Long exact global sheaf Ext sequence in the first variable

Using AC and the in-run theorem lem-ringed-space-module-sheaves-enough-injectives of the cohomology-of-quasi-coherent-sheaves pair, which supplies an ${\mathcal{O}}_Y$-injective resolution for every ${\mathcal{O}}_Y$-module, fix one such resolution $\mathcal{G}\to I^{\bullet }$; by [F3] applied in the abelian category of ${\mathcal{O}}_Y$-modules the complex ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })$ has $q$th term ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^q)$ and differential $d^q\circ (-)$, and by [F4] its $q$th cohomology is ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q(\mathcal{F},\mathcal{G})$.

For every $p$ the module $I^p$ is injective, so by [F1] every morphism $M\to I^p$ defined on a subobject of $E$ extends to $E$; applying this to the subobject ${\mathcal{F}}^{\mathrm{\prime }}\subseteq \mathcal{F}$ gives exactness of $0\to \mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^p)\to \mathrm{Hom}(\mathcal{F},I^p)\to \mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }},I^p)\to 0$: surjectivity of the last map is the extension property applied to ${\mathcal{F}}^{\mathrm{\prime }}\subseteq \mathcal{F}$, injectivity of the first is immediate from the epimorphism $\mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$, and exactness in the middle follows because a morphism $\mathcal{F}\to I^p$ killing ${\mathcal{F}}^{\mathrm{\prime }}$ factors through the quotient $\mathcal{F}\mathrm{/}{\mathcal{F}}^{\mathrm{\prime }}\cong {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$.

The three complexes $\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet })$, $\mathrm{Hom}(\mathcal{F},I^{\bullet })$ and $\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }},I^{\bullet })$ are concentrated in degrees $q\ge 0$, their differentials are post-composition with the differential $d^q$ of $I^{\bullet }$, so the degreewise exact sequence of step 1.2 commutes with those differentials; hence $0\to \mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet })\to \mathrm{Hom}(\mathcal{F},I^{\bullet })\to \mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }},I^{\bullet })\to 0$ is a short exact sequence of cochain complexes.

By [F2] the sequence of step 2.1 has a natural long exact sequence $\cdots \to H^q(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet }))\to H^q(\mathrm{Hom}(\mathcal{F},I^{\bullet }))\to H^q(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }},I^{\bullet }))\stackrel{{\mathrm{\partial }}^q}{\to }{H}^{q+1}(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet }))\to \cdots$, and substituting the identification of [F4] turns its terms into ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},\mathcal{G})$, ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q(\mathcal{F},\mathcal{G})$ and ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q({\mathcal{F}}^{\mathrm{\prime }},\mathcal{G})$.

Since the complexes are concentrated in degrees $q\ge 0$, the terms $H^{-1}$ in the long exact sequence of step 3.1 vanish, so the sequence begins $0\to H^0(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet }))\to H^0(\mathrm{Hom}(\mathcal{F},I^{\bullet }))\to H^0(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }},I^{\bullet }))\to H^1(\mathrm{Hom}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},I^{\bullet }))\to \cdots$; by the degree-zero clause of [F4] the first three terms are ${\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }},\mathcal{G})$, ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},\mathcal{G})$ and ${\mathrm{Hom}}_{{\mathcal{O}}_Y}({\mathcal{F}}^{\mathrm{\prime }},\mathcal{G})$, which is the displayed beginning of the statement.

Naturality in $\mathcal{G}$ and resolution independence hold as follows: a morphism $u:\mathcal{G}\to {\mathcal{G}}^{\mathrm{\prime }}$ with injective resolutions $I^{\bullet }$, $J^{\bullet }$ admits a coaugmentation-preserving comparison map $I^{\bullet }\to J^{\bullet }$ extending $u$ by [F6], and post-composition with it is a cochain map $\mathrm{Hom}(\mathcal{F},I^{\bullet })\to \mathrm{Hom}(\mathcal{F},J^{\bullet })$ inducing maps on cohomology that intertwine the connecting maps of [F2]; two choices of comparison map are cochain-homotopic by [F5], whose Dependent Choice hypothesis is licensed by [F6], so the induced maps on cohomology agree and the sequence depends on $\mathcal{G}$ and not on the resolution.

Naturality in the short exact sequence holds because a morphism of short exact sequences of ${\mathcal{O}}_Y$-modules induces a morphism of the degreewise exact sequences of complexes built in step 2.1, and [F2] provides the induced morphism of long exact sequences; the connecting maps ${\mathrm{\partial }}^q$ are then those supplied by [F2] composed with the identifications of [F4]. The statement claims the long exact sequence and its naturality, and no splitting or vanishing beyond degree zero, so nothing further is asserted.