MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES
|
|
- Francis Collins
- 5 years ago
- Views:
Transcription
1 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES JESSE BURKE AND MARK E. WALKER Abstract. W study matrix factorizations of locally fr cohrnt shavs on a schm. For a schm that is projctiv ovr an affin schm, w show that homomorphisms in th homotopy catgory of matrix factorizations may b computd as th hyprcohomology of a crtain mapping complx. Using this xplicit dscription, w giv anothr proof 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. W also 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 givn 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 (1.1) [MF (Q, f)] = cokr D b (R)/ Prf(R) =: D sing (R) 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 sing (R) th singularity catgory of R, following [5]. Th quivalnc (1.1) is inducd by snding a matrix factorization (A, B) to th imag of th R-modul cokr A in D sing (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 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 (1.1) 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 (1.1). Thrfor, bfor w dscrib our contributions to this qustion, lt us dscrib what is known. Th right hand sid of (1.1) 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 by [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 sing (Y ). Thr is a functor (1.2) [MF (X, L, W )] naiv D sing (Y ) that snds a matrix factorization (E 1 1 E E1 L) to cokr 1. 2 Mathmatics Subjct Classification. 14F5, 13D9, 13D2. 1
2 2 JESSE BURKE AND MARK E. WALKER 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 (1.2) is ssntially surjctiv. Indd, as in th affin cas, vry objct of D sing (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 (1.2) 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 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 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 [4, Thorm 3.5], [7, Thorm 3.14] show that (1.2) inducs a functor [MF (X, L, W )] D sing (Y ) which is an quivalnc. In fact [4] dos not assum X is rgular and shows that th inducd functor 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 pairing on ths Hom-sts 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 usd a slightly diffrnt dfinition of matrix factorization). W also dscrib th imag of this fully faithful functor compltly:
3 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 3 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 sing (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 sing (Y ) such that i C is prfct on X. In particular, if X is rgular thn cokr is an quivalnc. W cam to ths rsults studying affin complt intrsction rings. Lt Q b a rgular ring and f 1,..., f c a rgular squnc of lmnts in Q. 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 sing (R) = D sing (Y ). Composing this with th quivalnc of Thorm 2 w obtain an quivalnc D sing (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. 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. 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:
4 4 JESSE BURKE AND MARK E. WALKER 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 ) using th canonical isomorphisms and 1 Hom(E i, F j (1)) = Hom(E i, F j )(1) = Hom(E i ( 1), F j ) Hom(E, F 1 ) (1) = Hom(E 1, F ( 1)) 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 s : E F 1 and t : E 1 (1) F as in th diagram blow 1 E 1 E E 1 (1) s t g F 1 F F 1 (1) f 1 f g 1 g 1(1) such that 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 ndo-functor 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).
5 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 5 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 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. Thr is a modl structur on th catgory of matrix factorizations of arbitrary quasicohrnt shavs in which th wak quivalncs ar thos dfind abov; s [3]. This modl structur dos not play a rol in this papr. 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 i i cokr( 1 ) = cokr( 1 ) via th canonical map. 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.
6 6 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 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,T1) 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 Th homotopy catgory of matrix factorizations, writtn [M F (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 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 )]. Exampl If X is affin, L = O X and W is a non-zro-divisor, thn [MF (X, L, W )] = [MF (X, L, W )] naiv by Proposition Anothr vrsion of th homotopy catgory Th aim of this sction is to dscrib anothr triangulatd 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 quasicohrnt shaf F on X, lt Γ(U, F) dnot th cochain complx givn by th usual Cch construction.
7 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 7 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 Thr is a convrgnt spctral squnc H q (X, Hom MF (E, F)) = H q (Γ(U, Hom MF (E, F))). 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. H n (X, Hom MF (E, F)) H n (X, Hom MF (E, F )) Dfinition 3.1. Dfin th catgory [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 [1]. Thr is a canonical functor (3.3) [MF (X, L, W )] naiv [MF (X, L, W )] H 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 3.4. 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 Lmma 3.5. 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
8 8 JESSE BURKE AND MARK E. WALKER 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 3.6. 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 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 E q 1 1 (1) E q 2 ( 1) E q 2 1 E q 2 E q 2 1 (1).... 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 )
9 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 9 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. 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 complx 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.
10 1 JESSE BURKE AND MARK E. WALKER 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 (4.4) 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 (4.4) 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 4.5. 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. Proof of Thorm 4.2. W nd to prov H q Γ(X, Hom MF (E, F)) H q (X, Hom MF (E, F)) (4.6) Hom [MF ] (E, F) Hom [MF ]H (E, F) 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 (4.6) is onto. Suppos α Hom [MF ] (E, F) is in th krnl of (4.6). W may rprsnt α by a diagram of strict morphisms E G s F β with s a wak quivalnc. Lt G G b a wak quivalnc for th pair G, F givn 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) =
11 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 11 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-zrodivisor. 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 sing (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 sing (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(M, ). 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 ).
12 12 JESSE BURKE AND MARK E. WALKER 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, (5.3) Hom MF (E, F) Hom OX (E, i N ), which is natural in both variabls, givn by th diagram Hom(E, F 1) Hom(E 1, F )( 1) (,p ) 1 Hom(E, F ) Hom(E 1, F 1) (p,) Hom(E 1, i N )( 1) Hom(E, i N ) whr p : F i N is th canonical map. (Rcall that [ ] 1 (f1 ) = and = 1 (f ) Hom(E, F 1)(1) Hom(E 1, F ) (,p ) 1 Hom(E 1, i N ) [ ] (f ).) 1 (f 1 ) h 1 (1) (1), Sinc Hom OX (E, i N ) is canonically isomorphic to i Hom OY (i E, N ), it suffics to prov (5.3) is a quasi-isomorphism. Sinc Hom OX (E, ) and Hom OX (E 1, ) ar xact functors and F 1 F i N is an xact squnc, th krnl of (5.3) 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 1, F 1 )( 1). Hom(E 1, F 1 ) Hom(E 1, F 1 ) [ ] 1 Hom(E, F 1 ) Hom(E 1, F 1 ) [ ] Hom(E 1, F 1 )( 1) Hom(E 1, F 1 ) dtrmin a contracting homotopy for this krnl, proving that (5.6) 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 5.4. 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
13 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 13 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, i C ) = H i+1 Γ(Y, C). Lmma 5.5. 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 sing (Y ) (5.6) Hom D b (Y )(cokr E, N [m]) Hom Dsing(Y )(cokr E, N [m]) is an isomorphism for m. Proof. W adapt th proof of [5, 1.21]. (W cannot apply this rsult dirctly sinc Y nd not b Gornstin.) W first show (5.6) is onto for m. An lmnt of Hom Dsing(Y )(cokr E, N [m]) is rprsntd by a diagram in D b (Y ) (5.7) cokr(e) s A f N [m] 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 fist 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 Dsing(Y )(cokr E, N [m]) rprsntd by (5.7) is also rprsntd by a diagram of th form cokr(e) F k [ k 1] f g N [m] for k k. 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]) =
14 14 JESSE BURKE AND MARK E. WALKER 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 (5.7) is thus actually rprsntd by a map in D b (Y ). W hav provn that (5.6) 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 sing (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 sing (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 (5.6) is on-to-on for m. Rmark 5.8. 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 5.9. 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, that is an isomorphism for m. H m Γ(Y, Hom OY (i E, N )) Hom Dsing(Y )((cokr E)[ m], N ) 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 sing (Y ). By [7, 3.12], thr is a functorial isomorphism cokr(e[ m]) = (cokr E)[ m] in D sing (Y ). This givs th map. For m w hav th following chain of isomorphisms: Hom D b (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 5.4. On can chck that th diagram = Hom D b (Y )(cokr E[ m], N ) H m Γ(Y, Hom OY (i E, N )) Hom Dsing (Y )((cokr E)[ m], N ) commuts, whr th vrtical map is inducd by th canonical functor D b (Y ) D sing (Y ) and th diagonal map is dfind abov. Th rsult now follows from Lmma 5.5, which shows that th vrtical arrow is an isomorphism for m.
15 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 15 Thorm 5.1. 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 Dsing(Y )(cokr E[ m], N ) that is natural in both E and N, and maks th following diagram commut: (5.11) H m Γ(Y, Hom OY (i E, N )) H m (Y, Hom OY (i E, N )) = Hom Dsing(Y )(cokr E[ m], N ). Th vrtical map in th diagram is th canonical on and th diagonal map is th map dfind in Lmma 5.9. Proof. Considr th vrtical and diagonal maps in (5.11), both of which ar natural in both argumnts. By Lmma 5.9 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 5.4, 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 (5.11) is an isomorphism for q. W hav provn that thr is an intgr M such that thr xists an isomorphism (5.12) H m (Hom OY (i E, N )) Hom Dsing(Y )(cokr E[ m], N ) causing (5.11) 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 Dsing(Y )(cokr E[ m] P j, N ) for all m M 2j and for ach j = 1,..., n, making th vidnt analogu of (5.11) 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 ) HomDsing(Y )((cokr E O(j) (n j) )[ m], N ) for all m M 2j, making th vidnt analogu of (5.11) commut. Th claim follows immdiatly by dscnding induction on j.
16 16 JESSE BURKE AND MARK E. WALKER But P 1 = O X and so i E P 1 = i E, from which w dduc that isomorphisms as in (5.12) 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 sing (Y ). This assignmnt is natural in E, so givs a functor cokr : [MF (X, L, W )] naiv D sing (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 sing (Y ). 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 sing (Y ) [MF (X, L, W )] naiv [MF (X, L, W )] cokr D sing (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 sing (Y ) Proof. W will show thr is a fully faithful functor cokr : [MF ] H D sing (Y ) such that cokr H = cokr, whr H : [MF ] naiv [MF ] H is th functor dfind in (3.3). 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 sing (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.
17 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 17 To dfin th functor cokr, w st cokr (E) = cokr E. By Lmma 5.2(3) thr is a quasiisomorphism Hom MF (E, F) i Hom OY (i E, cokr F) that is natural in both argumnts, and thus thr is a natural isomorphism By Thorm 5.1 thr is an isomorphism H (X, Hom MF (E, F)) = H (Y, Hom OY (i E, cokr F)). H (Y, Hom OY (i E, cokr F)) = Hom Dsing(Y )(cokr E, cokr F) 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 Dsing(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 Dsing (Y )(cokr E, cokr F) Th right triangl is commutativ by Thorm 5.1. 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 th 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)) cokr H Γ(Y, Hom OY (i E, cokr F)) Hom Dsing(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 prov 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 6.7 will show that th objcts dfind blow ar xactly th cokrnls of matrix factorizations in D sing (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.
18 18 JESSE BURKE AND MARK E. WALKER 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 sing(y X) := RPrf(Y X). Prf(Y ) D rl sing(y X) D sing (Y ) is fully faithful and w may thus idntify D rl sing (Y X) with a full subcatgory of D sing(y ). (Th radr may wish to compar this dfinition with [8], whr a diffrnt dfinition of rlativ singularity catgory is givn.) Th following stablishs on half of Thorm 6.7 (undr mildr assumptions): Lmma 6.6. 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 Dsing rl (Y X), thr xists a matrix factorization E and an isomorphism G = cokr E in Dsing rl (Y X). Proof. Lt F b a right boundd complx of locally fr cohrnt shavs on Y that maps quasiisomorphically 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 Dsing rl (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 Dsing rl (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 with E, E 1 locally fr on X. In th diagram E 1 α E i M 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 6.7. 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 Dsing rl (Y X) of D sing(y ) and th inducd map is an quivalnc of triangulatd catgoris: [MF (X, L, W )] = D rl sing(y X)
19 MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES 19 In particular, if X is rgular, thn w hav an quivalnc of triangulatd catgoris cokr : [MF (X, L, W )] = D sing (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 Dsing rl (Y X). On th othr hand, for any objct F in Dsing rl (Y X), thr is a matrix factorization E such that cokr E = F by Lmma 6.6. Th final assrtion holds sinc vry boundd complx of cohrnt shavs on a rgular schm is prfct. Rfrncs [1] Ragnar-Olaf Buchwitz. Maximal Cohn-Macaulay moduls and Tat-cohomology ovr Gornstin rings. Unpublishd manuscript, availabl at [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: [4] Dmitri O. Orlov. Matrix factorizations for nonaffin LG-modls. Math. Ann. To appar. [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): , 26. [7] Alxandr Polishchuk and Arkady Vaintrob. Matrix factorizations and singularity catgoris for stacks. arxiv: [8] Lonid Positslski. Cohrnt analogus of matrix factorizations and rlativ singularity catgoris. arxiv: [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 Lctur Nots in Math., Vol Springr, Brlin, [1] Ian Shipman. A gomtric approach to Orlov s thorm. arxiv: Dpartmnt of Mathmatics, Univrsität Bilfld, 3351 Bilfld, Grmany. addrss: jburk@math.uni-bilfld.d Dpartmnt of Mathmatics, Univrsity of Nbraska, Lincoln, NE addrss: mwalkr5@math.unl.du
MATRIX FACTORIZATIONS OVER PROJECTIVE SCHEMES
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
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationWhat is a hereditary algebra?
What is a hrditary algbra? (On Ext 2 and th vanishing of Ext 2 ) Claus Michal Ringl At th Münstr workshop 2011, thr short lcturs wr arrangd in th styl of th rgular column in th Notics of th AMS: What is?
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationν a (p e ) p e fpt(a) = lim
THE F -PURE THRESHOLD OF AN ELLIPTIC CURVE BHARGAV BHATT ABSTRACT. W calculat th F -pur thrshold of th affin con on an lliptic curv in a fixd positiv charactristic p. Th mthod mployd is dformation-thortic,
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationMutually Independent Hamiltonian Cycles of Pancake Networks
Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks Chng-Kuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C discipl@ms0urlcomtw Hua-Min Huang Dpartmnt of Mathmatics
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationSpectral Synthesis in the Heisenberg Group
Intrnational Journal of Mathmatical Analysis Vol. 13, 19, no. 1, 1-5 HIKARI Ltd, www.m-hikari.com https://doi.org/1.1988/ijma.19.81179 Spctral Synthsis in th Hisnbrg Group Yitzhak Wit Dpartmnt of Mathmatics,
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationSome remarks on Kurepa s left factorial
Som rmarks on Kurpa s lft factorial arxiv:math/0410477v1 [math.nt] 21 Oct 2004 Brnd C. Kllnr Abstract W stablish a connction btwn th subfactorial function S(n) and th lft factorial function of Kurpa K(n).
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More informationMatrix factorizations over projective schemes
Jesse Burke (joint with Mark E. Walker) Department of Mathematics University of California, Los Angeles January 11, 2013 Matrix factorizations Let Q be a commutative ring and f an element of Q. Matrix
More informationLOCAL STRUCTURE OF ALGEBRAIC MONOIDS
LOCAL STRUCTURE OF ALEBRAIC MONOIDS MICHEL BRION Abstract. W dscrib th local structur of an irrducibl algbraic monoid M at an idmpotnt lmnt. Whn is minimal, w show that M is an inducd varity ovr th krnl
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park
Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp. 147 153 ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction
More informationREPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS
REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS JIAQI JIANG Abstract. This papr studis th rlationship btwn rprsntations of a Li group and rprsntations of its Li algbra. W will mak th corrspondnc in two
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationNIL-BOHR SETS OF INTEGERS
NIL-BOHR SETS OF INTEGERS BERNARD HOST AND BRYNA KRA Abstract. W study rlations btwn substs of intgrs that ar larg, whr larg can b intrprtd in trms of siz (such as a st of positiv uppr dnsity or a st with
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationINCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS. xy 1 (mod p), (x, y) I (j)
INCOMPLETE KLOOSTERMAN SUMS AND MULTIPLICATIVE INVERSES IN SHORT INTERVALS T D BROWNING AND A HAYNES Abstract W invstigat th solubility of th congrunc xy (mod ), whr is a rim and x, y ar rstrictd to li
More informationLecture 6.4: Galois groups
Lctur 6.4: Galois groups Matthw Macauly Dpartmnt of Mathmatical Scincs Clmson Univrsity http://www.math.clmson.du/~macaul/ Math 4120, Modrn Algbra M. Macauly (Clmson) Lctur 6.4: Galois groups Math 4120,
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More informationGROUP EXTENSION HOMOMORPHISM MATRICES. A. M. DuPre. Rutgers University. October 1, 1993
GROUP EXTENSION HOMOMORPHISM MATRICES A. M. DuPr Rutgrs Univrsity Octobr 1, 1993 Abstract. If i1 j1 1! N 1?! G 1?! H 1! 1 i2 j2 1! N 2?! G 2?! H 2! 1 f ar short xact squncs of groups, thn w associat to
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationarxiv: v4 [math.at] 3 Aug 2017
AN EQUIVARIANT TENSOR PRODUCT ON MACKEY FUNCTORS arxiv:1508.04062v4 [math.at] 3 Aug 2017 KRISTEN MAZUR Abstract. For all subgroups H of a cyclic p-group w dfin norm functors that build a -Macky functor
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationPlatonic Orthonormal Wavelets 1
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 4, 351 35 (1997) ARTICLE NO. HA9701 Platonic Orthonormal Wavlts 1 Murad Özaydin and Tomasz Przbinda Dpartmnt of Mathmatics, Univrsity of Oklahoma, Norman, Oklahoma
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationCategory Theory Approach to Fusion of Wavelet-Based Features
Catgory Thory Approach to Fusion of Wavlt-Basd Faturs Scott A. DLoach Air Forc Institut of Tchnology Dpartmnt of Elctrical and Computr Enginring Wright-Pattrson AFB, Ohio 45433 Scott.DLoach@afit.af.mil
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationFrom Elimination to Belief Propagation
School of omputr Scinc Th lif Propagation (Sum-Product lgorithm Probabilistic Graphical Modls (10-708 Lctur 5, Sp 31, 2007 Rcptor Kinas Rcptor Kinas Kinas X 5 ric Xing Gn G T X 6 X 7 Gn H X 8 Rading: J-hap
More informationGEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES. Eduard N. Klenov* Rostov-on-Don, Russia
GEOMETRICAL PHENOMENA IN THE PHYSICS OF SUBATOMIC PARTICLES Eduard N. Klnov* Rostov-on-Don, Russia Th articl considrs phnomnal gomtry figurs bing th carrirs of valu spctra for th pairs of th rmaining additiv
More informationTOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM
TOPOLOGICAL CYCLIC HOMOLOGY VIA THE NORM VIGLEIK ANGELTVEIT, ANDREW J. BLUMBERG, TEENA GERHARDT, MICHAEL A. HILL, TYLER LAWSON, AND MICHAEL A. MANDELL Abstract. W dscrib a construction of th cyclotomic
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationSeparating principles below Ramsey s Theorem for Pairs
Sparating principls blow Ramsy s Thorm for Pairs Manul Lrman, Rd Solomon, Hnry Towsnr Fbruary 4, 2013 1 Introduction In rcnt yars, thr has bn a substantial amount of work in rvrs mathmatics concrning natural
More informationAbstract Interpretation. Lecture 5. Profs. Aiken, Barrett & Dill CS 357 Lecture 5 1
Abstract Intrprtation 1 History On brakthrough papr Cousot & Cousot 77 (?) Inspird by Dataflow analysis Dnotational smantics Enthusiastically mbracd by th community At last th functional community... At
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationPROBLEM SET Problem 1.
PROLEM SET 1 PROFESSOR PETER JOHNSTONE 1. Problm 1. 1.1. Th catgory Mat L. OK, I m not amiliar with th trminology o partially orr sts, so lt s go ovr that irst. Dinition 1.1. partial orr is a binary rlation
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationHardy-Littlewood Conjecture and Exceptional real Zero. JinHua Fei. ChangLing Company of Electronic Technology Baoji Shannxi P.R.
Hardy-Littlwood Conjctur and Excptional ral Zro JinHua Fi ChangLing Company of Elctronic Tchnology Baoji Shannxi P.R.China E-mail: fijinhuayoujian@msn.com Abstract. In this papr, w assum that Hardy-Littlwood
More informationIterative Algebras for a Base
Elctronic Nots in Thortical Computr Scinc 122 (2005) 147 170 www.lsvir.com/locat/ntcs Itrativ Algbras for a Bas JiříAdámk 1,2 Stfan Milius 2 Institut of Thortical Computr Scinc, Tchnical Univrsity, Braunschwig,
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More information1 General boundary conditions in diffusion
Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS DONE RIGHT KYLE ORMSBY
LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS DONE RIGHT KYLE ORMSBY INTRODUCTION TO THE PROBLEM Considr a continous function F : R n R n W will think of F as a vctor fild, and can think of F x as a vlocity
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationOn the construction of pullbacks for safe Petri nets
On th construction of pullbacks for saf Ptri nts Eric Fabr Irisa/Inria Campus d Bauliu 35042 Rnns cdx, Franc Eric.Fabr@irisa.fr Abstract. Th product of saf Ptri nts is a wll known opration : it gnralizs
More informationsurface of a dielectric-metal interface. It is commonly used today for discovering the ways in
Surfac plasmon rsonanc is snsitiv mchanism for obsrving slight changs nar th surfac of a dilctric-mtal intrfac. It is commonl usd toda for discovring th was in which protins intract with thir nvironmnt,
More informationFinding low cost TSP and 2-matching solutions using certain half integer subtour vertices
Finding low cost TSP and 2-matching solutions using crtain half intgr subtour vrtics Sylvia Boyd and Robrt Carr Novmbr 996 Introduction Givn th complt graph K n = (V, E) on n nods with dg costs c R E,
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationCLONES IN 3-CONNECTED FRAME MATROIDS
CLONES IN 3-CONNECTED FRAME MATROIDS JAKAYLA ROBBINS, DANIEL SLILATY, AND XIANGQIAN ZHOU Abstract. W dtrmin th structur o clonal classs o 3-connctd ram matroids in trms o th structur o biasd graphs. Robbins
More informationSquare of Hamilton cycle in a random graph
Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationUNTYPED LAMBDA CALCULUS (II)
1 UNTYPED LAMBDA CALCULUS (II) RECALL: CALL-BY-VALUE O.S. Basic rul Sarch ruls: (\x.) v [v/x] 1 1 1 1 v v CALL-BY-VALUE EVALUATION EXAMPLE (\x. x x) (\y. y) x x [\y. y / x] = (\y. y) (\y. y) y [\y. y /
More informationObjective Mathematics
x. Lt 'P' b a point on th curv y and tangnt x drawn at P to th curv has gratst slop in magnitud, thn point 'P' is,, (0, 0),. Th quation of common tangnt to th curvs y = 6 x x and xy = x + is : x y = 8
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationDivision of Mechanics Lund University MULTIBODY DYNAMICS. Examination Name (write in block letters):.
Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions
More information