# The Huneke-Wiegand conjecture and middle terms of almost split sequences

## TL;DR

This paper proves the Huneke-Wiegand conjecture for certain Gorenstein local domains by analyzing the structure of almost split sequences and Cohen-Macaulay modules, showing nonfree modules with multiple indecomposable summands have nonvanishing self-extensions.

## Contribution

It establishes the conjecture for a class of modules based on their middle terms in almost split sequences in Gorenstein domains.

## Key findings

- Modules with multiple indecomposable summands in the middle of almost split sequences have nonvanishing self-extensions.
- The Huneke-Wiegand conjecture holds for these modules in Gorenstein local domains.
- Provides new insights into the structure of Cohen-Macaulay modules and their extensions.

## Abstract

Let R be a Gorenstein local domain of dimension one. We show that a nonfree maximal Cohen--Macaulay R-module M possessing more than one nonfree indecomposable summand in the middle term of the almost split sequence ending in M has a nonvanishing self extension. In other words, we show that the Huneke--Wiegand conjecture is affirmative for such R-modules M.