The Hilbert-Kunz functions of two-dimensional rings of type ADE

TL;DR

This paper calculates the Hilbert-Kunz functions for two-dimensional ADE type rings using representation theory and matrix factorizations, providing explicit formulas for these algebraic invariants.

Contribution

It introduces a novel approach combining matrix factorizations and syzygy modules to compute Hilbert-Kunz functions for ADE rings.

Findings

Explicit formulas for Hilbert-Kunz functions of ADE rings

Connection between Cohen-Macaulay modules and Hilbert-Kunz invariants

Method applicable to other classes of rings

Abstract

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.