Serre Duality for Noncommutative Projective Schemes

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 3, March 1997, Pages 697–707 S 0002-9939(97)03782-9

SERRE DUALITY FOR NONCOMMUTATIVE PROJECTIVE SCHEMES

AMNON YEKUTIELI AND JAMES J. ZHANG

(Communicated by Ken Goodearl)

Abstract. We prove the Serre duality theorem for the noncommutative projective scheme proj A when A is a graded noetherian PI ring or a graded noetherian AS-Gorenstein ring.

0. Introduction

Let k be a field and let A = i 0 Ai be an N-graded right noetherian k-algebra. ≥ In this paper we always assume that A is locally finite in the sense that each Ai is L finite dimensional over k.IfA0=k,thenAis called connected graded. We denote by Gr A the category of graded right A-modules and by gr A the subcategory

consisting of noetherian right A-modules. The augmentation ideal A 1 = i 1 Ai ≥ ≥ is denoted by m.LetM= iMibe a graded right A-module and x a homogeneous element of M.Wesayxis m-torsion if xmn =0forsomen.Allm-torsion elementsL form a submodule of M,L which is denoted by τ(M). The module M is call m- torsion (respectively m-torsion-free)ifτ(M)=M (respectively τ(M)= 0 ). Let Tor A denote the subcategory of Gr A of all m-torsion modules and tor {A }denote the intersection of Tor A and gr A. The (degree) shift of M, denoted by s(M), is n deﬁned by s(M)i = Mi+1, and we use M(n)forthen-th power of a shift s (M). Given an N-graded right noetherian ring A, the noncommutative projective scheme of A is deﬁned as Proj A := (QGr A, ,s)whereQGrAis the quotient category A Gr A/Tor A, is the image of AA in QGr A and s is the degree shift. Sometime it is easier toA work on noetherian objects, and the triple proj A := (qgr A, ,s) is also called the projective scheme of A,whereqgrA=grA/tor A.(See[AZ]A for more details.) The canonical functor from Gr A to QGr A (and from gr A to qgr A) is denoted by π.IfM Gr A, we will use the corresponding calligraphic ∈ letter for π(M) if no confusion occurs. For example, = π(AA). M 0 A Let X =projA. The global section is H (X, )=HomQGr A( , )andthe N A N cohomology is Hi(X, )=Exti ( , ) for all i>0. In the commutative case, N QGr A A N

Received by the editors September 20, 1995 and, in revised form, January 24, 1996. 1991 Mathematics Subject Classiﬁcation. Primary 14A22, 16W50, 16E30. Key words and phrases. noncommutative projective scheme, Serre duality theorem, Watts’ theorem, dualizing sheaf, balanced dualizing complex. The ﬁrst author is supported by an Allon Fellowship and is incumbent of the Anna and Maurice Boukstein Career Development Chair. The second author is supported by the NSF.

c 1997 American Mathematical Society

697

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the Serre duality theorem [H2, III.7.6] says there is an object ω0 in qgr A such that

0 0 d (0-1) θ :Homqgr A( ,ω ) H ( X, )∗ M −→ M

is a natural isomorphism for all qgr A. Here ∗ is the vector space dual and d is the cohomological dimension ofM∈X deﬁned by

cd(X)=max i Hi(X, ) =0forsome QGr A . { | M 6 M∈ } It is easy to see that such ω0 is unique, and it is called a dualizing sheaf on X =projA. In this paper we will prove a version of the Serre Duality Theorem for d noncommutative projective schemes proj A. The ﬁrst idea is to study H (X, )∗ as a functor from qgr A to Mod k. In general Mod A is the category of (un-− graded) right A-modules. Using a graded version of Watts’ theorem we will prove (0-1) for noncommutative rings under some hypotheses. These hypotheses can be checked for a class of rings including noetherian PI rings. The second idea is to use the balanced dualizing complex which was introduced and studied in [Ye]. The existence of balanced dualizing complex implies (0-1) holds for some ω0 constructed from the dualizing complex. Since noetherian AS-Gorenstein rings admit balanced dualizing complexes, (0-1) holds for such rings. In general we should not expect a dualizing sheaf ω0 to exist in qgr A,andwe can ask if (0-1) holds for some ω0 in QGr A. This was answered aﬃrmatively by P. Jørgensen recently in [Jø]. However when one uses the duality, there will be many advantages if ω0 is in qgr A. Hence it is still important to work out for what other classes of graded rings the dualizing sheaf ω0 is in qgr A.

1. Watts’ Theorem Let ( , ,s) be a triple consisting of a k-linear category , an object in C A C A C and an automorphism s of . Both (gr A, AA,s)and(qgrA, ,s) are examples of such triples. All functorsC in this paper will preserve the k-linearA structure. As the degree shift, sn will be denoted by (n) for all n Z. Let Γ denote the representing functorM Hom (s i , ).M Note that in∈ [AZ, Sec. 4], Γ is the i Z − ∈i C A − functor i Hom ( ,s ( )). But it is easy to see that these two Γs are naturally ∈Z C AL − isomorphic, and hence we will not distinguish them. For in ,Γ( )isaZ- graded kL-module with degree i part being Hom ( ( i), M). ByC compositionM of C A − M morphisms, Γ( )isaZ-graded k-algebra and Γ( )isaZ-graded right Γ( )- A i M A module with multiplication mr = ms− (r) for all m Hom ( ( i), )andr Hom ( ( j), ). ∈ C A − M ∈ If CFAis− a covariantA functor from to Mod k,thenFdenotes the functor F (si( )). If F is a contravariantC functor from to Mod k,thenF denotes i Z ∈ i− q C i F (s− ( )). In the case of a bi-functor Ext ( , ), we have L ∈Z − C M N L q q q Ext ( , ) = Ext ( , (i)) = Ext ( ( i), ). C M N ∼ C M N ∼ C M − N i i M∈Z M∈Z Now let F : Mod k be a contravariant functor. We use F( )fortheZ- C−→ A graded k-module F ( ( i)). For every x F ( ( i)) and r Γ( )j = i Z ∈ A − ∈ A − i ∈ A Hom ( ( j), ), we deﬁne a right Γ( )-action by xr = F (s− (r))(x). Clearly C A − A L A

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xr F ( ( i j)) and, for every w Γ( )l, ∈ A − − ∈ A i i j x(rw)=F(s− (rw))(x)=F(s− (rs− (w)))(x) i i j i j i = F (s− (r)s− − (w))(x)=F(s−− (w))F (s− (r))(x) =(xr)w. Hence F ( ) has a natural graded right Γ( )-module structure. A A Proposition 1.1. Let ( , ,s) be a triple as above, and let F be a contravariant functor from to ModCk.A Then there is a natural transformation σ : F C −→ HomGr A(Γ( ),F( )) such that σ ( i) are isomorphisms for all i Z,whereA= Γ( ). − A A − ∈ A Proof. Let be an object in . For every x F ( )andmj Hom ( ( j), ), M C ∈ M ∈ C A − M we deﬁne σ : F ( ) HomGr A(Γ( ),F( )) by M M −→ M A x σ (x):mj F (mj)(x) . 7−→ { M 7−→ } We claim that σ (x)isanA-homomorphism. For every r Ai =Hom ( ( i), ), M ∈ C A− A σ (x)mapsmjrto M j j F (mj r)(x)=F(mjs− (r))(x)=F(s− (r))F (mj )(x)

=(F(mj)(x))r = σ (x)(mj )r. M Hence σ (x)isanA-homomorphism and consequently σ is well deﬁned. For everyM morphism f : in , consider the followingM diagram: M−→Nσ C F ( ) N HomGr A(Γ( ),F( )) N −−−−→ N A

F (f) HomGr A(Γ(f),F ( )) A  σ  F ( ) M HomGr A(Γ( ),F( )). yM −−−−→ yM A For y F ( )andm Γ( )j, ∈ N ∈ M HomGr A(Γ(f),F( ))σ (y): m σ (y)(fm)=F(fm)(y) A N 7−→ N and σ F (f)(y): m F (m)(F (f)(y)) = F (fm)(y). M 7−→ Hence σ is a natural transformation. If = ( i), Γ( ( i)) = Γ( )( i). For every x F ( ( i)), σ ( i)(x)isan M A − A − ∼ A − ∈ A − A − A-homomorphism sending 1 to x. Hence σ ( i) is an isomorphism for all i. A − The next lemma can be proved easily by using the Five Lemma [Ro, 3.32] as in the proof of Watts’ original theorem [Ro, 3.33]. Lemma 1.2. Let F and H be two contravariant left exact functors from to an- C other category .Suppose i I is a set of generators such that every object D {G } M∈C is ﬁnitely presented by i I.Ifσ:F H is a natural transformation and σ i is an isomorphism for all{G i} I,thenσ−→is a natural isomorphism from F to H. G ∈ Now we apply Proposition 1.1 and Lemma 1.2 to (gr A, AA,s). Theorem 1.3 (Watts’ Theorem for gr A). Let A be a graded right noetherian algebra. Let F be a contravariant left exact functor from gr A to Mod k. Then F = HomGr A( ,B) for a Z-graded (not necessarily noetherian) right A-module ∼ − B = F (AA).

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Proof. First, the functor Γ is naturally isomorphic to the identity functor and the algebra A is naturally isomorphic to Γ(AA). Hence, by Proposition 1.1, σ : F −→ HomGr A( ,B) is a natural transformation. Note that HomGr A( ,B)isaleft − − exact functor. Since A is right noetherian, A(i) i Z is a set of generators and every object in gr A is finitely presented.{ By Proposition| ∈ } 1.1, the hypotheses of Lemma 1.2 hold. Therefore F = HomGr A( ,B). ∼ − Similarly a version of Watts’ theorem holds for the triple (qgr A, ,s). By A [AZ, 4.5], if A satisfies the condition χ1,thenAcan be recovered from the triple i (qgr A, ,s)uptom-torsion. Recall that A satisfies χi if Ext (A/m,M)are A gr A finite dimensional over k for all graded noetherian right A-modules M.Ifχiholds for all i,thenwesayAsatisfies χ.

Theorem 1.4 (Watts’ Theorem for qgr A). Let A be an N-graded right noetherian algebra satisfying χ1.LetFbe a contravariant left exact functor from qgr A to Mod k.IfB=F( ) 0is a noetherian right A-module, then F ∼= Homqgr A( , ) where = π(B). A≥ − B B Proof. Since A satisﬁes χ1, by [AZ, 4.6(2)] we may assume that A =Γ( ) 0.By A≥ Proposition 1.1, σ : F HomGr Γ( )(Γ( ),F( )) is a natural transformation. Clearly the following is−→ also a naturalA transformation:− A

HomGr Γ( )(Γ( ),F( )) HomGr A(Γ( ) 0,F( ) 0) HomQGr A(πΓ( ), ). A − A → − ≥ A ≥ → − B Since πΓ ∼= Idqgr A [AZ, 4.5] and B is noetherian, we have a natural transformation

η :HomGr Γ( )(Γ( ),F( )) HomQGr A( , )=Homqgr A( , ). A − A −→ − B − B Since A satisﬁes χ1, by [AZ, 3.13], for i 0, Homqgr A( ( i), ) = Bi = F ( ( i)) = HomGr Γ( )(Γ( ( i)),F( )). A − B ∼ A − ∼ A A − A Hence η ( i) is an isomorphism for i 0. Thus ησ is a natural transformation A − from F to Homqgr A( , ) such that (ησ) ( i) is an isomorphism for i 0. For every p, ( −i) Bi p is a set ofA generators− for the noetherian category {A − | ≥ } qgr A. Then the hypotheses of Lemma 1.2 hold and therefore F = Homqgr A( , ). ∼ − B

2. Duality theorems Let A be a graded right noetherian algebra and let M and N be graded right noetherian A-modules. Suppose the projective dimension of M, denoted by pd(M), is d< .ThenExti (M, ) = 0 for all i d + 1. Hence a short exact sequence ∞ gr A − ≥ 0 K L N 0 −→ −→ −→ −→ yields a long exact sequence in Mod k

0 Homgr A(M,K) Homgr A(M,L) Homgr A(M,N) −→ −→ −→ −→ Ext1 (M,K) Ext1 (M,L) Ext1 (M,N) gr A −→ gr A −→ gr A −→ ··· ··· Extd (M,K) Extd (M,L) Extd (M,N) 0 . gr A −→ gr A −→ gr A −→

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Applying the contravariant exact functor V V ∗ =Homk(V,k), we obtain 7−→ 0 Homgr A(M,K)∗ Homgr A(M,L)∗ Homgr A(M,N)∗ ←− ←− ←− ←− 1 1 1 Ext (M,K)∗ Ext (M,L)∗ Ext (M,N)∗ gr A ←− gr A ←− gr A ←− ··· ··· d d d Ext (M,K)∗ Ext (M,L)∗ Ext (M,N)∗ 0 . gr A ←− gr A ←− gr A ←− d Thus Extgr A(M, )∗ is a contravariant left exact functor from gr A to Mod k.By −d Theorem 1.3, Extgr A(M, )∗ ∼= HomGr A( ,B) for some graded right A-module − −d i B. By the long exact sequence above, Extgr−A(M, )∗ i 0 is a δ-functor in i { − | ≥ } the sense of [H2, p.205]. Since Extgr A( ,B) i 0 is a universal δ-functor [H2, III.1.4], there is a natural transformation{ − | ≥ } i i d i θ :Ext ( ,B) Ext − (M, )∗ Gr A − −→ gr A − with θ0 being the inverse of the natural isomorphism given in Theorem 1.3 for d the functor F =Extgr A(M, )∗. Hence we have proved part (a) of the following theorem. − Theorem 2.1. Let M be a noetherian graded right A-module with pd(M)=d. (a) For each i 0, there is a natural transformation ≥ i i d i θ :Ext ( ,B) Ext − (M, )∗ Gr A − −→ gr A − where B is a graded right A-module and θ0 is the inverse of the natural isomorphism given in Theorem 1.3. i d i (b) These θ are isomorphisms for all i if and only if Extgr−A(M,A(n)) = 0 for all i 1 and all n Z. ≥ ∈ Proof. (b) If θi is an isomorphism and i 1, then ≥ Extd i (M,A(n)) Exti (A(n),B) =0. gr−A ∼= Gr A ∗ d i Conversely suppose that Extgr−A(M,A(n)) = 0 for all n Z and all i 1. Since ∈ d i≥ A(n) n Z is a set of generators for gr A, by [H2, p.206], Extgr−A(M, )∗ is { | ∈ } d i − coeﬀaceable for all i 1. By [H2, III 1.3A], Extgr−A(M, )∗ i 0 is univer- ≥ i { − | ≥ } sal. The derived functor ExtGr A( ,B) i 0 is always a universal δ-functor [H2, III.1.4]. Two universal{ δ-functors− are| naturally≥ } isomorphic if θ0 is a natural isomorphism. From now on we consider the triple (qgr A, ,s). A Theorem 2.2. Let A be a graded right noetherian ring satisfying χ1.Let be in i M qgr A with max i Extqgr A( , ) =0for some = d< . { | d M N 6 N} ∞ (a) If B = n 0 Extqgr A( , ( n))∗ is a noetherian graded right A-module, then there is a natural≥ transformationM A − L i i d i θ :Ext ( , ) Ext − ( , )∗ qgr A − B −→ qgr A M − for all i,withθ0 being the inverse of the natural isomorphism given in Theorem 1.4. (b) Suppose A satisﬁes χ. Then θi is a natural isomorphism for all i if and only d i if Ext − ( , ( n)) = 0 for all i 1 and all n 0. qgr A M A − ≥

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Proof. The proof of part (a) is very similar to the proof of Theorem 2.1(a), namely using the argument before Theorem 2.1. The only difference is that in this case we d use Theorem 1.4, instead of Theorem 1.3, for the functor F =Extqgr A( , )∗. For part (b) we need the following modification. M − Suppose A satisfies χ.Ifθiis an isomorphism and i 1, then, by Serre’s finiteness theorem [AZ, 7.5], ≥

d i i i Ext − ( , ( n))∗ = Ext ( ( n), ) = H (X, (n)) = 0 qgr A M A − ∼ qgr A A − B ∼ B d i for all n 0. Conversely if Extqgr− A( , ( n))∗ =0foralli 1andalln 0, d i M A − ≥ then Extqgr− A( , )∗ i 0 is a universal δ-functor because ( n) n p is a set{ of generatorsM − for| qgr≥ A}for every p. Two universal δ-functors{A − are| naturally≥ } isomorphic, and hence each θi is a natural isomorphism.

Let M = i Mi be a Z-graded module. We say M is left bounded (respectively right bounded)ifMi =0foralli 0 (respectively i 0). If A satisﬁes χ, then by [AZ,L 7.5], each Hq(X, (i)) is ﬁnite dimensional over k and Hd(X, ):= q A A iH(X, (i)) is right bounded for all q 1. For a graded module M = i Mi, A q ≥ let M ∗ denote i M ∗ i. Hence H (X, )∗ is left bounded for all q 1. Now we proveL the Serre duality− theorem. A ≥ L L Theorem 2.3 (The Serre Duality for proj A). Let A be a graded right noetherian ring satisfying χ1, and assume that cd(projA)=d< . d ∞ d 1. Suppose H (X, )∗ 0 is a noetherian right A-module. Denote π(H (X, )∗) A ≥ A by ω0. (a) For each i, there is a natural transformation

i i 0 d i θ :Ext ( ,ω ) H − ( X, )∗ qgr A − −→ − where θ0 is a natural isomorphism. (b) Suppose A satisﬁes χ. Then θi are natural isomorphisms for all i if and d i 0 only if H − (X, ) is ﬁnite dimensional for d>i 1and H (X, ) is left bounded. 2. Conversely,A if there is a natural isomorphism≥ A

0 0 d θ :Homqgr A( ,ω ) H ( X, )∗ − −→ − 0 d for an object ω in qgr A,thenH(X, )∗ 0 is a noetherian right A-module and 0 d A ≥ ω = π(H (X, )∗). ∼ A Proof. Part 1 is an immediate consequence of Theorem 2.2. It remains to prove 0 0 part 2. Since A satisﬁes χ1 and ω is an object in qgr A, by [AZ, 4.5], Γ(ω ) 0 is 0 0 ≥ a noetherian right A-module and ω ∼= π(Γ(ω ) 0). Then part 2 follows from the isomorphism ≥

0 0 d d Γ(ω ) = Hom( ( i),ω )= H (X, ( i))∗ =H (X, )∗. ∼ A − ∼ A − A i i M M

Following the Serre duality theorem in the commutative case [H2, III.7.6] and Theorem 2.3, it is reasonable to make the following deﬁnition.

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Definition 2.4. (a) Let X =projAbe a projective scheme with cd(X)=d.An object ω0 in qgr A is called a dualizing sheaf of X if there is a natural isomorphism 0 0 d θ :Homqgr A( ,ω ) H ( X, )∗. − −→ − (b) Suppose X has a dualizing sheaf ω0.WesayXis classical Cohen-Macaulay if i i 0 d i θ :Ext ( ,ω ) H − ( X, )∗ qgr A − −→ − are natural isomorphisms for all i. It is easy to see that a dualizing sheaf is unique, up to isomorphism, if it exists. By the definition, existence of a dualizing sheaf is independent of the choice of the shift operator. If proj A has a dualizing sheaf ω0, then, for each i,there i i 0 d i is a natural transformation θ :Extqgr A( ,ω ) H − ( X, )∗. Under some hypotheses Theorem 2.3 gives a sufficient and− necessary−→ condition− for a dualizing sheaf to exist. Namely, to show the existence of a dualizing sheaf is equivalent to d showing that H (X, )∗ 0 is a noetherian right A-module. By [AZ, 7.9], if A is A ≥ d noetherian and satisfies χ,thenH(X, )∗ is a noetherian left A-module. If A is commutative, then left and right moduleA structures are the same. Hence Theorem 2.3 re-proves the Serre duality theorem in the commutative case. 3. PI rings

d It is unknown if H (X, )∗ is a noetherian right A-module for all noetherian locally finite algebras satisfyingA χ. This problem is solved for the following case. Let A be a noetherian graded algebra such that each m-torsion-free prime factor of A has a homogeneous normal element of positive degree. By [AZ, 8.12(2)], A satisfies χ and cd(projA) Kdim(AA) 1 < where Kdim is Krull (Rentschler- Gabriel) dimension. Familiar≤ examples− of such∞ rings are noetherian PI rings and quantum matrix algebras. A bimodule is called noetherian if it is both left and right noetherian. Theorem 3.1. Let A be a graded left and right noetherian algebra such that each m-torsion-free prime factor of A has a normal element of positive degree. Let M i be a graded noetherian A-bimodule. Then H (X, )∗ are noetherian A-bimodules for all i>0. M i Proof. By [AZ, 7.9], H (X, )∗ are left noetherian for i>0. We need to show M i that these are right noetherian. By [AZ, 7.2.(2)], it suffices to show that Hm(M)∗ i are right noetherian for all i,whereHm(M) is defined by i i n i (3-1) Hm(M) = lim Extgr A(A/m ,M)= lim Extgr A(A/A n,M). n ∼ n ≥ →∞ →∞ i Since limn is exact, Hm(M)∗ i Z is a δ-functor in the sense of [H2, p. 205]. By [AZ,→∞ 7.2 (2) and{ 3.6(3)], we| have∈ the} following statements: i (i) Hm( )∗ =0foralli<0andi>cd(projA)+1. 0 − (ii) Hm(N)∗ = τ(N)∗ and hence it is always finite dimensional. 0 i (iii) If N is finite dimensional, then Hm(N)∗ = N ∗ and Hm(N)∗ =0foralli 1. i ≥ (iv) Hm(N)∗ is left bounded for all i and all noetherian A-modules N. i Next we use induction on Kdim(MA)toprovethatHm(M)∗ are right noetherian for all i.IfKdim(MA) = 0, i.e., M is finite dimensional, then the statement is obvious. Now let M be a bimodule with Kdim(MA)=n>0 and suppose the

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statement holds for all noetherian bimodules N with Kdim(NA)

(4-2) HomQGr A( , ) = lim HomGr A(M n,N) ∼ n ≥ M N →∞ for all M gr A and N Gr A (see [AZ, (2.2.1)]). By the exactness of the functors ∈ ∈ M M n and limn , and arguing as in the proof of [Ye, 2.2], we see that the 7−→ ≥ →∞

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+ derived bi-functor R limn HomGr· A(M · n,N·) is deﬁned for N · D (Gr A)and →∞ ≥ ∈ M· D(Gr A). We may assume that N · is a bounded below complex of injectives [Ye,∈ 4.2], and then the R can be omitted. Now ﬁx N ·. By [H1, I.7.1(i)] the isomorphism (4-2) implies an isomorphism

RHomQGr· A( ·, ·) = R lim HomGr· A(M · n,N·) ∼ n ≥ M N →∞ b for M · D (Gr A). The lemma is proved by passing to cohomologies. ∈ f

Observe that by taking M · = A in (4-1) we get q q H (X, ·) = lim ExtGr A(A n,N·). ∼ n ≥ N →∞ q q+1 By [AZ, 7.2(2)], H (X, ) is isomorphic to the local cohomology Hm (M)for q M q 1, where Hm( ) is defined by (3-1). By [Ye, 4.17 and 4.18], if A has a balanced dualizing≥ complex− (see [Ye, 3.3 and 4.1] for the definition), then there is a version of the local duality theorem. We will use the local duality theorem to prove the Serre duality theorem for such A. Theorem 4.2. Let A be a connected graded left and right noetherian ring and let X =projA. Suppose that A has a balanced dualizing complex R·. b 1. Let M · Df (Gr A) and q Z. Then ∈ ∈ q q (4-3) H (M ) Ext− (M ,R) , m · ∼= Gr A · · ∗ and this is a locally finite and right bounded module. 2. Let · = π(R·) and · = π(M ·). Then R M q q+1 (q+1) H (X, ·)∗ = lim H (M · )∗ = Ext− ( ·, ·) ∼ n m n ∼ QGr A M →∞ ≥ M R q (4-4) =Ext− ( ·, ·[ 1]). QGr A M R − In this sense, we call ·[ 1] a dualizing complex for proj A. R − i 3. A satisfies the condition χ and cd(X)=d,where (d+1) = min i H ( ·) = 0 . − { | R 6 } 0 (d+1) 0 4. Let ω =H− ( ·). Then ω is the dualizing sheaf for proj A. 5. The following conditionsR are equivalent: (a) X =projAis classical Cohen-Macaulay. 0 (b) ω [d +1] is isomorphic to · in D(QGr A). q R (c) H (R·) is m-torsion for q> (d+1). − Remark. The dualizing complex R· is a complex of bimodules, but when passing to QGr A we forget the left module structure of R·. q Proof. 1. This is [Ye, 4.18]. Note that ExtGr− A(M ·,R·) is a noetherian left A- q module, so Hm(M ·) is a locally finite, right bounded module. 2. The exact sequence

0 A n A A/A n 0 , −→ ≥ −→ −→ ≥ −→ when considered as a triangle in D(Gr A), yields a long exact sequence

q q q+1 q+1 H (M·)0 ExtGr A(A n,M·) ExtGr A(A/A n,M·) H (M ·)0 ···→ → ≥ → ≥ → →···

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b q for any M · D (Gr A), where H (M ·)0 is the degree zero part of the q-th coho- ∈ f mology of M ·. First taking limn andthenreplacingM· by M · l in the above, we have →∞ ≥ q q q +1 q +1 H (M· l)0 H ( X, ·) H m (M · l) H (M · l)0 ···−→ ≥ −→ M −→ ≥ −→ ≥ −→··· where · = π(M ·)=π(M·l). Taking k-linear duals and using the fact that M ≥ q lim H (M · l)0∗ =0, l ≥ →∞ we obtain the first isomorphism of (4-4). The second one is a consequence of part 1 and Lemma 4.1. q 3. By part 1, for every noetherian right module M,Hm(M) is right bounded for all q. By [AZ, 3.8(3)], A satisfies χ. By [Ye, 4.2], we may assume R· is a minimal injective complex (of right A- i i i +1 modules), i.e., each R is an injective hull of ker(∂i), where ∂i : R R is the −→ coboundary map of R·.Sinceπis exact and has a right adjoint functor [AZ, (2.2.2)], + · = π(R·) is a minimal injective complex in D (QGr A). By the definition of d, R q (d+1) q we have H ( ·) = 0 for all q< (d+1),andH− ( ·) = 0. Hence =0for all q< (d+R 1) and, by (4-4), − R 6 R − q (q+1) H (X, ) = Ext− ( , ·)∗ =0. M ∼ QGr A M R + Therefore cd(X) d. By the definition of dualizing complex, R· Df (Gr A), + ≤ 0 (d+1) ∈ whence · D (QGr A)andω := H− ( ·) qgr A. By (4-3) R ∈ f R ∈ 0 (d+1) d (4-5) HomQGr A( ,ω )=Ext− ( , ·) = H (X, )∗. M ∼ QGr A M R ∼ M Thus Hd(X, ω0) =0andcd(X)=d. 4. Follows immediately6 from (4-5). 5. (a) (b) By Definiton 2.4, X =projAis classical Cohen-Macaulay if ⇔ i 0 d i and only if ExtQGr A( ,ω ) ∼= H − (X, )∗ for all i and . By (4-4), this is equivalent to M M M i 0 i (d+1) i (4-6) Ext ( ,ω )=Ext − ( , ·)=Ext ( , ·[ (d +1)]) QGr A M ∼ QGr A M R QGr A M R − for all i and . But (4-6) holds if and only if ·[ (d + 1)] is a minimal injective M0 0 R − resolution of ω . This is saying that ω [d + 1] is quasi-isomorphic to ·. 0 R q (b) (c) The complex ω [d+1] is quasi-isomorphic to · if and only if H ( ·)= ⇔ q q R q R 0 for all q> (d+1).ButH ( ·)=π(H (R·)). Therefore H ( ·)=0ifandonly q − R R if H (R·)ism-torsion.

Theorem 4.2.2 tells that if A has a balanced dualizing complex R·,thenprojA + has a dualizing complex ·[ 1] D (QGr A). P. Jørgensen in [Jø, 3.3] proved R − ∈ f recently that a dualizing complex always exists in D+(QGr A). The question of + when a dualizing complex exists in Df (QGr A) is still open. It was proved in [Ye] that the following algebras have balanced dualizing complexes: (a) graded noetherian AS-Gorenstein rings; (b) graded noetherian rings ﬁnite over their centers; (c) twisted homogeneous coordinate rings. Finally, we state an immediate corollary of Theorem 4.2 for AS-Gorenstein rings. Recall that a connected graded ring A with injective dimension d + 1 is called AS- Gorenstein if ExtA· (k, A) ∼= k(e)[ (d +1)]forsomee Z(see [Ye, 4.13(ii)]). By [Ye, 4.14] (which holds in this case− too; the proof is exactly∈ the same as the one of

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σ [Ye, 4.14]), R· = A( e)[d + 1] is a balanced dualizing complex over A for some algebra automorphism−σ. Corollary 4.3. Let A be a connected graded left and right noetherian AS-Goren- stein ring of injective dimension d +1,andletX=projA. 1. A satisfies the condition χ. 2. cd(X)=d,andω0= ( e)is the dualizing sheaf for X. 3. X is classical Cohen-Macaulay.A− Remark. Presumably the Brown Representability Theorem, as used in [Jø], would 0 imply that if the functor (Γm( ))∗,whereΓm( ) is defined to be Hm( ) [Ye, page 56], has finite cohomological dimension,− then− it is representable on D−(Gr A)by (RΓm(A))∗. However this still wouldn’t make (RΓm(A))∗ into a balanced dualizing complex in the sense of [Ye]. We thank the referee for calling attention to this point. References

[AZ] M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. in Math. 109 (1994), 228–287. MR 96a:14004 [H1] R. Hartshorne, Residues and Duality, Lecture Notes in Math. 20, Springer-Verlag, Berlin, 1966. MR 36:5145 [H2] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. MR 57:3116 [Jø] P. Jørgensen, Serre-Duality for Tails(A), Proc. Amer. Math. Soc., this issue. [Ro] J. J. Rotman, An Introduction to Homological Algebra, Academic Press, New York, 1979. MR 80k:18001 [Ye] A. Yekutieli, Dualizing complexes over noncommutative graded algebras,J.Alg.153 (1992), 41–84. MR 94a:16077

(A. Yekutieli) Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel E-mail address: [email protected]

(J.J.Zhang)Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195 E-mail address: [email protected]

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