Some Remarks on D-Koszul Algebras

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1 International Journal of Algebra, Vol. 4, 2010, no. 24, Some Remarks on D-Koszul Algebras Chen Pei-Sen Yiwu Industrial and Commercial College Yiwu, Zhejiang, , P.R. China Abstract. In this short paper, some new necessary and sufficient conditions for a standard graded k-algebra A = i 0 A i to be D-Koszul are given. Mathematics Subject Classification: 16S37, 16W50, 16E30, 16E40 Keywords: Koszul algebras, D-Koszul algebras, Yoneda algebras, Yoneda coalgebras 1. Introduction Koszul algebras were first introduced by Priddy in 1970 (see [11]), which is a class of quadratic algebras possessing a lot of beautiful homological properties. In the past 40 years, a lot of extensions of Koszul algebras have been done. In particular, Berger first introduced the notion of non-quadratic Koszul algebras (see [1]) in 2001 with the motivation of cubic Artin-Schelter regular algebras. In 2004, Zhang et al generalized the notion to the non-local case and called this class of algebras D-Koszul algebras (see [4]) with the motivation of quivers in representation theory. It turned out that D-Koszul algebras also possess a lot of nice properties similar to Koszul algebras and have a lot of applications in different branches of mathematics. See more extensions of Koszul objects, we refer to ([5]-[10]) for the further details. The present paper is focus on some new conditions such that a given standard graded algebra to be D-Koszul. In particular, we obtain the following results: Theorem 1.1. The following statements are equivalent for a standard graded algebra A. (1) A is a D-Koszul algebra; (2) GR δ A(A 0 ) = 0; (3) A is a D-Koszul module over A e ; (4) The multiplications µ : Ext 2 A(A 0, A 0 ) Ext n 2 A (A 0, A 0 )+Ext 1 A(A 0, A 0 ) Ext n 1 A (A 0, A 0 ) Ext n A(A 0, A 0 ) (n 1) are surjective, and Ext 2 A(A 0, A 0 ) = Ext p A (A 0, A 0 ) D ;

2 1178 Chen Pei-Sen (5) The comultiplications : Tor A n (A 0, A 0 ) Tor A 1 (A 0, A 0 ) Tor A n 1(A 0, A 0 ) +Tor A 2 (A 0, A 0 ) Tor A n 2(A 0, A 0 ) (n p+1) are injective, and Tor A 2 (A 0, A 0 ) = Tor A p (A 0, A 0 ) D. In particular, if A is a standard graded algebra with pure resolution, then the above statements are equivalent that the Ext module i 0 Exti A(M, A 0 ) is generated by Ext 0 A(M, A 0 ) as a graded i 0 Exti A(A 0, A 0 )-module, where M is a D-Koszul module. For the sake of simplicity, let s give some pre-knowledge. Throughout the whole paper, k denotes an fixed field, N and Z denote the sets of natural numbers and integers, respectively. The positively graded k- algebra A = i 0 A i will be called standard provided A 0 = k k, a finite product of k; A i A j = A i+j for all 0 i, j < ; dim k A i <. The graded Jacobson radical of the standard graded algebra A is obvious i 1 A i, denoted by J. Let Gr(A) and gr(a) denote the categories of graded A-modules and finitely generated graded A-modules, respectively. Definition 1.2. Let A be a standard graded algebra and M a finitely generated graded A-module. Let P n d n d 1 P 0 d 0 M 0 be a minimal graded projective resolution of M. Then M is called a D-Koszul module if and only if each P n is generated in degree δ(n) for all n 0, where { nd n 0(mod2) δ(n) = 2 (n 1)D + 1 n 1(mod2) 2 and D 2 an integer. In particular, if the trivial A-module A 0 is D-Koszul, then A is called a D-Koszul algebra. 2. On the new characterizations 2.1. Characterized by the generalized Castelnuovo-Mumford regularity. Definition 2.1. Let M = i 0 M i be a finitely generated graded module over A. The classical Castelnuovo-Mumford regularity of M is defined as denoted by R A (M). Call inf{j i : Tor A i (M, A 0 ) j = 0, for all i 0}, inf{ j f(i) : Tor A i (M, A 0 ) j = 0, for all i 0} the generalized Castelnuovo-Mumford regularity of M, denoted by GR f A (M), where stands for the absolute value and f : N N is a set function.

3 Some remarks on D-Koszul algebras 1179 Theorem 2.2. Let A be a standard graded algebra. Then A is a D-Koszul algebra if and only if GR δ A(A 0 ) = 0. Proof. ( ) Let A be a D-Koszul algebra. Then the trivial A-module A 0 admits a minimal graded projective resolution P n d n d 1 P 0 d 0 A 0 0 with each P n generated in degree δ(n). Therefore, for all i 0, Tor A i (A 0, A 0 ) = Tor A i (A 0, A 0 ) δ(i), which implies that GR δ A(A 0 ) = 0. ( ) Assume that GR δ A(A 0 ) = 0, that is, inf{ δ(i) j : Tor A i (A 0, A 0 ) j } = 0. Therefore, for all i 0, P i can be generated by homogeneous elements of degree δ(i), where P i is the i th term in the graded projective resolution of the trivial A-module A 0. Thus, A isa D-Koszul algebra Characterized by the (co)-multiplications. First letis recall bar and co-bar resolution. Let A be a standard graded algebra. Then A 0, the trivial A-module, possesses a canonical graded projective resolution: Bar n (A) n Bar 1 (A) 1 Bar 0 (A) 0 A 0 0, where for all n 0, Bar n (A) := A A0 J n and the differential n : A A0 J n A A0 J n 1 is defined by n 1 n(a 0 a 1 a n ) := ( 1) i a 0 a i a i+1 a n, (a 0 A, a 1,, a n J). i=0 Note that A 0 A Bar n (A) = A 0 A A A0 J n = J n for all n 0, we get the following complex with J n n J 2 2 J 1 1 J 0 0 A 0 n 1 n (a 1 a 2 a n ) := ( 1) i a 1 a i a i+1 a n, (a 1,, a n J). Now it is trivial that i=0 Tor A n (A 0, A 0 ) = ker n /Im n+1. Lemma 2.3. Using the above notations. T (A) := n 0 TorA n (A 0, A 0 ) is a bigraded coalgebra with the comultiplication = n,i n,i, where n,i is induced by n,i : J n J i J n i via n,i (a 1 a n ) = (a 1 a i ) (a i+1 a n ).

4 1180 Chen Pei-Sen Proof. It is easy to check that = n,i n,i provides a comultiplicative structure for the complex J and preserves kernels and images. Thus (J,, ) is a differential graded coalgebra and T (A) a graded coalgebra. Note that now A is a standard graded algebra, which implies that T (A) a bigraded coalgebra. The cobar complex is the cochain complexcob (A) defined by Cob n (A) := Hom A (J n, A 0 ) for all n 0, where the differential n+1 : Cob n (A) Cob n+1 (A) is the pullback of. Clearly, for all n 0, we have Ext n A(A 0, A 0 ) = ker n+1/im n. Lemma 2.4. Using the above notations. E(A) := n 0 Extn A(A 0, A 0 ) is a bigraded algebra with the multiplication µ = i,n µ i,n i, where µ i,n i is induced by µ i,n i : Cob i (A) Cob n i (A) Cob n (A) via µ i,n i (f g)(a 1 a 2 ) := f(a 1 ) g(a 2 ). Proof. It is easy to check that µ = n,i µ i,n i provides a multiplicative structure for the complex Cob (A) and preserves kernels and images. Thus (Cob (A),, µ) is a differential graded algebra and E(A) a graded algebra. Note that now A is a standard graded algebra, which implies that E(A) a bigraded algebra. We usually call T (A) the Yoneda coalgebra of A, and E(A) the Yoneda algebra of A. Lemma 2.5. The map µ n,i : Cob n i (A) Cob i (A) Cob n (A) and n,i : J n J n i J i are dual to one another. Proof. Let f 1 f i Cob i (A), g 1 g n i Cob n i (A) and a 1 a n J n. Then ((f 1 f i ) (g 1 g n i ))(a 1 a n ) = ((f 1 f i ) (g 1 g n i )) (a 1 a n ) = (f 1 f i )(a 1 a i )(g 1 g n i )(a i+1 a n ) = µ((f 1 f i ) (g 1 g n i ))(a 1 a n ). Therefore, we are done. Theorem 2.6. Let A be a standard graded algebra. Then the following statements are equivalent: (1) A is a D-Koszul algebra; (2) The multiplications µ : Ext 2 A(A 0, A 0 ) Ext n 2 A (A 0, A 0 )+Ext 1 A(A 0, A 0 ) Ext n 1 A (A 0, A 0 ) Ext n A(A 0, A 0 ) (n 1) are surjective, and Ext 2 A(A 0, A 0 ) = Ext p A (A 0, A 0 ) D ; (3) The comultiplications : Tor A n (A 0, A 0 ) Tor A 1 (A 0, A 0 ) Tor A n 1(A 0, A 0 ) +Tor A 2 (A 0, A 0 ) Tor A n 2(A 0, A 0 ) (n p+1) are injective, and Tor A 2 (A 0, A 0 ) = Tor A p (A 0, A 0 ) D.

5 Some remarks on D-Koszul algebras 1181 Proof. By (Theorem 4.1, [4]), we have that A is a D-Koszul algebra iff the Yoneda algebra n 0 Extn A(A 0, A 0 ) is minimally generated by Ext 0 A(A 0, A 0 ), Ext 1 A(A 0, A 0 ), Ext 2 A(A 0, A 0 ) and Ext 2 A(A 0, A 0 ) = Ext p A (A 0, A 0 ) D. Therefore, (1) (2) is immediate by induction on n. By Lemma 2.5, we have µ and are dual to each other, which establishes the equivalence of conditions (2) and (3) Characterized by the homological properties. Lemma 2.7. ([7]) Let A be a standard graded algebra and A e := A k A op its enveloping algebra. Let r be the graded Jacobson radical of A e and f : P Q be a homomorphism of finitely generated A e -projective modules. Then Imf rq if and only if for each simple A-module S, we have Im(f A 1 S ) J(Q A S). Proof. This result can be found in [7]. For the convenience of the reader, we also give the proof here. ( ) Let Q = Av k wa be an indecomposable A e -module. Assume f is an epimorphism, so f is a splittable epimorphism and by tensoring it with any A-module we get an epimorphism of A-modules. In particular, if we choose S = Aw/rw, we get an epimorphism f A 1 S : P A S Av. ( ) Assume that Imf rq. Since for each simple A-module T Aw/rw we have Q A T = 0, it is enough to prove that if S = Aw/rw, then Im(f A 1 S ) Av. Consider the following commutative diagram P f Q α P A S f A1 S Q A S, where α and β are the splittable A-epimorphisms given by the split exact sequences in the category of finitely generated A-modules 0 Av k wj Av k wa β Av 0 for β and similarly defined for α. Not that β 1 (v) = v w+av wj, thus each element in the preimage of v is a A e -generator for the module Q = Av k wa. If f A 1 S is an epimorphism, then βf is an epimorphism and β 1 (v) Imf 0, which implies that Imf contains an A e -generator of the cyclic module Q, so f is an epimorphism. Theorem 2.8. Let A be a standard graded algebra and A e its enveloping algebra. Then A is a D-Koszul algebra if and only if A is a D-Koszul module over A e. Proof. If P = Av k wa is an indecomposable A e -projective module and M a A-module, then P A M = (Av) dim wm as an A-module since Av k wa A M = β

6 1182 Chen Pei-Sen Av k wm. In particular, if M = S a simple A-module, then as A-modules we have P A S = Av if ws 0 and P A S = 0 otherwise. Let P : P n P 0 A 0 be a graded projective A e -resolution of A. Then by Lemma 2.7, P is minimal if and only if P A A 0 : P n A A 0 A A 0 P 0 A A 0 A 0 0 is a minimal graded projective resolution of A 0. Further, for all i 0, P i is generated in degree s as a graded A e -module if and only if P i A A 0 is generated in degree s as a graded A-module. Now we finish the proof. Theorem 2.9. Let A be a standard graded algebra with a pure resolution and M a D-Koszul module. Then the Ext module i 0 Exti A(M, A 0 ) is generated by Ext 0 A(M, A 0 ) as a graded i 0 Exti A(A 0, A 0 )-module if and only if A is a D-Koszul algebra. Proof. Let P and Q be the minimal graded projective resolutions of A 0 and M, respectively. By hypothesis, for all n 0, Q n is generated in degree δ(n). ( ) For all n 1, then Ext n A(M, A 0 ) = Ext n A(A 0, A 0 ) Ext 0 A(M, A 0 ) by hypothesis. Note that A is a positively graded algebra with a pure resolution, which implies that Ext n A(A 0, A 0 ) = Ext n A(A 0, A 0 ) s for some natural number s. Now observing that Ext n A(M, A 0 ) = Ext n A(M, A 0 ) δ(n) since M is a D-Koszul module. Thus Ext n A(A 0, A 0 ) = Ext n A(A 0, A 0 ) δ(n) for all n 0, which implies easily that A is a D-Koszul algebra. ( ) Suppose that A is a D-Koszul algebra. Then as a trivial A-module, A 0 admits a minimal graded projective resolution P n P 0 A 0 0 such that each projective module P n is generated in degree δ(n) for all n 0. Note that M is a D-Koszul module. Thus M has a minimal graded projective resolution Q n Q 1 Q 0 M 0 such that each projective module Q n is generated in degree δ(n) for all n 0. Then by Proposition 3.5 of [4], we have Ext i A(M, A 0 ) = Ext i A(A 0, A 0 ) Ext 0 A(M, A 0 ) for all i 0. That is, i 0 Exti A(M, A 0 ) is generated by Ext 0 A(M, A 0 ). Now it is easy to see that Theorem 1.1 is a summary of the above obtained results.

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