8 Chapter 1. Affine Toric Varieties V1 × V2 should be an affine variety with projections πi : V1 × V2 Vi such that whenever we have a diagram W φ 1 φ2 ν V1 ×V2 π 1 π 2 V1 V2 where φi :W →Vi are morphisms from an affine variety W, there should be a unique morphism ν (the dotted arrow) that makes the diagram commute, i.e., πi ν = φi. For the coordinate rings, this means that whenever we have a diagram C[V2] π∗ 2 φ∗ 2 C[V1] π∗ 1 φ∗ 1 C[V1 ×V2] ν∗ C[W] with C-algebra homomorphisms φ∗ i : C[Vi] C[W], there should be a unique C- algebra homomorphism ν∗ (the dotted arrow) that makes the diagram commute. By the universal mapping property of the tensor product of C-algebras, C[V1]⊗C C[V2] has the mapping properties we want. Since C[V1] ⊗C C[V2] is a finitely generated C-algebra with no nilpotents (see the appendix to this chapter), it is the coordinate ring C[V1 ×V2]. For more on tensor products, see [10, pp. 24–27] or [89, A2.2]. Example 1.0.10. Let V be an affine variety. Since Cn = Spec(C[y1,...,yn]), the product V ×Cn has coordinate ring C[V] ⊗C C[y1,...,yn] = C[V][y1,...,yn]. If V is contained in Cm with I(V) = f1,..., fs⊆ C[x1,...,xm], it follows that I(V ×Cn) = f1,..., fs⊆ C[x1,...,xm,y1,...,yn]. For later purposes, we also note that the coordinate ring of V × (C∗)n is C[V] ⊗C C[y±1,...,y±1] 1 n = C[V][y±1,...,y±1]. 1 n Given affine varieties V1 and V2, we note that the Zariski topology on V1 ×V2 is usually not the product of the Zariski topologies on V1 and V2. Example 1.0.11. Consider C2 = C × C. By definition, a basis for the product of the Zariski topologies consists of sets U1 ×U2 where Ui are Zariski open in C. Such a set is the complement of a union of collections of “horizontal” and “vertical” lines
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