A bound on the Hartshorne-Speiser-Lyubeznik number of semigroup rings

TL;DR

This work provides an explicit, computable bound on the Hartshorne-Speiser-Lyubeznik number of local cohomology modules $H^\ell_\mathfrak{m}(k[Q])$ for affine pointed semigroup rings $k[Q]$ in positive characteristic. The authors develop a combinatorial framework based on the semigroup geometry, including invariants $N_Q^{(\ell)}$ and $m_{\mathcal{H}}$ associated to faces $\mathcal{H}$ of $Q$, and prove that for $p> N_Q^{(\ell)}$ one has $\mathrm{HSL}(H^\ell_\mathfrak{m}(k[Q])) \le \max_{\mathcal{H},\dim\mathcal{H}\ge \ell} \lceil \log_p m_{\mathcal{H}} \rceil$. This yields a dichotomy for large $p$ (HSL number is $0$ or $1$) and provides computable consequences for Frobenius test exponents in Cohen-Macaulay and weakly $F$-nilpotent settings, with explicit examples. The results connect Frobenius actions on local cohomology to semigroup saturation data, enabling practical bounds for $F$-singularity invariants in semigroup rings.

Abstract

In this paper we prove an explicit, computable upper bound on the Hartshorne-Speiser-Lyubeznik number of the local cohomology of a pointed, affine semigroup ring over a perfect field of positive characteristic. This bound depends only on the characteristic of the ring and properties of the semigroup.

Figures (1)

• Figure : $Q$

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Corollary 1.7
• Example 1.8
• Proposition 2.1
• Lemma 3.2