TheoremStatement: Literature-sourcedProof: AI-adaptedjudge 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.

Universal property of scheme reduction

Statement

For every morphism $f : T \to X$ from a reduced scheme $T$ , there is a unique morphism $\overset{ˉ}{f} : T \to X_{r e d}$ whose composite with $X_{r e d} \to X$ is $f$ .

Facts & Assumptions

Given: A reduced scheme $T$ and a morphism $f : T \to X$ .

Proof

technique · direct

1.1given

On affine opens, the map on coordinate rings sends every nilpotent of the target ring to a nilpotent of the reduced source ring, hence to zero.

2.1step 1.1

It therefore factors uniquely through the quotient by the nilradical on each affine chart, producing compatible local maps to $X_{r e d}$ .

3.1step 2.1∎

The local maps glue uniquely, and their uniqueness on charts proves uniqueness globally.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Schemes, Section 12