By Karl-Heinz Fieseler and Ludger Kaup

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**Extra info for Algebraic Geometry [Lecture notes]**

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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 . 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.

If we then declare these functions to be regular: O(Sp(A)) := {f : Sp(A) −→ k; f ∈ A}, we obtain, due to the fact that mx = {0}, m= x∈X m∈Sp(A) an isomorphism ∼ = ΦA : A −→ O(Sp(A)), f → f . A comment on notation: Though the elements in the set Sp(A) are ideals, one usually prefers a geometric notation and denotes x ∈ Sp(A) its points, such that, strictly speaking, x = mx . Finally, given an algebra homomorphism σ : B → A between reduced affine algebras B and A, the inverse image σ −1 (m) of a maximal ideal m → A is again maximal, since k → B/σ −1 (m) → A/m ∼ = k implies B/σ −1 (m) ∼ = k.

Open and closed subspaces of an algebraic variety are separated. 4. Affine and more generally quasi-affine varieties are separated. 5. In the general case of a prevariety X with an affine open cover X = r i=1 Ui we thus get that r ∆ → Ui × Ui i=1 is a closed subspace of the open set 52 r i=1 Ui × Ui ⊂ X × X and that the diagonal map δ : X −→ ∆, x → (x, x), is an isomorphism of prevarieties. Hence X is separated iff ∆ij := ∆ ∩ (Ui × Uj ) is a closed subset of Ui × Uj for i = j. 6. Assume X = ri=1 Ui is an open affine cover of the prevariety X.