Multiplicity bounds in prime characteristic

TL;DR

This paper extends bounds on the multiplicity of certain local rings in prime characteristic, providing new upper bounds for $F$-injective and Cohen-Macaulay rings, including characteristic zero cases with dense $F$-injective type.

Contribution

It generalizes existing multiplicity bounds from $F$-pure to $F$-injective and Cohen-Macaulay rings, and extends these bounds to characteristic zero with dense $F$-injective type.

Findings

Bound on multiplicity for $F$-injective rings in prime characteristic.

Upper bounds for Cohen-Macaulay rings based on dimension, embedding dimension, and HSL numbers.

Extension of bounds to characteristic zero with dense $F$-injective type.

Abstract

We extend a result by Huneke and Watanabe bounding the multiplicity of $F$-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen-Macaulay ring of prime characteritic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen-Macaulay rings in characteristic zero which have dense $F$-injective type.