4 Chapter 1. Affine Toric Varieties • A point p of an affine variety V gives the maximal ideal { f ∈ C[V] | f (p) = 0} ⊆ C[V], and all maximal ideals of C[V] arise this way. Coordinate rings of affine varieties can be characterized as follows (Exercise 1.0.1). Lemma 1.0.1. A C-algebra R is isomorphic to the coordinate ring of an affine variety if and only if R is a finitely generated C-algebra with no nonzero nilpotents, i.e., if f ∈ R satisfies f = 0 for some ≥ 1, then f = 0. To emphasize the close relation between V and C[V], we sometimes write (1.0.1) V = Spec(C[V]). This can be made canonical by identifying V with the set of maximal ideals of C[V] via the fourth bullet above. More generally, one can take any commutative ring R and define the affine scheme Spec(R). The general definition of Spec uses all prime ideals of R, not just the maximal ideals as we have done. Thus some authors would write (1.0.1) as V = Specm(C[V]), the maximal spectrum of C[V]. Readers wishing to learn about affine schemes should consult [90] and [131]. The Zariski Topology. An affine variety V ⊆ Cn has two topologies we will use. The first is the classical topology, induced from the usual topology on Cn. The second is the Zariski topology, where the Zariski closed sets are subvarieties of V (meaning affine varieties of Cn contained in V ) and the Zariski open sets are their complements. Since subvarieties are closed in the classical topology (polynomials are continuous), Zariski open subsets are open in the classical topology. Given a subset S ⊆ V, its closure S in the Zariski topology is the smallest subvariety of V containing S. We call S the Zariski closure of S. It is easy to give examples where this differs from the closure in the classical topology. Affine Open Subsets and Localization. Some Zariski open subsets of an affine variety V are themselves affine varieties. Given f ∈ C[V] \{0}, let Vf = {p ∈ V | f (p) = 0} ⊆ V. Then Vf is Zariski open in V and is also an affine variety, as we now explain. Let V ⊆ Cn have I(V) = f1,..., fs and pick g ∈ C[x1,...,xn] representing f . Then Vf = V \ V(g) is Zariski open in V. Now consider a new variable y and let W = V( f1,..., fs,1 − gy) ⊆ Cn ×C. Since the projection map Cn ×C → Cn maps W bijectively onto Vf , we can identify Vf with the affine variety W ⊆ Cn ×C. When V is irreducible, the coordinate ring of Vf is easy to describe. Let C(V) be the field of fractions of the integral domain C[V]. Recall that elements of C(V) give rational functions on V. Then let (1.0.2) C[V] f = {g/ f ∈ C(V) | g ∈ C[V], ≥ 0}.

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