Cohomological patterns of coherent sheaves over projective schemes

Abstract

We study the sets P(X, ℱ) = (i,n) ∈ ℕ0 × ℤ Hi(X, ℱ(n)) ≠0}, where X is a projective scheme over a noetherian ring R0 and where ℱ is a coherent sheaf of OX-modules. In particular we show that P(X, ℱ) is a so called tame combinatorial pattern if the base ring R0 is semilocal and of dimension ≤ 1. If X = ℙR0d is a projective space over such a base ring R0, 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 R0 and prove some stability results for the R0-associated primes of the R0-modules Hi(X, ℱ (n)). © 2001 Elsevier Science B.V. All rights reserved.