12 1. Aﬃne and Projective Varieties variety structure and its coordinate ring is C[Vf] = C[X1,...,Xn, 1/f], i.e. the ring C[X1,...,Xn] localized in the multiplicative system of the powers of f. More generally, for V an aﬃne variety, f ∈ C[V ], the algebra of regular functions on the principal open set Vf := {x ∈ V : f(x) = 0} is the algebra C[V ]f, i.e. the algebra C[V ] localized in the multiplicative system of the powers of f. We note that arbitrary open sets of an aﬃne variety cannot be given the structure of an aﬃne variety. (See Exercise 11.) Now let V ⊂ An C , W ⊂ Am C be arbitrary aﬃne varieties. A map ϕ : V → W is a morphism of aﬃne varieties if for x = (x1, . . . , xn) ∈ V , ϕ(x1,...,xn) = (ϕ1(x), . . . , ϕm(x)), for some ϕi ∈ C[V ]. A morphism ϕ : V → W is con- tinuous for the Zariski topologies involved. Indeed if Z ⊂ W is the set of zeros of polynomial functions fi on W , then ϕ−1(Z) is the set of zeros of the polynomial functions fi◦ϕ on V . With a morphism ϕ : V → W , a C-algebra morphism ϕ∗ : C[W ] → C[V ] is associated, defined by ϕ∗(f) = f ◦ ϕ. Note that if y1,...,ym are the coordinate functions on W , we have ϕj = ϕ∗(yj) hence ϕ is recovered from ϕ∗. If X is a third aﬃne variety and ψ : W → X a morphism, we clearly have (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗. Proposition 1.1.20. If ϕ : V → W is a morphism of aﬃne varieties for which ϕ(V ) is dense in W , then ϕ∗ is injective. Proof. Let f, g ∈ C[W ] such that ϕ∗(f) = ϕ∗(g). We consider the subset {y ∈ W : f(y) = g(y)} of W . It is clearly closed. On the other hand it contains ϕ({x ∈ V : f(ϕ(x)) = g(ϕ(x))}) = ϕ(V ) hence it is dense in W . So it is equal to W . The morphism ϕ : V → W is an isomorphism if there exists a morphism ψ : W → V such that ψ ◦ ϕ = IdV and ϕ ◦ ψ = IdW , or equivalently ϕ∗ : C[W ] → C[V ] is an isomorphism of C-algebras (with its inverse being ψ∗). We say that the varieties V, W defined over the same field C are isomorphic if there exists an isomorphism ϕ : V → W . We will often need to consider maps on an irreducible aﬃne variety V which are not everywhere defined, so we introduce the following concept. Definition 1.1.21. a) If V is an irreducible aﬃne variety, a rational map ϕ : V → An C is an n-tuple (ϕ1, . . . , ϕn) of rational functions ϕ1,...,ϕn ∈ C(V ). The map ϕ is called regular at a point P of V if all ϕi are regular at P . The domain of definition dom(ϕ) is the set of all regular points of ϕ, i.e. dom(ϕ) = ∩n i=1 dom(ϕi).

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