# Powers of the maximal ideal and vanishing of (co)homology

## TL;DR

This paper demonstrates that positive powers of the maximal ideal in a Noetherian local ring are Tor-rigid and strongly-rigid, providing new characterizations of regularity and addressing a longstanding conjecture.

## Contribution

It proves that all positive powers of the maximal ideal are Tor-rigid and strongly-rigid, offering new insights into regularity and the torsion conjecture of Huneke and Wiegand.

## Key findings

- Positive powers of the maximal ideal are Tor-rigid.
- These ideals are also strongly-rigid.
- The results give new characterizations of regularity.

## Abstract

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.