Homologically smooth connected cochain DGAs

TL;DR

This paper investigates conditions under which a connected cochain DG algebra is homologically smooth, linking it to properties like the singularity category, cone length, and global dimension, and explores implications for modules over such algebras.

Contribution

It provides new criteria for homological smoothness of connected cochain DG algebras and examines properties of modules when the algebra is Gorenstein.

Findings

Homologically smooth DGAs are characterized via singularity category and cone length.

Cohomologically finite modules over smooth DGAs are shown to be compact.

In Gorenstein cases, Castelnuovo-Mumford regularity relates to depth and Ext-regularity.

Abstract

Let $\mathscr{A}$ be a connected cochain DG algebra such that $H(\mathscr{A})$ is a Noetherian graded algebra. We give some criteria for $\mathscr{A}$ to be homologically smooth in terms of the singularity category, the cone length of the canonical module $k$ and the global dimension of $\mathscr{A}$. For any cohomologically finite DG $\mathscr{A}$-module $M$, we show that it is compact when $\mathscr{A}$ is homologically smooth. If $\mathscr{A}$ is in addition Gorenstein, we get $$\mathrm{CMreg}M = \mathrm{depth}_{\mathscr{A}}\mathscr{A} + \mathrm{Ext.reg}\, M<\infty,$$ where $\mathrm{CMreg}M$ is the Castelnuovo-Mumford regularity of $M$, $\mathrm{depth}_{\mathscr{A}}\mathscr{A}$ is the depth of $\mathscr{A}$ and $ \mathrm{Ext.reg}\, M$ is the Ext-regularity of $M$.