Sheaves of smooth-function modules are cohomologically acyclic

Fix an open $U\subseteq M$ and write ${\mathcal{A}}_U={\mathcal{C}}_U^{\mathrm{\infty }}$; restriction makes $\mathcal{F}{\mathrm{\mid }}_U$ an ${\mathcal{A}}_U$-module sheaf by [F8]. If $U=\mathrm{\varnothing }$, [F7] makes every global-section group zero, so exactness is immediate; suppose $U\mathrm{\ne }\mathrm{\varnothing }$. To prove that $\mathrm{\Gamma }(U,-)$ is exact on these module sheaves, it suffices by left exactness in [F6] to prove surjectivity on global sections for an epimorphism $p:\mathcal{G}\to \mathcal{H}$. Fix $s\in \mathcal{H}(U)$. If $s$ has a global lift $t\in \mathcal{G}(U)$, take $g=t$. Otherwise index all local lift data by pairs $(V,t)$ with $V\subseteq U$ open, $t\in \mathcal{G}(V)$, and $p(t)=s{\mathrm{\mid }}_V$. Their domains cover $U$ by [F5]. Use [F4] to choose a smooth partition of unity $({\rho }_{(V,t)})$ subordinate to this indexed cover. On $V$, the product ${\rho }_{(V,t)}t$ extends by zero to a section of $\mathcal{G}(U)$: use zero on $U\setminus \mathrm{supp}{\rho }_{(V,t)}$; these definitions agree on the overlap because ${\rho }_{(V,t)}=0$ there. The extended sections are locally finite, so their local finite sums glue to $g\in \mathcal{G}(U)$ by the sheaf axiom, and $p(g)={\sum }_{(V,t)}{\rho }_{(V,t)}s=s$. Thus $\mathrm{\Gamma }(U,-)$ is exact on ${\mathcal{A}}_U$-module sheaves.

By [F2], choose the supplied injective resolution of the restricted module $0\to \mathcal{F}{\mathrm{\mid }}_U\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$ in $\mathrm{Mod}({\mathcal{A}}_U)$, using [F8]. Its underlying sequence of abelian sheaves is exact by [F2]. Each ${\mathcal{I}}^j$ is flasque as an abelian sheaf by [F3], so it is $\mathrm{\Gamma }(U,-)$-acyclic; the successive cycles are abelian sheaves and therefore lie in the domain of the supplied abelian-sheaf cohomology data [F6]. By [F4], AC supplies DC, so the acyclic-resolution theorem identifies $H^q(U,\mathcal{F}{\mathrm{\mid }}_U)$ with the cohomology of $\mathrm{\Gamma }(U,{\mathcal{I}}_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet })$. Step 1.1 makes this complex exact in every positive degree, hence the cohomology vanishes for $q>0$. Since $U$ was arbitrary, the theorem follows. Full AC is used for the two injective-resolution data; its consequences ${\mathrm{A}\mathrm{C}}_{\omega }$ and DC are used in [F4] and the acyclic-resolution comparison, respectively.