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 Römer. 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.

Final version

References in corpus (1)

• On the regularity over positively graded algebras