DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

De rham cohomology ring

Definition

The de Rham cohomology ring is $H_{d R}^{∙} (M) = ⨁_{k \in Z} H_{d R}^{k} (M)$ with $[α] [β] = [α \land β]$ and unit $[1]$ .

By Wedge product descends to de rham cohomology it is a unital graded-commutative real algebra. On the empty manifold it is the zero algebra, with $1 = 0$ ; this convention allows the zero algebra among unital algebras.

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 17, pp.441–443, Proposition 17.2 and Corollary 17.3; local quotient calculations below.

Depends on

Wedge product descends to de rham cohomology

Used by

Pullback is a homomorphism of de rham cohomology algebras Proposition

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)