A CRITERION FOR GORENSTEIN ALGEBRAS TO BE REGULAR

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1 PROCEEDINGS OF THE MERICN MTHEMTICL SOCIETY Volume 139, Number 5, May 2011, Pages S (2010) rticle electronically published on October 4, 2010 CRITERION FOR GORENSTEIN LGEBRS TO BE REGULR X.-F. MO ND Q.-S. WU (Communicated by Birge Huisgen-Zimmermann) bstract. In this paper we give a criterion for a left Gorenstein algebra to be S-regular. Let be a left Gorenstein algebra such that the trivial module k admits a finitely generated minimal free resolution. Then is S-regular if and only if its left Gorenstein index is equal to infi Ext depth Furthermore, is Koszul S-regular if and only if its left Gorenstein index is depth = infi Ext depth s applications, we prove that the category of S-regular algebras is a tensor category and that a left Noetherian p-koszul, left Gorenstein algebra is S-regular if and only if it is p-standard. This generalizes a result of Dong and the second author. Introduction connected graded algebra is called left (resp. right) Gorenstein if the Ext-group from the trivial left (resp. right) module to itself is 1-dimensional. Gorenstein algebras have a multitude of connections to algebraic topology, representation theory and non-commutative algebraic geometry. In the last twenty years, Gorenstein algebras have been intensively studied in literature. S-regular (resp. S-Gorenstein) algebras, which were introduced by rtin and Schelter ([S]), are Gorenstein algebras with finite global (resp. injective) dimension. S-regular algebras are thought to be the coordinate rings of the corresponding non-commutative projective spaces in the non-commutative projective geometry. One of the central questions in noncommutative projective geometry is to classify non-commutative projective spaces, or equivalently, to classify the corresponding S-regular algebras. In [DW], Dong and the second author proved that any Noetherian Koszul standard S-Gorenstein algebra is S-regular by using Catelnuovo-Mumford regularity. The motivation of this paper is to study when left Gorenstein algebras are S-regular. The following result is Theorem 2.2 and Corollary 2.4. Theorem. Let be a left Gorenstein algebra such that the trivial module k admits a finitely generated minimal free resolution. Then is S-regular if and only if Gor = infi Ext depth Received by the editors November 6, 2009 and, in revised form, May 9, Mathematics Subject Classification. Primary 16E65, 16W50, 16E30, 16E10, Key words and phrases. Connected graded algebra, Gorenstein algebra, S-Gorenstein algebra, S-regular algebra c 2010 merican Mathematical Society Reverts to public domain 28 years from publication

2 1544 X.-F. MO ND Q.-S. WU Furthermore, is Koszul and S-regular if and only if Gor =depth = infi Ext depth For the definition of Gor, see Definition 2.1. Note that k admits a finitely generated minimal free resolution of graded -modules if is a left Noetherian graded algebra. By using the above theorem, we prove that the tensor product of two S-regular graded algebras is also S-regular, and we prove the following proposition in Proposition 3.3. Proposition B. Let be a left Noetherian p-koszul left Gorenstein algebra (p 2). Then is S-regular if and only if is p-standard. This proposition is a generalization of [DW, Theorem 4.10]. The p-standard left Gorenstein algebra is defined in Definition Preliminaries Throughout this paper, k is a fixed field. k-algebra is called N-graded if has a k-vector space decomposition = i 0 i such that i j i+j for all i, j 0. n N-graded algebra is called connected graded if 0 = k. In the following, will always be a connected graded k-algebra if no special assumption is emphasized and m will be its maximal graded ideal i>0 i. left-module M, with a decomposition M = i Z M i of k-vector spaces, is called graded if i M j M i+j for all i, j. Graded right -modules are defined similarly. Obviously the residue field k has a trivial graded left or right -module structure via the canonical surjection ε : k. The opposite algebra of is denoted by op. ny graded right -module can be identified with a graded left op -module. graded-module M is called locally finite if each graded piece M i is a finitedimensional k-vector space; M is called bounded below if M i =0fori 0. Let M and N be two graded left -modules. n -module homomorphism f : M N is said to be a graded homomorphism of degree l if f(m i ) N i+l for all i Z. Let Gr be the category of graded left -modules and graded homomorphisms of degree 0, and let Hom Gr (, ) be the hom-functor in the category Gr. If M Gr, M(n) isthen-th shift of M, with M(n) i = M n+i. The graded vector space of all graded -homomorphisms from M to N is denoted by Hom (M,N) = Hom Gr (M,N(n)). n= For any complex X of graded -modules and d Z, wedenotex[d] asthed-th twisting of X such that (X[d]) i = X d+i. The derived category of Gr is denoted by D(Gr). The full subcategories of D(Gr) consisting of objects which are cohomologically bounded below, bounded above and bounded respectively are denoted by D + (Gr), D (Gr) andd b (Gr). The right derived functor of Hom (, ) is denoted by RHom (, ), and the left derived functor of is denoted by L. The Ext and Tor are defined as Ext i (X, Y )=H i (RHom (X, Y )) and Tor i (X, Y )=H i (X L Y ). bounded above complex L of graded free left -modules is called minimal if d i L (Li ) ml i+1. In this case, the differentials in Hom (L, k) andk L are zero.

3 CRITERION FOR GORENSTEIN LGEBRS TO BE REGULR 1545 bounded below complex I of graded injective -modules is called minimal if ker(d i I )isgradedessentialinii for each i Z. In this case, the complex Hom (k, I) has zero differential. For any graded -module M, its depth is depth M =infi Ext i (k, M) 0}. The projective dimension and injective dimension of M are defined respectively as pd M =supi Ext i (M,N) 0forsomeN Gr}, id M =supi Ext i (N,M) 0forsomeN Gr}. If M is bounded below, then pd M =supi Ext i (M,k) 0} and pd M coincides with the usual (ungraded) projective dimension of M by the existence of minimal free resolution. The graded global dimension of, defined via the (graded) projective dimensions of graded modules, coincides with the usual global dimension of, and it is well known that gl.dim() =pd k =pd opk. 2. Main theorem Let be a connected graded algebra. We say that is left (resp. right) Gorenstein if dim k Ext (k, ) =1(resp. dim k Ext op(k, ) = 1), where Ext (k, ) = i Z Exti (k, ). For a left Gorenstein graded algebra, there is some integer l such that (1) Ext i 0, i depth (k, ) =, k(l), i =depth. left (resp. right) Gorenstein graded algebra is called left (resp. right) S- Gorenstein (S stands for rtin-schelter) if id < (resp. id op < ). If, further, gl.dim <, thenwesay is left (resp. right) S-regular. The usual definition of S-regular algebras in the literature in the late 1980s and early 1990s requires the algebra to have finite Gelfand-Kirillov dimension. Note that our definition of S-regular algebra differs from this. By [SZ, Theorem 2.4], any left (or right) Noetherian connected graded algebra with finite global dimension has finite Gelfand-Kirillov dimension. Hence there is no difference between our definition and the original definition of S-regular algebra for left (or right) Noetherian connected graded algebras. Definition 2.1. Let be a left Gorenstein algebra. The number l as in (1) is called the left Gorenstein index of, denoted by Gor. For right Gorenstein graded algebras, we can define right Gorenstein index similarly. Note that the left (resp. right) Gorenstein index was called the rtin- Schelter index in [LPWZ]. For any left Gorenstein graded algebra, Gor is closely related to s S-regularity. The following theorem gives a criterion for a left Gorenstein graded algebra to be S-regular. Theorem 2.2. Let be a left Gorenstein algebra such that the trivial graded -module k admits a minimal free resolution consisting of finitely generated - modules. Then the following are equivalent: (1) is left S-regular. (2) is S-regular. (3) Gor infi Ext depth

4 1546 X.-F. MO ND Q.-S. WU (4) The d-th part F d of the minimal free resolution of k has a free basis concentrated in degrees Gor,whered =depth. (5) The d-th part F d of the minimal free resolution of k is generated by one element in degree Gor,whered =depth. If one of the above holds, then Gor = infi Ext depth Proof. (1) (5). Let be a left S-regular algebra with global dimension d. Then k admits a minimal free resolution ε 0 F d Fd 1 F1 F0 k 0, where each F i is finitely generated and F 0 =. Since pd k =gl.dim = d, it is easy to see that depth =pd k = d. Let Gor = l. Then 0 (F 0 ) ( ) ( ) (F d ) k(l) 0 is a finitely generated minimal free resolution of k (l), where M =Hom (M,) for any graded -module M. Hence (F d ) = (l) andf d = ( l). (5) (4). Obvious. (4) (3). It follows from Ext d (k, k) =Hom (F d,k). (3) (2). Let be a left Gorenstein algebra with depth = d and Gor = l infi Ext depth By the assumption, k admits a finitely generated minimal free resolution ε (2) F i F1 k 0, where F i = V i,eachv i is a finite-dimensional graded space and V d is concentrated in degrees l. The short exact sequence (3) 0 m ι ε k 0 induces the following long exact sequence: Ext d (k, m) l Ext d Ext (k, ) d (k,ε) l l Ext d (k, k) l. s a sub-quotient of Hom (F d, m), Ext d (k, m) is concentrated in degrees > l. Hence Ext d (k, m) l = 0. Since Ext d (k, ) l = k 0,Ext d (k, ε) l 0. There exists f Hom (F d,) such that f =0andεf 0. shom (F d,k) = Hom k (V d,k), there exists v V d such that εf(v) = 1. This implies that f(v) =1 for some v V d and f : F d is surjective. Therefore, f splits and F d = kerf ( kv). Since kerf is graded free, kerf = X d for some graded space X d.thenf d = (kv X d ). Since f =0,wehave (F d+1 ) X d. Let Q i =Hom (F i,)andδ =Hom (,). Since (2) is a finitely generated minimal free resolution, the complex (Q,δ): 0 Q 0 δ Q 1 δ δ Q i δ consisting of finitely generated graded free right -modules is minimal. Obviously, Q d = (kf ) (Homk (X d,k) ) and δ(f) =0. Since is left Gorenstein, by (1), there is an integer d such that H i (Q,δ)=Ext i 0, i d, (k, ) = kf, i = d.

5 CRITERION FOR GORENSTEIN LGEBRS TO BE REGULR 1547 In the following, we show by induction on j (0 j d) thatq d j admits a decomposition Q d j =(W d j R d j ) with W d = kf, R d =Hom k (X d,k), δ(w d j ) W d j+1, δ(r d j ) R d j+1, j =1,,d. Suppose this is true for j 1. We choose a graded free right -module decomposition Q d j =(E F S) of Q d j such that (4) δ(e) W d j+1, δ(f ) R d j+1, and dim k (E F ) is maximal among all such graded free right -module decompositions of Q d j satisfying (4). Since Q d j is finitely generated, such a maximal decomposition exists. We claim that S = 0. Suppose on the contrary that S 0. SinceS is finite-dimensional, there is a non-zero element x S with the minimal degree, i.e., x =min s s S}. Let δ(x) =δ 1 (x)+δ 2 (x), with δ 1 (x) W d j+1, δ 2 (x) R d j+1. Then δ(δ 1 (x)) = δ(δ 2 (x)) (W d j+2 ) (R d j+2 ) = 0. Hence both δ 1 (x) and δ 2 (x) are cocycles. When j =1,δ 1 (x) kf m and δ 2 (x) Hom k (X d,k) m by the minimality of (Q,δ). Since H d (Q,δ)=kf, bothδ 1 (x) andδ 2 (x) are coboundaries. When j 2, δ 1 (x) andδ 2 (x) are coboundaries since H d j+1 (Q,δ)=0. Hence there exist α 1,α 2 Q d j such that δ(α 1 )=δ 1 (x) andδ(α 2 )=δ 2 (x). For any α T, wedenoteα = α + α with α T k and α T m. It follows from the minimality of x that α 1 (E F ) m. Clearly, ᾱ 1 either belongs to E F or it does not. If ᾱ 1 E F,then(k(x α 1 ) ) ((E F S/kx) ) = 0. Indeed, if for any a, (x α 1 ) a (E F S/kx), thenwehavex a = (x α 1 ) a +ᾱ 1 a + α 1 a ((E F S/kx) ) (kx ) =0. Hence (k(x α 1 ) E F S/kx) =(E F S). Butthenδ(x α 1 )=δ 2 (x) R d j+1 and dim k (k(x α 1 ) E F )=dim k (E F ) + 1, which contradicts the maximality property of the chosen decomposition. Inthecaseofᾱ 1 E F,letᾱ 1 = e 1 + f 1 + s 1,wheree 1 E,f 1 F and 0 s 1 S. Then(kα 1 ) ((E F ) ) = 0. Indeed, if α 1 a (E F ), then ᾱ 1 a = α 1 a α 1 a (E F ), which implies that s 1 a =(ᾱ 1 e 1 f 1 ) a ((E F ) ) (S ) =0. Hence(kα 1 E F ) is a graded free submodule of (E F S). Sinces 1 = α 1 α 1 e 1 f 1 (kα 1 E F ), thereexists a homomorphism of graded right -modules g :(E F S) (kα 1 E F ) g(e) =e, for any e E, g(f) =f, for any f F, such that g(s) =0, for any s S/ks 1, g(s 1 )=s 1. Then g(α 1 )=α 1,andg is a surjective homomorphism of graded right -modules which splits. It is easy to see that the inclusion map from (kα 1 E F ) to

6 1548 X.-F. MO ND Q.-S. WU (E F S) is a right inverse of g. Then (E F S) =(kα 1 E F ) ker(g). Since ker(g) is bounded below, it is free. Now δ(α 1 ) W d j+1 and dim k (kα 1 E F )=dim k (E F )+1. This also contradicts the maximality property of the chosen decomposition. Therefore, S =0. Now let W d j = E and R d j = F.ThenQ d j admits a decomposition Q d j =(W d j R d j ), with δ(w d j ) W d j+1 and δ(r d j ) R d j+1. By induction, we have proved that for any j =0,,d, Q d j admits a decomposition Q d j =(W d j R d j ), with δ(w d j ) W d j+1 and δ(r d j ) R d j+1 for all j 1. For any i =0,,d,letP i = W i. Then the subcomplex (P,δ): 0 P 0 δ P 1 δ δ P d 1 δ P d 0 of (Q,δ) satisfies that H i (P 0, i d,,δ)= kf, i = d. Thisshowsthat(P,δ) is a minimal free resolution of k. Hence is a left Sregular graded algebra with gl.dim = d. Thisimpliesthat ε 0 F d Fd 1 F1 k 0 is a finitely generated minimal free resolution of k. The left Gorensteinness of implies that 0 (F 0 ) ( ) ( ) (F d ) k(l) 0 is a finitely generated minimal free resolution of k(l) asaright-module. Hence Ext i op(k(l),)= 0, i d, and Ext i k, i = d op(k, ) = 0, i d, k(l), i = d, so is S-regular. (2) (1). Obvious. The proof of (3) (2) in Theorem 2.2 is modified from the proof of [FM, Theorem 1]. For any left Noetherian connected graded algebra, k admits a finitely generated minimal free resolution of graded -modules. So we have the following corollary. Corollary 2.3. Let be a left Noetherian, left Gorenstein algebra. Then is S-regular if and only if Gor = infi Ext depth connected graded algebra is called Koszul if for each i 0, the i-th part F i of the minimal free resolution of k isgeneratedindegreei. For more details about Koszul algebras, please see [BGS] and [Sm].

7 CRITERION FOR GORENSTEIN LGEBRS TO BE REGULR 1549 Corollary 2.4. Let be a left Gorenstein algebra such that the trivial graded - module k admits a finitely generated minimal free resolution. Then is Koszul and S-regular if and only if Gor =depth = infi Ext depth Proof. If is Koszul and S-regular, then by Theorem 2.2 and the Koszulity of, Gor =depth = infi Ext depth Conversely, if Gor =depth = infi Ext depth (k, k) i 0}, then is S-regular by Theorem 2.2. We claim that is Koszul. Let d =gl.dim. Then Ext i 0, i d, (k, ) = k(d), i = d. By assumption, d = infi Ext d Hence the trivial module k admits a finitely generated minimal free resolution ε (5) 0 F d Fd 1 F1 k 0, where F d is generated in degrees d. On the other hand, each F i is generated in degrees i by the minimality of (5). Hence F d isgeneratedindegreed. We show by induction on n, n =1,,d,thatF d n is generated in degree d n. Suppose this is true for n 1. Then Ext d n+1 (k, m) is concentrated in degrees >n d 1, since it is a sub-quotient of Hom (F d n+1, m). By the long exact sequence Ext i (k, m) j Ext i (k, ) j Ext i (k, k) j Ext i+1 (k, m) j induced from the short exact sequence (3), we conclude that Ext d n (k, k) isconcentrated in degrees >n d 1. So F d n is generated in degrees d n. Onthe other hand, F d n is generated in degrees d n by the minimality of (5). Hence F d n is generated in degree d n. By induction, every F i is generated in degree i. Therefore is Koszul. 3. pplications In [DW, Theorem 4.10], Dong and Wu proved that any Noetherian Koszul standard S-Gorenstein algebra is S-regular by using Catelnuovo-Mumford regularity. Now we generalize it to the higher Koszul case. First, we recall the definition of p-koszul algebras ([Be], [HL], [YZ]). Let p>1 be an integer. Denote α p : N N as the map pq, n =2q, α p (n) = pq +1, n =2q +1. Definition 3.1. connected graded algebra is called p-koszul if for each i 0, the graded vector space Ext i (k, k) is concentrated in degree α p (i). connected graded algebra is p-koszul if and only if the i-th part F i of the minimal free resolution of k is generated in degree α p (i) foreachi 0. If p =2, then a p-koszul algebra is just a Koszul algebra. Definition 3.2. left Gorenstein algebra is p-standard if Gor =α p (depth ).

8 1550 X.-F. MO ND Q.-S. WU Obviously, 2-standard left Gorenstein algebra is just the standard left Gorenstein algebra as defined in [DW]. Proposition 3.3. Let be a left Noetherian, p-koszul algebra. Then is p- standard left Gorenstein if and only if is S-regular. Proof. If is p-standard left Gorenstein, then Gor = α p (depth ). Since is p-koszul, the graded vector space Ext depth (k, k) is concentrated in degree α p (depth ). Hence Gor = infi Ext depth By Corollary 2.3, is S-regular. Conversely, if is S-regular, then Gor = infi Ext depth (k, k) i 0} by Corollary 2.3. Since is p-koszul, the graded vector space Ext depth (k, k) is concentrated in degree α p (depth ). Hence, Gor = α p (depth ). This implies that is p-standard left Gorenstein. It is easy to see that Proposition 3.3 is a generalization of [DW, Theorem 4.10]. In the remainder of this section, we prove that the tensor product of two Sregular algebras is S-regular. First, we have the following proposition on the left Gorensteinness of algebras under tensor product. Proposition 3.4. Let and B be two left Gorenstein algebras. If either is finite-dimensional or B k has a finitely generated minimal free resolution of finite length, then B is left Gorenstein with depth B B =depth +depth B B and Gor B B =Gor +Gor B B. Proof. Let (F,δ )and(g,δ B ) be minimal free resolutions of k and B k respectively. Then the tensor product (P,δ)of(F,δ )and(g,δ B ) is a minimal complex of graded B-modules, and it is a minimal free resolution of k as a graded B-module. By assumption, we have dim k H(Hom (F,)) = dim k H(Hom B (G,B)) = 1, Hom (F,) k [ depth ](Gor ) and Hom B (G,B) k B [ depth B B](Gor B B). Here we use the twisting of complexes and the shift of graded modules. The readers can see the explanations of this two notations in Section 1. If either dim k < or (G,δ B ) is a bounded complex of finitely generated B-modules, then Hom B (G, B) = Hom B (G,B). Hence H(RHom B (k, B)) = H(Hom B (F G, B)) = H(Hom (F, Hom B (G, B))) = H(Hom (F, Hom B (G,B))) = H(Hom (F,[ depth B B](Gor B B))) = k[ depth depth B B](Gor +Gor B B).

9 CRITERION FOR GORENSTEIN LGEBRS TO BE REGULR 1551 Hence B is left Gorenstein with depth B B =depth +depth B B and Gor B B =Gor +Gor B B. For any connected graded algebra, dim k Ext (k, k) < if and only if has finite global dimension and k has a finitely generated minimal free resolution. If is left Gorenstein with finite global dimension, i.e., is left S-regular, then k has a finitely generated minimal free resolution by [SZ, Proposition 3.1], and so Ext (k, k) is finite dimensional. Using Proposition 3.4, we can prove the following proposition. Proposition 3.5. If and B are two S-regular algebras, then B is S-regular with gl.dim( B) =gl.dim +gl.dimb. Proof. Since and B are S-regular, both k and B k admit bounded finitely generated minimal free resolutions (F,δ )and(g,δ B ) respectively. Therefore dim k Ext (k, k) and dim k Ext B(k, k) are finite, and (P,δ)=F G is a minimal free resolution of k as a graded B-module. Let gl.dim = d and gl.dim B = d B.Then d + d B = infi P i 0} =gl.dim( B). Since and B are S-regular, by Theorem 2.2, d =depth, Gor = infi Ext d (k, k) i 0} and d B =depth B B, Gor B B = infi Ext d B B By Proposition 3.4, B is left Gorenstein with depth B B =depth +depth B B = d + d B and Gor B B =Gor +Gor B B = infi Ext d (k, k) i 0} infi Ext d B B On the other hand, infi Ext d +d B B (k, k) i 0} = infi Hom B (F d G d B,k) i 0} = infi Ext d (k, k) i 0} infi Ext d B B Hence Gor B B = infi Ext d +d B B Proposition 3.5 indicates that the category of S-regular algebras is a tensor category. cknowledgments The first author was supported by the China Postdoctoral Science Foundation (No ). The second author was supported by the NSFC (key project ) and the Doctorate Foundation (No ), Ministry of Education of China. The authors thank the referee for suggestions on the paper.

10 1552 X.-F. MO ND Q.-S. WU References [S] M. rtin and W.F. Schelter, Graded algebras of global dimension 3, dv. Math. 66 (1987), MR (88k:16003) [Be] R. Berger, Koszulity for nonquadratic algebras, J. lgebra 239 (2001), MR (2002d:16034) [BGS].. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. mer. Math. Soc. 9 (1996), MR (96k:17010) [DW] Z.-C. Dong and Q.-S. Wu, Non-commutative Castelnuovo-Mumford regularity and Sregular algebras, J. lgebra 322 (2009), MR (2010g:16019) [FM] Y. Félix and. Murillo, Gorenstein graded algebras and the evaluation map, Canad. Math. Bull. 41 (1998), MR (99c:57069) [HL] J.-W. He and D.-M. Lu, Higher Koszul algebras and -infinity algebras, J. lgebra 293 (2005), MR (2006m:16030) [LWZ] D.-M. Lu, Q.-S. Wu and J.J. Zhang, Homological integral of Hopf algebras, Trans. mer. Math. Soc. 359 (2007), MR (2008f:16083) [Men] C. Menini, Cohen-Macaulay and Gorenstein finitely graded rings, Rend. Sem. Mat. Univ. Padova 79 (1988), MR (89i:13030) [LPWZ] D.-M. Lu, J.-H. Palmieri, Q.-S. Wu and J.-J. Zhang, Koszul equivalences in -algebras, New York J. Math. 14 (2008), MR (2010b:16017) [Sm] S.P. Smith, Some finite-dimensional algebras related to elliptic curves, CMS Conf. Proc., Vol. 19, , mer. Math. Soc., MR (97e:16053) [SZ] D.R. Stephenson and J.J. Zhang, Growth of graded Noetherian rings, Proc. mer. Math. Soc. 125 (1997), MR (97g:16033) [YZ] Y. Ye and P. Zhang, Higher Koszul complexes, Sci. China Ser. 46 (2003), MR (2004f:16016) Institute of Mathematics, Fudan University, Shanghai , People s Republic of China address: @fudan.edu.cn Current address: Department of Mathematics, Shanghai University, , People s Republic of China address: xuefengmao@shu.edu.cn Institute of Mathematics, Fudan University, Shanghai , People s Republic of China address: qswu@fudan.edu.cn