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Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\mathcal{A}$. What is the motivation of this definition (I'm assuming this is from scheme theory), and how does this capture smoothness for $D^b(X)$ where $X$ is a smooth projective variety over $\mathbb{F}$? Also, what happens if $X$ is singular?

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    $\begingroup$ Well, if $A$ is an good ol' algebra, this means that $A$ has finite projective dimension as an $A$-bimodule and, additionally, that it has a projective resolution by finitely generated projectives. The first condition is one of regularity and the second one one of finiteness. The corresponding notion of smoothness is a standard one that shows up all over the place. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 30, 2016 at 5:45
  • $\begingroup$ Thank you. Looking around mathoverflow leads to this post mathoverflow.net/questions/20675/… $\endgroup$
    – 54321user
    Commented Oct 1, 2016 at 22:53

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