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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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    Thanks @ZhenLin it seems that I may have found a way to compute it, thanks2012-08-09

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