A Note on Support in Triangulated Categories

MIFP–08–05 Octob er 30, 2018 A Note on Supp ort in T riangulated Categories Aaron Bergma n a a Ge or ge P. & Cynthia W. Mitchel l Institute for F undamental Physics T exas A&M University Col le ge Station, TX 77843-4242 Email: abergman @physics.tam u.edu Abstract In this note, I define a notion of a compactly supp orted ob ject in a triangulated category . I pro v e a n um b er of prop ositions relating this to traditional notions of supp ort and giv e an application to the theory of deriv ed Morita equiv alence. I also discuss a connection to sup ersymm etric g a uge theories arising from D-branes at a singularit y . 1. In tro d u ction 1.1. Ov er view Let C b e a triangulated category with small copro ducts. W e will alwa ys work o v er a field k , and all dimensions will b e as ve ctor spaces ov er that field. Recall the following definition: Definition 1. A n obje ct E ∈ C is c al le d c omp act i f Hom( E , − ) c ommutes with c opr o ducts. The full sub category of compact o b jects is triangulated, and w e will denote it as C c . If C ∼ = D (QCoh( X )) for some v ariet y X , then it is w ell-known (see, for example, [ 1 ]) that the compact ob jects a re those complexes of shea v es lo cally quasi-isomorphic to bounded complexes of lo cally free coheren t s hea ves 1 . If, on the other hand, C ∼ = D ( A − Mo d) for an algebra A , then the compact ob jects are those complexes of mo dules quasi-isomorphic to b ounded complexes of finitely gene rated pro jectiv e modules 2 [ 2 ]. In b oth cas es, these are often termed p erfe c t ob jects. W e will denote the tria ng ulated subcategory of compact ob j ects as D c ( A − Mo d) or D c (QCoh( X )). W e will also sometimes refer to an ob ject in an ab elian category as compact or p erfect if its image in the deriv ed category is compact. In this note, w e will consider a notion of supp ort for an ob ject in a triangulated category . In particular, w e define Definition 2. An obje ct E ∈ C is said to b e c omp actly supp orte d if, fo r al l c omp act F ∈ C , ∞ X i = −∞ dim Hom( F , E [ i ]) < ∞ . This is equiv alen t to sa ying that the homological functor C ( − , E ) is of finite type when restricted t o the compact sub category . Giv en a triangulated category , w e can consider the full sub category of compactly supp orted ob jects whic h w e will denote by C cs . F or our examples, w e will de note these sub categories as D cs (QCoh( X )) and D cs ( A − Mo d). Prop osition 1. The ful l sub c ate gory of c omp actly supp o rte d o b j e cts is a triangulate d sub c at- e gory. 1 If X is noether ian, Coh( X ) is an abelia n categ ory , and D (Coh( X )) is equiv ale n t to the full sub category of D (QCoh( X )) whos e coho mology sheaves are coherent. It is straightforward to see that the co ho mology sheav es of a n y compact complex are cohere nt, so the compa ct ob jects ca n b e considered as a sub categor y of D (Co h( X )). 2 Here A − Mo d is the a belian catego ry o f possibly infinitely g enerated modules. If A is no etherian, finitely generated mo dules for m a n ab elian categor y which we will de no te A − mod, and a remark simila r to the previous foo tnote applies. 1 Pr o of. It is ob viously closed under shifts and finite sums. That it is closed under extensions can b e seen from the long exact sequence in Homs arising from a distinguished triangle. F rom the p oint of vie w o f noncomm utative geometry , the more relev an t ob ject is often the compact sub category of a giv en t r iangulated categor y . Th us, it is in teresting to consider the resulting sub categories of compactly supp orted compact ob jects. Then, the a bov e definition reduces to the stateme n t that C c ( − , E ) is of finite ty p e. If C c cs has a generator, then it is a compact algebraic k -linear space in the sense of Ko n tsevic h 3 [ 3 ]. T o relate this to more usual senses of supp ort, w e ha v e: Prop osition 2. L et X b e a no etherian variety and E a b ounde d c omplex of c oher ent sh e aves such that, for al l i ∈ Z , the supp ort of H i ( E ) i s pr op er. Then E is c om p ac tly supp orte d in D (Coh( X )) . Prop osition 3. L et A b e an algebr a over a field, and E a b ounde d c omple x of mo dules. If dim H i ( E ) < ∞ for all i ∈ Z , then E is c omp a ctly supp orte d in D ( A − Mo d) . These tw o prop ositions will b e prov en in section 2 , along with a c haracterization of compact supp ort when C has a compact generator. T o av oid confusion (or p erhaps to foster it), w e will a lwa ys use ‘pro p er supp ort’ to refer to the supp ort of a sheaf b eing a prop er subsc heme, and ‘compact sup p ort’ to refer to t he homo lo gical notio n defined ab o v e. With more restrictiv e assumptions, we can prov e con v erses of the ab o v e propositions. Prop osition 4. L et A b e an algebr a ov e r a field. Then D c cs ( A − Mo d) is e quiva l e nt to the ful l sub c ate g o ry o f D c ( A − Mo d ) c onsisting of obje cts whose c ohomol o gy mo dules have fi n ite dimension. Prop osition 5. L et X b e a smo oth no etherian variety. Then D c cs (Coh( X )) is e q uivalent to the ful l sub c ate gory of D c (Coh( X )) c o n sisting of obje cts whose c ohomolo gy she ave s have pr op er s upp ort. F or b oth prop ositions, w e can reduce to the case of sp ecific ob jects in the ab elian category through the use of t he hypercohomology sp ectral seque nce and rep eated truncations. F or the la tter prop osition, w e also require the follow ing lemma: 3 More prop erly , in o rder to be a linear space, C should b e enhanced over dg-V ect, and we can apply our criterion to the dimensions of the coho mo logies o f the Hom-complexes. 2 Lemma 1. L et X b e a smo oth no etherian variety. Then a c o her en t she af E has pr op er supp ort if a n d only if, for al l c o her en t she aves F , dim Ext i X ( F , E ) < ∞ for a ll i ∈ N . This w ill follo w from applying the hypothesis to the structure she a v es of reduced curv es in the supp ort o f E and applying the v a luativ e criteria of pro perness. 1.2. Ph ysics motiv ation The questions addressed in this note arise in the study of D-branes at singularities in string theory . In particular, let X b e a del P ezzo surface and K X b e the total space of the c anonical bundle on X . K X has a trivial cano nical bundle and is thus a no n-compact Calabi-Y au. It serv es as a local mo del for a singularity as one can collapse the z ero section to obta in a singular cone. In string theory , this collapse can b e in terpreted as a deformation of the K¨ ahler metric and is, as suc h, not visible to the top olo gical B-mo del string. Th us, to make sense of the top ological B-mo del on the singular space, it suffices to study it on this smo oth resolution. As has b een conjectured in [ 4 , 5 , 6 ], the top ological B-mo del is describ ed b y the (A ∞ enhancemen t of the) bounded deriv ed category of coheren t shea v es 4 . Using the resulting dictionary , w e can turn man y phy sical results in to mathematical theorems and vice vers a. F or example, t he inde p endence of the top olog ical B-mo del of the choice of resolution of the singularity b ecomes a statemen t ab out equiv alences o f deriv ed categories. In particular, Bridgeland [ 7 ] has shown that the deriv ed categories of coherent sheav es of differen t crepan t resolutions of a pro jectiv e t hr ee-fo ld with terminal singularities a re equiv alent. T o study D -branes lo cated at a singularit y , w e need to study t he stabilit y of skyscraper shea ve s lo cated o n the zero section of K X . This is done in [ 8 ] following results of Bridgeland [ 9 ]. In particular, we will let T be a lo cally free generator 5 of D (Coh( K X )) such that Ext i ( T , T ) = 0 for i 6 = 0. Let A = End ( T ) op . Then Rick ard’s derive d Morita equiv alence [ 2 ] tells us that t here is an equiv alence o f categories: D b (Coh( K X )) ∼ = D b ( A − mo d) (1.1) where the latter category is the b ounded deriv ed category of finitely generated A - mo dules. 4 When our target is smo oth, this is equiv alent to the perfect sub categor y o f the deriv ed category . 5 Such generato rs ma y b e obtained from pulling ba c k stro ng, s imple exceptiona l collections on X . See, for example, [ 10 , 11 ]. 3 Here, the bounded deriv ed categor ies arise as the p erfect sub categories considered ab o ve as K X is smo oth no etherian and A is no etherian of finite global dimension. A is a Calabi-Y au algebra [ 12 , 13 ] and can b e represen ted as the path algebra of a quiv er with relations deriv ed from a p oten tial. This data, along with the dimension v ector obtained from the mo dule corresp onding to a skyscrap er sheaf, defines the holomorphic data of an N = 1 quiv er gauge theory . This gauge theory is the w orldv olume theory tha t liv es on a D3-brane lo cated at the “tip” o f the cone, i.e., the D3-brane is g iv en b y the em b edding of R 4 in t o to R 4 × C ( X ) where C ( X ) is the cone obtained by collapsing the zero section of K X , and w e em b ed at the tip of C ( X ). It is an in teresting question to ask what is the proper “ph ysical” category to describe the top ological B-mo del on these noncompact spaces. The cat ego ry D b (Coh( K X )) is un- suitable because, among other things, it is not Calabi-Y au. In particular, there is no Serre functor as K X is noncompact. In the mathematics lit era t ure, it is common to study the full su b category of o b jects whose cohomolog y shea v es hav e reduced support along the zero section of K X . Under the equiv alence ( 1.1 ), this maps to the full sub category of nilp oten t mo dules [ 10 ]. This is a Calabi-Y au category and has man y nice mathematical prop erties, but its physic al significance is still somewhat m ysterious. Another obv ious cat ego ry to consider is D b prop (Coh( K X )), the full sub category with prop er su pp orts for the cohomology she a v es. This is also Calabi-Y au in t he sense that it ha s a Serre functor e quiv alent t o the shift b y three functor. It w as conjectured in [ 8 ] that this is equiv alen t to the category D b fd ( A − Mo d) con- sisting of ob jects whose cohomolo gy mo dules hav e finite dimension v ector. Using the ab ov e prop ositions, we can characterize these sub categories intrins ically a s the full sub categories with compact supp ort. Th us, we hav e Corollary 1. L e t X b e a smo oth, no etherian va ri e ty, and let T b e a c om p ac t gener ator of D ( C oh ( X )) such that Hom( T , T [ i ]) = 0 for i 6 = 0 and E nd ( T ) is l e ft-no etheria n and of finite glob al dimension. Then D b prop ( C oh ( X )) ∼ = D b fd (End( T ) op − mo d) . 1.3. Ac kno wledgmen ts I w ould like to thank Da vid Ben-Zvi for his help with the man uscript and for many other suggestions during the writing of this note. I w o uld a lso lik e to thank Zac h T eitler for a suggestion tha t led to a pro of of lemma 1 and T om Nevins for helpful e-mails. This r esearch w as supp orted by t he Natio nal Science F oundation under G ran t No. PHY-0505757, and by T exas A&M Univ ersit y . 4 2. Pro ofs of pr op ositions 2 and 3 F or conv enience, w e restate prop osition 2 : Prop osition 2. L et X b e a no etherian variety and E a b ounde d c omplex of c oher ent sh e aves such that, for al l i ∈ Z , the supp ort of H i ( E ) i s pr op er. Then E is c om p ac tly supp orte d in D (Coh( X )) . Pr o of. W e b egin with the case where E is a sheaf. W e can consider E as p ∗ E where p : Supp E → X is the em b edding of the supp ort with the natural subsc heme structure. Then, for a p erfect F w e can compute Hom D b ( X ) ( F , E [ i ]) ∼ = Hom D b (Coh(Supp E )) ( L p ∗ F , E [ i ]) ∼ = H i Supp E ( R H om ( L p ∗ F , E )) . Deriv ed pullbacks preserv e perfectness (cf. [ 14 ] 2.5.1) . As X is finite type, an y p erfect complex is bo unded, and lo cal Homs fro m p erfect complexes preserv e b oundedness, so R H om ( L p ∗ F , E ) is a b ounded complex of shea ve s. Since Supp E is finite dimens ional a nd no etherian, there are o nly a finite num b er of non-zero terms in the h yp ercohomo lo gy s p ec- tral sequenc e, a nd they a r e a ll finite dimensional as Supp E is prop er. Thus , E is compactly supp orted. F or E a b ounded complex, the result follo ws from examining the analog of the h yp ercohomology sp ectral sequence for the h yp erext functor Ext i ( F • , − ) and applying it to E , r educing to the previous case applied to the finite nu m b er of non-zero cohomology shea ve s. The pro of of prop osition 3 is similar: Prop osition 3. L et A b e an algebr a over a field, and E a b ounde d c omple x of mo dules. If dim H i ( E ) < ∞ for all i ∈ Z , then E is c omp a ctly supp orte d in D ( A − Mo d) . Pr o of. W e again star t with the case of a finite dimensional mo dule, M . W e can write its dimension as dim Hom A − Mo d ( A, M ). Sinc e an y finitely generated pro jectiv e module is a direct summand in a free mo dule on a finite set, we immediately hav e that dim Ho m A − Mo d ( P , M ) < ∞ for all finitely generated pro jectiv e mo dules, P . Since a p erfect complex of mo dules is quasiisomorphic to a b ounded complex of finitely generated pro jectiv e mo dules, w e see that 5 an y finite dimensional mo dule is compactly supp orted. W e c an again extend this to bounded complexes b y usual sp ectral sequence arg uments. In the case w here a triangulated category has a compact generator there is useful c har- acterization of c ompact supp ort. As ab ov e, w e let C denote our triangulated c ategory . An ob j ect A is called a compact g enerator if it is compact a nd if Hom C ( A , E [ i ]) = 0 for all i implies that E ∼ = 0. A theorem of Rav enel and Neeman [ 15 ] implies that the smallest tria n- gulated sub category of C con taining A and whic h is closed under direct summands is C c . In this situatio n, there is the following criterion fo r compact supp ort: Lemma 2. An obje ct, E , in a trian g ulate d c ate gory, C , with c omp ac t gener a tor, A , is c om- p actly supp orte d if and only if ∞ X i = −∞ dim Hom C ( A , E [ i ]) < ∞ . Pr o of. One direction is ob vious. T o see the other direction, let B b e the full sub category o f ob j ects in F ∈ C suc h that ∞ X i = −∞ dim Ho m C ( F , E [ i ]) < ∞ . By assumption, A is an ob ject in B . F urthermore, as in pr o po sition 1 , it follo ws fro m the lo ng exact sequence a ssociated to Hom C ( − , E [ i ]) that B is a triangulated sub category . Finally , it is ob viously closed under direct summands. Th us, b y Ra v enel and Neeman’s theorem, B m ust contain C c , and E is compactly supp orted. W e can now giv e a n essen tially equiv alen t proof o f prop osition 3 by noting that D ( A − Mo d) is generated by the free mo dule A (for a longer discussion, see , for example, [ 16 ]) and com- puting ∞ X i = −∞ dim Ho m( A, E [ i ]) = ∞ X i = −∞ dim H i ( E ) < ∞ . (2.1) In fact, this su ffices to also pro ve pr o po sition 4 , but w e pro v e it b y another metho d in the follo wing section that gene ralizes to a pro of of prop osition 5 . In addition, a theorem of Bondal and v an den Bergh [ 17 ] s tates that, for an y quasi- compact separated sch eme, X, D (QCoh( X )) is generated b y a single p erfect complex, giving an oft en simple criterion for compact supp ort. 6 3. Pro of of pr op osition 4 W e b egin with the following lemma: Lemma 3. L et A b e an algebr a over a field , P a pr oje ctive mo dule and E a b ounde d c omplex of mo d ules. Then ∞ X i = −∞ dim Hom D b ( A − Mod) ( P , E [ i ]) < ∞ implies that ∞ X i = −∞ dim Hom A − Mo d ( P , H i ( E )) < ∞ . Pr o of. Because the assumptions o f the lemma are inv ariant under s hifts and E is b ounded, w e can assume tha t τ ≥ 0 E ∼ = E where τ is the usual truncation functor. The re exists a distinguished triangle τ ≤ 0 E − → E − → τ ≥ 1 E . By a ssumption τ ≤ 0 E ∼ = τ ≤ 0 τ ≥ 0 E ∼ = H 0 ( E ). W e can apply the cohomolo gy functor Hom( P , − ) to get a long exact sequenc e. Since P is in D ≤ 0 and all three elemen ts in t he tria ngle are in D ≥ 0 , all the negative Homs v anish. T h us, w e can write 0 − → Hom( P , H 0 ( E )) − → Hom( P , E ) − → Hom( P , τ ≥ 1 E ) − → Hom( P , H 0 ( E )[1]) − → Hom( P , E [1]) − → Hom( P , ( τ ≥ 1 E )[1]) − → . . . . As Hom( P , E ) is finite dimensional, exactness implies that Hom( P , H 0 ( E )) is finite di- mensional. F urthermore, Hom( P , H 0 ( E )[ i ]) = 0 for i > 0 as P is pro jectiv e. Th us, Hom( P , ( τ ≥ 1 E )[ i ]) is also finite dimensional for all i . W e can now repeat this arg umen t for τ ≥ 1 E and pro ceed inductiv ely to see that Hom( P , H i ( E )) is finite dimens ional for all i . Note that the long exact sequenc e in fact implies that Hom( P , E [ i ]) ∼ = Hom( P , τ ≥ i E [ i ]) ∼ = Hom( P , H i ( E )) whic h can also b e computed direc tly . Th us, the su ms are equal, and w e repro duce the fact ( 2.1 ) no ted in the previous section. Com bining these results, we hav e: Prop osition 4. L et A b e an algebr a ov e r a field. Then D c cs ( A − Mo d) is e quiva l e nt to the ful l sub c ate g o ry o f D c ( A − Mo d ) c onsisting of obje cts whose c ohomol o gy mo dules have fi n ite dimension. 7 Pr o of. Giv en an ob ject in D c ( A − Mo d), it is quasiisomorphic to a bo unded complex of mo dules. If, furthermore, it has finite dimensional cohomology mo dules, prop osition 3 tells us tha t it has compact supp ort. No w assume that E is compactly supp orted. Then ∞ X i = −∞ dim Ho m( A, E [ i ]) < ∞ . By lemma 3 , the sum of the dimens ions of the cohomology mo dules is finite, and w e are done. 4. Pro of of pr op osition 5 In order to pr ov e prop osition 5 , we first pro v e lemm a 1 whic h w e restate he re. Lemma 1. L et X b e a n o etheria n variety. Then a c oher e n t she af E has pr op er s upp ort if and on l y if, for al l c oher en t she aves F and i ∈ N , dim Ext i X ( F , E ) < ∞ . Pr o of. W e can compute the Ext gro up by the lo cal to g lo bal sp ectral sequence with E pq 2 = H p ( E xt q ( F , E ) ) . Since E has prop er suppo rt, the lo cal Ext also do es, and th us the dimensions of the coho- mology groups a re finite dimensional. T his is a n upp er-righ t quadrant sp ectral sequence, so only a finite num b er of terms con tribute to any giv en Ext i X ( F , E ) , and w e are done. F or the other direction, w e apply the v aluativ e criterion for pro p erness. Since X is no etherian, E = Supp E is a closed subsc heme. Let R b e a discrete v aluation ring and K its field of fractions. W e are given a map fr om Sp ec K in to E . The closure of the imag e of this in X map is a closed sub v ariet y , f : C → E , with the reduced indu ced structure. C is a curv e and is th us (cf. [ 18 ] Ex. IV.1.4) either prop er or affine. W e can restrict E to C , and w e hav e by assumption that dim H 0 C ( f ! E ) = dim Hom X ( O C , E ) < ∞ . Since E is a closed subsc heme of X , C lies in the supp ort of E and th us f ! E = H om ( O C , E ) is globally suppor ted on C . Ho w eve r, if C w ere affine, t his w ould mean that the cohomolo g y 8 w ould b e infinite dimensional, so C mus t b e prop er. Therefore, there exis ts a map from Sp ec R to E , and t h us E is prop er. Note that to pro ve a sheaf has pro per supp ort, w e only need to test the Homs from the structure shea ves of reduced curv es. This will b e the only pa r t of the argumen t needed in what follo ws. Next w e prov e a lemma analo gous to lemma 3 : Lemma 4. L et X b e a no etherian variety and E a b ounde d c o m plex of she aves. Assume that ∞ X i = −∞ dim Hom D b (Coh( X )) ( O C , E [ i ]) < ∞ for al l O C the stru ctur e she a v es of r e duc e d curves i n the supp ort of E . Then, ∞ X i = −∞ dim Hom Coh( X ) ( O C , H i ( E )) < ∞ . Pr o of. As in the pro o f of lemma 3 , w e hav e a long exact sequence: 0 − → Hom( O C , H 0 ( E )) − → Hom( O C , E ) − → Hom( O C , τ ≥ 1 E ) − → Hom( O C , H 0 ( E )[1]) − → Hom( O C , E [1]) − → Hom( O C , ( τ ≥ 1 E )[1]) − → . . . . F rom the b eginning of this sequence, w e ha v e that Hom( O C , H 0 ( E )) is finite dimens ional. By the argumen t in lemma 1 , H 0 ( E ) has prop er supp ort a nd, hence, Hom( O C , H 0 ( E )[ i ]) is finite dimensional for all i . Th us, w e aga in see that τ ≥ 1 E satisfies our hy p othesis, and we pro ceed as in lemma 3 . Finally , we dispatch with prop osition 5 : Prop osition 5. L et X b e a smo oth no etherian variety. Then D c cs (Coh( X )) is e q uivalent to the ful l sub c ate gory of D c (Coh( X )) c o n sisting of obje cts whose c ohomolo gy she ave s have pr op er s upp ort. Pr o of. Since X is smo oth, all p erfect complexes are b ounded. 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