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A finite-family basis for the generated cdh topology

PDF p. 5: Finite refinements and integer sections

Statement

Let $B$ consist of finite families obtained from finite Nisnevich families and singleton cdp maps by finitely many covering-family compositions, including identities and the empty family over the empty scheme. A sieve is cdh covering exactly when it contains a family in $B$ . In particular every covering sieve contains a finite covering family of arrows individually belonging to it. A sieve is closed under precomposition; it is not assumed closed under taking coproducts of arrows.

Facts & Assumptions

Given: The site and its topology generated by the two kinds of families, as in Completely decomposed proper covers and the cdh topology.

Goal: Identify its covering sieves by an explicitly finite basis.

[F1]

Finite subfamilies of Nisnevich covers on the finite-type site reduces every generating Nisnevich family to a finite subfamily.

[F2]

Pullback and composition of the cdh generating families proves the base-change/composition closure of $B$ ; composites of finite families are finite.

[F3]

The sieve axioms for a topology and the meaning of least generated topology are those of Scheme sites and sheafification.

Proof

1.1F2F3

Define $J_{B} (X)$ to be the sieves containing a $B$ -family. The identity family proves maximality, and pulling back the contained family proves pullback stability by [F2]. For transitivity, let $S$ contain a finite $B$ -family ${f_{i} : X_{i} \to X}$ and let $R$ be a sieve such that $f^{*} R$ is in $J_{B}$ for every $f \in S$ . For each of these finitely many $f_{i}$ choose a finite $B$ -family ${g_{i j}}$ in $f_{i}^{*} R$ . The finite composite family ${f_{i} g_{i j}}$ belongs to $B$ and to $R$ . Hence $R \in J_{B}$ . Thus $J_{B}$ is a topology.

2.1F1F2F3step 1.1∎

It contains every generating cdp sieve and every generating Nisnevich sieve by [F1]. Therefore the least generated cdh topology is contained in $J_{B}$ . Conversely any topology containing the generators contains every $B$ -family sieve by repeated transitivity, and hence every sieve containing such a family by upward closure (itself an immediate consequence of transitivity). Apply this to the cdh topology to get the reverse inclusion. The equivalence and finiteness assertion follow. Only individual family arrows have been placed in the sieve.

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