The Stacks project

Remark 60.8.6 (Comparison with Zariski site). In Situation 60.7.5. The functor ( 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

I understand why is cocontinuous, but why is it continuous?

For example, suppose and . Take with the trivial PD structure, and with the standard PD structure on , so is the closed point. Then should send the cover to , which doesn't seem like a cover to me.

Did I miss something?

Comment #3500 by on

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

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  • 2 comment(s) on Section 60.8: The big crystalline site

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