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Closure of modules theorem

  1. Sep 9, 2008 #1

    mathwonk

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    can anyone give me a precise statement of the "closure of modules" theorem in several complex variables? it says something like: a criterion for the germ of a function to belong to the stalk of an ideal at a point, is that the function can be uniformly approximated on neighborhoods of that point by functions whose stalks do lie in the ideal. I have forgotten my gunning and rossi.
    so it is something like, if for every e>0 there is a nbhd V of p and a function in the ideal I which is uniformly closer to f than e on V, then f belongs to I.

    ????? And are there some bounds on the coefficients of the generators of the ideal in terms of given bounds on the coefficients of the approximating functions????
     
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  3. Sep 10, 2008 #2

    mathwonk

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    well i will answer my own question. the hypothesis i gave seems too weak. rather one apparently needs to assume that there is some compact nbhd of p on which f can be approximated arbitrarily well in the sup norm by functions with germs in the ideal.

    then one finds a nbhd where there are functions defined whose germs generate the ideal, and then one takes a sequence of approximations, which means a sequence of coefficient functions in terms of these fixed generators, and then one finds a convergent subsequence of coefficient functions on some smaller nbhd using a normal family argument.

    then the bound on the limiting coefficients comes from the bounds on the function and its approximations.

    apparently this stuff on several complex variables is not so well represented on the web, wikipedia and so on... a pity. although there was a book by joe taylor for sale on amazon with a selective excerpt visible.
     
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