# methods for computing cohomology from data of an exact sequence

Let X be a topological space and $$A$$ be a ring. Suppose we have an exact sequence of sheaves of $$A$$-modules on X, $$0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$$, suppose we are given a left-exact functor $$\phi$$ in the category of sheaves of $$A$$-modules, and $$R\phi$$ its derived functor. Suppose now $$R\phi(G),R\phi(H)$$ are computable. Is there a general method to compute cohomology $$R\phi(F)$$ from that of $$R\phi(G),R\phi(H)$$ ?

In the derived category $$F$$ is qis to the complex $$G\longrightarrow H$$, then I tried to compute by hand the resolution of this complex and try to get some information applying the $$R\phi$$ functor. In the case I am working on, it was sufficient, but I am wondering if there is a general method.

• $\phi$ is an endofunctor of the sheaf category? – Kevin Carlson Apr 14 at 7:28
• @KevinCarlson I updated my post to precise this point – epsilones Apr 14 at 8:25
• @QiaochuYuan - I think I didn't understand your answer since I am not looking at computing the derived functor of the composition of two functors, each if which we would know the derived functor. Could you be more precise ? – epsilones Apr 14 at 8:30
• I don't know how much that helps, but if the cohomolgies of $R\phi (G), R\phi (H)$ are nice (lots of zeroes or whatever) then the cohomology long exact sequence can provide information on the cohomology of $R\phi (F)$ – Max Apr 14 at 12:44