A diagonal bound for cohomological postulation numbers of projective schemes

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