The Stacks project

Lemma 15.47.4. Let $p$ be a prime number. Let $B$ be a domain with $p = 0$ in $B$. Let $f \in B$ be an element which is not a $p$th power in the fraction field of $B$. If $B$ is of finite type over a Noetherian complete local ring, then there exists a derivation $D : B \to B$ such that $D(f)$ is not zero.

Proof. Let $R$ be a Noetherian complete local ring such that there exists a finite type ring map $R \to B$. Of course we may replace $R$ by its image in $B$, hence we may assume $R$ is a domain of characteristic $p > 0$ (as well as Noetherian complete local). By Algebra, Lemma 10.154.11 we can write $R$ as a finite extension of $k[[x_1, \ldots , x_ n]]$ for some field $k$ and integer $n$. Hence we may replace $R$ by $k[[x_1, \ldots , x_ n]]$. Next, we use Algebra, Lemma 10.114.7 to factor $R \to B$ as

\[ R \subset R[y_1, \ldots , y_ d] \subset B' \subset B \]

with $B'$ finite over $R[y_1, \ldots , y_ d]$ and $B'_ g \cong B_ g$ for some nonzero $g \in R$. Note that $f' = g^{pN} f \in B'$ for some large integer $N$. It is clear that $f'$ is not a $p$th power in the fraction field of $B'$. If we can find a derivation $D' : B' \to B'$ with $D'(f') \not= 0$, then Lemma 15.47.1 guarantees that $D = g^ MD'$ extends to $B$ for some $M > 0$. Then $D(f) = g^ ND'(f) = g^ MD'(g^{-pN}f') = g^{M - pN}D'(f')$ is nonzero. Thus it suffices to prove the lemma in case $B$ is a finite extension of $A = k[[x_1, \ldots , x_ n]][y_1, \ldots , y_ m]$.

Assume $B$ is a finite extension of $A = k[[x_1, \ldots , x_ n]][y_1, \ldots , y_ m]$. Denote $L$ the fraction field of $B$. Note that $\text{d}f$ is not zero in $\Omega _{L/\mathbf{F}_ p}$, see Algebra, Lemma 10.152.2. We apply Lemma 15.45.5 to find a subfield $k' \subset k$ of finite index such that with $A' = k'[[x_1^ p, \ldots , x_ n^ p]][y_1^ p, \ldots , y_ m^ p]$ the element $\text{d}f$ does not map to zero in $\Omega _{L/K'}$ where $K'$ is the fraction field of $A'$. Thus we can choose a $K'$-derivation $D' : L \to L$ with $D'(f) \not= 0$. Since $A' \subset A$ and $A \subset B$ are finite by construction we see that $A' \subset B$ is finite. Choose $b_1, \ldots , b_ t \in B$ which generate $B$ as an $A'$-module. Then $D'(b_ i) = f_ i/g_ i$ for some $f_ i, g_ i \in B$ with $g_ i \not= 0$. Setting $D = g_1 \ldots g_ t D'$ we win. $\square$


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