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Example 60.22.2. Let $A = \mathbf{Z}_ p$ with divided power ideal $(p)$ endowed with its unique divided powers $\gamma $. Let $C = \mathbf{F}_ p[x_1, \ldots , x_ r]$. We choose the presentation

\[ C = P/J = P/pP\quad \text{with}\quad P = \mathbf{Z}_ p[x_1, \ldots , x_ r] \]

Note that $pP$ has divided powers by Divided Power Algebra, Lemma 23.4.2. Hence setting $D = P^\wedge $ with divided power ideal $(p)$ we obtain a situation as in Section 60.17. We conclude that $R\Gamma (\text{Cris}(X/S), \mathcal{O}_{X/S})$ is represented by the complex

\[ D \to \Omega ^1_ D \to \Omega ^2_ D \to \ldots \to \Omega ^ r_ D \]

see Proposition 60.21.3. Assuming $r > 0$ we conclude the following

  1. The cristalline cohomology of the cristalline structure sheaf of $X = \mathbf{A}^ r_{\mathbf{F}_ p}$ over $S = \mathop{\mathrm{Spec}}(\mathbf{Z}_ p)$ is zero except in degrees $0, \ldots , r$.

  2. We have $H^0(\text{Cris}(X/S), \mathcal{O}_{X/S}) = \mathbf{Z}_ p$.

  3. The cohomology group $H^ r(\text{Cris}(X/S), \mathcal{O}_{X/S})$ is infinite and is not a torsion abelian group.

  4. The cohomology group $H^ r(\text{Cris}(X/S), \mathcal{O}_{X/S})$ is not separated for the $p$-adic topology.

While the first two statements are reasonable, parts (3) and (4) are disconcerting! The truth of these statements follows immediately from working out what the complex displayed above looks like. Let's just do this in case $r = 1$. Then we are just looking at the two term complex of $p$-adically complete modules

\[ \text{d} : D = \left( \bigoplus \nolimits _{n \geq 0} \mathbf{Z}_ p x^ n \right)^\wedge \longrightarrow \Omega ^1_ D = \left( \bigoplus \nolimits _{n \geq 1} \mathbf{Z}_ p x^{n - 1}\text{d}x \right)^\wedge \]

The map is given by $\text{diag}(0, 1, 2, 3, 4, \ldots )$ except that the first summand is missing on the right hand side. Now it is clear that $\bigoplus _{n > 0} \mathbf{Z}_ p/n\mathbf{Z}_ p$ is a subgroup of the cokernel, hence the cokernel is infinite. In fact, the element

\[ \omega = \sum \nolimits _{e > 0} p^ e x^{p^{2e} - 1}\text{d}x \]

is clearly not a torsion element of the cokernel. But it gets worse. Namely, consider the element

\[ \eta = \sum \nolimits _{e > 0} p^ e x^{p^ e - 1}\text{d}x \]

For every $t > 0$ the element $\eta $ is congruent to $\sum _{e > t} p^ e x^{p^ e - 1}\text{d}x$ modulo the image of $\text{d}$ which is divisible by $p^ t$. But $\eta $ is not in the image of $\text{d}$ because it would have to be the image of $a + \sum _{e > 0} x^{p^ e}$ for some $a \in \mathbf{Z}_ p$ which is not an element of the left hand side. In fact, $p^ N\eta $ is similarly not in the image of $\text{d}$ for any integer $N$. This implies that $\eta $ “generates” a copy of $\mathbf{Q}_ p$ inside of $H^1_{\text{cris}}(\mathbf{A}_{\mathbf{F}_ p}^1/\mathop{\mathrm{Spec}}(\mathbf{Z}_ p))$.

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