The Stacks project

Remark 60.24.10 (Rlim). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent in $A/I$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. Let $X$ be a scheme over $S_0$ with $p$ locally nilpotent on $X$. Let $\mathcal{F}$ be any $\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^ e) \subset I$ is preserved by $\gamma $, see Divided Power Algebra, Lemma 23.4.5. Set $S_ e = \mathop{\mathrm{Spec}}(A/p^ eA)$ for $e \gg 0$. Then $\text{Cris}(X/S_ e)$ is a full subcategory of $\text{Cris}(X/S)$ and we denote $\mathcal{F}_ e$ the restriction of $\mathcal{F}$ to $\text{Cris}(X/S_ e)$. Then

\[ R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits _ e R\Gamma (\text{Cris}(X/S_ e), \mathcal{F}_ e) \]

Hints: Suffices to prove this for $\mathcal{F}$ injective. In this case the sheaves $\mathcal{F}_ e$ are injective modules too, the transition maps $\Gamma (\mathcal{F}_{e + 1}) \to \Gamma (\mathcal{F}_ e)$ are surjective, and we have $\Gamma (\mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ e \Gamma (\mathcal{F}_ e)$ because any object of $\text{Cris}(X/S)$ is locally an object of one of the categories $\text{Cris}(X/S_ e)$ by definition of $\text{Cris}(X/S)$.

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Your email address will not be published.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