The Stacks project

Lemma 10.160.10. Let $(R, \mathfrak m)$ be a Noetherian complete local ring. Assume $R$ is regular.

  1. If $R$ contains either $\mathbf{F}_ p$ or $\mathbf{Q}$, then $R$ is isomorphic to a power series ring over its residue field.

  2. If $k$ is a field and $k \to R$ is a ring map inducing an isomorphism $k \to R/\mathfrak m$, then $R$ is isomorphic as a $k$-algebra to a power series ring over $k$.

Proof. In case (1), by the Cohen structure theorem (Theorem 10.160.8) there exists a coefficient ring which must be a field mapping isomorphically to the residue field. Thus it suffices to prove (2). In case (2) we pick $f_1, \ldots , f_ d \in \mathfrak m$ which map to a basis of $\mathfrak m/\mathfrak m^2$ and we consider the continuous $k$-algebra map $k[[x_1, \ldots , x_ d]] \to R$ sending $x_ i$ to $f_ i$. As both source and target are $(x_1, \ldots , x_ d)$-adically complete, this map is surjective by Lemma 10.96.1. On the other hand, it has to be injective because otherwise the dimension of $R$ would be $< d$ by Lemma 10.60.13. $\square$

Comments (2)

Comment #1853 by Matthieu Romagny on

Typo in (2) : isomrphism --> isomorphism

Comment #1892 by on

This already got fixed in commit 368a679d67605624062d35132d96d4bc5b32c73d which you can find here.

There are also:

  • 2 comment(s) on Section 10.160: The Cohen structure theorem

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