# On the structure of the Sally module and the second normal Hilbert coefficient

## TL;DR

This paper characterizes the structure of the Sally module and the Hilbert function of the associated graded ring in Cohen-Macaulay local rings when the second normal Hilbert coefficient is nearly minimal, advancing understanding of Hilbert coefficients and related conjectures.

## Contribution

It provides a complete description of the Sally module and Hilbert function in cases of almost minimal second normal Hilbert coefficient, linking to Itoh's conjecture.

## Key findings

- Explicit structure of Sally module in the specified case
- Complete description of the Hilbert function of the associated graded ring
- Insights into the vanishing of the third Hilbert coefficient

## Abstract

The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the Hilbert function of the associated graded ring of the normal filtration. A deep analysis of the vanishing of the third Hilbert coefficient has been necessary. This study is related to a long-standing conjecture stated by S. Itoh.