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Then its vanishing set f −1 (0) ⊆ U is closed as f is continuous and {0} ⊂ A1 (k) is closed. Therefore if the restriction of f to U is zero, f is zero because U is dense in U . This shows that restriction maps are injective. The axiom of gluing implies therefore OX (U ∪ V ) = OX (U ) ∩ OX (V ) for all open subsets U, V ⊆ X. 18) Closed subprevarieties. Let X be a prevariety and let Z ⊆ X be an irreducible closed subset. We want to deﬁne on Z the structure of a prevariety. For this we have to deﬁne functions on open subsets U of Z.

Hint: Consider for an ideal b ⊆ A[T ] the chain of ideals ai ⊆ A, where ai is the ideal generated by the leading coeﬃcients of all polynomials in b of degree ≤ i. 2. Show that I(An (k)) = 0 without using Hilbert’s Nullstellensatz. 3♦. Determine all irreducible Hausdorﬀ spaces. Determine all noetherian Hausdorﬀ spaces. Show that a topological space is noetherian if and only if every open subspace is quasi-compact. 4♦. , for all x, y ∈ X there exist open neighborhoods U of x and V of y with y ∈ /U and x ∈ / V ).

Xi ) of the ﬁeld K(X0 , . . , Xn ). 2) can also be described as follows. Let fg ∈ F with f, g ∈ K[X0 , . . , Xn ]d for some d. Set f˜ = Xfd and g˜ = Xgd . i i f f˜ Xn 0 Then f˜, g˜ ∈ K[ X Xi , . . , Xi ] and Φi ( g ) = g ˜. 20) Deﬁnition of the projective space Pn (k). The projective space Pn (k) is an extremely important prevariety within algebraic geometry. Many prevarieties of interest are subprevarieties of the projective space. Moreover, the projective space is the correct environment for projective geometry which remedies the “defect” of aﬃne geometry of missing points at inﬁnity.