Koszul differential graded algebras and BGG correspondence

K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPON DENCE J.-W. HE AND Q.-S. WU Abstract. The concept of Koszu l differen tial graded algebra (Koszul DG alge- bra) is introduced. Koszul DG algebras exist extensiv ely , and hav e nice prop er- ties similar to the classic Koszul algebras. A DG v ersion of the Koszul duality is prov ed. When the Koszul DG algebra A is AS-regular, the Ext-algebra E of A is F rob enius. In this case, similar to the classical BGG correspondence, there is an equiv alence b etw een the stable category of finitely generated left E -mo dules, and the quotien t triangulated category of the full triangulated subcategory of the derive d category of ri gh t DG A -mo dules consisting of all compact DG m odules modulo the full triangulated sub category consisting of al l the right DG mo dules with finite dimensional cohomology . The classi cal BGG corresp ondence can be derive d from the DG version. Introduction In his b o ok [Ma] Manin presented an op en question: How to gener alize the Ko szulity to differential g raded (DG for short) algebra s? A ttempts hav e b een made by s e veral authors as in [PP] a nd [Be]. In their terminology , a DG a lgebra is said to b e Ko szul if the underlying gra de d algebra is Koszul. Koszul DG algebras in their sense are applied to dis cuss configura tion spa ces. In this pap er, we take a differe n t p oint of vie w . Let k b e a field. A co nnec ted DG algebra over k is a p os itively gr aded k -alg ebra A = ⊕ n ≥ 0 A n with A 0 = k suc h that there is a differential d : A → A o f degr e e 1 which is also a graded de r iv atio n. A connected DG algebra A is s aid to b e a Koszul D G algebr a if the minimal semifree resolution of the tr ivial DG mo dule A k has a semifree basis co nsisting of ho mogeneous elements o f degr ee zero (Definition 2.1 ). Our definition o f Kos zul DG alg ebra is a natural ge ner alization o f the usual Kos zul alg ebra. As we will s e e in Section 2, a connected gra ded algebra regar ded as a DG a lg ebra with zer o differential is a Ko szul DG algebra if and o nly if it is a Ko szul algebr a in the usual sense. Examples of Koszul DG a lgebras ca n b e found in v ar ious fields. F or example, let M b e a co nnected n -dimensional C ∞ manifold, a nd let ( A ∗ ( M ) = L n i =0 A i ( M ) , d ) b e the de Rham complex of M , then ( A ∗ ( M ) , d ) is a co mm utative DG algebr a and by de Rham theore m ([M]) the 0- th cohomo lo gy group H 0 ( A ∗ ( M )) ∼ = R . Hence the DG alg ebra A ∗ ( M ) has 2000 Mathematics Subje ct Classific ation. Pri mary 16E45, 16E10. Key wor ds and phr ases. Differential graded algebra, Derived category , Koszul algebra, BGG correspondence. 1 2 J.-W. HE AND Q.-S. WU a minimal mo del A ([KM]) or Sulliv a n mo del ([FHT2]), w hich is cer ta inly a connected DG algebra . If the manifold M has so me further prop erties (e.g., M = T n the n - dimensional torus), then the de Rham cohomolo gy alg ebra H ( A ∗ ( M )) is a Ko szul algebra. Hence the cohomo logy algebra o f its minimal mo del (or Sulliv an mo del) A is Kosz ul as A is quas i-isomorphic to A ∗ ( M ). Then A is a K oszul DG algebra by Prop ositio n 2 .3. Mor e exa mples of Koszul DG a lgebra will b e given in Section 2. In fact, we will s ee that a ny Koszul a lgebra c a n b e viewed a s the cohomolo gy algebra of some K oszul DG a lgebra. Bernstein-Gelfand-Gelfand in [BGG] established an equiv a lence b etw een the sta- ble categor y of finitely ge ner ated g raded mo dules over the exterio r a lgebra V V with V = k x 0 ⊕ k x 1 ⊕ · · · ⊕ k x n , and the b ounded derived categor y of cohere n t sheaves on the pro jective space P n . This e q uiv alence is now ca lle d the BGG c orr esp ondenc e . BGG co r resp ondence has b een generalize d to no ncommutativ e pr o jective g eometry by several authors . Let R b e a (nonco mm utative) Kos z ul alg e bra. If R is AS-r egular, Jørgens en prov ed in [Jo ] that there is a n equiv a lence b etw een the s table catego r y ov er the graded F rob enius alg e bra E ( R ) = E x t ∗ R ( k , k ) and the derived catego ry of the noncommutativ e ana logue QGr( R ) of the quasi- coherent sheaves ov er R ; Mart ´ ınez Villa-Saor ´ ın prov ed in [MS] that the stable catego ry of the finite dimensional mo d- ules over E ( R ) is equiv a lent to the b ounded derived ca tegory of the noncommutativ e analogue qgr R of the coherent sheav es ov er R . Mor i in [Mo] prov ed a similar version under a mor e g eneral condition. O ne of our purp ose s in this pa per is to es tablish a DG version of the BGG cor resp ondence. In some sp ecial case, the DG version of the BGG corres p o ndenc e coincide s with the classical o ne as establis hed in [BGG] and [MS]. The pap er is orga nized as follows. In Section 1, we g ive so me prelimina r ies and fix some notations for the pap er. In Section 2, we first pro p o s e a definition for K oszul DG a lgebras (Definition 2.1), then give some examples a nd discuss some ba sic prop er ties of Koszul DG alg ebras. F o r any connected DG algebra A , we prove that if the cohomolo g y alg ebra H ( A ) is Koszul in the usual sense, then A is a Kos z ul DG alg e bra (Pr o p o sition 2.3). The conv erse is not true in general. In Section 3 , we discuss the structure of the E xt-algebra s of Koszul DG a lgebras. F o r any K o szul DG algebra A , we prove that the E xt-algebra E = Ext ∗ A ( A k , A k ) of A is an aug ment ed, filtered alg ebra. Mor eov er, if H ( A ) is a Koszul algebr a, then the asso ciated graded algebr a g r ( E ) is isomor phic to the dual Koszul alge br a ( H ( A )) ! (Theorem 3.3). If further, A k is co mpact, then E is a finite dimensional lo ca l alg ebra; when H ( A ) is Ko szul, the filtratio n on E is exa ctly the J acobson radical filtration (Theorem 3.5). Using ba r and cobar constructions , we prove the following version of the Koszul duality on the E x t-algebra s (Theor em 3.8): Theorem [Koszul Duality on Ext-a lgebra]. Let A b e a Kosz ul DG algebr a a nd E b e its Ex t- a lgebra. If A k is compact, then E xt ∗ E ( E k , E k ) ∼ = H ( A ). K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 3 As a co rollar y , we show that the Ext-a lgebra of a Ko szul DG algebra A with A k compact is s trongly qua si-Koszul ([GM ]) if a nd only if its cohomolo gy a lgebra H ( A ) is a Koszul algebra . In Section 4, by using Lef` evre-Haseg aw a’s theorem in [Le , Ch.2] (see Theo rem 4.1), we esta blish a DG version of K oszul equiv alence and duality (Theorems 4.4 a nd 4.7). Theorem [K oszul equiv a lence and dua lit y]. Let A b e a Ko s zul DG a lgebra and E be its Ext-a lg ebra. Suppo se A k is c o mpact. Then there is a n equiv ale nce of triangu- lated categor ies b etw een D + ( E ) and D + dg ( A op ); a nd there is a duality of tria ngulated categorie s b etw ee n D b (mo d- E op ) a nd D c ( A op ). Here D + ( E ) is the the derived catego ry of b ounded b e low co chain complex e s o f left E -mo dules; D b (mo d- E ) (resp. D b (mo d- E op )) is the b ounded derived categ o ry of finitely gener ated left (resp. right) E -mo dules; D dg ( A op ) (resp. D + dg ( A op )) is the derived categor y of right DG A -mo dules (res p. b ounded b elow rig ht DG A -mo dules ), and D c ( A op ) is the full triangula ted sub category of D + dg ( A op ) consisting of all the compact o b jects. As a corolla ry , we s how that each finite dimensional lo cal algebr a with residue field k can b e viewed as the Ext-a lgebra o f some Koszul DG alg ebra. As a result, we see that the cohomolog y algebr a o f a Ko szul DG alg ebra may not b e Koszul. In Section 5 , we in tro duce the concept of AS-regula r DG algebr a. B ased on the r e- sult obtained in Section 4 , we s how that the Ext-alge br a of an AS-regular Koszul DG algebra is F rob enius (Prop ostion 5.4 and Corolla ry 5.5). W e then prov e a corresp on- dence b etw e e n some quotient catego ry of the derived catego ry of a Ko szul AS-r egular DG algebr a and the stable categor y of its E x t-algebra , which is similar to the classical BGG corres po ndence (Theorems 5.7 a nd 5.8). Theorem [BGG Co rresp ondence]. Let A b e a Koszul DG AS-regula r a lgebra with Ext-algebr a E = Ext ∗ A ( k , k ). Then there is a duality of triangulated c ategories b et ween mo d- E op and D c ( A op ) / D f d ( A op ) and a n eq uiv alence of triangulated categor ies b etw een mo d- E and D c ( A op ) / D f d ( A op ) . Here mo d- E op (resp. mo d- E ) is the stable ca tegory of finitely ge ne r ated right (r esp. left) E - mo dules . D f d ( A op ) is the full triangula ted sub categor y of the derived ca tegory of right DG A -mo dules consisting of all the DG mo dules with finite dimensiona l c o - homology . The r esults ab ove ar e generalized to Adams connec ted DG algebr as in Section 6. W e show that the nonco mm utative BGG corr esp ondence b etw een the tria ng ulated categorie s es tablished in [J o] and [MS] can b e deduced fro m the BGG corr esp ondence on Adams c o nnected DG algebras (Theor em 6.8 ). 4 J.-W. HE AND Q.-S. WU 1. Preliminaries Throughout, k is a fie ld and all algebra s ar e k -algebra s; una dorned ⊗ means ⊗ k and Hom mea ns Hom k . By a gra de d algebr a we mean a Z -gr a ded a lgebra. An augmente d gra ded a lg ebra is a gr aded alg ebra A with an augmentation map ε : A → k which is a gra ded algebra morphism. A p ositively gra ded alge bra A = L n ≥ 0 A n with A 0 = k is called a c onne cte d gr aded algebr a. L e t M and N b e graded A -mo dules. Hom A ( M , N ) is the set o f all graded A -mo dule morphisms. If L is a graded vector space, L # = Hom ( L, k ) is the gr aded vector spa c e dual. By a (co chain) D G algebr a we mean a gra ded algebra A = L n ∈ Z A n with a differ- ent ial d : A → A of deg ree 1, which is also a gr aded deriv ation. An augment e d D G algebr a is a DG a lgebra A such that the under lying g raded alg ebra is augmented with augmentation map ε : A → k s a tisfying ε ◦ d = 0. ker ε is called the augmented idea l of A . A c onne cte d DG a lgebra is a DG algebr a such that the underlying gr aded algebr a is connected. An y grade d alg ebra ca n b e viewed a s a DG alg ebra with differ ent ial d = 0; in this case it is called a DG algebra with trivial differential. Let ( A, d A ) be a DG a lgebra. A left differ en tial gr ade d mo dule over A (DG A - mo dule for s hort) is a left gr aded A -mo dule M with a differential d M : M → M of degree 1 such that d M satisfies the graded L e ibnitz r ule d M ( a m ) = d A ( a ) m + ( − 1) | a | a d M ( m ) for a ll gr aded elements a ∈ A, m ∈ M . A right DG mo dule over A is defined similar ly . W e denote A op as the opp osite DG algebra of A , whose pro duct is defined as a · b = ( − 1) | a |·| b | ba for all graded elements a, b ∈ A . Right DG mo dules ov er A can b e identified with DG A op -mo dules. Dually , by a (co chain) D G c o algebr a we mean a gr aded coa lgebra C = L n ∈ Z C n with a differential d : C → C of degr e e 1, which is als o a g raded co deriv ation. A c o augmente d DG c o algebr a is a DG co a lgebra C with a gra ded coalgebr a ma p η : k − → C , ca lle d coaugmentation ma p, such that d ◦ η = 0 . If C is a coaugmented DG coalgebr a, then C has a decomp osition C = k ⊕ ¯ C , where ¯ C is the kernel of the counit ε C , which is isomor phic to the co kernel e C of η . There is a copro duct ¯ ∆ : ¯ C → ¯ C ⊗ ¯ C defined by ¯ ∆( c ) = ∆( c ) − 1 ⊗ c − c ⊗ 1, such that ( ¯ C , ¯ ∆) is a co a lgebra without counit. ∆ induces a copr o duct e ∆ over e C . ( ¯ C , ¯ ∆) a nd ( e C , e ∆) are isomorphic a s coalg ebras. A coaugmented DG coalg ebra C is c o c omplete if, for any ho mogeneous element x ∈ ¯ C , there is an int eger n such that ¯ ∆ n ( x ) = ( ¯ ∆ ⊗ 1 ⊗ n − 1 ) ◦ · · · ◦ ( ¯ ∆ ⊗ 1 ) ◦ ¯ ∆( x ) = 0. A right DG C -como dule N is a gr aded right C -como dule with a graded co deriv atio n d N (i.e. ρ N d N = ( d N ⊗ 1 + 1 ⊗ d C ) ρ N ) of degr ee 1. A co co mplete right DG C -co mo dule is defined simila rly ([Le]). K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 5 F o r the s ta ndard facts ab out DG mo dules, semifree mo dules and semifree re s olu- tions of DG mo dules, etc, refer to [AFH] and [FHT2]. A DG A -mo dule M is said to be b ounde d b elow if M n = 0 for n ≪ 0. Let A b e a DG algebr a, M and N b e left DG A -mo dules, W b e a right DG A -mo dule. F ollowing [KM] a nd [W e ], the differential Ext and T o r a re defined as Ext n A ( M , N ) = H n (RHom A ( M , N )) and T o r n A ( W , M ) = H n ( W ⊗ L A M ) for a ll n ∈ Z . Let s b e the susp ensio n map (shifting map) with ( sX ) n = X n − 1 for any co chain complex X . Thus s i : X → s i X is of degr ee i for any i ∈ Z . 1.1. Bar constructions. Let A be a n aug ment ed DG a lg ebra w ith differential d . Let I ( A ) = · · · ⊕ A − 1 ⊕ ¯ A 0 ⊕ A 1 ⊕ · · · b e its a ugmented ideal. Let B ( A ) = T ( s − 1 ( I ( A ))) = k ⊕ s − 1 ( I ( A )) ⊕ s − 1 ( I ( A )) ⊗ s − 1 ( I ( A )) ⊕ [ s − 1 ( I ( A ))] ⊗ 3 ⊕ · · · . The homogeneo us element s − 1 a 1 ⊗ s − 1 a 2 ⊗· · ·⊗ s − 1 a n of B ( A ) is written as [ a 1 | a 2 | · · · | a n ] for ho mogenous ele ments a 1 , · · · , a n ∈ I ( A ). The co pr o duct ∆ : B ( A ) → B ( A ) ⊗ B ( A ) is defined by ∆([ a 1 | a 2 | · · · | a n ]) = 1 ⊗ [ a 1 | a 2 | · · · | a n ] + [ a 1 | a 2 | · · · | a n ] ⊗ 1 + P 1 ≤ i ≤ n − 1 [ a 1 | · · · | a i ] ⊗ [ a i +1 | · · · | a n ] , and define a counit ε : B ( A ) → k by ε | k = 1 k and ε ([ a 1 | · · · | a n ]) = 0 for n ≥ 1 . It is easy to check that ( B ( A ) , ∆ , ε ) is a co augmented gra ded coalge bra. Define δ 0 : B ( A ) → B ( A ) by δ 0 ([ a 1 | · · · | a n ]) = − n X i =1 ( − 1) ω i [ a 1 | · · · | d ( a i ) | · · · | a n ] , and define δ 1 : B ( A ) → B ( A ) b y δ 1 ([ a 1 ]) = 0 and δ 1 ([ a 1 | · · · | a n ]) = n X i =2 ( − 1) ω i [ a 1 | · · · | a i − 1 a i | · · · | a n ] , where ω i = P j 0 . A DG A -mo dule is semifr e e if and only if it has a semibas is ([AFH, P rop osition 2.5]). W e now give a definition of the K o szulity for DG algebra s. Definition 2.1. A connected DG algebra A is called a left Koszu l DG algebra if the trivial DG mo dule A k has a minimal semifree resolution ε : P → A k such that the semibasis of P consists of ele ments of degr ee zero . Rig ht Koszul DG algebra is defined similar ly . The next pro po sition tells us that a connected DG alg ebra is left Ko szul if and only it is right Kos zul. Prop ositio n 2.2. L et A b e a c onne cte d DG algebr a. The fol lowing stat emen t s ar e e quivalent. (i) A is a left Koszul DG algebr a; (ii) Ext n A ( A k , A k ) = 0 for al l n 6 = 0 ; K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 9 (iii) T or n A ( k A , A k ) = 0 for al l n 6 = 0 ; (iv) A is a right Koszul DG algebr a. Pr o of. Using the minimal semifree reso lution of the trivial mo dule.  Let R b e a co nnected graded a lgebra. Supp os e that · · · → Q n → · · · → Q 1 → Q 0 → R k → 0 is a minimal free resolution of the trivia l mo dule R k . If we consider R as a DG algebra with triv ia l differential, and view · · · → Q n → Q n − 1 → · · · as a do uble co mplex by using the sign tr ick, then the asso cia ted total complex (that is , Q 0 ⊕ Q 1 [ − 1] ⊕ · · · ⊕ Q n [ − n ] ⊕ · · · ) is a minimal semifree res olution of the trivia l DG mo dule R k . Therefor e R is a Koszul algebra in the usua l sense if and only if it is a K oszul DG a lg ebra with trivial differential. Prop ositio n 2.3. L et A b e a c onne cte d D G algebr a. If t he c ohomolo gy algebr a H ( A ) is a Koszul algebr a, then A is a Koszul DG algebr a. Pr o of. W e use the Eilenberg- Mo ore sp ectral sequence ([FHT2], [KM]) E p,q 2 = T o r p,q H ( A ) ( k , k ) = T o r H ( A ) − p ( k , k ) q = ⇒ T or p + q A ( k , k ) , where q is the grading induced by the gr adings on H ( A ) a nd H ( A ) k . This is a conv er- gent b ounded be low cohomolo gy sp ectral sequence. Since H ( A ) is a Koszul algebr a , E p,q 2 = 0 for p + q 6 = 0. Thus T or n A ( k , k ) = 0 for all n 6 = 0.  Before pro ceeding to discuss further prop erties of Koszul DG alg ebras, we give so me examples here. Example 2. 4. L et A b e the gr ade d algebr a k h x, y i / ( y 2 , y x ) , wher e | x | = | y | = 1 . L et d ( x ) = xy and d ( y ) = 0 . Then d induc es a differ ential d over A and A is a DG algebr a. It is not har d to che ck that H ( A ) = k ⊕ k y , which is a Koszul algebr a. Henc e by Pr op osition 2.3, A is a Koszul DG algebr a. The following example shows that Koszul DG algebr as with nontrivial differ ent ials exist extensively . Example 2. 5. Each Koszul algebr a R is the c ohomolo gy algebr a of a c ert ain Koszul DG algebr a with nontrivial differ ential. In fact, R c an b e viewe d as a c onne cte d DG algebr a with a trivial differ ential. Then by L emma 1.4 , Ω B ( R ) is quasi-isomorphic to R as DG algebr as. Henc e H (Ω B ( R )) ∼ = H ( R ) ∼ = R . Cle arly Ω B ( R ) is a c onne cte d DG algebr a with a nontrivial differ ential. By Pr op osition 2.3, Ω B ( R ) is a K oszu l DG algebr a. The converse of Prop o s ition 2.3 is not true, as we will see at the end of Section 4. How ev er we hav e the following prop os itio n. 10 J.-W. HE AND Q.-S. WU Prop ositio n 2.6. L et A b e a Koszul DG algebr a. If the glob al dimension gldim H ( A ) ≤ 2 , then H ( A ) is a Koszul algebr a. Pr o of. Let Q • : 0 − → Q 2 − → Q 1 − → H ( A ) − → k − → 0 be a minimal free res olution o f the trivial mo dule H ( A ) k . It is direct to chec k that in this cas e the E ilenberg-Mo or e reso lution ([FHT2], [KM]) of the trivial DG mo dule A k arising from Q • can b e chosen to b e minimal. If A is Koszul, then the minimal free resolution Q • m ust b e linea r a nd hence H ( A ) is Ko s zul.  The Koszulity o f DG a lgebras is pres erved under tak ing qua si-isomor phisms. Lemma 2.7. [KM, Pro p o s ition 4.2] L et A and B b e D G algebr as. If t her e is a quasi-isomorphi sm of DG algebr as f : A − → B , then t he r est riction of f induc es an e quivalenc e of triangulate d c ate gories f ∗ : D ( B ) − → D ( A ) with the inverse functor B ⊗ L A − . The same is tru e for D ( B op ) and D ( A op ) .  Prop ositio n 2. 8. L et A and B b e c onne cte d DG algebr as. Supp ose that ther e is a quasi-isomorphi sm of DG algebr as f : A − → B . If A (r esp. B ) is a Koszul DG algebr a, then so is B (r esp. A ). Pr o of. If A is a Koszul DG algebra , then Ext n A ( A k , A k ) = 0 for a ll n 6 = 0, that is, Hom D ( A ) ( A k , A k [ n ]) = 0 for all n 6 = 0. Hence Hom D ( B ) ( B k , B k [ n ]) ∼ = Hom D ( A ) ( f ∗ ( B k ) , f ∗ ( B k )[ n ]) = Hom D ( A ) ( A k , A k [ n ]) = 0 for a ll n 6 = 0. Hence Ext n B ( B k , B k ) = 0 for all n 6 = 0, a nd B is Ko s zul.  3. The Ex t-algebra of a Koszul D G algebra In this section, we study the structure of the Ext-a lg ebra of a Ko szul DG algebra . W e pr ov e a version of the Koszul duality on Ext- a lgebra for Kosz ul DG algebr as. Let P b e a semifree DG A -mo dule with a semifre e filtration 0 ⊆ P (0) ⊆ P (1) ⊆ · · · ⊆ P ( n ) ⊆ · · · . W e may adjust the semifree filtration of P to get a st andar d filtr ation of P a s in the following. Let E be a semifree basis o f P . Then as a gr aded A -mo dule, P = A ⊗ k E , where k E = ⊕ e ∈ E k e is a graded k -vector s pace. Set inductively , V ≤ 0 = V (0 ) = { v ∈ k E | d ( v ) = 0 } and F (0) = A ⊗ V (0) ⊆ P , V ≤ 1 = { v ∈ k E | d ( v ) ∈ F (0) } and F (1) = A ⊗ V ≤ 1 ⊆ P, V ≤ n = { v ∈ k E | d ( v ) ∈ F ( n − 1) } and F ( n ) = A ⊗ V ≤ n ⊆ P. Let V ( n ) be a s ubspace of V ≤ n such that V ≤ n = V ≤ n − 1 ⊕ V ( n ). Then for any 0 6 = v ∈ V ( n ), d ( v ) ∈ F ( n − 1 ) \ F ( n − 2) . K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 11 Obviously , ∪ n ≥ 0 F ( n ) = P a nd F ( n ) /F ( n − 1) ∼ = A ⊗ V ( n ) is a free DG mo dule ov er a basis of co cycles. Hence 0 ⊆ F (0) ⊆ F (1) ⊆ · · · ⊆ F ( n ) ⊆ · · · is a new semifree filtr ation on P , which is c alled the standar d semifr e e filtr ation o f P asso ciated to the semibasis E . As we will see in nex t example, the standar d semifree filtration de p ends on the choice of the semibasis. Example 3.1. Let A b e a connected DG algebra such that there is an element a ∈ A 1 with d A ( a ) 6 = 0 . Let P = Ae 0 ⊕ Ae 1 as a graded free A -mo dule with deg ( e i ) = i for i = 0 , 1. Define d ( e 0 ) = 0 and d ( e 1 ) = d A ( a ) e 0 . Then P is a semifree DG A - mo dule with a semifree filtration P : 0 ⊆ P (0) ⊆ P (1) = P where P (0 ) = Ae 0 and P (1) = Ae 0 ⊕ Ae 1 = Ae 0 ⊕ A ( e 1 − a e 0 ) = P . Then E = { e 0 , e 1 } and E ′ = { e 0 , e 1 − ae 0 } are t wo semibas is of the semifree DG mo dule P . Asso ciated to the s emibasis E , the standard filtration is the o riginal o ne P : 0 ⊆ P (0) ⊆ P (1) = P. Asso ciated to the semiba sis E ′ , the standa r d filtra tio n is F : 0 ⊆ F (0) = Ae 0 ⊕ A ( e 1 − a e 0 ) = P . The main reason to in tro duce the standar d filtration is that DG morphism pr eserves the standard filtra tion as in the following lemma, which is needed in the pr o of of Theorem 3.3. Lemma 3 . 2. L et A b e a c onne cte d DG algebr a, M and N b e minimal semifr e e DG A -mo dules with the s t andar d filtr ation 0 ⊆ M (0) ⊆ M (1) ⊆ · · · and 0 ⊆ N (0) ⊆ N (1) ⊆ · · · r esp e ctively. If t he semib asis of M and N c onsist of elements of de gr e e 0, then any DG mo dule m orphism f : M → N pr eserves the filt r ation. Pr o of. Assume that ther e ar e gra ded vector spaces U ( i ) a nd W ( i ) fo r i ≥ 0 such that M ( i ) / M ( i − 1) = A ⊗ U ( i ) and N ( i ) / N ( i − 1) = A ⊗ W ( i ). F or any u ∈ U (0), f ( u ) ∈ ⊕ i ≥ 0 W ( i ) and d ( f ( u )) = 0 since f is a co chain map. Let f ( u ) = v i 0 + · · · + v i t with 0 6 = v i j ∈ W ( i j ) for 0 ≤ j ≤ t a nd i 0 < i 1 < · · · < i t . Supp ose that t ≥ 1 . By the definition of standard filtration of N , d ( v i j ) ∈ N ( i j − 1) and d ( v i j ) / ∈ N ( i j − 2). How ev er, 0 = d ( f ( u )) = d ( v i 0 + · · · + v i t ) = d ( v i 0 + · · · + v i t − 1 ) + d ( v i t ) . It follows that d ( v i t ) = − d ( v i 0 + · · · + v i t − 1 ) ∈ N ( i t − 1 − 1 ) ⊆ N ( i t − 2 ), a co nt radiction. Hence t = 0 and f ( u ) ∈ W (0), which implies f ( M (0)) ⊆ N (0). Now supp ose f ( M ( n )) ⊆ N ( n ). Let ¯ M = M / M ( n ) and ¯ N = N / N ( n ). Then f induces a DG morphism ¯ f : ¯ M → ¯ N . ¯ M a nd ¯ N ar e minimal semifree mo dules with 12 J.-W. HE AND Q.-S. WU standard semifr e e filtration ¯ M (0) = M ( n + 1) / M ( n ) ⊆ ¯ M (1) = M ( n + 2) / M ( n ) ⊆ · · · and ¯ N (0) = N ( n + 1) / N ( n ) ⊆ ¯ N (1) = N ( n + 2 ) / N ( n ) ⊆ · · · resp ectively . B y the previo us narra tives, we hav e ¯ f ( ¯ M (0)) ⊆ ¯ N (0), which in turn implies f ( M ( n + 1)) ⊆ N ( n + 1 ).  Theorem 3. 3 . L et A b e a Koszul DG algebr a. Then (i) the Ext-algebr a E = Ext ∗ A ( A k , A k ) of A is an augmente d algebr a; (ii) ther e is a filtr ation F : E = F 0 ⊇ F 1 ⊇ · · · ⊇ F n ⊇ · · · on E such that E is a filt er e d algebr a. Mor e over, if H ( A ) is a Koszul algebr a, then the asso ciate d gr ade d algebr a g r F ( E ) is isomorphi c to t he dual Koszul algebr a ( H ( A )) ! . Pr o of. (i) and the first par t of (ii) may b e prov ed by us ing the bar constructio n of A . W e g ive a direct pro of here for later use. Let ε : P − → A k b e a minimal semifree resolutio n of the triv ial DG mo dule A k . Suppo se that 0 ⊆ P (0) ⊆ P (1) ⊆ · · · ⊆ P ( n ) ⊆ · · · is a standa rd semifree filtra tion of P ass o ciated to so me s emibasis. W e hav e g raded vector spa ces V (0) , V (1) , · · · , V ( n ) , · · · such that P (0) = A ⊗ V (0) and P ( n ) /P ( n − 1) = A ⊗ V ( n ) for all n ≥ 1. By the minimality o f P , it is easy to see that V (0) = k . Since A is Ko szul, E = E xt ∗ A ( A k , A k ) = E xt 0 A ( A k , A k ) = Y i ≥ 0 V ( i ) ∗ = k ⊕ Y i ≥ 1 V ( i ) ∗ . Define a decr easing filtra tion F on E by F : F 0 = E and F n = Y i ≥ n V ( i ) ∗ for n ≥ 1 . W e claim that E is a filtered algebra with this filtration. F or a ny x ∈ F n = Q i ≥ n V ( i ) ∗ and y ∈ F m = Q i ≥ m V ( i ) ∗ , we still use x to denote the corr esp onding DG mo dule morphism x : P / P ( n − 1) − → A k , a nd y the cor resp onding DG mo dule mor phism y : P / P ( m − 1 ) − → A k . Since P /P ( n − 1) is semifree, there is a DG mo dule morphis m f x : P /P ( n − 1) − → P such tha t ε ◦ f x = x ([AFH, Lemma 6.5.3 ]. Let g b e the comp osition P π − → P /P ( n − 1) f x − → P , where π is the natura l pr o jection map. By Le mma 3 .2, f x preserves the filtratio n, hence g ( P ( n − 1)) = 0 and g ( P ( n + i )) ⊆ P ( i ) for all i ≥ 0 . Let h b e the c o mpo sition P π − → P /P ( m − 1) y − → k . K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 13 By definition, the pr o duct y · x in the a lgebra E is the restr iction of h ◦ g to L i ≥ 0 V ( i ). Since h ◦ g ( P ( n + m − 1)) ⊆ h ( P ( m − 1)) = 0, it follows that y · x ∈ Q i ≥ n + m V ( i ) ∗ . Hence E is a filtered a lgebra with filtra tion { F n } . Define a map ǫ : E − → k by ǫ | k = id k and ǫ | F 1 = 0. Since F 1 is an ideal, ǫ is a n algebra morphism, hence an augmentation map. (i) is prov ed. Now we prove the second part of (ii). Suppo s e that H ( A ) is a Ko szul algebr a. The trivial H ( A )-mo dule H ( A ) k has a linear pro jective r esolution · · · − → H ( A ) ⊗ V ′ ( n ) δ n − → · · · δ 2 − → H ( A ) ⊗ V ′ (1) δ 1 − → H ( A ) ⊗ V ′ (0) δ 0 − → H ( A ) k − → 0 . The Eilenberg - Mo ore r esolution ([FHT2, Pr op osition 20.1 1]) P ′ of the DG mo dule A k ar ising from the pr evious resolutio n o f H ( A ) k is minimal. Hence P ∼ = P ′ as DG A -mo dules since A is connected and then V ( i ) ∼ = V ′ ( i ) as vector spac e s for all i ≥ 0. F o r conv enience, we identify V ( i ) with V ′ ( i ) for all i ≥ 0 a nd P with P ′ . By the construction of the filtration F on E , we get F n /F n − 1 ∼ = V ( n ) ∗ for a ll n ≥ 0. Hence we hav e (1) g r F ( E ) ∼ = M n ≥ 0 V ( n ) ∗ ∼ = Ext ∗ H ( A ) ( k , k ) as graded vector spa ces. Pick elements x ∈ V ( n ) ∗ and y ∈ V ( m ) ∗ . As we know, x and y c a n be extend to b e DG mo dule maps (also denoted by x a nd y resp ectively) P / P ( n − 1 ) x − → A k and P /P ( m − 1) y − → A k . As b efore, there are filtration-pr eserving DG mo dule mo rphisms f x : P /P ( n − 1) − → P and f y : P /P ( m − 1) − → P such that ε ◦ f x = x a nd ε ◦ f y = y . Let g b e the comp osition of the DG mo dule morphisms g : P π − → P /P ( n − 1) f x − → P π − → P /P ( m − 1) y − → k . Then the pro duct y · x ∈ V ( n + m ) ∗ of x and y in g r F ( E ) is the restric tio n of g to V ( n + m ). Since it is filtration-pr eserving, f x induces a mor phism of sp ectral sequences E p,q ∗ ( f x ) : E p,q ∗ ( P / P ( n − 1)) − → E p,q ∗ ( P ) . In particular, E p,q 1 ( P / P ( n − 1)) = H p + q A ⊗ V ( n − p ) and E p,q 1 ( P ) = H p + q A ⊗ V ( − p ) for all p ≤ 0 and p + q ≥ 0. Now we reg ard x ∈ V ( n ) ∗ and y ∈ V ( m ) ∗ as elements in Ext ∗ H ( A ) ( k , k ). Let h − p = L p ≥− q E p,q 1 ( f x ). Then we g et a commutativ e diag ram / / H ( A ) ⊗ V ( n + m ) h m   δ n + m / / · · · δ n +2 / / H ( A ) ⊗ V ( n + 1) h 1   δ n +1 / / H ( A ) ⊗ V ( n ) h 0   η x $ $ I I I I I I I I I I / / 0 / / H ( A ) ⊗ V ( m ) η y   δ m / / · · · δ 2 / / H ( A ) ⊗ V (1) δ 1 / / H ( A ) ⊗ V (0) δ 0 / / k / / 0 k where η x and η y are graded H ( A )-mo dule morphisms induced by x a nd y . T o avoid the p ossible co nfusion, we temp orar ily denote the Y o neda pro duct on Ex t ∗ H ( A ) ( k , k ) by y ∗ x . By the de finitio n of Y oneda pro duct, y ∗ x is equal to the restrictio n of 14 J.-W. HE AND Q.-S. WU η y ◦ h m to V ( n + m ). Let τ m : L m i =0 V ( i ) → V ( m ) be the pro jection map. F or any v ∈ V ( n + m ), η y ◦ h m ( v ) = η y  E − ( n + m ) ,n + m 1 ( f x )( v )  = η y ◦ τ m ◦ f x ( v ) = g ( v ) . Hence y ∗ x = y · x , that is , the pro ducts on g r F ( E ) and Ext ∗ H ( A ) ( k , k ) coincide under the isomor phism in (1). Since H ( A ) is Ko szul, Ext ∗ H ( A ) ( k , k ) ∼ = ( H ( A )) ! . Hence g r F ( E ) ∼ = ( H ( A )) ! .  Let A b e a connected DG algebra . When the trivial mo dule A k lies in D c ( A ), the DG a lgebra A usua lly has go o d prop er ties. The following pro p o s ition is clear . Prop ositio n 3.4. L et A b e a c onne cte d DG algebr a. If A k ∈ D c ( A ) and H ( A ) is a Koszul algebr a, then gldim H ( A ) < ∞ .  Theorem 3. 5 . L et A b e a c onne cte d D G algebr a. Supp ose A k ∈ D c ( A ) . (i) If A is a Koszu l DG algebr a, then the Ex t-algebr a E = Ext 0 A ( A k , A k ) is a finite dimensional lo c al algebr a with E /J = k , wher e J is t he Jac obson r adic al of E . (ii) If H ( A ) is a Koszul algebr a, then g r ( E ) = ( H ( A )) ! , wher e g r ( E ) is the gr ade d algebr a asso ciate d with the ra dic al filtr ation of the lo c al algebr a E . Pr o of. W e use the nota tions in the pro of of Theorem 3.3. (i) Since A k ∈ D c ( A ), there is an integer m such that P ( m ) / P ( m − 1) 6 = 0 and P ( i ) /P ( i − 1 ) = 0 for a ll i > m . Hence the filtra tio n F : E = F 0 ⊇ F 1 ⊇ F 2 ⊇ · · · stops at the m -th step. By Theorem 3.3, E is a filtered alg ebra, hence for x ∈ F 1 , x m +1 = 0. Thus E is a lo ca l alg ebra with Jaco bson ra dic a l J = F 1 and E /J ∼ = k . (ii) If H ( A ) is a Ko szul algebra, then by Pro po sition 3 .4, gldim H ( A ) < ∞ . Assume that gldim H ( A ) = n . Then the filtration F stops at the n -th s tep, and J n +1 = 0. By Theorem 3.3, g r F ( E ) ∼ = ( H ( A )) ! . If we can s how J i = F i for all 1 ≤ i ≤ n , then we a re done. Since V ( j ) = 0 for j ≥ n + 1, E = k ⊕ V (1) ∗ ⊕ · · · ⊕ V ( n ) ∗ and F i = L n j = i V ( j ) ∗ . By Theorem 3.3, g r F ( E ) is gener ated in deg r ee 1, so ( F 1 ) n = F n = V ( n ) ∗ , that is, J n = F n . Similarly , since V ( n ) ∗ = J n ⊆ J n − 1 , V ( n − 1 ) ∗ ⊆ ( F 1 ) n − 1 + V ( n ) ∗ = J n − 1 and F n − 1 = V ( n − 1) ∗ ⊕ V ( n ) ∗ ⊆ J n − 1 . On the other hand, J n − 1 ⊆ V ( n − 1) ∗ ⊕ V ( n ) ∗ . Hence F n − 1 = J n − 1 . An easy induction shows tha t J i = F i for a ll 1 ≤ i ≤ n .  W e nex t pr ov e a theor em similar to the K oszul duality for K o szul algebr as [BGS]. Let A be an aug ment ed DG algebra , and let R = B ( A ) b e its bar construction. Lemma 3.6 . [FHT2, P . 272] The map ϕ : R # − → E nd A ( A ⊗ R ) define d by ϕ ( f )(1[ a 1 | · · · | a n ]) = n X i =0 ( − 1) | f | ω i 1[ a 1 | · · · | a i ] f ([ a i +1 | · · · | a n ]) is a quasi-isomorphism of DG algebr as, wher e ω i = | a 1 | + · · · + | a i | − i .  K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 15 Lemma 3.7. L et A b e a Koszul DG algebr a and E b e its Ext-algebr a. If A k ∈ D c ( A ) , then E # is a c o augmente d c o algebr a and ther e is a quasi-isomorphism of DG algebr as ψ : Ω( E # ) − → A. Pr o of. Let R = B ( A ) b e the bar constr uction o f A . Then R is a coaugmented DG coalgebr a, and is concentrated in non-nega tive degre es. The graded vector space dual R # is a n aug ment ed DG alg ebra. It follows from Lemma 1 .4 and Lemma 3.6 that E ∼ = H (End A ( A ⊗ R )) ∼ = H ( R # ). Since A is Ko szul, E is co ncentrated in deg ree zero . The last isomor phism implies that H i ( R ) = 0 for all i > 0. Then there is natura lly a quasi-isomo rphism of coaugmented DG coalgebr as Z 0 ( R ) − → R , which induces a quasi-iso morphism of augmented DG algebras R # − → ( Z 0 ( R )) # . Therefore E ∼ = H ( R # ) ∼ = ( Z 0 ( R )) # as augmented alge br as. Since A k ∈ D c ( A ), E is a finite dimensional alg ebra. Hence E # ∼ = Z 0 ( R ) as coaug ment ed co algebra s, a nd there is a quasi-isomo rphism o f c o augmented DG co algebra s E # − → R. This induces a quasi-iso morphism of DG algebra s ξ : Ω( E # ) − → Ω( R ) = Ω B ( A ) . There is also a quas i-isomorphism o f DG alg ebras ζ : Ω B ( A ) − → A by Lemma 1.3. Hence the comp osition (2) Ω( E # ) ξ − → Ω B ( A ) ζ − → A gives a q uasi-isomor phism o f DG alg ebras ψ = ζ ◦ ξ : Ω( E # ) − → A . The pro o f is completed.  Theorem 3.8 (Koszul Duality on Ext-alg ebra) . L et A b e a Koszul DG algebr a and E b e its Ext -algebr a. If A k ∈ D c ( A ) , t hen Ext ∗ E ( E k , E k ) ∼ = H ( A ) . Pr o of. By Lemma 3.6, Ω( E # ) = B ( E ) # is qua si-isomor phic to E nd E ( E ⊗ B ( E )). Hence Ext ∗ E ( E k , E k ) ∼ = H (E nd E ( E ⊗ B ( E ))) ∼ = H (Ω( E # )) . It fo llows from Lemma 3.7 that Ex t ∗ E ( E k , E k ) ∼ = H ( A ).  As an application of Theor em 3.8, we have the following tw o coro llaries, which establish rela tions b etw een K oszul DG algebr as and (strong ly ) quasi- Koszul a lgebras. Corollary 3.9. L et A b e a Koszul D G algebr a. If A k ∈ D c ( A ) , then the fol lowing ar e e quivalent: (i) The Ext -algebr a E of A is a quasi-Koszul algebr a; (ii) H ( A ) is gener ate d in de gr e e 1. 16 J.-W. HE AND Q.-S. WU Pr o of. By The o rem 3.5, E is a finite dimensional lo cal algebra with the residue field k . The equiv alence of (i) and (ii) follows fro m Theore m 3.8 a nd [GM, Theo rem 4 .4 ].  Corollary 3.10. L et A b e a Koszul D G algebr a. If A k ∈ D c ( A ) , then the fol lowing ar e e quivalent: (i) The Ext -algebr a E of A is a str ongly quasi-Koszul algebr a; (ii) H ( A ) is a Koszul algebr a. Pr o of. (i) ⇒ (ii). By [GM, Theor em 6.1 ] a nd its pro of, Ext ∗ E ( E k , E k ) is a Kos zul algebra. Theorem 3.8 implies that H ( A ) ∼ = Ext ∗ E ( E k , E k ) is a Koszul a lgebra. (ii) ⇒ (i). Applying Theo rem 3.8 a gain, Ext ∗ E ( E k , E k ) ∼ = H ( A ) is a Koszul algebra. By [GM, Theo rem 9 .1], E is a strong ly quasi-K oszul alge bra.  4. Koszul Dual ity Let B be an augmented DG algebra and C be a coaugmented DG coalgebra . Lef` evre-Hasegaw a ([Le, Pro p o s ition 2.2.4 .1]) establis he d an equiv alence b et ween the derived categor y D ( B ) a nd the so called co derived ca tegory D ( C ) when B and C satisfy certain c o nditions. Thank s for the result of Lef` evre-Hasegawa we ca n prove a version o f Kos zul Duality ([BGS]) for Ko s zul DG alge br as. Let ( B , m B , d B ) be an a ugmented DG alg ebra with a n augmentation map ε B : B → k , and ( C, ∆ , d C ) b e a coaugmented DG coalgebr a with a c oaugmentation map η C : k → C . A gr aded linear map τ : C → B of degree 1 is called a twisting c o chain from C to B ([HMS], [Le]) if ε B ◦ τ ◦ η C = 0 , and m B ◦ ( τ ⊗ τ ) ◦ ∆ + d B ◦ τ + τ ◦ d C = 0 . Let Ω( C ) b e the c o bar constructio n of C . The twisting co chains from C to B a re one to one c o rresp onding to the DG a lgebra morphisms from Ω( C ) to B . There is a c anonic al t wisting c o chain τ 0 : C → Ω( C ) given by τ 0 ( c ) = [ c ] for any c ∈ ¯ C a nd τ 0 ( k ) = 0. Let τ : C → B be a t wisting co chain. F or any right DG C -co mo dule N , the twiste d tensor pr o duct N ⊗ τ B ([Le], [K e2]) is the rig h t DG B -mo dule defined by (i) N ⊗ τ B = N ⊗ B as a rig ht gr aded B -mo dule; (ii) the differential δ = d N ⊗ 1 + 1 ⊗ d B + (1 ⊗ m B )(1 ⊗ τ ⊗ 1)( ρ N ⊗ 1 ) , i.e. δ ( n ⊗ a ) = d ( n ) ⊗ a + ( − 1) | n | n ⊗ d ( a ) + X ( n ) ( − 1) | n (0) | n (0) ⊗ τ ( n (1) ) a, for a ny homo geneous elements n ∈ N and a ∈ B . K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 17 Dually , for any DG B - mo dule M , the twiste d t ensor pr o duct M ⊗ τ C is the r ight DG C -c omo dule defined by (i) M ⊗ τ C = M ⊗ C as a vector space; (ii) the differe ntial δ = d M ⊗ 1 + 1 ⊗ d C − ( m M ⊗ 1 )(1 ⊗ τ ⊗ 1)(1 ⊗ ∆), i.e. δ ( m ⊗ c ) = d ( m ) ⊗ c + ( − 1) | m | m ⊗ d ( c ) − X ( c ) ( − 1) | m | mτ ( c (1) ) ⊗ c (2) , for a ny homo geneous elements m ∈ M and c ∈ C . Let DGmo d- B be the categ ory of r ight DG B -mo dules a nd DGcom- C be the cate- gory of right DG C -como dules. Then there is a pa ir o f a djoint functor s ( L, R ) ([Ke2], [Le]): DGcom- C L = −⊗ τ B − − − − − − → ← − − − − − − R = −⊗ τ C DGmod- B . Let C b e a co complete DG coalg ebra, and DGcomc- C b e the categor y o f co complete right DG C -como dules. F or any M , N ∈ DGcomc- C , a DG como dule mo rphism f : M → N is called a we ak e quivalenc e rel ate d t o τ ([Ke2],[Le]) if L ( f ) : L M → LN is a quasi-iso morphism. Note that a weak equiv alence r elated to τ 0 ( B = Ω( C )) is a quasi-isomo rphism. B ut the conv erse is not tr ue in gener al ([Ke 2]). Let K ( C ) b e the homotopy category o f DGcomc- C . Equipp ed with the natura l exact triangles, K ( C ) is a triangulated ca tegory . Let W b e the c lass of weak equiv alences in the categor y K ( C ). Then W is a multiplicativ e sy stem. The co derived categ ory D dg ( C ) of C is defined to be K ( C )[ W − 1 ], the lo ca lization o f K ( C ) a t the class W of weak equiv a lences ([K e 2], [Le]). Le t D dg ( B op ) b e the derived catego ry o f r ight DG B -mo dules. The following theorem is prov ed by Lef` evre-Hasegawa in [Le, Ch.2], a nd also ca n be found in [Ke2]. Theorem 4.1. L et C b e a c o c omplete DG c o algebr a, B an augmen t e d DG algebr a and τ : C → B b e a twisting c o chain. Then the fol lowing ar e e quivalent: (i) The map τ induc es a quasi-isomorphism Ω( C ) → B ; (ii) The adjunction map B ⊗ τ C ⊗ τ B → B is a quasi-isomorphism; (iii) The fun ctors L and R induc e an e quivalenc e of t riangulate d c ate gories (also denote d by L and R ) D dg ( C ) L ⇄ R D dg ( B op ) .  Now let A be a Koszul DG a lgebra. Supp ose A k ∈ D c ( A ). By Theore m 3 .5, its E xt- algebra E is a finite dimensio nal lo cal alg ebra with the r esidue field k . Hence the vector space dual E ∗ = E # is a c oaugmented coa lgebra which is of co urse co complete. Hence all the DG E ∗ -como dules ar e co complete. Let C = E ∗ and B = Ω( C ). Clearly , B is 18 J.-W. HE AND Q.-S. WU a co nnected DG algebra , and the ca nonical twisting co chain τ 0 : C → Ω( C ) s atisfies the condition (i) in the Theorem 4.1. Hence we hav e the following equiv a lence of triangulated ca tegories D dg ( E ∗ ) L ⇄ R D dg (Ω( E ∗ ) op ) . Let D + dg (Ω( E ∗ ) op ) b e the derived catego ry o f a ll b o unded b elow r ight DG Ω( E ∗ )- mo dules, that is, consis ting of ob jects M with M n = 0 for n ≪ 0. Since Ω( E ∗ ) is connected, it is not hard to see that D + dg (Ω( E ∗ ) op ) is a full triangulated sub c ategory of D dg (Ω( E ∗ ) op ). Similarly , let K + dg ( C ) b e the homotopy ca tegory of b ounded b elow DG co complete como dules , and let D + dg ( E ∗ ) b e the lo caliza tion of K + dg ( E ∗ ) at the clas s of weak equiv alences W + in K + dg ( E ∗ ) ( W + is als o a multiplicativ e system). O ne can chec k that D + dg ( E ∗ ) is a full triangulated sub categ ory of D dg ( E ∗ ). Restr icting L and R to the sub catego r ies D + dg ( E ∗ ) a nd D + dg (Ω( E ∗ ) op ) r esp ectively , we get the following prop osition. Prop ositio n 4. 2. L et A b e a Koszul DG algebr a and E b e its Ext -algebr a. If A k ∈ D c ( A ) , t hen the fol lowing is an e quivalenc e of triangulate d c ate gories D + dg ( E ∗ ) L ⇄ R D + dg (Ω( E ∗ ) op ) .  Since E ∗ is concentrated in degree zer o, a DG E ∗ -como dule is exa ctly a co chain complex of E ∗ -como dules. Hence K + dg ( E ∗ ) = K + ( E ∗ ), the homotopy ca teg ory of bo unded b elow co chain complexes o f right E ∗ -como dules. It is not har d to see that the class W + of weak equiv a lences related to τ 0 is ex actly the cla s s of quasi-iso morphisms. Hence D + dg ( E ∗ ) = K + dg ( E ∗ )[( W + ) − 1 ] = D + ( E ∗ ), the derived categor y of b ounded below co chain co mplexes o f right E ∗ -como dules. By P rop osition 4.2 we hav e the following pro po sition. Prop ositio n 4. 3. L et A b e a Koszul DG algebr a and E b e its Ext -algebr a. If A k ∈ D c ( A ) , t hen ther e is an e quivalenc e of triangulate d c ate gories (we use the same nota- tions of the e quivalent functors as in Pr op. 4.2.) D + ( E ∗ ) L ⇄ R D + dg (Ω( E ∗ ) op ) .  Since E is a finite dimensional algebra , the categ ory of left E - mo dules is isomor phic to the categor y of right E ∗ -como dules ([Mo n, 1 .6 .4]). Hence there is an equiv alence of tria ngulated ca tegories D + ( E ) F ⇄ G D + ( E ∗ ) , where D + ( E ) is the der ived categor y of b ounded b elow c o chain complexes of le ft E -mo dules. K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 19 By Lemma 3.7, there is a quas i-isomorphism of DG algebr a ϕ : Ω( E ∗ ) − → A . Hence by Lemma 2.7, the following gives an equiv alence o f tr ia ngulated categ ories D + ( A op ) ϕ ∗ − − − − − − − → ← − − − − − − − −⊗ L Ω( E ∗ ) A D + (Ω( E ∗ ) op ) . Let Φ = ( − ⊗ L Ω( E ∗ ) A ) ◦ L ◦ F and Ψ = G ◦ R ◦ ϕ ∗ . W e hav e the following theor e m. Theorem 4.4 (Ko szul Equiv alence) . Le t A b e a Koszul DG algebr a and E b e its Ext-algebr a. If A k ∈ D c ( A ) , t hen we have an e quivalenc e of triangulate d c ate gories D + ( E ) Φ − → ← − Ψ D + dg ( A op ) . It is easy to s ee tha t Φ( E k ) = L ( k E ∗ ) ⊗ L Ω( E ∗ ) A = Ω( E ∗ ) ⊗ L Ω( E ∗ ) A = A A . T em- po rarily write h E k i the full triangula ted sub c ategory of D + ( E ) genera ted by E k . By restricting Φ and Ψ, we get a n equiv alence o f tr ia ngulated categorie s h E k i Φ res − − → ← − − − Ψ res D c ( A ) . Lemma 4.5. h E k i = D b (mo d- E ) , wher e mo d- E is the c ate gory of fin itely gener ate d left E - mo dules. Pr o of. It suffices to show that a ll the finitely g enerated E -mo dules ar e in h E k i . Since E is finite dimensional, any finitely g e nerated E -mo dule is finite dimensional. Clearly , all 1-dimensiona l mo dules are in h E k i . Let N b e a finite dimensional mo dule. Since so c(N) 6 = 0, we have an exact sequence 0 − → E k − → N − → N / E k − → 0 . Since dim N / E k < dim N , an induction o n the dimension of N implies that N lies in h E k i . Hence all finitely genera ted E -mo dules are in h E k i .  Corollary 4 .6. L et A b e a Koszul D G algebr a and E b e its Ext-algebr a. If A k ∈ D c ( A ) , t hen we have an e quivalenc e of triangulate d c ate gories D b (mo d- E ) Φ res − − → ← − − Ψ res D c ( A op ) . Since E is finite dimensional, the vector space dual ( ) ∗ induces a duality of trian- gulated catego ries D (mo d- E ) ( ) ∗ − − → ← − − ( ) ∗ D (mo d- E op ) . Now, we ar e able to give a version of the Koszul duality for K oszul DG a lgebras. Theorem 4.7 (Kos zul Duality) . L et A b e a Koszu l DG algebr a and E b e its Ext- algebr a. Supp ose A k ∈ D c ( A ) . Then ther e is a duality of triangulate d c ate gories D b (mo d- E op ) F − → ← − G D c ( A op ) . 20 J.-W. HE AND Q.-S. WU It is easy to s ee that (3) F ( k E ) = Φ( E k ) = A A and F ( E E ) = Φ( E E ∗ ) = L (( E ∗ ) E ∗ ) ⊗ L Ω( E ∗ ) A ( a ) ∼ = k Ω( E ∗ ) ⊗ L Ω( E ∗ ) A ∼ = k A , (4) where the iso morphism ( a ) holds, b eca use L (( E ∗ ) E ∗ ) = Ω( E ∗ ; E ∗ ) which is quas i- isomorphic to k Ω( E ∗ ) as a DG Ω( E ∗ )-mo dule by the narra tive b elow Lemma 1 .5. F r om the pro o f of ab ove results, we hav e proved in fact the following result. Corollary 4.8. L et R b e a finite dimensional lo c al algebr a with the r esidue fi eld k . Then ther e is a duality of triangulate d c ate gories D b ( R op ) ⇄ D c (Ω( R ∗ ) op ); and u nder this duality, t he trivial mo dule k R c orr esp onds to Ω( R ∗ ) and R R to k Ω( R ∗ ) .  The following corolla ry was indicated in [Ke2] a nd [Le]. As an application of Corol- lary 4.8, we give a pro of here. Corollary 4.9. L et R b e a finite dimensional lo c al algebr a with the r esidue fi eld k . Then t he c onne cte d DG algebr a Ω( R ∗ ) is a Koszul DG algebr a. Mor e over, t he Ex t- algebr a Ext ∗ Ω( R ∗ ) ( k , k ) is isomorphic t o R . Pr o of. By Corollar y 4 .8, k Ω( R ∗ ) is compact, and Ext n Ω( R ∗ ) op ( k , k ) = Hom D c (Ω( R ∗ ) op ) ( k , k [ − n ]) ∼ = Hom D b ( R op ) ( R [ n ] , R ) = 0 if n 6 = 0. Ther efore Ω( R ∗ ) is a Ko szul DG algebra . Moreover, the following a re algebra isomorphisms Ext ∗ Ω( R ∗ ) ( k , k ) ∼ = Ext ∗ Ω( R ∗ ) op ( k , k ) op ∼ = Ext ∗ R op ( R, R ) ∼ = R.  In pa r ticular, by Corollar y 4.9, if k is algebr a ically closed, then any finite dimen- sional lo ca l alg e br a can b e viewed as the Ext-algebr a of some Koszul DG algebra . Example 4.1 0. No w let V = k x ⊕ k y ⊕ k z and R = T ( V ) /T ≥ 4 ( V ). Clear ly , R is a finite dimensional lo ca l algebra . Then B = Ω( R ∗ ) is a Kosz ul DG algebra with Ext ∗ B ( B k , B k ) = R . Since g r ( R ) ∼ = R is not a Ko s zul alg e bra, s o R is not a strongly quasi-Ko szul alg ebra. By Cor ollary 3.10, the coho mology H ( B ) ca n not be a Koszul algebra. Hence the conv erse of Pro po sition 2.3 is not true. K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 21 5. BGG correspond ence In [BGG], Bernstein-Gelfand-Gelfa nd established an equiv a lence o f ca tegories grmo d-Λ( V ) ∼ = D b (Coh P n ) where grmo d-Λ( V ) is the stable categ ory of finitely genera ted gra ded mo dules over the exterior algebra Λ( V ) of a n ( n + 1)-dimensional space V = k x 0 ⊕ k x 1 ⊕ · · · ⊕ k x n , and D b (Coh P n ) is the b o unded derived c a tegory of coher ent sheav es over the n -dimensio nal pro jective space P n . This equiv alence is now called the BGG c orr esp ondenc e in liter- ature. A sketc h o f the pro of of the BGG corre s po ndence ca n b e found a ls o in [GMa, P .273, Ex.1 ]. The BGG corr e spo ndence ha s b een g eneralized to noncommutativ e pro- jective geometry by several a utho r s ([Jo], [MS], [Mo]). Let R b e a Kos zul no etheria n AS-Gorenstein algebr a with finite g lobal dimensio n. Then its Ext-a lgebra E ( R ) is a F r ob enius algebr a ([Sm]). A version of the no ncommut ative BGG corr e spo ndence was prov ed in [MS], which was sta ted as grmo d- E ( R ) ∼ = D b (qgr R op ) , where grmo d- E ( R ) is the stable ca tegory of finitely ge ne r ated graded mo dules ov er E ( R ) and qgr R op is the quotient ca tegory grmo d- R op / tors R op . Le t D b f d (grmo d- R op ) be the full s ubca tegory of D b (grmo d- R op ) co nsisting o f o b jects X with finite dimen- sional cohomolo g y g roups. It is well known that ([Mi]) D b (qgr R op ) = D b (grmo d- R op ) / D b f d (grmo d- R op ) . Hence the ab ov e BGG corre s po ndence can be stated as (5) grmo d- E ( R ) ∼ = D b (grmo d- R op ) / D b f d (grmo d- R op ) . In this sec tio n, we deduce a cor resp ondence similar to (5) for AS-Gorenstein Kos zul DG algebr as. First of all we recall the definition of AS-Gor enstein DG a lgebra. Let A b e a connected DG a lgebra. W e say that A is right AS-Gor enstein (AS stands fo r Artin- Schelt er) if RHom A op ( k , A ) ∼ = s n k for s o me integer n ([FHT1], [LP WZ],[LPWZ2]); A is right AS-r e gular if A is right AS-Gorenstein and k A ∈ D c ( A op ). Similarly , w e define left AS-Gorenstein DG algebra and left AS-re gular algebra . W e call A is AS- Go renstein (resp., regula r) if A is b oth left and r ig ht AS-Gor enstein (res p., reg ular). Prop ositio n 5.1. L et A b e a c onne cte d D G algebr a. If t he c ohomolo gy algebr a H ( A ) is a left AS -Gor enst ein algebr a, t hen A is a left A S -Gor enst ein DG algebr a. Pr o of. Consider the Eilenberg -Mo ore sp ectral sequence ([KM]) E p,q 2 = Ext p H ( A ) ( k , H ( A )) q = ⇒ E x t p + q A ( k , A ) where the index p in Ext p H ( A ) ( k , H ( A )) q is the us ual homologica l degr ee and q is the grading induced from the g radings of A k and H ( A ) . If the cohomo lo gy sp ectral 22 J.-W. HE AND Q.-S. WU sequence is r e g ular, then it is complete convergen t ([W e]). If H ( A ) is AS-Gorenstein, then by definition there exist so me integers d and l such tha t Ext n H ( A ) ( k , H ( A )) =    0 n 6 = d k [ l ] n = d. Then it is ro utine to see that Ext n A ( k , A ) =    0 n 6 = d + l k n = d + l . Hence RHom A ( k , A ) ∼ = s n k for n = d + l .  W e don’t know whether the conv erse o f Pr op osition 5.1 is true o r not. If A is a connected gra ded algebra, viewed a s a DG algebra with trivial differential, then A is an AS-Go renstein DG alg ebra if a nd only if A satisfies AS-Gorenstein condition in the usual s e nse. The AS-Go renstein pr op erty is inv aria n t under quas i- isomorphism. Prop ositio n 5.2 . L et f : A → A ′ b e a qu asi-isomorph ism of c onne ct e d DG algebr as. Then A is left AS-Gor enstein (AS-re gular) if and only if A ′ is. Pr o of. The pr o of is s imilar to that of P rop osition 2 .8.  Lemma 5.3 . L et A b e an AS-r e gu lar DG algebr a. Su pp ose RHom A op ( k , A ) ∼ = s l k for some inte ger l . L et P → k A b e a minimal semifr e e reso lution of k A with a semifr e e filtr ation 0 ⊆ P (0) ⊆ P (1) ⊆ · · · ⊆ P ( n ) such that P ( n ) = P and P ( n ) / P ( n − 1) 6 = 0 . Then P ( n ) /P ( n − 1) = A [ − l ] . Pr o of. There are finite dimensional gr aded vector s pa ces V (0) , V (1) , · · · , V ( n ) such that P ( i ) /P ( i − 1) = V ( i ) ⊗ A for all 0 ≤ i ≤ n ( P ( − 1) = 0 ). As graded A - mo dules P = L n i =0 V ( i ) ⊗ A . Hence Hom A ( P, A ) = L n i =0 A ⊗ V ( i ) # as gr aded left A -mo dules. Let { x 1 , · · · , x t } be a homogene o us bas is o f V ( n ). Let d b e the differential of Hom A ( P, A ) induced by the differentials of P and A . F or a n y 1 ≤ s ≤ t , define a graded r ight A -mo dule mor phism f s : P = ⊕ n i =0 V ( i ) ⊗ A − → A by sending x s to the identit y of A , x j to zero for j 6 = s , and se nding V ( r ) to zero for all r < n . One ca n see that f 1 , · · · , f t so defined are co cycles of the co chain complex Hom A ( P, A ). Since P is minima l, d ( g )( x j ) = ( d A g − ( − 1) | g | g d P )( x j ) ∈ A ≥ 1 for any homogeneous element g ∈ Hom A ( P, A ) and x j . Hence a ny f j (1 ≤ j ≤ t ) ca n not b e a cob oundary . By hypothesis RHo m A op ( k , A ) ∼ = s l k , which force s dim V ( n ) = 1 and the degree o f no n-zero elements in V ( n ) is − l , that is , P ( n ) /P ( n − 1) ∼ = s − l A .  The following prop ositio n is a sp ecial ca se o f [LPWZ2, Theor em 9.8 ]. Prop ositio n 5. 4. L et A b e a K oszu l DG algebr a with Ext -algebr a E = Ext ∗ A ( k , k ) . Then A is right AS-r e gular if and only if E is F r ob enius . K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 23 Pr o of. Supp o se that A is right AS-r egular. Then k A ∈ D c ( A op ), which is equiv a lent to A k ∈ D c ( A ). Hence E is finite dimensional. Since A is Koszul, RHom A op ( k , A ) ∼ = k by Lemma 5 .3. By Theo rem 4 .7, Ext n E op ( k , E ) = Hom D b (mo d- E op ) ( k , E [ − n ]) ∼ = Hom D c ( A op ) ( F ( E ) , F ( k )[ − n ]) ∼ = Hom D c ( A op ) ( k , A [ − n ]) = Ext n A op ( k , A ) . Hence Ext n E op ( k , E ) = 0 for n 6 = 0. Let 0 − → E E − → I 0 − → I 1 − → · · · be a minimal injective res o lution of E E . Since E is finite dimens io nal and lo ca l, a ll the injective mo dules I n ’s are finite dimensio nal. Hence 0 = Ext n E op ( k , A ) ∼ = socI n for all n ≥ 1, and Hom E op ( k , E ) = k . W e get I n = 0 fo r a ll n ≥ 1 a nd I 0 = E ∗ . Ther e fore we hav e a r ight E - mo dule isomorphism E ∼ = E ∗ , that is, E is a F rob enius algebr a. Conv ersely , if E is F rob enius, then it is finite dimensional, and hence k A ∈ D c ( A op ). Since E itself is injective a nd loca l, it follows Ext n E op ( k , E ) = 0 for n ≥ 1 and Ext 0 E op ( k , E ) = k . Hence Ext n A op ( k , A ) ∼ = Ext n E op ( k , E ) = 0 for n 6 = 0 and E xt 0 A op ( k , A ) ∼ = k . Then RHom A op ( k , A ) ∼ = k , and hence A is AS-r egular.  In [LPWZ2, Theor e m 9 .8], a more general ca se of the ab ove pr o p o sition is prov ed with s ome lo ca lly finite conditions. Corollary 5.5. L et A b e a Koszul DG algebr a. Then A is right AS-r e gular if and only if A is left AS- re gular. Pr o of. Note that E op ∼ = Ext ∗ A op ( k , k ) .  Next we ar e going to deduce a re sult similar to the cla ssical BGG co rresp ondence. Lemma 5.6. L et A b e a c onne ct e d DG algebr a such that k A ∈ D c ( A op ) . Then the ful l triangulate d su b c ate gory h k A i of D c ( A op ) gener ate d by k A , is e qual to D f d ( A op ) , the ful l sub c ate gory of D c ( A op ) c onsisting of DG mo dules M such that dim H ( M ) < ∞ . Pr o of. F or any DG mo dule M , temp ora rily we write ℓ ( M ) = sup { i | H i ( M ) 6 = 0 } − inf { i | H i ( M ) 6 = 0 } and λ ( M ) = sup { i | M i 6 = 0 } − inf { i | M i 6 = 0 } . W e prov e the lemma by an induction on ℓ ( M ). Let M be a DG A -mo dule with dim H ( M ) < ∞ . Without los s of generality , we may assume that H i ( M ) = 0 for i < 0 or i > n for n = ℓ ( M ). Since A is connected, by s uitable trunca tions, we may as s ume that M is concentrated in degrees 0 ≤ i ≤ n. If ℓ ( M ) = 0, then M is isomorphic in D c ( A op ) to a DG mo dule N with λ ( N ) = 0, which is a direct sum of finite c o pies o f k A , and hence is in h k A i . Now supp ose that ea ch DG mo dule M with 24 J.-W. HE AND Q.-S. WU dim H ( M ) < ∞ and ℓ ( M ) < n is in h k A i . If M is a DG mo dule with dim H ( M ) < ∞ and ℓ ( M ) = n , without lo ss o f generality , we may assume M 0 6 = 0 a nd M n 6 = 0, and M i = 0 for i < 0 and i > n . Then the vector space M n has a decomp os itio n M n = d ( M n − 1 ) ⊕ K for some subspace K . Since dim H ( M ) < ∞ , then dim K < ∞ . T a king K as a DG A -mo dule concentrated on degree ze r o, then we have an exact sequence of DG mo dules 0 − → s n K − → M − → M /s n K − → 0 . Now ℓ ( s n K ) = 0 a nd ℓ ( M /s n K ) ≤ n − 1. B y the induction hypothesis, b oth s n K a nd M /s n K ar e ob jects in h k A i , a nd hence M is in h k A i . Therefore h k A i = D f d ( A op ).  Theorem 5.7 (BGG Corr esp ondence) . L et A b e a Koszul D G AS-r e gular algebr a with Ext-algebr a E = Ext ∗ A ( k , k ) . Then ther e is a duality of triangulate d c ate gories mo d- E op ⇄ D c ( A op ) / D f d ( A op ) . Pr o of. By the Koszul Duality (Theorem 4.7), There is a dua lity of triangula ted cate- gories D b (mo d- E op ) ⇄ D c ( A op ); and under this duality the ob ject E E ∈ D b (mo d E op ) is cor resp onding to the ob ject k A ∈ D c ( A op ) by (4). Hence ther e is a duality D b (mo d- E op ) / h E E i ⇄ D c ( A op ) / h k A i , where h E E i is the full tr ia ngulated sub category o f D b (mo d- E op ) generated by E E . Since E is a finite dimensiona l lo cal algebra with E /J ( E ) ∼ = k (Theorem 3.5), all finitely gener a ted pro jective E - mo dules are free. Ther efore h E E i = D b (pro j E op ), where pro j E op is the catego ry of a ll finitely genera ted rig h t pro jective E -mo dules. Hence D b (mo d- E op ) / h E E i = D b (mo d- E op ) / D b (pro j E op ) . By P rop osition 5 .4, E is F rob enius, and hence ([Be 1 ]) D b (mo d- E op ) / D b (pro j E op ) ∼ = mo d- E op . On the o ther hand, by Lemma 5.6 D c ( A op ) / h k A i = D c ( A op ) / D f d ( A ) . In summar y , there is an a duality of triangulated c a tegories mo d- E op ⇄ D c ( A op ) / D f d ( A op ) .  Since E is finite dimensional, ther e is a n equiv alence form of the BGG Corres po n- dence. Theorem 5. 8. L et A b e a Koszul D G AS-r e gular alge br a with Ext-algebr a E = Ext ∗ A ( k , k ) . Then ther e is an e quivalenc e of t riangulate d c ate gories mo d- E ∼ = D c ( A op ) / D f d ( A op ) . K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 25 6. BGG correspond ence on Adams connected DG algebras Many examples of the DG algebr a fr o m algebra ic geometry and algebraic top ology admit an extra grading . Let A = ⊕ i,j ∈ Z A i j be a bigra ded spa ce. An element a ∈ A i j is of degree ( i, j ). The second gr a ding is usually called A dams gr ading ([K M], [LPWZ]). A DG algebr a ( A, d ) is called a DG a lgebra with Adams grading if A is bigraded a nd the differential d is of degree (1 , 0) (i.e., d prese r ves Adams gr ading). A DG mo dule ov er a DG algebra with Adams g rading is bigra ded and the differential preser ves the second grading. A DG algebr a A with Adams grading is augmente d if there is a n augmentation map ε : A → k of degree (0 , 0). A DG algebra with Ada ms gr a ding A is sa id to b e A dams c onne cte d if (1) A i j = 0 fo r i < 0 or j < 0, and (2 ) A 0 0 = k , A 0 j = 0 and A i 0 = 0 for i, j 6 = 0. All Adams connected DG alg ebras ar e augmented. Similarly , we define coaug men ted DG coa lgebras with Adams gr ading. In this section, all the DG alg e br as and DG coalg ebras inv olved ar e with Adams grading. F o r simplicity , we call a DG algebra (coalgebr a) with Adams gr ading an A dams DG algebr a (c o algebr a) . It is no t har d to see that the bar (cobar ) construction (see Section 1) of an (a ) (co)augmented Adams DG algebr a (coalg ebra) is an Adams DG coalgebr a (algebr a). The canonical twisting co chain (see Section 4) τ 0 from a co co mplete Adams DG co al- gebra C to Ω( C ) is of degree (1 , 0). Let A b e an Adams DG a lg ebra, and let AC dg ( A ) ( AC dg ( A op )) b e the category of left (right) DG A -mo dules with morphisms o f degr ee (0 , 0). W e use s i = [ i ] to de no te the i -th shift functor on the first gra ding and use s − j = ( j ) to denote the j -th shift functor on the Adams gra ding. Let AD dg ( A ) b e the derived catego ry of AC dg ( A ). Denote AD c ( A ) ( AD c ( A op )) a s the full tria ngulated sub catego ry of AD dg ( A ) ( AD dg ( A op )) generated by A A ( A A ). Let M and N be o b jects in AC dg ( A ), we us e E xt i,j A ( M , N ) = Hom AD dg ( A ) ( M , N [ − i ]( j )) to denote the der ived functor . Then E xt ∗ , ∗ A ( M , N ) = M i,j ∈ Z E xt i,j A ( M , N ) is a bigr aded space. In particular , if A is an augmented Adams DG algebr a, then E = E xt ∗ , ∗ A ( k , k ) is a bigr aded algebr a. F o r conv enience, we usually write E i j = E xt i,j A ( k , k ). The results obtained in previous se ctions can b e easily g eneralized to Adams DG algebras . Hence in this section, we only state the r esults without giving pro ofs. More general re sults can b e found in [LPWZ2, Sec tion 10], with some lo ca lly finite condi- tions. Let A b e an Adams co nnected DG alg ebra and M be a bo unded b elow DG mo dule ov er A . Then there is a minima l semifree resolutio n (the co nstruction is s imilar to [KM, Theorem IV.3.7]) P → M in AC dg ( A ) (see a lso [MW]). 26 J.-W. HE AND Q.-S. WU Definition 6.1 . Let A b e a n Adams connected DG algebra . It is called a Ko szul Adams DG alg e bra if E xt i, ∗ A ( k , k ) = L j ∈ Z E xt i,j A ( k , k ) = 0 for a ll i 6 = 0. It is not hard to see that, if A is a Kos z ul Adams DG algebr a , then its Ext-a lgebra E = E xt ∗ , ∗ A ( k , k ) has the prop er ty that E i j = 0 for i 6 = 0 or j > 0. Hence E is a negatively g raded alg ebra. Comparing with Theorem 3.3, we have the following. Prop ositio n 6.2. L et A b e a Koszul A dams DG algebr a, and let S j = E 0 − j . Then S = ⊕ j ≥ 0 S j is a c onne cte d gr ade d algebr a. If in addition A k ∈ AD c dg ( A ) , t hen S is a finite dimensional gr ade d algebr a. W e a lso have the following for m of Lef` evre-Haseg aw a’s Theore m. Theorem 6.3. L et C b e a c o c omplete A dams DG c o algebr a, B an augmen t e d A dams DG algebr a and τ : C → B is a twisting c o chain of de gr e e (1 , 0) . The fol lowing ar e e quivalent (i) The map τ induc es a quasi-isomorphism Ω( C ) → B ; (ii) The adjunction map B ⊗ τ C ⊗ τ B → B is a quasi-isomorphi sm; (iii) Ther e is an e quivalenc e of triangulate d c ate gories AD dg ( C ) ⇄ AD dg ( B op ) wher e AD dg ( C ) is the c o derive d c ate gory over the c o c omplete A dams DG al- gebr a C .  If A k ∈ AD c ( A ), then E is finite dimensio nal, and hence the gr aded vector space dual E # is finite dimensional c o algebra . By applying the a b ove theo rem and notice that a DG como dule over the Adams DG co algebra E # is exactly a co mplex of graded como dules over E # , we have the following prop o s ition which is analog ous to Theo rem 4.7. Prop ositio n 6.4. L et A b e a Koszul A dams D G algebr a. If A k ∈ AD c ( A ) , then ther e is a duality of triangulate d c ate gories D b (grmo d- E op ) F − → ← − G AD c ( A op ) .  It is conv enient for us to dea l with the p ositively gr aded a lgebra S , rather than the nega tively graded a lgebra E . The ab elian ca tegory grmo d- E op is eq uiv alent to grmo d- S op of finitely gener a ted right S -mo dules. W e hav e the following Kosz ul dua lit y theorem of Adams DG alg ebras. K OSZUL DIFFERENTIAL GRADED ALGEBRAS AND BGG CORRESPONDENCE 27 Theorem 6 . 5. L et A b e a K oszul A dams DG algebr a. L et S b e the gr ade d algebr a such that S j = E xt 0 , − j A ( k , k ) . If A k ∈ AD c ( A ) , t hen ther e is a duality of triangulate d c ate gories D b (grmo d- S op ) ψ − → ← − φ AD c ( A op ) .  T o es ta blish a version of the BGG cor resp ondence, we need the concept of AS- Gorenstein Adams DG algebra which is first intro duced in [LPWZ]. Definition 6.6. Let A b e an Adams connected DG alge br a. It is ca lled an AS - Gor enstein Adams DG algebr a if RHom A op ( k , A ) ∼ = k [ r ]( s ). Moreover if k A ∈ AD c ( A op ), then A is called a n AS- r e gular Adams DG alg ebra. The following prop os itio n is pr ov ed in [LPWZ] by using A ∞ -algebra . Also one can give a pr o of by using Theo rem 6 .5. Prop ositio n 6.7 . L et A b e a Koszul A dams DG algebr a. Then A is A S-r e gular if and only if its Ext-algebr a E is F r ob enius. Now we can state the BGG cor resp ondence on Adams DG algebra s. Theorem 6 .8. L et A b e a K oszu l AS-r e gu lar A dams DG algebr a and S b e the c on- ne ct e d gr ade d such that S j = E xt 0 , − j A ( k , k ) . Then ther e is an duality of t riangulate d c ate gories (6) grmo d- S op ∼ = AD c ( A op ) / AD f d ( A op ) , wher e AD f d ( A op ) is the fu l l triangulate d sub c ate gory of AD c ( A op ) c onsisting of obje ct s M such that dim H ( M ) < ∞ . Now let R be a no ether ian connected gra ded alge br a. Let A b e the Adams con- nected DG algebr a with trivial differential by taking A i i = R i and A i j = 0 if i 6 = j . If R is a K oszul algebr a, then it is not ha r d to see that A is a Koszul Adams DG alg e- bra. Moreov er, E xt 0 , − j A ( k , k ) = R ! j for all j ≥ 0, i.e., S = R ! = E ( R ) = E x t ∗ R ( k , k ). Suppo se that gl . dim R < ∞ . Then S = E ( R ) is finite dimensional. Since E ( R ) is finite dimensional gr mo d- E ( R ) op is dual to grmo d- E ( R ). Hence D b (grmo d- S op ) = D b (grmo d- E ( R ) op ) is dual to D b (grmo d- E ( R )). Le t us inspe c t the categor y AD c ( A op ) in Theor em 6 .5. Since the differential of A is trivia l and A is co nc e n trated in the di- agonal of the first quadra nt, the triangulated c a tegory AD dg ( A op ) is naturally equiv a - lent to the der ived catego r y D (Grmo d- R op ) of the categ ory Grmo d- R of right g raded R -mo dules. Under this equiv alence, A A is co rresp onding to R R in D (Grmo d- R op ). Hence AD c ( A op ) is equiv alen t to the full triangula ted sub c a tegory o f D (Grmo d- R op ) generated by R R (closed under the shifts on the grading o f R R ), which is equiv alent to D b (pro j R op ) , the b ounded derived ca teg ory o f finitely gener ated gra ded pro jective right R -mo dules. Since R is no etheria n and has finite glo bal dimensio n, D b (pro j R op ) is eq uiv alent to D b (grmo d- R op ) , the b ounded de r ived categor y o f finitely g enerated 28 J.-W. HE AND Q.-S. WU graded rig ht R - mo dules . In summary we have the equiv alence (which is established in [BGS]) of tria ngulated ca tegories if R is no etheria n and o f finite g lobal dimension D b (grmo d- E ( R )) ∼ = D b (grmo d- R op ) . Moreov er, we a ssume tha t R is a no etherian Ko szul AS-r egular a lgebra. Then the Adams connected DG algebr a A is Koszul Adams AS-re gular DG a lgebra. Hence in the left hand of (6), grmo d- S op is dua l to grmo d- E ( R ). Since AD c ( A op ) is equiv a- lent to D b (grmo d- R op ), the full triang ula ted sub category AD f d ( A op ) is equiv alen t to D b f d (grmo d- R op ), the triangulated s ubca tegory consisting o f ob jects X such that H X is finite dimensiona l. Hence in the r ight hand o f (6), AD c ( A op ) / AD f d ( A op ) ∼ = D b (grmo d- R op ) / D b f d (grmo d- R op ) which is equiv a le n t to D b (qgr R op ) by [Mi]. In s ummary we get the BGG co rresp on- dence esta blished in [MS ] grmo d- E ( R ) ∼ = D b (qgr R op ) . 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