Some Properties of Top Graded Local Cohomology Modules
نویسنده
چکیده
Let R = ⊕ d∈N0 Rd be a positively graded commutative Noetherian ring which is standard in the sense that R = R0[R1], and set R+ := ⊕ d∈N Rd, the irrelevant ideal of R. (Here, N0 and N denote the set of non-negative and positive integers respectively; Z will denote the set of all integers.) Let M = ⊕ d∈Z Md be a non-zero finitely generated graded R-module. This paper is concerned with the behaviour of the graded components of the graded local cohomology modules H R+(M) (i ∈ N0) of M with respect to R+. It is known (see [BS, 15.1.5]) that there exists r ∈ Z such that H R+(M)d = 0 for all i ∈ N0 and all d ≥ r, and that H R+(M)d is a finitely generated R0-module for all i ∈ N0 and all d ∈ Z. The first part (§1) of this paper deals with the case in which R = R0[U1, . . . , Us]/I, where U1, . . . , Us are indeterminates of degree one, and I ⊂ R0[U1, . . . , Us] is a homogeneous ideal. The main theorem of that section is that for d ≥ s, all the associated primes of H R+(R)−d contain a certain ideal of R0 called the “content” of I (see Definition 1.3.) This result provides an affirmative answer, in a special
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