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

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.