The structure of the Sally module of integrally closed ideals

TL;DR

This paper provides a detailed algebraic description of the Sally module for integrally closed ideals in Cohen-Macaulay rings when the first two Hilbert coefficients satisfy a specific near-extremal equality, revealing their homological and numerical properties.

Contribution

It offers a complete algebraic structure of the Sally module under a specific equality involving Hilbert coefficients, advancing understanding of integrally closed ideals.

Findings

Complete description of Sally module structure under the given equality.

Characterization of homological and numerical invariants of the associated graded ring.

Examples illustrating the theoretical results.

Abstract

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen-Macaulay local ring $A$ satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1, $ where $Q$ is a minimal reduction of $I$, and $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the first two Hilbert coefficients of $I, $ respectively the multiplicity and the Chern number of $I$. This almost extremal value of $\mathrm{e}_1(I) $ with respect classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.