Some asymptotic properties of the Rees powers of a module

TL;DR

This paper investigates the asymptotic behavior of Rees powers of modules over commutative rings, extending known results from ideal theory to modules, including inequalities related to analytic spread.

Contribution

It introduces new asymptotic properties of Rees powers of modules, generalizing classical ideal results to module settings.

Findings

Proves asymptotic properties of Rees powers of modules

Extends Burch's inequality for analytic spread to modules

Provides new insights into the structure of Rees modules

Abstract

Let R be a commutative ring and let G be a free R-module with positive rank e. For any R-submodule E of G we may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n of the image, that we call the n-th Rees powers of E. In this work we prove some asymptotic properties of the modules E_n, which extend well known similar ones for the case of ideals, among them Burch's inequality for the analytic spread.