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 10.152.4. Let $k$ be a field of characteristic $p > 0$. The following are equivalent:

  1. the field extension $K/k$ is separable (see Definition 10.41.1), and

  2. the map $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K/\mathbf{F}_ p}$ is injective.

Proof. Write $K$ as a directed colimit $K = \mathop{\mathrm{colim}}\nolimits _ i K_ i$ of finitely generated field extensions $k \subset K_ i$. By definition $K$ is separable if and only if each $K_ i$ is separable over $k$, and by Lemma 10.130.4 we see that $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K/\mathbf{F}_ p}$ is injective if and only if each $K_ i \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K_ i/\mathbf{F}_ p}$ is injective. Hence we may assume that $K/k$ is a finitely generated field extension.

Assume $k \subset K$ is a finitely generated field extension which is separable. Choose $x_1, \ldots , x_{r + 1} \in K$ as in Lemma 10.41.3. In this case there exists an irreducible polynomial $G(X_1, \ldots , X_{r + 1}) \in k[X_1, \ldots , X_{r + 1}]$ such that $G(x_1, \ldots , x_{r + 1}) = 0$ and such that $\partial G/\partial X_{r + 1}$ is not identically zero. Moreover $K$ is the field of fractions of the domain. $S = K[X_1, \ldots , X_{r + 1}]/(G)$. Write

\[ G = \sum a_ I X^ I, \quad X^ I = X_1^{i_1}\ldots X_{r + 1}^{i_{r + 1}}. \]

Using the presentation of $S$ above we see that

\[ \Omega _{S/\mathbf{F}_ p} = \frac{ S \otimes _ k \Omega _ k \oplus \bigoplus \nolimits _{i = 1, \ldots , r + 1} S\text{d}X_ i }{ \langle \sum X^ I \text{d}a_ I + \sum \partial G/\partial X_ i \text{d}X_ i \rangle } \]

Since $\Omega _{K/\mathbf{F}_ p}$ is the localization of the $S$-module $\Omega _{S/\mathbf{F}_ p}$ (see Lemma 10.130.8) we conclude that

\[ \Omega _{K/\mathbf{F}_ p} = \frac{ K \otimes _ k \Omega _ k \oplus \bigoplus \nolimits _{i = 1, \ldots , r + 1} K\text{d}X_ i }{ \langle \sum X^ I \text{d}a_ I + \sum \partial G/\partial X_ i \text{d}X_ i \rangle } \]

Now, since the polynomial $\partial G/\partial X_{r + 1}$ is not identically zero we conclude that the map $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{S/\mathbf{F}_ p}$ is injective as desired.

Assume $k \subset K$ is a finitely generated field extension and that $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K/\mathbf{F}_ p}$ is injective. (This part of the proof is the same as the argument proving Lemma 10.43.1.) Let $x_1, \ldots , x_ r$ be a transcendence basis of $K$ over $k$ such that the degree of inseparability of the finite extension $k(x_1, \ldots , x_ r) \subset K$ is minimal. If $K$ is separable over $k(x_1, \ldots , x_ r)$ then we win. Assume this is not the case to get a contradiction. Then there exists an element $\alpha \in K$ which is not separable over $k(x_1, \ldots , x_ r)$. Let $P(T) \in k(x_1, \ldots , x_ r)[T]$ be its minimal polynomial. Because $\alpha $ is not separable actually $P$ is a polynomial in $T^ p$. Clear denominators to get an irreducible polynomial

\[ G(X_1, \ldots , X_ r, T) = \sum a_{I, i} X^ I T^ i \in k[X_1, \ldots , X_ r, T] \]

such that $G(x_1, \ldots , x_ r, \alpha ) = 0$ in $L$. Note that this means $k[X_1, \ldots , X_ r, T]/(G) \subset L$. We may assume that for some pair $(I_0, i_0)$ the coefficient $a_{I_0, i_0} = 1$. We claim that $\text{d}G/\text{d}X_ i$ is not identically zero for at least one $i$. Namely, if this is not the case, then $G$ is actually a polynomial in $X_1^ p, \ldots , X_ r^ p, T^ p$. Then this means that

\[ \sum \nolimits _{(I, i) \not= (I_0, i_0)} x^ I\alpha ^ i \text{d}a_{I, i} \]

is zero in $\Omega _{K/\mathbf{F}_ p}$. Note that there is no $k$-linear relation among the elements

\[ \{ x^ I\alpha ^ i \mid a_{I, i} \not= 0 \text{ and } (I, i) \not= (I_0, i_0)\} \]

of $K$. Hence the assumption that $K \otimes _ k \Omega _{k/\mathbf{F}_ p} \to \Omega _{K/\mathbf{F}_ p}$ is injective this implies that $\text{d}a_{I, i} = 0$ in $\Omega _{k/\mathbf{F}_ p}$ for all $(I, i)$. By Lemma 10.152.2 we see that each $a_{I, i}$ is a $p$th power, which implies that $G$ is a $p$th power contradicting the irreducibility of $G$. Thus, after renumbering, we may assume that $\text{d}G/\text{d}X_1$ is not zero. Then we see that $x_1$ is separably algebraic over $k(x_2, \ldots , x_ r, \alpha )$, and that $x_2, \ldots , x_ r, \alpha $ is a transcendence basis of $L$ over $k$. This means that the degree of inseparability of the finite extension $k(x_2, \ldots , x_ r, \alpha ) \subset L$ is less than the degree of inseparability of the finite extension $k(x_1, \ldots , x_ r) \subset L$, which is a contradiction. $\square$


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