Lemma 63.13.1. Let $p$ be a prime number. Let $S$ be a scheme over $\mathbf{F}_ p$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ S$-module viewed as an $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$. Let $F : \mathcal{E} \to \mathcal{E}$ be a homomorphism of abelian sheaves on $S_{\acute{e}tale}$ such that $F(a e) = a^ pF(e)$ for local sections $a$, $e$ of $\mathcal{O}_ S$, $\mathcal{E}$ on $S_{\acute{e}tale}$. Then

\[ \mathop{\mathrm{Coker}}(F - 1 : \mathcal{E} \to \mathcal{E}) \]

is zero and

\[ \mathop{\mathrm{Ker}}(F - 1 : \mathcal{E} \to \mathcal{E}) \]

is a constructible abelian sheaf on $S_{\acute{e}tale}$.

**Proof.**
We may assume $S = \mathop{\mathrm{Spec}}(A)$ where $A$ is an $\mathbf{F}_ p$-algebra and that $\mathcal{E}$ is the quasi-coherent module associated to the free $A$-module $Ae_1 \oplus \ldots \oplus Ae_ n$. We write $F(e_ i) = \sum a_{ij} e_ j$.

Surjectivity of $F - 1$. It suffices to show that any element $\sum a_ i e_ i$, $a_ i \in A$ is in the image of $F - 1$ after replacing $A$ by a faithfully flat étale extension. Observe that

\[ F(\sum x_ ie_ i) - \sum x_ i e_ i = \sum x_ i^ p a_{ij} e_ j - \sum x_ i e_ i \]

Consider the $A$-algebra

\[ A' = A[x_1, \ldots , x_ n]/(a_ i + x_ i - \sum \nolimits _ j a_{ji} x_ j^ p) \]

A computation shows that $\text{d}x_ i$ is zero in $\Omega _{A'/A}$ and hence $\Omega _{A'/A} = 0$. Since $A'$ is of finite type over $A$, this implies that $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ is unramified and hence is quasi-finite. Since $A'$ is generated by $n$ elements and cut out by $n$ equations, we conclude that $A'$ is a global relative complete intersection over $A$. Thus $A'$ is flat over $A$ and we conclude that $A \to A'$ is étale (as a flat and unramified ring map). Finally, the reader can show that $A \to A'$ is faithfully flat by verifying directly that all geometric fibres of $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ are nonempty, however this also follows from Étale Cohomology, Lemma 59.63.2. Finally, the element $\sum x_ i e_ i \in A'e_1 \oplus \ldots \oplus A'e_ n$ maps to $\sum a_ i e_ i$ by $F - 1$.

Constructibility of the kernel. The calculations above show that $\mathop{\mathrm{Ker}}(F - 1)$ is represented by the scheme

\[ \mathop{\mathrm{Spec}}(A[x_1, \ldots , x_ n]/(x_ i - \sum \nolimits _ j a_{ji} x_ j^ p)) \]

over $S = \mathop{\mathrm{Spec}}(A)$. Since this is a scheme affine and étale over $S$ we obtain the result from Étale Cohomology, Lemma 59.73.1.
$\square$

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