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

with algebra structure defined in the following way

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

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

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

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

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

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

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

Comment #4399 by Johan on