A study of the cohomological rigidity property

TL;DR

This paper investigates the cohomological rigidity of tensor products over commutative Noetherian rings, establishing conditions for vanishing of local cohomology, and explores implications for module freeness and vector bundle splitting.

Contribution

It introduces new conditions linking local cohomology vanishing to tensor product properties and connects these to Tate homology and vector bundle criteria.

Findings

Vanishing of a single local cohomology module can imply the vanishing of all lower modules.

Provides bounds for the depth of tensor products of modules.

Offers criteria for module freeness over complete intersection rings.

Abstract

In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersection rings. Along the way, we also give a splitting criteria for vector bundles on smooth complete intersections.