Generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex

TL;DR

This paper studies the generalized Hilbert-Kunz function of the Rees algebra of the face ring of a simplicial complex, showing it is polynomial for large s and providing explicit calculations and computational tools.

Contribution

It proves the polynomiality of the generalized Hilbert-Kunz function for the Rees algebra of face rings and offers explicit computations and a Macaulay2 implementation.

Findings

The Hilbert-Kunz function is polynomial for large s.

Explicit calculations of the Hilbert-Kunz function in various examples.

Provision of a Macaulay2 code for computing the Hilbert-Kunz function.

Abstract

Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and ${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function $HK(s)=\ell({\mathcal R}(\mathfrak n)/(\mathfrak n, \mathfrak n t)^{[s]})$ is given by a polynomial for all large $s.$ We calculate it in many examples and also provide a Macaulay2 code for computing $HK(s).$