LemmaStatement: AI-generatedProof: AI-generated

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.

Gluing compatible local determinant homomorphisms

PDF p. 9: Surjectivity of the determinant on K1

Statement

Let $X$ be a ringed space covered by open sets $U_{i}$ . Let $G$ and $G_{i}$ be groups with homomorphisms $r_{i} : G \to G_{i}$ , and let $d_{i} : G_{i} \to O_{X} (U_{i})^{\times}$ be homomorphisms. Assume that, for every $g \in G$ , the sections $d_{i} (r_{i} (g))$ agree on every overlap. Then there is a unique homomorphism $d : G \to O_{X} (X)^{\times}$ whose restriction to each $U_{i}$ is $d_{i} r_{i}$ .

Facts & Assumptions

Given: The cover, groups, maps and overlap compatibility in the statement.

Goal: Construct the unique global homomorphism $d$ .

[F1]

Compatible local units glue uniquely, and gluing respects multiplication. Compatible local units glue uniquely

Proof

technique · Glue pointwise and use uniqueness

1.1F1given

For each $g \in G$ , the assumed overlap compatibility and [F1] give a unique global unit $d (g)$ restricting to $d_{i} (r_{i} (g))$ on every $U_{i}$ .

2.1F1step 1.1∎

For $g, h \in G$ , the restrictions of $d (g h)$ and $d (g) d (h)$ agree on every $U_{i}$ because $d_{i}$ and $r_{i}$ are homomorphisms. Uniqueness in [F1] gives $d (g h) = d (g) d (h)$ . The same uniqueness proves that any map with the specified restrictions equals $d$ .

Depends on

Compatible local units glue uniquely

Used by · 1

Units split the K one determinant

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.