Upper bound of multiplicity in Cohen-Macaulay rings of prime characteristic

TL;DR

This paper establishes an upper bound on the multiplicity of Cohen-Macaulay rings in prime characteristic, relating it to Frobenius test exponent, dimension, embedding dimension, and type, extending previous Gorenstein ring results.

Contribution

It provides a new upper bound for multiplicity in Cohen-Macaulay rings in prime characteristic, generalizing earlier results for Gorenstein rings.

Findings

Upper bound expressed in terms of Frobenius test exponent, dimension, embedding dimension, and type.

Extension of known bounds from Gorenstein to Cohen-Macaulay rings.

Applicable to local rings of prime characteristic with Cohen-Macaulay property.

Abstract

Let $(R, \frak m)$ be a local ring of prime characteristic $p$ and of dimension $d$ with the embedding dimension $v$, type $s$ and the Frobenius test exponent for parameter ideals $\mathrm{Fte}(R)$. We will give an upper bound for the multiplicity of Cohen-Macaulay rings in prime characteristic in terms of $\mathrm{Fte}(R),d,v$ and $s$. Our result extends the main results for Gorenstein rings due to Huneke and Watanabe.