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arXiv · 2608.09701

Tor and Ext vanishing results for commutative Artinian rings

Bernhard Böhmler·Rene Marczinzik
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Abstract

We give a negative answer to a question of Avramov, Buchweitz and \c{S}ega by constructing a commutative local finite-dimensional non-Gorenstein algebra $R$ with $\operatorname{Ext}_R^1(D(R),R)=0$; this question is related to the first Tachikawa conjecture. We also give a counterexample to a conjecture of Huneke, \c{S}ega and Vraciu on Tor vanishing over commutative finite-dimensional algebras. Finally, we construct a finite-dimensional commutative local self-injective algebra $R$ over $\mathbb{F}_2$ and an indecomposable non-projective $R$-module $M$ such that $\operatorname{Ext}_R^1(M,M)=\operatorname{Ext}_R^2(M,M)=0$, related to the second Tachikawa conjecture and answering a question of Dao.

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Bernhard Böhmler, Rene Marczinzik. 2026-08-10. Tor and Ext vanishing results for commutative Artinian rings. https://arxiv.org/abs/2608.09701

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