The Stacks project

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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GKU. Beware of the difference between the letter 'O' and the digit '0'.