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Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, respectively below) derived categories. What can one say about the existence of an adjunction $LF\dashv RG$?

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A hypothesis I've found useful before with this question is that if the categories are Grothendieck abelian categories and $F$ is exact, then $LF$ and $RG$ are adjoint on bounded below derived categories. This follows immediately because $G$ preserves injective objects. – Moosbrugger Nov 29 '11 at 20:28
Is the category of finite dimensional modules over an algebra over a field a Grothendieck Abelian category? Does the same statement hold for the bounded above derived category, i.e. with projective objects? – Eitan Chatav Nov 29 '11 at 20:42
The adjunction does always exist. It is a general fact, see MR2323740 (2008f:18011) Maltsiniotis, Georges Le théorème de Quillen, d'adjonction des foncteurs dérivés, revisité. (French) [Quillen's adjunction theorem for derived functors revisited] C. R. Math. Acad. Sci. Paris 344 (2007), no. 9, 549–552. – Fernando Muro Nov 29 '11 at 21:39
@Fernando: I don't see how to deduce that the adjunction always exists from the theorem you cited. The hypotheses are not dissimilar from what I wrote in my post above. (Note the author's use of the adjective "absolute"). – Moosbrugger Nov 30 '11 at 2:01
@Eitan: No -- a Grothendieck abelian category is like the category of all modules over a ring. It has infinite direct sums, filtered colimits preserve monomorphisms, and it satisfies a (very mild) set-theoretic condition. But it doesn't really matter -- all that's important is the existence of enough injectives (which the Grothendieck condition implies). So the dual thing is satisfied if there are enough projectives. – Moosbrugger Nov 30 '11 at 2:01

It seems this question was considered in:

William W Adams, Marc A Rieffel. Adjoint functors and derived functors with an application to the cohomology of semigroups Journal of Algebra, V. 7, N 1, 1967, 25-34

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I can't find it online. – Eitan Chatav Nov 29 '11 at 20:36
I believe this paper shows something slightly difference, but related, since in the end it uses preservation of injective objects. There are a couple of versions in the paper depending on which categories have enough injectives or projectives. Roughly it is concerned with a pair of adjoint exact functors $F,G$ and a third half exact functor $H$ such that say $H\circ G$ makes sense and it gives conditions so that $R(H\circ G)=H\circ RG$. A typical special case is Shapiro's lemma: if $A$ is a subgroup of $B$, then $H^n(A,M)=H^n(B,Coind(M))$. – Benjamin Steinberg Nov 29 '11 at 23:09

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