Cohomology of Coherent Sheaves and Series of Supernatural Bundles

We show that the cohomology table of any coherent sheaf on projective space is a convergent—but possibly infinite—sum of positive real multiples of the cohomology tables of what we call supernatural sheaves. Introduction Let K be a field, and let F be a coherent sheaf on P = P K . The cohomology table of F is the collection of numbers γ(F) = (γi,d) with γi,d = dimH (P,F(d)), which we think of as an element of the real vector space ∏∞ d=−∞ R . In Eisenbud-Schreyer [2009] we characterized the cohomology tables of vector bundles on P (up to a positive rational multiple) as the finite positive rational linear combinations of cohomology tables of supernatural bundles, which we described explicitly. In this paper we treat the cohomology tables of all coherent sheaves. These are given by infinite sums: Theorem 0.1. The cohomology table of any coherent sheaf on P can be written as a convergent series, with positive real coefficients, of cohomology tables of supernatural bundles supported on linear subspaces. We actually prove a more precise result, which includes a uniqueness statement. To state it we recall some ideas from Eisenbud-Schreyer [2009]. A sheaf F on P has supernatural cohomology if, for each integer d, the cohomology H (F(d)) is nonzero for at most one value of i and, in addition, the