5 - G-spaces and quotients

Summary

One aspect of the theory of linear algebraic groups which has been missing up to now is that of a quotient group. We need to first see how to give the structure of variety to a quotient and it will become clear that we cannot limit ourselves to affine varieties. Thus, we begin by recalling some basic aspects of the general theory of varieties and morphisms.

Actions of algebraic groups

In group theory, it is often helpful to consider actions of groups, for example the action of a group on itself by conjugation. We will find it necessary to consider actions of linear algebraic groups on affine and projective varieties.

For this recall that projective n-space ℙⁿ may be defined as the set of equivalence classes of kⁿ⁺¹ \ {(0, 0, …, 0)} modulo the diagonal action of k^× by multiplication. Taking common zeros of a collection of homogeneous polynomials in k[T₀, T₁, …, T_n] as closed sets defines a topology on ℙⁿ. A projective variety is then a closed subset of ℙⁿ equipped with the induced topology.

The k-algebra of regular functions on an affine variety here needs to be replaced by a sheaf of functions, as follows. First, for X an irreducible affine variety and x ∈ X, let I(x) ◃k[X] be the ideal of functions vanishing at x and let O_x be the localization of k[X] with respect to the prime ideal I(x).