# Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the sheaves $K^{\bullet}$ are acyclic, so I'm trying use spectral sequences. I have the term

$''E^{p,q}_2 =$ The p-th cohomology group of the complex

$H^q_d(C^\bullet(K^0)(X)) \xrightarrow{\delta} H^q_d(C^\bullet(K^1)(X)) \xrightarrow{\delta} H^q_d(C^\bullet(K^2)(X)) \xrightarrow{\delta} \cdots$,

now since the sheaves $K^{\bullet}$ are acyclic it follows that $''E^{p,q}_2$ converges to $0$ for $q \geq 1$ and $q < 0$ and all $p$, so $''E^{p,q}_\infty = Gr^p \mathbb{H}^{p + q} = F^p \mathbb{H}^{p+q}/F^{p + 1} \mathbb{H}^{p+q} = 0$ for $q \geq 1$ and $q < 0$ and all $p$ (Here the $F^q$'s are the members of the finite filtration $0 = F^{N} \mathbb{H}^{\bullet} \subset \cdots \subset F^{n+1} \mathbb{H}^{\bullet} \subset F^{n} \mathbb{H}^{\bullet} \subset F^{n-1} \mathbb{H}^{\bullet} \subset \cdots \subset F^0 \mathbb{H}^{\bullet} = \mathbb{H}^{\bullet}$), since I'm working with finite-dimensional vector spaces all this leads to

$\mathbb{H}^{n}(X,K)^{\bullet} = \bigoplus_{p+q=n,p+q\leq N} ''E^{p,q}_\infty$.

But to compute this I need to know the terms $''E^{p,0}_\infty$ for all $p$, does anyone know how to go about computing these terms or any other method to obtain the Hypercohomology of $K^{\bullet}$, or really, what do I do?

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Because your sheaves are acyclic, the spectral sequence degenerates on page 2. (As you say, $E^{p,q}_2 = 0$ for all $q > 0$.) It follows that $E^{p,q}_{\infty} = E^{p,q}_2$ for all $p, q$ – but that just means the hypercohomology of the complex of sheaves is isomorphic to the cohomology of the complex of global sections! –  Zhen Lin Jul 28 '12 at 2:40
Thanks, yes I know, and that's actually what I need the hypercohomology for, that's my end goal: to compute the cohomology of the complex of global sections by computing the hypercohomology. I do it because I asked what I should do to obtain the cohomology of the complex of global sections in case the only piece (or one of the few) information I had is that the sheaves are acyclic and they told me I should turn to hypercohomology, so kind of a circular reference, I don't know if there are other techniques to compute hypercohomology maybe? Thanks –  Richard Jennings Jul 29 '12 at 19:22
Perhaps that's all they mean: hypercohomology coincides with ordinary cohomology in this case. This spectral sequence is derived from the spectral sequence of the double complex $C^\bullet(K^\bullet)(X)$, so you could in principle compute the other spectral sequence of the same double complex, which has $E^{p,q}_2 = H^p(X, H^q(K^\bullet))$. But somewhere along the line you have to know something about your complex $K^\bullet$... –  Zhen Lin Jul 30 '12 at 1:29
Thanks @ZhenLin it seems that I may have found a way to compute it, thanks –  Richard Jennings Aug 9 '12 at 21:25