复形的C-Gorenstein投射维数

Complex C-Gorenstein Projection Dimension

设R是一个有单位元的结合环,C是一个关于直和封闭且包含所有投射模的左R-模类。介绍左R-模复形的C-Gorenstein投射维数的概念,它是复形的Gorenstein投射维数的一个推广。利用环模理论和同调代数的方法,讨论复形X的C-Gorenstein投射维数C-Gpd(X)与其每个层次上模Xm的C-Gorenstein投射维数C-Gpd(X^m)之间的关系,给出复形X的C-Gorenstein投射维数小于等于n的若干等价刻画。证明了C-Gpd(X)=sup{C-Gpd(X^m) m∈Ζ},且当C-Gpd(X)=n(n≥1)时,存在复形短正合列0→H→G→X→0和0→X→H’→G’→0,其中G,G’为C-Gorenstein投射复形,H的投射维数小于等于n-1且H’的投射维数小于等于n。

Let R be an associative ring with unit cells,C is a class of left R-modules that is closed under direct sums and contains all projective modules. The paper introduces the notion of C-Gorenstein projective dimensions of complexes of left R-modules,it is a generalization of Gorenstein projective dimension of complex. Using theories of rings and modules,and methods of homological algebras,relations between C-Gorenstein projective dimension C-Gpd(X) of complex X and C-Gorenste in projective dimension C-Gpd(X^m) of every term module Xmare discussed,some equivalent characterizations of Gorenstein projective dimension of complex X less than or equal to n are given. It proves C-Gpd(X)= sup{C-Gpd(X^m)| m ∈Z},and there exist short exact sequences of complexes 0 → H → G → X → 0 and 0 → X → H’→ G’→ 0,with G,G’ C-Gorenstein projection complexes,projection dimension of H’ is less than or equal to n-1 and projection dimension of H’ less than or equal to n,when C-Gpd(X)= n(n≥1).