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The weak normalization of an affine semigroup

Kyle Maddox, Srishti Singh

TL;DR

The paper introduces the $p$-weak normalization ${^{*p}}A$ of an affine semigroup $A$ as a computable intermediary between seminormalization and normalization, encoding Frobenius-structure data in characteristic $p$ and clarifying $F$-singularities of the related semigroup ring $k[A]$. It provides a geometric description ${^{*p}}A=\bigcup_{F\text{ face of }C(A)}\big(\widetilde{A_F/p^{N_0}}\cap \operatorname{int}(F)\big)$ and proves $^*k[A]=k[^{*p}A]$, with only finitely many possible ${^{*p}}A$ across primes. The work connects ${^{*p}}A$ to $F$-injectivity and $F$-nilpotence, showing ${^{*p}}A={^{+}A}$ iff $k[A]$ is $F$-injective and ${^{*p}}A=\overline{A}$ iff $k[A]$ is $F$-nilpotent, and derives a uniform bound $ ext{Fte}(I)\le N_0$ for all ideals $I$, where $p^{N_0}{^{*p}}A\subset A$. An explicit algorithm and implementations in SageMath/Macaulay2 are provided to compute ${^{*p}}A$ and Frobenius closures, enabling practical exploration of Frobenius-related singularities in affine semigroup rings.

Abstract

In this article, we define and explore the weak normalization of an affine semigroup. In particular, for a fixed prime integer, we provide a geometric description of the weak normalization of an affine semigroup with respect to that prime, which corresponds to the weak normalization of the affine semigroup ring over a field of that prime characteristic, similar to the description of the seminormalization of an affine semigroup given by Reid-Roberts. We then use this description to understand the singularities of an affine semigroup ring defined over a field of prime characteristic and provide several examples. In particular, we demonstrate that all affine semigroup rings defined over fields of prime characteristic have a uniform upper bound on the Frobenius test exponent of all ideals, which provides a large and important class of examples with a positive answer to a question of Katzman-Sharp on uniformity of Frobenius test exponents. Finally, we provide an algorithm and implementation to compute the weak normalization of an affine semigroup, as well as the Frobenius test exponent and Frobenius closures of ideals in the affine semigroup ring.

The weak normalization of an affine semigroup

TL;DR

The paper introduces the -weak normalization of an affine semigroup as a computable intermediary between seminormalization and normalization, encoding Frobenius-structure data in characteristic and clarifying -singularities of the related semigroup ring . It provides a geometric description and proves , with only finitely many possible across primes. The work connects to -injectivity and -nilpotence, showing iff is -injective and iff is -nilpotent, and derives a uniform bound for all ideals , where . An explicit algorithm and implementations in SageMath/Macaulay2 are provided to compute and Frobenius closures, enabling practical exploration of Frobenius-related singularities in affine semigroup rings.

Abstract

In this article, we define and explore the weak normalization of an affine semigroup. In particular, for a fixed prime integer, we provide a geometric description of the weak normalization of an affine semigroup with respect to that prime, which corresponds to the weak normalization of the affine semigroup ring over a field of that prime characteristic, similar to the description of the seminormalization of an affine semigroup given by Reid-Roberts. We then use this description to understand the singularities of an affine semigroup ring defined over a field of prime characteristic and provide several examples. In particular, we demonstrate that all affine semigroup rings defined over fields of prime characteristic have a uniform upper bound on the Frobenius test exponent of all ideals, which provides a large and important class of examples with a positive answer to a question of Katzman-Sharp on uniformity of Frobenius test exponents. Finally, we provide an algorithm and implementation to compute the weak normalization of an affine semigroup, as well as the Frobenius test exponent and Frobenius closures of ideals in the affine semigroup ring.
Paper Structure (8 sections, 13 theorems, 20 equations)

This paper contains 8 sections, 13 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Affine semigroups
  4. Purely inseparable extensions and weak normalization
  5. A brief primer on F-singularities
  6. The p-weak normalization of an affine semigroup
  7. Seminormality, Weak Normality, and F-singularities
  8. An algorithm to compute the p-weak normalization

Key Result

Theorem A

Let $A\subset \mathbb{N}^n$ be an affine semigroup and fix a prime $p$. Recall there is an $N_0$ such that $p^{N_0}({^{*p}}\space A) \subset A$. Then

Theorems & Definitions (37)

  • Definition : \ref{['defn: p-weak normalization']}
  • Theorem A: \ref{['thm: geometric description of p-weak normalization']}
  • Theorem B: \ref{['cor: wkNiffFinj']}, \ref{['cor: weak normalization equals normalization if and only if F-nilpotent']}
  • Theorem C: \ref{['thm: affine semigroup rings have uniformly bounded fte']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 27 more