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I have a problem in understanding the proof of the following theorem:

Let $I\subseteq\mathbb{C}[[x_1,...,x_n]]$ be an ideal. Then there exists a $k\in\mathbb{N}$ and a linear coordinate change $\phi:\mathbb{C}[[x_1,...,x_n]]\to\mathbb{C}[[x_1,...,x_n]]$ such that $\mathbb{C}[[x_1,...,x_k]]\subseteq\mathbb{C}[[x_1,...,x_n]]/\phi(I)$ and $\mathbb{C}[[x_1,...,x_n]]/\phi(I)$ is finite as $\mathbb{C}[[x_1,...,x_k]]$-module.

As for the proof, assume $I\neq0$. Let $0\neq f\in I$, then one finds a linear coordinate change $\phi_1:\mathbb{C}[[x_1,...,x_n]]\to\mathbb{C}[[x_1,...,x_n]]$ such that $\phi_1(f)$ is $x_n$-regular. By the Weierstraß Preparation Theorem, there is a unit $u$ and a Weierstraß polynomial $p$ w.r.t. $x_n$ such that $u\phi_1(f)=p$. In particular, $\mathbb{C}[[x_1,...,x_{n-1}]]\hookrightarrow\mathbb{C}[[x_1,...,x_n]]/p$ is finite. Hence $\mathbb{C}[[x_1,...,x_{n-1}]]\to\mathbb{C}[[x_1,...,x_n]]/\phi_1(I)$ is finite.

The bold part is where I am still stuck. First, finite means the right hand side is a finitely generated module over the left hand side, is this correct (or is it over the image of the lhs)? Assuming this, the first part is due to $p$ being a Weierstraß polynomial w.r.t. $x_n$, i.e. $p$ has terms containing $x_n$ only up to a certain order.

Edit: If $f=\sum_{\mu\geq m}f_\mu$ is the homogeneous decomposition of $f$ with $f_m\neq 0$, take any $(a_1,...,a_{n-1})\in\mathbb{C}^{n-1}$ with $f_m(a_1,...,a_{n-1},1)\neq 0$, and define $\phi_1(x_i):=x_i+a_i x_n$ for i, $\phi_1(x_n):=x_n$. Then $\phi_1$ is what we wanted. But still, I don't get why this $\phi_1$ seems to work for all $f\neq 0$ in $I$ simultaneously. Or is this not what we need for the proof to work? How can I 'patch' these to get the result about $\mathbb{C}[[x_1,...,x_n]]/\phi_1(I)$?

And does it have anything to say that once the arrow $\hookrightarrow$ is used (which indicates injectivity to me), and in the next sentence it's just a 'normal' map?

Thank you in advance!

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    The second map is finite because it is the composition of the first one with the canonical surjection $C[[x_1,...,x_n]]/(p) \to C[[x_1, ..., x_n]]/(\phi_1(I)$. As both maps are finite, the composition is finite.2012-08-26

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About the first question: A ring $B$ is finite over a ring $A$ iff $B$ is a finite $A$-module. Now, let $A = \mathbb C[[x_1,\ldots,x_{n-1}]]$. Then $C[[x_1,\ldots,x_n]]/\langle p \rangle \cong \bigoplus_{i=0}^{\deg p - 1} A x_n^i$ as $A$-module, so it is obviously finite over $A$.

For the second part, I'm not 100% sure, but I think you simply apply the $\phi_1$ that was used for the construction of $p$ above and apply it to all elements of the ideal. Since it is linear, the result is again an ideal.

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    Ah, right, power series ring. The CA lecture notes only cover the polynomial ring. Sorry about that.2012-04-16