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Math Help - inverse limit

  1. #1
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    inverse limit

    \varprojlim (\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)^k)\cong\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)???
    Why??
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by KaKa View Post
    \varprojlim (\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)^k)\cong\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)???
    Why??
    This notation means nothing to me, does it to anyone else? If so could they explain it please?

    CB
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  3. #3
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    Quote Originally Posted by KaKa View Post

    \varprojlim (\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)^k)\cong\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)???
    Why??
    that cannot be right! are you sure the right hand side of the isomorphism is not \mathbb{C}[[x_1, \cdots , x_n]] instead ?
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  4. #4
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    well, i thought i'd get an answer to my question from Kaka! anyway, here's why i said your isomorphism is not correct:

    it's fairly easy to prove that if R is a noetherian ring and I=(a_1, \cdots, a_r) an ideal of R, then \varprojlim \frac{R}{I^k} \cong \frac{R[[y_1, \cdots , y_r ]]}{(y_1-a_1, \cdots , y_r - a_r)}.

    now if you apply this fact to R=\mathbb{C}[[x_1, \cdots , x_n]] and I=(f_1, \cdots , f_r), we'll have:

    \varprojlim \frac{\mathbb{C}[[x_1, \cdots, x_n]]}{(f_1, \cdots , f_r)^k} \cong \frac{\mathbb{C}[[x_1, \cdots , x_n, y_1, \cdots , y_r ]]}{(y_1-f_1, \cdots , y_r -f_r)} \cong \mathbb{C}[[x_1, \cdots , x_n, f_1, \cdots , f_r]]

    =\mathbb{C}[[x_1, \cdots , x_n]].
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