LemmaStatement: AI-generatedProof: AI-generated

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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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Tensor products multiply frame transitions

PDF p. 4: Weight one units and Picard identification

Statement

Let $L, M$ be locally free rank-one modules over a sheaf of commutative rings. On a common trivializing cover, if their frames have transition units $g_{i j}$ and $h_{i j}$ , then the tensor-product frames of $L \otimes M$ have transition units $g_{i j} h_{i j}$ . The frame of the unit module has transition units $1$ .

Facts & Assumptions

Given: Local frames $e_{i}$ of $L$ and $f_{i}$ of $M$ on a common cover.

Goal: Compute tensor-product transitions.

[F1]

Frame transitions are defined and satisfy the cocycle equation by Frames and their transition units.

Proof

1.1F1construct

The tensor product of the two local isomorphisms from the structure sheaf identifies $L \otimes M$ locally with the structure sheaf; its frame is $e_{i} \otimes f_{i}$ .

2.1F1step 1.1algebra∎

On an overlap, $e_{j} \otimes f_{j} = (g_{i j} e_{i}) \otimes (h_{i j} f_{i}) = g_{i j} h_{i j} (e_{i} \otimes f_{i})$ . The standard frame $1$ of the unit module has transition $1$ .

Depends on

Frames and their transition units

Used by · 1

Units and Picard groups in Nisnevich degree zero and one

Dependency tree · 3 levels

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