# A problem on Čech cohomology

Let $X$ be a topological space and $$0\to\mathcal{F}^{\prime\prime}\to\mathcal{F}\to\mathcal{F}^\prime\to 0$$ be an exact sequence of sheaves on $X$. How can I show that $$H^1(X,\mathcal{F}^{\prime\prime})\to H^1(X,\mathcal{F})\to H^1(X,\mathcal{F}^{\prime})$$ is exact?

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Are your maps going the wrong way? –  Eric O. Korman Mar 26 '12 at 19:35
No. I copied the text from Liu's Arithmetical geometry book and the maps goes to the same direction as in the book. –  Amateur Mar 26 '12 at 19:40
Have you tried explicit computations with cocycles? –  Blah Mar 26 '12 at 20:24
No. Cohomology is relative new topic for me so I haven't found suitable tools I can use. I couldn't find cocycles in the table of contents so I was wondering are those required as prerequisites before reading the book. –  Amateur Mar 26 '12 at 20:53
O. Forster: Lectures on Riemann Surfaces , Theorem 15.12 –  Georges Elencwajg Mar 26 '12 at 21:10

First, to appraise the direction of the maps, whether as mnemonic or proof: the alleged exact fragment should be fitting into a long exact sequence involving (we know thanks to Grothendieck's Tohoku paper) the global sections function $\Gamma$, which would be the $H^o$ of which the $H^1$'s are (right) derived functors. The direction of the arrows can be remembered/determined by thinking about the $H^o$'s as "global functions" while the sheaves are "germs" (local fragments). Indeed, inclusions of a local "type" of "function" lead to same-direction maps on global sections ($H^o$'s), and, necessarily, on all $H^i$'s.