Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn

By Ulrich Gortz, Torsten Wedhorn

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The above construction X → (X, OX ) defines a fully faithful functor (Irreducible affine algebraic sets) → (Spaces with functions over k). 23 Prevarieties We have seen that we can embed the category of irreducible affine algebraic sets into the category of spaces with functions. Of course we do not obtain all spaces with functions in this way. We will now define prevarieties as those connected spaces with functions that can be glued together from finitely many spaces with functions attached to irreducible affine algebraic sets.

Let g ∈ OX (D(f )) and set a = { h ∈ Γ(X) ; hg ∈ Γ(X) }. Obviously a is an ideal of Γ(X) and we have to show that f ∈ rad(a). By Hilbert’s Nullstellensatz we have rad(a) = I(V (a)). Therefore it suffices to show f (x) = 0 for all x ∈ V (a). , x ∈ D(f ). As g ∈ OX (D(f )), we find / mx , with g = gg12 . Thus g2 ∈ a and as g2 (x) = 0 we have x ∈ / V (a). 41. If X is an irreducible affine algebraic set, U ⊆ X open, and f ∈ OX (U ), there do not necessarily exist g, h ∈ Γ(X) with f = hg ∈ K(X) and h(x) = 0 for all x ∈ U .

As f |U ∩Ui ∈ OUi (U ∩ Ui ), the function f has locally the form g ˜ ˜ ∈ k[T0 , . . , Ti , . . , Tn ]. 2) yields the desired ˜, h ˜ with g h form of f . Conversely, let f be an element of the right hand side. We fix i ∈ {0, . . , n}. Thus locally on U ∩ Ui the function f has the form hg with g, h ∈ k[X0 , . . , Xn ]d for some d. 2) we obtain that f has locally the form h˜g˜ ˜ ∈ k[T0 , . . , Ti , . . , Tn ]. This shows f |U ∩U ∈ OU (U ∩ Ui ). 60. Let i ∈ {0, . . , n}. The bijection Ui → An (k) induces an isomorphism ∼ (Ui , OPn (k)|Ui ) → An (k).

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