I am reading the Chapter IX of Algebraic Geometry II by Mumford. In the first section of the chapter is about Mori’s existence theorem. In Proposition 1.2, take $\mathscr{O}$ to be the stalk of a locally noetherian scheme and $\mathscr{m}$ to be its maximal ideal. Then the writer said that the $\mathscr{m} $-adic completion $\hat{\mathscr{O}}=A/a$ with $a\subset M^2$ where $A$ is a formal power series ring with its maximal ideal $M$.
The following is what I thought: since $\mathscr{O}$ is noetherian, we have $\mathscr{m}=(a_1, a_2, ...... , a_n)$, then $\hat{\mathscr{O}}=\mathscr{O}[[X_1,......, X_n]]/(X_1-a_1,......,X_n-a_n)$. But the other thing does not really fit.
Did I miss something important?