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arXiv · 1802.00307

Applications and homological properties of local rings with decomposable maximal ideals

Saeed Nasseh·Sean Sather-Wagstaff·Ryo Takahashi·Keller VandeBogert
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Abstract

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen-Macaulay fiber products of finite Cohen-Macaulay type.

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Saeed Nasseh, Sean Sather-Wagstaff, Ryo Takahashi, Keller VandeBogert. 2018-02-01. Applications and homological properties of local rings with decomposable maximal ideals. https://arxiv.org/abs/1802.00307

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