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16 1. Affine and Projective Varieties If Q = (q1, . . . , qn) ∈ V , we have gi(q1,...,qn) = 0 and so qi is a solution of the equation fi(qi) = 0, where fi(X) := XN +f i N−1 (y1, . . . , yr) XN−1+···+f i 1 (y1, . . . , yr) X+f i 0 (y1, . . . , yr). Now, as V is irreducible, we can consider its function field C(V ) and see fi(X) as a polynomial in C(V )[X]. Then each fi has a finite number of roots. Thus for any point P = (y1, . . . , yr) ∈ Ar we have only finitely many points Q ∈ V with ϕ(Q) = P . To show that ϕ−1(P ) is always nonempty, it is enough to show that for every point P = (p1, . . . , pr) ∈ Ar, we have (1.2) IP := I(V ) + (y1 − p1,...,yr − pr) = C[X1,...,Xn] as, by Hilbert’s Nullstellensatz, this will imply ϕ−1(P ) = V(IP ) = ∅. As- sertion (1.2) is equivalent to (y1 − p1,...,yr − pr) = C[x1,...,xn]. Since (y1−p1, . . . , yr−pr) is a maximal ideal in C[y1,...,yr], in particular a proper ideal, we can apply Nakayama’s lemma (see [Ma] p. 11) to the C[y1,...,yr]- finite algebra C[V ] and obtain (y1 − p1,...,yr − pr) = C[x1,...,xn]. The preceding proposition gives that an affine irreducible variety V of dimension r can be seen as a finite covering of the affine space Ar. We now consider extension of scalars for affine varieties. Let V ⊂ An C be an affine variety and L an algebraically closed field containing C. We shall denote by VL the affine variety contained in An L defined by VL = V(IL) for IL = I(V )L[X1, . . . , Xn]. We call VL the variety obtained from V by extending scalars to L. The coordinate ring of VL is L[V ] = L ⊗ C[V ]. It is clear that if V, W are affine varieties defined over C, we have V W ⇒ VL WL. The next proposition gives the converse of this implication. For its proof we are following a suggestion of Jakub Byszewski. Proposition 1.1.29. Let K, L be algebraically closed fields with K ⊂ L. Let V, W be affine algebraic varieties defined over K. Let VL,WL be the varieties obtained from V, W by extending scalars to L. If VL and WL are isomorphic, then V and W are isomorphic. Proof. Let A = K[V ], B = K[W ]. These are finitely generated K-algebras. Let us write A = K[x1,...,xn],B = K[y1,...,ym]. The isomorphism be- tween VL and WL induces an L-algebra isomorphism f : A ⊗K L = L[x1,...,xn] ∼ → B ⊗ L = L[y1,...,ym].