DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

The diagonal morphism

Definition

For $X \to S$ , the diagonal morphism is the unique $Δ_{X / S} : X \to X \times_{S} X$ satisfying ${pr}_{1} Δ_{X / S} = {id}_{X} = {pr}_{2} Δ_{X / S}$ . It exists by Existence of all scheme fibre products. For any test scheme $T$ , it takes an $S$ -morphism $a : T \to X$ to the compatible pair $(a, a)$ .

Depends on

Existence of all scheme fibre products

Used by

The doubled-origin diagonal is not closed Counterexample
Separated morphism of schemes Definition
The cotangent complex of a morphism of schemes Definition
The diagonal of the affine line Example
Affine morphisms are separated Lemma
Chow lemma for proper Noetherian schemes Lemma
Every scheme diagonal is an immersion Lemma
Fpqc descent of properness components Lemma
Monomorphisms and diagonals Lemma
Properness survives composition Lemma
Quasi-separatedness and the diagonal Lemma
Separated minimal union and translations Lemma
Separated morphisms compose Lemma
The diagonal commutes with base change Lemma
The graph is a pullback of the diagonal Lemma
The relative projective-space diagonal is closed Lemma
Affine-overlap criterion for separatedness Theorem
An unramified morphism has an open diagonal Theorem
Equalizers into separated schemes are closed Theorem
Formal unramifiedness iff Omega vanishes Theorem
High powers of an ample line bundle embed a proper scheme Theorem
Valuative uniqueness detects separatedness Theorem

Dependency tree · two levels

3 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

Stacks 26.21 introductory definition (standard reference, not scraped)