14 1. Affine and Projective Varieties c) There are nonempty open sets V0 V and W0 W such that the re- striction ϕ|V 0 : V0 W0 is an isomorphism. Proof. a) b) The rational map ψ : W V such that ϕ ψ = IdW and ψ ϕ = IdV is regular in a dense open set of W hence ϕ is dominant. As ϕ(V ) is dense in W , ϕ∗ is injective. Analogously, ψ∗ is injective. b) a) We define ψ : W V by taking ψ := (ϕ∗−1(xi))i for xi the restriction to V of the coordinate functions. By construction we have ϕ◦ψ = IdW and ψ ϕ = IdV . a) c) Let ϕ = (ϕ1, . . . , ϕn) and ϕi = fi/F for fi,F C[V ]. Then ϕi C[VF ], for VF the principal open set defined by the nonvanishing of F . Analogously, if ψ = (ψ1, . . . , ψn) and ψi = gi/G for gi,G C[W ], then ψi C[WG]. By taking V0 = VF ψ(dom ψ) and W0 = WG ϕ(dom ϕ), we obtain the result. c) b) ϕ is dominant as W0 ϕ(dom ϕ) and ϕ∗ is an isomorphism from C(W ) = C(W0) onto C(V ) = C(V0). We now prove that every irreducible affine variety is birationally equiv- alent to a hypersurface in some affine space. This fact will be used to determine the dimension of the tangent spaces of a variety. Proposition 1.1.27. Every irreducible affine variety V is birationally equiv- alent to a hypersurface in some affine space An. Proof. The field C(V ) is finitely generated over C. Then by Proposition 1.1.8, there exist elements x1,...,xd in C(V ), algebraically independent over C, such that C(V ) is algebraic over C(x1,...,xd). Since we are assuming char C = 0, we can apply the primitive element theorem and obtain C(V ) = C(x1,...,xd)(xd+1) for some element xd+1 algebraic over C(x1,...,xd). We then have an algebraic dependence relation f(x1,...,xd,xd+1) = 0, with f C[X1,...,Xd+1]. Let W be the hypersurface in An, with n = d + 1, defined by the vanishing of the polynomial f. Then by construction C(W ) C(V ), so V and W are birationally equivalent by Proposition 1.1.26. We shall now introduce the notion of dimension of an affine variety. If X is a noetherian topological space, we define the dimension of X to be the supremum of all integers n such that there exists a chain Z0 Z1 · · · Zn of distinct irreducible closed subsets of X. We define the dimension of an affine variety to be its dimension as a topological space. Clearly the dimension of an affine variety is the maximum of the dimensions of its irreducible components. For a ring A, we define the Krull dimension dim A of A to be the supremum of all integers n such that there exists a chain P0
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