Remark 60.8.6 (07ID): Comparison with Zariski site

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 60: Crystalline Cohomology
Section 60.8: The big crystalline site
Remark 60.8.6: Comparison with Zariski site (cite)

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Remark 60.8.6 (Comparison with Zariski site). In Situation 60.7.5. The functor (60.8.1.1) is cocontinuous (details omitted) and commutes with products and fibred products (Lemma 60.8.2). Hence we obtain a morphism of topoi

\[ U_{X/S} : (X/S)_{\text{CRIS}} \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{Zar}) \]

from the big crystalline topos of $X/S$ to the big Zariski topos of $X$. See Sites, Section 7.21.

Comments (2)

Comment #3457 by ZY on August 03, 2018 at 12:37

I understand why $U_{X/S}$ is cocontinuous, but why is it continuous?

For example, suppose $S = \text{Spec} (A, I, \gamma) =\text{Spec } (\mathbb Z/p^2, 0, 0)$ and $X = \text{Spec } A$ . Take $U_1 = T_1 = \text{Spec } \mathbb Z/p^2$ with the trivial PD structure, and $T_2 = (\text{Spec } \mathbb Z/p^2, p, \delta)$ with the standard PD structure on $p$ , so $U_2 \hookrightarrow T_2$ is the closed point. Then $U_{X/S}$ should send the cover $\{ T_2 \rightarrow T_1 \}$ to $\{U_2 \rightarrow U_1\}$ , which doesn't seem like a cover to me.

Did I miss something?

Comment #3500 by Johan on August 05, 2018 at 13:21

Yes, this is a mistake. Thanks very much for pointing this out. I have fixed this here.

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