Higher direct image of a sheaf

Let $(f,f^{\mathrm{♯}}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ be a morphism of ringed spaces and let $\mathcal{F}$ be an ${\mathcal{O}}_X$-module (Modules on a ringed space). The direct image presheaf $f_{\ast }\mathcal{F}$ is defined on opens $V\subseteq Y$ by $(f_{\ast }\mathcal{F})(V)=\mathcal{F}(f^{-1}(V))$ (Direct image of a sheaf along a continuous map); when $\mathcal{F}$ is a sheaf of modules, this presheaf is again a sheaf of modules on $Y$ (Direct image preserves sheaves and objectwise algebraic structure), and for a morphism of ringed spaces it is an ${\mathcal{O}}_Y$-module, because $f_{\ast }$ is right adjoint to the inverse image functor $f^{\ast }$ on modules (Pullback of modules is left adjoint to pushforward); a right adjoint is left exact and additive (A right adjoint is left exact and a left adjoint is right exact).

Fix the supplied functorial injective resolution datum $I$ on $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ of Enough injective sheaves of modules: it assigns to every ${\mathcal{O}}_X$-module $\mathcal{F}$ one specific injective resolution $0\to \mathcal{F}\to I^0(\mathcal{F})\to I^1(\mathcal{F})\to \cdots$ (Injective resolutions in an abelian category) with deleted complex $I(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}$ (Right derived objects relative to supplied injective resolution data). For every $q\in \mathbb{Z}$ define the $q$-th higher direct image sheaf

$R^q{f}_{\ast }\mathcal{F}:=R_I^q{f}_{\ast }(\mathcal{F})=H^q(f_{\ast }I(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}),$

the $q$-th right derived object of the additive left exact functor $f_{\ast }:\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)\to \mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_Y)$ relative to the fixed datum $I$ (Right derived objects relative to supplied injective resolution data), the cohomology object being taken in the abelian category of ${\mathcal{O}}_Y$-modules. We also keep the convention $R^q{f}_{\ast }\mathcal{F}=0$ for $q<0$.

By the derived-object convention this makes $R^q{f}_{\ast }\mathcal{F}$ a specific ${\mathcal{O}}_Y$-module for every $\mathcal{F}$ and $q$, depending on the fixed datum $I$; in degree zero the left exactness of $f_{\ast }$ supplies a canonical natural isomorphism $R^0{f}_{\ast }\mathcal{F}\stackrel{\sim }{\to }{f}_{\ast }\mathcal{F}$ (The zero-th right derived functor of a left exact functor recovers the functor, which assumes the Axiom of Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)), so the $q=0$ term is the ordinary direct image. The Axiom of Choice supplies the functorial injective resolution datum of $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$ (The Axiom of Choice) and implies the Dependent Choice required by the cited degree-zero and comparison theorems (AC implies DC implies countable choice). If $J$ is any other supplied injective resolution datum on $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$, then the functors $R_I^q{f}_{\ast }$ and $R_J^q{f}_{\ast }$ are naturally isomorphic (Two supplied injective resolution data define naturally isomorphic right derived functors), so the higher direct image is independent of the resolution up to a canonical comparison isomorphism.