Exterior powers and Tor-persistence

TL;DR

This paper investigates conditions under which commutative Noetherian rings are Tor-persistent, using exterior powers, and proves all local rings with cube-zero maximal ideal are Tor-persistent, also offering a new proof of Tachikawa Conjecture in certain graded rings.

Contribution

It introduces new methods using exterior powers to address Tor-persistence and proves all local rings with maximal ideal cubed zero are Tor-persistent, advancing understanding of the open question.

Findings

Every local ring with $rak{m}^3=0$ is Tor-persistent.

Provides partial answers to the open question on Tor-persistence.

Offers a new proof of the Tachikawa Conjecture for certain graded rings.

Abstract

A commutative Noetherian ring $R$ is said to be Tor-persistent if, for any finitely generated $R$-module $M$, the vanishing of $\operatorname{Tor}_i^R(M,M)$ for $i\gg 0$ implies $M$ has finite projective dimension. An open question of Avramov, et. al. asks whether any such $R$ is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring $(R,\mathfrak{m})$ with $\mathfrak{m}^3=0$ is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2.