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,\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.