On the derived DG functors

Abstract

Assume that abelian categories $A, B$ over a field admit countable direct limits and that these limits are exact. Let $F: D^+_{dg}(A) --> D^+_{dg}(B)$ be a DG quasi-functor such that the functor $Ho(F): D^+(A) \to D^+(B)$ carries $D^{\geq 0}(A)$ to $D^{\geq 0}(B)$ and such that, for every $i>0$, the functor $H^i F: A \to B$ is effaceable. We prove that $F$ is canonically isomorphic to the right derived DG functor $RH^0(F)$. We also prove a similar result for bounded derived DG categories in a more general setting. We give an example showing that the corresponding statements for triangulated functors are false. We prove a formula that expresses Hochschild cohomology of the categories $ D^b_{dg}(A)$, $ D^+_{dg}(A) $ as the $Ext$ groups in the abelian category of left exact functors $A \to Ind B$ .