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arXiv · 2007.09174

Exterior powers and Tor-persistence

Justin Lyle·Jonathan Montaño·Keri Sather-Wagstaff
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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.

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Justin Lyle, Jonathan Montaño, Keri Sather-Wagstaff. 2020-07-17. Exterior powers and Tor-persistence. https://arxiv.org/abs/2007.09174

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