Motive

geometry roughly= cohomology

 

examples

circle S^1

Betti cohomolgy (singular cohomology)

H^0(S^1,Z)=Z

H^1(S^1,Z)=Z

 

\mathbb{G}_m = \mathbb{C}^{x} = \mathbb{C}/{0} same homotopy class as S^1

Betti cohomology is same

H^0(\mathbb{G}_m,Z)=Z

H^1(\mathbb{G}_m,Z)=Z , this is dual to H_1(\mathbb{G}_m,Z)

 

de Rham cohomology

H^0_{dR}(\mathbb{G}_m)=\mathbb{C}

H^1_{dR}(\mathbb{G}_m)=\mathbb{C}