By Karl-Heinz Fieseler and Ludger Kaup

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

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The implication ”=⇒” makes use of the continuity of ϕ, while for ”⇐=” consider the compact set K consisting of the points yn and the accumulation points of that sequence. One could thus say, that ϕ : X −→ Y is proper iff the space X has ”no holes over Y ”. A closed subset Z → X of an affine variety X is again an affine variety with the regular function algebra O(Z) := O(X)|Z ∼ = O(X)/I(Z). But what can we do in order to give open subsets U ⊂ X also some structure, say, to make them objects in the category T A?

Evaluating qλ f that equality at any point x ∈ Z gives f (q) (x) = 0 for all q ∈ N resp. f (q) ∈ I(Z). 7. , fr ) := {[t]; f1 (t) = ... , Tn ]. Proof. Apply Prop. 6 to the affine cone C(X) → k n+1 . Note that homogeneous polynomials are functions only on k n+1 and not on Pn , but since such a polynomial either vanishes identically on a punctured line k ∗ · t (t = 0) or has no zeros there, the above description of a projective variety makes sense nevertheless. 8. , Tn ]. , Tn ] f ∈a satisfies N (a) = C(Y ).

Thus V ⊂ Z is open, as desired. 1. The dimension of an algebraic variety X at a point a ∈ X is defined as dima X := max{n ∈ N; ∃ X0 = {a} X1 Xn , Xi → X irreducible}, ... while dim X := max dima X. 2. 1. Since a maximal strictly increasing chain of irreducible subvarieties starts with a point, we have dim X = max{n ∈ N; ∃ X0 X1 ... Xn , Xi → X irreducible}. 2. If a ∈ U ⊂ X with an open subset U ⊂ X, then dima U = dima X. This follows from the fact that Y → Z := Y defines a bijection between the irreducible subvarieties Y → U containing a and the irreducible subvarieties Z → X containing a, the inverse map being given by Z → Z ∩ U.

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

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