Global sections on module sheaves have ordinary sheaf cohomology as derived functors

For each point $x\in X$, choose an embedding $M_x\hookrightarrow I_x$ into an injective ${\mathcal{A}}_x$-module by [F1]. The skyscraper sheaf $i_{x\ast }{I}_x$ has sections $I_x$ on opens containing $x$ and zero on opens not containing $x$, with $\mathcal{A}(U)$ acting through ${\mathcal{A}}_x$. The natural bijection ${\mathrm{Hom}}_{\mathcal{A}}(N,i_{x\ast }{I}_x)\cong {\mathrm{Hom}}_{{\mathcal{A}}_x}(N_x,I_x)$ shows $i_{x\ast }{I}_x$ is injective, since taking stalks is exact by [F2]. The product of these injectives is injective: Hom into the product is the product of Hom functors, and products of exact sequences of abelian groups are exact under the stated Axiom of Choice. The diagonal map $M\to {\prod }_{x\in X}{i}_{x\ast }{I}_x$ is injective on each stalk by construction. Thus $\mathrm{Mod}(\mathcal{A})$ has enough injectives.

Let $j:U\hookrightarrow X$ be open and $M$ an $\mathcal{A}{\mathrm{\mid }}_U$-module. Define $(j_{!}M)(V)$ as the sections of $M(V\cap U)$ whose support is closed in $V$, using Extension by zero for abelian sheaves on an open subspace. For $a\in \mathcal{A}(V)$ and such a section $s$, let $as$ be the action of $a{\mathrm{\mid }}_{V\cap U}$; if $s$ vanishes on $W\cap U$ near a point of $V\setminus U$, then so does $as$, so the support condition is preserved. The stalks are $M_x$ for $x\in U$ and zero otherwise by the underlying abelian construction. The abelian-sheaf adjunction extends a module map on $U$ uniquely to an abelian-sheaf map on $X$; the extension is $\mathcal{A}$-linear on stalks in $U$ by the original module-linearity and off $U$ because its source stalk is zero. Thus $j_{!}$ is left adjoint to restriction in module sheaves. The stalk description and [F2] show that it is exact.

Let $I$ be an injective $\mathcal{A}$-module sheaf and let $U\subseteq V$ be open. A section $s\in I(U)$ corresponds by evaluation at $1$ to a morphism $j_{!}{\mathcal{A}}_U\to I{\mathrm{\mid }}_V$. The natural map $j_{!}{\mathcal{A}}_U\to {\mathcal{A}}_V$ is a monomorphism: on stalks it is the identity for points of $U$ and the map $0\to {\mathcal{A}}_x$ outside $U$. Restriction preserves injectivity here because its exact left adjoint is extension by zero [F3, step 1.2]. Injectivity of $I{\mathrm{\mid }}_V$ extends the morphism across $j_{!}{\mathcal{A}}_U\hookrightarrow {\mathcal{A}}_V$. The extension is determined by the image $t$ of $1\in \mathcal{A}(V)$, and $t{\mathrm{\mid }}_U=s$. Every restriction $I(V)\to I(U)$ is therefore surjective, so $I$ is flasque.

Take an injective resolution $0\to M\to I^0\to I^1\to \cdots$ in $\mathrm{Mod}(\mathcal{A})$, which exists by step 1.1. Its global-sections complex computes module-derived global sections by definition. Each underlying abelian sheaf $I^p$ is flasque by step 2.1 and hence $\mathrm{\Gamma }$-acyclic by [F4]. Use the supplied abelian-sheaf datum of [F5], whose domain is all of $\mathrm{A}\mathrm{b}(X)$: the underlying $M$, every $I^p$, and every cycle are abelian sheaves and therefore belong to that domain. Forgetting the module action preserves exactness by [F2], so this is a $\mathrm{\Gamma }$-acyclic resolution relative to that datum. The acyclic-resolution theorem now identifies its global-sections cohomology with $H^q(X,M)$; its Dependent Choice hypothesis follows from the assumed AC. Comparison of resolutions makes the identification natural in $M$, and it preserves degree zero.