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\input{preamble}
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\begin{document}
\title{Crystalline Cohomology}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter is based on a lecture series given by Johan de Jong
held in 2012 at Columbia University.
The goals of this chapter are to give a quick introduction to
crystalline cohomology. A reference is the book \cite{Berthelot}.
\medskip\noindent
We have moved the more elementary purely algebraic discussion of divided
power rings to a preliminary chapter as it is also useful
in discussing Tate resolutions in commutative algebra.
Please see Divided Power Algebra, Section \ref{dpa-section-introduction}.
\section{Divided power envelope}
\label{section-divided-power-envelope}
\noindent
The construction of the following lemma will be dubbed the
divided power envelope. It will play an important role later.
\begin{lemma}
\label{lemma-divided-power-envelope}
Let $(A, I, \gamma)$ be a divided power ring.
Let $A \to B$ be a ring map. Let $J \subset B$ be an ideal
with $IB \subset J$. There exists a homomorphism of
divided power rings
$$
(A, I, \gamma) \longrightarrow (D, \bar J, \bar \gamma)
$$
such that
$$
\Hom_{(A, I, \gamma)}((D, \bar J, \bar \gamma), (C, K, \delta)) =
\Hom_{(A, I)}((B, J), (C, K))
$$
functorially in the divided power algebra $(C, K, \delta)$ over
$(A, I, \gamma)$. Here the LHS is morphisms of divided
power rings over $(A, I, \gamma)$ and the RHS is morphisms of
(ring, ideal) pairs over $(A, I)$.
\end{lemma}
\begin{proof}
Denote $\mathcal{C}$ the category of divided power rings
$(C, K, \delta)$. Consider the functor
$F : \mathcal{C} \longrightarrow \textit{Sets}$ defined by
$$
F(C, K, \delta) =
\left\{
(\varphi, \psi)
\middle|
\begin{matrix}
\varphi : (A, I, \gamma) \to (C, K, \delta)
\text{ homomorphism of divided power rings} \\
\psi : (B, J) \to (C, K)\text{ an }
A\text{-algebra homomorphism with }\psi(J) \subset K
\end{matrix}
\right\}
$$
We will show that
Divided Power Algebra, Lemma \ref{dpa-lemma-a-version-of-brown}
applies to this functor which will
prove the lemma. Suppose that $(\varphi, \psi) \in F(C, K, \delta)$.
Let $C' \subset C$ be the subring generated by $\varphi(A)$,
$\psi(B)$, and $\delta_n(\psi(f))$ for all $f \in J$.
Let $K' \subset K \cap C'$ be the ideal of $C'$ generated by
$\varphi(I)$ and $\delta_n(\psi(f))$ for $f \in J$.
Then $(C', K', \delta|_{K'})$ is a divided power ring and
$C'$ has cardinality bounded by the cardinal
$\kappa = |A| \otimes |B|^{\aleph_0}$.
Moreover, $\varphi$ factors as $A \to C' \to C$ and $\psi$ factors
as $B \to C' \to C$. This proves assumption (1) of
Divided Power Algebra, Lemma \ref{dpa-lemma-a-version-of-brown}
holds. Assumption (2) is clear
as limits in the category of divided power rings commute with the
forgetful functor $(C, K, \delta) \mapsto (C, K)$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-limits} and its proof.
\end{proof}
\begin{definition}
\label{definition-divided-power-envelope}
Let $(A, I, \gamma)$ be a divided power ring.
Let $A \to B$ be a ring map. Let $J \subset B$ be an ideal
with $IB \subset J$. The divided power algebra $(D, \bar J, \bar\gamma)$
constructed in Lemma \ref{lemma-divided-power-envelope}
is called the {\it divided power envelope of $J$ in $B$
relative to $(A, I, \gamma)$} and is denoted $D_B(J)$ or $D_{B, \gamma}(J)$.
\end{definition}
\noindent
Let $(A, I, \gamma) \to (C, K, \delta)$ be a homomorphism of divided
power rings. The universal property of
$D_{B, \gamma}(J) = (D, \bar J, \bar \gamma)$ is
$$
\begin{matrix}
\text{ring maps }B \to C \\
\text{ which map }J\text{ into }K
\end{matrix}
\longleftrightarrow
\begin{matrix}
\text{divided power homomorphisms} \\
(D, \bar J, \bar \gamma) \to (C, K, \delta)
\end{matrix}
$$
and the correspondence is given by precomposing with the map $B \to D$
which corresponds to $\text{id}_D$. Here are some properties of
$(D, \bar J, \bar \gamma)$ which follow directly from the universal
property. There are $A$-algebra maps
\begin{equation}
\label{equation-divided-power-envelope}
B \longrightarrow D \longrightarrow B/J
\end{equation}
The first arrow maps $J$ into $\bar J$ and $\bar J$ is the kernel
of the second arrow. The elements $\bar\gamma_n(x)$ where $n > 0$
and $x$ is an element in the image of $J \to D$ generate $\bar J$
as an ideal in $D$ and generate $D$ as a $B$-algebra.
\begin{lemma}
\label{lemma-divided-power-envelop-quotient}
Let $(A, I, \gamma)$ be a divided power ring.
Let $\varphi : B' \to B$ be a surjection of $A$-algebras with kernel $K$.
Let $IB \subset J \subset B$ be an ideal. Let $J' \subset B'$
be the inverse image of $J$. Write
$D_{B', \gamma}(J') = (D', \bar J', \bar\gamma)$.
Then $D_{B, \gamma}(J) = (D'/K', \bar J'/K', \bar\gamma)$
where $K'$ is the ideal generated by the elements $\bar\gamma_n(k)$
for $n \geq 1$ and $k \in K$.
\end{lemma}
\begin{proof}
Write $D_{B, \gamma}(J) = (D, \bar J, \bar \gamma)$.
The universal property of $D'$ gives us a homomorphism $D' \to D$
of divided power algebras. As $B' \to B$ and $J' \to J$ are surjective, we
see that $D' \to D$ is surjective (see remarks above). It is clear that
$\bar\gamma_n(k)$ is in the kernel for $n \geq 1$ and $k \in K$, i.e.,
we obtain a homomorphism $D'/K' \to D$. Conversely, there exists a divided
power structure on $\bar J'/K' \subset D'/K'$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-kernel}.
Hence the universal property of $D$ gives an inverse $D \to D'/K'$ and we win.
\end{proof}
\noindent
In the situation of Definition \ref{definition-divided-power-envelope}
we can choose a surjection $P \to B$ where $P$ is a polynomial
algebra over $A$ and let $J' \subset P$ be the inverse image of $J$.
The previous lemma describes $D_{B, \gamma}(J)$ in terms of
$D_{P, \gamma}(J')$. Note that $\gamma$ extends to a divided power
structure $\gamma'$ on $IP$ by
Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends}. Hence
$D_{P, \gamma}(J') = D_{P, \gamma'}(J')$ is an example of a special
case of divided power envelopes we describe in the following lemma.
\begin{lemma}
\label{lemma-describe-divided-power-envelope}
Let $(B, I, \gamma)$ be a divided power algebra. Let $I \subset J \subset B$
be an ideal. Let $(D, \bar J, \bar \gamma)$ be the divided power envelope
of $J$ relative to $\gamma$. Choose elements $f_t \in J$, $t \in T$ such
that $J = I + (f_t)$. Then there exists a surjection
$$
\Psi : B\langle x_t \rangle \longrightarrow D
$$
of divided power rings mapping $x_t$ to the image of $f_t$ in $D$.
The kernel of $\Psi$ is generated by the elements $x_t - f_t$ and
all
$$
\delta_n\left(\sum r_t x_t - r_0\right)
$$
whenever $\sum r_t f_t = r_0$ in $B$ for some $r_t \in B$, $r_0 \in I$.
\end{lemma}
\begin{proof}
In the statement of the lemma we think of $B\langle x_t \rangle$
as a divided power ring with ideal
$J' = IB\langle x_t \rangle + B\langle x_t \rangle_{+}$, see
Divided Power Algebra, Remark \ref{dpa-remark-divided-power-polynomial-algebra}.
The existence of $\Psi$ follows from the universal property of
divided power polynomial rings. Surjectivity of $\Psi$ follows from
the fact that its image is a divided power subring of $D$, hence equal to $D$
by the universal property of $D$. It is clear that
$x_t - f_t$ is in the kernel. Set
$$
\mathcal{R} = \{(r_0, r_t) \in I \oplus \bigoplus\nolimits_{t \in T} B
\mid \sum r_t f_t = r_0 \text{ in }B\}
$$
If $(r_0, r_t) \in \mathcal{R}$ then it is clear that
$\sum r_t x_t - r_0$ is in the kernel.
As $\Psi$ is a homomorphism of divided power rings
and $\sum r_tx_t = r_0 \in J'$
it follows that $\delta_n(\sum r_t x_t - r_0)$ is in the kernel as well.
Let $K \subset B\langle x_t \rangle$ be the ideal generated by
$x_t - f_t$ and the elements $\delta_n(\sum r_t x_t - r_0)$ for
$(r_0, r_t) \in \mathcal{R}$.
To show that $K = \Ker(\Psi)$ it suffices to show that
$\delta$ extends to $B\langle x_t \rangle/K$. Namely, if so the universal
property of $D$ gives a map $D \to B\langle x_t \rangle/K$
inverse to $\Psi$. Hence we have to show that $K \cap J'$ is
preserved by $\delta_n$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-kernel}.
Let $K' \subset B\langle x_t \rangle$ be the ideal
generated by the elements
\begin{enumerate}
\item $\delta_m(\sum r_t x_t - r_0)$ where $m > 0$ and
$(r_0, r_t) \in \mathcal{R}$,
\item $x_{t'}^{[m]}(x_t - f_t)$ where $m > 0$ and $t', t \in I$.
\end{enumerate}
We claim that $K' = K \cap J'$. The claim proves that $K \cap J'$
is preserved by $\delta_n$, $n > 0$ by the criterion of
Divided Power Algebra, Lemma \ref{dpa-lemma-kernel} (2)(c)
and a computation of $\delta_n$
of the elements listed which we leave to the reader.
To prove the claim note that $K' \subset K \cap J'$.
Conversely, if $h \in K \cap J'$ then, modulo $K'$ we can write
$$
h = \sum r_t (x_t - f_t)
$$
for some $r_t \in B$. As $h \in K \cap J' \subset J'$
we see that $r_0 = \sum r_t f_t \in I$. Hence $(r_0, r_t) \in \mathcal{R}$
and we see that
$$
h = \sum r_t x_t - r_0
$$
is in $K'$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-divided-power-envelope-add-variables}
Let $(A, I, \gamma)$ be a divided power ring.
Let $B$ be an $A$-algebra and $IB \subset J \subset B$ an ideal.
Let $x_i$ be a set of variables. Then
$$
D_{B[x_i], \gamma}(JB[x_i] + (x_i)) = D_{B, \gamma}(J) \langle x_i \rangle
$$
\end{lemma}
\begin{proof}
One possible proof is to deduce this from
Lemma \ref{lemma-describe-divided-power-envelope}
as any relation between $x_i$ in $B[x_i]$ is trivial.
On the other hand, the lemma follows from the universal property
of the divided power polynomial algebra and the universal property of
divided power envelopes.
\end{proof}
\noindent
Conditions (1) and (2) of the following lemma hold if $B \to B'$ is flat
at all primes of $V(IB') \subset \Spec(B')$ and is very closely related
to that condition, see
Algebra, Lemma \ref{algebra-lemma-what-does-it-mean}.
It in particular says that taking the divided power
envelope commutes with localization.
\begin{lemma}
\label{lemma-flat-base-change-divided-power-envelope}
Let $(A, I, \gamma)$ be a divided power ring.
Let $B \to B'$ be a homomorphism of $A$-algebras.
Assume that
\begin{enumerate}
\item $B/IB \to B'/IB'$ is flat, and
\item $\text{Tor}_1^B(B', B/IB) = 0$.
\end{enumerate}
Then for any ideal $IB \subset J \subset B$ the canonical map
$$
D_B(J) \otimes_B B' \longrightarrow D_{B'}(JB')
$$
is an isomorphism.
\end{lemma}
\begin{proof}
Set $D = D_B(J)$ and denote $\bar J \subset D$ its divided power ideal
with divided power structure $\bar\gamma$. The universal property of
$D$ produces a $B$-algebra map $D \to D_{B'}(JB')$, whence a map as in
the lemma. It suffices to show that
the divided powers $\bar\gamma$ extend to $D \otimes_B B'$ since then
the universal property of $D_{B'}(JB')$ will produce a map
$D_{B'}(JB') \to D \otimes_B B'$ inverse to the one in the lemma.
\medskip\noindent
Choose a surjection $P \to B'$ where $P$ is a polynomial algebra over $B$.
In particular $B \to P$ is flat, hence $D \to D \otimes_B P$ is flat by
Algebra, Lemma \ref{algebra-lemma-flat-base-change}.
Then $\bar\gamma$ extends to $D \otimes_B P$ by
Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends}; we
will denote this extension
$\bar\gamma$ also. Set $\mathfrak a = \Ker(P \to B')$ so that
we have the short exact sequence
$$
0 \to \mathfrak a \to P \to B' \to 0
$$
Thus $\text{Tor}_1^B(B', B/IB) = 0$ implies that
$\mathfrak a \cap IP = I\mathfrak a$.
Now we have the following commutative diagram
$$
\xymatrix{
B/J \otimes_B \mathfrak a \ar[r]_\beta &
B/J \otimes_B P \ar[r] &
B/J \otimes_B B' \\
D \otimes_B \mathfrak a \ar[r]^\alpha \ar[u] &
D \otimes_B P \ar[r] \ar[u] &
D \otimes_B B' \ar[u] \\
\bar J \otimes_B \mathfrak a \ar[r] \ar[u] &
\bar J \otimes_B P \ar[r] \ar[u] &
\bar J \otimes_B B' \ar[u]
}
$$
This diagram is exact even with $0$'s added at the top and the right.
We have to show the divided powers on the ideal
$\bar J \otimes_B P$ preserve the ideal
$\Im(\alpha) \cap \bar J \otimes_B P$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-kernel}.
Consider the exact sequence
$$
0 \to \mathfrak a/I\mathfrak a \to P/IP \to B'/IB' \to 0
$$
(which uses that $\mathfrak a \cap IP = I\mathfrak a$ as seen above).
As $B'/IB'$ is flat over $B/IB$ this sequence remains exact after
applying $B/J \otimes_{B/IB} -$, see
Algebra, Lemma \ref{algebra-lemma-flat-tor-zero}. Hence
$$
\Ker(B/J \otimes_{B/IB} \mathfrak a/I\mathfrak a \to
B/J \otimes_{B/IB} P/IP) =
\Ker(\mathfrak a/J\mathfrak a \to P/JP)
$$
is zero. Thus $\beta$ is injective. It follows that
$\Im(\alpha) \cap \bar J \otimes_B P$ is the
image of $\bar J \otimes \mathfrak a$. Now if
$f \in \bar J$ and $a \in \mathfrak a$, then
$\bar\gamma_n(f \otimes a) = \bar\gamma_n(f) \otimes a^n$
hence the result is clear.
\end{proof}
\noindent
The following lemma is a special case of
\cite[Proposition 2.1.7]{dJ-crystalline} which in turn is a
generalization of \cite[Proposition 2.8.2]{Berthelot}.
\begin{lemma}
\label{lemma-flat-extension-divided-power-envelope}
Let $(B, I, \gamma) \to (B', I', \gamma')$ be a homomorphism of
divided power rings. Let $I \subset J \subset B$ and
$I' \subset J' \subset B'$ be ideals. Assume
\begin{enumerate}
\item $B/I \to B'/I'$ is flat, and
\item $J' = JB' + I'$.
\end{enumerate}
Then the canonical map
$$
D_{B, \gamma}(J) \otimes_B B' \longrightarrow D_{B', \gamma'}(J')
$$
is an isomorphism.
\end{lemma}
\begin{proof}
Set $D = D_B(J)$ and denote $\bar J \subset D$ its divided power ideal
with divided power structure $\bar\gamma$. The universal property of
$D$ produces a homomorphism of divided power rings $D \to D_{B'}(J')$,
whence a map as in the lemma. It suffices to show that
there exist divided powers on the image of
$D \otimes_B I' + \bar J \otimes_B B' \to D \otimes_B B'$
compatible with $\bar \gamma$ and $\gamma'$ since then
the universal property of $D_{B'}(J')$ will produce a map
$D_{B'}(J') \to D \otimes_B B'$ inverse to the one in the lemma.
\medskip\noindent
Choose elements $f_t \in J$ which generate $J/I$. Set
$\mathcal{R} = \{(r_0, r_t) \in I \oplus \bigoplus\nolimits_{t \in T} B
\mid \sum r_t f_t = r_0 \text{ in }B\}$ as in the proof of
Lemma \ref{lemma-describe-divided-power-envelope}. This lemma shows that
$$
D = B\langle x_t \rangle/ K
$$
where $K$ is generated by the elements $x_t - f_t$ and
$\delta_n(\sum r_t x_t - r_0)$ for $(r_0, r_t) \in \mathcal{R}$.
Thus we see that
\begin{equation}
\label{equation-base-change}
D \otimes_B B' = B'\langle x_t \rangle/K'
\end{equation}
where $K'$ is generated by the images in $B'\langle x_t \rangle$
of the generators of $K$ listed above. Let $f'_t \in B'$ be the image
of $f_t$. By assumption (1) we see that the elements $f'_t \in J'$
generate $J'/I'$ and we see that $x_t - f'_t \in K'$. Set
$$
\mathcal{R}' =
\{(r'_0, r'_t) \in I' \oplus \bigoplus\nolimits_{t \in T} B'
\mid \sum r'_t f'_t = r'_0 \text{ in }B'\}
$$
To finish the proof we have to show that
$\delta'_n(\sum r'_t x_t - r'_0) \in K'$ for
$(r'_0, r'_t) \in \mathcal{R}'$, because then the presentation
(\ref{equation-base-change}) of $D \otimes_B B'$ is identical
to the presentation of $D_{B', \gamma'}(J')$ obtain in
Lemma \ref{lemma-describe-divided-power-envelope} from the generators $f'_t$.
Suppose that $(r'_0, r'_t) \in \mathcal{R}'$. Then
$\sum r'_t f'_t = 0$ in $B'/I'$. As $B/I \to B'/I'$ is flat by
assumption (1) we can apply the equational criterion of flatness
(Algebra, Lemma \ref{algebra-lemma-flat-eq}) to see
that there exist an $m > 0$ and
$r_{jt} \in B$ and $c_j \in B'$, $j = 1, \ldots, m$ such
that
$$
r_{j0} = \sum r_{jt} f_t \in I \text{ for } j = 1, \ldots, m,
\quad\text{and}\quad
r'_t = \sum c_j r_{jt}.
$$
Note that this also implies that $r'_0 = \sum c_j r_{j0}$.
Then we have
\begin{align*}
\delta'_n(\sum r'_t x_t - r'_0)
& =
\delta'_n(\sum c_j (\sum r_{jt} x_t - r_{j0})) \\
& =
\sum c_1^{n_1} \ldots c_m^{n_m}
\delta_{n_1}(\sum r_{1t} x_t - r_{10}) \ldots
\delta_{n_m}(\sum r_{mt} x_t - r_{m0})
\end{align*}
where the sum is over $n_1 + \ldots + n_m = n$. This proves what we want.
\end{proof}
\section{Some explicit divided power thickenings}
\label{section-explicit-thickenings}
\noindent
The constructions in this section will help us to define the connection
on a crystal in modules on the crystalline site.
\begin{lemma}
\label{lemma-divided-power-first-order-thickening}
Let $(A, I, \gamma)$ be a divided power ring. Let $M$ be an $A$-module.
Let $B = A \oplus M$ as an $A$-algebra where $M$ is an ideal of square zero
and set $J = I \oplus M$. Set
$$
\delta_n(x + z) = \gamma_n(x) + \gamma_{n - 1}(x)z
$$
for $x \in I$ and $z \in M$.
Then $\delta$ is a divided power structure and
$A \to B$ is a homomorphism of divided power rings from
$(A, I, \gamma)$ to $(B, J, \delta)$.
\end{lemma}
\begin{proof}
We have to check conditions (1) -- (5) of
Divided Power Algebra, Definition \ref{dpa-definition-divided-powers}.
We will prove this directly for this case, but please see the proof of
the next lemma for a method which avoids calculations.
Conditions (1) and (3) are clear. Condition (2) follows from
\begin{align*}
\delta_n(x + z)\delta_m(x + z)
& =
(\gamma_n(x) + \gamma_{n - 1}(x)z)(\gamma_m(x) + \gamma_{m - 1}(x)z) \\
& = \gamma_n(x)\gamma_m(x) + \gamma_n(x)\gamma_{m - 1}(x)z +
\gamma_{n - 1}(x)\gamma_m(x)z \\
& =
\frac{(n + m)!}{n!m!} \gamma_{n + m}(x) +
\left(\frac{(n + m - 1)!}{n!(m - 1)!} +
\frac{(n + m - 1)!}{(n - 1)!m!}\right)
\gamma_{n + m - 1}(x) z \\
& =
\frac{(n + m)!}{n!m!}\delta_{n + m}(x + z)
\end{align*}
Condition (5) follows from
\begin{align*}
\delta_n(\delta_m(x + z))
& =
\delta_n(\gamma_m(x) + \gamma_{m - 1}(x)z) \\
& =
\gamma_n(\gamma_m(x)) + \gamma_{n - 1}(\gamma_m(x))\gamma_{m - 1}(x)z \\
& =
\frac{(nm)!}{n! (m!)^n} \gamma_{nm}(x) +
\frac{((n - 1)m)!}{(n - 1)! (m!)^{n - 1}}
\gamma_{(n - 1)m}(x) \gamma_{m - 1}(x) z \\
& = \frac{(nm)!}{n! (m!)^n}(\gamma_{nm}(x) + \gamma_{nm - 1}(x) z)
\end{align*}
by elementary number theory. To prove (4) we have to see that
$$
\delta_n(x + x' + z + z')
=
\gamma_n(x + x') + \gamma_{n - 1}(x + x')(z + z')
$$
is equal to
$$
\sum\nolimits_{i = 0}^n
(\gamma_i(x) + \gamma_{i - 1}(x)z)
(\gamma_{n - i}(x') + \gamma_{n - i - 1}(x')z')
$$
This follows easily on collecting the coefficients of
$1$, $z$, and $z'$ and using condition (4) for $\gamma$.
\end{proof}
\begin{lemma}
\label{lemma-divided-power-second-order-thickening}
Let $(A, I, \gamma)$ be a divided power ring. Let $M$, $N$ be $A$-modules.
Let $q : M \times M \to N$ be an $A$-bilinear map.
Let $B = A \oplus M \oplus N$ as an $A$-algebra with multiplication
$$
(x, z, w)\cdot (x', z', w') = (xx', xz' + x'z, xw' + x'w + q(z, z') + q(z', z))
$$
and set $J = I \oplus M \oplus N$. Set
$$
\delta_n(x, z, w) = (\gamma_n(x), \gamma_{n - 1}(x)z,
\gamma_{n - 1}(z)w + \gamma_{n - 2}(x)q(z, z))
$$
for $(x, z, w) \in J$.
Then $\delta$ is a divided power structure and
$A \to B$ is a homomorphism of divided power rings from
$(A, I, \gamma)$ to $(B, J, \delta)$.
\end{lemma}
\begin{proof}
Suppose we want to prove that property (4) of
Divided Power Algebra, Definition \ref{dpa-definition-divided-powers}
is satisfied. Pick $(x, z, w)$ and $(x', z', w')$ in $J$.
Pick a map
$$
A_0 = \mathbf{Z}\langle s, s'\rangle \longrightarrow A,\quad
s \longmapsto x,
s' \longmapsto x'
$$
which is possible by the universal property of divided power
polynomial rings. Set $M_0 = A_0 \oplus A_0$ and
$N_0 = A_0 \oplus A_0 \oplus M_0 \otimes_{A_0} M_0$.
Let $q_0 : M_0 \times M_0 \to N_0$ be the obvious map.
Define $M_0 \to M$ as the $A_0$-linear map which sends
the basis vectors of $M_0$ to $z$ and $z'$. Define $N_0 \to N$
as the $A_0$ linear map which sends the first two basis vectors
of $N_0$ to $w$ and $w'$ and uses
$M_0 \otimes_{A_0} M_0 \to M \otimes_A M \xrightarrow{q} N$
on the last summand. Then we see that it suffices to prove the
identity (4) for the situation $(A_0, M_0, N_0, q_0)$.
Similarly for the other identities. This reduces us to the case
of a $\mathbf{Z}$-torsion free ring $A$ and $A$-torsion free modules.
In this case all we have to do is show that
$$
n! \delta_n(x, z, w) = (x, z, w)^n
$$
in the ring $A$, see Divided Power Algebra, Lemma \ref{dpa-lemma-silly}.
To see this note that
$$
(x, z, w)^2 = (x^2, 2xz, 2xw + 2q(z, z))
$$
and by induction
$$
(x, z, w)^n = (x^n, nx^{n - 1}z, nx^{n - 1}w + n(n - 1)x^{n - 2}q(z, z))
$$
On the other hand,
$$
n! \delta_n(x, z, w) = (n!\gamma_n(x), n!\gamma_{n - 1}(x)z,
n!\gamma_{n - 1}(x)w + n!\gamma_{n - 2}(x) q(z, z))
$$
which matches. This finishes the proof.
\end{proof}
\section{Compatibility}
\label{section-compatibility}
\noindent
This section isn't required reading; it explains how our discussion
fits with that of \cite{Berthelot}.
Consider the following technical notion.
\begin{definition}
\label{definition-compatible}
Let $(A, I, \gamma)$ and $(B, J, \delta)$ be divided power rings.
Let $A \to B$ be a ring map. We say
{\it $\delta$ is compatible with $\gamma$}
if there exists a divided power structure $\bar\gamma$ on
$J + IB$ such that
$$
(A, I, \gamma) \to (B, J + IB, \bar \gamma)\quad\text{and}\quad
(B, J, \delta) \to (B, J + IB, \bar \gamma)
$$
are homomorphisms of divided power rings.
\end{definition}
\noindent
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring.
Let $A \to C$ be a ring map with $p$ nilpotent in $C$.
Assume that $\gamma$ extends to $IC$ (see
Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends}).
In this situation, the (big affine) crystalline site of
$\Spec(C)$ over $\Spec(A)$
as defined in \cite{Berthelot}
is the opposite of the category of systems
$$
(B, J, \delta, A \to B, C \to B/J)
$$
where
\begin{enumerate}
\item $(B, J, \delta)$ is a divided power ring with $p$ nilpotent in $B$,
\item $\delta$ is compatible with $\gamma$, and
\item the diagram
$$
\xymatrix{
B \ar[r] & B/J \\
A \ar[u] \ar[r] & C \ar[u]
}
$$
is commutative.
\end{enumerate}
The conditions
``$\gamma$ extends to $C$ and $\delta$ compatible with $\gamma$''
are used in \cite{Berthelot} to insure that
the crystalline cohomology of $\Spec(C)$ is the same as the crystalline
cohomology of $\Spec(C/IC)$. We will avoid this issue
by working exclusively with $C$ such that $IC = 0$\footnote{Of course there
will be a price to pay.}. In this case,
for a system $(B, J, \delta, A \to B, C \to B/J)$ as above,
the commutativity of the displayed diagram above implies $IB \subset J$ and
compatibility is equivalent to the condition that
$(A, I, \gamma) \to (B, J, \delta)$ is a homomorphism of divided
power rings.
\section{Affine crystalline site}
\label{section-affine-site}
\noindent
In this section we discuss the algebraic variant of the crystalline site.
Our basic situation in which we discuss this material will be as
follows.
\begin{situation}
\label{situation-affine}
Here $p$ is a prime number, $(A, I, \gamma)$ is a divided power
ring such that $A$ is a $\mathbf{Z}_{(p)}$-algebra, and $A \to C$ is a
ring map such that $IC = 0$ and such that $p$ is nilpotent in $C$.
\end{situation}
\noindent
Usually the prime number $p$ will be contained in the
divided power ideal $I$.
\begin{definition}
\label{definition-affine-thickening}
In Situation \ref{situation-affine}.
\begin{enumerate}
\item A {\it divided power thickening} of $C$ over $(A, I, \gamma)$
is a homomorphism of divided power algebras $(A, I, \gamma) \to (B, J, \delta)$
such that $p$ is nilpotent in $B$ and a ring map $C \to B/J$ such that
$$
\xymatrix{
B \ar[r] & B/J \\
& C \ar[u] \\
A \ar[uu] \ar[r] & A/I \ar[u]
}
$$
is commutative.
\item A {\it homomorphism of divided power thickenings}
$$
(B, J, \delta, C \to B/J) \longrightarrow (B', J', \delta', C \to B'/J')
$$
is a homomorphism $\varphi : B \to B'$ of divided power $A$-algebras such
that $C \to B/J \to B'/J'$ is the given map $C \to B'/J'$.
\item We denote $\text{CRIS}(C/A, I, \gamma)$ or simply $\text{CRIS}(C/A)$
the category of divided power thickenings of $C$ over $(A, I, \gamma)$.
\item We denote $\text{Cris}(C/A, I, \gamma)$ or simply $\text{Cris}(C/A)$
the full subcategory consisting of $(B, J, \delta, C \to B/J)$ such that
$C \to B/J$ is an isomorphism. We often denote such an object
$(B \to C, \delta)$ with $J = \Ker(B \to C)$ being understood.
\end{enumerate}
\end{definition}
\noindent
Note that for a divided power thickening $(B, J, \delta)$ as above
the ideal $J$ is locally nilpotent, see
Divided Power Algebra, Lemma \ref{dpa-lemma-nil}.
There is a canonical functor
\begin{equation}
\label{equation-forget-affine}
\text{CRIS}(C/A) \longrightarrow C\text{-algebras},\quad
(B, J, \delta) \longmapsto B/J
\end{equation}
This category does not have equalizers or fibre products in general.
It also doesn't have an initial object ($=$ empty colimit) in general.
\begin{lemma}
\label{lemma-affine-thickenings-colimits}
In Situation \ref{situation-affine}.
\begin{enumerate}
\item $\text{CRIS}(C/A)$ has products,
\item $\text{CRIS}(C/A)$ has all finite nonempty colimits and
(\ref{equation-forget-affine}) commutes with these, and
\item $\text{Cris}(C/A)$ has all finite nonempty colimits and
$\text{Cris}(C/A) \to \text{CRIS}(C/A)$ commutes with them.
\end{enumerate}
\end{lemma}
\begin{proof}
The empty product is $(C, 0, \emptyset)$. If $(B_t, J_t, \delta_t)$ is a
family of objects of $\text{CRIS}(C/A)$ then we can form the product
$(\prod B_t, \prod J_t, \prod \delta_t)$ as in
Divided Power Algebra, Lemma \ref{dpa-lemma-colimits}.
The map $C \to \prod B_t/\prod J_t = \prod B_t/J_t$ is clear.
\medskip\noindent
Given two objects $(B, J, \gamma)$ and $(B', J', \gamma')$ of
$\text{CRIS}(C/A)$ we can form a cocartesian diagram
$$
\xymatrix{
(B, J, \delta) \ar[r] & (B'', J'', \delta'') \\
(A, I, \gamma) \ar[r] \ar[u] & (B', J', \delta') \ar[u]
}
$$
in the category of divided power rings. Then we see that we have
$$
B''/J'' = B/J \otimes_{A/I} B'/J' \longleftarrow C \otimes_{A/I} C
$$
see Divided Power Algebra, Remark \ref{dpa-remark-forgetful}.
Denote $J'' \subset K \subset B''$
the ideal such that
$$
\xymatrix{
B''/J'' \ar[r] & B''/K \\
C \otimes_{A/I} C \ar[r] \ar[u] & C \ar[u]
}
$$
is a pushout, i.e., $B''/K \cong B/J \otimes_C B'/J'$.
Let $D_{B''}(K) = (D, \bar K, \bar \delta)$
be the divided power envelope of $K$ in $B''$ relative to
$(B'', J'', \delta'')$. Then it is easily verified that
$(D, \bar K, \bar \delta)$ is a coproduct of $(B, J, \delta)$ and
$(B', J', \delta')$ in $\text{CRIS}(C/A)$.
\medskip\noindent
Next, we come to coequalizers. Let
$\alpha, \beta : (B, J, \delta) \to (B', J', \delta')$ be morphisms of
$\text{CRIS}(C/A)$. Consider $B'' = B'/ (\alpha(b) - \beta(b))$. Let
$J'' \subset B''$ be the image of $J'$. Let
$D_{B''}(J'') = (D, \bar J, \bar\delta)$ be the divided power envelope of
$J''$ in $B''$ relative to $(B', J', \delta')$. Then it is easily verified
that $(D, \bar J, \bar \delta)$ is the coequalizer of $(B, J, \delta)$ and
$(B', J', \delta')$ in $\text{CRIS}(C/A)$.
\medskip\noindent
By Categories, Lemma \ref{categories-lemma-almost-finite-colimits-exist}
we have all finite nonempty colimits in $\text{CRIS}(C/A)$. The constructions
above shows that (\ref{equation-forget-affine}) commutes with them.
This formally implies part (3) as $\text{Cris}(C/A)$ is the fibre category
of (\ref{equation-forget-affine}) over $C$.
\end{proof}
\begin{remark}
\label{remark-completed-affine-site}
In Situation \ref{situation-affine} we denote
$\text{Cris}^\wedge(C/A)$ the category whose objects are
pairs $(B \to C, \delta)$ such that
\begin{enumerate}
\item $B$ is a $p$-adically complete $A$-algebra,
\item $B \to C$ is a surjection of $A$-algebras,
\item $\delta$ is a divided power structure on $\Ker(B \to C)$,
\item $A \to B$ is a homomorphism of divided power rings.
\end{enumerate}
Morphisms are defined as in Definition \ref{definition-affine-thickening}.
Then $\text{Cris}(C/A) \subset \text{Cris}^\wedge(C/A)$ is the full
subcategory consisting of those $B$ such that $p$ is nilpotent in $B$.
Conversely, any object $(B \to C, \delta)$ of $\text{Cris}^\wedge(C/A)$
is equal to the limit
$$
(B \to C, \delta) = \lim_e (B/p^eB \to C, \delta)
$$
where for $e \gg 0$ the object $(B/p^eB \to C, \delta)$ lies
in $\text{Cris}(C/A)$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}.
In particular, we see that $\text{Cris}^\wedge(C/A)$ is a full subcategory
of the category of pro-objects of $\text{Cris}(C/A)$, see
Categories, Remark \ref{categories-remark-pro-category}.
\end{remark}
\begin{lemma}
\label{lemma-list-properties}
In Situation \ref{situation-affine}.
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 \ref{dpa-lemma-extend-to-completion}.
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
\begin{enumerate}
\item $p^eD \subset \bar J$ and $p^eD^\wedge \subset \bar J^\wedge$
are preserved by divided powers,
\item $D^\wedge/p^eD^\wedge = D/p^eD = D_e$ as divided power rings,
\item $(D_e, \bar J_e, \bar\gamma)$ is an object of $\text{Cris}(C/A)$,
\item $(D^\wedge, \bar J^\wedge, \bar\gamma^\wedge)$ is equal to
$\lim_e (D_e, \bar J_e, \bar\gamma)$, and
\item $(D^\wedge, \bar J^\wedge, \bar\gamma^\wedge)$ is an object of
$\text{Cris}^\wedge(C/A)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows from
Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}.
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 \ref{dpa-lemma-extend-to-completion}.
Part (5) is clear from the definitions.
\end{proof}
\begin{lemma}
\label{lemma-set-generators}
In Situation \ref{situation-affine}.
Let $P$ be a polynomial algebra over $A$ and let
$P \to C$ be a surjection of $A$-algebras with kernel $J$.
With $(D_e, \bar J_e, \bar\gamma)$ as in Lemma \ref{lemma-list-properties}:
for every object $(B, J_B, \delta)$ of $\text{CRIS}(C/A)$ there
exists an $e$ and a morphism $D_e \to B$ of $\text{CRIS}(C/A)$.
\end{lemma}
\begin{proof}
We can find an $A$-algebra homomorphism $P \to B$
lifting the map $C \to B/J_B$. By our definition of
$\text{CRIS}(C/A)$ we see that $p^eB = 0$ for
some $e$ hence $P \to B$ factors as $P \to P_e \to B$.
By the universal property of the divided power envelope we
conclude that $P_e \to B$ factors through $D_e$.
\end{proof}
\begin{lemma}
\label{lemma-generator-completion}
In Situation \ref{situation-affine}.
Let $P$ be a polynomial algebra over $A$ and let
$P \to C$ be a surjection of $A$-algebras with kernel $J$.
Let $(D, \bar J, \bar\gamma)$ be the $p$-adic completion of
$D_{P, \gamma}(J)$. For every object $(B \to C, \delta)$ of
$\text{Cris}^\wedge(C/A)$ there
exists a morphism $D \to B$ of $\text{Cris}^\wedge(C/A)$.
\end{lemma}
\begin{proof}
We can find an $A$-algebra homomorphism $P \to B$ compatible
with maps to $C$. By our definition of
$\text{Cris}(C/A)$ we see that $P \to B$ factors as
$P \to D_{P, \gamma}(J) \to B$. As $B$ is $p$-adically complete
we can factor this map through $D$.
\end{proof}
\section{Module of differentials}
\label{section-differentials}
\noindent
In this section we develop a theory of modules of differentials
for divided power rings.
\begin{definition}
\label{definition-derivation}
Let $A$ be a ring. Let $(B, J, \delta)$ be a divided power ring.
Let $A \to B$ be a ring map. Let $M$ be an $B$-module.
A {\it divided power $A$-derivation} into $M$ is a map
$\theta : B \to M$ which is additive, annihilates the elements
of $A$, satisfies the Leibniz rule
$\theta(bb') = b\theta(b') + b'\theta(b)$ and satisfies
$$
\theta(\delta_n(x)) = \delta_{n - 1}(x)\theta(x)
$$
for all $n \geq 1$ and all $x \in J$.
\end{definition}
\noindent
In the situation of the definition, just as in the case of usual
derivations, there exists a {\it universal divided power $A$-derivation}
$$
\text{d}_{B/A, \delta} : B \to \Omega_{B/A, \delta}
$$
such that any divided power $A$-derivation $\theta : B \to M$ is equal to
$\theta = \xi \circ d_{B/A, \delta}$ for some $B$-linear map
$\Omega_{B/A, \delta} \to M$. If $(A, I, \gamma) \to (B, J, \delta)$
is a homomorphism of divided power rings, then we can forget the
divided powers on $A$ and consider the divided power derivations of
$B$ over $A$. Here are some basic properties of the divided power
module of differentials.
\begin{lemma}
\label{lemma-omega}
Let $A$ be a ring. Let $(B, J, \delta)$ be a divided power ring and
$A \to B$ a ring map.
\begin{enumerate}
\item Consider $B[x]$ with divided power ideal $(JB[x], \delta')$
where $\delta'$ is the extension of $\delta$ to $B[x]$. Then
$$
\Omega_{B[x]/A, \delta'} =
\Omega_{B/A, \delta} \otimes_B B[x] \oplus B[x]\text{d}x.
$$
\item Consider $B\langle x \rangle$ with divided power ideal
$(JB\langle x \rangle + B\langle x \rangle_{+}, \delta')$. Then
$$
\Omega_{B\langle x\rangle/A, \delta'} =
\Omega_{B/A, \delta} \otimes_B B\langle x \rangle \oplus
B\langle x\rangle \text{d}x.
$$
\item Let $K \subset J$ be an ideal preserved by $\delta_n$ for
all $n > 0$. Set $B' = B/K$ and denote $\delta'$ the induced
divided power on $J/K$. Then $\Omega_{B'/A, \delta'}$ is the quotient
of $\Omega_{B/A, \delta} \otimes_B B'$ by the $B'$-submodule generated
by $\text{d}k$ for $k \in K$.
\end{enumerate}
\end{lemma}
\begin{proof}
These are proved directly from the construction of $\Omega_{B/A, \delta}$
as the free $B$-module on the elements $\text{d}b$ modulo the relations
\begin{enumerate}
\item $\text{d}(b + b') = \text{d}b + \text{d}b'$, $b, b' \in B$,
\item $\text{d}a = 0$, $a \in A$,
\item $\text{d}(bb') = b \text{d}b' + b' \text{d}b$, $b, b' \in B$,
\item $\text{d}\delta_n(f) = \delta_{n - 1}(f)\text{d}f$, $f \in J$, $n > 1$.
\end{enumerate}
Note that the last relation explains why we get ``the same'' answer for
the divided power polynomial algebra and the usual polynomial algebra:
in the first case $x$ is an element of the divided power ideal and hence
$\text{d}x^{[n]} = x^{[n - 1]}\text{d}x$.
\end{proof}
\noindent
Let $(A, I, \gamma)$ be a divided power ring. In this setting the
correct version of the powers of $I$ is given by the divided powers
$$
I^{[n]} = \text{ideal generated by }
\gamma_{e_1}(x_1) \ldots \gamma_{e_t}(x_t)
\text{ with }\sum e_j \geq n\text{ and }x_j \in I.
$$
Of course we have $I^n \subset I^{[n]}$. Note that $I^{[1]} = I$.
Sometimes we also set $I^{[0]} = A$.
\begin{lemma}
\label{lemma-diagonal-and-differentials}
Let $(A, I, \gamma) \to (B, J, \delta)$ be a homomorphism
of divided power rings. Let $(B(1), J(1), \delta(1))$ be the coproduct
of $(B, J, \delta)$ with itself over $(A, I, \gamma)$, i.e.,
such that
$$
\xymatrix{
(B, J, \delta) \ar[r] & (B(1), J(1), \delta(1)) \\
(A, I, \gamma) \ar[r] \ar[u] & (B, J, \delta) \ar[u]
}
$$
is cocartesian. Denote $K = \Ker(B(1) \to B)$.
Then $K \cap J(1) \subset J(1)$ is preserved by the divided power
structure and
$$
\Omega_{B/A, \delta} = K/ \left(K^2 + (K \cap J(1))^{[2]}\right)
$$
canonically.
\end{lemma}
\begin{proof}
The fact that $K \cap J(1) \subset J(1)$ is preserved by the divided power
structure follows from the fact that $B(1) \to B$ is a homomorphism of
divided power rings.
\medskip\noindent
Recall that $K/K^2$ has a canonical $B$-module structure.
Denote $s_0, s_1 : B \to B(1)$ the two coprojections and consider
the map $\text{d} : B \to K/K^2 +(K \cap J(1))^{[2]}$ given by
$b \mapsto s_1(b) - s_0(b)$. It is clear that $\text{d}$ is additive,
annihilates $A$, and satisfies the Leibniz rule.
We claim that $\text{d}$ is an $A$-derivation.
Let $x \in J$. Set $y = s_1(x)$ and $z = s_0(x)$.
Denote $\delta$ the divided power structure on $J(1)$.
We have to show that $\delta_n(y) - \delta_n(z) = \delta_{n - 1}(y)(y - z)$
modulo $K^2 +(K \cap J(1))^{[2]}$ for $n \geq 1$. We will show this
by induction on $n$. It is true for $n = 1$.
Let $n > 1$ and that it holds for all smaller values.
Note that
$$
\delta_n(z - y) =
\sum\nolimits_{i = 0}^n (-1)^{n - i}\delta_i(z)\delta_{n - i}(y)
$$
is an element of $K^2 +(K \cap J(1))^{[2]}$. From this and induction
we see that working modulo $K^2 +(K \cap J(1))^{[2]}$ we have
\begin{align*}
& \delta_n(y) - \delta_n(z) \\
& =
\delta_n(y) +
\sum\nolimits_{i = 0}^{n - 1} (-1)^{n - i}\delta_i(z)\delta_{n - i}(y) \\
& =
\delta_n(y) + (-1)^n\delta_n(y) +
\sum\nolimits_{i = 1}^{n - 1}
(-1)^{n - i}(\delta_i(y) - \delta_{i - 1}(y)(y - z))\delta_{n - i}(y)
\end{align*}
Using that $\delta_i(y)\delta_{n - i}(y) = \binom{n}{i} \delta_n(y)$
and that $\delta_{i - 1}(y)\delta_{n - i}(y) =
\binom{n - 1}{i} \delta_{n - 1}(y)$
the reader easily verifies that this expression comes out to give
$\delta_{n - 1}(y)(y - z)$ as desired.
\medskip\noindent
Let $M$ be a $B$-module. Let $\theta : B \to M$ be a divided power
$A$-derivation.
Set $D = B \oplus M$ where $M$ is an ideal of square zero. Define a
divided power structure on $J \oplus M \subset D$ by setting
$\delta_n(x + m) = \delta_n(x) + \delta_{n - 1}(x)m$ for $n > 1$, see
Lemma \ref{lemma-divided-power-first-order-thickening}.
There are two divided power algebra homomorphisms $B \to D$: the first
is given by the inclusion and the second by the map $b \mapsto b + \theta(b)$.
Hence we get a canonical homomorphism $B(1) \to D$ of divided power
algebras over $(A, I, \gamma)$. This induces a map $K \to M$
which annihilates $K^2$ (as $M$ is an ideal of square zero) and
$(K \cap J(1))^{[2]}$ as $M^{[2]} = 0$. The composition
$B \to K/K^2 + (K \cap J(1))^{[2]} \to M$ equals $\theta$ by construction.
It follows that $\text{d}$
is a universal divided power $A$-derivation and we win.
\end{proof}
\begin{remark}
\label{remark-filtration-differentials}
Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided
power structure on $B$. The universal module $\Omega_{B/A, \delta}$
comes with a little bit of extra structure, namely the $B$-submodule
$N$ of $\Omega_{B/A, \delta}$ generated by $\text{d}_{B/A, \delta}(J)$.
In terms of the isomorphism given in
Lemma \ref{lemma-diagonal-and-differentials}
this corresponds to the image of
$K \cap J(1)$ in $\Omega_{B/A, \delta}$. Consider the $A$-algebra
$D = B \oplus \Omega^1_{B/A, \delta}$ with ideal $\bar J = J \oplus N$
and divided powers $\bar \delta$ as in the proof of the lemma.
Then $(D, \bar J, \bar \delta)$ is a divided power ring
and the two maps $B \to D$ given by $b \mapsto b$ and
$b \mapsto b + \text{d}_{B/A, \delta}(b)$
are homomorphisms of divided power rings over $A$. Moreover, $N$
is the smallest submodule of $\Omega_{B/A, \delta}$ such that this is true.
\end{remark}
\begin{lemma}
\label{lemma-diagonal-and-differentials-affine-site}
In Situation \ref{situation-affine}.
Let $(B, J, \delta)$ be an object of $\text{CRIS}(C/A)$.
Let $(B(1), J(1), \delta(1))$ be the coproduct of $(B, J, \delta)$
with itself in $\text{CRIS}(C/A)$. Denote
$K = \Ker(B(1) \to B)$. Then $K \cap J(1) \subset J(1)$
is preserved by the divided power structure and
$$
\Omega_{B/A, \delta} = K/ \left(K^2 + (K \cap J(1))^{[2]}\right)
$$
canonically.
\end{lemma}
\begin{proof}
Word for word the same as the proof of
Lemma \ref{lemma-diagonal-and-differentials}.
The only point that has to be checked is that the
divided power ring $D = B \oplus M$ is an object of $\text{CRIS}(C/A)$
and that the two maps $B \to C$ are morphisms of $\text{CRIS}(C/A)$.
Since $D/(J \oplus M) = B/J$ we can use $C \to B/J$ to view
$D$ as an object of $\text{CRIS}(C/A)$
and the statement on morphisms is clear from the construction.
\end{proof}
\begin{lemma}
\label{lemma-module-differentials-divided-power-envelope}
Let $(A, I, \gamma)$ be a divided power ring. Let $A \to B$ be a ring
map and let $IB \subset J \subset B$ be an ideal. Let
$D_{B, \gamma}(J) = (D, \bar J, \bar \gamma)$ be the divided power envelope.
Then we have
$$
\Omega_{D/A, \bar\gamma} = \Omega_{B/A} \otimes_B D
$$
\end{lemma}
\begin{proof}
We will prove this first when $B$ is flat over $A$. In this case $\gamma$
extends to a divided power structure $\gamma'$ on $IB$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends}.
Hence $D = D_{B', \gamma'}(J)$ is equal to a quotient of
the divided power ring $(D', J', \delta)$ where $D' = B\langle x_t \rangle$
and $J' = IB\langle x_t \rangle + B\langle x_t \rangle_{+}$
by the elements $x_t - f_t$ and $\delta_n(\sum r_t x_t - r_0)$, see
Lemma \ref{lemma-describe-divided-power-envelope} for notation
and explanation. Write $\text{d} : D' \to \Omega_{D'/A, \delta}$
for the universal derivation. Note that
$$
\Omega_{D'/A, \delta} =
\Omega_{B/A} \otimes_B D' \oplus \bigoplus D' \text{d}x_t,
$$
see Lemma \ref{lemma-omega}. We conclude that $\Omega_{D/A, \bar\gamma}$
is the quotient of $\Omega_{D'/A, \delta} \otimes_{D'} D$ by the submodule
generated by $\text{d}$ applied to the generators of the
kernel of $D' \to D$ listed above, see Lemma \ref{lemma-omega}.
Since $\text{d}(x_t - f_t) = - \text{d}f_t + \text{d}x_t$
we see that we have $\text{d}x_t = \text{d}f_t$ in the quotient.
In particular we see that $\Omega_{B/A} \otimes_B D \to \Omega_{D/A, \gamma}$
is surjective with kernel given by the images of $\text{d}$
applied to the elements $\delta_n(\sum r_t x_t - r_0)$.
However, given a relation $\sum r_tf_t - r_0 = 0$ in $B$ with
$r_t \in B$ and $r_0 \in IB$ we see that
\begin{align*}
\text{d}\delta_n(\sum r_t x_t - r_0)
& =
\delta_{n - 1}(\sum r_t x_t - r_0)\text{d}(\sum r_t x_t - r_0)
\\
& =
\delta_{n - 1}(\sum r_t x_t - r_0)
\left(
\sum r_t\text{d}(x_t - f_t) + \sum (x_t - f_t)\text{d}r_t
\right)
\end{align*}
because $\sum r_tf_t - r_0 = 0$ in $B$. Hence this is already zero in
$\Omega_{B/A} \otimes_A D$ and we win in the case that $B$ is flat over $A$.
\medskip\noindent
In the general case we write $B$ as a quotient of a polynomial ring
$P \to B$ and let $J' \subset P$ be the inverse image of $J$. Then
$D = D'/K'$ with notation as in
Lemma \ref{lemma-divided-power-envelop-quotient}.
By the case handled in the first paragraph of the proof we have
$\Omega_{D'/A, \bar\gamma'} = \Omega_{P/A} \otimes_P D'$. Then
$\Omega_{D/A, \bar \gamma}$ is the quotient of $\Omega_{P/A} \otimes_P D$
by the submodule generated by $\text{d}\bar\gamma_n'(k)$ where $k$
is an element of the kernel of $P \to B$, see
Lemma \ref{lemma-omega} and the description of $K'$ from
Lemma \ref{lemma-divided-power-envelop-quotient}. Since
$\text{d}\bar\gamma_n'(k) = \bar\gamma'_{n - 1}(k)\text{d}k$ we see
again that it suffices to divided by the submodule generated by
$\text{d}k$ with $k \in \Ker(P \to B)$ and since $\Omega_{B/A}$
is the quotient of $\Omega_{P/A} \otimes_A B$ by these elements
(Algebra, Lemma \ref{algebra-lemma-differential-seq}) we win.
\end{proof}
\begin{remark}
\label{remark-absolute-de-rham-complex}
Let $B$ be a ring. Write $\Omega_B = \Omega_{B/\mathbf{Z}}$
for the absolute\footnote{This
actually makes sense: if $\Omega_B$ is the module of differentials
where we only assume the Leibniz rule and not the vanishing of $\text{d}1$,
then the Leibniz rule gives $\text{d}1 = \text{d}(1 \cdot 1) =
1 \text{d}1 + 1 \text{d}1 = 2 \text{d}1$ and hence
$\text{d}1 = 0$ in $\Omega_B$.} module of differentials of $B$.
Let $\text{d} : B \to \Omega_B$ denote the universal derivation. Set
$\Omega_B^i = \wedge^i_B(\Omega_B)$ as in
Algebra, Section \ref{algebra-section-tensor-algebra}.
The absolute {\it de Rham complex}
$$
\Omega_B^0 \to \Omega_B^1 \to \Omega_B^2 \to \ldots
$$
Here $\text{d} : \Omega_B^p \to \Omega_B^{p + 1}$
is defined by the rule
$$
\text{d}\left(b_0\text{d}b_1 \wedge \ldots \wedge \text{d}b_p\right) =
\text{d}b_0 \wedge \text{d}b_1 \wedge \ldots \wedge \text{d}b_p
$$
which we will show is well defined; note that
$\text{d} \circ \text{d} = 0$ so we get a complex.
Recall that $\Omega_B$ is the $B$-module generated by
elements $\text{d}b$ subject to the relations
$\text{d}(a + b) = \text{d}a + \text{d}b$ and
$\text{d}(ab) = b\text{d}a + a\text{d}b$
for $a, b \in B$. To prove that our map is well defined for $p = 1$
we have to show that the elements
$$
a\text{d}(b + c) - a\text{d}b - a\text{d}c
\quad\text{and}\quad
a\text{d}(bc) - ac\text{d}b - ab\text{d}c,\quad a,b,c \in B
$$
are mapped to zero by our rule. This is clear by direct computation
(using the Leibniz rule). Thus we get a map
$$
\Omega_B \otimes_\mathbf{Z} \ldots \otimes_\mathbf{Z} \Omega_B
\longrightarrow
\Omega_B^{p + 1}
$$
defined by the formula
$$
\omega_1 \otimes \ldots \otimes \omega_p
\longmapsto
\sum (-1)^{i + 1}
\omega_1 \wedge \ldots \wedge \text{d}(\omega_i) \wedge \ldots \wedge \omega_p
$$
which matches our rule above on elements of the form
$b_0\text{d}b_1 \otimes \text{d}b_2 \otimes \ldots \otimes \text{d}b_p$.
It is clear that this map is alternating. To finish we have to show
that
$$
\omega_1 \otimes \ldots \otimes f\omega_i \otimes \ldots \otimes \omega_p
\quad\text{and}\quad
\omega_1 \otimes \ldots \otimes f\omega_j \otimes \ldots \otimes \omega_p
$$
are mapped to the same element. By $\mathbf{Z}$-linearity and
the alternating property, it is enough to show this for $p = 2$, $i = 1$,
$j = 2$, $\omega_1 = a_1 \text{d}b_1$ and $\omega_2 = a_2 \text{d}b_2$.
Thus we need to show that
\begin{align*}
& \text{d}fa_1 \wedge \text{d}b_1 \wedge a_2\text{d}b_2
- fa_1 \text{d}b_1 \wedge \text{d}a_2 \wedge \text{d}b_2 \\
& =
\text{d}a_1 \wedge \text{d}b_1 \wedge fa_2\text{d}b_2
- a_1 \text{d}b_1 \wedge \text{d}fa_2 \wedge \text{d}b_2
\end{align*}
in other words that
$$
(a_2 \text{d}fa_1 + fa_1 \text{d}a_2 - fa_2 \text{d}a_1 - a_1 \text{d}fa_2)
\wedge \text{d}b_1 \wedge \text{d}b_2 = 0.
$$
This follows from the Leibniz rule.
\end{remark}
\begin{lemma}
\label{lemma-de-rham-complex}
Let $B$ be a ring. Let $\pi : \Omega_B \to \Omega$ be a surjective $B$-module
map. Denote $\text{d} : B \to \Omega$ the composition of $\pi$ with
$\text{d}_B : B \to \Omega_B$. Set $\Omega^i = \wedge_B^i(\Omega)$.
Assume that the kernel of $\pi$ is generated, as a $B$-module,
by elements $\omega \in \Omega_B$ such that
$\text{d}_B(\omega) \in \Omega_B^2$ maps to zero in $\Omega^2$.
Then there is a de Rham complex
$$
\Omega^0 \to \Omega^1 \to \Omega^2 \to \ldots
$$
whose differential is defined by the rule
$$
\text{d} : \Omega^p \to \Omega^{p + 1},\quad
\text{d}\left(f_0\text{d}f_1 \wedge \ldots \wedge \text{d}f_p\right) =
\text{d}f_0 \wedge \text{d}f_1 \wedge \ldots \wedge \text{d}f_p
$$
\end{lemma}
\begin{proof}
We will show that there exists a commutative diagram
$$
\xymatrix{
\Omega_B^0 \ar[d] \ar[r]_{\text{d}_B} &
\Omega_B^1 \ar[d]_\pi \ar[r]_{\text{d}_B} &
\Omega_B^2 \ar[d]_{\wedge^2\pi} \ar[r]_{\text{d}_B} &
\ldots \\
\Omega^0 \ar[r]^{\text{d}} &
\Omega^1 \ar[r]^{\text{d}} &
\Omega^2 \ar[r]^{\text{d}} &
\ldots
}
$$
the description of the map $\text{d}$ will follow from the construction
of $\text{d}_B$ in Remark \ref{remark-absolute-de-rham-complex}.
Since the left most vertical arrow is an isomorphism we have
the first square. Because $\pi$ is surjective, to get the second
square it suffices to show that $\text{d}_B$ maps the kernel
of $\pi$ into the kernel of $\wedge^2\pi$. We are given that any element
of the kernel of $\pi$ is of the form
$\sum b_i\omega_i$ with $\pi(\omega_i) = 0$ and
$\wedge^2\pi(\text{d}_B(\omega_i)) = 0$.
By the Leibniz rule for $\text{d}_B$ we have
$\text{d}_B(\sum b_i\omega_i) = \sum b_i \text{d}_B(\omega_i) +
\sum \text{d}_B(b_i) \wedge \omega_i$. Hence this maps to zero
under $\wedge^2\pi$.
\medskip\noindent
For $i > 1$ we note that $\wedge^i \pi$ is surjective with
kernel the image of $\Ker(\pi) \wedge \Omega^{i - 1}_B
\to \Omega_B^i$. For $\omega_1 \in \Ker(\pi)$ and
$\omega_2 \in \Omega^{i - 1}_B$ we have
$$
\text{d}_B(\omega_1 \wedge \omega_2) =
\text{d}_B(\omega_1) \wedge \omega_2 - \omega_1 \wedge \text{d}_B(\omega_2)
$$
which is in the kernel of $\wedge^{i + 1}\pi$ by what we just proved above.
Hence we get the $(i + 1)$st square in the diagram above.
This concludes the proof.
\end{proof}
\begin{remark}
\label{remark-divided-powers-de-rham-complex}
Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided power
structure on $B$. Set
$\Omega_{B/A, \delta}^i = \wedge^i_B \Omega_{B/A, \delta}$
where $\Omega_{B/A, \delta}$ is the target of the universal divided power
$A$-derivation $\text{d} = \text{d}_{B/A} : B \to \Omega_{B/A, \delta}$.
Note that $\Omega_{B/A, \delta}$ is the quotient of $\Omega_B$ by the
$B$-submodule generated by the elements
$\text{d}a = 0$ for $a \in A$ and
$\text{d}\delta_n(x) - \delta_{n - 1}(x)\text{d}x$ for $x \in J$.
We claim Lemma \ref{lemma-de-rham-complex} applies.
To see this it suffices to verify the elements
$\text{d}a$ and $\text{d}\delta_n(x) - \delta_{n - 1}(x)\text{d}x$
of $\Omega_B$ are mapped to zero in $\Omega^2_{B/A, \delta}$.
This is clear for the first, and for the last we observe that
$$
\text{d}(\delta_{n - 1}(x)) \wedge \text{d}x
= \delta_{n - 2}(x) \text{d}x \wedge \text{d}x = 0
$$
in $\Omega^2_{B/A, \delta}$ as desired. Hence we obtain a
{\it divided power de Rham complex}
$$
\Omega^0_{B/A, \delta} \to \Omega^1_{B/A, \delta} \to
\Omega^2_{B/A, \delta} \to \ldots
$$
which will play an important role in the sequel.
\end{remark}
\begin{remark}
\label{remark-connection}
Let $B$ be a ring. Let $\Omega_B \to \Omega$ be a quotient satisfying
the assumptions of Lemma \ref{lemma-de-rham-complex}.
Let $M$ be a $B$-module. A {\it connection} is an additive map
$$
\nabla : M \longrightarrow M \otimes_B \Omega
$$
such that $\nabla(bm) = b \nabla(m) + m \otimes \text{d}b$
for $b \in B$ and $m \in M$. In this situation we can define maps
$$
\nabla : M \otimes_B \Omega^i \longrightarrow M \otimes_B \Omega^{i + 1}
$$
by the rule $\nabla(m \otimes \omega) = \nabla(m) \wedge \omega +
m \otimes \text{d}\omega$. This works because if $b \in B$, then
\begin{align*}
\nabla(bm \otimes \omega) - \nabla(m \otimes b\omega)
& =
\nabla(bm) \otimes \omega + bm \otimes \text{d}\omega
- \nabla(m) \otimes b\omega - m \otimes \text{d}(b\omega) \\
& =
b\nabla(m) \otimes \omega + m \otimes \text{d}b \wedge \omega
+ bm \otimes \text{d}\omega \\
& \ \ \ \ \ \ - b\nabla(m) \otimes \omega - bm \otimes \text{d}(\omega)
- m \otimes \text{d}b \wedge \omega = 0
\end{align*}
As is customary we say the connection is {\it integrable} if and
only if the composition
$$
M \xrightarrow{\nabla} M \otimes_B \Omega^1
\xrightarrow{\nabla} M \otimes_B \Omega^2
$$
is zero. In this case we obtain a complex
$$
M \xrightarrow{\nabla} M \otimes_B \Omega^1
\xrightarrow{\nabla} M \otimes_B \Omega^2
\xrightarrow{\nabla} M \otimes_B \Omega^3
\xrightarrow{\nabla} M \otimes_B \Omega^4 \to \ldots
$$
which is called the de Rham complex of the connection.
\end{remark}
\begin{remark}
\label{remark-base-change-connection}
Let $\varphi : B \to B'$ be a ring map. Let $\Omega_B \to \Omega$ and
$\Omega_{B'} \to \Omega'$ be quotients satisfying the assumptions of
Lemma \ref{lemma-de-rham-complex}. Assume that the map
$\Omega_B \to \Omega_{B'}$,
$b_1\text{d}b_2 \mapsto \varphi(b_1)\text{d}\varphi(b_2)$ fits into a
commutative diagram
$$
\xymatrix{
B \ar[r] \ar[d] & \Omega_B \ar[r] \ar[d] & \Omega \ar[d]^{\varphi} \\
B' \ar[r] & \Omega_{B'} \ar[r] & \Omega'
}
$$
In this situation, given any pair $(M, \nabla)$ where $M$ is a $B$-module
and $\nabla : M \to M \otimes_B \Omega$ is a connection
we obtain a {\it base change} $(M \otimes_B B', \nabla')$ where
$$
\nabla' :
M \otimes_B B'
\longrightarrow
(M \otimes_B B') \otimes_{B'} \Omega' = M \otimes_B \Omega'
$$
is defined by the rule
$$
\nabla'(m \otimes b') =
\sum m_i \otimes b'\text{d}\varphi(b_i) + m \otimes \text{d}b'
$$
if $\nabla(m) = \sum m_i \otimes \text{d}b_i$. If $\nabla$ is integrable,
then so is $\nabla'$, and in this case there is a canonical map of
de Rham complexes
\begin{equation}
\label{equation-base-change-map-complexes}
M \otimes_B \Omega^\bullet
\longrightarrow
(M \otimes_B B') \otimes_{B'} (\Omega')^\bullet =
M \otimes_B (\Omega')^\bullet
\end{equation}
which maps $m \otimes \eta$ to $m \otimes \varphi(\eta)$.
\end{remark}
\begin{lemma}
\label{lemma-differentials-completion}
Let $A \to B$ be a ring map and let $(J, \delta)$ be a divided power
structure on $B$. Let $p$ be a prime number. Assume that $A$ is a
$\mathbf{Z}_{(p)}$-algebra and that $p$ is nilpotent in $B/J$. Then
we have
$$
\lim_e \Omega_{B_e/A, \bar\delta} =
\lim_e \Omega_{B/A, \delta}/p^e\Omega_{B/A, \delta} =
\lim_e \Omega_{B^\wedge/A, \delta^\wedge}/p^e \Omega_{B^\wedge/A, \delta^\wedge}
$$
see proof for notation and explanation.
\end{lemma}
\begin{proof}
By Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}
we see that $\delta$ extends
to $B_e = B/p^eB$ for all sufficiently large $e$. Hence the first limit
make sense. The lemma also produces a divided power structure $\delta^\wedge$
on the completion $B^\wedge = \lim_e B_e$, hence the last limit makes
sense. By Lemma \ref{lemma-omega}
and the fact that $\text{d}p^e = 0$ (always)
we see that the surjection
$\Omega_{B/A, \delta} \to \Omega_{B_e/A, \bar\delta}$ has kernel
$p^e\Omega_{B/A, \delta}$. Similarly for the kernel of
$\Omega_{B^\wedge/A, \delta^\wedge} \to \Omega_{B_e/A, \bar\delta}$.
Hence the lemma is clear.
\end{proof}
\section{Divided power schemes}
\label{section-divided-power-schemes}
\noindent
Some remarks on how to globalize the previous notions.
\begin{definition}
\label{definition-divided-power-structure}
Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings
on $\mathcal{C}$. Let $\mathcal{I} \subset \mathcal{O}$ be a
sheaf of ideals. A {\it divided power structure $\gamma$} on $\mathcal{I}$
is a sequence of maps $\gamma_n : \mathcal{I} \to \mathcal{I}$, $n \geq 1$
such that for any object $U$ of $\mathcal{C}$ the triple
$$
(\mathcal{O}(U), \mathcal{I}(U), \gamma)
$$
is a divided power ring.
\end{definition}
\noindent
To be sure this applies in particular to sheaves of rings on
topological spaces. But it's good to be a little bit more general
as the structure sheaf of the crystalline site lives on a... site!
A triple $(\mathcal{C}, \mathcal{I}, \gamma)$ as in the
definition above is sometimes called a {\it divided power topos}
in this chapter. Given a second $(\mathcal{C}', \mathcal{I}', \gamma')$ and
given a morphism of ringed topoi
$(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{C}'), \mathcal{O}')$
we say that $(f, f^\sharp)$ induces a {\it morphism of divided
power topoi} if $f^\sharp(f^{-1}\mathcal{I}') \subset \mathcal{I}$
and the diagrams
$$
\xymatrix{
f^{-1}\mathcal{I}' \ar[d]_{f^{-1}\gamma'_n} \ar[r]_{f^\sharp} &
\mathcal{I} \ar[d]^{\gamma_n} \\
f^{-1}\mathcal{I}' \ar[r]^{f^\sharp} & \mathcal{I}
}
$$
are commutative for all $n \geq 1$. If $f$ comes from a morphism of
sites induced by a functor $u : \mathcal{C}' \to \mathcal{C}$ then
this just means that
$$
(\mathcal{O}'(U'), \mathcal{I}'(U'), \gamma')
\longrightarrow
(\mathcal{O}(u(U')), \mathcal{I}(u(U')), \gamma)
$$
is a homomorphism of divided power rings for all $U' \in \Ob(\mathcal{C}')$.
\medskip\noindent
In the case of schemes we require the divided power ideal to be
{\bf quasi-coherent}. But apart from this the definition is exactly
the same as in the case of topoi. Here it is.
\begin{definition}
\label{definition-divided-power-scheme}
A {\it divided power scheme} is a triple $(S, \mathcal{I}, \gamma)$
where $S$ is a scheme, $\mathcal{I}$ is a quasi-coherent sheaf of
ideals, and $\gamma$ is a divided power structure on $\mathcal{I}$.
A {\it morphism of divided power schemes}
$(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ is
a morphism of schemes $f : S \to S'$ such that
$f^{-1}\mathcal{I}'\mathcal{O}_S \subset \mathcal{I}$ and such that
$$
(\mathcal{O}_{S'}(U'), \mathcal{I}'(U'), \gamma')
\longrightarrow
(\mathcal{O}_S(f^{-1}U'), \mathcal{I}(f^{-1}U'), \gamma)
$$
is a homomorphism of divided power rings for all $U' \subset S'$ open.
\end{definition}
\noindent
Recall that there is a 1-to-1 correspondence between quasi-coherent
sheaves of ideals and closed immersions, see
Morphisms, Section \ref{morphisms-section-closed-immersions}.
Thus given a divided power scheme $(T, \mathcal{J}, \gamma)$ we
get a canonical closed immersion $U \to T$ defined by $\mathcal{J}$.
Conversely, given a closed immersion $U \to T$ and a divided power
structure $\gamma$ on the sheaf of ideals $\mathcal{J}$ associated
to $U \to T$ we obtain a divided power scheme $(T, \mathcal{J}, \gamma)$.
In many situations we only want to consider such triples
$(U, T, \gamma)$ when the morphism $U \to T$ is a thickening, see
More on Morphisms, Definition \ref{more-morphisms-definition-thickening}.
\begin{definition}
\label{definition-divided-power-thickening}
A triple $(U, T, \gamma)$ as above is called a {\it divided power thickening}
if $U \to T$ is a thickening.
\end{definition}
\noindent
Fibre products of divided power schemes exist when one of the
three is a divided power thickening. Here is a formal statement.
\begin{lemma}
\label{lemma-fibre-product}
Let $(U', T', \delta') \to (S'_0, S', \gamma')$ and
$(S_0, S, \gamma) \to (S'_0, S', \gamma')$ be morphisms of
divided power schemes. If $(U', T', \delta')$ is a divided power
thickening, then there exists a divided power scheme $(T_0, T, \delta)$
and
$$
\xymatrix{
T \ar[r] \ar[d] & T' \ar[d] \\
S \ar[r] & S'
}
$$
which is a cartesian diagram in the category of divided power schemes.
\end{lemma}
\begin{proof}
Omitted. Hints: If $T$ exists, then $T_0 = S_0 \times_{S'_0} U'$
(argue as in Divided Power Algebra, Remark \ref{dpa-remark-forgetful}).
Since $T'$ is a divided power thickening, we see that $T$
(if it exists) will be a divided power thickening too.
Hence we can define $T$ as the scheme with underlying topological
space the underlying topological space of $T_0 = S_0 \times_{S'_0} U'$
and as structure sheaf on affine pieces the ring given
by Lemma \ref{lemma-affine-thickenings-colimits}.
\end{proof}
\noindent
We make the following observation. Suppose that $(U, T, \gamma)$
is triple as above. Assume that $T$ is a scheme over $\mathbf{Z}_{(p)}$
and that $p$ is locally nilpotent on $U$. Then
\begin{enumerate}
\item $p$ locally nilpotent on $T \Leftrightarrow U \to T$
is a thickening (see Divided Power Algebra, Lemma \ref{dpa-lemma-nil}), and
\item $p^e\mathcal{O}_T$ is locally on $T$ preserved by $\gamma$
for $e \gg 0$ (see
Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}).
\end{enumerate}
This suggest that good results on divided power thickenings will be
available under the following hypotheses.
\begin{situation}
\label{situation-global}
Here $p$ is a prime number and $(S, \mathcal{I}, \gamma)$ is a divided power
scheme over $\mathbf{Z}_{(p)}$. We set $S_0 = V(\mathcal{I}) \subset S$.
Finally, $X \to S_0$ is a morphism of schemes such that $p$ is
locally nilpotent on $X$.
\end{situation}
\noindent
It is in this situation that we will define the big and small
crystalline sites.
\section{The big crystalline site}
\label{section-big-site}
\noindent
We first define the big site. Given a divided power scheme
$(S, \mathcal{I}, \gamma)$ we say $(T, \mathcal{J}, \delta)$ is
a divided power scheme over $(S, \mathcal{I}, \gamma)$ if
$T$ comes endowed with a morphism $T \to S$ of divided power
schemes. Similarly, we say a divided power thickening $(U, T, \delta)$
is a divided power thickening over $(S, \mathcal{I}, \gamma)$
if $T$ comes endowed with a morphism $T \to S$ of divided power
schemes.
\begin{definition}
\label{definition-divided-power-thickening-X}
In Situation \ref{situation-global}.
\begin{enumerate}
\item A {\it divided power thickening of $X$ relative to
$(S, \mathcal{I}, \gamma)$} is given by a divided power thickening
$(U, T, \delta)$ over $(S, \mathcal{I}, \gamma)$
and an $S$-morphism $U \to X$.
\item A {\it morphism of divided power thickenings of $X$
relative to $(S, \mathcal{I}, \gamma)$} is defined in the obvious
manner.
\end{enumerate}
The category of divided power thickenings of $X$ relative to
$(S, \mathcal{I}, \gamma)$ is denoted $\text{CRIS}(X/S, \mathcal{I}, \gamma)$
or simply $\text{CRIS}(X/S)$.
\end{definition}
\noindent
For any $(U, T, \delta)$ in $\text{CRIS}(X/S)$
we have that $p$ is locally nilpotent on $T$, see discussion after
Definition \ref{definition-divided-power-thickening}.
A good way to visualize all the data associated to $(U, T, \delta)$
is the commutative diagram
$$
\xymatrix{
T \ar[dd] & U \ar[l] \ar[d] \\
& X \ar[d] \\
S & S_0 \ar[l]
}
$$
where $S_0 = V(\mathcal{I}) \subset S$. Morphisms of $\text{CRIS}(X/S)$
can be similarly visualized as huge commutative diagrams. In particular,
there is a canonical forgetful functor
\begin{equation}
\label{equation-forget}
\text{CRIS}(X/S) \longrightarrow \Sch/X,\quad
(U, T, \delta) \longmapsto U
\end{equation}
as well as its one sided inverse (and left adjoint)
\begin{equation}
\label{equation-endow-trivial}
\Sch/X \longrightarrow \text{CRIS}(X/S),\quad
U \longmapsto (U, U, \emptyset)
\end{equation}
which is sometimes useful.
\begin{lemma}
\label{lemma-divided-power-thickening-fibre-products}
In Situation \ref{situation-global}.
The category $\text{CRIS}(X/S)$ has all finite nonempty limits,
in particular products of pairs and fibre products.
The functor (\ref{equation-forget}) commutes with limits.
\end{lemma}
\begin{proof}
Omitted. Hint: See Lemma \ref{lemma-affine-thickenings-colimits}
for the affine case. See also
Divided Power Algebra, Remark \ref{dpa-remark-forgetful}.
\end{proof}
\begin{lemma}
\label{lemma-divided-power-thickening-base-change-flat}
In Situation \ref{situation-global}. Let
$$
\xymatrix{
(U_3, T_3, \delta_3) \ar[d] \ar[r] & (U_2, T_2, \delta_2) \ar[d] \\
(U_1, T_1, \delta_1) \ar[r] & (U, T, \delta)
}
$$
be a fibre square in the category of divided power thickenings of
$X$ relative to $(S, \mathcal{I}, \gamma)$. If $T_2 \to T$ is
flat and $U_2 = T_2 \times_T U$, then $T_3 = T_1 \times_T T_2$ (as schemes).
\end{lemma}
\begin{proof}
This is true because a divided power structure extends uniquely
along a flat ring map. See
Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends}.
\end{proof}
\noindent
The lemma above means that the base change of a flat morphism
of divided power thickenings is another flat morphism, and in
fact is the ``usual'' base change of the morphism. This implies
that the following definition makes sense.
\begin{definition}
\label{definition-big-crystalline-site}
In Situation \ref{situation-global}.
\begin{enumerate}
\item A family of morphisms $\{(U_i, T_i, \delta_i) \to (U, T, \delta)\}$
of divided power thickenings of $X/S$ is a {\it Zariski, \'etale, smooth,
syntomic, or fppf covering} if and only if the family of morphisms
of schemes $\{T_i \to T \}$ is one.
\item The {\it big crystalline site} of $X$ over $(S, \mathcal{I}, \gamma)$,
is the category $\text{CRIS}(X/S)$ endowed with the Zariski topology.
\item The topos of sheaves on $\text{CRIS}(X/S)$ is denoted
$(X/S)_{\text{CRIS}}$ or sometimes
$(X/S, \mathcal{I}, \gamma)_{\text{CRIS}}$\footnote{This clashes with
our convention to denote the topos associated to a site $\mathcal{C}$
by $\Sh(\mathcal{C})$.}.
\end{enumerate}
\end{definition}
\noindent
There are some obvious functorialities concerning these topoi.
\begin{remark}[Functoriality]
\label{remark-functoriality-big-cris}
Let $p$ be a prime number.
Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a
morphism of divided power schemes over $\mathbf{Z}_{(p)}$.
Set $S_0 = V(\mathcal{I})$ and $S'_0 = V(\mathcal{I}')$.
Let
$$
\xymatrix{
X \ar[r]_f \ar[d] & Y \ar[d] \\
S_0 \ar[r] & S'_0
}
$$
be a commutative diagram of morphisms of schemes and assume $p$ is
locally nilpotent on $X$ and $Y$. Then we get a continuous and
cocontinuous functor
$$
\text{CRIS}(X/S) \longrightarrow \text{CRIS}(Y/S')
$$
by letting $(U, T, \delta)$ correspond to $(U, T, \delta)$
with $U \to X \to Y$ as the $S'$-morphism from $U$ to $Y$.
Hence we get a morphism of topoi
$$
f_{\text{CRIS}} : (X/S)_{\text{CRIS}} \longrightarrow (Y/S')_{\text{CRIS}}
$$
see Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.
\end{remark}
\begin{remark}[Comparison with Zariski site]
\label{remark-compare-big-zariski}
In Situation \ref{situation-global}.
The functor (\ref{equation-forget}) is continuous, cocontinuous, and
commutes with products and fibred products.
Hence we obtain a morphism of topoi
$$
U_{X/S} : (X/S)_{\text{CRIS}} \longrightarrow \Sh((\Sch/X)_{Zar})
$$
from the big crystalline topos of $X/S$ to the big Zariski topos of $X$.
See Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.
\end{remark}
\begin{remark}[Structure morphism]
\label{remark-big-structure-morphism}
In Situation \ref{situation-global}.
Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$.
If we assume that $p$ is locally nilpotent on $S_0$ (which is always
the case in practice) then we obtain a situation as in
Definition \ref{definition-divided-power-thickening-X} with $S_0$ instead
of $X$. Hence we get a site $\text{CRIS}(S_0/S)$. If $f : X \to S_0$ is
the structure morphism of $X$ over $S$, then we get a commutative diagram
of morphisms of ringed topoi
$$
\xymatrix{
(X/S)_{\text{CRIS}}
\ar[r]_{f_{\text{CRIS}}} \ar[d]_{U_{X/S}} &
(S_0/S)_{\text{CRIS}} \ar[d]^{U_{S_0/S}} \\
\Sh((\Sch/X)_{Zar}) \ar[r]^{f_{big}} & \Sh((\Sch/S_0)_{Zar}) \ar[rd] \\
& & \Sh((\Sch/S)_{Zar})
}
$$
by Remark \ref{remark-functoriality-big-cris}. We think of the composition
$(X/S)_{\text{CRIS}} \to \Sh((\Sch/S)_{Zar})$ as the structure morphism of
the big crystalline site. Even if $p$ is not locally nilpotent on $S_0$
the structure morphism
$$
(X/S)_{\text{CRIS}} \longrightarrow \Sh((\Sch/S)_{Zar})
$$
is defined as we can take the lower route through the diagram above. Thus it
is the morphism of topoi corresponding to the cocontinuous
functor $\text{CRIS}(X/S) \to (\Sch/S)_{Zar}$ given by the rule
$(U, T, \delta)/S \mapsto T/S$, see
Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.
\end{remark}
\begin{remark}[Compatibilities]
\label{remark-compatibilities-big-cris}
The morphisms defined above satisfy numerous compatibilities. For example,
in the situation of Remark \ref{remark-functoriality-big-cris}
we obtain a commutative diagram of ringed topoi
$$
\xymatrix{
(X/S)_{\text{CRIS}} \ar[d] \ar[r] & (Y/S')_{\text{CRIS}} \ar[d] \\
\Sh((\Sch/S)_{Zar}) \ar[r] & \Sh((\Sch/S')_{Zar})
}
$$
where the vertical arrows are the structure morphisms.
\end{remark}
\section{The crystalline site}
\label{section-site}
\noindent
Since (\ref{equation-forget}) commutes with products and fibre
products, we see that looking at those $(U, T, \delta)$ such that
$U \to X$ is an open immersion defines a full
subcategory preserved under fibre products (and more generally
finite nonempty limits). Hence the following
definition makes sense.
\begin{definition}
\label{definition-crystalline-site}
In Situation \ref{situation-global}.
\begin{enumerate}
\item The (small) {\it crystalline site} of $X$ over
$(S, \mathcal{I}, \gamma)$, denoted $\text{Cris}(X/S, \mathcal{I}, \gamma)$
or simply $\text{Cris}(X/S)$ is the full subcategory of $\text{CRIS}(X/S)$
consisting of those $(U, T, \delta)$ in $\text{CRIS}(X/S)$ such that
$U \to X$ is an open immersion. It comes endowed with the Zariski topology.
\item The topos of sheaves on $\text{Cris}(X/S)$ is denoted
$(X/S)_{\text{cris}}$ or sometimes
$(X/S, \mathcal{I}, \gamma)_{\text{cris}}$\footnote{This clashes with
our convention to denote the topos associated to a site $\mathcal{C}$
by $\Sh(\mathcal{C})$.}.
\end{enumerate}
\end{definition}
\noindent
For any $(U, T, \delta)$ in $\text{Cris}(X/S)$ the morphism $U \to X$
defines an object of the small Zariski site $X_{Zar}$ of $X$. Hence
a canonical forgetful functor
\begin{equation}
\label{equation-forget-small}
\text{Cris}(X/S) \longrightarrow X_{Zar},\quad
(U, T, \delta) \longmapsto U
\end{equation}
and a left adjoint
\begin{equation}
\label{equation-endow-trivial-small}
X_{Zar} \longrightarrow \text{Cris}(X/S),\quad
U \longmapsto (U, U, \emptyset)
\end{equation}
which is sometimes useful.
\medskip\noindent
We can compare the small and big crystalline sites, just like
we can compare the small and big Zariski sites of a scheme, see
Topologies, Lemma \ref{topologies-lemma-at-the-bottom}.
\begin{lemma}
\label{lemma-compare-big-small}
Assumptions as in Definition \ref{definition-divided-power-thickening-X}.
The inclusion functor
$$
\text{Cris}(X/S) \to \text{CRIS}(X/S)
$$
commutes with finite nonempty limits, is fully faithful, continuous,
and cocontinuous. There are morphisms of topoi
$$
(X/S)_{\text{cris}} \xrightarrow{i} (X/S)_{\text{CRIS}}
\xrightarrow{\pi} (X/S)_{\text{cris}}
$$
whose composition is the identity and of which the first is induced
by the inclusion functor. Moreover, $\pi_* = i^{-1}$.
\end{lemma}
\begin{proof}
For the first assertion see
Lemma \ref{lemma-divided-power-thickening-fibre-products}.
This gives us a morphism of topoi
$i : (X/S)_{\text{cris}} \to (X/S)_{\text{CRIS}}$ and a left adjoint
$i_!$ such that $i^{-1}i_! = i^{-1}i_* = \text{id}$, see
Sites, Lemmas \ref{sites-lemma-when-shriek},
\ref{sites-lemma-preserve-equalizers}, and
\ref{sites-lemma-back-and-forth}.
We claim that $i_!$ is exact. If this is true, then we can define
$\pi$ by the rules $\pi^{-1} = i_!$ and $\pi_* = i^{-1}$
and everything is clear. To prove the claim, note that we already know
that $i_!$ is right exact and preserves fibre products (see references
given). Hence it suffices to show that $i_! * = *$ where $*$ indicates
the final object in the category of sheaves of sets.
To see this it suffices to produce a set of objects
$(U_i, T_i, \delta_i)$, $i \in I$ of $\text{Cris}(X/S)$ such that
$$
\coprod\nolimits_{i \in I} h_{(U_i, T_i, \delta_i)} \to *
$$
is surjective in $(X/S)_{\text{CRIS}}$ (details omitted; hint: use that
$\text{Cris}(X/S)$ has products and that the functor
$\text{Cris}(X/S) \to \text{CRIS}(X/S)$ commutes with them).
In the affine case this
follows from Lemma \ref{lemma-set-generators}. We omit the proof
in general.
\end{proof}
\begin{remark}[Functoriality]
\label{remark-functoriality-cris}
Let $p$ be a prime number.
Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$
be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$.
Let
$$
\xymatrix{
X \ar[r]_f \ar[d] & Y \ar[d] \\
S_0 \ar[r] & S'_0
}
$$
be a commutative diagram of morphisms of schemes and assume $p$ is
locally nilpotent on $X$ and $Y$. By analogy with
Topologies, Lemma \ref{topologies-lemma-morphism-big-small} we define
$$
f_{\text{cris}} : (X/S)_{\text{cris}} \longrightarrow (Y/S')_{\text{cris}}
$$
by the formula $f_{\text{cris}} = \pi_Y \circ f_{\text{CRIS}} \circ i_X$
where $i_X$ and $\pi_Y$ are as in Lemma \ref{lemma-compare-big-small}
for $X$ and $Y$ and where $f_{\text{CRIS}}$ is as in
Remark \ref{remark-functoriality-big-cris}.
\end{remark}
\begin{remark}[Comparison with Zariski site]
\label{remark-compare-zariski}
In Situation \ref{situation-global}.
The functor (\ref{equation-forget-small}) is continuous, cocontinuous, and
commutes with products and fibred products.
Hence we obtain a morphism of topoi
$$
u_{X/S} : (X/S)_{\text{cris}} \longrightarrow \Sh(X_{Zar})
$$
relating the small crystalline topos of $X/S$ with
the small Zariski topos of $X$.
See Sites, Section \ref{sites-section-cocontinuous-morphism-topoi}.
\end{remark}
\begin{lemma}
\label{lemma-localize}
In Situation \ref{situation-global}.
Let $X' \subset X$ and $S' \subset S$ be open subschemes such that
$X'$ maps into $S'$. Then there is a fully faithful functor
$\text{Cris}(X'/S') \to \text{Cris}(X/S)$
which gives rise to a morphism of topoi fitting into the commutative
diagram
$$
\xymatrix{
(X'/S')_{\text{cris}} \ar[r] \ar[d]_{u_{X'/S'}} &
(X/S)_{\text{cris}} \ar[d]^{u_{X/S}} \\
\Sh(X'_{Zar}) \ar[r] & \Sh(X_{Zar})
}
$$
Moreover, this diagram is an example of localization of morphisms of
topoi as in Sites, Lemma \ref{sites-lemma-localize-morphism-topoi}.
\end{lemma}
\begin{proof}
The fully faithful functor comes from thinking of
objects of $\text{Cris}(X'/S')$ as divided power
thickenings $(U, T, \delta)$ of $X$ where $U \to X$
factors through $X' \subset X$ (since then automatically $T \to S$
will factor through $S'$). This functor is clearly cocontinuous
hence we obtain a morphism of topoi as indicated.
Let $h_{X'} \in \Sh(X_{Zar})$ be the representable sheaf associated
to $X'$ viewed as an object of $X_{Zar}$. It is clear that
$\Sh(X'_{Zar})$ is the localization $\Sh(X_{Zar})/h_{X'}$.
On the other hand, the category $\text{Cris}(X/S)/u_{X/S}^{-1}h_{X'}$
(see Sites, Lemma \ref{sites-lemma-localize-topos-site})
is canonically identified with $\text{Cris}(X'/S')$ by the functor above.
This finishes the proof.
\end{proof}
\begin{remark}[Structure morphism]
\label{remark-structure-morphism}
In Situation \ref{situation-global}.
Consider the closed subscheme $S_0 = V(\mathcal{I}) \subset S$.
If we assume that $p$ is locally nilpotent on $S_0$ (which is always
the case in practice) then we obtain a situation as in
Definition \ref{definition-divided-power-thickening-X} with $S_0$ instead
of $X$. Hence we get a site $\text{Cris}(S_0/S)$. If $f : X \to S_0$
is the structure morphism of $X$ over $S$, then we get a
commutative diagram of ringed topoi
$$
\xymatrix{
(X/S)_{\text{cris}}
\ar[r]_{f_{\text{cris}}} \ar[d]_{u_{X/S}} &
(S_0/S)_{\text{cris}} \ar[d]^{u_{S_0/S}} \\
\Sh(X_{Zar}) \ar[r]^{f_{small}} & \Sh(S_{0, Zar}) \ar[rd] \\
& & \Sh(S_{Zar})
}
$$
see Remark \ref{remark-functoriality-cris}. We think of the composition
$(X/S)_{\text{cris}} \to \Sh(S_{Zar})$ as the structure morphism of the
crystalline site. Even if $p$ is not locally nilpotent on $S_0$
the structure morphism
$$
\tau_{X/S} : (X/S)_{\text{cris}} \longrightarrow \Sh(S_{Zar})
$$
is defined as we can take the lower route through the diagram above.
\end{remark}
\begin{remark}[Compatibilities]
\label{remark-compatibilities}
The morphisms defined above satisfy numerous compatibilities. For example,
in the situation of Remark \ref{remark-functoriality-cris}
we obtain a commutative diagram of ringed topoi
$$
\xymatrix{
(X/S)_{\text{cris}} \ar[d] \ar[r] & (Y/S')_{\text{cris}} \ar[d] \\
\Sh((\Sch/S)_{Zar}) \ar[r] & \Sh((\Sch/S')_{Zar})
}
$$
where the vertical arrows are the structure morphisms.
\end{remark}
\section{Sheaves on the crystalline site}
\label{section-sheaves}
\noindent
Notation and assumptions as in Situation \ref{situation-global}.
In order to discuss the small and big crystalline sites of $X/S$
simultaneously in this section we let
$$
\mathcal{C} = \text{CRIS}(X/S)
\quad\text{or}\quad
\mathcal{C} = \text{Cris}(X/S).
$$
A sheaf $\mathcal{F}$ on $\mathcal{C}$ gives rise to
a {\it restriction} $\mathcal{F}_T$ for every object $(U, T, \delta)$
of $\mathcal{C}$. Namely, $\mathcal{F}_T$ is the Zariski sheaf on
the scheme $T$ defined by the rule
$$
\mathcal{F}_T(W) = \mathcal{F}(U \cap W, W, \delta|_W)
$$
for $W \subset T$ is open. Moreover, if $f : T \to T'$ is a morphism
between objects
$(U, T, \delta)$ and $(U', T', \delta')$ of $\mathcal{C}$, then there
is a canonical {\it comparison} map
\begin{equation}
\label{equation-comparison}
c_f : f^{-1}\mathcal{F}_{T'} \longrightarrow \mathcal{F}_T.
\end{equation}
Namely, if $W' \subset T'$ is open then $f$ induces a morphism
$$
f|_{f^{-1}W'} :
(U \cap f^{-1}(W'), f^{-1}W', \delta|_{f^{-1}W'})
\longrightarrow
(U' \cap W', W', \delta|_{W'})
$$
of $\mathcal{C}$, hence we can use the restriction mapping
$(f|_{f^{-1}W'})^*$ of $\mathcal{F}$ to define a map
$\mathcal{F}_{T'}(W') \to \mathcal{F}_T(f^{-1}W')$.
These maps are clearly compatible with further restriction, hence
define an $f$-map from $\mathcal{F}_{T'}$ to $\mathcal{F}_T$ (see
Sheaves, Section \ref{sheaves-section-presheaves-functorial}
and especially
Sheaves, Definition \ref{sheaves-definition-f-map}).
Thus a map $c_f$ as in (\ref{equation-comparison}).
Note that if $f$ is an open immersion, then $c_f$ is an
isomorphism, because in that case $\mathcal{F}_T$ is just
the restriction of $\mathcal{F}_{T'}$ to $T$.
\medskip\noindent
Conversely, given Zariski sheaves $\mathcal{F}_T$ for every object
$(U, T, \delta)$ of $\mathcal{C}$ and comparison maps
$c_f$ as above which (a) are isomorphisms for open immersions, and (b)
satisfy a suitable cocycle condition, we obtain a sheaf on
$\mathcal{C}$. This is proved exactly as in
Topologies, Lemma \ref{topologies-lemma-characterize-sheaf-big}.
\medskip\noindent
The {\it structure sheaf} on $\mathcal{C}$ is the sheaf
$\mathcal{O}_{X/S}$ defined by the rule
$$
\mathcal{O}_{X/S} :
(U, T, \delta)
\longmapsto
\Gamma(T, \mathcal{O}_T)
$$
This is a sheaf by the definition of coverings in $\mathcal{C}$.
Suppose that $\mathcal{F}$ is a sheaf of $\mathcal{O}_{X/S}$-modules.
In this case the comparison mappings (\ref{equation-comparison})
define a comparison map
\begin{equation}
\label{equation-comparison-modules}
c_f : f^*\mathcal{F}_T \longrightarrow \mathcal{F}_{T'}
\end{equation}
of $\mathcal{O}_T$-modules.
\medskip\noindent
Another type of example comes by starting with a sheaf
$\mathcal{G}$ on $(\Sch/X)_{Zar}$ or $X_{Zar}$ (depending on whether
$\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$).
Then $\underline{\mathcal{G}}$ defined by the rule
$$
\underline{\mathcal{G}} :
(U, T, \delta)
\longmapsto
\mathcal{G}(U)
$$
is a sheaf on $\mathcal{C}$. In particular, if we take
$\mathcal{G} = \mathbf{G}_a = \mathcal{O}_X$, then we obtain
$$
\underline{\mathbf{G}_a} :
(U, T, \delta)
\longmapsto
\Gamma(U, \mathcal{O}_U)
$$
There is a surjective map of sheaves
$\mathcal{O}_{X/S} \to \underline{\mathbf{G}_a}$ defined by the
canonical maps $\Gamma(T, \mathcal{O}_T) \to \Gamma(U, \mathcal{O}_U)$
for objects $(U, T, \delta)$. The kernel of this map is denoted
$\mathcal{J}_{X/S}$, hence a short exact sequence
$$
0 \to
\mathcal{J}_{X/S} \to
\mathcal{O}_{X/S} \to
\underline{\mathbf{G}_a} \to 0
$$
Note that $\mathcal{J}_{X/S}$ comes equipped with a canonical
divided power structure. After all, for each object $(U, T, \delta)$
the third component $\delta$ {\it is} a divided power structure on the
kernel of $\mathcal{O}_T \to \mathcal{O}_U$. Hence the (big)
crystalline topos is a divided power topos.
\section{Crystals in modules}
\label{section-crystals}
\noindent
It turns out that a crystal is a very general gadget. However, the
definition may be a bit hard to parse, so we first give the definition
in the case of modules on the crystalline sites.
\begin{definition}
\label{definition-modules}
In Situation \ref{situation-global}.
Let $\mathcal{C} = \text{CRIS}(X/S)$ or $\mathcal{C} = \text{Cris}(X/S)$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on $\mathcal{C}$.
\begin{enumerate}
\item We say $\mathcal{F}$ is {\it locally quasi-coherent} if for every
object $(U, T, \delta)$ of $\mathcal{C}$ the restriction $\mathcal{F}_T$
is a quasi-coherent $\mathcal{O}_T$-module.
\item We say $\mathcal{F}$ is {\it quasi-coherent} if it is quasi-coherent
in the sense of
Modules on Sites, Definition \ref{sites-modules-definition-site-local}.
\item We say $\mathcal{F}$ is a {\it crystal in $\mathcal{O}_{X/S}$-modules}
if all the comparison maps (\ref{equation-comparison-modules}) are
isomorphisms.
\end{enumerate}
\end{definition}
\noindent
It turns out that we can relate these notions as follows.
\begin{lemma}
\label{lemma-crystal-quasi-coherent-modules}
With notation $X/S, \mathcal{I}, \gamma, \mathcal{C}, \mathcal{F}$
as in Definition \ref{definition-modules}. The following are equivalent
\begin{enumerate}
\item $\mathcal{F}$ is quasi-coherent, and
\item $\mathcal{F}$ is locally quasi-coherent and a crystal in
$\mathcal{O}_{X/S}$-modules.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (1). Let $f : (U', T', \delta') \to (U, T, \delta)$ be an object of
$\mathcal{C}$. We have to prove (a) $\mathcal{F}_T$ is a quasi-coherent
$\mathcal{O}_T$-module and (b) $c_f : f^*\mathcal{F}_T \to \mathcal{F}_{T'}$
is an isomorphism. The assumption means that we can find a covering
$\{(T_i, U_i, \delta_i) \to (T, U, \delta)\}$ and for each $i$
the restriction of $\mathcal{F}$ to $\mathcal{C}/(T_i, U_i, \delta_i)$
has a global presentation. Since it suffices to prove (a) and (b)
Zariski locally, we may replace $f : (T', U', \delta') \to (T, U, \delta)$
by the base change to $(T_i, U_i, \delta_i)$ and assume that $\mathcal{F}$
restricted to $\mathcal{C}/(T, U, \delta)$ has a global
presentation
$$
\bigoplus\nolimits_{j \in J}
\mathcal{O}_{X/S}|_{\mathcal{C}/(U, T, \delta)} \longrightarrow
\bigoplus\nolimits_{i \in I}
\mathcal{O}_{X/S}|_{\mathcal{C}/(U, T, \delta)} \longrightarrow
\mathcal{F}|_{\mathcal{C}/(U, T, \delta)}
\longrightarrow 0
$$
It is clear that this gives a presentation
$$
\bigoplus\nolimits_{j \in J} \mathcal{O}_T \longrightarrow
\bigoplus\nolimits_{i \in I} \mathcal{O}_T \longrightarrow
\mathcal{F}_T
\longrightarrow 0
$$
and hence (a) holds. Moreover, the presentation restricts to $T'$
to give a similar presentation of $\mathcal{F}_{T'}$, whence (b) holds.
\medskip\noindent
Assume (2). Let $(U, T, \delta)$ be an object of $\mathcal{C}$.
We have to find a covering of $(U, T, \delta)$ such that $\mathcal{F}$ has a
global presentation when we restrict to the localization of $\mathcal{C}$
at the members of the covering. Thus we may assume that $T$ is affine.
In this case we can choose a presentation
$$
\bigoplus\nolimits_{j \in J} \mathcal{O}_T \longrightarrow
\bigoplus\nolimits_{i \in I} \mathcal{O}_T \longrightarrow
\mathcal{F}_T
\longrightarrow 0
$$
as $\mathcal{F}_T$ is assumed to be a quasi-coherent $\mathcal{O}_T$-module.
Then by the crystal property of $\mathcal{F}$ we see that this pulls back
to a presentation of $\mathcal{F}_{T'}$ for any morphism
$f : (U', T', \delta') \to (U, T, \delta)$ of $\mathcal{C}$.
Thus the desired presentation of $\mathcal{F}|_{\mathcal{C}/(U, T, \delta)}$.
\end{proof}
\begin{definition}
\label{definition-crystal-quasi-coherent-modules}
If $\mathcal{F}$ satisfies the equivalent conditions of
Lemma \ref{lemma-crystal-quasi-coherent-modules}, then
we say that $\mathcal{F}$ is a
{\it crystal in quasi-coherent modules}.
We say that $\mathcal{F}$ is a {\it crystal in finite locally free modules}
if, in addition, $\mathcal{F}$ is finite locally free.
\end{definition}
\noindent
Of course, as Lemma \ref{lemma-crystal-quasi-coherent-modules} shows, this
notation is somewhat heavy since a quasi-coherent module is always a crystal.
But it is standard terminology in the literature.
\begin{remark}
\label{remark-crystal}
To formulate the general notion of a crystal we use the language
of stacks and strongly cartesian morphisms, see
Stacks, Definition \ref{stacks-definition-stack} and
Categories, Definition \ref{categories-definition-cartesian-over-C}.
In Situation \ref{situation-global} let
$p : \mathcal{C} \to \text{Cris}(X/S)$ be a stack.
A {\it crystal in objects of $\mathcal{C}$ on $X$ relative to $S$}
is a {\it cartesian section} $\sigma : \text{Cris}(X/S) \to \mathcal{C}$,
i.e., a functor $\sigma$ such that $p \circ \sigma = \text{id}$
and such that $\sigma(f)$ is strongly cartesian for all
morphisms $f$ of $\text{Cris}(X/S)$. Similarly for the big crystalline site.
\end{remark}
\section{Sheaf of differentials}
\label{section-differentials-sheaf}
\noindent
In this section we will stick with the (small) crystalline site
as it seems more natural. We globalize
Definition \ref{definition-derivation} as follows.
\begin{definition}
\label{definition-global-derivation}
In Situation \ref{situation-global} let
$\mathcal{F}$ be a sheaf of $\mathcal{O}_{X/S}$-modules on
$\text{Cris}(X/S)$. An
{\it $S$-derivation $D : \mathcal{O}_{X/S} \to \mathcal{F}$}
is a map of sheaves such that for every object $(U, T, \delta)$ of
$\text{Cris}(X/S)$ the map
$$
D : \Gamma(T, \mathcal{O}_T) \longrightarrow \Gamma(T, \mathcal{F})
$$
is a divided power $\Gamma(V, \mathcal{O}_V)$-derivation where $V \subset S$
is any open such that $T \to S$ factors through $V$.
\end{definition}
\noindent
This means that $D$ is additive, satisfies the Leibniz rule, annihilates
functions coming from $S$, and satisfies $D(f^{[n]}) = f^{[n - 1]}D(f)$
for a local section $f$ of the divided power ideal $\mathcal{J}_{X/S}$.
This is a special case of a very general notion which we now describe.
\medskip\noindent
Please compare the following discussion with
Modules on Sites, Section \ref{sites-modules-section-differentials}. Let
$\mathcal{C}$ be a site, let $\mathcal{A} \to \mathcal{B}$ be a
map of sheaves of rings on $\mathcal{C}$, let $\mathcal{J} \subset \mathcal{B}$
be a sheaf of ideals, let $\delta$ be a divided power structure on
$\mathcal{J}$, and let $\mathcal{F}$ be a sheaf of $\mathcal{B}$-modules.
Then there is a notion of a {\it divided power $\mathcal{A}$-derivation}
$D : \mathcal{B} \to \mathcal{F}$. This means that $D$ is $\mathcal{A}$-linear,
satisfies the Leibniz rule, and satisfies
$D(\delta_n(x)) = \delta_{n - 1}(x)D(x)$ for local sections $x$ of
$\mathcal{J}$. In this situation there exists a
{\it universal divided power $\mathcal{A}$-derivation}
$$
\text{d}_{\mathcal{B}/\mathcal{A}, \delta} :
\mathcal{B}
\longrightarrow
\Omega_{\mathcal{B}/\mathcal{A}, \delta}
$$
Moreover, $\text{d}_{\mathcal{B}/\mathcal{A}, \delta}$ is the composition
$$
\mathcal{B}
\longrightarrow
\Omega_{\mathcal{B}/\mathcal{A}}
\longrightarrow
\Omega_{\mathcal{B}/\mathcal{A}, \delta}
$$
where the first map is the universal derivation constructed in the proof
of Modules on Sites, Lemma \ref{sites-modules-lemma-universal-module}
and the second arrow is the quotient by the submodule generated by
the local sections
$\text{d}_{\mathcal{B}/\mathcal{A}}(\delta_n(x)) -
\delta_{n - 1}(x)\text{d}_{\mathcal{B}/\mathcal{A}}(x)$.
\medskip\noindent
We translate this into a relative notion as follows. Suppose
$(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to
(\Sh(\mathcal{C}'), \mathcal{O}')$ is a morphism of ringed topoi,
$\mathcal{J} \subset \mathcal{O}$ a sheaf of ideals, $\delta$ a
divided power structure on $\mathcal{J}$, and $\mathcal{F}$ a sheaf
of $\mathcal{O}$-modules. In this situation we say
$D : \mathcal{O} \to \mathcal{F}$ is a divided power $\mathcal{O}'$-derivation
if $D$ is a divided power $f^{-1}\mathcal{O}'$-derivation as defined above.
Moreover, we write
$$
\Omega_{\mathcal{O}/\mathcal{O}', \delta} =
\Omega_{\mathcal{O}/f^{-1}\mathcal{O}', \delta}
$$
which is the receptacle of the universal divided power
$\mathcal{O}'$-derivation.
\medskip\noindent
Applying this to the structure morphism
$$
(X/S)_{\text{Cris}} \longrightarrow \Sh(S_{Zar})
$$
(see Remark \ref{remark-structure-morphism}) we recover the notion of
Definition \ref{definition-global-derivation} above.
In particular, there is a universal divided power derivation
$$
d_{X/S} : \mathcal{O}_{X/S} \to \Omega_{X/S}
$$
Note that we omit from the notation the decoration indicating the
module of differentials is compatible with divided powers (it seems
unlikely anybody would ever consider the usual module of differentials
of the structure sheaf on the crystalline site).
\begin{lemma}
\label{lemma-module-differentials-divided-power-scheme}
Let $(T, \mathcal{J}, \delta)$ be a divided power scheme.
Let $T \to S$ be a morphism of schemes.
The quotient $\Omega_{T/S} \to \Omega_{T/S, \delta}$
described above is a quasi-coherent $\mathcal{O}_T$-module.
For $W \subset T$ affine open mapping into $V \subset S$ affine open
we have
$$
\Gamma(W, \Omega_{T/S, \delta}) =
\Omega_{\Gamma(W, \mathcal{O}_W)/\Gamma(V, \mathcal{O}_V), \delta}
$$
where the right hand side is
as constructed in Section \ref{section-differentials}.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-module-of-differentials}
In Situation \ref{situation-global}.
For $(U, T, \delta)$ in $\text{Cris}(X/S)$ the restriction
$(\Omega_{X/S})_T$ to $T$ is $\Omega_{T/S, \delta}$ and the restriction
$\text{d}_{X/S}|_T$ is equal to $\text{d}_{T/S, \delta}$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-module-of-differentials-on-affine}
In Situation \ref{situation-global}.
For any affine object $(U, T, \delta)$ of $\text{Cris}(X/S)$
mapping into an affine open $V \subset S$ we have
$$
\Gamma((U, T, \delta), \Omega_{X/S}) =
\Omega_{\Gamma(T, \mathcal{O}_T)/\Gamma(V, \mathcal{O}_V), \delta}
$$
where the right hand side is
as constructed in Section \ref{section-differentials}.
\end{lemma}
\begin{proof}
Combine Lemmas \ref{lemma-module-differentials-divided-power-scheme} and
\ref{lemma-module-of-differentials}.
\end{proof}
\begin{lemma}
\label{lemma-describe-omega-small}
In Situation \ref{situation-global}.
Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$.
Let
$$
(U(1), T(1), \delta(1)) = (U, T, \delta) \times (U, T, \delta)
$$
in $\text{Cris}(X/S)$. Let $\mathcal{K} \subset \mathcal{O}_{T(1)}$
be the quasi-coherent sheaf of ideals corresponding to the closed
immersion $\Delta : T \to T(1)$. Then
$\mathcal{K} \subset \mathcal{J}_{T(1)}$ is preserved by the
divided structure on $\mathcal{J}_{T(1)}$ and we have
$$
(\Omega_{X/S})_T = \mathcal{K}/\mathcal{K}^{[2]}
$$
\end{lemma}
\begin{proof}
Note that $U = U(1)$ as $U \to X$ is an open immersion and as
(\ref{equation-forget-small}) commutes with products. Hence we see that
$\mathcal{K} \subset \mathcal{J}_{T(1)}$. Given this fact the lemma follows
by working affine locally on $T$ and using
Lemmas \ref{lemma-module-of-differentials-on-affine} and
\ref{lemma-diagonal-and-differentials-affine-site}.
\end{proof}
\noindent
It turns out that $\Omega_{X/S}$ is not a crystal in quasi-coherent
$\mathcal{O}_{X/S}$-modules. But it does satisfy two closely
related properties (compare with
Lemma \ref{lemma-crystal-quasi-coherent-modules}).
\begin{lemma}
\label{lemma-omega-locally-quasi-coherent}
In Situation \ref{situation-global}.
The sheaf of differentials $\Omega_{X/S}$ has the following two
properties:
\begin{enumerate}
\item $\Omega_{X/S}$ is locally quasi-coherent, and
\item for any morphism $(U, T, \delta) \to (U', T', \delta')$
of $\text{Cris}(X/S)$ where $f : T \to T'$ is a closed immersion
the map $c_f : f^*(\Omega_{X/S})_{T'} \to (\Omega_{X/S})_T$ is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) follows from a combination of
Lemmas \ref{lemma-module-differentials-divided-power-scheme} and
\ref{lemma-module-of-differentials}.
Part (2) follows from the fact that
$(\Omega_{X/S})_T = \Omega_{T/S, \delta}$
is a quotient of $\Omega_{T/S}$ and that $f^*\Omega_{T'/S} \to \Omega_{T/S}$
is surjective.
\end{proof}
\section{Two universal thickenings}
\label{section-universal-thickenings}
\noindent
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 \ref{section-explicit-thickenings}.
\begin{remark}
\label{remark-first-order-thickening}
In Situation \ref{situation-global}.
Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$.
Write $\Omega_{T/S, \delta} = (\Omega_{X/S})_T$, see
Lemma \ref{lemma-module-of-differentials}.
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 \ref{lemma-divided-power-first-order-thickening}.
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 $\Omega_{T/S, \delta} \to \mathcal{O}_{T'}$.
\end{remark}
\begin{remark}
\label{remark-second-order-thickening}
In Situation \ref{situation-global}.
Let $(U, T, \delta)$ be an object of $\text{Cris}(X/S)$.
Write $\Omega_{T/S, \delta} = (\Omega_{X/S})_T$, see
Lemma \ref{lemma-module-of-differentials}.
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 \ref{lemma-divided-power-second-order-thickening}.
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 \ref{remark-first-order-thickening}.
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.
\end{remark}
\section{The de Rham complex}
\label{section-de-Rham}
\noindent
In Situation \ref{situation-global}.
Working on the (small) crystalline site, we define
$\Omega^i_{X/S} = \wedge^i_{\mathcal{O}_{X/S}} \Omega_{X/S}$
for $i \geq 0$. The universal $S$-derivation $\text{d}_{X/S}$ gives
rise to the {\it de Rham complex}
$$
\mathcal{O}_{X/S} \to \Omega^1_{X/S} \to \Omega^2_{X/S} \to \ldots
$$
on $\text{Cris}(X/S)$, see
Lemma \ref{lemma-module-of-differentials-on-affine} and
Remark \ref{remark-divided-powers-de-rham-complex}.
\section{Connections}
\label{section-connections}
\noindent
In Situation \ref{situation-global}.
Given an $\mathcal{O}_{X/S}$-module $\mathcal{F}$ on $\text{Cris}(X/S)$
a {\it connection} is a map of abelian sheaves
$$
\nabla :
\mathcal{F}
\longrightarrow
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}
$$
such that $\nabla(f s) = f\nabla(s) + s \otimes \text{d}f$
for local sections $s, f$ of $\mathcal{F}$ and $\mathcal{O}_{X/S}$.
Given a connection there are canonical maps
$
\nabla :
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^i_{X/S}
\longrightarrow
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^{i + 1}_{X/S}
$
defined by the rule $\nabla(s \otimes \omega) =
\nabla(s) \wedge \omega + s \otimes \text{d}\omega$
as in Remark \ref{remark-connection}. We say the connection is
{\it integrable} if $\nabla \circ \nabla = 0$. If $\nabla$ is integrable
we obtain the {\it de Rham complex}
$$
\mathcal{F} \to
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^1_{X/S} \to
\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega^2_{X/S} \to \ldots
$$
on $\text{Cris}(X/S)$. It turns out that any crystal in
$\mathcal{O}_{X/S}$-modules comes equipped with a canonical
integrable connection.
\begin{lemma}
\label{lemma-automatic-connection}
In Situation \ref{situation-global}.
Let $\mathcal{F}$ be a crystal in $\mathcal{O}_{X/S}$-modules
on $\text{Cris}(X/S)$. Then $\mathcal{F}$ comes equipped with a
canonical integrable connection.
\end{lemma}
\begin{proof}
Say $(U, T, \delta)$ is an object of $\text{Cris}(X/S)$.
Let $(U, T', \delta')$ be the infinitesimal thickening of $T$
by $(\Omega_{X/S})_T = \Omega_{T/S, \delta}$
constructed in Remark \ref{remark-first-order-thickening}.
It comes with projections $p_0, p_1 : T' \to T$
and a diagonal $i : T \to T'$. By assumption we get
isomorphisms
$$
p_0^*\mathcal{F}_T \xrightarrow{c_0}
\mathcal{F}_{T'} \xleftarrow{c_1}
p_1^*\mathcal{F}_T
$$
of $\mathcal{O}_{T'}$-modules. Pulling $c = c_1^{-1} \circ c_0$
back to $T$ by $i$ we obtain the identity map
of $\mathcal{F}_T$. Hence if $s \in \Gamma(T, \mathcal{F}_T)$
then $\nabla(s) = p_1^*s - c(p_0^*s)$ is a section of
$p_1^*\mathcal{F}_T$ which vanishes on pulling back by $\Delta$. Hence
$\nabla(s)$ is a section of
$$
\mathcal{F}_T
\otimes_{\mathcal{O}_T}
\Omega_{T/S, \delta}
$$
because this is the kernel of $p_1^*\mathcal{F}_T \to \mathcal{F}_T$
as $\Omega_{T/S, \delta}$ is the kernel of
$\mathcal{O}_{T'} \to \mathcal{O}_T$ by construction.
\medskip\noindent
The collection of maps
$$
\nabla : \Gamma(T, \mathcal{F}_T) \to
\Gamma(T, \mathcal{F}_T \otimes_{\mathcal{O}_T} \Omega_{T/S, \delta})
$$
so obtained is functorial in $T$ because the construction of $T'$
is functorial in $T$. Hence we obtain a connection.
\medskip\noindent
To show that the connection is integrable we consider the
object $(U, T'', \delta'')$ constructed in
Remark \ref{remark-second-order-thickening}.
Because $\mathcal{F}$ is a sheaf we see that
$$
\xymatrix{
q_0^*\mathcal{F}_T \ar[rr]_{q_{01}^*c} \ar[rd]_{q_{02}^*c} & &
q_1^*\mathcal{F}_T \ar[ld]^{q_{12}^*c} \\
& q_2^*\mathcal{F}_T
}
$$
is a commutative diagram of $\mathcal{O}_{T''}$-modules. For
$s \in \Gamma(T, \mathcal{F}_T)$ we have
$c(p_0^*s) = p_1^*s - \nabla(s)$. Write
$\nabla(s) = \sum p_1^*s_i \cdot \omega_i$ where $s_i$ is a local section
of $\mathcal{F}_T$ and $\omega_i$ is a local section of $\Omega_{T/S, \delta}$.
We think of $\omega_i$ as a local section of the structure
sheaf of $\mathcal{O}_{T'}$ and hence we write product instead of tensor
product. On the one hand
\begin{align*}
q_{12}^*c \circ q_{01}^*c(q_0^*s) & =
q_{12}^*c(q_1^*s - \sum q_1^*s_i \cdot q_{01}^*\omega_i) \\
& =
q_2^*s - \sum q_2^*s_i \cdot q_{12}^*\omega_i -
\sum q_2^*s_i \cdot q_{01}^*\omega_i +
\sum q_{12}^*\nabla(s_i) \cdot q_{01}^*\omega_i
\end{align*}
and on the other hand
$$
q_{02}^*c(q_0^*s) = q_2^*s - \sum q_2^*s_i \cdot q_{02}^*\omega_i.
$$
From the formulae of Remark \ref{remark-second-order-thickening} we see
that
$q_{01}^*\omega_i + q_{12}^*\omega_i - q_{02}^*\omega_i = \text{d}\omega_i$.
Hence the difference of the two expressions above is
$$
\sum q_2^*s_i \cdot \text{d}\omega_i -
\sum q_{12}^*\nabla(s_i) \cdot q_{01}^*\omega_i
$$
Note that
$q_{12}^*\omega \cdot q_{01}^*\omega' = \omega' \wedge \omega =
- \omega \wedge \omega'$ by the definition of the multiplication on
$\mathcal{O}_{T''}$. Thus the expression above is $\nabla^2(s)$ viewed
as a section of the subsheaf $\mathcal{F}_T \otimes \Omega^2_{T/S, \delta}$ of
$q_2^*\mathcal{F}$. Hence we get the integrability condition.
\end{proof}
\section{Cosimplicial algebra}
\label{section-cosimplicial}
\noindent
This section should be moved somewhere else. A
{\it cosimplicial ring} is a cosimplicial object
in the category of rings. Given a ring $R$, a
{\it cosimplicial $R$-algebra} is a cosimplicial object in the
category of $R$-algebras. A {\it cosimplicial ideal} in a cosimplicial
ring $A_*$ is given by an ideal $I_n \subset A_n$ for all $n$ such
that $A(f)(I_n) \subset I_m$ for all $f : [n] \to [m]$ in $\Delta$.
\medskip\noindent
Let $A_*$ be a cosimplicial ring. Let $\mathcal{C}$ be the category
of pairs $(A, M)$ where $A$ is a ring and $M$ is a module over $A$.
A morphism $(A, M) \to (A', M')$ consists of a ring map $A \to A'$ and
an $A$-module map $M \to M'$ where $M'$ is viewed as an $A$-module
via $A \to A'$ and the $A'$-module structure on $M'$. Having said this
we can define a {\it cosimplicial module $M_*$ over $A_*$} as a cosimplicial
object $(A_*, M_*)$ of $\mathcal{C}$ whose first entry is equal to $A_*$.
A {\it homomorphism $\varphi_* : M_* \to N_*$ of cosimplicial modules over
$A_*$} is a morphism $(A_*, M_*) \to (A_*, N_*)$ of cosimplicial objects
in $\mathcal{C}$ whose first component is $1_{A_*}$.
\medskip\noindent
A {\it homotopy} between homomorphisms $\varphi_*, \psi_* : M_* \to N_*$
of cosimplicial modules over $A_*$ is a homotopy between the associated
maps $(A_*, M_*) \to (A_*, N_*)$ whose first component is the
trivial homotopy (dual to
Simplicial, Example \ref{simplicial-example-trivial-homotopy}).
We spell out what this means. Such a homotopy is a homotopy
$$
h : M_* \longrightarrow \Hom(\Delta[1], N_*)
$$
between $\varphi_*$ and $\psi_*$ as homomorphisms of cosimplicial abelian
groups such that for each $n$ the map
$h_n : M_n \to \prod_{\alpha \in \Delta[1]_n} N_n$ is $A_n$-linear.
The following lemma is a version of
Simplicial, Lemma \ref{simplicial-lemma-functorial-homotopy}
for cosimplicial modules.
\begin{lemma}
\label{lemma-homotopy-tensor}
Let $A_*$ be a cosimplicial ring. Let $\varphi_*, \psi_* : K_* \to M_*$
be homomorphisms of cosimplicial $A_*$-modules.
\begin{enumerate}
\item
\label{item-tensor}
If $\varphi_*$ and $\psi_*$ are homotopic, then
$$
\varphi_* \otimes 1, \psi_* \otimes 1 :
K_* \otimes_{A_*} L_* \longrightarrow M_* \otimes_{A_*} L_*
$$
are homotopic for any cosimplicial $A_*$-module $L_*$.
\item
\label{item-wedge}
If $\varphi_*$ and $\psi_*$ are homotopic, then
$$
\wedge^i(\varphi_*), \wedge^i(\psi_*) :
\wedge^i(K_*) \longrightarrow \wedge^i(M_*)
$$
are homotopic.
\item
\label{item-base-change}
If $\varphi_*$ and $\psi_*$ are homotopic, and $A_* \to B_*$
is a homomorphism of cosimplicial rings, then
$$
\varphi_* \otimes 1, \psi_* \otimes 1 :
K_* \otimes_{A_*} B_* \longrightarrow M_* \otimes_{A_*} B_*
$$
are homotopic as homomorphisms of cosimplicial $B_*$-modules.
\item
\label{item-completion}
If $I_* \subset A_*$ is a cosimplicial ideal, then the induced
maps
$$
\varphi^\wedge_*, \psi^\wedge_* :
K_*^\wedge \longrightarrow M_*^\wedge
$$
between completions are homotopic.
\item Add more here as needed, for example symmetric powers.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $h : M_* \longrightarrow \Hom(\Delta[1], N_*)$ be the given
homotopy. In degree $n$ we have
$$
h_n = (h_{n, \alpha}) :
K_n \longrightarrow
\prod\nolimits_{\alpha \in \Delta[1]_n} K_n
$$
see Simplicial, Section \ref{simplicial-section-homotopy-cosimplicial}.
In order for a collection of $h_{n, \alpha}$ to form a homotopy,
it is necessary and sufficient if for every $f : [n] \to [m]$ we
have
$$
h_{m, \alpha} \circ M_*(f) = N_*(f) \circ h_{n, \alpha \circ f}
$$
see
Simplicial, Equation (\ref{simplicial-equation-property-homotopy-cosimplicial}).
We also should have that $\psi_n = h_{n, 0 : [n] \to [1]}$ and
$\varphi_n = h_{n, 1 : [n] \to [1]}$.
\medskip\noindent
In each of the cases of the lemma we can produce the corresponding maps.
Case (\ref{item-tensor}). We can use the homotopy $h \otimes 1$ defined
in degree $n$ by setting
$$
(h \otimes 1)_{n, \alpha} = h_{n, \alpha} \otimes 1_{L_n} :
K_n \otimes_{A_n} L_n
\longrightarrow
M_n \otimes_{A_n} L_n.
$$
Case (\ref{item-wedge}). We can use the homotopy $\wedge^ih$ defined
in degree $n$ by setting
$$
\wedge^i(h)_{n, \alpha} = \wedge^i(h_{n, \alpha}) :
\wedge_{A_n}(K_n)
\longrightarrow
\wedge^i_{A_n}(M_n).
$$
Case (\ref{item-base-change}). We can use the homotopy $h \otimes 1$ defined
in degree $n$ by setting
$$
(h \otimes 1)_{n, \alpha} = h_{n, \alpha} \otimes 1 :
K_n \otimes_{A_n} B_n
\longrightarrow
M_n \otimes_{A_n} B_n.
$$
Case (\ref{item-completion}). We can use the homotopy $h^\wedge$ defined
in degree $n$ by setting
$$
(h^\wedge)_{n, \alpha} = h_{n, \alpha}^\wedge :
K_n^\wedge
\longrightarrow
M_n^\wedge.
$$
This works because each $h_{n, \alpha}$ is $A_n$-linear.
\end{proof}
\section{Crystals in quasi-coherent modules}
\label{section-quasi-coherent-crystals}
\noindent
In Situation \ref{situation-affine}.
Set $X = \Spec(C)$ and $S = \Spec(A)$. We are going to
classify crystals in quasi-coherent modules on $\text{Cris}(X/S)$.
Before we do so we fix some notation.
\medskip\noindent
Choose a polynomial ring $P = A[x_i]$ over $A$ and a surjection $P \to C$
of $A$-algebras with kernel $J = \Ker(P \to C)$. Set
\begin{equation}
\label{equation-D}
D = \lim_e D_{P, \gamma}(J) / p^eD_{P, \gamma}(J)
\end{equation}
for the $p$-adically completed divided power envelope.
This ring comes with a divided power ideal $\bar J$ and divided power
structure $\bar \gamma$, see Lemma \ref{lemma-list-properties}.
Set $D_e = D/p^eD$ and denote $\bar J_e$ the image of $\bar J$ in $D_e$.
We will use the short hand
\begin{equation}
\label{equation-omega-D}
\Omega_D = \lim_e \Omega_{D_e/A, \bar\gamma} =
\lim_e \Omega_{D/A, \bar\gamma}/p^e\Omega_{D/A, \bar\gamma}
\end{equation}
for the $p$-adic completion of the module of divided power differentials,
see Lemma \ref{lemma-differentials-completion}.
It is also the $p$-adic completion of
$\Omega_{D_{P, \gamma}(J)/A, \bar\gamma}$
which is free on $\text{d}x_i$, see
Lemma \ref{lemma-module-differentials-divided-power-envelope}.
Hence any element of $\Omega_D$ can be written uniquely as a sum
$\sum f_i\text{d}x_i$ with for all $e$ only finitely many $f_i$
not in $p^eD$. Moreover, the maps
$\text{d}_{D_e/A, \bar\gamma} : D_e \to \Omega_{D_e/A, \bar\gamma}$
fit together to define a divided power $A$-derivation
\begin{equation}
\label{equation-derivation-D}
\text{d} : D \longrightarrow \Omega_D
\end{equation}
on $p$-adic completions.
\medskip\noindent
We will also need the ``products $\Spec(D(n))$ of $\Spec(D)$'', see
Proposition \ref{proposition-compute-cohomology} and its proof for an
explanation. Formally these are defined as follows. For $n \geq 0$ let
$J(n) = \Ker(P \otimes_A \ldots \otimes_A P \to C)$ where
the tensor product has $n + 1$ factors. We set
\begin{equation}
\label{equation-Dn}
D(n) = \lim_e
D_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n))/
p^eD_{P \otimes_A \ldots \otimes_A P, \gamma}(J(n))
\end{equation}
equal to the $p$-adic completion of the divided power envelope.
We denote $\bar J(n)$ its divided power ideal and $\bar \gamma(n)$
its divided powers. We also introduce $D(n)_e = D(n)/p^eD(n)$ as well
as the $p$-adically completed module of differentials
\begin{equation}
\label{equation-omega-Dn}
\Omega_{D(n)} = \lim_e \Omega_{D(n)_e/A, \bar\gamma} =
\lim_e \Omega_{D(n)/A, \bar\gamma}/p^e\Omega_{D(n)/A, \bar\gamma}
\end{equation}
and derivation
\begin{equation}
\label{equation-derivation-Dn}
\text{d} : D(n) \longrightarrow \Omega_{D(n)}
\end{equation}
Of course we have $D = D(0)$. Note that the rings $D(0), D(1), D(2), \ldots$
form a cosimplicial object in the category of divided power rings.
\begin{lemma}
\label{lemma-structure-Dn}
Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}).
The coprojection $P \to P \otimes_A \ldots \otimes_A P$,
$f \mapsto f \otimes 1 \otimes \ldots \otimes 1$
induces an isomorphism
\begin{equation}
\label{equation-structure-Dn}
D(n) = \lim_e D\langle \xi_i(j) \rangle/p^eD\langle \xi_i(j) \rangle
\end{equation}
of algebras over $D$ with
$$
\xi_i(j) = x_i \otimes 1 \otimes \ldots \otimes 1 -
1 \otimes \ldots \otimes 1 \otimes x_i \otimes 1 \otimes \ldots \otimes 1
$$
for $j = 1, \ldots, n$.
\end{lemma}
\begin{proof}
We have
$$
P \otimes_A \ldots \otimes_A P = P[\xi_i(j)]
$$
and $J(n)$ is generated by $J$ and the elements $\xi_i(j)$.
Hence the lemma follows from
Lemma \ref{lemma-divided-power-envelope-add-variables}.
\end{proof}
\begin{lemma}
\label{lemma-property-Dn}
Let $D$ and $D(n)$ be as in (\ref{equation-D}) and (\ref{equation-Dn}).
Then $(D, \bar J, \bar\gamma)$ and $(D(n), \bar J(n), \bar\gamma(n))$
are objects of $\text{Cris}^\wedge(C/A)$, see
Remark \ref{remark-completed-affine-site}, and
$$
D(n) = \coprod\nolimits_{j = 0, \ldots, n} D
$$
in $\text{Cris}^\wedge(C/A)$.
\end{lemma}
\begin{proof}
The first assertion is clear. For the second, if $(B \to C, \delta)$ is an
object of $\text{Cris}^\wedge(C/A)$, then we have
$$
\Mor_{\text{Cris}^\wedge(C/A)}(D, B) =
\Hom_A((P, J), (B, \Ker(B \to C)))
$$
and similarly for $D(n)$ replacing $(P, J)$ by
$(P \otimes_A \ldots \otimes_A P, J(n))$. The property on coproducts follows
as $P \otimes_A \ldots \otimes_A P$ is a coproduct.
\end{proof}
\noindent
In the lemma below we will consider pairs $(M, \nabla)$ satisfying the
following conditions
\begin{enumerate}
\item
\label{item-complete}
$M$ is a $p$-adically complete $D$-module,
\item
\label{item-connection}
$\nabla : M \to M \otimes^\wedge_D \Omega_D$ is a connection, i.e.,
$\nabla(fm) = m \otimes \text{d}f + f\nabla(m)$,
\item
\label{item-integrable}
$\nabla$ is integrable
(see Remark \ref{remark-connection}), and
\item
\label{item-topologically-quasi-nilpotent}
$\nabla$ is {\it topologically quasi-nilpotent}: If we write
$\nabla(m) = \sum \theta_i(m)\text{d}x_i$ for some operators
$\theta_i : M \to M$, then for any $m \in M$ there are only finitely
many pairs $(i, k)$ such that $\theta_i^k(m) \not \in pM$.
\end{enumerate}
The operators $\theta_i$ are sometimes denoted
$\nabla_{\partial/\partial x_i}$ in the literature.
In the following lemma we construct a functor from crystals in quasi-coherent
modules on $\text{Cris}(X/S)$ to the category of such pairs. We will show
this functor is an equivalence in
Proposition \ref{proposition-crystals-on-affine}.
\begin{lemma}
\label{lemma-crystals-on-affine}
In the situation above there is a functor
$$
\begin{matrix}
\text{crystals in quasi-coherent} \\
\mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S)
\end{matrix}
\longrightarrow
\begin{matrix}
\text{pairs }(M, \nabla)\text{ satisfying} \\
\text{(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})}
\end{matrix}
$$
\end{lemma}
\begin{proof}
Let $\mathcal{F}$ be a crystal in quasi-coherent modules on $X/S$.
Set $T_e = \Spec(D_e)$ so that $(X, T_e, \bar\gamma)$ is an object
of $\text{Cris}(X/S)$ for $e \gg 0$. We have morphisms
$$
(X, T_e, \bar\gamma) \to (X, T_{e + 1}, \bar\gamma) \to \ldots
$$
which are closed immersions. We set
$$
M =
\lim_e \Gamma((X, T_e, \bar\gamma), \mathcal{F}) =
\lim_e \Gamma(T_e, \mathcal{F}_{T_e}) = \lim_e M_e
$$
Note that since $\mathcal{F}$ is locally quasi-coherent we have
$\mathcal{F}_{T_e} = \widetilde{M_e}$. Since $\mathcal{F}$ is a
crystal we have $M_e = M_{e + 1}/p^eM_{e + 1}$. Hence we see that
$M_e = M/p^eM$ and that $M$ is $p$-adically complete.
\medskip\noindent
By Lemma \ref{lemma-automatic-connection} we know that $\mathcal{F}$
comes endowed with a canonical integrable connection
$\nabla : \mathcal{F} \to \mathcal{F} \otimes \Omega_{X/S}$.
If we evaluate this connection on the objects $T_e$ constructed above
we obtain a canonical integrable connection
$$
\nabla : M \longrightarrow M \otimes^\wedge_D \Omega_D
$$
To see that this is topologically nilpotent we work out what this means.
\medskip\noindent
Now we can do the same procedure for the rings $D(n)$.
This produces a $p$-adically complete $D(n)$-module $M(n)$. Again using
the crystal property of $\mathcal{F}$ we obtain isomorphisms
$$
M \otimes^\wedge_{D, p_0} D(1) \rightarrow M(1)
\leftarrow M \otimes^\wedge_{D, p_1} D(1)
$$
compare with the proof of Lemma \ref{lemma-automatic-connection}.
Denote $c$ the composition from left to right. Pick $m \in M$.
Write $\xi_i = x_i \otimes 1 - 1 \otimes x_i$.
Using (\ref{equation-structure-Dn}) we can write uniquely
$$
c(m \otimes 1) = \sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]}
$$
for some $\theta_K(m) \in M$ where the sum is over multi-indices
$K = (k_i)$ with $k_i \geq 0$ and $\sum k_i < \infty$. Set
$\theta_i = \theta_K$ where $K$ has a $1$ in the $i$th spot and
zeros elsewhere. We have
$$
\nabla(m) = \sum \theta_i(m) \text{d}x_i.
$$
as can be seen by comparing with the definition of
$\nabla$. Namely, the defining equation is
$p_1^*m = \nabla(m) - c(p_0^*m)$ in Lemma \ref{lemma-automatic-connection}
but the sign works out because in the Stacks project we consistently use
$\text{d}f = p_1(f) - p_0(f)$ modulo the ideal of the diagonal squared,
and hence $\xi_i = x_i \otimes 1 - 1 \otimes x_i$ maps to $-\text{d}x_i$
modulo the ideal of the diagonal squared.
\medskip\noindent
Denote $q_i : D \to D(2)$ and $q_{ij} : D(1) \to D(2)$ the coprojections
corresponding to the indices $i, j$. As in the last paragraph of the proof of
Lemma \ref{lemma-automatic-connection}
we see that
$$
q_{02}^*c = q_{12}^*c \circ q_{01}^*c.
$$
This means that
$$
\sum\nolimits_{K''} \theta_{K''}(m) \otimes \prod {\zeta''_i}^{[k''_i]}
=
\sum\nolimits_{K', K} \theta_{K'}(\theta_K(m))
\otimes \prod {\zeta'_i}^{[k'_i]} \prod \zeta_i^{[k_i]}
$$
in $M \otimes^\wedge_{D, q_2} D(2)$ where
\begin{align*}
\zeta_i & = x_i \otimes 1 \otimes 1 - 1 \otimes x_i \otimes 1,\\
\zeta'_i & = 1 \otimes x_i \otimes 1 - 1 \otimes 1 \otimes x_i,\\
\zeta''_i & = x_i \otimes 1 \otimes 1 - 1 \otimes 1 \otimes x_i.
\end{align*}
In particular $\zeta''_i = \zeta_i + \zeta'_i$ and we have that
$D(2)$ is the $p$-adic completion of the divided power polynomial
ring in $\zeta_i, \zeta'_i$ over $q_2(D)$, see Lemma \ref{lemma-structure-Dn}.
Comparing coefficients in the expression above it follows immediately that
$\theta_i \circ \theta_j = \theta_j \circ \theta_i$
(this provides an alternative proof of the integrability of $\nabla$) and that
$$
\theta_K(m) = (\prod \theta_i^{k_i})(m).
$$
In particular, as the sum expressing $c(m \otimes 1)$ above has to converge
$p$-adically we conclude that for each $i$ and each $m \in M$ only a finite
number of $\theta_i^k(m)$ are allowed to be nonzero modulo $p$.
\end{proof}
\begin{proposition}
\label{proposition-crystals-on-affine}
The functor
$$
\begin{matrix}
\text{crystals in quasi-coherent} \\
\mathcal{O}_{X/S}\text{-modules on }\text{Cris}(X/S)
\end{matrix}
\longrightarrow
\begin{matrix}
\text{pairs }(M, \nabla)\text{ satisfying} \\
\text{(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})}
\end{matrix}
$$
of Lemma \ref{lemma-crystals-on-affine}
is an equivalence of categories.
\end{proposition}
\begin{proof}
Let $(M, \nabla)$ be given. We are going to construct
a crystal in quasi-coherent modules $\mathcal{F}$.
Write $\nabla(m) = \sum \theta_i(m)\text{d}x_i$.
Then $\theta_i \circ \theta_j = \theta_j \circ \theta_i$ and we
can set $\theta_K(m) = (\prod \theta_i^{k_i})(m)$ for any multi-index
$K = (k_i)$ with $k_i \geq 0$ and $\sum k_i < \infty$.
\medskip\noindent
Let $(U, T, \delta)$ be any object of $\text{Cris}(X/S)$ with $T$ affine.
Say $T = \Spec(B)$ and the ideal of $U \to T$ is $J_B \subset B$.
By Lemma \ref{lemma-set-generators} there exists an integer $e$ and a morphism
$$
f : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)
$$
where $T_e = \Spec(D_e)$ as in the proof of
Lemma \ref{lemma-crystals-on-affine}.
Choose such an $e$ and $f$; denote $f : D \to B$ also the corresponding
divided power $A$-algebra map. We will set $\mathcal{F}_T$ equal to the
quasi-coherent sheaf of $\mathcal{O}_T$-modules associated to the $B$-module
$$
M \otimes_{D, f} B.
$$
However, we have to show that this is independent of the choice of $f$.
Suppose that $g : D \to B$ is a second such morphism. Since $f$ and $g$
are morphisms in $\text{Cris}(X/S)$ we see that the image of
$f - g : D \to B$ is contained in the divided power ideal $J_B$.
Write $\xi_i = f(x_i) - g(x_i) \in J_B$. By analogy with the proof
of Lemma \ref{lemma-crystals-on-affine} we define an isomorphism
$$
c_{f, g} : M \otimes_{D, f} B \longrightarrow M \otimes_{D, g} B
$$
by the formula
$$
m \otimes 1 \longmapsto
\sum\nolimits_K \theta_K(m) \otimes \prod \xi_i^{[k_i]}
$$
which makes sense by our remarks above and the fact that $\nabla$
is topologically quasi-nilpotent (so the sum is finite!).
A computation shows that
$$
c_{g, h} \circ c_{f, g} = c_{f, h}
$$
if given a third morphism
$h : (U, T, \delta) \longrightarrow (X, T_e, \bar\gamma)$.
It is also true that $c_{f, f} = 1$.
Hence these maps are all isomorphisms and we see that
the module $\mathcal{F}_T$ is independent of the choice of $f$.
\medskip\noindent
If $a : (U', T', \delta') \to (U, T, \delta)$ is a morphism of affine objects
of $\text{Cris}(X/S)$, then choosing $f' = f \circ a$ it is clear
that there exists a canonical isomorphism
$a^*\mathcal{F}_T \to \mathcal{F}_{T'}$. We omit the verification that this
map is independent of the choice of $f$. Using these maps as the restriction
maps it is clear that we obtain a crystal in quasi-coherent modules
on the full subcategory of $\text{Cris}(X/S)$ consisting of affine objects.
We omit the proof that this extends to a crystal on all of
$\text{Cris}(X/S)$. We also omit the proof that this procedure is a functor
and that it is quasi-inverse to the functor constructed in
Lemma \ref{lemma-crystals-on-affine}.
\end{proof}
\begin{lemma}
\label{lemma-crystals-on-affine-smooth}
In Situation \ref{situation-affine}.
Let $A \to P' \to C$ be ring maps with $A \to P'$ smooth and $P' \to C$
surjective with kernel $J'$. Let $D'$ be the $p$-adic completion of
$D_{P', \gamma}(J')$. There are homomorphisms of divided power $A$-algebras
$$
a : D \longrightarrow D',\quad b : D' \longrightarrow D
$$
compatible with the maps $D \to C$ and $D' \to C$ such that
$a \circ b = \text{id}_{D'}$. These maps induce
an equivalence of categories of pairs $(M, \nabla)$ satisfying
(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})
over $D$ and pairs $(M', \nabla')$ satisfying
(\ref{item-complete}), (\ref{item-connection}),
(\ref{item-integrable}), and (\ref{item-topologically-quasi-nilpotent})
over $D'$. In particular, the equivalence of categories of
Proposition \ref{proposition-crystals-on-affine}
also holds for the corresponding functor towards pairs over $D'$.
\end{lemma}
\begin{proof}
We can pick the map $P = A[x_i] \to C$ such that it factors through
a surjection of $A$-algebras $P \to P'$ (we may have to increase the
number of variables in $P$ to do this). Hence we obtain a surjective
map $a : D \to D'$ by functoriality of divided power envelopes and
completion. Pick $e$ large enough so that $D_e$ is a divided power
thickening of $C$ over $A$. Then $D_e \to C$ is a surjection whose kernel
is locally nilpotent, see Divided Power Algebra, Lemma \ref{dpa-lemma-nil}.
Setting $D'_e = D'/p^eD'$
we see that the kernel of $D_e \to D'_e$ is locally nilpotent.
Hence by Algebra, Lemma \ref{algebra-lemma-smooth-strong-lift}
we can find a lift $\beta_e : P' \to D_e$ of the map $P' \to D'_e$.
Note that $D_{e + i + 1} \to D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1}$
is surjective with square zero kernel for any $i \geq 0$ because
$p^{e + i}D \to p^{e + i}D'$ is surjective. Applying the usual lifting
property (Algebra, Proposition \ref{algebra-proposition-smooth-formally-smooth})
successively to the diagrams
$$
\xymatrix{
P' \ar[r] & D_{e + i} \times_{D'_{e + i}} D'_{e + i + 1} \\