This chapter contains preliminaries for the general study of reductive groups in the next chapter. In particular, we show that every split reductive group of semisimple rank 1 is isomorphic to exactly one of the following groups:
Group varieties of semisimple rank 0
Recall (19.20) that the rank of a group variety is the dimension of a maximal torus, and that this does not change under extension of the base field. The semisimple rank of G is the rank of the largest semisimple quotient of Gka.
THEOREM 20.1. Let G be a connected group variety over a field k.
(a) G has rank 0 if and only if it is unipotent.
(b) G has semisimple rank 0 if and only if it is solvable.
(c) G is reductive of semisimple rank 0 if and only if it is a torus.
PROOF. We may suppose that k is algebraically closed.
(a) This is a restatement of Proposition 16.60.
(b) If G is solvable, then G D R.G/, and so it has semisimple rank 0. Conversely, if G has semisimple rank 0, then G=R.G/ is unipotent (by (a)) and semisimple (19.2), and hence trivial. Thus, and so G is solvable.
(c) A torus is certainly reductive of semisimple rank 0. Conversely, if G is reductive of semisimple rank 0, then it is solvable. As Ru.G/ D e, this implies that G is a torus (16.33d).
Let C be a regular complete algebraic curve over k. The local ring at a point is a discrete valuation ring containing k with field of fractions, and every such discrete valuation ring arises from a unique P. Therefore, we can identify with the set of such discrete valuation rings in endowed with the topology for which the proper closed subsets are the finite sets. For an open subset U, the ring. Thus, we can recover C from its function field. In particular, two regular complete connected curves over k are isomorphic if they have isomorphic function fields.