Extremal behavior of reduced type of one dimensional rings

TL;DR

This paper investigates the invariant reduced type in one-dimensional complete local rings, exploring its extremal values, relation to valuation semigroups, and implications for Cohen-Macaulay and reflexive modules, with applications to numerical semigroup rings.

Contribution

It analyzes the extremal values of the reduced type invariant and connects it to valuation semigroups, providing classifications for rings with maximal or minimal reduced type.

Findings

Characterization of extremal reduced type values in various classes of rings

Relation between reduced type and valuation semigroup

Finiteness results for categories of Cohen-Macaulay and reflexive modules

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.