The Stacks Project


Tag 07LG

Chapter 51: Crystalline Cohomology > Section 51.21: Cohomology in the affine case

Proposition 51.21.3. Assumptions as in Proposition 51.21.1 but now assume that $\mathcal{F}$ is a crystal in quasi-coherent modules. Let $(M, \nabla)$ be the corresponding module with connection over $D$, see Proposition 51.17.4. Then the complex $$ M \otimes^\wedge_D \Omega^*_D $$ computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

Proof. We will prove this using the two spectral sequences associated to the double complex $K^{*, *}$ with terms $$ K^{a, b} = M \otimes_D^\wedge \Omega^a_{D(b)} $$ What do we know so far? Well, Lemma 51.19.3 tells us that each column $K^{a, *}$, $a > 0$ is acyclic. Proposition 51.21.1 tells us that the first column $K^{0, *}$ is quasi-isomorphic to $R\Gamma(\text{Cris}(X/S), \mathcal{F})$. Hence the first spectral sequence associated to the double complex shows that there is a canonical quasi-isomorphism of $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ with $\text{Tot}(K^{*, *})$.

Next, let's consider the rows $K^{*, b}$. By Lemma 51.17.1 each of the $b + 1$ maps $D \to D(b)$ presents $D(b)$ as the $p$-adic completion of a divided power polynomial algebra over $D$. Hence Lemma 51.20.2 shows that the map $$ M \otimes^\wedge_D\Omega^*_D \longrightarrow M \otimes^\wedge_{D(b)} \Omega^*_{D(b)} = K^{*, b} $$ is a quasi-isomorphism. Note that each of these maps defines the same map on cohomology (and even the same map in the derived category) as the inverse is given by the co-diagonal map $D(b) \to D$ (corresponding to the multiplication map $P \otimes_A \ldots \otimes_A P \to P$). Hence if we look at the $E_1$ page of the second spectral sequence we obtain $$ E_1^{a, b} = H^a(M \otimes^\wedge_D\Omega^*_D) $$ with differentials $$ E_1^{a, 0} \xrightarrow{0} E_1^{a, 1} \xrightarrow{1} E_1^{a, 2} \xrightarrow{0} E_1^{a, 3} \xrightarrow{1} \ldots $$ as each of these is the alternation sum of the given identifications $H^a(M \otimes^\wedge_D\Omega^*_D) = E_1^{a, 0} = E_1^{a, 1} = \ldots$. Thus we see that the $E_2$ page is equal $H^a(M \otimes^\wedge_D\Omega^*_D)$ on the first row and zero elsewhere. It follows that the identification of $M \otimes^\wedge_D\Omega^*_D$ with the first row induces a quasi-isomorphism of $M \otimes^\wedge_D\Omega^*_D$ with $\text{Tot}(K^{*, *})$. $\square$

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 3964–3974 (see updates for more information).

    \begin{proposition}
    \label{proposition-compute-cohomology-crystal}
    Assumptions as in Proposition \ref{proposition-compute-cohomology}
    but now assume that $\mathcal{F}$ is a crystal in quasi-coherent modules.
    Let $(M, \nabla)$ be the corresponding module with connection over $D$, see
    Proposition \ref{proposition-crystals-on-affine}. Then the complex
    $$
    M \otimes^\wedge_D \Omega^*_D
    $$
    computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
    \end{proposition}
    
    \begin{proof}
    We will prove this using the two spectral sequences associated to the
    double complex $K^{*, *}$ with terms
    $$
    K^{a, b} = M \otimes_D^\wedge \Omega^a_{D(b)}
    $$
    What do we know so far? Well, Lemma \ref{lemma-vanishing}
    tells us that each column $K^{a, *}$, $a > 0$ is acyclic.
    Proposition \ref{proposition-compute-cohomology} tells us that
    the first column $K^{0, *}$ is quasi-isomorphic to
    $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.
    Hence the first spectral sequence associated to the double complex
    shows that there is a canonical quasi-isomorphism of
    $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ with
    $\text{Tot}(K^{*, *})$.
    
    \medskip\noindent
    Next, let's consider the rows $K^{*, b}$. By
    Lemma \ref{lemma-structure-Dn}
    each of the $b + 1$ maps $D \to D(b)$ presents $D(b)$ as the $p$-adic
    completion of a divided power polynomial algebra over $D$.
    Hence Lemma \ref{lemma-relative-poincare} shows that the map
    $$
    M \otimes^\wedge_D\Omega^*_D
    \longrightarrow
    M \otimes^\wedge_{D(b)} \Omega^*_{D(b)} = K^{*, b}
    $$
    is a quasi-isomorphism. Note that each of these maps defines the {\it same}
    map on cohomology (and even the same map in the derived category) as
    the inverse is given by the co-diagonal map $D(b) \to D$ (corresponding
    to the multiplication map $P \otimes_A \ldots \otimes_A P \to P$).
    Hence if we look at the $E_1$ page of the second spectral sequence
    we obtain
    $$
    E_1^{a, b} = H^a(M \otimes^\wedge_D\Omega^*_D)
    $$
    with differentials
    $$
    E_1^{a, 0} \xrightarrow{0}
    E_1^{a, 1} \xrightarrow{1}
    E_1^{a, 2} \xrightarrow{0}
    E_1^{a, 3} \xrightarrow{1} \ldots
    $$
    as each of these is the alternation sum of the given identifications
    $H^a(M \otimes^\wedge_D\Omega^*_D) = E_1^{a, 0} = E_1^{a, 1} = \ldots$.
    Thus we see that the $E_2$ page is equal $H^a(M \otimes^\wedge_D\Omega^*_D)$
    on the first row and zero elsewhere. It follows that the identification
    of $M \otimes^\wedge_D\Omega^*_D$ with the first row induces a
    quasi-isomorphism of $M \otimes^\wedge_D\Omega^*_D$ with
    $\text{Tot}(K^{*, *})$.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 07LG

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?