Imported derived sections on a small site

PDF p. 4: Derived Nisnevich and cdh sections

Assume the Axiom of Choice and fixed size bounds, and let $\mathcal{C}$ be a small site with $U$ an object. The category of abelian sheaves on $\mathcal{C}$ is abelian. Evaluation at $U$ has a right derived functor on the unbounded derived category, computed by evaluating a K-injective resolution. For a sheaf $F$ placed in degree zero, this derived value may also be computed by an injective resolution concentrated in nonnegative degrees.

An injective sheaf is an object $I$ such that maps into $I$ extend across monomorphisms. A quasi-isomorphism is a chain map inducing isomorphisms on all cohomology objects. A complex $I^{\bullet }$ is K-injective if every chain map from an acyclic complex to it is chain-homotopic to zero. A K-injective resolution is a quasi-isomorphism into such a complex. The derived category is obtained from the category of complexes by formally inverting quasi-isomorphisms. A right derived functor here means the functor on that localization supplied by the cited construction.

The cited Stacks results supply the abelian category, the unbounded K-injective resolution and derived-functor construction on a ringed site, and the ordinary nonnegative injective-resolution computation. Apply the ringed-site result to the constant integer sheaf: its modules are precisely abelian sheaves. These are imported established results, not locally reconstructed proofs. The hypotheses are a small site, the indicated size conventions and Choice; $U$ must be an object of the site. Lemma 13.31.4 applies to the nonnegative complex of injectives in this abelian category and identifies an ordinary injective resolution as a K-injective resolution. Thus its evaluation computes the same unbounded derived value. The foundational existence proofs remain imported.