1.1. Affine varieties 15 P1 · · · Pn of distinct prime ideals of A. If V An C is an affine variety, by Proposition 1.1.13, irreducible closed subsets of V correspond to prime ideals of C[X1,...,Xn] containing I(V ) and these in turn correspond to prime ideals of C[V ]. Hence the dimension of V is equal to the Krull dimension of its coordinate ring C[V ]. We now recall that dim C[X1,X2,...,Xn] = n ([Ma] § 14, Theorem 22) which, by the preceding, implies dim An C = n. We recall as well that if a noetherian ring R is integral over a noetherian subring S, then dim S = dim R ([Ma] § 13, Theorem 20). Now in the situation of Noether’s normalization lemma (Proposition 1.1.8) and with the same notations there, we have dim R = r. Hence the dimension of a finitely generated integral domain R over C is equal to the transcendence degree of its fraction field over C. (See Remark 1.1.9.) We then obtain that if V is irreducible, the dimension of V is equal to the Krull dimension of C[V ] and equal to the transcendence degree trdeg[C(V ) : C] of the function field C(V ) of V over C. We now give a geometric interpretation of Noether’s normalization lemma. Let V An be an affine irreducible variety. We consider the ring C[V ] = C[X1,...,Xn]/I(V ) and denote by xi the image of Xi in C[V ], 1 i n. Then C[V ] = C[x1,...,xn] is a finitely generated integral domain over C. By Proposition 1.1.8, there exist elements y1,...,yr C[V ], with r n, al- gebraically independent over C such that C[V ] is integral over C[y1,...,yr]. The elements yi lift to elements yi C[X1,...,Xn], which define a morphism ψ = (y1, . . . , yr) : An Ar. The restriction ϕ of ψ to V is independent of the choice of the yi as yi mod I(V ) = yi. We will now show that the fibers of ϕ are finite and nonempty. Proposition 1.1.28. Let ϕ : V Ar be defined as above. For each P Ar, the fiber ϕ−1(P ) is finite and nonempty. Proof. We first prove that ϕ−1(P ) is finite. As C[V ] is integral over the ring C[y1,...,yr], there exist an integer N and polynomials f i j , 0 j N 1, 1 i n such that xN i + f i N−1 (y1, . . . , yr) xN−1 i + · · · + f i 1 (y1, . . . , yr) xi + f i 0 (y1, . . . , yr) = 0, 1 i n. So, we have XN i + f i N−1 (y1, . . . , yr) XN−1 i + · · · + f i 1 (y1, . . . , yr) Xi + f i 0 (y1, . . . , yr) =: gi(X1,...,Xn) I(V ).
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