By Karl-Heinz Fieseler and Ludger Kaup

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**Example text**

1. A projective variety is an algebraic variety isomorphic to a closed subspace X → Pn . 2. There are two ways to relate affine varieties to projective varieties: 1. To every embedded affine variety Y → kn ∼ = U 0 ⊂ Pn we can associate its projective closure Y → Pn = U0 ∪ P(0 × k n ). 2. The affine cone over a projective variety X → Pn is the affine variety C(X) := π −1 (X) = π −1 (X) ∪ {0} ⊂ k n+1 , where π : (k n+1 )∗ −→ Pn denotes the quotient morphism. A warning: The affine cone C(X) does not only depend on X as algebraic variety, but as well on the chosen embedding X → Pn !

5. Let us now describe a functor Sp : FRA −→ AV inverse to O : AV −→ FRA. For a reduced affine k-algebra A we set Sp(A) := {m → A, max. ideal} , the “(maximal) spectrum” of the ring A. , Tn ] −→ A, denote a → k[T ] its kernel and X := N (a) → k n its zero set. Then X −→ Sp(A), x → mx is a bijection and hence can be used to define a topology and regular functions on Sp(A). Indeed these data depend only on A: The closed sets are the sets N (b) := {m ∈ Sp(A); b ⊂ m} with an ideal b → A. It remains to determine O(Sp(A)): For every m ∈ Sp(A) the map ı : k = k·1 → A −→ A/m is an isomorphism.

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.