δ-Koszul对象的若干注记

Some Notes on δ-Koszul Objects

本文首先给出了δ-分解决定的代数的存在性的一个充分条件和一个必要条件,它与2005年Green和Marcos提出的关于δ-Koszul代数的三个问题有关.2007年,对有限生成分次模范畴中的短正合列ζ:0→K→M→N→0,Cheng和Ye证明了:若K,M是δ-Koszul模,则N也是δ-Koszul模;若K,N是δ-Koszul模,则M也是δ-Koszul模,并且给出反例说明即使M,N是δ-Koszul模,K也未必是δ-Koszul模.本文的另一主要内容是讨论了当M,N是δ-Koszul模时,K是δ-Koszul模的一些条件.

A sufficient and a necessary condition for the existence of δ-resolution deter- mined algebras are given in this paper, which is related to one of the three questions raised by Green and Marcos in 2005. Cheng and Ye proved in 2007 that for a short exact sequence ： 0 → K → M → N→ 0, in the category of finitely generated graded modules N is a δ-Koszul module provided that K and M are δ-Koszul modules; M is a δ-Koszul modules provided that K and N are δ-Koszul modules. Moreover, they gave counterexamples to show that K is not a δ-Koszul modules in general even though M and N are δ-Koszul modules. Another result of this paper is to discuss the conditions such that K is a δ-Koszul module provided that M and N are δ-Koszul modules.