nLab 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 $(Sp_k A, Sp_k \Delta_k)$ 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 $O(\alpha_k)=hom(\alpha_k, O_k)=\alpha_k(k[t])$

Example

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

Example

There is a group homomorphism

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

its kernel we denote by $n\mu_k:=ker(n)$. We have $n\mu_k(R)=\{x\in R|x^n=1\}$ 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.

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

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