Local-section formula for derived direct image

Let $0\to \mathcal{F}\to I^0\to I^1\to \cdots$ be the resolution supplied by the datum $I$ of [F1]. Fix open $U\subseteq V\subseteq X$. The map $j_{U!}{\mathcal{O}}_U\to j_{V!}{\mathcal{O}}_V$ of [F2] is a monomorphism of ${\mathcal{O}}_X$-modules. Since $I^p$ is injective in $\mathrm{M}\mathrm{o}\mathrm{d}({\mathcal{O}}_X)$, every morphism $j_{U!}{\mathcal{O}}_U\to I^p$ extends across this monomorphism to $j_{V!}{\mathcal{O}}_V$. Under the natural identifications of [F2], this is precisely surjectivity of the restriction $I^p(V)\to I^p(U)$. Thus every $I^p$ is flasque as an underlying sheaf of abelian groups.

For every open $U\subseteq X$ the restricted sheaf $I^p{\mathrm{\mid }}_U$ is flasque: for open $W\subseteq V\subseteq U$ the restriction $(I^p{\mathrm{\mid }}_U)(V)\to (I^p{\mathrm{\mid }}_U)(W)$ is the restriction $I^p(V)\to I^p(W)$ of the flasque sheaf $I^p$, which is surjective.

For every open $V\subseteq S$ the restricted complex $I^{\bullet }{\mathrm{\mid }}_{f^{-1}V}$ is exact at every positive term with $\mathcal{F}{\mathrm{\mid }}_{f^{-1}V}$ as its degree-zero cohomology sheaf: exactness of $0\to \mathcal{F}\to I^0\to I^1\to \cdots$ is checked on stalks [F8], and a point $x\in f^{-1}V$ has the same stalks in $f^{-1}V$ as in $X$. Hence $0\to \mathcal{F}{\mathrm{\mid }}_{f^{-1}V}\to I^0{\mathrm{\mid }}_{f^{-1}V}\to I^1{\mathrm{\mid }}_{f^{-1}V}\to \cdots$ is a resolution by flasque sheaves.

By [F3] each restricted term $I^p{\mathrm{\mid }}_{f^{-1}V}$ is $\mathrm{\Gamma }$-acyclic, so the resolution of step 3.1 is an acyclic resolution of $\mathcal{F}{\mathrm{\mid }}_{f^{-1}V}$; the functorial injective resolution datum on abelian sheaves over the space $f^{-1}V$ is defined on the whole category [F5], so every syzygy of that resolution lies in its domain, and the left exactness of the global-sections functor [F10] lets [F4] apply to $\mathrm{\Gamma }(f^{-1}V,-)$, giving canonical isomorphisms $H^q(f^{-1}V,\mathcal{F}{\mathrm{\mid }}_{f^{-1}V})\cong H^q(\mathrm{\Gamma }(f^{-1}V,I^{\bullet }))$ for every $q\ge 0$, the right-hand side being the cohomology of the complex of sections.

Let $C^{\bullet }=f_{\ast }{I}^{\bullet }$ be the complex of ${\mathcal{O}}_S$-modules with $C^p(V)=\mathrm{\Gamma }(f^{-1}V,I^p)$ for open $V\subseteq S$ [F9]. Its cohomology presheaf is $V\mapsto H^q(C^{\bullet }(V))=H^q(\mathrm{\Gamma }(f^{-1}V,I^{\bullet }))$, and the cohomology sheaf $H^q(C^{\bullet })$ of [F6] is the sheafification of this presheaf: indeed $Z^q=\ker d^q$ has $Z^q(V)=\ker (d^q\text{\hspace{0.17em}⁣}:C^q(V)\to C^{q+1}(V))$ because kernels of sheaf morphisms are computed on opens, $B^q$ is the sheafification of the presheaf $V\mapsto \mathrm{im}(d^{q-1}\text{\hspace{0.17em}⁣}:C^{q-1}(V)\to C^q(V))$ by [F6], and passing to the quotient and to stalks, which commute, shows that the sheafified presheaf cohomology and $\mathrm{coker}(B^q\to Z^q)$ have the same stalk at every point; a morphism of sheaves with bijective stalks is an isomorphism, so the two agree.

Combining steps 4.1 and 4.2, the sheaf $R^q{f}_{\ast }\mathcal{F}=H^q(f_{\ast }{I}^{\bullet })$ is canonically isomorphic to the sheafification of the presheaf $V\mapsto H^q(f^{-1}V,\mathcal{F}{\mathrm{\mid }}_{f^{-1}V})$. The isomorphism is natural in $V$ because every map used above is induced by restriction of sections along inclusions of opens, and for $q<0$ both sides are zero by [F1] and [F5].

Finally the presheaf $V\mapsto H^q(f^{-1}V,\mathcal{F}{\mathrm{\mid }}_{f^{-1}V})$ is a presheaf of ${\mathcal{O}}_S$-modules: for $V^{\mathrm{\prime }}\subseteq V$ restriction along $f^{-1}{V}^{\mathrm{\prime }}\subseteq f^{-1}V$ is additive and compatible with the ring maps, and the action of $a\in {\mathcal{O}}_S(V)$ is induced by the action of its image in ${\mathcal{O}}_X(f^{-1}V)$. The action maps are morphisms of presheaves into the sheaf $R^q{f}_{\ast }\mathcal{F}$, so by the universal property of sheafification [F7] they descend uniquely to an ${\mathcal{O}}_S$-module structure on the sheafification, and the isomorphism of step 5.1 is ${\mathcal{O}}_S$-linear. This proves the statement, and AC and DC are used exactly through the injective resolution datum [F1], the acyclicity [F3] and the acyclic resolution theorem [F4].