## Tag `07MU`

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

Remark 54.24.9 (Base change isomorphism). The map (54.24.8.1) is an isomorphism provided all of the following conditions are satisfied:

- $p$ is nilpotent in $A'$,
- $\mathcal{F}'$ is a crystal in quasi-coherent $\mathcal{O}_{X'/S'}$-modules,
- $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,
- $X = X' \times_{S'_0} S_0$,
- $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,
- $X' \to S'_0$ is a local complete intersection morphism (see More on Morphisms, Definition 36.51.2; this holds for example if $X' \to S'_0$ is syntomic or smooth),
- $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition 15.57.1; this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat).
Hints: Condition (1) means that in the arguments below $p$-adic completion does nothing and can be ignored. Using condition (3) and Mayer Vietoris (see Remark 54.24.2) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \mathop{\rm Spec}(C')$ and we are given a presentation $C' = A'/I'[x_1, \ldots, x_n]/(\bar f'_1, \ldots, \bar f'_c)$ where $\bar f'_1, \ldots, \bar f'_c$ is a Koszul-regular sequence in $A'/I'$. (This means that smooth locally $\bar f'_1, \ldots, \bar f'_c$ forms a regular sequence, see More on Algebra, Lemma 15.27.17.) We choose a lift of $\bar f'_i$ to an element $f'_i \in A'[x_1, \ldots, x_n]$. By (4) we see that $X = \mathop{\rm Spec}(C)$ with $C = A/I[x_1, \ldots, x_n]/(\bar f_1, \ldots, \bar f_c)$ where $f_i \in A[x_1, \ldots, x_n]$ is the image of $f'_i$. By property (7) we see that $\bar f_1, \ldots, \bar f_c$ is a Koszul-regular sequence in $A/I[x_1, \ldots, x_n]$. The divided power envelope of $I'A'[x_1, \ldots, x_n] + (f'_1, \ldots, f'_c)$ in $A'[x_1, \ldots, x_n]$ relative to $\gamma'$ is $$ D' = A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c \rangle/(\xi_i - f'_i) $$ see Lemma 54.2.4. Then you check that $\xi_1 - f'_1, \ldots, \xi_n - f'_n$ is a Koszul-regular sequence in the ring $A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. Similarly the divided power envelope of $IA[x_1, \ldots, x_n] + (f_1, \ldots, f_c)$ in $A[x_1, \ldots, x_n]$ relative to $\gamma$ is $$ D = A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle/(\xi_i - f_i) $$ and $\xi_1 - f_1, \ldots, \xi_n - f_n$ is a Koszul-regular sequence in the ring $A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. It follows that $D' \otimes_{A'}^\mathbf{L} A = D$. Condition (2) implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla)$ consisting of a $D'$-module with connection, see Proposition 54.17.4. Then $M = M' \otimes_{D'} D$ corresponds to the pullback $\mathcal{F}$. By assumption (5) we see that $M'$ is a flat $D'$-module, hence $$ M = M' \otimes_{D'} D = M' \otimes_{D'} D' \otimes_{A'}^\mathbf{L} A = M' \otimes_{A'}^\mathbf{L} A $$ Since the modules of differentials $\Omega_{D'}$ and $\Omega_D$ (as defined in Section 54.17) are free $D'$-modules on the same generators we see that $$ M \otimes_D \Omega^\bullet_D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{D'} D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{A'}^\mathbf{L} A $$ which proves what we want by Proposition 54.21.3.

The code snippet corresponding to this tag is a part of the file `crystalline.tex` and is located in lines 4740–4811 (see updates for more information).

```
\begin{remark}[Base change isomorphism]
\label{remark-base-change-isomorphism}
The map (\ref{equation-base-change-map}) is an isomorphism provided
all of the following conditions are satisfied:
\begin{enumerate}
\item $p$ is nilpotent in $A'$,
\item $\mathcal{F}'$ is a crystal in quasi-coherent
$\mathcal{O}_{X'/S'}$-modules,
\item $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,
\item $X = X' \times_{S'_0} S_0$,
\item $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,
\item $X' \to S'_0$ is a local complete intersection morphism (see
More on Morphisms, Definition \ref{more-morphisms-definition-lci}; this
holds for example if $X' \to S'_0$ is syntomic or smooth),
\item $X'$ and $S_0$ are Tor independent over $S'_0$ (see
More on Algebra, Definition \ref{more-algebra-definition-tor-independent};
this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat).
\end{enumerate}
Hints: Condition (1) means that in the arguments below $p$-adic completion
does nothing and can be ignored.
Using condition (3) and Mayer Vietoris (see
Remark \ref{remark-mayer-vietoris}) this reduces to the case
where $X'$ is affine. In fact by condition (6), after shrinking
further, we can assume that $X' = \Spec(C')$ and we are given a presentation
$C' = A'/I'[x_1, \ldots, x_n]/(\bar f'_1, \ldots, \bar f'_c)$
where $\bar f'_1, \ldots, \bar f'_c$ is a Koszul-regular sequence in $A'/I'$.
(This means that smooth locally $\bar f'_1, \ldots, \bar f'_c$ forms
a regular sequence, see More on Algebra,
Lemma \ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}.)
We choose a lift of
$\bar f'_i$ to an element $f'_i \in A'[x_1, \ldots, x_n]$. By (4) we see that
$X = \Spec(C)$ with $C = A/I[x_1, \ldots, x_n]/(\bar f_1, \ldots, \bar f_c)$
where $f_i \in A[x_1, \ldots, x_n]$ is the image of $f'_i$.
By property (7) we see that $\bar f_1, \ldots, \bar f_c$ is a Koszul-regular
sequence in $A/I[x_1, \ldots, x_n]$. The divided power envelope of
$I'A'[x_1, \ldots, x_n] + (f'_1, \ldots, f'_c)$ in $A'[x_1, \ldots, x_n]$
relative to $\gamma'$ is
$$
D' = A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c \rangle/(\xi_i - f'_i)
$$
see Lemma \ref{lemma-describe-divided-power-envelope}. Then you check that
$\xi_1 - f'_1, \ldots, \xi_n - f'_n$ is a Koszul-regular sequence in the
ring $A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$.
Similarly the divided power envelope of
$IA[x_1, \ldots, x_n] + (f_1, \ldots, f_c)$ in $A[x_1, \ldots, x_n]$
relative to $\gamma$ is
$$
D = A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle/(\xi_i - f_i)
$$
and $\xi_1 - f_1, \ldots, \xi_n - f_n$ is a Koszul-regular sequence in the
ring $A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$.
It follows that $D' \otimes_{A'}^\mathbf{L} A = D$. Condition (2)
implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla)$
consisting of a $D'$-module with connection, see
Proposition \ref{proposition-crystals-on-affine}.
Then $M = M' \otimes_{D'} D$ corresponds to the pullback $\mathcal{F}$.
By assumption (5) we see that $M'$ is a flat $D'$-module, hence
$$
M = M' \otimes_{D'} D = M' \otimes_{D'} D' \otimes_{A'}^\mathbf{L} A
= M' \otimes_{A'}^\mathbf{L} A
$$
Since the modules of differentials $\Omega_{D'}$ and $\Omega_D$
(as defined in Section \ref{section-quasi-coherent-crystals})
are free $D'$-modules on the same generators we see that
$$
M \otimes_D \Omega^\bullet_D =
M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{D'} D =
M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{A'}^\mathbf{L} A
$$
which proves what we want by
Proposition \ref{proposition-compute-cohomology-crystal}.
\end{remark}
```

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