How to define the canonical Godement resolution Good afternoon :
I whould like to know how to define the canonical Godement resolution of a flasque sheaf.
Thanks a lot.
 A: Let $\mathscr F$ be a sheaf on a topological space $(X,\tau)$. One can define the Godement sheaf of $\mathscr F$ as follows: for any $U\in\tau$, set
$$
\mathcal G^0\mathscr F(U)=\prod_{x\in U}\mathscr F_x,
$$
and for any inclusion $V\to U$ look at the restriction
$$
\mathcal G^0\mathscr F(U)\to\mathcal G^0\mathscr F(V),\,\,\,\,\,\,\,\,s\mapsto s|_V.
$$
Note that this is a restriction in the usual sense, as an element $s\in\mathcal G^0\mathscr F(U)$ is an actual function $s:U\to\coprod_{x\in U} \mathscr F_x$.
Here are some important facts:


*

*The (covariant) functor $\mathcal G^0:\textrm{Sh}(X)\to \textrm{Sh}(X)$ is exact. (easy)

*The Godement sheaf $\mathcal G^0\mathscr F$ is flasque for every $\mathscr F\in \textrm{Ob Sh}(X)$. (easy)

*There is a natural monomorphism $0\longrightarrow \mathscr F\overset{\eta}{\longrightarrow} \mathcal G^0\mathscr F$.

*There is a flasque resolution $\mathcal G^\bullet\mathscr F$ of $\mathscr F$:
$$
0\longrightarrow \mathscr F\overset{\eta}{\longrightarrow} \mathcal G^0\mathscr F \overset{d^0}{\longrightarrow} \mathcal G^1\mathscr F
\overset{d^1}{\longrightarrow} \mathcal G^2\mathscr F
\longrightarrow\dots\overset{d^{q-1}}{\longrightarrow}G^q\mathscr F.
\,\,\,\,\,\,\,\,\,\,\,\,\,(\star)
$$


Proof of 3.
Let $U\in\tau$. Define
$$
\eta_U:\mathscr F(U)\to\prod_{x\in U} \mathscr F_x,\,\,\,\,\,\,\,\,s\mapsto (s_x)_{x\in U}.
$$
Very explicitly, $\eta_U(s)$ is the function
$$
\eta_U(s):U\to\coprod_{x\in U} 
\mathscr F_x,\,\,\,\,\,\,\,\,\,x\mapsto s_x.
$$
We want to show that $\eta_U$ is injective (so that $\eta$ is injective too). Let $t\in\ker\eta_U$. Then $t_x=0$ for any $x\in U$.
Thus for every $x\in U$ there exists an open subset $W_x$ such that $t|_{W_x}=0$. The $W_x$'s cover $U$, so by a sheaf axiom
we get $t=0$ and $\eta_U$ is injective. 
Proof of 4.
Prove existence and exactness of $(\star)$ by induction on $q$, the step $q=0$ being clear by 3 and 2.
Now suppose we got $(\star)$ as we want, up to $q$. Let us define:


*

*$\mathcal G^{q+1}\mathscr F:=\mathcal G^0\textrm{coker}(d^{q-1})$; this definition is a good candidate because this sheaf is flasque by 2. Now the arrows:

*Define $d^q:\mathcal G^{q}\mathscr F\to\mathcal G^{q+1}\mathscr F$ to be the composition
$$
\mathcal G^{q}\mathscr F\to \textrm{coker}(d^{q-1})\to \mathcal G^0(\textrm{coker}(d^{q-1})).
$$
The second arrow is injective, so we get the desired exactness of $(\star)$.

