Question

Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem: Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of $F$. Then $H^{q}(X,\mathcal{F}) \cong H^{q}(\mathcal{L}^{\bullet}({X}))$.

Now let's say I want to compute the cohomology of the complex $\mathcal{L}^{\bullet}({X})$ but my cochain complex

$0\rightarrow \mathcal{F} \rightarrow \mathcal{L}^{1} \rightarrow \mathcal{L}^{2} \rightarrow \mathcal{L}^{3} \cdots $

is not exact, i.e. $\mathcal{L}^{\bullet}$ is not a resolution of $\mathcal{F}$. In my problem (the problem I'm working on) the sheaf $\mathcal{F}$ and all the sheaves $\mathcal{L}^{q}$ are acyclic. How do I go about solving computing the cohomology of $\mathcal{L}^{\bullet}({X})$? I'm a little bit lost here, any tips, ideas or techniques as to how to proceed?

Answer 1

Given the assumed acyclicity, what you are computing is the hypercohomology of the complex ${\cal L}^*$.

Answer 2

To complement Steven Landsburg's answer, you have a spectral sequence $$E_2^{pq}= H^p(X,\mathcal{H}^q(\mathcal{L}^\bullet)) \Rightarrow \mathbb{H}^{p+q}(X,\mathcal{L}^\bullet)$$ where the thing on the right is hypercohomology, i.e. the group you're after, and $$\mathcal{H}^q(\mathcal{L}^\bullet) =\ker[\mathcal{L}^q\to \mathcal{L}^{q+1}]/image(\ldots)$$ are the cohomology sheaves. This should get you started. Also keep in mind that $\mathcal{H}^0(\mathcal{L}^\bullet)=\mathcal{F}$ assuming exactness at the first step.