Sheaf cohomology classes as derived morphisms

For every abelian sheaf $\mathcal{F}$ the datum gives a coaugmented complex $0\to \mathcal{F}\to I^0(\mathcal{F})\to I^1(\mathcal{F})\to \cdots$ which is exact at every displayed term with every $I^n(\mathcal{F})$ injective [F3, F4], so read as a map of complexes ${\eta }_{\mathcal{F}}:\mathcal{F}\to I^{\bullet }(\mathcal{F})$ it is a quasi-isomorphism [F4]; the complex $I^{\bullet }(\mathcal{F})$ is bounded below with injective terms [F4], hence K-injective by [F5], dependent choice being available from the Axiom of Choice [F5].

For every $p$ composition with $Q({\eta }_{\mathcal{F}})$ is a bijection ${\mathrm{Hom}}_D({\mathbb{Z}}_X[-p],\mathcal{F})⟶{\mathrm{Hom}}_D({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F})),$ because $Q({\eta }_{\mathcal{F}})$ is invertible in $D(\mathrm{A}\mathrm{b}(X))$ [F9] and composition with an isomorphism is bijective.

Since $I^{\bullet }(\mathcal{F})$ is K-injective [step 1.1], the localization map $Q:{\mathrm{Hom}}_K({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F}))⟶{\mathrm{Hom}}_D({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F}))$ is bijective [F6].

The natural isomorphism of [F7], read through the reindexing convention that makes $K(\mathrm{A}\mathrm{b}(X))$ the published homotopy category, identifies the homotopy classes of cochain maps ${\mathbb{Z}}_X[-p]\to I^{\bullet }(\mathcal{F})$ with the zeroth cohomology of the cochain Hom complex: ${\mathrm{Hom}}_K({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F}))\cong H^0({\underset{}{\mathrm{Hom}}}^{\bullet }({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F})))\mathrm{.}$

By [F8] the degree-$n$ term of ${\underset{}{\mathrm{Hom}}}^{\bullet }({\mathbb{Z}}_X[-p],I^{\bullet }(\mathcal{F}))$ is ${\prod }_m\mathrm{Hom}(({\mathbb{Z}}_X[-p]{)}^m,I^{m+n}(\mathcal{F})),$ and $({\mathbb{Z}}_X[-p]{)}^m={\mathbb{Z}}_X^{m-p}$ is nonzero only for $m=p$, where it is ${\mathbb{Z}}_X$, so the term is $\mathrm{Hom}({\mathbb{Z}}_X,I^{n+p}(\mathcal{F}))$; its differential is $(du{)}^p=d_I{u}^p-(-1{)}^n{u}^{p+1}{d}_{{\mathbb{Z}}_X[-p]}^p$, and $d_{{\mathbb{Z}}_X[-p]}^p=(-1{)}^{-p}{d}_{{\mathbb{Z}}_X}^0=0$ while $u^{p+1}=0$ because $({\mathbb{Z}}_X[-p]{)}^{p+1}=0$ [F8], so $(du)=d_I\circ u$ in every degree. Applying [F10] in every degree identifies this Hom complex with the full graded complex having degree-$n$ term $\mathrm{\Gamma }(X,I^{n+p}(\mathcal{F}))$ for all $n\in \mathbb{Z}$ (zero when $n+p<0$), and differential $\mathrm{\Gamma }(X,d^{n+p})$. The degreewise sign twist $u\mapsto (-1{)}^{pn}u$ identifies it with the standard shifted complex $\mathrm{\Gamma }(X,I^{\bullet }(\mathcal{F}))[p]$ when that shift uses differential $(-1{)}^{p}d$ [F13]. In particular degree $-1$ is retained when $p>0$ and contributes the boundaries in degree zero.

Taking zeroth cohomology in [step 4.1] gives the quotient of the $0$-cocycles of that complex by the $0$-coboundaries [F11], that is $\ker (\mathrm{\Gamma }(X,d^p))/\mathrm{im}(\mathrm{\Gamma }(X,d^{p-1}))=H^p(\mathrm{\Gamma }(X,I^{\bullet }(\mathcal{F}{)}_{\mathrm{d}\mathrm{e}\mathrm{l}}))=H^p(X,\mathcal{F}),$ the middle expression being $H^p$ of the deleted resolution, whose entries in degrees $\ge 0$ are the $I^n(\mathcal{F})$ and which has zero differential in negative degrees [F1, F4].

Let $p=0$ and let $\phi :{\mathbb{Z}}_X\to \mathcal{F}$ be a morphism with ${\phi }_X(1_X)=s$. Under [step 2.1] $\phi$ goes to $Q({\eta }_{\mathcal{F}}\circ \phi )$, whose preimage under the bijection of [step 2.2] is the homotopy class of the cochain map ${\eta }_{\mathcal{F}}\circ \phi :{\mathbb{Z}}_X\to I^{\bullet }(\mathcal{F})$ [step 1.1]; under [step 3.1] and [step 4.1] this class corresponds to the class of the cocycle $\mathrm{\Gamma }(X,{\eta }_{\mathcal{F}})(s)=({\eta }_{\mathcal{F}}\circ \phi {)}_X(1_X)\in \mathrm{\Gamma }(X,I^0(\mathcal{F})),$ which is a cocycle because ${\eta }_{\mathcal{F}}\circ \phi$ is a cochain map [F13] and ${\mathbb{Z}}_X$ has zero differentials in nonzero degrees [F8]. The canonical isomorphism of [F12] identifies $H^0(X,\mathcal{F})$ with the kernel of $\mathrm{\Gamma }(X,I^0(\mathcal{F}))\to \mathrm{\Gamma }(X,I^1(\mathcal{F}))$ through exactly this map $\mathrm{\Gamma }(X,{\eta }_{\mathcal{F}})$, so the composite of clause 1 with it sends $\phi$ to ${\phi }_X(1_X)$; by [F10] the map $\phi \mapsto {\phi }_X(1_X)$ is a bijection onto $\mathrm{\Gamma }(X,\mathcal{F})$, and the constant section $1_X$ is the image of the identity of ${\mathbb{Z}}_X$. The description of ${\mathbb{Z}}_X(U)$ by locally constant functions [F14] and the additivity of $\mathrm{\Gamma }(X,-)$ [F15] are the ingredients of [F10] used here.

Composing the bijection of [step 2.1] with the inverse of the bijection of [step 2.2], the isomorphism of [step 3.1] and the equality of [step 5.1] gives the canonical isomorphism of clause 1. Naturality: a morphism $\psi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves comes with the cochain map $I(\psi ):I^{\bullet }(\mathcal{F})\to I^{\bullet }(\mathcal{G})$ supplied by the functorial datum [F3] and commuting with the coaugmentations, so $I(\psi )\circ {\eta }_{\mathcal{F}}={\eta }_{\mathcal{G}}\circ \psi$; every arrow used in [step 2.1], [step 2.2], [step 3.1], [step 4.1] and [step 5.1] is given by composition with $I(\psi )$ and $\mathrm{\Gamma }(X,I(\psi ))$, and the levelwise bijections ${\mathrm{\Phi }}_{J^n}$ of [F10] are natural in the sheaf variable, so the square of clause 1 commutes, the right-hand vertical map being $H^p(X,\psi )$ [F2]. Additivity: the bijections of [step 2.1] and [step 2.2] are composition with fixed morphisms and hence additive, the identification of [step 3.1] is an isomorphism of abelian groups, the levelwise maps of [F10] are additive [F10], and $H^0$ is additive, so the composite isomorphism is additive; alternatively additivity of the cup-style constructions follows from bilinearity of composition in the localization [F9].

Clause 1 is [step 6.1] with [step 5.1], and clause 2 is [step 5.2]. The Axiom of Choice is used only to obtain the functorial injective resolution datum [F3] and, through the Axiom of Dependent Choice [F5], the K-injectivity of the bounded-below injective complexes; no other selection occurs. ∎