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External normality and perfect-field smooth-locus inputs

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

Statement

Import the two source results recorded above: Noetherian normal schemes decompose into their finitely many clopen integral normal components and satisfy (R1); reduced finite-type schemes over a perfect field have identical regular and smooth loci, and that locus is open. Here codimension of an irreducible closed subset with generic point x is dim⁡OX,x, as in the cited Serre-condition convention.

Source contract

Given: A Noetherian normal scheme; for the smooth-locus assertion a reduced locally algebraic scheme over a perfect field.

Goal: Import the exact geometric consequences of normality used before constructing parameters.

[F1] Stacks Lemma28.7.5 and Lemma28.12.5 give decomposition and (R1).

[F2] Stacks Lemma33.25.8 gives the stated equality and openness over a perfect field.

External input

Import [F1]–[F2] with their exact hypotheses and reading limits. The five local lemmas apply these foundations and carry their perfectness/Noetherian conditions; they do not purport to prove the whole normality criterion or geometric-regularity theory.

Depends on

Used by · 6

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, normal components and Serre normality criterion · Lemma28.7.5 (0357), complete statement/proof; also Section28.12, Lemma28.12.5, https://stacks.math.columbia.edu/tag/033P

A normal scheme whose quasi-compact opens have finitely many irreducible components is a disjoint union of integral normal schemes. A locally Noetherian normal scheme has Serre condition(R1).

Hypotheses: The component decomposition needs the stated local finiteness of components, satisfied by a Noetherian scheme; the Serre criterion is locally Noetherian. No characteristic restriction.

Reading limits: Read the actual full Lemma28.7.5 and proof and the actual Section28.12 definition of(R1), codimension convention, and Lemma28.12.5 with its one-line proof reference. The commutative-algebra proof cited by that criterion was not newly reconstructed.

The Stacks Project, regular and smooth loci over perfect fields · Lemma33.25.8 and its complete proof; Lemmas33.25.3–33.25.4 for smoothness/regularity context.

For a reduced locally algebraic scheme over a perfect field, its smooth and regular loci coincide and form a dense open subscheme.

Hypotheses: Perfect field and reduced locally algebraic scheme. Algebraically closed k is perfect; normal schemes here are reduced and finite type. This assertion is not extended to imperfect fields.

Reading limits: Read the full indicated statement and printed proof, including its exact perfect-field hypothesis. Also read the section warning about nonperfect fields and positive-characteristic differential criteria. No claim to reconstruct all cited geometric-regularity foundations.

Sources