The weak normalization of an affine semigroup

TL;DR

This paper introduces the concept of weak normalization for affine semigroups, providing geometric descriptions, analyzing singularities, and offering algorithms to compute weak normalization and Frobenius-related invariants in prime characteristic.

Contribution

It defines weak normalization for affine semigroups, describes it geometrically, and develops algorithms for computing weak normalization and Frobenius invariants in prime characteristic.

Findings

Affine semigroup rings over prime fields have bounded Frobenius test exponents.

The weak normalization relates to seminormalization, aiding singularity analysis.

Algorithms for computing weak normalization and Frobenius invariants are provided.

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.