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Every qcqs cdh generator is an arc covering family

PDF p. 6: Proposition 2.6 proof

Statement

On the completed qcqs prime-field category every Nisnevich covering family and every abstract blowup family {Z→X,Y→X} is an arc cover. Here Z→X is a finitely presented closed immersion, Y→X is finitely presented proper and an isomorphism outside Z. Consequently the source cdh topology is coarser than the arc topology, including on arbitrary nonnoetherian objects.

Facts & Assumptions

Given: The hypotheses in the statement.

Goal: Establish the stated exact map or descent conclusion.

[F1]

Exact valuation inputs for the topology comparison supplies the exact valuative facts. [F2] The source qcqs cdh site fixes the cdh generators; the arc definition is recorded in that input.

Proof

1.1F1F2

For a Nisnevich family and a rank-at-most-one valuation ring V mapping to X, choose a member with a point over the image of the closed point; its pullback to Spec⁡V is étale and has a point over the closed point. Choose an affine neighborhood Spec⁡B of that point. By the exact valuation input [F1], its local ring W=Bm is an extension of valuation rings with the same value group as V. It therefore has the same rank, and Spec⁡W supplies the required lift into that cover member. For a family on a qcqs object take a finite subfamily, which exists since étale images are open.

1.2F1F2

For an abstract blowup family, if the generic point of Spec⁡V maps into Z, the pulled-back ideal is zero in Frac⁡V and therefore zero in the domain V, so the entire map factors through Z. Otherwise the generic point has the unique lift to Y over the open complement. Properness and the valuative criterion in [F1] extend it to Spec⁡V→Y. Thus the abstract family even permits the identity extension V→V.

2.1F1F2step 1.1step 1.2∎

These valuation tests are the defining tests for arc covers; the source topology is generated by the listed families. Both tests are stable under the source base changes, so every generated cdh covering sieve is arc covering. No finite-type or dimension bound on X was used.

Depends on

Used by · 2

Dependency tree · 3 levels

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