24 1. Aﬃne and Projective Varieties there exists an open neighborhood U of P and homogeneous polynomials f(X0,X1,...,Xn),g(X0,X1,...,Xn) of the same degree such that F = f/g in U and g(P ) = 0. Then we also have in U , F = f(1,Y1,...,Yn)/g(1,Y1,...,Yn), where Yi = Xi/X0, 1 ≤ i ≤ n are the aﬃne coordinates on U0. Hence F ∈ OAn(U). Reciprocally, if F ∈ OAn(U), on an open neighborhood U of P we have F = f/g and g(P ) = 0, where f, g ∈ C[Y1,...,Yn]. Then we also have in U , F = X max(deg f,deg g) 0 f(X1/X0,...,Xn/X0) Xmax(deg f,deg g) 0 g(X1/X0,...,Xn/X0) , which is the quotient of two homogeneous polynomials in X0,X1,...,Xn of the same degree hence F ∈ OPn(U). Analogously, if V ⊂ Pn is a projective variety, Vi = V ∩ Ui, we have OV (U) = OV i (U) for every open subset U of V contained in Vi.

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