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
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.
