The Stacks project

Remark 60.13.2. In Situation 60.7.5. Let $(U, T, \delta )$ be an object of $\text{Cris}(X/S)$. Write $\Omega _{T/S, \delta } = (\Omega _{X/S})_ T$, see Lemma 60.12.3. We also write $\Omega ^2_{T/S, \delta }$ for its second exterior power. We explicitly describe a second order thickening $T''$ of $T$. Namely, set

\[ \mathcal{O}_{T''} = \mathcal{O}_ T \oplus \Omega _{T/S, \delta } \oplus \Omega _{T/S, \delta } \oplus \Omega ^2_{T/S, \delta } \]

with algebra structure defined in the following way

\[ (f, \omega _1, \omega _2, \eta ) \cdot (f', \omega _1', \omega _2', \eta ') = (ff', f\omega _1' + f'\omega _1, f\omega _2' + f'\omega _2, f\eta ' + f'\eta + \omega _1 \wedge \omega _2' + \omega _1' \wedge \omega _2). \]

Let $\mathcal{J} \subset \mathcal{O}_ T$ be the ideal sheaf of the closed immersion $U \to T$. Let $\mathcal{J}''$ be the inverse image of $\mathcal{J}$ under the projection $\mathcal{O}_{T''} \to \mathcal{O}_ T$. Define a divided power structure on $\mathcal{J}''$ by setting

\[ \delta _ n''(f, \omega _1, \omega _2, \eta ) = (\delta _ n(f), \delta _{n - 1}(f)\omega _1, \delta _{n - 1}(f)\omega _2, \delta _{n - 1}(f)\eta + \delta _{n - 2}(f)\omega _1 \wedge \omega _2) \]

see Lemma 60.3.2. There are three ring maps $q_0, q_1, q_2 : \mathcal{O}_ T \to \mathcal{O}_{T''}$ given by

\begin{align*} q_0(f) & = (f, 0, 0, 0), \\ q_1(f) & = (f, \text{d}f, 0, 0), \\ q_2(f) & = (f, \text{d}f, \text{d}f, 0) \end{align*}

where $\text{d} = \text{d}_{T/S, \delta }$. Note that all three are compatible with the divided power structures on $\mathcal{J}$ and $\mathcal{J}''$. There are three ring maps $q_{01}, q_{12}, q_{02} : \mathcal{O}_{T'} \to \mathcal{O}_{T''}$ where $\mathcal{O}_{T'}$ is as in Remark 60.13.1. Namely, set

\begin{align*} q_{01}(f, \omega ) & = (f, \omega , 0, 0), \\ q_{12}(f, \omega ) & = (f, \text{d}f, \omega , \text{d}\omega ), \\ q_{02}(f, \omega ) & = (f, \omega , \omega , 0) \end{align*}

These are also compatible with the given divided power structures. Let's do the verifications for $q_{12}$: Note that $q_{12}$ is a ring homomorphism as

\begin{align*} q_{12}(f, \omega )q_{12}(g, \eta ) & = (f, \text{d}f, \omega , \text{d}\omega )(g, \text{d}g, \eta , \text{d}\eta ) \\ & = (fg, f\text{d}g + g \text{d}f, f\eta + g\omega , f\text{d}\eta + g\text{d}\omega + \text{d}f \wedge \eta + \text{d}g \wedge \omega ) \\ & = q_{12}(fg, f\eta + g\omega ) = q_{12}((f, \omega )(g, \eta )) \end{align*}

Note that $q_{12}$ is compatible with divided powers because

\begin{align*} \delta _ n''(q_{12}(f, \omega )) & = \delta _ n''((f, \text{d}f, \omega , \text{d}\omega )) \\ & = (\delta _ n(f), \delta _{n - 1}(f)\text{d}f, \delta _{n - 1}(f)\omega , \delta _{n - 1}(f)\text{d}\omega + \delta _{n - 2}(f)\text{d}(f) \wedge \omega ) \\ & = q_{12}((\delta _ n(f), \delta _{n - 1}(f)\omega )) = q_{12}(\delta '_ n(f, \omega )) \end{align*}

The verifications for $q_{01}$ and $q_{02}$ are easier. Note that $q_0 = q_{01} \circ p_0$, $q_1 = q_{01} \circ p_1$, $q_1 = q_{12} \circ p_0$, $q_2 = q_{12} \circ p_1$, $q_0 = q_{02} \circ p_0$, and $q_2 = q_{02} \circ p_1$. Thus $(U, T'', \delta '')$ is an object of $\text{Cris}(X/S)$ and we get morphisms

\[ \xymatrix{ T'' \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & T' \ar@<1ex>[r] \ar@<-1ex>[r] & T } \]

of $\text{Cris}(X/S)$ satisfying the relations described above. In applications we will use $q_ i : T'' \to T$ and $q_{ij} : T'' \to T'$ to denote the morphisms associated to the ring maps described above.


Comments (2)

Comment #4219 by Dario Weißmann on

Typo in the definition of the multiplication: has too many s

There is also an issue how the multiplication formula is displayed. It overlaps with the sidebar. Works fine in the pdf version though.

There are also:

  • 2 comment(s) on Section 60.13: Two universal thickenings

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 07J3. Beware of the difference between the letter 'O' and the digit '0'.