Derived global sections on a scheme site

PDF p. 4: Derived Nisnevich and cdh sections

Assume the Axiom of Choice and fixed universe size bounds. For each $\tau \in \{\mathrm{N}\mathrm{i}\mathrm{s},\mathrm{c}\mathrm{d}\mathrm{h},\mathrm{e}\mathrm{t}\}$, fix a small scheme site ${\mathcal{C}}_{\tau }$ equipped with the indicated Nisnevich, cdh, or étale topology, and an object represented by the scheme $X$. The choice of site and topology is part of the data; these names alone do not identify a big site with a small one.

An abelian sheaf is a contravariant functor from the site to abelian groups whose underlying set-valued presheaf satisfies the sheaf condition. Let $D(\mathrm{Ab}({\mathcal{C}}_{\tau }))$ be its unbounded derived category: complexes are localized at the chain maps inducing isomorphisms on all cohomology sheaves. For $C$ in this category define $R{\mathrm{\Gamma }}_{\tau }(X,C)$ as the right derived functor of evaluation at $X$, with value in the derived category $D(\mathrm{Ab})$ of complexes of abelian groups. Concretely, for a representative complex choose a quasi-isomorphism to a K-injective complex $I^{\bullet }$ and take the derived-category class of $I^{\bullet }(X)$. K-injective means that every chain map from an acyclic complex to $I^{\bullet }$ is chain-homotopic to zero. Set $H_{\tau }^n(X,C)=H^n(R{\mathrm{\Gamma }}_{\tau }(X,C))$, where $H^n$ is the kernel of the degree-$n$ differential modulo the preceding image.

For an ordinary abelian sheaf $F$ regarded as a complex concentrated in degree zero, $H_{\tau }^n(X,F)=0$ for $n<0$. This convention does not assert a comparison between the three topologies or an identification between sheafification and derived localization.

Scheme sites and sheafification

Imported derived sections on a small site

The already stated Choice premise is inherited by the selected K-injective construction and its derived evaluation. The Axiom of Choice

The chosen site and object make evaluation meaningful, and the size bounds specify the ambient categories. Imported derived sections on a small site supplies the abelian category, existence and choice-independent derived-functor interpretation of K-injective evaluation, and a nonnegative injective resolution computing the value for $F[0]$. Applying Negative cohomology of a nonnegative evaluated complex to that resolution proves the negative-degree assertion. These inputs apply to each chosen topology separately; no comparison among them is used.