Rigidity of Ext and Tor with coefficients in residue fields of a commutative noetherian ring

TL;DR

This paper proves that vanishing of certain Tor and Ext groups with residue field coefficients at a prime ideal implies their vanishing in all higher degrees, revealing a rigidity property in homological algebra over noetherian rings.

Contribution

It establishes new rigidity theorems for Ext and Tor functors with residue field coefficients, extending understanding of their vanishing behavior in commutative noetherian rings.

Findings

Vanishing of Tor in a certain degree implies vanishing in all higher degrees.

Similar rigidity results are proved for Ext groups.

Applications to homological dimension theory are discussed.

Abstract

Let p be a prime ideal in a commutative noetherian ring R. It is proved that if an R-module M satisfies Tor^R_n(k(p),M) = 0 for some n \geq dim R_p, where k(p) is the residue field at p, then Tor^R_i(k(p),M) = 0 holds for all i \geq n. Similar rigidity results concerning Ext_R^*(k(p),M) are proved, and applications to the theory of homological dimensions are explored.