INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

38 Mezzetti9.1. Definition. φ is a regular map or a morphism if(i) φ is continuous;(ii) φ preserves regular functions, i.e. for all U ⊂ Y (U open and non–empty) andfor all f ∈ O(U), then f ◦ φ ∈ O(φ −1 (U)):X↑φ −1 (U)φ−→φ|−→Y↑Uf→KNote that:a) for all X the identity map 1 X : X → X is regular;b) for all X, Y , Z and regular maps X φ → Y , Y ψ → Z, the composite map ψ ◦ φ isregular.An isomorphism of varieties is a regular map which possesses regular inverse,i.e. a regular φ : X → Y such that there exists a regular ψ : Y → X verifyingthe conditions ψ ◦ φ = 1 X and φ ◦ ψ = 1 Y . In this case X and Y are said to beisomorphic, and we write: X ≃ Y .If φ : X → Y is regular, there is a natural K–homomorphism φ ∗ : O(Y ) →O(X), called the comorphism associated to φ, defined by: f → φ ∗ (f) := f ◦ φ.The construction of the comorphism is functorial, which means that:a) 1 ∗ X = 1 O(X);b) (ψ ◦ φ) ∗ = φ ∗ ◦ ψ ∗ .This implies that, if X ≃ Y , then O(X) ≃ O(Y ). In fact, if φ : X → Y is anisomorphism and ψ is its inverse, then φ ◦ψ = 1 Y , so (φ ◦ψ) ∗ = ψ ∗ ◦φ ∗ = (1 Y ) ∗ =1 O(Y ) and similarly ψ ◦ φ = 1 X implies φ ∗ ◦ ψ ∗ = 1 O(X) .9.2. Examples.1) The homeomorphism φ i : U i → A n of Proposition 3.2 is an isomorphism.2) There exist homeomorphisms which are not isomorphisms. Let Y = V (x 3 −y 2 ) ⊂ A 2 . We have seen (see Exercise 7.2) that K[X] ≄ K[A 1 ], hence Y is notisomorphic to the affine line. Nevertheless, the following map is regular, bijectiveand also a homeomorphism (see Exercise 7.1):φ : A 1 → Y such that t → (t 2 , t 3 );{ yφ −1 : Y → A 1 is defined by (x, y) → x if x ≠ 00 if (x, y) = (0, 0).9.3. Proposition. Let φ : X → Y ⊂ A n be a map. Then φ is regular if and onlyif φ i := t i ◦ φ is a regular function on X, for all i = 1, . . ., n (where t 1 , . . ., t n arethe coordinate functions on Y ).Proof. If φ is regular, then φ i = φ ∗ (t i ) is regular by definition.