On unmixed and equi-dimensional associated graded rings

2024/05/31 by Tony J. Puthenpurakal, Puthenpurakal, Tony J., Samarendra Sahoo +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2405.20647

Abstract

Let (A,\mathfrakm) be an analytically un-ramified Noetherian local ring of dimension d ≥ 1, I a regular \mathfrakm-primary ideal of A and let I be integral closure ideal of I. If A is of characteristic p > 0 then let I^* denote the tight closure of I. Let G_I(A)=\bigoplus_{n≥ 0}Iⁿ/Iⁿ⁺¹ be the associated graded ring of A with respect to I. Assume G_I(A) is unmixed and equi-dimensional. We show that either the function P_I : n↦ λ(Iⁿ/Iⁿ) is a polynomial type of degree d-1 or Iⁿ=Iⁿ for all n≥ 1. We prove an analogus result for the tight closure filtration if A is of characteristic p > 0. When A is generalized Cohen-Macaulay and I is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of I and the tight closure filtration of I.

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