A problem on Čech cohomology

Question

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?

Answer 1

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.

Second, a more serious question is about the proposed proof method. The asserted exactness is certainly "standard", and provable in several ways, in various contexts and spirits. The genuinely "Cech" approach is possibly more down-to-earth, but has all the commensurate gritty issues. The "derived functor" approach is indirect, maybe "more expensive", but does succeed in showing that whatever work one does to prove such a thing will have proven many similar results at the same time. Still, perhaps the questioner is being asked to do a particular thing in a particular style... and should ascertain the constraints. (E.g., if there is no constraint to do a "Cech"-type computation, and depending on one's ulterior goals, one might look at derived functors rather than...)

And/but if "cocycles" really are unfamiliar to the questioner, probably it is worthwhile to experience first-hand that viewpoint. Georges Elencwajg's suggestion of Forster's "R.S." is possibly as good as any as an introduction. It is true, I think, that many of the specifics there are not essential to the question at hand, but they are equally interesting, and should be in the mind of any serious mathematician, so it'd be worthwhile.

I would advocate viewing quasi-classical "cocycle computations" as somewhat archaic, at least insofar as such things have been assimilated into very-smoothly functioning more-modern machines of various sorts. An interesting transitional stage, but not final.