(s, t, d)-bi-koszul algebras

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1 Science in China Series A: Mathematics Nov., 2009, Vol. 52, No. 11, math.scichina.com (s, t, d)-bi-koszul algebras SI JunRu Department of Mathematics, Hangzhou Dianzi University, Hangzhou , China ( Abstract The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called (s, t, d)-bi-koszul algebras as the generalization of bi-koszul algebras. An (s, t, d)-bi-koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an (s, t, d)-bi-koszul algebra is discussed. Based on it, the notion of strongly (s, t, d)-bi-koszul algebras is raised and their homological properties are further discussed. Keywords: bi-koszul algebra, periodic algebra, pure resolution MSC(2000): 16E05, 16E40, 16S37, 16W50 1 Introduction To study homological properties of a graded algebra A, we often try to find its projective resolution and compute out its Koszul dual. Projective resolutions of A are not determined uniquely, but the minimal resolution of A (if it exists) is unique up to isomorphism and can be understood as an invariant of A. The simple case for a minimal resolution is that in it every projective module is generated in one degree. We call A an algebra with pure resolution. This kind of algebras includes Koszul algebras, d-koszul algebras, piecewise-koszul algebras and so on. Many of them have significant applications in algebraic topology, algebraic geometry, quantum group and Lie algebra (see [1 6], etc.). But in fact, there are a large number of algebras with non-pure resolutions. Based on it, the notion of bi-koszul algebras was introduced in [7]. Roughly speaking, a bi-koszul algebra is a 1-generated graded algebra with (d, d +1)-degree relations and has a non-pure resolution of period 3 and jumping degree 2d. Some of homological properties of bi-koszul algebras were discussed in [7]. For a bi-koszul algebra, the two degrees of relations and the jumping degree are all related to a fixed integer d. There is a natural question: how would such algebra be if the three degrees are not related to one fixed number? We define this kind of algebras (s, t, d)-bi-koszul algebras. An (s, t, d)-bi-koszul algebra also has a non-pure resolution of period 3, but it has (s, t)-degree relations and the jumping degree d in its minimal resolution. We introduce the concept of periodic algebras, to which (s, t, d)-bi-koszul algebras belong, and recall the properties of periodic algebras with pure resolutions. Especially, the 3-periodic Received September 4, 2008; accepted April 26, 2009 DOI: /s Corresponding author This work was supported by National Natural Science Foundation of China (Grant No. Natural Science Foundation of Zhejiang Province of China (Grant No. J ) ) and the Citation: Si J R. (s, t, d)-bi-koszul algebras. Sci China Ser A, 2009, 52(11): , DOI: /s

2 2420 Si J R algebras with pure resolutions can be compared with (s, t, d)-bi-koszul algebras. Similarly to Koszul algebras and so on, in the definition of the (s, t, d)-bi-koszul algebra A, the trivial module A 0 can be seen as either a left A-module or a right A-module. The category of all (s, t, d)-bi-koszul modules is extension-closed. There is a criteria theorem for a graded algebra to be (s, t, d)-bi-koszul in terms of its Koszul dual. Based on it, we raise the notion of the strongly (s, t, d)-bi-koszul algebra (in fact, left strongly) whose Koszul dual is 0, 1, 2, 3-generated as a graded algebra. We introduce the strongly (s, t, d)-bi-koszul module as well, which is defined over an (s, t, d)-bi-koszul algebra (not necessarily strongly) and whose Koszul dual is 0-generated as a graded module. The category of all strongly (s, t, d)-bi-koszul modules is also extension-closed. The homological properties of strongly (s, t, d)-bi-koszul algebras (or modules) are further discussed. From them, we can obtain quadratic algebras (or modules); especially, Koszul algebras (or modules) under some condition. A construction is given to obtain an (s, t, d)-bi-koszul algebra (or module) from two 3-periodic algebras (or modules) with pure resolutions. In fact, the (s, t, d)-bi-koszul algebra (or module) constructed here is strongly and from it, we naturally get a Koszul algebra (or module). 2 Periodic algebras Let F be a field. Throughout the paper, we assume that A is a 1-generated positively graded F-algebra with A 0 semisimple. Denote by J the graded Jacobson radical of A. So J = A 1 A 2. Let Gr(A) (orgr(a o )) denote the category of graded left (or right) A-modules with degree 0 morphisms. For any M Gr(A), we denote the n-th shift of M by M(n)whereM(n) j = M j+n. Recall the graded Tor and Ext functor. Tor A is the derived functor of the graded A functor, where M A N := ( ) M i A0 N j n Z i+j=n for any M Gr(A o )andn Gr(A), and Ext A is the derived functor of the graded Hom A functor, where Hom A (M,N) := Hom Gr(A) (M,N(n)) n Z for any M,N Gr(A). Tor A (M,N) andext A (M,N) are both bigraded vector spaces with the (i, j) components Tor A i (M,N) j and Ext i A (M,N) j, respectively. We denote E(A) :=Ext A(A 0,A 0 ), E(M) :=Ext A(M,A 0 ), which are called the Koszul dual (or Ext-algebra) of the algebra A and the Koszul dual of the module M Gr(A), respectively. Clearly, E(A) is a bigraded algebra with Ej i (A) := Ext i A (A 0,A 0 ) j and E(M) is a bigraded left E(A)-module with E i j (M) :=Exti A (M,A 0) j. In this section, we give the definition of periodic algebras and recall the properties of periodic algebras with pure resolutions. Definition 2.1. Let a function Δ:N N N (finite product of t times) and Δ(n) denote both the image (x 1,...,x t ) and the set {x 1,...,x t }. Here, N N is a Z-module with natural operations.

3 (s, t, d)-bi-koszul algebras 2421 (1) Let M Gr(A). Assume the minimal projective resolution of M is Q : Q n Q 1 Q 0 M 0. If every Q n is generated in Δ(n) for any n 0, wecallδ or {Δ(n)} the resolution degree distribution of M. Assume the minimal projective resolution of A 0 Gr(A) is P : P n P 1 P 0 A 0 0. If every P n is generated in Δ(n) for any n 0, wecallδ or {Δ(n)} the resolution degree distribution of A. (2) If M Gr(A) has the resolution degree distribution Δ with Δ(n) only one natural number for all n 0, we say that M is a module with pure resolution. If A has the resolution degree distribution Δ with Δ(n) only one natural number for all n 0, we call A an algebra with pure resolution. (3) If there exists a minimal positive integer p such that Δ(p + n) =Δ(p) +Δ(n) for any n 0, we say that M is a p-periodic module. If the resolution degree distribution of A satisfies this equation, we call A a p-periodic algebra. Assume that Δ is the resolution degree distribution of M (or A), then min Δ(n) < min Δ(n+ 1), for any n 0. Assume Δ is the resolution degree distribution of A, thenδ(0)=0, Δ(1) = 1 under the assumption that A is 1-generated. Periodic algebras (or modules) with pure resolutions were discussed in a large volume of literature, such as Koszul algebras (cf. [1, 6]), d-koszul algebras (cf. [2, 4]), piecewise-koszul algebras (cf. [3, 5]) and so on. This kind of algebras has many nice homological properties. We cite several as follows. Proposition 2.2. Let A be a p-periodic algebra with pure resolution, and M Gr(A) a p-periodic module with pure resolution. Assume that A and M have the same resolution degree distribution. Then (i) E(A) is generated by E 0 (A), E 1 (A),..., E p (A); especially speaking, for any n 0 and i= 1,...,p, E pn+i (A) =E i (A)E pn (A) =E pn (A)E i (A); (ii) E(M) is 0-generated as a graded left E(A)-module; that is, for any n 0, E n (M) = E n (A)E 0 (M); (iii) E [0] (A) := n 0 Epn (A) is a Koszul algebra; (iv) E [0] (M) := n 0 Epn (M) is a Koszul module as an E [0] (A)-module. Proof. Let Δ be the resolution degree distribution of A and M. ThenΔ(np + i) =Δ(np)+ Δ(i) =nδ(p)+δ(i) for any n 0andi =1,...,p. By [4, Proposition 3.6], we get (i). By [4, Proposition 3.5], we get (ii). The last two statements come from [8, Theorem 4.2]. The Koszul dual of a periodic algebra with pure resolution must be generated in finite degrees, but conversely, an algebra with pure resolution whose Koszul dual is generated in finite degrees may not be a periodic algebra. Example 2.3. Let A be a (D 1,D 2,D 3 )-stacked monomial algebra with D 2 2andD 3 =0 in [9]. Then A is a δ-koszul algebra (see [8]) and E(A) is0, 1, 2, 3-generated. But A is not a periodic algebra.

4 2422 Si J R Consider the p-periodic algebra A with a pure resolution while the period p is small. When p = 1, for any n 0, Δ(1 + n) =Δ(1)+Δ(n). We can compute out Δ(n) =n for any n 0. Then A is a Koszul algebra. When p = 2, for any n 0, Δ(2 + n) =Δ(2)+Δ(n). Denote d := Δ(2). We can compute out that nd, if n 0(mod2), 2 Δ(n) = (n 1)d +1, if n 1(mod2). 2 Then A is a d-koszul algebra. When p = 3, for any n 0, Δ(3 + n) =Δ(3)+Δ(n). Denote τ := Δ(2) and d := Δ(3). The resolution degree distribution can be determined by τ and d. We specially write it down as a definition. Definition 2.4. Let A be a graded algebra and M Gr(A). WecallM a (τ,d)-type Koszul module, if the resolution degree distribution of M is Δ:N N defined as n d, if n 0(mod3), 3 n 1 Δ(n) = d +1, if n 1(mod3), 3 n 2 d + τ, if n 2(mod3), 3 with d>τ>1. If the resolution degree distribution of A is defined as Δ above, we call A a (τ,d)-type Koszul algebra. A is a (τ,d)-type Koszul algebra if and only if A is a 3-periodic algebra with a pure resolution. Now, A is a 1-generated graded algebra with τ-degree relations and has the jumping degree d in its minimal resolution. Note that if τ =2,thenA is a piecewise-koszul algebra (cf. [5]). The minimal case is τ =2,d= 3, and now A is a Koszul algebra. If d<2τ 1, by comparing the degree, A has the global dimension at most 3. 3 (s, t, d)-bi-koszul algebras In this section, we turn to discuss a kind of 3-periodic algebras with non-pure resolutions. Such algebras are in close contact with (τ,d)-type Koszul algebras. From now on, denote the graded algebra A := T A 0 A 1 (I,J) a quotient of a tensor algebra, where A 0 is a semisimple F-algebra, A 1 is a A 0 -A 0 -bimodule and (I,J) is the two-sided ideal of T A0 A 1 generated by I A s 1 and J A t 1 for two fixed integers t>s>1. We assume that I 0andJ 0. Definition 3.1. Let A be a graded algebra and M Gr(A). We define a function Δ:N N N satisfying n (d, d), if n 0(mod3), 3 n 1 Δ(n) = (d, d)+(1, 1), if n 1(mod3), 3 n 2 (d, d)+(s, t), if n 2(mod3), 3

5 (s, t, d)-bi-koszul algebras 2423 with 1 <s<tand d>s. Here, Δ satisfies the natural addition and scalar multiplication. We call M an (s, t, d)-bi-koszul module, if its resolution degree distribution is the above {Δ(n)}. We call A an (s, t, d)-bi-koszul algebra, if its resolution degree distribution is the above {Δ(n)}. An (s, t, d)-bi-koszul algebra A is a 1-generated graded algebra with (s, t)-degree relations and is a 3-periodic algebra with the jumping degree d, but not has a pure resolution. If t = s+1 and d =2s, thena is a bi-koszul algebra which was discussed in [7]. Remark 3.2. Let A be an (s, t, d)-bi-koszul algebra and M Gr(A) bean(s, t, d)-bi-koszul module. (1) Assume that N Gr(A) is generated in two degrees (i, j) withi<j;thatis,forany n i, A n i N i, if n<j, N n = A n j N j, if n j. Here, we promise four cases: N i = N j = 0 which implies N =0;N i =0andN j 0which implies N is j-generated; N i 0andN j = A j i N i which implies N is i-generated; N i 0and N j A j i N i which is called strictly (i, j)-generated. (2) Ω 2 (A 0 ) is strictly (s, t)-generated; Ω 3n+2 (A 0 ) may be generated in one degree for any n 1; Ω 3n+2 (M) may be generated in one degree for any n 0. Therefore, Ω 3 (A 0 )andm may have pure resolutions. Example 3.3. The following examples are all (s, t, d)-bi-koszul algebras in the three cases d = t, d<tand d>t. (1) A = F x,y,z (y 2,xyz) whose minimal resolution is A( 5) A( 4) A( 3) A( 2, 3) A( 1) 3 A F 0. Here, s =2,t= d =3. (2) A = F x,y,z (y 2,xyz 2 ) whose minimal resolution is A( 5) A( 4) A( 3) A( 2, 4) A( 1) 3 A F 0. Here, s =2,t=4,d=3andt>d. (3) A = F x,y,z,g,h (xy,xzx,gxz,hg) whose minimal resolution is 0 A( 6) A( 5) 2 A( 4) 3 A( 2) 2 A( 3) 2 A( 1) 5 A F 0. Here, s =2,t=3,d=4andt<d. In this paper, we only consider 1 <s<t<d. For example, all bi-koszul algebras belong to such kind of (s, t, d)-bi-koszul algebras. The follows are some properties of (s, t, d)-bi-koszul algebras which are so easy to prove that we omit it here. Accordingly, (s, t, d)-bi-koszul modules have the similar properties. Lemma 3.4. The following statements are equivalent: (i) A is an (s, t, d)-bi-koszul algebra;

6 2424 Si J R (ii) E i (A) =EΔ(i) i (A); (iii) Tor A i (A 0,A 0 )=Tor A i (A 0,A 0 ) Δ(i). Proposition 3.5. A is an (s, t, d)-bi-koszul algebra if and only if the minimal projective resolution of the trivial module A 0 Gr(A o ) satisfies the degree distribution {Δ(n)} defined in Definition 3.1. Proof. Use Tor A (A 0,A 0 ), by Lemma 3.4. The above proposition points out that when justifing an (s, t, d)-bi-koszul algebra, we can consider either A 0 Gr(A) ora 0 Gr(A o ). Proposition 3.6. Assume that A is an (s, t, d)-bi-koszul algebra. Then Ω 3n (A 0 )(nd) is an (s, t, d)-bi-koszul module, for any n 1. We give a property of (s, t, d)-bi-koszul modules which needn t be defined over an (s, t, d)- bi-koszul algebra. Proposition 3.7. Let 0 K M N 0 be a short exact sequence in Gr(A). Assume that K and N are both (s, t, d)-bi-koszul modules. Then M is also an (s, t, d)-bi-koszul module. Moreover, there is a short exact sequence 0 E(N) E(M) E(K) 0 as E(A)- modules. Proof. Apply ExtA (,A 0) to the short exact sequence 0 K M N 0toobtaina long exact sequence. Compare the second degree. This result shows that the category of all (s, t, d)-bi-koszul modules is closed under the extension. We want to construct an (s, t, d)-bi-koszul algebra (or module) by two 3-periodic algebras (or modules) with pure resolutions. Let B and C be connected positively graded algebras and M Gr(B),N Gr(C). Assume that M 0 = N0 = H, whereh is a fixed vector space. We recall some operations in [6]. Definition 3.8. The direct sum of algebras B C is the graded algebra with (B C) 0 = F and (B C) i = B i C i for any i>0, where the products B + C + = C + B + =0. The free product of algebras B C is the associative algebra generated freely by B and C; that is, B C = B+ i1 (C + B + ) j C+ i2. j 0;i 1,i 2 {0,1} Justify that B C can be seen as a left or right B-module and a left or right C-module. Definition 3.9. The direct sum of modules M H N := H M + N + where B N + = C M + =0; The free product of modules M H N := H (B C) B M + (B C) C N +. Clearly, M H N can be seen as a graded left B C-module, and M H N can be seen as a graded left B C-module. The following lemma comes from [6]. Lemma (i) E(B C) = E(B) E(C) as bigraded algebras; (ii) E(M N) = E(M) E(N) as bigraded modules over E(B C) or E(B) E(C). Theorem Assume that B is an (s, d)-type Koszul algebra and C is a (t, d)-type Koszul algebra. Then B C is an (s, t, d)-bi-koszul algebra.

7 (s, t, d)-bi-koszul algebras 2425 Proof. By Lemma 3.10, we have E(B C) = E(B) E(C). By the definition of, (E(B) E(C)) 0 = F and (E(B) E(C)) i = E i (B) E i (C) for any i>0. Comparing the second degree, we get B C is an (s, t, d)-bi-koszul algebra. Similarly to the proof of Theorem 3.11, we get the modular case. Theorem Assume that M Gr(B) is an (s, d)-type Koszul module and N Gr(C) is a (t, d)-type Koszul module. Then M N Gr(B C) is an (s, t, d)-bi-koszul module. Sometimes, B C can be represented explicitly. Also see [1]. Assume that B and C are two connected graded algebras with finite-dimensional generator space and relation space. We can denote B := F x 1,...,x n (f 1,...,f α ), C := F y 1,...,y m (g 1,...,g β ), where x i = y j =1foranyi =1,...,n, j =1,...,mand f i = s, g j = t for any i =1,...,α, j =1,...,β.Then B C = F x 1,...,x n,y 1,...,y m (f 1,...,f α,g 1,...,g β ). 4 Generation of Koszul dual Generation of Koszul dual often becomes an important criterion for a graded algebra to be Koszul or d-koszul. In this section, we discuss the generation of Koszul dual for an (s, t, d)-bi- Koszul algebra (or module). Denote by Gr 0 (A) the category of 0-generated graded left A-modules with degree 0 morphisms. Assume that P : P n+1 P n P 1 P 0 A 0 0 is the minimal projective resolution of A 0 Gr(A) and Q : Q n+1 Q n Q 1 Q 0 M 0 is the minimal projective resolution of M Gr(A). Proposition 4.1. Let M Gr 0 (A). Assume that P j is strictly generated in degrees {x 1,j,..., x tj,j} with x 1,j < <x tj,j for j = n, n +1.Then (i) E n x 1,n (M) =Ex n 1,n (A)E 0 (M); (ii) E n x j,n (M) = Ex n j,n (A)E 0 (M) E n x j,n (JM) as left E 0 (A)-modules, if j 2 and x j,n x k,n+1 for all k. Proof. We know that P j is generated in degrees {x 1,j,...,x tj,j} if and only if E j (A) is supported in degrees {x 1,j,...,x tj,j}. If the lowest degree of E j (A) isx 1,j, then the lowest degree of E j (JM) x 1,j + 1 (see [4]). Consider the natural short exact sequence 0 JM M M/JM 0. Applying Ext A (,A 0) to it, we get a long exact sequence E n (M/JM) fn E n (M) E n (JM) E n+1 (M/JM) fn+1 E n+1 (M) E n+1 (JM).

8 2426 Si J R Comparing the second degree, we get short exact sequences E n x 1,n (M/JM) f1,n E n x 1,n (M) 0, E n x j,n (M/JM) fj,n E n x j,n (M) E n x j,n (JM) 0, where j 2andx j,n x k,n+1 for all k. SinceImf n = E n (A)E 0 (M) in[7],wegettheresult. Theorem 4.2. A is an (s, t, d)-bi-koszul algebra if and only if E(A) begins with E 0 = E0 0(A), E 1 = E1(A), 1 E 2 = Es 2 (A) Et 2 (A), E 3 = Ed 3 (A) and for any n 1, (i) E 3n (A) =E 3 (A)E 3(n 1) (A), (ii) E 3n+1 (A) =E 1 (A)E 3n (A), (iii) E 3n+2 (A) = E 2 (A)E 3n (A) E 2 nd+t (JΩ3n (A 0 )) as left E 0 (A)-modules, where Ω 3n (A 0 ) is the 3n-th syzygy of A 0 Gr(A). Proof. Assume that A is an (s, t, d)-bi-koszul algebra. Clearly, E(A) begins with E 0 = E0 0(A), E 1 = E1(A), 1 E 2 = Es 2 (A) Et 2 (A), E 3 = Ed 3(A). Consider E3n+i (A) =E i (Ω 3n (A 0 )) for any n 1andi =1, 2, 3. Denote M := Ω 3n (A 0 )(nd) whichisan(s, t, d)-bi-koszul module by Proposition 3.6. By Proposition 4.1, we get E 3 (M) =E 3 (A)E 0 (M), E 1 (M) =E 1 (A)E 0 (M), E 2 (M) = E 2 (A)E 0 (M) E 2 t (JM). Replace M by Ω 3n (A 0 )(nd). We get (i), (ii) and (iii). Conversely, it is obvious. Theorem 4.3. A is an (s, t, d)-bi-koszul algebra if and only if E(A) begins with E 0 = E0(A), 0 E 1 = E1 1(A), E2 = Es 2(A) E2 t (A), E3 = Ed 3 (A) and for any n 1, (i) E 3n (A) =E 3(n 1) (A)E 3 (A), (ii) E 3n+1 (A) =E 3n (A)E 1 (A), (iii) E 3n+2 (A) = E 3n (A)E 2 (A) E 2 nd+t ( Ω 3n (A 0 ) J) as right E 0 (A)-modules, where Ω 3n (A 0 ) is the 3n-th syzygy of A 0 Gr(A o ). Proof. By Proposition 3.5, take A 0 Gr(A o ). There is another result similar to Proposition 4.1. By the same proof of Theorem 4.2, we get the result. Corollary 4.4. Assume that A is an (s, t, d)-bi-koszul algebra. Then for any n 1, E 3n+1 (A) =E 1 (A)E 3n (A) =E 3n (A)E 1 (A), E 3n+2 nd+s (A) =E2 s (A)E 3n (A) =E 3n (A)Es 2 (A), E 3n+3 (A) =E 3 (A)E 3n (A) =E 3n (A)E 3 (A). Proof. Combine Theorems 4.2 with 4.3. Theorem 4.5. Let A be an (s, t, d)-bi-koszul algebra and M Gr(A). ThenM is an (s, t, d)-bi-koszul module if and only if for any n 0, (i) E 3n (M) =E 3n (A)E 0 (M), (ii) E 3n+1 (M) =E 3n+1 (A)E 0 (M), (iii) E 3n+2 (M) = E 3n+2 (A)E 0 (M) E 3n+2 nd+t (JM) as left E0 (A)-modules. Proof. Directly come from Proposition 4.1. Remark 4.6. Let A be an (s, t, d)-bi-koszul algebra and M Gr(A) an(s, t, d)-bi-koszul module. Assume that P and Q are minimal resolutions of A and M, respectively. By Proposition 4.1, we get the following results.

9 (s, t, d)-bi-koszul algebras 2427 (1) If P 3n+2 is (nd + t)-generated, then Q 3n+2 is only (nd + t)-generated; (2) If P 3n+2 is generated in one degree nd + s, then E 3n+2 (A) =E 2 (A)E 3n (A) =E 3n (A)E 2 (A); (3) If Q 3n+2 is generated in one degree nd + s, thene 3n+2 (M) =E 3n+2 (A)E 0 (M). Observe that the Koszul dual of an (s, t, d)-bi-koszul algebra (or module) may not be generated in finite degrees. But we can obtain subalgebra (or submodule) of the Koszul dual which is generated in finite degrees. Define E [P ] (A) := (E 3n (A) E 3n+1 (A)), E [N] (A) := E 3n+2 (A), n 0 n 0 E [P ] (M) := n 0 (E 3n (M) E 3n+1 (M)), E [N] (M) := n 0 E 3n+2 (M). Proposition 4.7. Let A be an (s, t, d)-bi-koszul algebra and M Gr(A) an (s, t, d)-bi-koszul module. Assume that s 3. Then the following statements are true: (i) E [P ] (A) is a subalgebra of E(A) and 0, 1, 3-generated; (ii) E [N] (A) is a graded left (or right) E [P ] (A)-module; (iii) E [P ] (M) is a graded left E [P ] (A)-module and 0-generated; (iv) E [N] (M) is a graded left E [P ] (A)-module. Proof. Assume n, m 0. By E 3n+1 (A)E 3m+1 (A) = 0 and Corollary 4.4, we get (i). To prove (ii), use E 3n+1 (A)E 3m+2 (A) =E 3n+2 (A)E 3m+1 (A) =0. ByE 3n+1 (A)E 3m+1 (M) = 0, we get (iii). To prove (iv), use E 3n+1 (A)E 3m+2 (M) =0. In the above proposition, note that E [P ] (M) can be seen as a submodule of E(M) over E [P ] (A). Remark 4.8. Under the assumption of Remark 4.6, let k be a fixed integer. If for any n k, P 3n+2 is generated in one degree nd + s, then for any n k and i =1, 2, 3, E 3n+i (A) =E i (A)E 3n (A) =E 3n (A)E i (A). If for any n k, Q 3n+2 is generated in one degree nd + s, then for any m 3k, E m (M) =E m (A)E 0 (M). The Koszul dual of such (s, t, d)-bi-koszul algebra (or module) is generated in finite degrees from E 3k (A) (ore 3k (M)). Now, we consider the case that the Koszul dual itself is generated in finite degrees. Definition 4.9. We call A astrongly(s, t, d)-bi-koszul algebra, if the Koszul dual E(A) begins with E 0 = E0(A), 0 E 1 = E1(A), 1 E 2 = Es 2 (A) Et 2 (A), E 3 = Ed 3 (A) and for any n 1, i =1, 2, 3, E 3n+i (A) =E i (A)E 3n (A). We can compare this definition with Proposition 2.2(i) for p = 3. Clearly, a strongly (s, t, d)- bi-koszul algebra is an (s, t, d)-bi-koszul algebra and its Koszul dual is 0, 1, 2, 3-generated. In fact, the above notion is a left strongly (s, t, d)-bi-koszul algebra such that in Theorem 4.2 the

10 2428 Si J R obstruction E 2 nd+t (JΩ3n (A 0 )) vanishes, for any n 1. Correspondingly, we have the notion of a right strongly (s, t, d)-bi-koszul algebra from Theorem 4.3. Proposition Assume that E 3 (A)E 2 (A) =E 2 (A)E 3 (A). Then A is a left strongly (s, t, d)-bi-koszul algebra if and only if A is a right strongly (s, t, d)-bi-koszul algebra. Proof. We only need to show that E 3n+2 (A) =E 2 (A)E 3n (A) =E 3n (A)E 2 (A) for any n 1. In fact, E 3n+2 (A)=E 2 (A)E 3n (A) =E 2 (A)E 3 (A) E 3 (A) = E 3 (A)E 2 (A)E 3 (A) E 3 (A) = E 3 (A) E 3 (A)E 2 (A) =E 3n (A)E 2 (A). We complete the proof. Definition Let A be an (s, t, d)-bi-koszul algebra and M Gr(A). WecallM a strongly (s, t, d)-bi-koszul module, if the Koszul dual E(M) is 0-generated as a graded left E(A)- module; that is, E n (M) =E n (A)E 0 (M), for any n 0. Note that the definition above comes from Theorem 4.5 in which the obstruction E 3n+2 nd+t (JM) vanishes for any n 0. And naturally, a strongly (s, t, d)-bi-koszul module is defined over an (s, t, d)-bi-koszul algebra. Here, the algebra needn t be strongly. Remark Under the assumption of Remark 4.6, if for any n 1, P 3n+2 is generated in one degree nd + s, thena is a strongly (s, t, d)-bi-koszul algebra. Note that now A is either a left or a right strongly (s, t, d)-bi-koszul algebra. If for any n 0, Q 3n+2 is generated in one degree nd + s, thenm is a strongly (s, t, d)-bi-koszul module. We give some examples which are strongly (s, t, d)-bi-koszul algebras. Example (1) Strongly bi-koszul algebras; especially, AS-regular algebras of global dimension 4, which have 2 generators or 3 generators (cf. [10]); (2) Any (s, t, d)-bi-koszul algebra of global dimension at most 4. There are other examples. We continue to observe the algebra B C and the module M N obtained in Section 3. Firstly, we give a lemma which comes from [6]. Lemma Both B and C are Koszul algebras if and only if B C is a Koszul algebra. If both B and C are Koszul algebras, then both M and N are Koszul modules if and only if M H N is a Koszul module. Proposition The algebra B C obtained in Theorem 3.11 is a strongly (s, t, d)-bi- Koszul algebra and E [0] (B C) is a Koszul algebra. Proof. By Proposition 2.2, for any n 1andi =1, 2, 3, E 3n+i (B C) = E 3n+i (B) E 3n+i (C) =E i (B)E 3n (B) E i (C)E 3n (C) =(E i (B) E i (C))(E 3n (B) E 3n (C)) = E i (B C)E 3n (B C). So B C is a strongly (s, t, d)-bi-koszul algebra. We know that E [0] (B C) = E [0] (B) E [0] (C). Since E [0] (B) ande [0] (C) arebothkoszul algebras by Proposition 2.2, we get that E [0] (B C) is a Koszul algebra by Lemma 4.14.

11 (s, t, d)-bi-koszul algebras 2429 Proposition Assume that B and C are algebras defined in Theorem Then the module M N obtained in Theorem 3.12 is a strongly (s, t, d)-bi-koszul module over B C and E [0] (M N) is a Koszul module over E [0] (B C). Proof. The proof is similar to that of Proposition We discuss some properties of a strongly (s, t, d)-bi-koszul algebra (or module). The following result shows that the category of all strongly (s, t, d)-bi-koszul modules is closed under the extension. Proposition Let A be an (s, t, d)-bi-koszul algebra. Assume that 0 K M N 0 is a short exact sequence in Gr(A). If K and N are both strongly (s, t, d)-bi-koszul modules, then M is also a strongly (s, t, d)-bi-koszul module. Proof. By Proposition 3.7. Theorem Let A be an (s, t, d)-bi-koszul algebra and M Gr(A) an (s, t, d)-bi-koszul module. Then M is a strongly (s, t, d)-bi-koszul module if and only if Ω 2 (JM)(d) is an (s, t, d)- bi-koszul module. Proof. Assume that M is a strongly (s, t, d)-bi-koszul module. Apply ExtA (,A 0)tothe natural exact sequence 0 JM M M/JM 0. We obtain a long exact sequence δn 1 E n (M/JM) fn E n (M) gn E n (JM) δ n E n+1 (M/JM) fn+1 E n+1 (M) gn+1 E n+1 (JM) We know that M is a strongly (s, t, d)-bi-koszul module if and only if E n (M) =E n (A)E 0 (M) = Im(f n ) for any n 0. So f n are all surjective. We get g n =0andδ n are all injective. For any n 0, there exists a short exact sequence 0 E n (JM) δn E n+1 (M/JM) fn+1 E n+1 (M) 0. Comparing the second degree, we get Ω 2 (JM)(d) isan(s, t, d)-bi-koszul module. Conversely, we still get short exact sequences above, so f n are all surjective. That is, E n (M) =E n (A)E 0 (M). Therefore, M is a strongly (s, t, d)-bi-koszul module. We give some applications of Theorem Proposition Assume that A is an (s, t, d)-bi-koszul algebra. Then A is a strongly (s, t, d)-bi-koszul algebra if and only if for any n 1, Ω 3n (A 0 )(nd) is a strongly (s, t, d)-bi- Koszul module in Gr(A). Proof. Assume that A is a strongly (s, t, d)-bi-koszul algebra. Then for any m 0, E m (Ω 3n (A 0 )(nd)) = E 3n+m (A)(nd) =E m (A)E 3n (A)(nd) =E m (A)E 0 (Ω 3n (A 0 )(nd)). So we get the result. Conversely, assume any n 1. If M := Ω 3n (A 0 )(nd) isastrongly(s, t, d)- bi-koszul module in Gr(A), then Ω 2 (JM)(d) isan(s, t, d)-bi-koszul module by Theorem So E 2 (JΩ 3n (A 0 )) is only supported in degree nd + d. Hence, E 2 nd+t (JΩ3n (A 0 )) = 0. Thus, A is a strongly (s, t, d)-bi-koszul algebra. By this proposition, we know that if A is a strongly (s, t, d)-bi-koszul algebra, then Ω 2 (J)(d) is a strongly (s, t, d)-bi-koszul module.

12 2430 Si J R The following result directly comes from the proof of Theorem Corollary Let A be an (s, t, d)-bi-koszul algebra and M Gr(A) astrongly(s, t, d)- bi-koszul module. Then for any n 0, there is a short exact sequence 0 E n (JM) E n+1 (M/JM) E n+1 (M) 0. Let A be an (s, t, d)-bi-koszul algebra. For any M Gr(A), denote E [0] (M) := E 3n (M), E [2] (M) := E 3n+2 (M), n 0 n 0 which are regraded as E [0] (M) n := E 3n (M) ande [2] (M) n := E 3n+2 (M), respectively. Denote E [0] (A) := n 0 E3n (A) which is regraded as E [0] (A) n := E 3n (A). Proposition Let A be a strongly (s, t, d)-bi-koszul algebra and M Gr(A) astrongly (s, t, d)-bi-koszul module. Then E [0] (A) is a quadratic algebra and E [0] (M) is a quadratic module over E [0] (A). Proof. Clearly, E [0] (M/JM)andE [0] (M) are both 0-generated as E [0] (A)-modules. Since M is a strongly (s, t, d)-bi-koszul module, we get N := Ω 2 (JM)(d) isan(s, t, d)-bi-koszul module by Theorem So E 3n (N) =E 3n (A)E 0 (N); that is, E 3n+2 (JM)=E 3n (A)E 2 (JM). Therefore, E [2] (JM), E [0] (M/JM) ande [0] (M) are all 0-generated as E [0] (A)-modules. Considering the short exact sequence 0 E [2] (JM)( 1) E [0] (M/JM) E [0] (M) 0 by Corollary 4.20, which is the projective cover of E [0] (M) overe [0] (A), we have that E [0] (M) is a quadratic module. We know that the first term of the minimal resolution of E [0] (A) 0 = E [0] (A 0 ) 0 over E [0] (A) is 0 E 3n (A)( 1) E [0] (A) E [0] (A) 0 0. n 1 Clearly, for any n 0, E n+1 (A) =E n (Ω 1 (A 0 )) = E n (J). So E 3n (A) = E 3n+2 (J) = E 3n (Ω 2 (J)) = E [0] (Ω 2 (J)). n 1 n 0 n 0 Denote L := Ω 2 (J)(d) which is a strongly (s, t, d)-bi-koszul module by Proposition We get a short exact sequence by the above analysis 0 E [2] (JL)( 1) E [0] (L/JL) E [0] (L) 0, which is the projective cover of E [0] (L) overe [0] (A). Replacing L by Ω 2 (J)(d), we get 0 E [2] (JΩ 2 (J))( 1) E [0] (Ω 2 (J)/JΩ 2 (J)) E [0] (Ω 2 (J)) 0. So the first several terms of the minimal resolution of E [0] (A) can be written as 0 E [2] (JΩ 2 (J))( 2) E [0] (Ω 2 (J)/JΩ 2 (J))( 1) E [0] (A) E [0] (A) 0 0.

13 (s, t, d)-bi-koszul algebras 2431 Therefore, E [0] (A) is a quadratic algebra. Let A be an (s, t, d)-bi-koszul algebra, and M Gr(A). Denote that M (0) := M and M (i) := Ω 2 (JM (i 1) )(d) for any i 1. By the proof of Proposition 4.21, we know that if M (i) are all strongly (s, t, d)-bi-koszul modules, then E [0] (M) is a Koszul module. If M := Ω 2 (J)(d) satisfies that all M (i) are strongly (s, t, d)-bi-koszul modules, then E [0] (A) is a Koszul algebra. In Definition 4.11, a strongly (s, t, d)-bi-koszul module always is not necessarily defined over a strongly (s, t, d)-bi-koszul algebra. When A is a strongly (s, t, d)-bi-koszul algebra and M Gr(A) astrongly(s, t, d)-bi-koszul module, we have for any n 0andi =1, 2, 3, References E 3n+i (M) =E i (A) E 3 (A) E 3 (A) E 0 (M). }{{} n 1 Backelin J, Froberg R. Koszul algebras, Veronese subrings and rings with linear resolutions. Rev Roumaine Math Pures Appl, 30: (1985) 2 Berger R. Koszulity of nonquadratic algebras. J Algebra, 239: (2001) 3 Brenner S, Butler M C R, King A D. Periodic algebras which are almost Koszul. Algebra Rep Theory, 5: (2002) 4 Green E L, Marcos E N, Martinez-Villa R, et al. D-Koszul algebras. J Pure Appl Algebra, 193: (2004) 5 Lü J F, He J W, Lu D M. Piecewise-Koszul algebras. Sci China Ser A, 50: (2007) 6 Polishchuk A, Positselski L. Quadratic algebras. In: University Lecture Series, Vol. 37. Providence, RI: American Mathematical Society, Lu D M, Si J R. Koszulity of algebras with non-pure resolutions. Comm Algebra, (in press) 8 Green E L, Marcos E. δ-koszul algebras. Comm Algebra, 33: (2005) 9 Green E L, Snashall N. Finite generation of Ext for a generalization of D-Koszul algebras. J Algebra, 295: (2006) 10 Lu D M, Palmieri J H, Wu Q S, et al. Regular algebras of dimension 4 and their A -Ext-algebras. Duke Math J, 137(3): (2007)