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Extremal behavior of reduced type of one dimensional rings

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arxiv 2306.17069 v1 pith:TK6P3HBZ submitted 2023-06-29 math.AC math.RA

Sarasij Maitra , Vivek Mukundan This is my paper · ORCID
classification math.ACmath.RA
keywords ringsgorensteinreducedsemigrouptypecategorycompleteinvariant
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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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups

    math.AC 2026-01 accept novelty 7.0 of 10

    In shifted numerical semigroups, being nearly Gorenstein or almost symmetric eventually repeats with period r_k, via a corrected pseudo-Frobenius bijection.

  2. Existence of a far-flung Gorenstein numerical semigroup attaining the Herzog--Kumashiro--Stamate bound

    math.AC 2026-07 accept novelty 6.0 of 10

    For every t≥2 an extremal set A of size t produces a far-flung Gorenstein semigroup S(n(A),A) of type t attaining the multiplicity bound n(t); for t≥5 the same family also satisfies res(S)>l(S).

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