The Stacks project

Lemma 33.21.1. Let $k$ be a field. Let $X$ be a locally Noetherian scheme over $k$. Let $x \in X$ be a point with residue field $\kappa $. There is an isomorphism
\begin{equation} \label{varieties-equation-complete-local-ring} \kappa [[x_1, \ldots , x_ n]]/I \longrightarrow \mathcal{O}_{X, x}^\wedge \end{equation}

inducing the identity on residue fields. In general we cannot choose ( to be a $k$-algebra isomorphism. However, if the extension $\kappa /k$ is separable, then we can choose ( to be an isomorphism of $k$-algebras.

Proof. The existence of the isomorphism is an immediate consequence of the Cohen structure theorem1 (Algebra, Theorem 10.160.8).

Let $p$ be an odd prime number, let $k = \mathbf{F}_ p(t)$, and $A = k[x, y]/(y^2 + x^ p - t)$. Then the completion $A^\wedge $ of $A$ in the maximal ideal $\mathfrak m = (y)$ is isomorphic to $k(t^{1/p})[[z]]$ as a ring but not as a $k$-algebra. The reason is that $A^\wedge $ does not contain an element whose $p$th power is $t$ (as the reader can see by computing modulo $y^2$). This also shows that any isomorphism ( cannot be a $k$-algebra isomorphism.

If $\kappa /k$ is separable, then there is a $k$-algebra homomorphism $\kappa \to \mathcal{O}_{X, x}^\wedge $ inducing the identity on residue fields by More on Algebra, Lemma 15.38.3. Let $f_1, \ldots , f_ n \in \mathfrak m_ x$ be generators. Consider the map

\[ \kappa [[x_1, \ldots , x_ n]] \longrightarrow \mathcal{O}_{X, x}^\wedge ,\quad x_ i \longmapsto f_ i \]

Since both sides are $(x_1, \ldots , x_ n)$-adically complete (the right hand side by Algebra, Lemmas 10.96.3) this map is surjective by Algebra, Lemma 10.96.1 as it is surjective modulo $(x_1, \ldots , x_ n)$ by construction. $\square$

[1] Note that if $\kappa $ has characteristic $p$, then the theorem just says we get a surjection $\Lambda [[x_1, \ldots , x_ n]] \to \mathcal{O}_{X, x}^\wedge $ where $\Lambda $ is a Cohen ring for $\kappa $. But of course in this case the map factors through $\Lambda /p\Lambda [[x_1, \ldots , x_ n]]$ and $\Lambda /p\Lambda = \kappa $.

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