Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

F-Splittings of seminormal monoid algebras

Milena Hering, Kevin Tucker

TL;DR

This work provides a complete combinatorial framework for analyzing Frobenius-splitting phenomena in seminormal affine monoid algebras. By describing $\mathrm{Hom}_R(F^e_*R,R)$ via monomial maps $\pi_a$ indexed by $a\in \frac{1}{q}M$ and exploiting faces, RUFs, and $p$-faces, the authors obtain explicit formulas for the $F$-splitting ratio, splitting dimension, and splitting prime, and relate these to the normalization. They extend the normal-case picture to the seminormal setting and derive a precise description of Cartier-fixed ideals and the test ideal, including a canonical description of the non-$F$-pure ideal, all in terms of monomial ideals tied to the face structure. The results give a practical toolkit for computing $F$-splitting invariants and test ideals in seminormal toric contexts, with the Whitney umbrella as an illustrative example. Overall, the paper advances a combinatorial approach to non-normal $F$-singularities, enabling systematic verification and exploration of conjectures in this setting.

Abstract

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential splittings of iterates of Frobenius for seminormal monoid algebras. This allows us to give an easy formula for the F-splitting ratio of such rings as well as to compute the ideals stable under the Cartier algebra, including the test ideal.

F-Splittings of seminormal monoid algebras

TL;DR

This work provides a complete combinatorial framework for analyzing Frobenius-splitting phenomena in seminormal affine monoid algebras. By describing via monomial maps indexed by and exploiting faces, RUFs, and -faces, the authors obtain explicit formulas for the -splitting ratio, splitting dimension, and splitting prime, and relate these to the normalization. They extend the normal-case picture to the seminormal setting and derive a precise description of Cartier-fixed ideals and the test ideal, including a canonical description of the non--pure ideal, all in terms of monomial ideals tied to the face structure. The results give a practical toolkit for computing -splitting invariants and test ideals in seminormal toric contexts, with the Whitney umbrella as an illustrative example. Overall, the paper advances a combinatorial approach to non-normal -singularities, enabling systematic verification and exploration of conjectures in this setting.

Abstract

We compute a number of invariants of singularities defined via the Frobenius morphism for seminormal affine toric varieties over fields of characteristic p > 0. Our main technical tool is a combinatorial description of the potential splittings of iterates of Frobenius for seminormal monoid algebras. This allows us to give an easy formula for the F-splitting ratio of such rings as well as to compute the ideals stable under the Cartier algebra, including the test ideal.
Paper Structure (12 sections, 18 theorems, 30 equations, 1 figure)

This paper contains 12 sections, 18 theorems, 30 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Seminormality and weak normality
  4. Definition of F-splitting ratio used in Section \ref{['sec:Fsplittingratio']}
  5. Cartier algebra and test ideals used in Section \ref{['sec:testideals']}.
  6. Seminormal Monoid Algebras
  7. The set of homomorphisms from F*R to R.
  8. The action of Frobenius on R and a description of F*R
  9. Notation and review of the normal case.
  10. The seminormal case
  11. Computation of the F-splitting ratio and the F-splitting prime.
  12. Test ideals of seminormal monoid algebras

Key Result

Theorem 1.1

Let $R$ be a seminormal monoid algebra. Suppose $e$ is so that $p^e$ strictly bounds the $p$-power torsion of $M \cap \langle D \rangle / M_D$ for any face $D \prec C$. We have that $\pi_a \in \mathop{\mathrm{\rm Hom}}\nolimits_R(F^e_*R,R)$ if and only if the following conditions hold. Moreover, for any $p$-face $D$, the image of any $\pi_a\in \mathop{\mathrm{\rm Hom}}\nolimits_R(F^e_*R,R)$ does

Figures (1)

  • Figure 1: The monoid $S$ Example \ref{['ex:running']}.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1
  • Example 1.3
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 1
  • Remark 2.5
  • ...and 37 more