# Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F \circ G)$ that happens under various circumstances.

Questions:

1.) Suppose I have an additive functor $F$ between abelian categories (say with enough injectives and projectives). If $F$ is left or right exact I can consider its right or left derived functors respectively. With the derived category formalism is there a useful notion of a derived functors if $F$ is neither left nor right exact? Perhaps such a notion could associate to a short exact sequence a long exact sequence (that continues in both directions).

2.) I've heard that with the formalism of derived categories one is able to compose left and right derived functors. I assume that you have to work in an unbounded derived category somehow, because right and left derived functors are defined on different derived categories. This seems to be entirely omitted from Hartshorne. Is it useful to consider such compositions (say in algebraic geometry, or etale cohomology)?

References will be greatly appreciated!

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You can consider the right and left derived functors of an arbitrary additive functor, not necessarily left or right exact; IIRC it will just in general fail to be the case that the zeroth derived functor coincides with the original functor. – Qiaochu Yuan Apr 9 '13 at 5:56
An example of a "useful" composition of left and right derived functors is the direct image of D-modules, which is a $R\pi_*$ applied to a derived tensor product. – Ketil Tveiten Apr 9 '13 at 8:30