The structure of Sally modules of rank one

TL;DR

This paper provides a comprehensive structure theorem for Sally modules of rank one associated with certain $km$-primary ideals in Cohen-Macaulay local rings, specifically when a particular Hilbert coefficient equality holds.

Contribution

It establishes a complete structure theorem for Sally modules of rank one under a specific Hilbert coefficient condition, advancing understanding of their algebraic properties.

Findings

Characterization of Sally modules when $ ext{e}_1(I)= ext{e}_0(I)- ext{length}_A(A/I)+1$

Explicit description of the module structure in this setting

Connections between Hilbert coefficients and Sally module structure

Abstract

A complete structure theorem of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is given, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert coefficients of $I$.