Improved Bounds for the s-multiplicity

TL;DR

This paper develops new formulas relating s-multiplicity and h-functions for product and idealization rings in prime characteristic, providing improved bounds and insights into longstanding conjectures.

Contribution

It introduces formulas connecting s-multiplicity and h-functions for product and idealization rings, advancing understanding of their invariants in prime characteristic.

Findings

Derived formulas for s-multiplicity of product rings

Connected s-multiplicity of idealization rings to original modules

Provided new estimates for key conjectures in the field

Abstract

Let (RmR), (SmS) and (TmT) be Noetherian local rings sharing the same residue eld k and prime characteristic p > 0. We establish some formulas relating the h-function and s-multiplicity of the ber product R T S in terms of the h-functions and s-multiplicities of R, T and S. Furthermore, we derive formulas that connect the h-function and s-multiplicity of the idealization ring R M to the corresponding invariants of R and M, where M is a nitely generated R-module. As applications of these results, we derive new estimates for the Taylor-Miller question and the Watanabe-Yoshida conjecture concerning s-multiplicity.