In this chapter, we will need some additional algebraic geometry.

Definition 16.1. Let X,Y be irreducible affine varieties, φ: X → Y a dominant morphism. Then

1. (i)φ is finite if K[X] is an integral extension of φ*(K[Y]) ≡ K[Y].

2. (ii) φ is birational if there exists a non–empty open subset U of Y such that φ | φ–1 (U) : φ–1 (U) →U is an isomorphism.

Remark 16.2.

1. (i) If φ is finite, then it is closed (and hence surjective) and the inverse image of any point in Y is a finite set. See [34; Proposition 4.2].

2. (ii) Let be connected regular monoids with zero, a dominant homomorphism. If φ is finite, the clearly φ–1 (0) = {0}. The converse has been shown by Renner [91; Proposition 3.4.13], Equivalently φ is idempotent separating (Theorem 10.12). The homomorphism of Example 3.12 is not finite.

Definition 16.3. Let X be an irreducible affine variety. Then

1. (i) X is normal if K[X] is integrally closed, i.e., the integral closure of K[X] in K(X) is K[X].

2. (ii) The normalization of X is where is an irreducible normal affine variety, is a finite birational morphism.

Remark 16.4. Normality is a local property: X is normal if and only if for all x ε X, the ring 0x – {f/g|f,g ε K[X], g(x) ≠ 0} is integrally closed. In particular if X is normal, then any affine open subset of X is also normal.