Applications and homological properties of local rings with decomposable maximal ideals

TL;DR

This paper constructs specific local Cohen-Macaulay rings with decomposable maximal ideals to explore their homological properties, revealing cases where localizations behave differently regarding Auslander's conditions and semidualizing modules.

Contribution

It introduces new examples of Cohen-Macaulay rings with decomposable maximal ideals exhibiting unique homological behaviors not seen in prior work.

Findings

Constructed rings satisfy UAC but localizations do not satisfy AC.

Created rings with a fixed number of semidualizing modules whose localizations have exponentially more.

Characterized Cohen-Macaulay fiber products of finite Cohen-Macaulay type.

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.