关于Gorenstein投射维数的一个注记

本文研究了Gorenstein投射维数的相关问题.利用经典同调维数的研究方法,给出了Gorenstein投射维数有限模的Gorenstein投射维数的一个刻画,并利用这一结果证明了Gorenstein完全环和Artin环的Gorenstein整体维数分别由各自的循环模和单模的Gorenstein投射维数来确定.这些结论丰富了Gorenstein同调代数理论.     

弱直投射模和弱直内射模

引入了弱直投射和弱直内射模的概念，给出了它们的一些性质。使用弱直投射和弱直内射模刻划了遗传环、半遗传环、半单环和ＱＦ－环。     

Gr-凝聚环的小有限分次投射维数gr.fp.dimR

文 [1],[2 ]分别研究了Gr NoetherGr 局部 (半局部 )环的同调维数 ,本文主要进一步讨论Gr 凝聚Gr 半局部环的同调性质 .在§ 1中 ,主要刻画交换Gr 凝聚Gr 半局部环R的分次弱整体维数gr.gl.w .dimR ;在§ 2中 ,定义了分次环R的小有限分次投射维数gr.fp .dimR .刻画了gr.fp .dimR =gr .gl.w .dimR的Gr 凝聚环 .由于Gr Noether环是Gr 凝聚的 ,因而本文所得的结果对于Gr Noether环是自然成立的 .同时 ,本文所得的结果 ,也可视为文 [4 ]关于一般交换凝聚环相应结论的推广 .     

复形的$D_C$-投射维数

设$R$是交换环, $C$是半对偶$R$-模. 本文引入并研究同调下有界复形的$D_C$-投射维数.     

关于R—投射模是投射模的环

本文研究了所有Ｒ—投射模都是投射模的环（ＲＰ—环），得出了它的几个等价条件，证明了：Ｓ＝Ｒｎ为ＲＰ—环当且仅当Ｒ为ＲＰ—环；∑ｎｉ＝１Ｒｉ为ＲＰ—环当且仅当每个Ｒｉ为ＲＰ—环．讨论了ＲＰ—环的左投射维数．     

RP—环

本文定义了ＲＰ－环,讨论了它的性质和等价条件,得出了Ｓ＝Ｍｎ（Ｒ）是ＲＰ－环妥且仅当Ｒ是ＲＰ－环;Ｒｉ是ＲＰ－环当且仅当每个Ｒｉ是ＲＰ－环．最后讨论了ＲＰ－环的同态象和左投射维数．     

一类弱整体维数为2的局部环上的Bass-Quillen问题II

设R是U2环,即(R,m)是局部GCD整环,且存在u∈m-m2,使得R/(u)是赋值环,且Ru是Bézout整环.本文证明了若R是带有正规元素为u的U2环,且dim(R/(u))=1,则每个有限生成投射R[X1,...,Xn]模是自由模.由此得到了若R是广义伞环,则每个有限生成投射R[X1,...,Xn]模是自由模.     

弱$r$-Clean环

As generalization of r-clean rings and weakly clean rings, we define a ring R is weakly r-clean if for any a∈R there exist an idempotent e and a regular element r such that a = r + e or a = r-e. Some properties and examples of weakly r-clean rings are given. Furthermore, we prove the weakly clean rings and weakly r-clean rings are equivalent for abelian rings.     

一类弱整体维数为2的局部环上的Bass-Quillen问题

设局部GCD整环(R,m)满足: 存在u∈m-m², 使得R/(u)是赋值环, 且R_u是 Bézout整环, 则R叫做 U₂环, u叫做一个正规元素. 证明了若R是U₂环, 则R与一元多项式环R[X]都是凝聚环; 且若u是R的正规元素, dim(R/(u))=1, 则每个有限生成投射R[X]-模是自由模.     

关于R—投射模是投射模的环

本文研究了所有R-投射模都是投射模的环（RP-环），得出了它的几个等价条件，证明了：S=Rn为RP-环当且仅当R为RP-环；n↑∑↑n=1Ri为RP-环当且仅当每个Ri为RP-环。讨论了RP-环的左投射维数。     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

Weak Dimension of FP-injective Modules Over Chain Rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.     

自同态与环的弱维数

利用模的自同态研究ｖｏｎ Ｎｅｕｍａｎｎ正则环与环的弱维数,给出了ｖｏｎ Ｎｅｕｍａｎｎ正则环的新的刻划,同时也刻划了弱维数≤ｎ（ｎ≥０）的环。     

一类Morita Contexts的同调维数

本文得到了一类Mortia Contexts的总体维数与环R的总体维数的相等关系,并且将这个结论推广到n×n矩阵环。     

弱总体维数和模的自同态

本文用模的自同态,给出弱总体维数≤ｎ的环的特征,其中ｎ≥０．设Ｒ为环,部分地回答了下列问题：何时任意有限表现Ｒ－模Ｍ有无穷分解：０→Ｍ→Ｆ₀→Ｆ₁→…→Ｆ_n→…,其中每个Ｆ_i均是有限生成投射的,ｉ＝１,２,…？     

Generalized local duality,canonical modules,and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.