Homologically smooth connected cochain DGAs

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 $M$.