‎let $m$ be a non-zero finitely generated module over a commutative noetherian local ring $(r,mathfrak{m})$ with $dim_r(m)=t$‎. ‎let $i$ be an ideal of $r$ with $grade(i,m)=c$‎. ‎in this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎the main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎tor^{r}_c(k,h^c_i(m))to kotim...

Let R be a commutative Noetherian ring and let M be a nitely generated R-module. If I is an ideal of R generated by M-regular sequence, then we study the vanishing of the rst Tor functors. Moreover, for Artinian modules and coregular sequences we examine the vanishing of the rst Ext functors.

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

In this paper we make some computations in homological algebra using Gröbner bases for modules over polynomials rings with coefficients in a Noetherian commutative ring. In particular, we show easy procedures for computing the Ext and Tor modules.

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity Ext and Tor, with respect to homological degree, over complete intersection rings. derive from a theorem Gulliksen linearity result for modules in high degrees. show similar under additional hypothesis that enough Tor are supported dimension at most one; we then provide examples showing could be pretty hectic when latter ...

We prove rigidity type results on the vanishing of stable Ext and Tor for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of prerigidity, which generalizes phenomena for modules of finite complete intersection dimension and complexity one. Using this concept, we prove results on length and vanishing of homolo...

‎Let $M$ be a non-zero finitely generated module over a commutative Noetherian local ring $(R,mathfrak{m})$ with $dim_R(M)=t$‎. ‎Let $I$ be an ideal of $R$ with $grade(I,M)=c$‎. ‎In this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎The main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎Tor^{R}_c(k,H^c_I(M))to kotim...

Recently, Yekutieli introduced projective dimension, injective dimension and flat of DG-modules by generalizing the characterization ordinary modules vanishing Ext or Tor-groups. In this paper, we introduce a DG-version projective, resolution for over connective DG-algebra which are different from known resolutions. An important feature these resolutions is that, roughly speaking, “length” give...