The Stacks project

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.

Comments (1)

Comment #8734 by Eiki on

Some possibly pedantic comments; "two ring maps " aren't they technically maps of sheaves where on the level of sections they are described as above?

If we are later calling to be the maps on the corresponding schemes, maybe we want to give it some other name (probably rather than here)?

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