Sheaf Ext of coherent modules

Assume the Axiom of Choice (The Axiom of Choice). Let $(Y,{\mathcal{O}}_Y)$ be a ringed space and let $\mathcal{F},\mathcal{G}$ be ${\mathcal{O}}_Y$-modules. An ${\mathcal{O}}_Y$-injective resolution of $\mathcal{G}$ is an exact sequence $0⟶\mathcal{G}⟶I^0\stackrel{d^0}{\to }{I}^1\stackrel{d^1}{\to }\cdots$ of ${\mathcal{O}}_Y$-modules in which every $I^p$ is an injective object of the category of ${\mathcal{O}}_Y$-modules. The in-run theorem lem-ringed-space-module-sheaves-enough-injectives of batch 9 supplies such a resolution for every $\mathcal{G}$, and the published comparison results Injective comparison maps exist and Injective comparison maps are unique up to cochain homotopy (applied in the abelian category of ${\mathcal{O}}_Y$-modules) make the constructions below independent of the supplied resolution up to a canonical isomorphism, as recorded in the final paragraph of this definition. Their Dependent Choice hypothesis follows from the declared Axiom of Choice by AC implies DC implies countable choice.

Two complexes are attached to such a resolution:

• Global Ext. The complex of abelian groups ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })$ has degree-$p$ term ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^p)$ and differential $d^p\circ (-)$. Its cohomology in the sense of Cohomology object of a cochain complex is written ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^q(\mathcal{F},\mathcal{G}):=H^q({\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })),q\ge 0.$
• Sheaf Ext. The complex of ${\mathcal{O}}_Y$-modules $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })$, with internal Hom as in The internal Hom sheaf of two module sheaves and the same differential, has cohomology sheaves written $\mathcal{E}x{t}_{{\mathcal{O}}_Y}^q(\mathcal{F},\mathcal{G}):=H^q(\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })),q\ge 0.$

The terms of ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })$ are the global sections of the terms of $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},I^{\bullet })$, but the two constructions are different functors: taking global sections does not commute with taking cohomology, so the global Ext for $q>0$ is not in general the group of global sections of the sheaf Ext.

Because ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},-)$ is left exact, the degree-zero terms are identified with the ordinary Hom modules: ${\mathrm{Ext}}_{{\mathcal{O}}_Y}^0(\mathcal{F},\mathcal{G})\cong {\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},\mathcal{G})$ and $\mathcal{E}x{t}_{{\mathcal{O}}_Y}^0(\mathcal{F},\mathcal{G})\cong \mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},\mathcal{G})$, the isomorphisms being induced by the coaugmentation $\mathcal{G}\to I^0$.

Independence of the resolution. The subscript-free notation is justified as follows. Let $\mathcal{G}\to J^{\bullet }$ be a second ${\mathcal{O}}_Y$-injective resolution. The comparison theorem Injective comparison maps exist produces a coaugmentation-preserving cochain map $I^{\bullet }\to J^{\bullet }$, and Injective comparison maps are unique up to cochain homotopy shows that any two such maps are cochain-homotopic; applying the additive functors ${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{F},-)$ and $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(\mathcal{F},-)$ to such a homotopy produces a homotopy of the resulting complexes of abelian groups, respectively of sheaves of ${\mathcal{O}}_Y$-modules, because these functors are additive and preserve the homotopy relation. Cochain-homotopic maps induce the same map on cohomology, so the groups and sheaves above are well-defined up to canonical isomorphism independent of the resolution. This is the same well-definedness mechanism as in the abstract construction of Ext via an injective resolution of the second variable, specialized to ${\mathcal{O}}_Y$-modules.

Local computation. If $\mathcal{F}$ admits a resolution $\cdots \to F_1\to F_0\to \mathcal{F}\to 0$ by finite locally free ${\mathcal{O}}_Y$-modules, then $\mathcal{E}x{t}_{{\mathcal{O}}_Y}^q(\mathcal{F},\mathcal{G})$ is computed by the complex $\mathcal{H}o{m}_{{\mathcal{O}}_Y}(F_{\bullet },\mathcal{G})$; this local computation, together with its compatibility with the injective resolution defining sheaf Ext, is proved in Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension in the smooth regular-immersion case where it is used, and is not assumed here.