Asymptotic growth of saturated powers and epsilon multiplicity

TL;DR

This paper investigates the asymptotic behavior of saturated powers of modules over local domains, establishing the existence of certain limits and the epsilon multiplicity under mild conditions.

Contribution

It proves the existence of limits related to saturated powers and epsilon multiplicity for modules over local domains, extending previous limsup results.

Findings

Limit of saturation quotient exists as k approaches infinity.

Epsilon multiplicity of E exists as a limit under mild conditions.

Asymptotic growth rate characterized by k^{d+e-1}.

Abstract

Asymptotic properties of saturated powers of modules over a local domain R are studied. Under mild conditions, it is shown that the limit as k goes to infinity of the quotient of the saturation of the k-th power of a module E by the k-th power of E, when divided by k^{d+e-1}, exists. Here d is the dimension of R and e is the rank of E. We deduce that under these assumptions, the epsilon multiplicity of E, defined by Ulrich and Validashti as a limsup, actually exists as a limit.