§1.0. Background: Affine Varieties 5 In Exercise 1.0.3 you will prove that Spec(C[V] f ) is the affine variety Vf . Example 1.0.2. The n-dimensional torus is the affine open subset (C∗)n = Cn \ V(x1 ···xn) ⊆ Cn, with coordinate ring C[x1,...,xn]x 1 ···xn = C[x±1,...,x±1]. 1 n Elements of this ring are called Laurent polynomials. ♦ The ring C[V] f from (1.0.2) is an example of localization. In Exercises 1.0.2 and 1.0.3 you will show how to construct this ring for all affine varieties, not just irreducible ones. The general concept of localization is discussed in standard texts in commutative algebra such as [10, Ch. 3] and [89, Ch. 2]. Normal Affine Varieties. Let R be an integral domain with field of fractions K. Then R is normal, or integrally closed, if every element of K which is integral over R (meaning that it is a root of a monic polynomial in R[x]) actually lies in R. For example, any UFD is normal (Exercise 1.0.5). Definition 1.0.3. An irreducible affine variety V is normal if its coordinate ring C[V] is normal. For example, Cn is normal since its coordinate ring C[x1,...,xn] is a UFD and hence normal. Here is an example of a nonnormal affine variety. Example 1.0.4. Let C = V(x3 −y2) ⊆ C2. This is an irreducible plane curve with a cusp at the origin. It is easy to see that C[C] = C[x,y]/ x3 −y2 . Now let ¯and ¯be the cosets of x and y in C[C] respectively. This gives ¯/¯ ∈ C(C). A computation shows that ¯/¯ / C[C] and that (¯/¯)2 = ¯. Consequently C[C] and hence C are not normal. We will see below that C is an affine toric variety. ♦ An irreducible affine variety V has a normalization defined as follows. Let C[V] = {α ∈ C(V) : α is integral over C[V]}. We call C[V] the integral closure of C[V]. One can show that C[V] is normal and (with more work) finitely generated as a C-algebra (see [89, Cor. 13.13]). This gives the normal affine variety V = Spec(C[V] ). We call V the normalization of V. The natural inclusion C[V] ⊆ C[V] = C[V ] corresponds to a map V → V . This is the normalization map. Example 1.0.5. We saw in Example 1.0.4 that the curve C ⊆ C2 defined by x3 = y2 has elements ¯, ¯ ∈ C[C] such that ¯/¯ / C[C] is integral over C[C]. In Exer- cise 1.0.6 you will show that C[¯/¯] ⊆ C(C) is the integral closure of C[C] and that the normalization map is the map C → C defined by t → (t2,t3). ♦

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