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Demazure, lectures on p-divisible groups, II.3, affine k-groups

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Let AM kA\in M_k, let Δ:AA kA\Delta:A\to A\otimes_k A.

Then the kk-group (Sp kA,Sp kΔ k)(Sp_k A, Sp_k \Delta_k) is called affine k-group.

A kk-biring is defined to be a kk-module which is a kk-ring and a kk-coring such that AAAA\to A\otimes A is a kk-ring morphism and AAAA\otimes A\to A is a kk-coring morphism.

Examples

Example

The additive kk-group α k\alpha_k assigns to a kk-ring its underlying additive group. We have O(α k)=K[t]O(\alpha_k)=K[t] since by the Yoneda lemma we have O(α k)=hom(α k,O k)=α k(k[t])O(\alpha_k)=hom(\alpha_k, O_k)=\alpha_k(k[t])

Example

The multiplicative kk-group μ k\mu_k assigns to a kk-ring the multiplicative group of its invertible elements. We have O(μ k)=K[tt 1]O(\mu_k)=K[tt^{-1}].

Example

There is a group homomorphism

n:=n μ k:{μ k μ k x x nn:=n_{\mu_k}:\begin{cases} \mu_k&\to& \mu_k \\ x&\to &x^n \end{cases}

its kernel we denote by nμ k:=ker(n)n\mu_k:=ker(n). We have nμ k(R)={xRx n=1}n\mu_k(R)=\{x\in R|x^n=1\} and O(nμ k)=k[t]/(t n1)O(n\mu_k)=k[t]/(t^n -1). If kk is a field and nn is not 00 in kk the kk-group nμ kn\mu_k is étale since t n1t^n-1 is a separable polynomial. nμ k(k s)n\mu_k(k_s) is the Galois module of the nn-th root of unity.

Example

Let kk be a field of prime characteristic pp, let rr be an integer. Then there is a kk-group morphism.

p r:=p α k r:{α k α k x x p rp^r:=p^r_{\alpha_k}:\begin{cases} \alpha_k&\to& \alpha_k \\ x&\to &x^{p^r} \end{cases}

We have ker(p rα k)(R)={xRx p r=0}ker(p^r\alpha_k)(R)=\{x\in R | x^{p^r}=0\} and O(ker(p rα k))=K[t]/t p rO(ker(p^r \alpha_k))=K[t]/t^{p^r}. For any field KK we have ker(p α k r)(K)={0}ker(p^r_{\alpha_k})(K)=\{0\}.

Revised on May 27, 2012 13:26:52 by Stephan Alexander Spahn (79.227.168.80)