Algebraic Geometry [Lecture notes] - download pdf or read online

By Karl-Heinz Fieseler and Ludger Kaup

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21. For a principal open subset Xf ⊂ X of an affine variety X we have Xf ∼ = Sp(O(X)f ). 22. If X → k m and Y → k n are affine varieties, so is X × Y → k m × k n = k m+n . Note that the homomorphism O(X) ⊗ O(Y ) −→ O(X × Y ) induced by the bilinear map O(X) × O(Y ) −→ O(X × Y ), (f, g) → pr∗X (f ) · pr∗Y (g) is an isomorphism (since linear independent functions fi ∈ O(X) and gj ∈ O(Y ) give linear independent functions pr∗X (fi ) · pr∗Y (gj )); hence the affine variety X × Y depends only on the affine varieties X and Y and not on the chosen embeddings.

E. A∼ = O(X), iff it is affine and reduced. 5. e. if √ 0=0 holds in R. 4. An affine algebra A ∼ = k[T ]/a is reduced iff a = a, the latter being equivalent to a = I(X) for X = N (a). 6. An algebraic set X → k n is irreducible iff O(X) is an integral domain iff any two non-empty open subsets have points in common. Let us now consider the following category T A of Topological spaces with a distinguished Algebra of regular functions: 1. e. a function f ∈ A is invertible iff it has no zeros. Functions f ∈ A are also referred to as regular functions on X.

Since g1 g4 = g2 g3 we may define a regular function f ∈ O(U ) by f |X1 := g3 g4 , f |X1 := , g1 g2 but there is no representation f = g/h with regular functions g, h ∈ O(X) with a denominator nowhere vanishing on U , or equivalently, there is no proper principal open subset Xh ⊃ U . Indeed O(U )∗ = k ∗ . , t4 ) → (t1 , t2 ). It has fibers ϕ−1 (t1 , t2 ) = k(t1 , t2 ) ∼ = k for (t1 , t2 ) ∈ k 2 \ {0} and ϕ−1 (0, 0) = 0 × 0 × k 2 ∼ = k 2 , furthermore a section σ : U −→ X, namely σ(t1 , t2 ) := (t1 , t2 , 0, 0).

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

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