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 R\"omer. For a bounded complex of finitely generated graded $R$-modules $L$, we prove the equality $\text{reg}~ L=\max_{i\in \mathbb Z} {\text{reg}~ H_i(L)-i}$ given the condition $\text{depth}~ H_i(L)\ge \dim H_{i+1}(L)-1$ 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.