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
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.