Báo cáo toán học: Weakly d-Koszul Modules

A là một đại số d-Koszul và M ∈ gr (A), chúng tôi cho thấy rằng M là một yếu mô-đun khi và chỉ khi E (G (M)) = ⊕ n ≥ 0 Ext n (G (M), A0)tạo ra ở mức độ 0 E phân loại một (A) mô-đun-. Hơn nữa, mối quan hệ giữa các module yếu d-Koszul Koszul d-mô-đun và các mô-đun Koszul được thảo luận.
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Báo cáo toán học: " Weakly d-Koszul Modules " Vietnam Journal of Mathematics 34:3 (2006) 341–351 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 Weakly d-Koszul Modules Jia-Feng Lu and Guo-Jun Wang Department of Mathematics, Zhejiang University, Hangzhou 310027, China Received January 12, 2006 Revised March 27, 2006Abstract. Let A be a d-Koszul algebra and M ∈gr (A), we show that M is a weakly module if and only if E (G(M ))=⊕n≥0 Ext n (G(M ),A0 ) is generated in degree 0 asd-Koszul Aa graded E (A)-module. Moreover, relations among weakly d-Koszul modules, d-Koszulmodules and Koszul modules are discussed. We also show that the Koszul dual of aweakly d-Koszul module M : E (M )=⊕n≥0 Ext n (M,A0 ) is finitely generated as a graded AE (A)-module.2000 Mathematics Subject Classification: 16E40, 16E45, 16S37, 16W50.Keywords: d-Koszul algebras, d-Koszul modules, weakly d-Koszul modules.1. IntroductionThis paper is a continuation work of [9]. The concept of weakly d-Koszul module,which is a generalizaion of d-Koszul module, is firstly introduced in [9]. Thisclass of modules resemble classical d-Koszul modules in the way that they admitsa tower of d-Koszul modules. It is well known that both Koszul modules andd-Koszul modules are pure and they have many nice homological properties.From [9], we know that although weakly d-Koszul modules are not pure, theyhave many perfect properties similar to d-Koszul modules. Using Koszul dual to characterize Koszul modules is another effective aspect.For Koszul and d-Koszul modules, we have the following well known results from[4] and [6]. • Let A be a Koszul algebra and M ∈ grs (A). Then M is a Koszul module if and only if the Koszul dual E (M ) = ⊕n≥0Ext n (M, A0 ) is generated in A degree 0 as a graded E (A)-module.342 Jia-Feng Lu and Guo-Jun Wang • Let A be a d-Koszul algebra and M ∈ grs (A). Then M is a d-Koszul module if and only if the Koszul dual E (M ) = ⊕n≥0Ext n (M, A0 ) is generated in A degree 0 as a graded E (A)-module. It is a pity that we cannot get the similar result for weakly d-Koszul modulethough it is a generalizaion of d-Koszul module. We only have a necessarycondition for weakly d-Koszul modules (see [9]): • Let M be a weakly d-Koszul module with homogeneous generators being of degrees d0 and d1 (d0 < d1). Then E (M ) is generated in degrees 0 as a graded E (A)-module. One of the aims of this paper is to get a similar equivalent description forweakly d-Koszul modules. In order to do this, we cite the notion of the associatedgraded module of a module, denoted by G(M ), the formal definition will be givenlater. If we replace the weakly d-Koszul module M by G(M ), we can get thesimilar result: • Let A be a d-Koszul algebra and M ∈ gr(A). Then M is a weakly d-Koszul module if and only if E (G(M )) = ⊕n≥0 Ext n (G(M ), A0) is generated in A degree 0 as a graded E (A)-module. From this point of view, weakly d-Koszul modules have a close relation be-tween classical d-Koszul modules and Koszul modules. It is well known that to determine whether the Koszul dual E (M ) is finitelygenerated or not is very difficult in general. In this paper, we show that E (M )is finitely generated as a graded E (A)-module for a weakly d-Koszul module M ,which is an application of Theorem 2.5 [9] and another main result of this paper. The paper is organized as follows. In Sec. 2, we introduce so ...