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.

Matching families of abelian presheaves

PDF p. 4: Derived Nisnevich and cdh sections

Statement

Let $C$ be a small site and $F$ a presheaf of abelian groups. For a sieve $R$ on an object $U$ , the compatible families of sections of $F$ indexed by arrows in $R$ form an abelian group under componentwise addition. If $F$ is a sheaf, restriction from $F (U)$ to this group is an isomorphism for every covering sieve $R$ .

A presheaf of abelian groups is a contravariant functor to abelian groups. A compatible family assigns $s_{f} \in F (V)$ to every $f : V \to U$ in $R$ , with $F (g) (s_{f}) = s_{f g}$ for every $g : W \to V$ .

Facts & Assumptions

Given: A small site, an abelian presheaf $F$ , an object $U$ , and a sieve $R$ on $U$ .

Goal: Prove the group and sheaf assertions.

[F1]

A sheaf has a unique amalgamation for each compatible family on a covering sieve. Scheme sites and sheafification

Proof

technique · Check the matching equations componentwise

1.1givenconstruct

The zero family is compatible, and sums and inverses of compatible families remain compatible because every restriction map of $F$ is a group homomorphism. The abelian group laws hold componentwise.

2.1F1step 1.1∎

Restriction sends a section on $U$ to a compatible family and is a homomorphism. When $R$ covers $U$ , existence and uniqueness of amalgamations in [F1] make it bijective, hence an isomorphism.

Depends on

Scheme sites and sheafification

Used by · 1

Pointwise kernels of abelian sheaves

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.