A diagonal bound for cohomological postulation numbers of projective schemes

Let X be a projective scheme over a field K and let be a coherent sheaf of -modules. We show that the cohomological postulation numbers of , e.g., the ultimate places at which the cohomological Hilbert functions start to be polynomial for n 0, are (polynomially) bounded in terms of the cohomology diagonal of . As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions if runs through all coherent sheaves of -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 of a coherent sheaf of -modules is (polynomially) bounded in terms of the cohomology diagonal of .