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.

Closed graphs over separated targets

Statement

Let $g : X \to Y$ be an $S$ -morphism of schemes and suppose that $Y \to S$ is separated. Then the graph $Γ_{g} : X \to X \times_{S} Y$ is a closed immersion. No separatedness hypothesis on $X \to S$ is required.

Facts & Assumptions

Given: An $S$ -morphism $g : X \to Y$ with $Y \to S$ separated, and the graph $Γ_{g}$ .

[F1]

The graph morphism is $Γ_{g} = ({id}_{X}, g) : X \to X \times_{S} Y$ , supplied by Existence of all scheme fibre products; its first projection is the identity and its second projection is $g$ . (The graph morphism over a base)

[F2]

With $H = (g {pr}_{X}, {pr}_{Y}) : X \times_{S} Y \to Y \times_{S} Y$ , the square with top arrow $Γ_{g}$ , bottom arrow $Δ_{Y / S}$ , left arrow $g$ and right arrow $H$ is Cartesian; thus $Γ_{g}$ is the base change of $Δ_{Y / S}$ along $H$ . (The graph is a pullback of the diagonal)

[F3]

A morphism $Y \to S$ is separated when $Δ_{Y / S}$ is a closed immersion. (Separated morphism of schemes)

[F4]

Closed immersions remain closed immersions after arbitrary base change; this is asserted with no flatness or finiteness hypothesis. (Base change of immersions)

Proof

technique · direct

1.1

By [F1] the graph $Γ_{g} : X \to X \times_{S} Y$ is the morphism $({id}_{X}, g)$ , and [F2] exhibits it as the base change of $Δ_{Y / S} : Y \to Y \times_{S} Y$ along $H : X \times_{S} Y \to Y \times_{S} Y$ , the square in [F2] being Cartesian.

F1F2given

1.2

Since $Y \to S$ is separated, [F3] says that $Δ_{Y / S}$ is a closed immersion.

F3given

2.1

The base change of the closed immersion $Δ_{Y / S}$ along $H$ is a closed immersion by [F4]; by step 1.1 that base change is $Γ_{g}$ , so $Γ_{g}$ is a closed immersion.

F4step 1.1step 1.2

3.1

The argument used only the separatedness of $Y \to S$ and the Cartesian square of [F2]; nothing was assumed about $X \to S$ , which may even be nonseparated.

step 2.1∎

Depends on

Used by

The graph of a polynomial map as a closed subscheme Example

Dependency tree · two levels

17 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 26.21.10, printed p.41 (standard reference, not scraped)
Vakil, The Rising Sea, Section 11.3.6, printed p.309 (standard reference, not scraped)