2017/03/28 by Hailong Dao, Dao, Hailong, Ilya Smirnov +1
Mathematics · #13D40 #13H15 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1703.09427
openalex publication_date 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (R, \mathfrak m) be a Noetherian local ring and I a \mathfrak m-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if R has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. In addition, we can classify when the inequality always hold for a Cohen-Macaulay R of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on pg ideals by Okuma-Watanabe-Yoshida, multiplicity of the fiber cone, and the h-vector of the associated graded ring.