Betti numbers of the Frobenius powers of the maximal ideal over a general hypersurface

TL;DR

This paper investigates the stability of Betti numbers in the minimal resolutions of Frobenius powers of the maximal ideal over general hypersurfaces in three variables in positive characteristic, revealing structural and numerical properties.

Contribution

It introduces a new method to determine generator degrees of defining ideals for certain Gorenstein algebras and analyzes the Betti number stability in Frobenius powers.

Findings

Proves stability of Betti numbers in the periodic tails of resolutions.

Provides the Hilbert-Kunz function for the hypersurface ring.

Determines Castelnuovo-Mumford regularity of Frobenius power quotients.

Abstract

The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from its socle degree, where $\mathfrak{c}$ is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert-Kunz function of the hypersurface ring, as well as the Castelnuovo-Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.