Perfect Derived Categories of Positively Graded DG Algebras

PERFECT DERIVED CA TEGORIES OF POSITIVEL Y GRADED DG ALGEBRAS OLAF M. SCHN ¨ URER Abstract. W e inv estigate t he p erfect de r iv ed category dgP er( A ) of a posi - tiv ely graded different ial graded (dg) algebra A whose degree zero part i s a dg subalgebra and semis i mple as a ring. W e int r oduce an equiv alen t subcat- egory of dgPer( A ) whose ob jects are easy to describ e, define a t-structure on dgP er ( A ) and s tudy its heart. W e show that dgPe r ( A ) i s a Kr ull-Remak- Sc hmidt category . Then we co nsi der the heart in the case that A is a Koszul ring with differen tial zero satisfying some finitene ss conditions. Contents 1. Int r oductio n 1 2. Preliminarie s 4 3. Different ia l Graded Mo dules 5 4. Filtered DG Modules 6 5. t-Structure 10 6. Heart 12 7. Homotopically Minimal DG Modules 16 8. Indecomp osables 17 9. Koszul-Duality 19 References 23 1. Introduction Let k be a co mm utative ring and A = ( A = L i ∈ Z A i , d ) a differential gr aded k -algebra ( = dg algebra). Let dgDer( A ) b e the deriv ed category o f dg (right) mo dules ov er A (= A -mo dules), and dgPer ( A ) the p erfect de r ived ca tegory , i. e. the smallest s tr ict f ull triang ulated sub catego ry of dgDer( A ) containing A and closed under forming direct s ummands; the ob jects of dgPer ( A ) ar e precisely the compact ob jects of dgDer( A ) (see [Kel94, Kel98]). The aim of this ar ticle is to provide some descr iption of dgPer ( A ) and to define a t-structure on this category if A = ( A, d ) is a dg alg ebra satisfying the following conditions: (P1) A is positively graded, i. e. A i = 0 for i < 0; Date : October 2008, revised January 2010. 2000 Mathematics Subje ct Classific ation. 18E30, 16D90. Key wor ds and phr ases. Di fferen tial Gr aded M odule, DG Mo dule, t-Structure, Heart, Koszul Duality . Supported by a grant of the state of Baden-W¨ urttem b erg. 1 2 OLAF M. S CHN ¨ URER (P2) A 0 is a semisimple ring; (P3) the differential of A v anishes o n A 0 , i. e. d ( A 0 ) = 0. A t the end of this int r oductio n we explain our main motiv ation for studying such per fect derived categorie s. The o nly r elated and in fact motiv ating description (with a definition o f a t-str uc tur e) we know of ca n b e found in [BL94, 11]; the dg a lgebra ( R [ X 1 , . . . , X n ] , d = 0) co nsidered there is a po lynomial alge br a with generators in strictly p ositive even degrees . W e give an account of the r esults o f this article, alwa ys assuming that A is a dg algebra satisfying the conditions (P1)-(P3 ). Alternativ e de scriptions of the p erfect de riv ed category. The semisimple ring A 0 has only a finite n umber of non-isomor phic simple (right) modules ( L x ) x ∈ W . W e view A 0 as a dg subalgebr a A 0 of A and the L x as A 0 -mo dules concentrated in degree zero . Let dgP rae( A ) b e the smallest strict full triangula ted sub categor y of dg Der( A ) that contains all A -mo dules b L x := L x ⊗ A 0 A . (The name dg P rae was c hosen b ecause this categor y seemed to b e a precurs or of the p erfect der ived category : It is not requir ed to be clo sed under taking direct summands. But indeed it is closed under this op eratio n, cf. Theorem 1.) Define dgFilt( A ) to b e the full sub catego ry of dgDer( A ) whose o b jects are A -modules M admitting a finite filtr ation 0 = F 0 ( M ) ⊂ F 1 ( M ) ⊂ · · · ⊂ F n ( M ) = M by dg submo dules with sub q uotien ts F i ( M ) /F i − 1 ( M ) ∼ = { l i } b L x i for suitable l 1 ≥ l 2 ≥ · · · ≥ l n and x i ∈ W ; here { 1 } deno tes the shift functor. W e hav e inclusio ns dgFilt( A ) ⊂ dgP rae( A ) ⊂ dgPer ( A ). Theorem 1 (cf. Theo rems 1 3 and 16) . (1) dgPra e ( A ) is close d under taking dir e ct summands, i. e. dgPrae( A ) = dgPer( A ) . (2) The inclusion dgFilt( A ) ⊂ dgPer ( A ) is an e quivalenc e of c ate gori es. The pro of of this theorem relies o n the existence of a b ounded t-s tructure (as describ ed b elow) on dgPrae( A ). The ob jects of dgFilt( A ) can b e characterized as the homo topically pro jectiv e ob jects in dg Per( A ) that are homotopica lly minimal, cf. Prop osition 3 1. The equi- v alence dgFilt( A ) ⊂ dg Per( A ) enables us to prove tha t dgPer( A ) is a Kr ull- Remak- Schm idt category , cf. Pr opo sition 37. t-structure. Let dgPer ≤ 0 (and dgPer ≥ 0 ) b e the full sub categor ies of dgPer( A ) consisting of ob j ects M such that H i ( M L ⊗ A A 0 ) v anishes for i > 0 (for i < 0, resp ectively); here (? L ⊗ A A 0 ) is the left derived f unctor of the extension of scalars functor (? ⊗ A A 0 ). Let dgMo d( A ) b e the ab elian catego ry o f A -modules and dgFlag( A ) the full subcatego ry consisting of ob jects tha t ha ve an b L x -flag, i. e. a finite filtra tion with subquotients isomorphic to ob jects of { b L x } x ∈ W (without shifts). Theorem 2 (cf. Theo rem 16 and Propo sitions 20, 22) . (1) (dgPer ≤ 0 , dgPer ≥ 0 ) defines a b ou n de d (henc e non-degenerate) t-structur e on dgPer ( A ) . PERFECT DERIVED CA TEGORIES 3 (2) The he art ♥ of this t-stru ctur e is e quivalent to dgFlag ( A ) . Mor e pr e ci sely, dgFlag( A ) is a ful l ab elian sub c ate gory of dgMod( A ) and the obvious func- tor dgMo d( A ) → dgDer( A ) induc es an e quiva lenc e dgFlag ( A ) ∼ − → ♥ . (3) Any obje ct in ♥ has finite length, and the simple obje cts in ♥ ar e (u p to isomorphi sm ) the { b L x } x ∈ W . The tr uncation functor s o f this t-structure have a very simple description on dgFilt( A ). Koszul case. Assume now that the differential of A v anishes and that the un- derlying graded ring A is a Koszul ring (cf. [BGS96]). Le t E ( A ) := Ext • A ( A 0 , A 0 ) be the gr aded ring of self-extensions o f the right A -mo dule A 0 = A/ A + (in the category of (no n-graded) A -mo dules). Theorem 3 (cf. Theorem 39) . Assume that A is a Koszul ring with a Koszul r esolution of finite length with finitely gener ate d c omp onents. L et A = ( A, d = 0 ) . Then the he art ♥ of dgPer( A ) is e quiva lent to the op p osite c ate gory of t he c ate gory of finitely gener ate d left E ( A ) -m o dules. Motiv ation. W e b ecame interested in pe r fect derived categor ies when we studied the Borel-equiv aria n t der ived categor y of sheav es on the flag v ar iet y of a complex reductive g roup. W e sho w in [Sch07] (or [Sc h]) that this categ ory with its p erverse t-structure is t-equiv alent to the p erfect deriv ed catego r y dgPer( E ) for some dg algebra E (that meets the conditio ns (P1)-(P3)). Mor e precisely , E is the graded a l- gebra o f self-ex tensions of the direct sum of the simple equiv ariant p erverse s heav es and ha s differential d = 0. F or the pro of o f this eq uiv alence we need the t-s tructures (dgPer ≤ 0 , dgPer ≥ 0 ) introduced a bove on several categor ie s of the form dgPer( A ). There ar e s imilar equiv alences be tw ee n equiv ariant derived catego r ies of sheav es and categories o f the form dgPra e ( E ) = dgPer( E ), see [BL94, Lun95, Gui05]. The strategy to obtain these equiv alences is quite general (see [Lun95, 0.3]), the tricky po in t howev er is to e s tablish the fo rmality of some dg alg ebra B who s e p erfect derived catego ry is eq uiv alen t to the considered categ o ry of shea ves. F ormalit y means that B and its co homology H ( B ) (a dg algebra with differential z e ro) can b e connected b y a sequence of quasi-is omorphisms of dg algebra s . Since ea ch quasi- isomorphism of dg a lgebras induces an e quiv alence betw een their (p erfect) deriv ed categorie s, formality ena bles us to co nsider the more a c cessible dg algebra H ( B ) instead of B . Mor eov er, usually H ( B ) identifies with the extension a lg ebra of some nice ob ject from the g eometric side. This general s trategy sho ws tha t ca tegories of the form dg Prae( A ) = dgPer( A ) are natural ca ndidates for descr ibing certain tria ngulated c a tegories. Ov erview. This article is orga nized as follo ws. In Chapters 2 a nd 3 we introduce our notation, prov e some basic r esults on gra ded modules and rec all some results on dg mo dules. W e show the equiv alence dgFilt( A ) ⊂ dg Prae( A ) in Chapter 4. In the following t wo chapters we define the t-structur e on dgPer( A ), show that dgPrae( A ) = dgPer ( A ) and desc r ibe the heart. W e give alternative c har acteriza- tions o f the ob jects of dgFilt( A ) in Chapter 7. Chapter 8 contains so me results on indecomp osable s , a Fitting le mma for o b jects of dgFilt( A ) and the pro of that dgPer ( A ) is a Kr ull-Remak-Schmidt categor y . The last chapter (which is indepen- dent of Cha pter 8) co ncerns the case that A is a Koszul ring. 4 OLAF M. S CHN ¨ URER Ac kno wle dgmen ts. Mo st o f the re sults of this article (apart from some improve- men ts and Chapters 7 and 8) can be found in the a lgebraic pa rt of my thesis [Sch07]. I am grateful to my advisor W olfgang So ergel who introduced me to the dg world. I would like to thank Peter Fiebig, Bernhar d Ke lle r, Henning K rause, Catharina Str oppel and Geor die Williamson for helpful discussions and interest. I wish to thank the r eferee for carefully re ading the manuscript and for questions and comments. 2. Preliminaries W e int r o duce our notation and prov e some easy and probably well-known state- men ts that are crucial for the rest o f this ar ticle. W e fix some commutativ e ring k . All rings and algebras (= k - a lgebras) are ass umed to b e asso ciative and unital. If A is a r ing, we denote the category of right A -mo dules by Mo d( A ) and the full sub c ategory of finitely genera ted mo dules by mo d( A ). If A is g raded (= Z -gr aded) we wr ite gMo d( A ) for the ca tegory o f graded mo dules a nd g mod( A ) for the full sub c ategory of finitely generated gra ded mo dules. Let A = L i ∈ N A i be a p ositively graded k -algebra. Let A + = L i> 0 A i . The pro jection A → A/ A + = A 0 gives rise to the extensio n of scalars functor gMo d( A ) → gMo d( A 0 ) , M 7→ M := M / M A + = M ⊗ A A 0 . Lemma 4 . L et f : M → N b e a morphism in g mod( A ) . If f : M → N is an isomorphi sm and N is flat as an A -mo dule, then f is an isomorphism. Pr o of. W e use the follo wing trivial observ ation: If a graded A -module X with X i = 0 for i ≪ 0 s atisfies X = X/X A + = 0, then X = 0. Since (? ⊗ A A 0 ) is r ig h t ex a ct we obtain cok f = cok f = 0, and o ur observ ation implies co k f = 0. Applying (? ⊗ A A 0 ) to the shor t exac t sequence (ker f , M , N ) and using the flatness o f N , we see that 0 = ker f . Our observ ation shows that ker f = 0. So f is a n isomorphism.  W e a ls o hav e the extensio n of scalar s functor coming from the inclusio n A 0 ⊂ A , (1) pro d A A 0 : gMo d( A 0 ) → gMo d( A ) , M 7→ c M := M ⊗ A 0 A. W e often view A 0 -mo dules as graded A 0 -mo dules co nce n tra ted in degree zero. Assume now that A 0 is a semisimple r ing. Then A 0 has only a finite num ber of non-isomor phic simple (rig h t) mo dules ( L x ) x ∈ W . In particular we o bta in pr o j ective graded A -mo dules b L x = L x ⊗ A 0 A . Lemma 5. L et x , y ∈ W . If f : b L x → b L y is a non-zer o morphism in gMo d( A ) , it is an isomorp hism and x = y . Pr o of. Since b L x and b L y are g enerated in degree zer o (= genera ted as an A -mo dule by their degree zer o comp onents), f : L x → L y is non-zero, hence an isomor phism and x = y . Now use Lemma 4.  Let gpro j( A ) b e the full sub catego ry o f pro jectiv e ob jects in gmo d( A ). It is clear that a n y finite direc t sum o f shifted ob jects b L x is in gpro j( A ). The conv er s e is also true. W e include the proo f fo r completeness. Lemma 6. Each obje ct of gpro j( A ) is isomorphic to a fin ite dir e c t sum of shifte d b L x , for x ∈ W . PERFECT DERIVED CA TEGORIES 5 Pr o of. Let P b e in gpro j( A ). W e view the canonical morphism P → P as a morphism of A 0 -mo dules. Since A 0 is semisimple, there is a splitting σ : P → P . Since pro d A A 0 is left a djoin t to the res tr iction functor gMo d( A ) → gMo d( A 0 ) coming from the inclusio n A 0 ⊂ A , we get a morphism b σ : P ⊗ A 0 A → P in gmo d( A ). Since P is A -flat, we deduce from Lemma 4 that b σ is an isomo rphism. But P ⊗ A 0 A has the requir ed for m.  3. Differential G raded Modules W e review the language of differential gr aded (dg) mo dules ov er a dg algebra (see [Kel94, Ke l9 8, BL94]). Let A = ( A = L i ∈ Z A i , d ) b e a differen tial graded k -alg ebra (= dg a lgebra). A dg (righ t) mo dule ov er A will also be called an A -module o r a dg mo dule if there is no doubt ab out the dg alg ebra. W e often write M for a dg mo dule ( M , d M ). W e consider the ca tegory dg Mo d( A ) of dg mo dules, the homoto p y catego ry dgHot( A ) and the derived catego ry dg Der( A ) of dg mo dules. (In [K el94], these categor ies are denoted by C ( A ), H ( A ) and D ( A ) resp ectively .) W e often omit A from the notation. W e deno te the s hift functor on a ll these categor ies (and on gMo d( A ) and gmo d( A )) by M 7→ { 1 } M , e. g. ( { 1 } M ) i = M i +1 , d { 1 } M = − d M . W e define { n } = { 1 } n for n ∈ Z . The ho mo top y ca tegory dg Hot( A ) with t he shift functor { 1 } and the distin- guished triangles iso morphic to standard triangle s (= ma pping cones of morphisms) is a tr ia ngulated categ ory . An y shor t exact sequence 0 → K → M → N → 0 of A -mo dules that is A -split (= it s plits in g Mo d( A )) can b e co mpleted to a distin- guished triangle K → M → N → { 1 } K in dgHot( A ). The catego ry dg Der( A ) inherits the triang ula tion from dgHot( A ). Since dgDer( A ) ha s infinite direct sums, every idemp otent in dgDer( A ) splits (see [B N93, Prop. 3.2 ]). A dg mo dule P is calle d hom otopically pro jectiv e ([Kel98]), if it satisfies one of the following equiv alent co nditio ns ([BL94, 10.12 .2.2]): (a) Hom dgHot ( P, ?) = Hom dgDer ( P, ?), i. e. for all dg mo dules M , the canonical map Hom dgHot ( P, M ) → Hom dgDer ( P, M ) is an iso morphism. (b) F or eac h a c yclic dg mo dule M , w e have Hom dgHot ( P, M ) = 0 . In [Kel94, 3.1 ] such a mo dule is sa id to hav e pr op erty (P) , in [BL94, 10.1 2 .2] the term K -pr oje ctive is used. Since ev aluation a t 1 ∈ A 0 is an isomo rphism (2) Hom dgHot ( A , M ) ∼ − → H 0 ( M ) , A and ea c h dir ect summand of A is homotopica lly pro jective. Let dg Hotp( A ) b e the full subc ategory of dgHot( A ) consisting of homotopically pro jectiv e dg mo dules (this is the categ ory H p in [K e l94, 3.1]). It is a triangula ted sub c ategory of dgHot( A ) and clo sed under taking direct summands. The quotien t functor dgHot ( A ) → dgDer( A ) ind uces a tr iangulated equiv alence ([Kel94, 3.1, 4.1]) (3) dgHotp( A ) ∼ − → dgDer( A ) . The ca tegory dg P er( A ) (dgPer ( A ) resp ectively) is defined to b e the sma llest full triangulated subcateg ory of dgHot( A ) (of dgDer( A ) resp ectively) that contains A and is closed under taking direct s ummands. The category dg P er( A ) o f p erfect dg mo dules is a sub catego ry of dgHotp( A ). The categ ory dgPer( A ) is ca lled the p erfect 6 OLAF M. S CHN ¨ URER derived catego r y o f A . E quiv alence (3) restricts to a triangulated equiv alence dg P er( A ) ∼ − → dgPer( A ) . In the following we prefer to work in the homotopy category dgHot( A ) and leav e it to the rea der to transfer our results fro m dg P er( A ) to dgPer( A ). Note tha t the ob jects of dg P er( A ) are precisely the homotopically pro jectiv e o b jects o f dgPer( A ). Remark 7. Let B be a dg algebra concentrated in deg ree zero , i. e. a dg alge- bra whose underlying graded alg ebra is conc e ntrated in degree zero. Then B is necessarily of the form B = ( B = B 0 , d = 0) for some alg ebra B . In this cas e dgMo d( B ) is the categ ory of complex e s in Mo d( B ), dgHot( B ) is the corres ponding homo to p y catego ry and dg Der( B ) is the deriv ed catego ry o f the ab elian catego ry Mo d( B ) . The ob jects of the categor y dg P er( B ) ar e the o b jects of dgHotp( B ) that are isomor phic to b ounded complexes of finitely g e ne r ated pr o j ec - tive B -modules (see for example [BN93, Prop. 3.4]). ✸ 4. Fil te red DG Modules In this chapter we int r oduce a certain categor y dg F ilt of filtere d dg mo dules (for a suitable dg algebra ). Later on w e will see that this category is equiv alent to dg P er. Let A = ( A = L i ∈ Z A i , d ) b e a dg algebra. In the res t of this article w e always assume that A satisfies the following conditions: (P1) A is positively graded, i. e. A i = 0 for i < 0 ; (P2) A 0 is a semisimple ring; (P3) the differential of A v anishes o n A 0 , i. e. d ( A 0 ) = 0. Then A 0 has only a finite num b er of no n-isomorphic s imple (r ight ) modules ( L x ) x ∈ W , and A 0 is a dg subalgebra A 0 of A . As in the gra de d setting, the inclusion A 0 ֒ → A and the pr o j ectio n A → A / A + = A 0 give rise to extension of scalars functor s dgMo d( A ) → dgMo d( A 0 ) , M 7→ M := M / M A + = M ⊗ A A 0 , dgMo d( A 0 ) → dgMo d( A ) , M 7→ c M := M ⊗ A 0 A . W e often view A 0 -mo dules a s dg A 0 -mo dules concentrated in deg ree zero. In this manner, we obtain A -mo dules b L x = L x ⊗ A 0 A . Since each L x is a direct s ummand of the A 0 -mo dule A 0 , each b L x is a direct summand of the dg mo dule A 0 ⊗ A 0 A = A . Examples 8. Let R b e a semisimple ring (and a k -algebr a). Then the dg alge- bra ( R = R 0 , d = 0) sa tisfies the co nditio ns (P1)-(P 3). An y dg a lgebra that is concentrated in degree ze r o and satisfie s condition (P2) is of this for m. Let A = L i ∈ N A 0 be a p ositively graded alge bra with A 0 a semisimple ring. Then the dg algebr a ( A, d = 0) satisfies the co nditio ns (P1)-(P3). Any dg algebra with v anishing differential that satisfies conditions (P1 )-(P2) is of this form. F or example A could b e a p olynomial algebr a k [ X 1 , . . . , X n ] ov er a field k with homog eneous generator s X i of strictly p ositive degrees. In this case there is o nly one simple k - mo dule L = k , and b L = k [ X 1 , . . . , X n ]. A mor e genera l example for s uch an A would be a q uiv er algebra (over a fie ld k ) of a quiver with finitely many vertices and arrows of strictly p ositive degr ees, or a quo tien t of s uc h an algebr a by a homogeneous ide a l (whic h is ass umed to be zero in degree zero). Each vertex x o f the quiver gives rise to an idemp otent e x in A and to a dg module b L x = e x A . PERFECT DERIVED CA TEGORIES 7 Examples of dg algebr as satisfying conditions (P1)-(P3 ) with non-v anishing dif- ferential a rise for ex ample in rational homoto p y theor y (see [DGMS75] or [BT82, § 19]). ✸ W e consider the following full sub categ o ries of dgHot( A ): • dg F ilt ( A ): Its ob jects are A -mo dules M admitting a finite filtr ation 0 = F 0 ( M ) ⊂ F 1 ( M ) ⊂ · · · ⊂ F n ( M ) = M by dg submo dules with sub quotients F i ( M ) /F i − 1 ( M ) ∼ = { l i } b L x i in dgMo d( A ) for suitable l 1 ≥ l 2 ≥ · · · ≥ l n and x i ∈ W . W e ca ll such a filtration a n b L -filtration . • dg P ra e: This is the smalles t strict (= clos e d under isomorphisms) full triangulated sub categ o ry of dgHot( A ) that contains all ob jects ( b L x ) x ∈ W . These tw o sub categor ies c o rresp ond under the equiv alence (3) to the categories dgFilt( A ) and dgPrae( A ) o f the intro duction. Remark 9. The condition l 1 ≥ l 2 ≥ · · · ≥ l n is e ssent ia l for the definition of ob jects of dg F ilt ( A ). It means that F 1 ( M ) ∼ = { l 1 } b L x 1 is gener ated in degr ee − l 1 (as a graded A -mo dule), then F 2 ( M ) is an extension of { l 2 } b L x 2 (whic h is genera ted in degr ee − l 2 ≥ − l 1 ) by F 1 ( M ) and so on. Since { l 2 } b L x 2 is pro jectiv e a s a g raded A -mo dule, we obtain F 2 ( M ) ∼ = { l 1 } b L x 1 ⊕ { l 2 } b L x 2 in g Mo d( A ). T he same reaso ning yields by induction (4) F i ( M ) ∼ = { l 1 } b L x 1 ⊕ · · · ⊕ { l i } b L x i in gMo d( A ); note that the differential of F i ( M ) does no t necessarily resp ect such a dir ect sum decomp osition. Let us draw some modules in dg F ilt( A ). W e picture a mo dule b L x { l } (with x ∈ W , l ∈ Z ) as follows: (5) degree: − l − 2 − l − 1 − l − l + 1 − l + 2 − l + 3 . . . b L x { l } : • ◦ / / ◦ / / ◦ / / . . . A white ( ◦ ) or black ( • ) bea d in a column lab eled i re presents ( L x { l } ) i . If there is no bea d in column i , we hav e ( L x { l } ) i = 0. Since L x { l } is genera ted by its co mponent in deg ree − l (the black b ead), there are o nly b eads in degr ees ≥ − l . (The action of A is not explicitly drawn in this picture: E le men ts of A j map elements of a b ead to the b ead tha t is j steps to the r ight .) There is an ar row b etw een tw o b eads if the differential b etw een the corres p onding comp onents is p ossibly 6 = 0 . Since the differential has deg ree 1, all ar rows go from a column to its right neighbo ur. Note that there is no arr ow starting at the blac k b ead since b L x is a direct s ummand of A and d ( A 0 ) = 0. Now w e can dr aw pictures of more gene r al ob jects. F or exa mple, let M b e an ob ject of dg F ilt( A ) that a dmits an b L -filtration with four s teps 0 = F 0 ⊂ F 1 ⊂ F 2 ⊂ F 3 ⊂ F 4 = M and sub quotients F 1 ∼ = { 2 } b L x 1 , F 2 /F 1 ∼ = { 1 } b L x 2 , F 3 /F 2 ∼ = { 1 } b L x 3 , F 4 /F 3 ∼ = {− 2 } b L x 4 for so me x i ∈ W . As in (4) we identify M = F 4 as a graded A -mo dule with { 2 } b L x 1 ⊕ { 1 } b L x 2 ⊕ { 1 } b L x 3 ⊕ {− 2 } b L x 4 such that F i gets iden tified with the first i 8 OLAF M. S CHN ¨ URER summands. Here is our picture o f M : (6) degree: − 3 − 2 − 1 0 1 2 3 4 5 . . . {− 2 } b L x 4 : • A A A A A   0 0 0 0 0 0 0 0 0   * * * * * * * * * * * * * * ◦ / / A A A A A   0 0 0 0 0 0 0 0 0   * * * * * * * * * * * * * * ◦ / / A A A A A   0 0 0 0 0 0 0 0 0   * * * * * * * * * * * * * * ◦ / / " " E E E E E E   2 2 2 2 2 2 2 2 2 2   , , , , , , , , , , , , , , . . . { 1 } b L x 3 : • " " E E E E E   3 3 3 3 3 3 3 3 3 ◦ / / A A A A A   0 0 0 0 0 0 0 0 0 ◦ / / A A A A A   0 0 0 0 0 0 0 0 0 ◦ / / A A A A A   0 0 0 0 0 0 0 0 0 ◦ / / A A A A A   0 0 0 0 0 0 0 0 0 ◦ / / A A A A A   0 0 0 0 0 0 0 0 0 ◦ / / " " E E E E E E   2 2 2 2 2 2 2 2 2 2 . . . { 1 } b L x 2 : • " " E E E E E ◦ / / A A A A A ◦ / / A A A A A ◦ / / A A A A A ◦ / / A A A A A ◦ / / A A A A A ◦ / / " " E E E E E E . . . { 2 } b L x 1 : • ◦ / / ◦ / / ◦ / / ◦ / / ◦ / / ◦ / / ◦ / / . . . The direct sum of t he firs t i rows f r om below (without the ar rows) repres e nts the module F i as a graded A -mo dule. The arrows repr esent the differential o f M : Elements of a b ead b are mapp ed to the direct sum of those b eads that a r e e ndpoints of arrows sta r ting at b . Since M is genera ted by the black b eads a s an A -mo dule, t he dg submo dule M A + is represented by the white beads , a nd the quotient mo dule M = M / M A + is represented b y the black b eads (without ar rows): The differential on M v anishes. ✸ Remark 10. Let R = ( R = R 0 , d = 0) b e a dg algebra concentrated in deg ree zero. Recall from Rema r k 7 that each ob ject of dg P er( R ) is is o morphic to a bo unded complex o f finitely gene r ated pro jective R -mo dules. Let M = ( M = L j ∈ Z M j , d ) be such a complex. Assume that M j = 0 for | j | ≥ N . Define sub complexes G i ( M ) of M by G i ( M ) j = ( M j if j ≥ N − i , 0 otherwise. This defines a finite filtration 0 = G 0 ( M ) ⊂ G 1 ( M ) ⊂ · · · ⊂ G 2 N ( M ) = M of M with subq uotien ts G i ( M ) /G i − 1 ( M ) ∼ = { p i } M N − i with p i = i − N (here we consider M N − i as an R -mo dule sitting in degree zero ). Note that p 1 ≤ p 2 ≤ · · · ≤ p 2 N . Now as s ume that R is a semisimple ring with simple mo dules ( L x ) x ∈ W ; then we obtain from the ab ov e filtration ( G i ( M )) i a finite increasing filtration ( G ′ i ( M )) n i =0 with s ub quotients G ′ i ( M ) /G ′ i − 1 ( M ) ∼ = { q i } L x i with q 1 ≤ q 2 ≤ · · · ≤ p n and x i ∈ W . So we ha ve oppos ite inequalities compared to the definition of the categor y dg F ilt( R ). The o b jects of dg F ilt ( R ) are pr ecisely the b ounded complexes of finitely generated (pro jectiv e) R -mo dules with differential zero (cf. Picture (6)) Nevertheless the inclusion dg F ilt( R ) ⊂ dg P er( R ) is an equiv alence of ca tegories: This is well known s inc e R is a semisimple ring by ass umption. W e will prov e this for arbitr ary A satisfying (P1)-(P3 ) in Theorems 13 and 1 6. ✸ Lemma 11. We have inclusions dg F ilt ⊂ dg P rae ⊂ dg P er ⊂ dgHotp ⊂ dgHot . If ( M , d ) is in dg F ilt , the underlying gr ade d mo dule M is in gpro j( A ) . Remark 12. W e pro ve later on that the inclusion dg F ilt ⊂ dg P rae is an equiv a- lence (cf. Theorem 13). Moreo ver, dg P rae is in fact closed under forming direct summands, which implies dg P r ae = dg P er (cf. Theorem 16). ✸ PERFECT DERIVED CA TEGORIES 9 Pr o of. As a rig h t mo dule ov er itself, A 0 is isomor phic to a finite dir ect sum of simple mo dules L x . Hence A is isomo rphic to a finite dir ect sum of mo dules b L x . In particular, each b L x is in dg P er and homotopically pro jectiv e (see (2)). This sho ws dg P ra e ⊂ dg P er. As a gra ded A -mo dule, each b L x is pro jectiv e. So any b L -filtration of a dg mo dule ( M , d ) yields several A -split short exact s equences in dgMo d; in particula r the graded mo dule M is in gpro j( A ) (as already seen in Remar k 9 ). Moreover these A - split ex a ct sequences b ecome distinguished triangles in dgHo t and show that every ob ject of dg F ilt lies in dg P rae. The rema ining inclusions ar e obvious from the definitions.  Let N b e a module ov er a ring. If N ha s a comp osition series (of finite length) we denote its length by λ ( N ); o therwise we define λ ( N ) = ∞ . If ( M , d ) is in dg F ilt, M is a finitely g enerated mo dule over the semisimple ring A 0 , a nd its leng th λ ( M ) obviously coincides with the length of a n y b L -filtration of ( M , d ). If M is a graded A -mo dule, we define M ≤ i := X j ≤ i M j A to b e the g r aded submo dule of M that is generated by the degree ≤ i par ts. This defines a n increasing filtration . . . ⊂ M ≤ i ⊂ M ≤ i +1 ⊂ . . . of M . If M is the underlying graded mo dule o f a dg mo dule in dg F ilt, the different entries in this filtration define a filtration that is coarser than any b L -filtration. Define M 0. Recall that the cone C ( f ) is given by the gr aded (r ight ) A -mo dule Y ⊕ { 1 } X with differential d C ( f ) =  d Y f 0 − d X  . The case λ ( X ) = 1 : Using the shift { 1 } we may assume that X = b L x for some x ∈ W . Choo se an b L -filtration ( F i ( Y )) n i =0 of Y and abbreviate F i = F i ( Y ). Assume that F i /F i − 1 ∼ = { l i } b L x i , for l 1 ≥ l 2 ≥ · · · ≥ l n and x i ∈ W . If the ima ge of f is co ntained in Y < 0 , let s ∈ { 0 , . . . , n } with F s = Y < 0 . Then G i = ( F i ⊕ 0 if i ≤ s , F i − 1 ⊕ { 1 } b L x if i > s . defines a n b L -filtration ( G i ) n +1 i =0 of the cone C ( f ) = Y ⊕ { 1 } b L x . Th us C ( f ) is in dg F ilt. Now assume im f 6⊂ Y < 0 . Let t ∈ { 0 , n } b e minimal with im f ⊂ F t . Then t ≥ 1 and l t = 0, since X is generated b y its degree zer o pa rt. The c ompo sition X = b L x f − → F t ։ F t /F t − 1 ∼ = b L x t 10 OLAF M. S CHN ¨ URER is non- zero and an isomorphis m by Lemma 5. Hence f induces a n iso morphism f : b L x ∼ − → im f , and its mapping cone V = (im f ) ⊕ { 1 } b L x is an acyclic dg submo dule of C ( f ). The inverse of the is o morphism im f ∼ − → F t /F t − 1 splits the short exact sequence ( F t − 1 , F t , F t /F t − 1 ). This implies that, for 1 ≤ i ≤ n , (7) F i ∩ ( F i − 1 + im f ) = ( F i − 1 if i 6 = t , F i if i = t . The filtration ( F i + V ) i of C ( f ) induces a filtra tion of C ( f ) /V with s uccessive sub q uotien ts F i + V F i − 1 + V ∼ ← F i F i ∩ ( F i − 1 + V ) ∼ ← F i F i ∩ ( F i − 1 + im f ) Using (7), we s ee that this filtr ation is an b L -filtration of C ( f ) /V , hence C ( f ) /V is in dg F ilt . The sho rt exact sequence ( V , C ( f ) , C ( f ) /V ) o f A -mo dules and the a cyclicity of V show that C ( f ) → C ( f ) / V is a quas i-isomorphism. Since b oth C ( f ) and C ( f ) /V are homo to pically pro jectiv e, it is an isomorphism in dgHot. Hence C ( f ) is isomor phic to an ob ject of dg F ilt. Reduction to the case λ ( X ) = 1 : This follows fr o m [BBD82, 1.3.10] but let me include the proo f for conv enience. Let U := F 1 ( X ) b e the first step of an b L -filtration o f X , u : U → X the inclusion and p : X → X/U the pro jection. Then the sho r t exact sequence 0 → U u − → X p − → X/U → 0 is A -split a nd defines a dis tinguished triangle ( U, X , X/U ) in dg Hot. W e apply the o cta hedr al axiom to the maps u and f a nd get the dotted ar r ows in the following commutativ e dia gram. { 1 } U { 1 } u   ? ? ? ? ? { 1 } X { 1 } p   ? ? ? ? ? { 1 } X/ U X/U " "   C ( f ◦ u )   ? ?      C ( f ) < < ? ?      X p ? ?      f   ? ? ? ? ? ? Y < < ? ?      U u ? ?      f ◦ u = = The four paths with the b ended a rrows are disting uis hed triangle s . By the leng th 1 case we may r eplace C ( f ◦ u ) b y an iso morphic ob ject e C ( f ◦ u ) of dg F ilt. So C ( f ) is is o morphic to the c o ne o f a morphism X /U → e C ( f ◦ u ) and hence, by induction, isomo rphic to an ob ject of dg F ilt.  5. t-Str ucture Using the equiv alence dg F ilt ⊂ dg P r a e, we define a b ounded t-str ucture (see [BBD82]) on dg P rae. As a co rollary we obtain that dg P rae coincides with dg P er. Let A as before satisfy (P1)-(P3). If M is a graded A 0 -mo dule, we define its support s upp M by supp M := { i ∈ Z | M i 6 = 0 } PERFECT DERIVED CA TEGORIES 11 If I ⊂ Z is a subset we define the full sub category (8) dg P er I := { M ∈ dg P er | supp H ( M ) ⊂ I } . By replacing dg P er b y dg P rae or dg F ilt, we define dg P rae I and dg F ilt I . W e write dg P er ≤ n , dg P er ≥ n , dg P er n instead of dg P er ( −∞ ,n ] , dg P er [ n, ∞ ) , dg P er [ n,n ] resp ectively , a nd similarly for dg P rae and dg F ilt. Remark 14 . In (8) w e could also write M L ⊗ A A 0 instead of M = M ⊗ A A 0 , since each ob ject of dg P er is homoto pica lly pro jective, hence homoto pically flat (here (? L ⊗ A A 0 ) : dgDer( A ) → dgDer( A 0 ) is the derived functor of the extension of scalars functor). In fact this is the cor rect definitio n if one works in dgDer instead of dgHotp (a s we do in the introduction). ✸ Remark 15. It is instructive to c o nsider an ob ject M of dg F ilt (and to ha ve a picture in mind as picture (6) in Remark 9). Note that the differential of M v anishes: This is expla ined at the end o f Remark 9 in a n example, but the ar gumen t generalizes immediately to a n arbitrary ob ject of dg F ilt. This implies tha t M and H ( M ) coincide a nd in particula r hav e the s a me suppor t. So M lies in dg P r ae ≤ n (in dg P rae ≥ n ) if and only if it is generated in degr ees ≤ n (in degr ees ≥ n ) as a graded A -mo dule. ✸ Theorem 16 (t-structur e) . L et A b e a dg algebr a satisfying (P1)-(P3). Then dg P ra e = dg P er( A ) and (dg P er ≤ 0 , dg P er ≥ 0 ) defines a b ounde d (in p articular n on- de gener ate) t-structur e on dg P er( A ) . Remark 17. In the sp ecial ca se that A is a p olyno mia l ring ( R [ X 1 , . . . , X n ] , d = 0) with g enerators in strictly pos itiv e even degrees and differential zer o, this t-s tructure coincides with the t-structure defined in [BL94, 11.4 ]. ✸ Pr o of. W e first prov e that (dg P rae ≤ 0 , dg P ra e ≥ 0 ) defines a b ounded t-str ucture o n dg P ra e and ha ve to ch eck the three defining pr op erties ([BBD82, 1 .3 .1]). (a) Hom dgHot ( X, Y ) = 0 for X ∈ dg P rae ≤ 0 and Y ∈ dg P rae ≥ 1 : By Theorem 13 w e ma y assume that X , Y are in dg F ilt. But then even a n y morphism of the underlying graded A -mo dules is zer o. (b) dg P ra e ≤ 0 ⊂ dg P r ae ≤ 1 and dg P r ae ≥ 1 ⊂ dg P r ae ≥ 0 : O b vio us . (c) If X is in dg P ra e there is a distinguished triang le ( M , X , N ) with M in dg P ra e ≤ 0 and N in dg P rae ≥ 1 : W e may a ssume that X is in dg F ilt. The graded A -submo dule X ≤ 0 of X is a n A -submo dule, since it app ears in any b L -filtration of X . The short exact seq uence X ≤ 0 → X → X/X ≤ 0 is A -split and hence defines a distinguished triangle ( X ≤ 0 , X , X / X ≤ 0 ) in dgHot. All terms of this triangle are in dg F ilt, X ≤ 0 is in dg P r ae ≤ 0 and X/X ≤ 0 in dg P ra e ≥ 1 . It is o b vio us that any ob j ect of dg F ilt is c o n tained in dg F ilt [ a,b ] , for some in teg ers a ≤ b . Hence our t-structure is bounded. W e cla im tha t dg P ra e ⊂ dg Hot( A ) is closed under ta king dir ect summands. This will imply that dg P ra e = dg P er. But our cla im is an application of the ma in result of [LC0 7]: A triangulated category with a bo unded t-structur e is Kar oubian (= idemp otent -s plit = idempotent complete) (cf. [BS01]).  12 OLAF M. S CHN ¨ URER The t-structure (dg P er ≤ 0 , dg P er ≥ 0 ) on dg P er yields truncatio n functors τ ≤ n and τ ≥ n ([BBD82, 1.3 .3 ]). F or ob jects M in the equiv alen t subca teg ory dg F ilt, w e ca n assume that these truncation functors ar e given by M 7→ M ≤ n and M 7→ M / M 0. Cho ose i ∈ I s uch that 0 ( M ≤ i ( M . Then we ha ve a short exact sequence 0 → M ≤ i → M → M / M ≤ i → 0 in dgMo d, and f induces an endomo rphism of this sequence. By induction w e hav e Fitting deco mp ositions for M ≤ i and M / M ≤ i , and Lemma 35 s hows that these decomp ositions yield a Fitting decomp osition of M .  Prop osition 37 . (a) L et M b e an inde c o mp osable obje ct of dg F ilt (cf. Cor ol- lary 34). Then any element of End dgMo d ( M ) is either an isomorphism or nilp otent, and the nilp otent en domorphisms form an ide al in E nd dgMo d ( M ) . The same statement holds for E nd dgHot ( N ) if N is an inde c omp osable obje ct of dg P er . In p articular b oth endomorphi sm rings ar e lo c al rings. (b) Each obje ct of dg F ilt de c omp oses into a finite dir e ct sum of inde c omp osables of dg F ilt . Same for dg P er . (c) dg F ilt and dg P er ar e Krul l-R emak-Schmidt c ate gories. Pr o of. The first statement of (a) is a standard consequence of the Fitting decom- po sition. The second statement then follows fr o m Prop ositio n 33 and Corolla ry 34. Let us prov e (b). If X is a n ob ject of dg F ilt with a direct sum decomp osition X ∼ = Y ⊕ Z in dg Mod, then Y and Z ar e in dg F ilt by Corollar y 32, and λ ( X ) = λ ( Y ) + λ ( Z ). Moreover λ ( X ) is finite, a nd λ ( X ) = 0 if and only if X = 0. This s hows that a n y ob ject X of dg F ilt ha s a finite direc t sum decomp osition X ∼ = M 1 ⊕ · · · ⊕ M n in dgMo d where all M i are in dg F ilt and indeco mp osa ble in dgMo d. All M i are als o indeco mposa ble in dg F ilt (Coro llary 34). The analog s ta temen t for dg P er is true thanks to the equiv alence dg F ilt ⊂ dg P er (Theorems 13 a nd 16). Part (c) is a consequenc e of (a ) and (b).  9. Koszul-Duality W e study the case that the dg algebra A is a Ko szul ring with differential zero. Under some finiteness conditions w e show that the heart of the t- s tructure on dg P er( A ) is g overned by the dual Koszul ring. This is a shadow of the usual Koszul e quiv alence (see Remark 42 b elow). W e ass ume that k = Z in this chapter. Let A b e a Ko szul ring (see [BGS96]). It g iv es rise to a dg algebra A = ( A, d = 0) with differential zero. This dg algebra satisfies the co nditions (P1)-(P3). Since A is Koszul there is a resolutio n P ։ A 0 , . . . → P − 2 d − 2 − − → P − 1 d − 1 − − → P 0 ։ A 0 , of the graded (right) A -mo dule A 0 = A / A + such that each P − i is a pro jective ob ject in gMo d( A ) and gener ated in deg ree i , P − i = P i − i A . Such a reso lution is unique up to unique isomorphis m. Note that multiplication defines an is omorphism (11) P i − i ⊗ A 0 A ∼ − → P − i 20 OLAF M. S CHN ¨ URER (it is s urjectiv e, splits b y the pro jectivity o f P − i , a nd any splitting is surjective, since it is an isomor phis m in degree i ). Since the differ en tial of A v anishes, we can view ea ch P − i as a dg A -mo dule with differential zero . The maps d − i are then morphisms of dg mo dules. On the graded A -mo dule K := K ( A ) := M n ∈ N { n } P − n = P 0 ⊕ { 1 } P − 1 ⊕ . . . we define a differ e ntial d K by (12) d K ( p 0 , p − 1 , p − 2 , . . . ) := (( − 1 ) − 1 d − 1 ( p − 1 ) , ( − 1) − 2 d − 2 ( p − 2 ) , ( − 1) − 3 d − 3 ( p − 3 ) , . . . ) . This defines an o b ject K ( A ) = ( K , d K ) of dgMo d( A ). I t can be seen as an iterated mapping cone of the morphisms ( − 1) − i d − i of dg mo dules (if there ar e o nly finitely many summands). (W e chose the signs from the pr o o f of Theorem 39. If we omit all the s ig ns ( − 1) − i in (12), we obtain an isomorphic dg mo dule.) W e say that P is of finite length if P − j = 0 fo r j ≫ 0, and that P ha s finitely generated comp onents if ea ch P − i is a finitely g enerated A -mo dule. These tw o conditions mean pr ecisely that K is a finitely generated A -mo dule. Prop osition 3 8. L et A b e a Koszu l ring, A = ( A, d = 0 ) , and P ։ A 0 a r esolution as ab ove. Assume that P is of finite length with fin itely gener ate d c omp onents. Then K ( A ) is in dg F ilt 0 and an inje ctiv e obje ct of t he he a rt ♥ . Pr o of. Since P has finite length, the filtr a tion 0 ⊂ P 0 ⊂ P 0 ⊕ { 1 } P − 1 ⊂ · · · ⊂ K by A -submo dules stabilize s after finitely ma n y steps. Since all sub quotients { i } P − i hav e differential zero and a re finitely generated by their degr e e zero part, they hav e an b L -filtration (use (11)). Hence K ( A ) is in dg F ilt 0 ⊂ ♥ . Since P ։ A 0 is a reso lution, the cohomology H ( K ( A )) is A 0 , sitting in degree zero. Hence K ( A ) is an injectiv e ob ject of ♥ b y Pr opo sition 2 4.  F or a po sitively graded ring A form the g raded ring E ( A ) := Ext • Mo d( A ) ( A 0 , A 0 ) of self-extensions of the right A -mo dule A 0 = A/ A + (w e consider self-ex tensions in Mo d( A ), no t in gMo d( A )). This ring is called the Ko szul dual ring of A (this definition is slightly differ e n t fr o m the definition in [BGS96]: they consider se lf- extensions of the left A -mo dule A 0 ). Theorem 39. L et A b e a Koszul ring, A = ( A, d = 0 ) , and P ։ A 0 a r esolution as ab ove. A ssume that P is of finite length with finitely gener ate d c omp onents. Then: (a) The functor K := Hom ♥ (? , K ( A )) : ♥ → (End ♥ ( K ( A )) - Mo d) op induc es an e quivalenc e K : ♥ ∼ − → (End ♥ ( K ( A )) - mo d) op b etwe en ♥ and the opp osite c ate gory of the c ate gory of finitely gener ate d (left) End ♥ ( K ( A )) -mo dules. (b) End ♥ ( K ( A )) is isomorphic to E ( A ) = Ext • Mo d( A ) ( A 0 , A 0 ) . Example 40. Let A = C [ X ] b e a p olynomial ring with X ho mogeneous of degr e e one. This is a Kosz ul ring with Ko szul dua l ring the ring of dual num ber s B := E ( A ) = C [ ε ] / ( ε 2 ). The r esolution P ։ A 0 is given by . . . → 0 → {− 1 } A X − → A ։ C , a nd K ( A ) is A ⊕ A a s a g raded A -mo dule with differential  0 − X 0 0  . Theorem 39 PERFECT DERIVED CA TEGORIES 21 says that End ♥ ( K ( A )) is isomorphic to B (which is of co urse eas y to chec k, cf . Example 28), and that there is an equiv alence ♥ ∼ − → ( B - mo d) op . More g enerally , we could ta k e A = S V the symmetric algebra o f some finite dimensional vector space V (with all elements of V in degree one). Then E ( A ) = V V ∗ is the exterior algebra on the dual space of V , a nd we obta in an equiv alence ♥ ∼ − → ( V V ∗ - mo d) op . ✸ Pr o of. Part (a ) follows from Prop osition 38, the fact that A = P 0 is an A -submo dule of K ( A ) and Pro pos ition 26. F or the pro of of (b) w e need the following genera l construction. Let S = ( S, d S ) = ( . . . → S i d i − → S i +1 → . . . ) and T = ( T , d T ) b e complexes in so me ab elian cate- gory . Then w e define a complex of ab elian groups H om ( S, T ) as follows: Its i -th comp onent is H om i ( S, T ) = Y s + t = i Hom( S − s , T t ) , and its differ e ntial is g iven b y d f = d T ◦ f − ( − 1) i f ◦ d S for f homogeneo us o f degree i . Note tha t E nd ( S ) := H om ( S, S ) with the obvious comp osition b ecomes a dg algebra. W e now prov e part (b): Let v : gMo d( A ) → Mod( A ) b e the “for getting the grading” functor. W e denote the induced functor from the categ ory of complexes in gMo d( A ) to that of complexes in Mod( A ) b y the same letter. If M and N are in gMo d( A ), the map M n ∈ Z Hom gMo d( A ) ( M , { n } N ) → Hom Mo d( A ) ( v ( M ) , v ( N )) (13) f = ( f n ) 7→ X f n is alwa ys injective. It is bijective if M is finitely generated as an A -mo dule. Since v ( P ) ։ v ( A 0 ) is a pr o jective resolution in Mo d( A ), the co homology of R := E nd ( v ( P )) is E ( A ). The i -th comp onent of R is R i = Y s ∈ Z Hom Mo d( A ) ( v ( P − s ) , v ( P − s + i )) = M s ∈ Z Hom Mo d( A ) ( v ( P − s ) , v ( P − s + i )) v ( P ) ha s finite length, ∼ ← M s,n ∈ Z Hom gMo d( A ) ( P − s , { n } P − s + i ) P − s finitely gen., (1 3), = M n ∈ Z Y s ∈ Z Hom gMo d( A ) ( P − s , { n } P − s + i ) P has finite length, = M n ∈ Z H om i gMo d( A ) ( P, { n } P ) . So if we define R n = H om gMo d( A ) ( P, { n } P ) fo r n ∈ Z , we ge t an isomor phism o f complexes R ∼ ← M n ∈ Z R n . 22 OLAF M. S CHN ¨ URER Note that H i ( R n ) = Ext i gMo d( A ) ( A 0 , { n } A 0 ) v anishes if i 6 = − n (P ro of (cf. [BGS96, Prop. 2.1.3]): Ev en the complex H om gMo d( A ) ( P, { n } A 0 ) v anishes in a ll degrees 6 = − n ). Hence we obtain E i ( A ) = H i ( R ) ∼ ← M n ∈ Z H i ( R n ) = H i ( R − i ) . In order to compute H i ( R − i ), we observe tha t R j − i = Y s ∈ Z Hom gMo d( A ) ( P − s , {− i } P − s + j ) = M s ∈ Z Hom gMo d( A ) ( P − s , {− i } P − s + j ) v anishes if j < i , since P − s = P s − s A is g enerated in degree s a nd ( {− i } P − s + j ) s = P s − i − s + j v anishes for s − i < s − j . This implies E i ( A ) = H i ( R − i ) = ker( R i − i → R i +1 − i ) , so E ( A ) = L i ∈ Z E i ( A ) is a subs et of L i ∈ Z R i − i . O n the o ther hand we hav e End ♥ ( K ( A )) = End dgMo d( A ) ( K ( A )) ⊂ End gMo d( A ) ( K ( A )) = M i,s ∈ Z Hom gMo d( A ) ( { s } P − s , { s − i } P − s + i ) = M i ∈ Z R i − i . In order to identify these tw o subsets of L i ∈ Z R i − i , let f = ( f i ) i ∈ Z = ( f i s ) i,s ∈ Z be an element of L i ∈ Z R i − i with f i s ∈ Hom gMo d( A ) ( { s } P − s , { s − i } P − s + i ) . Then it is easy to chec k tha t f ∈ E ( A ) if and only if d − s + i ◦ f i s − ( − 1 ) i f i s − 1 ◦ d − s = 0 for all i , s ∈ Z , if and only if f ∈ End ♥ ( K ( A )).  Remark 41. Since the graded A -mo dule K ( A ) is a (finite) dir ect sum, its e ndo- morphism r ing End gMo d( A ) ( K ( A )) consists of matrice s and ca n b e equipp ed with a “diag o nal” Z -gr ading such that the piece Hom gMo d( A ) ( { s } P − s , { t } P − t ) has deg ree s − t . Thes e ma trices are not upper triang ular in g e neral. The par- ticular form of K ( A ) implies that E nd ♥ ( K ( A )) ⊂ E nd gMo d( A ) ( K ( A )) is a g r aded subalgebra . In fact it co nsists of upp er triangular matrices, i. e . it is positively (= non-negatively) gra ded: This can b e directly de duce d from the pro of of Theorem 39, since the extension algebra E ( A ) is p ositively gra de d. A more ge ne r al a rgument rests upon Remark 29: The explicit description of the so cle filtration of an ob ject of dgFlag there and the particular form of K ( A ) show tha t so c i +1 K ( A ) = P 0 ⊕ · · · ⊕ { i } P − i . PERFECT DERIVED CA TEGORIES 23 Now endomo rphisms of K ( A ) in ♥ resp ect the so cle filtration and are therefor e upper triangular. ✸ Remark 42. A. B eilinson, V. Ginzburg , and W. So ergel prov e an equiv alence o f triangulated catego ries K : D b (gmo d( B )) ∼ − → D b (gmo d( A )) for B a Koszul ring satisfying some finiteness conditions and A = E ( B ) its Koszul dual ring (see [BGS9 6, Thm. 1 .2.6]; we hav e adapted their s tatemen t to our setting, in par ticula r we use our definition of the Kosz ul dual ring using right mo dules). The finiteness c o nditions are: B is a finitely ge nerated B 0 -mo dule from the right and from the left, A = E ( B ) is right No etherian, and (this condition seems to be missing in [BGS96]) the r ight B - module ∗ B := Hom Mo d( B 0 ) ( B , B 0 ) is finitely generated); here D b (gmo d( B )) is the bounded derived category of the catego ry of finitely gener ated g raded left B -mo dules, and similarly for D b (gmo d( A )). The sta nda rd t-structure o n D b (gmo d( B )) with hea rt gmo d( B ) corres ponds (un- der this equiv alence) to some non-standa r d t-structure o n D b (gmo d( A )). Hence gmo d( B ) is equiv alent to the heart of this non- standard t-structure. Now assume that the K o szul r ing A = E ( B ) s atisfies the finiteness conditions of Theorem 39 and consider A = ( A, d = 0 ). (F or example, one could take the exterior a lg ebra B = V V of a finite dimensional vector space V , a nd A = S V ∗ the symmetric alge bra on the dual space of V .) There is a natural forgetful functor D b (gmo d( B )) → D b (mo d( B )) (induced by gmo d( B ) → mo d( B )), and ther e is a functor D b (gmo d( A )) → dg Per ( A ). F r om the ab ov e equiv alence b et ween the hearts it is reasona ble to exp ect (as a sha dow of K oszul duality) a n equiv alence mo d( B ) ∼ = ♥ , where ♥ is the heart o f our t-structure o n dgPer( A ). 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