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arXiv · 1705.05714

Totally reflexive modules over rings that are close to Gorenstein

Andrew R. Kustin·Adela Vraciu
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Abstract

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the canonical module of $R$ has a direct summand $T$ which is a test module for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$, for some finitely generated $R$-module $N$, then $N$ is free.

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Andrew R. Kustin, Adela Vraciu. 2017-05-16. Totally reflexive modules over rings that are close to Gorenstein. https://arxiv.org/abs/1705.05714

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