LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Equality on a smaller neighbourhood defines the germ equivalence relation

Statement

Let $F$ be a presheaf on a topological space $X$ and let $x \in X$ . For open neighbourhoods $U, V$ of $x$ and sections $s \in F (U)$ , $t \in F (V)$ , define $(U, s) \sim_{x} (V, t)$ when there exists an open neighbourhood $W$ of $x$ with $W \subseteq U \cap V$ and $s ∣_{W} = t ∣_{W} .$ Then $\sim_{x}$ is an equivalence relation.

Facts & Assumptions

Given: A presheaf $F$ on $X$ and a point $x \in X$ .

[F1]

The stalk is built from pairs $(U, s)$ modulo equality on a smaller neighbourhood of $x$ (The stalk of a presheaf at a point).

[F2]

An equivalence relation is one that is reflexive, symmetric, and transitive (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

Proof

technique · direct

1.1F1F2

Reflexivity holds because if $(U, s)$ is any pair, then $s ∣_{U} = s ∣_{U}$ , so $(U, s) \sim_{x} (U, s)$ . Symmetry holds because $s ∣_{W} = t ∣_{W}$ implies $t ∣_{W} = s ∣_{W}$ on the same neighbourhood $W$ .

1.2F1given

Suppose $(U, s) \sim_{x} (V, t)$ and $(V, t) \sim_{x} (Z, r)$ . Choose neighbourhoods $W_{1} \subseteq U \cap V$ and $W_{2} \subseteq V \cap Z$ with $s ∣_{W_{1}} = t ∣_{W_{1}}$ and $t ∣_{W_{2}} = r ∣_{W_{2}}$ . Then $W : = W_{1} \cap W_{2}$ is an open neighbourhood of $x$ contained in $U \cap Z$ , and on $W$ one has $s ∣_{W} = t ∣_{W} = r ∣_{W}$ . Hence $(U, s) \sim_{x} (Z, r)$ .

2.1F2step 1.1step 1.2∎

Steps 1.1 and 1.2 show that $\sim_{x}$ is reflexive, symmetric, and transitive. By [F2], it is an equivalence relation.

Depends on

Used by

Cited to discharge well-definedness by The stalk of a presheaf at a point.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Sheaves on Spaces, Section 11 (standard reference, not scraped)