2024/05/16 by Antonino Ficarra, Ficarra, Antonino, Cleto B. Miranda-Neto +3
Mathematics · #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2405.10192
openalex publication_date 2024/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.