The Stacks project

Lemma 60.25.4. Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma )$ be a divided power scheme over $\mathbf{Z}_{(p)}$ with $p \in \mathcal{I}$. We set $S_0 = V(\mathcal{I}) \subset S$. Let $f : X' \to X$ be an iterated $\alpha _ p$-cover of schemes over $S_0$ with constant degree $q$. Let $\mathcal{F}$ be any crystal in quasi-coherent sheaves on $X$ and set $\mathcal{F}' = f_{\text{cris}}^*\mathcal{F}$. In the distinguished triangle

\[ Ru_{X/S, *}\mathcal{F} \longrightarrow f_*Ru_{X'/S, *}\mathcal{F}' \longrightarrow E \longrightarrow Ru_{X/S, *}\mathcal{F}[1] \]

the object $E$ has cohomology sheaves annihilated by $q$.

Proof. Note that $X' \to X$ is a homeomorphism hence we can identify the underlying topological spaces of $X$ and $X'$. The question is clearly local on $X$, hence we may assume $X$, $X'$, and $S$ affine and $X' \to X$ given as a composition

\[ X' = X_ n \to X_{n - 1} \to X_{n - 2} \to \ldots \to X_0 = X \]

where each morphism $X_{i + 1} \to X_ i$ is an $\alpha _ p$-cover. Denote $\mathcal{F}_ i$ the pullback of $\mathcal{F}$ to $X_ i$. It suffices to prove that each of the maps

\[ R\Gamma (\text{Cris}(X_ i/S), \mathcal{F}_ i) \longrightarrow R\Gamma (\text{Cris}(X_{i + 1}/S), \mathcal{F}_{i + 1}) \]

fits into a triangle whose third member has cohomology groups annihilated by $p$. (This uses axiom TR4 for the triangulated category $D(X)$. Details omitted.)

Hence we may assume that $S = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(C)$, $X' = \mathop{\mathrm{Spec}}(C')$ and $C' = C[z]/(z^ p - c)$ for some $c \in C$. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$. Let $D$ be the $p$-adically completed divided power envelop of $\mathop{\mathrm{Ker}}(P \to C)$ in $P$ as in ( Set $P' = P[z]$ with surjection $P' \to C'$ mapping $z$ to the class of $z$ in $C'$. Choose a lift $\lambda \in D$ of $c \in C$. Then we see that the $p$-adically completed divided power envelope $D'$ of $\mathop{\mathrm{Ker}}(P' \to C')$ in $P'$ is isomorphic to the $p$-adic completion of $D[z]\langle \xi \rangle /(\xi - (z^ p - \lambda ))$, see Lemma 60.25.3 and its proof. Thus we see that the result follows from this lemma by the computation of cohomology of crystals in quasi-coherent modules in Proposition 60.21.3. $\square$

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