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Regularity and dimensions for finite-type cycle arguments

PDF p. 13: Proof of Proposition 6.1: ordinary CH3 identification and degree-seven vanishing

Statement

Let k be a field. A smooth k-scheme is regular, and every regular local ring is a domain. If T is an integral finite-type k-scheme and D⊂T is integral and closed, then dim⁡T=trdeg⁡kk(T) and codim⁡TD=dim⁡T−dim⁡D. Every nonempty open of T has the same dimension as T. Every irreducible component of an effective Cartier divisor on T has dimension dim⁡T−1.

If p:P→Q is a proper dominant morphism of integral finite-type k-schemes, then dim⁡P=dim⁡Q+trdeg⁡k(Q)k(P). None of these assertions requires separatedness over k.

External input

Import the stated regularity, domain and dimension results from the specified Stacks lemmas. The affine dimension and local codimension formulas apply on affine neighbourhoods of the relevant generic points; all nonempty affine charts of an integral scheme have the same function field. Their formulas therefore retain their scope on nonseparated schemes. The field base satisfies Situation 42.7.1. The application uses only finite-type schemes. The cited source statements and indicated proofs were read; their deeper prerequisites are accepted external inputs, not reconstructed local proofs. The version is the live tag text retrieved on 2026-11-09; no pinned repository commit is asserted.

Depends on

Used by · 1

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

Stacks Project, smooth schemes over fields · Lemma 33.25.3; live tag text inspected 2026-11-09

A smooth scheme over a field is regular.

Hypotheses: Any field; no separatedness assumption.

Reading limits: Read the complete statement and its short proof; the referenced algebraic regularity criterion was not reconstructed.

Stacks Project, regular local rings · Lemma 10.106.2; live section text inspected 2026-11-09

Every regular local ring is a domain.

Hypotheses: Regular local means noetherian regular local.

Reading limits: Read the lemma and its displayed proof; the associated-graded and intersection prerequisites were not reconstructed.

Stacks Project, finite-type dimension formulas · Lemmas 10.116.1-10.116.3; live section text inspected 2026-11-09

For finite-type domains over a field, dimension is transcendence degree; specialization strictly lowers it, and the local dimension formula identifies codimension with the dimension difference.

Hypotheses: Finite-type algebras over any field.

Reading limits: Read these three statements and proofs; their normalization and maximal-chain prerequisites were not reconstructed.

Stacks Project, dimension formula for a proper dominant map · Lemma 29.53.4; live tag text inspected 2026-11-09

For a closed dominant locally finite-type morphism of integral schemes with locally noetherian target, dimension equals target dimension plus function-field transcendence degree.

Hypotheses: Proper maps in the application are closed; all schemes are finite type over a field.

Reading limits: Read the complete statement and proof and the statement and short proof of its cited Lemma 29.53.1 (02JU); deeper algebraic dimension prerequisites were not reconstructed.

Stacks Project, dimensions of Cartier divisor components · Lemma 42.16.1; live tag text inspected 2026-11-09

Components of an effective Cartier divisor on an integral scheme have dimension one less in the base dimension function.

Hypotheses: Situation 42.7.1, applied to Spec k with dimension function zero; finite-type integral schemes.

Reading limits: Read the complete lemma and proof; the principal ideal dimension theorem remains an imported prerequisite.

Sources