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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.


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up vote 4 down vote accepted

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!

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In particular, to compute the hypercohomology groups of a complex like $\Omega^\bullet$ you need more than an acyclic resolution of each of the $\Omega^k$. Usually this is done using what's called Cartan-Eilenberg resolutions. – Mariano Suárez-Alvarez Jun 17 '11 at 16:10
@Akhil Mathew: Thank you very much, this is exactly what I was looking for! Thanks! – Benjamin Jun 17 '11 at 16:48
why do you restrict yourself to complexes which are bounded from one side? the current technology allows to derive a functor on the entire derived category. – the L Jun 17 '11 at 17:37
@anonymous. Are you thinking about Spaltenstein, N. (1988), "Resolutions of unbounded complexes", Compositio Mathematica 65 (2): 121–154 ? Yes, there is no bounded hypothesis in this paper, but, if I'm not wrong, the "derived" functor Spaltenstein constructs is just well-defined on the derived category. I mean, if I didn't miss the point when I read it, Spaltenstein doesn't prove that his derived functors have the universal property they should have (to be Kan extensions, which is true for bounded complexes). – a.r. Jun 17 '11 at 17:46
@Agustí: But this follows immediately from Spaltenstein's Proposition 1.4 (d) combined with the lemmas in Keller's section 14 in derived categories and their uses (link in my comment to your answer). – t.b. Jun 18 '11 at 6:23

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

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Thank you very much for all these references! – Benjamin Jun 17 '11 at 19:22
@Benjamin. You're welcome. I think you should vote Akhil's answer too, in addition to chose it as the best one. – a.r. Jun 18 '11 at 6:11
Other modern references: Kashiwara-Shapira (both: Sheaves on Manifolds and Categories and Sheaves). Weibel should also be mentioned and if you read French, I'd recommend Illusie's article Catégories dérivées et dualité, travaux de J.-L. Verdier. A very fine exposition is also contained in Keller's Derived categories and their uses – t.b. Jun 18 '11 at 6:11

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