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.

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Local sections of a torsor give a cocycle

PDF p. 4: Weight one units and Picard identification

Statement

Let $G$ be a sheaf of abelian groups on a site with fibre products. A $G$ -torsor is a sheaf of sets with a $G$ -action which is locally nonempty and simply transitive on sections whenever nonempty. If $s_{i}$ are sections over a covering ${U_{i} \to U}$ and $s_{j} = g_{i j} s_{i}$ on $U_{i} \times_{U} U_{j}$ , then $g_{i j} g_{j k} = g_{i k}$ on triple fibre products. Replacing $s_{i}$ by $s_{i}^{'} = h_{i} s_{i}$ replaces $g_{i j}$ by $g_{i j}^{'} = h_{j} g_{i j} h_{i}^{- 1}$ .

Facts & Assumptions

Given: A $G$ -torsor and the displayed local sections.

Goal: Prove the cocycle and change-of-section formulas.

[F1]

Restrictions and fibre products are those of the site in Scheme sites and sheafification.

Proof

1.1F1given

Simple transitivity gives a unique $g_{i j}$ on each double fibre product. On a triple fibre product, $s_{k} = g_{j k} g_{i j} s_{i}$ and also $s_{k} = g_{i k} s_{i}$ ; uniqueness and commutativity give the cocycle equation.

2.1step 1.1algebra∎

Substitution gives $s_{j}^{'} = h_{j} g_{i j} h_{i}^{- 1} s_{i}^{'}$ . Uniqueness gives the stated replacement formula.

Depends on

Scheme sites and sheafification

Used by · 2

Dependency tree · 3 levels

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