Derived cdh localization

PDF p. 6: Lcdh and truncation in Proposition 2.6

Assume the Axiom of Choice and the fixed universe bounds of Derived global sections on a scheme site. Fix the small cdh site $\mathcal{C}$ of Completely decomposed proper covers and the cdh topology. A presheaf of abelian groups is a contravariant functor $\mathcal{C}\to \mathrm{Ab}$. A complex-valued presheaf means a cochain complex of such presheaves. Its derived category $D({\mathrm{PSh}}_{\mathrm{Ab}}(\mathcal{C}))$ is obtained by inverting objectwise quasi-isomorphisms: maps inducing isomorphisms on cohomology at every object of $\mathcal{C}$.

Let $a$ be cdh sheafification of abelian presheaves and $i$ the inclusion of abelian sheaves into abelian presheaves. For a complex $C$, apply $a$ degreewise, choose a quasi-isomorphism $aC\to I$ to a K-injective complex of cdh sheaves, and define $L_{\mathrm{c}\mathrm{d}\mathrm{h}}C$ to be the object represented by the underlying presheaf complex $iI$. A K-injective complex is one for which every chain map from an acyclic complex is chain-homotopic to zero. The unit $C\to L_{\mathrm{c}\mathrm{d}\mathrm{h}}C$ is induced by $C\to iaC\to iI$ in the derived category of presheaves.

A morphism of complexes is a cdh local equivalence if its degreewise sheafification is a quasi-isomorphism of complexes of cdh sheaves. An object $E$ of the derived presheaf category is cdh-local if, for every cdh local equivalence $u:A\to B$, precomposition with $u$ induces a bijection $\mathrm{Hom}(B,E)\to \mathrm{Hom}(A,E)$. Equivalently, $E$ is cdh-local exactly when its unit $E\to L_{\mathrm{c}\mathrm{d}\mathrm{h}}E$ is an isomorphism. Here equivalence means isomorphism in the indicated derived category, not an objectwise isomorphism of complexes.

For $X$ an object of the chosen site, evaluation of $L_{\mathrm{c}\mathrm{d}\mathrm{h}}C$ at $X$ is represented by $I(X)$ and hence is $R{\mathrm{\Gamma }}_{\mathrm{c}\mathrm{d}\mathrm{h}}(X,aC)$ in $D(\mathrm{Ab})$ as defined in Derived global sections on a scheme site. This convention uses unbounded derived sheaves and their derived sections; it is the hypercomplete derived-sheaf convention. No assertion is made that ordinary Čech descent alone implies locality. The truncation ${\tau }^{\le n}C$ is the objectwise cohomological truncation, with degree $n$ equal to the kernel of $d^n$, degrees below $n$ unchanged and degrees above $n$ zero. Sheafification of cohomology presheaves, derived localization, truncation and evaluation are distinct operations; a proposed interchange or comparison with another topology requires its own hypotheses.

The already stated Choice premise is used through the imported K-injective resolution-existence construction in Imported derived sections on a small site. The Axiom of Choice

The cdh site and its topology are fixed by Completely decomposed proper covers and the cdh topology. The plus construction and its universal property in Scheme sites and sheafification apply to abelian-group objects: matching families, restrictions and equivalence on a common covering subsieve are compatible with addition. Thus $a$ is left adjoint to $i$ on abelian groups. It is exact. Indeed, at each object the plus construction is a filtered colimit, over covering sieves ordered by refinement, of matching-family groups. Matching families preserve finite limits, and filtered colimits of abelian groups commute with finite limits; hence $a$ preserves kernels. If a map of presheaves is objectwise surjective, every section of the sheafification of its target locally lifts: represent the section locally by a target-presheaf section and lift that section objectwise. A locally surjective map of sheaves is an epimorphism, since two sheaf morphisms agreeing after local lifts agree on a cover and therefore agree globally. Applying this to the objectwise cokernel factorization shows that $a$ also preserves cokernels.

The supplier Imported derived sections on a small site gives K-injective resolutions of unbounded complexes of abelian sheaves under the stated size and choice assumptions. Since $a$ is exact, it sends acyclic presheaf complexes to acyclic sheaf complexes. The ordinary adjunction therefore identifies chain maps from an acyclic presheaf complex to $iI$, up to homotopy, with those from its sheafification to $I$; these vanish. Thus $iI$ is K-injective as a presheaf complex. The adjunction on complexes consequently induces the derived adjunction $a:D({\mathrm{PSh}}_{\mathrm{Ab}}(\mathcal{C}))\rightleftarrows D({\mathrm{Sh}}_{\mathrm{Ab}}(\mathcal{C})):Ri$, where $Ri$ is represented by $iI$. The counit $aRi\to \mathrm{i}\mathrm{d}$ is an isomorphism, because $aiI=I$ and $aC\to I$ is a quasi-isomorphism. Hence $Ri$ is fully faithful and $Ria$ is a localization, independent up to unique isomorphism of the chosen resolution.

For a local equivalence $u$, the derived adjunction makes precomposition by $u$ bijective on maps into any $RiF$. Conversely, the unit of any object is a local equivalence because applying $a$ identifies it with the resolution map $aC\to I$. If an object is local, applying its defining bijection to its unit produces a retraction; naturality of the unit and the isomorphism of the unit on objects in the image of $Ri$ show that this retraction is also an inverse. This proves the stated characterization. Finally, evaluation of $iI$ at $X$ is $I(X)$, and Derived global sections on a scheme site identifies it with derived cdh sections. The definition makes no comparison with Čech descent or with another site.