Regularity bounds for complexes and their homology

Abstract

Let R be a standard graded algebra over a field k. We prove an Auslander-Buchsbaum formula for the absolute Castelnuovo-Mumford regularity, extending important cases of previous works of Chardin and Romer. For a bounded complex of finitely generated graded R-modules L, we prove the equality reg L = max(i is an element of Z){reg H-i(L) - i} given the condition depth Hi(L) >= dim Hi+1(L) - 1 for all i < sup L. As applications, we recover previous bounds on regularity of Tor due to Caviglia, Eisenbud-Huneke-Ulrich, among others. We also obtain strengthened results on regularity bounds for Ext and for the quotient by a linear form of a module.