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.

Units in a sheaf of rings form a sheaf

PDF p. 4: Weight one units and Picard identification

Statement

Let $O$ be a sheaf of commutative rings on a site. The presheaf $U \mapsto O (U)^{\times}$ is a sheaf of abelian groups, where $O (U)^{\times}$ denotes the multiplicatively invertible elements.

Facts & Assumptions

Given: A sheaf of commutative rings $O$ on a site.

Goal: Prove the sheaf condition for its units.

[F1]

A sheaf has unique gluing for compatible local sections, as in Scheme sites and sheafification.

Proof

1.1F1construct

For a covering of $U$ , compatible local units glue by [F1] to a section $a \in O (U)$ . Their inverses are compatible, because restriction preserves inverses, and glue to a section $b$ . The restrictions of both $a b$ and $1$ agree on the cover, so $a b = 1$ by uniqueness. Thus $a$ is a unit.

2.1F1step 1.1∎

Uniqueness of a glued unit follows from uniqueness in $O$ . Multiplication and inversion commute with restriction, and commutativity makes the unit groups abelian. Hence the units form a sheaf of abelian groups.

Depends on

Scheme sites and sheafification

Used by · 2

Dependency tree · 3 levels

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