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Restriction of hyperdescent and arbitrary abstract blowup squares

PDF p. 6: Proposition 2.6 proof

Statement

Let $F = Q$ or $F = F_{p}$ for a prime $p$ , let $Λ$ be a finite commutative ring, and put $E_{Λ} (X) = R Γ_{e t} (X, {\underline{Λ}}_{X})$ , with its actual derived pullback maps and the constant coefficient sheaf in degree zero. On the entire site of quasi-compact quasi-separated $F$ -schemes, with the specified small-skeleton universe convention, $E_{Λ}$ satisfies cdh hyperdescent and Nisnevich descent. Here the cdh topology is generated by Nisnevich covers and abstract blowup families ${Z \to X, Y \to X}$ with $Z \to X$ a finitely presented closed immersion and $Y \to X$ finitely presented proper, inducing an isomorphism away from $Z$ . The functor sends every such abstract blowup square to a cartesian square in $D (Λ)$ and takes the empty scheme to zero. No invertibility assumption on the order of $Λ$ is imposed.

Facts & Assumptions

Given: The prime field $F$ , finite commutative ring $Λ$ , actual étale cohomology functor $E_{Λ}$ , and exact qcqs site in the statement.

Goal: Establish hyperdescent, Nisnevich descent, abstract blowup square descent and the zero empty-scheme value for this functor.

[F1]

The universe convention and the exact qcqs prime-field site are fixed in The source qcqs cdh site.

[F2]

Every specified cdh generator is an arc cover, so the source cdh topology is coarser than the arc topology. Every qcqs cdh generator is an arc covering family

[F3]

For finite commutative $Λ$ , the actual complex $E_{Λ}$ with its derived pullbacks satisfies arc hyperdescent on all qcqs schemes. From constructible-category hyperdescent to the actual cohomology complex

[F4]

The EM qcqs-square criterion, with the necessary zero value on the empty scheme, is imported in Exact hypercomplete arc descent for finite constructible coefficients. This criterion concerns ordinary sheaves and is used after hyperdescent has been established.

Proof

1.1F1F2F3

The source category of qcqs $F$ -schemes is stable under the fiber products in its covers and hypercovers. In a cdh hypercover each matching map is locally surjective for the cdh topology. By the generator comparison the same map is locally surjective for the finer arc topology. Thus it is an arc hypercover; apply the endomorphism-hyperdescent supplier to this diagram of underlying schemes, with the finite ring $Λ$ fixed above. This proves cdh hyperdescent for the actual functor and its pullback maps, and in particular Nisnevich descent.

2.1F1F4step 1.1∎

In particular the functor is an ordinary cdh sheaf. The étale topos of the empty scheme has only the zero sheaf of $Λ$ -modules, so $E_{Λ} (\emptyset) = 0$ . The qcqs-square criterion therefore applies to its underlying $D (Z)$ -valued functor and makes every specified abstract blowup square cartesian. Restriction of scalars $D (Λ) \to D (Z)$ preserves limits, being a right adjoint, and detects equivalences on cohomology, so the square is also cartesian in $D (Λ)$ . This supplies all the source abstract squares with both finite-presentation conventions; no finite-type or separatedness hypothesis on the schemes, or restriction to ordinary ideal blowups, has been introduced.

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