On the derived DG functors

TL;DR

This paper proves that certain DG quasi-functors between derived categories are isomorphic to derived functors under specific conditions, and provides formulas for Hochschild cohomology in terms of Ext groups.

Contribution

It establishes conditions under which DG quasi-functors are equivalent to derived functors and offers new formulas for Hochschild cohomology of derived categories.

Findings

DG quasi-functors are isomorphic to derived functors under specific conditions

Provides formulas for Hochschild cohomology as Ext groups in functor categories

Shows that analogous statements for triangulated functors are false

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 $A \to Ind B$ .