Alphabeta MathHow 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
Definition
The real de Rham cohomology is , with as in Closed and exact differential forms.
This is Cohomology object of a cochain complex in real vector spaces. Only a closed form represents a class . For closed forms , equality means precisely for some -form . Addition and real scalar multiplication are induced by those of forms. All groups on the empty manifold are zero.
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
Used by
- Closed defining forms have vanishing Godbillon-Vey class Corollary
- De rham cohomology of a finite disjoint union is the direct sum Corollary
- Connections can change a representative without changing its class Counterexample
- The Godbillon-Vey class of a codimension-one foliation Definition
- Explicit Godbillon-Vey rescaling calculation Example
- The de rham cohomology class of a form is defined without closedness False statement
- First Chern form agrees with the topological line class Lemma
- The Euler characteristic of the structure sheaf is one minus the genus Lemma
- The period pairing is well defined and computed by integration Lemma
- Zero and out of range de rham cohomology Proposition
- Compact Kähler Hodge decomposition from elliptic harmonic forms Theorem
- Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
- Mayer vietoris sequence in de rham cohomology Theorem
- Poincare lemma for differential forms on star shaped domains Theorem
- Pullback induces a well defined map on de rham cohomology Theorem
- The de Rham map on cohomology is well defined Theorem
- The symplectic period formula for integrals of wedge products Theorem
- Wedge product descends to de rham cohomology Theorem
- Zero th de rham cohomology is locally constant functions Theorem
Dependency tree · two levels
7 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)
- Nigel Hitchin, Differentiable Manifolds (2014) (standard reference, not scraped)