Bounds on Dao numbers and applications to regular local rings

TL;DR

This paper investigates bounds on Dao numbers in Noetherian local rings, providing new characterizations of regularity and improving previous bounds using tools like Castelnuovo-Mumford regularity and reduction numbers.

Contribution

It answers a question of H. Dao by establishing bounds on Dao numbers and offers new criteria for regularity of local rings based on Dao number properties.

Findings

Established bounds on Dao numbers in local rings.

Characterized regularity via Dao number conditions.

Improved previous results on Dao number bounds.

Abstract

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo-Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. Finally, as an application of the theory of Dao numbers, we provide new characterizations of when $R$ is regular; for instance, we show that this holds if and only if the maximal ideal of $R$ can be generated by a $d$-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.