Some computations of generalized Hilbert-Kunz function and multiplicity

TL;DR

This paper computes the generalized Hilbert-Kunz invariant for specific modules over various classes of rings in characteristic p, including hypersurfaces, toric rings, and F-regular rings, advancing understanding of their algebraic properties.

Contribution

It introduces methods to compute the generalized Hilbert-Kunz invariant for modules over hypersurfaces, toric, and F-regular rings, expanding the class of rings with known invariants.

Findings

Computed invariants for hypersurfaces of finite representation type

Derived invariants for toric rings

Analyzed invariants for F-regular rings

Abstract

Let $R$ be a local ring of characteristic $p>0$ which is $F$-finite and has perfect residue field. We compute the generalized Hilbert-Kunz invariant for certain modules over several classes of rings: hypersurfaces of finite representation type, toric rings, $F$-regular rings.