Elimination Theory, Generic Freeness, and the Dimension of Fibers

Abstract

In this chapter we shall study the following question: Given a homomorphism φ :R → S of Noetherian rings such that S is a finitely generated R-algebra, how do the “fibers” S ⊗ _P K(R/P) vary as we vary the prime P of R? If S is flat over R, then as we have seen, there is some sense in which the fibers vary continuously. The main result below, Grothendieck’s generic freeness lemma, a consequence of the Noether normalization theorem, implies that if R ⊂ S are domains, then flatness always holds over a nonempty open set of R, so that “most” fibers share common properties.