Homologically Smooth Connected Cochain DGAs

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