Canonical map from fixed-cover Čech to sheaf cohomology

Since $u$ is a quasi-isomorphism [F1], its induced map $H^p(u):H^p(\mathrm{\Gamma }(X,G^{\bullet }))\to H^p(\mathrm{Tot}D)$ is an isomorphism in every degree, so ${\phi }_{\mathcal{U}}^p:=H^p(u{)}^{-1}\circ H^p(w)$ is a well-defined homomorphism ${\check{H}}^p(\mathcal{U},\mathcal{F})\to H^p(X,\mathcal{F})$ once ${\check{H}}^p(\mathcal{U},\mathcal{F})=H^p(C^{\bullet }(\mathcal{U},\mathcal{F}))$ and the identification of $H^p(X,\mathcal{F})$ with $H^p(\mathrm{\Gamma }(X,G^{\bullet }))$ of [F3] are used; the maps $u$ and $w$ are the concrete restriction and $\epsilon$-maps of [F2]. This defines the comparison map of the statement.

Let $\varphi :\mathcal{F}\to \mathcal{G}$ be a morphism of abelian sheaves. By functoriality of the Godement construction [F4] the morphism $\varphi$ induces cochain maps $G^{\bullet }(\varphi ):G^{\bullet }(\mathcal{F})\to G^{\bullet }(\mathcal{G})$ commuting with the augmentations, hence a cochain map $\mathrm{\Gamma }(X,G^{\bullet }(\varphi )):\mathrm{\Gamma }(X,G^{\bullet }(\mathcal{F}))\to \mathrm{\Gamma }(X,G^{\bullet }(\mathcal{G}))$, a map of double complexes $D(\varphi ):D(\mathcal{F})\to D(\mathcal{G})$ by the same componentwise formula, and a cochain map $C^{\bullet }(\mathcal{U},\varphi )$ on Čech cochains; by construction $D(\varphi )\circ u_{\mathcal{F}}=u_{\mathcal{G}}\circ \mathrm{\Gamma }(X,G^{\bullet }(\varphi ))$ and $D(\varphi )\circ w_{\mathcal{F}}=w_{\mathcal{G}}\circ C^{\bullet }(\mathcal{U},\varphi )$, because both sides send a section or a Čech cochain to the family of its restrictions composed with $\varphi$. Taking $H^p$ and inverting the isomorphisms $H^p(u_{\mathcal{F}})$, $H^p(u_{\mathcal{G}})$, and using that $H^p(X,\varphi )$ corresponds to $H^p(\mathrm{\Gamma }(X,G^{\bullet }(\varphi )))$ under the natural identification of [F3], gives ${\phi }_{\mathcal{U}}^p(\mathcal{G})\circ {\check{H}}^p(\mathcal{U},\varphi )=H^p(X,\varphi )\circ {\phi }_{\mathcal{U}}^p(\mathcal{F})$, which is assertion 1.

Let $c:J\to I$ be a refinement function from $\mathcal{V}=(V_j{)}_{j\in J}$ to $\mathcal{U}$ with cochain map $c^{\mathrm{♯}}$ [F5]; the same formula, applied with the coefficient sheaf $G^q$ in place of $\mathcal{F}$, gives cochain maps $c^{\mathrm{♯}}:C^p(\mathcal{U},G^q)\to C^p(\mathcal{V},G^q)$, well defined because the needed inclusions $V_{j_0}\cap \cdots \cap V_{j_p}\subseteq U_{c(j_0)}\cap \cdots \cap U_{c(j_p)}$ hold, and these are natural in the coefficient, so they assemble into a map of double complexes $D(c):D(\mathcal{U})\to D(\mathcal{V})$ commuting with both $h$ and $v$, hence into a map of total complexes. By the definitions of the two augmentations one has $D(c)\circ u_{\mathcal{U}}=u_{\mathcal{V}}$ as maps $\mathrm{\Gamma }(X,G^{\bullet })\to \mathrm{Tot}D(\mathcal{V})$, since a global section restricts to the same family over either cover, and $D(c)\circ w_{\mathcal{U}}=w_{\mathcal{V}}\circ c^{\mathrm{♯}}$, since both sides send a Čech cochain $\alpha$ to the family over $\mathcal{V}$ of the sections $\epsilon (\alpha )$ restricted along the containments. Therefore ${\phi }_{\mathcal{V}}^p\circ H^p(c^{\mathrm{♯}})=H^p(u_{\mathcal{V}}{)}^{-1}{H}^p(w_{\mathcal{V}})H^p(c^{\mathrm{♯}})=H^p(u_{\mathcal{V}}{)}^{-1}{H}^p(D(c))H^p(w_{\mathcal{U}})=H^p(u_{\mathcal{V}}{)}^{-1}{H}^p(u_{\mathcal{V}})H^p(u_{\mathcal{U}}{)}^{-1}{H}^p(w_{\mathcal{U}})={\phi }_{\mathcal{U}}^p$, and by [F6] the outcome is independent of the chosen refinement function, which is assertion 2.

Let $s\in \mathrm{\Gamma }(X,\mathcal{F})$ and view it as an element of $\mathrm{\Gamma }(X,G^0)$ through the augmentation. Both $u(s)$ and $w(s)$ are the element of $D^{0,0}={\prod }_{i\in I}{G}^0(U_i)$ with components $s{\mathrm{\mid }}_{U_i}$, in the first case by [F2] applied to the global section $s$ of $G^0$, in the second case because $\epsilon$ induces the identity of $\mathrm{\Gamma }(X,\mathcal{F})$ on global sections; this element is a cocycle of $\mathrm{Tot}D$ in degree zero, since its Čech differential vanishes by gluing and its vertical differential is $d^0\epsilon (s)=0$. Under the isomorphism ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathrm{\Gamma }(X,\mathcal{F})$ of [F7] the class of $s$ is $H^0(w)$ of the class of $s$ in $H^0(\mathrm{\Gamma }(X,G^{\bullet }))$, and under the isomorphism $H^0(X,\mathcal{F})\cong \mathrm{\Gamma }(X,\mathcal{F})$ of [F8] the class of $s$ corresponds to $H^0(u)$ of the same class; hence ${\phi }_{\mathcal{U}}^0=H^0(u{)}^{-1}{H}^0(w)$ fixes the class of every global section, which is assertion 3.

The comparison map is defined in [step 1.1]; its naturality in the sheaf is [step 2.1], its compatibility with refinement and the independence of the refinement function is [step 2.2], and its degree-zero identification is [step 2.3]. This proves assertions 1, 2 and 3. The assumption of [F9] entered only through the Godement resolution and the quasi-isomorphism statement [F1] for $u$, both of which are theorems of this page under the same hypothesis. ∎