Motive

geometry roughly= cohomology

 

example

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,\mathbb{Z})=Z , this is dual to H_1(\mathbb{G}_m,Z) we can call the generator as \(\gamma_0^{\vee}\) where \(\gamma_0\) is the homology generator.

 

de Rham cohomology

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

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

 

De Rham isomorphism

H^1(\mathbb{G}_m,Z) \times H^1_{dR}(\mathbb{G}_m) \to \mathbb{C} is a perfect pairing

(\gamma,\omega) \to \int_{\gamma}\omega

i.e. H^1_{dR}(\mathbb{G}_m) = ^1(\mathbb{G}_m,Z)\otimes_{\mathbb{Z}}\mathbb{C}

 

question. under this isomorphism, \frac{dz}{z} = c\times \(\gamma_0^{\vee}\)  what is c?

c = \int_{\gamma_0}\frac{dz}{z} = 2\pi i

 

Etale cohomology

exponential map : \mathbb{C}\to \mathbb{C}^{*}

 

H^1(\mathbb{G}_m,\mathbb{Z}) = Hom(\pi_1(\mathbb{C}^{*}),\mathb{Z})

 

 

cyclotomic character