2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127665Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \oplus_{n \geq 0}I^n/\m I^n$ be the fiber-cone of $I$. We prove certain relations among the Hilbert coefficients of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.To appear in Canadian Journal of Math. Many examples added in section 5 on analytic spread three. Introduction is longer and more describtive. An appendix on filter regular sequences and minimal reductions addedCommutative Algebra13H10, 13H15, 13A30 (Primary) 13C15, 13A02 (Secondary)text