Good evening,
I have two questions concerning Cech cohomology of presheaves.
(1) Let $X$ be a topological space and $0\to\mathcal{F}\to\mathcal{G} \to \mathcal{H}\to 0$ a short exact sequence of sheaves of abelian groups on $X.$ Does there exist a long exact sequence for the Cech cohomology of these sheaves as the case of sheaf cohomology?
(2) Let $\mathcal{F}$ be a presheaf of abelian groups on $X.$ Do we have the equality $\check{H}^p(X,\mathcal{F}) = H^p(X,\mathcal{F}^+)$ where the group on the left is Cech cohomology and the one on the right is the sheaf cohomology, and $\mathcal{F}^+$ is the associated sheaf of the presheaf $\mathcal{F}.$ For example, if $\mathcal{F}$ generates a zero sheaf, is $\check{H}^p(X,\mathcal{F}) = 0$ true for $p>0$ ?
Does anyone know something about these?
Thanks in advance.
EDIT : the same question as (2) but for the Cech cohomology : do we have the equality $\check{H}^p(X,\mathcal{F}) = \check{H}^p(X,\mathcal{F}^+)$?
Best Answer
In a paracompact space, the answer to (1) is yes, and is done in Godement's book Topologie algébrique et théorie de faisceaux. Your last question in (2) is Theorem 5.10.2 there, and (2) is Theorem 5.10.1, and the question in youe edit is the Corollary to theorem 5.10.2. (You can replace the hypothesis on the space by some technology—«cohomolgy with supports in a paracompatifying family»...)
In other words, it is probably a good idea to read chapter 5 in that book! :D