Flasque abelian sheaves are Γ-acyclic

Write ${\mathcal{F}}_U:=\mathcal{F}{\mathrm{\mid }}_U$ for the restriction of $\mathcal{F}$ to the open subspace $U$. Then ${\mathcal{F}}_U$ is flasque on $U$: for open $W\subseteq W^{\mathrm{\prime }}\subseteq U$ one has ${\mathcal{F}}_U(W)=\mathcal{F}(W)$ and ${\mathcal{F}}_U(W^{\mathrm{\prime }})=\mathcal{F}(W^{\mathrm{\prime }})$ with the same restriction map, which is surjective because $\mathcal{F}$ is flasque; and ${\mathcal{F}}_U$ is a sheaf of abelian groups on $U$.

By [F6] the Axiom of Choice gives DC, and by [F5] the category $\mathrm{A}\mathrm{b}(U)$ has enough injectives with a supplied functorial injective resolution datum $I_U$; apply the datum to ${\mathcal{F}}_U$ to get a specific injective resolution $0\to {\mathcal{F}}_U\stackrel{\eta }{\to }{\mathcal{I}}^0\stackrel{d}{\to }{\mathcal{I}}^1\to \cdots$ on $U$, whose cokernel we write $\mathcal{Q}:=\mathrm{coker}(\eta )={\mathcal{I}}^0\mathrm{/}{\mathcal{F}}_U$. By [F8] the sequence $0\to {\mathcal{F}}_U\to {\mathcal{I}}^0\to \mathcal{Q}\to 0$ is short exact; by [F1] the sheaf ${\mathcal{I}}^0$ is flasque, so [F2] applies to this sequence and shows that ${\mathcal{I}}^0(W)\to \mathcal{Q}(W)$ is surjective for every open $W\subseteq U$, and that $\mathcal{Q}$ is flasque. [F1, F2, F5, F6, F8, step 1.1]

Using [F3] on the short exact sequence of step 2.1 gives an exact sequence $H^0(U,{\mathcal{I}}^0)\to H^0(U,\mathcal{Q})\to H^1(U,{\mathcal{F}}_U)\to H^1(U,{\mathcal{I}}^0)$; here $H^1(U,{\mathcal{I}}^0)=0$ by [F4] applied to the injective object ${\mathcal{I}}^0$ and the datum $I_U$, and the first map is surjective because by [F7] it is, up to the canonical identification $H^0=\mathrm{\Gamma }$, the map ${\mathcal{I}}^0(W)\to \mathcal{Q}(W)$ with $W=U$, which step 2.1 shows to be surjective. Hence $H^1(U,{\mathcal{F}}_U)=0$. [F3, F4, F7, step 2.1]

For $q\ge 1$ the same long exact sequence is, around degree $q+1$, $H^q(U,{\mathcal{I}}^0)\to H^q(U,\mathcal{Q})\stackrel{\mathrm{\partial }}{\to }{H}^{q+1}(U,{\mathcal{F}}_U)\to H^{q+1}(U,{\mathcal{I}}^0)$, and both outer groups vanish by [F4] because ${\mathcal{I}}^0$ is injective [F5]. Hence for every $q\ge 1$ the connecting map is an isomorphism $H^q(U,\mathcal{Q})\stackrel{\sim }{\to }{H}^{q+1}(U,{\mathcal{F}}_U)$; for $q\ge 1$ this is a dimension shift from the flasque quotient $\mathcal{Q}$ found in step 2.1. [F3, F4, F5, step 2.1]

I claim that $H^q(U,\mathcal{E})=0$ for every $q>0$ and every flasque abelian sheaf $\mathcal{E}$ on $U$. The case $q=1$ is step 3.1, applied to $\mathcal{E}$ in place of ${\mathcal{F}}_U$ (the argument of step 2.1 and step 3.1 uses only the flasqueness of $\mathcal{E}$, the existence of a supplied injective resolution and the lifting property [F2]). For $q>1$ apply the same construction to $\mathcal{E}$: with $0\to \mathcal{E}\to {\mathcal{I}}^0\to {\mathcal{Q}}_{\mathcal{E}}\to 0$ and ${\mathcal{Q}}_{\mathcal{E}}$ flasque, [F2], step 3.2 applied to $\mathcal{E}$ give $H^q(U,\mathcal{E})\cong H^{q-1}(U,{\mathcal{Q}}_{\mathcal{E}})$, and $H^{q-1}(U,{\mathcal{Q}}_{\mathcal{E}})=0$ because $q-1\ge 1$ and ${\mathcal{Q}}_{\mathcal{E}}$ is flasque, by induction on $q$. This proves the claim. [F1, F2, F3, F4, F5, F6, F7, step 2.1, step 3.1, step 3.2]

Applying the claim of step 4.1 to the flasque sheaf ${\mathcal{F}}_U=\mathcal{F}{\mathrm{\mid }}_U$ on $U$ gives $H^q(U,\mathcal{F}{\mathrm{\mid }}_U)=0$ for every $q>0$; since $U\subseteq X$ was an arbitrary open subspace, this is the statement. [step 1.1, step 4.1] ∎