Upper semi-continuity of the Hilbert-Kunz multiplicity

TL;DR

This paper proves that the Hilbert-Kunz multiplicity exhibits upper semi-continuity in certain algebraic structures, enhancing understanding of its behavior in algebraic geometry and commutative algebra.

Contribution

It establishes the upper semi-continuity of the Hilbert-Kunz multiplicity in F-finite rings and related algebraic structures, a previously unproven property.

Findings

Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings.

The result applies to algebras of essentially finite type over excellent local rings.

Provides new insights into the behavior of Hilbert-Kunz multiplicity in algebraic geometry.

Abstract

We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.