Chapter 8 developed geometric invariant theory. More accurately it developed the simplified, bare-bones version which suffices for us. In Section 8.8, especially in Theorem 8.8.6, we stated the main results of this simplified version of geometric invariant theory. Now we have to prove these claims. This chapter is devoted to the proofs.

In previous chapters the opening paragraphs consisted of a summary of the main results. In this chapter such a summary would be out of place, for the simple reason that it already occurred in Theorem 8.8.6. As we learned, in Theorem 8.8.6, the main results of this chapter tell us that certain open subsets of the ringed space (X/G, OX/G) are schemes of finite type over ℂ. The results then go on to describe the coherent sheaves on the quotient, and the fact that exact sequences of coherent G–sheaves on X yield exact sequences of coherent sheaves on the open subset V ⊂ X/G.

Since there is little point in repeating the statement of Theorem 8.8.6 in detail, we will spend the remainder of these opening paragraphs telling the reader what else can be found in the chapter. In the course of the proof we do introduce certain ideas which might be worth summarizing.

We spend Sections 9.2 and 9.3 discussing the linear representations of affine group schemes (G, OG). Once again this is a subject of tremendous independent interest.