nLab
rational cohomology

Contents

Context

Rational homotopy theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

By rational cohomology one usually means ordinary cohomology with rational number coefficients, denoted H (,)H^\bullet\big(-, \mathbb{Q}\big).

Hence, with the pertinent conditions on the domain space XX satisfied, its rational cohomology H (,)H^\bullet\big(-, \mathbb{Q}\big) is what is computed by the Cech cohomology or singular cohomology or sheaf cohomology of XX with coefficients in \mathbb{Q}.

Properties

Universal coefficient theorem

Example

(universal coefficient theorem in rational cohomology)

For rational numbers-coefficients \mathbb{Q}, the Ext groups Ext 1(;)Ext^1(-;\mathbb{Q}) vanish, and hence the universal coefficient theorem identifies rational cohomology groups with the dual vector space of the rational vector space of rational homology groups:

H (;)Hom (H (;);). H^\bullet \big( -; \, \mathbb{Q} \big) \;\; \simeq \;\; Hom_{\mathbb{Z}} \Big( H_\bullet\big(-;\,\mathbb{Q} \big); \, \mathbb{Q} \Big) \,.

(e.g. Moerman 15, Cor. 1.2.1)

PL de Rham theorem

Last revised on December 6, 2020 at 09:57:05. See the history of this page for a list of all contributions to it.