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.

Sheafhood of algebraic-structure valued presheaves is detected on underlying sets

Statement

Let $R$ be a ring, and let $F$ be a presheaf of groups, rings, or left $R$ -modules on a topological space $X$ . Then $F$ is a sheaf in the corresponding algebraic category if and only if its underlying presheaf of sets is a sheaf.

Facts & Assumptions

Given: A ring $R$ and a presheaf $F$ of groups, rings, or left $R$ -modules on $X$ .

[F1]

Such a presheaf is, by definition, a set-valued presheaf together with objectwise algebraic operations preserved by restriction maps; it is called a sheaf exactly when the underlying set-valued presheaf is a sheaf (Presheaves and sheaves of groups, rings, and modules).

[L1]

The sheaf condition itself is the locality and unique-gluing condition for the underlying sets of sections (A sheaf on a topological space).

Proof

technique · direct

1.1

If $F$ is a sheaf of groups, rings, or left $R$ -modules, then [F1] already says that its underlying set-valued presheaf is a sheaf.

2.1

Conversely, assume the underlying set-valued presheaf is a sheaf. The sets $F (U)$ already carry the given group, ring, or left $R$ -module structures, and the restriction maps already preserve those structures by [F1]. Since [L1] tests only locality and gluing of the underlying sections, the assumed setwise sheaf condition is exactly the required algebra-valued sheaf condition.

F1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Sections 5, 9, and 10 (standard reference, not scraped)