Alphabeta Math
TheoremStatement: Literature-sourcedProof: Literature-sourced
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.

External finite-type dimension, rational-point and etale-fiber inputs

PDF p. 15: Theorem6.7: rank-three Bertini functional

Statement

Import the exact finite-type geometric facts in the source record: dimension/codimension and image-closure inequalities for finite-type field schemes; product dimensions; the Jacobson and closed-point residue-field facts; and the implication ΩB/L=0⇒B finite étale for B finite type over a field L. Consequently over algebraically closed L, Spec⁡B is a finite disjoint union of reduced rational points. No characteristic-zero generic-smoothness theorem is part of this input.

Source contract

Given: Finite-type schemes over a field, with the irreducible hypotheses of the dimension statements applied componentwise; a finite-type field algebra for the last implication.

Goal: Import the ordinary geometric foundations used in the locally authored parameter argument.

[F1] Stacks33.20.3–33.20.5 provide the exact dimension facts;33.14.1 provides the closed-point facts.

[F2] Stacks10.143.3(7) and10.143.4 provide the field-algebra differential and finite étale conclusions.

External input

Import [F1]–[F2] with the source record's hypotheses and honest proof-reading limits. These foundations do not import the proposition being authored: finite first-jet generation, the determinant-incidence dimension and simultaneous avoidance are proved in the five local items. In particular a smooth total incidence scheme is not being used to assume separability of its projection.

Depends on

Used by · 13

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

The Stacks Project, finite-type dimensions and image closures · Lemmas33.20.3(2),(4),(6),(10), 33.20.4(2) and33.20.5, full indicated proofs; also Lemma33.20.2 for zero-dimensional context.

Finite-type dimension/codimension is computed by transcendence degrees; image closure has dimension at most the source; product dimensions add. These give dimensions of locally trivial affine bundles and bounds for projected incidence schemes.

Hypotheses: Locally algebraic schemes over a field; source irreducible in33.20.4(2), applied componentwise when necessary; finite-type schemes in this application have finitely many components and finite dimension.

Reading limits: Read actual full Lemmas33.20.2–33.20.5 statements and displayed proofs, including the transcendence-degree proof of33.20.4(2). No reading of all underlying commutative-algebra dimension references is claimed.

The Stacks Project, closed points on finite-type field schemes · Lemma33.14.1, full statement/proof.

Schemes locally of finite type over a field are Jacobson and closed-point residue fields are algebraic over that field. Thus nonempty finite-type opens over algebraically closed k have k-rational closed points.

Hypotheses: Locally finite type over a field; algebraic closure is used only to identify closed-point residue fields with k.

Reading limits: Read the complete actual lemma and its proof reduction to the Hilbert Nullstellensatz. The complete earlier Nullstellensatz proof was not reconstructed.

The Stacks Project, etale finite-type algebras over fields · Definition10.143.1; Lemma10.143.3(7), proof paragraph(7); Lemma10.143.4, full statement/proof. Stable individual tags00U2 and00U3.

For a finite-type algebra B over any field L, vanishing Omega_(B/L) implies B is etale over L. Such an algebra is a finite product of finite separable field extensions. This fixes actual smooth scheme structure, including geometric reducedness, from the differential calculation.

Hypotheses: Finite-type algebra over a field, all characteristics. This is not a claim that Omega=0 implies etale for arbitrary base rings. The empty algebra corresponds to the empty scheme.

Reading limits: Read the complete definitions, statement of10.143.3, proof paragraph(7), and complete proof of10.143.4. Also inspected the complete proof of regular-local Lemma10.106.3 (00NQ) and the etale morphism definitions/fiber statement in Section29.37 (02GH), but those extra passages are not needed as additional imports. General smooth-algebra foundations referenced in paragraph(7) remain imported.

Sources