1.2. Abstract aﬃne varieties 17 Then f(xi) is a polynomial in y1,...,ym with coeﬃcients in L, 1 ≤ i ≤ n, f −1 (yj) is a polynomial in x1,...,xn with coeﬃcients in L, 1 ≤ j ≤ m. Let S ⊂ L be the K-algebra generated by the coeﬃcients of all these polynomials. Then S is a finitely generated K-algebra and we have an isomorphism (1.3) A ⊗K S ∼ → B ⊗K S. As K is algebraically closed, using Proposition 1.1.11, we have either S = K, in which case we obtain A A ⊗K K B ⊗K K B hence V W as wanted or S is not a field. In this latter case, let M be a maximal ideal in S. As S is a finitely generated K-algebra and K is algebraically closed, we have S/M K (by Hilbert Nullstellensatz 1.1.6). Tensoring the isomorphism (1.3) by S/M over S, we obtain A ⊗K S ⊗S S/M ∼ → B ⊗K S ⊗S S/M ↓ ↓ A ⊗K K = A ∼ → B ⊗K K = B. Hence V W . 1.2. Abstract aﬃne varieties So far we have considered aﬃne varieties as closed subsets of aﬃne spaces. We shall now see that they can be defined in an intrinsic way (i.e. not depending on an embedding in an ambient space) as topological spaces endowed with a sheaf of functions satisfying the properties of the regular functions. Definition 1.2.1. A sheaf of functions on a topological space X is a func- tion F which assigns to every nonempty open subset U ⊂ X a C-algebra F(U) of C-valued functions on U such that the following two conditions hold: a) If U ⊂ U are two nonempty open subsets of X and f ∈ F(U ), then the restriction f|U belongs to F(U). b) Given a family of open sets Ui,i ∈ I, covering U and functions fi ∈ F(Ui) for each i ∈ I, such that fi and fj agree on Ui ∩ Uj, for each pair of indices i, j, there exists a function f ∈ F(U) whose restriction to each Ui equals fi. Definition 1.2.2. A topological space X together with a sheaf of functions OX is called a geometric space. We refer to OX as the structure sheaf of the geometric space X.

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