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
33.21.1.1
\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 (33.21.1.1) to be a k-algebra isomorphism. However, if the extension \kappa /k is separable, then we can choose (33.21.1.1) 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 pth power is t (as the reader can see by computing modulo y^2). This also shows that any isomorphism (33.21.1.1) 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
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