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Integral components of a smooth affine pure-dimensional scheme

Statement

Assume AC. Let $k$ be a field and $X$ a nonempty smooth affine finite-type $k$ -scheme whose irreducible components all have dimension $d$ , where $d \geq 0$ is an integer. Dimension is the supremum of lengths of strict chains of nonempty irreducible closed subsets, with a one-member chain of length zero. Then $X$ is a finite disjoint union $X = ⨆_{a = 1}^{m} X_{a}$ , $m \geq 1$ , of nonempty open and closed integral smooth affine finite-type $k$ -schemes of dimension $d$ . These pieces are precisely its irreducible components with their open subscheme structures. Here integral means reduced and irreducible, and open and closed means that both a subset and its complement are open.

External input

Two established geometric facts are used: smooth schemes over a field are normal (Stacks Project, Lemma 33.25.4, tag 056T), and a normal Noetherian scheme is a finite disjoint union of integral normal schemes (Lemma 28.7.6, tag 033M; its component identification is explained in Lemma 28.7.5). Normal means that every local ring is a domain integrally closed in its fraction field. The finite-type and affine refinements below are proved locally from the declared library interfaces.

Facts & Assumptions

Given: AC, $k$ , $X$ , $d$ and the dimension convention of the statement.

Goal: Produce the asserted finite disjoint integral affine decomposition.

[F1]

The field, base-scheme, affine, smooth and finite-type conventions are supplied by Field, Schemes and morphisms over a base, Affine schemes and their coordinate rings, Smooth morphism of schemes and Locally finite type and finite type morphisms. Smoothness is preserved on open subschemes. Irreducible components are maximal irreducible subsets, and integral schemes are nonempty reduced irreducible schemes. Irreducible components of a topological space, Integral schemes

[F2]

Under AC, an affine finite-type scheme $X = Spec A$ over a field has finitely generated coordinate $k$ -algebra $A$ . A field is Noetherian; a finitely generated algebra over a Noetherian ring is Noetherian; a single affine chart with Noetherian coordinate ring makes a scheme Noetherian. An affine scheme of finite type over a field has a finitely generated coordinate ring, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes

[F3]

The two imported normality and decomposition facts are precisely those recorded in External input. They impose no characteristic restriction. For a Noetherian normal scheme its irreducible components are integral open and closed subschemes, and there are only finitely many of them.

[F4]

Under AC, a clopen subset of $Spec A$ is $D (1 - e)$ for an idempotent $e \in A$ , meaning $e^{2} = e$ . The open subscheme $D (f)$ is $Spec A_{f}$ , where $A_{f}$ denotes localization by the powers of $f$ . Under AC every affine scheme is quasi-compact. If $A = k [b_{1}, \dots, b_{s}]$ , this localization is generated over $k$ by the images of the $b_{i}$ and $f^{- 1}$ . A clopen decomposition of the spectrum comes from a nontrivial idempotent, A principal localization identifies its spectrum with a distinguished open, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Every affine scheme is quasi-compact

[A1]

AC is inherited by the affine-coordinate and clopen-idempotent suppliers. The Axiom of Choice

Proof

1.1F1F2F3A1given

Write $X = Spec A$ by [F1]. By [F2], $A$ is a finitely generated $k$ -algebra and is Noetherian, so $X$ is Noetherian. Smoothness and the first imported fact in [F3] make $X$ normal. The second imported fact decomposes $X$ into finitely many nonempty integral open subschemes $X_{a}$ , exactly its irreducible components. Each complement is the union of the other open pieces, hence each piece is also closed. Nonemptiness of $X$ makes the number $m$ of pieces positive.

2.1F1F4A1step 1.1given∎

Fix a piece $X_{a}$ . By [F4] its clopen underlying subset is $D (1 - e_{a})$ for an idempotent $e_{a} \in A$ , and its open scheme structure is $Spec A_{1 - e_{a}}$ . If $b_{1}, \dots, b_{s}$ generate $A$ over $k$ , their images together with $(1 - e_{a})^{- 1}$ generate this localization, so it is finite type over $k$ ; it is locally of finite type on this chart and is quasi-compact by [F4], hence finite type by [F1]. Restricting the original smooth structure morphism gives smoothness by [F1]. Its underlying space is the original irreducible component with the subspace topology, so its chain dimension is $d$ by the given purity assumption. Its integrality was obtained in step 1.1. This proves all the claimed properties, for every piece.

Depends on

Used by · 1

Mohan Kumar–Murthy rank-three Chow classification

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

The Stacks Project, Schemes smooth over fields · Lemma 33.25.4 (056T), statement and proof.

A scheme smooth over a field is geometrically regular, geometrically normal and geometrically reduced; in particular it is normal.

Hypotheses: A smooth morphism X to Spec(k), k any field; no irreducibility, connectedness or characteristic restriction.

Reading limits: Read the complete displayed lemma and its short proof on the tag page. Only normality over k is used; the cited algebraic proofs inside Stacks were not reconstructed.

The Stacks Project, Normal schemes · Definition 28.7.1; Lemma 28.7.5 and its proof; Lemma 28.7.6 (tag 033M) and its proof.

A normal Noetherian scheme is a finite disjoint union of integral normal schemes; the summands are its irreducible components with open scheme structures. Normality means all local rings are integrally closed domains.

Hypotheses: The decomposition is applied only to a Noetherian normal scheme. The more general Lemma 28.7.5 requires finitely many irreducible components on every quasi-compact open.

Reading limits: Read Definition 28.7.1 and Lemmas 28.7.5–28.7.6 with their proofs in the section, and the separate tag 033M statement and proof. The cited commutative algebra decomposition proof was not reconstructed.

Sources

The Stacks Project, Schemes smooth over fields · Lemma 33.25.4 (056T), statement and proof.
The Stacks Project, Normal schemes · Definition 28.7.1; Lemma 28.7.5 and its proof; Lemma 28.7.6 (tag 033M) and its proof.