Dao's question on the asymptotic behaviour of fullness

TL;DR

This paper investigates the asymptotic behavior of certain ideal properties in local rings with infinite residue fields, providing bounds and formulas using advanced algebraic tools.

Contribution

It offers new bounds and formulas for the asymptotic invariants of $ ext{full}$, $ ext{weakly } ext{full}$, and related properties of ideals in local rings.

Findings

Established upper bounds for asymptotic invariants.

Derived formulas for specific cases of these invariants.

Utilized reduction numbers, Ratliff-Rush closure, and Castelnuovo-Mumford regularity.

Abstract

For a local ring $(R, \M)$ of infinite residue field and positive depth, we address the question raised by H. Dao on how to control the asymptotic behaviour of the $\M$-full, full, and weakly $\M$-full properties of certain ideals (such notions were first investigated by D. Rees and J. Watanabe), by means of bounding appropriate numbers which express such behaviour. We establish upper bounds, and in certain cases even formulas for such invariants. The main tools used in our results are reduction numbers along with Ratliff-Rush closure of ideals, and also the Castelnuovo-Mumford regularity of the Rees algebra of $\M$.