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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.

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  • $\begingroup$ $\phi$ is an endofunctor of the sheaf category? $\endgroup$ – Kevin Carlson Apr 14 at 7:28
  • $\begingroup$ @KevinCarlson I updated my post to precise this point $\endgroup$ – epsilones Apr 14 at 8:25
  • $\begingroup$ @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 ? $\endgroup$ – epsilones Apr 14 at 8:30
  • $\begingroup$ 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 $R\phi (F)$ $\endgroup$ – Max Apr 14 at 12:44

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