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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 is an arc cover. Here is a finitely presented closed immersion, is finitely presented proper and an isomorphism outside . 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.
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
For a Nisnevich family and a rank-at-most-one valuation ring mapping to , choose a member with a point over the image of the closed point; its pullback to is étale and has a point over the closed point. Choose an affine neighborhood of that point. By the exact valuation input [F1], its local ring is an extension of valuation rings with the same value group as . It therefore has the same rank, and 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.
For an abstract blowup family, if the generic point of maps into , the pulled-back ideal is zero in and therefore zero in the domain , so the entire map factors through . Otherwise the generic point has the unique lift to over the open complement. Properness and the valuative criterion in [F1] extend it to . Thus the abstract family even permits the identity extension .
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 was used.
Depends on
Dependency tree · 3 levels
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