Proposition 19

Let $A$ be a commutative ring and $m$ a maximal ideal of $A$. The Hausdorff completion $\hat{A}$ of $A$ with respect to the $m$-adic topology is a local ring whose maximal ideal is $m$.

If $a = \bigcap_{k \geq 1} m^k$, $\hat{A}$ is the completion of the Hausdorff ring $A/a$ associated with $A$ and, as $m/a$ is maximal in $A/a$, we may assume that $A$ is Hausdorff with respect to the $m$-adic topology. As $A/m$ and $\hat{A}/\hat{m}$ are isomorphic rings (no. 12, formula (21)), $m$ is maximal in $\hat{A}$. As the topology on $\hat{A}$ is defined by the filtration $(m^n)^*$ (no. 12), the proposition will be a consequence of the following lemma: