On the vanishing of Ext and Tor

TL;DR

This paper establishes conditions under which Ext and Tor functors vanish and explores their implications for module morphisms over noetherian and Gorenstein rings, revealing deep connections between these homological invariants.

Contribution

It proves new equivalences and vanishing results for Ext and Tor functors related to module morphisms over noetherian and Gorenstein rings, connecting these concepts.

Findings

Vanishing of Ext^1(f, -) iff vanishing of Tor_1(f, -)

Equivalence of Ext^1(f, -) being epic and Tor_1(f, -) being monic

Simultaneous vanishing of Tor_1, Ext^1, and Ext^1 over Gorenstein projective modules

Abstract

This paper contains two theorems concerning the vanishing of natural transformations of (co)homology functors. Precisely, assume that $R$ is a right noetherian ring and $f: M\rt N$ is a morphism of finitely generated right $R$-modules. The first theorem proves that the natural transformation $\Ext^1(f, -)$ vanishes over the category of finitely generated right $R$-modules if and only if $\Tor_1(f, -)$ vanishes over the category of finitely generated left $R$-modules. As a corollary of this result, we establish that $\Ext^1(f, -)$ is epic if and only if $\Tor_1(f, -)$ is monic. The second theorem shows that if $R$ is left and right noetherian and $M, N$ are Gorenstein projective, then the natural transformations $\Tor_1(f, -)$, $\Ext^1(-, f)$ and $\Ext^1(f, -)$ vanish over the category of finitely generated Gorenstein projective modules, simultaneously. This, in particular, yields that over Gorenstein projective modules, the notions of phantom morphisms and $\Ext$-phantom morphisms coincide. Also, it is proved that if $R$ is $n$-Gorenstein, then for any integer $i>n$, the natural transformations $\Ext^{i}(f, -)$, $\Ext^{i}(-, f)$ and $\Tor_{i}(f, -)$ vanish over finitely generated modules, simultaneously. As an interesting consequence, we show that under the same assumptions, $\Ext^i(-, f)$ is epic (resp. monic) if and only if $\Ext^i(f, -)$ is monic (resp. epic) if and only if $\Tor_i(f, -)$ is epic (resp. monic).