Group Schemes and Affine Group Schemes

In this post we motivate the notion of affine group schemes, and discuss the various equivalent ways of defining them.

Motivation

Group schemes are an extremely powerful, and beautiful set of objects. For some reason, they are often times not discussed in basic algebraic geometry courses. One reason this may happen is because while the definition of group schemes is relatively simple, and one can easily deduce their basic properties. But, quickly the easy proofs give way to deep facts, and complicated machinery (e.g. root systems and representations), making them too much a diversion from a standard course. This is a shame though, since group schemes offer some of the most illuminating examples of many of the natural properties of schemes.

Simply, a group scheme is a group object in the category of schemes (usually over some fixed scheme $S$). So, it is a scheme $G$, equipped with morphisms $m:G\times G\to G$, $i:G\to G$, and $e:\ast\to G$. These morphisms are suppose to represent the multiplication of the group, the inverse function of the group (i.e. $x\mapsto x^{-1}$), and the morphism picking out the identity element of the group. So, in this way, group schemes are the schemes, as Lie groups are to manifolds.

The similarity between group schemes and Lie groups doesn’t just end there. The place that group schemes occupy in algebraic geometry, much mirrors that of Lie groups within differential geometry. One one hand, they are extremely simple. The homogeneity properties they possess from their simply transitive self-action forces them to be extremely symmetrical. This, in turn, forces them to automatically have many desirable properties. On the other hand, group schemes comprise some of the most complicated schemes one may encounter in a first course on schemes, for example $\text{GL}_n$ (to be defined later).

The other source of similarity between group schemes and Lie groups is via the utility they have with respect to other objects. Said less cryptically, one of the most useful properties that group schemes have is they act on other schemes. Usually one can phrase some important identification between points of a scheme, by saying that they are precisely the objects in the orbits of some action of a group scheme on the scheme. Or, one may be able to express algebro-geometric symmetry via the action of some group scheme on the scheme.

The last similarity between Lie groups and group schemes, one which is of particular importance to us right now, is the huge difference between compact and non-compact ones. Anyone who has taken a course on Lie groups is well aware of the huge difference in the theory that one encounters when trying to transition between non-compact Lie groups and their compact counterparts. This difference may be even greater in theory of group schemes, where the two different cases form two entirely different subjects.

Namely, while there is not a (useful) topological notion of compactness for schemes, there is a useful replacement–properness. Thus, one might expect that the world of group schemes may be carved into its proper and non-proper pieces. But, unlike topological spaces which exist in the binariness that is compact and non-compact, the world of schemes is much more nuanced. Namely, non-proper schemes can be so disparate (an obvious non-trivial dichotomy being separated vs. non-separated) from one another, that we want to consider a slightly more specific class. Namely, the affine group schemes. These are the two subjects that the study of group schemes is usually divided between–the affine group schemes (linear group schemes) and the proper group schemes (abelian varieties).

Now, while abelian varieties (which include elliptic curves) are undoubtedly important, in this (and hopefully subsequent) posts, we shall discuss linear group schemes. The reason for the modifier ‘linear’ because of an important theorem (one that is easy to prove [two lines] once we have the machinery] which says that every affine group scheme (over a field) $G$ embeds into $\text{GL}_n$. These comprise many of the important groups that show up, for example, in representation theory. They are also the groups which most effectively act on natural algebro-geometric objects.

Group Objects

One of the other aspects of the theory of group schemes which differs greatly from the theory of Lie groups, is the huge presence of the functorial point of view. Namely, group schemes over $S$ will be merely schemes $X/S$ together with a factorization of $h_X:(\textbf{Sch}/S)^\text{op}\to\mathbf{Set}$ as $F\circ G$ where $G$ is some functor $(\textbf{Sch}/S)^\text{op}\to \mathbf{Grp}$ and $F:\mathbf{Grp}\to\mathbf{Set}$ is the forgetful functor.

This actually holds more generally though, and so we quickly review this idea. Let us begin by recalling that if $\mathscr{C}$ is a category with finite products and a final object $\ast$, then a group object in $\mathscr{C}$ is an object $G$ of $\mathscr{C}$ together with morphisms $m:G\times G\to G$ (where the product is taken in $\mathscr{C}$!), $i:G\to G$, and $e:\ast\to G$ such that the following identities hold

1. (Associativity) $m\circ(m\times\text{id})=m\circ (\text{id}\times m)$
2. (Inverses) $m\circ(\text{id}\times i)=m\circ (i\times\text{id})=e\circ (\alpha)$ (where $\alpha:G\to \ast$)
3. (Identity) $m\circ(\text{id}\times e)=m\circ(e\times\text{id})=\text{id}\circ\beta$ (where $\beta$ is the unique isomorphism $\ast\times G$).

As should be inferred from the labels, these conditions are just saying that the morphisms coming with a group object should satisfy the usual axioms that make a group. One interesting thing to note is that unlike the case of groups, here we actually specify the inverse operation as well as the identity element, opposed to just claiming their existence. This is important because we want the inverse operation, for example, to actually be a morphism in the category.

Now, the group objects in a category $\mathscr{C}$ form a category themselves. In particular, if $(G,m,i,e)$ and $(G',m',i',e')$ are two group objects of $\mathscr{C}$, then a morphism from $(G,m,i,e)$ to $(G',m',i',e')$ should just be a morphism $f:G\to G'$ in $\mathscr{C}$ such that $m'\circ (f\times f)=f\circ m$, where the morphism $f\times f:G\times G\to G'\times G'$ is the obvious one. Or, in other words, morphisms should just be morphisms of the underlying $\mathscr{C}$-objects that commute with respective multiplications.

Brief Discussion of Yoneda’s Lemma

Now, we’d like to recast the notion of group objects in purely functorial terms. If you believe that this seems like an obnoxious goal (which, on the face of things, isn’t unreasonable), you will be quickly become a convert. In fact, group objects are more often than not presented to us in functorial form–it is the above definition which is less natural. But, let’s put the cart before the horse.

To make rigorous our functorial definition of group objects, we need to recall the ubiquitous Yoneda lemma. For this, let’s recall that if $\mathscr{C}$ is a category, and $X$ an object of $\mathscr{C}$, then $h_X$ denotes the contravariant functor $\mathscr{C}\to\mathbf{Set}$ (i.e. a presheaf on $\mathscr{C}$ in modern parlance) given by $\text{Hom}_\mathscr{C}(-,X)$ (with the obvious action on arrows).

Note that if $f:X\to Y$ is a morphism, one obtains a natural transformation $h_f:h_X \to h_Y$ as follows. For an object $Z\in\mathscr{C}$, we obtain a morphism $h_X(Z)\to h_Y(Z)$ by taking $\alpha:Z\to X$ to $f\circ \alpha:Z\to Y$. One can quickly check that this is, indeed, a natural transformation.

In this way, we obtain a functor $h:\mathscr{C}\to\widehat{\mathscr{C}}$, where $\widehat{\mathscr{C}}$ is the functor category $\mathbf{Set}^{\mathscr{C}^\text{op}}$ (i.e. the category of presheaves on $\mathscr{C}$). Yoneda’s lemma then tells us something interesting about this functor.

Theorem1 (Yoneda’s Lemma): Let $\mathscr{C}$ be an object of $X$, and $F$ an object of $\widehat{\mathscr{C}}$. Then, the map $\text{Hom}_{\widehat{\mathscr{C}}}(h_X,F)\to F(X)$ taking $\eta:h_X\to F$ to $\eta(X)(\text{id}_X)$ is a bijection.

Why this has impact on our functor $h:\mathscr{C}\to \widehat{\mathscr{C}}$ comes from taking $F=h_Y$. Then, this says that the map $\text{Hom}_{\widehat{\mathscr{C}}}(h_X,h_Y)$ sending $\eta:h_X\to h_Y$ to $\eta(X)(\text{id}_X)$ is a bijection to $h_Y(X)=\text{Hom}_\mathscr{C}(X,Y)$.

But, if $f:X\to Y$ is an element of $\text{Hom}_\mathscr{C}(X,Y)$, and $h_f:h_X\to h_Y$ is the natural transformation we defined above, then what is $h_f(X)(\text{id}_X)$? Well, $h_f(X)$ is the map $\text{Hom}_\mathscr{C}(X,X)\to\text{Hom}_\mathscr{C}(X,Y)$ taking $g$ to $f\circ g$. Thus, $h_f(X)(\text{id}_X)=f\circ \text{id}_X=f$. Thus, we see that $f\mapsto h_f$ is the right inverse to the bijection defined in Yoneda’s lemma, and so must be a bijection itself.

This then tells us that our functor $h:\mathscr{C}\to\widehat{\mathscr{C}}$ is an embedding (i.e. fully faithful). Thus, to give a morphism $f:X\to Y$ is the same thing as to give a natural transformation $h_X\to h_Y$.

Group Objects as Functors

Now that we have recalled the statement of Yoneda’s lemma we can effectively discuss how to categorify the definition of group object.

Note first that if $X$ and $Y$ are objects of a category $\mathscr{C}$, then we can canonically identify the functors $h_{X\times Y}$ and $h_X\times h_Y$. Indeed, the natural isomorphism $\eta:h_{X\times Y}\overset{\approx}{\to}h_X\times h_Y$, on an object $Z$ takes $\text{Hom}_\mathscr{C}(Z,X\times Y)\to \text{Hom}_\mathscr{C}(X,Z)\times\text{Hom}_\mathscr{C}(Y,Z)$ as $\alpha\mapsto (pr_X\circ \alpha,pr_Y\circ \alpha)$ where $pr_X:X\times Y\to X$ and $pr_Y:X\times Y\to Y$ are the projection maps associated to $X\times Y$.

So, with this in mind, let’s see what Yoneda’s lemma says about the definition of a group object. Well, to give a morphism $m:G\times G\to G$ is the same as giving a morphism $h_m:h_{G\times G}\to h_G$, or with the identification we discussed in the last paragraph, $h_m:h_G\times h_G\to h_G$. Similarly, to give a morphism $i:G\to G$ is the same as giving a morphism $h_i:h_G\to h_G$. And finally, giving a morphism $e:\ast\to G$ is the same thing as giving a morphism $h_e:h_\ast\to h_G$.

But, a group object was more than an object equipped with some maps. These maps needed to satisfy certain equational properties. For example, we required, for example, that $m\circ (m\times\text{id})=(m\times\text{id})\circ m$ (the associativity axioms). But, $m\circ (m\times\text{id})$ and $(m\times\text{id})\circ m$ are both elements of $\text{Hom}_\mathscr{C}(G\times G,G)$. But, by Yoneda’s lemma (the injectivity part) to check that two morphisms $G\times G\to G$ are the same, we need only check that their associated natural transformations $h_{G\times G}\to h_G$ are the same.

But, to do this, we need only check that for any object $Z$ of $\mathscr{C}$ the two set maps $h_{G\times G}(Z)\to h_G(Z)$ are equal. But, the equality of the set maps $h_{m\circ (m\times\text{id})}(Z)$ and $h_{(m\times\text{id})\circ m}(Z)$ is clearly equivalent to the associativity of the set maps $h_m(Z)$. In fact, using exactly the same observations, one sees that the maps $m,i$, and $e$ satisfying the axioms of a group object is equivalent to the statement that $h_m(Z)$, which is a binary function $h_X(Z)\times h_X(Z)\to h_X(Z)$, defines a group structure on $h_X(Z)$ with inverse map given by $h_i(Z)$, and identity element $h_e(Z)$ (note that since $\ast$ is a terminal object $h_\ast$ is always just a point, so $h_e(Z)$ is constant, so just picks out the identity element).

Moreover, note that if $\alpha:Z\to Z'$ is a morphism in $\mathscr{C}$, then by definition we obtain a set map $h_X(Z')\to h_X(Z)$ defined by

$h_X(Z')\ni (\gamma:Z'\to X)\mapsto (\gamma\circ \alpha:Z'\to X)\in h_X(Z)$

Note then that

$h_m(Z)(\gamma,\gamma')\circ\alpha=h_m(Z')(\gamma\circ\alpha,\gamma\circ\alpha')$

purely because $h_m$ is a natural transformation $h_G\times h_G\to h_G$. But, since $h_m(Z)$ is the multiplication map for the group structure on $h_X(Z)$, and $h_m(Z')$ is similarly the multiplication structure on $h_X(Z')$, we find that $h_X(\alpha):h_X(Z')\to h_X(Z)$ is a group map.

Thus, we see that the group object $(G,m,i,e)$ actually defines a functor $G:\mathscr{C}^\text{op}\to\mathbf{Grp}$ whose underlying set functor (i.e. the composition of this functor with the forgetful functor $\mathbf{Grp}\to\mathbf{Set}$ is just $h_G:\mathscr{C}^\text{op}\to\mathbf{Set}$.

Tracing the argument backwards shows that the converse is true, and thus we obtain the following theorem:

Theorem 2: To give a group object $G$ in $\mathscr{C}$ is the same as giving a factorization of the functor $h_G:\mathscr{C}^\text{op}\to\mathbf{Set}$ through the forgetful functor $\mathbf{Grp}\to\mathbf{Set}$.

Note that various factorizations of $h_G$ through the forgetful functor correspond to putting different group structures on the same object $G$ of $\mathscr{C}$.

Moreover, we can actually take Theorem 2 one step further. Namely, what does it mean to give a morphism from a group object $(G,i,m,e)$ to a group object $(G',m',i',e')$? It means to give a morphism $f:G \to G'$ in $\mathscr{C}$ such that $m'\circ (f\times f)= f \circ m$. But, by Yoneda’s lemma, and the above observation, this is equivalent to requiring that the induced map $h_f:h_G\to h_{G'}$ satisfy $h_{m'}\circ h_{f\times f}=h_f\circ h_m$. But, this is precisely the statement that for each object $Z$ of $\mathscr{C}$ the induced set map $h_f(Z):h_G(Z)\to h_{G'}(Z)$ is actually a map of groups. Thus, we have actually defined a natural transformation $G\to G'$ (where these are the associated functors to group). Moreover, once again by Yoneda, all such morphisms of group objects arise in this way:

Theorem 3: If $(G,m,i,e)$ and $(G',m',i',e')$ are group objects in $\mathscr{C}$, with associated factorization $h_G=\text{Forget}\circ F_G$ and $h_{G'}=\text{Forget}\circ F_{G'}$, then to give a morphism of group objects $(G,m,i,e)\to (G',m',i',e')$ is the same thing as giving a natural transformation $F_G\to F_{G'}$.

Because of these theorems, we will often times abuse notation (an abuse which, as they usually are, is extremely useful) and denote the group object in $\mathscr{C}$ and the associated functor $\mathscr{C}^\text{op}\to\mathbf{Grp}$ both by the symbol $G$. So, for example, instead of the group $h_G(X)$, we will now write $G(X)$.

Another way to phrase the above is that a group object in $\mathscr{C}$ is that it is a representable functor $\mathscr{C}^\text{op}\to\mathbf{Grp}$ (or, more technically, representable when composed with the forgetful functor to $\mathbf{Set}$), and to to give a morphism between group objects is simply to give a natural transformation between these functors.

Also, from now on, as is customary, we shall drop any reference to the operations of a group object $(G,m,i,e)$ and refer to it only as $G$, unless confusion would arise.

Group Schemes

Now that we have set up all of this notation, it is very simple to define group schemes. Namely, for a scheme $S$, we define a group scheme over $S$, to be a group object in $\mathbf{Sch}/S$. We shall, almost always, only talk about group schemes over an affine scheme $S=\text{Spec}(R)$ (often times when $R$ is a field), in which case we shall also refer to group schemes ‘over $R$‘.

Note then that the morphisms of a group $G$ object take the form of $S$-morphisms $m:G\times_S G\to G$, $i:G\to G$, and $e:S\to G$ (i.e. a section of the structure morphism $G\to S$).

One nice property about group schemes, is that it suffices to actually give their values on affines. Roughly, this is because affine schemes over $S$ are “dense” in $\mathbf{Sch}/S$. To make this rigorous, let us define, for a scheme $S$, the category $\mathbf{Aff}/S$ to be the full subcategory of affine schemes in $\mathbf{Sch}/S$. The claim is then the following:

Theorem 4: The following sets of data are equivalent, for an object $G$ in $\mathbf{Sch}/S$:

1. Giving a factorization of $G$ as $\text{Forget}\circ F$ for some functor $F:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Set}$.
2. Giving a factorization of $G\mid_{(\mathbf{Aff}/S)^\text{op}}:(\mathbf{Aff}/S)^\text{op}\to\mathbf{Set}$ as $\text{Forget}\circ F$, for some functor $F:(\mathbf{Aff}/S)^\text{op}\to\mathbf{Set}$.

Proof: We clearly have a map from the data in 1. to the data in 2., just by taking a factorization $G=\text{Forget}\circ F$ and mapping it to $G\mid_{(\mathbf{Aff}/S)^\text{op}}=\text{Forget}\circ F\mid_{(\mathbf{Aff}/S)^\text{op}}$. We proceed to show that this is “injective” and “surjective”.

For injectivity, we prove something stronger, in the form of the following lemma:

Lemma 5: Let $G:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Set}$ and $H:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Set}$ be sheaves in the Zariski topology (you can just think of $G$ and $H$ as representable, if that’s easier). Then, any natural isomorphism $\eta:G\mid_{(\mathbf{Aff}/S)^\text{op}}\xrightarrow{\approx} H\mid_{(\mathbf{Aff}/S)^\text{op}}$ can be lifted to an isomorphism $\widetilde{\eta}:G\xrightarrow{\approx}H$. Moreover, if $G$ and $H$ are given a factorization through $\mathbf{Grp}$, and $\eta$ is a natural isomorphism of functors to $\mathbf{Grp}$, then so is $\widetilde{\eta}$.

Proof: We need to define $\eta(X):G(X)\to H(X)$ for an arbitrary $S$-scheme $X$. To do this, let $\{U_i\}$ be an open cover of $X$ by affine open subschemes, and for each pair $(i,j)$, let $\{U_{ijk}\}$ be an open cover of $U_{ij}:=U_i\cap U_j$ by affine open subschemes. We then have the following diagram

$\displaystyle \begin{matrix}G(X) & \to & \prod_i G(U_i) & \overset{\xrightarrow{ }}{\to}& \prod_{i,j,k}G(U_{ijk})\\ & & \downarrow & & \downarrow\\ H(X) & \to & \prod_i H(U_i) & \overset{\xrightarrow{ }}{\to} & \prod_{i,j,k}H(U_{ijk})\end{matrix}\quad{\mathbf{(\ast)}}$

where the first map in the top row is the obvious one, and the second is the composition of the following arrows

$\displaystyle \prod_i G(U_i)\overset{\xrightarrow{ }}{\to}\prod_{i,j}G(U_{i,j})\to \prod_{i,j,k}G(U_{i,j,k})$

where the first pair of arrows is the usual restrictions, and the second arrow is the product of the usual arrows $G(U_{ij})\to\prod_k G(U_{ijk})$. So, in essence, it takes a tuple $(f_i)$ and sends it along the first arrow to the tuple which, in the $ijk$ entry has $(f_i\mid_{U_{ij}})\mid_{U_{ijk}}$ and along the second arrow sends it to the tuple which in the $ijk$ entry has $(f_j\mid_{U_{ij}})\mid_{U_{ijk}}$. The maps for $H$ are defined similarly. The vertical arrows are the arrows $\eta(U_{ij})$ and $\eta(U_{ijk})$.

Now, since

$\displaystyle G(X)\to\prod_i G(U_i)\to\prod_{i,j}G(U_{ij})$

is an equalizer, and $\prod_{i,j}G(U_{i,j})\to\prod_{i,j,k}G(U_{ijk})$ is injective, we see that

$\displaystyle G(X) \to \prod_i G(U_i) \overset{\xrightarrow{ }}{\to}\prod_{i,j,k}G(U_{ijk})$

is also an equalizer. The analogous statement for $H$ is also true. Thus, we obtain an arrow $G(X)\to H(X)$ in $(\ast)$ by the universal property of equalizers, and the fact that the right-hand squares in $(\ast)$ commute (this is just the product of the naturality squares for $\eta$).

One can check, by refining both open covers, that the map $\eta(X):G(X)\to H(X)$ is independent of open cover. And, of course, since each $\eta(U_{ij})$ And $\eta(U_{ijk})$ is an isomorphism, so is the map $\eta(X):G(X)\to H(X)$. Lastly, to check the naturality of $\eta$ follows by definition, and the fact that the $\eta$ are natural on the affine schemes.

Finally, if $G$ and $H$ were group functors, and $\eta$ was a natural transformation of group functors, then it’s clear that the constructed maps $\eta(X)$ are group maps, and thus the lifted natural transformation $\widetilde{\eta}$ is actually an isomorphism of group functors as desired. $\blacksquare$

We must now show the surjectivity of this map. Namely, let’s suppose that we have put a group structure on $G(U)$ for every affine scheme $U$. For an aribtrary $S$-scheme $X$ define the group structure on $G(X)$ to be $\varprojlim G(U_i)$, where $\{U_i\}$ is the set of all affine open subschemes of $X$, made into a directed set by definiing $U_i\to U_j$ to be $U_j\hookrightarrow U_i$ (the opposite of inclusion maps). We must now show that this group structure is functorial in $G$, in the sense that for every arrow $f:X\to Y$, the map $G(Y)\to G(X)$ is a group map, where each are given the above described group structure. But, for each affine open $V_j$ of $Y$, we obtain a map $G(V_j)\to \varprojlim G(U_\ell)$, where $\{U_\ell\}$ are the affine open subschemes of $f^{-1}(V_j)$ which is a group map. One then recovers the induced map

$G(Y)=\varprojlim G(V_j)\to \varprojlim G(U_i)=G(X)$

as the inverse limit of $G(V_j)\to \varprojlim G(U_\ell)$ over all $j$.

To conclude we need to check that this functor $X\mapsto \varprojlim G(U_j)$ agrees with the original group structure when restricted to $\mathbf{Aff}/S$. But, this is just because $G\mid_{(\mathbf{Aff}/S)^\text{op}}:(\mathbf{Aff}/S)^\text{op}\to\mathbf{Grp}$ is a sheaf. Thus,

$G(U)=\text{Hom}(U,G)=\text{Hom}(\varinjlim U_i,G))=\varprojlim \text{Hom}(U_i,G)=\varprojlim G(U_i)$

for every affine scheme $U$, and that this isomorphism is an isomorphism of groups, and is functorial in $U$. $\blacksquare$

While this example isn’t the most useful in practice, it is extremely philosophically appealing. It really does tell us that group functors are really determined by their action on affine schemes.

We shall a group scheme an affine group scheme if the underlying scheme is affine.

Some Key Examples

Now that we know how to define group schemes, let’s give some good examples of them to keep in mind. To keep things simple, we will assume, in what follows, that $S=\text{Spec}(R)$, for some ring $R$. That said, all parts that make sense for a general base (e.g. the functorial definitions of $\mathbb{G}_a$ and $\mathbb{G}_m$) work over any base.

$\text{ }$

For perhaps the simplest example, let’s consider the functor $\mathbb{G}_a:(\textbf{Sch}/S)^\text{op}\to\mathbf{Grp}$ (or $\mathbb{G}_{a,S}$ if we’re emphasizing the base) which on objects sends $X\to S$ to $(\mathcal{O}_X(X),+)$ (i.e. just thinking about $\mathcal{O}_{X}(X)$ as an additive group). On morphisms $\mathbb{G}_a$ sends an $S$-morphism $X\to Y$ to the associated map of abelian groups $\mathcal{O}_Y(Y)\to \mathcal{O}_X(X)$. It is clear to see that this is functorial.

But, to conclude that $\mathbb{G}_a$ is actually an affine group scheme over $S$ we need to show that $\mathbb{G}_a$, composed with the forgetful functor $\mathbf{Grp}\to\mathbf{Set}$, is representable. To do this, merely note that $\mathbb{G}_a$ is the same thing as $h_{\mathbb{A}^1_S}$. Thus, $\mathbb{G}_a$ is an affine group scheme over $S$, with representing object $\mathbb{A}^1_S$.

For completeness, we should mention what the explicit operations on $\mathbb{A}^1_S$ are. To this end, we need to define maps $m:\mathbb{A}^1_S\times\mathbb{A}^1_S\to\mathbb{A}^1_S$, $i:\mathbb{A}^1_S\to\mathbb{A}^1_S$, and $e:S\to\mathbb{A}^1_S$ (note that $S$ [or more technically $S\xrightarrow{\text{id}}S$] is the terminal object]. Well, to give such an $m$ is the same thing as giving an $R$-algebra map $m^\ast:R[t]\to R[x,y]$, $i$ is the same as giving a $R$-algebra map $i^\ast:R[t]\to R[t]$, and $e$ is the same thing as giving a $R$-algebra map $e^\ast:R[t]\to R$. So, let $m^\ast$ be defined by $t\mapsto x+y$, $i^\ast$ as $t\mapsto -t$, and $e^\ast(t)=0$. One can easily check that the desired relations hold, and that this does, in fact, define the same affine group scheme as $\mathbb{G}_a$.

We shall $\mathbb{G}_a$ the additive group over $S$ (or sometimes just ‘over $R$‘).

$\text{ }$

The next fundamental example is the next obvious choice after the additive group. Define the functor $\mathbb{G}_m:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Grp}$ on objects by $\mathbb{G}_m(X)=\mathcal{O}_X(X)^\times$ (with the usual product), and for an $S$-morphism $X\to Y$ associate the group map $\mathcal{O}_Y(Y)^\times\to \mathcal{O}_X(X)^\times$ coming from the associated ring map $\mathcal{O}_Y(Y)\to \mathcal{O}_X(X)$. It is clear to see that $\mathbb{G}_m$ really is a functor.

To see that $\mathbb{G}_m$ is an affine group scheme, we need to show that its composition with the forgetful functor is representable. But, one can quickly check that this is the case, in particular, that $\mathbb{G}_m$ is isomorphic to $h_{\mathbb{A}^1_S-\{0\}}$ (where $\mathbb{A}^1_S-\{0\}$ is more precisely written as $\text{Spec}(R[x,y]/(xy-1))$).

Once again, let’s actually say what the group operation on $X:=\mathbb{A}^1_S-\{0\}$ is. Once again, we need to define maps $m:X\times X\to X$, $i:X\to X$, and $e:S\to X$. As before, this is equivalent to defining $k$-algebra map $m^\ast:R[t,t^{-1}]\to R[x,y,x^{-1},y^{-1}]$, $i^\ast:R[t,t^{-1}]\to R[t,t^{-1}]$, and $e^\ast:R[t,t^{-1}]\to R$. Well, define $m^\ast$ to be such that $t\mapsto xy$, $i^\ast$ as such that $t\mapsto t^{-1}$, and $e^\ast$ as such that $t\mapsto 1$. Once again, one quickly checks that the group structure, and maps, that $h_X$ then define, as a functor to $\mathbf{Grp}$, coincide with $\mathbb{G}_m$.

We call this the multiplicative group over $S$ (or sometimes just ‘over $k$‘). What the above allows us to do

$\text{ }$

As a generalization of the last example, we consider higher dimensional analogues of the multiplicative group. Namely, let $\text{GL}_n$ be the functor $(\mathbf{Sch}/S)^\text{op}\to\mathbf{Grp}$ which associates to every scheme $X$, the group $\text{GL}_n(\mathcal{O}_X(X))$, and which associates to a scheme map $X\to Y$, the associated group map $\text{GL}_n(\mathcal{O}_Y(Y))\to\text{GL}_n(\mathcal{O}_X(X))$ afforded to us by the ring map $\mathcal{O}_Y(Y)\to\mathcal{O}_X(X)$. One quickly checks that this is functorial.

For the third time, to check that $\text{GL}_n$ really does define an affine group scheme over $S$, we need to show that the composition of $\text{GL}_n$ with the forgetful functor is representable. But, as one can quickly check, the scheme $\text{GL}_n$

$\text{Spec}(R[x_{1,1},x_{1,2},\ldots,x_{n,n},\frac{1}{\det [x_{i,j}]})$

represents $\text{GL}_n$. Thus, $\text{GL}_n$ really is an affine group scheme over $S$.

Now as is becoming clear, we define the operations on $\text{GL}_n$, we need to define $R$-algebra maps

$m^\ast:R[t_{1,1},\ldots,t_{n,n},\frac{1}{\det [t_{i,j}]}]\to R[x_{1,1},\ldots,x_{n,n},y_{1,1},\ldots,y_{n,n},\frac{1}{\det [x_{i,j}]},\frac{1}{\det [y_{i,j}]}]$

, $i^\ast:R[t_{1,1},\ldots,t_{n,n},\frac{1}{\det [t_{i,j}]}]\to R[t_{1,1},\ldots,t_{n,n},\frac{1}{\det [t_{i,j}]}]$, and $e^\ast:R[t_{1,1},\ldots,t_{n,n},\frac{1}{\det [t_{i,j}]}]\to R$. We define these as follows:

$m^\ast:t_{i,j}\mapsto \sum_{k=1}^{n}x_{i,k}y_{k,j}$

$i^\ast:t_{i,j}\mapsto \frac{1}{\det[t_{i,j}]} a_{i,j}$

where $a_{i,j}$ is the $(i,j)^\text{th}$ entry of the adjugate matrix of $[t_{i,j}]$ (which is a polynomial in $t_{i,j}$!), and

$e^\ast: t_{i,j}\mapsto \begin{cases}1 & \mbox{if}\quad i=j\\ 0 & \mbox{if}\quad i\ne j\end{cases}$

as expected.

$\text{ }$

Using the exact same idea as the previous example, one can define $\text{SL}_n:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Set}$. One can show that it is represented by $\text{Spec}(R[x_{i,j},\det([x_{i,j}])-1])$.

$\text{ }$

As the next example, we can consider the functor ${\boldsymbol {\mu}}_n:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Grp}$ defined on objects by

${\boldsymbol \mu}_n(X):=\left\{r\in \mathcal{O}_X(X)^\times:r^n=1\right\}$

and on morphisms by sending $f:X\to Y$ the natural map ${\boldsymbol \mu}_n(Y)\to{\boldsymbol \mu}_n(X)$ afforded to us by the ring map $\mathcal{O}_Y(Y)\to\mathcal{O}_X(X)$.

One can easily check that ${\boldsymbol \mu}_n$ is representable, with representing scheme $\text{Spec}(R[t]/(t^n-1))$.

The explicit group operations on ${\boldsymbol \mu}_n$ are given by

$m^\ast:R[t]/(t^n-1)\to R[x,y]/(x^n-1,y^n-1):t\mapsto xy$,

$i:R[t]/(t^n-1)\to R[t]/(t^n-1):t\mapsto t^{-1}$,

and,

$e:R[t]/(t^n-1)\to R:t\mapsto 1$

as one would expect. As one can probably guess, in this way, we have actually shown that ${\boldsymbol \mu}_n$ is a subgroup scheme of $\mathbb{G}_m$.

$\text{ }$

For this example, we assume that $R$ is actually a field of characteristic $p>0$. We can then define, for each $n\in\mathbb{N}$, an additive analogue of ${\boldsymbol \mu}_{p^n}$. Namely, define a functor ${\boldsymbol \alpha}_{p^n}:(\mathbf{Sch}/S)^\text{op}\to\mathbf{Grp}$ defined on objects by

${\boldsymbol \alpha}_{p^n}(X):=\left\{r\in \mathcal{O}_X(X):r^{p^n}=0\right\}$

and on morphisms in the obvious way.

To show that this is actually a group scheme, we need to show that its composition with the forgetful functor is representable. But, this is easily achieved. Namely, one just checks that $\text{Spec}{(k[t]/(t^{p^n})}$ represents ${\boldsymbol \alpha}_{p^n}$, as a functor to $\mathbf{Set}$.

Finally, every elliptic curve $E/S$ is a proper group scheme over $S$. For a discussion of group schemes see Ben Moonen’s book on Abelian Varieties.