A computation concerning relative Hilbert-Kunz multiplicities

TL;DR

This paper presents detailed computations of a specific example related to Hilbert-Kunz multiplicities, illustrating a case where the converse of a main theorem holds outside previously known scenarios.

Contribution

It provides an explicit example demonstrating the converse of a main theorem in Hilbert-Kunz theory, expanding understanding of its applicability.

Findings

The example confirms the converse holds in a new, previously unrecognized case.

Computations serve as a reference for future research in Hilbert-Kunz multiplicities.

The paper clarifies the boundaries of existing theorems through concrete example.

Abstract

In preparing the paper "Some extensions of Hilbert-Kunz multiplicity", we had occasion to perform an intricate set of computations pertaining to a single illustrative example. In the end, we have decided not to include the computations in the main paper, but instead to publish them here for later reference. In particular, the current preprint gives an example where the converse to one of our main theorems holds, even though it does not fit into any of the previously known cases where a converse holds.