Bounds on Dao numbers and applications to regular local rings
Antonino Ficarra, Cleto B. Miranda-Neto, Douglas S. Queiroz
TL;DR
The paper bound ${\mathfrak d}_3(I)$ and related Dao numbers for ideals in Noetherian local or graded rings with infinite residue field and positive depth, by linking fullness properties to Castelnuovo–Mumford regularity and reduction numbers. It proves a sharp upper bound ${\mathfrak d}_3(I) \le \mathrm{reg}_{\mathcal{R}(\mathfrak m)}\mathcal{R}(\mathfrak m, I)$, extending prior results and clarifying when stronger bounds hold. The work yields new characterizations of regular local rings, showing equivalences among being regular, having the maximal ideal generated by a $d$-sequence, and vanishing Dao numbers for reductions, with further implications for the Zariski–Lipman conjecture via a proposed derivations-based criterion. It also discusses partial results and conjectures in higher dimension, including the conjectured relation ${\mathfrak d}_3(I)=\mathrm{r}_I(\mathfrak m)$ for minimal reductions and the role of $s(\mathfrak m)$ and $\widetilde{\mathfrak m^k}$ in these 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.