By Ulrich Gortz, Torsten Wedhorn

ISBN-10: 3834806765

ISBN-13: 9783834806765

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**Extra resources for Algebraic geometry 1: Schemes**

**Sample text**

A) Let char k = 2 and let Z1 = V (U (T − 1) − 1) and Z2 = V (Y 2 − X 2 (X + 1)) be closed subsets of A2 (k). Show that (t, u) → (t2 − 1, t(t2 − 1)) deﬁnes a bijective morphism Z1 → Z2 which is not an isomorphism. (b) Show that the morphism A1 (k) → V (Y 2 − X 3 ) ⊂ A2 (k), t → (t2 , t3 ) is a bijective morphism that is not an isomorphism. 13. Show that for n ≥ 2 the open subprevariety An (k) \ {0} ⊂ An (k) is not an aﬃne variety. Is A1 (k) \ {0} aﬃne? 14. Let X be a prevariety and let Y be an aﬃne variety.

Xn ) for all x0 , . . , xn ∈ k, 0 = λ ∈ K. (b) Let a ⊆ K[X0 , . . , Xn ] be an ideal. Show that the following assertions are equivalent. (i) The ideal a is generated by homogeneous elements. (ii) For every f ∈ a all its homogeneous components are again in a. (iii) We have a = d≥0 a ∩ K[X0 , . . , Xn ]d . An ideal satisfying these equivalent conditions is called homogeneous. (c) Show that intersections, sums, products, and radicals of homogeneous ideals are again homogeneous. (d) Show that a homogeneous ideal p ⊆ K[X0 , .

Show that the aﬃne algebraic set V (Y 2 − X 3 − X) ⊂ A2 (k) is irreducible and in particular connected. Sketch the set { (x, y) ∈ R2 ; y 2 = x3 + x } and show that it is connected. 9♦. Describe the union of the n coordinate axes in An (k) as an algebraic set. 10. Identifying A1 (k) × A1 (k) and A2 (k) as sets, show that the Zariski topology on A2 (k) is strictly ﬁner than the product topology. 11. We identify the space M2 (k) of 2 × 2-matrices over k with A4 (k) (with a b coordinates a, b, c, d).

### Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn

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