On the second rigidity theorem of Huneke and Wiegand

TL;DR

This paper demonstrates that a previously claimed depth formula theorem by Huneke and Wiegand is false in general, providing counterexamples and highlighting its impact on related literature.

Contribution

It offers the first known counterexamples to the flawed claim and clarifies the incorrectness of the original theorem in the context of commutative algebra.

Findings

Counterexamples show the claim is false in general

The original depth formula claim is invalid

Several papers relied on the flawed claim

Abstract

In 2007 Huneke and Wiegand announced in an erratum that one of the conclusions of their depth formula theorem is flawed due to an incorrect convention for the depth of the zero module. Since then, the deleted claim has remained unresolved. In this paper we give examples to prove that the deleted claim is false, in general. Moreover, we point out several places in the literature which relied upon this deleted claim or the initial argument from 2007.