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arXiv · 2306.17069

Extremal behavior of reduced type of one dimensional rings

Sarasij Maitra·Vivek Mukundan
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Abstract

Let $R$ be a domain that is a complete local $\mathbb{k}$ algebra in dimension one. In an effort to address the Berger's conjecture, a crucial invariant reduced type $s(R)$ was introduced by Huneke et. al. In this article, we study this invariant and its max/min values separately and relate it to the valuation semigroup of $R$. We justify the need to study $s(R)$ in the context of numerical semigroup rings and consequently investigate the occurrence of the extreme values of $s(R)$ for the Gorenstein, almost Gorenstein, and far-flung Gorenstein complete numerical semigroup rings. Finally, we study the finiteness of the category $\text{CM}(R)$ of maximal Cohen Macaulay modules and the category $\text{Ref}(R)$ of reflexive modules for rings which are of maximal/minimal reduced type and provide many classifications.

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Sarasij Maitra, Vivek Mukundan. 2023-06-29. Extremal behavior of reduced type of one dimensional rings. https://arxiv.org/abs/2306.17069

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