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 10.155.10. Let $R$ be a Noetherian normal domain with fraction field $K$ of characteristic $p > 0$. Let $a \in K$ be an element such that there exists a derivation $D : R \to R$ with $D(a) \not= 0$. Then the integral closure of $R$ in $L = K[x]/(x^ p - a)$ is finite over $R$.

Proof. After replacing $x$ by $fx$ and $a$ by $f^ pa$ for some $f \in R$ we may assume $a \in R$. Hence also $D(a) \in R$. We will show by induction on $i \leq p - 1$ that if

\[ y = a_0 + a_1x + \ldots + a_ i x^ i,\quad a_ j \in K \]

is integral over $R$, then $D(a)^ i a_ j \in R$. Thus the integral closure is contained in the finite $R$-module with basis $D(a)^{-p + 1}x^ j$, $j = 0, \ldots , p - 1$. Since $R$ is Noetherian this proves the lemma.

If $i = 0$, then $y = a_0$ is integral over $R$ if and only if $a_0 \in R$ and the statement is true. Suppose the statement holds for some $i < p - 1$ and suppose that

\[ y = a_0 + a_1x + \ldots + a_{i + 1} x^{i + 1},\quad a_ j \in K \]

is integral over $R$. Then

\[ y^ p = a_0^ p + a_1^ p a + \ldots + a_{i + 1}^ pa^{i + 1} \]

is an element of $R$ (as it is in $K$ and integral over $R$). Applying $D$ we obtain

\[ (a_1^ p + 2a_2^ p a + \ldots + (i + 1)a_{i + 1}^ p a^ i)D(a) \]

is in $R$. Hence it follows that

\[ D(a)a_1 + 2D(a) a_2 x + \ldots + (i + 1)D(a) a_{i + 1} x^ i \]

is integral over $R$. By induction we find $D(a)^{i + 1}a_ j \in R$ for $j = 1, \ldots , i + 1$. (Here we use that $1, \ldots , i + 1$ are invertible.) Hence $D(a)^{i + 1}a_0$ is also in $R$ because it is the difference of $y$ and $\sum _{j > 0} D(a)^{i + 1}a_ jx^ j$ which are integral over $R$ (since $x$ is integral over $R$ as $a \in R$). $\square$


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