Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension

TL;DR

This paper establishes lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension, showing that non-regular rings must have multiplicities exceeding a certain dimension-dependent threshold.

Contribution

It provides the first dimension-dependent lower bounds for Hilbert-Kunz multiplicities in local rings, advancing understanding of their behavior in non-regular cases.

Findings

Non-regular rings have Hilbert-Kunz multiplicities strictly greater than 1.

Lower bounds depend only on the dimension of the ring.

Results apply to formally unmixed local rings in positive characteristic.

Abstract

Let $(R,\m)$ be a formally unmixed local ring of positive prime characteristic and dimension $d$. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if $R$ is not regular, there exists a lower bound, strictly greater than one, depending only on $d$, for its Hilbert-Kunz multiplicity.