16
$\begingroup$

Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.

Can we reconstruct $X$ from its small crystalline topos $((X/W(k))_{\mathrm{cris}}, \mathcal{O}_{X/W(k)})$ considered with the structure morphism to $((\mathrm{Spec}\:k/W(k))_{\mathrm{cris}}, \mathcal{O}_{\mathrm{Spec}\:k/W(k)})$?

Can we at least find the Hodge numbers?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

Edit: This answer is probably wrong, sorry. The issue is indicated [in bold] below.

Yes, we can reconstruct $X$ (as a $k$-scheme).

It would be kind of trivial if you had asked for the small Zariski topos instead of the small crystalline topos since the small Zariski topos $\mathrm{Sh}(X)$ is just the space $X$ viewed as a topos (generalized space). So the question is how to reconstruct the small Zariski topos (including the structure sheaf) from the small crystalline topos.

Claim. Let $X$ be a scheme over an arbitrary base scheme $S$. Then $\mathrm{Sh}(X)$ (or rather $X$ itself, as a locale) is the localic reflection of $(X / S)_{\mathrm{cris}}$.

Proof. The opens of the localic reflection are given by the subterminal objects of the topos in question, i.e. by the subsheaves $\mathcal{F}$ of the terminal sheaf on the small crystalline site of $X$ over $S$. Whenever $\mathcal{F}(U \hookrightarrow T)$ is inhabited for an $S$-PD-thickening $T$ of an open subscheme $U$ of $X$, then also $\mathcal{F}(U \rightarrow U) =: \mathcal{F}(U)$ is inhabited, since there is a morphism from $(U \rightarrow U)$ to $(U \hookrightarrow T)$. [But this morphism does not cover $(U \hookrightarrow T)$ and there is in general no morphism in the opposite direction. So $\mathcal{F}(U)$ inhabited probably doesn't imply $\mathcal{F}(U \hookrightarrow T)$ inhabited.] Also, for a cover $U_i$ of $U$, the sheaf condition for $\mathcal{F}$ says that $\mathcal{F}(U)$ is inhabited if the $\mathcal{F}(U_i)$ are. In summary, a subsheaf of the terminal sheaf is precisely given by an open of $X$. (There is an isomorphism of frames.) $\blacksquare$

Since $\mathcal{O}_{X/S}(U \rightarrow U) = \mathcal{O}_X(U)$ we also have the structure sheaf.

Note that we don't even need the structure morphism you specified, only the one to $\mathrm{Sh}(\mathrm{Spec}\:k)$.

Improve this answer
$\endgroup$
6
  • 1
    $\begingroup$ why doesn't this argument apply to the étale topos? $\endgroup$
    – user164740
    Commented Oct 1, 2020 at 20:00
  • 2
    $\begingroup$ @JoeT If I'm not mistaken the localic reflection of the petit étale topos is the petit Zariski topos, so it does work. But the localic reflection of the gros Zariski topos is something else entirely. (Too many points!) $\endgroup$
    – Zhen Lin
    Commented Oct 1, 2020 at 22:28
  • $\begingroup$ @JoeT Thinking about your comment made me realize that my above argument is probably wrong since F(U) inhabited might not imply F(U --> T) inhabited. Seems like I answered too hastily. :-/ $\endgroup$
    – Matthias Hutzler
    Commented Oct 1, 2020 at 22:34
  • $\begingroup$ @ZhenLin The answer to this other question implies that it doesn't work in the ètale case. $\endgroup$
    – Matthias Hutzler
    Commented Oct 1, 2020 at 22:36
  • $\begingroup$ @MatthiasHutzler The question there is different, by my understanding. It asks for a reconstruction of $(X, O_X)$, or at least some cohomological invariants, from the petit étale topos as an abstract topos (so, without the structure sheaf). $\endgroup$
    – Zhen Lin
    Commented Oct 1, 2020 at 22:46

You must log in to answer this question.