International Journal of Algebra, Vol. 5, 20, no. 3, 605-60 A Note on Piecewise-Koszul Complexes Pan Yuan Yiwu Industrial and Commercial College Yiwu, Zhejiang 322000 P.R. China mathpanyuan@gmail.com mathpanyuan@26.com Abstract. The notion of piecewise-koszul complex was first introduced by Lü and Si in [2]. In order to make the complex exact, the authors defined the so-called special piecewise-koszul algebras. In this short note, we prove that for a positively graded algebra A, A is a special piecewise-koszul algebra if and only if A is a Koszul algebra, which shows that we have to change the definition of piecewise-koszul complex for a non-koszul piecewise-koszul algebra. Mathematics Subject Classification: 6S37, 6W50 Keywords: Koszul algebras, piecewise-koszul algebras, piecewise-koszul complexes. Introduction Piecewise-Koszul algebras, first introduced by Lü, He and Lu in [], were another quadratic homogeneous positively graded algebras in general. Recently, Lü and Si defined the so-called piecewise-koszul complex related to a piecewise-koszul algebra. Further, in order to make the complex exact, the authors defined the so-called special piecewise-koszul algebras. In this short note, we prove that A is a special piecewise-koszul algebra if and only if A is a Koszul algebra, which shows that we have to change the definition of piecewise-koszul complex in [2] for a non-koszul piecewise-koszul algebra. Now let us recall some notations and definitions. Throughout, let Z and N denote the set of integers and natural numbers, K denotes a fixed field, and all the graded K-algebras A = A i are assumed with the following properties: (a) A 0 = K K K, finite copies of K; (b) A is generated in degrees 0 and ; that is, A i A j = A i+j for all 0 i, j < ; Research is supported by Yiwu Industrial And Commercial College(Grant 2009) and the foundation of Educatioal Department of Zhejiang Province in China(Grant Y2006432)
606 Pan Yuan (c) all A i are of finite dimension as a K-space. Definition.. [] Let M = M i be a finitely generated graded A- module. We call M a piecewise-koszul module provided that M admits a minimal graded projective resolution F : F n F F 0 M 0 with each F n generated in degree δp(n), d where the set function δp d : N N is defined by nd, if n 0(p), p (n )d δp(n) d +, if n (p), = p (n p+)d + p, if n p (p). p and d p > 2. In particular, if the trivial A-module A 0 is a piecewise-koszul module, then A = A i is called a piecewise-koszul algebra. Definition.2. [2] Let A = T A0 (A )/ R be a quadratic algebra, define the so called piecewise-koszul complex PK : PK pm+p pm+p PK pm pm PK pm pm PK PK 0 as follows. For any i 0, PK i is a graded projective A-module given by PK i = A A0 PK i δ d p(i), where PK i δp(i) is an A 0-A d 0 -bimodule concentrated in degree δp(i), d i.e., for i 3, PK i δp(i) = A u d R A v A δd p (i). u+v+2=δp d(i) In particular, we have The differential of PK, i : via PK 0 = A, PK = A A, PK 2 = A R. PK i PK i is the restriction of the map i : A A δd p (i) A A δd p (i ) a a a 2 a δ d p (i) aa a δ d p (i) δ d p(i ) a δ d p (i) δ d p(i )+ a δ d p (i). More precisely, pm (a a a 2 a δ d p (pm)) = aa a 2 a d p+ a d p+2 a δ d p (pm); pm+ (a a a 2 a δ d p (pm+)) = aa a 2 a δ d p (pm+); pm+2 (a a a 2 a δ d p (pm+2)) = aa a 2 a δ d p (pm+2); pm+p (a a a 2 a δ d p (pm+p )) = aa a 2 a δ d p (pm+p ), where a A and a i A, i =, 2,.
A note on piecewise-koszul complexes 607 Definition.3. [2] Let A be a piecewise-koszul algebra, writing A as T A0 (A )/ R. If we have (A n R) (R A n ) = (A i R A j ) i+j=n for all n d 2, then A will be called a special piecewise-koszul algebra. 2. The main result and proof We begin with recalling one of the main results of [2], we also give the proof here for the completeness. Lemma 2.. Let A be a special piecewise-koszul algebra and E(A) its Yoneda- Ext algebra. Let B =,j Z Bi j, where B i = B i δ d p(i) := A! δ d p(i) and Bi j = 0 if j δ d p(i). Then E(A) = B as bigraded K-algebras. Proof. Firstly we claim: Ext i A(A 0, A 0 ) = B i δ d p(i), i 0. In fact, since A is a special piecewise-koszul algebra, by Theorem 2.5 of [2], the piecewise-koszul complex PK of A is a projective resolution of the trivial A-module A 0. Hence Ext i A(A 0, A 0 ) = ker(hom A(PK i, A 0 ) Hom A (PK i+, A 0 )) Im(Hom A (PK i, A 0 ) Hom A (PK i, A 0 )) = Hom A (PK i, A 0 ) = Hom A (A A0 PK i δ d p (i), A 0) = Hom A0 (PK i δ d p (i), Hom A(A, A 0 )) = Hom A0 (PK i δ d p(i), A 0) = Hom A0 ((A! δ d p (i)), A 0 ) = A! δ d p (i) = Bi δ d p (i). In order to finish the proof, define a multiplication on B as follows: for ψ B i and ϕ B j, { ψ ϕ, at least one of i and j is of the form kp, k Z, ψ ϕ = 0, otherwise, where is the product of the dual algebra A!. Under the multiplication, it is easy to see that B is a bigraded algebra. Note that we have the following isomorphisms, and A! δ d p (n) = (PKn δ d p(n) ), Ext n A(A 0, A 0 ) = A! δ d p (n) Ext n A(A 0, A 0 ) = Hom A (A A0 PK n δ d p(n), A 0).
608 Pan Yuan Let ϕ Ext n A(A 0, A 0 ) = A! δ d p (n), ψ Extm A (A 0, A 0 ) = A! δ d p (m). Let ϕ : A A 0 PK n δ d p (n) A 0 and ψ : A A0 PK m δ d p(m) A 0 be the A-module morphisms induced by ϕ and ψ, respectively. Now showing that ψ ϕ = ψ ϕ, where denotes the product of Ext A(A 0, A 0 ), the product of A!. Now consider the following diagram n+m+ A PK n+m δ d P (n+m) n+m n+ A PK n δp d (n) ϕ m ϕ 0 ϕ m+ A PK m m δ A PK 0 P d (m) δp d (0 ψ A 0 where ϕ i for i = 0,,..., m is defined as follows. (i) If n = pk. Define n ɛ A 0 0 ϕ l (a v v δ d P (n+l)) = a v v δ d P (l)ϕ(v δ d P (l)+ v δ d P (n+l)), where l = 0,,, m. Clearly, l ϕ l = ϕ l n+l. Thus, ψϕ m (a v v δ d P (n+m)) =ψ(a v v δ d P (m))ϕ(v δ d P (m)+ v δ d P (m+n)) =a ψ(a v v δ d P (m))ϕ(v δ d P (m)+ v δ d P (m+n)) =aψ ϕ(v v δ d P (n+m)). (ii) If n = pk + i, i =, 2,, p. If l = pk + j, j =, 2,, p. Then define ϕ l (a v v δ d P (n+l)) = av v d p+ v δ d P (n+l) δp d (n) ϕ(v δ d P (n+l) δp d (n)+ v δ d P (n+l)). Similarly, l ϕ l = ϕ l n+l. If ψ ϕ 0, then δp(m d + n) = δp(m) d + δp(n). d Since n = pk + i, i =, 2,, p, clearly m = pk. By the computations similar to (i), we have ψ ϕ = ψ ϕ. Therefore, E(A) = B as bigraded K-algebras. Lemma 2.2. ([3]) Let A be a positively graded algebra and M = M i be a finitely generated module over A. Then we have the following isomorphisms: Ext i A(A 0, A 0 ) i = (q A)!, Ext i A(M, A 0 ) i = (q M)! qa, where q A and q M are the quadratic parts of A and M, respectively.
A note on piecewise-koszul complexes 609 Proposition 2.3. Let A be a piecewise-koszul algebra and E(A) its Yoneda- Ext algebra. Then A! = Ext A(A 0, A 0 ) = Ext A(A 0, A 0 ). Proof. By the definition of piecewise-koszul algebra, we have that A is a quadratic algebra. Thus q A = A, which implies easily that ( q A)! = A!. By Lemma 2.2, we have A! = Exti A(A 0, A 0 ) i. Note again that A is piecewise-koszul, we have Ext A(A 0, A 0 ) = Ext A(A 0, A 0 ). Therefore, we finish the proof. Now we can state and prove our main result: Theorem 2.4. Let A = A i be a graded algebra. Then A is a special piecewise-koszul algebra if and only if A is a Koszul algebra. Proof. ( ) By Lemma 2.4 of [7], for any Koszul algebra A = T A0 (A )/ R, we have (A n R) (R A n ) = (A i R A j ) i+j=n for all n 0. That is, Koszul algebras are certainly special piecewise-koszul algebras. ( ) Suppose that A is a special piecewise-koszul algebra, then by Lemma 2., we have Ext i A(A 0, A 0 ) = A! δp d(i) as bigraded algebras. Note that A is a piecewise-koszul algebra, by Lemma 2.2, we have Note also that A! = Ext A(A 0, A 0 ) = Ext A(A 0, A 0 ). A! δ d p(i) A! and Ext i A(A 0, A 0 ) i Ext i A(A 0, A 0 ), which implies that A! = A! as graded algebras, which forces δp(i) d δd p(i) = i for all i 0. That is, A is a Koszul algebra. Remark 2.5. Theorem 2.4 tells us that for a non-koszul piecewise-koszul algebra A, if we adopt the Definition.2, we can t get the exactness of piecewise- Koszul complex PK A 0 0. That is, we have to change Definition.2 for the case of a non-koszul piecewise-koszul algebra.
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