tailieunhanh - 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. | Vietnam Journal of Mathematics 34 3 2006 341-351 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 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 2006 Abstract. Let A be a d-Koszul algebra and MEgr A we show that M is a weakly d-Koszul module if and only if E G M n 0Ext A G M A0 is generated in degree 0 as a graded _E A -module. Moreover relations among weakly d-Koszul modules d-Koszul modules and Koszul modules are discussed. We also show that the Koszul dual of a weakly d-Koszul module M E M n 0Ext A M A0 is finitely generated as a graded _E A -module. 2000 Mathematics Subject Classification 16E40 16E45 16S37 16W50. Keywords d-Koszul algebras d-Koszul modules weakly d-Koszul modules. 1. Introduction This 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 . This class of modules resemble classical d-Koszul modules in the way that they admits a tower of d-Koszul modules. It is well known that both Koszul modules and d-Koszul modules are pure and they have many nice homological properties. From 9 we know that although weakly d-Koszul modules are not pure they have 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 e grs A . Then M is a Koszul module if and only if the Koszul dual E M ny0Ext A M Ao is generated in 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 G grs A . Then M is a d-Koszul module if and only if the Koszul dual E M ny0Ext rA M A0 is generated in degree 0 as a graded E A -module. It is a pity that we cannot get the similar result for weakly d-Koszul module though