Relationship between sheaf cohomology and $H^n(F)$

Question

If $F$ is a complex of sheaves in the derived category of sheaves, what is the relation (if there is any) between $H^n(X,F)=H^n(R\Gamma(X,F))$ and $H^n(F)$?

Answer 1

Give a look at "hyperhomology": it is a special case of derived functors in the context of chain complexes. There is a spectral sequence in hyperhomology that can be stated also without explicitly using this language.

In particular, you could derive yourself the result by using a Cartan-Eilenberg (injective) resolution for the complex.

Here it is: there is a spectral sequence with $E_1$ page equal to

$$ E_1^{p, q} := H^p(X, H^q(F)) $$

That converges to $H^{p+q}(X, F) $. Note that here there are two notions of (co) homology at work: one that it is about $F$ being a chain complex, the other one $F$ being "coefficients for $X$ cohomology". As you can see, this spectral sequence tells us that if you know only the homology sheaves $H^n(F) $, you can still get much of the derived functor $R\Gamma(X, F) $: its cohomology will be filtered by the objects you will find at the infinity page of the spectral sequence.

If you are for example over a field and every homology at play has finite dimension, in principle you can determine exactly the homology of $R\Gamma(X, F) $ since you will know the dimension over the field.