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?

  • 1
    $\begingroup$ He might be referring to the Cohen structure theorem (en.wikipedia.org/wiki/Cohen_structure_theorem). This essentially says that any complete Noetherian local ring is the quotient of a formal power series ring. However, this will not be the formal power series ring $\mathcal{O}[[X_1, \cdots, X_n]]$, think about what happens when you take the stalk at the origin of the affine line. $\endgroup$ – user45878 Apr 6 '18 at 14:04

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