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Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring which containsa 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 complexityor curvature, e.g., $ M $ can be a nonzero direct summand of some syzygy moduleof the residue field $ R/\mathfrak{m} $. It is shown that the following areequivalent: (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 canonicalmodule 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 ofsuch local rings is important.

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