Cohomological patterns of coherent sheaves over projective schemes

Abstract

We study the sets P(X, F)={(i,n)∈ N 0× Z | H i(X, F(n))≠0} , where X is a projective scheme over a noetherian ring R 0 and where F is a coherent sheaf of O X -modules. In particular we show that P(X, F) is a so called tame combinatorial pattern if the base ring R 0 is semilocal and of dimension ⩽1. If X= P d R 0 is a projective space over such a base ring R 0, the possible sets P(X, F) are shown to be precisely all tame combinatorial patterns of width ⩽ d. We also discuss the “tameness problem” for arbitrary noetherian base rings R 0 and prove some stability results for the R 0-associated primes of the R 0-modules H i(X, F(n)) .