1.1. Affine varieties 9 dense. A subset of a topological space is reducible (resp. irreducible) if it is so as a topological space with the induced topology. Recall as well that a Noetherian topological space can be written as a union of its irreducible components, i.e. its finitely many maximal irreducible subsets. If a subset is irreducible, its closure is also so irreducible components are closed. For closed subsets in An C irreducibility is characterized in terms of the corresponding ideal by the following proposition. Proposition 1.1.13. A closed set V in An C is irreducible if and only if its ideal I(V ) is prime. In particular, An C itself is irreducible. Proof. Write I = I(V ). Suppose that V is irreducible and let f1,f2 C[X1,...,Xn] such that f1f2 I. Then each P V is a zero of f1 or f2 hence V V(I1) V(I2), for Ii the ideal generated by fi,i = 1, 2. Since V is irreducible, it must be contained within one of these two sets, i.e. f1 I or f2 I, and I is prime. Reciprocally, assume that I is prime but V = V1 V2, with V1,V2 closed in V . If none of the Vi’s covers V , we can find fi I(Vi) but fi I, i = 1, 2. But f1f2 vanish on V , so f1f2 I, contradicting that I is prime. Example 1.1.14. As C[X1,X2,...,Xn] is a unique factorization domain, for f C[X1,...,Xn] \ C, the irreducible components of the hypersurface V(f) in An are just the hypersurfaces defined as the zero sets of the irre- ducible factors of f. Example 1.1.15. For the closed set V(X1 X2) An, V(X1 X2) = V(X1) V(X2) is the decomposition as the union of its irreducible components which are coordinate hyperplanes. For the closet set V(X2 −1) A2 C , V(X2 −1) = V(X 1) V(X + 1) is the descomposition as the union of its irreducible components which are points. From Hilbert’s Nullstellensatz and Proposition 1.1.13, we obtain that V and I set the following bijective correspondences. {radical ideals of C[X1,X2,...,Xn]} {closed sets in An C }, {prime ideals of C[X1,X2,...,Xn]} {irreducible closed sets in An C }, {maximal ideals of C[X1,X2,...,Xn]} {points in An C }. A principal open set of An C is the set of non-zeros of a single polynomial. We note that principal open sets are a basis of the Zariski topology. The
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