Dao numbers and the asymptotic behaviour of fullness

2024/02/08 by Antonino Ficarra, Ficarra, Antonino
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2402.05555

openalex publication_date 2024/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the present paper, we study the Dao numbers \mathfrakd₁(I),\mathfrakd₂(I) and \mathfrakd₃(I) of an ideal I of a Noetherian local ring (R,\mathfrakm,K) or a standard graded Noetherian K-algebra. They are defined as the smallest ℓ≥0 such that I\mathfrakm^k is \mathfrakm-full, full, weakly \mathfrakm-full, respectively, for all k≥ℓ. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra R(\mathfrakm). If R is a Koszul algebra, we prove that the Dao numbers are less or equal to reg_gr_\mathfrakm(R)gr_\mathfrakm(I), where gr_\mathfrakm(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.

Related

• Bounds on Dao numbers and applications to regular local rings
• Dao's question on the asymptotic behaviour of fullness
• Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules
• The Proj of the Rees algebra of a graded family of ideals
• Castelnuovo-Mumford regularity for complexes and weakly Koszul modules