UC Berkeley

The Castelnuovo–Mumford regularity of a sheaf on projective space is an integer that describes the vanishing of higher cohomology of twists of the sheaf. Regularity can be computed from the degrees of the syzygies of the corresponding graded module. Maclagan and Smith definedan analogous invariant for sheaves on smooth projective toric varieties, where the regularity is no longer directly bounded by Betti numbers. We investigate the relationship between regularity and Betti numbers in a number of situations that generalize classical results, such as the regularity of powers of ideals and the Betti numbers of truncations of modules.