On the derived DG functors

Assume that abelian categories ${\cA}$, ${\cB}$ over a field admit countable direct limits and that these limits are exact. Let $\cF: D^+_{dg}({\cA}) \to D^+_{dg}( {\cB})$ be a DG quasi-functor such that the functor $Ho(\cF): D^+({\cA}) \to D^+({\cB})$ carries $D^{\geq 0}({\cA})$ to $D^{\geq 0}({\cB})$ and such that, for every $i >0$, the functor $H^i \cF: \cA \to \cB$ is effaceable. We prove that $\cF$ is canonically isomorphic to the right derived DG functor $RH^0(\cF)$. We also prove a similar result for bounded derived DG categories and a formula that expresses Hochschild cohomology of the categories $ D^b_{dg}({\cA})$, $ D^+_{dg}({\cA}) $ as the $Ext$ groups in the abelian category of left exact functors $\cA \to Ind \cA$ . The proofs are based on a description of Drinfeld’s category of quasi-functors as the derived category of a certain category of sheaves.

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Published 1 January 2010