8 1. Aﬃne and Projective Varieties (1.1) 1 = m i=1 gifi + g0(Tf − 1), for some gi ∈ C[X1,X2,...,Xn,T ], fi ∈ I. Let T r be the highest power of T appearing in the polynomials gi, for 0 ≤ i ≤ m. Multiplying (1.1) by f r gives f r = m i=1 Gifi + G0(Tf − 1), where the Gi = f r gi are polynomials in X1,...,Xn,Tf. Considering this last equality modulo Tf − 1, we obtain the relationship f r ≡ m i=1 hifi mod(Tf − 1), where hi(X1,...,Xn) := Gi(X1,...,Xn, 1), 1 ≤ i ≤ m. Now the natural morphism C[X1,X2,...,Xn] → C[X1,X2,...,Xn,T ] C[X1,X2,...,Xn,T ]/(Tf−1) is injective. So we have the equality f r = m i=1 hifi in C[X1,...,Xn]. Thus f r ∈ I. ✷ Remark 1.1.12. In the proof of Hilbert’s Nullstellensatz, the hypothesis C algebraically closed was only used to prove a). Then b) was proved assuming a) and c) was proved assuming b). For any field C it can be proved that the three statements are equivalent. Indeed assuming c), if M is a maximal ideal of C[X1,X2,...,Xn], we have I(V(M)) = √ M = M C[X1,X2,...,Xn] hence V(M) = ∅. If P ∈ V(M), then M ⊂ I(P ), and as M is maximal, M = I(P ). As a consequence of Hilbert’s Nullstellensatz, we have that V and I set a bijective correspondence between the collection of all radical ideals of C[X1,...,Xn] and the collection of all aﬃne varieties of An C which inverts inclusion. Recall that a nonempty topological space X is said to be reducible if it can be written as a union of two closed proper subsets. It is irreducible if it is not reducible, or equivalently, if all nonempty open subsets of X are

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