Exact linear hyperdescent and derived-localization input

PDF p. 6: Proposition 2.6 proof

For a small Grothendieck site $\mathcal{C}$ and a commutative unital ring $\mathrm{\Lambda }$, the localization of presheaves of unbounded complexes of $\mathrm{\Lambda }$-modules at maps inducing isomorphisms on all associated cohomology sheaves has as its local objects exactly the objects satisfying descent for every hypercover. Its homotopy category is $D(\mathrm{Sh}(\mathcal{C},\mathrm{\Lambda }))$. Equivalently its reflector is $Ria$, represented by evaluating a K-injective sheaf resolution after exact degreewise sheafification. Hypercover descent means that evaluation on an augmented hypercover is the derived product totalization of the cosimplicial evaluation diagram.

Import the stated established hypercover criterion and derived-sheaf model comparison. The source uses homological complexes; reindexing to cochains converts its homology sheaves into cohomology sheaves without changing which maps are local equivalences. The completed general site-derived input supplies K-injective resolutions on the chosen small site.