Tor and Ext vanishing results for commutative Artinian rings

Abstract

We give a negative answer to a question of Avramov, Buchweitz and Şega 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, Şega 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.

Disclosure

“nal commutative local Gorenstein F2 -algebra R admitting an indecomposable non-projective R-module M such that ExtiR (M, M ) = 0 for i = 1, 2. We display the concrete algebra R and module M , which were also found with the assistance of ChatGPT, in Section 3, together with Magma code verifying the theorem. This also resolves a problem in the literature that had remained open for nearly 20 years. In [LH, Theorem 4.6] a result was presented that would imply that if R is a commutati”

PDF page 2

Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:08:06.746953Z

Structural counts