LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The actual smooth open pair and its choice scope

PDF p. 13: Proposition 6.1 support argument

Statement

Assume AC. Let $X$ be an affine finite-type threefold over algebraically closed $k$ , let $U = X_{s m o o t h}$ , and let $T \subset X$ be closed with $T \subset U$ . Put $V = U ∖ T$ and $W = X ∖ T$ . All four schemes are qcqs and $U, V$ are smooth separated finite type over the perfect field $k$ . They are regular Noetherian as schemes over the prime field. The two opens $U, W$ cover $X$ and their intersection is $V$ , giving a Nisnevich two-open covering square. If $T$ is an integral curve, it is affine; it lies in one of the open-and-closed relative-dimension strata of $U$ .

Facts & Assumptions

Given: The hypotheses in the statement, with the Axiom of Choice and fixed universe bounds.

Goal: Construct the stated actual maps and prove their consequence.

[F1]

Article scheme conventions, Motivic and filtered K theory with support and Nisnevich covers and distinguished squares fix the finite-type, size, AC and two-open conventions. Basic smoothness here includes its regularity over a perfect field and locally constant relative dimension.

Proof

1.1F1

Finite-type affine $X / k$ is Noetherian and separated. Every open is quasi-compact in a Noetherian space, and its induced scheme is separated finite type. Thus the four objects lie in the completed qcqs category. The smooth locus and its open $V$ are smooth over $k$ , hence regular Noetherian. An algebraically closed field is perfect: in positive characteristic each polynomial $t^{p} - a$ has a root, so Frobenius is surjective; characteristic zero is perfect. Forgetting the map to $k$ does not change regularity or Noetherianness.

2.1F1step 1.1∎

The inclusion $T \subset U$ gives $X = U \cup W$ and $U \cap W = V$ . Open immersions are étale, and each point has a residue-preserving lift to one of these opens, so the family is Nisnevich. A closed subscheme of affine $X$ is affine, giving the curve assertion. For a smooth morphism to a field the relative dimension is locally constant; the finitely many dimension strata of $U$ are open and closed. The connected integral curve lies in one such stratum. These are component/dimension bookkeeping facts, with no new comparison theorem assumed.

Depends on

Used by · 4

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