Algorithms in Real Algebraic Geometry by Denis S. Arnon, Bruno Buchberger

By Denis S. Arnon, Bruno Buchberger

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For a better understanding of the map a → VK (a) from the set of ideals in K[X1 , . . , Xn ] to the set of affine algebraic sets in K n , we introduce a map IK in the opposite direction. For this to every subset E ⊆ K n , we associate the ideal IK (E) := { F ∈ K[X1 , . . , Xn ] | F (a) = 0 for all a ∈ E } = a∈E ma , which is even a radical ideal in K[X1 , . . , Xn ] . For an affine algebraic set V ⊆ K n , the ideal IK (V ) is called the i d e a l o f V . The proofs of the following rules are simple verifications.

Xn ] with F ⊇ G . (6) VK (1) = ∅ and VK (0) = K n . Therefore the affine algebraic sets in K n form the closed sets of a topology on K n . This topology is called the Z a r i s k i t o p o l o g y on K n . The open sets are the complements K n \ VK (Fj , j ∈ J ) =: DK (Fj , j ∈ J ) = j ∈J DK (Fj ) . In particular, DK (F ) = { a ∈ K n | F (a) = 0 } = K n \ VK (F ) for every polynomial F ∈ K[X1 , . . , Xn ] . These open subsets are called the d i s t i n g u i s h e d o p e n s u b s e t s in K n .

7. Proposition If R is a Noetherian ring then X = Spec R is a Noetherian topological space. More generally: If a = ni=1 Rfi is a finitely generated ideal in a commutative ring R, then D(a) = ni=1 D(fi ) is quasi-compact. 8. Example The ring Q [ Xi : i ∈ N ]/(Xi2 : i ∈ N) is not Noetherian, but its spectrum is singleton and hence Noetherian. 20 (2)). Further, if R is an integral domain, then any non-empty open subset contains the point corresponding to the zero prime ideal, in particular, U ∩ V = ∅ for arbitrary non-empty open subsets U, V ⊆ Spec R.

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