The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.152.7. Let $k \subset K$ be an extension of fields.

  1. If $K$ is purely transcendental over $k$, then $K$ is formally smooth over $k$.

  2. If $K$ is separable algebraic over $k$, then $K$ is formally smooth over $k$.

  3. If $K$ is separable over $k$, then $K$ is formally smooth over $k$.

Proof. For (1) write $K = k(x_ j; j \in J)$. Suppose that $A$ is a $k$-algebra, and $I \subset A$ is an ideal of square zero. Let $\varphi : K \to A/I$ be a $k$-algebra map. Let $a_ j \in A$ be an element such that $a_ j \mod I = \varphi (x_ j)$. Then it is easy to see that there is a unique $k$-algebra map $K \to A$ which maps $x_ j$ to $a_ j$ and which reduces to $\varphi $ mod $I$. Hence $k \subset K$ is formally smooth.

In case (2) we see that $k \subset K$ is a colimit of étale ring extensions. An étale ring map is formally étale (Lemma 10.146.2). Hence this case follows from Lemma 10.146.3 and the trivial observation that a formally étale ring map is formally smooth.

In case (3), write $K = \mathop{\mathrm{colim}}\nolimits K_ i$ as the filtered colimit of its finitely generated sub $k$-extensions. By Definition 10.41.1 each $K_ i$ is separable algebraic over a purely transcendental extension of $k$. Hence $K_ i/k$ is formally smooth by cases (1) and (2) and Lemma 10.136.3. Thus $H_1(L_{K_ i/k}) = 0$ by Lemma 10.152.6. Hence $H_1(L_{K/k}) = 0$ by Lemma 10.132.9. Hence $K/k$ is formally smooth by Lemma 10.152.6 again. $\square$


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