Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and close matters (like spelling out the compact objects, or perhaps discussing pairing with derived category of $A^{op}$-modules into $Vect$, etc.)?

Thank you, Sasha

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  • $\begingroup$ What's the connection between the title and the questions? $\endgroup$ – Fernando Muro Jan 11 '16 at 17:33
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    $\begingroup$ @FernandoMuro: I edited the title; Is it better now? $\endgroup$ – Sasha Jan 11 '16 at 17:42

The fact that $\mathbf{D}(A)$ is compactly generated follows from the fact that $A$ (and its suspensions) generate $\mathbf{D}(A)$ and that the smallest thick subcategory containing $A$ is $\mathrm{Perf}(A)$, in other words, compact objects are prefect complexes, i.e. bounded complex of projective $A$-modules. This goes back to Rickard and Thomason-Trobaugh.

The fact that $\mathbf{D}(A)$ is cocomplete follows from existence and exactness of coproducts in the category of $A$-modules.

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