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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A sheaf restricts injectively to a covering singleton

PDF p. 5: HK18 Theorem 1.4

Statement

Let $F$ be a set-valued sheaf on a scheme site, and suppose $u : U \to V$ generates a covering sieve. Then the restriction $F (V) \to F (U)$ is injective. A sheaf has unique amalgamations of compatible sections on each covering sieve.

Facts & Assumptions

Given: $F$ , $u$ and the covering sieve generated by $u$ .

Goal: Prove injectivity of restriction.

[F1]

Sections of a sheaf agreeing on every arrow in a covering sieve are equal, by uniqueness of amalgamation. Scheme sites and sheafification

Proof

technique · Restrict along the generated sieve

1.1given

If $a, b \in F (V)$ have equal restrictions to $U$ , then their restrictions along any arrow factoring through $u$ are equal by functoriality. These arrows form the sieve generated by $u$ .

2.1F1step 1.1∎

That sieve is covering, so [F1] gives $a = b$ . Hence restriction is injective.

Depends on

Scheme sites and sheafification

Used by · 2

Dependency tree · 3 levels

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