Rigidity of Ext and Tor via flat-cotorsion theory

TL;DR

This paper demonstrates that under certain conditions, the vanishing of Ext modules at a high degree implies their vanishing at all higher degrees, and it also advances results on Tor-rigidity using minimal semi-flat-cotorsion replacements.

Contribution

It improves bounds on Ext and Tor-rigidity results by leveraging the existence of minimal semi-flat-cotorsion replacements in the derived category.

Findings

Ext vanishing at degree n implies vanishing at all higher degrees for modules over certain rings.

Enhanced bounds on Tor-rigidity results.

Utilization of minimal semi-flat-cotorsion replacements to achieve these improvements.

Abstract

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.