nLab rational cohomology

Idea

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

Hence, with the pertinent conditions on the domain space $X$ satisfied, its rational cohomology $H^\bullet\big(-, \mathbb{Q}\big)$ is what is computed by the Cech cohomology or singular cohomology or sheaf cohomology of $X$ 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 $Ext^1(-;\mathbb{Q})$ 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:

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

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

PL de Rham theorem

• PL de Rham theorem

Related concepts

• rational vector space

• integral cohomology, real cohomology, complex cohomology

• rational Cohomotopy

• rational homotopy theory

• Chern character, Chern-Dold character

• rational equivariant K-theory

• equivariant Chern character