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18 1. Affine and Projective Varieties Definition 1.2.3. Let (X, OX) and (Y, OY ) be geometric spaces. A mor- phism ϕ : (X, OX) → (Y, OY ) is a continuous map ϕ : X → Y such that for every open subset U of Y and every f ∈ OY (U), the function ϕ∗(f) := f ◦ ϕ belongs to OX(ϕ−1(U)). Example 1.2.4. Let X be an affine variety. To each nonempty open set U ⊂ X we assign the ring OX(U) of regular functions on U. Then (X, OX) is a geometric space. Moreover the two notions of morphism agree. Let (X, OX) be a geometric space and let Z be a subset of X with the induced topology. We can make Z into a geometric space by defining OZ(V ) for an open set V ⊂ Z as follows: a function f : V → C is in OZ(V ) if and only if there exists an open covering V = ∪iVi in Z such that for each i we have f|V i = gi|V i for some gi ∈ OX(Ui) where Ui is an open subset of X containing Vi. It is not difficult to check that OZ is a sheaf of functions on Z. We will refer to it as the induced structure sheaf and denote it by OX |Z . Note that if Z is open in X then a subset V ⊂ Z is open in Z if and only if it is open in X, and OX(V ) = OZ(V ). Let X be a topological space and let X = ∪iUi be an open cover. Given sheaves of functions OU i on Ui for each i, which agree on each Ui ∩ Uj, we can define a natural sheaf of functions OX on X by ”gluing” the OU i . Let U be an open subset in X. Then OX(U) consists of all functions on U, whose restriction to each U ∩ Ui belongs to OU i (U ∩ Ui). Let (X, OX) be a geometric space. If x ∈ X we denote by vx the map from the set of C-valued functions on X to C obtained by evaluation at x: vx(f) = f(x). Definition 1.2.5. A geometric space (X, OX) is called an abstract affine variety if the following three conditions hold. a) OX(X) is a finitely generated C-algebra, and the map from X to the set HomC(OX(X), C) of C-algebra morphisms defined by x → vx is a bijection. b) For each f ∈ OX(X), f = 0, the set Xf := {x ∈ X : f(x) = 0} is open, and every nonempty open set in X is a union of some Xf’s. c) OX(Xf) = OX(X)f, where OX(X)f denotes the C-algebra OX(X) lo- calized at f.