# Thread: inverse limit

1. ## 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??

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

CB

3. Originally Posted by KaKa

$\varprojlim (\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)^k)\cong\mathbb{C}[[x_1,...,x_n]]/(f_1,...,f_r)$???
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 ?

4. 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:

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

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