Alphabeta MathHow 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.
External geometric inputs for affine reduction
PDF p. 4: Smooth comparison with Bloch cycles
Statement
A scheme smooth over a field is regular and normal. If it is also noetherian, its irreducible components are disjoint normal integral open-and-closed subschemes. Consequently a smooth finite-type scheme over a field admits a cover by integral smooth affine opens, each separated and equidimensional.
External input
Import the two cited Stacks results with their stated hypotheses. A noetherian space has finitely many irreducible components, so their open complements are finite unions of closed components. Affine opens inside the components give the stated cover: smoothness restricts to opens, affine schemes are separated, and an integral scheme has one irreducible component. This last sentence is the application of the imported results.
Depends on
Dependency tree · 1 level
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.
External dependencies
A scheme smooth over a field is regular and geometrically normal, hence normal.
Hypotheses: X is smooth over a field k.
Reading limits: Read both statements and their displayed proofs; the linked commutative-algebra inputs were not reconstructed.
A normal scheme whose quasi-compact opens have finitely many irreducible components is a disjoint union of normal integral schemes.
Hypotheses: Normality and the stated finiteness condition on quasi-compact opens.
Reading limits: Read the statement and displayed proof; its affine normal-ring input was not reconstructed.
Sources
- The Stacks Project, Schemes smooth over fields · Lemmas 33.25.3 and 33.25.4.
- The Stacks Project, Normal schemes · Lemma 28.7.5.