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Seminormalization and the finite-type interface

PDF p. 5: Seminormalization in Lemma 2.4

Statement

For a scheme $X$ , the category of universal homeomorphisms $Y \to X$ inducing isomorphisms on residue fields has an initial object $X^{s n} \to X$ , with seminormal source; its construction is functorial in $X$ . A scheme is seminormal precisely when every such morphism with reduced source is an isomorphism. If $X$ is locally of finite type over a field, then $X^{s n} \to X_{r e d}$ is finite, and a normal $X$ is seminormal. Here seminormal means that every point has an affine neighbourhood whose ring has the unique square-cube root property stated in the interface definition.

External input

The universal construction, its functoriality, the characterization and normality implication are imported from the cited Stacks passages. For finite type over a field, the cited relative construction places the seminormalization algebra inside the finite normalization algebra on affine opens; those rings are noetherian, so the intermediate algebra is finite. The reduction has the same underlying points and residue fields as $X$ . This input is not a locally reconstructed proof of the cited results.

Depends on

Used by · 1

Seminormality and seminormalization

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, Absolute weak normalization and seminormalization · Lemma 29.48.7 (3)–(4) and its functoriality assertion

The category of universal homeomorphisms to a scheme inducing residue-field isomorphisms has an initial object with seminormal source, functorial in the scheme; an object is seminormal exactly when the initial arrow to it is an isomorphism.

Hypotheses: The base is an arbitrary scheme; seminormality is the affine-local square-cube condition.

Reading limits: Statement and displayed proof inspected; the source explicitly omits the proof of the seminormal parts (3) and (4). No local reconstruction is claimed.

The Stacks Project, Weak normalization · Lemmas 29.56.3 and 29.56.7, and the implication following Definition 29.56.9

The relative seminormalization is constructed by intermediate subalgebras of the normalization; it equals the absolute seminormalization when quasi-compact opens have finitely many irreducible components. Normal schemes in that scope are seminormal.

Hypotheses: For the comparison, every quasi-compact open has finitely many irreducible components; the relative construction uses a quasi-compact, quasi-separated, dominant morphism.

Reading limits: The stated passages and their displayed proofs were inspected; the proof of the residue-field version in Lemma 29.56.3 is omitted in the source. The wider normalization theory was not reconstructed.

The Stacks Project, Normalization · Lemma 33.27.1(1)–(2)

The normalization of a scheme locally of finite type over a field is finite and factors through its reduction.

Hypotheses: The scheme is locally of finite type over a field.

Reading limits: The statement and displayed proof were inspected; its cited prerequisites were not independently reconstructed.

The Stacks Project, Absolute weak normalization and seminormalization · Lemma 29.48.9

A scheme is seminormal if and only if every universal homeomorphism to it inducing residue-field isomorphisms and having reduced source is an isomorphism.

Hypotheses: The target is any scheme.

Reading limits: Statement and displayed proof inspected; cited prerequisites were not independently reconstructed.

Sources

The Stacks Project, Absolute weak normalization and seminormalization · Lemma 29.48.7 (3)–(4) and its functoriality assertion
The Stacks Project, Weak normalization · Lemmas 29.56.3 and 29.56.7, and the implication following Definition 29.56.9
The Stacks Project, Normalization · Lemma 33.27.1(1)–(2)
The Stacks Project, Absolute weak normalization and seminormalization · Lemma 29.48.9