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Tag 07MN

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

Remark 51.24.4 (Alternating Čech complex). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a separated quasi-compact scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

Choose a finite affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$ and a total ordering on $\Lambda$. Write $U_\lambda = \mathop{\rm Spec}(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct an alternating Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

We are going to use the notation introduced in Remark 51.24.3. Denote $\Omega_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed module of differentials of $D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power structure. Let $\nabla$ be the integrable connection on $M_{\lambda_0 \ldots \lambda_n}$ coming from Proposition 51.17.4. Consider the double complex $M^{\bullet, \bullet}$ with terms $$ M^{n, m} = \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n} M_{\lambda_0 \ldots \lambda_n} \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}} \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}. $$ For the differential $d_1$ (increasing $n$) we use the usual Čech differential and for the differential $d_2$ we use the connection, i.e., the differential of the de Rham complex. We claim that $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet}) $$ Here $\text{Tot}(-)$ denotes the total complex associated to a double complex, see Homology, Definition 12.22.3.

Hints: We have $$ R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet) $$ by Proposition 51.23.1. The right hand side of the formula is simply the alternating Čech complex for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ (which induces an open covering of the final sheaf of $\text{Cris}(X/S)$) and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$, see Proposition 51.21.3. Now the result follows from a general result in cohomology on sites, namely that the alternating Čech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here).

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    \begin{remark}[Alternating {\v C}ech complex]
    \label{remark-alternating-cech-complex}
    Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
    ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and
    $S_0 = \Spec(A/I)$. Let $X$ be a separated quasi-compact scheme
    over $S_0$ such that $p$ is locally nilpotent on $X$. Let
    $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.
    
    \medskip\noindent
    Choose a finite affine open covering
    $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$
    and a total ordering on $\Lambda$.
    Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra
    $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$.
    Having fixed these choices we can construct an alternating
    {\v C}ech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
    
    \medskip\noindent
    We are going to use the notation introduced in
    Remark \ref{remark-cech-complex}.
    Denote $\Omega_{\lambda_0 \ldots \lambda_n}$
    the $p$-adically completed module of differentials of
    $D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power
    structure. Let $\nabla$ be the integrable connection on
    $M_{\lambda_0 \ldots \lambda_n}$ coming from
    Proposition \ref{proposition-crystals-on-affine}.
    Consider the double complex $M^{\bullet, \bullet}$ with
    terms
    $$
    M^{n, m} =
    \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n}
    M_{\lambda_0 \ldots \lambda_n}
    \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}}
    \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}.
    $$
    For the differential $d_1$ (increasing $n$) we use the usual
    {\v C}ech differential and for the differential $d_2$ we use
    the connection, i.e., the differential of the de Rham complex.
    We claim that
    $$
    R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet})
    $$
    Here $\text{Tot}(-)$ denotes the total complex associated to a
    double complex, see
    Homology, Definition \ref{homology-definition-associated-simple-complex}.
    
    \medskip\noindent
    Hints: We have
    $$
    R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S),
    \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet)
    $$
    by Proposition \ref{proposition-compare-with-de-Rham}.
    The right hand side of the formula is simply the alternating {\v C}ech complex
    for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$
    (which induces an open covering of the final sheaf of $\text{Cris}(X/S)$)
    and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$,
    see Proposition \ref{proposition-compute-cohomology-crystal}.
    Now the result follows from a general result in cohomology on sites,
    namely that the alternating {\v C}ech complex computes the cohomology
    provided it gives the correct answer on all the pieces (insert future
    reference here).
    \end{remark}

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