Bass Numbers in the Graded Case, a-Invariant Formulas, and an Analogue of Faltingsʹ Annihilator Theorem

Let M = n Mn be a finitely generated graded R-module. The first main result of the paper presents a refinement of the theory of the Bass numbers of M with respect to an irrelevant graded prime ideal of R. This refinement is related to the siting (in terms of degrees) of *indecomposable *injective direct summands, having associated prime , of terms in the minimal *injective resolution of M. The second main result has a consequence which can be described in terms of the Castelnuovo regularity r of M: any *indecomposable *injective direct summand, having irrelevant associated prime, of any term in the minimal *injective resolution of M, must vanish in all degrees greater than r. Results of this type provide “uniform upper bounds” for the “degrees of influence” of such irrelevant primes, and these bounds are exploited in the later sections of the paper to (a) generalize known “a-invariant formulas” due to E. Hyry and N. V. Trung; (b) to throw new light on the fact that, for an ideal 0 of R0, the sequence (gradeMn 0)n is ultimately constant; and (c) to describe the ultimate constant value of that sequence by means of an analogue of Faltingsʹ annihilator theorem for local cohomology.