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

By Karl-Heinz Fieseler and Ludger Kaup

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Extra info for Algebraic Geometry [Lecture notes]

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E. if we use C ∼ = R2 and Cn ∼ = R2n . Note that for Zariski open U ⊂ Cn we have inclusions O(U ) H(U ) E(U ). 3. Let U ⊂ X be an open subset of the affine variety X = Sp(A). e. h ∈ O(X) does not vanish on U , but in general a regular function f ∈ O(U ) need not admit a representation f = g/h on the entire set U . If U = V1 ∪ ... ∪ Vr with special open sets Vi = Xhi , we can write f = gi /hi on each Vi , such that for irreducible X, we may write r O(U ) = O(Vi ) ⊂ Q(A). , hr ). In particular U ⊂ Xh .

1. A complex analytic space (or complex analytic variety) X is a C-ringed Hausdorff space admitting an open cover X = i∈I Ui with open subspaces Ui ∼ = Zi → Wi , where Zi is an analytic subset of ni the open subset Wi ⊂ C . 2. A complex n-manifold is a complex analytic space X = i∈I Ui with open subspaces Ui ∼ = Wi , the Wi being open subspaces Wi ⊂ Cn . 3. A Riemann surface is a connected complex 1-manifold. 9. 1. An affine variety X → Cn is an analytic subset of Cn , hence a complex analytic space, denoted Xh .

Let X be an affine variety. For principal open subsets U ⊂ X we have already defined O(U ). For arbitrary open U ⊂ X we set then O(U ) := {f : U −→ k; f |V ∈ O(V ) for all principal open subsets V ⊂ U } and obtain a structure sheaf on X: Let V ⊂ U = i∈I Ui be a principal open subset and f : U −→ k with f |Ui ∈ O(Ui ). Since V is quasicompact, we may refine V = i∈I V ∩ Ui by a finite covering V = V1 ∪ ... ∪ Vr with principal open subsets Vj ⊂ V . 23 tells us that f |V ∈ O(V ). Since that holds for any principal open subset V ⊂ U , we get f ∈ O(U ).

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