A naive question about the Godement resolution So, I have read several accounts on Godement resolution, including an exposition by Godement himself in "Topologie algébrique et théorie des faisceaux". And there is still one thing that confuses me immensely.
By definition of the Godement sheaf $\mathcal{G}_{\mathcal{F}}$ for a sheaf $\mathcal{F}$ on a space $X$ we have $$
\mathcal{G}_{\mathcal{F}}(U)=\prod_{x\in U} \mathcal{F}_x,\quad U\in \operatorname{Op}(X).
$$
If I understand correctly for Hausdorff spaces we have ${(\mathcal{G}_{\mathcal{F}})}_x=\mathcal{F}_x.$ (For two-point space with non-discrete topology and a constant sheaf this is not the case.)
Now to define higher terms of the Godement resolution we have to apply $\mathcal{G}$ to the cokernel of the canonical map
$$
j:\mathcal{F}\to \mathcal{G}_{\mathcal{F}},\quad \mathcal{F}(U)\ni s\mapsto \prod_{x\in U}res^U_x(s)\in \mathcal{G}_{\mathcal{F}}(U)
$$
However, by the remark above it seems to me that the sheaf $\operatorname{coker}(j)$ has trivial stalks and hence is trivial for Hausdorff spaces.
What am I missing? Perhaps my misunderstanding stems from me not understanding the concept of the cokernel sheaf? Also, I would be very grateful if someone could explain how to explicitly define stalks of the sheaf $\mathcal{G}_{\mathcal{F}}^1$ in terms of the initial sheaf $\mathcal{F}.$
 A: The stalks of the sheaf $\mathcal{G}_{\mathcal{F}}$ are way bigger than the stalks of $\mathcal{F}$.
Say for example $\mathcal{F}$ is the sheaf of smooth functions on $\mathbb{R}$.  Let's look at the point $x=0$.
Now a section of $\mathcal{F}$ over $(-1,1)$ is a smooth function on $(-1,1)$.  A section of $\mathcal{G}_{\mathcal{F}}$ over $(-1,1)$ is a choice for every $t\in (-1,1)$ of a germ of a smooth function at $t$, and these choices are completely arbitrary.
Now, what is the stalk of $\mathcal{F}$ at $0$? It consists of germs of smooth functions around $0$.  What about an element of stalk of $\mathcal{G}_{\mathcal{F}}$ at $0$?  Need to choose a germ of a smooth function at each point around $0$. But we don't need these germs to be chosen as the germ of the same function. So for instance, i can take the germs to be of the function $0$ at each $t\ne 0$, and the germ of the function $f(t) = t$ at $t=0$. This does not come from $\mathcal{F}$ in any way. You can say: wait, at $0$ it's OK, yeah, but it has to be OK around $0$.
