1 Introduction
It was pointed out to us by Zheng that the proof of [Reference Chiarellotto and LazdaCL18, Theorem 6.1] is invalid. The problem is in the final step of the proof on p. 237, where we showed that there was an exact sequence
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU1.png?pub-status=live)
and claimed to deduce $\ell$-independence of
$H_{\ell }^{i}(X)$ from
$\ell$-independence of all the other terms
$H_{\ell }^{i+n}(X_{n})$. Of course, this deduction does not work, since there might be infinitely many such other terms.
In their paper [Reference Lu and ZhengLZ19], Lu and Zheng provide (amongst other things) an alternative proof of this $\ell$-independence result, at least for
$\ell \neq p$, see Theorem 1.4(2). In this corrigendum we will explain how to fix the proof of [Reference Chiarellotto and LazdaCL18, Theorem 6.1] by instead proving a stronger version of [Reference Chiarellotto and LazdaCL18, Corollary 5.5] where the semistable hypothesis is removed. In particular, this includes the case
$\ell =p$.
Notation and conventions
We will use notation from [Reference Chiarellotto and LazdaCL18] freely.
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
$R$.
Proposition 2.1 (Illusie, Nakayama [Reference NakayamaNak98, Appendix A.4]).
If ${\mathcal{X}},{\mathcal{X}}^{\prime }$ are strictly semistable schemes over
$R$, and
$g:X_{1}\rightarrow X_{1}^{\prime }$ is an isomorphism between their mod
$\unicode[STIX]{x1D70B}^{2}$-reductions, then
$g$ induces a canonical isomorphism
$g:(X,{\mathcal{M}}_{X})\overset{{\sim}}{\rightarrow }(X^{\prime },{\mathcal{M}}_{X^{\prime }})$ of log schemes over
$k^{\times }$.
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:
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU2.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU3.png?pub-status=live)
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU4.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU5.png?pub-status=live)
of log structures by mapping
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU6.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU7.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU8.png?pub-status=live)
of $S$- and
$R$-schemes respectively such that the diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU9.png?pub-status=live)
commutes. Then by Proposition 2.1 we obtain isomorphisms
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU10.png?pub-status=live)
of log schemes over $k_{S}^{\times }$, as well as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU11.png?pub-status=live)
of log schemes over $k^{\times }$. The above commutative diagrams of strictly semistable schemes induce commutative diagrams
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU12.png?pub-status=live)
of log schemes. Note that the morphism of punctured points along the bottom of each square is given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU13.png?pub-status=live)
where $u\in S^{\ast }$ is such that
$\unicode[STIX]{x1D70B}=u\unicode[STIX]{x1D70B}_{S}^{e}$.
Proposition 2.2. The diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU14.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU15.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU16.png?pub-status=live)
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU17.png?pub-status=live)
respectively, and the morphisms ${\mathcal{M}}_{X}\rightarrow {\mathcal{M}}_{Y}$ and
${\mathcal{M}}_{X}^{\prime }\rightarrow {\mathcal{M}}_{Y}^{\prime }$ are defined by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU18.png?pub-status=live)
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU19.png?pub-status=live)
respectively. Hence in the diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU20.png?pub-status=live)
the composite $f\circ g_{X}$ is given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU21.png?pub-status=live)
and the composite $g_{Y}\circ f$ is given by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU22.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU23.png?pub-status=live)
in ${\mathcal{O}}_{Y_{e}}$ and so deduce that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU24.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU25.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU26.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU27.png?pub-status=live)
and hence a $G_{F_{S}}$-equivariant morphism
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU28.png?pub-status=live)
in cohomology, for any $\ell \neq p$. On the other hand we have a natural
$G_{F_{S}}$-equivariant map
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU29.png?pub-status=live)
and by [Reference NakayamaNak98, Proposition 4.2] equivariant isomorphisms
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU30.png?pub-status=live)
Proposition 3.1. The diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU31.png?pub-status=live)
commutes.
Proof. Consider the commutative diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU32.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU33.png?pub-status=live)
is given as the composite
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU34.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU35.png?pub-status=live)
in log crystalline cohomology, and the morphism ${\mathcal{Y}}_{F_{S}}\rightarrow {\mathcal{X}}_{F}$ induces a morphism
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU36.png?pub-status=live)
in Robba-ring valued rigid cohomology. Then following [Reference Chiarellotto and LazdaCL18, Proposition 5.4] we can construct isomorphisms
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU37.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU38.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU39.png?pub-status=live)
by smooth and proper base change in log-crystalline cohomology, as well as isomorphisms of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FB})$-modules
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU40.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU41.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU42.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU43.png?pub-status=live)
commutes.
Proof. Given the construction of the horizontal isomorphisms outlined above, it suffices to show that the diagram
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU44.png?pub-status=live)
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.
Definition 4.1. We say that $X$ is globally defined if there exist a smooth curve
$C/k$, a
$k$-valued point
$c\in C(k)$, a smooth and proper morphism
$\mathbf{X}\rightarrow (C\setminus \{c\})$ and an isomorphism
$F\cong \widehat{k(C)}_{c}$ such that
$\mathbf{X}_{F}\cong X$.
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU45.png?pub-status=live)
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:
– a 2-truncated augmented simplicial scheme
which is a proper hypercover after base changing to$$\begin{eqnarray}{\mathcal{X}}_{1}\rightrightarrows {\mathcal{X}}_{0}\rightarrow {\mathcal{X}}\end{eqnarray}$$
$F$;
– 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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU47.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU48.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU49.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU50.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU51.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU52.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU53.png?pub-status=live)
agrees with the modulo $t^{n}$-reduction of
$\unicode[STIX]{x1D704}^{\ast }$ via the isomorphism
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU54.png?pub-status=live)
sending $t_{c}$ to
$t$.
Proof. Since $R$ is smooth, we may choose étale co-ordinates around the image
$\unicode[STIX]{x1D704}^{\ast }(s)$ of the closed point of
$\text{Spec}({\mathcal{O}}_{F})$ under
$\unicode[STIX]{x1D704}^{\ast }$. This induces an étale map
$\text{Spec}(R)\rightarrow \mathbb{A}_{k}^{n}$ for some
$n$, and it is a simple exercise to prove the corresponding claim for
$\mathbb{A}_{k}^{n}$. We then just take the pull-back to
$\text{Spec}(R)$.◻
The canonical inclusion $\unicode[STIX]{x1D704}$ induces similar inclusions
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU55.png?pub-status=live)
for $\#\in \{\text{un},s,\emptyset \}$, as well as
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU56.png?pub-status=live)
for all $j$, and again for
$\#\in \{\text{un},s,\emptyset \}$. We will need the following form of Krasner’s lemma [Sta18,
§0BU9].
Lemma 4.4. Let $K$ be a local field, with ring of integers
${\mathcal{O}}_{K}$, and let
$P(x)$ be an Eisenstein polynomial over
${\mathcal{O}}_{K}$. Let
$L$ be the corresponding finite totally ramified extension, and let
$\unicode[STIX]{x1D6FC}$ be a root of
$P$ in
$L$. Then for any
$m\geqslant 1$ there exists an
$n\geqslant 2$ such that any
$Q(x)\in {\mathcal{O}}_{K}[x]$ congruent to
$P$ modulo
$\mathfrak{m}_{K}^{n}$ is Eisenstein, and
$L$ contains a root
$\unicode[STIX]{x1D6FD}$ of
$Q$ such that
$L=K(\unicode[STIX]{x1D6FD})$ and
$\unicode[STIX]{x1D6FC}\equiv \unicode[STIX]{x1D6FD}$ modulo
$\mathfrak{m}_{L}^{m}$.
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU57.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU58.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU59.png?pub-status=live)
is coCartesian. We can then extend this to a homomorphism
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU60.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU61.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU62.png?pub-status=live)
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:
–
${\mathcal{Z}}_{0}$ 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
${\mathcal{O}}_{F_{1,j}}$;
– there is an isomorphism
of 2-truncated simplicial schemes, such that$$\begin{eqnarray}\left[{\mathcal{X}}_{1}\rightrightarrows {\mathcal{X}}_{0}\right]\otimes _{{\mathcal{O}}_{F}}{\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }\left[{\mathcal{Z}}_{1}\rightrightarrows {\mathcal{Z}}_{0}\right]\otimes _{{\mathcal{O}}_{F}}{\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
is in fact an isomorphism of$$\begin{eqnarray}{\mathcal{X}}_{0}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{0}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
${\mathcal{O}}_{F_{0}}/(t^{m})$-schemes, and
is obtained as a disjoint union of isomorphisms$$\begin{eqnarray}{\mathcal{X}}_{1}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{1}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
of$$\begin{eqnarray}{\mathcal{X}}_{1,j}\otimes {\mathcal{O}}_{F}/t^{m}\overset{{\sim}}{\rightarrow }{\mathcal{Z}}_{1,j}\otimes {\mathcal{O}}_{F}/t^{m}\end{eqnarray}$$
${\mathcal{O}}_{F_{1,j}}/(t^{m})$-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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU67.png?pub-status=live)
of 2-truncated simplicial log schemes over $k^{\times }$. Now by [Reference Chiarellotto and LazdaCL18, Propositions 5.3, 5.4] there are isomorphisms
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU68.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU69.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU70.png?pub-status=live)
(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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU71.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20200617023242326-0309:S0010437X20007228:S0010437X20007228_eqnU72.png?pub-status=live)
for all primes $\ell$.
Proof. By taking $\widetilde{F}_{1}/F$ a sufficiently large finite extension such that all of the
$F_{1,j}$ embed into
$\widetilde{F}_{1}$ and applying [Reference de JongdJ96, Theorem 4.1], we can extend
$X_{1}\rightrightarrows X_{0}\rightarrow X$ to a full proper hypercover
$X_{\bullet }\rightarrow X$ such that for
$n\geqslant 2$ there exists a finite extension
$F_{n}/\widetilde{F}_{1}$ with
$X_{n}$ smooth over
$F_{n}$. Now applying [Reference Chiarellotto and LazdaCL18, Lemma 6.4] we can see that the terms in
$i$th column of the resulting spectral sequence have to be ‘quasi-pure’ of weight
$i$. Therefore the spectral sequence degenerates exactly as in the proof of [Reference Chiarellotto and LazdaCL18, Theorem 6.1], and the proposition follows.◻
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
Remark 4.6. Note the use of the finite field hypothesis (via a weight argument) in the proof of Proposition 4.5. It might be possible to relax the assumption to $k$ perfect using a more sophisticated argument.
Acknowledgements
Both authors would like to thank W. Zheng for pointing out the error in [Reference Chiarellotto and LazdaCL18], as well as L. Illusie for useful discussions concerning log structures, in particular the proof of Proposition 2.1. We would also like to thank the anonymous referee for a careful reading of an earlier version of the manuscript, in particular for correcting a mistake in our use of alterations.