PC-内射性及其相关同调问题研究

发布时间：2019-04-22 20:58

【摘要】：本文主要研究了模的伪凝聚性和PC-内射性确定的同调维数及其在形式三角矩阵环上的应用.设凡是任何环,若R-模N的每个有限生成子模是有限表现的,则称N是伪凝聚模. R-模L称为PC-内射模是指对任何伪凝聚模N,有 Ext1/R(N,L) = 0.本文主要结果如下:设R是Noether环,则R-模L是PC-内射模当且仅当L是内射模;设R是凝聚环,则R-模L是PC-内射模当且仅当对任何伪凝聚模N及任何正整数k ≥ 1,有ExtRk(N,L) = 0;本文还引入了模的PC-内射维数和环的整体PC-内射维数(PC-dim(R))的概念,证明了若R是凝聚环,则有w.gl.dim(R)≤PC-dim(R)≤gl.dim(R)≤PC-dim(R) + 1.随后,本文又给出凝聚环上PC-内射维数的换环定理及其相关的维数公式,证明了若R是凝聚环,则有PC-dim(R[x])=PC-dim(R)+ 1.设A,B是任何环,M是A-B-双模,则称T=(?)是形式三角矩阵环.最后,本文对形式三角矩阵环上的PC-内射模结构进行刻画,并计算了形式三角矩阵环的整体PC-内射维数,得到了若T是右凝聚环,M是有限表现右A-模,则有Max{PC-dim(A),PC-dim(B)}≤PC-dim(T) ≤ 1 + Max{1 + PC-dim(A),PC-dim(B)};若T是 Noether 环,则有]Max{gl.dim(A),gl.dim(B)}≤gl.dim(T)≤1+Max{1+gl.dim(A),gl.dim(B)};特别地,若T是Noether环,M是平坦右A-模,则有Max{gl.dim(A),gl.dim(B)}≤gl.dim(T) 1 + Max{gl.dim(A), gl.dim(B)}。
[Abstract]:In this paper, we mainly study the homological dimension of the pseudo-cohesion and PC- injectivity of modules and their applications in formal triangular matrix rings. If every finitely generated submodule of R-module N is finitely represented, then N is called a pseudo-coherent module. The R-module L is called PC- injective module, which means that for any pseudo-condensed module N, there is Ext1/R (N, L) = 0. The main results are as follows: let R be a Noether ring, then R-module L is an PC- injective module if and only if L is an injective module; Let R be a coherent ring, then R-module L is a PC- injective module if and only if there is ExtRk (N, L) = 0 for any pseudo-coherent module N and any positive integer k 鈮，

本文编号：2463191