formal power series ring over field is m-adic complete

Question

I am currently reading Serge Lang "Algebra" and I am reading the chapter about Power Series. It says that the power series ring $$ R=k[[X_1,...,X_n]] $$ over a field $k$ is a local ring with maximal ideal $m=(X_1,...,X_n)$. So far so fine.

Then the writer says that it's immediat that $R$ is complete with respect to the $m$-adic topology and I am stuck right here. It must be very obvious what the unique limit for a Cauchy sequence with respect to the $m$-adic topology looks like but I don't get it.

Any hints?

Answer 1

Consider the ring of polynomials $S=k[x_1,\ldots,x_n]$ over a field $k$. Then $\mathfrak{m}:=(x_1,\ldots ,x_n)$ is a maximal ideal in $S$, such that the $\mathfrak{m}$-adic completion of $S$ can be viewed as just the formal power series $R=k[[x_1,\ldots,x_n]]$. So $R$ is complete. To see this, define $f\colon \widehat{S_{\mathfrak{m}}}\rightarrow R$ by $$ f\mapsto (f+\mathfrak{m},f+\mathfrak{m}^2,f+\mathfrak{m}^3,\cdots ) $$ The preimage of any $(f_1+\mathfrak{m}, f_2+\mathfrak{m}^2,\ldots)$ can be computed as $f_1 +(f_2 -f_1)+(f_3 -f_2)\cdots$, and this is trivially a homomorphism. Hence these rings are isomorphic.

Answer 2

Saying that a sequence $(F_N)$ of is a Cauchy sequence can be translated into the following (we denote $F_N:=\sum_{k\in\mathbb{N}^n}a_{N,k}X^{k_1}\dots X^{k_n}$) :

For any $K\geq 0$, there exists $N_0$, such that for any $k\in\mathbb{N}^n$ verifying $k_i\leq K$ and for any $N,M\geq N_0$, we have $a_{N,k}=a_{M,k}$.

This is how the limit appears.