Cohomological patterns of coherent sheaves over projective schemes

We study the sets image, where X is a projective scheme over a noetherian ring R0 and where image is a coherent sheaf of image-modules. In particular we show that image is a so called tame combinatorial pattern if the base ring R0 is semilocal and of dimension less-than-or-equals, slant1. If image is a projective space over such a base ring R0, the possible sets image are shown to be precisely all tame combinatorial patterns of width less-than-or-equals, slantd. 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 image.