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

Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which contains a regular sequence $\underline{x}=x_1,\dots ,x_d\in \mathfrak{m}\setminus {\mathfrak{m}}^2$ 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.

13 pages, Comments are welcome