Leray acyclic-cover comparison

The hypothesis is exactly the acyclicity condition of [F4]: every nonempty finite intersection $W$ of members of $\mathcal{U}$ satisfies $H^q(W,\mathcal{F}{\mathrm{\mid }}_W)=0$ for $q>0$. By [F1] the map $w:C^{\bullet }(\mathcal{U},\mathcal{F})\to \mathrm{Tot}D$ is therefore a quasi-isomorphism, and the same statement [F1] gives that $u:\mathrm{\Gamma }(X,G^{\bullet })\to \mathrm{Tot}D$ is a quasi-isomorphism; by [F3] the induced maps $H^p(u)$ and $H^p(w)$ are isomorphisms in every degree.

The comparison map is ${\phi }_{\mathcal{U}}^p=H^p(u{)}^{-1}\circ H^p(w)$ under the identification $H^p(X,\mathcal{F})\cong H^p(\mathrm{\Gamma }(X,G^{\bullet }))$ by [F2]; a composite of two isomorphisms of abelian groups is an isomorphism, and $H^p(u{)}^{-1}$ exists as an isomorphism by [step 1.1]. Hence ${\phi }_{\mathcal{U}}^p$ is an isomorphism for every $p\ge 0$. The listed extra properties are the corresponding assertions of Canonical map from fixed-cover Čech to sheaf cohomology, namely naturality in $\mathcal{F}$, compatibility with refinement together with independence of the refinement function, and the degree-zero identification with the identity of $\mathrm{\Gamma }(X,\mathcal{F})$. ∎