The Stacks project

Remark 60.5.4. In Situation 60.5.1 we denote $\text{Cris}^\wedge (C/A)$ the category whose objects are pairs $(B \to C, \delta )$ such that

  1. $B$ is a $p$-adically complete $A$-algebra,

  2. $B \to C$ is a surjection of $A$-algebras,

  3. $\delta $ is a divided power structure on $\mathop{\mathrm{Ker}}(B \to C)$,

  4. $A \to B$ is a homomorphism of divided power rings.

Morphisms are defined as in Definition 60.5.2. Then $\text{Cris}(C/A) \subset \text{Cris}^\wedge (C/A)$ is the full subcategory consisting of those $B$ such that $p$ is nilpotent in $B$. Conversely, any object $(B \to C, \delta )$ of $\text{Cris}^\wedge (C/A)$ is equal to the limit

\[ (B \to C, \delta ) = \mathop{\mathrm{lim}}\nolimits _ e (B/p^ eB \to C, \delta ) \]

where for $e \gg 0$ the object $(B/p^ eB \to C, \delta )$ lies in $\text{Cris}(C/A)$, see Divided Power Algebra, Lemma 23.4.5. In particular, we see that $\text{Cris}^\wedge (C/A)$ is a full subcategory of the category of pro-objects of $\text{Cris}(C/A)$, see Categories, Remark 4.22.5.


Comments (0)

There are also:

  • 5 comment(s) on Section 60.5: Affine crystalline site

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Your email address will not be published.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