Bass numbers in the graded case, a-invariant formulas, and an analogue of Faltings' annihilator theorem

Let R = + (n greater than or equal to 0) R-n be a positively graded commuta tive Noetherian ring. A graded R-module is called *indecomposable (respecti vely *injective) if it is indecomposable (respectively injective) in the ca tegory of graded R-modules. A graded prime ideal of R is called irrelevant if it contains R+ := + (n>0) R-n. Let M = +(n is an element of Z) M-n 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 o f R. This refinement is related to the siting (in terms of degrees) of *ind ecomposable *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 s ummand, having irrelevant associated prime, of any term in the minimal *inj ective 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 sect ions 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 c(0) of R-0, the sequence (grade(Mn) c(0))(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. (C) 1999 Ac ademic Press.