an a djoin t pair o f functors, f ∗ : Qcoh G Y → Qcoh G X ( F , θ ) 7→ ( f ∗ F , (1 × f ) ∗ θ ) , f ∗ : Qcoh G X → Qcoh G Y ( F , θ ) 7→ ( f ∗ F , (1 × f ) ∗ θ ) . Remark 2.6. The definition of f ∗ and f ∗ are sensible (as interpreted through natural iso- morphisms) as σ X , π X are flat a nd the squares G × X G × Y X Y 1 × f σ X σ Y f G × X G × Y X Y 1 × f π X π Y f are Cartesian. Definition 2.7. Give n an affine algebraic group, G , w e let b G := Hom alg grp ( G, G m ) . The finitely-generated Ab elian group, b G , is called the group of characte rs of G . As b G is Ab elian, we shall use additive notation for group structure on b G . F or a character, χ ∈ b G , w e let K χ denote the k ernel of χ . W e also get an auto-equiv alence ( χ ) : Qcoh G X → Qcoh G X E 7→ E ⊗ O X p ∗ L χ where p : X → Sp ec k is the structure map and L χ is the ob ject of Qcoh G (Sp ec k ) corre- sp onding to χ . Lemma 2.8. L et G act on X and Y . Assume we ha v e an e quivariant morphism, f : X → Y . F o r E ∈ Qcoh G Y lo c al ly-fr e e and F ∈ Qcoh G X , ther e is a natur al isomorphism f ∗ F ⊗ O X E ∼ = f ∗ ( F ⊗ O X f ∗ E ) . Pr o of. This follows f r om the usual pro jection formula a pplied b oth to E and θ .  W e will a lso need a more general pull-bac k functor. Definition 2.9. Let H and G be affine algebraic groups and let X and Y be se para t ed sc hemes of finite t yp e equipp ed with a ctio ns, σ H,X : H × X → X and σ G,Y : G × Y → Y . Let ψ : H → G b e a homomorphism of alg ebraic groups. A ψ -equiv ariant morphism , o r a mo rphism equiva riant with resp ect to ψ , is a morphism of sc hemes, f : X → Y , suc h that diagram KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 9 H × X G × Y X Y ψ × f σ H,X σ G,Y f comm utes. Giv en a ψ - equiv a rian t morphism, f , w e can define the pull-bac k functor, f ∗ : Qcoh G Y → Qcoh H X ( F , θ ) 7→ ( f ∗ F , ( ψ × f ) ∗ θ ) . In the case tha t X = Y , w e denote this functor by Res ψ . If , in addition, ψ : H → G is a closed subgroup, the pull-back is called t he restriction f unctor a nd denoted by Res G H . Remark 2.10. While there is a bit of notational conflict here, we will alwa ys try to eliminate this confusion with exp osition. Definition 2.11. Let G and H b e affine a lg ebraic gr oups, X and Y separated sc hemes of finite t yp e equipped with actions G × X → X and H × Y → Y . Let π 1 : X × Y → X and π 2 : X × Y → Y b e the tw o pro j ections. The pro jection, π 1 , is equiv arian t with resp ect to the pro jection G × H → G while π 2 is equiv ar ia n t with respect to the pro j ection G × H → H . Let E ∈ Qcoh G X and F ∈ Qcoh H Y . The exterior p ro duct o f E and F is the quasi-coherent G × H -equiv aria n t sheaf E ⊠ F := π ∗ 1 E ⊗ O X × Y π ∗ 2 F . Let H b e a closed subgroup of G and let σ : H × X → X b e an action of G on X . The pro duct, G × X , carries an action of H defined b y τ : H × G × X → G × X ( h, g , x ) 7→ ( m ( g , i ( h )) , σ ( h, x )) . Lemma 2.12. The fp p f quotient of G × X by H exists as a sep ar a te d algebr aic sp ac e of finite typ e o ver k . It i s denote d by G H × X . Pr o of. By Artin’s Theorem, see [Ana73, Theorem 3.1.1], G H × X exists as a separated algebraic space of finite ty p e.  Let ι : X → G H × X be the inclusion, x 7→ ( e, x ). This is equiv arian t with resp ect to the inclusion of H in G . Lemma 2.13. The pul l-b ack functor, ι ∗ : Qcoh G ( G H × X ) → Qcoh H X , is an e q uiva lenc e. Mor e over, it in duc es an e quiva lenc e b etwe en the sub c ate go ries o f c oher ent e quivari a nt she aves and an e quivalenc e b etwe en the sub c ate gories of lo c al ly-fr e e e quivaria n t sh e aves. Pr o of. This is an immediate consequence of faithfully-flat descen t, see [Tho97, Lemma 1.3].  Definition 2.14. Let H b e a close d subgroup of G and assume w e hav e an action, σ : G × X → X . The a ctio n, σ , descen ds to a G - equiv a rian t morphism, α : G H × X → X . The induction functor , Ind G H : Qcoh H X → Qcoh G X 10 BALLARD, F A VERO, A N D KA TZAR KO V is defined t o b e t he comp osition, α ∗ ◦ ( ι ∗ ) − 1 . Lemma 2.15. L et H b e a close d sub gr oup of G and ass ume w e have an a ction, σ : G × X → X . The functor, Ind G H , is right adjoint to the r estriction, Res G H , and Res G H ∼ = ι ∗ ◦ α ∗ . Pr o of. Note that t he iden tity map on X can b e f actored as X ι → G H × X α → X . Th us, R es G H = ι ∗ ◦ α ∗ whic h is left adjoin t to α ∗ ◦ ( ι ∗ ) − 1 .  Lemma 2.16. L et H b e a close d sub gr oup of G and let X b e a sep ar ate d scheme of finite typ e e quipp e d with an action, σ : G × X → X . L et p : G/H × X → X b e the pr oje ction onto X . a) The H -cr osse d p r o duct, G H × X , is a scheme, G -e quivariantly iso morphic to G/H × X , with the diago n al G -action. b) The functor, Res G H , is exa c t. c) F or E ∈ Qcoh H X and F ∈ Qcoh G X lo c al ly-fr e e, ther e is the fol lowi n g pr oje ction formula, i.e. a natur al isomorphism, Ind G H ( E ⊗ O X Res G H F ) ∼ = Ind G H E ⊗ O X F . d) Ther e is a na tur al iso morphism Ind G H ◦ Res G H ∼ = p ∗ p ∗ of functors. e) If we, additional ly, assume that G/H is affine, then Ind G H is exact. I n p articular, if H is n o rmal, then Ind G H is exact. Pr o of. F or a), as w e are ov er k , the quotien t of G b y H , as a fppf sheaf, exists as a quasi- pro jectiv e sc heme. By [W at79, Theorem 16.1], one can find a G -represen tatio n, V , with a subspace, W , whose stabilizer is exactly H . Let n = dim W . P assing to the Gra ssmannian, G ( n, V ), H is exhibited as the stabilizer of a closed p o int and by [DeGa70, I I I, § 3, Prop o sition 5.2] is represen table by sc heme with a lo cally-closed em b edding into G ( n, V ) . No w, the H - crossed pro duct, G H × X , is G -equiv ariantly isomorphic to the pro duct, G/H × X , with the diagonal G -a ctio n, via the isomorphism Φ : G H × X → G/H × X ( g , x ) 7→ ( g H , g x ) . F or α : G H × X → X , w e ha ve α = p ◦ Φ. F or b), recall that Res G H ∼ = ι ∗ ◦ α ∗ . Th en, Res G H ∼ = ι ∗ ◦ Φ ∗ ◦ p ∗ . Both ι ∗ and Φ ∗ are equiv alences so b oth are exact while p ∗ is exact as G/H is flat ov er k . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 11 F or c), let E ∈ Qcoh H X and F ∈ Qcoh G X with F lo cally-free. Since ι ∗ is an equiv alence, w e can write E = ι ∗ E ′ for E ′ ∈ Qcoh G G H × X , Ind G H ( E ⊗ O X Res G H F ) ∼ = α ∗ ( ι ∗ ) − 1 ( E ⊗ O X ι ∗ α ∗ F ) ∼ = α ∗ ( ι ∗ ) − 1 ( ι ∗ E ′ ⊗ O X ι ∗ α ∗ F ) ∼ = α ∗ ( E ′ ⊗ O G H × X α ∗ F ) ∼ = α ∗ E ′ ⊗ O X F ∼ = Ind G H E ⊗ O X F where w e used the pro jection formula fo r α and the fa ct the ι ∗ is a monoidal functor. F or d), we ha ve isomorphisms Ind G H ◦ Res G H ∼ = α ∗ ◦ ( ι ∗ ) − 1 ◦ ι ∗ ◦ α ∗ ∼ = α ∗ ◦ α ∗ ∼ = p ∗ ◦ Φ ∗ ◦ Φ ∗ ◦ p ∗ ∼ = p ∗ ◦ p ∗ . W e used the fact tha t Φ ∗ ∼ = (Φ ∗ ) − 1 as Φ is an isomorphism. F or e), the ma p p is affine so p ∗ is exact. Consequen tly , Ind G H ∼ = α ∗ ◦ ( ι ∗ ) − 1 ∼ = p ∗ ◦ Φ ∗ ◦ ( ι ∗ ) − 1 is a comp osition of exact functors. If H is normal, then G/H is an affine algebraic group, [W at79, Theorem 16.3].  Remark 2.17. Notice that when H is not normal w e may only consider G/H as a sc heme with an action o f G and no t as an affine algebraic group. F urthermore, G/H p ossesse s a transitiv e G -action a nd, since the base field has characteristic zero, is generically smo ot h. Consequen tly , G/H is a smo oth v a riet y . Lemma 2.18. L et H b e a close d normal sub gr o up of G . Assume that G/H i s Ab elian. Then, ther e is a natur al isom o rphism Ind G H ◦ Res G H E ∼ = M χ ∈ [ G/H E ( χ ) wher e we view χ as a ch a r acter of G via the homomorphism G → G/H . Pr o of. By Lemma 2 .1 6, w e hav e an isomorphism Ind G H ◦ Res G H ∼ = p ∗ p ∗ where p : G/H × X → X is the pro jection. Th us, Ind G H ◦ Res G H E ∼ = p ∗ p ∗ E ∼ = Γ( G/H , O G/H ) ⊗ k E . Since G/H is Ab elian, Γ( G/H , O G/H ) ∼ = k [ [ G/H ] and Γ( G/H , O G/H ) ⊗ k E ∼ = M χ ∈ [ G/H E ( χ ) .  12 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 2.19. L et ψ : G → H b e a flat homom orphism of affine algebr aic gr oups. L et G act on the alge b r aic varieties Z and X and H act on the algebr a ic varieties Y and W . Assume we have a Cartesian squar e Z Y X W u ′ v ′ v u wher e u ′ and u ar e ψ -e quivariant whi l e v ′ is G -e quivariant and v is H -e quivaria n t. Assume that u is flat. Then, we hav e a natur al isomorphism of functors u ∗ ◦ v ∗ ∼ = v ′ ∗ ◦ u ′∗ : Qcoh H Y → Qcoh G X . Pr o of. F or a H - equiv arian t quasi-coheren t sheaf, ( E , θ ), w e ha v e u ∗ v ∗ E ∼ = v ′ ∗ u ′∗ E via flat base c hange. W e also hav e a Cart esian diagr a m G × Z H × Y G × X H × W ψ × u ′ 1 G × v ′ 1 H × v ψ × u and ψ × u is flat. So ( ψ × u ) ∗ (1 × v ) ∗ ∼ = (1 × v ′ ) ∗ ( ψ × u ) ∗ via flat base c hange, ag a in. Using this fact on θ , w e get an equiv arian t isomorphism b et w een u ∗ v ∗ E and v ′ ∗ u ′∗ E .  Definition 2.20. Let X b e a separated sche me of finite type ov er k . Let σ : G × X → X act on X and N b e a closed no r mal subgro up of G suc h that σ | N × X : N × X → X is the trivial action. Consider a quasi-coheren t G - equiv arian t sheaf ( E , θ ) and the restriction of θ to N θ | N × X : σ | ∗ N × X E ∼ = π ∗ E → π ∗ E . Via adjunction, w e ha v e a morphism, E u E → Γ( N , O N ) ⊗ k E 1 Γ( N , O N ) ⊗ k E − π ∗ θ | N × X → Γ( N , O N ) ⊗ k E . where u : Id → π ∗ π ∗ is the unit o f adjunction. Let E N b e the ke rnel of this total mo r phism. Then, θ preserv es E N and the pair ( E N , θ | σ ∗ E N ) is a G -equiv ariant sheaf that natura lly de- scends to a quasi-coheren t G/ N -equiv ar ia n t sheaf on X . Denote the functor b y ( • ) N : Qcoh G X → Qcoh G/ N X . W e shall o f ten, interc hang eably , view E N as a G -equiv ariant sheaf or as a G/ N -equiv ariant sheaf without additional no t a tional adornmen t. Remark 2.21. The lo cal sections of the sheaf F N o v er an o p en subset U ⊆ X are F N ( U ) = { f ∈ F ( U ) | θ F n ( f ) = f , ∀ n ∈ N } . In fact, this description can b e tak en as a definition o f F N . Lemma 2.22. The functor ( • ) N is right adjoint to Res π for the quotient ho momorphism, π : G → G/ N . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 13 Pr o of. Let φ : Res π E → F be a G -equiv ar ia n t morphism. Since θ Res π E n = θ E π ( n ) = 1 E , N acts trivially o n Res π . As φ is G -equiv ariant, w e hav e θ F n ◦ φ = φ ◦ θ Res π E n = φ for all n ∈ N , and the image of Res π E under φ m ust lie in F N . So a ny G -equiv aria n t morphism fro m Res π E factors through F N uniquely . Of course, any G -equiv ariant morphism, Res π E → F N , induces a G - equiv arian t mor phism, Res π E → F , via comp o sition with the inclusion, F N → F . Hence , we hav e an isomorphism Hom Qcoh G X (Res π E , F ) ∼ = Hom Qcoh G X (Res π E , F N ) . As b oth Res π E and F N are N -inv ariant, an y G - equiv a rian t mor phism, Res π E → F N , uniquely descends to a G / N -equiv a rian t morphism. So, Hom Qcoh G X (Res π E , F N ) ∼ = Hom Qcoh G/ N X ( E , F N ) .  Lemma 2.23. F or any F 1 ∈ Qcoh G/ N X and F 2 ∈ Qcoh G X , ther e is a natur al isomorphism of G -e quivariant she aves (Res π F 1 ⊗ O X F 2 ) N ∼ = F 1 ⊗ O X F N 2 . Pr o of. Since Res π F 1 is completely N - in v a r ia n t, w e ha v e an isomorphism θ Res π F 1 ⊗ O X F 2 n := θ Res π F 1 n ⊗ O X θ F 2 n ∼ = 1 F 1 ⊗ θ F 2 n . for all n ∈ N . Th us, θ Res π F 1 ⊗ O X F 2 n is the iden tity on a lo cal section f 1 ⊗ f 2 if a nd only if θ F 2 n is the identit y on f 2 . Th e result follo ws f rom Remark 2.21.  Corollary 2.24. L et N b e a close d norm a l sub gr oup of an affine algebr aic sub gr oup G . Assume that G acts on X and G/ N acts on Y . L et f : X → Y b e a morphism e quivariant with r esp e ct to the q uotient homo m orphism π : G → G/ N . We have the p ul lb ack f ∗ : Qcoh G/ N Y → Qcoh G X . Consider Y with the induc e d G action to have the pushforwar d f ∗ : Qcoh G X → Qcoh G Y . The c omp o s ition, ( f ∗ ) N , is right adjoint to f ∗ . Pr o of. The functor f ∗ is the comp osition of f ∗ : Qcoh G Y → Qcoh G X and Res π . As w e ha v e adjunctions, f ∗ ⊣ f ∗ and Res N G ⊣ ( • ) N , the latter b y Lemma 2 .22, w e get the desired statemen t.  Lemma 2.25. L et G act on X and Y . L et N b e a close d normal sub gr oup which acts trivial ly on X and Y an d let f : X → Y b e a G -e quivariant morphism. F or any E ∈ Qcoh G X , ther e is a natur al isom orphism ( f ∗ E ) N ∼ = f ∗ E N . Pr o of. By definition, ( f ∗ E ) N is the kerne l of the comp osition f ∗ E → Γ ( N , O N ) ⊗ k f ∗ E 1 Γ( N , O N ) ⊗ k f ∗ E − π Y ∗ (1 G × f ) ∗ θ | N × X → Γ( N , O N ) ⊗ k f ∗ E where π Y : G × Y → Y is the pro jection. The ab ov e is f ∗  E → Γ( N , O N ) ⊗ k E 1 Γ( N , O N ) ⊗ k E − π X ∗ θ | N × X → Γ( N , O N ) ⊗ k E  where π X : G × X → X is the pro jection. This is the definition of f ∗ E N .  14 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 2.26. L et N b e a close d normal sub gr oup of G . L et G act on X and Y with N acting trivial ly on b oth X and Y . L e t f : X → Y b e a flat G -e quivariant morphis m . F or e ach E ∈ Qcoh G Y , ther e i s a na tur al is o morphism of G -e quivariant she aves f ∗ E N ∼ = ( f ∗ E ) N . Pr o of. By definition, ( f ∗ E ) N is the kerne l of the comp osition f ∗ E → Γ( N , O N ) ⊗ k f ∗ E 1 Γ( N , O N ) ⊗ k f ∗ E − π X ∗ (1 G × f ) ∗ θ | N × X → Γ( N , O N ) ⊗ k f ∗ E where π X : G × X → X is the pro j ection. Therefore, by flat base c hange this is equal to the k ernel of the comp o sition f ∗ E → Γ ( N , O N ) ⊗ k f ∗ E 1 Γ( N , O N ) ⊗ k f ∗ E − f ∗ π Y ∗ θ | N × X → Γ( N , O N ) ⊗ k f ∗ E where π Y : G × Y → Y is the pro jection. Since f is fla t, this is isomorphic to f ∗ applied to the ke rnel of the comp osition E → Γ ( N , O N ) ⊗ k E 1 Γ( N , O N ) ⊗ k E − π Y ∗ θ | N × X → Γ( N , O N ) ⊗ k E . This k ernel is t he definition of E N .  Definition 2.27. Let f : X → Y b e a morphism of separated sc hemes of finite ty p e. W e sa y that X p ossesses an f -a mple f amily of line bundles if there is a set of inv ertible shea ve s, L α , α ∈ A , suc h that for an y quasi-coheren t sheaf, E , the natural morphism M α ∈ A L α ⊗ O X f ∗ f ∗ ( L ∨ α ⊗ O X E ) → E is an epimorphism. If f : X → Sp ec k is the structure morphism, we shall simply r efer to the set L α , α ∈ A as a n ample family . Whe n X p ossess an ample family it is called diviso rial . If X and Y p o ssess an action of G , f is G -equiv ariant, and eac h L α admits an equiv ariant structure, then w e will say that the f -ample family is equiv a rian t. Remark 2.28. This is one of the m ultitude of equiv a len t definitions of an f -ample f amily [Ill71, Prop osition 2.2 .3 ]. Let us r ecall the fo llo wing r esult of Thomason. Theorem 2.29. L et X b e a normal schem e of finite typ e acte d on b y an affine algebr aic gr oup G . Assume that X is div isorial. Then, X p ossesses an e q uivariant ample family. In p articular, fo r a n y c oher ent G -e q uiva ri a nt sh e af, E , ther e exists a lo c al ly-fr e e G -e quivariant she af of finite r ank, V , and a n epimorphism, V → E . Pr o of. The conclusion is true replacing G b y the connected comp o nen t of the iden tity , G 0 , b y [Tho97 , Lemma 2 .10]. Applying [Tho97, Lemma 2.1 4] sho ws it is also true f or G .  Remark 2.30. In what follows, w e often assume that a sc heme is divisorial a nd implicitly use the theorem ab ov e to obtain an equiv ariant ample family . W e finish the section b y recalling a simple fact ab out the global dimensions of catego r ies of equiv ariant shea v es. Let G b e an a ffine alg ebraic group and let X b e a separated sc heme of finite type. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 15 Definition 2.31. Recall that the global dimension of an Ab elian category , A , is the maximal n suc h that Ext n A ( A, B ) is nonzero for some pair of ob jects, A and B , of A . Let g ldim A denote the g lo bal dimension of A . Let A b e an Ab elian category and let A b e an ob ject. The p roj ective dimension of A is p dim A := min { s | Ext s A ( A, • ) = 0 } . It is defined to b e infinite if no suc h s exists. The ob ject, A , is said to ha v e lo cally-finite p rojective dimension if for eac h A ′ ∈ A , there exists an s 0 suc h that Ext s A ( A, A ′ ) = 0 for all s ≥ s 0 . Note that t he global dimension of A is sup A p dim A. Lemma 2.32. L et E b e a quasi-c oher ent G -e quivaria n t she af. If E has lo c al ly-finite pr oj e ctive dimension as an obje ct of Q coh X , then it has lo c al ly-finite pr oje ctive dime n sion as an obje ct of Qcoh G X . Mor e ov er, we ha v e the fol l o wing in e qualities, p dim( E , θ ) ≤ p dim E + gldim Qcoh G Sp ec k gldim Qcoh G X ≤ gldim Qcoh X + gldim Qcoh G Sp ec k . In p articular, if X is smo oth, then gldim Qcoh G X is fin ite. Pr o of. Since, by definition, Hom Qcoh G X ( E , F ) = Hom Qcoh X ( E , F ) G , there is a sp ectral sequence E p,q 2 : Ext p Qcoh X ( E , F ) R q G ⇒ Ext p + q Qcoh G X ( E , F ) . Let p 0 := sup { p | Ext p Qcoh X ( E , F ) 6 = 0 } . As Ext q Qcoh G Spec k ( k , M ) = M R q G , w e see that Ext r Qcoh G X ( E , F ) v a nishes for r > gldim Qcoh X + g ldim Qcoh G Sp ec k ≥ p 0 + gldim Qcoh G Sp ec k . This give s the stated inequalit y . Cho ose a closed embedding of G ⊂ GL n . Th en, M G ∼ = (Ind GL n G M ) GL n and the functor of GL n -in v arian ts is exact. Th us, M R q G = 0 for q > dim G L n /G as Ind GL n G is the comp osition of ( ι ∗ ) − 1 and the pushforw ar d o f GL n /G → Sp ec k . Sinc e Ext s Qcoh G Spec k ( V , W ) ∼ = Ext s Qcoh G Spec k ( k , Hom k ( V , W )) ∼ = Hom k ( V , W ) R q G the global dimension of Qcoh G Sp ec k is finite. Th us, if E has lo cally-finite pro jectiv e dimension as an ob ject o f Qcoh X , then it has lo cally-finite pro jectiv e dimension a s an ob j ect of Qcoh G X . If X is smo oth, it is w ell-kno wn tha t gldim Qcoh X = dim X .  16 BALLARD, F A VERO, A N D KA TZAR KO V Remark 2.33. In g eneral, the g lo bal dimension of Qcoh G X can b e strictly smaller than the global dimension of Qcoh X . Indeed, Qcoh G G , with the left action of G on itself, is equiv alent to Qcoh Sp ec k and, therefore, m ust hav e global dimension zero. W e t ha nk Kuznetsov for p oin ting this o ut. 3. Equiv ariant f a ctoriza tions Let G b e an affine algebraic group a nd let X b e a smo oth v ariety equipp ed with an action σ : G × X → X . Let w ∈ Γ ( X , L ) G b e a G -inv aria n t section of an inv ertible equiv ar ia n t sheaf, L . Definition 3.1. The dg-category of facto rizations of w , is denoted b y F act ( X , G, w ). The ob jects of F act ( X , G, w ) are pairs, E − 1 E 0 E − 1 ⊗ O X L φ E 0 φ E − 1 of morphisms in Qcoh G X , satisfying φ E − 1 ◦ φ E 0 = w ( φ E 0 ⊗ L ) ◦ φ E − 1 = w . W e denote suc h an ob ject b y ( E − 1 , E 0 , φ E − 1 , φ E 0 ) or simply b y E when there is no confusion. The morphism complex b et w een tw o ob jects, E and F , as a graded v ector space, can b e described as follow s. F or n = 2 l , we ha ve Hom n F act ( X ,G,w ) ( E , F ) = Hom Qcoh G X ( E − 1 , F − 1 ⊗ O X L l ) ⊕ Hom Qcoh G X ( E 0 , F 0 ⊗ O X L l ) and for n = 2 l + 1, w e ha v e Hom n F act ( X ,G,w ) ( E , F ) = Hom Qcoh G X ( E 0 , F − 1 ⊗ O X L l +1 ) ⊕ Hom Qcoh G X ( E − 1 , F 0 ⊗ O X L l ) The differen tia l applied to ( f − 1 , f 0 ) ∈ Hom n F act ( X,G,w ) ( E , F ) = (  ( f 0 ◦ φ E 0 − ( φ F 0 ⊗ O X L l ) ◦ f − 1 , ( f − 1 ⊗ O X L ) ◦ φ E − 1 − ( φ F − 1 ⊗ O X L l ) ◦ f 0  if n = 2 l  ( f 0 ◦ φ E 0 + ( φ F − 1 ⊗ O X L l ) ◦ f − 1 , ( f − 1 ⊗ O X L ) ◦ φ E − 1 + ( φ F 0 ⊗ O X L l +1 ) ◦ f 0  if n = 2 l + 1 . Giv en a n additiv e sub category of Qcoh G X , we can fo r m a corresp onding dg - sub category of Fact ( X, G, w ) b y requiring t he comp onen ts, E − 1 and E 0 , to b e ob jects from that additive sub category . Definition 3.2. Denote b y fact ( X , G, w ), V ect ( X, G, w ), vect ( X , G, w ), and Inj ( X, G, w ), resp ectiv ely , the full dg-sub category of F act ( X , G, w ) whose comp onen ts, resp ectiv ely , are coheren t, lo cally-f ree, lo cally-free of finite rank, and injectiv e as quasi-coheren t G -equiv a rian t shea v es. Remark 3.3. Catego r ies o f pro jective factorizations only prov e useful when X is affine and G is reductiv e. Then, an y lo cally-free G -equiv ariant sheaf of finite rank is pro jective . Definition 3.4. The shift , denoted b y [1], sends a factorizatio n, E , to the f a ctorization, E [1] := ( E 0 , E − 1 ⊗ O X L , − φ E 0 , − φ E − 1 ⊗ O X L ) . Lemma 3.5. We have an e quality Hom n F act ( X ,G,w ) ( E , F ) = Hom 0 F act ( X ,G,w ) ( E , F [ n ]) . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 17 Pr o of. This is a straigh tf o rw ard c hec k and is suppressed.  One can pa ss to an a sso ciat ed Ab elian category . It has the same ob j ects as F a ct ( X , G, w ), but morphisms b et we en E and F are closed degree-zero morphisms in Hom F act ( X ,G,w ) ( E , F ). Denote this category by Z 0 F act ( X , G, w ). The category , Z 0 F act ( X , G, w ), with comp onen t- wise k ernels and cokerne ls is an Ab elian category . Definition 3.6. Give n a complex of ob jects from Z 0 F act ( X , G, w ), · · · → E b f b → E b +1 f b +1 → · · · f t − 1 → E t → · · · , the totalization , T , is the factorization T − 1 := M i =2 l E i − 1 ⊗ O X L − l ⊕ M i =2 l − 1 E i 0 ⊗ O X L − l T 0 := M i =2 l E i 0 ⊗ O X L − l ⊕ M i =2 l +1 E i − 1 ⊗ O X L − l φ T 0 :=         . . . 0 0 0 0 . . . − φ E − 1 − 1 0 0 0 0 f − 1 0 φ E 0 0 0 0 0 0 f 0 − 1 − φ E 1 − 1 ⊗ L − 1 0 0 0 0 . . . . . .         φ T − 1 :=         . . . 0 0 0 0 . . . − φ E − 1 0 ⊗ L 0 0 0 0 f − 1 − 1 ⊗ L φ E 0 − 1 0 0 0 0 f 0 0 − φ E 1 0 0 0 0 0 . . . . . .         F or a n y closed morphism of cohomological degree zero, f : E → F , in F act ( X , G, w ), w e can form the cone factorizat io n, C ( f ), a s the to t a lization of the complex E f → F where F is in degree zero. Prop osition 3.7. The homo topy c ate gory, [ F act ( X , G, w )] , is a triangulate d c ate gory. Pr o of. The translatio n is [1] and the class of tria ng les is giv en by sequences o f morphisms E f → F → C ( f ) → E [1] . The pro o f no w runs completely analogously to pro ving that the homotopy category of chain complexes of an Ab elian category is triangulated. It is therefore suppressed.  Definition 3.8. (P ositselski) Let Acyc ( X , G, w ) denote the full sub category of ob jects o f F act ( X , G, w ) consisting of totalizations of b ounded exact complexes from Z 0 F act ( X , G, w ). Ob jects of Acyc ( X , G, w ) are called a cyclic . Similarly , let acyc ( X , G, w ) denote the sub cate- gory of totalizations of b ounded exact complexes o f coheren t factorizations. W e will also need the analog s for factorizations with lo cally-free comp onen ts. T he full sub- category of ob jects o f V ect ( X , G, w ) consisting of to talizations of b ounded exact complexes 18 BALLARD, F A VERO, A N D KA TZAR KO V from Z 0 V ect ( X , G, w ) is denoted b y AcycV ect ( X , G, w ). Similarly , let acycvect ( X , G, w ) de- note the sub category of to t a lizations of b ounded ex act complexe s of coheren t lo cally-free factorizations. Definition 3.9. (Pos itselski) The absolute derived category of [ Fact ( X, G, w )] is the V erdier quotien t of [ Fact ( X, G, w )] b y [ Acyc ( X , G, w )], D abs [ F act ( X , G, w )] := [ F act ( X , G, w )] / [ Acyc ( X, G, w )] . The absolute derived catego ry of [ fa ct ( X , G, w )] is the V erdier quotient of [ fact ( X , G, w )] by [ acyc ( X , G, w )], D abs [ fact ( X , G, w )] := [ fact ( X , G , w ) ] / [ acyc ( X, G, w )] . The absolute derived category of [ V ect ( X , G, w )] is the V erdier quotien t of [ V ect ( X , G, w )] b y [ AcycV ect ( X , G, w )] D abs [ V ect ( X , G, w )] := [ V ect ( X , G, w )] / [ AcycV ect ( X , G, w )] . The absolute derived category of [ vect ( X, G, w )] is the V erdier quotien t of [ vect ( X , G, w )] b y [ acycvect ( X , G, w )], D abs [ vect ( X , G, w )] := [ vect ( X , G, w ) ] / [ acycvect ( X, G, w )] . W e sa y that tw o fa ctorizations are quasi-isomo rphic if they are isomorphic in the appropri- ate absolute deriv ed category . W e will a lso use v ersions of these categories with supp ort conditions. Let Z b e a closed G -in v arian t subset of X and set U := X \ Z . Let j : U → X b e the inclusion. Definition 3.10. The category , D abs Z [ F act ( X , G, w )], is the k ernel of the functor, j ∗ : D abs Z [ F act ( X , G, w )] → D abs Z [ F act ( U, G, w | U )] . Define D abs Z [ fact ( X , G, w )], D abs Z [ V ect ( X , G, w )], D abs Z [ vect ( X , G, w )] a nalogously . Let us r ecall some useful facts, due essen tia lly to Pos itselski, ab out D abs [ F act ( X , G, w )]. Prop osition 3.11. F actorizations with inj e ctive c om p onen ts ar e right ortho gonal to acyclic c omp l e xes in [ F act ( X , G, w )] . Mor e ov e r, the c omp osition, [ Inj ( X , G, w )] → [ F act ( X , G, w )] → D abs [ F act ( X , G, w )] is an e quivalenc e. Pr o of. This is a v ersion of [P os09, Theorem 3.6] o f Posits elski. In this g eneralit y , it is a sp ecial case of [BDFIK1 2, Lemma 2.22 and Corollary 2.23].  Definition 3.12. W e let I nj coh ( X , G, w ) b e the full dg-category o f F act ( X , G, w ) consisting of factorizations that hav e injectiv e comp onen ts and t hat are quasi-isomorphic to a factor izat io n with coheren t comp onents. Corollary 3.13. The c omp osition, [ Inj coh ( X , G, w )] → [ F act ( X , G, w )] → D abs [ fact ( X , G, w )] is an e quivalenc e. Pr o of. This is an immediate cor o llary of Pro p osition 3 .1 1.  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 19 Prop osition 3.14. The natur al functor, D abs [ V ect ( X , G, w )] → D abs [ F act ( X , G, w )] , is an e quivalenc e as is the natur al functor, D abs [ vect ( X , G, w )] → D abs [ fact ( X , G, w )] . Mor e over, if X is affine and G is r e ductive, fac toriz a tions with lo c al ly-fr e e c omp on ents ar e left ortho gonal to a c yclic c om p lexes in [ F act ( X , G, w )] and the c o m p ositions [ V ect ( X , G, w )] → [ F act ( X , G, w )] → D abs [ F act ( X , G, w )] [ vect ( X, G, w )] → [ fact ( X, G, w )] → D abs [ fact ( X , G, w )] ar e e quivalen c es. Pr o of. W e first c hec k that any factorization is quasi-isomorphic t o a lo cally-free factor izat io n. Moreo v er, if the or ig inal factor izat io n is coheren t, then the lo cally-fr ee factor ization can b e c hosen to hav e finite rank. The argumen t is con tained in the pro of of [Pos09, Theorem 3 .6]. Let E b e a f a ctorization. By Theorem 2.29, w e can find lo cally- f ree G - equiv a rian t shea v es, V − 1 and V 0 and epimorphisms V − 1 f − 1 → E − 1 V 0 f 0 → E 0 . F orm the factorization, G + ( V ), V 0 ⊗ O X L − 1 ⊕ V − 1   0 1 w 0   → V − 1 ⊕ V 0   0 w 1 0   → V 0 ⊕ V − 1 ⊗ O X L . The maps, V 0 ⊗ O X L − 1 ⊕ V − 1 (0 f − 1 ) → E − 1 V − 1 ⊕ V 0 (0 f 0 ) → E 0 , giv e an epimorphism in Z 0 F act ( X , G, w ). Th us, for an y fa ctorization, there exists a factor- ization with lo cally-free comp o nen ts mapping epimorphically onto it . W e can construct an exact complex o f ob j ects of Z 0 F act ( X , G, w ) · · · → V s → · · · → V 1 → E → 0 where each V j is a factorization with lo cally-free comp o nen ts. Let K s b e the k ernel of V s → V s − 1 for s > dim X . Since X is smo oth, the comp onents of K s are lo cally-free. Th us, w e ha v e an exact sequence 0 → K s → V s → · · · → V 1 → E → 0 . In D abs [ fact ( X , G, w )], we hav e an isomorphism T → E where T is the totalization of K s → V s → · · · → V 1 . The fa cto r ization, T , ha s lo cally-f r ee comp onen ts. 20 BALLARD, F A VERO, A N D KA TZAR KO V Th us, t he natural functors, D abs [ V ect ( X , G, w )] → D abs [ F act ( X , G, w )] D abs [ vect ( X , G, w )] → D abs [ fact ( X , G, w )] , are essen tially surjectiv e. W e next c hec k fully-faithfulness. F or fully-faithfulness, it suffices to show that giv en a short exact sequence 0 → E 3 → E 2 → E 1 → 0 (3.1) there exists a factorization, S ∈ AcycV ect ( X , G , w ) , that is isomorphic to the totalizat io n, T , of (3.1) in D abs [ F act ( X , G, w )]. Moreov er, if E i are all coheren t, then S can b e tak en to ha v e finite rank. Using what w e hav e already prov en, w e can find a lo cally-free factorization V 1 1 and an epimorphism V 1 1 → E 1 . Next c ho o se a lo cally-free fa ctorization V 2 1 and an epimorphism onto the fib er pro duct V 2 1 → E 2 × E 1 V 1 1 . Let V 3 1 b e the k ernel of the ma p V 2 1 → E 2 × E 1 V 1 1 → V 1 1 . Th ere is a comm utative diagram 0 E 3 E 2 E 1 0 0 V 3 1 V 2 1 V 1 1 0 with the v ertical morphisms b eing epimorphisms. Replacing ( 3 .1) the k ernels of t he v ertical morphisms, rep eating the arg umen t, and iterating, w e get a n exact sequence of short exact sequence s 0 0 0 0 E 3 E 2 E 1 0 0 V 3 1 V 2 1 V 1 1 0 . . . . . . . . . 0 V 3 s V 2 s V 1 s 0 0 0 0 where eac h V i j is lo cally-free, and of finite rank if eac h E i is coheren t. The long exact sequence of short exact sequence s give s rise to a long exact sequence of the totalizations of t hese short exact sequences 0 → T s → T s − 1 → · · · → T 1 → T → 0 . Eac h T j lies in AcycV ect ( X , G, w ), or in acycvect ( X , G, w ) if each E i is coheren t. Th us, the totalization of T s → T s − 1 → · · · → T 1 lies in AcycV ect ( X , G, w ), or in acy cv ect ( X, G, w ) if eac h E i is coheren t, and is isomorphic t o T in D abs [ F act ( X , G, w )]. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 21 If w e assume that X is a ffine and G is reductiv e, t hen an y G -equiv a rian t lo cally-free sheaf is pro jectiv e. The res ult in this case is a v ersion of [P o s09, Theorem 3 .6] of P ositselski. F or this generality , w e arg ue as follows . By [BDFIK12, L emma 2.22], factorizations with pro jectiv e comp o nen ts are left or thogonal to a cyclic factorizations. Th us, the comp ositions [ V ect ( X , G, w )] → [ F act ( X , G, w )] → D abs [ F act ( X , G, w )] [ vect ( X, G, w )] → [ fact ( X, G, w )] → D abs [ fact ( X , G, w )] are fully-faithful. As w e hav e already seen they are essen tially surjectiv e, they mus t b oth b e equiv alences.  F or a definition of a compactly-generated tria ng ulated cat ego ry a nd compact g enerato r s, refer to Section 4. Prop osition 3.15. The triangulate d c a te gory, D abs [ F act ( X , G, w )] , is c omp actly-gener ate d. The obje cts of D abs [ fact ( X , G, w )] ar e a set of c omp act gener ators. Pr o of. The pro of of this fact is a rep etition of the argumen t of [P os09, Theorem 3 .1 1.2] using the fact that a n y quasi-coheren t G -equiv ar ia n t sheaf on X , hence any fa ctorization, is a union o f its coheren t subshea v es [Tho97, Lemma 1.4]. More precisely , one can use Lemma 4.7 (which is a consequence of Thomason’s result) and follo w Posits elski’s argument v erbatim.  Remark 3.16. It is a subtle pro blem to determine whether or not all compact ob jects o f D abs [ F act ( X , G, w )] are isomorphic to ob jects of D abs [ fact ( X , G, w )]. By Prop osition 3.15 and [Nee92, Theorem 2.1], ev ery compact ob j ect is a summand of a n ob ject of D abs [ fact ( X , G, w )] under a splitting in D abs [ F act ( X , G, w )]. How ev er, those summands ma y not b e represen table b y coheren t fa ctorizations. See [O r l1 1] fo r an in v estigatio n of the relationship with comple- tions of X . T o handle the p ossible idemp ot ent incompleteness of our factorizations categories, w e mak e the following definitions. Definition 3.17. Let Inj coh ( X , G, w ) b e the full dg - sub category of Inj ( X, G, w ) consisting of factorizations which are compact in [ Inj ( X , G, w )] ∼ = D abs [ F act ( X , G, w )]. Let vect ( X, G, w ) b e the full dg-sub category of V ect ( X , G, w ) consisting of factorizations whic h are compact in D abs [ V ect ( X , G, w )] ∼ = D abs [ F act ( X , G, w )]. Let D abs [ fact ( X , G, w )] denote the idemp otent-completion of D abs [ fact ( X , G, w )]. Note that b y Prop osition 3.1 1 , w e ha ve  Inj coh ( X , G, w )  ∼ = D abs [ fact ( X , G, w ) ] . If X is affine and G is r eductiv e, b y Prop osition 3.14 , w e ha ve [ vect ( X , G, w )] ∼ = D abs [ fact ( X , G, w )] F rom a complex on the zero lo cus of w , one can form a factorization. Definition 3.18. Let Y b e the zero lo cus of w in X . Denote b y Q coh G Y the dg-category of c hain complexes o f quasi-coheren t G - equiv arian t shea v es on Y . 22 BALLARD, F A VERO, A N D KA TZAR KO V W e hav e the dg -functor, see [P os11, Section 3.7], Υ : Qcoh G Y → F a ct ( X , G, w ) C 7→ ( M l ∈ Z i ∗ C 2 l − 1 ⊗ O X L − l , M l ∈ Z i ∗ C 2 l ⊗ O X L − l , ⊕ l ∈ Z i ∗ d 2 l − 1 C ⊗ O X L − l , ⊕ l ∈ Z i ∗ d 2 l C ⊗ O X L − l ) , In the case that C is a coherent G -equiv ariant sheaf and the context allow s, w e will denote Υ C simply by C Note that Υ C is the totalization of the c hain complex · · · → Υ C b → · · · → Υ C t → · · · . It is clear that Υ tak es b o unded acyclic c hain complexes in Qcoh G Y to acyclic c hain com- plexes on [ F act ( X , G, w )]. Thus , Υ descends to a functor Υ : D b (Qcoh G Y ) → D abs [ F act ( X , G, w )] . Moreo v er, Υ take s b ounded complexes of coheren t shea v es to coheren t factorizations so it induces a functor Υ : D b (coh G Y ) → D abs [ fact ( X , G, w )] . No w, we giv e an explicit construction, due essen tially to Eisen bud [Eis80, Section 7], of a factorization asso ciated to certain in v arian t closed subsc hemes in the zero lo cus of w . Consider an equiv a r ia n t morphism E s → O X with E lo cally-free of finite rank. W e notationally identify s with the correspo nding g lobal section of E ∨ . F urther, a ssume there exists an equiv a rian t morphism t : O X → E ⊗ O X L making the diagram E O X E ⊗ O X L O X ⊗ O X L s t s ⊗ O X L w w comm ute. Definition 3.19. The Koszul factorization asso ciat ed to the data ( E , s, t ) is defined a s K − 1 ( s, t ) := M l ≥ 0 (Λ 2 l +1 E ) ⊗ O X L l K 0 ( s, t ) := M l ≥ 0 (Λ 2 l E ) ⊗ O X L l φ K 0 , φ K − 1 := • y s + • ∧ t. Prop osition 3.20. Assume t ha t ( E , s, t ) as ab o v e exist. L et O Z s b e the c okernel of s . If rank E = co dim Z s , then K ( s, t ) is quasi-iso morphic to the f actorization, Υ O Z s . L et O Z t ∨ b e the c ok ernel of E ∨ ⊗ O X L ∨ t ∨ → O X . If ra nk E = co dim Z t ∨ , then K ( s, t ) is quasi-isomorphic to the factorization Υ  O Z t ∨ ⊗ O X Λ rk E E [ − rk E ]  . Pr o of. Eac h is a straightforw ard application of [BDFIK12, Lemma 3.4], see also [Bec12 , Section 3.2].  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 23 Lemma 3.21. We have an is o m orphism of factorizations, K ( s, t ) ∨ ∼ = K ( t ∨ , s ∨ ) . Pr o of. This is immediate fro m the definitions.  W e describe some functors asso ciated with natural op erations o n factorizations, mirroring those discussed in Section 2. Definition 3.22. Let X b e a smo oth v ariety equipp ed with an action of G . Assume w e ha v e w , v ∈ Γ( X , L ) G . W e define a dg- functor, ⊗ O X : Fact ( X, G, w ) ⊗ k F act ( X , G, v ) → Fact ( X, G, w + v ) , b y setting ( E ⊗ O X F ) − 1 := E − 1 ⊗ O X F 0 ⊕ E 0 ⊗ O X F − 1 ( E ⊗ O X F ) 0 := E 0 ⊗ O X F 0 ⊕ E − 1 ⊗ O X F − 1 ⊗ O X L φ E ⊗ O X F 0 :=  φ E 0 ⊗ O X 1 F 0 1 E 0 ⊗ O X φ F 0 − 1 E − 1 ⊗ O X φ F − 1 φ E − 1 ⊗ O X 1 F − 1  φ E ⊗ O X F − 1 :=  φ E − 1 ⊗ O X 1 F 0 − 1 E − 1 ⊗ O X φ F 0 ⊗ O X L 1 E 0 ⊗ O X φ F − 1 φ E 0 ⊗ O X L ⊗ O X 1 F − 1  Giv en α : E → E ′ [ r ] and β : F → F ′ [ s ], one has α ⊗ O X β : E ⊗ O X E ′ → F ⊗ O X F ′ [ r + s ] defined by α ⊗ O X β =                                α − 1 ⊗ β 0 0 0 α 0 ⊗ β − 1 ! , α 0 ⊗ β 0 0 0 α − 1 ⊗ β − 1 ⊗ L !! r , s ev en 0 α 0 ⊗ β − 1 − α − 1 ⊗ β 0 0 ! , 0 − α − 1 ⊗ β − 1 ⊗ L α 0 ⊗ β 0 0 !! r ev en , s o dd α − 1 ⊗ β 0 0 0 α 0 ⊗ β − 1 ! , α 0 ⊗ β 0 0 0 α − 1 ⊗ β − 1 ⊗ L !! r o dd , s ev en 0 − α 0 ⊗ β − 1 α − 1 ⊗ β 0 0 ! , 0 α − 1 ⊗ β − 1 ⊗ L − α 0 ⊗ β 0 0 !! r , s o dd F or a lo cally-free f a ctorization, V , the functor, V ⊗ O X • : [ F act ( X , G, v )] → [ F act ( X , G, w + v )] , preserv es acyclic complexes and descends to a functor. V ⊗ O X • : D abs [ F act ( X , G, v )] → D abs [ F act ( X , G, w + v )] . F or E ∈ F act ( X , G, w ), w e define E L ⊗ O X • := V ⊗ O X • . where V is a lo cally-free fa ctorization quasi-isomorphic to E . 24 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 3.23. The func tor, E L ⊗ O X • : D abs [ F act ( X , G, v )] → D abs [ F act ( X , G, w + v )] is wel l-define d , i.e. it do es not dep end on the cho ic e of r epr esentative o f the quasi-isomorphism class. Pr o of. By Prop osition 3.1 4, inclusion of Vec t ( X , G, v ) into Fact ( X, G, v ) induces an equiv a- lence D abs [ V ect ( X , G, v ) ] → D abs [ F act ( X , G, v )] . W e may therefore view the derived f unctor o n the absolute deriv ed category of lo cally-free factorizations, E L ⊗ O X • : D abs [ V ect ( X , G, v )] → D abs [ V ect ( X , G, w + v )] . Since tensoring with a lo cally-free sheaf is exact, tensoring with a lo cally-fr ee factorization preserv es acyclic factorizations and we ha v e natural quasi-isomorphisms E ⊗ O X W ∼ = V ⊗ O X W =: E L ⊗ O X W . when W is lo cally-free and V is lo cally-free and quasi-isomorphic to E .  Definition 3.24. Let X b e a smo oth v ariety equipp ed with an action of G . Assume w e ha v e w ∈ Γ( X, L ) G . Let p : X → Sp ec k b e the structure morphism. L et ( C , d ) b e a b ounded complex of vec to r spaces. Le t E ∈ F act ( X , G, w ). Define a factorization E ⊗ k C by E ⊗ k C := E ⊗ O X p ∗ (Υ C ) . Denote the corresp onding f unctor by E ⊗ k • : Qcoh b (Sp ec k ) → F act ( X , G, w ) . This functor tak es an exact c hain complex to an acyclic factorization in Fact ( X, G, w ). Th us, it descends to a functor E ⊗ k • : D b (Qcoh Sp ec k ) → D abs [ F act ( X , G, w )] . Next w e giv e a v ersion o f sheaf Hom. Definition 3.25. Let X b e a smo oth v ariety equipp ed with an action of G . Assume w e ha v e sections, w , v ∈ Γ( X , L ) G . W e define a dg-functor, H om X : Fact ( X, G, w ) op ⊗ k F act ( X , G, v ) → F act ( X , G , v − w ) , b y setting H om X ( E , F ) − 1 := H om X ( E − 1 , F 0 ) ⊗ O X L − 1 ⊕ H om X ( E 0 , F − 1 ) H om X ( E , F ) 0 := H om X ( E 0 , F 0 ) ⊕ H om X ( E − 1 , F − 1 ) φ H om X ( E , F ) 0 :=  ( • ) ◦ φ E − 1 φ F 0 ◦ ( • ) ( φ F − 1 ⊗ O X L − 1 ) ◦ ( • ) ( • ) ◦ φ E 0  φ H om X ( E , F ) − 1 :=  − ( • ) ◦ φ E 0 φ F 0 ◦ ( • ) φ F − 1 ◦ ( • ) − ( • ) ◦ ( φ E − 1 ⊗ O X L − 1 )  Giv en α : E → E ′ [ r ] and β : F → F ′ [ s ], one has H om X ( α, β ) : H om X ( E ′ , F ) → H om X ( E , F ′ )[ r + s ] KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 25 defined by  β 0 ◦ ( • ) ◦ ( α − 1 ⊗ L − l +1 ) 0 0 β − 1 ◦ ( • ) ◦ ( α 0 ⊗ L − l )  ,  β 0 ◦ ( • ) ◦ ( α 0 ⊗ L − l ) 0 0 β − 1 ◦ ( • ) ◦ ( α − 1 ⊗ L − l )  if r = 2 l , s = 2 j ,  0 β − 1 ◦ ( • ) ◦ ( α 0 ⊗ L − l ) − β 0 ◦ ( • ) ◦ ( α − 1 ⊗ L − l +1 ) 0  ,  0 − β − 1 ◦ ( • ) ◦ ( α − 1 ⊗ L − l ) β 0 ◦ ( • ) ◦ ( α 0 ⊗ L − l ) 0  if r = 2 l , s = 2 j + 1 ,  − β 0 ◦ ( • ) ◦ ( α 0 ⊗ L − l ) 0 0 β − 1 ◦ ( • ) ◦ ( α − 1 ⊗ L − l )  ,  β 0 ◦ ( • ) ◦ ( α − 1 ⊗ L − l ) 0 0 − β − 1 ◦ ( • ) ◦ ( α 0 ⊗ L − l − 1 )  if r = 2 l + 1 , s = 2 j , a nd  0 β − 1 ◦ ( • ) ◦ ( α − 1 ⊗ L − l ) β 0 ◦ ( • ) ◦ ( α 0 ⊗ L − l ) 0  ,  0 β − 1 ◦ ( • ) ◦ ( α 0 ⊗ L − l − 1 ) β 0 ◦ ( • ) ◦ ( α − 1 ⊗ L − l ) 0  if r = 2 l + 1 , s = 2 j + 1. F or a lo cally-free f a ctorization, V , the functor, H om X ( V , • ) : [ F act ( X , G, v )] → [ F act ( X , G, v − w )] , preserv es acyclic complexes and descends to a functor. H om X ( V , • ) : D abs [ F act ( X , G, v )] → D abs [ F act ( X , G, v − w )] . F or E ∈ F act ( X , G, w ), w e define R H om X ( E , • ) := H o m X ( V , • ) . where V is a lo cally-free fa ctorization quasi-isomorphic to E . Lemma 3.26. The func tor, R H om X ( E , • ) : D abs [ F act ( X , G, v )] → D abs [ F act ( X , G, v − w )] is wel l-define d , i.e. it do es not dep end on the cho ic e of r epr esentative o f the quasi-isomorphism class. Pr o of. The pro o f is completely analogous to that of Lemma 3.23 a nd is therefore suppress ed.  Prop osition 3.27. L et X b e a smo o th variety e quipp e d with an action of an affine a l g ebr aic gr oup G . L et w , v ∈ Γ( X, L ) G b e i n variant se ctions of an inve rtible e quivariant she af, L . F o r E ∈ F act ( X , G, w ) , F ∈ F act ( X , G, v ) and G ∈ F act ( X , G, w + v ) , ther e ar e natur al isomorphisms Hom F act ( X,G,w + v ) ( E ⊗ O X F , G ) ∼ = Hom F act ( X ,G,w ) ( E , H o m X ( F , G )) . Pr o of. W e first c hec k this for Hom 0 . W e ha ve Hom 0 F act ( X ,G,w + v ) ( E ⊗ O X F , G ) := Hom Qcoh G X (( E ⊗ O X F ) − 1 , G − 1 ) ⊕ Hom Qcoh G X (( E ⊗ O X F ) 0 , G 0 ) := Hom Qcoh G X ( E − 1 ⊗ O X F 0 , G − 1 ) ⊕ Hom Qcoh G X ( E 0 ⊗ O X F − 1 , G − 1 ) ⊕ Hom Qcoh G X ( E 0 ⊗ O X F 0 , G 0 ) ⊕ Hom Qcoh G X ( E − 1 ⊗ O X F − 1 ⊗ O X L , G 0 ) . 26 BALLARD, F A VERO, A N D KA TZAR KO V Applying Hom- tensor adjunction for G -equiv ar ia n t shea v es, w e hav e an isomorphism ∼ = Hom Qcoh G X ( E − 1 , H om X ( F 0 , G − 1 )) ⊕ Hom Qcoh G X ( E 0 , H om X ( F − 1 , G − 1 )) ⊕ Hom Qcoh G X ( E 0 , H om X ( F 0 , G 0 )) ⊕ Hom Qcoh G X ( E − 1 , H om X ( F − 1 ⊗ O X L , G 0 )) =: Ho m Qcoh G X ( E − 1 , H om X ( F , G ) − 1 ) ⊕ Hom Qcoh G X ( E 0 , H om X ( F , G ) 0 ) =: Hom 0 F act ( X,G,w + v ) ( E , H o m X ( F , G )) . Since Hom 0 ( • , • [ n ]) = Hom n ( • , • ), this defines the natura l transformat io n on the whole morphism space of Fact . It is straigh tforward to c hec k that these maps comm ute with the differen tials.  Corollary 3.28. We have an adjoin t p air of d erive d f unc tors • L ⊗ O X F : D abs [ F act ( X , G, w )] → D abs [ F act ( X , G, w + v )] R H om X ( F , • ) : D abs [ F act ( X , G, w + v )] → D abs [ F act ( X , G, w )] . Pr o of. This fo llo ws b y replacing the first en try in a morphism space b y a lo cally-free factor- ization and the second by an injectiv e f actorization and applying Prop o sition 3.27.  Definition 3.29. W e will fo cus on a part icular case of sheaf-Hom. Consider the factorizat io n Υ O X of 0 ∈ Γ( X , L ) G . Denote it b y O X . W e g et functors ( • ) ∨ := H om X ( • , O X ) : F act ( X , G, w ) op → F act ( X , G, − w ) ( • ) L ∨ := R H om X ( • , O X ) : D abs [ F act ( X , G, w )] op → D abs [ F act ( X , G, − w )] . Lemma 3.30. The func tor, ( • ) L ∨ : D abs [ fact ( X , G, w )] op → D abs [ fact ( X , G, − w )] , is an e quivalenc e. Pr o of. It is simple to che ck that for a lo cally-free fa cto r izat io n of finite rank, F , w e ha ve a natural isomorphism F ∼ = F ∨∨ . An y ob ject of D abs [ fact ( X , G, w ) op ] is quasi-isomorphic to a lo cally-free fa cto r ization o f finite rank by Prop o sition 3.14.  Lemma 3.31. L et V ∈ v ect ( X , G, w ) . Then, ther e is an isomorp h ism V ∨ ⊗ O X • ∼ = H om X ( V , • ) . Similarly, for E ∈ fact ( X , G, w ) , ther e i s an isomorp h ism E L ∨ L ⊗ O X • ∼ = R H om X ( E , • ) . Pr o of. The first isomorphism follow s immediately fr o m insp ection o f the definitions. The second is a quic k consequence o f the first.  Assume we ha ve t w o smo oth v arieties, X and Y , b ot h carrying a G -action, a nd a mor- phism, f : X → Y . Let w ∈ Γ( Y , L ) G . W e ha ve pull-bac k and pushforw ard f unctors. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 27 Definition 3.32. f ∗ : Fact ( Y , G, w ) → F act ( X , G, f ∗ w ) ( E − 1 , E 0 , φ E − 1 , φ E 0 ) 7→ ( f ∗ E − 1 , f ∗ E 0 , f ∗ φ E − 1 , f ∗ φ E 0 ) and f ∗ : F act ( X , G, f ∗ w ) → F act ( X , G, w ) ( F − 1 , F 0 , φ F − 1 , φ F 0 ) 7→ ( f ∗ F − 1 , f ∗ F 0 , f ∗ φ F − 1 , f ∗ φ F 0 ) . Note that by the pro jection formula f ∗ ( F ⊗ O X f ∗ L ) ∼ = ( f ∗ F ) ⊗ O X L under whic h f ∗ ( f ∗ w ) corresp onds to w so this is w ell-defined. Definition 3.33. F or a factorization, E , of 0 ∈ Γ( X, L ) G . W e let the unfolding of E b e the complex a E ∈ Qcoh G ( X ) with ( a E ) j = ( E − 1 ⊗ L l j = 2 l − 1 E 0 ⊗ L l j = 2 l . W e shall also use a sligh tly differen t v ersion o f the pushforw ard. Let X b e equipp ed with an action o f G and consider the structure morphism, p : X → Sp ec k . It is G -equiv aria nt if w e equip Sp ec k with the trivial action. Then, w e ha ve a pushforw ard p ∗ : Fact ( X, G, 0) → Qcoh G (Sp ec k ) F 7→ p ∗ ( a F ) where p ∗ : Qcoh G ( X ) → Qcoh G (Sp ec k ) is the usual pushforward of equiv ar ian t shea ve s. Lemma 3.34. L et E , F ∈ Fact ( X, G, w ) . T hen, we have an isomorphi s m of c omplexes ( p ∗ H om X ( E , F )) G ∼ = Hom F act ( X,G,w ) ( E , F ) . Pr o of. This is immediate fro m the definitions.  Lemma 3.35. Push-forwar d, f ∗ , is right adjoint to pul l - b ack, f ∗ . Pr o of. Applying the standard adjunction b etw een f ∗ and f ∗ for equiv arian t shea v es to the comp onen ts of the factorization gives the statemen t.  W e also define their derive d analogs. Definition 3.36. Define the left-deriv ed functor of f ∗ b y L f ∗ : D abs [ F act ( Y , G, w )] → D abs [ F act ( X , G, f ∗ w )] E 7→ f ∗ V where V is a f actorization with lo cally-free comp o nen ts quasi-isomorphic to E . Define the r ig h t-deriv ed functor of f ∗ b y R f ∗ : D abs [ F act ( X , G, f ∗ w )] → D abs [ F act ( X , G, w )] E 7→ f ∗ I where I is a factorization with injective comp onen ts quasi-isomorphic to E . Lemma 3.37. Both L f ∗ and R f ∗ ar e w e l l-define d, i.e. they do not dep en d on the choic es of r epr esentatives of a quasi-isomorphis m class . 28 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. The deriv ed push-forw ar d is well-defin ed by Prop osition 3.11 since [ Inj ( X , G, f ∗ w )] ∼ = D abs [ F act ( X , G, f ∗ w )]. The deriv ed pull-bac k functor, f ∗ , is w ell-defined b y Prop osition 3.14 since D abs [ V ect ( X , G, w )] ∼ = D abs [ F act ( X , G, w )] and f ∗ preserv es acyclic complexes of lo cally-free shea v es.  Lemma 3.38. F or e ach, E ∈ D abs [ F act ( Y , G, w )] and F ∈ D abs [ F act ( X , G, f ∗ w )] , ther e is a natur al isomorphism R f ∗ F L ⊗ O Y E ∼ = R f ∗ ( F L ⊗ O X L f ∗ E ) . Pr o of. This follo ws from replacing E b y a factor izat io n with lo cally-free comp onen ts, F by a factorization with injectiv e comp o nen ts, and applying the pro jection formula, Lemma 2.8, to the comp o nen ts of the f a ctorizations.  W e also hav e an extension of pullbac k to allo w for a gro up homomorphism. Definition 3.39. Assume we hav e tw o smo oth v arieties, X and Y , and tw o affine algebraic groups, G and H . Let ψ : G → H b e a homomorphism and assume that G acts on X while H acts on Y . Let f : X → Y b e a ψ -equiv a rian t morphism. Let w ∈ Γ( Y , L ) H so t hat f ∗ w ∈ Γ( X, f ∗ L ) G . W e hav e a functor, f ∗ : Fact ( Y , H , w ) → F act ( X , G, f ∗ w ) ( E − 1 , E 0 , φ E − 1 , φ E 0 ) 7→ ( f ∗ E − 1 , f ∗ E 0 , f ∗ φ E − 1 , f ∗ φ E 0 ) . The left-deriv ed functor of f ∗ is L f ∗ : D abs [ F act ( Y , H , w )] → D abs [ F act ( X , G, f ∗ w )] E 7→ f ∗ V where V is a f actorization with lo cally-free comp o nen ts quasi-isomorphic to E . Lemma 3.40. The functor, L f ∗ , is wel l-defin e d, i.e . it do es not dep end on the choic e of r epr esentatives of a quasi-isomorphis m class . Pr o of. The pro of is completely analogous to that o f Lemma 3.37 .  W e also extend the restriction and induction functors. Definition 3.41. Let X b e a smo oth v ar iety equipp ed with a n action of an affine alg ebraic group, G . Let w ∈ Γ( G, L ) G . Let ψ : H → G b e a closed subgroup of G . Res G H : Fact ( X, G, w ) → F act ( X , H , w ) ( E − 1 , E 0 , φ E − 1 , φ E 0 ) 7→ (Res G H E − 1 , Res G H E 0 , Res G H φ E − 1 , Res G H φ E 0 ) and Ind G H : Fact ( X, H , w ) → F a ct ( X , G, w ) ( F − 1 , F 0 , φ F − 1 , φ F 0 ) 7→ (Ind G H F − 1 , Ind G H F 0 , Ind G H φ F − 1 , Ind G H φ F 0 ) . The action o n morphisms is clear. The restriction functor, Res G H , is exact so it immediately descends to Res G H : D abs [ F act ( X , G, w )] → D abs [ F act ( X , H , w )] . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 29 The induction functor, Ind G H , is left-exact so w e ha v e its right-deriv ed functor, R Ind G H : D abs [ F act ( X , H , w )] → D abs [ F act ( X , G, w )] E 7→ f ∗ I where I is a factorization with injective comp onen ts quasi-isomorphic to E . Lemma 3.42. The func tor, Res G H , is left adjoint to the functor, Ind G H . Pr o of. This is an immediate consequence of Lemma 2.15.  Corollary 3.43. We have an adjoin t p air of f unc tors, Res G H : D abs [ F act ( X , G, w )] → D abs [ F act ( X , H , w )] R Ind G H : D abs [ F act ( X , H , w )] → D abs [ F act ( X , G, w )] . Pr o of. This is an immediate consequence of Lemma 3.42.  Lemma 3.44. F or e ach, E ∈ D abs [ F act ( X , G, w )] and F ∈ D abs [ F act ( X , H , w )] , ther e is a natur al isomorphism R Ind G H F L ⊗ O Y E ∼ = R Ind G H ( F L ⊗ O X Res G H E ) . Pr o of. This follo ws from replacing F b y a factorization with injectiv e comp onen ts and ap- plying the pro jection f orm ula, Lemma 2.16, to the comp onen ts of the factorizations.  Finally , we extend the functor of inv a r ian ts. Definition 3.45. Let N b e a closed normal subgroup of G . Let X b e a smo oth v ariety equipped with an action of G o n whic h N acts tr ivially . Let L b e an inv ertible G / N - equiv arian t sheaf. Note that L inherits a G -equiv ariant structure. Consider a section w ∈ Γ( X , L ) G ∼ = Γ( X , L ) G/ N . W e define ( • ) N : Fact ( X, G, w ) → F act ( X , G/ N , w ) ( E − 1 , E 0 , φ E − 1 , φ E 0 ) 7→ ( E N − 1 , E N 0 , ( φ E − 1 ) N , ( φ E 0 ) N ) . The deriv ed functor of inv ariants is ( • ) R N : D abs [ F act ( X , G, w )] → D abs [ F act ( X , G/ N , w )] E 7→ I N where I is a factorization that has injectiv e comp onen ts a nd that is quasi-isomorphic to E . Definition 3.46. Let L b e an inv ertible equiv ariant sheaf o n X and let w ∈ Γ( X , L ) G . Let V( L ) := Sp ec X (Sym L ) denote the geometric vec tor bundle asso ciated to L . It carr ies an action of G × G m where G acts via the equiv a rian t structure on L and G m dilates the fibers of the bundle . The section, w , defines a regular function, f w ∈ Γ(V ( L ) , O V( L ) (1)) G × G m where (1) denotes the pro jection c haracter, G × G m → G m . Finally , let U( L ) denote the complemen t of t he zero section in V( L ). Let π : U( L ) → X denote the pro jection. It is equiv a rian t with resp ect to the pro jection, G × G m → G m . 30 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 3.47. The p ul l b a ck functor, π ∗ : Qcoh G X → Qcoh G × G m U( L ) , is an e quivalenc e. Mor e over, π ∗ induc es e quival e n c es b etwe en sub c ate go ries of c oher ent and lo c al ly-fr e e e quivariant she aves. Pr o of. The v ariet y , U( L ), is a G m -torsor ov er X . Th us, the fppf quotien t of U( L ) by G m is X . The statemen t of the lemma is a consequence of faithfully-flat descen t. In other words, the global quotien t stac k [U( L ) / G m ] is represen ted b y X , and therefore they hav e the same sheaf theory .  Lemma 3.48. The p ul l b a ck functor, π ∗ : Fact ( X, G, w ) → F act (U( L ) , G × G m , f w ) , is an e quivalenc e of dg-c ate gories. Mor e ove r, π ∗ r estricts to e quivalenc es, π ∗ : Inj ( X , G, w ) → Inj (U( L ) , G × G m , f w ) , π ∗ : V ect ( X , G, w ) → V ect (U( L ) , G × G m , f w ) , π ∗ : fact ( X , G, w ) → fact (U( L ) , G × G m , f w ) , π ∗ : vect ( X , G, w ) → vect (U( L ) , G × G m , f w ) . Pr o of. This is an immediate consequence of Lemma 3.47.  The following definitions seem to ha ve no na t ur a l extension to t he case of general equi- v ariant line bundles. They will b e essen tia l later in the pap er. Definition 3.49. Let X and Y b e smo oth v arieties and let w ∈ Γ( X , O X ) and v ∈ Γ( Y , O Y ). W e set w ⊞ v := w ⊗ 1 + 1 ⊗ v ∈ Γ( X, O X ) ⊗ k Γ( Y , O Y ) ∼ = Γ( X × Y , O X × Y ) . W e will hav e to deal with tw o p oten tials, w , v ∈ Γ( X , O X ), that are semi-in v arian t with resp ect to different characters of differen t groups. The largest group fo r whic h w ⊞ v is semi-in v ariant is a s follows . Definition 3.50. L et G and H b e affine alg ebraic groups and let χ : G → G m and χ ′ : H → G m b e c haracters. Define a c haracter of G × H b y χ ′ − χ : G × H → G m ( g , h ) 7→ χ ( g ) − 1 χ ′ ( h ) . Let G × G m H b e the ke rnel of χ ′ − χ o r equiv a lently the fib er pro duct of G and H ov er G m . Definition 3.51. Let X b e a smo oth v ar iety equipp ed with a n action of an affine alg ebraic group, G , and let Y b e a smo oth v ar iet y eq uipp ed with a n action of an affine alg ebraic group, H . Let χ : G → G m and χ ′ : H → G m b e c haracters. L et w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H . W e hav e a dg- functor ⊠ : F act ( X , G, w ) ⊗ k F act ( Y , H , v ) → F act ( X × Y , G × G m H , w ⊞ v ) called the exterior p ro duct . It is defined a s E ⊠ F := Res G × H G × G m H ( π ∗ 1 E ) ⊗ O X × Y Res G × H G × G m H ( π ∗ 2 F ) KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 31 Explicitly , w e hav e ( E ⊠ F ) − 1 := Res G × H G × G m H ( E − 1 ⊠ F 0 ⊕ E 0 ⊠ F − 1 ) ( E ⊠ F ) 0 := Res G × H G × G m H ( E 0 ⊠ F 0 ⊕ E − 1 ( χ ) ⊠ F − 1 ) Lemma 3.52. Assume that χ ′ − χ is not torsion. L et E 1 ∈ D abs [ fact ( X , G, w )] , F 1 ∈ D abs [ fact ( Y , H, v )] a nd let E 2 ∈ D abs [ F act ( X , G, w )] , F 2 ∈ D abs [ F act ( Y , H , v )] . T aking exterior pr o ducts induc es a natur al i s o morphism: ⊠ : M t ∈ Z Hom D abs [ Fac t ( X,G,w )] ( E 1 , E 2 [ − t ]) ⊗ k Hom D abs [ Fac t ( Y ,H ,v )] ( F 1 , F 2 [ t ]) → Hom D abs [ Fac t ( X × Y , G × G m H,w ⊞ v )] ( E 1 ⊠ F 1 , E 2 ⊠ F 2 ) . Pr o of. W e ma y assume that that all factorizatio ns are lo cally-free in o rder to simplify nota- tion for the deriv ed functors in the pro of. W e suppress t he subscripts on Hom’s and tensor pro ducts to help control notational g irth. W e hav e the f ollo wing chain of isomorphisms Hom( E 1 ⊠ F 1 , E 2 ⊠ F 2 ) := Hom(Res G × H G × G m H ( π ∗ 1 E 1 ) ⊗ Res G × H G × G m H ( π ∗ 2 F 1 ) , Res G × H G × G m H ( π ∗ 1 E 2 ) ⊗ Res G × H G × G m H ( π ∗ 2 F 2 ) ∼ = Hom(Res G × H G × G m H ( π ∗ 1 E 1 ) , H om (Res G × H G × G m H ( π ∗ 2 F 1 ) , Res G × H G × G m H ( π ∗ 1 E 2 ) ⊗ Res G × H G × G m H ( π ∗ 2 F 2 ))) ∼ = Hom(Res G × H G × G m H ( π ∗ 1 E 1 ) , Res G × H G × G m H ( π ∗ 1 E 2 ) ⊗ H om (Res G × H G × G m H ( π ∗ 2 F 1 ) , Res G × H G × G m H ( π ∗ 2 F 2 ))) ∼ = Hom(Res G × H G × G m H ( π ∗ 1 E 1 ) , Res G × H G × G m H ( π ∗ 1 E 2 ) ⊗ Res G × H G × G m H π ∗ 2 H om ( F 1 , F 2 )) ∼ = Hom( π ∗ 1 E 1 , Ind G × H G × G m H (Res G × H G × G m H ( π ∗ 1 E 2 ) ⊗ Res G × H G × G m H π ∗ 2 H om ( F 1 , F 2 ))) ∼ = Hom( π ∗ 1 E 1 , π ∗ 1 E 2 ⊗ Ind G × H G × G m H Res G × H G × G m H π ∗ 2 H om ( F 1 , F 2 )) ∼ = Hom( π ∗ 1 E 1 , π ∗ 1 E 2 ⊗ M l ∈ Z π ∗ 2 H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) (3.2) The second line is by definition. The third line is Corollary 3.28 i.e. tensor-Hom adjunction. The fourth line can b e seen by app ealing to Lemma 3.31 and asso ciativit y of tensor pro duct using the fact tha t F 1 is lo cally-free of finite rank to pull out a dual and put it back in. Note that Res G × H G × G m H comm utes with duals so the or der of op erations is not germane. The fifth line follo ws from the fact that the functors Res and π ∗ i are b oth monoidal, so t hey comm ute with ⊗ and H om . The sixth line uses the adjunction of Corolla ry 3.4 3. Since w e hav e assumed that χ ′ − χ is not torsion, w e hav e an isomorphism G × H /G × G m H ∼ = G m . As this quotien t is affine, Ind is exact and R Ind G H ∼ = Ind G H . The sev en th line is the pro jection form ula for the induction functor, Lemma 3.4 4. The eigh th line uses Lemma 2.18. Let q : Y → Sp ec k and p : X → Sp ec k b e t he structure morphisms. Contin uing with the isomorphisms from Equation (3.2) and using morphism spaces in D abs [ F act ( X , G, w )], we 32 BALLARD, F A VERO, A N D KA TZAR KO V ha v e Hom( E 1 ⊠ F 1 , E 2 ⊠ F 2 ) ∼ = Hom( E 1 , ( R π 1 ∗ π ∗ 1 E 2 ⊗ M l ∈ Z π ∗ 2 H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) R H ) ∼ = Hom( E 1 , E 2 ⊗ ( R π 1 ∗ M l ∈ Z π ∗ 2 H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) R H ) ∼ = Hom( E 1 , E 2 ⊗ ( R π 1 ∗ π ∗ 2 M l ∈ Z H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) R H ) ∼ = Hom( E 1 , E 2 ⊗ ( p ∗ R q ∗ M l ∈ Z H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) R H ) ∼ = Hom( E 1 , E 2 ⊗ k M l ∈ Z Hom( F 1 , F 2 [2 l ])( − lχ ) ⊕ Ho m ( F 1 , F 2 [2 l + 1 ])( − lχ )[ − 1]) ∼ = Hom( E 1 , M t ∈ Z E 2 [ − t ] ⊗ k Hom( F 1 , F 2 [ t ])) ∼ = M t ∈ Z Hom( E 1 , E 2 [ − t ]) ⊗ k Hom( F 1 , F 2 [ t ]) . The first line use s that the righ t adjoin t to π ∗ 1 is the comp osition ( R π 1 ∗ ) R H b y Corol- lary 2.24. The second line mora lly uses the pro jection form ula. Ho wev er, w e ha v e not pro- vided a pro jection form ula in this general con text. W e can w ork a round this b y deriving the t w o pro jection form ulas from Lemmas 2.8 a nd 2 .16 and rewriting the functor π ∗ 1 = ( π ′ 1 ) ∗ ◦ Res r where r : G × H → G denotes the pro jection homomo r phism and π ′ 1 : X × Y → X denotes the G × H equiv ariant pro jection where H a cts trivially on X . The fourth line uses fla t base c hange, L emma 2 .19. The fifth line uses Lemma 2.26 to pull the in v arian ts inside p ∗ . The sixth line comes from substitution of the isomorphism, ( R q ∗ M l ∈ Z H om ( F 1 , F 2 )( l ( χ ′ − χ )) ) R H ∼ = M l ∈ Z Hom( F 1 , F 2 [2 l ])( − lχ ) ⊕ Ho m( F 1 , F 2 [2 l + 1 ])( − lχ )[ − 1] . (3.3) Equation (3.3) is a consequenc e of Lemma 3.34 and the iden tity ( χ ′ ) = [2]. The sixth line uses that E 2 ⊗ • comm utes with copro ducts a nd a straigh tf o rw ard iden tification of the twis ts with shifts using ( χ ) = [2]. The final line fo llows since E 1 is a coheren t factorizatio n. By Prop osition 3.15 it is a compact ob ject, and therefore, Hom( E 1 , • ) comm utes with copro ducts. The tota l isomorphism gives an in ve rse to ⊠ .  Finally , let us define a v ersion of an in tegral transformation for fa ctorizations. Definition 3.53. Let P ∈ F act ( X × Y , G × G m H , ( − w ) ⊞ v ). Equip Y with the trivial G action to giv e it a G × H action in full. View π 2 : X × Y → Y as G × H -equiv ariant. Set Φ X → Y P : Fact ( X, G, w ) → F act ( Y , H , v ) E 7→  π 2 ∗ ( π ∗ 1 E ⊗ O X × Y Ind G × H G × G m H P )  G . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 33 W e will a lso denote t he asso ciated functor on the deriv ed categories by Φ X → Y P : D abs [ F act ( X , G, w )] → D abs [ F act ( Y , H , v )] E 7→  R π 2 ∗ ( L π ∗ 1 E L ⊗ O X × Y Ind G × H G × G m H P )  R G . The ob j ect P is called the k ernel of Φ X → Y P . View F as a factorization of 0 ∈ Γ( X , O X ( χ )) G as Υ F . Define the factorizatio n, ∇ ( F ) := Ind G × G m G G ∆ ∗ F := Υ Ind G × G m G G ∆ ∗ F . Set ∇ := ∇ ( O X ) . Lemma 3.54. Ther e is a natur al tr ansfo rmation of dg-functors Φ ∇ ( F ) → • ⊗ O X F inducing an isom orphism of derive d functors, Φ ∇ ( F ) ∼ = • L ⊗ O X F : D abs [ F act ( X , G, w )] → D abs [ F act ( X , G, w )] . In p articular, ∇ is the kernel of the identity functor. Pr o of. F or any E ∈ F act ( X , G, w ), we ha v e a nat ura l morphism  π 2 ∗  π ∗ 1 E ⊗ O X × X Ind G × G G ∆ ∗ F  G 7→  π 2 ∗ Ind G × G G ∆ ∗  ∆ ∗ Res G × G G π ∗ 1 E ⊗ O X F  G ∼ =  π 2 ∗ Ind G × G G ∆ ∗ ( E ⊗ O X F )  G ∼ = E ⊗ O X F The first line is from the pro jection fo r m ula for ∆ ∗ , ∆ ∗ , Lemma 2.8, a nd Res G × G G , Ind G × G G , Lemma 2 .1 6, applied comp o nen t-wise to a factorization. The second line comes from the isomorphism ∆ ∗ Res G × G G π ∗ 1 ∼ = ∆ ∗ π ∗ 1 ∼ = ( π 1 ◦ ∆) ∗ ∼ = Id where for t he first isomorphism w e view π 1 as G -equiv ar ia n t with resp ect to the diagonal action of G on X × X . F or the thir d line, we use tha t ( π 2 ∗ Ind G × G G ∆ ∗ ) G ∼ = Id as the functor, ( π 2 ∗ Ind G × G G ∆ ∗ ) G , is righ t adjoin t to ∆ ∗ Res G × G G π ∗ 2 ∼ = Id. Com bining the natural morphisms giv es the na tural transformation Φ ∇ ( F ) → • ⊗ O X F . The statement f o r t he deriv ed functors follows via the same a rgumen t, replacing the usual functors by their deriv ed v ersions, and noting that deriv ed pro jection fo r mula is an isomor- phism by Lemma 3.38.  Lemma 3.55. L et p : X → Sp ec k b e the structur e map. Ther e is a natur al tr ansfo rm ation of dg-functors ( p ∗ ∆ ∗ ( E ∨ ⊠ F )) G → Hom F act ( X ,G,w ) ( E , F ) inducing a natur al is o morphism ( R p ∗ L ∆ ∗ ( E ∨ ⊠ F )) R G ∼ = R Hom F act ( X ,G,w ) ( E , F ) if we as s ume E ∈ D abs [ fact ( X , G, w )] . 34 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. W e hav e ( p ∗ ∆ ∗ E ∨ ⊠ F ) G =  p ∗ ∆ ∗  Res G × G G × G m G π ∗ 1 E ∨ ⊗ O X × X Res G × G G × G m G π ∗ 2 F  G ∼ = ( p ∗ ( E ∨ ⊗ O X F )) G → ( p ∗ H om X ( E , F )) G ∼ = Hom F act ( X ,G,w ) ( E , F ) . The first line is b y definition. The second line follows from by distributing ∆ ∗ across the tensor pro duct then observing tha t w e ha ve an is omor phism Res G × G G × G m G π ∗ 1 ∼ = ( π ′ 1 ) ∗ where π ′ 1 : X × X → X is eq uiv arian t with resp ect to the first pro jection G × G m G → G and similarly , R es G × G G × G m G π ∗ 2 ∼ = ( π ′ 2 ) ∗ . Finally , π 1 ◦ ∆ ∼ = π 2 ◦ ∆ ∼ = 1 X . The t hir d line follo ws fr o m the natural map E ∨ ⊗ O X F → H om X ( E , F ) . The fourt h line is induced from the isomorphism of functors ( p ∗ H om X ( E , F )) G ∼ = Hom F act ( X ,G,w ) ( E , F ) of Lemma 3.3 4. The statemen t fo r the deriv ed functors follow s via analogous argumen ts replacing the usual functors by their deriv ed v ersion and using Lemma 3 .31 to kno w tha t the natural map E L ∨ ⊗ O X F → R H om X ( E , F ) is an isomorphism if E is coheren t.  Definition 3.56. The trace functo r on D abs [ F act ( X × X, G × G m G, ( − w ) ⊞ w )] is the functor L T r := ( R p ∗ L ∆ ∗ ) R G : D abs [ F act ( X × X , G × G m G, ( − w ) ⊞ w )] → D b (Qcoh Sp ec k ) . Lemma 3.57. Assume that ( G × G m G ) /G ∼ = K χ is finite. Ther e is an isomorph i s m of functors T r ∼ = R Hom F act ( X × X ,G × G m G, ( − w ) ⊞ w ) ( ∇ L ∨ , • ) on D abs [ fact ( X × X, G × G m G, ( − w ) ⊞ w )] . Pr o of. As ( G × G m G ) /G ∼ = K χ is finite, Ind G × G m G G preserv es coheren t G -equiv ariant shea v es. F or coheren t fa ctorizations, dualization is an an ti-equiv alence by Lemma 3.30. No w, we ha ve L T r = ( R p ∗ L ∆ ∗ ) R G ∼ = R Hom F act ( X ,G, 0) ( O X , L ∆ ∗ ( • )) ∼ = R Hom F act ( X ,G, 0) ( L ∆ ∗ ( • ) L ∨ , O X ) ∼ = R Hom F act ( X × X ,G × G m G,w ⊞ ( − w )) (( • ) L ∨ , Ind G × G m G G ∆ ∗ O X ) ∼ = R Hom F act ( X × X ,G × G m G, ( − w ) ⊞ w ) ( ∇ L ∨ , • ) . The first line is a definition. The second line is from the isomorphism of functors, R Hom F act ( X ,G, 0) ( O X , • ) ∼ = ( R p ∗ ) R G . The third line uses that ∆ ∗ comm utes with duals and • is assumed to b e coheren t. The fourth line is adjunction betw een Ind G × G m G G ∆ ∗ and L ∆ ∗ . The fif t h line uses dualization, coherence of • , and the definition o f ∇ .  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 35 In the pro cess o f pro ving a generation statemen t for categories of factorizations, w e will w an t to mak e use of some geometry . As suc h, w e need an alternate, more geometric, c harac- terization of t hese f actorization categories. This c hara cterization is due, in v ario us generalit y , to Eisen bud [Eis80], Buc hw eitz [Buc86], Orlo v [Orl09, Orl1 2], P olishc huk - V ain trob [PV10 ], and [Pos 11 ]. Let us r ecall the definition of the singularit y category . Definition 3.58. Let Y b e a sc heme of finite ty p e ov er k and let G b e an affine algebraic group acting on Y . Assume that Y has enough lo cally-free G -equiv arian t shea ve s. The G - equiv arian t singularit y category , or G -equiv ar ia n t stable category , of Y is the V erdier quotien t D sg G ( Y ) := D b (coh G Y ) / p erf G Y where p erf G Y is the thick sub category of lo cally-free G - equiv arian t shea v es of finite rank on Y . Let Z be a closed G -in v a rian t subset of Y , then we let D sg Z,G ( Y ) b e the k ernel o f the functor, j ∗ : D sg G ( Y ) → D sg G ( U ). Assume w e hav e a smo oth v ariety X equipp ed with an action of an affine algebraic group G and an inv a rian t section w ∈ Γ( X, L ) G for an in v ertible equiv a rian t sheaf, L . Se t Y = Z w to b e the v a nishing lo cus of w . Let i : Y → X denote the inclusion. Lemma 3.59. The scheme, Y , has enough lo c al ly-fr e e G -e quivariant she aves. Mor e o ver, every c oher ent G -e q uivariant sh e af on Y admi ts an epimorphis m fr om i ∗ V wher e V is lo c al ly- fr e e of fi nite r ank. Pr o of. As X is smo oth, it has enough lo cally-free G -equiv ar ia n t shea v es b y Theorem 2.29. Giv en an y coheren t G -equiv a rian t sheaf on Y , E , we can find a lo cally-free G -equiv ariant sheaf of finite rank, V , and an epimorphism, ψ : V → i ∗ E . T he morphism, i ∗ ψ : i ∗ V → i ∗ i ∗ E ∼ = E , remains an epimorphism as i ∗ is right exact.  Consider the functor, cok : [ vect ( X, G, w )] → D sg G ( Y ) E 7→ cok φ E 0 . Lemma 3.60. Assume that w is not ide n tic al ly zer o on an y c omp onent of X . The functor, cok , is wel l-define d an d exact. Pr o of. This is a sp ecial case of [PV10 , Lemma 3.1 2].  Lemma 3.61. Assume that w is no t id entic al ly zer o on any c omp on e nt of X . L et Z b e a close d G -inva riant subset of Y . The functor, cok , desc ends to the absolute derive d c ate gory, cok : D abs Z [ vect ( X , G, w )] → D sg Z,G ( Y ) . Pr o of. Let us treat the situation Z = Y first. In the case where G is trivial, this is [Orl12, Prop osition 3.2]. The same argumen t w orks with the inclusion of G . W e recall the argumen t for the conv enience of the reader. Let 0 → G q → E p → F → 0 36 BALLARD, F A VERO, A N D KA TZAR KO V b e an exact sequence of factorizations and let T b e the tota lization. Recall that T − 1 := G − 1 ⊗ O X L ⊕ E 0 ⊕ F − 1 T 0 := G 0 ⊗ O X L ⊕ E − 1 ⊗ O X L ⊕ F 0 φ T 0 :=   φ G 0 ⊗ O X L 0 0 q − 1 ⊗ O X L − φ E − 1 0 0 p 0 φ F 0   φ T − 1 :=   φ G − 1 ⊗ O X L 0 0 q 0 ⊗ O X L − φ E 0 ⊗ O X L 0 0 p − 1 ⊗ O X L φ F − 1   Consider the asso ciated exact sequence ov er coh G X 0 → G 0   q 0 φ G − 1   → E 0 ⊕ G − 1 ⊗ O X L   p 0 0 − φ E − 1 q − 1 ⊗ O X L   → F 0 ⊕ E − 1 ⊗ O X L  φ F − 1 p 1 ⊗ O X L  → F − 1 ⊗ O X L → 0 and let U b e the cok ernel of the map G 0 → E 0 ⊕ G − 1 ⊗ O X L . Let ( α 0 , α 1 ) : G − 1 ⊗ O X L ⊕ E 0 → U b e the epimorphism a nd  β 0 β 1  : U → F 0 ⊕ E − 1 ⊗ O X L b e the monomorphism. W e hav e a comm utativ e diagram E 0 ⊕ G − 1 ⊗ O X L U ⊕ G 0 ⊗ O X L F − 1 ⊕ E 0 ⊕ G − 1 ⊗ O X L F 0 ⊕ E − 1 ⊗ O X L ⊕ G 0 ⊗ O X L F − 1 F − 1 ⊗ O X L  α 0 α 1 0 φ G 0 ⊗ O X L    0 0 1 E 0 0 0 1 G − 1 ⊗ O X L     β 0 0 β 1 0 0 1 G 0 ⊗ O X L   φ T 0  1 F − 1 0 0   φ F − 1 p 1 ⊗ O X L 0  w with columns b eing short exact sequences. Th us, we ha v e an exact sequence of cok ernels, as coheren t shea v es on Y , 0 → cok  α 0 α 1 0 φ G 0 ⊗ O X L  → cok φ T 0 → i ∗ F − 1 ⊗ O X L → 0 . As i ∗ ( F − 1 ⊗ O X L ) is trivial in D sg G ( Y ), w e ha ve an isomorphism cok  α 0 α 1 0 φ G 0 ⊗ O X L  ∼ = cok φ T 0 in D sg G ( Y ). W e a lso hav e a commutativ e diagram KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 37 G 0 G 0 ⊗ O X L E 0 ⊕ G − 1 ⊗ O X L U ⊕ G 0 ⊗ O X L U U w  q 0 φ G − 1   0 1 G 0 ⊗ O X L   α 0 α 1 0 φ G 0 ⊗ O X L   α 0 α 1   1 U 0  1 U with columns b eing short exact sequences. Th us, we hav e an isomorphism of coheren t shea v es i ∗ G 0 ∼ = cok  α 0 α 1 0 φ G 0 ⊗ O X L  . Th us, cok  α 0 α 1 0 φ G 0 ⊗ O X L  ∼ = cok φ T 0 is trivial in D sg G ( Y ). This pro v es the statement when Z = Y . No w the general case follo ws from the case where Z = Y . Indeed, it is clear tha t cok comm utes with restriction to op en subsets. Th us, w e ha v e a comm utativ e diagram of functors D abs [ vect ( X , G, w )] D sg G ( Y ) D abs [ vect ( U, G, w | U )] D sg G ( Y ∩ U ) cok j ∗ j ∗ cok where j : U = X \ Z → X is the inclusion. Th us, cok induces a functor b et we en the k ernels of j ∗ . On the facto r ization side, this is D abs Z [ vect ( X , G, w )] while on the singularit y side this is D sg Z,G ( Y ).  Definition 3.62. Define the functor L cok : D abs Z [ fact ( X , G, w )] → D sg Z,G ( Y ) E 7→ cok V where V is a f actorization that has lo cally-free comp onen ts and is quasi-isomorphic to E . In the o ther direction, w e use the functor Υ. Lemma 3.63. Assume that w is not ide n tic al ly zer o on an y c omp onent of X . The functor, Υ , desc ends further to a functor Υ : D sg G ( Y ) → D abs [ fact ( X , G, w ) ] . Mor e over, if Z is a clo se d G -i n variant subset of Y , then Υ induc es a functor Υ : D sg Z,G ( Y ) → D abs Z [ fact ( X , G, w )] . Pr o of. W e treat the first statemen t. W e need to c hec k that Υ annihilates p erf G Y . By Lemma 3.59, it suffices to show that it annihilates i ∗ V fo r V a lo cally-free G -equiv aria nt 38 BALLARD, F A VERO, A N D KA TZAR KO V sheaf of finite rank on X . F or a coheren t G -equiv ariant sheaf on X , E , define a factor ization, H E := ( E , E , w , 1 E ). There is a short exact sequence 0 → H V ⊗ L − 1 → H V → Υ( i ∗ V ) → 0 . Th us, Υ( i ∗ V ) is quasi-isomorphic to t he cone H V ⊗ L − 1 → H V . It is straigh tforward to see that any H V is con tra ctible. Th us, Υ( i ∗ V ) is zero in D abs [ fact ( X , G, w )]. It is clear that Υ comm utes with restriction to op en subsets. Th us, w e hav e a commutativ e diagram of functors D sg G ( Y ) D abs [ vect ( X, G, w )] D sg G ( U ) D abs [ vect ( V , G, w | V )] Υ j ∗ U j ∗ V Υ where j U : U = Y \ Z → X and j V : V = X \ Z → X are the inclusions. Th us, Υ induces a functor b et we en the k ernels of j ∗ U and j ∗ V . On the fa cto r izat io n side, this is D abs Z [ vect ( X , G, w )] while on the singularit y side this is D sg Z,G ( Y ).  Prop osition 3.64. L et X b e a smo o th variety e quipp e d with an action of an affine a l g ebr aic gr oup G and an in variant se ction w ∈ Γ( X , L ) G for an inve rtible e quivariant she af, L . L et Y b e the vani s h ing lo cus of w and let Z b e a close d G -invariant s ubse t of Y . Assume that w is not iden tic al ly zer o on any c o mp one nt o f X . The functor, Υ : D sg Z,G ( Y ) → D abs Z [ fact ( X , G, w )] , is essen tial ly surje ctive. Pr o of. Let us c heck that Υ ◦ L cok ∼ = Id. Recall that, for a c oherent G -equiv ariant sheaf on X , E , w e define a factorization, H E := ( E , E , w , 1 E ). There is a short exact sequence of factorizations 0 → H V − 1 → V → Υ cok V → 0 for a fa ctorization with lo cally- f ree comp o nents. As H V is con tractible, V is quasi-isomorphic to Υ cok V .  Remark 3.65. One can pro v e that Υ is an equiv alence b y using argumen ts in the pro of [P os11, Theorem 2.7] and accounting f or a group action, see also [Orl12, Theorem 3.5] and [PV10, Theorem 3.14]. W e skip this, a s o nly essen tial surjectivity is necessary for the generation arg umen ts o f Section 4. W e finish b y recording an observ atio n concerning how Υ in teracts with exterior pro ducts. Lemma 3.66. L et X and Y b e smo oth v a rieties and let G and H b e affine algebr aic gr oups acting on, r esp e ctively, X and Y . L et w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H for char- acters χ : G → G m and χ ′ : H → G m . L et i w : Z w → X b e the zer o lo cus of w , i v : Z v → Y b e the zer o lo cus of v , and i w ⊞ v : Z w ⊞ v → X × Y b e the zer o lo cus of w ⊞ v . F o r any E ∈ coh G Z w and F ∈ coh H Z v , ther e ar e natur al isomorphisms of G × G m H - e quivarian t fa c torizations of w ⊞ v , (Υ E ) ⊠ (Υ F ) ∼ = Υ Res G × H G × G m H ( i w ∗ E ⊠ i v ∗ F ) . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 39 Pr o of. It is straigh tforward to c hec k that b oth of these factorizations are Υ Res G × H G × G m H ( π ∗ 1 i w ∗ E ⊗ O X × Y π ∗ 2 i v ∗ F ) .  4. Genera tion of equiv ariant de rived ca te gories T o iden tify the in ternal Hom dg-catego ries for equ iv arian t factorizations, w e will need to pro v e a g eneration statemen t for o ur candidate categories. In this se ction, w e la y the groundw ork and establish results to whic h w e will app eal in Section 5. F or a singular v ariet y , X , equipp ed with a G - a ction, w e w ant to find a nice set of gen- erators fo r the bounded deriv ed category of coheren t G -equiv ariant shea v es, D b (coh G X ). One natural appro a c h would b e to study generation in a compactly-generated triangulated category whose category of compact ob jects is exactly D b (coh G X ). Suc h categories do exist. Since D b (coh G X ) admits an enhancemen t to a dg-category , we could use the deriv ed cate- gory of dg- mo dules ov er the enhancemen t. Or, a more geometric construction due to Krause, [Kra05], uses the homoto p y category of injectiv e complexes of quasi-coheren t shea v es, in the non-equiv ariant setting. This could b e extended to handle our situation. Ho w ev er, this is not the approa ch w e c ho ose. Instead, w e follow the metho d of Rouquier in [Rou08] and fo cus on D b (Qcoh G X ), the b ounded deriv ed category of all quasi-coheren t G -equiv ariant sheav es on X . The category , D b (Qcoh G X ), is not compactly-g enerated as it do es not p ossess all copro ducts. Ho w ev er, the definition of a compact ob ject is still v alid and useful fo r D b (Qcoh G X ). Indeed [Ro u0 8, Prop osition 6.15] implies that the categor y of compact ob jects of D b (Qcoh G X ) is exactly D b (coh G X ). A further adv antage of studying D b (Qcoh G X ) comes fro m the fact t ha t lo cal cohomology of a coherent G -equiv ariant, or quasi-coheren t sheaf, is a lw a ys b ounded and quasi-coheren t, though usually nev er coheren t. Let us r ecall some notions of generation. Definition 4.1. Giv en a triangulat ed category , T , w e say that a sub cat ego ry , S , is thick if it is triangulated and closed under summands. Let S ′ b e another sub category . W e sa y that a sub categor y , S , generates S ′ , if the smallest full triangulated subcatego ry of T con taining S , a nd closed under finite copro ducts and summands, contains S ′ . If S ′ = T , we shall often say that S generates. W e say that S generate s S ′ up to infinite cop ro ducts if the smallest full triangulated sub- category of T con taining S , and closed under arbitrary copro ducts and summands, contains S ′ . If S ′ = T , we shall often say that S generates up to infinite copro ducts. In addition, recall that an ob ject C of T is called compact if Hom T ( C , • ) comm utes with all copro ducts. A triangulat ed category , T , is compactly- generated if it is co-complete, the compact o b jects form a set, and Hom T ( C , X ) = 0 for all compact ob jects, C , of T implies that X ∼ = 0. The follow ing is a no w-standard result on compactly-generated triangulat ed categories. Lemma 4.2. Assume T is a c o-c omplete triangulate d c ate gory an d the c omp act obj e cts in T fo rm a s et. Then, T is c om p actly-gene r ate d if and only if the c omp act obje cts gene r ate up to infinite c opr o ducts. Pr o of. See [Nee92] for a pro of.  The follow ing result generalizes one direction of Lemma 4.2. 40 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 4.3. L et T b e a triangulate d c ate gory. L et C , C ′ b e a sub c ate gory of c omp act obje cts of T . If C g ener ates C ′ up to in finite c op r o ducts, then C gener ates C ′ . Pr o of. See [BV03, Prop osition 2.2.4] or [R o u08, Corollary 3.13].  Let X b e a separated, reduced sc heme of finite t yp e o v er k a nd let G b e a n affine algebraic group acting on X , σ : G × X → X . W e record some generation r esults ab out the category , D b (coh G X ). Their statemen ts a nd pro ofs are in the sty le of Rouquier, [Ro u0 8], see also the argumen ts in [LP11 ]. Definition 4.4. Let E b e a quasi-coheren t G -equiv arian t sheaf on X . Let Z b e a G - in v aria n t subsc heme of X determined by a sheaf of ideals, I Z . W e sa y that E is scheme-theo retically supp o rted on Z if I Z E = 0. W e sa y that E is set-theo retically supp orted on Z if j ∗ E = 0 for the inclusion j : X \ Z → X . Let D b Z (Qcoh G X ) be the triang ula t ed sub catego ry o f D b (Qcoh G X ) consisting o f com- plexes whose cohomo lo gy shea ve s are set-theoretically supp orted on Z . Remark 4.5. Let l : Z → X b e the inclusion of Z into X . Then, a quasi-coheren t G - equiv arian t sheaf is sc heme-theoretically supp orted on Z if and only if it is in the essen tial image of l ∗ . Lemma 4.6. L et Z b e a G -invaria nt cl o se d s ubscheme of X . L et S , S ′ b e s ub c ate gories of D b Z (coh G X ) . If S gener ates S ′ up to infinite c opr o ducts in D b Z (Qcoh G X ) , then S gener ates S ′ . Pr o of. W e apply Lemma 4.3. The compact ob jects of D b Z (Qcoh G X ) are exactly the ob jects of D b Z (coh G X ) b y Prop osition 6.15 of [Rou08].  W e also record the following useful statement. Lemma 4.7. A ny quasi- c oher ent G -e quivariant she af, E , is gener ate d up to infi nite c opr o d- ucts by its c oher ent G - e quivariant subshe aves. Pr o of. An y quasi-coheren t G -equiv aria n t sheaf, E , is the colimit of it s coheren t G - equiv arian t subshea ves , see [Tho97, Lemma 1.4]. Th e colimit fits to in to an exact sequence , 0 → M F ⊂E F coheren t F → M F ⊂E F coherent F → colim F ∼ = E → 0 . Here the morphism, M F ⊂E F coherent F → M F ⊂E F coheren t F , is defined as f o llo ws. Giv en t w o coheren t equiv ariant subshea v es, F and F ′ , the morphism F → F ′ equals      0 if F 6⊆ F ′ − i if i : F ֒ → F ′ is a prop er inclusion 1 if F = F ′ . As suc h, E is isomorphic t o a cone o v er an endomorphism of a copro duct of coheren t equiv ari- an t shea v es. Thus , E is generated, up to infinite copro ducts, b y its coheren t G -equiv ariant subshea ves .  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 41 Let Z b e a G - in v aria n t closed subset of X and l : Z → X b e the inclusion. Lemma 4.8. The c ate go ry, D b Z (Qcoh G X ) , is gener ate d up to infinite c opr o ducts by the image of l ∗ : D b (coh G Z ) → D b (coh G X ) . Pr o of. If we can generate the cohomology shea ves of a b ounded complex, then w e can gener- ate said complex. So w e may reduce to g enerating all quasi-coheren t G -equiv ariant shea v es that are set-theoretically supp orted on Z . By Lemma 4.7, it suffices to generate all coheren t G -equiv ariant shea v es t ha t are set-theoretically suppor t ed on Z . Ho w ev er, for a coheren t sheaf set-theoretically supp or t ed on Z , there is an n suc h that I n Z E = 0. Thus , w e hav e a filtration 0 = I n Z E ⊂ I n − 1 Z E ⊂ · · · ⊂ I Z E ⊂ E . There are exact triangles I s Z E → I s − 1 Z E → F s → I s Z E [1] with F s sc heme-theoretically supp orted on Z . Th us, w e see can generate a coherent G - equiv arian t sheaf using coheren t G -equiv ar ian t shea v es sc heme-theoretically supp orted o n Z finishing the argumen t.  Before contin uing with t he course of the argumen t, let us recall the defin itio n of local cohomology for equiv a r ian t sh eav es. F or the arg umen ts of this section, lo cal cohomolo gy complexes provide an efficien t means of chopping complexes up with r esp ect to their supp ort. Let Z b e a G - in v ar ia n t closed subset of X and E b e a quasi-coheren t G -equiv a r ian t sheaf on X . Set H Z E ( U ) := { e ∈ Γ( U, E ) | ∃ n, I n Z e = 0 } Q Z E := j ∗ j ∗ E where j : X \ Z → X is the inclusion of the complemen t o f Z . There is a left exact sequence 0 → H Z E → E → Q Z E . Moreo v er, if E is flasque, there is a short exact sequence 0 → H Z E → E → Q Z E → 0 . The quasi-coheren t sheaf, H Z E , inherits the G -equiv ariant structure of E . Let R H Z : D b (Qcoh G X ) → D b (Qcoh G X ) R Q Z : D b (Qcoh G X ) → D b (Qcoh G X ) b e the asso ciated right-deriv ed functors. Note that there is a triangle of exact functors R H Z → Id → R Q Z → R H Z [1] . (4.1) W e no w use the ab o v e discussion to reduce generation arguments to the G -inv ar ia n t irre- ducible case. Lemma 4.9. L et X = Z 1 ∪ Z 2 b e the de c omp osition of X into two G -invariant close d subsets, Z 1 and Z 2 . L et l i : Z i → X denote the inclusion of Z i into X . The obje c ts in the essential image of the pushforwar d, l i ∗ : D b (coh G Z i ) → D b (coh G X ) , for i = 1 , 2 gener ate D b (Qcoh G X ) up to infinite c opr o ducts. 42 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. W e app eal to the exact triangle in Equation ( 4 .1) to see tha t E is generated b y R Q Z 1 E and R H Z 1 E . Note that R Q Z 1 E is supp orted o n the complemen t o f Z 1 . As X = Z 1 ∪ Z 2 , R Q Z 1 E is set-theoretically supp orted on Z 2 while R H Z 1 E is set-theoretically supp orted on Z 1 . Apply ing Lemma 4.8, finishes the argument.  W e will need to pa ss to the singular lo cus so we record a simple lemma. Lemma 4.10. L et σ : G × X → X b e an action of an affine algebr aic gr oup, G , on a r e duc e d, sep ar ate d scheme of fini te typ e, X . L et Sing X denote the close d subset o f X defi n e d by Sing X := { x ∈ X | O X,x is not r e gular } . Equip Sing X with the r e duc e d, induc e d structur e she af. Then, the action of G on X r estricts to Sing X . Pr o of. It suffices to ve rif y t ha t σ g := σ ( g , • ) : X → X preserv es Sing X for eac h g ∈ G . Ho w ev er, σ g is an automorphism of X and hence mus t preserv e Sing X .  W e will need to use normality of a v ariet y whic h is not guara n teed by the assumptions of the pro ceeding lemmas. W e take a momen t to commen t on lifting the action of G to the normalization in an equiv aria n t manner. Lemma 4.11. L et ν : e X → X b e the normalization of X . Ther e is a unique a ction of G on e X mak i n g ν G -e quivariant. Pr o of. Since G is smo oth, G × e X is normal. The map σ ◦ (1 × ν ) : G × e X → X is dominant and therefore fa ctors uniquely t hr o ugh ν . Let ˜ σ : G × e X → e X b e the unique lift. The uniqueness of the lift also allows one to verify tha t e σ is an a ction of G on X . With this lift, ν : e X → X b ecomes G -equiv aria n t.  Lemma 4.12. L et f : X → Y b e a G -e quivariant morphis m such that X p ossesses an f -ampl e fam i l y of e quivariant line bund les, L α , α ∈ A . Th e ful l sub c ate gory of D b (coh G X ) c onsi s ting of obje cts of the form L α ⊗ f ∗ E for E ∈ coh G Y and α ∈ A gener ates al l c oher ent G -e quivariant she av es of lo c a l ly-finite pr oje ctive dimension in Qcoh X . Mor e o v e r, if Y p ossesse s enough lo c al ly-fr e e G -e quiva ri a nt she av e s of fi n ite r ank, then the ful l sub c ate gory of D b (coh G X ) c onsisting of obje cts o f the form L α ⊗ f ∗ V for V ∈ coh G Y l o c al ly-fr e e and α ∈ A gener ates a l l c oher ent G -e quivariant she a v es of lo c al ly- finite pr oje ctive dimension in Qcoh X . Pr o of. Let E b e a coheren t G -equiv ariant sheaf of lo cally-finite pro j ective dimension in Qcoh X . There is a finite set A ′ ⊆ A suc h that M α ∈ A ′ L α ⊗ f ∗ f ∗ ( L − 1 α ⊗ E ) → E is an epimorphism as L α is an f -ample family . F or eac h α , there exists a coherent G - equiv arian t subsheaf, F α , of f ∗ ( L − 1 α ⊗ E ) suc h that the restriction of t he co-unit morphism remains an epimorphism M α ∈ A ′ L α ⊗ f ∗ F α → E . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 43 If w e assume t ha t Y p ossesses enough lo cally-free G -equiv ariant shea v es of finite rank, there is a lo cally-free G -equiv ariant sheaf, V α , on Y and an epimorphism, V α → F α . Pulling bac k and comp osing, w e hav e a n epimorphism M α ∈ A ′ L α ⊗ f ∗ V α → E . T aking ke rnels a nd iterating this pro cess w e may construct an exact sequence · · · → G s → · · · → G 1 → E → 0 where eac h G i is a sum o f ob jects of the L α ⊗ f ∗ F α for some finite set of α ∈ A ′ . Moreov er, if Y p ossesses enough lo cally-free equiv ariant shea ve s, w e may tak e E to b e lo cally-free. Let K s b e the ke rnel of G s → G s − 1 . W e ha ve a short exact sequence 0 → G s → · · · → G 1 → E → 0 . This represen ts an elemen t of Ext s Qcoh G X ( E , K s ) . As E is an ob j ect of lo cally-finite pro jectiv e dimension in Qcoh X , f rom Lemma 2.32, there is an s 0 suc h that Ext s Qcoh G X ( E , K s ) = 0 for s ≥ s 0 . T ak e s lar g er than s 0 . Th en, there is a quasi-isomorphism, K s [ s ] ⊕ E ≃ G s → · · · → G 1 . Th us, E is generated b y o b jects o f the form L α ⊗ f ∗ E for E ∈ coh G Y and α ∈ A . If Y ha s enough equiv arian t lo cally-free shea ves , then E is generated b y ob j ects of t he form L α ⊗ f ∗ E for E ∈ coh G Y lo cally-free and α ∈ A .  Next, w e demonstrate ho w to pro duce a set o f generators from a set of generators o f the singular lo cus of X . Lemma 4.13. L et X b e a divisorial variety. L et Sing X b e the singular lo cus of X with its r e duc e d, induc e d structur e she af. L et l : Sing X → X denote the in clusion. L et Y b e a close d subset of X that is G -invariant. Then, the sub c ate gory, whose obje cts ar e • ν ∗ V wher e V is a lo c al ly-fr e e G -e quivariant she aves of fi nite r ank on e X plus • the obje cts in the essential image o f the pushfo rw ar d, l ∗ : D b (coh G Y ∩ Sing X ) → D b (coh G X ) , gener a te the sub c ate g o ry D b Y (Qcoh G X ) up to infinite c opr o ducts. Mor e over, if one assumes that X has en ough lo c al ly-fr e e G -e quivariant she aves, then the sub c ate gory, whose obje cts ar e • lo c al ly-fr e e G -e quiva ri a n t sh e aves of finite r ank on X , plus • the obje cts in the essential image o f the pushfo rw ar d, l ∗ : D b (coh G Y ∩ Sing X ) → D b (coh G X ) , gener a tes D b Y (Qcoh G X ) up to infinite c opr o ducts. 44 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. T o generate a b ounded complex, it suffices to g enerate its cohomology sheav es. There- fore, w e ma y reduce to generating quasi-coheren t G -equiv ariant shea v es set-theoretically sup- p orted o n Y up to infinite copro ducts. By Lemma 4.7, it then suffices to generate coheren t G -equiv ariant subshea v es set-theoretically supp orted o n Y up to infinite copro ducts. Let E b e a coheren t G - equiv arian t sheaf. Complete the unit of the adjunction, E → ν ∗ ν ∗ E to an exact triangle E → ν ∗ ν ∗ E → D → E [1] . Since ν is an isomorphism on U , D is set-theoretically supp orted on Sing X ∩ Y . Since D is coherent it is generated b y the essen tial image of l ∗ b y Lemmas 4.6 and 4.8 . Th us, to generate E it suffices to generate ν ∗ ν ∗ E . Note also that if E is a lo cally-free sheaf of finite rank, then w e generate ν ∗ ν ∗ E as w e are allow ed to use E . Set Z = ν − 1 (Sing X ) and U = e X \ Z . If V is a lo cally-free G -equiv a r ian t sheaf of finite rank on e X , t hen w e hav e an exact tria ngle, R H Z ∩ ν − 1 ( Y ) V → V → R Q Z ∩ ν − 1 ( Y ) V → R H Z ∩ ν − 1 ( Y ) V [1] . Applying ν ∗ , w e hav e another exact triang le, ν ∗ R H Z ∩ ν − 1 ( Y ) V → ν ∗ V → ν ∗ R Q Z ∩ ν − 1 ( Y ) V → ν ∗ R H Z ∩ ν − 1 ( Y ) V [1] . The set-theoretic supp ort of ν ∗ R H Z ∩ ν − 1 ( Y ) V is con tained in Sing X ∩ Y a s ν ( Z ) = Sing X . By Lemma 4.8 , ν ∗ R H Z ∩ ν − 1 ( Y ) V is generated up to infinite copro ducts b y the essen tial image of l ∗ . Thus, ν ∗ R Q Z ∩ ν − 1 ( Y ) V is generated up to infinite copro ducts by t he full sub category consisting of ν ∗ V where V is a lo cally- f ree G -equiv ariant on e X and the essen tial imag e of l ∗ . Let E b e a coheren t G -equiv ariant sheaf o n X supp orted on Y . W e hav e a triang le, R H Z ∩ ν − 1 ( Y ) ν ∗ E → ν ∗ E → R Q Z ∩ ν − 1 ( Y ) ν ∗ E → R H Z ∩ ν − 1 ( Y ) ν ∗ E [1] , Applying ν ∗ , w e get a no ther triangle, ν ∗ R H Z ∩ ν − 1 ( Y ) ν ∗ E → ν ∗ ν ∗ E → ν ∗ R Q Z ∩ ν − 1 ( Y ) ν ∗ E → ν ∗ R H Z ∩ ν − 1 ( Y ) ν ∗ E [1] , w e see that to generate ν ∗ ν ∗ E it suffices to generate ν ∗ R H Z ∩ ν − 1 ( Y ) ν ∗ E and ν ∗ R Q Z ∩ ν − 1 ( Y ) ν ∗ E . The complex, ν ∗ R H Z ∩ ν − 1 ( Y ) ν ∗ E , is set-theoretically supp orted on Sing X ∩ Y . By Lemma 4.8 , ν ∗ R H Z ∩ ν − 1 ( Y ) ν ∗ E is generated up to infinite copro ducts b y t he essen tial image of l ∗ . Th us, w e reduce to generating ν ∗ R Q Z ∩ ν − 1 ( Y ) ν ∗ E . As ν is an affine morphism, the pullback o f an ample family remains an ample family . Using Theorem 2.29, w e may construct an exact complex · · · → V s → · · · → V 2 → V 1 → ν ∗ E → 0 . where each V i is a lo cally-free G -equiv a rian t sheaf of finite rank. Apply j ∗ where j : U = e X \ ( Z ∩ ν − 1 ( Y )) → e X is the inclusion. As j ∗ is exact, the complex · · · → j ∗ V s → · · · → j ∗ V 2 → j ∗ V 1 → j ∗ ν ∗ E → 0 remains exact. Let K s b e the k ernel of the ma p j ∗ V s → j ∗ V s − 1 . Th e exact sequenc e 0 → K s → j ∗ V s → · · · → j ∗ V 2 → j ∗ V 1 → j ∗ ν ∗ E → 0 represen ts an elemen t of Ext s Qcoh G U ( j ∗ ν ∗ E , K s ). As j ∗ ν ∗ E is supp o r t ed on the smo oth subset, U ⊃ ˜ X \ Z , this v anishes f o r s ≥ s 0 , for some s 0 , by Lemma 2.3 2. Conseque ntly , there is a quasi-isomorphism j ∗ ν ∗ E ⊕ K s [ s ] ≃ j ∗ V s → · · · → j ∗ V 2 → j ∗ V 1 . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 45 Applying ν ∗ R j ∗ , w e see that ν ∗ R Q Z ∩ ν − 1 ( Y ) ν ∗ E is generated by ν ∗ R Q Z ∩ ν − 1 ( Y ) V i for 1 ≤ i ≤ s . W e ha v e already o bserv ed that w e can generate ν ∗ R Q Z ∩ ν − 1 ( Y ) V up to infinite copro ducts when V is lo cally-free of finite rank. W e conclude that w e can generate ν ∗ R Q Z ∩ ν − 1 ( Y ) ν ∗ E using the sub cat ego ry consisting of ν - pushforw ards of G -equiv a r ia n t in ve rtible sheav es o n e X and the essen tia l image of l ∗ up to infinite copro ducts finishing the argument. If we assume that X has enough lo cally-free G -equiv a rian t sheav es, then w e can rep eat the previous argumen t replacing e X by X .  Corollary 4.14. Assume that X p ossesse s enough lo c al ly-fr e e G -e quivariant she ave s. L et Sing X b e the singular lo cus of X with its r e duc e d, induc e d structur e sh e af. L et l : Sing X → X denote the inclusion. L et Y b e a close d subset. The sub c a te gory, D b Y (coh G X ) , is gener ate d by al l lo c al ly-fr e e G -e quiva riant she aves of finite r ank o n X a n d al l obje cts in the esse n tial image of l ∗ : D b (coh G Sing X ∩ Y ) → D b (coh G X ) . Pr o of. The second part of Lemma 4.13 states that the sub catego r y consisting of a ll lo cally- free coheren t G -equiv ariant shea ves and the essen tial imag e of l ∗ generates D b Y (Qcoh G X ) up to infinite copro ducts. Thus , by Lemma 4.6, the sub category consisting o f all lo cally-free G - equiv arian t shea v es of finite rank on X and the essen tial image of l ∗ generates D b Y (coh G X ).  Remark 4.15. One ma y use induction b y iterativ ely passing to singular lo ci to pro duce a sligh tly smaller generating sub category for D b (coh G X ). Definition 4.16. Assume that X has enough G - equiv arian t lo cally-free shea ve s. Let U b e an op en G -in v arian t subset of X and let Pe rf U,G X b e the full sub cat ego ry o f D b (Qcoh G X ) whose restriction to D b (Qcoh G U ) is quasi-isomorphic to a b ounded complex of lo cally-free G -equiv ariant shea v es. L et p erf U,G X b e the sub category of P erf U,G X consisting of complexes quasi-isomorphic to b ounded complexes o f coheren t shea v es. Lemma 4.17. Assume that X has enough G -e quivarian t lo c al ly-fr e e she aves. The c ate g o ry, P erf U,G X , is gener ate d up to infinite c opr o ducts by lo c al ly-fr e e G - e quivariant she aves of finite r ank a n d the image of l ∗ : D b (coh G Y ) → D b (coh G X ) wher e Y = X \ U . Pr o of. Let E ∈ P erf U,G X . Using the assumption that G has enough lo cally-free G -equiv a rian t shea v es and a standard arg umen t (see for the example the pro of of Lemma 4.12), w e ma y construct a b ounded complex of lo cally-free shea v es P and a morphism P → E whose cone is a quasi-coheren t sheaf that is lo cally-free on U . Since, b y Lemma 4.7, we ma y generate b ounded complexes of lo cally-free shea ve s P up to infinite copro ducts with lo cally- free coherent shea v es, it suffices to generate this cone. W e con tinue with the assumption that E is a quasi-coheren t sheaf. There is a n exact triangle R H Z E → E → R Q Z E → R H Z E [1] . It suffices to generate R H Z E and R Q Z E . W e can generate R H Z E up to infinite copro ducts b y the image of l ∗ b y Lemma 4.8. Th us, we reduce to g enerating R Q Z E . 46 BALLARD, F A VERO, A N D KA TZAR KO V Using the assumption of ha ving enough G -equiv a rian t lo cally-free shea ves , we ma y con- struct an exact complex · · · → V s → · · · → V 1 → E → 0 with V i b eing lo cally-free G - equiv a rian t shea v es. Apply j ∗ to get an exact complex · · · → j ∗ V s → · · · → j ∗ V 1 → j ∗ E → 0 . Let K s b e the k ernel of the ma p j ∗ V s → j ∗ V s − 1 . Th e exact sequenc e 0 → K s → j ∗ V s → · · · → j ∗ V 1 → j ∗ E → 0 represen ts an elemen t of Ext s Qcoh G U ( j ∗ E , K s ). As j ∗ E is p erfect, this v anishes for s ≥ s 0 , for some s 0 , b y Lemma 2.3 2. Assuming s ≥ s 0 , there is a quasi-isomorphism j ∗ E ⊕ K s [ s ] ≃ j ∗ V s → · · · → j ∗ V 2 → j ∗ V 1 . Pushing this forw ard via R j ∗ sho ws that R Q Z E is g enerated by R Q Z V for V lo cally-free. Th us, w e reduce to generating R Q Z V for V lo cally-free. But, for suc h a V , there is an exact triangle, R H Z V → V → R Q Z V → R H Z V [1] . and w e ma y generate R H Z V and V up t o infinite copro ducts b y lo cally-free G -equiv aria nt shea v es of finite rank and the imag e of l ∗ b y Lemma 4.8.  Corollary 4.18. Assume that X has e n ough G -e quivariant lo c al l y-fr e e she aves. The c a t- e gory, p erf U,G X , is gener a te d by lo c al ly-fr e e G -e quivariant she aves of finite r ank and the image of l ∗ : D b (coh G Y ) → D b (coh G X ) wher e Y = X \ U . Pr o of. This follows f r om Lemma 4.17 by applying Lemma 4.6.  The follow ing lemma shows that generator s restrict under c hanging of t he group. Lemma 4.19. L et X b e a sep ar ate d , r e duc e d, divis orial scheme of finite typ e e quipp e d with a G action. Assume that G/H is a ffi n e. Then, D b (coh H X ) is gener ate d by the essential image of Res G H : D b (coh G X ) → D b (coh H X ) . Pr o of. Recall that Res G H factors as ι ∗ ◦ α ∗ where α : G H × X → X is induced by the action of G on X and ι : X → G H × X is induced b y the unit elem ent of G . The functor, ι ∗ : D b (coh G G H × X ) → D b (coh H X ), is an equiv alence b y Lemma 2.13 so it suffices to sho w that the image of α ∗ : D b (coh G X ) → D b (coh G G H × X ) generates. W e f a ctor α as G H × X G/H × X X α Φ p KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 47 where Φ : G H × X → G/H × X ( g , x ) 7→ ( g H , σ ( g , x )) and p is the pro j ection. The morphism, Φ, is an isomorphism so we reduce to c hec king that the image of p ∗ : D b (coh G X ) → D b (coh G G/H × X ) generates. Let us handle the case that dim X = 0. Under our standing assumptions X is reduced, therefore X is smo oth. Since G/H is affine, O G/H is ample a nd is naturally equiv ariant. Lemma 4.12 applies directly to show that the essen tial image of p ∗ generates No w assume we ha v e prov en the statemen t for X with all comp onen ts of X having dimen- sion < n a nd a ssume that dim X = n . By Corollary 4.14, D b (coh G G/H × X ) is generated by ν ′ ∗ V where V a r e lo cally-free G -equiv ariant shea v es of finite rank, ν ′ : ^ G/H × X → G/H × X is the nor ma lizat io n, and the essen tial image of l ∗ : D b (coh G Sing G/H × X ) → D b (coh G G/H × X ) . Since G/H is smoot h Sing ( G/H × X ) = G/H × Sing X . Applying the induction h yp o t hesis, the essen tial image o f p ∗ : D b (coh G Sing X ) → D b (coh G Sing G/H × X ) generates. Th us, the essen tial image of p ∗ : D b (coh G X ) → D b (coh G G/H × X ) generates the essen tial image of l ∗ . W e are left to generate the coherent G - equiv arian t shea v es, ν ′ ∗ V , f o r V lo cally-free G -equiv aria n t shea v es o f finite rank on the normalization. Since G/H is smo oth, ^ G/H × X ∼ = G/H × e X . W e hav e a comm utativ e diagram. G/H × e X G/H × X e X X 1 × ν ˜ p p ν Applying Lemma 4 .12, since O G/H × e X is e p -ample, an y lo cally-free G -equiv ariant sheaf of finite rank, V , is generated b y the essen tial image of e p ∗ . Therefore, ν ′ ∗ V = (1 × ν ) ∗ V is g enerated b y the essen tial image o f (1 × ν ) ∗ ◦ e p ∗ . As p is flat, (1 × ν ) ∗ ◦ e p ∗ ∼ = p ∗ ◦ ν ∗ . Th us, ν ′ ∗ V is generated by the essen tial image o f p ∗ finishing the pro of.  The next prop osition demonstrates that exterior pro ducts g enerate in the equiv ariant setting. Prop osition 4.20. L et G and H b e affine algebr aic gr oups, an d X and Y b e sep ar ate d, r e duc e d, divis o rial schemes of fi nite typ e e quipp e d w ith actions G × X → X and H × Y → Y . The sub c ate gory c onsisting of E ⊠ F for E ∈ coh G X and F ∈ coh H Y gener ates D b (Qcoh G × H X × Y ) up to infinite c opr o ducts. 48 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. By Lemma 4.7, it suffices to generate all coheren t G × H -equiv aria nt shea v es up to infinite copro ducts. W e pro ceed by induction o n the dimension of X × Y . Assume t hat dim X × Y = 0. The morphism, h := f × g : X × Y → Sp ec k × Sp ec k ∼ = Sp ec k , coming from the pro duct of the structure maps, f : X → Sp ec k and g : Y → Sp ec k , is affine and G × H -equiv ariant therefore O X × Y is ample. By Lemma 2.32, an y ob ject of coh X × Y has lo cally-finite pro jectiv e dimension since X × Y is smo oth. Apply ing Lemma 4.12, w e see that the essen tial image of h ∗ generates D b (coh G × H X × Y ). Moreo v er, h ∗ ( E ⊠ F ) ∼ = f ∗ E ⊠ g ∗ F . So v alidit y of the claim in the case X = Y = Spec k implies v a lidity o f the claim for all X × Y of dimension zero. F or a finite dimensional G × H represen tation, the ev aluation morphism Hom Qcoh H Spec k (Res G × H H V , V ) ⊗ k Res G × H H V → V is an epimorphism. Here, Hom Qcoh H (Res G × H H V , V ) is a G -represen tation. By Lemma 2.32 the category of G -represen tations has finite global dimension. Th us, there are enough exterior pro ducts to resolv e any G × H -represen tation finishing the base case of the induction. Assume w e hav e prov en the statemen t whenev er dim X × Y < n and let us treat a pro duct with dim X × Y = n . F rom Lemma 4 .1 3, D b (Qcoh G × H X × Y ) is generated up to infinite copro ducts b y ν ∗ V for lo cally-free G × H -equiv a r ia n t shea v es of finite rank on the normalization ^ X × Y and the essen t ia l image of l ∗ : D b (coh G × H Sing X × Y ) → D b (coh G × H X × Y ) . The singular lo cus of X × Y is the unio n of t w o closed subsets: (Sing X ) × Y and X × Sing Y . F rom Lemma 4.9, D b (Qcoh G × H Sing ( X × Y )) is g enerated up to infinite copro ducts b y the images of D b (coh G × H (Sing X ) × Y ) and D b (coh G × H X × Sing Y ) under pushforw ard. Using the induction h yp othesis, exterior pro ducts generate b oth D b (coh G × H (Sing X ) × Y ) and D b (coh G × H X × Sing Y ). Th us, the essen tial image of l ∗ is generated up to infinite copro ducts b y exterior pro ducts. Ne xt, w e turn to lo cally-free equiv arian t sheav es pushed f o rw ard from the normalization. The normalization of X × Y is the pro duct of the normalizations, e X × e Y , [E G A IV.2, Corollary 6.14.3]. W e ha v e assumed t ha t X a nd Y ha ve ample fa milies. Since normalization is affine, e X and e Y ha ve ample fa milies giv en b y the pullbacks from X and Y , respectiv ely . The exterior pro duct o f ample families is again an ample family . Since e X × e Y is normal, taking sufficien t p o w ers of eac h line bundle, w e get an ample f a mily where all the line bundles admit equiv a rian t structures, [Tho97 , Lemma 2.10]. Th us, for any lo cally-free G - equiv arian t sheaf, V , there is an exact sequence o f equiv ariant sheav es · · · → F s → · · · → F 1 → V → 0 where each F i is an exterior pro duct. The lo cally-fr ee sheaf V has lo cally-finite pro jectiv e dimension, and th us is a summand of the complex F s → · · · → F 1 for s sufficien tly large. W e see that exterior pro ducts generate all lo cally-free equiv a r ia n t shea v es on e X × e Y . Pushing f o rw ard an exterior pro duct under the normalization, map KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 49 e X × e Y → X × Y , yields another exterior pro duct via the pro jection formula and flat base c hange. Thus , exterior pro ducts also generate ν ∗ V for lo cally-free G × H -equiv arian t shea v es of finite ra nk, V , on the normalization. This finishes the pro of.  Corollary 4.21. L et G and H b e affine algebr aic gr oups, X and Y sep ar ate d, r e duc e d schemes of finite t yp e e quipp e d with action s G × X → X and H × Y → Y . Th e sub c at- e gory c onsis ting of E ⊠ F for E ∈ coh G X and F ∈ coh H Y gener ates D b (coh G × H X × Y ) . Pr o of. This follows f r om Prop osition 4.20 by applying Lemma 4.6.  Next, w e turn our a tten tion to sho wing that exterior pro ducts of factorizations g enerate the a ppropriate category . W e will demonstrate suc h generation for exterior pro ducts in the singularit y category and then use that to pass to f a ctorizations. Lemma 4.22. L et X and Y b e smo oth v a rieties and let G and H b e affine algebr aic gr oups acting o n, r esp e ctively, X and Y . L et w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H for char ac ters χ : G → G m and χ ′ : H → G m . L et i w : Z w → X b e the zer o lo cus of w , i v : Z v → Y b e the zer o lo cus of v , and i w ⊞ v : Z w ⊞ v → X × Y b e the zer o lo cus of w ⊞ v . L et l : Sing Z w × Sing Z v → Z w ⊞ v b e the inclusion. Obje c ts of the form l ∗ Res G × H G × G m H ( E ⊠ F ) for E ∈ coh G Sing Z w and F ∈ coh H Sing Z v gener a te D sg Z w × Z v ,G × G m H ( Z w ⊞ v ) . Pr o of. By Coro llary 4.18, the in ve rse image of D sg Z w × Z v ,G × G m H ( Z w ⊞ v ) in D b (coh G × G m H Z w ⊞ v ) is generated by lo cally-free G - equiv ariant shea v es and ob jects of D b (coh G × G m H Z w ⊞ v ) set- theoretically supp orted o n Z w × Z v . By Corolla ry 4.14, lo cally-free G -equiv ar ia n t shea ve s of finite rank on Z w ⊞ v and ob jects in the image of l ∗ for the inclusion l : Sing Z w ⊞ v ∩ ( Z w × Z v ) → Z w ⊞ v generate D b Z w × Z v (coh G × G m H Z w ⊞ v ). So, in com binatio n, w e can gener- ate D sg Z w × Z v ,G × G m H ( Z w ⊞ v ) using the essen tial image of l ∗ . It remains t o c hec k t ha t exterior pro ducts generate D b (coh G × H Sing Z w ⊞ v ∩ ( Z w × Z v )). Note that Sing Z w ⊞ v ∩ ( Z w × Z v ) = Sing Z w × Sing Z v . By Lemma 4.19, the essen tia l image of Res G × H G × G m H : D b (coh G × H Sing Z w × Sing Z v ) → D b (coh G × G m H Sing Z w × Sing Z v ) generates. Notice also that Sing Z w × Sing Z v is divisorial simply by pulling bac k the ample family . Henc e, w e may apply Corollary 4.21, to see that D b (coh G × G m H Sing Z w × Sing Z v ) is generated b y E ⊠ F for E ∈ coh G Z w and F ∈ coh H Z v .  Lemma 4.23. L et X and Y b e smo oth v a rieties and let G and H b e affine algebr aic gr oups acting on, r esp e ctively, X and Y . L et w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H for char- acters χ : G → G m and χ ′ : H → G m . L et i w : Z w → X b e t he zer o lo cus of w and le t i v : Z v → Y b e the zer o lo cus of v . The derive d c ate gory of c oher en t fa ctorizations supp orte d on Z w × Z v , D abs Z w × Z v [ fact ( X × Y , G × G m H , w ⊞ v )] , is gener ate d b y exterior pr o d ucts. Pr o of. By Lemma 4 .22, ob jects of the form l ∗ Res G × H G × G m H E ⊠ F for E ∈ coh G Sing Z w and F ∈ coh H Sing Z v generate D sg Z w × Z v ,G × G m H ( Z w ⊞ v ). By Lemma 3.66, for an y E ∈ coh G Z w and F ∈ coh H Z v , there a re natural isomorphisms of G × G m H -equiv aria nt factorizations of w ⊞ v , (Υ E ) ⊠ (Υ F ) ∼ = Υ Res G × H G × G m H ( i w ∗ E ⊠ i v ∗ F ) . Finally , by Prop osition 3.64, Υ is ess entially surjectiv e. Th us, (Υ E ) ⊠ (Υ F ) for E ∈ coh G Z w and F ∈ coh H Z v generate D abs Z w × Z v [ fact ( X × Y , G × G m H , w ⊞ v )].  50 BALLARD, F A VERO, A N D KA TZAR KO V 5. Bimodule and functor ca tegorie s for equiv ariant f actoriza tions 5.1. Morita pro ducts and functor categories for factor ization categories. W e now turn to studying tensor pro ducts and internal-homomorphism dg- categories of factorization categories in t he homoto p y category o f dg-cat ego ries, Ho(dg- cat k ). The main r eferences for bac kground are [Kel06, T o¨ e07 ]. Definition 5.1. A dg-functor, f : C → D , is a quasi-equiva l ence if H • ( f ) : H • (Hom C ( c, c ′ )) → H • (Hom D ( f ( c ) , f ( c ′ ))) is an isomorphism fo r all c, c ′ ∈ C and [ f ] : [ C ] → [ D ] is essen tially surjectiv e. Let Ho(dg-cat) k denote the lo calization of dg-cat k at the class of quasi-equiv a lences. This category is called the homotop y catego ry of dg-categories . If C and D are quasi-equiv a len t, w e shall write C ≃ D . Definition 5.2. Let D b e a dg-category . The categor y of left D -mo dules, denoted D -Mo d, is the dg-categor y of dg- functors, D → C ( k ) where C ( k ) is the dg- category of chain complexes of v ector spaces ov er k . The category of right D -mo dules is the category of left D op -mo dules. Eac h ob ject d ∈ D pro vides a represen table right mo dule h d : D op → C ( k ) d ′ 7→ Hom D ( d ′ , d ) . W e denote t he dg-Y oneda em b edding by h : D → D op -Mo d. The V erdier quotient of [ D -Mo d] by the sub category of a cyclic mo dules is called the derived catego ry of D-mo dules and is denoted b y D [ D -Mo d]. The smallest thic k sub categor y of D[ D op -Mo d] con taining the image o f [ h ] is called the catego ry of p erfect D-mo dules and is denoted by p erf ( D ). Remark 5.3. Throughout the pap er, with the exception of the pro of of Coro llary 5.18, w e will take C to b e a quasi-small dg- category . A dg- category D is quasi-small if [ D ] is essen tially small. In t his case, w e can c ho ose a small full sub category of D quasi-equiv alen t to D a nd w ork with that sub category to define categories of mo dules and bimo dules. This sidesteps certain set-theoretic issues in the quasi-small case. How ever, doing this in eac h example is tedious and not edifying. So w e will suppress these arg uments throughout the pap er. When C is not quasi-small, but only U - small, one only considers U -small dg-mo dules. W e suppress any of the set-theoretic issues as w e do not ascend to a higher univ erse in the pro o f of Corollary 5.18. Definition 5.4. Let C and D be tw o dg-categories. A quasi-functo r a : C → D is a dg - functor a : C → D op -Mo d suc h that for eac h c ∈ C , a ( c ) is quasi-isomorphic to h d for some d ∈ D . Note that a quasi- functor corresp onds to a bimo dule a ∈ C ⊗ k D op -Mo d. Also not e, that any quasi-functor induces a functor on homotopy categories whic h we denote b y [ a ] : [ C ] → [ D ]. In particular, it make s sense to extend the definition of quasi-equiv alence to quasi-functors. Lemma 5.5. The isomorphism classes of morphisms fr om C to D i n Ho(dg-cat) k ar e in bije ction with isomorphism class e s of quasi-functors fr om C to D viewe d as obje cts of D[ C ⊗ k D op -Mo d] . In p articular, two dg- c ate gories ar e quasi-e quivalent if a n d only if they a r e r elate d by a quasi-functor that is a quasi-e quivalenc e. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 51 Pr o of. This is an immediate consequence of the internal Hom constructed b y T¨ oen f o r Ho(dg-cat) k , [T o¨ e07, Theorem 6.1].  The follow ing prov ides a useful language to k eep track of dg-catego r ies. Definition 5.6. Let T b e a triangulated category . An enhancement of T is a dg-category , C , and an exact equiv alence ǫ : [ C ] → T . W e recall the following result concerning dg-quotien ts. Theorem 5.7. L et C b e a smal l dg-c a te gory and let D b e a ful l dg-sub c ate gory. The r e exis ts a dg-c ate gory C / D , unique in Ho(dg-cat k ) , and dg-func tor ξ : C → C / D such that for any morphism η : C → A in Ho( dg -cat k ) with η | D = 0 ther e e xists a m o rphism λ : C / D → A with η ∼ = λ ◦ ξ . Pr o of. This is [Dri04, Theorem 3 .4 ]. The ob j ects of C / D in [Dri04, Section 3] are exactly the ob jects of C . Note that w e use that k is a field here.  Let X b e a smo oth v ariety , G b e an a ffine algebraic group acting on X , L b e an in v ertible G -equiv ariant sheaf on X , and w ∈ Γ( X, L ) G . Definition 5.8. Let D abs vect ( X, G, w ) denote t he dg-quotient a s in Theorem 5.7 of v ect ( X, G, w ) b y acy cv ect ( X , G, w ). Corollary 5.9. The dg-quotient D abs vect ( X, G, w ) is an en hanc e m ent of D abs [ fact ( X , G, w )] . Pr o of. The result is an immediate consequence of Theorem 5.7 and Prop osition 3.14.  Definition 5.10. W e will need the follo wing factorization of 0 . Let J b e a n injectiv e resolution of O X and consider the factorization, I O := Υ J , of 0. Prop osition 5.11. Th e dg-c ate gory Inj ( X , G , w ) is a n enhan c emen t of D abs [ F act ( X , G, w )] . The dg-c ate gory Inj coh ( X , G, w ) is an enhanc ement of D abs [ fact ( X , G, w ) ] . T her e is an iso- morphism in Ho( dg -cat k ) b etwe en Inj coh ( X , G, − w ) an d D abs vect ( X, G, w ) op . If X is affine and G is r e ductive, then, additional ly, V ect ( X , G, w ) is a n enhanc e m ent of D abs [ F act ( X , G, w )] and vect ( X , G, w ) is an enhanc ement of D abs [ fact ( X , G, w )] . Pr o of. The first t w o statements follow from Prop osition 3.11. While the final t w o follow from Prop osition 3.14. F or the third statemen t, consider the dg-functor, H om X ( • , I O ) : vect ( X , G, w ) op → Inj coh ( X , G, − w ) , whic h sends t he sub category acycvect ( X , G, w ) op to acyclic factorizations with injective com- p onen ts. Th us, the induced functor H om X ( • , I O ) : acycvect ( X , G, w ) op → Inj coh ( X , G, − w ) v anishes o n homotopy categories. By [Dri04, Theorem 1.6.2] and Lemma 3.30, Inj coh ( X , G, − w ) is a dg-quotien t of v ect ( X, G, w ) op b y acycvect ( X , G, w ) op . By uniqueness, there is an iso- morphism in Ho(dg-cat k ) b et wee n Inj coh ( X , G, − w ) and D abs vect ( X , G, w ) op .  Corollary 5.12. Ther e is an isomorphism in Ho(dg - cat k ) , Inj coh ( X , G, − w ) ∼ = Inj coh ( X , G, w ) op . 52 BALLARD, F A VERO, A N D KA TZAR KO V Pr o of. The dg- functor H om X ( • , O X ) : vect ( X , G, w ) op → vect ( X, G, − w ) is a n equiv alence of dg-categories that preserv es the sub categories of acyclic lo cally-free factorizations. Th us, it induces a quasi-equiv a lence D abs vect ( X , G, w ) op ∼ = D abs vect ( X , G, − w ) . Applying Prop osition 5.1 1 finishes the argumen t.  Definition 5.13. Let Inj Z ( X , G, w ) b e t he full sub cat ego ry of Inj ( X , G, w ) consisting of factorizations acyclic off of Z . Let Inj coh ,Z ( X , G, w ) b e the full sub cat ego ry of Inj coh ( X , G, w ) consisting of factorizatio ns acyclic off of Z . Let Inj coh ,Z ( X , G, w ) b e the full sub category of Inj ( X , G, w ) consisting of factor izat io ns acyclic o ff of Z and compact in [ Inj Z ( X , G, w )]. Let V ect Z ( X , G, w ) b e the full subcategory of V ect ( X , G, w ) consisting factorizations acyclic off of Z . Let vect Z ( X , G, w ) b e the full sub category o f vect ( X , G, w ) consisting factorizations acyclic off of Z . Let V ect Z ( X , G, w ) b e the f ull sub category of V ect ( X , G, w ) consisting o f factorizations a cyclic off of Z and compact in D abs [ V ect Z ( X , G, w )]. Corollary 5.14. The dg-c ate gory Inj Z ( X , G, w ) is an enh a nc eme nt of D abs Z [ F act ( X , G, w )] . The dg-c ate gory Inj coh ,Z ( X , G, w ) is an enhanc ement of D abs Z [ fact ( X , G, w )] . If X is affi n e and G is r e ductive, then, additional ly, V ect Z ( X , G, w ) is an enh anc em ent of D abs Z [ F act ( X , G, w )] and vect Z ( X , G, w ) is a n enha n c emen t of D abs Z [ fact ( X , G, w )] . Mor e over, Inj coh ,Z ( X , G, w ) is quasi-e quiva l e n t to vect Z ( X , G, w ) . Pr o of. This is an immediate consequenc e of Prop osition 5.1 1 giv en the definitions ab o ve .  Theorem 5.15. L et X and Y b e sm o oth varieties and le t G and H b e affine algebr aic gr oups ac ting on, r esp e ctively, X and Y . L et w ∈ Γ( X, O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H for char ac ters χ : G → G m and χ ′ : H → G m . L et i w : Z w → X b e the zer o lo cus of w and let i v : Z v → Y b e the zer o lo cus of v . Assume that χ ′ − χ is not torsion . The dg-functor, λ w ⊞ v : I nj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) → ( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d I 7→ Hom F act ( X × Y ,G × G m H,w ⊞ v ) ( • ⊠ • , I ) . induc es a n e quivalenc e ǫ w ⊞ v : D abs Z w × Z v [ F act ( X × Y , G × G m H , w ⊞ v )] → D(( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d) satisfying ǫ w ⊞ v ( E ⊠ F ) ∼ = h E ⊗ k F . If, in addition, X and Y ar e affine and G a nd H a r e r e ductive, then the dg-functor λ w ⊞ v : Vec t Z w × Z v ( X × Y , G × G m H , w ⊞ v ) → ( vect ( X , G, w ) ⊗ k vect ( Y , H , v )) op -Mo d) V 7→ Hom F act ( X × Y ,G × G m H,w ⊞ v ) ( • ⊠ • , V ) . induc es a n e quivalenc e ǫ w ⊞ v : D abs Z w × Z v [ F act ( X × Y , G × G m H , w ⊞ v )] → D(( v ect ( X , G, w ) ⊗ k vect ( Y , H , v )) op -Mo d) satisfying ǫ w ⊞ v ( E ⊠ F ) ∼ = h E ⊗ k F . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 53 Pr o of. W e just need to c hec k that the induced functor, ǫ w ⊞ v : D abs Z w × Z v [ F act ( X × Y , G × G m H , w ⊞ v )] ∼ = [ Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v )] [ λ w ⊞ v ] → [ I nj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d] → D(( I nj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d) is an equiv alence. Note that ǫ w ⊞ v comm utes with copro ducts since the exterior pro ducts, E ⊠ F , are compact in D abs [ F act ( X × Y , G × G m H , w ⊞ v )] w hen E ∈ Inj coh ( X , G, w ) and F ∈ Inj coh ( Y , H, v ). The triangula t ed category , [ I nj Z w × Z v ( X × Y , G × G m H , w ⊞ v )], is com- pactly g enerated by Prop osition 3.15 and the ob jects, h E ⊗F , for a E ∈ Inj coh ( X , G, w ) and F ∈ Inj coh ( Y , H, v ), form a compact set of generators for the catego ry , D(( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d). Th us to chec k that ǫ w ⊞ v is an equiv alence it suffices to c hec k that it takes a compact generating set to a compact generating set and is fully-faithful on those sets. Let us first sho w that there is a quasi-isomorphism of bimo dules h E ⊗F := Ho m Inj coh( X,G,w ) ( • , E ) ⊗ k Hom Inj coh( Y ,H ,v ) ( • , F ) ≃ Hom F act ( X × Y ,G × G m H,w ⊞ v ) ( • ⊠ • , I E ⊠ F ) where we ha ve a morphism of factorizatio ns E ⊠ F → I E ⊠ F whose cone is acyclic a nd where the comp onents of I E ⊠ F ha v e injectiv e comp onents . W e ha v e the natural morphism Hom Inj coh( X,G,w ) ( • , E ) ⊗ k Hom Inj coh( Y ,H,v ) ( • , F ) ⊠ → Hom F act ( X × Y ,G × G m H,w ⊞ v ) ( • ⊠ • , E ⊠ F ) → Hom F act ( X × Y ,G × G m H,w ⊞ v ) ( • ⊠ • , I E ⊠ F ) . where the later morphism is giv en b y comp osing with E ⊠ F → I E ⊠ F . By Lemma 3.52, this is a quasi-isomorphism. Again, app ealing to Lemma 3.52 sho ws that ǫ w ⊞ v is fully- faithful on exterior pro ducts. It remains to che ck t ha t exterior pro ducts are g enerators for D abs Z w × Z v [ F act ( X × Y , G × G m H , w ⊞ v )], but this is Lemma 4 .2 3. The stateme nts with X and Y affine and G and H reductiv e follo w via an analo gous argumen t. Indeed, in t his case , taking G inv ariants is exact and lo cally-free ob jects are pro jectiv e so we can work with lo cally-free ob jects in the exact same manner.  Definition 5.16. Let C b e a dg- category . The category C -Mo d p ossesses the structure of a mo del c at ego ry w ith f : F → G b eing a fibratio n, resp ectiv ely a w eak equiv alence, if f ( c ) : F ( c ) → G ( c ) is an epimorphism in eac h degree, resp ective ly a quasi-isomorphism, for eac h c ∈ C . This determines t he cofibrations: they are those morphisms satisfying the left lifting prop ert y with resp ect t o all acyclic fibrations, i.e. those maps that a re fibratio ns and w eak equiv alences. An y ob ject of C -Mo d is fibran t. W e let b C be the sub category of cofibran t ob jects in C op -Mo d. The dg- category b C is a n enhancemen t of D[ C op -Mo d]. W e let b C pe b e the full sub-dg-category of b C consisting of all ob j ects that are compact in D[ C op -Mo d]. As an y represen ta ble dg-mo dule is cofibrant, w e ha v e a dg-functor h : C → b C pe . F ollowin g the lead of T¨ oen, we introduce the follow ing pro duct. Assume that C is small and let D b e a nother small dg-category ov er k . The Morita product of C a nd D is C ⊛ D := \ ( C ⊗ k D ) pe 54 BALLARD, F A VERO, A N D KA TZAR KO V view ed as an ob ject of Ho(dg- cat k ). Because w e view it a s a n ob ject of Ho(dg-cat k ), it is unique up to quasi-equiv alence. Remark 5.17. The cofibran t ob jects of C op -Mo d are exactly the summands o f semi-free dg-mo dules [FHT01]. One can chec k that summands of semi-free dg-mo dules ha v e the appropriate lifting prop ert y . F urthermore, for any dg-mo dule, M , there exists a semi-free dg-mo dule, F , and an acyclic fibrat ion, F → M . If we assume that M is cofibrant, this m ust split. Corollary 5.18. L et X b e a s mo oth variety, G b e an affine algebr aic gr oup acting on X , L b e an invertible G -e quivariant she af on X , and w ∈ Γ( X, L ) G . L et Y b e a smo oth va ri e ty, H b e a n affine algebr aic gr oup acting X , L ′ b e an invertible H -e quivariant she af o n Y , and v ∈ Γ( Y , L ′ ) H . Ther e ar e is omorphisms in Ho(dg- cat k ) Inj (U( L ) × U( L ′ ) , G × H × G m , f w ⊞ f v ) ∼ = \ Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v ) and Inj coh ( X , G, w ) ⊛ Inj coh ( Y , H, v ) ∼ = Inj coh (U( L ) × U( L ′ ) , G × H × G m , f w ⊞ f v ) . Assume in addition that X and Y ar e affine and G and H a r e r e ductive. Then, ther e a r e isomorphisms in Ho(dg-cat k ) V ect (U( L ) × U( L ′ ) , G × H × G m , f w ⊞ f v ) ∼ = \ vect ( X , G, w ) ⊗ k vect ( Y , H , v ) and vect ( X, G, w ) ⊛ vect ( Y , H , v ) ∼ = vect (U( L ) × U( L ′ ) , G × H × G m , f w ⊞ f v ) . In the sp e cial c ase t hat L = O X ( χ ) and L ′ = O Y ( χ ′ ) for char acters χ : G → G m and χ ′ : H → G m , if we assume that χ or χ ′ is not torsion, then ther e a r e isomorphisms in Ho(dg-cat k ) Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) ∼ = \ Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v ) and Inj coh ( X , G, w ) ⊛ Inj coh ( Y , H, v ) ∼ = Inj coh Z w × Z v ( X × Y , G × G m H , w ⊞ v ) . Assume in addition that X and Y ar e affine and G and H a r e r e ductive. Then, ther e a r e isomorphisms in Ho(dg-cat k ) V ect Z w × Z v ( X × Y , G × G m H , w ⊞ v ) ∼ = \ vect ( X , G, w ) ⊗ k vect ( Y , H , v ) and vect ( X , G, w ) ⊛ vect ( Y , H , v ) ∼ = vect Z w × Z v ( X × Y , G × G m H , w ⊞ v ) . Pr o of. By Lemma 3 .4 8, w e hav e equiv alences of dg-catego ries F act ( X , G, w ) ∼ = F act (U( L ) , G × G m , f w ) F act ( Y , H , w ) ∼ = F act (U( L ′ ) , H × G m , f v ) . Replacing ( X , G, L , w ) and ( Y , H , L ′ , v ) b y (U( L ) , G × G m , O U( L ) (1) , f w ) and (U( L ′ ) , H × G m , O U( L ′ ) (1) , f v ), w e may a ssume that L and L ′ are (non-equiv arian tly) t rivial a s sheav es on X and Y , resp ectiv ely , and con tinue the argumen t. Finally , as f w and f v are b oth linear along the fibers, Euler’s form ula using the fiber c o ordinates shows that f w v anishes only along the singular lo cus of f w and similarly f o r f v . Th us, f w and f v b oth v anish along the KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 55 singular lo cus of f w ⊞ f v . Consequen tly , fa ctorizations supp orted aw ay fro m Z f w × Z f v are automatically acyclic. Th us, we are reduced to proving the sp ecial case of t he statemen t. W e hav e a dg- functor, λ w ⊞ v : I nj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) → ( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d and an inclusion \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) → ( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d . W e then ha v e a dg- functor a : Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) → ( \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d M 7→ Hom ( Inj coh ( X,G,w ) ⊗ k Inj coh ( Y ,H ,v )) op -Mo d ( • , M ) . F or any N ∈ Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v ), there exis ts an M ∈ \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) and a quasi-isomorphism f : M → N . T he induced natura l transformatio n Hom( • , f ) : Hom( • , M ) → Hom ( • , N ) is a quasi-isomorphism if w e restrict the argument to lie in \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ). Th us, a ( N ) is quasi-isomorphic to h M . Giv en M quasi-isomorphic to N and M ′ quasi- isomorphic to N ′ , w e ha ve natural isomorphisms Hom D[( \ Inj coh ( X,G, − w ) ⊗ k Inj coh ( Y ,H ,v )) op -Mo d] ( a ( N ) , a ( N ′ )) ∼ = Hom D[( \ Inj coh ( X,G, − w ) ⊗ k Inj coh ( Y ,H ,v )) op -Mo d] ( h M , h M ′ ) ∼ = Hom [ \ Inj coh ( X,G, − w ) ⊗ k Inj coh ( Y ,H ,v )] ( M , M ′ ) ∼ = Hom D[( Inj coh ( X,G,w ) ⊗ k Inj coh ( Y ,H ,v )) op -Mo d] ( M , M ′ ) ∼ = Hom D[( Inj coh ( X,G,w ) ⊗ k Inj coh ( Y ,H ,v )) op -Mo d] ( N , N ′ ) ∼ = Hom [ Inj Z w × Z v ( X × Y , G × G m H,w ⊞ v ] ( N , N ′ ) where the first isomorphism is due to the fact that a ( N ) is quasi-isomorphic to h M and a ( N ′ ) is quasi-isomorphic to h M ′ , the second uses the Y oneda embedding, the third uses that \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) is an enhancemen t of D[( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d], the fourth uses the assumed quasi-isomorphisms, and the final isomorphism uses that Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) is an enhancemen t of D[( Inj coh ( X , G, w ) ⊗ k Inj coh ( Y , H, v )) op -Mo d], i.e. The- orem 5.15. Th us, a is a quasi-functor inducing a quasi-equiv alence Inj Z w × Z v ( X × Y , G × G m H , w ⊞ v ) ≃ \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) . The isomorphism in Ho(dg-cat k ) Inj Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) ∼ = \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) induces an isomorphism b et w een the compact ob j ects, Inj coh ( X , G, w ) ⊛ Inj coh ( Y , H, v ) ∼ = Inj coh ,Z w × Z v ( X × Y , G × G m H , w ⊞ v ) . In the case tha t X and Y are affine and G and H are reductiv e, an analogous argumen t suffices. Indeed, as noted b efore, taking G in v arian ts is exact and lo cally-free ob jects are pro jectiv e so we can work with lo cally-free ob jects in the exact same manner.  56 BALLARD, F A VERO, A N D KA TZAR KO V Remark 5.19. In the case that L = O X ( χ ) and L ′ = O Y ( χ ′ ), the quotien t stac k [U( O X ( χ )) × U( O Y ( χ ′ )) / ( G × H × G m )] is isomorphic to [ X × Y × G m / ( G × H )] via the morphism φ : U( O X ( χ )) × U( O Y ( χ ′ )) ∼ = G m × X × G m × Y → X × Y × G m ( α, x, β , y ) 7→ ( x, y , α − 1 β ) . The quotien t stac k [ X × Y × G m / ( G × H )] is isomorphic to [ X × Y /G × G m H ] as the map ( G × H ) G × G m H × ( X × Y ) → X × Y × G m ( g , h, x, y ) 7→ ( x, y , χ ( g ) − 1 χ ′ ( h )) is an isomorphism assuming that χ ′ − χ : G × H → G m is not torsion. This g iv es a direct comparison for t he tw o LG mo dels describing the Morita pro duct in the case L = O X ( χ ) and L ′ = O Y ( χ ′ ). One of the many great results of [T o ¨ e07 ] is the fo llo wing. It provides a description of the con tin uous internal Hom dg-categor y in Ho( dg-cat k ). Theorem 5.20. L et C and D b e smal l dg-c ate g o ries ove r k . Then, ther e i s an iso morphism in Ho(dg-cat k ) R Hom c ( b C , b D ) ∼ = \ C op ⊗ k D . Given a mo d ule, F ∈ \ C op ⊗ k D , the c orr e s p ondin g dg-functor, Ψ F : C → b D , sends c ∈ C to F ( c , • ) ∈ b D . This uniquely determin e s a d g - functor, Ψ F : b C → b D , f o r which [Ψ F ] c ommutes with c opr o ducts. Pr o of. As stated, this result is [T o¨ e07 , Corollary 7.6].  Remark 5.21. T¨ oen’s result is more general. The field, k , can b e replaced b y a comm utativ e ring. The deriv atio n of the tensor pro duct, ⊗ k , is then required. Applying Theorem 5.20, w e can giv e the following description of the contin uous internal Hom dg-catego ry for equiv ariant factorizations. Theorem 5.22. L et X and Y b e sm o oth varieties and let G and H b e affine algeb r aic gr oups. Assume that G acts o n X and H a cts on Y . L et χ : G → G m and χ ′ : H → G m b e char acters and let w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H . Ther e is a n isomorphis m in Ho(dg-cat k ) R Hom c ( \ Inj coh ( X , G, w ) , \ Inj coh ( Y , H, v )) ∼ = Inj Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) such that the induc e d ma p on homo topy c ate gories c orr esp ond ing to I ∈ Inj Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) is Φ I . If X is affine and G is r e ductive, then ther e is an isomorphism in Ho(dg-cat k ) R Hom c ( \ vect ( X, G, w ) , \ vect ( Y , H , v )) ∼ = V ect Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) such that the induc e d map on homotopy c ate gories c orr esp on d ing to P ∈ V ect Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) is Φ P . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 57 Pr o of. W e hav e isomorphisms in Ho(dg- cat k ), R Hom c ( \ Inj coh ( X , G, w ) , \ Inj coh ( Y , H, v )) ∼ = \ Inj coh ( X , G, w ) op ⊗ k Inj coh ( Y , H, v ) ∼ = \ Inj coh ( X , G, − w ) ⊗ k Inj coh ( Y , H, v ) ∼ = Inj Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) . The first line follo ws from Theorem 5.20. The second line comes from Prop osition 5.1 1 whic h stat es that Inj coh ( X , G, w ) op is quasi-equiv alent to Inj coh ( X , G, − w ). The third line is an application of Corolla ry 5.18. Next, we need to c hec k that the induced functor o n homotopy categories f o r a giv en I ∈ Inj Z w × Z v ( X × Y , G × G m H , ( − w ) ⊞ v ) is Φ I up to isomorphism. Recall that the isomorphism of Inj coh ( X , G, w ) op and Inj coh ( X , G, − w ) follows from the diag ram of dg-f unctors Inj coh ( X , G, w ) op vect ( X , G, − w ) Inj coh ( X , G, − w ) vect ( X , G, w ) op H om X ( • , I O ) H om X ( • , I O ) H om X ( • , O X ) The induced dg- functor on t he image o f H om X ( • , I O ) : vect ( X , G, − w ) → Inj coh ( X , G, w ) op is H om X ( E , I O ) ⊗ J 7→ Hom F act ( H om X ( E ∨ , I O ) ⊠ J , I ) ∼ = 7→ Hom F act (Res G × H G × G m H π ∗ 2 J , H om X × Y (Res G × H G × G m H π ∗ 1 H om X ( E ∨ , I O ) , I )) ∼ = 7→ Hom F act (Res G × H G × G m H π ∗ 2 J , Res G × H G × G m H π ∗ 1 E ∨ ⊗ O X × Y I ) ∼ = 7→ Hom F act ( J , π 2 ∗ Ind G × H G × G m H  Res G × H G × G m H π ∗ 1 E ∨ ⊗ O X × Y I  ) ∼ = 7→ Hom F act ( J , π 2 ∗  π ∗ 1 E ∨ ⊗ O X × Y Ind G × H G × G m H I  ) . The first line uses t ensor-Ho m a dj unction, Prop osition 3.27. The second line uses the na tural isomorphism, Res G × H G × G m H π ∗ 1 E ∨ ⊗ O X × Y I → H om X × Y (Res G × H G × G m H π ∗ 1 H om X ( E ∨ , I O ) , I )). The third line uses the adjunctions, π ∗ 2 ⊣ π 2 ∗ and Res G × H G × G m H ⊣ Ind G × H G × G m H . The fourth line applies the pro jection formula, L emma 2.16. As H om X ( E , I O ) is quasi-isomorphic to E ∨ , from the aligned displa y , w e see that the induced map on t he homotopy categories is Φ I . The case of X affine and G r eductive is handled in a n analog ous, ev en simpler, ma nner.  Pr o of of T he or em 1.1. By L emma 3.48, we ha ve equiv alences o f dg-categories F act ( X , G, w ) ∼ = F act (U( L ) , G × G m , f w ) F act ( Y , H , w ) ∼ = F act (U( L ′ ) , H × G m , f v ) . Theorem 5.22 applied to (U( L ) , G × G m , f w ) and (U( L ′ ) , H × G m , f v ) giv es the statemen t.  58 BALLARD, F A VERO, A N D KA TZAR KO V 5.2. Ho c hsc hild in v arian ts. In this s ection, w e c ompute the Ho c hsc hild in v ariants in a simple case: G acting linearly on A n . W e start o ut a bit more generally . Let G act o n X and let w ∈ Γ( X, O X ( χ )) G . F or the whole of this section, w e assume Sing Z ( − w ) ⊞ w ⊆ Z w × Z w so w e ma y remo ve the supp ort restrictions in the results of Section 5.1. Definition 5.23. Let C b e a s mall dg-category . The Ho chschild cohomology of C is the graded ve ctor space M t ∈ Z Hom D( C op ⊗ C -Mo d) ( C , C [ t ]) . where C is the bimo dule giv en by C ( c, c ′ ) = Hom C ( c, c ′ ) . When C = Inj coh ( X , G, w ), we denote the Ho c hsc hild cohomology b y HH • ( X , G, w ). W e hav e a tra ce functor T r : C op ⊗ C → C ( k ) ( c, c ′ ) 7→ Hom C ( c, c ′ ) . This admits an extension to C ⊗ C op -Mo d by F 7→ F ⊗ C ⊗ C op C . The Ho chschild homology of C is defined to b e the homolog y o f C L ⊗ C op ⊗ C C . When C = Inj coh ( X , G, w ), we denote the Ho c hsc hild cohomology b y HH • ( X , G, w ). Lemma 5.24. L et X and Y b e smo oth v a rieties and let G and H b e affine algebr aic gr oups acting on, r esp e ctively, X and Y . L et w ∈ Γ( X , O X ( χ )) G and v ∈ Γ( Y , O Y ( χ ′ )) H for char- acters χ : G → G m and χ ′ : H → G m . Assume that Sing Z ( − w ) ⊞ w ⊆ Z w × Z w . We have is o m orphisms HH t ( X , G, w ) ∼ = Hom D abs [ Fac t ( X × Y ,G × G m G,w ⊞ ( − w )] ( ∇ , ∇ [ t ]) . We also ha v e isomorphisms HH t ( X , G, w ) ∼ = H t ( L T r ∇ ) wher e L T r is tr ac e functor on D abs [ F act ( X × Y , G × G m G, ( − w ) ⊞ w ] . Pr o of. By Theorem 5.15, w e hav e an equiv alence D abs [ F act ( X × Y , G × G m G, w ⊞ ( − w )] → D( Inj coh ( X , G, w ) op ⊗ Inj coh ( X , G, w ) -Mo d) P 7→ R Hom( • ⊠ • L ∨ , P ) . The assumption on the singular supp ort of Z ( − w ) ⊞ w allo ws us to remov e the supp or t condition. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 59 W e hav e natural quasi-isomorphisms R Hom( E ⊠ F L ∨ , ∇ ) = R Hom( E ⊠ F L ∨ , Ind G × G m G G ∆ ∗ O X ) ≃ R Hom( L ∆ ∗ Res G × G m G G E ⊠ F L ∨ , O X ) ≃ R Hom( E ⊗ F L ∨ , O X ) ≃ R Hom( E , F ) . The first line is the definition of ∇ . The second line is an application of the adjunctions Res G × G m G G ⊣ Ind G × G m G G , Corollar y 3.43, and L ∆ ∗ ⊣ ∆ ∗ , deriv ed fro m Lemma 3.35. The third line comes from t he iden tity L ∆ ∗ ◦ π ∗ i ∼ = Id fo r i = 1 , 2 . The final line is tensor-Hom adjunction, Corollary 3.28, and the assumption that F is quasi-isomorphic to a coheren t factorization so F L ∨ L ∨ ∼ = F . W e turn to the statemen t concerning Ho chs child homolog y . Unde r the equiv a lence D abs [ F act ( X × Y , G × G m G, ( − w ) ⊞ w ] → D ( Inj coh ( X , G, w ) ⊗ Inj coh ( X , G, w ) op -Mo d) the categorical trace corresp onds to the trace functor ( R p ∗ L ∆ ∗ ) R G b y Lemma 3.55.  Remark 5.25. As transp osing the t wo copies of X induces an equiv alence D abs [ F act ( X × X , G × G m G, w ⊞ ( − w )] ∼ = D abs [ F act ( X × X , G × G m G, ( − w ) ⊞ w ] whic h preserv es the diagonal, w e can compute Ho c hsc hild in v arian ts in either deriv ed cate- gory of factorizations. The Ho c hsc hild cohomology is a subalgebra o f a la rger algebra. Definition 5.26. The extended Hochschild cohomology o f ( X, G, w ) is the b G × Z -graded k -algebra M ρ ∈ b G,t ∈ Z Hom D abs [ fact ( X × X,G × G m G, ( − w ) ⊞ w )] ( ∇ , ∇ ( ρ )[ t ]) . W e denote t he extended Ho c hsc hild cohomology b y HH • e ( X , G, w ). Remark 5.27. The ring HH • e ( X , G, w ) is a f a ctorization a nalog of g eneralized Ho c hsc hild cohomology of a v ariet y X with supp o rt in T ∈ D b (coh X × X ) and co efficien ts in E ∈ D b (coh X × X ), HH • T ( X , E ) defined by Kuznetso v [Kuz10]. Here, w e take E to b e the diagonal and T to b e the kerne ls of t wist functors. Lemma 5.28. Ther e is a natur al isomorphism , HH t ( X , G, w ) → HH (0 ,t ) e ( X , G, w ) . Pr o of. This is clear.  T o compute HH • e ( X , G, w ), w e first mus t identify the complex L ∆ ∗ Ind G × G m G G ∆ ∗ O X of coheren t G -equiv ar ian t shea v es on X . Let K χ b e the k ernel of χ . 60 BALLARD, F A VERO, A N D KA TZAR KO V Lemma 5.29. Ther e is a G × G m G -e quivaria nt i s omorphism, Σ : G × G m G G × X × X → K χ × X × X ( g 1 , g 2 , x 1 , x 2 ) 7→ ( g 1 g − 1 2 , σ ( g 1 , x 1 ) , σ ( g 2 , x 2 )) , wher e G × G m G acts on K χ via ( g 1 , g 2 ) · g := g 1 g g − 1 2 . Pr o of. The inv erse morphism is K χ × X × X → G × G m G G × X × X ( g , x 1 , x 2 ) 7→ ( g , e, σ ( g − 1 , x 1 ) , x 2 ) .  Consider the G × G m G -equiv ariant sub v ariety defined b y O (∆) := { ( g , x 1 , x 2 ) | σ ( g , x 2 ) = x 1 } ⊂ K χ × X × X . Lemma 5.30. Unde r the c om p osition of the e quivale nc e of L em ma 2.13 and the e quivalenc e Σ ∗ , the G -e quivariant she af ∆ ∗ O X c orr esp onds to the structur e she af of O ( ∆) in K χ × X × X i.e. ι ∗ Σ ∗ O O (∆) ∼ = ∆ ∗ O X . Pr o of. Recall that the equiv alence o f Lemm a 2.13 is induced b y ι ∗ where ι : X × X → G × G m G G × X × X is the inclusion along the iden tit y . Note that Σ ◦ ι remains the inclusion along the iden tity , but now of X × X into K χ × X × X . Since b oth Σ ∗ and ι ∗ are equiv alences b efore deriving, they are exact. Th us, the statemen t of the lemma is equiv alent to che ckin g that the equation defining O (∆) restricts to the diagonal when w e restrict to { e } × X × X . This is clear.  F rom now on, w e assume that K χ is finite. Conside r the coheren t sheaf M g ∈ K χ O Γ t ( σ g ) where Γ t ( σ g ) := { ( x 1 , x 2 ) ∈ X × X | σ ( g , x 2 ) = x 1 } is the tr a nsp ose of the graph of σ g . Lemma 5.31. The c oher ent she af L g ∈ K χ O Γ t ( σ g ) p osse s ses a natur al G × G m G -e quivaria nt structur e such that ther e is an isomorph ism of c oher ent G × G m G -e quivaria nt s he aves Ind G × G m G G ∆ ∗ O X ∼ = p ∗ ( O O (∆) ) ∼ = M g ∈ K χ O Γ t ( σ g ) . wher e p : K χ × X × X → X × X is the pr oje ction. Pr o of. The second isomorphism is clear from the (no w) standing assumption that K χ is finite and induces the natural equiv a rian t structure on L g ∈ K χ O Γ t ( σ g ) . F or the first isomorphism, w e recall that, in general, Ind G H is the comp osition α ∗ ◦ ( ι ∗ ) − 1 where ι : X → G H × X is the inclusion alo ng the iden tity and α : G H × X → X is the morphism induced by the action of G on X . In our case, w e hav e the commutativ e diagram KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 61 G × G m G G × X × X K χ × X × X X × X Σ α p No w, by Lemma 5.3 0, w e hav e ( ι ∗ ) − 1 ∆ ∗ O X ∼ = Σ ∗ O O (∆) . Applying α ∗ to b oth sides w e get Ind G × G m G G ∆ ∗ O X ∼ = p ∗ ( O O (∆) ) where the simplification on the r ig h t hand side comes either by flat base c hang e for t he isomorphism Σ or b y using the isomorphism Σ − 1 ∗ = Σ ∗ .  F rom this p oint forw ar d, we restrict our a t t ention to X = A n equipped with a linear a ctio n of G suc h that K χ is finite. It is easy to see that this implies that G is reductiv e. W rite A n = Sp ec Sym( V ). Then, w e ha ve a right exact sequence V ⊗ k O A n × A n s → O A n × A n → ∆ ∗ O A n → 0 where the first morphism is v ⊗ f 7→ f ( v ⊗ 1 − 1 ⊗ v ) . The p otential ( − w ) ⊞ w v anishes on ∆ ∗ O A n . Since X is affine and G is reductiv e, lo cally-free coheren t equiv ariant shea v es are pro jectiv e ob jects. Th us, t here exists a morphism t : O A n × A n → V ⊗ k O A n × A n making the diagram V ⊗ k O A n × A n O A n × A n V ⊗ k O A n × A n O A n × A n s ( − w ) ⊞ w ( − w ) ⊞ w s t comm ute. Similarly , given g ∈ G , w e can t wist this diagram by σ g as follo ws. W e hav e a righ t exact sequence V ⊗ k O A n × A n s g → O A n × A n → O Γ t ( σ g ) → 0 where the first morphism is v ⊗ f 7→ f ( g − 1 · v ⊗ 1 − 1 ⊗ v ) . Here g − 1 · v is the elemen t of Sym V g iven by the automorphism of rings dual to σ g : A n → A n . F or g ∈ K χ , ( − w ) ⊞ w v anishes o n O Γ t ( σ g ) so there exists a t g : O A n × A n → V ⊗ k O A n × A n making the diagram 62 BALLARD, F A VERO, A N D KA TZAR KO V V ⊗ k O A n × A n O A n × A n V ⊗ k O A n × A n O A n × A n s g ( − w ) ⊞ w ( − w ) ⊞ w s g t g comm ute. Lemma 5.32. Ther e ar e quasi-iso m orphisms of G × G m G -e quivaria nt f a ctorizations, M g ∈ K χ K ( s g , t g ) ∼ = M g ∈ K χ O Γ t ( σ g ) ∼ = Ind G × G m G G ∆ ∗ O A n . Pr o of. The second isomorphism is a lready stated in Lemma 5 .3 1. The first quasi-isomorphism follo ws f rom an immediate application of Prop osition 3.20.  Since eac h K ( s g , t g ) is a factorization with lo cally-free comp onen ts, to compute L ∆ ∗ Ind G × G m G G ∆ ∗ O A n w e ma y compute ∆ ∗   M g ∈ K χ K ( s g , t g )   . W e record the following lemma as a reminder of the structure of ∆ ∗ K ( s g , t g ). Lemma 5.33. The fa ctorization ∆ ∗ K ( s g , t g ) has c omp onents ∆ ∗ K ( s g , t g ) − 1 = M l ≥ 0 Λ 2 l +1 V ⊗ k O A n ( l χ ) ∆ ∗ K ( s g , t g ) 0 = M l ≥ 0 Λ 2 l V ⊗ k O A n ( l χ ) and morphisms given by • y ∆ ∗ s g + • ∧ ∆ ∗ t g wher e ∆ ∗ s g : V ⊗ k O A n → O A n v ⊗ f 7→ f ( g − 1 · v − v ) . Pr o of. This is clear from the definition of the Koszul factorization, K ( s g , t g ).  Definition 5.34. Let g ∈ G . Set V g := { v ∈ V | g − 1 · v = v } . The ideal sheaf of ( A n ) g corresp onds to { g − 1 · f − f | f ∈ Sym V } . This determines a subspace W g ⊆ V . Note that there is a n equiv ariant splitting V = V g ⊕ W g . Let κ g : G → G m b e the c haracter corr espo nding to Λ dim W g W g . More precisely , O A n ( κ g ) is the in v ertible sheaf corresp onding to the free graded mo dule of r ank 1, Λ dim W g W g ⊗ k Sym V . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 63 Lemma 5.35. Ther e is a quasi-iso m orphism b etwe e n ∆ ∗ K ( s g , t g ) and the K o szul factoriza- tion i g ∗ K (0 , d w g ) wher e i g : ( A n ) g → A n is the inclusion, 0 is the mo rp h ism V g ⊗ k O ( A n ) g 0 → O ( A n ) g , and w g is the r estriction of w to ( A n ) g . Pr o of. Consider the pullbac k of s g and t g to ( A n ) g × ( A n ) g via i g × i g : ( A n ) g × ( A n ) g → A n × A n . W e hav e ( i g × i g ) ∗ s g ( v ) = v ⊗ 1 − 1 ⊗ v and a commutativ e diagram V g ⊗ k O ( A n ) g × ( A n ) g O ( A n ) g × ( A n ) g V g ⊗ k O ( A n ) g × ( A n ) g O ( A n ) g × ( A n ) g ( i g × i g ) ∗ s g ( − w g ) ⊞ w g ( − w g ) ⊞ w g ( i g × i g ) ∗ s g ( i g × i g ) ∗ t g Let ∆ g : ( A n ) g → ( A n ) g × ( A n ) g b e the diag onal em b edding. Then, ∆ ∗ g ( i g × i g ) ∗ t g = d w g . As the diag r am ( A n ) g ( A n ) g × ( A n ) g A n A n × A n ∆ g i g i g × i g ∆ comm utes, we hav e i ∗ g ∆ ∗ t g = ∆ ∗ g ( i g × i g ) ∗ t g = d w g while i ∗ g ∆ ∗ s g = ∆ ∗ g ( i g × i g ) ∗ s g = 0. Th us, i ∗ g ∆ ∗ K ( s g , t g ) ∼ = K (0 , d w g ) . No w, a sso ciated to the adjunction i ∗ g ⊣ i g ∗ , w e ha ve a morphism π : ∆ ∗ K ( s g , t g ) → i g ∗ i ∗ g ∆ ∗ K ( s g , t g ) ∼ = i g ∗ K (0 , d w g ) whic h w e claim is a quasi-isomorphism. T o ve rif y this claim, w e c hec k that the k ernel of π , ke r ( π ), is acyclic. The comp onen ts of k er( π ) are k er( π ) − 1 = M l ≥ 0 ,a> 0 a + b =2 l +1 Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) k er( π ) 0 = I ( A n ) g ⊕ M l ≥ 0 ,a> 0 a + b =2 l Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) . Let J j := ker( • y ∆ ∗ s g ) : Λ j W g ⊗ k O A n → Λ j − 1 W g ⊗ k O A n 64 BALLARD, F A VERO, A N D KA TZAR KO V and J 0 := I ( A n ) g . As ∆ ∗ s g v anishes on V g and ∆ ∗ t g has image in V g , w e ha ve a filtr a tion F j k er( π ). In the case j = 2 u , it is F j k er ( π ) − 1 = M b ≥ j,a> 0 a + b =2 l +1 Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) F j k er( π ) 0 = Λ j V g ⊗ k J j ( uχ ) ⊕ M b ≥ j,a> 0 a + b =2 l Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) . In the case j = 2 u + 1, it is F j k er( π ) − 1 = Λ j V g ⊗ k J j ( uχ ) ⊕ M b ≥ j,a> 0 a + b =2 l +1 Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) F j k er( π ) 0 = M b ≥ j,a> 0 a + b =2 l Λ a W g ⊗ k Λ b V g ⊗ k O A n ( l χ ) . The asso ciated graded factor ization, F j k er( π ) /F j +1 k er( π ), is the totalization of the exact sequence 0 → Λ j V g ⊗ k Λ dim W g W g ⊗ k O A n • y ∆ ∗ s g → · · · • y ∆ ∗ s g → Λ j V g ⊗ k Λ j +1 W g ⊗ k O A n • y ∆ ∗ s g → Λ j V g ⊗ k J j → 0 where the final term is in degree − dim W g . Th us, ker( π ) is filtered b y acyclic complexes and hence acyclic. This implies tha t π is a quasi-isomorphism a s desired.  Definition 5.36. Let κ : G → G m b e the c haracter corresp onding to Λ n V . Lemma 5.37. Assume that K χ is finite. Then, ther e is an isomorphism HH t ( A n , G, w ) ∼ = HH ( κ,n + t ) e ( A n , G, w ) . Pr o of. W e hav e, HH t ( A n , G, w ) ∼ = Hom((Ind G × G m G G ∆ ∗ O X ) ∨ , Ind G × G m G G ∆ ∗ O X [ t ]) ∼ = Hom( M g ∈ K χ K ( s g , t g ) ∨ , Ind G × G m G G ∆ ∗ O X [ t ]) ∼ = Hom( M g ∈ K χ K ( t ∨ g , s ∨ g ) , Ind G × G m G G ∆ ∗ O X [ t ]) ∼ = Hom( M g ∈ K χ O Γ t ( σ g ) ⊗ k Λ n V ∨ [ − n ] , Ind G × G m G G ∆ ∗ O X [ t ]) ∼ = Hom(Ind G × G m G G ∆ ∗ O X , Ind G × G m G G ∆ ∗ O X ⊗ k Λ n V [ t + n ]) = Hom(Ind G × G m G G ∆ ∗ O X , Ind G × G m G G ∆ ∗ O X ( κ )[ t + n ]) = HH ( κ,n + t ) e ( A n , G, w ) . All morphisms are computed in D abs [ fact ( A n × A n , G × G m G, ( − w ) ⊞ w )]. The first line follows from Lemma 3.57. The second line fo llo ws from Lemma 5.32. The third line is Lemma 3.21. The fourth line comes from Prop osition 3.20. The fifth line is another application of Lemma 5.32. The six line is b y definition a s is the sev en th line.  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 65 Definition 5.38. L et ( r 1 , . . . , r c ) b e a sequence of elemen ts of a comm utative ring, R . W e let H • ( r ) denote the coho mology of the Koszul complex for ( r 1 , . . . , r c ). W e call H • ( r ) the Koszul cohomology of ( r 1 , . . . , r c ). In the case, ( r 1 , . . . , r c ) = ( ∂ 1 w , . . . , ∂ n w ) for R = k [ x 1 , . . . , x n ], w e denote the Koszul cohomology by H • (d w ). The Jacobian algeb ra of w is H 0 (d w ) but we denote it by Jac( w ) for transparency . Theorem 5.39. L et G act line arly o n A n and let w ∈ Γ( A n , O A n ( χ )) G . Assume that K χ is finite and χ : G → G m is surje ctive. Then, HH ( ρ,t ) e ( A n , G, w ) ∼ =     M g ∈ K χ ,l ≥ 0 t − dim W g =2 u H 2 l (d w g )( ρ − κ g + ( u − l ) χ ) ⊕ M g ∈ K χ ,l ≥ 0 t − dim W g =2 u +1 H 2 l +1 (d w g )( ρ − κ g + ( u − l ) χ )     G If, additional ly, we assume the supp ort o f (d w ) is { 0 } , then we have HH ( ρ,t ) e ( A n , G, w ) ∼ =     M g ∈ K χ t − dim W g =2 u Jac( w g )( ρ − κ g + uχ ) ⊕ M g ∈ K χ t − dim W g =2 u +1 Jac( w g )( ρ − κ g + uχ )     G . Pr o of. W e hav e HH ( ρ,t ) e ( A n , G, w ) := Hom D abs [ fact ( A n × A n ,G × G m G, ( − w ) ⊞ w )] (Ind G × G m G G ∆ ∗ O A n , Ind G × G m G G ∆ ∗ O A n ( ρ )[ t ]) ∼ = Hom D abs [ fact ( A n × A n ,G, ( − w ) ⊞ w )] (Res G × G m G G Ind G × G m G G ∆ ∗ O A n , ∆ ∗ O A n ( ρ )[ t ]) ∼ = Hom D abs [ fact ( A n ,G, 0)] ( L ∆ ∗ Res G × G m G G Ind G × G m G G ∆ ∗ O A n , O A n ( ρ )[ t ]) ∼ = Hom( L ∆ ∗ Ind G × G m G G ∆ ∗ O A n , O A n ( ρ )[ t ]) ∼ = Hom( M g ∈ K χ i g ∗ K (0 , d w g ) , O A n ( ρ )[ t ]) ∼ = Hom( O A n , M g ∈ K χ i g ∗ K (0 , d w g ) ∨ ( ρ )[ t ]) . ∼ = Hom( O A n , M g ∈ K χ i g ∗ K (d w g , 0)( ρ − κ g )[ t − dim W g ]) . The first line is b y definition. The second line is adjunction for Res and Ind, Lemma 3.42. The third line applies the adjunction, L ∆ ∗ ⊣ ∆ ∗ , Lemma 3.35 . The fo urth line is a slight notational respite obtained by viewing ∆ as a n equiv ariant for the diagonal em b edding of G in to G × G m G . The fifth line is Lemma 5.3 5. The sixth line is just the equiv alence ( − ) ∨ . W e justify the sev enth line in the next para graph. Let e K (0 , d w g ) b e t he Koszul facto r izat io n on A n asso ciated to V g ⊗ k O A n 0 → O A n 66 BALLARD, F A VERO, A N D KA TZAR KO V and O A n d w g → V g ⊗ k O A n ( χ ) . Using con tra ction with morphism, W g ⊗ k O A n → O A n w ⊗ k f 7→ f w , w e ha v e a exact sequence of Koszul fa ctorizations, 0 → Λ dim W g W g ⊗ k e K (0 , d w g ) → · · · → W g ⊗ k e K (0 , d w g ) → e K (0 , d w g ) → i g ∗ K (0 , d w g ) → 0 Hence, K (0 , d w g ) ∨ is quasi-isomorphic to t he totalization of the complex 0 ← Λ dim W g W ∨ g ⊗ k e K (0 , d w g ) ∨ ← · · · ← W ∨ g ⊗ k e K (0 , d w g ) ∨ ← 0 This is, in turn quasi-isomorphic t o i g ∗ K (d w g , 0) ⊗ k Λ dim W g W ∨ g [ − dim W g ]. The factorization, K (d w g , 0)( ρ − κ g ), has comp onents K (d w g , 0)( ρ − κ g ) − 1 = M l ≥ 0 Λ 2 l +1 V ∨ g ⊗ k O ( A n ) g ( ρ − κ g − ( l + 1) χ ) K (d w g , 0)( ρ − κ g ) 0 = M l ≥ 0 Λ 2 l V ∨ g ⊗ k O ( A n ) g ( ρ − κ g − l χ ) with morphisms giv en b y con tra ctio n with d w g . The cohomology of K ( d w g , 0)( ρ − κ g ) is H 2 u ( K (d w g , 0)( ρ − κ g )) ∼ = M l ≥ 0 H 2 l (d w g )( ρ − κ g + ( u − l ) χ ) H 2 u +1 ( K (d w g , 0)( ρ − κ g )) ∼ = M l ≥ 0 H 2 l +1 (d w g )( ρ − κ g + ( u − l ) χ ) . Th us, we ha ve Hom( O A n , M g ∈ K χ i g ∗ K (d w g , 0)( ρ − κ g )[ t − dim W g ]) ∼ =     M g ∈ K χ ,l ≥ 0 t − dim W g =2 u H 2 l (d w g )( ρ − κ g + ( u − l ) χ ) ⊕ M g ∈ K χ ,l ≥ 0 t − dim W g =2 u +1 H 2 l +1 (d w g )( ρ − κ g + ( u − l ) χ )     G If (d w ) has supp ort { 0 } , then so do es (d w g ) for all g . So all Koszul complexes only hav e cohomology in homological degree zero.  Remark 5.40. By sp ecializing to appropr ia te graded pieces , one can use Theorem 5.39 to extract b oth HH • ( A n , G, w ) and HH • ( A n , G, w ). Corollary 5.41. L et A n = Spec(Sym V ) c arry a G m action with weig h t ( − 1) . L et w ∈ Sym V b e h omo ge ne ous of de gr e e d . Then, w e have isom o rphisms HH t ( A n , G m , w ) ∼ = ( Jac( w ) d ( n + t 2 ) − n t 6 = 0 Jac( w ) d ( n 2 ) − n ⊕ k ⊕ d − 1 t = 0 . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 67 Pr o of. W e ha v e κ = − n . In this case, K χ ∼ = Z /d Z . If g 6 = e , then V g = { 0 } , thus κ g = − n and dim W g = n . F or g = e , we ha v e κ e = 0 and dim W e = 0. Applying Lemma 5.3 7 and Theorem 5.39, w e ha ve HH t ( A n , G m , w ) ∼ = Jac( w ) − n + d ( n + t 2 ) ⊕ M g 6 = e Jac( w g ) d t 2 . W e hav e Jac( w g ) ∼ = k (0) so the lat t er term only contributes to t = 0.  Remark 5.42. This computation was first done by C˘ ald˘ araru and T u, [CT10, Example 6.4]. It is also p erformed, indep enden tly , by P olishc h uk and V a in trob [PV11 ]. 6. Implica tions for Hodge Theor y In this section, w e giv e tw o applications of the ideas and computations of the previous sections to Ho dge theory . T o fully state the results, w e recall some of the functoria lit y of Ho c hsc hild homology . R ecall that p erf ( C ) consists of all compact ob jects in D( C op -Mo d). Prop osition 6.1. L et C and D b e satur ate d dg-c ate gories over k . L et F b e an obje ct of p erf ( C op ⊗ D ) . Then, ther e is a hom o morphism of ve ctor sp ac es, F • : HH • ( C ) → HH • ( D ) . Mor e over, the as s i g nment, F 7→ F • , is natur al in the fol lowing sense . L et F 1 ∈ p erf ( B op ⊗ C ) and F 2 ∈ p erf ( C op ⊗ D ) and let F 2 ◦ F 1 denote the B - D bimo dule c orr es p onding to the tensor pr o duct F 1 L ⊗ C F 2 . Then, ( F 2 ◦ F 1 ) • ∼ = F 2 • ◦ F 1 • . Pr o of. This is [PV12, Lemma 1.2.1].  Definition 6.2. Let C and D be saturated dg-categor ies o v er k . Let F b e an ob j ect of p erf ( C op ⊗ k D ). W e will call the linear ma p, F • , the pushfo rw ard by F . F or an o b ject E ∈ p erf ( C ), w e get an induced map, E • : k [0] ∼ = HH • ( k ) → HH • ( C ) . The map, E • , is called the Chern characte r map and the elemen t E • (1) is called the Chern cha racter of E . The map E 7→ E • (1) is denoted b y ch. There is a lso a natural pairing on Ho chs ch ild homology . Prop osition 6.3. L et C b e satur ate d d g-c ate gory over k . Ther e is a natur al p airing h· , · i : HH • ( C ) ⊗ k HH • ( C ) → k satisfying χ  ⊕ i ∈ Z Hom perf ( C ) ( E 1 , E 2 [ i ])  = h ch( E 1 ) , ch( E 2 ) i for E 1 , E 2 ∈ p erf ( C ) . Pr o of. This pairing is constructed for smo oth a nd prop er dg-alg ebras in [Shk07, Section 1 .2]. In this case, the equalit y χ  ⊕ i ∈ Z Hom perf ( C ) ( E 1 , E 2 [ i ])  = h ch( E 1 ) , ch( E 2 ) i is a sp ecial case of [Shk07, Theorem 1.3.1]. The pa iring is also defined for a general saturated dg-category in [PV12, Section 1.2]. As an y saturated dg-category is Morita equiv alen t to a 68 BALLARD, F A VERO, A N D KA TZAR KO V smo oth and prop er dg- algebra, the natur a lit y of the pairing extends the result f r o m alg ebras to categories.  Definition 6.4. Let C b e a saturated dg-category . W e shall call the pairing h· , · i : HH • ( C ) ⊗ k HH • ( C ) → k the categoric al pairing o n Ho c hsc hild homology . W e will a lso need the follow ing result due to Polishc huk a nd V ain trob. Theorem 6.5. L et A n c arry a line ar action of G , an algebr aic gr oup, and let w ∈ Γ( A n , O A n ( χ )) G . Assume that K χ is finite and χ : G → G m is surje c tive. F urthermor e, assume that (d w ) is supp orte d at { 0 } ∈ A n . F or a c h ar acter, ρ : G → G m , the twist functor, ( ρ ) : D abs fact ( A n , G, w ) → D abs fact ( A n , G, w ) , induc es a pushforwar d map, ( ρ ) • : HH • ( A n , G, w ) → HH • ( A n , G, w ) which i s m ultiplic ation by ρ ( g ) − 1 on Jac( w g ) for g ∈ K χ . In other wor ds, the de c omp osition of The o r em 5.39 is exactly the eigensp ac e de c omp osition for the action of b G on HH • ( A n , G, w ) . Pr o of. This is part of [PV11, Theorem 2.6.1], a lb eit stated in the notation used in this pap er.  6.1. Another lo ok at Griffiths’ Theorem. In this section, w e r ecall a celebrated result of Griffiths, repro v ed and understo o d in catego rical language as a combin at ion of Theorem 5 .39, the Ho chsc hild-Kostant-Rosen b erg isomorphism, and a theorem of Orlo v [Orl09]. Definition 6.6. Let Z b e a smooth complex pro jectiv e h yp ersurface in P n − 1 C defined b y w ∈ C [ x 1 , . . . , x n ]. An elemen t of H 2( n − 2 − k ) ( Z ; C ) is called p rimitive if it cups trivially with H k , where H is the class of a hyperplane section. W e write H • prim ( Z ; C ) for the subspace of primitiv e classes. W e will write H • , • prim ( Z ) for the interse ctions of H • prim ( Z ; C ) with eac h bi-graded piece of the Dolb eault cohomolog y of Z . In our con text, by the Lefsc hetz Hyp erplane Theorem, all primitive cohomology classes lie in the middle dimensional cohomology , H n − 2 ( Z ; C ). F urthermore, all elemen t s ar e primitiv e when n is o dd. When n is ev en, all Dolb eault classes of type ( p, n − 2 − p ), H p,n − 2 − p ( Z ), with p 6 = n − 2 2 are primitiv e, while H n − 2 2 , n − 2 2 prim ( Z ) are j ust those classes lying in the kerne l of the cup pro duct with H . Th e following description is due to G riffiths. Theorem 6.7. Ther e is an isomorphism, H p,n − 2 − p prim ( Z ) ∼ = Jac( w ) d ( n − 1 − p ) − n . Pr o of. This is [Gri69, Theorem 8.1].  KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 69 Comparing G r iffiths’ result with Theorem 5.3 9 w e see a striking similarit y . Indeed, Jac( w ) d ( n − 1 − p ) − n , is also the summand of HH n − 2 − 2 p ( A n , G m , w ) correspo nding to g = e . This is not a coincidence. T o give a precise comparison, we will need to recall tw o results. Definition 6.8. L et Z b e a smo oth, pro jective v ariety . Let I nj coh ( Z ) denote the dg- category of b ounded b elow c hain complexes of injective shea ve s on Z with bo unded and coheren t cohomology . W e denote the Ho c hsc hild homology of Inj coh ( Z ) b y HH • ( Z ). Definition 6.9. The Mukai pairing on H • ( Z ; C ) is ( v , v ′ ) M := Z Z v ∨ · v ′ · td( Z ) where v ∨ = P p,q ( − 1) p v p,q if v = P p,q v p,q is the Ho dge decomp osition. The first result w e use is the Ho chsc hild-Kostant-Rosen b erg isomorphism. It allows one to rein terpret Dolb eault cohomology categorically . Theorem 6.10. L et Z b e sm o oth pr oje ctive variety. Ther e ar e n atur al i somorphisms, HH t ( Z ) ∼ = M q − p = t H q ( Z , Ω p Z ) ∼ = M q − p = t H p,q ( Z ) . We d enote the isomorphi s m by φ HKR : HH • ( Z ) → H • ( Z ; C ) . Under the HKR i s o morphism, we have h α, α ′ i = ( φ HKR ( α ) , φ HKR ( α ′ )) M . The Chern cha r acter a nd classic al Chern cha r acter agr e e under the HKR isomorphism φ HKR (c h( E )) = c h class ( E ) . F urthermor e, fo r an inte gr al functor, Φ K : D b (coh X ) → D b (coh X ) , the action of Φ K• under the HKR isomo rphism is the c ohomolo gic al inte gr al tr ansform, Φ H K , asso cia te d to ch class ( K ) ∈ H • ( X × Y ; C ) . Pr o of. The HKR isomorphism in the a ffine case is due to [HK R 62]. In this generalit y , it is due to Swan [Swa96, Corollary 2.6] and Kon tsevic h [Kon03], see also [Y ek02]. The preserv ation of the Chern c haracter w as stated in [Mar0 1 ] and prov en as [Cal05, Theorem 4.5]. The equalit y of the pa ir ing s is [Ra m1 0 , Theorem 1]. The equalit y φ HKR ◦ Φ K• = Φ H K ◦ φ HKR is a consequence of [Ra m10, Theorem 2 ] and the definition of Φ muk ∗ in [Ram10].  Definition 6.11. Let Z b e a smo oth, pro jectiv e v ariet y . Define the endofunctor, { 1 } := L O Z ◦ T O (1) : I nj coh ( Z ) → Inj coh ( Z ) , where T O (1) ( E ) := E ⊗ O Z O Z (1) and, for i ∈ Z , L O Z ( i ) ( E ) := Cone  Hom( e O Z ( i ) , E ) ⊗ k e O Z ( i ) → E  where e O Z ( i ) is an injectiv e resolution of O Z ( i ). Let ς ( O Z ( i )) : Id → L O Z ( i ) denote the induced natural transformation. 70 BALLARD, F A VERO, A N D KA TZAR KO V The second result we use is a theorem of Orlov [Orl09], generalized mildly t o accoun t for a larger gr ading group. Let G b e an Ab elian affine algebraic gro up acting on A n . W e a ssume that G has rank one so that G ∼ = G m × G tors for G tors a finite Ab elian group. Definition 6.12. W e say that G acts p ositively o n A n if with resp ect to t he induced G m - action all nonzero linear functions o n A n ha v e p o sitive degree. W e ha v e a G m -equiv ariant isomorphism ω A n ∼ = O A n ( N ) for N equal to the sum of the degrees of x i if A n = Sp ec k [ x 1 , . . . , x n ]. Theorem 6.13. L et w ∈ Γ( A n , O A n ( χ )) G for a char acter χ : G → G m with χ | G m = d > 0 . L et Y b e the z e r o lo cus of w on punctur e d affine sp a c e A n \ { 0 } . If G = G m and N = n , le t Z denote the pr oje ctive hyp ersurfac e determine d by w . Assume w i s not zer o and that Y is smo oth. F urther, assume that G acts p ositively. • If d < N , then ther e exists m o rphisms in Ho(dg- cat k ) Φ : Inj coh ( A n , G, w ) → Inj coh G ( Y ) Φ ! : I nj coh G ( Y ) → Inj coh ( A n , G, w ) and a s e m i-ortho go n al de c omp osi tion D b (coh G Y ) = * M α | G m = d − N O Y ( α ) , . . . , M α | G m = − 1 O Y ( α ) , [Φ] ([ Inj coh ( A n , G, w )]) + . Mor e over, if G = G m and N = n , ther e ar e quasi-isomorphism s of bim o dules Φ ! ◦ { 1 } ◦ Φ ∼ = (1) Φ ! ◦ Φ ∼ = ∇ and [Φ ! ] O Z ( i ) ∼ = 0 for d − N ≤ i ≤ − 1 . • If d = N , then ther e exists in v erse morphisms in Ho(dg-cat k ) Φ : Inj coh ( A n , G, w ) → Inj coh G ( Y ) Ψ : Inj coh G ( Y ) → Inj coh ( A n , G, w ) . If, in addition, G = G m and N = n , ther e is a quasi-isomorphism of bim o dules { 1 } ◦ Φ ∼ = Φ ◦ (1) . Mor e over, f o r e ach s ∈ k [ x 1 , . . . , x n ] homo gene ous o f de gr e e i , the natur a l tr ansfor- mations of exact functors, s : Id D abs [ fact ( A n , G m ,w )] → ( i ) s : Id D b (coh Z ) → T O Z ( i ) satisfy the identity Φ( s ) = ς ( O Z ) ◦ · · · ◦ ς ( O Z ( i − 1)) ◦ s : Id → Φ ◦ ( i ) ◦ Φ − 1 ∼ = L O Z ◦ · · · L O Z ( i − 1) ◦ T O Z ( i ) . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 71 • If d > N , then ther e exists m o rphisms in Ho(dg- cat k ) Ψ ! : I nj coh ( A n , G, w ) → Inj coh G ( Y ) Ψ : Inj coh G ( Y ) → Inj coh ( A n , G, w ) and a s e m i-ortho go n al de c omp osi tion D abs [ fact ( A n , G, w )] = * M α | G m = − 1+ d − N k ( α ) , . . . , M α | G m =0 k ( α ) , [Ψ]  [ Inj coh G ( Y )]  + . Mor e over, if G = G m and N = n , ther e ar e quasi-isomorphism s of bim o dules Ψ ! ◦ (1) ◦ Ψ ∼ = { 1 } Ψ ! ◦ Ψ ∼ = ∆ ∗ O Z . and [Ψ ! ] k ( j ) ∼ = 0 for d − N − 1 ≥ j ≥ 0 . Pr o of. In t he case that G m = G , in [Orl09, The orem 2.1 3], Orlo v constructs the triang u- lated functors and the semi-ortho g onal decomp ositions of the t riangulated categories. The isomorphisms on the lev el of triangulated functors w ere constructed in [BFK11, Prop osition 5.8]. Cald˘ a r˘ aru and T u [CT10, Theorem 5.9] lifted these functors to dg-functors b et w een appropriate enhancemen ts. W e indicate the extension t o G as in the statemen t of the theo- rem. Consider the follo wing diagram of dg-categories: Inj coh G , ≥ i ( U ) vect ( A n , G, w ) op -Mo d Inj coh G ( Y ) Υ i π i ω i Here U is zero lo cus o f w in A n . The dg-category Inj coh G , ≥ i ( U ) consists of b ounded b elo w complexes o f injectiv e G -equiv ariant sheav es on U whose cohomology lies in G m -degrees ≥ i , is b ounded, and finitely-generated. Let Υ i denote the restriction of Υ to Inj coh G , ≥ i ( U ) whic h is then naturally a dg-mo dule for v ect coh ( A n , G, w ). It is easy to see that Υ i is a quasi-functor. Finally , let π the restriction along the inclusion Y → U , π i the restriction of π to Inj coh G , ≥ i ( U ), and let ω i denote the functor , ω i ( F ) := M α ∈ b G α | G m ≥ i H 0 ( Y , F ( α )) . Note that, as ω i is righ t adjoint to π i at the lev el of the Ab elian categor y o f equiv ar ian t shea v es, the cor r esp o nding dg-functors are also adjoint. Next, define D i to b e the quasi-essen tial imag e of ω i , in particular D i is closed under quasi- isomorphism, and P ≥ i to b e t he full dg - sub category of Inj coh G , ≥ i ( U ) con taining the injectiv e resolutions of O U ( α ) for α | G m ≤ i . Finally , let T i b e the full dg-sub category con taining all F that satisfy H •  Hom Inj coh G , ≥ i ( U ) ( F , P )  = 0 for all P ∈ P ≥ i . 72 BALLARD, F A VERO, A N D KA TZAR KO V As π ◦ ω i = Id, the restriction of π i to D i is a quasi-equiv a lence and ω i its in ve rse. F ollo wing argumen ts of [Orl09], whic h we suppress, the restriction of Υ to T i is a quasi-equiv alence. Let ν i b e the inv erse to Υ | T i in Ho(dg- cat k ). One t hen sets Φ i := π ◦ ν i , Φ := Φ 1 Φ ! i := Υ ◦ ω i , Φ ! := Φ ! 1 Ψ i := Υ ◦ ω i , Ψ := Ψ 1 Ψ ! i := π ◦ ν i − d + n , Ψ ! := Ψ ! 1 . The pro ofs of the existence of the semi-orthogonal decomp ositions follow along the same argumen ts of [Orl09] using the fact that R Hom Qcoh U ( • , O U ) : D b (coh G U ) op → D b (coh G U ) is an equiv alence satisfying R Hom Qcoh U ( k , O U ) ∼ = k ( ν )[ − n ] for ν ∈ b G with ν | G m = N . In the case G m = G and n = N , w e hav e an equiv alence Q coh G Y ∼ = Qcoh Z . The statemen ts that [Φ ! ] O Z ( i ) ∼ = 0 for d − N ≤ i ≤ − 1 and [Ψ ! ] k ( j ) ∼ = 0 for d − N − 1 ≥ j ≥ 0 follow immediately fro m [O rl09]. The only remaining statemen t to chec k is that concerning the existence of quasi-isomorphisms b et w een the stated bimo dules. It suffices to show that the corresp onding dg-functors are naturally quasi-isomorphic. No w, consider the f ollo wing dg-functor, M : Inj coh G , ≥ i ( U ) → Inj coh G , ≥ 0 ( U ) E 7→ Cone  Hom Inj coh G , ≥ 1 ( U ) ( e O U , E (1)) ⊗ k ( e O U ev → E (1)  where ( e O U is an injectiv e resolution of O U . Note tha t w e hav e a natura l transformation η : (1) → M . Consider the diagram Inj coh G , ≥ i ( U ) Inj coh G , ≥ 0 ( U ) Inj coh ( Z ) Inj coh ( Z ) M ω 1 π L O Z ◦ T O Z (1) The comp osition equals ( π ◦ M ◦ ω i )( E ) := Cone  Hom Inj coh G , ≥ 0 ( U ) ( e O U , ω i E (1)) ⊗ k π e O U ev → ( π ◦ ω i )( E (1))  . Using the adjunction, π ⊣ ω i , and the iden tit y , π ◦ ω i ∼ = Id, the comp osition is isomorphic to Cone  Hom Inj coh ( Z ) ( e O Z , E (1)) ⊗ k e O Z ev → E (1)  = E { 1 } . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 73 Th us, we ha ve a natural isomorphism π ◦ M ◦ ω i ∼ = { 1 } . (6.1) W e will use this equation in b oth cases. No w, a ssume that d ≤ n and consider the comp osition Φ ! ◦ { 1 } ◦ Φ = Υ ◦ ω 1 ◦ { 1 } ◦ π ◦ ν 1 . W e can substitute Υ ◦ ω 1 ◦ { 1 } ◦ π ◦ ν 1 ∼ = Υ ◦ ω 1 ◦ π ◦ M ◦ ω 1 ◦ π ◦ ν 1 . Since the image of ν 1 lies in D 1 b y [Orl09], we ha v e ω 1 ◦ π ◦ ν 1 ∼ = ν 1 . One can c hec k, as in [BFK11, Lemma 5.7], that M ◦ ν 1 has quasi-essen tia l image in D 1 , th us w e ha v e a na tural quasi-isomorphism M ◦ ν 1 → ω 1 ◦ π ◦ M ◦ ν 1 . This give s a na t ur a l quasi-isomorphism Φ ! ◦ { 1 } ◦ Φ ≃ Υ ◦ M ◦ ν 1 . The comp osition Υ ◦ (1) ◦ ν 1 Υ( η ν 1 ) → Υ ◦ M ◦ ν 1 is a quasi-isomorphism for all ob jects as Υ( e O U ) is acyclic. Thus , using the ab o v e and Equation (6.1), w e ha v e a quasi-isomorphism Φ ! ◦ { 1 } ◦ Φ ≃ Υ ◦ M ◦ ν 1 ≃ Υ ◦ (1) ◦ ν 1 = ( 1 ) . No w, a ssume that d ≥ n and consider the comp osition Ψ ! ◦ (1) ◦ Ψ = π ◦ ν 1 − d + n ◦ (1) ◦ Υ ◦ ω 1 . One has a natural quasi-isomorphism (1) ◦ Υ = Υ ◦ (1) Υ( η ) → Υ ◦ M . Th us, π ◦ ν 1 − d + n ◦ (1) ◦ Υ ◦ ω 1 ∼ = π ◦ ν 1 − d + n ◦ Υ ◦ M ◦ ω 1 . As D 1 ⊂ T 1 − d + n b y [Orl0 9] and M ( D 1 ) lies in D 1 , w e ha ve π ◦ ν 1 − d + n ◦ Υ ◦ M ◦ ω 1 ∼ = π ◦ M ◦ ω 1 ∼ = { 1 } where the last quasi-isomorphism is Equation (6.1). Finally , let us assume that d = N = n a nd G = G m . Let s ∈ k [ x 1 , . . . , x n ] b e homogeneous of degree 1, the nat ura l transformations of exact functors, s : Id D abs [ fact ( A n , G m ,w )] → (1) s : Id D b (coh Z ) → T O Z (1) . Let E b e an ob ject of Inj coh ( Z ) and consider s : E → T O Z (1) ( E ). Applying ω 1 giv es a morphism ω 1 ( s ) : ω 1 ( E ) → ω 1 ( E ) ≥ 2 (1) . 74 BALLARD, F A VERO, A N D KA TZAR KO V Comp osing with the inclusion ω 1 ( E ) ≥ 2 (1) → ω 1 ( E )(1) equals s : ω 1 ( E ) → ω 1 ( E )(1) . Apply the dg-functor L O U ( I ) := Cone(Hom( e O U , I ) ⊗ k e O U → I ) . W e get a map ς ( O U ) ω 1 ( E ) ◦ s : ω 1 ( E ) → L O U ( ω 1 ( E )(1)) . Since ω 1 ( E ) ≥ 2 (1) is concen tra ted in homogeneous degrees ≥ 1, w e ha ve H • (Hom( e O U , ω 1 ( E ) ≥ 2 (1))) = H • (Hom( O U , ω 1 ( E ) ≥ 2 (1))) = 0 . Th us, ς ( O U ) ω 1 ( E ) ≥ 2 (1) : ω 1 ( E ) ≥ 2 (1) → L O U ( ω 1 ( E ) ≥ 2 (1)), is a quasi-isomorphism. Applying π , giv es Φ( s ) = ς ( O Z ) E ◦ s : E → L O Z ◦ T O Z (1) ( E ) on D b (coh Z ). It is straightforw ard to c hec k there are isomorphisms Φ ◦ ( i ) ◦ Φ − 1 ∼ = { i } ∼ = L O Z ◦ · · · ◦ L O Z ( i − 1) ◦ T O Z ( i ) . W e hav e tw o algebra homomorphisms S → M i ∈ Z Nat(Id , { i } ) where Nat denotes natural transformations. The first is giv en b y conjugation b y Φ while the second is s 7→ ς ( O Z ) ◦ · · · ◦ ς ( O Z ( i − 1)) ◦ s. for s ∈ S i . These agree on generators fo r S and hence a gree ov erall.  Remark 6.14. F rom the argumen ts ab o ve, it is clear tha t in the case G = G m and d = N = n , that Φ( k [1]) ∼ = O Z . Remark 6.15. One could also apply the results in [BF K12] on V GIT fo r equiv ariant fac- torizations. Or, o ne could directly lift the statemen ts of [Orl09] using t he results of [Ela11 ]. Remark 6.16. The case G 6 = G m will b e used in [BFK13]. Henceforth, w e will only apply Theorem 6.13 under the a ssumption that G = G m act in the usual manner on A n . Corollary 6.17. L et w b e a de gr e e d homo gene ous p olynomial in k [ x 1 , . . . , x n ] with its stan- dar d gr ading. L et Z b e the pr oje ctive hyp ersurfac e define d by w . Assume that Z i s smo oth. • If d < n , we have a c ommutative diag r am of ve ctor sp ac es HH • ( Z ) HH • ( Z ) HH • ( A n , G m , w ) HH • ( A n , G m , w ) { 1 } • Φ • Φ ! • (1) • KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 75 Mor e over, Φ ! • ◦ Φ • = 1 , the functor Φ ! • is right adjoint to Φ • under the c ate g o ric al p airing, a nd we have an ortho gona l de c omp osition HH • ( Z ) = Im Φ • ⊕ − 1 M j = d − n − 1 C · c h( O Z ( j )) . • If d = n , we have a c ommutative diag r am of ve ctor sp ac es HH • ( Z ) HH • ( Z ) HH • ( A n , G m , w ) HH • ( A n , G m , w ) { 1 } • Φ • Φ • (1) • and Φ • is an is o morphism. • If d > n , we have a c ommutative diag r am of ve ctor sp ac es HH • ( Z ) HH • ( Z ) HH • ( A n , G m , w ) HH • ( A n , G m , w ) { 1 } • Ψ • Ψ ! • (1) • Mor e over, Ψ ! • ◦ Ψ • = 1 , the functor Ψ ! • is right a d joint to Ψ • under the c ate goric al p airing, and we have an ortho gona l de c omp osition HH • ( A n , G m , w ) = Im Ψ • ⊕ d − n − 1 M j =0 C · c h( k ( j )) . Pr o of. All statemen ts but the adjunction a nd ortho gonal decomp osition ar e immediate con- sequence s of Theorem 6.1 3 and the functoria lit y fo r pushforw ar ds, [PV12, Section 1]. W e c hec k the adjunctions. W e only prov ide an argumen t f or the case d > n . The case d < n is analo gous. W e ha ve a splitting HH • ( A n , G m , w ) = Im Ψ • ⊕ ker Ψ ! • . (6.2) Coun ting dimensions, we also ha ve an or thogonal decomp osition HH • ( A n , G m , w ) = Im Ψ • ⊕ d − n − 1 M j =0 C · c h( k ( j )) . Th us, k er Ψ ! • = d − n − 1 M j =0 C · c h( k ( j )) 76 BALLARD, F A VERO, A N D KA TZAR KO V and the splitting of Equation (6.2 ) is orthog onal with respect to the Muk ai pairing. The adjunction now f o llo ws via a straigh tforward linear algebra argumen t.  Remark 6.18. F or the case, d ≤ n , the argument can be significantly simplified using [Kuz11, Theorem 7.1]. This result guarantee s a splitting of HH • ( X ) for an y semi-orthogonal decomp osition of D b (coh X ) at the tria ngulated lev el without ha ving to pro ve an ything at the lev el of dg- categories. F o r the sak e of this utility , w e will a pp eal to this result in Section 6.2 . Definition 6.19. Let T : V → V b e a linear endomorphism of a v ector space, V , o ve r C , and let λ ∈ C . W e denote the λ -eigenspace of T by E λ ( T ) . Lemma 6.20. Under the HKR iso morphism, The or em 6 . 1 0, ther e is an e quality φ HKR ( E 1 ( { 1 } • )) = H • prim ( Z ; C ) . Pr o of. Let us first observ e that φ − 1 HKR  H • prim ( Z ; C )  ⊆ E 1 ( { 1 } • ) . It easy to c hec k, cf. [Huy05, Exercise 5.37], that, for v ∈ H • ( Z ; C ), T H O Z (1) ( v ) = v · c h class ( O Z (1)) . If w e assume that v is primitiv e, then v · c h class ( O Z (1)) = v . It is also easy to v erify , cf.[Huy05, Exercise 8.15], that L H O Z ( v ) = v − (c h class ( O Z ) , v ) M c h class ( O Z ) . By definition, the pairing is expressed as (c h class ( O Z ) , v ) M = Z Z c h class ( O Z ) ∨ · v · td( Z ) = Z Z v · td( Z ) . As the T o dd class, td( Z ), is of the form 1 + H p ( H ) for some p olynomial p , a nd v is primitive , w e ha v e Z Z v · td( Z ) = Z Z v . Ho w ev er, b y the Lefsc hetz Hyp erplane Theorem, primitive classes cannot hav e top dimen- sional comp onents. Hence, Z Z v = 0 and L H O Z ( v ) = v . As the cohomolog ical integral transform { 1 } H corresp onds to { 1 } • under the HKR isomor- phism, Theorem 6.10, we see that φ − 1 HKR ( v ) ∈ E 1 ( { 1 } • ). Next, let L n − 2 i =0 C · H i b e the subspace of H • ( Z ; C ) corresp onding to p o we rs of the h yp er- plane class, H . As sume that v = P n − 2 i =0 a i H i lies in φ HKR ( E 1 ( { 1 } • )) = E 1 ( { 1 } H ). The n, a 0 = a 0 − (c h class ( O Z ) , v ) M a i = i X j =0 a j ( i − j )! , i > 0 . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 77 This immediately implies tha t v = 0. As the induced map on cohomology { 1 } H preserv es the splitting H • ( Z ; C ) = H • prim ( Z ; C ) ⊕ n − 2 M i =0 C · H i , w e see that E 1 ( { 1 } H ) = H • prim ( Z ; C ) and φ − 1 HKR (H • prim ( Z ; C )) = E 1 ( { 1 } • ) .  Theorem 6.21. L et w b e a homo gene ous p olynom ial of de gr e e d in C [ x 1 , . . . , x n ] . Assume that w defines a sm o oth pr oje ctive hyp ersurfac e, Z . • If we as s ume d ≤ n , then the line ar map, Φ • , induc es an isom o rphism, Φ • : E 1 ( { 1 } • ) → E 1 ((1) • ) . • If we as s ume d ≥ n , then the line ar map, Ψ ! • , induc es an isomorphism, Ψ ! • : E 1 ( { 1 } • ) → E 1 ((1) • ) . In p articular, Orlov’s the or em and the HKR isomo rp hism p r ovide iso morphisms, H p,n − 2 − p prim ( Z ) ∼ = Jac( w ) d ( n − 1 − p ) − n . Pr o of. Let us t reat the case d ≤ n first. Let v ∈ E 1 ( { 1 } • ). By Lemma 6.20, φ HKR ( v ) ∈ H • prim ( Z ; C ). Thus , v is orthog onal to ch( O Z ( j )) under the Muk ai pairing for eac h j ∈ Z . By Corollary 6.1 7, w e hav e an orthogo nal decomp osition HH • ( Z ) = Φ • HH • ( A n , G m , w ) ⊕ − 1 M j = d − n C · c h( O Z ( j )) . W rite v = Φ • v ′ ⊕ v ′′ with resp ect to this decomp osition. Th us, for j ∈ Z , 0 = ( φ HKR ( v ) , φ HKR (c h( O Z ( j ))) M = h v , c h( O Z ( j )) i = h v ′′ , c h ( O Z ( j )) i as φ HKR ( v ) ∈ H • prim ( Z ; C ) a nd φ HKR (c h( O Z ( j )) = c h class ( O Z ( j )) ∈ L n − 2 i =0 C · H i are or- thogonal with respect to the Muk ai pairing. Due to their exceptionalit y , the set of vec to r s c h( O Z ( d − n )) , . . . , c h( O Z ( − 1)) f orms an orthonormal basis for L − 1 j = d − n C · c h ( O Z ( j )). Con- sequen tly , v ′′ = 0 . Using Corollary 6.17 rep eatedly , w e hav e (1) • ( v ′ ) = Φ ! • { 1 } • Φ • v ′ = Φ ! • Φ • v ′ = v ′ i.e. v ′ ∈ E 1 ((1) • ). Th us, Φ • maps E 1 ((1) • ) monomorphically in to E 1 ( { 1 } • ). Coun ting dimensions, we see this is an isomorphism. No w, let us turn our at t ention to d ≥ n . By Theorem 6.13, w e ha v e an orthogonal decomp osition HH • ( A n , G m , w ) = Ψ • HH • ( Z ) ⊕ d − n − 1 M j =0 C · c h( k ( j )) . Assume that v ∈ E 1 ( { 1 } • ). W rite ((1) • ◦ Ψ • )( v ) = Ψ • v ⊕ v ′ 78 BALLARD, F A VERO, A N D KA TZAR KO V with resp ect to this decomp o sition. Let us compute h c h( k ( j ) ) , v ′ i for some j ∈ Z . By orthogonality , w e hav e h c h( k ( j ) ) , v ′ i = h c h( k ( j ) ) , ((1) • ◦ Ψ • )( v ) i . Since ( − 1) is in v erse to (1) and Ψ • ⊣ Ψ ! • , from Coro llary 6.17 , w e ha ve h c h( k ( j ) ) , ((1) • ◦ Ψ • )( v ) i = h (Ψ ! • ◦ ( − 1) • )(c h( k ( j )) ) , v i . Using the f unctoria l prop erties of pushforw ards, w e hav e (Ψ ! • ◦ ( − 1) • )(c h( k ( j )) ) = ch  Ψ ! ( k ( j − 1))  . It is easy to c hec k, in Orlo v’s equiv a lence, that Ψ ! k ( j − 1) lies in the smallest tr ia ngulated sub category of D b (coh Z ) generated b y the ob jects O Z ( j ), j ∈ Z . Note that w e do not need to pass to direct summands. Th us, c h(Ψ ! k ( j − 1)) ∈ n − 2 M j =0 C · c h( O Z ( j )) = φ − 1 HKR n − 2 M j =0 C · H j ! . By Lemma 6.2 0, φ HKR ( v ) ∈ H • prim ( Z ; C ). Th us, v is or t ho gonal to ch(Ψ ! k ( j − 1)) for any j ∈ Z . T herefore, v ′ = 0 and we hav e a well-define d monomorphism Ψ • : E 1 ( { 1 } • ) → E 1 ((1) • ) . Coun ting dimensions finishes the argumen t. No w, to see that H p,n − 2 − p prim ( Z ) ∼ = Jac( w ) d ( n − 1 − p ) − n , notice that b y Corollary 5.4 1 and Theorem 6 .5, we hav e an isomorphism E 1 ((1) • ) ∩ HH t ( A n , G m , w ) ∼ = Jac d ( n + t 2 ) − n . By Theorem 6.21, w e hav e an isomorphism E 1 ( { 1 } • ) ∩ HH t ( Z ) ∼ = E 1 ((1) • ) ∩ HH t ( A n , G m , w ) . F rom Lemma 6.20 and Theorem 6.1 0, w e hav e an isomorphism H • prim ( Z ) ∩ M q − p = t H p,q ( Z ) ∼ = E 1 ( { 1 } • ) ∩ HH t ( Z ) . Since we only hav e primitiv e cohomology in the middle degree, w e m ust hav e p + q = n − 2. Solving for t give s t = n − 2 − 2 p . Plugging in gives the statemen t.  Remark 6.22. One can also define H • prim ( Z ) as the orthogonal to P i ∈ Z k · ch( O Z ( i )) with resp ect to the categorical pairing. This extends Theorem 6.21 to o t her algebraically closed fields of characteristic zero. Remark 6.23. In a dditio n to ha ving in teresting Eigenspace s, the determinant of { 1 } • is the geometric gen us of t he h yp ersurface. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 79 Definition 6.24. Let Z b e a smo oth, pro jectiv e hypersurface. Let K b e the ob ject K := I ∆ ⊗ O Z × Z π ∗ 1 O Z (1)[1] . Define the g raded ring S ( Z ) := M i ≥ 0 Hom D b (coh Z × Z ) (∆ ∗ O Z , K ∗ i ) where K ∗ i denotes i -th self-conv olutio n K , cf. [Huy 05 , Section 5.1]. Lemma 6.25. Assume that Z is Cala b i-Y au. Ther e is an isomorph ism of functors { 1 } ∼ = Φ K : D b (coh Z ) → D b (coh Z ) and an inje ctive hom omorphism of gr ade d rings Jac( w ) → S ( Z ) wher e w is the defining p olynomial of Z . Pr o of. It is straightforw ard to chec k tha t w e hav e a quasi-isomorphism of kernels , K ∼ = { 1 } . Using Orlov’s equiv alence from Theorem 6.13, w e get a isomorphism of graded rings M i ≥ 0 Hom D abs [ fact ( A n × A n , G m × G m G m , ( − w ) ⊞ w )] ( ∇ , ∇ ( i )) → M i ≥ 0 Hom D b (coh Z × Z ) (∆ ∗ O Z , K ∗ i ) . There is a natural homomorphism of graded rings k [ x 1 , . . . , x n ] → M i ≥ 0 Hom D abs [ fact ( A n × A n , G m × G m G m , ( − w ) ⊞ w )] ( ∇ , ∇ ( i )) giv en b y multiplying b y a p olynomial. By Theorem 5.39, this induces a monomorphism Jac( w ) → M i ≥ 0 Hom D abs [ fact ( A n × A n , G m × G m G m , ( − w ) ⊞ w )] ( ∇ , ∇ ( i )) . The tota l comp osition is the desired homomorphism Jac( w ) → S ( Z ).  Remark 6.26. A natural question to ask of Griffiths’ Residue Theorem is: whe re do a ll the other g raded pieces of t he Jacobian algebra go? Lemma 6 .2 5 provid es the answ er in terms of the deriv ed category of Z for a Calabi- Y au h yp ersurface. The whole Jacobian algebra sits as a graded subring of morphisms in D b (coh Z × Z ) from the iden tit y functor to p o w ers of { 1 } . Certain p ow ers of { 1 } are shifts of the Serre functor. Those graded pieces of the Jacobian algebra then app ear in HH • ( Z ) ∼ = H • ( Z ; C ). In the F ano case, we ha ve to replace S ( Z ) with the graded alg ebra M i ≥ 0 Hom D b (coh Z × Z ) ( P , P ∗ { i } ∗ P ) where P = Φ ◦ Φ ! is the k ernel asso ciated to t he inclusion of D abs [ fact ( A n , G m , w )] → D b (coh Z ) as an admissible sub category , [Kuz11]. In the general type case, w e hav e different k ernels, K i = Ψ ! ◦ ( i ) ◦ Ψ, for eac h i . The natura l rep ository fo r the Jacobian algebra is the g raded v ector space M i ≥ 0 Hom D b (coh Z × Z ) (∆ ∗ O Z , K i ) . In eac h situation, w e ha v e a categorical realization of Griffiths’ fundamen ta l result that sees the entire Jacobian a lgebra. 80 BALLARD, F A VERO, A N D KA TZAR KO V 6.2. Using equiv arian t factorizations to study algebraic cycles. In this section we examine ho w algebraic classes b eha v e under v ariation of the group action. Using The o- rem 6 .5, the induction functor, and functoriality of push-forw ards, Prop osition 6.1, one can precisely relate the algebraic classes under induction and restriction of the group action. The follo wing is essen tially due to Polishc huk and V ain tro b. Prop osition 6.27. L et A n c arry a line ar action of G , an A b elian al g ebr aic gr oup, a n d let w ∈ Γ( A n , O A n ( χ )) G . Assume that K χ is finite and χ : G → G m is surje ctive. F urthermor e, assume that (d w ) is supp orte d at { 0 } ∈ A n . L et φ : H → G b e an inje ctive homom orphism of affine algebr aic g r oups and assume that χ ◦ φ is surje c tive. Con sider the functors, Ind G H : vect ( A n , H , w ) → vect ( A n , G, w ) Res G H : vect ( A n , G, w ) → vect ( A n , H , w ) , and the ind uc e d maps, Ind G H • : HH • ( A n , H , w ) → HH • ( A n , G, w ) Res G H • : HH • ( A n , G, w ) → HH • ( A n , H , w ) . The c omp osi tion is the line ar map satisfying Ind G H • ◦ Res G H • : HH • ( A n , G, w ) → HH • ( A n , G, w ) v 7→ ( | G/H | v v ∈ Jac( w g ) with g ∈ K χ ◦ φ 0 v ∈ Jac( w g ) with g 6∈ K χ ◦ φ . Pr o of. Let K denote the k ernel of b φ : b G → b H . F o r c ∈ K , c ( g ) = 1 if and only if g ∈ K χ ◦ φ . F rom Lemma 2.16, we hav e an isomorphism of functors, Ind G H ◦ Res G H ∼ = p ∗ p ∗ , where p : G/H × A n → A n is the pro jection. There fo re, Ind G H ◦ Res G H ∼ = L c ∈ K ( c ). Note that L c ∈ K ( c ) can b e factored as a comp osition vect ( A n , G, w ) κ → a c ∈ K vect ( A n , G, w ) ⊕ → vect ( A n , G, w ) where κ maps to the factor corresp onding to c b y the auto equiv alence, ( c ), and ⊕ is the functor that takes ` E c to ⊕E c . Here ` c ∈ K vect ( A n , G, w ) denotes the category whose ob- jects are | K | -tuples of ob jects from v ect ( A n , G, w ) and whose morphisms are | K | -tuples of morphisms vect ( A n , G, w ). Denote an o b ject of ` c ∈ K vect ( A n , G, w ) b y ⊕ c ∈ K E c e c where w e think of e c as orthog o nal idemp o t ents. A generator of vect ( A n , G, w ) exists by Lemma 4.14, Prop osition 3.64, and the assump- tion that the supp o rt of (d w ) is { 0 } . Cho ose a generator , G , and let A denote its dg- endomorphism complex. If w e ta k e ⊕G e c as our generator of ` c ∈ K vect ( A n , G, w ), w e see its dg-endomorphism complex is ˜ A = Ae 1 ⊕ · · · ⊕ Ae c where e c are (closed) ortho g onal idemp o- ten ts. It is easy to see tha t ˜ A L ⊗ ˜ A e ˜ A ∼ = ⊕ c ∈ K ( A L ⊗ A e A ) e c . Th us, HH • ( ` c ∈ K vect ( A n , G, w )) is isomorphic to ⊕ c ∈ K HH • ( A n , G, w ) e c . Theorem 6.5 sa ys that the action on the comp o nen t o f HH • ( ` c ∈ K vect ( A n , G, w )) corre- sp onding to Jac( w g ) is m ultiplication by c ( g ) − 1 . In terms of ˜ A and A , ⊕ : ` c ∈ K vect ( A n , G, w ) → vect ( A n , G, w ) corresp onds to the sum- ming map ˜ A → A whic h tak es ⊕ a c e c to P a c . It is easy to see the induced a ction on Ho c hsc hild homology is again summation. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 81 No w, w e see that if g ∈ K χ ◦ φ , then eac h c acts tr ivially and the summand corresp onding to Jac( w g ) gets m ultiplied by | K | = | G/H | . If g 6∈ K χ ◦ φ , then c ( g ) is nonzero and P c ∈ K c ( g ) = 0.  Next, w e prov e a lemma that allows us to lift algebraic cycles via induction. Lemma 6.28. L et A n c arry a line ar action of G , an Ab elian alge br aic gr oup, an d let w ∈ Γ( A n , O A n ( χ )) G . Assume that K χ is finite and χ : G → G m is surje ctive. F urthermor e, assume that (d w ) is supp orte d at { 0 } ∈ A n . Assume that the image o f the Chern char acter, c h : K 0 ( A nr , G, w ⊞ r ) → HH 0 ( A nr , G, w ⊞ r ) , sp ans , over C , for al l r ≥ 1 . F urthermor e, a s sume that (Jac( w g )( − κ e − κ g − uχ )) G = 0 for t 6 = 0 and g 6 = e wher e either 2 u = dim W g + t − n or 2 u + 1 = dim W g + t − n . Then, the image of the Chern char acter, c h : K 0 ( A nr , G × G m r , w ⊞ r ) → HH 0 ( A nr , G × G m r , w ⊞ r ) , also sp ans, over C . Pr o of. If C 1 , . . . , C n are saturated dg- categories, then it is stra ig h tforward to ve rify r O i =1 HH • ( C i ) ∼ = HH • ( C 1 ⊛ · · · ⊛ C r ) where the isomorphism is giv en b y taking tensor pro ducts ov er k . Th us, by Corollar y 5.18, HH • ( A nr , G × G m r , w ⊞ r ) ∼ = HH • ( A n , G, w ) ⊗ r . In particular, HH 0 ( A nr , G × G m r , w ⊞ r ) ∼ = M i 1 + ··· + i r =0 HH i 1 ( A n , G, w ) ⊗ k · · · ⊗ k HH i r ( A n , G, w ) . (6.3) T o v erify the claim, w e need to find a basis o f HH • ( A nr , G × G m r , w ⊞ r ) whic h are Chern c har- acters of ob jects of D abs [ fact ( A nr , G × G m r , w ⊞ r )]. W e pro ceed b y induction o n r . The base case, r = 1, is co v ered under the assumptions of the lemma. Assume the lemma is tr ue for all pro ducts of size < r , and consider the case o f r . Under the isomor phism of Equation (6.3 ) , it is enough to find a basis of decomp o sable v ectors, i.e. those expressible as tensor pro ducts of elemen ts of HH • ( A n , G, w ). Let v := v 1 ⊗ k · · · ⊗ k v n ∈ HH 0 ( A nr , G × G m r , w ⊞ r ) b e a decomp o sable vec tor. W e ha ve t w o cases: one, some v i ∈ HH 0 ( A n , G, w ) , and, tw o, no v i ∈ HH 0 ( A n , G, w ) . Let us consider case one first. In this case, v 1 ⊗ k · · · ⊗ k b v i ⊗ k · · · ⊗ k v n ∈ HH 0 (( A n ) × r − 1 , G × G m r − 1 , w ⊞ r − 1 ) , under the isomorphism of Equation (6.3). By induction, there exists a factorization, E ∈ D abs [ fact ( A n ( r − 1) , G × G m r − 1 , w ⊞ r − 1 )], with c h( E ) = v 1 ⊗ k · · · ⊗ k b v i ⊗ k · · · ⊗ k v n 82 BALLARD, F A VERO, A N D KA TZAR KO V and E ′ ∈ D abs [ fact ( A n , G, w )] with c h( E ′ ) = v i . Then, c h( E ⊠ E ′ ) = v 1 ⊗ k · · · ⊗ k v n . This cov ers the first case. Let us mov e to the second case. Not e that, since w e ha ve assumed (Jac( w g )( − κ e − κ g − uχ )) G = 0 for t 6 = 0 and g 6 = e , all of the v i ∈ HH 0 ( A n , G, w ) lie in the un twis ted sector corresp onding to g = e . Consider, the diagonal homomorphism, φ : G → G × G m r . By Prop osition 6.27, w e kno w the map,  Ind G × G m r G ◦ Res G × G m r G  • : HH 0 ( A nr , G × G m r , w ⊞ r ) → HH 0 ( A nr , G × G m r , w ⊞ r ) , applied to v is  Ind G × G m r G • ◦ Res G × G m r G •  ( v ) = | ( G × G m r ) /G | v . By assumption, w e can find an E ∈ D abs [ fact ( A ⊗ n , M , w ⊞ n )] with c h ( E ) = Res G × G m r G • ( v ). By Prop osition 6.1, w e get c h(Ind G × G m r G E ) = Ind G × G m r G • (c h( E )) =  Ind G × G m r G • ◦ Res G × G m r G •  ( v ) = | ( G × G m r ) /G | v . Th us, o v er C , w e can find a spanning set of decomp osable v ectors in the imag e of the Chern class map.  Remark 6.29. If w e could define an appropriate rationa l structure on t he Ho c hsc hild homol- ogy of vect ( A n , G, w ), the arg uments of Lemma 6 .28 w ould generalize t o sho w the fo llo wing statemen t. Assume that c h : K 0 ( A nr , G, w ⊞ r ) → HH 0 ( A nr , G, w ⊞ r ) Q , spans, ov er Q , for all r ≥ 1. F urthermore, assume that (Jac( w g )( − κ e − κ g − uχ )) G = 0 for t 6 = 0 a nd g 6 = e . Then, the image of the Chern character, c h : K 0 ( A nr , G × G m r , w ⊞ r ) → HH 0 ( A nr , G × G m r , w ⊞ r ) Q , also spans, ov er Q . As suc h, this giv es a b o otstrap pro cedure for pro ving the Ho dge conjec- ture for Morita pro ducts of factorization categories b y proving it for simpler grading gro ups. In fact, recen t work of Blanc [Bla12] may yield the a ppropriate rat ional structure. Corollary 6.30. Co n sider A n C with the standar d G m -action. L et w b e the F ermat cubic or quartic p olynomial. Then, the image of c h : K 0 ( A nr C , G × G m r m , w ⊞ r ) → HH 0 ( A nr C , G × G m r m , w ⊞ r ) sp ans over C . Pr o of. The result is a consequence of the splitting result for Ho c hsc hild homology of deriv ed categories under semi-orthogonal decomp osition, [K uz09, Theorem 7.3]. W e do this b y applying L emma 6.28 fo r G = G m . T o do so, we m ust che ck that c h : K 0 ( A nr C , G × G m r m , w ⊞ r ) → HH 0 ( A nr C , G × G m r m , w ⊞ r ) KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 83 spans. Appealing to Theorem 6.13, we hav e a semi-orthogonal decomp o sition, D b (coh Z w ⊞ r ) = hO Z w ⊞ r ( − r n + d ) , . . . , O Z w ⊞ r ( − 1) , D abs [ fact ( A nr C , G m , w ⊞ r )] i , where Z w ⊞ r is the asso ciated pro jectiv e hy p ersurface. Kuznetsov ’s result then states w e ha v e a decomp osition HH 0 ( Z w ⊞ r ) = − 1 M i = − r n + d C · c h( O Z w ⊞ r ( i )) ⊕ HH 0 ( A nr C , G m , w ⊞ r ) . Ran [Ran80] pr ov ed that for d = 3 , 4, the image of c h : K 0 (D b (coh Z w ⊞ r )) → HH 0 ( Z w ⊞ r ) spans HH 0 ( Z w ⊞ r ) o ve r C . Using Prop osition 6.1, we deduce that the image of c h : K 0 ( A nr C , G m , w ⊞ r ) → HH 0 ( A nr C , G m , w ⊞ r ) spans ov er C . The v anishing condition on the t wisted sectors of the Ho c hsc hild homology follo ws a s the fixed lo cus of an y g / ∈ G m is the origin of A n . This verifie s the h yp otheses of Lemma 6.28 so w e may conclude that the image of c h : K 0 ( A nr C , G × G m r m , w ⊞ r ) → HH 0 ( A nr C , G × G m r m , w ⊞ r ) spans ov er C f o r all r ≥ 1.  Remark 6.31. One may rephrase the conclusion of Corollary 6.3 0 as: the Ho dge conj ecture o v er Q is true f o r D abs fact ( A nr C , G × G m r m , w ⊞ r ). W e can apply Lemma 6.28 to repro v e the Ho dg e conjecture for arbitrary self-pro ducts of a certain K3 surface closely related to the F ermat cubic fo urfold. W e first recall a result of Kuznetso v. Prop osition 6.32. L et X b e the F erm at cubic fourfold in P 5 . Ther e e xists a u niq ue K3 surfac e, Y , such that ther e is a semi -ortho gona l de c omp osition, D b (coh X ) = hO X ( − 3) , O X ( − 2) , O X ( − 1) , D b (coh Y ) i . Pr o of. The F ermat cubic fourf old is a Pfaffian cubic. Thus , the existence of Y is consequence of Kuznetso v’s results on Homo lo gical Pro jectiv e D ua lit y , see [Kuz10 ] for the statemen t. As men tioned previously , Ran prov ed that the image o f c h : K 0 ( X ) → HH 0 ( X ) spans o ve r Q , [Ran80 ]. Using the splitting of Ho c hsc hild homolog y and naturality of push- forw ards in Ho chsc hild ho mology , w e deduce that the image of c h : K 0 ( Y ) → HH 0 ( Y ) spans o v er Q . In particular, since Y is a K3 surface, it m ust ha ve Picard rank 20 . If w e hav e t w o suc h K3’s surfaces, Y 1 and Y 2 , with D b (coh X ) = hO X ( − 3) , O X ( − 2) , O X ( − 1) , D b (coh Y 1 ) i = h O X ( − 3) , O X ( − 2) , O X ( − 1) , D b (coh Y 2 ) i . Then, w e m ust hav e an equiv alence, D b (coh Y 1 ) ∼ = D b (coh Y 2 ) . 84 BALLARD, F A VERO, A N D KA TZAR KO V Ho w ev er, K3 surfaces with Picard ra nk mor e than 11 do not hav e non- trivial F ourier-Muk ai partners [HLOY04, Corollary 2.7.1 ].  Corollary 6.33. L et Y b e the K3 surfac e app e arin g in Pr op osition 6.32. The Ho dge c o n je c- tur e holds for al l self-pr o ducts, Y × r , r ≥ 1 . Pr o of. By [Kuz11, Theorem 7.1], the pro jection functor, D b (coh X ) → D b (coh Y ), lifts to a dg- functor b etw een enhancemen t s. It is then straightforw ard to c hec k that Theorem 6.1 3 induces a quasi-equiv alence b et we en Inj coh ( A 6 C , G m , w ) a nd Inj coh ( Y ), where w = x 3 1 + · · · + x 3 6 . Th us, we ha ve quasi-equiv alences, Inj coh ( Y ) ⊛ r ≃ Inj coh ( A 6 C , G m , w ) ⊛ r ≃ Inj coh ( A 6 r C , G × G m r m , w ⊞ r ) . The final quasi-equiv alence is Corollary 5.18. T o¨ en, [T o¨ e07, Section 8], prov es that there is a quasi-equiv alence Inj coh ( Y ) ⊛ r ≃ Inj coh ( Y × r ) . W e know the Ho dge conjecture for I nj coh ( A 6 r C , G × G m r m , w ⊞ r ) is true b y Corollary 6.30 .  Remark 6.34. In the initial v ersion of this pap er, w e claimed that Corollary 6.33 was a new case of the Ho dge conjecture. After the first vers ion w as released, w e w ere inf o rmed b y P . Stellari t hat this case is already know n, see [RM08]. W e happily thank Stellari for this comm unication. Remark 6.35. Ran’s w ork w as extended b y N. Aoki, [Aok83]. Aoki’s w or k relies on that of T. Shio da, [Shi79]. Shioda prov es that the Ho dge conjecture holds for F ermat h yp ersurfaces as long as a certain arithmetic condition is satisfied. Aoki gives a rein terpretation o f this arithmetic condition. O ne can directly construct factorizations whose Chern characters span the classes, D m − 1 d , studied b y Shio da-Aoki. Using their arithmetic argumen t, one can then pro v e dir e ctly that the Ho dge conjecture holds for categories of G m -equiv ariant factorizations of F ermat p oten tials o f degree d in A 2 m C when d is prime, d = 4, or eve ry prime divisor o f d is greater than m + 1. W e omit the details, though they can b e found in the initial arXiv ed v ersion of this ar ticle. Reference s [Ana73] S. Anantharaman. Sch´ emas en gr oup es, esp ac es homo gnes et esp ac es alg ´ ebriques sur u ne b ase de dimension 1 . (F rench) Sur les g roup es alg´ ebriques, pp. 5-7 9. Bull. So c. Math. F rance, Mem. 33 , So c. Math. F rance, P ar is, 1973 . [Aok83] N. Aoki. On some arithmetic pr oblems r elate d to the Ho dge cycles on the F ermat varieties . Ma th. Ann. 26 6 (1983), no. 1 , 23-5 4. [BDFIK12] M. B a llard, D. Deliu, D. F avero, M.U. Isik, L. Katza rko v. R esolutions in factorization c ate gories . arXiv:121 2.326 4 . [BFK11] M. Ballard, D. F avero, L. Ka tzarko v. Orlov sp e ctr a: b ounds and gaps . Inv ent. Math. 189 (2012), no. 2, 359-43 0. [BFK12] M. Ballar d, D. F av ero, L. Ka tzarko v. V ariation of Ge ometric Invariant The ory quotients and de- rive d c ate gories , arXiv:120 3.6643. [BFK13] M. Ballard, D. F av ero, L. Katza r ko v. A c ate gory of kernels for e quivariant factorizations and its implic ations for H o dge the ory, II: furt her implic ations . ar Xiv:1310.2 656 . T o app ear in J. Math. P ur es Appl. [Bec12] H. Becker. Mo dels for s ingu larity c ate gories . arXiv:12 05.447 3 . KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 85 [BFN10] D. B en-Zvi, J. F rancis , D. Nadler. Inte gr al tr ansforms and Drinfeld c enters in derive d algebr aic ge ometry . J. Amer. Math. Soc. 23 (20 1 0), no. 4, 909 - 966. [Bla12] A. Blanc. T op olo gic al K-the ory and its Chern char acter for non-c ommutative sp ac es . arXiv:121 1.7360 . [Blu07] M. Blume. McKay c orr esp ondenc e and G-Hilb ert schemes , Ph.D. thesis, T ¨ ubingen, 2 007. Curr ently av ailable at http://tobias-lib.uni-tuebingen.de/ volltexte/2007/2 941/p df/diss.p df . [BV03] A. Bonda l, M. V an den B ergh. Gener ators and r epr esent ability of funct ors in c ommutative and non-c ommutative ge ometry . Mosc. Math. J. 3 (20 0 3), no.1, 1 -36. [Buc86] R.-O. Buch weitz. Maximal Cohen-Mac aulay m o dules and T ate-c ohomolo gy over Gor enst ein rings . preprint, 1 986. [CS10] A. Canonaco , P . Stella ri. Non-un iqueness of F ourier-Mukai kernels . Math. Z. 2 72 (2012), no. 1-2, 577-5 88. [Cal05] A. C˘ ald˘ ar aru. The Mukai p airing, II: The Ho chschild -K ost ant-R osenb er g isomorphism . Adv. Math. 194 (200 5), no. 1, 34-6 6. [CT10] A. C˘ ald˘ arar u, J. T u. Curve d A ∞ algebr as and L andau-Ginzbur g mo dels . arXiv:1007 .2679 [Dri04] V. Dr infeld. DG quotients of DG ca tegories. J. Algebra 2 72 (2004 ), no . 2 , 643 -691. [DeGa70] M. Demazure, P . Gabriel. Gr oup es a lg´ ebriques. T ome I: G´ eom´ etrie a lg´ ebrique, g´ en ´ e ralit´ es, gro up es commutatifs. (F r ench) Avec un appe ndice Corps de cla sses loca l par Michiel Ha z ewinkel. Masson & Cie, ´ Editeur, Paris; North- Ho lland Publishing Co ., Amster da m, 197 0. [Dyc11] T. Dyc kerhoff. Comp act gener ators in c ate gories of matrix factorizations . Duke Math. J. 15 9 (201 1), no. 2, 223-27 4. [DM12] T. Dyck erho ff, D. Murfet. The Kapustin-Li formula r evisite d . Adv. Math. 23 1 (2012), no. 3 -4, 18 58- 1885. [Eis80] D. Eisenbud. Homolo gic al algebr a on a c omplete interse ction, with an appli c ation to gr oup r epr esen- tations . T ra ns. Amer. Ma th. So c. 260 (1980 ), no. 1 , 35- 6 4. [Ela11] A. E lagin. Cohomolo gic al desc en t the ory for a morphism of stacks and for e quivariant derive d c at- e gories . (Russ ian) Mat. Sb. 2 02 (20 11), no. 4 , 31-64; translation in Sb. Math. 202 (20 11), no . 3- 4, 495-5 26 [FJR07] H. F an, T. J a rvis, Y. Ruan. The Witten e quation, m irr or symmetry and quantum singularity the ory . T o appe ar Annals of Ma th. [FHT01] Y. F ´ elix, S. Halp erin, J.-C. Thomas. Rational homotopy theory . Graduate T exts in Mathematics, 205. Springe r-V e r lag, New Y ork, 2001. [EGA IV.2 ] A. Grothendieck. ´ El´ emen ts de g´ eom ´ etr ie alg´ ebrique. IV. ´ Etude lo cale des sch´ ema s e t des mor- phismes de sch ´ emas. II. (F r ench) Inst. Hautes ´ Etudes Sci. Publ. Math. No. 24 1965. [Gri69] P . Griffiths. On the p erio ds of c ertain r ational inte gr als . Ann. of Math. (2) 9 0 (19 69), 460- 541. [HLOY04] S. Hos o no, B . Lian, K. Oguis o , S.-T. Y au. F ourier-Mukai numb er of a K3 surfac e . Algebr a ic structures and mo duli spaces, 1 7 7–19 2 , CRM Pro c. Lecture Notes, 38 , Amer. Math. So c., Pr ovidence, RI, 200 4 . [HKR62] G. Ho chschild, B. K ostant, A. Rosenberg. Differ ential forms on r e gular affine algebr as . T ra ns. Amer. Math. So c. 1 02 (1962 ) 383-4 0 8. [Huy05] D. Huybrech ts. F our ier-Muk ai T ra nsforms in Algebraic Geometr y . Oxford Mathema tical Mono- graphs. The Clarendon Pr ess, Oxfor d Universit y Pr ess, Ox fo rd, 2006 . [Ill71] L. Illusie. Existenc e de r´ esolutions glob ales . Th´ eorie des intersections et th´ eor ` e me de Riemann-Ro ch. S´ emina ir e de G´ eom´ etrie Alg´ ebrique du Bois-Ma rie 1 966-1 967 (SGA 6). Dirig´ e par P . Be rthelot, A. Grothendieck et L. Illusie. Avec la collab ora tion de D. F er r and, J. P . Jo uanolou, O. Jussila , S. Kleima n, M. Raynaud et J. P . Serre. Lecture No tes in Ma thema tics, V ol. 225. Springer- V erla g, Berlin-New Y ork, 1971. [KL03a] A. Kapustin, Y. Li. D-br anes in L andau-Ginzbur g mo dels and algebr aic ge ometry . J . High E ner gy Phys. 5 (2003), no. 12, 44 pp. (electr onic) [KL03b] A. Ka pustin, Y. Li. T op olo gic al c orr elators in L andau-Ginzbur g mo dels with b oundaries . Adv. Theor. Math. Phys. 7 (2 003), no . 4, 7 27-74 9. [KKP08 ] L. Katzarko v, M. Ko nt se vich, T. P antev, T. Ho dge the or etic asp e cts of mirr or symmet ry , in F ro m Ho dge theory to integrability and TQFT tt ∗ -geometry , 87- 174, Pr o c. Symp os. P ure Math., 7 8, Amer. Math. So c., Providence, RI, 20 08. 86 BALLARD, F A VERO, A N D KA TZAR KO V [Kel06] B. Keller , On differ ential gr ade d c ate gories . International Congress o f Mathematicians. V ol. I I, 151- 190, Eur . Math. Soc ., Z¨ urich, 2006. [KR08a] M. K hov anov, L. Rozansky . Matrix factorizations and link homolo gy . F und. Math. 199 (2008 ), no. 1, 1- 9 1. [KR08b] M. Khov anov, L. Roza nsky . Matrix factorizatio n s and link homolo gy. II . Geom. T op ol. 12 (200 8), no. 3, 1387-1 425. [Kon03] M. Ko nt sev ich. Deformation quantization of Poisson manifolds, I . Lett. Math. Phys. 66 (2003), no . 3, 15 7-216 . [Kra05 ] H. Kra use. The st able derive d c ate gory of a No etherian scheme . Co mpo s. Math. 141 (200 5), no. 5, 1128- 1162. [Kuz09] A. Kuznets ov. Ho chschild homolo gy and semiortho gonal de c omp ositions . arXiv:09 04.433 0 . [Kuz10] A. K uznetsov. Derive d c ate gories of cubic fourfolds . Coho mological and ge o metric appro a ches to rationality pr oblems, 21 9-243 , Prog r. Ma th., 28 2, Birk h¨ auser Bosto n, Inc., Bo ston, MA, 2010. [Kuz11] A. Kuznetsov. Base change for semiortho gonal de c omp ositions . Comp os. Math. 147 (2011 ), no. 3, 852-8 76. [LP11] K. Lin, D. Pomerleano. Glob al matr ix factorizations . arXiv:1 101.5 8 47 . [Mar01] N. Mark arian. Poinc ar´ e-Birkhoff-Witt isomorphism, Ho chschild homolo gy and Riemann-R o ch the o- r em . MP I 20 01-52 Pr eprint, 200 1. Currently av ailable a t htt p:// www.mpim-b onn.mpg.de/preblo b/ 1208. [MFK94] D. Mumford, J. F ogar ty , F. K irwan. Geometr ic in v ariant theory . Third edition. E rgebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics a nd Rela ted Areas (2)], 34 . Springer - V erla g, Berlin, 1 994. [Mur09] D. Murfet. R esidues and duality for singularity c ate gories of isolate d Gor enstein singularities . arXiv:091 2.162 9 . [Nee92] A. Neeman. The c onne ction b etwe en the K -t he ory lo c alization the or em of Thomason, T r ob augh and Y ao and t he s m ashing sub c ate gories of Bousfield and R avenel . Ann. Sci. cole No rm. Sup. (4) 25 (19 92), no. 5, 547-56 6. [Orl04] D. Orlov. T riangulate d c ate gories of singularities and D-br anes in Landau-Ginzbur g mo dels . T r. Mat. Inst. Steklov a 246 (20 0 4), 240- 2 62. [Orl06] D. Or lov, T riangulate d c ate gories of singularities, and e quivalenc es b etwe en L andau-Ginzbur g mo dels . (Russian) Mat. Sb. 197 (2006 ), no. 12, 117- 132; tr anslation in Sb. Math. 197 (2006), no. 11-12 , 1827- 1840. [Orl09] D. Orlov. Derive d c ate gories of c oher ent she aves and t r iangulate d c ate gories of singularities . Algebra , arithmetic, a nd geo metr y: in ho no r of Y u. I. Manin. V ol. I I, 503- 531, P rogr. Math., 270, Bir k h¨ auser Boston, Inc., Boston, MA, 2009 . [Orl11] D. Or lov. F ormal c ompletions and idemp otent c ompletions of t riangulate d c ate gories of singularities . Adv. Ma th. 226 (20 11), no.1, 206-217 . [Orl12] D. O rlov. Matrix factorizations for nonaffine LG-mo dels . Math. Ann. 353 (20 12), no. 1, 95- 1 08. [PV10] A. Polishch uk, A. V aintrob. Matrix factorizations and singularity c at e gories for stacks . arXiv:101 1.454 4 . [PV11] A. Polishch uk, A. V aintrob. Matrix fa ctorizations and Cohomolo gic al Field The ories . arXiv:110 5.290 3 . [PV12] A. Polishch uk, A. V aintrob. Chern char acters and Hirzebruch-Riemann-R o ch formula for matrix factorizations . Duke Math. J. 161 (2012), no. 10, 18 63-19 26. [Pos09] L. Positselski. Two kinds of derive d c ate gories, Koszul duality, and c omo dule-c ontr amo dule c orr e- sp ondenc e . arXiv:090 5.2621 . [Pos11] L. Positselski. Coher ent analo gues of matrix factorizations and r elative singularity c ate gories . arXiv:110 2.026 1 . [Pre11 ] A. Preygel. Thom-Seb astiani and duality for matrix factorizations . arXiv:11 01.583 4 . [Ram10] A. Ramado ss. The Mukai p airing and inte gr al t r ansforms in Ho chschild homolo gy . Mosc . Math. J. 10 (2010 ), no. 3 , 629 -645, 662- 663. [Ran80] Z. Ra n. Cycles on F ermat hyp ersurfac es . Compo sitio Math. 42 (1980/81 ), no. 1 , 121 -142. [RM08] J.J. Ram´ on Ma r ´ ı. O n the Ho dge c onje ctu r e for pr o ducts of c ertain surfac es . Collect. Ma th. 59 (200 8), no. 1, 1-26. [Rou08] R. Rouquier . Dimensions of triangulate d c ate gories . J. K -Theory 1 (2008), no.2, 193 -256. KERNELS FOR EQU I V ARIAN T F ACTORIZA TIONS AND HODGE THEOR Y 87 [Seg09] E. Sega l. The close d st ate sp ac e of affine L andau-Ginzbur g B-mo dels . arXiv:0904 .1 339 . [Shk07] D. Shklyarov. Hirzebruch-Riemman-R o ch for DG algebr as . a rXiv:071 0 .1937 . [Shi79] T. Shio da. The Ho dge c onje ctur e and the T ate c onje ctur e for F ermat varieties . Pro c. Japa n Acad. Ser. A Math. Sci. 55 (197 9), no. 3, 111- 114. [Swa96] R. Sw an. Ho chschild c ohomolo gy of quasipr oje ctive schemes . J. Pure Appl. Algebra 110 (1996), no. 1, 57 -80. [Tho97] R.W. Thomason. Equivariant r esolution, line arization, and Hilb ert’s fourte enth pr oblem over arbi- tr ary b ase schemes . Adv. Math. 6 5 (19 87), 16-3 4. [T o¨ e07] B. T o¨ en. The homotopy the ory of dg-c ate gories and derive d Morita the ory . Inv ent. Math. 167 (2007), no. 3, 615-66 7. [T ot0 4] B. T otaro. The r esolut ion pr op erty for schemes and s tacks . J . Reine Angew. Math. 577 (200 4), 1- 22. [T u10 ] J. T u. Matrix factorizatio ns via Koszul duality . arXiv:100 9.415 1. [V af91 ] C. V afa. T op olo gic al L andau-Ginzbur g mo dels . Mo d. P h ys . L e tt. A 6 (1 991), no.4, 337-34 6. [W at79 ] W. W aterhouse. Int r o duction to affine g roup schemes. Graduate T exts in Mathematics, 66 . Springer- V erla g, New Y ork -Berlin, 197 9. [Y ek0 2 ] A. Y ekutieli. The c ontinuous Ho chschild c o chain c omplex of a s cheme . Canad. J. Math. 5 4 (2002), no. 6, 1319-1 337. Ma tthew Balla rd University of South Carolina, Dep a r tment of Ma thema tics, Columbia, SC, U SA Email: ballard@math.sc.edu Da vid F a vero University of Al ber t a, Dep ar tment of Ma thema tics, Ed monton, AB, Canada Email: fa vero@gmail.com Ludmil Ka tzarko v University of Miami, Dep ar tment of Ma thema tics, Coral Ga bles, FL, USA Universit ¨ at von W ien, F akul t ¨ at f ¨ ur Ma thema tik, Wien, ¨ Osterreich Email: lk atzark@math.uci.edu