A filtration of the Sally module and the first normal Hilbert coefficient

The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results to characterize the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen–Macaulay local ring.