The Stacks project

Lemma 60.5.5. In Situation 60.5.1. Let $P \to C$ be a surjection of $A$-algebras with kernel $J$. Write $D_{P, \gamma }(J) = (D, \bar J, \bar\gamma )$. Let $(D^\wedge , J^\wedge , \bar\gamma ^\wedge )$ be the $p$-adic completion of $D$, see Divided Power Algebra, Lemma 23.4.5. For every $e \geq 1$ set $P_ e = P/p^ eP$ and $J_ e \subset P_ e$ the image of $J$ and write $D_{P_ e, \gamma }(J_ e) = (D_ e, \bar J_ e, \bar\gamma )$. Then for all $e$ large enough we have

  1. $p^ eD \subset \bar J$ and $p^ eD^\wedge \subset \bar J^\wedge $ are preserved by divided powers,

  2. $D^\wedge /p^ eD^\wedge = D/p^ eD = D_ e$ as divided power rings,

  3. $(D_ e, \bar J_ e, \bar\gamma )$ is an object of $\text{Cris}(C/A)$,

  4. $(D^\wedge , \bar J^\wedge , \bar\gamma ^\wedge )$ is equal to $\mathop{\mathrm{lim}}\nolimits _ e (D_ e, \bar J_ e, \bar\gamma )$, and

  5. $(D^\wedge , \bar J^\wedge , \bar\gamma ^\wedge )$ is an object of $\text{Cris}^\wedge (C/A)$.

Proof. Part (1) follows from Divided Power Algebra, Lemma 23.4.5. It is a general property of $p$-adic completion that $D/p^ eD = D^\wedge /p^ eD^\wedge $. Since $D/p^ eD$ is a divided power ring and since $P \to D/p^ eD$ factors through $P_ e$, the universal property of $D_ e$ produces a map $D_ e \to D/p^ eD$. Conversely, the universal property of $D$ produces a map $D \to D_ e$ which factors through $D/p^ eD$. We omit the verification that these maps are mutually inverse. This proves (2). If $e$ is large enough, then $p^ eC = 0$, hence we see (3) holds. Part (4) follows from Divided Power Algebra, Lemma 23.4.5. Part (5) is clear from the definitions. $\square$

Comments (0)

There are also:

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Your email address will not be published.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