Injective modules are flasque and Ext from the structure sheaf is cohomology

Extension by zero for modules. Let $o:U\hookrightarrow Y$ be an open inclusion and let $\mathcal{G}$ be an ${\mathcal{O}}_U$-module. Define the presheaf $o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathcal{G}$ on $Y$ by $(o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathcal{G})(W)=\{s\in \mathcal{G}(W\cap U):\mathrm{Supp}(s)\text{ is closed in }W\},W\subseteq Y\text{ open},$ with restriction maps those of $\mathcal{G}$ and with the ${\mathcal{O}}_Y(W)$-module structure induced by the ring map ${\mathcal{O}}_Y(W)\to {\mathcal{O}}_U(W\cap U)$ [F1]. The support condition is stable under multiplication by functions and under restrictions, and the presheaf is a sheaf because its sections are the sections of the abelian extension by zero $o_{!}\mathcal{G}$ of [F3] with the additional module structure: the underlying abelian sheaf of $o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathcal{G}$ is exactly $o_{!}\mathcal{G}$, and the module structure is well defined on the same section sets. Consequently the functor $o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}$ is exact on $\mathcal{O}$-modules, since the forgetful functor to abelian sheaves preserves kernels and cokernels [F1] and $o_{!}$ is exact on abelian sheaves [F3].

The Hom complex of the structure sheaf. For every ${\mathcal{O}}_Y$-module $\mathcal{H}$ the map ${\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y,\mathcal{H})⟶\mathrm{\Gamma }(Y,\mathcal{H}),\phi ⟼{\phi }_Y(1_Y),$ is a bijection: two morphisms with the same value at $1_Y$ agree on the unit section over every open and hence on all sections, and conversely a section $s\in \mathrm{\Gamma }(Y,\mathcal{H})$ defines a morphism whose value on $f\in {\mathcal{O}}_Y(W)$ is $f\cdot s{\mathrm{\mid }}_W$, with inverse given by the unit section. This bijection is natural in $\mathcal{H}$ and identifies the complex ${\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y,J^{\bullet })$ degreewise with the complex $\mathrm{\Gamma }(Y,J^{\bullet })$ of [F1], the differentials corresponding because both are postcomposition with the differentials of $J^{\bullet }$. Hence $H^q({\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y,J^{\bullet }))\cong H^q(\mathrm{\Gamma }(Y,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet }))$ for every $q\ge 0$.

The adjunction. The abelian-sheaf adjunction of [F3] sends a morphism $o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathcal{G}\to \mathcal{H}$ to its restriction over $U$. It restricts to an adjunction of $\mathcal{O}$-modules. Indeed an ${\mathcal{O}}_Y$-linear map restricts over $U$ to an ${\mathcal{O}}_U$-linear map. Conversely the abelian adjoint of an ${\mathcal{O}}_U$-linear map is ${\mathcal{O}}_Y$-linear stalkwise: at a point of $U$ the stalk map is the given ${\mathcal{O}}_U$-linear map, and at a point outside $U$ the source stalk of $o_{!}\mathcal{G}$ is zero; equality of the two candidate multiplication morphisms is detected on stalks. Thus ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}\mathcal{G},\mathcal{H})\cong {\mathrm{Hom}}_{{\mathcal{O}}_U}(\mathcal{G},\mathcal{H}{\mathrm{\mid }}_U)\mathrm{.}$ For $\mathcal{G}={\mathcal{O}}_U$ and $\mathcal{H}=\mathcal{I}$, evaluation at the unit section gives ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_U,\mathcal{I})\cong {\mathrm{Hom}}_{{\mathcal{O}}_U}({\mathcal{O}}_U,\mathcal{I}{\mathrm{\mid }}_U)\cong \mathcal{I}(U)\mathrm{.}$ This uses stalkwise module linearity, not surjectivity of ${\mathcal{O}}_Y(W)\to {\mathcal{O}}_U(W\cap U)$, which need not hold.

The comparison map is a monomorphism. For open $U\subseteq V\subseteq Y$ let $i:U\hookrightarrow V$ be the inclusion. The natural map $\alpha :o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_U⟶o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_V$ that extends a section of ${\mathcal{O}}_Y$ over $W\cap U$ with support closed in $W$ by zero across $W\cap (V\setminus U)$ is a morphism of ${\mathcal{O}}_Y$-modules, because extension by zero is ${\mathcal{O}}_Y(W)$-linear on the subsheaf of sections with closed support [F1, step 1.1]. Its section maps are injective: a section $s$ over $W\cap U$ with closed support in $W$, extended by zero over $W\cap (V\setminus U)$, has support closed in $W$ as well and restricts back to $s$. Hence $\ker (\alpha )=0$ by [F4], so $\alpha$ is a monomorphism.

Injective modules are flasque. Let $s\in \mathcal{I}(U)$. Under the bijection of step 2.1 for $U$ the element $s$ corresponds to some morphism $f:o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_U\to \mathcal{I}$. By step 2.2 the map $\alpha$ is a monomorphism, so [F2] applied to the subobject $\alpha :o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_U\rightarrowtail o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_V$ and the morphism $f$ provides $g:o_{!}^{\mathrm{m}\mathrm{o}\mathrm{d}}{\mathcal{O}}_V\to \mathcal{I}$ with $g\circ \alpha =f$. Let $t\in \mathcal{I}(V)$ correspond to $g$ under the bijection of step 2.1 for $V$. Precomposition with $\alpha$ corresponds under these two bijections to restriction along $U\subseteq V$, so $g\circ \alpha =f$ says $t{\mathrm{\mid }}_U=s$. Hence every section over $U$ extends to $V$, the restriction map $\mathcal{I}(V)\to \mathcal{I}(U)$ is surjective, and since $U\subseteq V$ were arbitrary $\mathcal{I}$ is flasque, which is clause 1.

Flasque injective resolutions compute cohomology. Let $\mathcal{G}$ be an ${\mathcal{O}}_Y$-module and let $\mathcal{G}\to J^{\bullet }$ be the ${\mathcal{O}}_Y$-injective resolution supplied by the functorial datum of [F6]. By step 3.1 every $J^p$ is flasque as an abelian sheaf, so $H^q(Y,J^p)=0$ for every $q>0$ by [F5]: each $J^p$ is acyclic for the global sections functor $\mathrm{\Gamma }(Y,-)$ on $\mathrm{A}\mathrm{b}(Y)$. The underlying abelian complex of $\mathcal{G}\to J^{\bullet }$ is therefore a $\mathrm{\Gamma }(Y,-)$-acyclic resolution of the abelian sheaf $\mathcal{G}$, with all its cycles lying in the class $\mathcal{D}$ of all abelian sheaves on $Y$, on which the supplied datum of [F7] is defined. The Axiom of Dependent Choice is available by [F9] and [A1], so [F8] gives a canonical isomorphism $R_I^q\mathrm{\Gamma }(Y,\mathcal{G})\cong H^q(\mathrm{\Gamma }(Y,J_{\mathrm{d}\mathrm{e}\mathrm{l}}^{\bullet })),q\ge 0.$

Conclusion. Combining steps 4.1 and 1.2 with the identification ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q({\mathcal{O}}_Y,\mathcal{G})=H^q({\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{O}}_Y,J^{\bullet }))$ of [F6] and $H^q(Y,\mathcal{G})=R_I^q\mathrm{\Gamma }(Y,\mathcal{G})$ of [F7] gives the canonical isomorphism ${\chi }_{\mathcal{G}}^q$ of clause 2 for every ${\mathcal{O}}_Y$-module $\mathcal{G}$ and every $q\ge 0$; in degree zero both bijections display the value at the unit section, which is the identification asserted in the statement. Naturality in $\mathcal{G}$ holds because the supplied resolution datum is functorial: a morphism $\psi :\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$ gives a cochain map $J^{\bullet }(\mathcal{G})\to J^{\bullet }({\mathcal{G}}^{\mathrm{\prime }})$ commuting with the coaugmentations, and the comparisons used in steps 4.1 and 1.2 are built from the datum and the fixed functor $\mathrm{\Gamma }(Y,-)$ and therefore intertwine the two $\chi$'s; the right-hand isomorphism of step 4.1 is the canonical comparison of the two acyclic resolutions, so the square commutes. Clause 1 is step 3.1. The Axiom of Choice [A1] is assumed in the statement and is used exactly through the functorial injective resolution data of [F6] and [F7] for modules and for abelian sheaves and, through the Dependent Choice instance of [F9], in the acyclic-resolution comparison of step 4.1; no further selection of resolutions, indices or sections is made, the charts and open sets being arbitrary parameters of the construction.