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

By Karl-Heinz Fieseler and Ludger Kaup

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2. The algebra O(X) is a finite dimensional k-vector space iff |X| < ∞. In that case we have a ring isomorphism O(X) ∼ = k r , where r := |X|. 10. Here are two examples of affine varieties X, which are given independent of any explicit embedding into some k n : 1. Closed subsets of an affine variety are again affine varieties, also called ”closed subvarieties”: A closed subset X → Y of an affine variety Y inherits the structure of an affine variety: Take O(X) := O(Y )|X ∼ = O(Y )/I(X) 32 with the ideal O(Y ) ← I(X) := {f ∈ O(Y ); f |X = 0} .

5. A k-ringed space (X, O = OX ) is a topological space together with a k-structure sheaf O. A morphism between k-ringed spaces (X, OX ) and (Y, OY ) is a continuous map ϕ : X −→ Y , such that ϕ∗ (OY (V )) ⊂ OX (ϕ−1 (V )) holds for all open subsets V ⊂ Y . We denote RS = RS k the category of k-ringed spaces. 6. The category AV of affine varieties is equivalent to a full subcategory of RS k . 44 Proof. 3. Now assume that ϕ : X −→ Y is a morphism of affine varieties. We have to show that ϕ is also a morphism in RS k .

An abstract version of k n : Every finite dimensional k-vector space V is in a natural way a to k dim V isomorphic affine variety: The algebra O(V ) is generated by the linear forms V −→ k, indeed O(V ) ∼ = S(V ∗ ) is isomorphic to the symmetric algebra over the dual vector space V ∗ . ) The topology on V is the coarsest topology such that all functions f ∈ O(V ) become continuous. 11. Here are some morphisms in T A : 1. The inclusion j : X → k n of an algebraic subset into k n is a morphism. 2.

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