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.

Frames and their transition units

PDF p. 4: Weight one units and Picard identification

Statement

Let $O$ be a sheaf of commutative rings on a site, and let $L$ be a sheaf of $O$ -modules locally isomorphic to $O$ . A frame over $U$ is an $O ∣_{U}$ -module isomorphism $O ∣_{U} \to L ∣_{U}$ . The frames form a torsor under $O^{\times}$ : they form a sheaf of sets, have local sections, and whenever two frames exist over the same object there is a unique unit carrying one to the other. If frames $e_{i}$ are chosen on a cover, their transition units $g_{i j}$ defined by $e_{j} = g_{i j} e_{i}$ satisfy $g_{i j} g_{j k} = g_{i k}$ on triple overlaps.

Facts & Assumptions

Given: $O$ and $L$ as stated.

Goal: Establish the frame torsor and its transition equation.

[F1]

The units form a sheaf by Units in a sheaf of rings form a sheaf.

Proof

1.1F1construct

A module map from $O ∣_{U}$ to $L ∣_{U}$ is determined by the image of $1$ . If it is an isomorphism, composing with another frame's inverse gives an automorphism of $O ∣_{U}$ , necessarily multiplication by a unique unit. Compatible maps glue as maps of sheaves; compatible local inverses glue as well, so frames satisfy the sheaf condition. Local freeness supplies local frames.

2.1step 1.1algebra∎

On a triple overlap, $e_{k} = g_{j k} e_{j} = g_{j k} g_{i j} e_{i}$ . Uniqueness of the unit in step 1.1 gives $g_{i j} g_{j k} = g_{i k}$ . Thus frames have the stated torsor and transition properties.

Depends on

Units in a sheaf of rings form a sheaf

Used by · 2

Dependency tree · 3 levels

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