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Let R=⊕n≥0Rn be a positively graded commutative Noetherian ring. A graded R-module is called *indecomposable (respectively *injective) if it is indecomposable (respectively injective) in the category of graded R-modules. A graded prime ideal of R is called irrelevant if it contains R+≔⊕n>0Rn.Let M=⊕n∈ZMn 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 p of R. This refinement is related to the siting (in terms of degrees) of *indecomposable *injective direct summands, having associated prime p, 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 c0 of R0, the sequence (gradeMnc0)n is ultimately constant; and (c) to describe the ultimate constant value of that sequence by means of an analogue of Faltings' annihilator theorem for local cohomology.

Let $ A$ itself, but with respect to an arbitrary ideal of $ A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true.