## Download Basic Algebraic Geometry 1: Varieties in Projective Space by Igor R. Shafarevich, Miles Reid PDF By Igor R. Shafarevich, Miles Reid

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Substituting our expression for y in this gives p(x, y) = p(x, xv) = xr (because p has no constant term), where r is a regular function on the curve, and hence u = xr/q = xu1 . If u1 (P ) = 0 then we can repeat the argument, getting u = x 2 u2 , and so on. We now prove that, provided u is not identically 0 on the curve, this process must stop after a finite number of steps. For this, return to the expression u = p/q, in which, by assumption, p is not divisible by f . 1). Suppose a = x k a0 with a0 (0) = 0.

Then Xi = Xi ∩ X = Xi ∩ Yj = j (Xi ∩ Yj ). j Since by assumption Xi is irreducible, we have Xi ∩ Yj = Xi for some j , that is, Xi ⊂ Yj . Repeating the argument with the Xi and Yj interchanged gives Yj ⊂ Xi for some i . Hence Xi ⊂ Yj ⊂ Xi , so that by the irredundancy of the representation, i = i and Yj = Xi . The theorem is proved. We now restate the condition that a closed set X is irreducible in terms of its coordinate ring k[X]. If X = X1 ∪ X2 is reducible then since X X1 there exists a polynomial F1 that is 0 on X1 but not 0 on X, and a similar polynomial F2 for X2 .

The facts we have just proved show that X → k[X] defines an equivalence of categories between closed subsets of affine spaces (with regular maps between them) and a certain subcategory of the category of commutative algebras over k (with algebra homomorphisms). What is this subcategory, that is, which algebras are of the form k[X]? 3 An algebra A over a field k is isomorphic to a coordinate ring k[X] of some closed subset X if and only if A has no nilpotents (that is f m = 0 implies that f = 0 for f ∈ A) and is finitely generated as an algebra over k.