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4 1. Affine and Projective Varieties If P = (x1, . . . , xn) ∈ An C , {P } = V(X1 − x1,...,Xn − xn) is an affine variety. The following two propositions are easy to prove. Proposition 1.1.2. Let S, S1,S2 denote subsets of An C , I1,I2 denote ideals of C[X1,...,Xn]. We have a) S1 ⊂ S2 ⇒ I(S1) ⊃ I(S2), b) I1 ⊂ I2 ⇒ V(I1) ⊃ V(I2), c) I(S) = C[X1,X2,...,Xn] ⇔ S = ∅. Proposition 1.1.3. The correspondence V satisfies the following equalities: a) An C = V(0), ∅ = V(C[X1,...,Xn]), b) If I and J are two ideals of C[X1,...,Xn], V(I) ∪ V(J) = V(I ∩ J), c) If {Iα} is an arbitrary collection of ideals of C[X1,...,Xn], ∩αV(Iα) = V( α Iα). We then have that affine varieties in An C satisfy the axioms of closed sets in a topology. This topology is called Zariski topology. Hilbert’s basis theorem implies the descending chain condition on closed sets and therefore the ascending chain condition on open sets. Hence An C is a Noetherian topo- logical space. This implies that it is quasicompact. However, the Hausdorff condition fails. Example 1.1.4. For a point P = (x1, x2,...,xn) ∈ An C , the ideal I(P ) = (X1−x1, X2−x2,...,Xn−xn) is maximal, as it is the kernel of the evaluation morphism vP : C[X1,X2,...Xn] → C f → f(P ). We recall that for an ideal I of a commutative ring A the radical √ I of I is defined by √ I := {a ∈ A : ar ∈ I for some r ≥ 1}. It is an ideal of A containing I. A radical ideal is an ideal which is equal to its radical. An ideal I of the ring A is radical if and only if the quotient ring A/I has no nonzero nilpotent elements. As examples of radical ideals, we have that a prime ideal is radical and ideals of the form I(S) for S ⊂ An C are radical ideals of C[X1,...,Xn]. Example 1.1.5. The ideal (X1 X2) is a radical ideal of C[X1,...,Xn] which is not prime. The ideal (X2−1) is a radical ideal of C[X] which is not prime.