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Finite refinement for integer cdh sections

PDF p. 5: Finite refinements and integer sections

Statement

On the specified site of separated finite-type schemes over a field $k$ , every cdh covering sieve $S$ on $X$ contains a finite covering family ${U_{i} \to X}$ . Its finite coproduct $W = ∐_{i} U_{i} \to X$ is a covering morphism and has an MVW two-stage refinement $V \overset{q}{\to} X^{'} \overset{p}{\to} X,$ where $p$ is completely decomposed proper and $q$ is Nisnevich, with a map $h : V \to W$ over $X$ . Splitting $V$ into the finite clopen pieces $V_{i} = h^{- 1} (U_{i})$ gives a finite Nisnevich family over $X^{'}$ whose composites to $X$ all belong to $S$ . Neither $W \to X$ nor $V \to X$ is asserted to belong to $S$ .

Every section of the constant integer cdh sheaf has finitely many constant local representatives on such a finite covering family, and therefore factors through the sheafification of a finite subset of $Z$ . If desired the above two-stage refinement may be chosen for that representing family, retaining its finitely many representative values on the split pieces. No conclusion for an arbitrary qcqs base outside this site is asserted.

Facts & Assumptions

Given: The exact finite-type cdh site, a covering sieve $S$ , and, for the last assertion, $s \in H_{c d h}^{0} (X, Z)$ with $Z$ meaning the sheafification of the constant presheaf.

Goal: Prove finite refinement and finite local values without a coproduct-in-sieve assumption.

[F1]

Finite subfamilies of Nisnevich covers on the finite-type site, Pullback and composition of the cdh generating families and A finite-family basis for the generated cdh topology prove that covering sieves contain finite composite generating families.

[F2]

Finite coproducts and refinement inside a sieve proves that their coproduct is a covering morphism and explains how any two-stage refinement is split back into arrows of the original sieve.

[F3]

External MVW proper-cover identification and two-stage refinement imports MVW12.26/12.28 and identifies its topology with the specified one, under exactly these hypotheses.

[F4]

Constant-sheaf sections have finite local representatives supplies local representability from the actual two-plus sheafification and the finite constant-set factorization. Scheme sites and sheafification fixes that convention.

Proof

1.1F1F2F3given

Apply [F1] to $S$ to obtain a finite $B$ -family ${U_{i} \to X}$ whose arrows individually lie in $S$ . By [F2] its finite coproduct $W \to X$ is a cdh covering morphism. All schemes remain separated of finite type over $k$ . Therefore [F3] applies to this morphism and gives $V \to X^{'} \to X$ and $h : V \to W$ with the claimed proper and Nisnevich conditions. Lemma12.26 in [F3] identifies that proper cover as completely decomposed.

2.1F2step 1.1

Apply [F2] to $h$ : the $V_{i}$ form a finite clopen decomposition of $V$ , their maps to $X^{'}$ are a Nisnevich family, and their composites to $X$ factor through the corresponding $U_{i} \to X$ . Those composites consequently lie in $S$ . This proves the sieve-level two-stage refinement in its precise family sense; no coproduct arrow has been added to $S$ .

3.1F1F2F4step 1.1step 2.1∎

For an integer section use [F4] to obtain its covering sieve of constant representatives, then a finite $B$ -family in that sieve. The finitely many representatives $n_{i}$ give its factorization through the constant-set sheaf of $D = {n_{i}}$ . Apply steps 1.1 and 2.1 to this family if a two-stage form is desired. Since $V_{i} \to X$ factors through $U_{i}$ , the representative on $V_{i}$ remains $n_{i}$ ; no new values appear. This proves the finite-values assertion and completes the proposition with the original cover-refinement claim preserved.

Depends on

Used by · 1

Integer cdh sections on seminormal schemes

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · Lemma2.4 proof, printed p.5, final paragraph