Algebraic Geometry [Lecture notes] by Karl-Heinz Fieseler and Ludger Kaup

By Karl-Heinz Fieseler and Ludger Kaup

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2. : ϕ∗ (B) ⊂ A . 7. 1. Any maximal ideal m ⊂ O(X) is of the form m = ma := {f ∈ O(X); f (a) = 0} with a unique point a ∈ X: Consider the restriction map := X : k[T ] −→ O(X). 19. Since I({a}) ⊃ I(X), we have a ∈ X and thus m = (I({a})) = ma . 2. e. , gr ∈ O(X) with ri=1 gi fi = 1. In particular regular functions without zeros are invertible. For convenience of notation we shall from now on denote the objects in T A simply by capital letters X, Y, ... and, in analogy to algebraic sets, O(X), O(Y ), ...

2 we have to check that the prevariety X ×Y is a product of X and Y in the category RS k . That follows from the fact, that given morphisms ϕ : Z −→ X and ψ : Z −→ Y , the map (ϕ, ψ) : Z −→ X × Y is a morphism as well, since the restrictions (ϕ, ψ)|(ϕ,ψ)−1 (Ui ×Vµ ) : (ϕ, ψ)−1 (Ui × Vµ ) −→ Ui × Vµ are. 4. A prevariety X is called separated or an algebraic variety (over k) if the diagonal ∆ ⊂ X × X is closed in X × X. 5. 1. , Sn − Tn ). 2. If ϕ : X −→ Y is an injective morphism from a prevariety X into an algebraic variety Y , then X is an algebraic variety as well: We have ∆X = (ϕ × ϕ)−1 (∆Y ).

3, part 2. 3. 1. Let X be an affine variety, U = Xf and V = Ug be principal open subsets of X resp. U . Then V is a principal open subset of X. 51 2. Let X, Y be affine varieties, f ∈ O(X), g ∈ O(Y ). Then (X × Y )f ⊗g ∼ = Xf × Yg . Proof. Do the first part yourself! For the second part, note that both the RHS and the LHS are affine varieties with the same underlying set, hence it suffices to check that the regular function algebras agree. 2 we have to check that the prevariety X ×Y is a product of X and Y in the category RS k .

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