DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Tensor product of sheaves of modules

Definition

Let $(X, O_{X})$ be a ringed space, and let $F, G$ be $O_{X}$ -modules. The tensor-product presheaf

F \otimes_{p, O_{X}} G

is defined by $(F \otimes_{p, O_{X}} G) (U) : = F (U) \otimes_{O_{X} (U)} G (U)$ for each open set $U \subseteq X$ , with restriction induced from the restriction maps of $F$ and $G$ .

The tensor-product sheaf is the sheafification of this presheaf: $F \otimes_{O_{X}} G : = a (F \otimes_{p, O_{X}} G) .$

Tensor product of sheaves of modules

Definition

Depends on

Dependency tree · 2 levels

Sources