Dao numbers and the asymptotic behaviour of fullness

Abstract

In the present paper, we study the Dao numbers ${\mathfrak{d}}_1(I),{\mathfrak{d}}_2(I)$ and ${\mathfrak{d}}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\ell \ge 0$ such that $I{\mathfrak{m}}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full, respectively, for all $k\ge \ell$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to ${\text{reg}}_{{\text{gr}}_{\mathfrak{m}}(R)}{\text{gr}}_{\mathfrak{m}}(I)$, where ${\text{gr}}_{\mathfrak{m}}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.

arXiv commentThis is the final version of the paper, accepted for publication in Journal of Algebra and its Applications, https://doi.org/10.1142/S021949882650132X