A solution to Itoh's conjecture for integral closure filtration

2024/04/11 by Tony J. Puthenpurakal, Puthenpurakal, Tony J.
Mathematics · #13D45 #13H15 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 13A30 #Secondary 13H10

paper · pdf · doi:10.48550/arxiv.2404.07638

openalex publication_date 2024/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (A,\mathfrakm) be an analytically unramified Cohen-Macaulay local ring of dimension d ≥ 3 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 in 1992 that if third Hilbert coefficient of G_\mathfraka(A)^* , i.e., e₃^\mathfraka^*(A) = 0 and A is Gorenstein then G_\mathfraka(A)^* is Cohen-Macaulay. In this paper we prove Itoh's conjecture (more generally for analytically unramified Cohen-Macaulay local rings).

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