LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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.

Open and closed immersions are separated

Statement

Every open immersion, every closed immersion and every immersion (locally closed immersion) of schemes is separated as a morphism.

Facts & Assumptions

Given: An immersion $f : Z \to X$ with factorization $f = j \circ i$ into a closed immersion $i : Z \to U$ and an open immersion $j : U ↪ X$ .

[F1]

An immersion is a morphism factoring as a closed immersion into an open subscheme of its target. (Immersion of schemes)

[F2]

Open immersions and closed immersions are monomorphisms; monomorphism means injectivity on morphism sets. (Immersions and affine localizations are monomorphisms)

[F3]

A monomorphism is separated as a morphism. (Monomorphisms and diagonals)

[F4]

Every immersion is a monomorphism, locally of finite type and separated. (Immersions are monomorphisms locally of finite type)

[F5]

If $Z \to U$ and $U \to X$ are separated morphisms, then $Z \to X$ is separated. (Separated morphisms compose)

[F6]

An open immersion identifies its source with an open subscheme of its target; a closed immersion has underlying map a homeomorphism onto a closed subset with surjective structure map. (Open immersions of schemes, Closed immersions of schemes)

[F7]

A morphism is separated when its diagonal is a closed immersion. (Separated morphism of schemes)

Proof

1.1

An open immersion is a monomorphism by [F2] and hence separated by [F3]; the same argument applies to a closed immersion.

F2F3F6

2.1

For the general immersion, [F1] provides the factorization $f = j \circ i$ with $j$ an open immersion and $i$ a closed immersion; by step 1.1 both $i : Z \to U$ and $j : U \to X$ are separated, so [F5] makes their composite $f$ separated.

F1F5step 1.1

3.1

Both statements also follow directly from [F4], which shows that an immersion is a monomorphism and therefore separated, in agreement with step 2.1; the diagonal of an immersed morphism is an isomorphism, hence a closed immersion by [F7].

F4F7step 2.1∎

Depends on

Used by

An open immersion has valuative uniqueness but not existence Example

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Schemes, Lemma 26.23.8, printed p.48 (standard reference, not scraped)
Vakil, The Rising Sea, Section 11.3.C, printed p.308 (standard reference, not scraped)