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arXiv · 1311.2572

Regularity bounds for complexes and their homology

Hop D. Nguyen
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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.

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Hop D. Nguyen. 2015-09-22. Regularity bounds for complexes and their homology. https://doi.org/10.1017/s0305004115000390

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