1.1. Affine varieties 7 Proposition 1.1.11. Let C be an arbitrary field, R = C[x1,...,xn] a finitely generated C-algebra. If R is a field, then it is algebraic over C. Proof. By Noether’s normalization Lemma 1.1.8, there exist elements y1,..., yr R, with r n, algebraically independent over C such that R is inte- gral over A := C[y1,...,yd], hence a finite A-algebra. This implies that A is a field. Indeed, a nonzero element α in A has an inverse α−1 in R. Writing down an integral dependence relation for α−1 over A, α−n + a1 α−n+1 + · · · + an−1 α−1 + an = 0, and multiplying it by αn−1, we obtain α−1 = −(a1 + · · · + an−1 αn−2 + an αn−1) A. But A can only be a field for d = 0, so R is a finite extension of C, hence algebraic over C. Proof of Hilbert’s Nullstellensatz. a) Let M be a maximal ideal in A. Then we have that K := A/M is a field, which is generated over C by the residue classes Xi mod M. By Proposition 1.1.11, K is algebraic over C and, as C is algebraically closed, the natural morphism of C-algebras ϕ : C C[X1,X2,...Xn] π C[X1,X2,...,Xn]/M = K is an isomorphism between C and K. Let xi := ϕ−1(Xi mod M), 1 i n. Then Xi xi Ker π = M and so, (X1 x1,X2 x2,...,Xn xn) M. As (X1 −x1,X2 −x2,...,Xn −xn) is a maximal ideal, we have equality. b) Let I A. There exists a maximal ideal M of A such that I M. From a) we have M = I(P ) for some point P An C , so {P } V(I(P )) V(I) hence V(I) is not empty. c) For an ideal I of A we want to prove that if f is an element in I(V(I)), then f r belongs to I for some integer r. We shall use Rabinowitsch’s trick, which consists of adding a variable T and considering the natural inclusion C[X1,X2,...,Xn] C[X1,X2,...,Xn,T ] and the ideal J = (I, Tf 1) of C[X1,X2,...,Xn,T ]. We clearly have V(J) = {(x1,x2,...,xn,y) = (P, y) An+1 C : P V(I) and yf(P ) = 1}. Projection onto the first n coordinates maps V(J) to the subset of V(I) of points P with f(P ) = 0. But f belongs to I(V(I)), so V(J) = ∅. By b), we then have J = C[X1,X2,...,Xn,T ]. In particular, 1 J, so we can write
Previous Page Next Page