By Ulrich Gortz, Torsten Wedhorn
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Extra resources for Algebraic geometry 1: Schemes
As a set this will be the set of prime ideals of A. This diﬀers from the approach in the ﬁrst chapter where the points of an aﬃne variety corresponded to the maximal ideals. g. for local rings). Moreover, if ϕ : A → B is a ring homomorphism, the inverse image of a maximal ideal in B is not necessarily a maximal ideal of A while the analogous statement for prime ideals is true. Thus working with prime ideals we obtain a functorial construction. This is the content of the ﬁrst part of this chapter.
I∈I V (a ∩ a ) = V (aa ) = V (a) ∪ V (a ). Proof. 2). The lemma shows that the subsets V (a) of Spec A form the closed sets of a topology on Spec A. This leads us to the following deﬁnition. 2. Let A be a ring. The set Spec A of all prime ideals of A with the topology whose closed sets are the sets V (a), where a runs through the set of ideals of A, is called the prime spectrum of A or simply the spectrum of A. The topology thus deﬁned is called the Zariski topology on Spec A. 42 2 Spectrum of a Ring If x is a point in Spec A, we will often write px instead of x when we think of x as a prime ideal of A.
Then a ϕ is a homeomorphism of Spec B onto the closed subset V (a) of Spec A. (2) Let S be a multiplicative subset of A and let ϕ : A → S −1 A =: B be the canonical homomorphism. Then a ϕ is a homeomorphism of Spec S −1 A onto the subspace of Spec A consisting of prime ideals p ⊂ A with S ∩ p = ∅. Proof. In both cases it is clear that a ϕ is injective with the stated image. Moreover in both cases a prime ideal q of B contains an ideal b of B, if and only if ϕ−1 (q) contains ϕ−1 (b). This shows that a ϕ(V (b)) = V (ϕ−1 (b)) ∩ Im(a ϕ).
Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn