Sheaf cohomology

Question

Let $\mathcal{F}$ be a sheaf over, say, a paracompact differentiable manifold $M$. Then to compute the cohomology $H(M,\mathcal{F})$ of $\mathcal{F}$, we can use any acyclic resolution or use Cech cohomology.

Now, in an article I am reading, some cohomology is defined as the cohomology of a certain complex of sheaves $\Omega^\bullet$, not just one sheaf. Does this mean that in order to compute such a cohomology, I need for instance an acyclic resolution for each $\Omega^k$?

Please tell me if I am saying something wrong.

Thanks!

Answer 1

I suspect (but am not sure) that your article may be referring to hypercohomology. The point of hypercohomology is that, given a functor $F: \mathcal{A} \to \mathcal{B}$ (say, left-exact, like the global section functor; let's also assume $\mathcal{A}$ has enough injectives), one can define the so-called "hyper-derived functors" $\mathbf{R}^i F$, each of which is a functor from complexes on $\mathcal{A}$ to $\mathcal{B}$. A short exact sequence of complexes leads to a long exact sequence of hypercohomology, just as with the ordinary derived functors.

The more modern way to think of hypercohomology is to use the derived category. The point is then that a functor $F: \mathcal{A} \to \mathcal{B}$ induces a total derived functor functor on the bounded-below derived categories $\mathbf{D}^+(\mathcal{A}) \to \mathbf{D}^+(\mathcal{B})$ (you can think of the derived category as localizing the category of chain complexes with respect to quasi-isomorphisms, though it's better to go first through the homotopy category). Then the hypercohomology functors are just defined by taking the $i$th cohomology of the total derived functor.

To compute this, you start with a bounded-below complex $K^\bullet$, find a quasi-isomorphism $K^\bullet \to I^\bullet$ where $I^\bullet $ consists of $F$-acyclic (say, injective) objects, and take $F(I^\bullet)$ as the output of the derived functor.

I could say more if you clarify that this is in fact what you are looking for!

Answer 2

You can do as Akhil says, and also as you suggested, if you take into account that the acyclic resolutions of each $\Omega^k$ cannot be taken independently. The right notion of resolution for a complex (one possibility at least) is that of Cartan-Eilenberg resolution, which you can find in the classical book "Homological Algebra" of Cartan and Eilenberg (in the chapter discussing hyperhomology), or also in Grothendieck's EGA III, "Étude cohomologique des faisceaux coherents" (in French). Or you can also look in Hartshorne's "Residues and duality". Another possibility: apply Godement's cosimplicial resolution directly to your complex $\Omega$ degree-wise.

EDIT. I forgot. A more modern reference: Gelfand-Manin, Methods of Homological Algebra