Question

I sometimes see cohomologies of complexes of sheaves. What is the definition of these? Say if $\mathcal F^* $ is a complex of sheaves on $C$, what is $H^i(C,\mathcal F^*) $?

Answer 1

There are two concepts defined for complexes of sheaves, both called "cohomology", which are related but different. The more basic concept is the kind of cohomology that is defined for any complex $A^\bullet$ of objects in an abelian category: $$\DeclareMathOperator{\im}{im}H^n(A^\bullet) = \ker d^\bullet / \im d^{\bullet - 1},$$ where $d^\bullet$ are the differentials. This is not the kind you're asking about. What you are seeing is the derived functor kind of cohomology, in this case the derived functor of global sections of sheaves. If $X$ is a topological space (or site) and $\def\sh#1{\mathcal{#1}}\sh{F}^\bullet$ is a complex of sheaves on $X$, then: $$H^n(X, \sh{F}^\bullet) = H^n \bigl(R\Gamma(X, \sh{F}^\bullet)\bigr).$$ The $H^n$ on the right is the "complex cohomology" and the $H^n$ on the left is what we are defining to be the "sheaf cohomology". Here, $R\Gamma(X, -)$ is the functor obtained in the manner described by Sandor in his answer: find (any) quasi-isomorphism of $\sh{F}^\bullet$ with a complex of injective sheaves $\sh{I}^\bullet$, and define: $$R\Gamma(X,\sh{F}^\bullet) = \Gamma(X,\sh{I}^\bullet);$$ in other words, just apply global sections directly to the complex of injectives. For why this works, and what it means, you'll need to read a book; Weibel is a great reference, though I really like Gelfand–Manin for learning. YMMV.

Answer 2

If $\mathscr F$ is a sheaf, you define cohomology by first taking an injective resolution. But that's really just a complex of injective sheaves quasi-isomorphic to $\mathscr F$. That does not need $\mathscr F$ to be a sheaf. If $F$ is a complex, then take a complex of injective sheaves quasi-isomorphic to $F$. From that point do the same as you do in the case of sheaves.

Answer 3

I apologize for the shameless self promotion, but you can also try these notes.