Cohomological patterns of coherent sheaves over projective schemes

We study the sets P(X, F) = {(i. n) is an element of N-0 x Z \ H-i (X, F(n)) not equal 0}, where X is a projective scheme over a noetherian ring R-0 and where F is a coherent sheaf of C-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 less than or equal to 1. If X = P-R0(d). 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 less than or equal to 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)). (C) 2001 Elsevier Science B.V. All rights reserved.