The Stacks project

Remark 60.26.3. Let $(\mathcal{E}, F)$ be an $F$-crystal as in Definition 60.26.2. In the literature the nondegeneracy condition is often part of the definition of an $F$-crystal. Moreover, often it is also assumed that $F \circ V = p^ n\text{id}$. What is needed for the result below is that there exists an integer $j \geq 0$ such that $\mathop{\mathrm{Ker}}(F)$ and $\mathop{\mathrm{Coker}}(F)$ are killed by $p^ j$. If the rank of $\mathcal{E}$ is bounded (for example if $X$ is quasi-compact), then both of these conditions follow from the nondegeneracy condition as formulated in the definition. Namely, suppose $R$ is a ring, $r \geq 1$ is an integer and $K, L \in \text{Mat}(r \times r, R)$ are matrices with $K L = p^ i 1_{r \times r}$. Then $\det (K)\det (L) = p^{ri}$. Let $L'$ be the adjugate matrix of $L$, i.e., $L' L = L L' = \det (L)$. Set $K' = p^{ri} K$ and $j = ri + i$. Then we have $K' L = p^ j 1_{r \times r}$ as $K L = p^ i$ and

\[ L K' = L K \det (L) \det (M) = L K L L' \det (M) = L p^ i L' \det (M) = p^ j 1_{r \times r} \]

It follows that if $V$ is as in Definition 60.26.2 then setting $V' = p^ N V$ where $N > i \cdot \text{rank}(\mathcal{E})$ we get $V' \circ F = p^{N + i}$ and $F \circ V' = p^{N + i}$.

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