Demo of the perfect pairing between characters and cocharacters of a torus

In this file, we present the pairing between characters and cocharacters.

TODO

Actually compute the pairing on an explicit torus to be the usual inner product on ℤⁿ.

Let S be a scheme and G be a group scheme over S.

We denote the characters of G by X(S, G) and the cocharacters by X*(S, G).

Let R be a domain and G be a split torus over R.

Then we have a pairing between X(Spec R, G) and X*(Spec R, G).

If furthermore G := 𝔾ₘ[Spec R, σ] is the explicit torus with finite dimensions given by σ, then this pairing is perfect.