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Compatible local torsor sections determine an isomorphism

PDF p. 4: Weight one units and Picard identification

Statement

Let $P, Q$ be torsors under a sheaf of abelian groups $G$ on a site, and suppose sections $p_{i} \in P (U_{i})$ and $q_{i} \in Q (U_{i})$ over a cover of the terminal object have the same transition units, with the convention of Local sections of a torsor give a cocycle. Then there is a unique equivariant isomorphism $P \to Q$ carrying every $p_{i}$ to $q_{i}$ .

Facts & Assumptions

Given: The two torsors and matching local sections.

Goal: Construct the equivariant isomorphism.

[F1]

Torsors are locally nonempty sheaves with simply transitive actions, as in Local sections of a torsor give a cocycle.

Proof

1.1F1construct

Over each $U_{i}$ , send $g p_{i}$ to $g q_{i}$ . Simple transitivity makes this a well-defined equivariant bijection, natural after restriction. On overlaps the common transition unit gives the same map from either index.

2.1F1step 1.1∎

The sheaf condition glues these local maps, and glues their local inverses, to mutually inverse global morphisms. Any equivariant map taking $p_{i}$ to $q_{i}$ agrees with the constructed map locally, hence globally.

Depends on

Local sections of a torsor give a cocycle

Used by · 1

Units and Picard groups in Nisnevich degree zero and one

Dependency tree · 3 levels

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