2.1

In this article, $p$ denotes a prime number, $k$ denotes a perfect field of characteristic $p$ , $\operatorname{W}$ the ring of Witt vectors of $k$ and $K$ the fraction field of $\operatorname{W}$ .

Let $\mathfrak{X}$ be an adic formal $\operatorname{W}$ -scheme. For any $n\geqslant 1$ , we denote by $\mathfrak{X}_{n}$ the reduction modulo $p^{n}$ of $\mathfrak{X}$ . If we use a gothic letter $\mathfrak{X}$ to denote an adic formal $\operatorname{W}$ -scheme, the corresponding roman letter $X$ will denote its special fiber $\mathfrak{X}_{1}$ .

We denote by $\mathbf{S}$ the category whose objects are adic formal $\operatorname{W}$ -schemes of finite type [Reference AbbesAbb10, 2.3.13] and morphisms are adic morphisms [Reference AbbesAbb10, 2.2.7]. By [Reference Grothendieck and DieudonnéEGAI, 6.1.5(v)], morphisms of $\mathbf{S}$ are of finite type. We denote by $\mathbf{S}^{\diamond }$ the full subcategory of $\mathbf{S}$ consisting of flat formal $\operatorname{W}$ -schemes of finite type.

2.2

Let $\mathscr{A}$ be an abelian category. We denote by $\mathscr{A}_{\mathbb{Q}}$ the category with the same objects as $\mathscr{A}$ such that the set of morphisms is given for any object $M,N$ of $\mathscr{A}$ by

$$\begin{eqnarray}\operatorname{Hom}_{\mathscr{A}_{\mathbb{Q}}}(M,N)=\operatorname{Hom}_{\mathscr{A}}(M,N)\,\otimes _{\mathbb{Z}}\,\mathbb{Q}.\end{eqnarray}$$

For any object $M$ of $\mathscr{A}$ , we denote its image in $\mathscr{A}_{\mathbb{Q}}$ by $M_{\mathbb{Q}}$ .

2.3

Let $\mathfrak{X}$ be an object of $\mathbf{S}$ . For any $\mathscr{O}_{\mathfrak{X}}$ -module $\mathscr{F}$ , we set $\mathscr{F}[\frac{1}{p}]=\mathscr{F}\,\otimes _{\mathbb{Z}_{p}}\,\mathbb{Q}_{p}$ .

We denote by $\mathbf{Coh}(\mathscr{O}_{\mathfrak{X}})$ (respectively $\mathbf{Coh}(\mathscr{O}_{\mathfrak{X}}[\frac{1}{p}])$ ) the category of coherent $\mathscr{O}_{\mathfrak{X}}$ -modules (respectively $\mathscr{O}_{\mathfrak{X}}[\frac{1}{p}]$ -modules). The canonical functor $\mathbf{Coh}(\mathscr{O}_{\mathfrak{X}})\rightarrow \mathbf{Coh}(\mathscr{O}_{\mathfrak{X}}[\frac{1}{p}])$ defined by $\mathscr{F}\mapsto \mathscr{F}[\frac{1}{p}]$ induces an equivalence of categories $\mathbf{Coh}(\mathscr{O}_{\mathfrak{X}})_{\mathbb{Q}}\xrightarrow[{}]{{\sim}}\mathbf{Coh}(\mathscr{O}_{\mathfrak{X}}[\frac{1}{p}])$  [Reference Abbes, Gros and TsujiAGT16, III.6.16].

2.4

Let $\mathscr{C}$ and $\mathscr{D}$ be two sites, $\widehat{\mathscr{C}}$ (respectively $\widehat{\mathscr{D}}$ ) the category of presheaves of sets on $\mathscr{C}$ (respectively $\mathscr{D}$ ) and $u:\mathscr{C}\rightarrow \mathscr{D}$ a functor. We have a functor

$$\begin{eqnarray}\widehat{u}^{\ast }:\widehat{\mathscr{D}}\rightarrow \widehat{\mathscr{C}},\quad \mathscr{G}\mapsto \widehat{u}^{\ast }(\mathscr{G})=\mathscr{G}\circ u.\end{eqnarray}$$

It admits a right adjoint $\widehat{u}_{\ast }:\widehat{\mathscr{C}}\rightarrow \widehat{\mathscr{D}}$  [Reference Artin, Grothendieck and VerdierSGA4, I 5.1].

If $u:\mathscr{C}\rightarrow \mathscr{D}$ is a cocontinuous (respectively continuous) functor and $\mathscr{F}$ (respectively $\mathscr{G}$ ) is a sheaf on $\mathscr{C}$ (respectively $\mathscr{D}$ ), then $\widehat{u}_{\ast }(\mathscr{F})$ (respectively $\widehat{u}^{\ast }(\mathscr{G})$ ) is a sheaf on $\mathscr{D}$ (respectively  $\mathscr{C}$ ) [Reference Artin, Grothendieck and VerdierSGA4, III 1.2, 2.1 and 2.2].

Let $\widetilde{\mathscr{C}}$ (respectively $\widetilde{\mathscr{D}}$ ) be the topos of the sheaves of sets on $\mathscr{C}$ (respectively $\mathscr{D}$ ) and $u:\mathscr{C}\rightarrow \mathscr{D}$ a cocontinuous functor. Then $u$ induces a morphism of topoi $g:\widetilde{\mathscr{C}}\rightarrow \widetilde{\mathscr{D}}$ defined by $g_{\ast }=\widehat{u}_{\ast }$ and $g^{\ast }=a\circ \widehat{u}^{\ast }$ , where $a$ is the sheafification functor (cf. [Reference Artin, Grothendieck and VerdierSGA4, III 2.3]).

2.6

In the following, $\mathfrak{S}$ denotes an object of $\mathbf{S}^{\diamond }$ and $X$ an $S$ -scheme.

We equip $p\mathscr{O}_{\mathfrak{S}}$ with the canonical PD-structure $\unicode[STIX]{x1D6FE}$ . Recall that the crystalline site $\operatorname{Crys}(X/\mathfrak{S})$ is defined as follows [Reference Berthelot and OgusBO15, 7.17]: an object is a quadruple $(U,T,\unicode[STIX]{x1D704},\unicode[STIX]{x1D6FF})$ consisting of an open subscheme $U$ of $X$ , a scheme $T$ over $\mathfrak{S}_{n}$ for some integer $n\geqslant 1$ , a closed immersion $\unicode[STIX]{x1D704}:U\rightarrow T$ and a PD-structure $\unicode[STIX]{x1D6FF}$ on $\operatorname{Ker}(\mathscr{O}_{T}\rightarrow \mathscr{O}_{U})$ compatible with $\unicode[STIX]{x1D6FE}$ . A morphism from $(U^{\prime },T^{\prime },\unicode[STIX]{x1D704}^{\prime },\unicode[STIX]{x1D6FF}^{\prime })$ to $(U,T,\unicode[STIX]{x1D704},\unicode[STIX]{x1D6FF})$ of $\operatorname{Crys}(X/\mathfrak{S})$ consists of an open immersion $U^{\prime }\rightarrow U$ and an $\mathfrak{S}$ -morphism $T^{\prime }\rightarrow T$ compatible with $\unicode[STIX]{x1D704}^{\prime },\unicode[STIX]{x1D704}$ and the PD-structures. A family of morphisms $\{(U_{i},T_{i})\rightarrow (U,T)\}_{i\in I}$ is a covering if each morphism $T_{i}\rightarrow T$ is an open immersion and $|T|=\bigcup _{i\in I}|T_{i}|$ . We denote by $(X/\mathfrak{S})_{\operatorname{crys}}$ the topos of sheaves of sets on $\operatorname{Crys}(X/\mathfrak{S})$ .

The presheaf of rings defined by $(U,T)\mapsto \unicode[STIX]{x1D6E4}(T,\mathscr{O}_{T})$ is a sheaf that we denote by $\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}}$ . For an $\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}}$ -module $\mathscr{F}$ and an object $(U,T)$ of $\operatorname{Crys}(X/\mathfrak{S})$ , we denote by $\mathscr{F}_{T}$ the evaluation of $\mathscr{F}$ at $(U,T)$  [Reference Berthelot and OgusBO15, 5.1].

2.8

Crystals have an equivalent description in terms of modules equipped with hyper-PD-stratification and of modules with integrable connection. Let us briefly recall these notions.

Let $\mathfrak{X}$ be an adic formal $\mathfrak{S}$ -scheme of finite type and $\mathfrak{X}^{2}=\mathfrak{X}\times _{\mathfrak{S}}\mathfrak{X}$ . Let $\operatorname{G}$ be an adic formal $\mathfrak{X}^{2}$ -scheme and let $q_{1},q_{2}:\operatorname{G}\rightarrow \mathfrak{X}$ be the canonical projections. A formal $\mathfrak{X}$ -groupoid structure over $\mathfrak{S}$ on $\operatorname{G}$ are three adic morphisms $\unicode[STIX]{x1D6FC}:\operatorname{G}\times _{\mathfrak{X}}\operatorname{G}\rightarrow \operatorname{G},\unicode[STIX]{x1D704}:\mathfrak{X}\rightarrow \operatorname{G}$ and $\unicode[STIX]{x1D702}:\operatorname{G}\rightarrow \operatorname{G}$ , where the fibered product $\operatorname{G}\times _{\mathfrak{X}}\operatorname{G}$ is taken on the left (respectively right) for the $\mathfrak{X}$ -structure defined by $q_{2}$ (respectively $q_{1}$ ), satisfying the compatibility conditions for groupoids (cf. [Reference XuXu19, 4.7]). We set $q_{13}=\unicode[STIX]{x1D6FC}$ and $q_{12},q_{23}:\operatorname{G}\times _{\mathfrak{X}}\operatorname{G}\rightarrow \operatorname{G}$ , the projections in the first and second components, respectively.

Let $\operatorname{G}$ be a formal $\mathfrak{X}$ -groupoid over $\mathfrak{S}$ and $M$ an $\mathscr{O}_{\mathfrak{X}}$ -module. An $\mathscr{O}_{\operatorname{G}}$ -stratification on $M$ is an $\mathscr{O}_{\operatorname{G}}$ -linear isomorphism $\unicode[STIX]{x1D700}:q_{2}^{\ast }(M)\xrightarrow[{}]{{\sim}}q_{1}^{\ast }(M)$ satisfying $\unicode[STIX]{x1D704}^{\ast }(\unicode[STIX]{x1D700})=\operatorname{id}_{M}$ and the cocycle condition $q_{12}^{\ast }(\unicode[STIX]{x1D700})\circ q_{23}^{\ast }(\unicode[STIX]{x1D700})=q_{13}^{\ast }(\unicode[STIX]{x1D700})$ .

2.9

Suppose that $X$ is smooth over $S$ and admits a smooth lifting $\mathfrak{X}$ over $\mathfrak{S}$ . We denote by $\text{P}_{\mathfrak{X}/\mathfrak{S}}$ the adic formal $\mathfrak{X}^{2}$ -scheme defined by the PD-envelope of the diagonal immersion $\mathfrak{X}\rightarrow \mathfrak{X}^{2}$ compatible with the canonical PD-structure $\unicode[STIX]{x1D6FE}$ (§ 2.6). By the universal property of the PD-envelope, the formal $\mathfrak{X}^{2}$ -scheme $\text{P}_{\mathfrak{X}/\mathfrak{S}}$ is equipped with a formal $\mathfrak{X}$ -groupoid structure (§ 2.8).

Given an object $\mathscr{F}$ of $\mathscr{C}(\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}})$ (Definition 2.7), the coherent $\mathscr{O}_{\mathfrak{X}}$ -module $\mathscr{F}_{\mathfrak{X}}=\mathop{\varprojlim }\nolimits_{n\geqslant 1}\mathscr{F}_{\mathfrak{X}_{n}}$ is equipped with an $\mathscr{O}_{\text{P}_{\mathfrak{ X}/\mathfrak{S}}}$ -stratification and then an integrable connection relative to $\mathfrak{S}$ . Moreover, the following categories are canonically equivalent (see [Reference Berthelot and OgusBO15, 6.6] and [Sta, 07JH]).

1. (i) The category $\mathscr{C}(\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}})$ .

2. (ii) The category of coherent $\mathscr{O}_{\mathfrak{X}}$ -modules equipped with an $\mathscr{O}_{\text{P}_{\mathfrak{ X}/\mathfrak{S}}}$ -stratification.

3. (iii) The category of coherent $\mathscr{O}_{\mathfrak{X}}$ -modules equipped with a topologically quasi-nilpotent integrable connection relative to $\mathfrak{S}$  [Reference Berthelot and OgusBO15, 6.1].

When $\mathscr{T}$ has enough points and, for every point $x$ of $\mathscr{T}$ , the stalk of $A$ at $x$ is a local ring, the locally projective $A$ -modules of finite type are locally free $A$ -modules of finite type [Reference Berthelot, Grothendieck and IllusieSGA6, I 2.15.1].

2.13

We denote by $u_{X/\mathfrak{S},\operatorname{crys}}:(X/\mathfrak{S})_{\operatorname{crys}}\rightarrow X_{\operatorname{zar}}$ the canonical morphism of topoi [Reference Berthelot and OgusBO15, 5.12] and by $g_{X/\mathfrak{S},\operatorname{crys}}$ the composition

$$\begin{eqnarray}g_{X/\mathfrak{S},\operatorname{crys}}:(X/\mathfrak{S})_{\operatorname{crys}}\rightarrow X_{\operatorname{zar}}\rightarrow \mathfrak{S}_{\operatorname{zar}},\end{eqnarray}$$

which is ringed by $\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}}$ and $\mathscr{O}_{\mathfrak{S}}$ . We call $\operatorname{R}^{\bullet }g_{X/\mathfrak{S},\operatorname{crys}\ast }(-)$ the relative crystalline cohomology.

For an isocrystal $\mathscr{E}=\mathscr{F}_{\mathbb{Q}}$ with $\mathscr{F}\in \mathbf{Ob}(\mathscr{C}(\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}}))$ , we set (§ 2.1)

$$\begin{eqnarray}\displaystyle \operatorname{R}^{q}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\mathscr{E})=\operatorname{R}^{q}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\mathscr{F})\biggl[\frac{1}{p}\biggr],\quad \operatorname{R}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\mathscr{E})=\operatorname{R}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\mathscr{F})\biggl[\frac{1}{p}\biggr]. & & \displaystyle \nonumber\end{eqnarray}$$

It is clear that the above definition is independent of the choice of $\mathscr{F}$ .

When $X$ is smooth and proper over $S$ and $\mathfrak{S}$ is separated, Shiho proved that the relative crystalline cohomology $\operatorname{R}^{q}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\mathscr{E})$ is a coherent $\mathscr{O}_{\mathfrak{S}}[\frac{1}{p}]$ -module [Reference ShihoShi07a, 1.15]. Moreover, he showed a base-change result for relative crystalline cohomology (cf. [Reference ShihoShi07a, 1.16 and 1.19]).

3.2

Let $f:(\mathfrak{T}^{\prime },u^{\prime })\rightarrow (\mathfrak{T},u)$ and $g:(\mathfrak{T}^{\prime \prime },u^{\prime \prime })\rightarrow (\mathfrak{T},u)$ be two morphisms of $\operatorname{Conv}(X/\mathfrak{S})$ . We denote by $\mathfrak{Z}$ the closed formal subscheme of $\mathfrak{T}^{\prime }\times _{\mathfrak{T}}\mathfrak{T}^{\prime \prime }$ defined by the ideal of $p$ -torsion elements of $\mathscr{O}_{\mathfrak{T}^{\prime }\times _{\mathfrak{T}}\mathfrak{T}^{\prime \prime }}$ . The fibered product of $f$ and $g$ in $\operatorname{Conv}(X/\mathfrak{S})$ is represented by $\mathfrak{Z}$ , which is flat over $\operatorname{W}$ , equipped with the composition $Z_{0}\rightarrow T_{0}^{\prime }\times _{T_{0}}T_{0}^{\prime \prime }\rightarrow X$ induced by $u^{\prime }$ and $u^{\prime \prime }$ .

If either $\mathfrak{T}^{\prime \prime }\rightarrow \mathfrak{T}$ or $\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ is flat, then $\mathfrak{Z}$ is equal to $\mathfrak{T}^{\prime }\times _{\mathfrak{T}}\mathfrak{T}^{\prime \prime }$ .

3.3

Let $\mathfrak{T}$ be an object of $\mathbf{S}$ . We denote by $\mathbf{Zar}_{/\mathfrak{T}}$ (respectively $\mathfrak{T}_{\operatorname{zar}}$ ) the Zariski site (respectively topos) of $\mathfrak{T}$ .

We say that a family of morphisms $\{f_{i}:\mathfrak{T}_{i}\rightarrow \mathfrak{T}\}_{i\in I}$ of $\mathbf{S}$ is an fppf covering if each morphism $f_{i}$ is flat and if $|\mathfrak{T}|=\bigcup _{i\in I}f_{i}(|\mathfrak{T}_{i}|)$ . Since $\mathfrak{T}$ is quasi-compact, each fppf covering of $\mathfrak{T}$ admits a finite fppf subcovering of $\mathfrak{T}$ . Note that fppf coverings in $\mathbf{S}$ are stable by base change and by composition.

We denote by $\mathbf{Fft}_{/\mathfrak{T}}$ the full subcategory of $\mathbf{S}_{/\mathfrak{T}}$ consisting of adic flat formal $\mathfrak{T}$ -schemes and by $\mathfrak{T}_{\operatorname{fppf}}$ the topos of sheaves of sets on $\mathbf{Fft}_{/\mathfrak{T}}$ , equipped with the topology generated by fppf coverings.

Given a morphism $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ of $\mathbf{S}$ , the canonical functor $\mathbf{Fft}_{/\mathfrak{T}}\rightarrow \mathbf{Fft}_{/\mathfrak{T}^{\prime }}$ (respectively $\mathbf{Zar}_{/\mathfrak{T}}\rightarrow \mathbf{Zar}_{/\mathfrak{T}^{\prime }}$ ) defined by $\mathfrak{Y}\mapsto \mathfrak{Y}\times _{\mathfrak{T}}\mathfrak{T}^{\prime }$ is continuous and left exact. For $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ , it induces the functorial morphism of topoi $f_{\unicode[STIX]{x1D70F}}:\mathfrak{T}_{\unicode[STIX]{x1D70F}}^{\prime }\rightarrow \mathfrak{T}_{\unicode[STIX]{x1D70F}}$ .

3.4

We say that a family of morphisms $\{(\mathfrak{T}_{i},u_{i})\rightarrow (\mathfrak{T},u)\}_{i\in I}$ of $\operatorname{Conv}(X/\mathfrak{S})$ is a Zariski (respectively fppf) covering if the family of morphisms $\{\mathfrak{T}_{i}\rightarrow \mathfrak{T}\}_{i\in I}$ of $\mathbf{S}$ is a Zariski (respectively fppf) covering. By § 3.3, Zariski (respectively fppf) coverings in $\operatorname{Conv}(X/\mathfrak{S})$ form a pretopology. For $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ , we denote by $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ the topos of sheaves of sets on $\operatorname{Conv}(X/\mathfrak{S})$ , equipped with the $\unicode[STIX]{x1D70F}$ -topology.

3.6

Let $(\mathfrak{T},u)$ be an object of $\operatorname{Conv}(X/\mathfrak{S})$ and $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ . The canonical functor

$$\begin{eqnarray}\mathbf{Fft}_{/\mathfrak{T}}\quad (\text{respectively }\mathbf{Zar}_{/\mathfrak{T}})\rightarrow \operatorname{Conv}(X/\mathfrak{S}),\quad (f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T})\mapsto (\mathfrak{T}^{\prime },u\circ f_{0})\end{eqnarray}$$

is cocontinuous and induces a morphism of topoi (§ 2.4)

(3.6.1) $$\begin{eqnarray}s_{\mathfrak{T}}:\mathfrak{T}_{\unicode[STIX]{x1D70F}}\rightarrow (X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}},\quad \forall \unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}.\end{eqnarray}$$

For any sheaf $\mathscr{F}$ of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , we set $\mathscr{F}_{\mathfrak{T}}=s_{\mathfrak{T}}^{\ast }(\mathscr{F})$ , called the evaluation of $\mathscr{F}$ at $\mathfrak{T}$ . For any morphism $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ of $\operatorname{Conv}(X/\mathfrak{S})$ , we have a canonical morphism (§ 3.3)

(3.6.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FD}_{f}:\mathscr{F}_{\mathfrak{T}}\rightarrow f_{\unicode[STIX]{x1D70F}\ast }(\mathscr{F}_{\mathfrak{T}^{\prime }})\end{eqnarray}$$

and we denote its adjoint by

(3.6.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{f}:f_{\unicode[STIX]{x1D70F}}^{\ast }(\mathscr{F}_{\mathfrak{ T}})\rightarrow \mathscr{F}_{\mathfrak{T}^{\prime }}.\end{eqnarray}$$

It is clear that $\unicode[STIX]{x1D6FE}_{\operatorname{id}}=\operatorname{id}$ . If $f$ is a morphism of $\mathbf{Fft}_{/\mathfrak{T}}$ (respectively $\mathbf{Zar}_{/\mathfrak{T}}$ ), $f_{\unicode[STIX]{x1D70F}}$ is the localization morphism at $\mathfrak{T}^{\prime }$ and then $\unicode[STIX]{x1D6FE}_{f}$ is an isomorphism. If $g:\mathfrak{T}^{\prime \prime }\rightarrow \mathfrak{T}^{\prime }$ is another morphism of $\operatorname{Conv}(X/\mathfrak{S})$ , one verifies that $\unicode[STIX]{x1D6FE}_{g\circ f}=\unicode[STIX]{x1D6FE}_{f}\circ f_{\unicode[STIX]{x1D70F}}^{\ast }(\unicode[STIX]{x1D6FE}_{g})$ .

3.8

Note that the fppf topology on $\operatorname{Conv}(X/\mathfrak{S})$ is finer than the Zariski topology. Equipped with the fppf topology on the source and the Zariski topology on the target, the identical functor $\operatorname{id}:\operatorname{Conv}(X/\mathfrak{S})\rightarrow \operatorname{Conv}(X/\mathfrak{S})$ is cocontinuous and induces a morphism of topoi (§ 2.4)

(3.8.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}:(X/\mathfrak{S})_{\operatorname{conv},\operatorname{fppf}}\rightarrow (X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}.\end{eqnarray}$$

If $\mathscr{F}$ is a sheaf of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{fppf}}$ , then $\unicode[STIX]{x1D6FC}_{\ast }(\mathscr{F})$ is equal to $\mathscr{F}$ as presheaves. If $\mathscr{G}$ is a sheaf of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ , then $\unicode[STIX]{x1D6FC}^{\ast }(\mathscr{G})$ is the sheafification of $\mathscr{G}$ with respect to the fppf topology.

3.9

Let $g:\mathfrak{S}^{\prime }\rightarrow \mathfrak{S}$ be a morphism of $\mathbf{S}^{\diamond }$ (§ 2.1), $X^{\prime }$ an $S^{\prime }$ -scheme and $f:X^{\prime }\rightarrow X$ a morphism compatible with $g$ , i.e. the diagram

(3.9.1)

is commutative. For any object $(\mathfrak{T},u)$ of $\operatorname{Conv}(X^{\prime }/\mathfrak{S}^{\prime })$ , $(\mathfrak{T},f\circ u)$ defines an object of $\operatorname{Conv}(X/\mathfrak{S})$ . We obtain a functor that we denote by

(3.9.2) $$\begin{eqnarray}\unicode[STIX]{x1D711}:\operatorname{Conv}(X^{\prime }/\mathfrak{S}^{\prime })\rightarrow \operatorname{Conv}(X/\mathfrak{S}),\quad (\mathfrak{T},u)\mapsto (\mathfrak{T},f\circ u).\end{eqnarray}$$

It is clear that $\unicode[STIX]{x1D711}$ commutes with the fibered product (§ 3.2).

3.11

By § 2.4 and Lemma 3.10, the functor $\unicode[STIX]{x1D711}$ (3.9.2) induces a morphism of topoi

(3.11.1) $$\begin{eqnarray}\displaystyle f_{\operatorname{conv},\unicode[STIX]{x1D70F}}:(X^{\prime }/\mathfrak{S}^{\prime })_{\operatorname{conv},\unicode[STIX]{x1D70F}}\rightarrow (X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}} & & \displaystyle\end{eqnarray}$$

such that the pullback functor is induced by the composition with $\unicode[STIX]{x1D711}$ . For a sheaf $\mathscr{F}$ of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ and an object $\mathfrak{T}$ of $\operatorname{Conv}(X^{\prime }/\mathfrak{S}^{\prime })$ , we have (Proposition 3.7)

(3.11.2) $$\begin{eqnarray}(f_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\mathscr{F}))_{\mathfrak{ T}}=\mathscr{F}_{\unicode[STIX]{x1D711}(\mathfrak{T})}.\end{eqnarray}$$

For any morphism $h$ of $\operatorname{Conv}(X^{\prime }/\mathfrak{S}^{\prime })$ , the transition morphism of $f_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\mathscr{F})$ associated to $g$ (Proposition 3.7) is equal to the transition morphism of $\mathscr{F}$ associated to $\unicode[STIX]{x1D711}(h)$ .

By considering inverse image functors, one verifies that the following diagram commutes (3.8.1):

(3.11.3)

3.12

Let $\mathfrak{T}$ be an object of $\mathbf{S}$ and $\mathscr{F}$ a coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module. Since $\mathfrak{T}$ is quasi-compact, by Gabber–Bosch–Görtz’s fppf descent for coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -modules [Reference AbbesAbb10, 5.11.11], the presheaf on $\mathbf{Fft}_{/\mathfrak{T}}$

$$\begin{eqnarray}(f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T})\mapsto \unicode[STIX]{x1D6E4}(\mathfrak{T}^{\prime },f_{\operatorname{ zar}}^{\ast }(\mathscr{F}))\end{eqnarray}$$

is a sheaf for the fppf topology. In particular, $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ defines a sheaf of rings of $\mathfrak{T}_{\operatorname{fppf}}$ that we still denote by $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ . We call abusively a coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module of $\mathfrak{T}_{\operatorname{fppf}}$ a sheaf of $\mathfrak{T}_{\operatorname{fppf}}$ associated to a coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module.

3.13

We define a presheaf of rings $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ on $\operatorname{Conv}(X/\mathfrak{S})$ by

(3.13.1) $$\begin{eqnarray}(\mathfrak{T},u)\mapsto \unicode[STIX]{x1D6E4}\biggl(\mathfrak{T},\mathscr{O}_{\mathfrak{T}}\biggl[\frac{1}{p}\biggr]\biggr).\end{eqnarray}$$

By fppf descent (§ 3.12), $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ is a sheaf for the fppf topology. Since the fppf topology is finer than the Zariski topology, it is also a sheaf for the Zariski topology.

For any object $(\mathfrak{T},u)$ of $\operatorname{Conv}(X/\mathfrak{S})$ , we have $(\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}])_{\mathfrak{T}}=\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ . If $\mathscr{F}$ is an $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -module of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , $\mathscr{F}_{\mathfrak{T}}$ is an $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module of $\mathfrak{T}_{\unicode[STIX]{x1D70F}}$ . For any morphism $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ of $\operatorname{Conv}(X/\mathfrak{S})$ , the transition morphism $\unicode[STIX]{x1D6FE}_{f}$ (Proposition 3.7) extends to an $\mathscr{O}_{\mathfrak{T}^{\prime }}[\frac{1}{p}]$ -linear morphism (§ 3.12)

(3.13.2) $$\begin{eqnarray}c_{f}:f_{\unicode[STIX]{x1D70F}}^{\ast }(\mathscr{F}_{\mathfrak{ T}})\rightarrow \mathscr{F}_{\mathfrak{T}^{\prime }}.\end{eqnarray}$$

In view of Proposition 3.7, we deduce the following description for $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules.

With the notation of § 3.11, the morphism $f_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ is ringed by $\mathscr{O}_{X^{\prime }/\mathfrak{S}}[\frac{1}{p}]$ and $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ . The inverse image functor of modules $f_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }$ sends coherent $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules (respectively crystals) to coherent $\mathscr{O}_{X^{\prime }/\mathfrak{S}^{\prime }}[\frac{1}{p}]$ -modules (respectively crystals).

3.16

Let $\mathscr{E}$ be a coherent crystal of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ . By fppf descent (§ 3.12), $\mathscr{E}$ is also a sheaf for the fppf topology. In particular, the direct image and inverse image functors of $\unicode[STIX]{x1D6FC}$ (§ 3.8) induce an equivalence between the category of coherent crystals of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ and the category of coherent crystals of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{fppf}}$ .

Following [Reference OgusOgu84, Reference ShihoShi07a], for $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ , a coherent crystal of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ is called a convergent isocrystal of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ . We denote the full subcategory of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules consisting of these objects by $\operatorname{Iso}^{\dagger }(X/\mathfrak{S})$ .

We say that a convergent isocrystal $\mathscr{E}$ is locally projective if for every object $\mathfrak{T}$ of $\operatorname{Conv}(X/\mathfrak{S})$ , $\mathscr{E}_{\mathfrak{T}}$ is locally projective of finite type (Definition 2.11).

Proposition 3.17 ([Reference OgusOgu07, 0.7.2], [Reference ShihoShi07a, 2.35]).

Suppose that $X$ is smooth over $S$ . There exists a canonical functor (Definition 2.7)

(3.17.1) $$\begin{eqnarray}\unicode[STIX]{x1D704}:\operatorname{Iso}^{\dagger }(X/\mathfrak{S})\rightarrow \mathscr{C}(\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}})_{\mathbb{ Q}}.\end{eqnarray}$$

The construction of $\unicode[STIX]{x1D704}$ , that we will briefly recall in § 3.20, is based on the following construction in formal geometry.

3.18

Let $\mathfrak{Y}$ be an object of $\mathbf{S}^{\diamond }$ and $\mathscr{A}$ an open ideal of finite type of $\mathfrak{Y}$ containing $p$  [Reference AbbesAbb10, 2.1.19]. We denote by $\mathfrak{Y}^{\prime }$ the admissible blow-up of $\mathscr{A}$ in $\mathfrak{Y}$  [Reference AbbesAbb10, 3.1.2]. The ideal $\mathscr{A}\mathscr{O}_{\mathfrak{Y}^{\prime }}$ is invertible [Reference AbbesAbb10, 3.1.4(i)] and $\mathfrak{Y}^{\prime }$ is flat over $\operatorname{W}$  [Reference AbbesAbb10, 3.1.4(ii)]. We denote by $\mathfrak{Y}_{(\mathscr{A}/p)}$ the maximal open formal subscheme of $\mathfrak{Y}^{\prime }$ on which

(3.18.1) $$\begin{eqnarray}(\mathscr{A}\mathscr{O}_{\mathfrak{X}^{\prime }})|\mathfrak{X}_{(\mathscr{A}/p)}=(p\mathscr{O}_{\mathfrak{X}^{\prime }})|\mathfrak{X}_{(\mathscr{A}/p)}\end{eqnarray}$$

and we call it the dilatation of $\mathscr{A}$ with respect to $p$  [Reference AbbesAbb10, 3.2.3.4 and 3.2.7]. Note that $\mathfrak{Y}_{(\mathscr{A}/p)}$ is the complement of $\operatorname{Supp}(\mathscr{A}\mathscr{O}_{\mathfrak{Y}^{\prime }}/p\mathscr{O}_{\mathfrak{Y}^{\prime }})$ in $\mathfrak{Y}^{\prime }$  [Reference Grothendieck and DieudonnéEGAI, 0.5.2.2].

Let $Z$ be a closed subscheme of $Y$ and $\mathscr{I}$ the ideal sheaf associated to the canonical morphism $Z\rightarrow \mathfrak{Y}$ . For any $n\geqslant 1$ , we denote by $\mathfrak{T}_{Z,n}(\mathfrak{Y})$ the dilatation of $\mathscr{I}^{n}+p\mathscr{O}_{\mathfrak{Y}}$ with respect to $p$  [Reference OgusOgu84, 2.5].Footnote ² By (3.18.1), there exists a morphism from the reduced subscheme of $(\mathfrak{T}_{Z,n}(\mathfrak{Y}))_{1}$ to $Z$ which fits into the following diagram:

(3.18.2)

In particular, $\mathfrak{T}_{Z,n}(\mathfrak{Y})$ defines an object of $\operatorname{Conv}(Z/\operatorname{W})$ .

The universal property of dilatation [Reference AbbesAbb10, 3.2.6] can be reinterpreted in the following way.

Proposition 3.19 [Reference XuXu19, 3.5].

Keep the notation and assumptions of § 3.18. Let $\mathfrak{T}$ be an adic flat formal $\operatorname{W}$ -scheme, $T^{(1)}$ the closed subscheme of $T$ defined by the ideal sheaf $\{x\in \mathscr{O}_{T}|x^{p}=0\}$ and $f:\mathfrak{T}\rightarrow \mathfrak{Y}$ an adic morphism. Suppose that there exists a morphism $T\rightarrow Z$ (respectively $T^{(1)}\rightarrow Z$ ) which fits into the following diagram:

Then there exists a unique adic morphism $g:\mathfrak{T}\rightarrow \mathfrak{T}_{Z,1}(\mathfrak{Y})$ (respectively $g:\mathfrak{T}\rightarrow \mathfrak{T}_{Z,p}(\mathfrak{Y})$ ) lifting  $f$ .

3.20

We briefly review the construction of $\unicode[STIX]{x1D704}$ (Proposition 3.17) in the case where $X$ is separated and admits a smooth lifting $\mathfrak{X}$ over $\mathfrak{S}$ .

We set $\text{Q}_{\mathfrak{X}/\mathfrak{S}}=\mathfrak{T}_{X,p}(\mathfrak{X}^{2})$ , the dilatation of the diagonal immersion $X\rightarrow \mathfrak{X}^{2}=\mathfrak{X}\times _{\mathfrak{S}}\mathfrak{X}$ (§ 3.18). Using its universal property (Proposition 3.19), one verifies that $\text{Q}_{\mathfrak{X}/\mathfrak{S}}$ is equipped with a formal $\mathfrak{X}$ -groupoid structure (§ 2.8) (cf. [Reference XuXu19, 4.11]).

Let $\mathscr{E}$ be an object of $\operatorname{Iso}^{\dagger }(X/\mathfrak{S})$ . The canonical morphisms $p_{1},p_{2}:\text{Q}_{\mathfrak{X}/\mathfrak{S}}\rightarrow \mathfrak{X}$ give rise to morphisms of $\operatorname{Conv}(X/\mathfrak{S})$ . Since $\mathscr{E}$ is a crystal, the composition of transition morphisms $\unicode[STIX]{x1D700}=c_{p_{1}}^{-1}\circ c_{p_{2}}$ defines an $\mathscr{O}_{\text{Q}_{\mathfrak{ X}/\mathfrak{S}}}$ -stratification on $\mathscr{E}_{\mathfrak{X}}$ (§ 2.8).

Using 3.19, the canonical morphism $\text{P}_{\mathfrak{X}/\mathfrak{S}}\rightarrow \mathfrak{X}^{2}$ induces a morphism of formal $\mathfrak{X}$ -groupoids $\text{P}_{\mathfrak{X}/\mathfrak{S}}\rightarrow \text{Q}_{\mathfrak{X}/\mathfrak{S}}$ (cf. [Reference XuXu19, 5.12]). By taking the inverse image, we obtain an $\mathscr{O}_{\text{P}_{\mathfrak{ X}/\mathfrak{S}}}$ -stratification on $\mathscr{E}_{\mathfrak{X}}$ and then a crystal of $\mathscr{O}_{X/\mathfrak{S}}^{\operatorname{crys}}$ -modules of finite presentation up to isogeny by § 2.9 and Proposition 2.10. The construction is clearly functorial.

3.21

There exists a canonical morphism of topoi $u_{X/\mathfrak{S}}:(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}\rightarrow X_{\operatorname{zar}}$ (cf. [Reference OgusOgu07, § 4]). Suppose that $\mathfrak{S}$ is separated and that $X$ admits a smooth lifting $f:\mathfrak{X}\rightarrow \mathfrak{S}$ . Let $\mathscr{E}$ be a convergent isocrystal of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ . By § 2.9 and Proposition 2.10, there exists an integrable connection on $\mathscr{E}_{\mathfrak{X}}$ relative to $\mathfrak{S}$ and we denote by $\mathscr{E}_{\mathfrak{X}}\,\otimes _{\mathscr{O}_{\mathfrak{X}}}\,\widehat{\unicode[STIX]{x1D6FA}}_{\mathfrak{X}/\mathfrak{S}}^{\bullet }$ the associated de Rham complex. Then there exists a canonical isomorphism in the derived category $\operatorname{D}(\mathfrak{X}_{\operatorname{zar}},f^{-1}(\mathscr{O}_{\mathfrak{S}}[\frac{1}{p}]))$ [Reference ShihoShi07a, 2.33]

(3.21.1) $$\begin{eqnarray}\operatorname{R}u_{X/\mathfrak{S}\ast }(\mathscr{E})\xrightarrow[{}]{{\sim}}\mathscr{E}_{\mathfrak{X}}\,\otimes _{\mathscr{O}_{\mathfrak{X}}}\,\widehat{\unicode[STIX]{x1D6FA}}_{\mathfrak{X}/\mathfrak{S}}^{\bullet }.\end{eqnarray}$$

Based on the above isomorphism, the finiteness and the base-change property of relative crystalline cohomology (§ 2.13), Shiho showed the following results.

Theorem 3.22 [Reference ShihoShi07a, 2.36].

Assume that $\mathfrak{S}$ is separated and that $X$ is smooth and proper over $S$ . Let $\mathscr{E}$ be a convergent isocrystal of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ and $g_{\operatorname{conv},\operatorname{zar}}:(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}\rightarrow (S/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ the functorial morphism. Then there exists a canonical isomorphism in the derived category of $\mathscr{O}_{\mathfrak{S}}[\frac{1}{p}]$ -modules (§ 2.13 and Proposition 3.17)

(3.22.1) $$\begin{eqnarray}(\operatorname{R}g_{\operatorname{conv},\operatorname{zar}\ast }(\mathscr{E}))_{\mathfrak{S}}\xrightarrow[{}]{{\sim}}\operatorname{R}g_{X/\mathfrak{S},\operatorname{crys}\ast }(\unicode[STIX]{x1D704}(\mathscr{E})).\end{eqnarray}$$

In particular, $(\operatorname{R}^{i}g_{\operatorname{conv},\operatorname{zar}\ast }(\mathscr{E}))_{\mathfrak{S}}$ is coherent for any $i\geqslant 0$ .

Theorem 3.24 [Reference ShihoShi07a, 2.38].

Let $\unicode[STIX]{x1D711}:\mathfrak{S}^{\prime }\rightarrow \mathfrak{S}$ be an adic morphism of adic separated flat formal $\operatorname{W}$ -schemes of finite type, $X^{\prime }=X\times _{\mathfrak{S}}\mathfrak{S}^{\prime }$ and $\unicode[STIX]{x1D711}_{\operatorname{conv},\operatorname{zar}}:(X^{\prime }/\mathfrak{S}^{\prime })_{\operatorname{conv},\operatorname{zar}}\rightarrow (X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ the functorial morphism of convergent topoi (3.11.1). Then, for a locally projective convergent isocrystal $\mathscr{E}$ of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{zar}}$ , we have a canonical isomorphism in the derived category of $\mathscr{O}_{\mathfrak{S}^{\prime }}[\frac{1}{p}]$ -modules:

(3.24.1) $$\begin{eqnarray}\operatorname{L}\unicode[STIX]{x1D711}_{\operatorname{zar}}^{\ast }((\operatorname{R}g_{\operatorname{ conv},\operatorname{zar}\ast }(\mathscr{E}))_{\mathfrak{S}})\xrightarrow[{}]{{\sim}}(\operatorname{R}g_{\operatorname{conv},\operatorname{zar}\ast }^{\prime }(\unicode[STIX]{x1D711}_{\operatorname{ conv},\operatorname{zar}}^{\ast }(\mathscr{E})))_{\mathfrak{ S}^{\prime }}.\end{eqnarray}$$

4.1

In this section, we keep the notation of § 3 and let $g:X\rightarrow Y$ denote a morphism of $S$ -schemes.

Let $\mathfrak{T}$ be an object of $\operatorname{Conv}(Y/\mathfrak{S})$ and $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ . By fppf descent for morphisms of formal $\operatorname{W}$ -schemes [Reference AbbesAbb10, 5.12.1], the presheaf associated to $\mathfrak{T}$ is a sheaf for the fppf (respectively Zariski) topology that we denote by $\widetilde{\mathfrak{T}}$ . We set $X_{T_{0}}=X\times _{Y}T_{0}$ and we denote by

$$\begin{eqnarray}\displaystyle & \displaystyle g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}}:(X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}\rightarrow (T_{0}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D714}_{\mathfrak{T}}:(X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}\rightarrow (X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}} & \displaystyle \nonumber\end{eqnarray}$$

the functorial morphisms of topoi (3.11.1).

Lemma 4.2 [Reference BerthelotBer74, V 3.2.2].

There exists a canonical equivalence of topoi:

(4.2.1) $$\begin{eqnarray}(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}/g_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\widetilde{\mathfrak{T}})}\xrightarrow[{}]{{\sim}}(X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}},\end{eqnarray}$$

which identifies the localization morphism and $\unicode[STIX]{x1D714}_{\mathfrak{T}}$ .

The lemma can be verified in the same way as [Reference BerthelotBer74, V 3.2.2].

Lemma 4.3 [Reference BerthelotBer74, V 3.2.3].

For any $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -module $E$ of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , there exists a canonical isomorphism in $\operatorname{D}^{+}(\mathfrak{T}_{\unicode[STIX]{x1D70F}},\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}])$ :

(4.3.1) $$\begin{eqnarray}(\operatorname{R}g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }(E))_{\mathfrak{T}}\xrightarrow[{}]{{\sim}}(\operatorname{R}g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E)))_{\mathfrak{ T}}.\end{eqnarray}$$

Proof. Let $E$ be an abelian sheaf of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ and $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ a morphism of $\operatorname{Conv}(Y/\mathfrak{S})$ (§ 4.1). The morphism $f$ induces a functorial morphism of topoi $\unicode[STIX]{x1D711}:(X_{T_{0}^{\prime }}/\mathfrak{T}^{\prime })_{\operatorname{conv},\unicode[STIX]{x1D70F}}\rightarrow (X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , which fits into the following commutative diagram:

(4.3.2)

We have $\unicode[STIX]{x1D714}_{\mathfrak{T}^{\prime }}=\unicode[STIX]{x1D714}_{\mathfrak{T}}\circ \unicode[STIX]{x1D711}$ . By Lemma 4.2, $\unicode[STIX]{x1D711}$ (respectively $f_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ ) coincides with the localization morphism on the sheaf $g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}}^{\ast }(\widetilde{\mathfrak{T}}^{\prime })$ (respectively $\widetilde{\mathfrak{T}}^{\prime }$ ). Then $g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E))$ is the sheaf associated to the presheaf on $\operatorname{Conv}(T_{0}/\mathfrak{T})$ :

$$\begin{eqnarray}(f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T})\mapsto \unicode[STIX]{x1D6E4}((X_{T_{0}^{\prime }}/\mathfrak{T}^{\prime })_{\operatorname{ conv},\unicode[STIX]{x1D70F}},\unicode[STIX]{x1D714}_{\mathfrak{T}^{\prime }}^{\ast }(E)).\end{eqnarray}$$

The sheaf $(g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }(E))_{\mathfrak{T}}$ is associated to the presheaf

$$\begin{eqnarray}(f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T})\mapsto \unicode[STIX]{x1D6E4}((X/\mathfrak{S})_{\operatorname{ conv},\unicode[STIX]{x1D70F}/g_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\widetilde{\mathfrak{T}}^{\prime })},E|_{g_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\widetilde{\mathfrak{T}}^{\prime })}).\end{eqnarray}$$

By Lemma 4.2, we deduce a canonical isomorphism of $\mathfrak{T}_{\unicode[STIX]{x1D70F}}$ :

(4.3.3) $$\begin{eqnarray}(g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }(E))_{\mathfrak{T}}\xrightarrow[{}]{{\sim}}(g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E)))_{\mathfrak{ T}}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D714}_{\mathfrak{T}}$ coincides with a localization morphism, if $I^{\bullet }$ is an injective resolution of $E$ , $\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(I^{\bullet })$ is an injective resolution of $\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E)$ . Then the isomorphism (4.3.1) follows from (4.3.3).◻

Remark 4.4. Keep the notation of Lemma 4.3. Let $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ be a morphism of $\operatorname{Conv}(Y/\mathfrak{S})$ . It induces morphisms of topoi (4.3.2). We consider canonical morphisms

(4.4.1) $$\begin{eqnarray}f_{\unicode[STIX]{x1D70F}}^{\ast }(\operatorname{R}^{i}g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E))_{\mathfrak{ T}})\rightarrow (f_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\operatorname{R}^{i}g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E))))_{\mathfrak{ T}^{\prime }}\xrightarrow[{}]{{\sim}}\operatorname{R}^{i}g_{X/\mathfrak{T}^{\prime },\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}^{\prime }}^{\ast }(E))_{\mathfrak{ T}^{\prime }},\end{eqnarray}$$

where the first morphism is the transition morphism of $\operatorname{R}^{i}g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(E))$ associated to $f$ and the second one is an isomorphism because $\unicode[STIX]{x1D714}_{\mathfrak{T}^{\prime }}=\unicode[STIX]{x1D714}_{\mathfrak{T}}\circ \unicode[STIX]{x1D711},\unicode[STIX]{x1D711},f_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ are localization morphisms [Reference Artin, Grothendieck and VerdierSGA4, V 5.1].

In view of the proof of Lemma 4.3, via (4.3.1), the above composition is compatible with the transition morphism of $\operatorname{R}^{i}g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }(E)$ associated to $f$ :

(4.4.2) $$\begin{eqnarray}f_{\unicode[STIX]{x1D70F}}^{\ast }((\operatorname{R}^{i}g_{\operatorname{ conv},\unicode[STIX]{x1D70F}\ast }(E))_{\mathfrak{T}})\rightarrow (\operatorname{R}^{i}g_{\operatorname{ conv},\unicode[STIX]{x1D70F}\ast }(E))_{\mathfrak{T}^{\prime }}.\end{eqnarray}$$

Corollary 4.5 [Reference Berthelot, Breen and MessingBBM82, 1.1.19].

Let $\mathscr{E}$ be a convergent isocrystal of $(X/\mathfrak{S})_{\operatorname{conv},\operatorname{fppf}}$ . We have (§ 3.8)

(4.5.1) $$\begin{eqnarray}\operatorname{R}^{i}\unicode[STIX]{x1D6FC}_{\ast }(\mathscr{E})=0,\quad \forall i\geqslant 1.\end{eqnarray}$$

The assertion can be verified in the same way as [Reference Berthelot, Breen and MessingBBM82, 1.1.19].

Corollary 4.6. Let $\mathfrak{S}^{\prime }\rightarrow \mathfrak{S}$ be a morphism of $\mathbf{S}^{\diamond }$ (§ 2.1), $Y^{\prime }$ an $S^{\prime }$ -scheme and $h:Y^{\prime }\rightarrow Y$ a morphism compatible with $S^{\prime }\rightarrow S$ . We set $X^{\prime }=X\times _{Y}Y^{\prime }$ and we denote by $g^{\prime }:X^{\prime }\rightarrow Y^{\prime }$ and $h^{\prime }:X^{\prime }\rightarrow X$ the canonical morphisms:

Then, for any $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -module $E$ of $(X/\mathfrak{S})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , the base-change morphism

(4.6.1) $$\begin{eqnarray}h_{\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\operatorname{R}g_{\operatorname{ conv},\unicode[STIX]{x1D70F}\ast }(E))\xrightarrow[{}]{{\sim}}\operatorname{R}g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }^{\prime }(h_{\operatorname{ conv},\unicode[STIX]{x1D70F}}^{\prime \ast }(E))\end{eqnarray}$$

is an isomorphism.

4.7

In the remainder of this section, we consider the case where $\mathfrak{S}=\operatorname{Spf}(\operatorname{W})$ and $X,Y$ are schemes over $S=\operatorname{Spec}(k)$ . We denote by $\operatorname{pConv}(X/\operatorname{W})$ the full subcategory of $\operatorname{Conv}(X/\operatorname{W})$ consisting of objects $(\mathfrak{T},u)$ such that $u$ can be lifted to a $k$ -morphism $\widetilde{u}:T\rightarrow X$ . Given an object $\mathfrak{T}$ of $\operatorname{pConv}(X/\operatorname{W})$ and a morphism $f:\mathfrak{T}^{\prime }\rightarrow \mathfrak{T}$ of $\operatorname{Conv}(X/\operatorname{W})$ , then $\mathfrak{T}^{\prime }$ is still an object of $\operatorname{pConv}(X/\operatorname{W})$ . Objects of $\operatorname{pConv}(X/\operatorname{W})$ are closely related to ‘ $p$ -adic enlargements’ in [Reference OgusOgu84].

We end this section by showing the following result.

Proof. We take again the notation of § 3.20 for $\mathfrak{Y}\rightarrow \operatorname{Spf}(\operatorname{W})$ and we set $\mathfrak{Z}=\text{Q}_{\mathfrak{Y}/\operatorname{W}}$ , which we consider as an object of $\operatorname{Conv}(Y/\operatorname{W})$ , and $\mathscr{F}=\operatorname{R}^{i}g_{\operatorname{conv},\operatorname{zar}\ast }(\mathscr{E})$ . By Lemma 4.3, we have canonical isomorphisms

(4.10.1) $$\begin{eqnarray}\mathscr{F}_{\mathfrak{Y}}\xrightarrow[{}]{{\sim}}(\operatorname{R}^{i}g_{X/\mathfrak{Y},\operatorname{zar}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{Y}}^{\ast }(\mathscr{E})))_{\mathfrak{ Y}},\quad \mathscr{F}_{\mathfrak{Z}}\xrightarrow[{}]{{\sim}}(\operatorname{R}^{i}g_{X/\mathfrak{Z},\operatorname{zar}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{Z}}^{\ast }(\mathscr{E})))_{\mathfrak{ Z}}.\end{eqnarray}$$

By Theorem 3.22, $\mathscr{F}_{\mathfrak{Y}}$ is coherent. The projections $p_{1},p_{2}:\mathfrak{Z}\rightarrow \mathfrak{Y}$ define two morphisms of $\operatorname{Conv}(Y/\operatorname{W})$ and induce two morphisms of topoi

$$\begin{eqnarray}\displaystyle (X_{Z_{0}}/\mathfrak{Z})_{\operatorname{conv},\operatorname{zar}}\xrightarrow[{}]{{\sim}}(X\times _{Y,p_{1}}Z/\mathfrak{Z})_{\operatorname{conv},\operatorname{zar}}\rightarrow (X/\mathfrak{Y})_{\operatorname{conv},\operatorname{zar}}, & & \displaystyle \nonumber\\ \displaystyle (X_{Z_{0}}/\mathfrak{Z})_{\operatorname{conv},\operatorname{zar}}\xrightarrow[{}]{{\sim}}(X\times _{Y,p_{2}}Z/\mathfrak{Z})_{\operatorname{conv},\operatorname{zar}}\rightarrow (X/\mathfrak{Y})_{\operatorname{conv},\operatorname{zar}}. & & \displaystyle \nonumber\end{eqnarray}$$

Since $X$ is smooth over $k$ , $\mathscr{E}$ is locally projective by Lemma 4.9. The projections $p_{1},p_{2}$ are rig-flat [Reference AbbesAbb10, 5.4.12]. By Theorem 3.24 and Remark 4.4, $p_{1},p_{2}$ induce isomorphisms

(4.10.2) $$\begin{eqnarray}p_{2}^{\ast }(\mathscr{F}_{\mathfrak{ Y}})\xrightarrow[{c_{p_{2}}}]{{\sim}}\mathscr{F}_{\mathfrak{Z}}\xleftarrow[{c_{p_{1}}}]{{\sim}}p_{1}^{\ast }(\mathscr{F}_{\mathfrak{ Y}}).\end{eqnarray}$$

By a standard argument, the isomorphism $c_{p_{1}}^{-1}\circ c_{p_{2}}$ defines an $\mathscr{O}_{\text{Q}_{\mathfrak{ Y}/\operatorname{W}}}$ -stratification on $\mathscr{F}_{\mathfrak{Y}}$ . Taking pullback by $\text{P}_{\mathfrak{Y}/\operatorname{W}}\rightarrow \text{Q}_{\mathfrak{Y}/\operatorname{W}}$ (Proposition 3.17), we obtain an $\mathscr{O}_{\text{P}_{\mathfrak{ Y}/\operatorname{W}}}$ -stratification on $\mathscr{F}_{\mathfrak{Y}}$ . The second assertion follows from Lemma 2.12.◻

4.11

In the following, we prove Proposition 4.8. By Lemma 4.9, $\mathscr{E}$ is locally projective. We set $\mathscr{F}_{\unicode[STIX]{x1D70F}}^{i}=\operatorname{R}^{i}g_{\operatorname{conv},\unicode[STIX]{x1D70F}\ast }(\mathscr{E})$ and $\mathscr{G}_{\unicode[STIX]{x1D70F}}^{i}=(\operatorname{R}^{i}g_{X/\mathfrak{T},\unicode[STIX]{x1D70F}\ast }(\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(\mathscr{E})))$ (§ 4.1). By Lemma 4.3, we have a canonical isomorphism

(4.11.1) $$\begin{eqnarray}\mathscr{F}_{\unicode[STIX]{x1D70F},\mathfrak{T}}^{i}\xrightarrow[{}]{{\sim}}\mathscr{G}_{\unicode[STIX]{x1D70F},\mathfrak{T}}^{i}.\end{eqnarray}$$

Proof of Proposition 4.8 for Zariski topology. (i) Since $(\mathfrak{T},u)$ is an object of $\operatorname{pConv}(Y/\operatorname{W})$ , we take a lifting $\widetilde{u}:T\rightarrow Y$ of $u$ and we set $X_{T}=X\times _{Y}T$ . Then we have a canonical equivalence $(X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}\xrightarrow[{}]{{\sim}}(X_{T}/\mathfrak{T})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ (Definition 3.1) and the assertion follows from Theorem 3.22.

(ii) The question being local, by Corollary 4.6, we may therefore assume that $Y$ is affine and admits a smooth lifting $\mathfrak{Y}$ over $\operatorname{W}$ . Then $\mathscr{F}_{\operatorname{zar},\mathfrak{Y}}^{i}$ and $\mathscr{G}_{\operatorname{zar},\mathfrak{Y}}^{i}$ are locally projective of finite type by Lemma 4.10.

We first prove the assertion (ii) for a morphism $h:\mathfrak{T}\rightarrow \mathfrak{Y}$ of $\operatorname{pConv}(Y/\operatorname{W})$ with target $\mathfrak{Y}$ . By Theorem 3.24, we have a spectral sequence

(4.11.2) $$\begin{eqnarray}\operatorname{E}_{2}^{i-j,j}=\operatorname{L}_{i-j}h_{\operatorname{zar}}^{\ast }(\mathscr{G}_{\operatorname{ zar},\mathfrak{Y}}^{j})\Rightarrow \mathscr{G}_{\operatorname{ zar},\mathfrak{T}}^{i}.\end{eqnarray}$$

Since each $\mathscr{G}_{\operatorname{zar},\mathfrak{Y}}^{j}$ is locally projective of finite type, we deduce that $\operatorname{E}_{2}^{i-j,j}=0$ for $i\neq j$ . Then the transition morphism of $\mathscr{F}_{\operatorname{zar},\mathfrak{Y}}^{i}$ associated to $f$ is an isomorphism by Remark 4.4 and (4.11.1).

Since the question is local, for a general morphism $f:(\mathfrak{T}^{\prime },u^{\prime })\rightarrow (\mathfrak{T},u)$ of $\operatorname{pConv}(Y/\operatorname{W})$ , we may assume that $u$ can be lifted to a morphism $h:\mathfrak{T}\rightarrow \mathfrak{Y}$ of $\operatorname{pConv}(Y/\operatorname{W})$ . By the previous result, $c_{h}$ and $c_{h\circ f}$ are isomorphisms. Then we deduce that $c_{f}$ is an isomorphism by Proposition 3.14(c).

4.12

Proof of Proposition 4.8 for fppf topology. We consider the presheaf $\mathscr{P}$ on $\operatorname{Conv}(Y/\operatorname{W})$ defined by

$$\begin{eqnarray}(\mathfrak{T},u)\mapsto \operatorname{H}^{i}((X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\operatorname{fppf}},\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(\mathscr{E})).\end{eqnarray}$$

By Corollary 4.5, the right-hand side is isomorphic to $\operatorname{H}^{i}((X_{T_{0}}/\mathfrak{T})_{\operatorname{conv},\operatorname{zar}},\unicode[STIX]{x1D714}_{\mathfrak{T}}^{\ast }(\unicode[STIX]{x1D6FC}_{\ast }(\mathscr{E})))$ . We set $\mathscr{F}_{\operatorname{zar}}^{i}=\operatorname{R}^{i}g_{\operatorname{conv},\operatorname{zar}\ast }(\unicode[STIX]{x1D6FC}_{\ast }(\mathscr{E}))$ . By Lemma 4.3, the fppf (respectively Zariski) sheaf associated to $\mathscr{P}$ is $\mathscr{F}_{\operatorname{fppf}}^{i}$ (respectively $\mathscr{F}_{\operatorname{zar}}^{i}$ ). Then we deduce a canonical isomorphism (3.8.1)

(4.12.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\ast }(\mathscr{F}_{\operatorname{ zar}}^{i})\xrightarrow[{}]{{\sim}}\mathscr{F}_{\operatorname{ fppf}}^{i}.\end{eqnarray}$$

Let $\mathfrak{T}$ be an object of $\operatorname{pConv}(Y/\operatorname{W})$ . By Proposition 4.8 for Zariski topology and fppf descent, we deduce that $\mathscr{F}_{\operatorname{fppf},\mathfrak{T}}^{i}$ is the fppf sheaf associated to the coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module $\mathscr{F}_{\operatorname{zar},\mathfrak{T}}^{i}$ (§ 3.12) and hence is coherent. Assertion (i) follows.

Since $\mathscr{F}_{\operatorname{fppf},\mathfrak{T}}^{i}$  is the fppf sheaf associated to $\mathscr{F}_{\operatorname{zar},\mathfrak{T}}^{i}$ , assertion (ii) follows from Proposition 4.8(ii) for Zariski topology and (4.12.1). ◻

5.1

In this section, $\mathfrak{S}$ denotes an adic flat formal $\operatorname{W}$ -scheme of finite type. We suppose that the Frobenius morphism $F_{S_{0}}:S_{0}\rightarrow S_{0}$ of the reduced subscheme of $S$ is flat (and hence faithfully flat). Let $X$ be an $S_{0}$ -scheme locally of finite type. We denote by $(-)^{\prime }$ the base-change functor $F_{S_{0}}$ and by $F_{X/S_{0}}:X\rightarrow X^{\prime }$ the relative Frobenius morphism of $X$ relative to $S_{0}$ . Then we have a commutative diagram

(5.1.1)

The following theorem is one of the main results in this section.

Theorem 5.2. Suppose that the Frobenius morphism $F_{S_{0}}:S_{0}\rightarrow S_{0}$ is flat. For every $S_{0}$ -scheme locally of finite type $X$ , the functorial morphism of convergent topoi (3.11.1) of $F_{X/S_{0}}$ induces an equivalence of topoi

(5.2.1) $$\begin{eqnarray}F_{X/S_{0},\operatorname{conv},\operatorname{fppf}}:(X/\mathfrak{S})_{\operatorname{conv},\operatorname{fppf}}\xrightarrow[{}]{{\sim}}(X^{\prime }/\mathfrak{S})_{\operatorname{ conv},\operatorname{fppf}}.\end{eqnarray}$$

Proof. The morphism $F_{X/S_{0},\operatorname{conv},\operatorname{fppf}}$ is induced by the functor (3.9.2):

(5.2.2) $$\begin{eqnarray}\unicode[STIX]{x1D70C}:\operatorname{Conv}(X/\mathfrak{S})\rightarrow \operatorname{Conv}(X^{\prime }/\mathfrak{S}),\quad (\mathfrak{T},u)\mapsto (\mathfrak{T},F_{X/S_{0}}\circ u).\end{eqnarray}$$

Note that $F_{X/S_{0}}\circ u=u^{\prime }\circ F_{T_{0}/S_{0}}$ .

By 3.10 and Lemmas 5.4 and 5.5 in the following, the functor $\unicode[STIX]{x1D70C}$ satisfies the conditions of Proposition 2.5. Then the theorem follows from Proposition 2.5.◻

Proof. The functor $\unicode[STIX]{x1D70C}$ is clearly faithful. We prove its fullness. Let $(\mathfrak{T}_{1},u_{1})$ and $(\mathfrak{T}_{2},u_{2})$ be two objects of $\operatorname{Conv}(X/\mathfrak{S})$ and $g:\unicode[STIX]{x1D70C}(\mathfrak{T}_{1},u_{1})\rightarrow \unicode[STIX]{x1D70C}(\mathfrak{T}_{2},u_{2})$ a morphism of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ . We set $g_{0}:T_{1,0}\rightarrow T_{2,0}$ , the induced morphism. To show that the morphism $\mathfrak{T}_{1}\rightarrow \mathfrak{T}_{2}$ defines a morphism of $\operatorname{Conv}(X/\mathfrak{S})$ which is sent to $g$ by $\unicode[STIX]{x1D70C}$ , it suffices to show that $u_{1}=u_{2}\,\circ \,g_{0}$ . Since $g$ is a morphism of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ , we have a commutative diagram

Then the assertion follows from Lemma 5.3 applied to $u_{1}$ and $u_{2}\circ g_{0}$ .◻

Lemma 5.5. (i) Let $(\mathfrak{T},u)$ be an object of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ such that $\mathfrak{T}$ is affine and that $u:T_{0}\rightarrow X^{\prime }$ factors through an affine open subscheme $U^{\prime }$ of $X^{\prime }$ . Then there exist an object $(\mathfrak{Z},v)$ of $\operatorname{Conv}(X/\mathfrak{S})$ and an fppf covering $\{f:\unicode[STIX]{x1D70C}(\mathfrak{Z},v)\rightarrow (\mathfrak{T},u)\}$ in $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ .

(ii) Keep the assumption and notation of (i). Let $g:(\mathfrak{T}_{1},u_{1})\rightarrow (\mathfrak{T},u)$ be a morphism of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ . Then there exist a morphism $h:(\mathfrak{Z}_{1},v_{1})\rightarrow (\mathfrak{Z},v)$ of $\operatorname{Conv}(X/\mathfrak{S})$ and an fppf covering $\{\unicode[STIX]{x1D711}:\unicode[STIX]{x1D70C}(\mathfrak{Z}_{1},v_{1})\rightarrow (\mathfrak{T}_{1},u_{1})\}$ such that the following diagram is Cartesian:

(5.5.1)

(iii) Every object of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ admits a Zariski covering whose objects satisfy the conditions of (i).

Proof. (i) We set $U=F_{X/S_{0}}^{-1}(U^{\prime })$ , which is an affine $S_{0}$ -scheme of finite type, and we take a closed $S_{0}$ -immersion $\unicode[STIX]{x1D704}_{0}:U\rightarrow Y_{0}=\operatorname{Spec}(\mathscr{O}_{S_{0}}[T_{1},\ldots ,T_{d}])$ . We put $\mathfrak{Y}=\operatorname{Spf}(\mathscr{O}_{\mathfrak{S}}\{T_{1},\ldots ,T_{d}\})$ and denote by $F:\mathfrak{Y}\rightarrow \mathfrak{Y}$ the $\mathfrak{S}$ -morphism defined by sending each $T_{i}$ to $T_{i}^{p}$ .

Note that $Y_{0}^{\prime }=Y_{0}$ and the restriction of $F$ on $Y_{0}$ is the same as the relative Frobenius morphism $F_{Y_{0}/S_{0}}$ . We have a commutative diagram

(5.5.2)

and a canonical morphism $U\rightarrow U^{\prime }\times _{Y_{0},F}Y_{0}$ . We denote the composition of $\unicode[STIX]{x1D704}_{0}^{\prime }:U^{\prime }\rightarrow Y_{0}$ and $Y_{0}\rightarrow \mathfrak{Y}$ by $\unicode[STIX]{x1D704}^{\prime }$ . Since $\mathfrak{Y}$ is smooth over $\mathfrak{S}$ , there exists an $\mathfrak{S}$ -morphism $\unicode[STIX]{x1D70F}:\mathfrak{T}\rightarrow \mathfrak{Y}$ lifting $\unicode[STIX]{x1D704}^{\prime }\circ u:T_{0}\rightarrow \mathfrak{Y}$ . We consider the following commutative diagram:

(5.5.3)

where $U_{0}\rightarrow U^{\prime }$ is induced by $F_{U/S_{0}}$ .

If the ideal sheaf associated to $\unicode[STIX]{x1D704}_{0}:U{\hookrightarrow}Y_{0}$ is locally generated by polynomials $\{f_{1},\ldots ,f_{n}\}$ of $\mathscr{O}_{S_{0}}[T_{1},\ldots ,T_{d}]$ , the ideal sheaf associated to $\unicode[STIX]{x1D704}_{0}^{\prime }\times _{Y_{0},F}Y_{0}:U^{\prime }\times _{Y_{0},F}Y_{0}{\hookrightarrow}Y_{0}$ (5.5.2) is locally generated by $\{f_{1}^{p},\ldots ,f_{n}^{p}\}$ . Then the canonical morphism $U\rightarrow U^{\prime }\times _{Y_{0},F}Y_{0}$ induces an isomorphism

$$\begin{eqnarray}U_{0}\xrightarrow[{}]{{\sim}}(U^{\prime }\times _{Y_{0},F}Y_{0})_{0}.\end{eqnarray}$$

By (5.5.3) and [Reference Grothendieck and DieudonnéEGAI, 4.5.11], we obtain an object $(\mathfrak{T}\times _{\mathfrak{Y},F}\mathfrak{Y},v)$ of $\operatorname{Conv}(X/\mathfrak{S})$ and a morphism $f:\unicode[STIX]{x1D70C}(\mathfrak{T}\times _{\mathfrak{Y},F}\mathfrak{Y},v)\rightarrow (\mathfrak{T},u)$ of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ . Since the reduction modulo $p$ of $F$ is faithfully flat of finite type [Reference IllusieIll96, 3.2], so is $F$ (cf. [Reference XuXu19, 7.2]). Then $f$ is an fppf covering (§ 3.4).

(ii) We denote by $(\mathfrak{Z}_{1},w)$ the fibered product $\unicode[STIX]{x1D70C}(\mathfrak{Z},v)\times _{(\mathfrak{T},u)}(\mathfrak{T}_{1},u_{1})$ in $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ . By applying Lemma 3.10 to the projection $(\mathfrak{Z}_{1},w)\rightarrow \unicode[STIX]{x1D70C}(\mathfrak{Z},v)$ , we obtain the Cartesian diagram (5.5.1). Since $\unicode[STIX]{x1D711}$ is the base change of $f$ , $\unicode[STIX]{x1D711}$ is an fppf covering.

(iii) Let $(\mathfrak{T},u)$ be an object of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ and $U^{\prime }$ an affine open subscheme of $X^{\prime }$ . We denote by $\mathfrak{T}_{U^{\prime }}$ the open formal subscheme of $\mathfrak{T}$ associated to the open subset $u^{-1}(|U^{\prime }|)$ of $|T_{0}|=|T|$ . The assertion follows by taking an affine covering of $\mathfrak{T}_{U^{\prime }}$ for every $U^{\prime }$ .◻

We deduce from Theorem 5.2 a new proof of Frobenius descent for convergent isocrystals (Proposition 5.7) and a comparison of de Rham complexes for the Frobenius descent (Corollary 5.9).

Proposition 5.7. Keep the assumption of Theorem 5.2. The inverse image and the direct image functors of $F_{X/S_{0},\operatorname{conv},\operatorname{zar}}$ induce equivalences of categories quasi-inverse to each other (Definition 3.15):

(5.7.1) $$\begin{eqnarray}\operatorname{Iso}^{\dagger }(X/\mathfrak{S})\rightleftarrows \operatorname{Iso}^{\dagger }(X^{\prime }/\mathfrak{S}).\end{eqnarray}$$

Proof. By § 3.16, convergent isocrystals are sheaves for fppf topology and we work with fppf topology in this proof. We write simply (5.2.1) for $F_{X/S_{0}}$ and we will show that the direct image and inverse image functors of $F_{X/S_{0}}$ send coherent crystals of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules to coherent crystals of $\mathscr{O}_{X^{\prime }/\mathfrak{S}}[\frac{1}{p}]$ -modules. The assertion for the inverse image follows from (3.11.2) and we will prove it for the direct image.

Let $\mathscr{F}$ be a coherent crystal of $\mathscr{O}_{X/\mathfrak{S}}[\frac{1}{p}]$ -modules and $(\mathfrak{T},u)$ an object of $\operatorname{Conv}(X^{\prime }/\mathfrak{S})$ . We first show that $(F_{X/S_{0}\ast }(\mathscr{F}))_{\mathfrak{T}}$ is coherent. By Lemma 5.5(iii), we may assume that $(\mathfrak{T},u)$ satisfies the conditions of Lemma 5.5(i). Then, by Lemmas 5.5(ii) and 5.6, there exist objects $\mathfrak{Z}$ and $\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}$ of $\operatorname{Conv}(X/\mathfrak{S})$ , an fppf covering $\{f:\unicode[STIX]{x1D70C}(\mathfrak{Z})\rightarrow \mathfrak{T}\}$ and two morphisms $p_{1},p_{2}:\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}\rightarrow \mathfrak{Z}$ such that $\unicode[STIX]{x1D70C}(\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z})=\unicode[STIX]{x1D70C}(\mathfrak{Z})\times _{\mathfrak{T}}\unicode[STIX]{x1D70C}(\mathfrak{Z})$ and that $\unicode[STIX]{x1D70C}(p_{1})$ and $\unicode[STIX]{x1D70C}(p_{2})$ are the canonical projections of $\unicode[STIX]{x1D70C}(\mathfrak{Z})\times _{\mathfrak{T}}\unicode[STIX]{x1D70C}(\mathfrak{Z})$ . In particular, the morphism of formal schemes $\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}\rightarrow \mathfrak{Z}$ attached to $p_{1}$ (respectively $p_{2}$ ) is the projection on the first (respectively second) component.

Since the adjunction morphism $F_{X/S_{0}}^{\ast }F_{X/S_{0}\ast }\rightarrow \operatorname{id}$ is an isomorphism (Theorem 5.2), we have (3.11.2)

(5.7.2) $$\begin{eqnarray}(F_{X/S_{0}\ast }(\mathscr{F}))_{\unicode[STIX]{x1D70C}(\mathfrak{Z})}=\mathscr{F}_{\mathfrak{Z}},\quad (F_{X/S_{0}\ast }(\mathscr{F}))_{\unicode[STIX]{x1D70C}(\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z})}=\mathscr{F}_{\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}}.\end{eqnarray}$$

Since $\mathscr{F}$ is a crystal, we have $\mathscr{O}_{\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}}$ -linear isomorphisms

(5.7.3) $$\begin{eqnarray}p_{2}^{\ast }(\mathscr{F}_{\mathfrak{ Z}})\xrightarrow[{{\sim}}]{c_{p_{2}}}\mathscr{F}_{\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z}}\xleftarrow[{{\sim}}]{c_{p_{1}}}p_{1}^{\ast }(\mathscr{F}_{\mathfrak{ Z}}).\end{eqnarray}$$

Then we obtain a descent datum $(\mathscr{F}_{\mathfrak{Z}},c_{p_{1}}^{-1}\circ c_{p_{2}})$ for the fppf covering $\{f:\mathfrak{Z}\rightarrow \mathfrak{T}\}$ . By fppf descent [Reference AbbesAbb10, 5.11.11], there exist a coherent $\mathscr{O}_{\mathfrak{T}}[\frac{1}{p}]$ -module $\mathscr{M}$ and a canonical $\mathscr{O}_{\mathfrak{Z}}$ -linear isomorphism $f^{\ast }(\mathscr{M})\xrightarrow[{}]{{\sim}}\mathscr{F}_{\mathfrak{Z}}$ .

On the other hand, since $F_{X/S_{0}\ast }(\mathscr{F})$ is a sheaf in fppf topology, there exists an exact sequence

(5.7.4) $$\begin{eqnarray}0\rightarrow (F_{X/S_{0}\ast }(\mathscr{F}))(\mathfrak{T})\rightarrow (F_{X/S_{0}\ast }(\mathscr{F}))(\unicode[STIX]{x1D70C}(\mathfrak{Z}))\rightarrow (F_{X/S_{0}\ast }(\mathscr{F}))(\unicode[STIX]{x1D70C}(\mathfrak{Z}\times _{\mathfrak{T}}\mathfrak{Z})).\end{eqnarray}$$

By (5.7.2), we deduce an $\mathscr{O}_{\mathfrak{T}}$ -linear isomorphism $\mathscr{M}\xrightarrow[{}]{{\sim}}(F_{X/S_{0}\ast }(\mathscr{F}))_{\mathfrak{T}}$ . In particular, $(F_{X/S_{0}\ast }(\mathscr{F}))_{\mathfrak{T}}$ is coherent. Hence, $F_{X/S_{0}\ast }(\mathscr{F})$ is coherent.

Following the same argument as in the second part of the proof of [Reference XuXu19, 9.13], we show that for every morphism $g$ of $\operatorname{Conv}(X/\mathfrak{S})$ , the transition morphism $c_{g}$ associated to $F_{X/S_{0}\ast }(\mathscr{F})$ is an isomorphism, i.e.  $F_{X/S_{0}\ast }(\mathscr{F})$ is a crystal.◻

Proposition 5.8. We consider the following diagram:

(5.8.1)

where vertical arrows are defined in § 3.21. Let $\mathscr{E}$ be a convergent isocrystal of $(X/\operatorname{W})_{\operatorname{conv},\operatorname{zar}}$ and denote the structure morphism $X\rightarrow S_{0}$ by $f$ . Then there exists a canonical isomorphism in the derived category $\operatorname{D}(X_{\operatorname{zar}},f^{-1}(\mathscr{O}_{\mathfrak{S}}))$ :

(5.8.2) $$\begin{eqnarray}F_{X/S_{0}\ast }(\operatorname{R}u_{X/\mathfrak{S}\ast }(\mathscr{E}))\xrightarrow[{}]{{\sim}}\operatorname{R}u_{X^{\prime }/\mathfrak{S}\ast }(F_{X/S_{0}\operatorname{conv},\operatorname{zar}\ast }(\mathscr{E})).\end{eqnarray}$$

Corollary 5.9. Keep the assumption of Proposition 5.8 and suppose that there exist smooth liftings $\mathfrak{X}$ of $X$ and $\mathfrak{X}^{\prime }$ of $X^{\prime }$ over $\mathfrak{S}$ . Let $f:X\rightarrow S_{0}$ be the canonical morphism. Then there exists a canonical isomorphism between the de Rham complexes of $\mathscr{E}$ and of $F_{X/S_{0},\operatorname{conv},\operatorname{zar}\ast }(\mathscr{E})$ in $\operatorname{D}(X_{\operatorname{zar}}^{\prime },f^{-1}(\mathscr{O}_{\mathfrak{S}}))$ :

(5.9.1) $$\begin{eqnarray}F_{X/S_{0}\ast }(\mathscr{E}_{\mathfrak{X}}\,\otimes _{\mathscr{O}_{\mathfrak{X}}}\,\widehat{\unicode[STIX]{x1D6FA}}_{\mathfrak{X}/\mathfrak{S}}^{\bullet })\xrightarrow[{}]{{\sim}}(F_{X/S_{0},\operatorname{conv},\operatorname{zar}\ast }\mathscr{E})_{\mathfrak{X}^{\prime }}\,\otimes _{\mathscr{O}_{\mathfrak{X}^{\prime }}}\,\widehat{\unicode[STIX]{x1D6FA}}_{\mathfrak{X}^{\prime }/\mathfrak{S}}^{\bullet }.\end{eqnarray}$$

Inspired by Ogus’ arguments in [Reference OgusOgu84], we use Proposition 4.8 and Dwork’s trick to prove Theorem 5.10. The proof of Proposition 4.8 relies on the local projectiveness of $\mathscr{E}$ , which is obtained from the smoothness of $X$ and of $Y$ (Lemma 4.9). In Theorem 8.2, we will improve the above theorem to the non-smooth case.

We first introduce certain subcategories of $\operatorname{Conv}(X/\operatorname{W})$ .

Given an object $(\mathfrak{T},u)$ of $\operatorname{Conv}^{(n)}(X/\operatorname{W})$ and a morphism $(\mathfrak{T}^{\prime },u^{\prime })\rightarrow (\mathfrak{T},u)$ of $\operatorname{Conv}(X/\operatorname{W})$ , then $(\mathfrak{T}^{\prime },u^{\prime })$ is also an object of $\operatorname{Conv}^{(n)}(X/\operatorname{W})$ . In particular, $T^{(0)}=T$ and $\operatorname{Conv}^{(0)}(X/\operatorname{W})$ coincides with $\operatorname{pConv}(X/\operatorname{W})$ (§ 4.7).

Proof. Let $(\mathfrak{T},u)$ be an object of $\operatorname{Conv}^{(n+1)}(X/\operatorname{W})$ and $\widetilde{u}:T^{(n+1)}\rightarrow X$ a lifting of $u$ . The absolute Frobenius morphism $T^{(n)}\rightarrow T^{(n)}$ factors through the closed subscheme $T^{(n+1)}$ and then the relative Frobenius morphism $F_{T^{(n)}/k}$ factors through $(T^{(n+1)})^{\prime }$ . We have a commutative diagram

(5.12.1)

Then the morphism $F_{X/k}\,\circ \,u$ can be lifted to a $k$ -morphism $T^{(n)}\rightarrow X^{\prime }$ and the lemma follows.◻

5.13

Proof of Theorem 5.10. By Corollary 4.5, it suffices to prove the assertion for fppf topology. There exists an object $\mathscr{G}$ of $\operatorname{Iso}^{\dagger }(X^{\prime }/\operatorname{W})$ with $F_{X/k,\operatorname{conv},\operatorname{fppf}}^{\ast }(\mathscr{G})\simeq \mathscr{E}$ (Proposition 5.7). If we set $\mathscr{F}=\operatorname{R}^{i}g_{\operatorname{conv},\operatorname{fppf}\ast }(\mathscr{E})$ and $\mathscr{H}=\operatorname{R}^{i}g_{\operatorname{conv},\operatorname{fppf}\ast }^{\prime }(\mathscr{G})$ , then we have $F_{Y/k,\operatorname{conv},\operatorname{fppf}}^{\ast }(\mathscr{H})\simeq \mathscr{F}$ by Theorem 5.2.

Each object (respectively morphism) of $\operatorname{Conv}(Y/\operatorname{W})$ belongs to a subcategory $\operatorname{Conv}^{(n)}(Y/\operatorname{W})$ (Definition 5.11) for some integer $n$ . We prove the following assertions by induction:

1. (i) for every object $\mathfrak{T}$ of $\operatorname{Conv}^{(n)}(Y/\operatorname{W})$ , $\mathscr{F}_{\mathfrak{T}}$ is coherent;

2. (ii) for every morphism $f$ of $\operatorname{Conv}^{(n)}(Y/\operatorname{W})$ , the transition morphism $c_{f}$ associated to $\mathscr{F}$ is an isomorphism.

The assertions for $n=0$ are proved in Proposition 4.8. Suppose that the assertions hold for $n\geqslant 0$ ; we prove them for $n+1$ . Let $(\mathfrak{T},u)$ be an object of $\operatorname{Conv}^{(n+1)}(Y/\operatorname{W})$ . By (3.11.2), we deduce that

$$\begin{eqnarray}\mathscr{H}_{\unicode[STIX]{x1D70C}(\mathfrak{T})}\xrightarrow[{}]{{\sim}}\mathscr{F}_{\mathfrak{T}}.\end{eqnarray}$$

By the induction hypotheses, for any object $\mathfrak{Z}$ of $\operatorname{Conv}^{(n)}(X^{\prime }/\operatorname{W})$ , $\mathscr{H}_{\mathfrak{Z}}$ is coherent. Then assertion (i) follows from Lemma 5.12 and the induction hypotheses.

Assertion (ii) can be verified in the same way by § 3.11 and Lemma 5.12. ◻

5.14

The Frobenius homomorphism $\unicode[STIX]{x1D70E}:\operatorname{W}\rightarrow \operatorname{W}$ induces a morphism of topoi $(X^{\prime }/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}\rightarrow (X/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ for $\unicode[STIX]{x1D70F}\in \{\operatorname{zar},\operatorname{fppf}\}$ . For any sheaf $\mathscr{E}$ of $(X/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ , we denote by $\mathscr{E}^{\prime }$ the inverse image of $\mathscr{E}$ to $(X^{\prime }/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ .

A convergent $F$ -isocrystal of $\operatorname{Conv}(X/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ is a pair $(\mathscr{E},\unicode[STIX]{x1D711})$ consisting of a convergent isocrystal $\mathscr{E}$ of $(X/\operatorname{W})_{\operatorname{conv},\unicode[STIX]{x1D70F}}$ and an isomorphism, called a Frobenius structure of $\mathscr{E}$ ,

(5.14.1) $$\begin{eqnarray}\unicode[STIX]{x1D711}:F_{X/k,\operatorname{conv},\unicode[STIX]{x1D70F}}^{\ast }(\mathscr{E}^{\prime })\xrightarrow[{}]{{\sim}}\mathscr{E}.\end{eqnarray}$$

Proof. By Theorem 5.10, it suffices to prove the assertion for fppf topology. Consider the isomorphism

(5.15.1) $$\begin{eqnarray}\operatorname{R}^{i}g_{\operatorname{ conv},\operatorname{fppf}\ast }(\unicode[STIX]{x1D711}):\operatorname{R}^{i}g_{\operatorname{ conv},\operatorname{fppf}\ast }(F_{X/k,\operatorname{conv},\operatorname{fppf}}^{\ast }(\mathscr{E}^{\prime }))\xrightarrow[{}]{{\sim}}\operatorname{R}^{i}g_{\operatorname{ conv},\operatorname{fppf}\ast }(\mathscr{E}).\end{eqnarray}$$

By Corollary 4.6 and Theorem 5.2, the left-hand side is isomorphic to

$$\begin{eqnarray}F_{Y/k,\operatorname{conv},\operatorname{fppf}}^{\ast }((\operatorname{R}^{i}g_{\operatorname{ conv},\operatorname{fppf}\ast }(\mathscr{E}))^{\prime }).\end{eqnarray}$$

Then the assertion follows. ◻

6.1

Recall that $\mathbf{S}$ denotes the category of adic formal $\operatorname{W}$ -schemes of finite type whose morphisms are $\operatorname{W}$ -morphisms of finite type (§ 2.1). The set $\mathbf{B}$ of admissible blow-ups in $\mathbf{S}$ forms a right multiplicative system in $\mathbf{S}$  [Reference AbbesAbb10, 4.1.4]. We denote by $\mathbf{R}$ the localized category of $\mathbf{S}$ relative to $\mathbf{B}$ . Objects of $\mathbf{R}$ are called coherent rigid spaces (over $K=\operatorname{W}[\frac{1}{p}]$ ).