Itoh's conjecture for normal ideals

2022/05/21 by Tony J. Puthenpurakal, Puthenpurakal, Tony J.
Mathematics · Medicine · #13D45 #13H15 #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Intracranial Aneurysms: Treatment and Complications #Primary 13A30 #Secondary 13H10

paper · pdf · doi:10.48550/arxiv.2205.10615

openalex publication_date 2022/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (A,\mathfrakm) be an analytically unramified Cohen-Macaulay local ring and let \mathfraka be an \mathfrakm-primary ideal in A. If I is an ideal in A then let I^* be the integral closure of I in A. Let G_\mathfraka(A)^* = \bigoplus_{n≥ 0} (\mathfrakaⁿ)^*/(\mathfrakaⁿ⁺¹)^* be the associated graded ring of the integral closure filtration of \mathfraka. Itoh conjectured that if e₃^\mathfraka^*(A) = 0 and A is Gorenstein then G_\mathfraka(A)^* is Cohen-Macaulay. In this paper we prove an important case of Itoh's conjecture: we show that if A is Cohen-Macaulay and if \mathfraka is normal (i.e., \mathfrakaⁿ is integrally closed for all n ≥ 1) with e₃^\mathfraka(A) = 0 then G_\mathfraka(A) is Cohen-Macaulay.

Related

• A solution to Itoh's conjecture for integral closure filtration
• On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups
• Integrally closed ideals of reduction number three
• Depth of associated graded rings via Hilbert coefficients of ideals
• Complete intersection Approximation, Dual Filtrations and Applications