A diagonal bound for cohomological postulation numbers of projective schemes

Abstract

Let X be a projective scheme over a field K and let F be a coherent sheaf of O X -modules. We show that the cohomological postulation numbers ν F i of F , e.g., the ultimate places at which the cohomological Hilbert functions n↦ dim K(H i(X, F(n)))=:h F i(n) start to be polynomial for n⪡0, are (polynomially) bounded in terms of the cohomology diagonal (h F i(−i)) i=0 dim( F) of F . As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions h F i if F runs through all coherent sheaves of O X -modules with fixed cohomology diagonal. In order to prove these results, we extend the regularity bound of Bayer and Mumford [Computational Algebraic Geometry and Commutative Algebra, Proc. Cortona, 1991, Cambridge Univ. Press, 1993, pp. 1–48] from graded ideals to graded modules. Moreover, we prove that the Castelnuovo–Mumford regularity of the dual F ∨ of a coherent sheaf of O P K r -modules F is (polynomially) bounded in terms of the cohomology diagonal of F .