Dao's question on the asymptotic behaviour of fullness

2023/08/08 by Cleto B. Miranda-Neto, Miranda-Neto, Cleto B., Douglas S. Queiroz +1
Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2308.03997

openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Bounds on Dao numbers and applications to regular local rings
• Dao numbers and the asymptotic behaviour of fullness
• The Rees Valuations of Complete Ideals in a Regular Local Ring
• Ratliff-Rush ideal and reduction numbers
• Asymptotic vanishing conditions which force regularity in local rings of prime characteristic