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 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 is tame, and its set of minimal associated primes is finite for all i.