Gorensteinness of short local rings in terms of the vanishing of Ext and Tor

TL;DR

This paper characterizes Gorenstein local rings through the vanishing of Ext and Tor for modules with maximal complexity, providing new criteria and insights into their structure.

Contribution

It establishes equivalences between Gorensteinness and the vanishing of Ext and Tor for specific modules, partially answering a question by Takahashi.

Findings

Gorensteinness characterized by Ext and Tor vanishing

Equivalent conditions for Gorenstein local rings

Analysis of vanishing of Ext for canonical modules

Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $ \underline{x} = x_1,\ldots,x_d \in \mathfrak{m} \smallsetminus \mathfrak{m}^2 $ such that $ \mathfrak{m}^3 \subseteq (\underline{x}) $. Let $ M $ be a finite $ R $-module with maximal complexity or curvature, e.g., $ M $ can be a nonzero direct summand of some syzygy module of the residue field $ R/\mathfrak{m} $. It is shown that the following are equivalent: (1) $R$ is Gorenstein, (2) $\mathrm{Ext}_R^{\gg 0}(M,R)=0$, and (3) $\mathrm{Tor}_{\gg 0}^R(M,\omega) = 0$, where $\omega$ denotes a canonical module of $R$. It gives a partial answer to a question raised by Takahashi. Moreover, the vanishing of $\mathrm{Ext}_R^{\gg 0}(\omega,N)$ for certain $ R $-module $ N $ is also analyzed. Finally, it is studied why Gorensteinness of such local rings is important.