okay little thread about group schemes for checking that I'm getting them so far... (it's rather technical )

as usual, feel free to add anything and correct my mistakes!

so first of all, what is a group scheme? there are several definitions but I will talk about two of them

let's begin with definition #1, which I find it the most general:

"a group scheme is a group object in the category of S-schemes."

So we need to understand what's a group object and why/what is a S-scheme. I'm going to start with the reasons for the later

The philosophy here is based in two considerations:

1) we want to make morphisms between schemes our main objects of study rather than schemes themselves

2) not only that, we always want to work over a fixed, base scheme S. This means,

that in the category of S-schemes, if we have a morphism between two schemes T->T' we also have morphisms T->S and T'->S so that the resulting diagram commutes.

This is a really useful point of view since you can think of the S-scheme morphism T->S as a collection of schemes

parametrized by the points of S. Also, it allows us to perform base change, by taking pullback over S. Explicitly: if T, T' are S-schemes, B=T \\times_S T' is the base change from S to T' of T.

These are the main reasons for working in the category of S-schemes

Let's move on to the "group object" part now.

Let C be a category with finite products and terminal object that I will denote by 1. Then a group object in C is exactly what one can expect: it's an object G of C together with three maps:

1) the multiplicative map m: G \\times G->G,
2) the identity map (section) e: 1->G,
3) the inversion map i: G->G,

such that they satisfy the usual group axioms, but we need to state them in terms of diagrams now. For example, asking for associativity is equivalent to ask for

the first diagram here to commute.
Similarly, identity corresponds to the second diagram commuting, and inversion to the third one.

Moreover, if G makes the fourth diagram commutative, we say that G is commutative. (\\tau is a map that switches factors)

okay, now that we have the ingredients for the definition of group scheme, the next step is to define what is a homomorphism between them:

if G and H are group schemes, a morphism G->H in C is nothing but a homomorphism of group schemes so that all the relevant diagrams commute

Now I'm going to give definition #2 as an excuse for giving another useful point of view:

"a group scheme (over S) is a contravariant functor from the category of S-schemes to the category of groups such that the underlying functor to the category of sets is representable."

this definition emphasizes the importance of the functor of points here.

In general, let X be an object of the category C. Define h_X(Y) = Hom_C(Y,X) for any object Y in C. h_X defines a contravariant functor from C to Set, and by Yoneda we know that X is completely determined

from the functor h_X.

Now let's suppose that G is a group object in C. Then the corresponding functor h_G inherits the group structure of G. Moreover, if we have a morphism f:Y->Y' in C, h_G induces a map f*:h_G(Y')->h_G(Y) which is a group homomorphism!

The converse is also true: if G is an object in C such that h_G is endowed with group structure and each such f* is a group homomorphism, then G is naturally a group object in C.

This fact means that providing G with group structure is the same as extending h_G to a functor

from C to the category of groups.

Anyway, this was the general setting but we are interested in the case C = category of S-schemes, in order to fit definition #2.

Let's give some examples of group schemes and see which scheme represents the functor h_G.

But first I'm just going to emphasize this important point again: natural transformations h_X->h_Y correspond bijectively with morphisms of S-schemes X->Y by what we've just been discussing.

Okay, let's look at the following example: the multiplicative group scheme, denoted G_m.

We can regard G_m as a commutative group variety (and an affine variety), in this case it's isomorphic to the nonzero points of A^1, and the group law G_m \\times G_m->G_m is just given by the multiplication (x,y) -> x*y.

If we regard it as a scheme over Z, G_m is by definition

Spec Z[x,x^-1].

Since Spec Z is final in the category of schemes, we can say that G_m over an arbitrary scheme S is the group scheme G_m \\times_Z S (remember base change discussed before?) which is, in particular, Spec R[x,x^-1] for any ring R.

Let's now consider a "sub"example of this.

Let T=Spec R for any ring R, let μ_n=Spec Z[x]/(x^n-1). The corresponding functor here is Hom(T, μ_n) = {x in R such that x^n=1}. This is nothing but the n-th roots of unity in R, which is a commutative group and so μ_n is naturally

a commutative group scheme.

I won't give any more examples but group schemes are a very useful tool to, in particular, study families of elliptic curves parametrized by points of a scheme.

There's a lot more to say about group schemes, like, their relation to Hopf algebras,

Cartier duality, the fact that the category of finite commutative group schemes over k is an abelian category, and so much more. But I haven't covered these interesting topics yet.

A good reference to read about this in more detail is Silverman's AEC II, chapter IV.

Thank you for reading and happy Valentine's day!