The Stacks project

Remark 60.24.1 (Higher direct images). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma ) \to (S', \mathcal{I}', \gamma ')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let

\[ \xymatrix{ X \ar[r]_ f \ar[d] & X' \ar[d] \\ S_0 \ar[r] & S'_0 } \]

be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.

Given an object $(U', T', \delta ')$ of $\text{Cris}(X'/S')$ set $U = X \times _{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \delta )$ the divided power scheme over $S$ such that

\[ \xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' } \]

is cartesian in the category of divided power schemes, see Lemma 60.7.4. There is an induced morphism $U \to T_0$ and we obtain a morphism $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see Remark 60.9.3. Let $\mathcal{F}_ U$ be the pullback of $\mathcal{F}$. Let $\tau _{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism. Then we have
\begin{equation} \label{crystalline-equation-identify-pushforward} \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} = R(T \to T')_*\left(R\tau _{U/T, *} \mathcal{F}_ U \right) \end{equation}

where the left hand side is the restriction (see Section 60.10).

Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma 7.30.3) of $\text{Cris}(X/S)$ at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta ')}$. Next, reduce the statement to the case where $\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.

Comments (0)

There are also:

  • 2 comment(s) on Section 60.24: Some further results

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Your email address will not be published.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 07MJ. Beware of the difference between the letter 'O' and the digit '0'.