# Integral $p$-adic cohomology theories

Tomoyuki Abe and Richard Crew

##  1 Introduction

Suppose $\Lambda$ is a complete local ring with residue field $k$. Can we
expect that there is a “reasonable” cohomology theory $H^{\cdot}(X,\Lambda)$
with its values in $\Lambda$-modules for separated schemes of finite type over
a field $k$? Here “reasonable” means, in a first approximation that it has all
the usual properties of integral Betti cohomology when $k={\mathbb{C}}$. For
example if $\Lambda={\mathbb{Z}}_{\ell}$ with $\ell$ different from
$p=\mathrm{char}(k)$, $\ell$-adic cohomology
$H^{\cdot}_{\mathrm{et}}(X,{\mathbb{Z}}_{\ell})$ is such a theory. If
$\Lambda=W$ is a Cohen ring of $k$, crystalline cohomology gives such a theory
when $X/k$ is proper and smooth, but not in more general situations: Berthelot
found that the torsion of $H^{1}_{\mathit{crys}}(X,W)$ could be infinite if
$X$ was even mildly singular. Of course if we replace $W$ by its fraction
field $K$, rigid cohomology has the desired properties.

In the article in which he first discussed such theories Grothendieck
emphasized the importance of $p$-torsion phenomena, which is only visible in
an “integral” $p$-adic theory such as crystalline cohomology:

> Such a theory should associate to each scheme $X$ of finite type over a
> perfect field $k$ of characteristic $p>0$, cohomology groups which are
> modules over an integral domain, whose quotient field is of characteristic
> $0$, and which satisfy all the desirable formal properties (functoriality,
> finite-dimensionality…). This cohomology should also, most importantly,
> explain torsion phenomena, and in particular $p$-torsion. ([9, §1.7],
> emphasis in the original).

The work of Illusie and others on the de Rham-Witt complex shows the immense
richness of $p$-torsion phenonmena, but of course only in the proper smooth
case.

When $k$ is perfect and $X/k$ is smooth Davis, Langer and Zink [6] have
defined an overconvergent version of the de Rham-Witt complex
$W^{\dagger}\Omega^{\cdot}_{X/W(k)}$ and constructed an isomorphism

$H^{\cdot}(X,W^{\dagger}\Omega^{\cdot}_{X/W(k)})\otimes{\mathbb{Q}}\simeq
H^{\cdot}_{\mathit{rig}}(X)$

when $X$ is quasiprojective. It’s evidently too much to hope that the
$W(k)$-modules $H^{\cdot}(X,W^{\dagger}\Omega^{\cdot}_{X/W(k)})$ are finitely
generated, and we will see that this is never true if $X$ is a smooth affine
curve. Davis, Langer and Zink made the more reasonable conjecture that the
image of $H^{\cdot}(X,W^{\dagger}\Omega^{\cdot}_{X/W(k)})$ in
$H^{\cdot}_{\mathit{rig}}(X)$ is finitely generated as a $W(k)$-module, but in
a recent preprint Ertl and Shiho [7] have produced counterexamples to this
assertion as well. In any case one loses all torsion information by replacing
$H^{\cdot}(X,W^{\dagger}\Omega^{\cdot}_{X/W(k)})$ by its image in
$H^{\cdot}_{\mathit{rig}}(X)$.

The main result of this paper is a nonexistence theorem for certain theories
of this sort. We assume that a theory has a comparison theorem with rigid
cohomology (or rigid cohomology with compact support), that the cohomology
groups are finitely generated $\Lambda$-modules, and – a natural but important
additional restriction – the theory is compatible with finite étale descent in
a manner to be explained later. The result is that there is no such theory
even for affine curves. We do not rule out the possibility of theories
satisfying weaker descent conditions, such as cdh-descent. In fact in another
recent preprint [8] Ertl, Shiho and Sprang construct a “good” integral
$p$-adic theory under certain assumptions concerning resolution of
singularities in positive characteristic. This construction uses cdh-descent
and so does not contradict our result if their hypotheses on resolution hold.
They also consider a theory based on simplicial generically étale hypercovers,
and show that it is not independent of the choice of hypercover, and thus does
not provide a theory compatible with finite étale descent. We should remark
finally that Bhargav Bhatt has also given a proof of the nonexistence of this
sort of theory (private communication) by considering Artin-Schreier covers of
the the affine line by itself; he shows that the existence of such a theory
would imply that the affine line has Euler characteristic $0$. The method of
the present paper on the other hand shows that the misbehavior of $p$-adic
cohomology theories is ubiquitous, at least as far as curves are concerned;
see the example at the end of section 4.

The original version of this result was inspired by discussions at the
conference “$p$-adic cohomology and arithmetic applications” at BIRS in
October 2017. Later versions were produced during visits of the second author
to the Kavli IPMU in November 2019 and the University of Rennes I in May 2022,
and he would like to thank both instutiions for their hospitality. The first
author is supported by JSPS KAKENHI Grant Numbers 16H05993, 18H03667,
20H01790.

##  2 Globally Perfect Models

In what follows $\Lambda$ is a complete noetherian local ring with residue
field $k$ and fraction field $K$. As usual $D_{\mathit{perf}}(\Lambda)$ is the
triangulated category of perfect complexes of $\Lambda$-modules, which since
$\Lambda$ is local means that an object of $D_{\mathit{perf}}(\Lambda)$ is
quasi-isomorphic to a bounded complex of free $\Lambda$-modules.

We have in mind the following requirements on a cohomology theory
$H^{\cdot}(X)$ on a subcategory of the category of $k$-schemes of finite type:

  * •

$H^{\cdot}(X)$ can be used to compute rigid cohomology (or rigid cohomology
with compact supports) via a suitable comparison theorem.

  * •

$H^{\cdot}(X)$ may be computed as the cohomology of an object of
$D_{\mathit{perf}}(\Lambda)$;

  * •

The $H^{\cdot}(X)$ are compatible with finite étale descent, in a sense to be
explained presently.

When $\Lambda$ is regular the second requirement reduces to the condition that
the $H^{n}(X)$ be finitely generated $\Lambda$-modules, and vanish for $|n|\gg
0$.

To explain the last condition we use the finite étale site, referring to the
book of Abbes, Gros and Tsuji [1, Ch. VI §9] for proofs of the assertions used
below. If $X$ is a scheme the site $\mathrm{Fet}(X)$ is the category of finite
étale morphisms $Y\to X$, and the coverings are surjective morphisms. The
associated topos will be written $X_{\mathrm{fet}}$. A morphism $\pi:Y\to X$
induces a morphism $\pi_{\mathrm{fet}}:Y_{\mathrm{fet}}\to X_{\mathrm{fet}}$
of topoi; in what follows we will abbreviate the associated functors
$\pi_{\mathrm{fet}}^{*}$ and $\pi_{\mathrm{fet}*}$ by $\pi^{*}$ and $\pi_{*}$
(in other words the latter refer to the morphism of finite étale topoi, not
étale topoi). There is also a projection $\rho_{X}:X_{\mathrm{et}}\to
X_{\mathrm{fet}}$ such that
$\pi_{\mathrm{fet}}\rho_{Y}\simeq\rho_{X}\pi_{\mathrm{et}}$ in
$\mathrm{Fet}(X)$ for any $\pi:Y\to X$. If $X$ is a coherent scheme with
finitely many components (e.g. if $X$ is of finite type over a field) the
inverse image $\rho_{X}^{*}:X_{\mathrm{fet}}\to X_{\mathrm{et}}$ is fully
faithful ([1, Prop. VI.9.18]).

###### 2.1 Definition

An object $M$ of $D^{+}(X_{\mathrm{fet}},\Lambda)$ is _globally perfect_ if
there are integers $a\leq b$ such that for all $\pi:Y\to X$ in
$\mathrm{Fet}(X)$, $R\Gamma(Y,\pi^{*}M)$ is an object of
$D^{[a,b]}_{\mathit{perf}}(\Lambda)$.

We denote by ${\operatorname{Mod}}^{\cdot}_{K}$ the category of
${\mathbb{Z}}$-graded $K$-vector spaces. The next definition formulates our
notion of what it means for a cohomology theory with values in $K$-vector
spaces to have a “good” integral model compatible with finite étale descent.

###### 2.2 Definition

Suppose $X$ is a $k$-scheme of finite type and
$H^{\cdot}:\mathrm{Fet}(X)\to{\operatorname{Mod}}^{.}_{K}$ is a functor. A
_globally perfect model of $H^{\cdot}$_ is a globally perfect $M$ in
$D^{+}(X_{\mathrm{fet}},\Lambda)$ and a functorial isomorphism

$H^{\cdot}(Y)\xrightarrow{\sim}H^{\cdot}(R\Gamma(Y,\pi^{*}M))\otimes_{\Lambda}K$

for all $Y\to X$ in $\mathrm{Fet}(X)$.

Note that if $\Lambda$ is regular the condition on $R\Gamma(Y,\pi^{*}M)$ is
equivalent to saying that the $H^{p}(Y,\pi^{*}M)$ are finitely generated and
vanish outside of a finite range depending only on $X$.

###### 2.3 Theorem

Suppose $\Lambda$ is noetherian, $X$ is _affine_ , $k_{\mathrm{et}}$ has
finite cohomological dimension, $M$ is a globally perfect object of
$D(X_{\mathrm{fet}},\Lambda)$ and $\pi:Y\to X$ is finite étale Galois with
group $G$. There is an object $M_{Y}$ of $D_{\mathit{perf}}(\Lambda[G])$ whose
image under the forgetful functor $D_{\mathit{perf}}(\Lambda[G])\to
D_{\mathit{perf}}(\Lambda)$ is isomorphic to $R\Gamma(Y,\pi^{*}M)$, and
$H^{\cdot}(M_{Y})\simeq H^{\cdot}(Y,\pi^{*}M)$ as $\Lambda[G]$-modules.

We could express the conclusion of the theorem by saying that
“$R\Gamma(Y,\pi^{*}M)$ with its $G$-action is a perfect complex of
$\Lambda[G]$-modules.”

For the proof of theorem 2.3 we will need some facts about cohomological
dimension. Recall that if $R$ is a ring and $X$ is a topos, the
$R$-cohomological dimension of $X$ is the smallest integer $d$ such that
$H^{n}(X,F)=0$ for all $n>d$ and every $R$-module $F$ in $X$. The
${\mathbb{Z}}$-cohomological dimension of $X$ will also be called simply the
cohomological dimension and will be denoted by $\mathrm{coh.dim}(X)$.

###### 2.4 Lemma

Suppose $k$ is a field of characteristic $p>0$ such that $k_{\mathrm{et}}$ has
finite cohomological dimension. Let $X$ be a $k$-scheme of finite type.

  * •

The cohomological dimension of $X_{\mathrm{et}}$ is finite.

  * •

If $X$ is affine the cohomological dimension of $X_{\mathrm{fet}}$ is finite.

Proof. If $k$ is separably closed, the first assertion is a theorem of Gabber
whose proof can be found in [4, §1.1]. The general case follows by the
Hochschild-Serre spectral sequence.

For the second we can assume that $X$ is connected; in this case [1, VI.9.8]
shows that $X_{\mathrm{fet}}$ is equivalent to the classifying topos
$B_{\pi_{1}(X)}$ and that

$H^{n}(X_{\mathrm{fet}},F)\simeq H^{n}(\pi_{1}(X),F)$

for all abelian sheaves $F$ on $X_{\mathrm{fet}}$. It thus suffices to show
that $H^{n}(\pi_{1}(X),F)$ vanishes for $|n|\gg 0$ and all continuous
$\pi_{1}(X)$-modules $F$.

Serre [10, Ch. 1 §2.2 Cor. 3] showed that $H^{n}(\pi_{1}(X),F)$ is torsion for
$n>0$, so $H^{n}(\pi_{1}(X),F)=0$ if $F$ is a ${\mathbb{Q}}$-module and $n>0$.
Again by Abbes-Gros-Tsuji [1, Prop. VI.9.12] $X_{\mathrm{fet}}$ is a coherent
topos and thus $H^{n}(\pi_{1}(X),\\_)$ commutes with filtered inductive
limits. A devissage using the exact sequence

$0\to F_{\mathit{tor}}\to F\to F\otimes{\mathbb{Q}}\to
F\otimes({\mathbb{Q}}/{\mathbb{Z}})\to 0$

reduces to the case when $F$ is torsion. Since $\pi_{1}(X)$ acts continuously
on $F$, the latter is a filtered inductive limit of $\pi_{1}(X)$-modules
corresponding to locally constant constructible torsion sheaves. So we can
assume that $F$ is constructible, and as $X$ is affine we can use a result of
Achinger [2, Thm. 1.1.1] showing that there are isomorphisms

$H^{\cdot}(X_{\mathrm{et}},F)\simeq H^{\cdot}(\pi_{1}(X),F).$

Thus the assertion follows from the étale case.

In what follows $\Lambda_{X}$ denotes the sheafification in $\mathrm{Fet}(X)$
of the constant presheaf with value $\Lambda$ (here $X$ is any scheme). If
$\pi:Y\to X$ is an object of $\mathrm{Fet}(X)$ then
$\Lambda_{Y}=\pi^{*}\Lambda_{X}$.

###### 2.5 Proposition

If $M$ is a globally perfect object of $D^{+}(X_{\mathrm{fet}},\Lambda)$ then
$M$ has finite Tor-dimension. Suppose conversely that $k_{\mathrm{et}}$ has
finite cohomological dimension, $X$ is affine and $M$ satisfies

  * •

$M$ has finite Tor-dimension, and

  * •

for every $\pi:Y\to X$ in $\mathrm{Fet}(X)$, $R\Gamma(Y,\pi^{*}M)$ is in
$D_{\mathit{perf}}(\Lambda)$.

Then $M$ is globally perfect.

Proof. Suppose first that $M$ is globally perfect, and let $a\leq b$ be as in
definition 2.1. If $\bar{x}$ runs through the set of geometric points of $X$,
the family of fiber functors associated to the points $\rho_{X}(\bar{x})$ of
$X_{\mathrm{fet}}$ is conservative by [1, Lemma VI.9.6]. Since
$(M\otimes_{\Lambda}N)_{x}\simeq M_{x}\otimes_{\Lambda}N$ for any such point
it suffices to show that $M_{x}$ has finite Tor-dimension as a
$\Lambda$-module for all $x$, but this is true since $M_{x}$ is a direct limit
of complexes in $D^{[a,b]}_{\mathit{ftd}}(\Lambda)$.

Suppose conversely that $X$ is affine, $k_{\mathrm{et}}$ has finite
cohomological dimension and $M$ satisfies the two conditions. We can then
apply lemma 2.4 to conclude that $X_{\mathrm{fet}}$ has finite cohomological
dimension. As in the proof of lemma 2.4 we can identify $X_{\mathrm{fet}}$
with $B_{\pi_{1}(X)}$ and $M$ with an object of $D(\Lambda[\pi_{1}(X)])$ whose
image in $D(\Lambda)$ belongs to $D^{[a,b]}_{\mathit{ftd}}(\Lambda)$. If $Y\to
X$ is in $\mathrm{Fet}(X)$, $\pi_{1}(Y)$ is a subgroup of $\pi_{1}(X)$ and
thus

$\mathrm{coh.dim}(Y_{\mathrm{fet}})\leq\mathrm{coh.dim}(X_{\mathrm{fet}})$

by [10, 3.3 Prop. 14]. Since $X_{\mathrm{fet}}$ has finite cohomological
dimension we can apply [SGA4, Exp. XVII Thm. 5.2.11] to conclude that
$R\Gamma(Y,M)$ is in $D^{[a,b+\mathrm{coh.dim}(X)]}_{\mathit{ftd}}(\Lambda)$,
and since $R\Gamma(Y,\pi^{*}M)$ is in $D_{\mathit{perf}}(\Lambda)$ by
hypothesis, $M$ is globally perfect.

###### 2.6 Lemma

For any $\pi:Y\to X$ in $\mathrm{Fet}(X)$ and $M$ in
$D^{+}(X_{\mathrm{fet}},\Lambda)$ there is a functorial isomorphism

$M\otimes_{\Lambda_{X}}^{L}R\pi_{*}\Lambda_{Y}\xrightarrow{\sim}R\pi_{*}\pi^{*}(M).$

Proof. The adjunction $\pi^{*}R\pi_{*}(\Lambda_{Y})\to\Lambda_{Y}$ yields a
morphism

$\displaystyle\pi^{*}(M\otimes_{\Lambda_{X}}^{L}R\pi_{*}(\Lambda_{Y}))$
$\displaystyle\simeq\pi^{*}(M)\otimes_{\Lambda_{Y}}^{L}\pi^{*}R\pi_{*}(\Lambda_{Y})$
$\displaystyle\to\pi^{*}(M)\otimes_{\Lambda_{Y}}^{L}\Lambda_{Y}\simeq\pi^{*}(M)$

and applying the adjunction to this yields the morphism in the lemma. It will
be an isomorphism if it is after pulling it back by a covering morphism
$g:W\to X$ in $\mathrm{Fet}(X)$. By [1, VI.9.4] $g^{*}$ has an exact left
adjoint, so $g^{*}$ sends injectives to injectives and it follows that $g^{*}$
and $R\pi_{*}$ commute. We may therefore replace $\pi:Y\to X$ by its base
change $W\times_{X}Y\to W$. Now there is a finite étale $g:W\to X$ such that
$W\times_{X}Y$ is a disjoint sum of copies of $W$, and in this case the
assertion is clear.

The same argument as in the proof of the last lemma shows:

###### 2.7 Lemma

Suppose $\pi:Y\to X$ is finite étale and Galois with group $G$. The natural
morphism

$\pi_{*}\Lambda_{Y}\to R\pi_{*}\Lambda_{Y}$

is an isomorphism, and $\pi_{*}\Lambda_{Y}$ with its natural $G$-action is a
flat $\Lambda_{X}[G]$-module.

Proof of theorem 2.3: By lemma 2.7. the sheaf

$\mathcal{G}:=\pi_{*}\Lambda_{Y}$

is a flat sheaf of $\Lambda_{X}[G]$-modules. Since $M$ is globally perfect it
has finite Tor-dimension as an object of $D(X_{\mathrm{fet}},\Lambda_{X})$ by
proposition 2.5. Therefore $M\otimes^{L}_{\Lambda_{X}}\mathcal{G}$ has finite
Tor-dimension as an object of $D(X_{\mathrm{fet}},\Lambda_{X}[G])$. Since
$X_{\mathrm{fet}}$ has finite cohomological dimension by lemma 2.4 we can
appeal once again to [SGA4, Exp. XVII Thm. 5.2.11] to conclude that

$M_{Y}:=R\Gamma(X_{\mathrm{fet}},M\otimes^{L}_{\Lambda_{X}}\mathcal{G})$

is in $D_{\mathit{ftd}}(\Lambda[G])$. By the same reasoning the image of
$M_{Y}$ in $D(\Lambda)$ lies in $D^{b}(\Lambda)$; then $M_{Y}$ is in
$D^{b}(\Lambda[G])$. Since $\Lambda$ is noetherian it follows that $M_{Y}$ is
an object of $D_{\mathit{perf}}(\Lambda[G])$.

On the other hand lemmas 2.7 and 2.6 show that

$M\otimes^{L}_{\Lambda_{X}}\mathcal{G}\simeq
M\otimes^{L}_{\Lambda_{X}}R\pi_{*}{\Lambda_{Y}}\simeq R\pi_{*}\pi^{*}M$

in $D(X_{\mathrm{fet}},\Lambda_{X})$, whence isomorphisms

$M_{Y}=R\Gamma(X_{\mathrm{fet}},M\otimes^{L}_{\Lambda_{X}}\mathcal{G})\simeq
R\Gamma(X_{\mathrm{fet}},R\pi_{*}\pi^{*}M)\simeq
R\Gamma(Y_{\mathrm{fet}},\pi^{*}M).$

Since these isomorphisms are functorial and the $G$-module structure of
$\mathcal{G}=\pi_{*}\Lambda_{Y}$ was induced by the action of $G$ on $Y$ the
induced isomorphisms

$H^{\cdot}(M_{Y})\simeq H^{\cdot}(Y_{\mathrm{fet}},\pi^{*}M)$

are isomorphisms of $\Lambda[G]$-modules.

##  3 The main theorem

Denote by $K_{0}$ the fraction field of a Cohen ring of $k$, and recall that
$K$ is the fraction field of $\Lambda$, which we can make into an extension
field of $K_{0}$. In what follows rigid cohomology has coefficients in
$K_{0}$.

###### 3.1 Lemma

Suppose $f:X\to Y$ is a morphism of schemes. If $M$ is a globally perfect
object of $D(X_{\mathrm{fet}},\Lambda)$, $Rf_{*}M$ is a globally perfect
object of $D(Y_{\mathrm{fet}},\Lambda)$.

Proof. If $V\to Y$ is in $\mathrm{Fet}(Y)$, $X\times_{Y}V\to X$ is in
$\mathrm{Fet}(X)$ and

$R\Gamma(V,Rf_{*}M)\simeq R\Gamma(X\times_{Y}V,M)$

and the assertion follows.

###### 3.2 Theorem

Suppose $k$ is a field of characteristic $p>0$ and $\Lambda$ is a complete
noetherian local ring with residue field $k$ and fraction field $K$. Suppose
$X$ is a smooth affine curve over $k$. Neither of the functors

$H^{\cdot}_{\mathit{rig}}(\\_)\otimes_{K_{0}}K,\
H^{\cdot}_{\mathit{rig},c}(\\_)\otimes_{K_{0}}K:\mathrm{Fet}(X)\to{\operatorname{Mod}}_{K}^{\cdot}$

have a globally perfect model.

Proof: By duality it suffices to show this for cohomology with compact
supports, so we treat the case of $H^{\cdot}_{\mathit{rig},c}(\\_)$. We first
reduce to the case when $k_{\mathrm{et}}$ has finite cohomological dimension.
By standard arguments there is a subfield $k_{0}\subseteq k$ finitely
generated over ${\mathbb{F}}_{p}$ and a smooth affine curve $X_{0}$ over
$k_{0}$ such that $X\simeq X_{0}\otimes_{k_{0}}k$. Suppose $M$ is a globally
perfect model of
$H^{\cdot}_{\mathit{rig},c}(\\_):\mathrm{Fet}(X)\to{\operatorname{Mod}}_{K}^{\cdot}$;
if $f:X\to X_{0}$ is the projection then

$R\Gamma(X_{0},Rf_{*}M)\simeq R\Gamma(X,M)$

and $Rf_{*}M$ is a globally perfect model of

$H^{\cdot}_{\mathit{rig},c}(\\_)\otimes_{W}K:\mathrm{Fet}(X_{0})\to{\operatorname{Mod}}_{K}^{\cdot}$

where $W$ is a Cohen ring for $k_{0}$. Thus it suffices to show that this
functor does not have a globally perfect model; i.e. we can replace $X/k$ by
$X_{0}/k_{0}$. Equivalently we can assume that $k_{\mathrm{et}}$ has finite
cohomological dimension, and as $X$ is affine we can apply theorem 2.3.

Any smooth affine curve $X$ has an Artin-Schreier cover $\pi:Y\to X$ that
ramifies at infinity. So it suffices to invoke the following lemma:

###### 3.3 Lemma

With the hypotheses of the theorem, suppose

  * •

$\pi:Y\to X$ is a finite étale Galois cover whose group $G$ is a $p$-group;

  * •

$M$ is a globally perfect model of the functor
$H^{\cdot}_{\mathit{rig}}(\\_)\otimes_{K_{0}}K:\mathrm{Fet}(X)\to{\operatorname{Mod}}_{K}^{\cdot}$
or of
$H^{\cdot}_{\mathit{rig},c}(\\_)\otimes_{K_{0}}K:\mathrm{Fet}(X)\to{\operatorname{Mod}}_{K}^{\cdot}$.

Then $\pi:Y\to X$ extends to a finite étale morphism
$\bar{\pi}:\bar{Y}\to\bar{X}$ of smooth projective curves.

Proof. We first consider the case of the functor
$H^{\cdot}_{\mathit{rig},c}(\\_)\otimes_{K_{0}}K$. If it has a globally
perfect module then by the previous theorem there is a perfect complex
$M_{Y}^{\cdot}$ of $\Lambda[G]$-modules such that

$H^{n}(M_{Y}^{\cdot}\otimes_{\Lambda}K)\simeq
H^{n}_{\mathit{rig},c}(Y)\otimes_{K_{0}}K$

and then

$H^{n}(M_{Y}^{\cdot})^{G}\simeq H^{n}_{\mathit{rig},c}(X)\otimes_{K_{0}}K$

since $\pi:Y\to X$ is finite étale.

Since $G$ is a $p$-group, the group ring $\Lambda[G]$ is local, so a finitely
generated projective $\Lambda[G]$-module is free. Therefore these last
isomorphisms imply that

$\chi_{c}(X)=|G|\chi_{c}(Y)$ (3.3.1)

where

$\chi_{c}(Z)=\sum_{i}(-1)^{i}\dim_{K_{0}}H^{i}_{\mathit{rig},c}(Z)$

for any separated $k$-scheme $Z$ of finite type.

Comparing 3.3.1 with the Grothendieck-Ogg-Shafarevich formula shows that the
ramification of $\pi:Y\to X$ is tame at infinity. But since $G$ is a
$p$-group, $\pi$ must be totally wild at infinity if it is ramified at all. It
is therefore unramified at infinity, which is the conclusion of the lemma.

Finally if $H^{\cdot}_{\mathit{rig}}(\\_)\otimes_{K_{0}}K$ has a global
perfect model the same argument shows that $\chi(Y)=|G|\chi(X)$ where $\chi$
is now the Euler characteristic for rigid cohomology without supports. But for
any smooth variety $Z$, $\chi(Z)=\chi_{c}(Z)$ and we conclude as before.

##  4 Examples

The “standard” cohomology theories can all be computed by complexes of étale
sheaves, and therefore satisfy étale cohomological descent. Thus when finite
they have globally perfect models. For the sake of completeness let’s review
the basic examples. The reader should have no problem thinking of global
perfect models for various sorts of cohomology theories: $\ell$-adic
cohomology with or without supports, crystalline cohomology of proper smooth
varieties (use the crystalline-étale topos $(X/V)_{\mathit{crys-et}}$ of
Berthelot-Breen-Messing [3], or else the de Rham-Witt complex). The case of
$p$-adic étale cohomology with compact supports (where $p$ is the residual
characteristic) was used in [5]

Suppose on the other hand that $k$ is perfect and
$W^{\dagger}\Omega^{\cdot}_{X/W}$ is the complex of Davis, Langer and Zink [6]
and let $\Lambda=W(k):=W$. If the $W$-modules
$H^{\cdot}(Y,W^{\dagger}\Omega^{\cdot}_{Y/W})$ were finitely generated for all
$Y\to X$ in $\mathrm{Fet}(X)$ then $W^{\dagger}\Omega^{\cdot}_{Y/W}$ would be
a global perfect model of
$H^{\cdot}_{\mathit{rig}}(\\_):\mathrm{Fet}(X)\to{\operatorname{Mod}}^{\cdot}_{K}$
where $K$ is the fraction field of $W=W(k)$. We conclude that for every smooth
affine curve $X$ there is a finite étale $Y\to X$ such that
$H^{\cdot}(Y,W^{\dagger}\Omega^{\cdot}_{Y/W})$ is not a finitely generated
$W$-module.

## References

  * [1] A. Abbes, M. Gros and T. Tsuji, The $p$-adic Simpson Correspondence, Ann. Math. Studies 193, Princeton Univ. Press 2016.
  * [2] P. Achinger, Wild Ramification and $K(\pi,1)$ Spaces, Inv. Math. 210 (2017) pp. 453–499
  * [3] P. Berthelot, L. Breen and W. Messing, Théorie de Dieudonné Cristalline II, Lecture Notes in Math. 930, Springer 1982.
  * [4] D.-C. Cisinski and F. Déglise, Étale Motives, Comp. Math. 152 no. 3 (2016) pp. 556–666.
  * [5] R. Crew, Etale $p$-covers in characteristic $p$, Comp. Math. 52 (1984) pp. 31–45.
  * [6] C. Davis., A. Langer and T. Zink, Overconvergent de Rham-Witt cohomology, Ann. Sci. ENS 44 (2011) p.197–262.
  * [7] V. Ertl and A. Shiho, On infiniteness of integral overconvergent de Rham-Witt cohomology modulo torsion, preprint arXiv:1812.03720
  * [8] V. Ertl, A. Shiho and J. Sprang, Integral $p$-adic cohomology theories for open and singular varieties, preprint https://arxiv.org/abs/2105.11009
  * [9] A. Grothendieck, Crystals and the De Rham Cohomology of Schemes, in Dix exposées sur la cohomologie des schémas, North-Holland 1968.
  * [10] J.-P. Serre, Cohomologie galoisienne, Lecture Notes in Math. 5, Springer 1965.
  * [SGA4] A. Grothendieck, M. Artin and J.-L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas vol. 3, Lecture Notes in Math. 307, Springer 1973.

Tomoyuki Abe:
Kavli Institute for the Physics and Mathematics of the Universe (WPI)
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan
e-mail<EMAIL_ADDRESS>
Richard Crew:
Department of Mathematics, University of Florida
358 Little Hall, Gainesville FL 32611
USA
e-mail<EMAIL_ADDRESS>