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Analytic and Algebraic Dependence of Meromorphic Functions - download pdf or read online

By Aldo Andreotti, Wilhelm Stoll

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Ran~ N S>~ < n. Hence T -c- kU~~ ( N x ) . 20 implies rank @INk ~ n - 1. dimension n - 1. 23, @(N k) is almost thin of If N k = Sk, then rank @IN k = rank @IS k G n - 1. 23 ~(Nk) Is almost thln of dimension n - 1. cause A is at most countable, the union T is almost thin. Be- and because Y has pure dimension n, S$ In a certain sense, this Lemma can be considered as a substitute for Sard's theorem for complex spaces. 2?. ~: X ~ Y Let X be an irreducible, be a surJectlve, If ~ is also q-flberlng, Proof.

Sup(rk]k Lemma x Then rankx~ r = sup(rk]k Suppose = r a n k x ~ k = r k. ¢ A). 18. (and only one|). Together, Hence Define rankx~ Because t = s u p { r ] C C X r) is defined. only in f i n i t e l y many T h e n C 1 = Xt+ 1 • a ¢ C with dimaX= t ~ d i m x X ~ s. then d l m x X -~ r for all x analytic s. subset Then subset of X. Since the C is a n a l y t i c = s = t, h e n c e x each p o i n t supremum of the ~ r} is of X is con- and thin in C. If x Therefore r. is f i n i t e and s = t and C - C 1 ~ C - D.

However, the natural map ~ ( U ) InJectlve and defines ~ ( U ) a subpresheaf of F ( ~ ) . as a subrlng of F(U,~0. A section I n , o v e r is Hence ~ i s the open set U ~ ~ is 60 called a meromorphlc function on U and ~ ( U ) of meromorphic functions on U. ~(U) is a field. Clearly, ~(U) ~ ( U ) (Observe, that f c ~ ~x. ). Moreover, ~(U)= is the ring If and only if U is irreducible, be the set of units in ~ ( U ) . x E U}. = F(U,~) Then ~(U). Let ~ * ( U ) ~*(U) = {f ~ ( U ) I f x + 0 if x with f + 0 may not be an unit in ~(U)N ~*(U).

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