Sheaf cohomology on non paracompact topological spaces

Question

I have some confusion on the subject of sheaf cohomology on non-paracompact topological spaces, i hope you can help me.

My reference is Godement's book "Topologie algebrique et theorie dex faisceaux".

I know that, given a sheaf $\mathcal{F}$ on $X$ and $\phi$ a family of supports on $X$, I can define the cohomology $H^n_{\phi}(X,\mathcal{F})$ as $H^n(\Gamma_\phi(\mathcal{C}^*(X,\mathcal{F}))$ where $\mathcal{C}^*(X,\mathcal{F})$ is the canonical sequence and $\Gamma_\phi(\mathcal{F})=\{s\in\mathcal{F}(X)|supp(s)\in \phi\}$.

When Godement writes $H^n(X,\mathcal{F})$, without the $\phi$ being indicated, i think he means $H^n(\Gamma(\mathcal{C}^*(X,\mathcal{F}))$.

My questions are the following:

1) What is the difference between $H^n(\Gamma_\phi(\mathcal{C}^*(X,\mathcal{F}))$ and $H^n(\Gamma(\mathcal{C}^*(X,\mathcal{F}))$? What is the utility of the family of supports?

2) I've heard that when $X$ is not paracompact then $H^n(\Gamma(\mathcal{C}^*(X,\mathcal{F}))$ (but not $H^n(\Gamma_\phi(\mathcal{C}^*(X,\mathcal{F}))$) fails to be "functorial" (i think in the sense that short exact sequences don't go to long exact sequences). Can you explain this to me? What exactly doesn't work for $H^n(\Gamma(\mathcal{C}^*(X,\mathcal{F}))$?

Thank you very much

Answer 1

I hope the following is an answer to (a part) of your wondering : Stefan Schroër constructed a non-paracompact Hausdorf space for which Cech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic ... See : Top. and its Appl., vol.160, issue 13, 15/8/2013 (1809-1815). On the other hand, I recommend B. Iversen's book , "Cohomology of Sheaves".

Answer 2

2) Are you sure you are not confusing the sheaf cohomology $H^n(\Gamma(C^*(X,\mathcal F))$ with Čech cohomology$\check H^n(X,\mathcal F)$? The sheaf cohomology has long exact sequences for arbitrary topological space and short exact sequence of sheaves. For paracompact spaces the two are equal, there is a proof of it in Godement's book.