Some properties of graded local cohomology modules

We consider a finitely generated graded module M over a standard graded commutative Noetherian ring R=⊕d⩾0Rd and we study the local cohomology modules HR+i(M) with respect to the irrelevant ideal R+ of R. We prove that the top nonvanishing local cohomology is tame, and the set of its minimal associated primes is finite. When M is Cohen-Macaulay and R0 is local, we establish new formulas for the index of the top, respectively bottom, nonvanishing local cohomology. As a consequence, we obtain that the (Sk)-loci of a Cohen-Macaulay R-module M, regarded as an R0-module, are open in Spec(R0). Also, when dim(R0)⩽2 and M is a Cohen-Macaulay R-module, we prove that HR+i(M) is tame, and its set of minimal associated primes is finite for all i.