# On an example concerning the second rigidity theorem

## TL;DR

This paper investigates conditions under which tensor products of modules are reflexive, focusing on Tor-rigidity and the Hochster--Huneke graph, ultimately supporting the Second Rigidity Theorem of Huneke and Wiegand.

## Contribution

It provides a new criterion that rules out certain examples of non-reflexive tensor products, strengthening the understanding of module rigidity.

## Key findings

- Identifies a condition preventing the specific example from occurring.
- Supports the validity of the Second Rigidity Theorem.
- Analyzes the structure of the Hochster--Huneke graph in this context.

## Abstract

In this paper we revisit an example of Celikbas and Takahashi concerning the reflexivity of tensor products of modules. We study Tor-rigidity and the Hochster--Huneke graph with vertices consisting of minimal prime ideals, and determine a condition with which the aforementioned example cannot occur. Our result, in particular, corroborates the Second Rigidity Theorem of Huneke and Wiegand.