## 60.13 Two universal thickenings

The constructions in this section will help us define a connection on a crystal in modules on the crystalline site. In some sense the constructions here are the “sheafified, universal” versions of the constructions in Section 60.3.

Remark 60.13.1. 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 explicitly describe a first order thickening $T'$ of $T$. Namely, set

$\mathcal{O}_{T'} = \mathcal{O}_ T \oplus \Omega _{T/S, \delta }$

with algebra structure such that $\Omega _{T/S, \delta }$ is an ideal of square zero. Let $\mathcal{J} \subset \mathcal{O}_ T$ be the ideal sheaf of the closed immersion $U \to T$. Set $\mathcal{J}' = \mathcal{J} \oplus \Omega _{T/S, \delta }$. Define a divided power structure on $\mathcal{J}'$ by setting

$\delta _ n'(f, \omega ) = (\delta _ n(f), \delta _{n - 1}(f)\omega ),$

see Lemma 60.3.1. There are two ring maps

$p_0, p_1 : \mathcal{O}_ T \to \mathcal{O}_{T'}$

The first is given by $f \mapsto (f, 0)$ and the second by $f \mapsto (f, \text{d}_{T/S, \delta }f)$. Note that both are compatible with the divided power structures on $\mathcal{J}$ and $\mathcal{J}'$ and so is the quotient map $\mathcal{O}_{T'} \to \mathcal{O}_ T$. Thus we get an object $(U, T', \delta ')$ of $\text{Cris}(X/S)$ and a commutative diagram

$\xymatrix{ & T \ar[ld]_{\text{id}} \ar[d]^ i \ar[rd]^{\text{id}} \\ T & T' \ar[l]_{p_0} \ar[r]^{p_1} & T }$

of $\text{Cris}(X/S)$ such that $i$ is a first order thickening whose ideal sheaf is identified with $\Omega _{T/S, \delta }$ and such that $p_1^* - p_0^* : \mathcal{O}_ T \to \mathcal{O}_{T'}$ is identified with the universal derivation $\text{d}_{T/S, \delta }$ composed with the inclusion $\Omega _{T/S, \delta } \to \mathcal{O}_{T'}$.

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.

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