Fix an example in crystalline

Thanks to Marco d'Addezio for pointing this out

Author: aisejohan

crystalline.tex

@@ -4228,13 +4228,19 @@ \section{Two counter examples}
 $$
 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 $H^1_{\text{cris}}(\mathbf{A}_{\mathbf{F}_p}^1/\Spec(\mathbf{Z}_p))$.
+$\text{d}$ which is divisible by $p^t$. Hence the cohomology
+class of $\eta$ in $H^1(\text{Cris}(X/S), \mathcal{O}_{X/S})$
+is contained in $\bigcap p^tH^1(\text{Cris}(X/S), \mathcal{O}_{X/S})$.
+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.
+Hence the cohomology class of $\eta$ is nonzero and we see that
+(4) is true. (In fact the cohomology class of $\eta$ is nontorsion.)
+On a positive note, the cohomology groups
+$H^i(\text{Cris}(X/S), \mathcal{O}_{X/S})$
+are derived complete $\mathbf{Z}_p$-modules as cohomology of
+a complex of $p$-adically complete modules, see
+More on Algebra, Section \ref{more-algebra-section-derived-completion}.
 \end{example}