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

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Let $A\in M_k$, let $\Delta :A\to A{\otimes }_{k}A$.

Then the $k$-group $({\mathrm{Sp}}_{k}A,{\mathrm{Sp}}_k{\Delta }_k)$ is called affine k-group.

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

Examples

Example

The additive $k$-group ${\alpha }_k$ assigns to a $k$-ring its underlying additive group. We have $O({\alpha }_k)=K[t]$ since by the Yoneda lemma we have $O({\alpha }_k)=\mathrm{hom}({\alpha }_k,O_k)={\alpha }_k(k[t])$

Example

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

Example

There is a group homomorphism

$$n:=n_{{\mu }_k}:\{\begin{array}{lll}{\mu }_k& \to & {\mu }_k\\ x& \to & x^n\end{array}$$n:=n_{\mu_k}:\begin{cases} \mu_k&\to& \mu_k \\ x&\to &x^n \end{cases}

its kernel we denote by $n{\mu }_k:=\ker (n)$. We have $n{\mu }_k(R)=\{x\in R\mid x^n=1\}$ and $O(n{\mu }_k)=k[t]/(t^n-1)$. If $k$ is a field and $n$ is not $0$ in $k$ the $k$-group $n{\mu }_k$ is étale since $t^n-1$ is a separable polynomial. $n{\mu }_k(k_s)$ is the Galois module of the $n$-th root of unity.

Example

Let $k$ be a field of prime characteristic $p$, let $r$ be an integer. Then there is a $k$-group morphism.

$$p^r:=p_{{\alpha }_k}^r:\{\begin{array}{lll}{\alpha }_k& \to & {\alpha }_k\\ x& \to & x^{p^r}\end{array}$$p^r:=p^r_{\alpha_k}:\begin{cases} \alpha_k&\to& \alpha_k \\ x&\to &x^{p^r} \end{cases}

We have $\ker (p^r{\alpha }_k)(R)=\{x\in R\mid x^{p^r}=0\}$ and $O(\ker (p^r{\alpha }_k))=K[t]/t^{p^r}$. For any field $K$ we have $\ker (p_{{\alpha }_k}^r)(K)=\{0\}$.