Complete intersection Approximation, Dual Filtrations and Applications

TL;DR

This paper introduces a two-step method involving complete intersection approximation and dual filtration analysis to study associated graded modules of Cohen-Macaulay modules, leading to new characterizations and proofs of conjectures.

Contribution

It develops a novel two-step approach to analyze associated graded modules, extending classical results and proving a conjecture related to normal ideals and Cohen-Macaulay properties.

Findings

Characterization of the $a$-invariant in terms of regular sequences.

Link between minimal multiplicity and the $a$-invariant for integrally closed ideals.

Proof that certain normal ideals with zero $e_3$-invariant have Cohen-Macaulay associated graded rings.

Abstract

We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ in a complete Noetherian local ring $(A,\mathfrak{m})$. The first step, we call it complete intersection approximation, enables us to reduce to the case when both $A$, $ G_\mathfrak{a}(A) = \bigoplus_{n \geq 0} \mathfrak{a}^n/\mathfrak{a}^{n+1} $ are complete intersections and $M$ is a maximal CM $A$-module. The second step consists of analyzing the classical filtration $\{Hom_A(M,\mathfrak{a}^n) \}_{\mathbb{Z}}$ of the dual $Hom_A(M,A)$. We give many applications of this point of view. For instance let $(A,\mathfrak{m})$ be equicharacteristic and CM. Let $a(G_\mathfrak{a}(A))$ be the $a$-invariant of $G_\mathfrak{a}(A)$. We prove: 1. $a(G_\mathfrak{a}(A)) = -\dim A$ iff $\mathfrak{a}$ is generated by a regular sequence. 2. If $\mathfrak{a}$ is integrally closed and $a(G_\mathfrak{a}(A)) = -\dim A + 1$ then $\mathfrak{a}$ has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of $h$-vectors. As another application we prove a conjecture of Itoh, if $A$ is a CM local ring and $\mathfrak{a}$ is a normal ideal with $e_3^\mathfrak{a}(A) = 0$ then $G_\mathfrak{a}(A)$ is CM.