# Affine Group Schemes (I.1) : Definition as a Representable Functor
###### tags: affine group schemes
*Reference : Waterhouse, Introduction to Affine Group Schemes*

Let $k$ be a field. Denote $\mathtt{Alg}_k$ as the category of $k$-algebras.

## Motivation through examples
Standard constructions of group schemes from f.g. $k$-algebras can be made functorial.
Moreover, it is also sometimes the case that these functors are even representable.

#### Example : Additive subgroup $\mathbf{G}_a$
> Define $\mathbf{G}_a(R)=(R,+)$. Then $\mathbf{G}_a$ is represented by $k[X]$ (that is : $\mathbf{G}_a\simeq\operatorname{Hom}_k(k[X],-)$).

#### Example : Multiplicative subgroups $\mathbf{G}_m$
> Define $\mathbf{G}_m(R)=(R^\times,\ast)$. Then $\mathbf{G}_m$ is represented by $k[X,Y]/(XY-1)$.

#### Example : Roots of unity $\mu_n$
> Define $\mu_m(R)=\{x\in R:x^n=1\}$. Then $\mu_m$ is represented by $k[X]/(X^n-1)$.

#### Example : Trivial groups $\ast$
> $R\mapsto\ast$ is represented by $k$.

#### Example : Matrix groups $M_{m,n}$
> $R\mapsto\mathbf{M}_{m,n}(R)$ is represented by $k[X_{11},X_{12},\dotsc,X_{mn}]$.

#### Example : Special linear group $\mathbf{SL}_n$
> $R\mapsto\mathbf{SL}_n(R)$ is represented by $k[X_{11},X_{12},\dotsc,X_{nn}]/(\operatorname{det}(X_{ij})-1)$.

#### Example : General linear group $\mathbf{GL}_n$
> $R\mapsto\mathbf{GL}_n(R)$ is represented by $k[X_{11},X_{12},\dotsc,X_{nn},\operatorname{det}(X_{ij})^{-1}]$.

## Yoneda Lemma

Consider the presheaf category $[\mathtt{Alg}_k,\mathtt{Set}]$.
Natural transformations between representable functors are characterized by Yoneda lemma :
in particular, they are given by morphisms between the representing objects.
The examples above are all representable functors in $[\mathtt{Alg}_k,\mathtt{Set}]$.

#### Example/Exercise : Standard inclusions
> What corresponds to the standard inclusions $\mu_n\to\mathbf{G}_m,\mathbf{SL}_n\to \mathbf{GL}_n\to \mathbf{M}_{nn}$ ?

#### Example : Binary product of representable functors
> Suppose $F,G$ are represented by $A,B$, then $F\times G$ is represented by $A\otimes_kB$ - as $A\otimes_kB$ is the coproduct of $A,B$ in the category $\mathtt{Alg}_k$.

#### Exercise : Pullback of representable functors
> Compute the representing object of a pullback of representable functors.

#### Remark : Uniqueness of representing objects
> Yoneda lemma also gives us the uniqueness of representing objects :
> if two thing represent the same thing, they are isomorphic.

## Hopf Algebra

### An interesting correspondence

Suppose we are given a group functor : a functor $\mathbf{G}:\mathtt{Alg}_k\to\mathtt{Set}$ that factors through the forgetful functor $\mathtt{Grp}\to\mathtt{Set}$, and suppose that $\mathbf{G}$ has a representative $A$.
A group structures on $\mathbf{G}$ is given by the following data of natural transformations :$$\mathbf{G}\times\mathbf{G}\to\mathbf{G},\quad\ast\to\mathbf{G},\quad\mathbf{G}\to\mathbf{G}$$ encoding multiplication, unit, inverse, satisfying some commutativity diagrams.
Now by Yoneda lemma, these would correspond to morphisms :$$\Delta:A\to A\otimes_kA,\quad\varepsilon:A\to k,\quad S:A\to A$$ encoding comultiplication, counit, coinverse, satisfying some dual commutativity diagrams.
Spelling out everything and using Yoneda lemma, we have the following correspondence :
> Group structures on $\mathbf{G}$ are in bijective correspondence with Hopf algebra structures on its representing object $A$.

#### Exercise :
> Work out $\Delta,\varepsilon,S$ for the case $\mathbf{G}=\mathbf{G}_a$.

## Base Change
Suppose we have a homomorphism $k'\to k$, then we have a functor $$[\mathtt{Alg}_{k'},\mathtt{Set}]\to[\mathtt{Alg}_{k},\mathtt{Set}],\qquad F\mapsto F_{k}$$ given by restriction of scalars.
#### Example :
> If $F\in[\mathtt{Alg}_{k'},\mathtt{Set}]$ is represented by some $A$, then $F_{k}$ is represented by $A\otimes_{k'}k$ by
> $$\operatorname{Hom}_{k'}(A,B)\simeq\operatorname{Hom}_{k'}(A\otimes_kk',B)$$

## Main definition
Finally, let us define the notion of an affine group scheme.
#### Definition
> An affine group scheme over $k$ is a representable functor in $[\mathtt{Alg}_k,\mathtt{Group}]$.