The structure of the Sally module of integrally closed ideals

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 ${\mathrm{e}}_1(I)={\mathrm{e}}_0(I)-{\mathrm{\ell }}_A(A\mathrm{/}I)+{\mathrm{\ell }}_A(I^2\mathrm{/}QI)+1,$ 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.