By Andreas Gathmann

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20. (a) Affine varieties are varieties. (b) Open, closed, and locally closed subprevarieties of varieties are varieties. We will therefore simply call them open, closed, and locally closed subvarieties, respectively. Proof. (a) If X ⊂ An then ∆X = V (x1 − y1 , . . , xn − yn ) ⊂ X × X, where x1 , . . , xn and y1 , . . , yn are the coordinates on the two factors, respectively. Hence ∆X is closed. 16). As ∆Y = i−1 (∆X ) and ∆X is closed by assumption, ∆Y is closed as well by the continuity of i.

16 (Products of prevarieties). Let X and Y be prevarieties. A product of X and Y is a prevariety P together with morphisms πX : P → X and πY : P → Y satisfying the following universal property: for any two morphisms fX : Z → X and fY : Z → Y from another prevariety Z there is a unique morphism f : Z → P such that πX ◦ f = fX and πY ◦ f = fY . 10, this means that giving a morphism to the product P is the same as giving a morphism to each of the factors X and Y . 17 (Existence and uniqueness of products).

Now let I OX,a be any ideal not contained in Ia . By definition, this means that there is an element gf ∈ I with f (a) = 0 and g(a) = 0. But then gf exists in OX,a as well. Hence 1 = f g · gf ∈ I, and we conclude that I = OX,a . 23. Let X ⊂ An be an affine variety, and let a ∈ X be a point. Show that OX,a ∼ = OAn ,a /I(X) OAn ,a , where I(X) OAn ,a denotes the ideal in OAn ,a generated by all quotients 1f for f ∈ I(X). 3. 24. Let F be a sheaf on a topological space X, and let a ∈ X. Show that the stalk Fa is a local object in the following sense: if U ⊂ X is an open neighborhood of a then Fa is isomorphic to the stalk of F |U at a on the topological space U.

### Algebraic Geometry by Andreas Gathmann

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