2 Log structures

We begin with a general result on semistable reduction and log schemes. Let $R$ be a complete discrete valuation ring (DVR) with perfect residue field $k$, $\unicode[STIX]{x1D70B}$ a uniformiser for $R$, and let ${\mathcal{X}}\rightarrow \text{Spec}(R)$ be a strictly semistable scheme. That is, ${\mathcal{X}}$ is Zariski locally étale over $R[x_{1},\ldots ,x_{n}]/(x_{1}\cdots x_{r}-\unicode[STIX]{x1D70B})$ for some $n,r$. There is a natural log structure ${\mathcal{M}}_{{\mathcal{X}}}$ on ${\mathcal{X}}$ given by functions invertible outside the special fibre $X$, and we let ${\mathcal{M}}_{X}$ denote the pull-back of this log structure to $X$. We will also write $X_{i}$ for the reduction of ${\mathcal{X}}$ modulo $\unicode[STIX]{x1D70B}^{i+1}$, and $k^{\times }$ for $k$ equipped with the log structure pulled back from the canonical log structure $R^{\times }$ on $R$.

Sketch of proof.

Use $g$ to identify $X_{1}$ and $X_{1}^{\prime }$, and thus $X$ and $X^{\prime }$. Let ${\mathcal{M}}_{X}$ and ${\mathcal{M}}_{X}^{\prime }$ be the log structures on $X$ coming from ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime }$ respectively.

Near a closed point of $X$ let $X^{(1)},\ldots ,X^{(r)}$ be the irreducible components of $X$, and pick $x_{1},\ldots ,x_{r}\in {\mathcal{O}}_{{\mathcal{X}}}$ such that $X^{(i)}=V(x_{i})$. Similarly pick $x_{1}^{\prime },\ldots ,x_{r}^{\prime }\in {\mathcal{O}}_{{\mathcal{X}}^{\prime }}$ such that $X^{(i)}=V(x_{i}^{\prime })$. Let $v\in {\mathcal{O}}_{{\mathcal{X}}}^{\ast }$ and $v^{\prime }\in {\mathcal{O}}_{{\mathcal{X}}^{\prime }}^{\ast }$ be such that $x_{1}\cdots x_{r}=v\unicode[STIX]{x1D70B}$ and $x_{1}^{\prime }\cdots x_{r}^{\prime }=v^{\prime }\unicode[STIX]{x1D70B}$. Then in a neighbourhood of $p$ the morphisms $({\mathcal{X}},{\mathcal{M}}_{{\mathcal{X}}})\rightarrow \text{Spec}(R^{\times })$ and $({\mathcal{X}}^{\prime },{\mathcal{M}}_{{\mathcal{X}}^{\prime }})\rightarrow \text{Spec}(R^{\times })$ can be described by the following diagrams:

Pulling back to $k$, we see that the morphisms $(X,{\mathcal{M}}_{X})\rightarrow \text{Spec}(k^{\times })$ and $(X,{\mathcal{M}}_{X}^{\prime })\rightarrow \text{Spec}(k^{\times })$ can be described by the diagrams

and

respectively, again in a neighbourhood of $p$. Since $V(x_{i})=V(x_{i}^{\prime })$ inside $X_{1}$, we must have $x_{i}=u_{i}x_{i}^{\prime }$ for some $u_{i}\in {\mathcal{O}}_{X_{1}}^{\ast }$, and so we can define an isomorphism

$$\begin{eqnarray}{\mathcal{M}}_{X}\overset{{\sim}}{\rightarrow }{\mathcal{M}}_{X}^{\prime }\end{eqnarray}$$

of log structures by mapping

$$\begin{eqnarray}(u,a_{1},\ldots ,a_{r})\mapsto (uu_{1}^{a_{1}}\cdots u_{r}^{a_{r}},a_{1},\ldots ,a_{r}).\end{eqnarray}$$

This is checked to be a morphism of log structures over $k^{\times }$ by using the above local descriptions. Note that any other choice $u_{i}^{\prime }$ must satisfy $(u_{i}-u_{i}^{\prime })x_{i}^{\prime }=0$ in ${\mathcal{O}}_{X_{1}}$, and hence we must have $u_{i}-u_{i}^{\prime }\in (\unicode[STIX]{x1D70B})$. In particular, the above isomorphism does not depend on the choice of $u_{i}$. By a similar argument, neither does it depend on the choice of $x_{i}$ and $x_{i}^{\prime }$, and so it glues to give a global isomorphism $(X,{\mathcal{M}}_{X})\cong (X,{\mathcal{M}}_{X}^{\prime })$ of log schemes over $k^{\times }$.◻

We will need to extend this result to cover morphisms between strictly semistable schemes over different bases. So suppose that $R\rightarrow S$ is a finite morphism of complete DVRs, with induced residue field extension $k\rightarrow k_{S}$. Let $\unicode[STIX]{x1D70B}_{S}$ be a uniformiser for $S$, and let $e=v_{\unicode[STIX]{x1D70B}_{S}}(\unicode[STIX]{x1D70B})$. We do not assume that the induced extension $Q(R)\rightarrow Q(S)$ of fraction fields is separable.

Suppose that we have strictly semistable schemes ${\mathcal{X}},{\mathcal{X}}^{\prime }$ over $R$ and ${\mathcal{Y}},{\mathcal{Y}}^{\prime }$ over $S$, and a pair of commutative diagrams

As before, let us write $Y_{j}$ for the reduction of ${\mathcal{Y}}$ modulo $\unicode[STIX]{x1D70B}_{S}^{j+1}$. Suppose that we have isomorphisms

$$\begin{eqnarray}g_{Y}:Y_{e}\overset{{\sim}}{\rightarrow }Y_{e}^{\prime },\quad g_{X}:X_{1}\overset{{\sim}}{\rightarrow }X_{1}^{\prime }\end{eqnarray}$$

of $S$- and $R$-schemes respectively such that the diagram

commutes. Then by Proposition 2.1 we obtain isomorphisms

$$\begin{eqnarray}g_{Y}:(Y,{\mathcal{M}}_{Y})\overset{{\sim}}{\rightarrow }(Y^{\prime },{\mathcal{M}}_{Y^{\prime }})\end{eqnarray}$$

of log schemes over $k_{S}^{\times }$, as well as

$$\begin{eqnarray}g_{X}:(X,{\mathcal{M}}_{X})\overset{{\sim}}{\rightarrow }(X^{\prime },{\mathcal{M}}_{X^{\prime }})\end{eqnarray}$$

of log schemes over $k^{\times }$. The above commutative diagrams of strictly semistable schemes induce commutative diagrams

of log schemes. Note that the morphism of punctured points along the bottom of each square is given by

$$\begin{eqnarray}\displaystyle k^{\ast }\oplus \mathbb{N} & \rightarrow & \displaystyle k_{S}^{\ast }\oplus \mathbb{N}\nonumber\\ \displaystyle (\unicode[STIX]{x1D706},a) & \mapsto & \displaystyle (\unicode[STIX]{x1D706}u^{a},ea),\nonumber\end{eqnarray}$$

where $u\in S^{\ast }$ is such that $\unicode[STIX]{x1D70B}=u\unicode[STIX]{x1D70B}_{S}^{e}$.

Proposition 2.2. The diagram

of log schemes commutes.

Proof. Let us use $g$ to identify $Y_{e}=Y_{e}^{\prime }$ and $Y=Y^{\prime }$, and let ${\mathcal{M}}_{Y}$ and ${\mathcal{M}}_{Y}^{\prime }$ be the log structures on $Y$ coming from ${\mathcal{Y}}$ and ${\mathcal{Y}}^{\prime }$ respectively. Similarly identify $X_{1}=X_{1}^{\prime }$ and $X=X^{\prime }$, and let ${\mathcal{M}}_{X}$ and ${\mathcal{M}}_{X}^{\prime }$ be the log structures on $X$ coming from ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime }$ respectively.

Locally on $X$ and $Y$, choose functions $y_{1},\ldots ,y_{s}\in {\mathcal{O}}_{{\mathcal{Y}}}$, $y_{1}^{\prime },\ldots ,y_{s}^{\prime }\in {\mathcal{O}}_{{\mathcal{Y}}^{\prime }}$ cutting out the irreducible components of $Y$, and functions $x_{1},\ldots ,x_{r}\in {\mathcal{O}}_{{\mathcal{X}}}$ and $x_{1}^{\prime },\ldots ,x_{r}^{\prime }\in {\mathcal{O}}_{{\mathcal{X}}^{\prime }}$ cutting out the irreducible components of $X$. Write

$$\begin{eqnarray}f^{\ast }(x_{i})=\unicode[STIX]{x1D6FC}_{i}y_{1}^{d_{i1}}\cdots y_{s}^{d_{is}},\quad {f^{\prime }}^{\ast }(x_{i}^{\prime })=\unicode[STIX]{x1D6FC}_{i}^{\prime }{y_{1}^{\prime }}^{d_{i1}^{\prime }}\cdots {y_{s}^{\prime }}^{d_{is}^{\prime }},\end{eqnarray}$$

since both $d_{ij}$ and $d_{ij}^{\prime }$ are given by the multiplicity of the $j$th irreducible component of $Y$ in the scheme theoretic preimage of the $i$th irreducible component of $X$ inside $Y_{e}$, we must have $d_{ij}=d_{ij}^{\prime }$. Moreover, since $V(f^{\ast }(x_{i}))\subset V(\unicode[STIX]{x1D70B}_{S}^{e})=V(y_{1}^{e}\cdots y_{s}^{e})$ we must have $d_{ij}\leqslant e$ for all $i,j$.

Now choose $u_{i}\in {\mathcal{O}}_{X_{1}}^{\ast }$ such that $x_{i}=u_{i}x_{i}^{\prime }$, and $v_{j}\in {\mathcal{O}}_{Y_{e}}^{\ast }$ such that $y_{j}=v_{j}y_{j}^{\prime }$. Then the isomorphisms of log structures induced by $g_{Y}$ and $g_{X}$ are given by

$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{Y}={\mathcal{O}}_{Y}^{\ast }\oplus \mathbb{N}^{s} & \rightarrow & \displaystyle {\mathcal{M}}_{Y}^{\prime }={\mathcal{O}}_{Y}^{\ast }\oplus \mathbb{N}^{s}\nonumber\\ \displaystyle (v,b_{1},\ldots ,b_{s}) & \mapsto & \displaystyle (vv_{1}^{b_{1}}\cdots v_{s}^{b_{s}},b_{1},\ldots ,b_{s})\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle {\mathcal{M}}_{X}={\mathcal{O}}_{X}^{\ast }\oplus \mathbb{N}^{r} & \rightarrow & \displaystyle {\mathcal{M}}_{X}^{\prime }={\mathcal{O}}_{X}^{\ast }\oplus \mathbb{N}^{r}\nonumber\\ \displaystyle (u,a_{1},\ldots ,a_{r}) & \mapsto & \displaystyle (uu_{1}^{a_{1}}\cdots u_{r}^{a_{r}},a_{1},\ldots ,a_{r})\nonumber\end{eqnarray}$$

respectively, and the morphisms ${\mathcal{M}}_{X}\rightarrow {\mathcal{M}}_{Y}$ and ${\mathcal{M}}_{X}^{\prime }\rightarrow {\mathcal{M}}_{Y}^{\prime }$ are defined by

$$\begin{eqnarray}(u,a_{1},\ldots ,a_{r})\mapsto \biggl(f^{\ast }(u)\unicode[STIX]{x1D6FC}_{1}^{a_{1}}\cdots \unicode[STIX]{x1D6FC}_{r}^{a_{r}},\mathop{\sum }_{i}d_{i1}a_{i},\ldots ,\mathop{\sum }_{i}d_{is}a_{i}\biggr)\end{eqnarray}$$

and

$$\begin{eqnarray}(u,a_{1},\ldots ,a_{r})\mapsto \biggl(f^{\ast }(u){\unicode[STIX]{x1D6FC}_{1}^{\prime }}^{a_{1}}\cdots {\unicode[STIX]{x1D6FC}_{r}^{\prime }}^{a_{r}},\mathop{\sum }_{i}d_{i1}a_{i},\ldots ,\mathop{\sum }_{i}d_{is}a_{i}\biggr)\end{eqnarray}$$

respectively. Hence in the diagram

the composite $f\circ g_{X}$ is given by

$$\begin{eqnarray}(u,a_{1},\ldots ,a_{r})\mapsto \biggl(f^{\ast }(u)(\unicode[STIX]{x1D6FC}_{1}^{\prime }f^{\ast }(u_{1}))^{a_{1}}\cdots (\unicode[STIX]{x1D6FC}_{r}^{\prime }f^{\ast }(u_{r}))^{a_{r}},\mathop{\sum }_{i}d_{i1}a_{i},\ldots ,\mathop{\sum }_{i}d_{is}a_{i}\biggr)\end{eqnarray}$$

and the composite $g_{Y}\circ f$ is given by

$$\begin{eqnarray}(u,a_{1},\ldots ,a_{r})\mapsto \biggl(f^{\ast }(u)(\unicode[STIX]{x1D6FC}_{1}v_{1}^{d_{11}}\cdots v_{s}^{d_{1s}})^{a_{1}}\cdots (\unicode[STIX]{x1D6FC}_{r}v_{1}^{d_{r1}}\cdots v_{s}^{d_{rs}})^{a_{r}},\mathop{\sum }_{i}d_{i1}a_{i},\ldots ,\mathop{\sum }_{i}d_{is}a_{i}\biggr).\end{eqnarray}$$

We thus need to show that $\unicode[STIX]{x1D6FC}_{i}^{\prime }f^{\ast }(u_{i})=\unicode[STIX]{x1D6FC}_{i}v_{1}^{d_{i1}}\cdots v_{s}^{d_{is}}$ in ${\mathcal{O}}_{Y}^{\ast }$ for all $i$. But now we write

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}y_{1}^{d_{i1}}\cdots y_{s}^{d_{is}}=f^{\ast }(x_{i})=f^{\ast }(u_{i}x_{i}^{\prime })=f^{\ast }(u_{i})\unicode[STIX]{x1D6FC}_{i}^{\prime }{y_{1}^{\prime }}^{d_{i1}}\cdots {y_{s}^{\prime }}^{d_{is}}\end{eqnarray}$$

in ${\mathcal{O}}_{Y_{e}}$ and so deduce that

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}v_{1}^{d_{i1}}\cdots v_{s}^{d_{is}}{y_{1}^{\prime }}^{d_{i1}}\cdots {y_{s}^{\prime }}^{d_{is}}=f^{\ast }(u_{i})\unicode[STIX]{x1D6FC}_{i}^{\prime }{y_{1}^{\prime }}^{d_{i1}}\cdots {y_{s}^{\prime }}^{d_{is}}.\end{eqnarray}$$

We deduce that the difference $\unicode[STIX]{x1D6FD}_{i}=\unicode[STIX]{x1D6FC}_{i}^{\prime }f^{\ast }(u_{i})-\unicode[STIX]{x1D6FC}_{i}v_{1}^{d_{i1}}\cdots v_{s}^{d_{is}}$ annihilates ${y_{1}^{\prime }}^{d_{i1}}\cdots {y_{s}^{\prime }}^{d_{is}}$ inside ${\mathcal{O}}_{Y_{e}}$, and since each $d_{ij}\leqslant e$ we deduce that in fact $\unicode[STIX]{x1D6FD}_{i}$ annihilates $\unicode[STIX]{x1D70B}_{S}^{e}$, and therefore must lie in $(\unicode[STIX]{x1D70B}_{S})$. Hence $\unicode[STIX]{x1D6FD}_{i}=0$ in ${\mathcal{O}}_{Y}$ and the proof is complete.◻

3 Functoriality of comparison isomorphisms

We will also need to know that the comparison isomorphisms [Reference Chiarellotto and LazdaCL18, Propositions 5.3, 5.4] are compatible with morphisms of semistable schemes over different bases. So let us suppose that we are again in the above set-up, where we have a commutative diagram

of strictly semistable schemes ${\mathcal{Y}}$ and ${\mathcal{X}}$ over $S$ and $R$ respectively, with $S$ the integral closure of $R$ in some finite extension of its fraction field. Let us assume that $R$, and hence $S$, is of equicharacteristic $p>0$, with fraction fields $F$ and $F_{S}$ respectively, whose absolute Galois groups we will denote by $G_{F}$ and $G_{F_{S}}$. Fix an embedding $F^{\text{sep}}{\hookrightarrow}F_{S}^{\text{sep}}$ of separable closures; note that this sends $F^{\text{tame}}$ into $F_{S}^{\text{tame}}$ and induces an injective homomorphism $G_{F_{S}}\rightarrow G_{F}$ with finite cokernel.

Let ${\mathcal{X}}^{\times }$ and ${\mathcal{Y}}^{\times }$ denote these semistable schemes endowed with their canonical log structures, and $X^{\times }$ and $Y^{\times }$ the corresponding log special fibres. We therefore have a commutative diagram

of log schemes. For every finite subextension $F\subset L\subset F^{\text{tame}}$, let $X_{L}^{\times }$ denote the corresponding base change of $X^{\times }$, and $X^{\times ,\text{tame}}$ the inverse limit of the étale topoi of all such $X_{L}^{\times }$; we have $Y^{\times ,\text{tame}}$ defined entirely similarly. Via the embedding $F^{\text{tame}}{\hookrightarrow}F_{S}^{\text{tame}}$ this induces a $G_{F_{S}}$-equivariant morphism of topoi

$$\begin{eqnarray}Y^{\times ,\text{tame}}\rightarrow X^{\times ,\text{tame}}\end{eqnarray}$$

and hence a $G_{F_{S}}$-equivariant morphism

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(X^{\times ,\text{tame}},\mathbb{Q}_{\ell })\rightarrow H_{\acute{\text{e}}\text{t}}^{i}(Y^{\times ,\text{tame}},\mathbb{Q}_{\ell })\end{eqnarray}$$

in cohomology, for any $\ell \neq p$. On the other hand we have a natural $G_{F_{S}}$-equivariant map

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{X}}\times _{R}F^{\text{sep}},\mathbb{Q}_{\ell })\rightarrow H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}\times _{S}F_{S}^{\text{sep}},\mathbb{Q}_{\ell }),\end{eqnarray}$$

and by [Reference NakayamaNak98, Proposition 4.2] equivariant isomorphisms

$$\begin{eqnarray}\displaystyle H_{\acute{\text{e}}\text{t}}^{i}(X^{\times ,\text{tame}},\mathbb{Q}_{\ell }) & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{X}}\times _{R}F^{\text{sep}},\mathbb{Q}_{\ell }),\nonumber\\ \displaystyle H_{\acute{\text{e}}\text{t}}^{i}(Y^{\times ,\text{tame}},\mathbb{Q}_{\ell }) & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}\times _{S}F_{S}^{\text{sep}},\mathbb{Q}_{\ell }).\nonumber\end{eqnarray}$$

Proposition 3.1. The diagram

commutes.

Proof. Consider the commutative diagram

of topoi as in [Reference NakayamaNak98, §3], where ${\mathcal{Y}}^{\times ,\text{tame}}$ and ${\mathcal{X}}^{\times ,\text{tame}}$ are defined by ‘base change’ along $F_{S}\rightarrow F_{S}^{\text{tame}}$ and $F\rightarrow F^{\text{tame}}$ respectively. Then the isomorphism

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(Y^{\times ,\text{tame}},\mathbb{Q}_{\ell })\overset{{\sim}}{\rightarrow }H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}\times _{S}F_{S}^{\text{sep}},\mathbb{Q}_{\ell })\end{eqnarray}$$

is given as the composite

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{i}(Y^{\times ,\text{tame}},\mathbb{Q}_{\ell })\overset{{\sim}}{\leftarrow }H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}^{\times ,\text{tame}},\mathbb{Q}_{\ell })\rightarrow H_{\acute{\text{e}}\text{t}}^{i}({\mathcal{Y}}\times _{S}F_{S}^{\text{sep}},\mathbb{Q}_{\ell })\end{eqnarray}$$

using the proper base change theorem in log-étale cohomology [Reference NakayamaNak97, Theorem 5.1], and there is a similar statement for ${\mathcal{X}}$. The claim then follows simply from commutativity of the above diagram of log schemes.◻

We will also need a version of this result for $p$-adic cohomology. Write $W=W(k)$, $W_{S}=W(k_{S})$, let $K=W[1/p]$, $K_{S}=W_{S}[1/p]$, and let ${\mathcal{R}}_{K}\supset {\mathcal{E}}_{K}^{\dagger }\subset {\mathcal{E}}_{K}$, and ${\mathcal{R}}_{K_{S}}\supset {\mathcal{E}}_{K_{S}}^{\dagger }\subset {\mathcal{E}}_{K_{S}}$ denote copies of the Robba ring, the bounded Robba ring and the Amice ring over $K$ and $K_{S}$ respectively. Lift the extension $F\rightarrow F_{S}$ to a finite flat morphism ${\mathcal{E}}_{K}^{\dagger }\rightarrow {\mathcal{E}}_{K_{S}}^{\dagger }$ which extends to finite flat morphisms ${\mathcal{R}}_{K}\rightarrow {\mathcal{R}}_{K_{S}}$ and ${\mathcal{E}}_{K}\rightarrow {\mathcal{E}}_{K_{S}}$. Then, as above, the morphism of log schemes $Y^{\times }\rightarrow X^{\times }$ induces a morphism

$$\begin{eqnarray}H_{\text{log}\text{-}\text{cris}}^{i}(X^{\times }/K^{\times })\rightarrow H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times })\end{eqnarray}$$

in log crystalline cohomology, and the morphism ${\mathcal{Y}}_{F_{S}}\rightarrow {\mathcal{X}}_{F}$ induces a morphism

$$\begin{eqnarray}H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K})\rightarrow H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}})\end{eqnarray}$$

in Robba-ring valued rigid cohomology. Then following [Reference Chiarellotto and LazdaCL18, Proposition 5.4] we can construct isomorphisms

$$\begin{eqnarray}\displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(X^{\times }/K^{\times })\otimes _{K}{\mathcal{R}}_{K} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K}),\nonumber\\ \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times })\otimes _{K_{S}}{\mathcal{R}}_{K_{S}} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}})\nonumber\end{eqnarray}$$

as follows. Let $t$ denote a co-ordinate on ${\mathcal{E}}_{K}^{\dagger }$ and $t_{S}$ a co-ordinate on ${\mathcal{E}}_{K_{S}}^{\dagger }$ such that $t\in W_{S}\unicode[STIX]{x27E6}t_{S}\unicode[STIX]{x27E7}$. Write $S_{K}=K\otimes W\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ and $S_{K_{S}}=K_{S}\otimes W_{S}\unicode[STIX]{x27E6}t_{S}\unicode[STIX]{x27E7}$. Equip $W\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ (respectively $W_{S}\unicode[STIX]{x27E6}t_{S}\unicode[STIX]{x27E7}$) with the log structure defined by the ideal $(t)\subset W\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ (respectively $(t_{S})\subset W\unicode[STIX]{x27E6}t_{S}\unicode[STIX]{x27E7}$) and define the log-crystalline cohomology groups

$$\begin{eqnarray}\displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{X}}^{\times }/S_{K}) & := & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{X}}^{\times }/W\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7})\otimes _{\mathbb{Z}}\mathbb{Q},\nonumber\\ \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{Y}}^{\times }/S_{K_{S}}) & := & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{Y}}^{\times }/W_{S}\unicode[STIX]{x27E6}t_{S}\unicode[STIX]{x27E7})\otimes _{\mathbb{Z}}\mathbb{Q};\nonumber\end{eqnarray}$$

these are naturally endowed with the extra structure of $\text{log}\text{-}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$-modules over $S_{K}$ and $S_{K_{S}}$ respectively. Moreover, we have isomorphisms of $\unicode[STIX]{x1D711}$-modules

$$\begin{eqnarray}\displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{X}}^{\times }/S_{K})\otimes _{S_{K},t\mapsto 0}K & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times }),\nonumber\\ \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{Y}}^{\times }/S_{K_{S}})\otimes _{S_{K_{S}},t_{S}\mapsto 0}K_{S} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times }),\nonumber\end{eqnarray}$$

by smooth and proper base change in log-crystalline cohomology, as well as isomorphisms of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$-modules

$$\begin{eqnarray}\displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{X}}^{\times }/S_{K})\otimes _{S_{K}}{\mathcal{R}}_{K} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K}),\nonumber\\ \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}({\mathcal{Y}}^{\times }/S_{K_{S}})\otimes _{S_{K_{S}}}{\mathcal{R}}_{K_{S}} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}}),\nonumber\end{eqnarray}$$

by [Reference Lazda and PálLP16, Proposition 5.45]. It therefore follows from the logarithmic form of Dwork’s trick [Reference KedlayaKed10, Corollary 17.2.4] that the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$-modules $H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K})$ and $H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}})$ are unipotent, that there are isomorphisms

$$\begin{eqnarray}\displaystyle \left(H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K})[\log t]\right)^{\unicode[STIX]{x1D6FB}=0} & \cong & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(X^{\times }/K^{\times }),\nonumber\\ \displaystyle \left(H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}})[\log t_{S}]\right)^{\unicode[STIX]{x1D6FB}=0} & \cong & \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times })\nonumber\end{eqnarray}$$

and moreover the connection $\unicode[STIX]{x1D6FB}$ on the rigid cohomology groups appearing on the left-hand side can be completely recovered from the monodromy operator $N$ on the right-hand side. This allows us to construct isomorphisms of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$-modules

$$\begin{eqnarray}\displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(X^{\times }/K^{\times })\otimes _{K}{\mathcal{R}}_{K} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{X}}_{F}/{\mathcal{R}}_{K}),\nonumber\\ \displaystyle H_{\text{log}\text{-}\text{cris}}^{i}(Y^{\times }/K_{S}^{\times })\otimes _{K_{S}}{\mathcal{R}}_{K_{S}} & \overset{{\sim}}{\rightarrow } & \displaystyle H_{\text{rig}}^{i}({\mathcal{Y}}_{F_{S}}/{\mathcal{R}}_{K_{S}})\nonumber\end{eqnarray}$$

where the left-hand side is endowed a natural connection coming from $N$; for more details see, for example, [Reference MarmoraMar08, §3.2].

Proposition 3.2. The diagram

commutes.

Proof. Given the construction of the horizontal isomorphisms outlined above, it suffices to show that the diagram

of log-crystalline cohomology groups commutes, which as in Proposition 3.1 simply follows from functoriality of log-crystalline cohomology. ◻

4 Cohomology and global approximations

Now suppose that $k$ is a finite field, $F=k(\!(t)\!)$, and $X/F$ is a smooth and proper variety.

We will prove the following strengthened version of [Reference Chiarellotto and LazdaCL18, Corollary 5.5].

Theorem 4.2. For any smooth and proper variety $X/F$ there exists a globally defined smooth and proper variety $Z/F$ such that

$$\begin{eqnarray}H_{\ell }^{i}(X)\cong H_{\ell }^{i}(Z)\end{eqnarray}$$

for all $\ell$ (including $\ell =p$).

Once we have shown this, the proof of [Reference Chiarellotto and LazdaCL18, Theorem 6.1] can then be completed using [Reference Chiarellotto and LazdaCL18, Proposition 5.8], exactly as in the proof of [Reference Chiarellotto and LazdaCL18, Theorem 5.1].

To prove Theorem 4.2, first of all choose a proper and flat model ${\mathcal{X}}$ for $X$ over the ring of integers ${\mathcal{O}}_{F}$. By [Reference de JongdJ96, Theorem 6.5] we may choose an alteration ${\mathcal{X}}_{0}\rightarrow {\mathcal{X}}$ and a finite extension $F_{0}/F$ such that ${\mathcal{X}}_{0}$ is strictly semistable over ${\mathcal{O}}_{F_{0}}$.

Next, we take the fibre product ${\mathcal{X}}_{0}\times _{{\mathcal{X}}}{\mathcal{X}}_{0}$, and let ${\mathcal{X}}_{1}^{\prime }$ denote the disjoint union of the reduced, irreducible components of ${\mathcal{X}}_{0}\times _{{\mathcal{X}}}{\mathcal{X}}_{0}$ which are flat over ${\mathcal{O}}_{F_{0}}$, or equivalently which map surjectively to $\text{Spec}({\mathcal{O}}_{F_{0}})$. Once more applying [Reference de JongdJ96, Theorem 6.5] to each of the connected components of ${\mathcal{X}}_{1}^{\prime }$ in turn enables us to produce:

1. – a 2-truncated augmented simplicial scheme

   $$\begin{eqnarray}{\mathcal{X}}_{1}\rightrightarrows {\mathcal{X}}_{0}\rightarrow {\mathcal{X}}\end{eqnarray}$$
   which is a proper hypercover after base changing to $F$;

2. – a collection $F_{1,1},\ldots ,F_{1,s}$ of finite field extensions of $F_{0}$

such that ${\mathcal{X}}_{1}$ is a disjoint union of schemes ${\mathcal{X}}_{1,j}$, for $1\leqslant j\leqslant s$, proper and strictly semistable over $\text{Spec}({\mathcal{O}}_{F_{1,j}})$.

Let $k_{0}$ denote the residue field of $F_{0}$, $k_{1,j}$ the residue field of $F_{1,j}$, and consider the intermediate extensions

$$\begin{eqnarray}F\subset F_{0}^{\text{un}}\subset F_{0}^{s}\subset F_{0}\subset F_{1,j}^{\text{un}}\subset F_{1,j}^{s}\subset F_{1,j},\end{eqnarray}$$

where $F_{0}^{\text{un}}/F$ and $F_{1,j}^{\text{un}}/F_{0}$ are separable and unramified, $F_{0}^{s}/F_{0}^{\text{un}}$ and $F_{1,j}^{s}/F_{1,j}^{\text{un}}$ are separable and totally ramified, and $F_{0}/F_{0}^{s}$ and $F_{1,j}/F_{1,j}^{s}$ are totally inseparable, of degree $p^{d_{0}}$ and $p^{d_{1,j}}$ respectively. Let $t$ denote a uniformiser for $F$, $t_{0}$ one for $F_{0}^{s}$, and let $P_{0}$ be the minimal polynomial of $t_{0}$ over $F_{0}^{\text{un}}$. Then $t_{0}^{\prime }:=t_{0}^{1/p^{d_{0}}}$ is a uniformiser for ${\mathcal{O}}_{F_{0}}$. Similarly, let $t_{1,j}$ be a uniformiser for $F_{1,j}^{s}$, and $P_{1,j}$ the minimal polynomial of $t_{1,j}$ over $F_{1,j}^{\text{un}}$. Then $t_{1,j}^{\prime }:=t_{1,j}^{1/p^{d_{1,j}}}$ is a uniformiser for ${\mathcal{O}}_{F_{1,j}}$.

Now choose a finitely generated sub-$k$-algebra $R\subset {\mathcal{O}}_{F}$, containing $t$, such that there exists a proper, flat scheme ${\mathcal{Y}}\rightarrow \text{Spec}(R)$ whose base change to ${\mathcal{O}}_{F}$ is exactly ${\mathcal{X}}$. By [Reference SpivakovskySpi99, Theorem 10.1], we may at any point increase $R$ to ensure that it is in fact smooth over $k$. Next, enlarge $R$ so that $R_{0}^{\text{un}}:=R\,\otimes _{k}\,k_{0}\subset {\mathcal{O}}_{F_{0}^{\text{un}}}$ contains all the coefficients of the minimal (Eisenstein) polynomial $P_{0}$ of $t_{0}$, and let $R_{0}^{s}$ denote the corresponding finite flat extension $R_{0}^{\text{un}}[x]/(P_{0})$ of $R_{0}^{\text{un}}$. We can thus consider $R_{0}^{s}\subset {\mathcal{O}}_{F_{0}^{s}}$ as a subring containing $t_{0}$, and we set $R_{0}=R_{0}^{s}[t_{0}^{\prime }]$. Hence we have $R_{0}\subset {\mathcal{O}}_{F_{0}}$ such that

$$\begin{eqnarray}R_{0}\otimes _{R}{\mathcal{O}}_{F}\overset{{\sim}}{\rightarrow }{\mathcal{O}}_{F_{0}}.\end{eqnarray}$$

Note also that $R_{0}$ is finite and flat over $R$; after localising $R$ within ${\mathcal{O}}_{F}$ we may in fact assume that $R_{0}$ is finite free over $R$.

Next we enlarge $R$ so that there exists a proper and flat morphism ${\mathcal{Y}}_{0}\rightarrow \text{Spec}(R_{0})$ whose base change to ${\mathcal{O}}_{F_{0}}$ is ${\mathcal{X}}_{0}$. Again, by further enlarging $R$ we may in addition assume that the map ${\mathcal{X}}_{0}\rightarrow {\mathcal{X}}$ arises from a proper surjective map

$$\begin{eqnarray}{\mathcal{Y}}_{0}\rightarrow {\mathcal{Y}}\end{eqnarray}$$

of $R$-schemes, and moreover that there exists an open cover of ${\mathcal{Y}}_{0}$ by schemes which are étale over $R_{0}[x_{1},\ldots ,x_{n}]/(x_{1}\cdots x_{r}-t_{0}^{\prime })$ for some $n,r$. In other words, ${\mathcal{Y}}_{0}$ is ‘strictly $t_{0}^{\prime }$-semistable’.

We now repeat this process to produce further finite free extensions $R_{0}\rightarrow R_{1,j}^{\text{un}}\rightarrow R_{1,j}^{s}\rightarrow R_{1,j}$ for all $j$, and an injection $R_{1,j}\subset {\mathcal{O}}_{F_{1,j}}$ containing the image of $t_{1,j}^{\prime }$ such that

$$\begin{eqnarray}R_{1,j}\otimes _{R}{\mathcal{O}}_{F}\overset{{\sim}}{\rightarrow }{\mathcal{O}}_{F_{1,j}}.\end{eqnarray}$$

We can also find proper, strictly $t_{1,j}^{\prime }$-semistable schemes ${\mathcal{Y}}_{1,j}\rightarrow \text{Spec}(R_{1,j})$ whose base change to ${\mathcal{O}}_{F_{1,j}}$ is ${\mathcal{X}}_{1,j}$, so that setting ${\mathcal{Y}}_{1}:=\coprod _{j}{\mathcal{Y}}_{1,j}$ (and again, possibly increasing $R$), we obtain a 2-truncated augmented simplicial scheme

$$\begin{eqnarray}{\mathcal{Y}}_{1}\rightrightarrows {\mathcal{Y}}_{0}\rightarrow {\mathcal{Y}}\end{eqnarray}$$

which becomes a proper hypercover over a dense open subscheme of $\text{Spec}(R)$, and whose base change to ${\mathcal{O}}_{F}$ is exactly our original 2-truncated augmented simplicial scheme

$$\begin{eqnarray}{\mathcal{X}}_{1}\rightrightarrows {\mathcal{X}}_{0}\rightarrow {\mathcal{X}}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D704}:R{\hookrightarrow}{\mathcal{O}}_{F}$ denote the canonical inclusion, and $\unicode[STIX]{x1D704}^{\ast }:\text{Spec}({\mathcal{O}}_{F})\rightarrow \text{Spec}(R)$ the induced morphism of schemes. Note that since $\unicode[STIX]{x1D704}^{\ast }$ maps the generic point of $\text{Spec}({\mathcal{O}}_{F})$ to that of $\text{Spec}(R)$, the map ${\mathcal{Y}}\rightarrow \text{Spec}(R)$ is generically smooth. We may thus choose an open subset $U\subset \text{Spec}(R)$ such that ${\mathcal{Y}}_{U}\rightarrow U$ is smooth, and such that the base change of $[{\mathcal{Y}}_{1}\rightrightarrows {\mathcal{Y}}_{0}\rightarrow {\mathcal{Y}}]$ to $U$ is a proper hypercover.

Lemma 4.3. For any $n\geqslant 0$ there exists a smooth curve $C/k$, a rational point $c\in C(k)$, a uniformiser $t_{c}$ at $c$, and a locally closed immersion $C\rightarrow \text{Spec}(R)$ such that $C\setminus \{c\}\subset U$, and the induced map

$$\begin{eqnarray}\text{Spec}({\mathcal{O}}_{C,c}/\mathfrak{m}_{c}^{n})\rightarrow \text{Spec}(R)\end{eqnarray}$$

agrees with the modulo $t^{n}$-reduction of $\unicode[STIX]{x1D704}^{\ast }$ via the isomorphism

$$\begin{eqnarray}\widehat{{\mathcal{O}}}_{C,c}\overset{{\sim}}{\rightarrow }{\mathcal{O}}_{F}\end{eqnarray}$$

sending $t_{c}$ to $t$.

The canonical inclusion $\unicode[STIX]{x1D704}$ induces similar inclusions

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{0}^{\#}:R_{0}^{\#}{\hookrightarrow}R_{0}^{\#}\otimes _{R}{\mathcal{O}}_{F}={\mathcal{O}}_{F_{0}^{\#}}\end{eqnarray}$$

for $\#\in \{\text{un},s,\emptyset \}$, as well as

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{1,j}^{\#}:R_{1,j}^{\#}{\hookrightarrow}R_{1,j}^{\#}\otimes _{R}{\mathcal{O}}_{F}={\mathcal{O}}_{F_{1,j}^{\#}}\end{eqnarray}$$

for all $j$, and again for $\#\in \{\text{un},s,\emptyset \}$. We will need the following form of Krasner’s lemma [Sta18, §0BU9].

We will use this as follows: given $n_{1}\geqslant \max _{j}\{[F_{1,j}:F]\}$ Lemma 4.4 shows that there exists some $n_{0}\geqslant \max \left\{2,[F_{0}:F]\right\}$ such that any polynomial $Q_{1,j}$ with coefficients in ${\mathcal{O}}_{F_{1,j}^{\text{un}}}$ which agrees with the minimal polynomial $P_{1,j}$ of $t_{1,j}$ modulo $(t_{0}^{\prime })^{n_{0}}$ is Eisenstein, and has a root in ${\mathcal{O}}_{F_{1,j}^{s}}$ which agrees with $t_{1,j}$ modulo $t_{1,j}^{n_{1}}$. Applying the lemma again shows the existence of some $n\geqslant 2$ such that any polynomial $Q_{0}$ with coefficients in ${\mathcal{O}}_{F_{0}^{\text{un}}}$ which agrees with $P_{0}$ modulo $t^{n}$ is Eisenstein, and has a root in ${\mathcal{O}}_{F_{0}^{s}}$ which agrees with $t_{0}$ modulo $t_{0}^{n_{0}}$. Now choose a $k$-algebra homomorphism $\unicode[STIX]{x1D706}:R\rightarrow {\mathcal{O}}_{F}$ as provided by Lemma 4.3, that is, factoring through the local ring of some smooth point on a curve inside $\text{Spec}(R)$ and agreeing with $\unicode[STIX]{x1D704}$ modulo $t^{n}$.

Since $\unicode[STIX]{x1D706}$ is a $k$-algebra homomorphism, we have a canonical isomorphism $R_{0}^{\text{un}}\otimes _{R,\unicode[STIX]{x1D706}}{\mathcal{O}}_{F}\overset{{\sim}}{\rightarrow }{\mathcal{O}}_{F_{0}^{\text{un}}}$, which therefore induces a homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{0}^{\text{un}}:R_{0}^{\text{un}}\rightarrow {\mathcal{O}}_{F_{0}^{\text{un}}}\end{eqnarray}$$

extending $\unicode[STIX]{x1D706}$ and which agrees with $\unicode[STIX]{x1D704}_{0}^{\text{un}}$ modulo $t^{n}$. Now let $Q_{0}=\unicode[STIX]{x1D706}_{0}^{\text{un}}(P_{0})$ denote the image under $\unicode[STIX]{x1D706}_{0}^{\text{un}}$ of the minimal polynomial $P_{0}$ of $t_{0}$; this is therefore a monic polynomial with coefficients in ${\mathcal{O}}_{F_{0}^{\text{un}}}$, which agrees with $P_{0}$ modulo $t^{n}$. Thus it is also Eisenstein, and by the choice of $n$ we know that ${\mathcal{O}}_{F_{0}^{s}}$ contains a root of $\unicode[STIX]{x1D706}_{0}^{\text{un}}(P_{0})$ which is congruent to $t_{0}$ modulo $t_{0}^{n_{0}}$ and generates ${\mathcal{O}}_{F_{0}^{s}}$ as an ${\mathcal{O}}_{F_{0}^{\text{un}}}$-algebra. This then allows us to extend $\unicode[STIX]{x1D706}_{0}^{\text{un}}$ to a homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{0}^{s}:R_{0}^{s}\rightarrow {\mathcal{O}}_{F_{0}^{s}}\end{eqnarray}$$

which agrees with $\unicode[STIX]{x1D704}_{0}^{s}$ modulo $t_{0}^{n_{0}}$, and since $\unicode[STIX]{x1D706}_{0}^{s}(t_{0})$ generates ${\mathcal{O}}_{F_{0}^{s}}$ as an ${\mathcal{O}}_{F_{0}^{\text{un}}}$-algebra, we deduce that the diagram

is coCartesian. We can then extend this to a homomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{0}:R_{0}\rightarrow {\mathcal{O}}_{F_{0}}\end{eqnarray}$$

agreeing with $\unicode[STIX]{x1D704}_{0}$ modulo $(t_{0}^{\prime })^{n_{0}}$, and forming a similar coCartesian diagram to $\unicode[STIX]{x1D706}_{0}^{s}$. We now play exactly the same game for all of the $R_{1,j}$, to produce $\unicode[STIX]{x1D706}_{1,j}:R_{1,j}\rightarrow {\mathcal{O}}_{F_{1,j}}$ extending all other $\unicode[STIX]{x1D706}_{0}$ and all previous $\unicode[STIX]{x1D706}_{1,j}^{\#}$, which agree with $\unicode[STIX]{x1D704}_{1,j}$ modulo $(t_{1,j}^{\prime })^{n_{1,j}}$, and which form coCartesian diagrams

Now let ${\mathcal{Z}}$ be the base change of ${\mathcal{Y}}$ to ${\mathcal{O}}_{F}$ via $\unicode[STIX]{x1D706}$; note that the generic fibre ${\mathcal{Z}}_{F}$ is globally defined by construction. Similarly define ${\mathcal{Z}}_{0}$ to be the base change of ${\mathcal{Y}}_{0}$ to ${\mathcal{O}}_{F_{0}}$ via $\unicode[STIX]{x1D706}_{0}$, ${\mathcal{Z}}_{1,j}$ the base change of ${\mathcal{Y}}_{1,j}$ to ${\mathcal{O}}_{F_{1,j}}$ via $\unicode[STIX]{x1D706}_{1,j}$, and ${\mathcal{Z}}_{1}:=\coprod _{j}{\mathcal{Z}}_{1,j}$, so we have a 2-truncated augmented simplicial scheme

$$\begin{eqnarray}{\mathcal{Z}}_{1}\rightrightarrows {\mathcal{Z}}_{0}\rightarrow {\mathcal{Z}}\end{eqnarray}$$

over ${\mathcal{O}}_{F}$, which gives a proper hypercover after base changing to $F$. For any $m\geqslant 2$ we can therefore take $n_{1}\geqslant m\max _{j}\{[F_{1,j}:F]\}$ to ensure:

1. – ${\mathcal{Z}}_{0}$ is a proper and strictly semistable scheme over ${\mathcal{O}}_{F_{0}}$, and each ${\mathcal{Z}}_{1,j}$ is a proper and strictly semistable scheme over ${\mathcal{O}}_{F_{1,j}}$;

2. – there is an isomorphism

   $$\begin{eqnarray}\left[{\mathcal{X}}_{1}\rightrightarrows {\mathcal{X}}_{0}\right]\otimes _{{\mathcal{O}}_{F}}{\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }\left[{\mathcal{Z}}_{1}\rightrightarrows {\mathcal{Z}}_{0}\right]\otimes _{{\mathcal{O}}_{F}}{\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
   of 2-truncated simplicial schemes, such that
   $$\begin{eqnarray}{\mathcal{X}}_{0}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{0}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
   is in fact an isomorphism of ${\mathcal{O}}_{F_{0}}/(t^{m})$-schemes, and
   $$\begin{eqnarray}{\mathcal{X}}_{1}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{1}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
   is obtained as a disjoint union of isomorphisms
   $$\begin{eqnarray}{\mathcal{X}}_{1,j}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{1,j}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
   of ${\mathcal{O}}_{F_{1,j}}/(t^{m})$-schemes.

Thus if we let ${\mathcal{X}}_{0,s}^{\times }$ and ${\mathcal{Z}}_{0,s}^{\times }$ denote the log schemes over $k_{0}^{\times }$ given by the special fibres of ${\mathcal{X}}_{0}$ and ${\mathcal{Z}}_{0}$, and ${\mathcal{X}}_{1,s}^{\times }$ and ${\mathcal{Z}}_{1,s}^{\times }$ the log schemes over $\coprod _{j=1}^{s}\text{Spec}(k_{1,j}^{\times })$ given by the special fibres of ${\mathcal{X}}_{1}$ and ${\mathcal{Z}}_{1}$, then by Proposition 2.2 there is an isomorphism

$$\begin{eqnarray}[{\mathcal{Z}}_{1,s}^{\times }\rightrightarrows {\mathcal{Z}}_{0,s}^{\times }]\cong [{\mathcal{X}}_{1,s}^{\times }\rightrightarrows {\mathcal{X}}_{0,s}^{\times }]\end{eqnarray}$$

of 2-truncated simplicial log schemes over $k^{\times }$. Now by [Reference Chiarellotto and LazdaCL18, Propositions 5.3, 5.4] there are isomorphisms

$$\begin{eqnarray}\displaystyle H_{\ell }^{i}({\mathcal{X}}_{0,F_{0}}) & \cong & \displaystyle H_{\ell }^{i}({\mathcal{Z}}_{0,F_{0}}),\nonumber\\ \displaystyle H_{\ell }^{i}({\mathcal{X}}_{1,F_{1,j}}) & \cong & \displaystyle H_{\ell }^{i}({\mathcal{Z}}_{1,j,F_{1,j}})\nonumber\end{eqnarray}$$

between the cohomology of the generic fibres of ${\mathcal{X}}_{0},{\mathcal{X}}_{1,j}$ and ${\mathcal{Z}}_{0},{\mathcal{Z}}_{1,j}$, as Weil–Deligne representations over $F_{0}$ and $F_{1,j}$ respectively. If we define the category

$$\begin{eqnarray}\text{Rep}_{\mathbb{Q}_{\ell }^{\prime }}(\text{WD}_{F_{1}}):=\mathop{\prod }_{j=1}^{s}\text{Rep}_{\mathbb{ Q}_{\ell }^{\prime }}(\text{WD}_{F_{1,j}})\end{eqnarray}$$

of Weil–Deligne representations over $F_{1}:=\prod _{j}F_{1,j}$ to be the product of the categories of Weil–Deligne representations over each $F_{1,j}$, then by Propositions 3.1 and 3.2, the diagram

(with horizontal arrows given by the differences of the two pullback maps) commutes via the restriction functor from Weil–Deligne representations over $F_{0}$ to Weil–Deligne representations over $F_{1}$.

Let $\text{Ind}_{F_{i}}^{F}$ denote a right adjoint to the restriction functor from Weil–Deligne representations over $F$ to those over $F_{i}$: on the separable part this is the normal induction of representations, on the inseparable part it is a quasi-inverse to Frobenius pull-back, and $\text{Ind}_{F_{1}}^{F}=\bigoplus _{j}\text{Ind}_{F_{1,j}}^{F}$. We therefore have a commutative diagram

and, in particular, the kernels of both horizontal maps are isomorphic as Weil–Deligne representations over $F$. The proof of Theorem 4.2 now boils down to the following claim.

Proposition 4.5. Let $X_{1}\rightrightarrows X_{0}\rightarrow X$ be a 2-truncated semisimplicial proper hypercover of a smooth and proper $F$-variety $X$, such that there exist finite field extensions $F_{0}/F$ and $F_{1,j}/F_{0}$ for $1\leqslant j\leqslant s$, with $X_{0}$ smooth over $F_{0}$, and $X_{1}=\coprod _{j}X_{1,j}$ with $X_{1,j}$ smooth over $F_{1,j}$. If we set $F_{1}=\prod _{j=1}^{s}F_{1,j}$, then

$$\begin{eqnarray}H_{\ell }^{i}(X)\cong \text{ker}(\text{Ind}_{F_{0}}^{F}H_{\ell }^{i}(X_{0})\rightarrow \text{Ind}_{F_{1}}^{F}H_{\ell }^{i}(X_{1}))\end{eqnarray}$$

for all primes $\ell$.

We now deduce from the proposition that $H_{\ell }^{i}(X)\cong H_{\ell }^{i}({\mathcal{Z}}_{F})$ as Weil–Deligne representations for all $i,\ell$, and by construction ${\mathcal{Z}}_{F}$ is globally defined. This completes the proof of Theorem 4.2