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I was wondering whether there is a characterization of perfect DG modules over a DG algebra as there is one for modules over a ring. Namely, an object in $D(R)$, where $R$ is a ring, is perfect if and only if it is isomorphic to a bounded complex of finitely generated projective modules. Is there any similar characterization when we take $R$ a dga and $D(R)$ the derived category of DG modules over it?

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Well by definition a DG module $X$ is perfect iff it is a direct summand of a module which is a finite iterated extension of free modules (it's then a nontrivial result that this property is the same as being compact). If $Y$ is a dg module and $Y'$ is an extension of $Y$ by $R[n],$ this extension is classified by an element $\xi\in Ext^{1-n}(R, Y) = H^{1-n}(Y),$ so (choosing a cycle representing $\xi$) we see $Y'$ can be represented by a cone of the corresponding explicit arrow $R[n-1]\to Y.$ This lets you view any dg module which is a finite iterated extension of free modules as the total module of a bicomplex whose row complex is free and finite-dimensional. Now a perfect complex is a direct summand (equivalently, retract) of such a complex.

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  • $\begingroup$ Great, thanks! Have you read at what I wrote instead? Does that also work? $\endgroup$
    – Federico Barbacovi
    Commented Sep 14, 2019 at 7:23
  • $\begingroup$ What you wrote is usually formulated as the fact that a DG module over a smooth, proper DG algebra is perfect if and only if it is perfect over the ground field. That is indeed true, though not all interesting algebras are smooth and perfect. $\endgroup$
    – Dmitry Vaintrob
    Commented Sep 14, 2019 at 8:00
  • $\begingroup$ I understand, thank you very much! $\endgroup$
    – Federico Barbacovi
    Commented Sep 14, 2019 at 8:04
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I add it as an answer because I think this might be a partial answer. Assume that

  • the DGA $R$ is perfect and that the resolution by projective bimodules is made of modules whose underlying complex is perfect over the base field $k$,

  • the underlying complex of $R$ is perfect over the base field.

Then, every module is quasi isomorphic to the convolution of the complex obtained tensoring the projective resolution with the module itself. In particular, as this convolution will still be a projective DG module, we have a cofibrant DG module quasi isomorphic to the DG module we started with. A result of Toën and Vaquié tells us that under the above hypotheses, a cofibrant DG module is perfect over $R$ if and only if the underlying complex is perfect over $k$. Therefore, if we assume the underlying complex of the DG module we started with was perfect over the base field, we deduce that the DG module is perfect over $R$, as the quasi isomorphic cofibrant object is. It should all be correct, but if someone spots any errors I'm happy to hear any suggestions.

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