1.1. Affine varieties 5 For an ideal I of C[X1,...,Xn], we can easily see the inclusion I I(V(I)). When the field C is algebraically closed, equality is given by the next theorem. Theorem 1.1.6. (Hilbert’s Nullstellensatz) Let C be an algebraically closed field and let A = C[X1,...,Xn]. Then the following hold: a) Every maximal ideal M of A is of the form M = (X1 x1,X2 x2,...,Xn xn) = I(P ), for some point P = (x1, x2,...,xn) in An C . b) If I is a proper ideal of A, then V(I) = ∅. c) If I is any ideal in A, then I = I(V(I)). Remark 1.1.7. Point b) justifies the name of the theorem, namely “theorem on the zeros”. To see the necessity of the condition C algebraically closed, we can consider the ideal (X2 + 1) in R[X]. For the proof of Hilbert’s Nullstellensatz we shall use the following result, which is valuable on its own. Proposition 1.1.8 (Noether’s normalization lemma). Let C be an arbitrary field, R = C[x1,...,xn] a finitely generated C-algebra. Then there exist elements y1,...,yr R, with r n, algebraically independent over C such that R is integral over C[y1,...,yr]. Proof. Let ϕ : C[X1,...,Xn] C[x1,...,xn] be the C-algebra morphism given by ϕ(Xi) = xi, 1 i n. Clearly, ϕ is an epimorphism. If it is an isomorphism, we just take yi := xi, 1 i n. If not, let f = ai 1 ...in Xi1 1 . . . Xin n be a nonzero polynomial in Ker ϕ. We introduce an or- der relation in the set of monomials by defining ai 1 ...in Xi1 1 . . . Xnni a i 1 ...i n X i 1 1 . . . Xnn i if and only if (i1, . . . , in) (i 1 , . . . , i n ), with respect to the lexicographical order, i.e. for some k {1,...,n}, we have il = i l if l k and ik i k . Let aj 1 ...jn Xj1 1 . . . Xnn j be the largest nonzero monomial in f. We can assume aj 1 ...jn = 1. Let now d be an integer greater than all the exponents of the n variables appearing in the nonzero monomials of f. We consider the polynomial h(X1,...,Xn) := f(X1 + Xdn−1,X n 2 + Xdn−2,...,X n n−1 + Xd,Xn). n The monomial ai 1 ...i n Xi1 1 . . . Xin n in f becomes under the change of variables ai 1 ...in Xn1dn−1+i2dn−2+···+in−1d+in+ i terms of lower degree in Xn hence h is
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