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Chapter 51: Crystalline Cohomology > Section 51.24: Some further results

Remark 51.24.9 (Base change isomorphism). The map (51.24.8.1) is an isomorphism provided all of the following conditions are satisfied:

  1. $p$ is nilpotent in $A'$,
  2. $\mathcal{F}'$ is a crystal in quasi-coherent $\mathcal{O}_{X'/S'}$-modules,
  3. $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,
  4. $X = X' \times_{S'_0} S_0$,
  5. $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,
  6. $X' \to S'_0$ 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),
  7. $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition 15.56.1; 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 51.24.2) 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 51.2.4. 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 51.17.4. 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 51.17) 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 51.21.3.

    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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