Homology, Homotopy and Applications, vol. 14(2), 212, pp.37 61 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES JESSE BURKE and MARK E. WALKER (communicatd by Charls A. Wibl) Abstract W study matrix factorizations of rgular global sctions of lin bundls on schms. If th lin bundl is vry ampl rlativ to a Nothrian affin schm w show that morphisms in th homotopy catgory of matrix factorizations may b computd as th hyprcohomology of a crtain mapping complx. Using this xplicit dscription, w prov an analogu of Orlov s thorm that thr is a fully faithful mbdding of th homotopy catgory of matrix factorizations into th singularity catgory of th corrsponding zro subschm. Morovr, w giv a complt dscription of th imag of this functor. 1. Introduction Givn an lmnt f in a commutativ ring Q, a matrix factorization of f is a pair of n n matrics (A, B) such that AB = f I n = BA. This construction was introducd by Eisnbud in [2] to study moduls ovr th factor ring R = Q/(f). H showd that if Q is a rgular local ring and f is nonzro, th minimal fr rsolution of vry finitly gnratd R-modul is vntually dtrmind by a matrix factorization [2, Thorm 6.1]. Buchwitz obsrvd in [1] (s also [5, 3.9]) that Eisnbud s Thorm implis that thr is an quivalnc [MF (Q, f)] = cokr D b (R)/ Prf(R) =: D sg (R) (1) btwn th homotopy catgory of matrix factorizations, which is dfind analogously to th homotopy catgory of complxs of moduls, and th quotint of th boundd drivd catgory of finitly gnratd R-moduls by prfct complxs. Rcall that a complx is prfct if it is isomorphic in D b (R) to a boundd complx of finitly gnratd projctiv R-moduls. W call D sg (R) th singularity catgory of R, following [5]. Th quivalnc (1) is inducd by snding a matrix factorization (A, B) to th imag of th R-modul cokr A in D sg (R). In this papr w study a schm thortic gnralization of matrix factorizations by rplacing Q with a Nothrian sparatd schm X, f by a global sction W of a lin bundl L on X, and R by th zro subschm i: Y X of W. A matrix Rcivd Fbruary 2, 212, rvisd May 8, 212; publishd on Octobr 8, 212. 2 Mathmatics Subjct Classification: 14F5, 13D9, 13D2. Ky words and phrass: matrix factorization, singularity catgory. Articl availabl at http://intlprss.com/hha/v14/n2/a3 and doi:1.431/hha.212.v14.n2.a3 Copyright c 212, Intrnational Prss. Prmission to copy for privat us grantd.
38 JESSE BURKE and MARK E. WALKER factorization of th tripl (X, L, W ) is a pair of locally fr shavs E 1, E on X and maps E 1 1 E E1 L such that 1 and ( 1 1 L ) ar both multiplication by W. Th goal of this papr is to xplor a gnralization of th quivalnc (1) to this schm-thortic stting. Th dfinition of matrix factorizations for schms givn hr was introducd in [7]; similar constructions hav bn studid in [3, 4, 8]. All of ths paprs hav in som way dalt with gnralizing (1). Thrfor, bfor w dscrib our contributions to this qustion, lt us dscrib what is known. Th right hand sid of (1) maks sns for any schm, in particular th zro subschm Y X of W. For th lft hand sid, on may mimic th affin cas and dfin morphisms analogously to th homotopy catgory of complxs of shavs. W writ this catgory as [MF (X, L, W )] naiv, for rasons that will b clar soon. For a matrix factorization (E 1 1 E E1 L), multiplication by W on cokr 1 is zro, and thus w may viw cokr 1 as an objct of D sg (Y ). Thr is a functor [MF (X, L, W )] naiv D sg (Y ) (2) that snds a matrix factorization (E 1 1 E E1 L) to cokr 1. Whn X is a rgular schm for which vry cohrnt shaf is th quotint of a locally fr shaf, and W is a rgular global sction of L (i.., W : O X L is injctiv), it is straightforward to s that th functor (2) is ssntially surjctiv. Indd, as in th affin cas, vry objct of D sg (Y ) is isomorphic to a cohrnt shaf M that is maximal Cohn-Macaulay (i.., for ach y Y, M y is a maximal Cohn-Macaulay (MCM) modul ovr th ring O Y,y ), and on may mimic th standard argumnt in th affin cas that associats a matrix factorization to a MCM modul: On first taks a surjction E i M with E locally fr on X. Th hypothss nsur that th krnl E 1 will also b locally fr. Multiplication by W dtrmins th vrtical maps in th diagram α E 1 E i G β W W W E 1 L E L i G L. α idl Sinc th right-most such map is th zro map, thr xists a diagonal arrow β causing both triangls to commut. W thus obtain a pair of locally fr cohrnt shavs E, E 1 on X and morphisms E 1 α β E E 1 L such that both compositions β α and (α id L ) β ar multiplication by W. This is a matrix factorization of th data (X, L, W ). In gnral (2) will not b an quivalnc. For obsrv that if th cokrnl of a matrix factorization is locally fr, thn it is trivial in th singularity catgory. Whn X is affin, locally fr shavs ar projctiv and th lifting proprty of such shavs allows on to construct a null-homotopy. But in th non-affin cas thr is no rason
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 39 such a null-homotopy should xist in gnral, and indd thr ar many xampls of matrix factorizations that ar non-zro in th naiv homotopy catgory but that hav a locally fr cokrnl; s Exampl 2.11. Whn X is not affin on may, as is don in [4, 7], tak th Vrdir quotint of th naiv homotopy catgory by objcts with a locally fr cokrnl. Lt us writ this catgory as [MF (X, L, W )]. Thn [7, Thorm 3.14], s also [4, Thorm 3.5], shows that (2) inducs an quivalnc [MF (X, L, W )] = D sg (Y ). In [4] th lin bundl L is assumd to b O X but th schm X is not assumd to b rgular. In that cas it is shown that th inducd functor [MF (X, L, W )] D sg (Y ) is fully faithful. S also [8] for a proof of a similar rsult using xotic drivd catgoris. A drawback of Vrdir quotints is that morphism sts in quotint catgoris can b difficult to comput. In this papr w offr a diffrnt approach to dscribing th catgory [MF (X, L, W )]. For vry pair of matrix factorizations E, F thr is mapping complx, dnotd by Hom MF (E, F), which is a twistd two-priodic complx of locally fr shavs on X; s Dfinition 2.3 for th prcis dscription. W dfin th catgory [MF (X, L, W )] H to hav objcts all matrix factorizations and for two such objcts E, F, morphisms btwn thm ar givn by Hom [MF ]H (E, F) = H Hom MF (E, F), whr H dnots hyprcohomology in dgr. Thr is a composition which is associativ and unital. Rcall that a schm X is projctiv ovr a ring Q if thr is a closd mbdding j : X P m Q for som m. In this cas, w say that O X(1) := j O P m Q (1) is th corrsponding vry ampl lin bundl. Our first main rsult shows that in this cas th two homotopy catgoris coincid. Thorm 1. Lt X b a schm that is projctiv ovr a Nothrian ring and L = O X (1) th corrsponding vry ampl lin bundl. For a global sction W of L thr is an quivalnc of catgoris [MF (X, L, W )] = [MF (X, L, W )] H. Using this concrt dscription of th morphisms in th homotopy catgory, w ar abl to giv anothr proof of (an analogu of) [4, Thorm 3.4] (which assumd that L = O X ) whn X is projctiv ovr a Nothrian ring and L = O X (1). Thorm 2. Lt X b a schm that is projctiv ovr a Nothrian ring of finit Krull dimnsion, L = O X (1) th corrsponding vry ampl lin bundl, and W a rgular global sction of L. Dfin i: Y X to b zro subschm of W. Thr is a functor cokr: [MF (X, L, W )] D sg (Y ), which snds a matrix factorization (E 1 1 E E1 L) to cokr 1, and which is fully faithful. Th ssntial imag is givn by th objcts C in D sg (Y ) such that i C is prfct on X. In particular, if X is rgular thn cokr is an quivalnc.
4 JESSE BURKE and MARK E. WALKER In [4] Orlov rmarks that it would b intrsting to undrstand th diffrnc btwn [MF (X, L, W )] and D sg (Y ). Th abov thorm shows that, whn X is projctiv ovr an affin schm and L = O X (1), th diffrnc is xactly th class of objcts in D sg (Y ) that ar not prfct ovr X. W cam to ths rsults studying affin complt intrsction rings. Lt Q b a rgular ring, f 1,..., f c a rgular squnc of lmnts in Q, and st R = Q/(f 1,..., f c ). Dfin X = P c 1 Q = Proj Q[T 1,..., T c ], L = O X (1), W = i f it i Γ(X, L) and Y X th zro subschm of W. Orlov showd in [6, Thorm 2.1] that thr is an quivalnc D sg (R) = D sg (Y ). Composing this with th quivalnc of Thorm 2 w obtain an quivalnc D sg (R) = [MF (X, L, W )]. In a companion papr to this on w will us this quivalnc and th xplicit dscription of th Hom-sts in [MF (X, L, W )] givn hr to study th cohomology of moduls ovr R. Acknowldgmnts Th authors thank Grg Stvnson for carfully rading a prliminary vrsion of this papr and offring hlpful commnts on it. 2. Th gnralizd catgory of matrix factorizations Throughout X will dnot a Nothrian sparatd schm and L a lin bundl on X. To simplify notation, vn if L is not vry ampl, for a quasi-cohrnt shaf G (or a complx of such) on X and intgr n, w will writ G(n) for G OX L n. (Rcall L n := Hom OX (L n, O X ) for n 1.) In particular, O(1) = L. Similarly, if f is morphism of (complxs of) quasi-cohrnt shavs, thn f(1) = f id L. Th following dfinition first appard in [7]. Dfinition 2.1. Lt W b a global sction of L. A matrix factorization E = (E 1 1 E E1 (1)) of th tripl (X, L, W ) consists of a pair of locally fr cohrnt shavs E 1, E on X and morphisms 1 : E 1 E and : E E 1 (1) such that 1 and 1 (1) ar multiplication by W. A strict morphism of matrix factorizations from (E 1 E E 1 (1)) to (F 1 F F 1 (1)) is a pair of maps E F, E 1 F 1 causing th vidnt pair of squars to commut. Matrix factorizations and strict morphisms of such form a catgory which w writ MF (X, L, W ) xact or just MF xact for short. Th largr catgory, with objcts matrix factorizations of arbitrary cohrnt shavs and arrows strict morphisms dfind in th sam way as abov, is an ablian catgory. Th catgory MF xact is a full subcatgory of this ablian catgory and is closd undr xtnsions, and hnc MF xact has th structur of an xact catgory in th sns of Quilln [9]. A squnc E E E in MF xact is a short xact squnc if it dtrmins a short xact squnc of locally fr cohrnt shavs in both dgrs.
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 41 Dfinition 2.2. A twistd priodic complx of locally fr cohrnt shavs for (X, L) is a chain complx C of locally fr cohrnt shavs on X togthr with a spcifid isomorphism α: C[2] = C(1), whr w us th convntion that C[2] i = C i+2. Th catgory TPC(X, L) has as objcts twistd priodic complxs and a morphism of such objcts is a chain map that commuts with th isomorphisms in th vidnt sns. Thr is an quivalnc TPC(X, L) = MF (X, L, ) xact givn by snding (C, α) to C 1 d C α 1 d C 1 (1). Th most important xampl of a twistd priodic complx, for us, is th following: Dfinition 2.3. Lt E = (E 1 1 f 1 f E E1 (1)) and F = (F 1 F F1 (1)) b matrix factorizations for (X, L, W ). W dfin th mapping complx of E, F, writtn Hom MF (E, F), to b th following twistd priodic complx of locally fr shavs: Hom(E, F 1 ) Hom(E, F ) ( Hom(E, F 1 ) ) ( 1) Hom(E 1, F ( 1)) 1 Hom(E 1, F 1 ) Hom(E 1, F ( 1)) (1) 1 (1). Hr, Hom dnots th shaf of homomorphisms btwn two cohrnt shavs on X and Hom(E, F ) Hom(E 1, F 1 ) lis in dgr. Th diffrntials ar givn by [ ] [ ] 1 (f1 ) = and (f ) =, (f ) (f 1 ) 1 using th canonical isomorphisms Hom(E i, F j (1)) = Hom(E i, F j )(1) = Hom(E i ( 1), F j ) 1 and Hom(E, F 1 ) (1) = Hom(E 1, F ( 1)) Hom(E, F 1 (1)) Hom(E 1, F ). On chcks that 1 and 1 (1) ar both, and hnc Hom MF (E, F) is in fact a twistd priodic complx. Not that thr is an isomorphism Hom MFxact (E, F) = Z (Γ(X, Hom MF (E, F))) whr Γ(X, Hom MF (E, F)) is th complx of ablian groups obtaind by applying th global sctions functor dgr-wis to Hom MF (E, F), and Z dnots th cycls in dgr. Dfinition 2.4. A strict morphism (g 1, g ): E F is nullhomotopic if thr ar maps
42 JESSE BURKE and MARK E. WALKER s: E F 1 and t: E 1 (1) F as in th diagram blow such that 1 E 1 E E 1 (1) s t g F 1 F F 1 (1) f 1 f g 1 g 1 (1) g 1 = s 1 + f ( 1) t( 1) and g = f 1 s + t. Two strict morphisms ar homotopic if thir diffrnc is nullhomotopic. Th naiv homotopy catgory of matrix factorizations, writtn [MF (X, L, W )] naiv, is th catgory with th sam objcts as MF (X, L, W ) xact and arrows givn by strict morphisms modulo homotopy. Equivalntly, for objcts E, F: Hom [MF ]naiv (E, F) = H Γ(X, Hom MF (E, F)). Dfinition 2.5. Th shift functor on [MF (X, L, W )] naiv, writtn [1], is th ndofunctor givn on objcts by ( ) ( ) E 1 1 E E1 (1) [1] = E E1 (1) 1(1) E (1). Th con of a strict morphism f = (g 1, g ): E F is th matrix factorization ( [ ] [ ] 1 ) (1) g f 1 g 1 (1) f con(f) = E F 1 E 1 (1) F E (1) F 1 (1). Thr ar maps F con(f) E[1] dfind in th usual mannr, and w dfin a distinguishd triangl to b a triangl in [MF ] naiv isomorphic to on of th form E f F con(f) E[1]. As rmarkd in [7], ths structurs mak [MF (X, L, W )] naiv into a triangulatd catgory, a fact on can chck dirctly by mimicking th proof for th homotopy catgory of complxs in an ablian catgory. Exampl 2.6. Lt Q b a Nothrian ring, X = Spc Q, L = O X, and W an lmnt of Q. Thn a matrix factorization is a pair of projctiv Q-moduls E 1, E and maps E 1 E such that composition in ithr dirction is multiplication by W. Th catgory [MF (X, L, W )] naiv is th homotopy catgory of matrix factorizations, as dfind for instanc in [5, 3.1], whr it is dnotd HMF (Q, W ). For a point x X, w may localiz an objct E of MF (X, L, W ) xact at x in th vidnt mannr to obtain an objct of MF (O X,x, L x, W x ) xact. It is clar that th
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 43 functor MF (X, L, W ) xact MF (O X,x, L x, W x ) xact, E E x prsrvs homotopis, and in this way w obtain a triangulatd functor [MF (X, L, W )] naiv [MF (O X,x, L x, W x )] naiv. Dfinition 2.7. A map of matrix factorizations E E in [MF (X, L, W )] naiv is a wak quivalnc if for ach x X, th map E x E x is an isomorphism in th catgory [MF (O X,x, L x, W x )] naiv. A matrix factorization F is locally contractibl if th uniqu map F is a wak quivalnc. This is quivalnt to th condition that F x is contractibl for all x X. Rmark 2.8. It is assrtd in [3, 2.6] that thr is a modl catgory structur for th vidnt gnralization of th notion of matrix factorizations for (X, L, W ) in which th objcts involv a pair of arbitrary quasi-cohrnt shavs. This largr catgory, which is closd undr all small coproducts, plays an important rol in th work of Positslski [8] and Lin and Pomrlano [3]. For a global sction W of a lin bundl L, th zro subschm of W is th subschm of X dtrmind by th idal shaf givn as th imag of W : L O X. Proposition 2.9. Assum X is a Nothrian schm, L is a lin bundl on X, and W is a rgular global sction of L, i.., th map W : O X L is injctiv. Lt i: Y X b th zro subschm of W. Considr th following conditions on a matrix factorization E = (E 1 1 E E1 (1)) : (1) E = in th catgory [MF (X, L, W )] naiv. (2) Th canonical surjction p: i E i cokr( 1 ) of cohrnt O Y -shavs splits. (3) E is locally contractibl. (4) i cokr( 1 ) is a locally fr cohrnt shaf on Y. In gnral, w hav 1 2 3 4. If X is affin thn all four conditions ar quivalnt. Rmark 2.1. Not that multiplication by W annihilats cokr( 1 ), and hnc via th canonical map. i i cokr( 1 ) = cokr( 1 ) Proof. W first obsrv that th proof of [5, 3.8] applis to show that (4) implis (1) whn X is affin. W nxt prov that (1) implis (2). Lt M = i cokr( 1 ) = cokr(i 1 ). If E =, thn E has a contracting homotopy, givn by s: E E 1 and t: E 1 (1) E satisfying 1 s + t = id and s(1) 1 (1) + t = id. As in th argumnt in th proof of [5, 3.7], sinc i (t 1 ) = and M is th cokrnl of i ( 1 ), thr is a map j : M i E such that jp = i (t ). But thn pjp = pi (t ) = pi (id 1 s) = p sinc pi ( 1 ) =. Sinc p is onto, w hav pj = id M.
44 JESSE BURKE and MARK E. WALKER If (i cokr E) x is a fr O Y,x modul for all x Y, thn, sinc (4) implis (1) in th affin cas, w conclud that E x = [MF (O X,x, L X, W x )]. This provs that (4) implis 3. It is clar that (2) implis (4) and hnc that (2) implis (3). Finally, if E x =, thn sinc (1) implis (2), w s that th map px : (i E ) x (i cokr( 1 )) x splits and hnc (i cokr( 1 )) x is fr. This provs (3) implis (4). Th following shows that a locally contractibl matrix factorization nd not b in [MF ] naiv. Exampl 2.11. Undr th assumption of Proposition 2.9, suppos that E = (E 1 E E 1 (1)) is a matrix factorization of (X, L, W ) such that i M := i cokr(e 1 E ) is a locally fr cohrnt O Y -shaf, but that th surjction i E i M dos not split. Thn by Proposition 2.9, E is locally contractibl, but E is not isomorphic to in [MF ] naiv. Such xampls ar common. For instanc, tak X = P 2 k = Proj k[t, T 1, T 2 ], L = O X (1) and W = T 2, so that Y = P 1 k = Proj k[t, T 1 ]. Thn lt M = O Y, E = O X ( 1) 2 and E i M b th composition of E can i O Y ( 1) 2 (T,T2) i O Y. Thn th krnl E 1 of E i M is locally fr and, using th argumnt found in th introduction, this lads to a matrix factorization of (X, L, W ) with i cokr(e 1 E ) = M. But th surjction dos not split. i E = O Y ( 1) 2 (T,T 1 ) O Y = M Th collction of locally contractibl objcts is th intrsction of th krnls of th triangulatd functors [MF (X, L, W )] naiv [MF (O X,x, L x, W x )], as x rangs ovr all points of X. Rcall that a triangulatd subcatgory of a triangulatd catgory is thick if it is closd undr dirct summands. Th krnl of any triangulatd functor is thick and an arbitrary intrsction of thick subcatgoris is thick. Thus th collction of locally contractibl objcts forms a thick subcatgory of [MF (X, L, W )] naiv. Th following catgory, whos dfinition is originally du to Orlov and appard, for xampl, in [7], is th cntral objct of study in this papr: Dfinition 2.12. Th homotopy catgory of matrix factorizations, writtn [MF (X, L, W )], is th Vrdir quotint of [MF (X, L, W )] naiv by th thick subcatgory of locally contractibl objcts: [MF (X, L, W )] = [MF (X, L, W )] naiv locally contractibl objcts. Rmark 2.13. A strict map E E is a wak quivalnc if and only if it fits into a distinguishd triangl E E F E [1] in [MF (X, L, W )] naiv such that F is locally contractibl. Thus wak quivalncs ar invrtibl in [MF (X, L, W )].
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 45 Exampl 2.14. If X is affin, L = O X and W is a non-zro-divisor, thn by Proposition 2.9. [MF (X, L, W )] = [MF (X, L, W )] naiv 3. Anothr vrsion of th homotopy catgory Th aim of this sction is to dscrib anothr catgory associatd to MF (X, L, W ), which w writ as [MF (X, L, W )] H. In th nxt sction, w prov that whn X is projctiv ovr a Nothrian ring and L is th corrsponding vry ampl lin bundl, [MF ] H and [MF ] ar quivalnt. Th advantag [MF ] H njoys ovr [MF ] is that its Hom sts ar mor xplicit. W mak a fixd choic of a finit affin opn covr U = {U 1,..., U m } of X, and for any quasi-cohrnt shaf F on X, lt Γ(U, F) dnot th cochain complx givn by th usual Cch construction. Sinc X is sparatd, th cohomology of th complx Γ(U, F) givs th shaf cohomology of F. W dfin Γ(U, Hom MF (E, F)) to b th total complx associatd to th bicomplx i Γ(U i, Hom MF (E, F)) i<j Γ(U i U i, Hom MF (E, F)) givn by applying th Cch construction dgr-wis. If G is anothr matrix factorization, thr is an vidnt morphism of chain complxs Γ(U, Hom MF (E, F)) Γ(U, Hom MF (F, G)) Γ(U, Hom MF (E, G)) which on can chck is associativ and unital. Thus MF (X, L, W ), with function spacs Γ(U, Hom MF (E, F)), is a DG catgory. W st H q (X, Hom MF (E, F)) = H q (Γ(U, Hom MF (E, F))). Thr is a convrgnt spctral squnc H p (X, H q (Hom MF (E, F))) = H p+q (X, Hom MF (E, F)), whr H q is th q-th cohomology shaf of a complx. In particular, if Hom MF (E, F) Hom MF (E, F ) is a quasi-isomorphism, thn th map is an isomorphism for all n. Dfinition 3.1. Dfin th catgory H n (X, Hom MF (E, F)) H n (X, Hom MF (E, F )) [MF (X, L, W )] H whos objcts ar matrix factorizations and whos morphisms ar Hom [MF ]H (E, F) = H (X, Hom MF (E, F)). Thus [MF ] H is th homotopy catgory associatd to th DG catgory abov. Rmark 3.2. This dfinition was inspird by Shipman s catgory of gradd D- Brans [11], who was in turn inspird by Sgal [1].
46 JESSE BURKE and MARK E. WALKER Thr is a canonical functor [MF (X, L, W )] naiv [MF (X, L, W )] H (3) that is th idntity on objcts and is givn on morphisms by th canonical map H Γ(X, Hom MF (E, F)) H (X, Hom MF (E, F)). Exampl 3.3. If X is affin this functor is an quivalnc sinc ach map H Γ(X, Hom MF (E, F)) H (X, Hom MF (E, F)) is an isomorphism. If w furthr assum L = O X and W is a non-zro-divisor of Q, thn both [MF (X, L, W )] naiv and [MF (X, L, W )] H ar quivalnt to [MF (X, L, W )] by Exampl 2.14. Lmma 3.4. If a strict morphism f : E E of matrix factorizations is a wak quivalnc, thn for all matrix factorizations F, th inducd map on mapping complxs Hom MF (E, F) Hom MF (E, F) is a quasi-isomorphism in T P C(X, L). In particular, th map in [MF (X, L, W )] H inducd by f : E E is an isomorphism. Proof. For matrix factorizations E, F and for all x X thr is an isomorphism Hom MF (X,L,W ) (E, F) x = HomMF (OX,x,L x,w x )(E x, F x ). Sinc w ar assuming E x E x is an isomorphism in [MF (O X,x, L x, W x )], it follows that Hom MF (X,L,W ) (E, F) x Hom MF (X,L,W ) (E, F) x is a quasi-isomorphism for all F and x. This provs Hom MF (E, F) Hom MF (E, F) is a quasi-isomorphism in T P C(X, L) and hnc that H n (X, Hom MF (E, F)) = H n (X, Hom MF (E, F)) is an isomorphism for all n. Th cas n = shows E and E co-rprsnt th sam functor on [MF (X, L, W )] H and hnc ar isomorphic. Th following is a formal consqunc of th lmma: Proposition 3.5. Th functor [MF (X, L, W )] naiv [MF (X, L, W )] H factors canonically as [MF (X, L, W )] naiv [MF (X, L, W )] [MF (X, L, W )] H. 4. Equivalnc of homotopy catgoris Th goal of this sction is to prov Thorm 1 of th introduction: th functor [MF (X, L, W )] [MF (X, L, W )] H is an quivalnc of catgoris whn X is projctiv ovr a Nothrian ring and L is th corrsponding vry ampl lin bundl. For such an X and L, w will oftn us Srr s
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 47 Vanishing Thorm: for any cohrnt shaf F on X and all n, H i (X, F(n)) = for i >. Givn a boundd complx of strict morphisms of matrix factorizations E := ( E p E p+1 E q), w dfin its total objct, writtn Tot(E ), as follows. W may visualiz E as a commutativ diagram E q ( 1) E q 1 E q E q 1 (1). E q 1 ( 1) E q 1 1 E q 1 1 (1) E q 1 E q 2 ( 1) E q 2 1 E q 2 1 (1) E q 2.... E p ( 1) E p 1 E p E p 1 (1) W W form Tot(E ) by taking dirct sums along lins of slop -1 in this diagram, and maps ar dfind just as for th usual total complx associatd to a bicomplx. Th rsulting chain of maps clarly satisfis th rquird twistd priodicity, making it an objct of MF (X, L, W ). A spcial cas of th Tot construction will b spcially usful. First, for a locally fr cohrnt shaf P on X and a matrix factorization E, lt P E dnot th matrix factorization obtaind by applying th functor P OX to th data dfining E. If P is a boundd complx of locally fr cohrnt shavs on X, thn P E = ( P p E P p+1 E P q E ) is a boundd complx of strict morphisms of matrix factorizations, and w may form its associatd total matrix factorization Tot(P E). W will nd th following lmma. Lmma 4.1. Lt X b a Nothrian sparatd schm, L any lin bundl on X and W a global sction of L. If P 1 P 2 is a quasi-isomorphism btwn boundd complxs of locally fr cohrnt shavs on X, thn Tot(P 1 E) Tot(P 2 E) is a wak quivalnc of matrix factorizations.
48 JESSE BURKE and MARK E. WALKER Proof. W may localiz at a point x, in which cas th assrtion bcoms: If W is an lmnt in a local ring Q, P 1 P 2 is a quasi-isomorphism of boundd complxs of fr Q-moduls of finit rank, and E is a matrix factorization of W ovr Q, thn Tot(P 1 E) Tot(P 2 E) is an isomorphism in th catgory [MF (Spc Q, O, W )]. In this stting, P 1 P 2 is a chain homotopy quivalnc (i.., thr is an invrs up to chain homotopy). It thrfor suffics to not that th functor Tot( E) from boundd complxs of fr moduls to matrix factorizations snds chain homotopis of complxs to homotopis of matrix factorizations. Thorm 4.2. Lt X b a schm that is projctiv ovr a Nothrian ring Q and L = O X (1) th corrsponding vry ampl lin bundl on X. For any global sction W of L, th canonical functor is an quivalnc. [MF (X, L, W )] [MF (X, L, W )] H Th proof of th thorm uss th following lmma. Lmma 4.3. Undr th assumptions of Thorm 4.2, for any pair of matrix factorizations E, F, thr is a wak quivalnc E E such that th canonical map is an isomorphism. Hom [MF ]naiv (E, F) = Hom [MF ]H (E, F) Proof. Say X is a closd subschm of P m Q = Proj Q[x,..., x m ] and L = O X (1) is th rstriction of O P m(1) to X. For ach positiv intgr j w hav a surjction O X ( j) k O X givn by th th st of monomials of dgr j in m + 1 variabls (so that k = k(j) is th numbr of such monomials). Lt P(j) b th associatd truncatd Koszul complx O X ( kj) (k k) OX ( (k 1)j) ( k k 1) OX ( 2j) (k 2) OX ( j) k, indxd cohomologically with O X ( nj) (k n) in dgr n + 1. Thn th vidnt map P(j) OX is a quasi-isomorphism, and so by Lmma 4.1, th inducd map E := Tot(P(j) E) Tot(O X E) = E is a wak quivalnc. W prov that for j this wak quivalnc has th dsird proprty. Writ E, E 1 and E, E 1 for th componnts of E and E. W hav that E = E ( j) k E 1 (1 2j) (k 2) E (1 3j) (k 3) E1 (2 4j) (k 4) and similarly for E 1. In particular, for any intgr N w may choos j so that E and E 1 ar dirct sums of locally fr cohrnt shavs of th form E ( a) and E 1 ( b) with a, b N.
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 49 Lt C dnot th twistd two-priodic complx Hom MF (E, F). By th abov, for any intgr N, w may pick j so that C and C 1 ar dirct sums of locally fr cohrnt shavs of th form Hom OX (E, F )(a), Hom OX (E 1, F )(b), Hom OX (E, F 1 )(c), or Hom OX (E 1, F 1 )(d) with a, b, c, d N. Thinking of C as an unboundd complx w hav C q = C ( q 2 ) if q is vn or C q = C 1 ( q 1 2 ) if q is odd. Thus, for any intgrs N and M, w may choos j sufficintly larg so that ach of C M, C M+1, is a dirct sum of locally fr cohrnt shavs as in (4) with a, b, c, d N. In particular, for any M, w may choos j sufficintly larg so that H p (X, C q ) = for all p > and all q M. Th rsult now follows from th spctral squnc E p,q 1 = H p (X, C q ) = H p+q (X, Hom MF (E, F)) using that X has boundd cohomological dimnsion for quasi-cohrnt shavs. In mor dtail, if X has cohomological dimnsion n, w may choos j sufficintly larg so that H p (X, C q ) = for all p > and q n 1. It follows that H (X, Hom MF (E, F)) = E, 2 = H Γ(X, Hom MF (E, F)). Rmark 4.4. Th proof of th Lmma actually shows that for any matrix factorizations E, F and any intgr M, thr is a wak quivalnc E E so that is an isomorphism for all q M. H q Γ(X, Hom MF (E, F)) H q (X, Hom MF (E, F)) Proof of Thorm 4.2. W nd to prov Hom [MF ] (E, F) Hom [MF ]H (E, F) (5) is an isomorphism for vry pair of matrix factorizations E, F. Givn such a pair, lt E E b a wak quivalnc as in Lmma 4.3 so that and considr th commutativ diagram Hom [MF ]naiv (E, F) Hom [MF ]naiv (E, F) = Hom [MF ]H (E, F), Hom [MF ] (E, F) = Hom [MF ]H (E, F) Hom [MF ]naiv (E, F) Hom [MF ] (E, F) Hom [MF ]H (E, F). Th middl and right-most vrtical arrows ar isomorphisms and th composition of th arrows along th bottom is an isomorphism. It follows that (5) is onto. Suppos α Hom [MF ] (E, F) is in th krnl of (5). W may rprsnt α by a diagram of strict morphisms E s G β F with s a wak quivalnc. Lt G G b a wak quivalnc for th pair G, F givn = (4)
5 JESSE BURKE and MARK E. WALKER by Lmma 4.3, so that Hom [MF ]naiv (G, F) = Hom [MF ]H (G, F) is an isomorphism. By prcomposing th abov diagram for α with th wak quivalnc G G, w may rprsnt α also as a diagram of strict morphisms of th form E G s β F with s a wak quivalnc. Sinc α is mappd to zro and s is mappd to an isomorphism, β is mappd to zro in [MF ] H. But thn sinc Hom [MF ]naiv (G, F) = Hom [MF ]H (G, F) is an isomorphism, w hav that β is zro alrady in [MF ] naiv and hnc also is zro in [MF ]. It follows that α = in [MF ]. 5. Hom-sts in th singularity catgory In this sction w rturn to th gnral situation, with X a Nothrian sparatd schm and L a lin bundl. But w mak th furthr assumption that W Γ(X, L) is a rgular sction, i.., W : O X L is injctiv. Equivalntly, for ach x X, th lmnt W x O X,x = Lx is a non-zro-divisor. Dfin Y to b th zro subschm of W (i.., by th idal givn as th imag of th injctiv map W : L O X ). Th singularity catgory of a schm Z is th Vrdir quotint D sg (Z) := D b (Z)/ Prf(Z), whr D b (Z) is th boundd drivd catgory of cohrnt shavs and Prf(Z) is th full subcatgory consisting of prfct complxs i.., thos complxs that ar locally quasi-isomorphic to boundd complxs of fr moduls of finit rank. This construction was introducd by Buchwitz [1] in th cas whn Z is affin and rdiscovrd by Orlov [5]. For a matrix factorization ( ) E = E 1 1 E E1 (1) dfin cokr(e) to b cokr( 1 ). Multiplication by W on th cohrnt shaf cokr(e) is zro and hnc cokr(e) may b rgardd as a cohrnt shaf on Y and thus as an objct of D sg (Y ). (Mor formally, th canonical map givs an isomorphism cokr( 1 ) = i i cokr( 1 ) and w dfin cokr(e) = i cokr( 1 ).) Dfinition 5.1. For a matrix factorization E = (E 1 1 E E1 (1)), i E is th chain complx of locally fr cohrnt shavs on Y i E ( 1) ( 1) i E 1 1 i E i E 1 (1) 1(1) i E (1). W writ Hom OY (M, N ) for th shaf of homomorphisms btwn two shavs M, N on Y (or th total product complx of th bicomplx of such if M or N is a complx) and Ext O Y (M, N ) for th right drivd functors of Hom OY (M, ).
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 51 Lmma 5.2. Assum X is a Nothrian sparatd schm, L is a lin bundl on X, and W is a rgular global sction of L. Lt E b a matrix factorization of (X, L, W ), st M = cokr E, and lt N b any cohrnt shaf on Y. Th following hold: (1) i E is an acyclic complx and th brutal truncation (i E) is a rsolution of M by locally fr cohrnt shavs on Y. (2) For all q 1, thr is an isomorphism Ext q O Y (M, N )(1) = Ext q+2 O Y (M, N ). (3) If N = cokr(f) for a matrix factorization F, thn thr is a quasi-isomorphism, natural in both E and F, Hom MF (E, F) i Hom OY (i E, N ). (4) If X is projctiv ovr a ring and L is th corrsponding vry ampl lin bundl, thn th dg map of th local-to-global spctral squnc Ext q O Y (M, N ) Γ(Y, Ext q O Y (M, N )) is an isomorphism for q. (Hr N can b an arbitrary cohrnt shaf.) Proof. It is clar that i E is a complx, sinc th composition of two adjacnt maps is multiplication by th imag of W in O Y (1), which is zro. Acyclicity may b chckd locally, and for any point x X, th complx (i E) x is a two-priodic complx inducd by a matrix factorization ovr O X,x of a non-zro divisor, and hnc is acyclic by [2, 5.1]. Th rst of part (1) now follows dirctly sinc M was dfind to b i cokr( 1 ). For part (2), thr is an isomorphism i E( 1) = i E[ 2], and hnc, using part (1), w hav Ext q O Y (M, N )(1) = H q (Hom OY (i E, N ))(1) = H q (Hom OY (i E( 1), N )) = H q (Hom OY (i E[ 2], N )) = H q+2 (Hom OY (i E, N )) = Ext q+2 O Y (M, N ), for q 1. For part (3), thr is a surjctiv map of chain complxs, Hom MF (E, F) Hom OX (E, i N ), (6) which is natural in both variabls, givn by th diagram Hom(E, F 1 ) Hom(E 1, F )( 1) 1 Hom(E, F ) Hom(E 1, F 1 ) Hom(E, F 1 )(1) Hom(E 1, F ) h 1 (1) (,p ) (p,) Hom(E 1, i N )( 1) Hom(E, i N ) (,p ) 1 Hom(E 1, i N ) (1), whr p: F i N is th canonical map. (Rcall that [ ] [ ] 1 (f1 ) = and (f ) =.) (f ) (f 1 ) 1 Sinc Hom OX (E, i N ) is canonically isomorphic to i Hom OY (i E, N ), it suffics to prov (6) is a quasi-isomorphism. Sinc Hom OX (E, ) and Hom OX (E 1, ) ar xact 1
52 JESSE BURKE and MARK E. WALKER functors and F 1 F i N is an xact squnc, th krnl of (6) is Th maps Hom(E, F 1 ) Hom(E 1, F 1 )( 1) Hom(E, F 1 ) Hom(E 1, F 1 )( 1) [ 1 ] 1 Hom(E, F 1 ) Hom(E 1, F 1 ) [ ] 1 Hom(E, F 1 ) Hom(E 1, F 1 ) [ ] 1 1 Hom(E, F 1 )( 1) Hom(E 1, F 1 ). [ ] Hom(E, F 1 )( 1) 1 Hom(E 1, F 1 ) dtrmin a contracting homotopy for this krnl, proving that (6) is a quasi-isomorphism. For part (4), considr th local-to-global spctral squnc E p,q 2 = H p (Y, Ext q O Y (M, N )) = Ext p+q O Y (M, N ). Sinc Ext 2q+2 O Y (M, N ) = Ext 2 O Y (M, N )(q) for all q and similarly for odd indics, th spctral squnc dgnrats for q by Srr s Vanishing Thorm. Rcall from 2.2 that a twistd priodic complx of shavs is a complx of locally fr cohrnt shavs C togthr with a spcifid isomorphism α: C[2] = C(1). Lmma 5.3. Lt Y b any schm that is projctiv ovr a Nothrian ring and L = O Y (1) th corrsponding vry ampl lin bundl. (W hav in mind th cas whn Y X is th zro subschm of W.) Lt C b a twistd priodic complx on Y. For i thr ar isomorphisms H i Γ(Y, C) = Γ(Y, H i (C)), whr Γ(Y, C) is Γ(Y, ) applid dgr-wis to C, and H i is th ith cohomology shaf of C. Proof. Lt B i and Z i b th imag and krnl shavs of i C, rspctivly. Sinc Z i = Z i 2 (1), B i = B i 2 (1) and L is vry ampl, th highr shaf cohomology groups of Z i, B i vanish for i by Srr Vanishing. It follows that, for i, th xact squncs Z i C i i C B i and B i Z i+1 H i+1 rmain xact upon applying Γ(, Y ) and thus Γ(Y, H i+1 ) = Γ(Y, Zi+1 ) Γ(Y, B i ) = kr Γ(Y, i+1 C ) Im Γ(Y, C i ) = H i+1 Γ(Y, C). Lmma 5.4. Assum X is a Nothrian sparatd schm with nough locally frs (i.., vry cohrnt shaf on X is th quotint of a locally fr cohrnt shaf), L is a lin bundl, and W is a rgular global sction of L. Lt E b a matrix factorization, lt Y X b th zro subschm of W, and lt N b any cohrnt shaf on Y. Th map inducd by th canonical functor D b (Y ) D sg (Y ) Hom Db (Y )(cokr E, N [m]) Hom Dsg(Y )(cokr E, N [m]) (7) is an isomorphism for m.
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 53 Proof. W adapt th proof of [5, 1.21]. (W cannot apply this rsult dirctly sinc Y nd not b Gornstin.) W first show (7) is onto for m. An lmnt of Hom Dsg(Y )(cokr E, N [m]) is rprsntd by a diagram in D b (Y ) cokr(e) s A f N [m] (8) such that con(s) is prfct. Th first stp is to show that w can rplac A by a cohrnt shaf that is th cokrnl of a matrix factorization. By Lmma 5.2(1) th shaf cokr E admits a right rsolution by locally fr cohrnt shavs on Y : cokr E Q 1 Q 2... whr Q 1 = i E 1 (1), Q 2 = i E (1),.... For any k 1 w thus hav a distinguishd triangl in D b (Y ) F k [ k 1] cokr(e) Q k F k [ k] whr F k = cokr(q k Q k+1 ). W claim thr is an intgr k such that for k k, th composition F k [ k 1] cokr(e) con(s) is zro. Sinc F k is th cokrnl of a matrix factorization, namly F k = cokr(e[k]), this claim follows from th mor gnral claim: givn a prfct complx P =... P n P n+1... thr is an intgr k such that Hom D b(cokr(f)[ k 1], P ) = for all matrix factorizations F and all intgrs k k. To prov this, obsrv first that, sinc X has nough locally frs, w may assum P is a boundd complx of locally fr cohrnt shavs. Now obsrv that Ext j (cokr(f), O Y ) = for all j > ; this is a local assrtion and locally it is th statmnt that th transpos of a matrix factorization is also a matrix factorization. It follows that Ext j Y (cokr F, Pn ) = Ext j Y (cokr F, O Y ) OY P n = for all j > and all n, and thn a standard argumnt givs th claim. (S th proof of [5, 1.18].) Considr th triangl A s cokr(e) con s A[1]. Sinc for k k th composition F k [ k 1] cokr(e) con(s) is zro, thr xists a map g : F k [ k 1] A that maks th lft triangl commut: F k [ k 1] g s A cokr(e) con(s) A[1]. Sinc con(f k [ k 1] cokr(e)) = Q k is prfct, composing with g shows that th lmnt of Hom Dsg (Y )(cokr E, N [m]) rprsntd by (8) is also rprsntd by a diagram of th form for k k. cokr(e) F k [ k 1] f g N [m]
54 JESSE BURKE and MARK E. WALKER Now considr th triangl Q k [ 1] F k [ k 1] cokr(e) Q k. For any k, thr is an intgr m = m (k) such that Hom Db (Y )(Q k [ 1], N [m]) = for all m m this holds bcaus Ext i (Q t, N ) = H i (Y, (Q t ) N ) = for all t and all i. It follows that th composition of Q k [ 1] F k [ k 1] f g N [m] is zro for m m and hnc th map F k [ k 1] N [m] factors through F k [ k 1] cokr(e). Th lmnt rprsntd by (8) is thus actually rprsntd by a map in D b (Y ). W hav provn that (7) is onto for k k and m m (k). Suppos now f : cokr(e) N [m] is a morphism in D b (Y ) that dtrmins th zro map in D sg (Y ). Thn thr is a map s: A cokr(e) such that con(s) is prfct and such that A cokr(e) f N [m] is th zro map i.., th imag of f in D sg (Y ) is rprsntd by a diagram of th form cokr(e) s A N [m]. Th argumnt abov shows that thr is an intgr k such that w may tak A = F k [ k 1] and con(s) = Q k for k k. In othr words, for k k, th map f factors as cokr(e) Q k N [m] in D b (Y ). But, as shown abov, for m m (k), w hav Hom D b (Y )(Q k, N [m]) =. This provs that (7) is on-to-on for m. Rmark 5.5. Any schm X that is projctiv ovr a ring Q will hav nough locally fr shavs. Indd, w may assum that X = P m Q = Proj Q[T 1,..., T m ] for som ring Q and m. Thn any cohrnt shaf F is isomorphic to M for som finitly gnratd gradd Q[T 1,..., T m ]-modul M. Thr xists a surjction E M with E a finitly gnratd gradd fr Q[T 1,..., T c ]-modul, and th associatd map Ẽ M givs th rquird surjction from a locally fr cohrnt shaf on X onto F. For th Lmma blow, it may hlp to kp in mind th cas whn N is th cokrnl of a matrix factorization F. In this cas, by Lmma 5.2(3) w hav that H m Γ(Y, Hom OY (i E, N )) = H m Γ(X, Hom MF (E, F)), and th right hand sid is th st of strict morphisms btwn E[ m] and F. Lmma 5.6. Lt X b a schm that is projctiv ovr a Nothrian ring, L = O X (1) th corrsponding vry ampl lin bundl, and W a rgular global sction of L. Lt E b a matrix factorization, lt Y X b th zro subschm of W, and lt N b any cohrnt shaf on Y. For vry m Z, thr is a map, natural with rspct to E and N, H m Γ(Y, Hom OY (i E, N )) Hom Dsg(Y )((cokr E)[ m], N ) that is an isomorphism for m. Proof. Any lmnt of H m Γ(Y, Hom OY (i E, N )) givs a morphism of shavs cokr(e[ m]) N which w considr as a morphism in D sg (Y ). By [7, 3.12], thr is a functorial isomorphism cokr(e[ m]) = (cokr E)[ m] in D sg (Y ). This givs th map.
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 55 For m w hav th following chain of isomorphisms: Hom Db (Y )(cokr E[ m], N ) = Ext m O Y (cokr E, N ) = Γ(Y, Ext m O Y (cokr E, N )) = Γ(Y, H m (Hom OY (i E, N ))) = H m Γ(Y, Hom OY (i E, N )). Th scond is givn by 5.2(4), th third by 5.2(1), and th fourth by 5.3. On can chck that th diagram Hom D b (Y )(cokr E[ m], N ) = H m Γ(Y, Hom OY (i E, N )) Hom Dsg(Y )((cokr E)[ m], N ) commuts, whr th vrtical map is inducd by th canonical functor D b (Y ) D sg (Y ) and th diagonal map is dfind abov. Th rsult now follows from Lmma 5.4, which shows that th vrtical arrow is an isomorphism for m. Thorm 5.7. Lt X b a schm that is projctiv ovr a Nothrian ring, L = O X (1) th corrsponding vry ampl lin bundl, and W a rgular global sction of L. Lt E b a matrix factorization, lt Y X b th zro subschm of W, and lt N b any cohrnt shaf on Y. For vry m Z thr is an isomorphism, H m (Y, Hom OY (i E, N )) = Hom Dsg(Y )(cokr E[ m], N ) that is natural in both E and N, and maks th following diagram commut: H m Γ(Y, Hom OY (i E, N )) H m (Y, Hom OY (i E, N )) = Hom Dsg (Y )(cokr E[ m], N ). (9) Th vrtical map in th diagram is th canonical on and th diagonal map is th map dfind in Lmma 5.6. Proof. Considr th vrtical and diagonal maps in (9), both of which ar natural in both argumnts. By Lmma 5.6 th diagonal map is an isomorphism for m. For th vrtical map, w us th spctral squnc E p,q 2 = H p (Y, H q (Hom OY (i E, N ))) = H p+q (X, Hom OY (i E, N )). Sinc E is a matrix factorization, w hav that H 2q+2 (Hom OY (i E, N )) = H 2 (Hom OY (i E, N ))(q) and similarly in th odd cas. Thus by Srr s Vanishing Thorm, th spctral squnc abov dgnrats for q, giving isomorphisms Γ(Y, H q (Hom OY (i E, N ))) = H q (Y, Hom OY (i E, N )). By Lmma 5.3, for q, thr is an isomorphism Γ(Y, H q (Hom OY (i E, N ))) = H q (Γ(Y, Hom OY (i E, N ))) which shows that th lft-hand map in (9) is an isomorphism for q.
56 JESSE BURKE and MARK E. WALKER W hav provn that thr is an intgr M such that thr xists an isomorphism H m (Hom OY (i E, N )) Hom Dsg(Y )(cokr E[ m], N ) (1) causing (9) to commut, for all m M. Sinc X is a subschm of Proj Q[x 1,..., x n ] for som Nothrian ring Q and intgr n, w hav an associatd Koszul xact squnc O X O X (1) n O X (2) (n 2) OX (n) (n n). Dfin P j for j = 1,..., n to b th krnl of th map O X (j) (n j) OX (j + 1) ( n j+1) in this squnc. W hav an xact squnc of locally fr shavs P j O(j) (n j) Pj+1 for ach j = 1,..., n, from which w obtain an xact squnc of matrix factorizations W claim thr xists an isomorphism E P j E O(j) (n j) E Pj+1. H Hom OY ((i E P j )[ m], N ) Hom Dsg (Y )(cokr E[ m] P j, N ) for all m M 2j and for ach j = 1,..., n, making th vidnt analogu of (9) commut. To s this, not that (i E)(j) = (i E)[2j]. Thus thr xists an isomorphism H Hom OY ((E O(j) (n j) )[ m], N ) HomDsg(Y )((cokr E O(j) (n j) )[ m], N ) for all m M 2j, making th vidnt analogu of (9) commut. Th claim follows immdiatly by dscnding induction on j. But P 1 = O X and so i E P 1 = i E, from which w dduc that isomorphisms as in (1) xist for all m M 2, having startd from th assumption that thr xistd isomorphisms for m M. Clarly such isomorphisms xist thn for all m. 6. Rlating matrix factorizations and th singularity catgory Th goal of this sction is to prov Thorm 2 of th introduction. W continu to assum X is a Nothrian sparatd schm, L is a lin bundl on X, and W Γ(X, L) is a global sction. Dfin Y X to b th zro subschm of W. For a matrix factorization ( ) E = E 1 1 E E1 (1), w viw cokr(e) := i cokr( 1 ) as an objct of D sg (Y ). This assignmnt is natural in E, so givs a functor cokr: [MF (X, L, W )] naiv D sg (Y ) By [7, 3.12] this is a triangulatd functor. E cokr(e). Lmma 6.1. For a Nothrian sparatd schm X, lin bundl L, and a rgular global sction W of L, if E is a locally contractibl matrix factorization, thn cokr(e) = in D sg (Y ).
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 57 Proof. It is nough to show that cokr(e) x is a fr O Y,x modul for all x Y. By assumption E x = HMF (O X,x, W x ) for all x X, and so th rsult follows from Proposition 2.9. By th univrsal proprty of localization w immdiatly obtain: Proposition 6.2. For a Nothrian sparatd schm X, a lin bundl L and a rgular global sction W of L, thr is a triangulatd functor such that th composition is th functor cokr. cokr: [MF (X, L, W )] D sg (Y ) [MF (X, L, W )] naiv [MF (X, L, W )] cokr D sg (Y ) Thorm 6.3. Lt X b a schm that is projctiv ovr a Nothrian ring, L = O X (1) th corrsponding vry ampl lin bundl, and W a rgular global sction of L. Dfin Y X to b th zro subschm of W. Thn th triangulatd functor is fully faithful. cokr: [MF (X, L, W )] D sg (Y ) Proof. W will show thr is a fully faithful functor cokr : [MF ] H D sg (Y ) such that cokr H = cokr, whr H : [MF ] naiv [MF ] H is th functor dfind in (3). To s this implis th Thorm, lt F : [MF ] naiv [MF ] b th canonical functor, and G: [MF ] [MF ] H th quivalnc of Thorm 4.2; not that H = G F. Considr th commutativ diagram: F [MF ] naiv G [MF ] [MF ] H = cokr cokr cokr D sg (Y ). Sinc cokr G F = cokr = cokr F, w hav that cokr G = cokr by th univrsal proprty of Vrdir quotints. Thus if cokr is fully faithful, so will b cokr. To dfin th functor cokr, w st cokr (E) = cokr E. By Lmma 5.2(3) thr is a quasi-isomorphism Hom MF (E, F) i Hom OY (i E, cokr F) that is natural in both argumnts, and thus thr is a natural isomorphism H (X, Hom MF (E, F)) = H (Y, Hom OY (i E, cokr F)). By Thorm 5.7 thr is an isomorphism H (Y, Hom OY (i E, cokr F)) = Hom Dsg(Y )(cokr E, cokr F)
58 JESSE BURKE and MARK E. WALKER that is natural in both argumnts. By composing ths, w obtain a natural map Hom [MF ]H (E, F) = H (X, Hom MF (E, F)) Hom Dsg(Y )(cokr E, cokr F) dtrmining th functor cokr, which is a fully faithful sinc this map is bijctiv. W hav lft to prov that cokr H = cokr. Both sids agr on objcts, but w nd to show thy induc th sam map on morphisms. Considr th diagram: H Γ(X, Hom MF (E, F)) H Γ(Y, Hom OY (i E, cokr F)) H H (X, Hom MF (E, F)) H (Y, Hom OY (i E, cokr F)) Hom Dsg (Y )(cokr E, cokr F). Th right triangl is commutativ by Thorm 5.7. Th lft hand squar is commutativ as th horizontal maps ar inducd by th map of chain complxs Hom MF (E, F) i Hom OY (i E, cokr F) and H ( ) H ( ) is a natural transformation. Th composition of th bottom arrows in th diagram is by dfinition th map on morphisms inducd by th functor cokr. Thus by th commutativity of th diagram, th map on morphisms inducd by cokr H is qual to th composition of th top two arrows of th diagram. To finish th proof it is now nough to show that th following diagram commuts: Hom [MF ] (E, F) = H Γ(X, Hom MF (E, F)) H Γ(Y, Hom OY (i E, cokr F)) cokr Hom Dsg(Y )(cokr E, cokr F). This follows from th dfinition of th map Hom MF (E, F) i Hom OY (i E, cokr F) givn in (th proof of) 5.2(3). Rmark 6.4. Thorm 6.3 has bn provd in [7, 3.14] in cas X is a smooth stack. Orlov provs an analogu in [4] without assuming X is projctiv ovr an affin schm. Howvr h assums that L = O X. His Thorm dos not sm to imply ours, nor dos ours imply his, as O X is not vry ampl in gnral. Lin and Pomrlano, in [3], giv a diffrnt proof of Orlov s Thorm in cas X is dfind ovr C and smooth. Finally Positslski, in [8], provs an analogu of Thorm 6.3 using xotic drivd catgoris, which dos not rquir X to b projctiv. H dos not rquir th cohrnt shavs in th dfinition of matrix factorization to b locally fr. Thorm 6.8 will show that th objcts dfind blow ar xactly th cokrnls of matrix factorizations in D sg (Y ), undr th assumptions of Thorm 6.3. Dfinition 6.5. Lt i: Y X b a closd immrsion of finit flat dimnsion. An objct F in D b (Y ) is rlativly prfct on Y if i F is prfct on X. W writ RPrf(Y X) for th full subcatgory of D b (Y ) whos objcts ar rlativly prfct on X.
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 59 Sinc i has finit flat dimnsion, Prf(Y ) is a thick subcatgory of RPrf(Y X). W dfin th rlativ singularity catgory of i to b th Vrdir quotint Th canonical functor D rl sg (Y X) := RPrf(Y X). Prf(Y ) D rl sg (Y X) D sg (Y ) is fully faithful and w may thus idntify D rl sg (Y X) with a full subcatgory of D sg (Y ). Rmark 6.6. In this sam contxt, a rlativ singularity catgory has also bn dfind by Positslski in [8]. In gnral ths two catgoris nd not b quivalnt, but w point out a rlation btwn th two in 6.9. Th following stablishs on half of Thorm 6.8 (undr mildr assumptions): Lmma 6.7. Assum X is a Nothrian sparatd schm of finit Krull dimnsion and that X has nough locally frs (i.., vry cohrnt shaf on X is th quotint of a locally fr cohrnt shaf). Lt L b a lin bundl on X, assum W a rgular global sction of L and lt i: Y X b th zro subschm of W. For vry objct G of D rl sg (Y X), thr xists a matrix factorization E and an isomorphism G = cokr E in D rl sg (Y X). Proof. Lt F b a right boundd complx of locally fr cohrnt shavs on Y that maps quasi-isomorphically to G. Such a complx xists sinc X, and hnc Y, has nough locally frs. Lt F k dnot th brutal truncation of F in dgr k. For any k, th con of th canonical map F F k is a prfct complx and hnc F F k is an isomorphism in D rl sg (Y X). Taking k, th complx F k is xact xcpt in dgr k, and hnc w hav rducd to th cas whr G = M[ k] for som cohrnt shaf M and intgr k. Sinc cokr is triangulatd, w may assum k =. Sinc i G is a prfct complx, i M is locally of finit projctiv dimnsion. In fact, for all x X, th projctiv dimnsion of i M x as a O X,x -modul is at most d := dim(x). Considr again a rsolution F of M by locally fr cohrnt shavs on Y. Sinc a locally fr cohrnt shaf on Y is locally of projctiv dimnsion on as a cohrnt shaf on X, a high nough syzygy of this rsolution of M will also b locally of projctiv dimnsion on on X. Spcifically, th only non-zro cohomology shaf of i F d will b locally of projctiv dimnsion on on X, and sinc F d = M = G in D rl sg (Y X), w may assum G = M whr M is a cohrnt shaf on Y such that i M is locally of projctiv dimnsion on on X. Now considr any surjction E i M with E a locally fr cohrnt shaf on X. Sinc i M is locally of projctiv dimnsion on, th krnl E 1 of this surjction is locally fr. That is, w hav a rsolution of th form E 1 α E i M
6 JESSE BURKE and MARK E. WALKER with E, E 1 locally fr on X. In th diagram α E 1 E β W W E 1 (1) α(1) E (1) i M W i M(1) th right-most map is th zro map, and hnc thr xists diagonal arrow β as shown causing both triangls to commut. This dtrmins a matrix factorization E with cokr(e) = M = G. Thorm 6.8. Lt X b a schm that is projctiv ovr a Nothrian ring of finit Krull dimnsion, L = O X (1) th corrsponding vry ampl lin bundl, and W a rgular global sction of L. Dfin Y X to b th zro subschm of W. Th functor cokr, dfind in Proposition 6.2, factors naturally through th subcatgory D rl sg (Y X) of D sg (Y ) and th inducd map is an quivalnc of triangulatd catgoris: [MF (X, L, W )] = D rl sg (Y X). In particular, if X is rgular, thn w hav an quivalnc of triangulatd catgoris cokr: [MF (X, L, W )] = D sg (Y ). Proof. For any matrix factorization E = (E 1 1 E E1 ) thr is a short xact squnc of cohrnt shavs on X E 1 1 E i cokr(e). Indd, sinc W is rgular sction, th composition 1 = W : E 1 E 1 (1) is injctiv and hnc 1 must b. Thus i cokr(e) is prfct and cokr factors through th subcatgory D rl sg (Y X). On th othr hand, for any objct F in D rl sg (Y X), thr is a matrix factorization E such that cokr E = F by Lmma 6.7. Th final assrtion holds sinc vry boundd complx of cohrnt shavs on a rgular schm is prfct. Rmark 6.9. Lt i: Y X b as abov. Lt T b th thick subcatgory of D sg (Y ) gnratd by th objcts Li M, whr M rangs ovr all objcts of D sg (X). Positslski dfins in [8] th rlativ singularity catgory of i: Y X, which w dnot p D rl sg (Y X), to b th Vrdir quotint D sg (Y )/T. It is asy to chck that D rl sg (Y X) is a subcatgory of T := { N D rl sg (Y ) Hom Dsg(Y )(M, N ) = for all objcts M T }. This implis, kping in mind th dfinition of morphisms in D sg (Y )/T, that th composition of th functors D rl sg (Y X) D sg (Y ) D sg (Y )/T =: p D rl sg (Y X) (11) is fully faithful. This fully faithful functor nd not b an quivalnc. Indd, lt k b a fild and lt Q = k[[x, y]]/(x 2 ) and R = Q/(y 2 ) = k[[x, y]]/(x 2, y 2 ). Lt Y := Spc R Spc Q =: X b th natural inclusion. If (11) wr an quivalnc, this would imply
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 61 that th smallst thick subcatgory of D sg (Y ) containing D rl sg (Y X) and T wr all of D sg (Y ). Howvr w claim that th rsidu fild k is not in th thick subcatgory gnratd by D rl sg (Y X) or T. Indd, it is asy to chck that objcts in ithr of ths catgoris hav priodic fr rsolutions, but th ranks of th fr moduls in a minimal fr rsolution of k grow linarly and so k cannot hav a priodic rsolution. Rfrncs [1] Ragnar-Olaf Buchwitz. Maximal Cohn-Macaulay moduls and Tatcohomology ovr Gornstin rings. Unpublishd manuscript, availabl at https://tspac.library.utoronto.ca/handl/187/16682, 1987. [2] David Eisnbud. Homological algbra on a complt intrsction, with an application to group rprsntations. Trans. Amr. Math. Soc., 26(1):35 64, 198. [3] Kvin H. Lin and Danil Pomrlano. Global matrix factorizations. arxiv:111.5847. [4] Dmitri O. Orlov. Matrix factorizations for nonaffin LG-modls. Math. Ann., 353(1):95 18, 212. [5] Dmitri O. Orlov. Triangulatd catgoris of singularitis and D-brans in Landau-Ginzburg modls. Tr. Mat. Inst. Stklova, 246(Algbr. Gom. Mtody, Svyazi i Prilozh.):24 262, 24. [6] Dmitri O. Orlov. Triangulatd catgoris of singularitis, and quivalncs btwn Landau-Ginzburg modls. Mat. Sb., 197(12):117 132, 26. [7] Alxandr Polishchuk and Arkady Vaintrob. Matrix factorizations and singularity catgoris for stacks. arxiv:111.4544. [8] Lonid Positslski. Cohrnt analogus of matrix factorizations and rlativ singularity catgoris. arxiv:112.261. [9] Danil Quilln. Highr algbraic K-thory. I. In Algbraic K-thory, I: Highr K-thoris (Proc. Conf., Battll Mmorial Inst., Sattl, Wash., 1972), pags 85 147. Lctur Nots in Math., Vol. 341. Springr, Brlin, 1973. [1] Ed Sgal. Th closd stat spac of affin Landau-Ginzburg B-modls. J. Noncommutativ Gomtry. To appar. [11] Ian Shipman. A gomtric approach to Orlov s thorm. Comp. Math. To appar. Jss Burk jburk@math.uni-bilfld.d Dpartmnt of Mathmatics, Univrsität Bilfld, 3351 Bilfld, Grmany Mark E. Walkr mwalkr5@unl.du Dpartmnt of Mathmatics, Univrsity of Nbraska, Lincoln, NE 68588 USA