Proposition 60.21.3. Assumptions as in Proposition 60.21.1 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 60.17.4. 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 60.19.3 tells us that each column $K^{a, *}$, $a > 0$ is acyclic. Proposition 60.21.1 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 60.17.1 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 60.20.2 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 $\text{Tot}(K^{*, *})$.
$\square$

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