# Ramified class field theory and duality over finite fields

Rahul Gupta, Amalendu Krishna  Fakultät für Mathematik, Universität
Regensburg, 93040, Regensburg, Germany<EMAIL_ADDRESS>regensburg.de Department of Mathematics, Indian Institute of Science,
Bangalore, 560012, India<EMAIL_ADDRESS>
###### Key words and phrases:

Milnor $K$-theory, Class field theory, $p$-adic étale cohomology, Cycles with
modulus

###### 2020 Mathematics Subject Classification:

Primary 14C25; Secondary 14F42, 19E15

> _Abstract._ We prove a duality theorem for the $p$-adic étale motivic
> cohomology of a variety $U$ which is the complement of a divisor on a smooth
> projective variety over ${\mathbb{F}}_{p}$. This extends the duality
> theorems of Milne and Jannsen-Saito-Zhao. The duality introduces a
> filtration on $H^{1}_{{\acute{e}t}}(U,{{\mathbb{Q}}}/{{\mathbb{Z}}})$. We
> identify this filtration to the classically known Matsuda filtration when
> the reduced part of the divisor is smooth. We prove a reciprocity theorem
> for the idele class groups with modulus introduced by Kerz-Zhao and Rülling-
> Saito. As an application, we derive the failure of Nisnevich descent for
> Chow groups with modulus.

###### Contents

  1. 1 Introduction
  2. 2 Relative Milnor $K$-theory
  3. 3 The idele class group of Kerz-Zhao
  4. 4 Comparison theorem for cohomology of $K$-sheaves
  5. 5 Relative logarithmic Hodge-Witt sheaves
  6. 6 Cartier map for twisted de Rham-Witt complex
  7. 7 The pairing of cohomology groups
  8. 8 Perfectness of the pairing
  9. 9 Reciprocity theorem for $C_{KS}(X|D)$
  10. 10 Counterexamples to Nisnevich descent

##  1\. Introduction

The objective of this paper is to study the duality and reciprocity theorems
for non-complete smooth varieties over finite fields and draw consequences.
Below we describe the contexts and statements of our main results.

###  1.1. The duality theorem

Let $k$ be a finite field of characteristic $p$ and $X$ a smooth projective
variety of dimension $d$ over $k$. Let
$W_{m}\Omega^{r}_{X,{\operatorname{log}}}$ be the logarithmic Hodge-Witt sheaf
on $X$, defined as the image of the dlog map from the Milnor $K$-theory sheaf
${\mathcal{K}}^{M}_{r,X}$ to the $p$-typical de Rham-Witt sheaf
$W_{m}\Omega^{r}_{X}$ in the étale topology. Milne [39] proved that there is a
perfect pairing of cohomology groups

(1.1) $H^{i}_{\acute{e}t}(X,W_{m}\Omega^{r}_{X,{\operatorname{log}}})\times
H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{X,{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{m}}.$

By [11, Theorem 8.4], there is an isomorphism
$H^{i}_{\acute{e}t}(X,W_{m}\Omega^{r}_{X,{\operatorname{log}}})\cong
H^{i+r}_{{\acute{e}t}}(X,{{\mathbb{Z}}}/{p^{m}}(r))$, where the latter is the
$p$-adic étale motivic cohomology due to Suslin-Voevodsky. Milne’s theorem can
thus be considered as a perfect duality for the $p$-adic étale motivic
cohomology of smooth projective varieties over $k$. The corresponding duality
for the $\ell$-adic étale cohomology is the classical Poincaré duality for
étale cohomology for a prime $\ell\neq p$.

If $U$ is a smooth quasi-projective variety over $k$ which is not complete,
then there is a perfect duality (see [43]) for the $\ell$-adic étale
cohomology in the form

(1.2) $H^{i}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{\ell^{m}}(r))\times
H^{2d+1-i}_{c}(U,{{\mathbb{Z}}}/{\ell^{m}}(d-r))\to{{\mathbb{Z}}}/{\ell^{m}},$

where $H^{i}_{c}(U,{{\mathbb{Z}}}/{\ell^{m}}(j))$ is the $\ell$-adic étale
cohomology of $U$ with compact support.

However, there is no known $p$-adic analogue of the étale cohomology of $U$
with compact support such that a $p$-adic analogue of (1.2) could hold.
Construction of this duality is yet an open problem. Recall that one of the
applications (which is the interest of this paper) of such a duality theorem
is to the study of the mod-$p$ étale fundamental group of $U$, which is in
general a very complicated object. A duality theorem such as above would allow
one to study this group in terms of a more tractable étale cohomology of $U$
with compact support.

In [22], Jannsen-Saito-Zhao proposed an approach in a special case when $U$ is
the complement of a divisor $D$ on a smooth projective variety $X$ over $k$
such that $D_{\rm red}$ is a simple normal crossing divisor. They constructed
a relative version of the logarithmic Hodge-Witt sheaves on $X$, denoted by
$W_{m}\Omega^{r}_{X|D,{\operatorname{log}}}$. Using these sheaves, they showed
that there is a semi-perfect pairing (see Definition 8.1)

(1.3)
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\times{\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{X|nD,{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{m}}$

of topological abelian groups, where the first group has discrete topology and
the second has profinite topology. This pairing is perfect when $i=1$ and
$r=0$.

In this paper, we propose a different approach to the $p$-adic duality for
$U$. This new approach has the advantage that it allows $D$ to be an arbitrary
divisor on $X$. This is possible by virtue of the choice of the relative
version of the logarithmic Hodge-Witt sheaves. Instead of using the Hyodo-Kato
de Rham-Witt sheaves with respect to a suitable log structure considered in
[22], we use a more ingenuous version of the relative logarithmic Hodge-Witt
sheaf, which we denote by $W_{m}\Omega^{r}_{(X,D),{\operatorname{log}}}$. The
latter is defined to be the kernel of the canonical surjection
$W_{m}\Omega^{r}_{X,{\operatorname{log}}}\twoheadrightarrow
W_{m}\Omega^{r}_{D,{\operatorname{log}}}$. Our main result on the $p$-adic
duality for $U$ is roughly the following. We refer to Theorem 8.6 for the
precise statement.

######  Theorem 1.1.

Let $X$ be a smooth projective variety of dimension $d$ over a finite field
$k$ of characteristic $p$. Let $D\subset X$ be an effective Cartier divisor
with complement $U$. Let $r,i\geq 0$ and $m\geq 1$ be integers. Then there is
a semi-perfect pairing of topological abelian groups

$H^{i}_{{\acute{e}t}}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\times{\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{m}}.$

This pairing is perfect if $D_{\rm red}$ is a simple normal crossing divisor,
$i=1,r=0$ and one of the conditions $\\{d\neq 2,k\neq{\mathbb{F}}_{2}\\}$
holds.

We show that Theorem 1.1 recovers the duality theorem of Jannsen-Saito-Zhao if
$D_{\rm red}$ is a simple normal crossing divisor.

As a consequence of Theorem 1.1, we obtain a filtration $\\{{\rm
fil}^{{\acute{e}t}}_{nD}H^{1}(K)\\}_{n}$, where $K$ is the function field of
$X$ and $H^{1}(K)$ is a shorthand for
$H^{1}_{\acute{e}t}(K,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ (see § 8.2). We let
${\rm fil}_{D}H^{1}(K)$ be the subgroup of $H^{1}(K)$ introduced in [15,
Definition 7.12]. This coincides with the filtration defined in [29,
Definition 2.9] by [16, Theorem 1.2]. ${\rm fil}_{D}H^{1}(K)$ can be described
as the subgroup of continuous characters of the absolute Galois group of $K$
whose Artin conductors (see [37, Definition 3.2.5]) at the generic points of
$D$ are bounded by the multiplicities of $D$ in $D_{\rm red}$. This is a more
intricate subgroup of $H^{1}(K)$ than ${\rm fil}^{{\acute{e}t}}_{D}H^{1}(K)$
because the latter can be described in terms of the simpler objects such as
the cohomology of the relative logarithmic Hodge-Witt sheaves. On the other
hand, ${\rm fil}_{D}H^{1}(K)$ determines the ramification theory of finite
étale coverings of $U$. It is therefore desirable to know if and when these
two filtrations agree. Our next result is the following.

######  Theorem 1.2.

Let $X$ be a smooth projective variety of dimension $d$ over a finite field
$k$ of characteristic $p$. Let $D\subset X$ be an effective Cartier divisor
such that $D_{\rm red}$ is regular. Assume that either $d\neq 2$ or
$k\neq{\mathbb{F}}_{2}$. Then

${\rm fil}_{D}H^{1}(K)={\rm fil}^{{\acute{e}t}}_{D}H^{1}(K).$

###  1.2. The reciprocity theorem

The purpose of reciprocity theorems in class field theory over a perfect field
is to be able to represent the abelianized étale fundamental groups of
varieties over the field in terms of idele class groups which are often
described in terms of explicit sets of generators and relations. Let us assume
that the base field $k$ is finite of characteristic $p$. In this case, such a
reciprocity theorem for smooth projective varieties over $k$ is due to Kato-
Saito [25] which describes the abelianized étale fundamental group in terms of
the Chow group of 0-cycles.

For the more intricate case of a smooth quasi-projective variety $U$ which is
not complete, an approach was introduced by Kato-Saito [26] whose underlying
idea is to study the so called ‘étale fundamental group of $X$ with modulus
$D$’, where $D\subset X$ is a fixed divisor which is supported on the
complement of $U$ in a normal compactification $U\subset X$. This group
characterizes the finite étale coverings of $U$ whose ramification along
$X\setminus U$ is bounded by $D$ in a certain sense. There are various ways to
make sense of this bound on the ramification, and they give rise to several
definitions of the étale fundamental group with modulus. It turns out that
depending on what one wants to do, each of these has certain advantage over
the others.

Kato and Saito were able to describe $\pi^{\rm ab}_{1}(U)$ in terms of the
limit (over $D$) of the idele class groups with modulus $H^{d}_{{\rm
nis}}(X,{\mathcal{K}}^{M}_{d,(X,D)})$, where $d=\text{\rm dim}(X)$. In [16,
Theorems 1.1], it was shown that $H^{d}_{{\rm
nis}}(X,{\mathcal{K}}^{M}_{d,(X,D)})$ describes $\pi^{{\rm adiv}}_{1}(X,D)$
for every $D$ if we let $\pi^{{\rm adiv}}_{1}(X,D)$ be the Pontryagin dual to
the Matsuda filtration ${\rm fil}_{D}H^{1}(K)$. It was also shown in loc. cit.
that $\pi^{{\rm adiv}}_{1}(X,D)$ coincides with the fundamental group with
modulus $\pi^{\rm ab}_{1}(X,D)$, introduced earlier by Deligne and Laumon [34]
if $X$ is smooth. The latter was shown to coincide (along the degree zero
parts) with the Chow group of 0-cycles with modulus ${\rm CH}_{0}(X|D)$ by
Kerz-Saito [29] (when $p\neq 2$) and Binda-Krishna-Saito [3].

If we use Kato’s Swan conductor instead of Matsuda’s Artin conductor to bound
the ramification in terms of a divisor supported away from $U$, we are led to
a different notion of the abelianized étale fundamental group with modulus
which we denote by $\pi^{{\rm abk}}_{1}(X,D)$. This is defined as the
Pontryagin dual to the subgroup ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)$. The latter
is the subgroup of continuous characters of the absolute Galois group of $K$
whose Swan conductors (defined by Kato [24]) at the generic points of $D$ are
bounded by the multiplicities of $D$ in $D_{\rm red}$.

One can now ask if $\pi^{{\rm abk}}_{1}(X,D)$ could be described by an idele
class group with modulus, similar to the $K$-theoretic idele class group of
Kato-Saito and the cycle-theoretic idele class group of Kerz-Saito. Our next
result solves this problem.

Let $\widehat{{\mathcal{K}}}^{M}_{r,X}$ be the improved Milnor $K$-theory
sheaf of Gabber and Kerz [27]. Let $\widehat{{\mathcal{K}}}^{M}_{r,X|D}$ be
the relative Milnor $K$-theory sheaf, defined locally as the image of the map
${\mathcal{K}}^{M}_{1,(X,D)}\otimes_{{\mathbb{Z}}}j_{*}{\mathcal{O}}^{\times}_{U}\otimes_{{\mathbb{Z}}}\cdots\otimes_{{\mathbb{Z}}}j_{*}{\mathcal{O}}^{\times}_{U}\to\widehat{{\mathcal{K}}}^{M}_{r,X}$.
We refer to Lemma 3.1 for the proof that this map is defined. This sheaf was
considered by Rülling-Saito [42] when $X$ is smooth and $D_{\rm red}$ is a
simple normal crossing divisor. There are degree maps $\deg\colon H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})\to{\mathbb{Z}}$ (see § 3.1) and
$\deg^{\prime}\colon\pi^{{\rm abk}}_{1}(X,D)\to\widehat{{\mathbb{Z}}}$. We let
$H^{d}_{\rm nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})_{0}={\rm Ker}(\deg)$
and $\pi^{{\rm abk}}_{1}(X,D)_{0}={\rm Ker}(\deg^{\prime})$.

######  Theorem 1.3.

Let $X$ be a normal projective variety of dimension $d$ over a finite field.
Let $D\subset X$ be an effective Cartier divisor whose complement is regular.
Then there is a continuous reciprocity homomorphism

$\rho^{\prime}_{X|D}\colon H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})\to\pi_{1}^{{\rm abk}}(X,D)$

with dense image such that the induced map

$\rho^{\prime}_{X|D}\colon H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})_{0}\xrightarrow{\cong}\pi_{1}^{{\rm
abk}}(X,D)_{0}$

is an isomorphism of finite groups.

Let $H^{i}_{\mathcal{M}}(X|D,{\mathbb{Z}}(j))$ be the motivic cohomology with
modulus. This is defined as the Nisnevich hypercohomology of the sheafified
cycle complex with modulus $z^{j}(X|D,2j-\bullet)$, introduced in [4]. Using
Theorem 1.3 and [42, Theorem 1], we obtain the following.

######  Corollary 1.4.

Assume in Theorem 1.3 that $X$ is regular and $D_{\rm red}$ is a simple normal
crossing divisor. Then there is an isomorphism of finite groups

$cyc^{\prime}_{X|D}\colon
H^{2d}_{\mathcal{M}}(X|D,{\mathbb{Z}}(d))_{0}\xrightarrow{\cong}\pi_{1}^{{\rm
abk}}(X,D)_{0}.$

This result can be viewed as the cycle-theoretic description of the
$\pi_{1}^{{\rm abk}}(X,D)$, analogous a similar result for $\pi_{1}^{{\rm
adiv}}(X,D)$, proven in [29], [3] and [16].

###  1.3. Failure of Nisnevich descent for Chow group with modulus

Recall that the classical higher Chow groups of smooth varieties satisfy the
Nisnevich descent in the sense that the canonical map ${\rm CH}^{i}(X,j)\to
H^{2i-j}_{\mathcal{M}}(X,{\mathbb{Z}}(i))$ is an isomorphism for $i,j\geq 0$,
where the latter is the Nisnevich hypercohomology of the sheafified (in
Nisnevich topology) Bloch’s cycle complex. However, this question for the
higher Chow groups with modulus is yet an open problem. Some cases of this
were verified by Rülling-Saito [42, Theorem 3]. As an application of Corollary
1.4, we prove the following result which provides a counterexample to the
Nisnevich descent for the Chow groups with modulus. This was one of our
motivations for studying the reciprocity for $\pi_{1}^{{\rm abk}}(X,D)$.

######  Theorem 1.5.

Let $X$ be a smooth projective surface over a finite field. Let $D\subset X$
be an effective Cartier divisor such that $D_{\rm red}$ is a simple normal
crossing divisor. Then the canonical map

${\rm CH}_{0}(X|D)\to H^{4}_{\mathcal{M}}(X,{\mathbb{Z}}(2))$

is not always an isomorphism.

Using Theorem 1.5, one can show that the Nisnevich descent for the Chow groups
with modulus fails over infinite fields too.

Recall that the map in Theorem 1.5 is known to be an isomorphism if $X$ is a
curve. This is related to the fact that for a Henselian discrete valuation
field $K$ with perfect residue field, the Swan conductor and the Artin
conductor agree for the characters of the absolute Galois group of $K$. In
this context, we also remark that one could attempt to define an analogue of
the Deligne-Laumon fundamental group with modulus $\pi^{\rm ab}_{1}(X,D)$ by
using the Brylinski-Kato filtration along all integral curves in $X$ not
contained in $D$. However, the resulting fundamental group will coincide with
$\pi^{{\rm adiv}}_{1}(X,D)$ and not yield anything new for the same reason as
above.

###  1.4. Overview of proofs

We give a brief overview of our proofs. The main idea behind the proof of our
duality theorem is the observation that the naive relative logarithmic Hodge-
Witt sheaves are isomorphic to the naive relative mod-$p$ Milnor $K$-theory
sheaves (in the sense of Kato-Saito) in the pro-setting. The hope that the
étale cohomology of relative mod-$p$ Milnor $K$-theory sheaves are the correct
objects to use for duality originated from our results in previous papers
which showed that the Nisnevich cohomology of the Kato-Saito relative Milnor
$K$-theory yields a reciprocity isomorphism without any condition on the
divisor.

To implement the above ideas, we need to prove many results about the relative
logarithmic Hodge-Witt sheaves and their relation with various versions of
relative Milnor $K$-theory. One of these is a pro-isomorphism between the
relative logarithmic Hodge-Witt sheaves and the twisted de Rham-Witt sheaves.
The latter are easier objects to work with because duality for them follows
from the Grothendieck-Serre coherent duality. The next step is to construct
higher Cartier operators on the twisted de Rham-Witt sheaves. This allows us
to construct the relative version of the two-term complexes considered by
Milne [39]. The proof of the duality theorem is then reduced to coherent
duality using an induction procedure.

To prove Theorem 1.2, we use our duality theorem and the reciprocity theorem
of [16] to reduce it to proving an independent statement that the top
Nisnevich cohomology of the relative Milnor $K$-theory sheaf coincides with
the corresponding étale cohomology if the reduced divisor is regular (see
Theorem 4.9).

To prove Theorem 1.3, we follow the strategy we used in the proof of a similar
result for $\pi^{{\rm adiv}}_{1}(X,D)$ in [16]. But there are some new
ingredients to be used. The first key ingredient is a result of Kerz-Zhao [30]
which gives an idelic presentation of $H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})$. The second key result is a
theorem of Kato [24] which gives a criterion for the characters of the
absolute Galois group of a Henselian discrete valuation field to annihilate
various subgroups of the Milnor $K$-theory of the field under a suitable
pairing (see Theorem 9.1). What remains after this is to prove Proposition
9.3.

We prove various results about relative Milnor $K$-theory in sections 2, 3 and
4. We study the relative logarithmic Hodge-Witt sheaves and prove some results
about their cohomology in § 5. We introduce the Cartier operators on the
twisted de Rham-Witt sheaves in § 6. We construct the pairing for the duality
theorem and prove its perfectness in sections 7 and 8. The reciprocity theorem
is proven in § 9 and the failure of Nisnevich descent for the Chow groups with
modulus is shown in § 10.

###  1.5. Notation

We shall work over a field $k$ of characteristic $p>0$ throughout this paper.
We let ${\operatorname{\mathbf{Sch}}}_{k}$ denote the category of separated
and essentially of finite type schemes over $k$. The product $X\times_{{\rm
Spec\,}(k)}Y$ in ${\operatorname{\mathbf{Sch}}}_{k}$ will be written as
$X\times Y$. We let $X^{(q)}$ (resp. $X_{(q)}$) denote the set of points on
$X$ having codimension (resp. dimension) $q$. We let
${\operatorname{\mathbf{Sch}}}_{k/{\rm zar}}$ (resp.
${\operatorname{\mathbf{Sch}}}_{k/{\rm nis}}$, resp.
${\operatorname{\mathbf{Sch}}}_{k/{\acute{e}t}})$ denote the Zariski (resp.
Nisnevich, resp. étale) site of ${\operatorname{\mathbf{Sch}}}_{k}$. We let
$\epsilon\colon{\operatorname{\mathbf{Sch}}}_{k/{\text{\'{e}t}}}\to{\operatorname{\mathbf{Sch}}}_{k/{\rm
nis}}$ denote the canonical morphism of sites. If ${\mathcal{F}}$ is a sheaf
on ${\operatorname{\mathbf{Sch}}}_{k/{\rm nis}}$, we shall denote
$\epsilon^{*}{\mathcal{F}}$ also by ${\mathcal{F}}$ as long as the usage of
the étale topology is clear in a context. For
$X\in{\operatorname{\mathbf{Sch}}}_{k}$, we shall let $\psi\colon X\to X$
denote the absolute Frobenius morphism.

For an abelian group $A$, we shall write
${\operatorname{Tor}}^{1}_{{\mathbb{Z}}}(A,{{\mathbb{Z}}}/n)$ as ${}_{n}A$ and
$A/{nA}$ as $A/n$. The tensor product $A\otimes_{{\mathbb{Z}}}B$ will be
written as $A\otimes B$. We shall let $A\\{p^{\prime}\\}$ denote the subgroup
of elements of $A$ which are torsion of order prime to $p$. We let $A\\{p\\}$
denote the subgroup of elements of $A$ which are torsion of order some power
of $p$.

##  2\. Relative Milnor $K$-theory

In this section, we recall several versions of relative Milnor $K$-theory
sheaves and establish some relations among their cohomology.

For a commutative ring $A$, the Milnor $K$-group $K^{M}_{r}(A)$ (as defined by
Kato [23]) is the $r$-th graded piece of the graded ring $K^{M}_{*}(A)$. The
latter is the quotient of the tensor algebra $T_{*}(A^{\times})$ by the two-
sided graded ideal generated by the homogeneous elements
$a_{1}\otimes\cdots\otimes a_{r}$ such that $r\geq 2$, and $a_{i}+a_{j}=1$ for
some $1\leq i\neq j\leq r$. The residue class of $a_{1}\otimes\cdots\otimes
a_{r}\in T_{r}(A^{\times})$ in $K^{M}_{r}(A)$ is denoted by the Milnor symbol
$\underline{a}=\\{a_{1},\ldots,a_{r}\\}$. Given an ideal $I\subset A$, the
relative Milnor $K$-theory $K^{M}_{*}(A,I)$ is defined as the kernel of the
restriction map $K^{M}_{*}(A)\to K^{M}_{*}(A/I)$.

Let $\widetilde{K}^{M}_{r}(A)$ denote the $r$-th graded piece of the graded
ring $\widetilde{K}^{M}_{*}(A)$, where the latter is the quotient of the
tensor algebra $T_{*}(A^{\times})$ by the two-sided graded ideal generated by
the homogeneous elements $a_{1}\otimes a_{2}$ such that $a_{1}+a_{2}=1$. We
let $\widetilde{K}^{M}_{*}(A,I)$ be the kernel of the restriction map
$\widetilde{K}^{M}_{*}(A)\to\widetilde{K}^{M}_{*}(A/I)$. It is clear that
there is a natural surjection
$\widetilde{K}^{M}_{*}(A)\twoheadrightarrow{K}^{M}_{*}(A)$. This is an
isomorphism if $A$ is a local ring with infinite residue field (see [27,
Proposition 2]). The following says that a similar thing holds also for the
relative $K$-theory.

######  Lemma 2.1.

Let $A$ be a local ring and let $I\subset A$ be an ideal. Then the following
hold.

  1. (1)

$K^{M}_{r}(A,I)$ and $\widetilde{K}^{M}_{r}(A,I)$ are generated by the Milnor
symbols $\\{a_{1},\ldots,a_{r}\\}$ such that $a_{i}\in K^{M}_{1}(A,I)$ for
some $1\leq i\leq r$.

  2. (2)

The canonical map $\widetilde{K}^{M}_{r}(A,I)\to K^{M}_{r}(A,I)$ is
surjective. This is an isomorphism if $A$ has infinite residue field.

###### Proof.

The assertion (1) for $K^{M}_{r}(A,I)$ is [26, Lemma 1.3.1] and the proof of
(1) for $\widetilde{K}^{M}_{r}(A,I)$ is completely identical to that of
$K^{M}_{r}(A,I)$. The second part of (2) follows from the corresponding result
in the non-relative case mentioned above. To prove the first part of (2), we
fix an integer $r\geq 1$. Let
$\underline{a}=\\{a_{1},\ldots,a_{r}\\}\in\widetilde{K}^{M}_{r}(A)$ be such
that $a_{i}\in\widetilde{K}^{M}_{1}(A,I)=K^{M}_{1}(A,I)$ for some $1\leq i\leq
r$. It is then clear that $\underline{a}\in\widetilde{K}^{M}_{r}(A,I)$. Using
this observation, our assertion follows directly from item (1) for
$K^{M}_{r}(A,I)$. ∎

For a ring $A$ as above, we let $\widehat{K}^{M}_{*}(A)$ denote the improved
Milnor $K$-theory defined independently by Gabber and Kerz [27]. For an ideal
$I\subset A$, we define $\widehat{K}^{M}_{*}(A,I)$ to be the kernel of the map
$\widehat{K}^{M}_{*}(A)\to\widehat{K}^{M}_{*}(A/I)$. Let $K_{*}(A)$ denote the
Quillen $K$-theory of $A$. We state the following facts as a lemma and refer
to [27] for their source.

######  Lemma 2.2.

There are natural maps

(2.1)
$K^{M}_{*}(A)\xleftarrow{\alpha_{A}}\widetilde{K}^{M}_{*}(A)\xrightarrow{\beta_{A}}\widehat{K}^{M}_{*}(A)\xrightarrow{\gamma_{A}}K_{*}(A),$

where $\alpha_{A}$ is always surjective and $\beta_{A}$ is surjective if $A$
is local. These two maps are isomorphisms if $A$ is either a field or a local
ring with infinite residue field.

Let $X$ be a Noetherian scheme and $\iota\colon D\hookrightarrow X$ a closed
immersion. We shall say that $(X,D)$ is a modulus pair if $D$ is an effective
Cartier divisor on $X$. This Cartier divisor may be empty. If ${\mathcal{P}}$
is a property of schemes, we shall say that $(X,D)$ satisfies ${\mathcal{P}}$
if $X$ does so. We shall say that $(X,D)$ has dimension $d$ if $X$ has Krull
dimension $d$.

Given a Noetherian scheme $X$ and an integer $r\geq 1$, let
${\mathcal{K}}^{M}_{r,X}$ denote the sheaf on $X_{\rm nis}$ whose stalk at a
point $x\in X$ is $K^{M}_{r}({\mathcal{O}}^{h}_{X,x})$. We let
${\mathcal{K}}^{M}_{r,(X,D)}$ be the kernel of the restriction map
${\mathcal{K}}^{M}_{r,X}\twoheadrightarrow{\mathcal{K}}^{M}_{r,D}:=\iota_{*}{\mathcal{K}}^{M}_{r,D}$.
We shall usually refer to ${\mathcal{K}}^{M}_{r,(X,D)}$ as the Kato-Saito
relative Milnor $K$-sheaves. We define the sheaves
$\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}$ and
$\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}$ in an analogous way. We let
${\mathcal{K}}_{r,X}$ be the Quillen $K$-theory sheaf on $X_{\rm nis}$.

######  Lemma 2.3.

Let $X$ be a reduced Noetherian scheme of Krull dimension $d$ and let
$D\subset X$ be a nowhere dense closed subscheme. Then the canonical map

$H^{d}_{\rm nis}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n)\to H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{r,(X,D)}}/n)$

is an isomorphism for every integer $n\geq 0$.

###### Proof.

We look at the commutative diagram

(2.2)
$\textstyle{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{{\mathcal{K}}}^{M}_{r,X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}^{M}_{r,(X,D)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}^{M}_{r,X}.}$

Lemma 2.1 says that the left vertical arrow is surjective. We have seen above
that the kernel of the right vertical arrow is supported on a nowhere dense
closed subscheme of $X$ (this uses reducedness of $X$). Hence, the same holds
for the left vertical arrow. But this easily implies the lemma when $n=0$.

For $n\geq 1$, we use the commutative diagram

$\textstyle{H^{d}_{\rm
nis}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{H^{d}_{\rm
nis}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\rm
nis}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{r,(X,D)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n}$$\textstyle{H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{r,(X,D)}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{r,(X,D)}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It is easily seen that the two rows are exact. The $n=0$ case shows that the
left and the middle vertical arrows are isomorphisms. It follows that the
right vertical arrow is also an isomorphism. ∎

For $D\subset X$ as above and $n\geq 1$, let

(2.3)
$\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}:={\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/{(\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}\cap
n\widetilde{{\mathcal{K}}}^{M}_{r,X})}={\rm
Image}(\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}\to{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/n).$

Since the map $n\widetilde{{\mathcal{K}}}^{M}_{r,X}\to
n\widetilde{{\mathcal{K}}}^{M}_{r,D}$ is surjective, it easily follows that
there are exact sequences

(2.4)
$0\to\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}\to{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/n\to{\widetilde{{\mathcal{K}}}^{M}_{r,D}}/n\to
0;$
${}_{n}{\widetilde{{\mathcal{K}}}^{M}_{r,D}}\to{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n\to\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}\to
0.$

We let
$\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}:={\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/{(\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}\cap
n\widehat{{\mathcal{K}}}^{M}_{r,X})}$. It is then clear that the two exact
sequences of (2.4) hold for the improved Milnor $K$-sheaves as well.

The following is immediate from (2.4).

######  Lemma 2.4.

Let $X$ be a Noetherian scheme of Krull dimension $d$ and let $D\subset X$ be
a nowhere dense closed subscheme. Then the canonical map

$H^{d}_{\rm nis}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n)\to H^{d}_{\rm
nis}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})$

is an isomorphism for all $n\geq 1$.

To proceed further, we need the following result about the sheaf cohomology.
Given a field $k$ and a topology $\tau$ on ${\rm Spec\,}(k)$, let
$cd_{\tau}(k)$ denote the cohomological dimension of $k$ for torsion
$\tau$-sheaves.

######  Lemma 2.5.

Let $k$ be a field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$. Let
${\mathcal{F}}$ be a torsion sheaf on $X_{\rm nis}$ such that
${\mathcal{F}}_{x}=0$ for every $x\in X_{(q)}$ with $q>0$. Then
$H^{q}_{\tau}(X,{\mathcal{F}})=0$ for $q>cd_{\tau}(k)$ and $\tau\in\\{{\rm
nis},{\acute{e}t}\\}$. In particular, given an exact sequence of Nisnevich
sheaves

$0\to{\mathcal{F}}\to{\mathcal{F}}^{\prime}\to{\mathcal{F}}^{\prime\prime}\to
0,$

the induced map $H^{q}_{\tau}(X,{\mathcal{F}}^{\prime})\to
H^{q}_{\tau}(X,{\mathcal{F}}^{\prime\prime})$ is an isomorphism for
$q>cd_{\tau}(k)$ and $\tau\in\\{{\rm nis},{\acute{e}t}\\}$.

###### Proof.

We fix $\tau\in\\{{\rm nis},{\acute{e}t}\\}$. Let $J$ be the set of finite
subsets of $X_{(0)}$. Given any $\alpha\in J$, let $S_{\alpha}\subset X$ be
the 0-dimensional reduced closed subscheme defined by $\alpha$ and let
$U_{\alpha}=X\setminus S_{\alpha}$. Let $\iota_{\alpha}\colon
S_{\alpha}\hookrightarrow X_{\alpha}$ and $j_{\alpha}\colon
U_{\alpha}\hookrightarrow X$ be the inclusions. As $J$ is cofiltered with
respect to inclusion, there is a cofiltered system of short exact sequences of
$\tau$-sheaves

(2.5)
$0\to(j_{\alpha})_{!}({\mathcal{F}}|_{U_{\alpha}})\to{\mathcal{F}}\to(\iota_{\alpha})_{*}({\mathcal{F}}|_{S_{\alpha}})\to
0.$

It is easily seen from our assumption that ${\underset{\alpha\in
J}{\varinjlim}}\ (j_{\alpha})_{!}({\mathcal{F}}|_{U_{\alpha}})=0$ so that the
map ${\mathcal{F}}\to{\underset{\alpha\in J}{\varinjlim}}\
(\iota_{\alpha})_{*}({\mathcal{F}}|_{S_{\alpha}})$ is an isomorphism. Since
$H^{*}_{\tau}(S_{\alpha},{\mathcal{F}}|_{S_{\alpha}})\cong
H^{*}_{\tau}(X,(\iota_{\alpha})_{*}({\mathcal{F}}|_{S_{\alpha}}))$, we are
done by [8, Proposition 58.89.6]. ∎

######  Lemma 2.6.

Let $k$ be a field and $(X,D)$ a reduced modulus pair in
${\operatorname{\mathbf{Sch}}}_{k}$. Assume that $X$ is of pure dimension
$d\geq 1$ and $\tau\in\\{{\rm nis},{\acute{e}t}\\}$. Let $n\geq 1$ be an
integer and

(2.6) $H^{d}_{\tau}(X,\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)})\to
H^{d}_{\tau}(X,\widehat{{\mathcal{K}}}^{M}_{r,(X,D)});$ (2.7)
$H^{d}_{\tau}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})\to
H^{d}_{\tau}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})$

the canonical maps. Then the following hold.

  1. (1)

If $k$ is infinite, then both maps are isomorphisms.

  2. (2)

If $d=1$ and $r\leq 1$, then both maps are isomorphisms.

  3. (3)

If $d\geq 2$ and $\tau={\rm nis}$, then both maps are isomorphisms.

  4. (4)

If $k$ is finite, $d\geq 3$ and $\tau={\acute{e}t}$, then (2.7) is an
isomorphism.

  5. (5)

If $k\neq{\mathbb{F}}_{2}$ is finite, $d=2,r\leq 2$ and $\tau={\acute{e}t}$,
then (2.7) is an isomorphism.

###### Proof.

The items (1) and (2) are clear from Lemma 2.2. To prove (3) and (4), we let
$n\geq 1$ and consider the commutative diagram

$\textstyle{H^{d-1}_{\tau}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d-1}_{\tau}(D,{\widetilde{{\mathcal{K}}}^{M}_{r,D}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(X,{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(D,{\widetilde{{\mathcal{K}}}^{M}_{r,D}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d-1}_{\tau}(X,{\widehat{{\mathcal{K}}}^{M}_{r,X}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d-1}_{\tau}(D,{\widehat{{\mathcal{K}}}^{M}_{r,D}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(X,{\widehat{{\mathcal{K}}}^{M}_{r,X}}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\tau}(D,{\widehat{{\mathcal{K}}}^{M}_{r,D}}/n)}$

with $d\geq 2$. The two rows are exact by (2.4). It suffices to show that all
vertical arrows except possibly the middle one are isomorphisms. But this
follows easily from Lemmas 2.2 and 2.5 if we observe that $cd_{\rm nis}(k)=0$
for $k$ arbitrary and $cd_{\acute{e}t}(k)=1$ for $k$ finite (see [2, Exposé
X]). The proof of (2.6) for $\tau={\rm nis}$ is identical if we observe that
Lemma 2.5 holds even if the sheaf ${\mathcal{F}}$ is not torsion when
$\tau={\rm nis}$.

We now prove (5). We only consider the case $r=2$ as the other cases are
trivial. We look at the commutative diagram of Nisnevich (or étale) sheaves

(2.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{X}}$$\textstyle{{\widetilde{{\mathcal{K}}}^{M}_{r,D}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{D}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widehat{{\mathcal{K}}}^{M}_{r,X}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widehat{{\mathcal{K}}}^{M}_{r,D}}/n\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We now recall from [27, Proposition 10(6)] that for a local ring $A$
containing $k$, the map $\widehat{K}^{M}_{2}(A)\to K_{2}(A)$ is an
isomorphism. Using this, it follows from [31, Corollary 3.3] that
$\widehat{K}^{M}_{2}(A)$ is generated by the Milnor symbols $\\{a,b\\}$
subject to bilinearity, Steinberg relation, and the relation
$\\{a,b\\}=-\\{b,a\\}$. Since ${\rm
Ker}(\widetilde{K}^{M}_{2}(A)\twoheadrightarrow\widehat{K}^{M}_{2}(A))$
surjects onto ${\rm
Ker}({\widetilde{K}^{M}_{2}(A)}/n\twoheadrightarrow{\widehat{K}^{M}_{2}(A)}/n)$,
it follows that the latter is generated by the images of the symbols of the
type $\\{a,b\\}+\\{b,a\\}$. Since $A^{\times}\to(A/I)^{\times}$ is surjective
for any ideal $I\subset A$, we conclude that the map

(2.9) ${\rm
Ker}({\widetilde{K}^{M}_{2}(A)}/n\twoheadrightarrow{\widehat{K}^{M}_{2}(A)}/n)\to{\rm
Ker}({\widetilde{K}^{M}_{2}(A/I)}/n\twoheadrightarrow{\widehat{K}^{M}_{2}(A/I)}/n)$

is surjective. We have therefore shown that ${\rm
Ker}(\alpha_{X})\twoheadrightarrow{\rm Ker}(\alpha_{D})$ in (2.8).
Subsequently, a diagram chase shows that

(2.10)
$\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}\to\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n}$

is a surjective map of Nisnevich (or étale) sheaves. We let ${\mathcal{F}}$
denote the kernel of this map. A diagram similar to (2.2) (with
${\mathcal{K}}^{M}_{r,X}$ replaced by $\widehat{{\mathcal{K}}}^{M}_{r,X}$) and
Lemma 2.2 together show that ${\mathcal{F}}_{x}=0$ for every $x\in X_{(q)}$
with $q>0$. We now apply Lemma 2.5 to finish the proof. ∎

Combining Lemmas 2.3, 2.4 and 2.6, we get the following key result that we
shall use.

######  Proposition 2.7.

Let $k$ be a field and $(X,D)$ a reduced modulus pair in
${\operatorname{\mathbf{Sch}}}_{k}$. Assume that $X$ is of pure dimension
$d\geq 2$. Then there are natural isomorphisms

$H^{d}_{\rm nis}(X,{\mathcal{K}}^{M}_{r,(X,D)})\xrightarrow{\cong}H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{r,(X,D)});$ $H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{r,(X,D)}}/n)\xrightarrow{\cong}H^{d}_{\rm
nis}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/n})$

for every integer $n\geq 1$. If $d=1$, these isomorphisms hold if either $k$
is infinite or $r\leq 1$.

##  3\. The idele class group of Kerz-Zhao

Let $A$ be a local domain with fraction field $F$ and let $I=(f)$ be a
principal ideal, where $f\in A$ is a non-zero element. Let $A_{f}$ denote the
localization $A[f^{-1}]$ obtained by inverting the powers of $f$ so that there
are inclusions of rings $A\hookrightarrow A_{f}\hookrightarrow F$. We let
$\widehat{K}^{M}_{1}(A|I)=K^{M}_{1}(A,I)$ and for $r\geq 2$, we let
$\widehat{K}^{M}_{r}(A|I)$ denote the image of the canonical map of abelian
groups

(3.1)
$K^{M}_{1}(A,I)\otimes(A_{f})^{\times}\otimes\cdots\otimes(A_{f})^{\times}\to
K^{M}_{r}(F),$

induced by the product in Milnor $K$-theory, where the tensor product is taken
$r$ times. These groups coincide with the relative Milnor $K$-groups of
Rülling-Saito (see [42, Definition 2.4 and Lemma 2.1]) when $A$ is regular and
${\rm Spec\,}(A/f)_{\rm red}$ is a normal crossing divisor on ${\rm
Spec\,}(A)$. We shall denote the associated sheaf on an integral scheme $X$
with an effective Cartier divisor $D$ by
$\widehat{{\mathcal{K}}}^{M}_{r,X|D}$.

We let $E_{r}(A,I)$ be the image of the canonical map ${K}^{M}_{r}(A,I)\to
K^{M}_{r}(F)$. This is same as the image of the map
$\widetilde{K}^{M}_{r}(A,I)\to K^{M}_{r}(F)$ by Lemma 2.1. The following
result will be important for us.

######  Lemma 3.1.

We have the following.

  1. (1)

The composite map

$\widehat{K}^{M}_{r}(A|I)\to K^{M}_{r}(F)\xrightarrow{\partial}{\underset{{\rm
ht}({\mathfrak{p}})=1}{\oplus}}\ K^{M}_{r-1}(k({\mathfrak{p}}))$

is zero. In particular,
$\widehat{K}^{M}_{r}(A|I)\subset\widehat{K}^{M}_{r}(A)$ if $A$ is regular and
equicharacteristic or a Henselian discrete valuation ring.

  2. (2)

There is an inclusion $E_{r}(A,I)\subseteq\widehat{K}^{M}_{r}(A|I)$ which is
an equality if $I$ is a prime ideal.

###### Proof.

We let ${\mathfrak{p}}\subset A$ be a prime ideal of height one such that
$f\in{\mathfrak{p}}$. To prove (1), it suffices to show that the composite
$\widehat{K}^{M}_{r}({A_{\mathfrak{p}}}|{I_{\mathfrak{p}}})\to
K^{M}_{r}(F)\xrightarrow{\partial}K^{M}_{r-1}(k({\mathfrak{p}}))$ is zero,
where $\partial$ is the boundary map defined in [23, § 1]. We can therefore
assume that $\text{\rm dim}(A)=1$ and $f\notin A^{\times}$. By the definition
of $\partial$, we can assume that $A$ is normal. But one easily checks in this
case that $\widehat{K}^{M}_{r}(A|I)\subset\widehat{K}^{M}_{r}(A)$ (see [42,
Proposition 2.8]) and $\partial(\widehat{K}^{M}_{r}(A))=0$ (see [7,
Proposition 2.7]). The second part of (1) follows from [27, Proposition 10]
and [35, Theorem 5.1].

We now prove (2). One checks using Lemma 2.1 that
$E_{r}(A,I)\subseteq\widehat{K}^{M}_{r}(A|I)$. Suppose now that $I$ is prime
and consider an element $\alpha=\\{1+af,u_{2},\ldots,u_{r}\\}$, where $a\in A$
and $u_{i}\in(A_{f})^{\times}$ for $i\geq 2$. Since $I$ is prime, it is easy
to check that $u_{i}=a_{i}f^{n_{i}}$ for $a_{i}\in A^{\times}$ and $n_{i}\geq
0$. Using the bilinearity and the Steinberg relations, it easily follows from
[23, Lemma 1] that $\alpha\in E_{r}(A,I)$. ∎

Suppose that $k$ is any field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ is
regular. Let $D\subset X$ be an effective Cartier divisor. It follows from
Lemma 3.1 that there are inclusions
$\widehat{{\mathcal{K}}}^{M}_{r,X|nD}\subset\widehat{{\mathcal{K}}}^{M}_{r,X}\supset\widehat{{\mathcal{K}}}^{M}_{r,(X,nD)}$
for $n\geq 1$. We let
$\overline{{\widehat{{\mathcal{K}}}^{M}_{r,X|D}}/m}={\widehat{{\mathcal{K}}}^{M}_{r,X|D}}/{(\widehat{{\mathcal{K}}}^{M}_{r,X|D}\cap
m\widehat{{\mathcal{K}}}^{M}_{r,X})}$.

######  Proposition 3.2.

Let $k$ be a field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ a regular
scheme of dimension $d\geq 1$. Let $D\subset X$ be a simple normal crossing
divisor. Let $\tau\in\\{{\rm nis},{\acute{e}t}\\}$ and $m\geq 1$. Then the
conclusions (1) $\sim$ (5) of Lemma 2.6 hold for the maps

$\\{H^{d}_{\tau}(X,\widetilde{{\mathcal{K}}}^{M}_{r,(X,nD)})\\}_{n}\to\\{H^{d}_{\tau}(X,\widehat{{\mathcal{K}}}^{M}_{r,X|nD})\\}_{n};$
$\\{H^{d}_{\tau}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{r,(X,nD)}}/m})\\}_{n}\to\\{H^{d}_{\tau}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{r,X|nD}}/m})\\}_{n}$

of pro-abelian groups.

###### Proof.

We let ${\mathcal{E}}_{r,(X,nD)}={\rm
Image}(\widetilde{{\mathcal{K}}}^{M}_{r,(X,nD)}\to\widehat{{\mathcal{K}}}^{M}_{r,X})$
and $\overline{{{\mathcal{E}}_{r,(X,nD)}}/m}={\rm
Image}(\widetilde{{\mathcal{K}}}^{M}_{r,(X,nD)}\to{\widehat{{\mathcal{K}}}^{M}_{r,X}}/m)$.
It is easy to check using [42, Proposition 2.8] that the inclusions
$\\{{\mathcal{E}}_{r,(X,nD)}\\}_{n}\hookrightarrow\\{\widehat{{\mathcal{K}}}^{M}_{r,X|nD}\\}_{n}$
and
$\\{\overline{{{\mathcal{E}}_{r,(X,nD)}}/m}\\}_{n}\hookrightarrow\\{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,X|nD}}/m}\\}_{n}$
are pro-isomorphisms. The rest of the proof is now completely identical to
that of Lemma 2.6. ∎

Let us now assume that $k$ is any field and
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ is an integral scheme of dimension
$d\geq 1$. We shall endow $X$ with its canonical dimension function $d_{X}$
(see [28]). Assume that $D\subset X$ is an effective Cartier divisor. Let $F$
denote the function field of $X$ and $U=X\setminus D$. We refer the reader to
[26, § 1] or [15, § 2] for the definition and all properties of Parshin chains
and their Milnor $K$-groups that we shall need.

We let ${\mathcal{P}}_{U/X}$ denote the set of Parshin chains on the pair
$(U\subset X)$ and let ${\mathcal{P}}^{{\operatorname{\rm max}}}_{U/X}$ be the
subset of ${\mathcal{P}}_{U/X}$ consisting of maximal Parshin chains. If $P$
is a Parshin chain on $X$ of dimension $d_{X}(P)$, then we shall consider
$K^{M}_{d_{X}(P)}(k(P))$ as a topological abelian group with its canonical
topology if $P$ is maximal (see [15, § 2,5]). Otherwise, we shall consider
$K^{M}_{d_{X}(P)}(k(P))$ as a topological abelian group with its discrete
topology. If $P=(p_{0},\ldots,p_{d})$ is a maximal Parshin chain on $X$, we
shall let $P^{\prime}=(p_{0},\ldots,p_{d-1})$.

Recall from [15, Definition 3.1] that the idele group of $(U\subset X)$ is
defined as

(3.2)
$I_{U/X}={\underset{P\in{\mathcal{P}}_{U/X}}{\bigoplus}}K^{M}_{d_{X}(P)}(k(P)).$

We consider $I_{U/X}$ as a topological group with its direct sum topology. We
let

(3.3) $I(X|D)={\rm
Coker}\left({\underset{P\in{\mathcal{P}}^{{\operatorname{\rm
max}}}_{U/X}}{\bigoplus}}\widehat{K}^{M}_{d_{X}(P)}({\mathcal{O}}^{h}_{X,P^{\prime}}|I_{D})\to
I_{U/X}\right),$

where $I_{D}$ is the extension of the ideal sheaf
${\mathcal{I}}_{D}\subset{\mathcal{O}}_{X}$ to
${\mathcal{O}}^{h}_{X,P^{\prime}}$ and the map on the right is induced by the
composition
$\widehat{K}^{M}_{d_{X}(P)}({\mathcal{O}}^{h}_{X,P^{\prime}}|I_{D})\hookrightarrow
K^{M}_{d_{X}(P)}(k(P))\to I_{U/X}$ for $P\in{\mathcal{P}}^{{\operatorname{\rm
max}}}_{U/X}$. We consider $I(X|D)$ a topological group with its quotient
topology. Recall from [15, § 3.1] that $I(X,D)$ is defined analogous to
$I(X|D)$, where we replace
$\widehat{K}^{M}_{d_{X}(P)}({\mathcal{O}}^{h}_{X,P^{\prime}}|I_{D})$ by
${K}^{M}_{d_{X}(P)}({\mathcal{O}}^{h}_{X,P^{\prime}},I_{D})$.

We let ${\mathcal{Q}}_{U/X}$ denote the set of all $Q$-chains on $(U\subset
X)$. Recall that the idele class group of the pair $(U\subset X)$ is the
topological abelian group

(3.4) $C_{U/X}={\rm
Coker}\left({\underset{Q\in{\mathcal{Q}}_{U/X}}{\bigoplus}}K^{M}_{d_{X}(Q)}(k(Q))\xrightarrow{\partial_{U/X}}I_{U/X}\right)$

with its quotient topology. We let

(3.5) $C(X|D)={\rm
Coker}\left({\underset{Q\in{\mathcal{Q}}_{U/X}}{\bigoplus}}K^{M}_{d_{X}(Q)}(k(Q))\to
I(X|D)\right).$

This is a topological group with its quotient topology. This idele class group
was introduced by Kerz-Zhao in [30, Definition 2.1.4]. We let $C(X,D)$ be the
cokernel of the map
${\underset{Q\in{\mathcal{Q}}_{U/X}}{\oplus}}K^{M}_{d_{X}(Q)}(k(Q))\to
I(X,D)$. One checks using part (2) of Lemma 3.1 that there is a commutative
diagram

(3.6)
$\textstyle{I_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X|D).}$

If $\text{\rm dim}(X)=d$, we let $C_{KS}(X|D)=H^{d}_{\rm
nis}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})$ and
$C^{{\acute{e}t}}_{KS}(X|D)=H^{d}_{\acute{e}t}(X,\widehat{{\mathcal{K}}}^{M}_{d,X|D})$.
We let $C_{KS}(X,D)=H^{d}_{\rm nis}(X,{{\mathcal{K}}}^{M}_{d,(X,D)})$ and
$C^{{\acute{e}t}}_{KS}(X,D)=H^{d}_{\acute{e}t}(X,{{\mathcal{K}}}^{M}_{d,(X,D)})$.

######  Lemma 3.3.

The canonical map $C_{KS}(X,D)\to C_{KS}(X|D)$ is surjective. It is an
isomorphism if $D\subset X$ is a reduced Cartier divisor.

###### Proof.

It is clear that the stalks of the kernel and cokernel of the map
${\mathcal{K}}^{M}_{r,(X,D)}\to\widehat{{\mathcal{K}}}^{M}_{r,X|D}$ vanish at
the generic point of $X$. It follows moreover from Lemma 3.1 that these stalks
vanish at all points of $X$ with dimension $d-1$ or more if $D$ is reduced. A
variant of Lemma 2.5 therefore implies the desired result. ∎

Let ${\mathcal{Z}}_{0}(U)$ be the free abelian group on $U_{(0)}$. It is clear
from the definitions of $C_{KS}(X|D)$ and $C(X|D)$ that there is a diagram

(3.7)
$\textstyle{{\mathcal{Z}}_{0}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{KS}(X|D),}$

where the left vertical arrow is induced by the inclusion of the Milnor
$K$-groups of the length zero Parshin chains on $(U\subset X)$ into $I_{U/X}$,
and the diagonal arrow is induced by the coniveau spectral sequence for
$C_{KS}(X|D)$. The following is [30, Theorem 3.1.1] whose proof is obtained by
simply repeating the proof of [28, Theorem 8.2] mutatis mutandis (see [15,
Theorem 3.8]).

######  Theorem 3.4.

Assume that $U$ is regular. Then one has an isomorphism

$\phi_{X|D}\colon C(X|D)\xrightarrow{\cong}C_{KS}(X|D)$

such that (3.7) is commutative.

###  3.1. Degree map for $C_{KS}(X|D)$

Let $k$ be any field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ an integral
scheme of dimension $d\geq 1$. Assume that $D\subset X$ is an effective
Cartier divisor. Let ${\rm CH}^{F}_{r}(X)$ denote the Chow group of
$r$-dimensional cycles on $X$ as in [9, Chapter 1]. It follows from Lemma 3.1
and [23, Proposition 1] that

(3.8) $\widehat{{\mathcal{K}}}^{M}_{r,X|D}\to{\underset{x\in
X^{(0)}}{\oplus}}(\iota_{x})_{*}(K^{M}_{r}(k(x)))\to{\underset{x\in
X^{(1)}}{\oplus}}(\iota_{x})_{*}(K^{M}_{r-1}(k(x)))\to\cdots\hskip 56.9055pt$
$\hskip 108.12054pt\cdots\to{\underset{x\in
X^{(d-1)}}{\oplus}}(\iota_{x})_{*}(K^{M}_{r-d+1}(k(x)))\xrightarrow{\partial}{\underset{x\in
X^{(d)}}{\oplus}}(\iota_{x})_{*}(K^{M}_{r-d}(k(x)))\to 0$

is a complex of Nisnevich sheaves on $X$. An elementary cohomological argument
(see [16, § 2.1]) shows that by taking the cohomology of the sheaves in this
complex, one gets a canonical homomorphism

(3.9) $\nu_{X|D}\colon C_{KS}(X|D)\to{\rm CH}^{F}_{0}(X).$

If $X$ is projective over $k$, there is a degree map $\deg\colon{\rm
CH}^{F}_{0}(X)\to{\mathbb{Z}}$. We let $\deg\colon C_{KS}(X|D)\to{\mathbb{Z}}$
be the composition $C_{KS}(X|D)\xrightarrow{\nu_{X|D}}{\rm
CH}^{F}_{0}(X)\xrightarrow{\deg}{\mathbb{Z}}$. We let $C_{KS}(X|D)_{0}={\rm
Ker}(\deg)$. We let $C_{KS}(X,D)_{0}$ be the kernel of the composite map
$C_{KS}(X,D)\twoheadrightarrow C_{KS}(X|D)\xrightarrow{\deg}{\mathbb{Z}}$.

##  4\. Comparison theorem for cohomology of $K$-sheaves

Let $k$ be a perfect field of characteristic $p>0$ and
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ an integral regular scheme of
dimension $d$. Let $D\subset X$ be an effective Cartier divisor. Let
$U=X\setminus D$ and $F=k(X)$. Let $m,r\geq 1$ be integers. A combination of
[22, Theorem 1.1.5] and [30, Theorems 2.2.2, 3.3.1] implies the following.

######  Theorem 4.1.

Assume that $k$ is finite and $D_{\rm red}$ is a simple normal crossing
divisor. Then the canonical map

$H^{d}_{\rm
nis}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|D}}/{p^{m}}})\to
H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|D}}/{p^{m}}})$

is an isomorphism.

######  Corollary 4.2.

Under the assumptions of Theorem 4.1, the canonical map

$\\{H^{d}_{\rm
nis}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})\\}_{n}\to\\{H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})\\}_{n}$

is an isomorphism if either $k\neq{\mathbb{F}}_{2}$ or $d\neq 2$.

###### Proof.

Combine Lemma 2.6, Proposition 3.2 and Theorem 4.1. ∎

Our goal in this section is to prove an analogue of Theorem 4.1 for the
cohomology of ${\mathcal{K}}^{M}_{d,(X,D)}$ when $D_{\rm red}$ is regular.
This will be used in the proof of Theorem 1.2.

###  4.1. Filtrations of Milnor $K$-theory

Let $A$ be a regular local ring essentially of finite type over $k$ and
$I=(f)$ a principal ideal, where $f\in A$ is a non-zero element such that
$R={A}/{I}$ is regular. Let $A_{f}$ denote the localization $A[f^{-1}]$
obtained by inverting the powers of $f$. Let $F$ be the quotient field of $A$
so that there are inclusions of rings $A\hookrightarrow A_{f}\hookrightarrow
F$. For an integer $n\geq 1$, let $I^{n}=(f^{n})$.

We let $E_{r}(A,I^{n})$ be the image of the canonical map
$\widetilde{K}^{M}_{r}(A,I^{n})\to\widehat{K}^{M}_{r}(A)\subset K^{M}_{r}(F)$.
One easily checks that there is a commutative diagram (see (2.3))

(4.1)
---
$\textstyle{\widehat{K}^{M}_{r}(A,I^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\widehat{K}^{M}_{r}(A,I^{n})}/{p^{m}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{\widehat{K}^{M}_{r}(A,I^{n})+p^{m}\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{K}^{M}_{r}(A,I^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\widetilde{K}^{M}_{r}(A,I^{n})}/{p^{m}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{E_{r}(A,I^{n})+p^{m}\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}^{M}_{r}(A|I^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\widehat{K}^{M}_{r}(A|I^{n})}/{p^{m}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{\widehat{K}^{M}_{r}(A|I^{n})+p^{m}\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)},}$

where the existence of the left-most bottom vertical arrow and the right-most
bottom horizontal inclusion follows from [42, Proposition 2.8] (see also [14,
Lemma 3.8]).

We consider the filtrations $F^{\bullet}_{m,r}(A)$ and $G^{\bullet}_{m,r}(A)$
of ${\widehat{K}^{M}_{r}(A)}/{p^{m}}$ by letting
$F^{n}_{m,r}(A)=\frac{E_{r}(A,I^{n})+p^{m}\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)}$
and
$G^{n}_{m,r}(A)=\frac{\widehat{K}^{M}_{r}(A|I^{n})+p^{m}\widehat{K}^{M}_{r}(A)}{p^{m}\widehat{K}^{M}_{r}(A)}$.
We then have a commutative diagram of filtrations

(4.2)
$\textstyle{E_{r}(A,I^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}^{M}_{r}(A|I^{\bullet})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{K}^{M}_{r}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\bullet}_{m,r}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G^{\bullet}_{m,r}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widehat{K}^{M}_{r}(A)}/{p^{m}}.}$

######  Lemma 4.3.

One has $\widehat{K}^{M}_{r}(A|I^{n+1})\subseteq E_{r}(A,I^{n})$ and
$G^{n+1}_{m,r}(A)\subseteq F^{n}_{m,r}(A)$. Furthermore,
$E_{r}(A,I)=\widehat{K}^{M}_{r}(A|I)$ and $F^{1}_{m,r}(A)=G^{1}_{m,r}(A)$.

###### Proof.

This is immediate from [42, Proposition 2.8]. ∎

We let $\overline{\widetilde{K}^{M}_{r}(A_{f})}$ be the image of the canonical
map $\widetilde{K}^{M}_{r}(A_{f})\to K^{M}_{r}(F)$. We let
$H^{n}_{m,r}(A)=\frac{\widehat{K}^{M}_{r}(A|I^{n})+p^{m}\overline{\widetilde{K}^{M}_{r}(A_{f})}}{p^{m}\overline{\widetilde{K}^{M}_{r}(A_{f})}}$.
Note that the canonical map
$\widetilde{K}^{M}_{r}(A)\twoheadrightarrow\widehat{K}^{M}_{r}(A)$ is
surjective. We therefore have a surjective map
$G^{n}_{m,r}(A)\twoheadrightarrow H^{n}_{m,r}(A)$.

######  Lemma 4.4.

The canonical map

$\frac{G^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\to\frac{H^{n}_{m,r}(A)}{H^{n+1}_{m,r}(A)}$

is an isomorphism.

###### Proof.

The map in question is clearly surjective since its both sides are quotients
of $\widehat{K}^{M}_{r}(A|I^{n})$. So we need to prove only the injectivity.
Using Lemma 4.3 and an easy reduction, one can see that it suffices to show
that the map

$\frac{\overline{\widetilde{K}^{M}_{r}(A_{f})}}{\widehat{K}^{M}_{r}(A)}\xrightarrow{p^{m}}\frac{\overline{\widetilde{K}^{M}_{r}(A_{f})}}{\widehat{K}^{M}_{r}(A)}$

is injective.

Since $\overline{\widetilde{K}^{M}_{r}(A_{f})}$ has no $p$-torsion, a snake
lemma argument shows that the previous injectivity is equivalent to the
injectivity of the map
${\widehat{K}^{M}_{r}(A)}/{p^{m}}\to{\overline{\widetilde{K}^{M}_{r}(A_{f})}}/{p^{m}}$.
We now look at the composition
${\widehat{K}^{M}_{r}(A)}/{p^{m}}\to{\overline{\widetilde{K}^{M}_{r}(A_{f})}}/{p^{m}}\to{K^{M}_{r}(F)}/{p^{m}}$.
It suffices to show that this composition is injective. But this is an easy
consequence of [27, Proposition 10(8)] and [11, Theorem 8.1]. ∎

We now consider the map

(4.3)
$\rho\colon\Omega^{r-1}_{R}\oplus\Omega^{r-2}_{R}\to{\widehat{K}^{M}_{r}(A|I^{n})}/{\widehat{K}^{M}_{r}(A|I^{n+1})}$

having the property that

$\rho(a{\rm dlog}b_{1}\wedge\cdots\wedge{\rm
dlog}b_{r-1},0)=\\{1+\widetilde{a}f^{n},\widetilde{b}_{1},\ldots,\widetilde{b}_{r-1}\\};$
$\rho(0,a{\rm dlog}b_{1}\wedge\cdots\wedge{\rm
dlog}b_{r-2})=\\{1+\widetilde{a}f^{n},\widetilde{b}_{1},\ldots,\widetilde{b}_{r-2},f\\},$

where $\widetilde{a}$ and $\widetilde{b}_{i}$’s are arbitrary lifts of $a$ and
$b_{i}$’s, respectively, under the surjection $A\twoheadrightarrow R$. It
follows from [5, §4, p. 122] that this map is well defined. It is clear from
(4.2) and Lemma 4.3 that $\rho$ restricts to a map

(4.4)
$\rho\colon\Omega^{r-1}_{R}\to{E_{r}(A,I^{n})}/{\widehat{K}^{M}_{r}(A|I^{n+1})}.$

This map is surjective by Lemma 2.1.

Let $\psi\colon R\to R$ be the absolute Frobenius morphism. Let
$Z_{i}\Omega^{r-1}_{R}$ be the unique $R$-submodule of
$\psi^{i}_{*}\Omega^{r-1}_{R}$ such that the inverse Cartier operator (see
[20, Chap. 0, § 2]) induces an $R$-linear isomorphism $C^{-1}\colon
Z_{i-1}\Omega^{r-1}_{R}\xrightarrow{\cong}{Z_{i}\Omega^{r-1}_{R}}/{d\Omega^{r-2}_{R}}$,
where we let $Z_{1}\Omega^{r-1}_{R}={\rm
Ker}(\Omega^{r-1}_{R}\to\Omega^{r}_{R})$. We let
$B_{1}\Omega^{r-1}_{R}=d\Omega^{r-2}_{R}$ and let $B_{i}\Omega^{r-1}_{R}$
($i\geq 2$) be the unique $R$-submodule of $\psi^{i}_{*}\Omega^{r-1}_{R}$ such
that $C^{-1}\colon
B_{i-1}\Omega^{r-1}_{R}\to{B_{i}\Omega^{r-1}_{R}}/{d\Omega^{r-2}_{R}}$ is an
isomorphism of $R$-modules.

######  Lemma 4.5.

Let $n=n_{1}p^{s}$, where $s\geq 0$ and $p\nmid n_{1}$. Then $\rho$ induces
isomorphisms

$\rho^{n}_{m,r}\colon\frac{\psi^{m}_{*}\Omega^{r-1}_{R}}{Z_{m}\Omega^{r-1}_{R}}\xrightarrow{\cong}\frac{F^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\
\ {\rm if}\ m\leq s;$
$\rho^{n}_{m,r}\colon\frac{\psi^{s}_{*}\Omega^{r-1}_{R}}{B_{s}\Omega^{r-1}_{R}}\xrightarrow{\cong}\frac{F^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\
\ {\rm if}\ m>s.$

###### Proof.

Once it exists, $\rho^{n}_{m,r}$ is clearly surjective in both cases. We
therefore need to prove its existence and injectivity. These are easy
consequences of various lemmas in [5, § 4] (see also [22, Proposition 1.1.9])
once we have Lemma 4.4. The proof goes as follows.

When $m\leq s$, we look at the diagram

(4.5)
$\textstyle{\frac{\psi^{m}_{*}\Omega^{r-1}_{R}}{Z_{m}\Omega^{r-1}_{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{n}_{m,r}}$$\textstyle{\frac{F^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{\psi^{m}_{*}\Omega^{r-1}_{R}}{Z_{m}\Omega^{r-1}_{R}}\oplus\frac{\psi^{m}_{*}\Omega^{r-2}_{R}}{Z_{m}\Omega^{r-2}_{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\frac{G^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{H^{n}_{m,r}(A)}{H^{n+1}_{m,r}(A)},}$

where the bottom right isomorphism is by Lemma 4.4. The right vertical arrow
is clearly injective. The composite arrow on the bottom is an isomorphism by
[5, (4.7), Remark 4.8]. It follows that the top horizontal arrow exists and it
is injective.

When $m>s$, we look at the commutative diagram

(4.6)
$\textstyle{\frac{\psi^{s}_{*}\Omega^{r-1}_{R}}{B_{s}\Omega^{r-1}_{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{n}_{m,r}}$$\textstyle{\frac{F^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{\psi^{s}_{*}\Omega^{r-1}_{R}}{B_{s}\Omega^{r-1}_{R}}\oplus\frac{\psi^{s}_{*}\Omega^{r-2}_{R}}{B_{s}\Omega^{r-2}_{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Coker}(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\frac{G^{n}_{m,r}(A)}{G^{n+1}_{m,r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\frac{H^{n}_{m,r}(A)}{H^{n+1}_{m,r}(A)},}$

where $\theta$ is defined in [5, Lemma 4.5]. The map $\rho$ on the bottom is
again an isomorphism. Hence, it suffices to show that the left vertical arrow
is injective. But this is immediate from the definition of $\theta$ (see loc.
cit.) since $C^{-s}$ is an isomorphism. ∎

###  4.2. The comparison theorem

Let $k$ be a perfect field of characteristic $p>0$. Let
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ be a regular scheme of pure dimension
$d\geq 1$ and let $D\subset X$ be a regular closed subscheme of codimension
one. Let ${\mathcal{F}}^{n}_{m,r,X}$ be the sheaf on $X_{{\rm nis}}$ such that
for any point $x\in X$, the stalk $({\mathcal{F}}^{n}_{m,r,X})_{x}$ coincides
with $F^{n}_{m,r}({\mathcal{O}}^{h}_{X,x})$, where
$I={\mathcal{I}}_{D}{\mathcal{O}}^{h}_{X,x}$. The corresponding étale sheaf is
defined similarly by replacing the Henselization ${\mathcal{O}}^{h}_{X,x}$ by
the strict Henselization ${\mathcal{O}}^{sh}_{X,x}$. We define the Nisnevich
and étale sheaves ${\mathcal{G}}^{n}_{m,r,X}$ in an identical way, using the
groups $G^{n}_{m,r}(A)$ for stalks.

######  Lemma 4.6.

The change of topology map

$H^{i}_{\rm nis}(X,{{\mathcal{F}}^{1}_{m,r,X}}/{{\mathcal{F}}^{n}_{m,r,X}})\to
H^{i}_{\acute{e}t}(X,{{\mathcal{F}}^{1}_{m,r,X}}/{{\mathcal{F}}^{n}_{m,r,X}})$

is an isomorphism for every $i\geq 0$.

###### Proof.

We consider the exact sequence of Nisnevich (or étale) sheaves

(4.7)
$0\to\frac{{\mathcal{F}}^{n}_{m,r,X}}{{\mathcal{G}}^{n+1}_{m,r,X}}\to\frac{{\mathcal{G}}^{1}_{m,r,X}}{{\mathcal{G}}^{n+1}_{m,r,X}}\to\frac{{\mathcal{F}}^{1}_{m,r,X}}{{\mathcal{F}}^{n}_{m,r,X}}\to
0,$

where we have replaced ${\mathcal{G}}^{1}_{m,r,X}$ by
${\mathcal{F}}^{1}_{m,r,X}$ in the quotient on the right using Lemma 4.3.
Working inductively on $n$, it follows from [22, Proposition 1.1.9] that the
middle term in (4.7) has identical Nisnevich and étale cohomology. It suffices
therefore to show that the same holds for the left term in (4.7) too.

Since this isomorphism is obvious for $i=0$, it suffices to show using the
Leray spectral sequence that ${\bf
R}^{i}\epsilon_{*}({{\mathcal{F}}^{n}_{m,r,X}}/{{\mathcal{G}}^{n+1}_{m,r,X}})=0$
for $i>0$. Equivalently, we need to show that if $A={\mathcal{O}}^{h}_{X,x}$
for some $x\in X$ and $f\in A$ defines $D$ locally at $x$, then
$H^{i}_{\acute{e}t}(A,{F^{n}_{m,r}(A)}/{G^{n+1}_{m,r}(A)})=0$ for $i>0$. But
this is immediate from Lemma 4.5. ∎

######  Lemma 4.7.

Assume that $k$ is finite. Then the change of topology map

$H^{d}_{\rm nis}(X,{{\mathcal{F}}^{n}_{m,d,X}})\to
H^{d}_{\acute{e}t}(X,{{\mathcal{F}}^{n}_{m,d,X}})$

is an isomorphism.

###### Proof.

We first assume $n=1$. Using Lemma 4.3, we can replace
${\mathcal{F}}^{1}_{m,d,X}$ by ${\mathcal{G}}^{1}_{m,d,X}$. By [22, Theorem
1.1.5] and [30, Theorems 2.2.2, Proposition 2.2.5], the dlog map
${\mathcal{G}}^{1}_{m,d,X}\to W_{m}\Omega^{d}_{X,{\operatorname{log}}}$ is an
isomorphism in Nisnevich and étale topologies. We conclude by [30, Theorem
3.3.1].

We now assume $n\geq 2$ and look at the exact sequence

$0\to{\mathcal{F}}^{n}_{m,d,X}\to{\mathcal{F}}^{1}_{m,d,X}\to\frac{{\mathcal{F}}^{1}_{m,d,X}}{{\mathcal{F}}^{n}_{m,d,X}}\to
0.$

We saw above that the middle term can be replaced by
$W_{m}\Omega^{d}_{X,{\operatorname{log}}}$. We can now conclude the proof by
Lemmas 4.6 and the independent statement Lemma 5.6 via the 5-lemma. ∎

######  Lemma 4.8.

Assume that $k$ is finite. Then the change of topology map

$H^{d}_{\rm
nis}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})\to
H^{d}_{\acute{e}t}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})$

is an isomorphism if either $d\neq 2$ or $k\neq{\mathbb{F}}_{2}$.

###### Proof.

In view of Lemma 4.7, it suffices to show that the canonical map (see (4.1))

(4.8)
$H^{d}_{\tau}(X,\overline{{\widetilde{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})\to
H^{d}_{\tau}(X,{{\mathcal{F}}^{n}_{m,d,X}})$

is an isomorphism for $\tau\in\\{{\rm nis},{\acute{e}t}\\}$. Using Lemma 2.5
and (4.1), the proof of this is completely identical to that of Lemma 2.6. ∎

The following is the main result we were after in this section. We restate all
assumptions for convenience.

######  Theorem 4.9.

Let $k$ be a finite field of characteristic $p$. Let
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ be a regular scheme of pure dimension
$d\geq 1$ and let $D\subset X$ be a regular closed subscheme of codimension
one. Assume that either $d\neq 2$ or $k\neq{\mathbb{F}}_{2}$. Then there is a
natural isomorphism

${H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{d,(X,nD)}})}/{p^{m}}\xrightarrow{\cong}H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})$

for every $m,n\geq 1$.

###### Proof.

Combine Lemmas 2.3, 2.4, 2.6 and 4.8. ∎

######  Corollary 4.10.

Under the assumptions of Theorem 4.9, there is a commutative diagram

(4.9) $\textstyle{{H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{d,(X,nD)}})}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,nD)}}/{p^{m}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\rm
nis}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|nD}}/{p^{m}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|nD}}/{p^{m}}})}$

for $m,n\geq 1$ in which the horizontal arrows are isomorphisms.

###### Proof.

In view of Theorems 4.1 and 4.9, we only have to explain the vertical arrows.
But their existence follows from Lemmas 2.3, 2.4 and 2.6, and the diagram
(4.1). ∎

######  Remark 4.11.

The reader may note that Theorem 4.1 holds whenever $D_{\rm red}$ is a simple
normal crossing divisor, but this is not the case with Theorem 4.9. The main
reason for this is that while $\widehat{K}^{M}_{d,X|D}$ satisfies cdh-descent
for closed covers [22], the same is not true for $\widehat{K}^{M}_{d,(X,D)}$.
This prevents us from using an induction argument on the number of irreducible
components of $D$.

##  5\. Relative logarithmic Hodge-Witt sheaves

In this section, we shall recall the relative logarithmic Hodge-Witt sheaves
and show that they coincide with the Kato-Saito relative Milnor $K$-theory
with ${{\mathbb{Z}}}/{p^{m}}$ coefficients in a pro-setting. We fix a field
$k$ of characteristic $p>0$ and let $X\in{\operatorname{\mathbf{Sch}}}_{k}$.

###  5.1. The relative Hodge-Witt sheaves

Let $\\{W_{m}\Omega^{\bullet}_{X}\\}_{m\geq 1}$ denote the pro-complex of de
Rham-Witt (Nisnevich) sheaves on $X$. This is a pro-complex of differential
graded algebras with the structure map $R$ and the differential $d$. Let
$[-]_{m}\colon{\mathcal{O}}_{X}\to W_{m}{\mathcal{O}}_{X}$ be the
multiplicative Teichmüller homomorphism. Recall that the pro-complex
$\\{W_{m}\Omega^{\bullet}_{X}\\}_{m\geq 1}$ is equipped with the Frobenius
homomorphism of graded algebras $F\colon W_{m}\Omega^{r}_{X}\to
W_{m-1}\Omega^{r}_{X}$ and the additive Verschiebung homomorphism $V\colon
W_{m}\Omega^{r}_{X}\to W_{m+1}\Omega^{r}_{X}$ . These homomorphisms satisfy
the following properties which we shall use frequently.

  1. (1)

$FV=p=VF$.

  2. (2)

$FdV=d$, $dF=pFd$ and $pdV=Vd$.

  3. (3)

$F(d[a])=[a^{p-1}]d[a]$ for all $a\in{\mathcal{O}}_{X}$.

  4. (4)

$V(F(x)y)=xV(y)$ and $V(xdy)=VxdVy$ for all $x\in W_{m}\Omega^{i}_{X}$, $y\in
W_{m}\Omega^{j}_{X}$.

Let ${\rm
fil}^{i}_{V}W_{m}\Omega^{r}_{X}=V^{i}W_{m-i}\Omega^{r}_{X}+dV^{i}W_{m-i}\Omega^{r-1}_{X},\
{\rm fil}^{i}_{R}W_{m}\Omega^{r}_{X}={\rm Ker}(R^{r-i})$ and ${\rm
fil}^{i}_{p}W_{m}\Omega^{r}_{X}={\rm Ker}(p^{r-i})$ for $0\leq i\leq r$. By
[19, Lemma 3.2.4] and [20, Proposition I.3.4], we know that ${\rm
fil}^{i}_{V}W_{m}\Omega^{r}_{X}={\rm
fil}^{i}_{R}W_{m}\Omega^{r}_{X}\subseteq{\rm fil}^{i}_{p}W_{m}\Omega^{r}_{X}$
and the last inclusion is an equality if $X$ is regular. We let
$ZW_{m}\Omega^{r}_{X}={\rm Ker}(d\colon W_{m}\Omega^{r}_{X}\to
W_{m}\Omega^{r+1}_{X})$ and $BW_{m}\Omega^{r}_{X}={\rm Image}(d\colon
W_{m}\Omega^{r-1}_{X}\to W_{m}\Omega^{r}_{X})$. We set
${\mathcal{H}}^{r}(W_{m}\Omega^{\bullet}_{X})={ZW_{m}\Omega^{r}_{X}}/{BW_{m}\Omega^{r}_{X}}$.

Let $\psi\colon X\to X$ denote the absolute Frobenius morphism. One then knows
that $d\colon\psi_{*}W_{1}\Omega^{r}_{X}\to\psi_{*}W_{1}\Omega^{r+1}_{X}$ is
${\mathcal{O}}_{X}$-linear. In particular, the sheaves
$\psi_{*}ZW_{1}\Omega^{r}_{X},\ \psi_{*}BW_{1}\Omega^{r}_{X}$ and
$\psi_{*}{\mathcal{H}}^{r}(W_{1}\Omega^{\bullet}_{X})$ are quasi-coherent on
$X_{\rm nis}$. If $X$ is $F$-finite, then $\psi$ is a finite morphism, and
therefore, $\psi_{*}ZW_{1}\Omega^{r}_{X},\ \psi_{*}BW_{1}\Omega^{r}_{X}$ and
$\psi_{*}{\mathcal{H}}^{r}(W_{1}\Omega^{\bullet}_{X})$ are coherent sheaves on
$X_{\rm nis}$.

Let $D\subset X$ be a closed subscheme defined by a sheaf of ideals
${\mathcal{I}}_{D}$ and let $nD\subset X$ be the closed subscheme defined by
${\mathcal{I}}^{n}_{D}$ for $n\geq 1$. We let
$W_{m}\Omega^{\bullet}_{(X,D)}={\rm
Ker}(W_{m}\Omega^{\bullet}_{X}\twoheadrightarrow\iota_{*}W_{m}\Omega^{\bullet}_{D})$,
where $\iota\colon D\hookrightarrow X$ is the inclusion. We shall often use
the notation $W_{m}\Omega^{\bullet}_{(A,I)}$ if $X={\rm Spec\,}(A)$ and
$D={\rm Spec\,}(A/I)$. If $D$ is an effective Cartier divisor on $X$ and
$n\in{\mathbb{Z}}$, we let $W_{m}{\mathcal{O}}_{X}(nD)$ be the Nisnevich sheaf
on $X$ which is locally defined as $W_{m}{\mathcal{O}}_{X}(nD)=[f^{-n}]\cdot
W_{m}{\mathcal{O}}_{X}\subset j_{*}W_{m}{\mathcal{O}}_{U}$, where
$f\in{\mathcal{O}}_{X}$ is a local equation of $D$ and $j\colon
U\hookrightarrow X$ is the inclusion of the complement of $D$ in $X$. One
knows that $W_{m}{\mathcal{O}}_{X}(nD)$ is a sheaf of invertible
$W_{m}{\mathcal{O}}_{X}$-modules. We let
$W_{m}\Omega^{\bullet}_{X}(nD)=W_{m}\Omega^{\bullet}_{X}\otimes_{W_{m}{\mathcal{O}}_{X}}W_{m}{\mathcal{O}}_{X}(nD)$.
It is clear that there is a canonical $W_{m}{\mathcal{O}}_{X}$-linear map
$W_{m}\Omega^{\bullet}_{X}(-D)\to W_{m}\Omega^{\bullet}_{(X,D)}$. The reader
is invited to compare the following with a weaker statement [10, Proposition
2.5].

######  Lemma 5.1.

If $(X,D)$ is a modulus pair and $m\geq 1$ an integer, then the canonical map
of pro-sheaves

$\\{W_{m}\Omega^{\bullet}_{X}(-nD)\\}_{n}\to\\{W_{m}\Omega^{\bullet}_{(X,nD)}\\}_{n}$

on $X_{\rm nis}$ is surjective. If $X$ is furthermore regular, then the map
$W_{m}\Omega^{\bullet}_{X}(-D)\to W_{m}\Omega^{\bullet}_{(X,D)}$ is injective.

###### Proof.

The first part follows directly from Lemma 5.2 below. We prove the second
part. Let $U=X\setminus D$ and $j\colon U\hookrightarrow X$ the inclusion.
Using the Néron-Popescu approximation and the Gersten resolution of
$W_{m}\Omega^{\bullet}_{X}$ (when $X$ is smooth), proven in [12, Proposition
II.5.1.2], it follows that the canonical map $W_{m}\Omega^{\bullet}_{X}\to
j_{*}W_{m}\Omega^{\bullet}_{U}$ is injective (see [33, Proposition 2.8]).
Since $W_{m}{\mathcal{O}}_{X}(-D)$ is invertible as a
$W_{m}{\mathcal{O}}_{X}$-module, there are natural isomorphisms
$j_{*}W_{m}\Omega^{\bullet}_{U}\otimes_{W_{m}{\mathcal{O}}_{X}}W_{m}{\mathcal{O}}_{X}(-D)\to
j_{*}(W_{m}\Omega^{\bullet}_{U}\otimes_{W_{m}{\mathcal{O}}_{U}}j^{*}W_{m}{\mathcal{O}}_{X}(-D))\cong
j_{*}W_{m}\Omega^{\bullet}_{U}$ (see [17, Exercise II.5.1(d)]). Using the
invertibility of $W_{m}{\mathcal{O}}_{X}(-D)$ again, we conclude that the
canonical map $W_{m}\Omega^{\bullet}_{X}(-D)\to
j_{*}W_{m}\Omega^{\bullet}_{U}$ is injective. Since this map factors through
$W_{m}\Omega^{\bullet}_{X}(-D)\to W_{m}\Omega^{\bullet}_{(X,D)}\hookrightarrow
W_{m}\Omega^{\bullet}_{X}$, the lemma follows. ∎

######  Lemma 5.2.

Let $A$ be a commutative ${\mathbb{F}}_{p}$-algebra and $I=(f)\subseteq A$ a
principal ideal. Let $m\geq 1$ be any integer and $n=p^{m}$. Then the
inclusion $W_{m}\Omega^{r}_{(A,I^{n})}\subseteq W_{m}\Omega^{r}_{A}$ of
$W_{m}(A)$-modules factors through
$W_{m}\Omega^{r}_{(A,I^{n})}\subseteq[f]\cdot W_{m}\Omega^{r}_{A}$ for every
$r\geq 0$.

###### Proof.

By [18, Lemma 1.2.2], $W_{m}\Omega^{r}_{(A,I)}$ is the $W_{m}(A)$-submodule of
$W_{m}\Omega^{r}_{A}$ generated by the de Rham-Witt forms of the type
$a_{0}da_{1}\wedge\cdots\wedge da_{r}$, where $a_{i}\in W_{m}(A)$ for all $i$
and $a_{i}\in W_{m}(I)$ for some $i$. We let
$\omega=a_{0}da_{1}\wedge\cdots\wedge da_{r}\in W_{m}\Omega^{r}_{(A,I^{n})}$
such that $a_{i}\in W_{m}(I^{n})$ for some $i\neq 0$. We can assume after a
permutation that $i=1$. We let $\omega^{\prime}=da_{2}\wedge\cdots\wedge
da_{r}$. We can write $a_{1}=\stackrel{{\scriptstyle
m-1}}{{\underset{i=0}{\sum}}}V^{i}([f^{n}]_{m-i}[a^{\prime}_{i}]_{m-i})$ for
some $a^{\prime}_{i}\in A$. It follows that $\omega=\stackrel{{\scriptstyle
m-1}}{{\underset{i=0}{\sum}}}a_{0}dV^{i}([f^{n}]_{m-i}[a^{\prime}_{i}]_{m-i})\wedge\omega^{\prime}$.

We fix an integer $0\leq i\leq m-1$ and let
$\omega_{i}=a_{0}dV^{i}([f^{n}]_{m-i}[a^{\prime}_{i}]_{m-i})\wedge\omega^{\prime}$.
It suffices to show that each $\omega_{i}\in W_{m}\Omega^{r}_{X}(-D)$, where
$X={\rm Spec\,}(A)$ and $D={\rm Spec\,}(A/I)$. However, we have

$\begin{array}[]{lll}\omega_{i}&=&a_{0}dV^{i}([f^{n}]_{m-i}[a^{\prime}_{i}]_{m-i})\wedge\omega^{\prime}\\\
&=&a_{0}d([f^{p^{m-i}}]_{m}V^{i}[a^{\prime}_{i}]_{m-i})\wedge\omega^{\prime}\\\
&=&[f^{p^{m-i}}]_{m}a_{0}dV^{i}[a^{\prime}_{i}]_{m-i}\wedge\omega^{\prime}+p^{m-i}[f^{p^{m-i}-1}]_{m}a_{0}V^{i}[a^{\prime}_{i}]_{m-i}d[f]_{m}\wedge\omega^{\prime}.\end{array}$

It is clear that the last term lies in $W_{m}\Omega^{r}_{X}(-D)$. This
finishes the proof. ∎

The following result will play a key role in our exposition.

######  Corollary 5.3.

If $(X,D)$ is a regular modulus pair, then the canonical map of pro-sheaves

$\\{W_{m}\Omega^{\bullet}_{X}(-nD)\\}_{n}\to\\{W_{m}\Omega^{\bullet}_{(X,nD)}\\}_{n}$

on $X_{\rm nis}$ is an isomorphism for every $m\geq 1$.

###  5.2. The relative logarithmic sheaves

In the remaining part of § 5, our default topology will be the étale topology.
All other topologies will be mentioned specifically. Let $k$ be a field of
characteristic $p>0$ and $D\subseteq X$ a closed immersion in
${\operatorname{\mathbf{Sch}}}_{k}$. Recall from [20] that
$W_{m}\Omega^{r}_{X,{\operatorname{log}}}$ is the étale subsheaf of
$W_{m}\Omega^{r}_{X}$ which is the image of the map ${\rm
dlog}\colon{\widetilde{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}}\to
W_{m}\Omega^{r}_{X}$, given by ${\rm dlog}(\\{a_{1},\ldots,a_{r}\\})={\rm
dlog}[a_{1}]_{m}\wedge\cdots\wedge{\rm dlog}[a_{r}]_{m}$. It is easily seen
that this map exists. One knows (e.g., see [36, Remark 1.6]) that this map in
fact factors through ${\rm
dlog}\colon{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}}\twoheadrightarrow
W_{m}\Omega^{r}_{X,{\operatorname{log}}}$. Moreover, this map is
multiplicative. The naturality of the dlog map gives rise to its relative
version

(5.1) ${\rm
dlog}\colon\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/{p^{m}}}\to
W_{m}\Omega^{r}_{(X,D),{\operatorname{log}}}:={\rm
Ker}(W_{m}\Omega^{r}_{X,{\operatorname{log}}}\twoheadrightarrow
W_{m}\Omega^{r}_{D,{\operatorname{log}}})$

which is a morphism of étale sheaves of
${\widehat{{\mathcal{K}}}^{M}_{*,X}}/{p^{m}}$-modules as $r\geq 0$ varies.

Recall that the Frobenius map $F\colon W_{m+1}\Omega^{r}_{X}\to
W_{m}\Omega^{r}_{X}$ sends ${\rm Ker}(R)$ into the subsheaf
$dV^{m-1}\Omega^{r-1}_{X}$ so that there is an induced map $\overline{F}\colon
W_{m}\Omega^{r}_{X}\to{W_{m}\Omega^{r}_{X}}/{dV^{m-1}\Omega^{r-1}_{X}}$. We
let $\pi\colon
W_{m}\Omega^{r}_{X}\to{W_{m}\Omega^{r}_{X}}/{dV^{m-1}\Omega^{r-1}_{X}}$ denote
the projection map. Since $R-F\colon W_{m+1}\Omega^{r}_{X}\to
W_{m}\Omega^{r}_{X}$ is surjective by [20, Proposition I.3.26], it follows
that $\pi-\overline{F}$ is also surjective. Indeed, this surjectivity is
proven for smooth schemes over $k$ in loc. cit., but then the claim follows
because $X$ can be seen locally as a closed subscheme of a regular scheme and
a regular scheme is a filtered inductive limit of smooth schemes over $k$, by
Néron–Popescu desingularisation. We thus get a commutative diagram

(5.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}\Omega^{r}_{X,{\operatorname{log}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}\Omega^{r}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi-\overline{F}}$$\textstyle{\frac{W_{m}\Omega^{r}_{X}}{dV^{m-1}\Omega^{r-1}_{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}\Omega^{r}_{D,{\operatorname{log}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}\Omega^{r}_{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi-\overline{F}}$$\textstyle{\frac{W_{m}\Omega^{r}_{D}}{dV^{m-1}\Omega^{r-1}_{D}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The middle and the right vertical arrows are surjective by definition and the
two rows are exact by loc. cit. and [40, Corollary 4.2]. The surjectivity of
the left vertical arrow follows by applying the dlog map to the surjection
$\widehat{{\mathcal{K}}}^{M}_{r,X}\twoheadrightarrow\widehat{{\mathcal{K}}}^{M}_{r,D}$.
Taking the kernels of the vertical arrows, we get a short exact sequence

(5.3) $0\to W_{m}\Omega^{r}_{(X,D),{\operatorname{log}}}\to
W_{m}\Omega^{r}_{(X,D)}\to\frac{W_{m}\Omega^{r}_{(X,D)}}{W_{m}\Omega^{r}_{(X,D)}\cap
dV^{m-1}\Omega^{r-1}_{X}}\to 0.$

The following property of the relative dlog map will be important to us.

######  Lemma 5.4.

Assume that $k$ is perfect and $X$ is regular. Then the map of étale pro-
sheaves

${\rm
dlog}\colon\\{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,nD)}}/{p^{m}}}\\}_{n}\to\\{W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}$

is an isomorphism for every $m,r\geq 1$.

###### Proof.

We consider the commutative diagram of pro-sheaves:

(5.4)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,nD)}}/{p^{m}}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
dlog}}$$\textstyle{{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
dlog}}$$\scriptstyle{\cong}$$\textstyle{\\{{\widehat{{\mathcal{K}}}^{M}_{r,nD}}/{p^{m}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
dlog}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}\Omega^{r}_{X,{\operatorname{log}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The middle vertical arrow is an isomorphism by the Bloch-Gabber-Kato theorem
for fields [5, Corollary 2.8] and the proof of the Gersten conjecture for
Milnor $K$-theory ([27, Proposition 10]) and logarithmic Hodge-Witt sheaves
([13, Thèoréme 1.4]). The right vertical arrow is an isomorphism by [36,
Theorem 0.3]. Since the rows are exact, it follows that the left vertical
arrow is also an isomorphism. This finishes the proof. ∎

We now prove a Nisnevich version of Lemma 5.4. For any $k$-scheme $X$, we let
$W_{m}\Omega^{r}_{X,{\operatorname{log}},{\rm nis}}$ be the image of the map
of Nisnevich sheaves ${\rm
dlog}\colon{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}}\to
W_{m}\Omega^{r}_{X}$, given by ${\rm dlog}(\\{a_{1},\ldots,a_{r}\\})={\rm
dlog}[a_{1}]_{m}\wedge\cdots\wedge{\rm dlog}[a_{r}]_{m}$ (see [36, Remark 1.6]
for the existence of this map). The naturality of this map gives rise to its
relative version

(5.5) ${\rm
dlog}\colon\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,D)}}/{p^{m}}}\to
W_{m}\Omega^{r}_{(X,D),{\operatorname{log}},{\rm nis}}:={\rm
Ker}(W_{m}\Omega^{r}_{X,{\operatorname{log}},{\rm nis}}\twoheadrightarrow
W_{m}\Omega^{r}_{D,{\operatorname{log}},{\rm nis}})$

if $D\subseteq X$ is a closed immersion.

######  Lemma 5.5.

Assume that $k$ is perfect and $X$ is regular. Then the map of Nisnevich pro-
sheaves

${\rm
dlog}\colon\\{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,nD)}}/{p^{m}}}\\}_{n}\to\\{W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}},{\rm
nis}}\\}_{n}$

is an isomorphism for every $m,r\geq 1$.

###### Proof.

The proof is completely identical to that of Lemma 5.4, where we only have to
observe that the middle and the right vertical arrows are isomorphisms even in
the Nisnevich topology, by the same references that we used for the étale
case. ∎

######  Lemma 5.6.

Assume that $k$ is a finite field and $X$ is regular of pure dimension $d\geq
1$. Then the canonical map

$H^{i}_{\rm nis}(X,{\widehat{{\mathcal{K}}}^{M}_{d,X}}/{p^{m}})\to
H^{i}_{\acute{e}t}(X,{\widehat{{\mathcal{K}}}^{M}_{d,X}}/{p^{m}})$

is an isomorphism for $i=d$ and surjective for $i=d-1$.

###### Proof.

We have seen in the proofs of Lemmas 5.4 and 5.5 that for every $r\geq 0$,
there are natural isomorphisms

(5.6) ${({\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}})}_{{\rm nis}}\cong
W_{m}\Omega^{r}_{X,{\operatorname{log}},{\rm nis}}\ \ {\rm and}\ \
{({\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}})}_{{\acute{e}t}}\cong
W_{m}\Omega^{r}_{X,{\operatorname{log}}}.$

Using these isomorphisms, the lemma follows from [30, Propositions 3.3.2,
3.3.3]. The only additional input one has to use is that
$H^{b}_{x}(X_{\acute{e}t},W_{m}\Omega^{d}_{X,{\operatorname{log}}})=0$ if
$x\in X^{(a)}$ and $b<a$ by [39, Corollary 2.2] so that the right hand side of
[loc. cit., (3.3.3)] is zero. This is needed in the proof of the $i=d-1$ case
of the lemma. ∎

Assume now that $k$ is a perfect field,
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ is a regular scheme of pure dimension
$d\geq 1$ and $D\subset X$ is a nowhere dense closed subscheme. We fix
integers $m\geq 1$ and $i,r\geq 0$. Under these assumptions, we shall prove
the following results.

######  Lemma 5.7.

There is a short exact sequence

$0\to
W_{m-i}\Omega^{r}_{X,{\operatorname{log}}}\xrightarrow{\underline{p}^{i}}W_{m}\Omega^{r}_{X,{\operatorname{log}}}\xrightarrow{R^{m-i}}W_{i}\Omega^{r}_{X,{\operatorname{log}}}\to
0$

of sheaves on $X$ in Nisnevich and étale topologies.

###### Proof.

The étale version of the lemma is already known by [6, Lemma 3]. To prove its
Nisnevich version, we can use the first isomorphism of (5.6). This reduces the
proof to showing that tensoring the exact sequence

$0\to{{\mathbb{Z}}}/{p^{m-i}}\xrightarrow{p^{i}}{{\mathbb{Z}}}/{p^{m}}\xrightarrow{\pi_{m-i}}{{\mathbb{Z}}}/{p^{i}}\to
0$

with ${\widehat{{\mathcal{K}}}^{M}_{r,X}}$ yields an exact sequence

(5.7)
$0\to{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m-i}}\xrightarrow{p^{i}}{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{m}}\xrightarrow{\pi_{m-i}}{\widehat{{\mathcal{K}}}^{M}_{r,X}}/{p^{i}}\to
0$

of Nisnevich sheaves on $X$. But this follows directly from the fact that
${\widehat{{\mathcal{K}}}^{M}_{r,X}}$ has no $p^{i}$-torsion (see [11, Theorem
8.1]). ∎

######  Lemma 5.8.

There is a short exact sequence

$0\to\\{W_{m-i}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\xrightarrow{\underline{p}^{i}}\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\xrightarrow{R^{m-i}}\\{W_{i}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\to
0$

of pro-sheaves on $D$ in Nisnevich and étale topologies.

###### Proof.

The argument below works for either of Nisnevich and étale topologies. We
shall prove the lemma by modifying the proof of [40, Theorem 4.6]. The latter
result says that there is an exact sequence

(5.8)
$\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\xrightarrow{{p}^{i}}\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\xrightarrow{R^{m-i}}\\{W_{i}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\to
0.$

It suffices therefore to show that the first arrow in this sequence has a
factorization

(5.9)
$\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\stackrel{{\scriptstyle
R^{i}}}{{\twoheadrightarrow}}\\{W_{m-i}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\stackrel{{\scriptstyle\underline{p}^{i}}}{{\hookrightarrow}}\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}.$

To prove this factorization, we look at the commutative diagram

(5.10)
$\textstyle{\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{p}^{i}}$$\textstyle{\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{W_{m}\Omega^{r}_{nD}\\}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{p}^{i}}$$\textstyle{\\{W_{m}\Omega^{r}_{nD}\\}_{n}.}$

It follows from [40, Proposition 2.14] that the map $p^{i}$ at the bottom has
a factorization (see § 5.1 for the definitions of various filtrations of
$W_{m}\Omega^{\bullet}_{nD}$)

(5.11) $\\{W_{m}\Omega^{r}_{nD}\\}_{n}\stackrel{{\scriptstyle
R^{i}}}{{\twoheadrightarrow}}\\{W_{m-i}\Omega^{r}_{nD}\\}_{n}\stackrel{{\scriptstyle\underline{p}^{i}}}{{\hookrightarrow}}\\{W_{m}\Omega^{r}_{nD}\\}_{n}.$

Since the image of $\\{W_{m}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}$ under
$R^{i}$ is $\\{W_{m-i}\Omega^{r}_{nD,{\operatorname{log}}}\\}_{n}$ (note the
surjectivity of the second arrow in (5.8)), it follows at once from (5.11)
that the map $p^{i}$ on the top in (5.10) has a factorization as desired in
(5.9). This concludes the proof. ∎

######  Proposition 5.9.

There is a short exact sequence

$0\to\\{W_{m-i}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}\xrightarrow{\underline{p}^{i}}\\{W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}\xrightarrow{R^{m-i}}\\{W_{i}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}\to
0$

of pro-sheaves on $X$ in Nisnevich and étale topologies.

###### Proof.

Combine the previous two lemmas and use that the maps
$W_{m}\Omega^{r}_{X,{\operatorname{log}}}\to
W_{m}\Omega^{r}_{nD,{\operatorname{log}}}$ and
$W_{m}\Omega^{r}_{X,{\operatorname{log}},{\rm nis}}\to
W_{m}\Omega^{r}_{nD,{\operatorname{log}},{\rm nis}}$ are surjective. ∎

Applying the cohomology functor, we get

######  Corollary 5.10.

There is a long exact sequence of pro-abelian groups

$\cdots\to\\{H^{j}_{\acute{e}t}(X,W_{m-i}\Omega^{r}_{(X,nD),{\operatorname{log}}})\\}_{n}\xrightarrow{\underline{p}^{i}}\\{H^{j}_{\acute{e}t}(X,W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}})\\}_{n}\hskip
85.35826pt$ $\hskip
213.39566pt\xrightarrow{R^{m-i}}\\{H^{j}_{\acute{e}t}(X,W_{i}\Omega^{r}_{(X,nD),{\operatorname{log}}})\\}_{n}\xrightarrow{\partial^{i}}\cdots$

The same also holds in the Nisnevich topology.

###  5.3. Some cohomology exact sequences

Let us now assume that $k$ is a finite field and
$X\in{\operatorname{\mathbf{Sch}}}_{k}$ is regular of pure dimension $d\geq
1$. For any $p^{m}$-torsion abelian group $V$, we let $V^{\star}={\rm
Hom}_{{{\mathbb{Z}}}/{p^{m}}}(V,{{\mathbb{Z}}}/{p^{m}})$. Let $D\subset X$ be
an effective Cartier divisor with complement $U$. We let
$F^{j}_{m,r}(n)=H^{j}_{\acute{e}t}(X,W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}})$
and $F^{j}_{m,r}(U)={\varprojlim}_{n}F^{j}_{m,r}(n)$. Each group
${(F^{j}_{m,r}(n))}^{\star}$ is a profinite abelian group (see [41, Theorem
2.9.6]).

######  Lemma 5.11.

The sequence

$\cdots\to F^{j}_{m-1,r}(U)\to F^{j}_{m,r}(U)\to F^{j}_{1,r}(U)\to
F^{j+1}_{m-1,r}(U)\to\cdots$

is exact.

###### Proof.

We first prove by induction on $m$ that
${\varprojlim}^{1}_{n}F^{j}_{m,r}(n)=0$. The $m=1$ case follows from Lemma
8.4. In general, we break the long exact sequence of Corollary 5.10 into short
exact sequences. This yields exact sequences of pro-abelian groups

(5.12) $0\to{\rm Image}(\partial^{1})\to\\{F^{j}_{m-1,r}(n)\\}_{n}\to{\rm
Ker}(R^{m-1})\to 0;$ (5.13) $0\to{\rm
Ker}(R^{m-1})\to\\{F^{j}_{m,r}(n)\\}_{n}\to{\rm Image}(R^{m-1})\to 0;$ (5.14)
$0\to{\rm Image}(R^{m-1})\to\\{F^{j}_{1,r}(n)\\}_{n}\to{\rm
Image}(\partial^{1})\to 0.$

Lemma 8.4 says that $\\{F^{j}_{1,r}(n)\\}_{n}$ is a pro-system of finite
abelian groups. This implies by (5.14) that each of ${\rm Image}(R^{m-1})$ and
${\rm Image}(\partial^{1})$ is isomorphic to a pro-system of finite abelian
groups. In particular, the pro-systems of (5.14) do not admit higher derived
lim functors. On the other hand, ${\varprojlim}^{1}_{n}F^{j}_{m-1,r}(n)=0$ by
induction. It follows from (5.12) that ${\varprojlim}^{1}_{n}{\rm
Ker}(R^{m-1})=0$. Using (5.13), we get
${\varprojlim}^{1}_{n}F^{j}_{m,r}(n)=0$.

It follows from what we have shown is that the above three short exact
sequences of pro-abelian groups remain exact after we apply the inverse limit
functor. But this implies that the long exact sequence of Corollary 5.10 also
remains exact after applying the inverse limit functor. This proves the lemma.
∎

######  Lemma 5.12.

There is a long exact sequence

$\cdots\to{\varinjlim}_{n}{(F^{j}_{i,r}(n))}^{\star}\xrightarrow{(R^{m-i})^{\star}}{\varinjlim}_{n}{(F^{j}_{m,r}(n))}^{\star}\xrightarrow{(\underline{p}^{i})^{\star}}{\varinjlim}_{n}{(F^{j}_{m-i,r}(n))}^{\star}\to\cdots$

of abelian groups for every $i\geq 0$.

###### Proof.

This is an easy consequence of Corollary 5.10 using the fact that the
Pontryagin dual functor (recalled in § 7.4) is exact on the category of
discrete torsion abelian groups (see [41, Theorem 2.9.6]) and the direct limit
functor is exact on the category of ind-abelian groups. We also have to note
that the Pontryagin dual of a discrete $p^{m}$-torsion abelian group $V$
coincides with $V^{\star}$. ∎

######  Lemma 5.13.

Assume further that $D\subset X$ is a simple normal crossing divisor. Then the
group ${\varprojlim}_{n}F^{d}_{m,d}(n)$ is profinite and the canonical map

${\varinjlim}_{n}{(F^{d}_{m,d}(n))}^{\star}\to({\varprojlim}_{n}F^{d}_{m,d}(n))^{\star}$

is an isomorphism of topological abelian groups if either
$k\neq{\mathbb{F}}_{2}$ or $d\neq 2$.

###### Proof.

By [26, Theorem 9.1], Proposition 2.7, Corollary 4.2 and Lemma 5.4,
$\\{F^{d}_{m,d}(n)\\}_{n}$ is isomorphic to a pro-abelian group
$\\{E_{n}\\}_{n}$ such that each $E_{n}$ is finite and the map $E_{n+1}\to
E_{n}$ is surjective. We can therefore apply Lemma 7.3. ∎

##  6\. Cartier map for twisted de Rham-Witt complex

In this section, we shall prove the existence and some properties of the
Cartier homomorphism for the twisted de Rham-Witt complex. We fix a perfect
field $k$ of characteristic $p>0$ and a modulus pair $(X,D)$ in
${\operatorname{\mathbf{Sch}}}_{k}$ such that $X$ is connected and regular of
dimension $d\geq 1$. Let $\iota\colon D\hookrightarrow X$ and $j\colon
U\hookrightarrow X$ be the inclusions, where $U=X\setminus D$. Let $F$ denote
the function field of $X$. We also fix integers $m\geq 1$, $n\in{\mathbb{Z}}$
and $r\geq 0$. Let $W_{m}{\mathcal{O}}_{X}(nD)$ be the Nisnevich sheaf on $X$
defined in § 5.1. We begin with following result describing the behavior of
the Frobenius and Verschiebung operators on
$W_{\bullet}{\mathcal{O}}_{X}(nD)$. We shall then use this result to describe
these operators on the full twisted de Rham-Witt complex. Until we talk about
the topology of $X$ again in this section, we shall assume it to be the
Nisnevich topology.

######  Lemma 6.1.

We have

  1. (1)

$F(W_{m+1}{\mathcal{O}}_{X}(nD))\subseteq W_{m}{\mathcal{O}}_{X}(pnD)$.

  2. (2)

$V(W_{m}{\mathcal{O}}_{X}(pnD))\subseteq W_{m+1}{\mathcal{O}}_{X}(nD)$.

  3. (3)

$R(W_{m+1}{\mathcal{O}}_{X}(nD))\subseteq W_{m}{\mathcal{O}}_{X}(nD)$.

###### Proof.

We can check the lemma locally at a point $x\in X$. We let
$A={\mathcal{O}}^{h}_{X,x}$ and let $f\in A$ define $D$ at $x$. The lemma is
easy to check when $n\leq 0$. We therefore assume $n\geq 1$. For $w\in
W_{m+1}(A)$, we have $F(w[f^{-n}])=F(w)\cdot F([f^{-n}])=F(w)\cdot[f^{-pn}]\in
W_{m}{\mathcal{O}}^{h}_{X,x}(pnD)$. This proves (1). For (2), we use the
projection formula to get $V(w^{\prime}[f^{-pn}])=V(w^{\prime}\cdot
F([f^{-n}]))=V(w^{\prime})\cdot[f^{-n}]\in W_{m+1}{\mathcal{O}}^{h}_{X,x}(nD)$
for $w^{\prime}\in W_{m}(A)$. The part (3) is obvious. ∎

######  Lemma 6.2.

For $i\geq 1$, there is a short exact sequence

$0\to
W_{m-i}{\mathcal{O}}_{X}(p^{i}nD)\xrightarrow{V^{i}}W_{m}{\mathcal{O}}_{X}(nD)\xrightarrow{R^{m-i}}W_{i}{\mathcal{O}}_{X}(nD)\to
0.$

###### Proof.

Since $R$ is a ring homomorphism on $j_{*}W_{m}{\mathcal{O}}_{U}$ on which it
is surjective, it follows that it is surjective on
$W_{m}{\mathcal{O}}_{X}(*D)$. Since the sequence

(6.1) $0\to
W_{m-i}{\mathcal{O}}_{X}\xrightarrow{V^{i}}W_{m}{\mathcal{O}}_{X}\xrightarrow{R^{m-i}}W_{i}{\mathcal{O}}_{X}\to
0$

is anyway exact, and since $V^{i}$ and $R$ are defined by Lemma 6.1, we only
have to show that $V^{i}(j_{*}W_{m-i}{\mathcal{O}}_{U})\cap
W_{m}{\mathcal{O}}_{X}(nD)=V^{i}(W_{m-i}{\mathcal{O}}_{X}(p^{i}nD))$. Arguing
by induction on $i\geq 1$, it suffices to check this for $i=1$.

Now, we let $w=(a_{1},\ldots,a_{m-1})\in j_{*}W_{m-1}{\mathcal{O}}_{U}$ be
such that
$V(w)=(0,a_{1},\ldots,a_{m-1})=(b_{1},\ldots,b_{m})\cdot[f^{-n}]=(b_{1}f^{-n},b_{2}f^{-pn},\ldots,b_{m}f^{-p^{m-1}n})$,
where $b_{i}\in{\mathcal{O}}_{X}$. This implies that $b_{1}=0$ and
$a_{i}=b_{i+1}f^{-p^{i}n}$ for $i\geq 1$. Hence,
$w=(b_{2}f^{-pn},\ldots,b_{m}f^{-p^{m-1}n})=(b_{2},\ldots,b_{m})\cdot[f^{-pn}]\in
W_{m-1}{\mathcal{O}}_{X}(pnD)$. ∎

We shall now extend Lemma 6.1 to the de Rham-Witt forms of higher degrees. One
knows using the Gersten conjecture for the de Rham-Witt sheaves due to Gros
[12] that the map $W_{m}\Omega^{r}_{X}\to W_{m}\Omega^{r}_{K}$ is injective.
It follows that the canonical map $W_{m}\Omega^{r}_{X}\to
j_{*}W_{m}\Omega^{r}_{U}$ is injective too. Using the invertibility of the
$W_{m}{\mathcal{O}}_{X}$-module $W_{m}{\mathcal{O}}_{X}(nD)$, we get an
injection

(6.2)
$\textstyle{W_{m}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}{\mathcal{O}}_{X}(nD)\otimes_{W_{m}{\mathcal{O}}_{X}}W_{m}\Omega^{r}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}{\mathcal{O}}_{X}(nD)\otimes_{W_{m}{\mathcal{O}}_{X}}j_{*}W_{m}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}W_{m}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{m}{\mathcal{O}}_{X}(nD)\cdot
j_{*}W_{m}\Omega^{r}_{U}.}$

######  Lemma 6.3.

Let $q\geq 1$ be a positive integer. Then the inclusions
$W_{m}{\mathcal{O}}_{X}(qnD)\hookrightarrow W_{m}{\mathcal{O}}_{X}(q(n+1)D)$
induce an isomorphism

${\underset{n\geq 1}{\varinjlim}}\
W_{m}{\mathcal{O}}_{X}(qnD)\xrightarrow{\cong}j_{*}W_{m}{\mathcal{O}}_{U}.$

###### Proof.

We shall prove the lemma by induction on $m$. For $m=1$, the statement is
clear. We now assume $m\geq 2$ and use the commutative diagram

(6.3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\underset{n\geq
1}{\varinjlim}}\
W_{m-1}{\mathcal{O}}_{X}(qpnD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\underset{n\geq
1}{\varinjlim}}\
W_{m}{\mathcal{O}}_{X}(qnD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{m-1}}$$\textstyle{{\underset{n\geq
1}{\varinjlim}}\
W_{1}{\mathcal{O}}_{X}(qnD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}W_{m-1}{\mathcal{O}}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{j_{*}W_{m}{\mathcal{O}}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{m-1}}$$\textstyle{j_{*}W_{1}{\mathcal{O}}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

It is easy to check that the bottom row is exact. The top row is exact by
Lemma 6.2. The right vertical arrow is an isomorphism by $m=1$ case and the
left vertical arrow is an isomorphism by induction on $m$ (with $q$ replaced
by $qp$). The lemma follows. ∎

######  Corollary 6.4.

For $q\geq 1$, the canonical map $W_{m}\Omega^{r}_{X}(qnD)\to
W_{m}\Omega^{r}_{X}(q(n+1)D)$ is injective and the map ${\underset{n\geq
1}{\varinjlim}}\ W_{m}\Omega^{r}_{X}(qnD)\to j_{*}W_{m}\Omega^{r}_{U}$ is an
isomorphism of $W_{m}{\mathcal{O}}_{X}$-modules.

###### Proof.

The injectivity claim follows from (6.2). To prove the isomorphism, we take
the tensor product with the $W_{m}{\mathcal{O}}_{X}$-module
$W_{m}\Omega^{r}_{X}$ on the two sides of the isomorphism in Lemma 6.3. This
yields an isomorphism ${\underset{n\geq 1}{\varinjlim}}\
W_{m}\Omega^{r}_{X}(qnD)\cong(j_{*}W_{m}{\mathcal{O}}_{U})\otimes_{W_{m}{\mathcal{O}}_{X}}W_{m}\Omega^{r}_{X}$.
On the other hand, the étale descent property (see [20, Proposition I.1.14])
of the de Rham-Witt sheaves says that the canonical map
$(j_{*}W_{m}{\mathcal{O}}_{U})\otimes_{W_{m}{\mathcal{O}}_{X}}W_{m}\Omega^{r}_{X}\to
j_{*}W_{m}\Omega^{r}_{U}$ is an isomorphism. The corollary follows. ∎

In view of Corollary 6.4, we shall hereafter consider the sheaves
$W_{m}\Omega^{r}_{X}(nD)$ as $W_{m}{\mathcal{O}}_{X}$-submodules of
$j_{*}W_{m}\Omega^{r}_{U}$.

######  Lemma 6.5.

The $V,F$ and $R$ operators of $j_{*}W_{m}\Omega^{r}_{U}$ satisfy the
following.

  1. (1)

$F(W_{m}\Omega^{r}_{X}(nD))\subseteq W_{m-1}\Omega^{r}_{X}(pnD)$.

  2. (2)

$V(W_{m}\Omega^{r}_{X}(pnD))\subseteq W_{m+1}\Omega^{r}_{X}(nD)$.

  3. (3)

$R(W_{m}\Omega^{r}_{X}(nD))\subseteq W_{m}\Omega^{r}_{X}(nD)$.

###### Proof.

We have $F([f^{-n}]\omega)=F(\omega)F([f^{-n}])\in W_{m-1}\Omega^{r}_{X}\cdot
W_{m-1}{\mathcal{O}}_{X}(pnD)=W_{m-1}\Omega^{r}_{X}(pnD)$, where the first
inclusion holds by Lemma 6.1. We have
$V([f^{-pn}]\omega)=V(F([f^{-n}])\omega)=[f^{-n}]V(\omega)\in
W_{m+1}{\mathcal{O}}_{X}(nD)\cdot
W_{m+1}\Omega^{r}_{X}=W_{m+1}\Omega^{r}_{X}(nD)$, where the second inclusion
holds again by Lemma 6.1. The last assertion is clear. ∎

######  Lemma 6.6.

The multiplication by $p$ map $p\colon W_{m+1}\Omega^{r}_{X}(nD)\to
W_{m+1}\Omega^{r}_{X}(nD)$ has a factorization

$W_{m+1}\Omega^{r}_{X}(nD)\stackrel{{\scriptstyle
R}}{{\twoheadrightarrow}}W_{m}\Omega^{r}_{X}(nD)\stackrel{{\scriptstyle\underline{p}}}{{\hookrightarrow}}W_{m+1}\Omega^{r}_{X}(nD)$

in the category of sheaves of $W_{m+1}{\mathcal{O}}_{X}$-modules. This
factorization is natural in $X$.

###### Proof.

Since $R$ is $W_{m+1}{\mathcal{O}}_{X}$-linear, we get an exact sequence (see
§ 5.1)

(6.4) $0\to(V^{m}W_{1}\Omega^{r}_{X}+dV^{m}W_{1}\Omega^{r-1}_{X})\cdot
W_{m+1}{\mathcal{O}}_{X}(nD)\to
W_{m+1}\Omega^{r}_{X}(nD)\xrightarrow{R}W_{m}\Omega^{r}_{X}(nD)\to 0.$

On the other hand, we note that $p\colon W_{m+1}\Omega^{r}_{X}\to
W_{m+1}\Omega^{r}_{X}$ is also a $W_{m+1}{\mathcal{O}}_{X}$-linear
homomorphism. Since $W_{m+1}{\mathcal{O}}_{X}(nD)$ is an invertible
$W_{m+1}{\mathcal{O}}_{X}$-module, we get an exact sequence

(6.5) $0\to{{\rm Ker}(p)}\cdot W_{m+1}{\mathcal{O}}_{X}(nD)\to
W_{m+1}\Omega^{r}_{X}(nD)\xrightarrow{p}W_{m+1}\Omega^{r}_{X}(nD).$

Since ${\rm Ker}(p)={\rm Ker}(R)$ as $X$ is regular (see § 5.1), we get ${{\rm
Ker}(p)}\cdot
W_{m+1}{\mathcal{O}}_{X}(nD)=(V^{m}W_{1}\Omega^{r}_{X}+dV^{m}W_{1}\Omega^{r-1}_{X})\cdot
W_{m+1}{\mathcal{O}}_{X}(nD)$. The first part of the lemma now follows. The
naturality is clear from the proof. ∎

######  Lemma 6.7.

The square

(6.6)
$\textstyle{W_{m}\Omega^{r}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{p}}$$\scriptstyle{j^{*}}$$\textstyle{W_{m+1}\Omega^{r}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{j_{*}W_{m}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{p}}$$\textstyle{j_{*}W_{m+1}\Omega^{r}_{U}}$

is Cartesian.

###### Proof.

We can check this locally. So let $x\in j_{*}W_{m}\Omega^{r}_{U}$ be such that
$\underline{p}(x)\in W_{m+1}\Omega^{r}_{X}$. Since $j$ is affine and
$W_{m}\Omega^{r}_{X}$ is an $W_{m}{\mathcal{O}}_{X}$-module, it follows that
$j_{*}W_{m+1}\Omega^{r}_{U}\xrightarrow{j_{*}R}j_{*}W_{m}\Omega^{r}_{U}$ is
surjective. We can therefore find $\tilde{x}\in j_{*}W_{m+1}\Omega^{r}_{U}$
such that $R(\tilde{x})=x$. In particular, $p\tilde{x}=\underline{p}(x)\in
W_{m+1}\Omega^{r}_{X}$. We thus get $VF(\tilde{x})=p\tilde{x}\in
W_{m+1}\Omega^{r}_{X}$. Since

$(j_{*}VW_{m}\Omega^{r}_{U})\cap W_{m+1}\Omega^{r}_{X}={\rm Ker}(F^{m}\colon
W_{m+1}\Omega^{r}_{X}\to j_{*}\Omega^{r}_{U})={\rm Ker}(F^{m}\colon
W_{m+1}\Omega^{r}_{X}\to\Omega^{r}_{X})$ $=VW_{m}\Omega^{r}_{X},$

it follows that there exists $y^{\prime}\in W_{m}\Omega^{r}_{X}$ such that
$VF(\tilde{x})=Vy^{\prime}$.

Since ${\rm Ker}(V\colon j_{*}W_{m}\Omega^{r}_{U}\to
j_{*}W_{m+1}\Omega^{r}_{U})=j_{*}FdV^{m}\Omega^{r-1}_{U}$ (see [20,
I.3.21.1.4]), it follows that there exists $z^{\prime}\in
j_{*}\Omega^{r-1}_{U}$ such that
$FdV^{m}(z^{\prime})=F(\tilde{x})-y^{\prime}$. Equivalently,
$F(\tilde{x}-dV^{m}(z^{\prime}))=y^{\prime}\in W_{m}\Omega^{r}_{X}$. Since

$j_{*}FW_{m+1}\Omega^{r}_{U}\cap W_{m}\Omega^{r}_{X}={\rm Ker}(F^{m-1}d\colon
W_{m}\Omega^{r}_{X}\to j_{*}\Omega^{r+1}_{U})={\rm Ker}(F^{m-1}d\colon
W_{m}\Omega^{r}_{X}\to\Omega^{r+1}_{X})$ $=FW_{m+1}\Omega^{r}_{X},$

we can find $y^{\prime\prime}\in W_{m+1}\Omega^{r}_{X}$ such that
$F(\tilde{x}-dV^{m}(z^{\prime}))=F(y^{\prime\prime})$.

Since ${\rm Ker}(F\colon j_{*}W_{m+1}\Omega^{r}_{U}\to
j_{*}W_{m}\Omega^{r}_{U})=j_{*}V^{m}\Omega^{r}_{U}$ (see [20, I.3.21.1.2]), we
can find $z^{\prime\prime}\in j_{*}\Omega^{r}_{U}$ such that
$V^{m}(z^{\prime\prime})=\tilde{x}-dV^{m}(z^{\prime})-y^{\prime\prime}$.
Equivalently,
$\tilde{x}-y^{\prime\prime}=V^{m}(z^{\prime\prime})+dV^{m}(z^{\prime})$. On
the other hand, we have

$V^{m}j_{*}\Omega^{r}_{U}+dV^{m}j_{*}\Omega^{r-1}_{U}={\rm Ker}(R\colon
j_{*}W_{m+1}\Omega^{r}_{U}\to j_{*}W_{m}\Omega^{r}_{U}).$

We thus get $x=R(\tilde{x})=R(y^{\prime\prime})$. Since $y^{\prime\prime}\in
W_{m+1}\Omega^{r}_{X}$, we get $x\in W_{m}\Omega^{r}_{X}$. This proves the
lemma. ∎

It is easy to see that for $n\in{\mathbb{Z}}$, the differential $d\colon
j_{*}W_{m}\Omega^{r}_{X}\to j_{*}W_{m}\Omega^{r+1}_{X}$ restricts to a
homomorphism $d\colon W_{m}\Omega^{r}_{X}(nD)\to
W_{m}\Omega^{r+1}_{X}((n+1)D)$ such that the composite map $d^{2}\colon
W_{m}\Omega^{r}_{X}(nD)\to W_{m}\Omega^{r+2}_{X}((n+2)D)$ is zero by Corollary
6.4. The map $d\colon W_{m}\Omega^{r}_{X}(p^{m}nD)\to
W_{m}\Omega^{r+1}_{X}((p^{m}n+1)D)$ actually factors through $d\colon
W_{m}\Omega^{r}_{X}(p^{m}nD)\to W_{m}\Omega^{r+1}_{X}(p^{m}nD)$ as one easily
checks. In particular, $W_{m}\Omega^{\bullet}_{X}(p^{m}nD)$ is a complex for
every $m\geq 1$ and $n\in{\mathbb{Z}}$.

Let

$Z_{1}W_{m}\Omega^{r}_{X}(nD)=(j_{*}Z_{1}W_{m}\Omega^{r}_{U})\cap
W_{m}\Omega^{r}_{X}(nD)=j_{*}{\rm Ker}(F^{m-1}d)\cap W_{m}\Omega^{r}_{X}(nD)$
$={\rm Ker}(F^{m-1}d\colon W_{m}\Omega^{r}_{X}(nD)\to j_{*}\Omega_{U})={\rm
Ker}(F^{m-1}d\colon
W_{m}\Omega^{r}_{X}(nD)\to\Omega^{r+1}_{X}(p^{m-1}(n+1)D)),$

where the third equality is by the left exactness of $j_{*}$ and the last
equality by Lemma 6.5. If $m=1$, we get

(6.7) $Z_{1}W_{1}\Omega^{r}_{X}(nD)=Z\Omega^{r}_{X}(nD)={\rm
Ker}(d\colon\Omega^{r}_{X}(nD)\to\Omega^{r+1}_{X}((n+1)D)).$

We let $B\Omega^{r}_{X}(nD)={\rm
Image}(d\colon\Omega^{r}_{X}((n-1)D)\to\Omega^{r}_{X}(nD))$.

######  Proposition 6.8.

There exists a homomorphism $C\colon Z_{1}W_{m}\Omega^{r}_{X}(pnD)\to
W_{m}\Omega^{r}_{X}(nD)$ such that the diagram

(6.8)
---
$\textstyle{Z_{1}W_{m}\Omega^{r}_{X}(pnD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\scriptstyle{C}$$\textstyle{W_{m+1}\Omega^{r}_{X}(nD)}$$\textstyle{W_{m}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{p}}$

is commutative. The map $C$ induces an isomorphism of
${\mathcal{O}}_{X}$-modules

$C\colon{\mathcal{H}}^{r}(\psi_{*}W_{1}\Omega^{\bullet}_{X}(pnD))\xrightarrow{\cong}W_{1}\Omega^{r}_{X}(nD).$

###### Proof.

We consider the diagram

(6.9)
$\textstyle{Z_{1}W_{m}\Omega^{r}_{X}(pnD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{W_{m}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{p}}$$\textstyle{W_{m+1}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}Z_{1}W_{m}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{C}$$\scriptstyle{V}$$\textstyle{j_{*}W_{m}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{p}}$$\textstyle{j_{*}W_{m+1}\Omega^{r}_{U},}$

where the vertical arrows are inclusions from Corollary 6.4. The right square
exists and commutes by Lemma 6.6. The big outer square clearly commutes. It
suffices therefore to show that the right square is Cartesian.

We have a commutative diagram of $W_{m+1}{\mathcal{O}}_{X}$-modules

(6.10)
$\textstyle{W_{m+1}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{W_{m+1}\Omega^{r}_{X}(nD)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{W_{m+1}\Omega^{r}_{X}(nD)}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{j_{*}W_{m+1}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{j_{*}W_{m+1}\Omega^{r}_{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{j_{*}{W_{m+1}\Omega^{r}_{U}}/p\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

The top sequence is clearly exact and the bottom sequence is exact by the
Serre vanishing because all sheaves are $W_{m+1}{\mathcal{O}}_{X}$-modules and
$j$ is an affine morphism. By Lemma 6.6, it suffices to show that the right
vertical arrow is injective.

To prove this injectivity, we note that the top row of (6.10) is same as the
sequence

$W_{m+1}\Omega^{r}_{X}\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD)\xrightarrow{p\otimes
1}W_{m+1}\Omega^{r}_{X}\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD)\to{W_{m+1}\Omega^{r}_{X}}/p\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD)\to
0,$

where ${\mathcal{O}}=W_{m+1}{\mathcal{O}}_{X}$. Similarly, we have

$j_{*}{W_{m+1}\Omega^{r}_{U}}/p\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD)\cong
j_{*}({W_{m+1}\Omega^{r}_{U}}/p\otimes_{W_{m+1}{\mathcal{O}}_{U}}j^{*}{\mathcal{O}}(nD))\cong$
$j_{*}({W_{m+1}\Omega^{r}_{U}}/p\otimes_{W_{m+1}{\mathcal{O}}_{U}}W_{m+1}{\mathcal{O}}_{U})\cong
j_{*}{W_{m+1}\Omega^{r}_{U}}/p,$

where the first isomorphism follows from the projection formula for
${\mathcal{O}}$-modules using the fact that ${\mathcal{O}}(nD)$ is an
invertible ${\mathcal{O}}$-module (see [17, Exercise II.5.1]).

Moreover, it is clear that the right vertical arrow in (6.10) is the map

${W_{m+1}\Omega^{r}_{X}}/p\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD)\xrightarrow{j^{*}\otimes
1}(j_{*}{W_{m+1}\Omega^{r}_{U}}/p)\otimes_{{\mathcal{O}}}{\mathcal{O}}(nD).$

Since ${\mathcal{O}}(nD)$ is an invertible ${\mathcal{O}}$-module, it suffices
to show that the map $j^{*}\colon{W_{m+1}\Omega^{r}_{X}}/p\to
j_{*}{W_{m+1}\Omega^{r}_{U}}/p$ is injective. But this follows from Lemma 6.7
since $j^{*}$ in (6.6) is injective. This proves the first part of the lemma.

We now prove the second part for which we can assume $m=1$. We know
classically that the map $C$ on $Z\Omega^{r}_{X}$ induces an
${\mathcal{O}}_{X}$-linear isomorphism
$C\colon{\mathcal{H}}^{r}(\psi_{*}\Omega^{\bullet}_{X})\xrightarrow{\cong}\Omega^{r}_{X}$.
Taking its inverse, we get an ${\mathcal{O}}_{X}$-linear isomorphism

$C^{-1}\colon\Omega^{r}_{X}\xrightarrow{\cong}{\mathcal{H}}^{r}(\psi_{*}\Omega^{\bullet}_{X}).$

Since $\psi_{*}\Omega^{\bullet}_{X}\in D^{+}({\rm Coh}_{X})$, we see that
${\mathcal{H}}^{r}(\psi_{*}\Omega^{\bullet}_{X})$ is a coherent
${\mathcal{O}}_{X}$-module. In particular, we get an isomorphism

$C^{-1}\colon\Omega^{r}_{X}(nD)\xrightarrow{\cong}{\mathcal{H}}^{r}(\psi_{*}\Omega^{\bullet}_{X})(nD).$

On the other hand, we have

${\mathcal{H}}^{r}(\psi_{*}\Omega^{\bullet}_{X})(nD)\cong{\mathcal{H}}^{r}((\psi_{*}\Omega^{\bullet}_{X})(nD))\cong{\mathcal{H}}^{r}(\psi_{*}(\Omega^{\bullet}_{X}\otimes_{{\mathcal{O}}_{X}}\psi^{*}({\mathcal{O}}(nD))))\cong{\mathcal{H}}^{r}(\psi_{*}(\Omega^{\bullet}_{X}(pnD))).$

This proves the second part. ∎

######  Lemma 6.9.

We have the following.

  1. (1)

${\rm Ker}(F^{m-1}\colon
W_{m}\Omega^{r}_{X}(nD)\to\Omega^{r}_{X}(p^{m-1}nD))=VW_{m-1}\Omega^{r}_{X}(pnD)$.

  2. (2)

$Z_{1}W_{m}\Omega^{r}_{X}(pnD)=FW_{m+1}\Omega^{r}_{X}(nD)$.

###### Proof.

We first prove (1). It is clear that the right hand side is contained in the
left hand side. We prove the other inclusion. It suffices to show that

$(j_{*}VW_{m-1}\Omega^{r}_{U})\cap W_{m}\Omega^{r}_{X}(nD)\subset
VW_{m-1}\Omega^{r}_{X}(pnD).$

We can check this locally. So let $D$ be defined by $f\in{\mathcal{O}}_{X}$
and let $y=Vx=[f^{-n}]\omega$, where $x\in j_{*}W_{m-1}\Omega^{r}_{U}$ and
$\omega\in W_{m}\Omega^{r}_{X}$. This yields
$\omega=[f^{n}]Vx=V(F([f^{n}])x)=V([f^{pn}]x)$. This implies that $\omega\in
W_{m}\Omega^{r}_{X}\cap j_{*}VW_{m-1}\Omega^{r}_{U}$. On the other hand, we
have

$W_{m}\Omega^{r}_{X}\cap j_{*}VW_{m-1}\Omega^{r}_{U}={\rm Ker}(F^{m-1}\colon
W_{m}\Omega^{r}_{X}\to j_{*}\Omega^{r}_{U})={\rm Ker}(F^{m-1}\colon
W_{m}\Omega^{r}_{X}\to\Omega^{r}_{X})$ $=VW_{m-1}\Omega^{r}_{X}.$

We thus get $\omega\in VW_{m-1}\Omega^{r}_{X}$. Let $y^{\prime}\in
W_{m-1}\Omega^{r}_{X}$ be such that $\omega=Vy^{\prime}$. This yields
$y=[f^{-n}]\omega=[f^{-n}]Vy^{\prime}=V(F([f^{-n}])y^{\prime})=V([f^{-pn}]y^{\prime})\in
VW_{m-1}\Omega^{r}_{X}(pnD)$. This proves (1).

We now prove (2). Since $FW_{m+1}\Omega^{r}_{X}(nD)\subset
j_{*}FW_{m+1}\Omega^{r}_{U}\cap W_{m}\Omega^{r}_{X}(pnD)$ by Lemma 6.5, it
follows that $FW_{m+1}\Omega^{r}_{X}(nD)\subset
Z_{1}W_{m}\Omega^{r}_{X}(pnD)$. We show the other inclusion.

We let $z\in Z_{1}W_{m}\Omega^{r}_{X}(pnD)$ so that $z=Fx=[f^{-pn}]\omega$ for
some $x\in j_{*}W_{m+1}\Omega^{r}_{U}$ and $\omega\in W_{m}\Omega^{r}_{X}$. We
can then write $\omega=[f^{pn}]Fx=F([f^{n}]x)$. This implies that $\omega\in
W_{m}\Omega^{r}_{X}\cap j_{*}FW_{m+1}\Omega^{r}_{U}$. On the other hand, we
have

$W_{m}\Omega^{r}_{X}\cap j_{*}FW_{m+1}\Omega^{r}_{U}={\rm Ker}(F^{m-1}d\colon
W_{m}\Omega^{r}_{X}\to j_{*}\Omega^{r+1}_{U})={\rm Ker}(F^{m-1}d\colon
W_{m}\Omega^{r}_{X}\to\Omega^{r+1}_{X})$ $=FW_{m+1}\Omega^{r}_{X}.$

We can thus write $\omega=Fx^{\prime}$ for some $x^{\prime}\in
W_{m+1}\Omega^{r}_{X}$. This gives
$z=[f^{-pn}]\omega=[f^{-pn}]Fx^{\prime}=F([f^{-n}])Fx^{\prime}=F([f^{-n}]x^{\prime})\in
FW_{m+1}\Omega^{r}_{X}(nD)$. This proves (2). ∎

##  7\. The pairing of cohomology groups

We fix a finite field $k$ of characteristic $p$ and an integral smooth
projective scheme $X$ of dimension $d\geq 1$ over $k$. Let $D\subset X$ be an
effective Cartier divisor with complement $U$. Let $\iota\colon
D\hookrightarrow X$ and $j\colon U\hookrightarrow X$ be the inclusions. In
this section, we shall establish the pairing for our duality theorem for the
$p$-adic étale cohomology of $U$. We fix integers $m,n\geq 1$ and $r\geq 0$.
We shall consider the étale topology throughout our discussion of duality.

###  7.1. The complexes

We consider the complex of étale sheaves

(7.1)
$W_{m}{\mathcal{F}}^{r,\bullet}_{n}=[Z_{1}W_{m}\Omega^{r}_{X}(nD)\xrightarrow{1-C}W_{m}\Omega^{r}_{X}(nD)].$

The differential of this complex is induced by the composition

$Z_{1}W_{m}\Omega^{r}_{X}(nD)\hookrightarrow
Z_{1}W_{m}\Omega^{r}_{X}(pnD)\xrightarrow{C}W_{m}\Omega^{r}_{X}(nD),$

where the last map is defined by virtue of Proposition 6.8.

We now consider the map $F\colon W_{m+1}\Omega^{r}_{X}(-nD)\to
W_{m}\Omega^{r}_{X}(-pnD)$ whose existence is shown in Lemma 6.5. We have
shown in (5.2) that $F({\rm
Ker}(R))=F((V^{m}W_{1}\Omega^{r}_{X}+dV^{m}W_{1}\Omega^{r-1}_{X})\cap
W_{m+1}\Omega^{r}_{X}(-nD))\subseteq dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{X}(-pnD)$. It follows that $F$ induces a map
$\overline{F}\colon
W_{m}\Omega^{r}_{X}(-nD)\to{W_{m}\Omega^{r}_{X}(-pnD)}/{(dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{X}(-pnD))}$. We denote the composition

$W_{m}\Omega^{r}_{X}(-nD)\to\frac{W_{m}\Omega^{r}_{X}(-pnD)}{dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{X}(-pnD)}\hookrightarrow\frac{W_{m}\Omega^{r}_{X}(-nD)}{dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{X}(-nD)}$

also by $\overline{F}$ and consider the complex of étale sheaves

(7.2)
$W_{m}{\mathcal{G}}^{r,\bullet}_{n}=\left[W_{m}\Omega^{r}_{X}(-nD)\xrightarrow{\pi-\overline{F}}\frac{W_{m}\Omega^{r}_{X}(-nD)}{dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{X}(-nD)}\right],$

where $\pi$ is the quotient map. We let

(7.3)
$W_{m}{\mathcal{H}}^{d,\bullet}=[W_{m}\Omega^{d}_{X}\xrightarrow{1-C}W_{m}\Omega^{d}_{X}],$

where the map $C$ (see Proposition 6.8 for $n=0$) is defined because
$Z_{1}W_{m}\Omega^{d}_{X}=W_{m}\Omega^{d}_{X}$.

###  7.2. The pairing of complexes

We consider the pairing

(7.4) $\langle,\rangle_{1}\colon Z_{1}W_{m}\Omega^{r}_{X}(nD)\times
W_{m}\Omega^{d-r}_{X}(-nD)\xrightarrow{\wedge}W_{m}\Omega^{d}_{X}$

by letting $\langle w_{1},w_{2}\rangle_{1}=w_{1}\wedge w_{2}$. This is defined
by the definition of the twisted de Rham-Witt sheaves.

We define a pairing $\langle,\rangle_{2}\colon
Z_{1}W_{m}\Omega^{r}_{X}(nD)\times W_{m}\Omega^{d-r}_{X}(-nD)\to
W_{m}\Omega^{d}_{X}$ by $\langle w_{1},w_{2}\rangle_{2}=-C(w_{1}\wedge
w_{2})$. We claim that $C(w_{1}\wedge w_{2})=0$ if $w_{2}\in
dV^{m-1}\Omega^{d-r-1}_{X}\cap W_{m}\Omega^{d-r}_{X}(-nD)$. To prove it, we
write $w_{1}=F(w^{\prime}_{1})$ for some $w^{\prime}_{1}\in
j_{*}W_{m+1}\Omega^{r}_{U}$. This gives $V(w_{1}\wedge
j^{*}w_{2})=V(F(w^{\prime}_{1})\wedge j^{*}w_{2})=w^{\prime}_{1}\wedge
j^{*}Vw_{2}=0$, where the last equality holds because
$VdV^{m-1}\Omega_{X}^{d-r-1}\subset pdV^{m}\Omega_{X}^{d-r-1}$ and
$p\Omega_{X}^{d-r-1}=0$. In particular, $j^{*}(w_{1}\wedge w_{2})\in{\rm
Ker}(V:W_{m}\Omega_{U}^{d}\to
W_{m+1}\Omega_{U}^{d})=FdV^{m}\Omega_{U}^{d-1}=dV^{m-1}\Omega_{U}^{d-1}={\rm
Ker}(C_{U})$, where $C_{U}$ is the Cartier map for $U$ (see [21, Chapitre III]
for the last equality). It follows that $j^{*}\circ C(w_{1}\wedge
w_{2})=C_{U}\circ j^{*}(w_{1}\wedge w_{2})=0$. Since
$W_{m}\Omega^{d}_{X}\hookrightarrow j_{*}W_{m}\Omega^{d}_{U}$, the claim
follows. Using the claim, we get a pairing

(7.5) $\langle,\rangle_{2}\colon
Z_{1}W_{m}\Omega^{r}_{X}(nD)\times\frac{W_{m}\Omega^{d-r}_{X}(-nD)}{dV^{m-1}\Omega^{d-r-1}_{X}\cap
W_{m}\Omega^{d-r}_{X}(-nD)}\to W_{m}\Omega^{d}_{X}.$

We define our third pairing of étale sheaves

(7.6) $\langle,\rangle_{3}\colon W_{m}\Omega^{r}_{X}(nD)\times
W_{m}\Omega^{d-r}_{X}(-nD)\to W_{m}\Omega^{d}_{X}$

by $\langle w_{1},w_{2}\rangle_{3}=w_{1}\wedge w_{2}$.

######  Lemma 7.1.

The above pairings of étale sheaves give rise to a pairing of two-term
complexes of sheaves

(7.7) $\langle,\rangle\colon W_{m}{\mathcal{F}}^{r,\bullet}_{n}\times
W_{m}{\mathcal{G}}^{d-r,\bullet}_{n}\to W_{m}{\mathcal{H}}^{d,\bullet}.$

###### Proof.

By [38, § 1, p.175], we only have to show that

(7.8) $(1-C)(w_{1}\wedge w_{2})=(1-C)(w_{1})\wedge
w_{2}-C(w_{1}\wedge(\pi-\overline{F})w_{2})$

for all $w_{1}\in Z_{1}W_{m}\Omega^{r}_{X}(nD)$ and $w_{2}\in
W_{m}\Omega^{r}_{X}(-nD)$. But this follows from the equalities

$\begin{array}[]{lll}(1-C)(w_{1}\wedge w_{2})&=&w_{1}\wedge
w_{2}-C(w_{1}\wedge w_{2})\\\ &=&w_{1}\wedge w_{2}-C(w_{1})\wedge C(w_{2})\\\
&=&w_{1}\wedge w_{2}-C(w_{1})\wedge w_{2}+C(w_{1})\wedge w_{2}-C(w_{1})\wedge
C(w_{2})\\\ &=&(1-C)(w_{1})\wedge w_{2}-C(w_{1})\wedge(C-1)(w_{2})\\\
&{=}^{\dagger}&(1-C)(w_{1})\wedge
w_{2}-C(w_{1}\wedge(\pi-\overline{F})(w_{2})),\end{array}$

where ${=}^{\dagger}$ holds because $C\circ\overline{F}={\rm id}$ (see [44,
Proposition 1.1.4]). ∎

###  7.3. The pairing of étale cohomology

Using Lemma 7.1, we get a pairing of hypercohomology groups

(7.9)
${\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\times{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{m}{\mathcal{G}}^{d-r,\bullet}_{n})\to{\mathbb{H}}^{d+1}_{\acute{e}t}(X,W_{m}{\mathcal{H}}^{d,\bullet}).$

By [44, Proposition 1.1.7] and [39, Corollary 1.12], there is a quasi-
isomorphism
$W_{m}\Omega^{d}_{X,{\operatorname{log}}}\xrightarrow{\cong}W_{m}{\mathcal{H}}^{d,\bullet}$,
and a bijective trace homomorphism ${\rm Tr}\colon
H^{d+1}_{\acute{e}t}(X,W_{m}\Omega^{d}_{X,{\operatorname{log}}})\xrightarrow{\cong}{{\mathbb{Z}}}/{p^{m}}$.
We thus get a pairing of ${{\mathbb{Z}}}/{p^{m}}$-modules

(7.10)
${\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\times{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{m}{\mathcal{G}}^{d-r,\bullet}_{n})\to{{\mathbb{Z}}}/{p^{m}}.$

Since this pairing is compatible with the change in values of $m$ and $n$, we
get a pairing of ind-abelian (in first coordinate) and pro-abelian (in second
coordinate) groups

(7.11)
$\\{{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\\}_{n}\times\\{{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{m}{\mathcal{G}}^{d-r,\bullet}_{n})\\}_{n}\to{{\mathbb{Z}}}/{p^{m}}.$

It follows from Corollary 6.4 that ${\varinjlim}_{n}\
W_{m}{\mathcal{F}}^{r,\bullet}_{n}\xrightarrow{\cong}j_{*}([Z_{1}W_{m}\Omega^{r}_{U}\xrightarrow{1-C}W_{m}\Omega^{r}_{U}])$.
Since $j$ is affine, we get

(7.12) ${\varinjlim}_{n}\
W_{m}{\mathcal{F}}^{r,\bullet}_{n}\xrightarrow{\cong}{\bf
R}j_{*}([Z_{1}W_{m}\Omega^{r}_{U}\xrightarrow{1-C}W_{m}\Omega^{r}_{U}])\cong{\bf
R}j_{*}(W_{m}\Omega^{r}_{U,{\operatorname{log}}}).$

In order to understand the pro-complex
$\\{W_{m}{\mathcal{G}}^{r,\bullet}_{n}\\}_{n}$, we let

$W_{m}\widetilde{{\mathcal{G}}}^{r,\bullet}_{n}=\left[W_{m}\Omega^{r}_{(X,nD)}\xrightarrow{\pi-\overline{F}}\frac{W_{m}\Omega^{r}_{(X,nD)}}{dV^{m-1}\Omega^{r-1}_{X}\cap
W_{m}\Omega^{r}_{(X,nD)}}\right].$

This complex is defined by the exact sequence (5.3). It follows then from
Corollary 5.3 that the canonical inclusion
$\\{W_{m}{\mathcal{G}}^{r,\bullet}_{n}\\}_{n}\hookrightarrow\\{W_{m}\widetilde{{\mathcal{G}}}^{r,\bullet}_{n}\\}_{n}$
is an isomorphism of pro-complexes of sheaves. Using the exact sequence (5.3)
and Lemma 5.4, we get that there is a canonical isomorphism of pro-complexes

(7.13) ${\rm
dlog}\colon\\{\overline{{\widehat{{\mathcal{K}}}^{M}_{r,(X,nD)}}/{p^{m}}}\\}_{n}\xrightarrow{\cong}\\{W_{m}\Omega^{r}_{(X,nD),{\operatorname{log}}}\\}_{n}\xrightarrow{\cong}\\{W_{m}{\mathcal{G}}^{r,\bullet}_{n}\\}_{n}.$

We conclude that (7.11) is equivalent to the pairing of ind-abelian (in first
coordinate) and pro-abelian (in second coordinate) groups

(7.14)
$\\{{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\\}_{n}\times\\{H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\\}_{n}\to{{\mathbb{Z}}}/{p^{m}}.$

###  7.4. Continuity of the pairing

Recall from § 5.3 that for a ${{\mathbb{Z}}}/{p^{m}}$-module $V$, one has
$V^{\star}={\rm Hom}_{{{\mathbb{Z}}}/{p^{m}}}(V,{{\mathbb{Z}}}/{p^{m}})$. For
any profinite or discrete torsion abelian group $A$, let $A^{\vee}={\rm
Hom}_{{\rm cont}}(A,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ denote the Pontryagin dual
of $A$ with the compact-open topology (see [15, § 7.4]). If $A$ is discrete
and $p^{m}$-torsion, then we have $A^{\vee}=A^{\star}$. We shall use the
following topological results.

######  Lemma 7.2.

Let $\\{A_{n}\\}_{n}$ be a direct system of discrete torsion topological
abelian groups whose limit is $A$ with the direct limit topology. Then the
canonical map $\lambda\colon A^{\vee}\to{\varprojlim}_{n}A^{\vee}_{n}$ is an
isomorphism of profinite groups.

###### Proof.

The fact that $\lambda$ is an isomorphism of abelian groups is well known and
elementary. We show that $\lambda$ is a homeomorphism. Since its source and
targets are compact Hausdorff, it suffices to show that it is continuous. We
let $B_{n}={\rm Image}(A_{n}\to A)$. Then, we have a surjection of finite ind-
groups $\\{A_{n}\\}_{n}\twoheadrightarrow\\{B_{n}\\}_{n}$ whose limits
coincides with $A$. We thus get maps
$A^{\vee}\xrightarrow{\lambda^{\prime}}{\varprojlim}_{n}B^{\vee}_{n}\xrightarrow{\lambda^{\prime\prime}}{\varprojlim}_{n}A^{\vee}_{n}$
such that $\lambda=\lambda^{\prime\prime}\circ\lambda^{\prime}$. Since
$\lambda^{\prime\prime}$ is clearly continuous, we have to only show that
$\lambda^{\prime}$ is continuous. We can therefore assume that each $A_{n}$ is
a subgroup of $A$. Lemma now follows from [41, Lemma 2.9.3 and Theorem 2.9.6].
∎

######  Lemma 7.3.

Let $\\{A_{n}\\}_{n}$ be an inverse system of discrete topological abelian
groups. Let $A$ be the limit of $\\{A_{n}\\}_{n}$ with the inverse limit
topology. Then any $f\in A^{\vee}$ factors through the projection
$\lambda_{n}\colon A\to A_{n}$ for some $n\geq 1$. In particular, the
canonical map ${\varinjlim}_{n}A^{\vee}_{n}\xrightarrow{\eta}A^{\vee}$ is
continuous and surjective. This map is an isomorphism if the transition maps
of $\\{A_{n}\\}_{n}$ are surjective.

###### Proof.

Since $f\colon A\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ is continuous and its target
is discrete, it follows that ${\rm Ker}(f)$ is open. By the definition of the
inverse limit topology, the latter contains ${\rm Ker}(\lambda_{n})$ for some
$n\geq 1$. Letting $A^{\prime}_{n}={A}/{{\rm Ker}(\lambda_{n})}$, this implies
that $f$ factors through $f^{\prime}_{n}\colon
A^{\prime}_{n}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. Since $A_{n}$ is discrete,
the map $(A_{n})^{\vee}\to(A^{\prime}_{n})^{\vee}$ is surjective by [15, Lemma
7.10]. Choosing a lift $f_{n}$ of $f^{\prime}_{n}$ under this surjection, we
see that $f$ factors through
$A\xrightarrow{\lambda_{n}}A_{n}\xrightarrow{f_{n}}{{\mathbb{Q}}}/{{\mathbb{Z}}}$.
The continuity of $\eta$ is equivalent to the assertion that the map
$A^{\vee}_{n}\to A^{\vee}$ is continuous for each $n$. But this is well known
since $\lambda_{n}$ is continuous. The remaining parts of the lemma are now
obvious. ∎

######  Remark 7.4.

The reader should note that the map ${\varinjlim}_{n}A^{\vee}_{n}\to A^{\vee}$
may not in general be injective even if each $A_{n}$ is finite.

For $n\geq 1$, we endow each of
${\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})$ and
$H^{i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})$ with
the discrete topology. We endow
${\varprojlim}_{n}H^{i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})$
with the inverse limit topology and
${\varinjlim}_{n}{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\cong
H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})$ with the
direct limit topology. Note that the latter topology is discrete.

If we let
$x\in{\varinjlim}_{n}{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})$,
then we can find some $n\gg 0$ such that $x=f_{n}(x^{\prime})$ for some
$x^{\prime}\in{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})$,
where
$f_{n}\colon{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})\to{\varinjlim}_{n}{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{m}{\mathcal{F}}^{r,\bullet}_{n})$
is the canonical map to the limit. This gives a map

(7.15) $\langle
x,\cdot\rangle\colon{\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{m}}$

which sends $y$ to $\langle x^{\prime},\pi(y)\rangle$ under the pairing
(7.10), where $\pi$ is the composite map

${\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\cong{\varprojlim}_{n}{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{m}{\mathcal{G}}^{d-r,\bullet}_{n})\xrightarrow{g_{n}}{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{m}{\mathcal{G}}^{d-r,\bullet}_{n}).$

One checks that (7.15) is defined. Since
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}$ is
profinite, this shows that the map (see Lemma 7.2)

(7.16)
$\theta_{m}\colon{\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\to
H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}$

is continuous. Since
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})$ is discrete,
the map

(7.17) $\vartheta_{m}\colon
H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\to({\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}}))^{\vee}$

is clearly continuous. We have thus shown that after taking the limits, (7.14)
gives rise to a continuous pairing of topological abelian groups

$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\times{\varprojlim}_{n}\
H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

Since each
$H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})$ is
$p^{m}$-torsion and discrete, we get the following.

######  Proposition 7.5.

There is a continuous pairing of topological abelian groups

(7.18)
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\times{\varprojlim}_{n}\
H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\to{{\mathbb{Z}}}/{p^{m}}.$

Equivalently, there is a continuous pairing of topological abelian groups

(7.19)
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\times{\varprojlim}_{n}\
H^{d+1-i}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d-r,(X,nD)}}/{p^{m}}})\to{{\mathbb{Z}}}/{p^{m}}.$

These pairings are compatible with respect to the canonical surjection
${{\mathbb{Z}}}/{p^{m}}\twoheadrightarrow{{\mathbb{Z}}}/{p^{m-1}}$ and the
inclusion ${{\mathbb{Z}}}/{p^{m-1}}\stackrel{{\scriptstyle
p}}{{\hookrightarrow}}{{\mathbb{Z}}}/{p^{m}}$.

It follows from (7.14) that the map $\vartheta_{m}$ in (7.17) has a
factorization

(7.20)
$H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})\xrightarrow{\theta^{\prime}_{m}}{\varinjlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})^{\vee}\stackrel{{\scriptstyle\vartheta^{\prime}_{m}}}{{\twoheadrightarrow}}\hskip
85.35826pt$ $\hskip
227.62204pt({\varprojlim}_{n}H^{d+1-i}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}}))^{\vee},$

where $\vartheta^{\prime}_{m}$ is the canonical map. This map is continuous
and surjective by Lemma 7.3. The map $\theta^{\prime}_{m}$ is continuous since
its source is discrete. Our goal in the next section will be to show that the
maps $\theta_{m}$ and $\theta^{\prime}_{m}$ are isomorphisms.

##  8\. Perfectness of the pairing

We continue with the setting of § 7. Our goal in this section is to prove a
perfectness statement for the pairing (7.14). To make our statement precise,
it is convenient to have the following definition.

######  Definition 8.1.

Let $\\{A_{n}\\}_{n}$ and $\\{B_{n}\\}_{n}$ be ind-system and pro-system of
$p^{m}$-torsion discrete abelian groups, respectively. Suppose that there is a
pairing

$(\star)\hskip
28.45274pt\\{A_{n}\\}_{n}\times\\{B_{n}\\}_{n}\to{{\mathbb{Z}}}/{p^{m}}.$

We shall say that this pairing is continuous and semi-perfect if the induced
maps

$\theta\colon{\varprojlim}_{n}B_{n}\to({\varinjlim}_{n}A_{n})^{\vee}\ \ {\rm
and}\ \
\theta^{\prime}\colon{\varinjlim}_{n}A_{n}\to{\varinjlim}_{n}B^{\vee}_{n}$

are continuous and bijective homomorphisms of topological abelian groups.
Recall here that
$({\varinjlim}_{n}A_{n})^{\vee}\cong{\varprojlim}_{n}A^{\vee}_{n}$ by Lemma
7.2. We shall say that $(\star)$ is perfect if it is semi-perfect, the
surjective map (see Lemma 7.3)
${\varinjlim}_{n}B^{\vee}_{n}\to({\varprojlim}_{n}B_{n})^{\vee}$ is bijective
and $\theta$ is a homeomorphism. Note that perfectness implies that
$\theta^{\prime}$ is also a homeomorphism.

The following is easy to check.

######  Lemma 8.2.

The pairing $(\star)$ is perfect if it is semi-perfect and $\\{B_{n}\\}_{n}$
is isomorphic to a surjective inverse system of compact groups.

Our goal is to show that (7.14) is semi-perfect by induction on $m\geq 1$. We
first consider the case $m=1$. We shall prove this case using our earlier