In other words, for every scheme one has the contravariant Hom functor.
Thus, a group scheme over is a functor such that the composition of with the forgetful functor is representable. Thus, in practice, and surely in this post, we will often times think about group schemes as just being group-valued contravariant functors on. We will, without comment, take for granted that these functors are, in fact, representable when thought about set-valued functors leaving the details of this to the reader. Let us now give some examples of group schemes over :. Just as an example, in the above, is represented by and the general linear group by. So, let us now define our actual term of interest.
The term is not a misnomer since:. Theorem 1: Let be a group scheme. Then, is a linear algebraic group if and only if there exists an embedding for some. Here the embedding means a closed embedding of groups which is just a group map whose underlying map of schemes is a closed embedding. Thus Theorem 1 tells us that linear algebraic groups are just closed subgroups of the general linear groups—they are groups with faithful representations.
While the proof of Theorem 1 is not too difficult, there is something semi-subtle happening. Indeed, one can define a group scheme over any scheme , and the definition of a linear algebraic group still stands—finite type and affine. Now, one could spend literally years trying to understand the geometry of a general group scheme, or even just of a linear algebraic group cf.
SGA 3 , but that is not our goal here. Thus, let us suffice ourselves with two theorems concerning such geometry. Theorem 2: Let be a linear algebraic group.
Then, is smooth if and only if is geometrically reduced. If has characteristic , then is automatically smooth. One nice corollary of the above is the following. There are many theorems below which, personally, I find very exciting. Conversely, if is geometrically reduced then is reduced. Tag V that there is a dense open for which is regular. But, then for any other closed point one sees that is a smooth neighborhood of , and thus is regular. Thus, all the closed points of are regular, thus is regular, thus is smooth.
The second claim is a famous theorem of Chevalley.
To summarize this proof: smoothness happens on a dense open for reduced schemes, but groups are homogenous every point has a neighborhood isomorphic to a neighborhood of every point so this smoothness propagates over the whole scheme. This line of argument is very common.
Now, this is one of the most complicated parts of the basic theory of group schemes—what does mean? This is, almost everywhere in scheme theory, a difficult question—quotients are hard. In this context, we have the following definition though. Given a map of group schemes over the kernel is the naive kernel i.
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Theorem 3: Let be a linear algebraic group over , and a closed subgroup. Then, exists. If is normal in then this quotient has the unique structure of a group scheme such that the canonical map is a group map. Let us explore the above point just a tiny bit since, frankly, it can be quite confusing.
In particular, let us focus our attention on what might be the most confusing application of Theorem 3. Namely, let us consider two group schemes over assumed to be characteristic for convenience. Now, let us begin by making the following surprising observation:. Namely, for a second, let us denote the group scheme as. Why is this true? For notational convenience i. Indeed, the non-trivial element of acts on this matrix by. That said, this matrix is not in the image of.
Thus, we see that is not a sheaf, and so, by fiat, not representable.
So, the question remains—what are the sections if not? Namely, we have a short exact sequence of fppf sheaves. Thus, we see that the natural map is an isomorphism whenever and. So, for example, we see that. Namely, calling it is wise since where this denominator is embedded as invertible scalar matrices. Again though, we have an exact sequence for all. That said, the above shows how subtle these notions are and how confusing the naming of them is. So, one should not sweat these details too much except to be very mindful that most group schemes like only necessarily give the correct group on geometric points on other points one has to make a cohomology computation.
Let us give some examples of these objects. Then, one can show that this is a -dimensional non-split i. One might recognize this construction since, in essence, this is just the standard embedding of into done on the level of schemes. Consider the following subgroup of. In fact, there is a pretty natural short exact sequence. So, how do tori help us understand number theory better?
Well, the key is the following two theorems which, at first, seem intimidating but are really not:. Theorem 4: Let be a field. Then, there is a canonical bijection between the isomorphism classes of -dimensional tori over and the set where has the discrete topology. Explicitly, this says that there is a bijection between -dimensional tori and the fields and the quadratic Galois extensions.
What is this bijection?
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Well, goes to the split torus and goes to. But, one can make an even more powerful statement which, to me, is a truly spectacular, beautiful result that one would not expect should exist:. Theorem 5: Let be a perfect field and a -dimensional connected smooth linear algebraic group. Then, is one of the following:. Corollary 6: Let be, as usual, a finite field. Then, up to isomorphism, there are only three -dimensional smooth connected linear algebraic groups over.here
They are , , and. OK, this is nice, but so what? So, finally, let us consider what is the beginning of many number theoretic jokes: so an elliptic curve over with bad reduction at walks into a bar…. But, really, let be an elliptic curve. Then, since has class number we can lift to a curve in a canonical way its minimal Weierstrass model. Thus, by Corollary 6 there are four choices for what is.
These four choices have names:. Before we begin in earnest, let us say, again, what the intuition is: reductive group schemes are those with nice representation theory. This manifests itself in characteristic zero in the following nice way:. Theorem 7: Let be characteristic. Then, a linear algebraic group is reductive if and only if is semisimple.