Constant sheaves on irreducible spaces are flasque and acyclic

For every open $U\subseteq X$ the isomorphism of [F1] identifies the group $A_X(U)$ with the group ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(U)$ of locally constant functions $U\to A$, with pointwise addition, and identifies the section ${\eta }_U(a)$ with the constant function with value $a$; the restrictions of the two sheaves correspond under the identification, because the isomorphism is one of sheaves.

Let $U\subseteq X$ be a nonempty open subset. By [F2] the subspace $U$ is irreducible, hence connected. Let $f:U\to A$ be locally constant. Each fibre $f^{-1}(a)\subseteq U$ is open, because every point of it has an open neighbourhood on which $f$ is constant with value $a$; the fibres are pairwise disjoint and cover $U$. Since $U$ is nonempty there is a point $x\in U$, and I claim $f$ is the constant function with value $f(x)$. Indeed, if there were $y\in U$ with $f(y)\mathrm{\ne }f(x)$, then $f^{-1}(f(x))$ and $U\setminus f^{-1}(f(x))$ would be open subsets of $U$: the first by local constancy, the second because it is the union of the open fibres $f^{-1}(a)$ over the values $a\mathrm{\ne }f(x)$. Both are nonempty, they are disjoint, and their union is $U$, so $(f^{-1}(f(x)),U\setminus f^{-1}(f(x)))$ would be a separation of $U$, contradicting connectedness by [F3]. Hence ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(U)$ consists of the constant functions, and the map $A\to {\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(U)$, $a\mapsto (x\mapsto a)$, is a bijection: it is surjective by what was just proved and injective because two constant functions with distinct values differ at every point of the nonempty set $U$.

For $U=\mathrm{\varnothing }$ the group ${\underset{‾}{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\varnothing })$ is the set of functions $\mathrm{\varnothing }\to A$, a singleton, and it is the zero group for the pointwise addition of [F1]; equivalently $A_X(\mathrm{\varnothing })$ is a singleton by [F6] applied to the sheaf $A_X$, hence the zero group as well.

Let $U\subseteq V\subseteq X$ be open subsets. If $U=\mathrm{\varnothing }$ then the restriction map $A_X(V)\to A_X(\mathrm{\varnothing })$ has zero target by [step 2.2] and is surjective. If $U\mathrm{\ne }\mathrm{\varnothing }$ then also $V\mathrm{\ne }\mathrm{\varnothing }$, and both $A_X(V)$ and $A_X(U)$ are identified, by [step 1.1] and [step 2.1], with $A$ through the constant functions, in such a way that the restriction map corresponds to the map sending the constant function with value $a$ on $V$ to its restriction on $U$, which is again the constant function with value $a$; so the restriction map is the identity of $A$, in particular surjective. Since $U\subseteq V$ were arbitrary open subsets, $A_X$ is flasque by [F4].

By [step 3.1] the sheaf $A_X$ of abelian groups on $X$ is flasque, so the acyclicity theorem [F5] applies with $U=X$ and gives $H^q(X,A_X)=0$ for every $q>0$. The Axiom of Choice is used at exactly this point, as the hypothesis [F7] of the acyclicity theorem; the computation of sections in [step 2.1] and the flasqueness in [step 3.1] use no choice principle, the identifications being those of the canonical isomorphism of [F1]. ∎