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 $\check{H}^p(X,\mathcal{F}) = \check{H}^p(X,\mathcal{F}^+)$?