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

This entry is about a section of the text

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

Then the $k$-group $\left({\mathrm{Sp}}_{k}A,{\mathrm{Sp}}_{k}{\Delta }_{k}\right)$ 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 $O\left({\alpha }_{k}\right)=K\left[t\right]$ since by the Yoneda lemma we have $O\left({\alpha }_{k}\right)=\mathrm{hom}\left({\alpha }_{k},{O}_{k}\right)={\alpha }_{k}\left(k\left[t\right]\right)$

###### Example

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

###### Example

There is a group homomorphism

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

its kernel we denote by $n{\mu }_{k}:=\mathrm{ker}\left(n\right)$. We have $n{\mu }_{k}\left(R\right)=\left\{x\in R\mid {x}^{n}=1\right\}$ and $O\left(n{\mu }_{k}\right)=k\left[t\right]/\left({t}^{n}-1\right)$. 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. $n{\mu }_{k}\left({k}_{s}\right)$ 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.

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

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

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