Let $X$ be a scheme or fs log scheme over a finite field. There seem to be several slightly different definitions of the (log) crystalline site of $X/S$ available in the literature, depending on whether you take the objects of $\mathrm{Cris}(X/S)$ to be:
- ($X$ a scheme) diagrams $X \leftarrow U\to T$ where $U\to T$ is a PD thickening over $S$ and $U\to X$ is a Zariski-open immersion [1];
- ($X$ a scheme) diagrams $X \leftarrow U\to T$ where $U\to T$ is a PD thickening over $S$ and $U\to X$ is an etale map;
- ($X$ a log scheme) diagrams $X \leftarrow U\to T$ where $U\to T$ is a log PD thickening over $S$ and $U\to X$ is an etale map on the level of underlying schemes (endowing $U$ with the pulled back log structure) [2];
- ($X$ a log scheme) diagrams $X \leftarrow U\to T$ where $U\to T$ is a log PD thickening over $S$ and $U\to X$ is a Kummer etale map [3].
In each case there is a corresponding Grothendieck topology, induced from the Zariski/etale/etale/Kummer etale topology.
My question is whether these differences in definitions matter when it comes to defining crystals or isocrystals (of coherent $\mathcal O_{X/S}$-modules) on these sites. That is, are the categories of crystals or isocrystals on sites 1. and 2. (respectively 3. and 4.) equivalent? A reference would be appreciated, particularly for the equivalence of 3. and 4. (which is the case I actually care about).
[1] P. Berthelot and A. Ogus: Notes on Crystalline Cohomology
[2] K. Kato: Logarithmic structures of Fontaine–Illusie.
[3] F. Andreatta and A. Iovita: Semistable Sheaves and Comparison Isomorphisms in the Semistable Case.
Best Answer
It seems that the categories of log-crystals in the strict etale and Kummer etale topologies are not actually equivalent to one another, contrary to what I expected.
Here's a sketch of a proof. Firstly, we'll restrict attention to the case that $X=S$ is log-smooth, so that the category of log-crystals on $X/S$ in the strict/Kummer etale topology is the same as the category of coherent sheaves in the strict/Kummer etale topology. So it suffices to find an example of an $X$ such that these categories $Coh(X_{et})$ and $Coh(X_{ket})$ of coherent sheaves are not equivalent.
For this, suppose that $2$ is invertible on $X$ and that $t\in\Gamma(X,M_X)$ is an element of the monoid-sheaf on $X$. We define $X_2=X\times_{Spec(\mathbb Z[t])}Spec(\mathbb Z[t^{1/2}])$ (pullback taken in the category of fs log schemes). The projection $\pi\colon X_2\to X$ is Kummer etale, and there is an involution of $X_2$ over $X$ given by $t^{1/2}\mapsto-t^{1/2}$.
Claim: $Coh(X_{ket})$ is equivalent via $\pi^*$ to the category of coherent sheaves $E$ on $X_{2,ket}$ together with an isomorphism $\phi\colon\iota^*E \xrightarrow\sim E$ satisfying $\phi\circ\iota^*\phi = 1_E$.
Proof of claim: The key point is that $X_2\times_XX_2=X_2\times\mu_2$, where the fibre product is taken in fs log schemes, and similarly for the triple fibre product. This then implies that specifying gluing data for a sheaf $E$ on $X_2$ for the map $X_2\to X$ is equivalent to specifying an equivariant $\mu_2$-action on $E$, i.e. an isomorphism $\phi$ as above.
The claim allows one to give plenty of examples of coherent sheaves on $X_{ket}$ which do not arise from $X_{et}$. For instance, suppose that $X=Spec(\mathbb F_p[t])$ is the affine line over $\mathbb F_p$ with divisorial log structure associated to the point $\{0\}$. Let $E_{2,Zar}$ be the coherent sheaf on $X_{2,Zar}$ associated to the $\mathbb F_p[t^{1/2}]$-module $M_2 = t^{1/2}\mathbb F_p[t^{1/2}]$, and let $E_2$ be the pullback of $E_{2,Zar}$ to $X_{2,ket}$. The semilinear automorphism $\psi$ of $M_2$ given by $t^{n/2}\mapsto (-1)^nt^{n/2}$ gives rise to an isomorphism $\phi\colon\iota^* E_2\xrightarrow\sim E_2$ as in the claim, and hence $E_2$ descends to a coherent sheaf $E$ on $X_{ket}$. But $E$ does not arise from a coherent sheaf on $X_{et}$. For, if it did, it would be the sheaf associated to a finitely generated $\mathbb F_p[t]$-module $M$, so we would have $M_2=\mathbb F_p[t^{1/2}]\otimes_{\mathbb F_p[t]}M$ with $\psi$ being the isomorphism $\iota^*\otimes1_M$. But this is not the case e.g. since $M_2$ is not generated as a $\mathbb F_p[t^{1/2}]$-module by its $\psi$-invariant subspace.