# Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn

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By Ulrich Gortz, Torsten Wedhorn

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As i Ui = Pn (k) there is at most one way to do this: We deﬁne the topology on Pn (k) by calling a subset U ⊆ Pn (k) open if U ∩Ui is open in Ui for all i. This deﬁnes a topology on Pn (k) as for all i = j the set Ui ∩Uj = D(Tj ) ⊆ Ui is open (we use here on Ui ∼ = An (k) the coordinates T0 , . . , Ti , . . , Tn ). With this deﬁnition, (Ui )0≤i≤n is an open covering of Pn (k). We still have to deﬁne functions on open subsets U ⊆ Pn (k). We set OPn (k) (U ) = { f ∈ Map(U, k) ; ∀i ∈ {0, . . , n} : f |U ∩Ui ∈ OUi (U ∩ Ui ) }.

4) A point x ∈ X is called a maximal point if its closure {x} is an irreducible component of X. 44 2 Spectrum of a Ring Thus a point η ∈ X is generic if and only if it is a generization of every point of X. As the closure of an irreducible set is again irreducible, the existence of a generic point implies that X is irreducible. 9. 8 have the following algebraic meaning. (1) A point x ∈ X is closed if and only if px is a maximal ideal. (2) A point η ∈ X is a generic point of X if and only if pη is the unique minimal prime ideal.

In particular, Pn (k) is not an aﬃne variety for n ≥ 1. Proof. , Xi Xi where the intersection is taken in K(Pn (k)). The last assertion follows because if Pn (k) were aﬃne, its set of points would be in bijection to the set of maximal ideals in the ring k = OPn (k) (Pn (k)). This implies that Pn (k) consists of only one point, so n = 0. 21) Projective varieties. 62. A prevariety is called a projective variety if it is isomorphic to a closed subprevariety of a projective space Pn (k). As in the aﬃne case, we speak of projective varieties rather than prevarieties.