observation that $W_{m}\Omega^{\bullet}_{X}(p^{m}nD)$ is a complex for every
$m\geq 1$ and $n\in{\mathbb{Z}}$.

In this case, we have $Z_{1}\Omega^{r}_{X}(pnD)={\rm
Ker}(d\colon\Omega^{r}_{X}(pnD)\to\Omega^{r+1}_{X}(pnD))$ by (6.7). We claim
that the inclusion
$d\Omega^{r-1}_{X}(-pnD)\hookrightarrow\Omega^{r}_{X}(-pnD)\cap
d\Omega^{r-1}_{X}$ is actually a bijection when $n\geq 1$. Equivalently, the
map
${\Omega^{r}_{X}(-pnD)}/{d\Omega^{r-1}_{X}(-pnD)}\to{\Omega^{r}_{X}}/{d\Omega^{r-1}_{X}}$
is injective. Since $\psi$ is identity on the topological space $X$, the
latter injectivity is equivalent to showing that the map

(8.1)
$\frac{(\psi_{*}\Omega^{r}_{X})(-nD)}{d(\psi_{*}\Omega^{r-1}_{X})(-nD)}\cong\frac{\psi_{*}\Omega^{r}_{X}(-pnD)}{d\psi_{*}\Omega^{r-1}_{X}(-pnD)}\to\frac{\psi_{*}\Omega^{r}_{X}}{d\psi_{*}\Omega^{r-1}_{X}}$

is injective. Using a snake lemma argument, it suffices to show that the map
$\psi_{*}Z_{1}\Omega^{r-1}_{X}\to\psi_{*}Z_{1}\Omega^{r-1}_{X}\otimes_{{\mathcal{O}}_{X}}{\mathcal{O}}_{nD}$
is surjective. But this is obvious.

######  Lemma 8.3.

For $n\in{\mathbb{Z}}$, the sheaves $\psi_{*}(Z_{1}\Omega^{r}_{X}(pnD))$ and
$\psi_{*}({\Omega^{r}_{X}(pnD)}/{d\Omega^{r-1}_{X}(pnD)})$ are locally free
${\mathcal{O}}_{X}$-modules. For $n\geq 1$, the map

$\psi_{*}(Z_{1}\Omega^{r}_{X}(pnD))\to{\mathcal{H}{om}}_{{\mathcal{O}}_{X}}(\psi_{*}({\Omega^{d-r}_{X}(-pnD)}/{d\Omega^{d-r-1}_{X}(-pnD)}),\Omega^{d}_{X}),$

induced by (7.5), is an isomorphism.

###### Proof.

We first note that

(8.2)
$\begin{array}[]{lll}\psi_{*}(Z_{1}\Omega^{r}_{X}(pnD))&\cong&\psi_{*}({\rm
Ker}(\Omega^{r}_{X}(pnD)\to\Omega^{r+1}_{X}(pnD)))\\\ &\cong&{\rm
Ker}(\psi_{*}\Omega^{r}_{X}(pnD)\to\psi_{*}\Omega^{r+1}_{X}(pnD))\\\
&\cong&{\rm
Ker}((\psi_{*}\Omega^{r}_{X})(nD)\to(\psi_{*}\Omega^{r+1}_{X})(nD))\\\
&\cong&({\rm Ker}(\psi_{*}\Omega^{r}_{X}\to\psi_{*}\Omega^{r+1}_{X}))(nD)\\\
&\cong&(\psi_{*}Z_{1}\Omega^{r}_{X})(nD).\end{array}$

Since $\psi_{*}$ also commutes with ${\rm Coker}(d)$, we similarly get

(8.3)
$\psi_{*}({\Omega^{r}_{X}(pnD)}/{d\Omega^{r-1}_{X}(pnD)})\cong({\psi_{*}\Omega^{r}_{X}}/{d\psi_{*}\Omega^{r-1}_{X}})(nD).$

In the exact sequence

$0\to\psi_{*}Z_{1}\Omega^{r}_{X}\to\psi_{*}\Omega^{r}_{X}\xrightarrow{d}\psi_{*}\Omega^{r+1}_{X}\to{\psi_{*}\Omega^{r+1}_{X}}/{d\psi_{*}\Omega^{r}_{X}}\to
0$

of ${\mathcal{O}}_{X}$-linear maps between coherent
${\mathcal{O}}_{X}$-modules, all terms are locally free by [38, Lemma 1.7]. It
therefore remains an exact sequence of locally free
${\mathcal{O}}_{X}$-modules after tensoring with ${\mathcal{O}}_{X}(nD)$ for
any $n\in{\mathbb{Z}}$. The first part of the lemma now follows by using (8.2)
and (8.3) (for different values of $r$).

To prove the second part, we can again use (8.2) and (8.3). Since
${\mathcal{O}}_{X}(nD)$ is invertible, it suffices to show that
$(\psi_{*}Z_{1}\Omega^{r}_{X})(nD)\to{\mathcal{H}{om}}_{{\mathcal{O}}_{X}}({\psi_{*}\Omega^{d-r}_{X}}/{d\psi_{*}\Omega^{d-r-1}_{X}},\Omega^{d}_{X}(nD))$
is an isomorphism. But the term on the right side of this map is isomorphic to
the sheaf
${\mathcal{H}{om}}_{{\mathcal{O}}_{X}}({\psi_{*}\Omega^{d-r}_{X}}/{d\psi_{*}\Omega^{d-r-1}_{X}},\Omega^{d}_{X})(nD)$
via the canonical isomorphism

${\mathcal{H}{om}}_{{\mathcal{O}}_{X}}({\mathcal{A}},{\mathcal{B}})\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}\xrightarrow{\cong}{\mathcal{H}{om}}_{{\mathcal{O}}_{X}}({\mathcal{A}},{\mathcal{B}}\otimes_{{\mathcal{O}}_{X}}{\mathcal{E}}),$

which locally sends $(f\otimes b)\to(a\mapsto f(a)\otimes b)$ if
${\mathcal{E}}$ is locally free. We therefore have to only show that the map
$\psi_{*}Z_{1}\Omega^{r}_{X}\to{\mathcal{H}{om}}_{{\mathcal{O}}_{X}}({\psi_{*}\Omega^{d-r}_{X}}/{d\psi_{*}\Omega^{d-r-1}_{X}},\Omega^{d}_{X})$
is an isomorphism. But this follows from [38, Lemma 1.7]. ∎

######  Lemma 8.4.

The groups
${\mathbb{H}}^{i}_{\acute{e}t}(X,W_{1}{\mathcal{F}}^{r,\bullet}_{n})$ and
$H^{i}_{{\acute{e}t}}(X,\Omega^{r}_{(X,nD),{\operatorname{log}}})$ are finite
for $i,r\geq 0$.

###### Proof.

Using (5.3), the finiteness of
$H^{i}_{{\acute{e}t}}(X,\Omega^{r}_{(X,nD),{\operatorname{log}}})$ is reduced
to showing that the group
$H^{i}_{{\acute{e}t}}(X,\frac{\Omega^{r}_{(X,nD)}}{\Omega^{r}_{(X,nD)}\cap
d\Omega^{r-1}_{X}})$ is finite. By (5.2), it suffices to show that
$H^{i}_{{\acute{e}t}}(Z,\frac{\Omega^{r}_{Z}}{d\Omega^{r-1}_{Z}})$ is finite
if $Z\in\\{X,nD\\}$. Since $k$ is perfect, the absolute Frobenius $\psi$ is a
finite morphism. In particular, $\psi_{*}$ is exact. It suffices therefore to
show that
$H^{i}_{{\acute{e}t}}(Z,\frac{\psi_{*}\Omega^{r}_{Z}}{\psi_{*}d\Omega^{r-1}_{Z}})$
is finite. But this is clear because $k$ is finite, $Z$ is projective over $k$
and ${\psi_{*}\Omega^{r}_{Z}}/{\psi_{*}d\Omega^{r-1}_{Z}}$ is coherent. The
finiteness of
${\mathbb{H}}^{i}_{\acute{e}t}(X,W_{1}{\mathcal{F}}^{r,\bullet}_{n})$ follows
by a similar argument because $\psi_{*}(Z_{1}\Omega^{r}_{X}(nD))$ is coherent.
∎

######  Lemma 8.5.

The pairing

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

is a perfect pairing of the ind-abelian and pro-abelian finite groups.

###### Proof.

The finiteness follows from Lemma 8.4. For perfectness, it suffices to prove
that without using the pro-systems, the pairing in question is a perfect
pairing of finite abelian groups if we replace $n$ by $pn$. We shall show the
latter. For an ${\mathbb{F}}_{p}$-vector space $V$, we let $V^{\star}={\rm
Hom}_{{\mathbb{F}}_{p}}(V,{\mathbb{F}}_{p})$.

Using the definitions of various pairings of sheaves and complexes of sheaves,
we have a commutative diagram of long exact sequences

(8.4)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}_{\acute{e}t}(X,Z_{1}\Omega^{r}_{X}(pnD))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i-1}_{\acute{e}t}(X,\Omega^{r}_{X}(pnD))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{1}{\mathcal{F}}^{r,\bullet}_{pn})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d+1-i}_{\acute{e}t}(X,\frac{\Omega^{d-r}_{X}(-pnD)}{d\Omega^{d-r-1}_{X}(-pnD)})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{d+1-i}_{\acute{e}t}(X,\Omega^{d-r}_{X}(-pnD))^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{1}{\mathcal{G}}^{d-r,\bullet}_{pn})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

The exactness of the bottom row follows from Lemma 8.4 and [15, Lemma 7.10].
Using Lemma 8.3 and Grothendieck duality for the structure map $X\to{\rm
Spec\,}({\mathbb{F}}_{p})$, the map

$H^{i}_{\acute{e}t}(X,\psi_{*}(Z_{1}\Omega^{r}_{X}(pnD)))\to
H^{d-i}_{\acute{e}t}(X,\psi_{*}(\frac{\Omega^{d-r}_{X}(-pnD)}{d\Omega^{d-r-1}_{X}(-pnD)}))^{\star}$

is an isomorphism. Since $\psi_{*}$ is exact, it follows that the map

(8.5) $H^{i}_{\acute{e}t}(X,Z_{1}\Omega^{r}_{X}(pnD))\to
H^{d-i}_{\acute{e}t}(X,\frac{\Omega^{d-r}_{X}(-pnD)}{d\Omega^{d-r-1}_{X}(-pnD)})^{\star}$

is an isomorphism. By the same reason, the map

(8.6) $H^{i}_{\acute{e}t}(X,\Omega^{r}_{X}(pnD))\to
H^{d-i}_{\acute{e}t}(X,\Omega^{d-r}_{X}(-pnD))^{\star}$

is an isomorphism. Using (8.4), (8.5) and (8.6), we conclude that the right
vertical arrow in (8.4) is an isomorphism. An identical argument shows that
the map
${\mathbb{H}}^{d+1-i}_{\acute{e}t}(X,W_{1}{\mathcal{G}}^{d-r,\bullet}_{pn})\to{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{1}{\mathcal{F}}^{r,\bullet}_{pn})^{\star}$
is an isomorphism. This finishes the proof of the perfectness of the pairing
of the hypercohomology groups. ∎

We can now prove the main duality theorem of this paper.

######  Theorem 8.6.

Let $k$ be a finite field and $X$ a smooth and projective scheme of pure
dimension $d\geq 1$ over $k$. Let $D\subset X$ be an effective Cartier divisor
with complement $U$. Let $m\geq 1$ and $i,r\geq 0$ be integers. Then

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

is a semi-perfect pairing of ind-abelian and pro-abelian groups.

###### Proof.

We have shown already (see (7.18)) that the pairing is continuous after taking
limits. We need to show that the maps $\theta_{m}$ (see (7.16)) and
$\theta^{\prime}_{m}$ (see (7.20)) are isomorphisms of abelian groups. We
shall prove this by induction on $m\geq 1$.

We first assume $m=1$. Then Lemma 8.5 implies that the map

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

is an isomorphism of pro-abelian groups. Taking the limit and using (7.13), we
get an isomorphism

$\theta_{1}\colon
F^{d+1-i}_{1,r}(U)\xrightarrow{\cong}{\varprojlim}_{n}{\mathbb{H}}^{i}_{\acute{e}t}(X,W_{1}{\mathcal{F}}^{r,\bullet}_{n})^{\vee}.$

But the term on the right is same as
$H^{i}_{\acute{e}t}(U,W_{1}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}$ by
(7.12) and Lemma 7.2. Lemma 8.5 also implies that $\theta^{\prime}_{1}$ is an
isomorphism. This proves $m=1$ case of the theorem.

We now assume $m\geq 2$ and recall the definitions of $F^{j}_{m,r}(n)$ and
$F^{j}_{m,r}(U)$ from § 5.3. We consider the commutative diagram

(8.7)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{d+1-i}_{m-1,r}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{m-1}}$$\textstyle{F^{d+1-i}_{m,r}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{m}}$$\textstyle{F^{d+1-i}_{1,r}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{1}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{i}_{\acute{e}t}(U,W_{m-1}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{\vee}}$$\textstyle{H^{i}_{\acute{e}t}(U,W_{m}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\underline{p}}^{m-1})^{\vee}}$$\textstyle{H^{i}_{\acute{e}t}(U,W_{1}\Omega^{r}_{U,{\operatorname{log}}})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots.}$

The top row is exact by Lemma 5.11. The bottom row is exact by Lemma 5.7 and
[15, Lemma 7.10]. The maps $\theta_{i}$ are isomorphisms for $i\leq m-1$ by
induction. We conclude that $\theta_{m}$ is also an isomorphism. An identical
argument, where we apply Lemma 5.12 instead of Lemma 5.11, shows that
$\theta^{\prime}_{m}$ is an isomorphism. This finishes the proof. ∎

######  Remark 8.7.

In [22, Theorem 4.1.4] and [44, Theorem 3.4.2], a pairing using a logarithmic
version of the pro-system
$\\{H^{d+1-r}_{\acute{e}t}(X,W_{m}\Omega^{d-r}_{(X,nD),{\operatorname{log}}})\\}_{n}$
is given under the assumption that $D_{\rm red}$ is a simple normal crossing
divisor. The authors in these papers say that their pairing is perfect.
Although they do not explain their interpretations of perfectness, what they
actually prove is the semi-perfectness in the sense of Definition 8.1,
according to our understanding.

######  Corollary 8.8.

The pairing of Theorem 8.6 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.

###### Proof.

Combine Theorem 8.6, Lemma 8.2, Corollary 4.2, Proposition 2.7, [26, Theorem
9.1] and the equivalence of (7.18) and (7.19). ∎

We let
$C^{{\acute{e}t}}_{KS}(X,D;m)=H^{d}_{\acute{e}t}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,D)}}/{p^{m}}})$.
We shall study the following special case in the next section.

######  Corollary 8.9.

Under the assumptions of Theorem 8.6, the map

$\theta_{m}\colon{\varprojlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m)\to{\pi^{\rm
ab}_{1}(U)}/{p^{m}}$

is a bijective and continuous homomorphism between topological abelian groups.
This is an isomorphism of topological groups under the assumptions of
Corollary 8.8.

###  8.1. The comparison theorem

We shall continue with the assumptions of Theorem 8.6. We fix an integer
$m\geq 1$. Let $\pi^{\rm ab}_{1}(U)$ be the abelianized étale fundamental
group of $U$ and $\pi^{\rm adiv}_{1}(X,D)$ the co-1-skeleton étale fundamental
group of $X$ with modulus $D$, introduced in [15, Definition 7.5]. The latter
characterizes finite abelian covers of $U$ whose ramifications are bounded by
$D$ at each of its generic point, where the bound is given by means of
Matsuda’s Artin conductor. There is a natural surjection $\pi^{\rm
ab}_{1}(U)\twoheadrightarrow\pi^{\rm adiv}_{1}(X,D)$.

Let $C_{KS}(X,D)=H^{d}_{\rm nis}(X,{\mathcal{K}}^{M}_{d,(X,D)})$ and
$C_{KS}(X,D;m)=H^{d}_{\rm
nis}(X,{{\mathcal{K}}^{M}_{d,(X,D)}}/{p^{m}})\cong{C_{KS}(X,D)}/{p^{m}}$. By
Proposition 2.7, we have

(8.8) $C_{KS}(X,D;m)\xrightarrow{\cong}H^{d}_{\rm
nis}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,(X,D)}}/{p^{m}}}).$

We let $\widetilde{C}_{U/X}=\varprojlim_{n}C_{KS}(X,D)$. The canonical map
${\widetilde{C}_{U/X}}/{p^{m}}\to\varprojlim_{n}C_{KS}(X,D;m)$ is an
isomorphism by [16, Lemma 5.9]. The groups $C_{KS}(X,D)$ and $C_{KS}(X,D;m)$
have the discrete topology while $\widetilde{C}_{U/X}$ and
${\widetilde{C}_{U/X}}/{p^{m}}$ have the inverse limit topology. The groups
$\pi^{\rm ab}_{1}(U)$ and $\pi^{\rm adiv}_{1}(X,D)$ have the profinite
topology. By [15, Theorem 1.2], there is a commutative diagram of continuous
homomorphisms of topological abelian groups

(8.9)
$\textstyle{\widetilde{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U/X}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X|D}}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X,D).}$

The horizontal arrows are injective with dense images. They become
isomorphisms after tensoring with ${{\mathbb{Z}}}/{p^{m}}$ by [16, Corollary
5.15].

We let
$\widetilde{C}^{{\acute{e}t}}_{U/X}(m)={\varprojlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m)$.
By Corollary 8.9, we have a bijective and continuous homomorphism between
topological abelian groups
$\rho^{{\acute{e}t}}_{U/X}\colon\widetilde{C}^{{\acute{e}t}}_{U/X}(m)\rightarrow{\pi^{\rm
ab}_{1}(U)}/{p^{m}}\cong
H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{p^{m}})^{\star}$. We therefore have a
diagram

(8.10)
$\textstyle{{\widetilde{C}_{U/X}}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U/X}}$$\scriptstyle{\eta}$$\textstyle{\widetilde{C}^{{\acute{e}t}}_{U/X}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{{\acute{e}t}}_{U/X}}$$\textstyle{{\pi^{\rm
ab}_{1}(U)}/{p^{m}}}$

of continuous homomorphisms, where $\eta$ is the change of topology
homomorphism. We wish to prove the following.

######  Proposition 8.10.

The diagram (8.10) is commutative.

###### Proof.

It follows from [16, Theorem 1.2] that ${\rho_{U/X}}$ is an isomorphism of
profinite groups. Since the image of the composite map
${\mathcal{Z}}_{0}(U)\xrightarrow{{\operatorname{\rm
cyc}}_{U/X}}{\widetilde{C}_{U/X}}/{p^{m}}\xrightarrow{\rho_{U/X}}{\pi^{\rm
ab}_{1}(U)}/{p^{m}}$ is dense by the generalized Chebotarev density theorem,
it follows that the image of ${\operatorname{\rm cyc}}_{U/X}$ is also dense in
${\widetilde{C}_{U/X}}/{p^{m}}$. Since ${\pi^{\rm ab}_{1}(U)}/{p^{m}}$ is
Hausdorff, it suffices to show that $\rho_{U/X}\circ{\operatorname{\rm
cyc}}_{U/X}=\rho^{{\acute{e}t}}_{U/X}\circ\eta\circ{\operatorname{\rm
cyc}}_{U/X}$. Equivalently, we have to show that for every closed point $x\in
U$, one has that $(\rho^{{\acute{e}t}}_{U/X}\circ\eta\circ{\operatorname{\rm
cyc}}_{U/X})([x])$ is the image of the Frobenius element under the map ${\rm
Gal}({\overline{k}}/{k(x)})\to{\pi^{\rm ab}_{1}(U)}/{p^{m}}$. But this is well
known (e.g., see [30, Theorem 3.4.1]). ∎

###  8.2. A new filtration of
$H^{1}_{\acute{e}t}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$

We keep the assumptions of Theorem 8.6. By (7.13) and Theorem 8.6, we have the
isomorphism of abelian groups

$\theta^{\prime}_{m}\colon
H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\xrightarrow{\cong}{\varinjlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m)^{\vee},$

where $H^{1}(U,{{\mathbb{Z}}}/{p^{m}})$ denote the étale cohomology
$H_{{\acute{e}t}}^{1}(U,{{\mathbb{Z}}}/{p^{m}})$. We let

${\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})=(\theta^{\prime}_{m})^{-1}({\rm
Image}(C^{{\acute{e}t}}_{KS}(X,nD;m)^{\vee}\to{\varinjlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m)^{\vee})).$

We set

(8.11) ${\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})={\varinjlim}_{m}{\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}}).$

It follows that $\\{{\rm
fil}^{{\acute{e}t}}_{nD}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})\\}_{n}$
defines an increasing filtration of
$H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$. This is an étale version of
the filtration ${\rm fil}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$
defined in [15, definition 7.12]. This new filtration is clearly exhaustive.
We do not know if ${\rm
fil}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$ and ${\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$ are
comparable in general. We can however prove the following.

######  Theorem 8.11.

Let $k$ be a finite field and $X$ a smooth and projective scheme of pure
dimension $d\geq 1$ over $k$. Let $D\subset X$ be an effective Cartier divisor
with complement $U$ such that $D_{\rm red}$ is a simple normal crossing
divisor. Assume that either $d\neq 2$ or $k\neq{\mathbb{F}}_{2}$. Then one has

${\rm fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\subseteq{\rm
fil}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\ \ {\rm and}\ \ {\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})\subseteq{\rm
fil}_{D}H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$

as subgroups of $H^{1}(U,{{\mathbb{Z}}}/{p^{m}})$ and
$H^{1}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$, respectively. These
inclusions are equalities if $D_{\rm red}$ is regular.

###### Proof.

The claim about the two inclusions follows directly from the definitions of
the filtrations in view of Proposition 8.10 and Corollary 8.8.

Moreover, Corollary 8.9 yields that $\rho^{{\acute{e}t}}_{U/X}$ is an
isomorphism of profinite topological groups. Hence, there exists a unique
quotient $\pi^{{\acute{e}t}}_{1}(X,nD;m)$ of ${\pi^{\rm ab}_{1}(U)}/{p^{m}}$
such that the diagram

(8.12)
$\textstyle{\widetilde{C}^{{\acute{e}t}}_{U/X}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{{\acute{e}t}}_{U/X}}$$\textstyle{{\pi^{\rm
ab}_{1}(U)}/{p^{m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{{\acute{e}t}}_{1}(X,nD;m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{{\acute{e}t}}_{X|nD}}$$\textstyle{{\pi^{{\rm
adiv}}_{1}(X,nD)}/{p^{m}}}$

commutes and the horizontal arrows are isomorphisms of topological groups. One
knows that ${\rm fil}_{nD}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})=({\pi^{{\rm
adiv}}_{1}(X,nD)}/{p^{m}})^{\vee}$. By Theorem 4.9, it is easy to see that
${\rm
fil}^{{\acute{e}t}}_{nD}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\cong\pi^{{\acute{e}t}}_{1}(X,nD;m)^{\vee}$
when $D_{\rm red}$ is regular.

More precisely, if $D_{\rm red}$ is a simple normal crossing divisor and
either $d\neq 2$ or $k\neq{\mathbb{F}}_{2}$, we have the following diagram.

(8.13)
---
$\textstyle{H^{1}U,{{\mathbb{Z}}}/{p^{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{\prime}}$$\scriptstyle{\cong}$$\textstyle{{\varinjlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{({\varprojlim}_{n}C^{{\acute{e}t}}_{KS}(X,nD;m))^{\vee}}$$\textstyle{{\rm
fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{{\acute{e}t}}_{KS}(X,nD;m)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{C_{KS}(X,nD;m)^{\vee}.}$

Assume now that $D_{{\rm red}}$ is regular. By Theorem 4.9, it then follows
that the transition maps in the pro-system
$\\{C^{{\acute{e}t}}_{KS}(X,nD;m))\\}_{n}$ are surjective. In particular, the
right top vertical arrow is injective and hence the middle horizontal dotted
arrow is a honest arrow such that the top square commutes. It follows from
Proposition 8.10, that the bottom square commutes as well. By the definition
of ${\rm fil}^{{\acute{e}t}}_{D}H^{1}(U,{{\mathbb{Z}}}/{p^{m}})$, it is clear
that the middle horizontal arrow is now surjective. To prove its injectivity,
it suffices to show that the right bottom vertical arrow is injective. But
this follows from Theorem 4.9. ∎

##  9\. Reciprocity theorem for $C_{KS}(X|D)$

In this section, we shall prove the reciprocity theorem for the idele class
group $C_{KS}(X|D)$. Before going into this, we recall the definition of some
filtrations of $H^{1}_{\acute{e}t}(K,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ for a
Henselian discrete valuation field $K$.

###  9.1. The Brylinski-Kato and Matsuda filtrations

Let $K$ be a Henselian discrete valuation field of characteristic $p>0$ with
ring of integers ${\mathcal{O}}_{K}$, maximal ideal ${\mathfrak{m}}_{K}\neq 0$
and residue field ${\mathfrak{f}}$. We let
$H^{q}(A)=H^{q}_{\acute{e}t}(A,{{\mathbb{Q}}}/{{\mathbb{Z}}}(q-1))$ for any
commutative ring $A$. The generalized Artin-Schreier sequence gives rise to an
exact sequence

(9.1) $0\to{{\mathbb{Z}}}/{p^{r}}\to
W_{r}(K)\xrightarrow{1-F}W_{r}(K)\xrightarrow{\partial}H^{1}_{{\text{\'{e}t}}}(K,{{\mathbb{Z}}}/{p^{r}})\to
0,$

where $F((a_{r-1},\ldots,a_{0}))=(a^{p}_{r-1},\ldots,a^{p}_{0})$. We write the
Witt vector $(a_{r-1},\ldots,a_{0})$ as $\underline{a}$ in short. Let
$v_{K}\colon K^{\times}\to{\mathbb{Z}}$ be the normalized valuation. Let
$\delta_{r}$ denote the composite map
$W_{r}(K)\xrightarrow{\partial}H^{1}_{{\text{\'{e}t}}}(K,{{\mathbb{Z}}}/{p^{r}})\hookrightarrow
H^{1}(K)$. Then one knows that $\delta_{r}=\delta_{r+1}\circ V$. For an
integer $m\geq 1$, let ${\rm ord}_{p}(m)$ denote the $p$-adic order of $m$ and
let $r^{\prime}=\min(r,{\rm ord}_{p}(m))$. We let ${\rm ord}_{p}(0)=-\infty$.

For $m\geq 0$, we let

(9.2) ${\rm fil}^{{\rm
bk}}_{m}W_{r}(K)=\\{\underline{a}|p^{i}v_{K}(a_{i})\geq-m\\};\ \ {\rm and}$
(9.3) ${\rm fil}^{{\rm ms}}_{m}W_{r}(K)={\rm fil}^{{\rm
bk}}_{m-1}W_{r}(K)+V^{r-r^{\prime}}({\rm fil}^{{\rm
bk}}_{m}W_{r^{\prime}}(K)).$

We have ${\rm fil}^{{\rm bk}}_{0}W_{r}(K)={\rm fil}^{{\rm
ms}}_{0}W_{r}(K)=W_{r}({\mathcal{O}}_{K})$. We let ${\rm fil}^{{\rm
bk}}_{-1}W_{r}(K)={\rm fil}^{{\rm ms}}_{-1}W_{r}(K)=0$.

For $m\geq 1$, we let

(9.4) ${\rm fil}^{{\rm
bk}}_{m-1}H^{1}(K)=H^{1}(K)\\{p^{\prime}\\}\bigoplus{\underset{r\geq
1}{\bigcup}}\delta_{r}({\rm fil}^{{\rm bk}}_{m-1}W_{r}(K))\ \ \mbox{and}$
${\rm fil}^{{\rm
ms}}_{m}H^{1}(K)=H^{1}(K)\\{p^{\prime}\\}\bigoplus{\underset{r\geq
1}{\bigcup}}\delta_{r}({\rm fil}^{{\rm ms}}_{m}W_{r}(K)).$

Moreover, we let ${\rm fil}^{{\rm ms}}_{0}H^{1}(K)={\rm fil}^{{\rm
bk}}_{-1}H^{1}(K)=H^{1}({\mathcal{O}}_{K})$, the subgroup of unramified
characters. The filtrations ${\rm fil}^{{\rm bk}}_{\bullet}H^{1}(K)$ and ${\rm
fil}^{{\rm ms}}_{\bullet}H^{1}(K)$ are due to Brylinski-Kato [24] and Matsuda
[37], respectively. We refer to [15, Theorem 6.1] for the following.

######  Theorem 9.1.

The two filtrations defined above satisfy the following relations.

  1. (1)

$H^{1}(K)={\underset{m\geq 0}{\bigcup}}{\rm fil}^{{\rm
bk}}_{m}H^{1}(K)={\underset{m\geq 0}{\bigcup}}{\rm fil}^{{\rm
ms}}_{m}H^{1}(K)$.

  2. (2)

${\rm fil}^{{\rm ms}}_{m}H^{1}(K)\subset{\rm fil}^{{\rm
bk}}_{m}H^{1}(K)\subset{\rm fil}^{{\rm ms}}_{m+1}H^{1}(K)$ for all $m\geq-1$.

  3. (3)

If $m\geq 1$ such that ${\rm ord}_{p}(m)=0$, then ${\rm fil}^{{\rm
bk}}_{m-1}H^{1}(K)={\rm fil}^{{\rm ms}}_{m}H^{1}(K)$. In particular, ${\rm
fil}^{{\rm bk}}_{0}H^{1}(K)={\rm fil}^{{\rm ms}}_{1}H^{1}(K)$, which is the
subgroup of tamely ramified characters.

For integers $m\geq 0$ and $r\geq 1$, let $U_{m}K^{M}_{r}(K)$ be the subgroup
$\\{1+{\mathfrak{m}}^{m}_{K},K^{\times},\ldots,K^{\times}\\}$ of
$K^{M}_{r}(K)$. We let $U^{\prime}_{m}K^{M}_{r}(K)$ be the subgroup
$\\{1+{\mathfrak{m}}^{m}_{K},{\mathcal{O}}^{\times}_{K},\ldots,{\mathcal{O}}^{\times}_{K}\\}$
of $K^{M}_{r}(K)$. It follows from [15, Lemma 6.2] that
$U_{m+1}K^{M}_{r}(K)\subseteq U^{\prime}_{m}K^{M}_{r}(K)\subseteq
U_{m}K^{M}_{r}(K)$ for every integer $m\geq 0$. If $K$ is a $d$-dimensional
Henselian local field (see [15, § 5.1]), there is a pairing

(9.5) $\\{,\\}\colon K^{M}_{d}(K)\times H^{1}(K)\to
H^{d+1}(K)\cong{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

The following result is due to Kato and Matsuda (when $p\neq 2$). We refer to
[15, Theorem 6.3] for a proof.

######  Theorem 9.2.

Let $\chi\in H^{1}(K)$ be a character. Then the following hold.

  1. (1)

For every integer $m\geq 0$, we have that $\chi\in{\rm fil}^{{\rm
bk}}_{m}H^{1}(K)$ if and only if $\\{\alpha,\chi\\}=0$ for all $\alpha\in
U_{m+1}K^{M}_{d}(K)$.

  2. (2)

For every integer $m\geq 1$, we have that $\chi\in{\rm fil}^{{\rm
ms}}_{m}H^{1}(K)$ if and only if $\\{\alpha,\chi\\}=0$ for all $\alpha\in
U^{\prime}_{m}K^{M}_{d}(K)$.

We have the inclusions $K\hookrightarrow K^{sh}\hookrightarrow\overline{K}$,
where $\overline{K}$ is a fixed separable closure of $K$ and $K^{sh}$ is the
strict Henselization of $K$. We shall use the following key result in the
proof of our reciprocity theorem.

######  Proposition 9.3.

For $m\geq 0$, the canonical square

$\textstyle{{\rm fil}^{{\rm
bk}}_{m}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}^{{\rm
bk}}_{m}H^{1}(K^{sh})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(K^{sh})}$

is Cartesian.

###### Proof.

Since ${\rm fil}^{{\rm bk}}_{\bullet}H^{1}(K)$ is an exhaustive filtration of
$H^{1}(K)$ by Theorem 9.1(1), it suffices to show that for every $m\geq 1$,
the square

$\textstyle{{\rm fil}^{{\rm
bk}}_{m-1}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}^{{\rm
bk}}_{m}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}^{{\rm
bk}}_{m-1}H^{1}(K^{sh})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}^{{\rm bk}}_{m}H^{1}(K^{sh})}$

is Cartesian. Equivalently, it suffices to show that for every $m\geq 1$, the
map

$\phi^{*}_{m}\colon{\rm gr}^{{\rm bk}}_{m}H^{1}(K)\to{\rm gr}^{{\rm
bk}}_{m}H^{1}(K^{sh}),$

induced by the inclusion $\phi\colon K\hookrightarrow K^{sh}$, is injective.

We fix $m\geq 1$. By [24, Corollary 5.2], there exists an injective (non-
canonical) homomorphism ${\rm rsw}_{\pi_{K}}\colon{\rm gr}^{{\rm
bk}}_{m}H^{1}(K)\hookrightarrow\Omega^{1}_{{\mathfrak{f}}}\oplus{\mathfrak{f}}$.
By Theorem 5.1 of loc. cit., this refined swan conductor ${\rm rsw}_{\pi_{K}}$
depends only on the choice of a uniformizer $\pi_{K}$ of $K$. Since
${\mathcal{O}}^{sh}_{K}$ is unramified over ${\mathcal{O}}_{K}$, we can choose
$\pi_{K}$ to be a uniformizer of $K^{sh}$ as well. It therefore follows that
for all $m\geq 1$, the diagram

$\textstyle{{\rm gr}^{{\rm
bk}}_{m}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
rsw}_{\pi_{K}}}$$\textstyle{\Omega^{1}_{{\mathfrak{f}}}\oplus{\mathfrak{f}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
gr}^{{\rm
bk}}_{m}H^{1}(K^{sh})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
rsw}_{\pi_{K}}}$$\textstyle{\Omega^{1}_{\overline{{\mathfrak{f}}}}\oplus\overline{{\mathfrak{f}}}}$

is commutative, where $\overline{{\mathfrak{f}}}$ is a separable closure of
${\mathfrak{f}}$. Since the horizontal arrows in the above diagram are
injective, it suffices to show that the natural map
$\Omega^{1}_{{\mathfrak{f}}}\to\Omega^{1}_{\overline{{\mathfrak{f}}}}$ is
injective. But this is clear. ∎

###  9.2. Logarithmic fundamental group with modulus

Let $k$ be a finite field of characteristic $p$ and $X$ an integral projective
scheme over $k$ of dimension $d\geq 1$. Let $D\subset X$ be an effective
Cartier divisor with complement $U$. Let $K=k(\eta)$ denote the function field
of $X$. We let $C=D_{\rm red}$. We fix a separable closure $\overline{K}$ of
$K$ and let $G_{K}$ denote the absolute Galois group of $K$. Recall the
following notations from [26, § 3.3] or [15, § 2.3].

Assume that $X$ is normal. Let $\lambda$ be a generic point of $D$. Let
$K_{\lambda}$ denote the Henselization of $K$ at $\lambda$. Let
$P=(p_{0},\ldots,p_{d-2},\lambda,\eta)$ be a Parshin chain on $(U\subset X)$.
Let $V\subset K$ be a $d$-DV which dominates $P$. Let
$V=V_{0}\subset\cdots\subset V_{d-2}\subset V_{d-1}\subset V_{d}=K$ be the
chain of valuation rings in $K$ induced by $V$. Since $X$ is normal, it is
easy to check that for any such chain, one must have
$V_{d-1}={\mathcal{O}}_{X,\lambda}$. Let $V^{\prime}$ be the image of $V$ in
$k(\lambda)$. Let $\widetilde{V}_{d-1}$ be the unique Henselian discrete
valuation ring having an ind-étale local homomorphism
$V_{d-1}\to\widetilde{V}_{d-1}$ such that its residue field $E_{d-1}$ is the
quotient field of $(V^{\prime})^{h}$. Then $V^{h}$ is the inverse image of
$(V^{\prime})^{h}$ under the quotient map
$\widetilde{V}_{d-1}\twoheadrightarrow E_{d-1}$. It follows that its function
field $Q(V^{h})$ is a $d$-dimensional Henselian discrete valuation field whose
ring of integers is $\widetilde{V}_{d-1}$ (see [26, § 3.7.2]). It then follows
that there are canonical inclusions of discrete valuation rings

(9.6)
${\mathcal{O}}_{X,\lambda}\hookrightarrow\widetilde{V}_{d-1}\hookrightarrow{\mathcal{O}}^{sh}_{X,\lambda}.$

Moreover, we have (see the proof of [26, Proposition 3.3])

(9.7)
${\mathcal{O}}^{h}_{X,P^{\prime}}\cong{\underset{V\in{\mathcal{V}}(P)}{\prod}}\widetilde{V}_{d-1},$

where ${\mathcal{V}}(P)$ is the set of $d$-DV’s in $K$ which dominate $P$. As
an immediate consequence of Proposition 9.3, we therefore get the following.

######  Corollary 9.4.

For every $m\geq 0$, the square

$\textstyle{{\rm fil}^{{\rm
bk}}_{m}H^{1}(K_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(K_{\lambda})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
fil}^{{\rm
bk}}_{m}H^{1}(Q(V^{h}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}(Q(V^{h}))}$

is Cartesian.

Let ${\rm Irr}_{C}$ denote the set of all generic points of $C$ and let
$C_{\lambda}$ denote the closure of an element $\lambda\in{\rm Irr}_{C}$. We
write $D={\underset{\lambda\in{\rm Irr}_{C}}{\sum}}n_{\lambda}C_{\lambda}$. We
can also write $D={\underset{x\in X^{(1)}}{\sum}}n_{x}\overline{\\{x\\}}$,
where $n_{x}=0$ for all $x\in U$. We allow $D$ to be empty in which case we
write $D=0$. For any $x\in X^{(1)}\cap C$, we let $\widehat{K}_{x}$ denote the
quotient field of the ${\mathfrak{m}}_{x}$-adic completion
$\widehat{{\mathcal{O}}_{X,x}}$ of ${\mathcal{O}}_{X,x}$. Let
${\mathcal{O}}^{sh}_{X,x}$ denote the strict Henselization of
${\mathcal{O}}_{X,x}$ and let $K^{sh}_{x}$ denote its quotient field. Then it
is clear from the definitions that there are inclusions

(9.8) $K\hookrightarrow K_{x}\hookrightarrow
K^{sh}_{x}\hookrightarrow\overline{K}\ \mbox{and}\ \ K\hookrightarrow
K_{x}\hookrightarrow\widehat{K}_{x}.$

######  Definition 9.5.

Let ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)$ denote the subgroup of characters
$\chi\in H^{1}(K)$ such that for every $x\in X^{(1)}$, the image $\chi_{x}$ of
$\chi$ under the canonical surjection $H^{1}(K)\twoheadrightarrow
H^{1}(K_{x})$ lies in ${\rm fil}^{{\rm bk}}_{n_{x}-1}H^{1}(K_{x})$. It is easy
to check that ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)\subset H^{1}(U)$. We let ${\rm
fil}^{{\rm
bk}}_{D}H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{m})=H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{m})\cap{\rm
fil}^{{\rm bk}}_{D}H^{1}(K)$.

Recall that $H^{1}(K)$ is a torsion abelian group. We consider it a
topological abelian group with discrete topology. In particular, all the
subgroups $H^{1}(U)$ (e.g., ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)$) are also
considered as discrete topological abelian groups. Recall from [15, Definition
7.12] that ${\rm fil}_{D}H^{1}(K)$ is a subgroup of $H^{1}(U)$ which is
defined similar to ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)$, where we only replace
${\rm fil}^{{\rm bk}}_{n_{\lambda}-1}H^{1}(K_{\lambda})$ by ${\rm fil}^{{\rm
ms}}_{n_{\lambda}}H^{1}(K_{\lambda})$.

######  Definition 9.6.

We define the quotient $\pi_{1}^{{\rm abk}}(X,D)$ of $\pi_{1}^{\rm ab}(U)$ to
be the Pontryagin dual of ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)\subset H^{1}(U)$,
i.e.,

$\pi_{1}^{{\rm abk}}(X,D):={\rm Hom}_{{\rm cont}}({\rm fil}^{{\rm
bk}}_{D}H^{1}(K),{\mathbb{Q}}/{\mathbb{Z}})={\rm Hom}({\rm fil}^{{\rm
bk}}_{D}H^{1}(K),{\mathbb{Q}}/{\mathbb{Z}}).$

Since $H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{p^{m}})=\
_{p^{m}}H^{1}(U,{{\mathbb{Q}}}/{{\mathbb{Z}}})$, it follows that

${\pi_{1}^{{\rm abk}}(X,D)}/{p^{m}}\cong{\rm Hom}_{{\rm cont}}({\rm fil}^{{\rm
bk}}_{D}H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{p^{m}}),{{\mathbb{Q}}}/{{\mathbb{Z}}}).$

Since ${\rm fil}^{{\rm bk}}_{D}H^{1}(K)$ is a discrete topological group, it
follows that $\pi_{1}^{{\rm abk}}(X,D)$ is a profinite group. Moreover, [41,
Theorem 2.9.6] implies that

(9.9) $\pi_{1}^{{\rm abk}}(X,D)^{\vee}\cong{\rm fil}^{{\rm bk}}_{D}H^{1}(K)\ \
{\rm and}\ \ ({\pi_{1}^{{\rm abk}}(X,D)}/{p^{m}})^{\vee}\cong{\rm fil}^{{\rm
bk}}_{D}H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{p^{m}}).$

######  Lemma 9.7.

The quotient map $\pi^{\rm ab}_{1}(U)\to\pi_{1}^{{\rm abk}}(X,D)$ factors
through $\pi_{1}^{{\rm adiv}}(X,D)$.

###### Proof.

This is a straightforward consequence of Theorem 9.1 and [15, Theorem 7.16]
once we recall that

(9.10) $\pi_{1}^{{\rm adiv}}(X,D)={\rm Hom}({\rm
fil}_{D}H^{1}(K),{\mathbb{Q}}/{\mathbb{Z}}).$

∎

######  Remark 9.8.

Using [1, § 3], one can mimic the construction of [15, § 7] to show that
$\pi_{1}^{{\rm abk}}(X,D)$ is isomorphic to the abelianization of the
automorphism group of the fiber functor of a certain Galois subcategory of the
category of finite étale covers of $U$. Under this Tannakian interpretation,
$\pi_{1}^{{\rm abk}}(X,D)$ characterizes the finite étale covers of $U$ whose
ramifications are bounded at each generic point of $D$ by means of Kato’s Swan
conductor.

###  9.3. Reciprocity for $C_{KS}(X|D)$

We shall continue with the setting of § 9.2. Before we prove the reciprocity
theorem for $C_{KS}(X|D)$, we recall the construction of the reciprocity map
for $C_{U/X}$ from [15, § 5.4]. We let $P=(p_{0},\ldots,p_{s})$ be any Parshin
chain on $X$ with the condition that $p_{s}\in U$. Let
$X(P)=\overline{\\{p_{s}\\}}$ be the integral closed subscheme of $X$ and let
$U(P)=U\cap X(P)$. We let $V\subset k(p_{s})$ be an $s$-DV dominating $P$.
Then $Q(V^{h})$ is an $s$-dimensional Henselian local field. By [15,
Proposition 5.1], we therefore have the reciprocity map

(9.11) $\rho_{Q(V^{h})}\colon K^{M}_{s}(Q(V^{h}))\to{\rm Gal}({{Q(V^{h})}^{\rm
ab}}/{Q(V^{h})})\cong\pi^{\rm ab}_{1}({\rm Spec\,}(Q(V^{h}))).$

Taking the sum of these maps over ${\mathcal{V}}(P)$ and using [15, Lemma
5.10], we get a reciprocity map

(9.12) $\rho_{k(P)}\colon K^{M}_{s}(k(P))\to\pi^{\rm ab}_{1}({\rm
Spec\,}(k(P)))\to\pi^{\rm ab}_{1}(U),$

where last map exists because $p_{s}\in U$. Taking sum over all Parshin chains
on the pair $(U\subset X)$, we get a reciprocity map $\rho_{U/X}\colon
I_{U/X}\to\pi^{\rm ab}_{1}(U)$. By [15, Theorem 5.13, Proposition 5.15], this
descends to a continuous homomorphism of topological groups

(9.13) $\rho_{U/X}\colon C_{U/X}\to\pi^{\rm ab}_{1}(U).$

Recall that $\pi_{1}^{{\rm adiv}}(X,D)_{0}$ is the kernel of the composite map
$\pi_{1}^{{\rm adiv}}(X,D)\twoheadrightarrow\pi_{1}^{\rm
ab}(X)\to\widehat{{\mathbb{Z}}}$. One defines $\pi_{1}^{{\rm abk}}(X,D)_{0}$
similarly. One of the main results of this paper is the following.

######  Theorem 9.9.

There is a continuous homomorphism

$\rho^{\prime}_{X|D}\colon C(X|D)\to\pi_{1}^{{\rm abk}}(X,D)$

with dense image such that the diagram

(9.14)
$\textstyle{{C}_{U/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{U/X}}$$\scriptstyle{p^{\prime}_{X|D}}$$\textstyle{\pi^{\rm
ab}_{1}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}_{X|D}}$$\textstyle{C(X|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime}_{X|D}}$$\textstyle{\pi^{{\rm
abk}}_{1}(X,D)}$

is commutative. If $X$ is normal and $U$ is regular, then
$\rho^{\prime}_{X|D}$ induces an isomorphism of finite groups

$\rho^{\prime}_{X|D}\colon C(X|D)_{0}\xrightarrow{\cong}\pi_{1}^{{\rm
abk}}(X,D)_{0}.$

###### Proof.

We first show the existence of $\rho^{\prime}_{X|D}$. Its continuity and
density of its image will then follow by the corresponding assertions for
$\rho_{U/X}$, shown in [15]. In view of (9.13), we only have to show that if
$\chi$ is a character of $\pi_{1}^{{\rm abk}}(X,D)$, then the composite
$\chi\circ q^{\prime}_{X|D}\circ\rho_{U/X}$ annihilates ${\rm
Ker}(p^{\prime}_{X|D})$. By (3.3), we only need to show that $\chi\circ
q^{\prime}_{X|D}\circ\rho_{U/X}$ annihilates the image of
$\widehat{K}^{M}_{d}({\mathcal{O}}^{h}_{X,P^{\prime}}|I_{D})\to C_{U/X}$,
where $P$ is any maximal Parshin chain on $(U\subset X)$. But the proof of
this is completely identical to that of [15, Theorem 8.1], only difference
being that we have to use part (1) of Theorem 9.2 instead of part (2).

We now assume that $X$ is normal and $U$ is regular. In this case, we can
replace $C(X|D)$ by $C_{KS}(X|D)$ by Theorem 3.4. We show that
$\rho^{\prime}_{X|D}$ is injective on all of $C_{KS}(X|D)$. By Lemmas 3.3 and
9.7, there is a diagram

(9.15) $\textstyle{{\rm fil}^{{\rm
bk}}_{D}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime\vee}_{X|D}}$$\scriptstyle{\alpha^{\vee}}$$\textstyle{C_{KS}(X|D)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{\vee}}$$\textstyle{{\rm
fil}_{D}H^{1}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\vee}_{X|D}}$$\textstyle{C_{KS}(X,D)^{\vee},}$

whose vertical arrows are injective. It is clear from the construction of the
reciprocity maps that this diagram is commutative.

To show that $\rho^{\prime}_{X|D}$ is injective, it suffices to show that
$\rho^{\prime\vee}_{X|D}$ is surjective (see [15, Lemma 7.10]). We fix a
character $\chi\in C_{KS}(X|D)^{\vee}$ and let
$\widetilde{\chi}=\beta^{\vee}(\chi)$. Since $\rho^{\vee}_{X|D}$ is surjective
by [16, Theorem 1.1], we can find a character $\chi^{\prime}\in{\rm
fil}_{D}H^{1}(K)$ such that
$\widetilde{\chi}=\rho^{\vee}_{X|D}(\chi^{\prime})$. We need to show that
$\chi^{\prime}\in{\rm fil}^{{\rm bk}}_{D}H^{1}(K)$.

We fix a point $x\in{\rm Irr}_{C}$ and let $\chi^{\prime}_{x}$ be the image of
$\chi^{\prime}$ in $H^{1}(K_{x})$. We need to show that
$\chi^{\prime}_{x}\in{\rm fil}^{{\rm bk}}_{n_{x}-1}H^{1}(K_{x})$, where
$n_{x}$ is the multiplicity of $D$ at $x$. By Corollary 9.4, it suffices to
show that for some maximal Parshin chain $P=(p_{0},\ldots,p_{d-2},x,\eta)$ on
$(U\subset X)$ and $d$-DV $V\subset K$ dominating $P$, the image of
$\chi^{\prime}_{x}$ in $H^{1}(Q(V^{h}))$ lies in the subgroup ${\rm fil}^{{\rm
bk}}_{n_{x}-1}H^{1}(Q(V^{h}))$. But this is proven by repeating the proof of
[16, Theorem 1.1] mutatis mutandis by using only one modification. Namely, we
need to use part (1) of Theorem 9.2 instead of part (2).

The surjectivity of $\rho^{\prime}_{X|D}$ on the degree zero part follows
because the top horizontal arrow in the commutative diagram

(9.16)
$\textstyle{C_{KS}(X,D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X|D}}$$\scriptstyle{\beta}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X,D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{C_{KS}(X|D)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\prime}_{X|D}}$$\textstyle{\pi^{{\rm
abk}}_{1}(X,D)_{0}}$

is surjective by [16, Theorem 1.1]. The finiteness of $C_{KS}(X|D)_{0}$
follows because $\beta$ is surjective by Lemma 3.3 and $C_{KS}(X,D)_{0}$ is
finite by [16, Theorem 4.8]. This concludes the proof. ∎

Using Theorem 9.9 and [15, Lemma 8.4], we get the following.

######  Corollary 9.10.

Under the additional assumptions of Theorem 9.9, the map

$\rho^{\prime}_{X|D}\colon{C_{KS}(X|D)}/m\to{\pi^{{\rm abk}}_{1}(X,D)}/m$

is an isomorphism of finite groups for every integer $m\geq 1$.

###  9.4. Filtration of Kerz-Zhao

Let $k$ be a finite field of characteristic $p$ and $X$ an integral regular
projective scheme over $k$ of dimension $d\geq 1$. Let $D\subset X$ be an
effective Cartier divisor with complement $U$ such that $D_{\rm red}$ is a
simple normal crossing divisor. Let $K$ denote the function field of $X$. By
[22, Theorems 1.1.5, 4.1.4], there is an isomorphism

(9.17) $\lambda_{m}\colon
H^{1}_{{\acute{e}t}}(U,{{\mathbb{Z}}}/{p^{m}})\xrightarrow{\cong}{\varinjlim}_{n}H^{d}_{{\acute{e}t}}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|nD}}/{p^{m}}})^{\vee}.$

Each group
$H^{d}_{{\acute{e}t}}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|nD}}/{p^{m}}})$
is finite by [30, Theorem 3.3.1] and Corollary 9.10. It follows that (9.17) is
an isomorphism of discrete torsion topological abelian groups.

Since
$H^{d}_{{\acute{e}t}}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|D}}/{p^{m}}})^{\vee}\hookrightarrow{\varinjlim}_{n}H^{d}_{{\acute{e}t}}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|nD}}/{p^{m}}})^{\vee}$,
we can define

${\rm fil}^{{\rm
bk}}_{D}H^{1}_{\acute{e}t}(U,{{\mathbb{Z}}}/{p^{m}}):=(\lambda_{m})^{-1}(H^{d}_{{\acute{e}t}}(X,\overline{{\widehat{{\mathcal{K}}}^{M}_{d,X|D}}/{p^{m}}})^{\vee}).$

This filtration was defined by Kerz-Zhao [30, Definition 3.3.7]. We let ${\rm
fil}^{{\rm
bk}}_{D}H^{1}_{\acute{e}t}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})={\varinjlim}_{m}{\rm
fil}^{{\rm bk}}_{D}H^{1}_{\acute{e}t}(U,{{\mathbb{Z}}}/{p^{m}})$ and ${\rm
fil}^{{\rm bk}}_{D,{\acute{e}t}}H^{1}(K)=H^{1}(K)\\{p^{\prime}\\}\oplus{\rm
fil}^{{\rm
bk}}_{D}H^{1}_{\acute{e}t}(U,{{\mathbb{Q}}_{p}}/{{\mathbb{Z}}_{p}})$.

Using [30, Theorem 3.3.1] and Corollary 9.10, we get the following logarithmic
version of Theorem 8.11 (without assuming that $D_{\rm red}$ is regular). This
identifies the filtration due to Kerz-Zhao to the one induced by the
Brylinski-Kato filtration.

######  Corollary 9.11.

As subgroups of $H^{1}(K)$, one has

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

##  10\. Counterexamples to Nisnevich descent

We shall now prove Theorem 1.5. Let $k$ be a finite field of characteristic
$p$ and $X$ an integral regular projective scheme of dimension two over $k$.
Let $C\subset X$ be an integral regular curve with complement $U$. Let
$K=k(X)$ and ${\mathfrak{f}}=k(\lambda)$, where $\lambda$ is the generic point
of $C$. Let $K_{\lambda}$ be the Henselian discrete valuation field with ring
of integers ${\mathcal{O}}^{h}_{X,\lambda}$ and residue field
${\mathfrak{f}}$. Fix a positive integer $m_{0}=p^{r}m^{\prime}$, where $r\geq
1$ and $p\nmid m^{\prime}$. Assume that $p\neq 2$. Then Matsuda has shown (see
the proof of [37, Proposition 3.2.7]) that there is an isomorphism

(10.1) $\eta\colon\frac{{\rm fil}^{{\rm ms}}_{m_{0}}H^{1}(K_{\lambda})}{{\rm
fil}^{{\rm
bk}}_{m_{0}-1}H^{1}(K_{\lambda})}\xrightarrow{\cong}B_{r}\Omega^{1}_{{\mathfrak{f}}},$

where $B_{\bullet}\Omega^{1}_{{\mathfrak{f}}}$ is an increasing filtration of
$\Omega^{1}_{{\mathfrak{f}}}$, recalled in the proof of Lemma 4.4. Suppose
that
$B_{1}\Omega^{1}_{{\mathfrak{f}}}=d({\mathfrak{f}})\subset\Omega^{1}_{{\mathfrak{f}}}$
is not zero. Then $B_{r}\Omega^{1}_{{\mathfrak{f}}}\neq 0$ for every $r\geq
1$. Hence, the left hand side of (10.1) is not zero. Since the restriction map
$\delta_{\lambda}\colon H^{1}(K)\to H^{1}(K_{\lambda})$ is surjective, it
follows that we can find a continuous character $\chi\in H^{1}(K)$ such that
$\delta_{\lambda}(\chi)\in{\rm fil}^{{\rm
ms}}_{m_{0}}H^{1}(K_{\lambda})\setminus{\rm fil}^{{\rm
bk}}_{m_{0}-1}H^{1}(K_{\lambda})$.

We let $U^{\prime}\subseteq U$ be the largest open subscheme where $\chi$ is
unramified and let $C^{\prime}=X\setminus U^{\prime}$ with the reduced closed
subscheme structure. Since $X$ is a surface, we can find a morphism $f\colon
X^{\prime}\to X$ which is a composition of a sequence monoidal transformations
such that the reduced closed subscheme $f^{-1}(C^{\prime})$ is a simple normal
crossing divisor. In particular, $E_{0}\to C$ is an isomorphism if we let
$E_{0}$ be the strict transform of $C$. We let $E=f^{-1}(C^{\prime})$ with
reduced structure. Note that there is a finite closed subset $T\subset
C^{\prime}$ such that $f^{-1}(X\setminus T)\to X\setminus T$ is an
isomorphism.

We let $E=E_{0}+E_{1}+\cdots+E_{s}$, where each $E_{i}$ is integral. We let
$\lambda_{i}$ be the generic point of $E_{i}$ so that $\lambda_{0}=\lambda$.
Let ${\rm ar}_{\lambda_{i}}\colon H^{1}(K)\to{\mathbb{Z}}$ be the Artin
conductor (see [37, Definition 3.2.5]). Theorem 9.1 implies that ${\rm
ar}_{\lambda_{0}}(\chi)=m_{0}$. We let $m_{i}={\rm ar}_{\lambda_{i}}(\chi)$
for $i\geq 1$ and define $D^{\prime}=\stackrel{{\scriptstyle
s}}{{\underset{i=0}{\sum}}}m_{i}E_{i}$. It is then clear that $\chi\in{\rm
fil}_{D}H^{1}(K)\setminus{\rm fil}^{{\rm bk}}_{D}H^{1}(K)$. We have thus found
a smooth projective integral surface $X^{\prime}$ and an effective Cartier
divisor $D^{\prime}\subset X^{\prime}$ with the property that $D^{\prime}_{\rm
red}$ is a simple normal crossing divisor and $\frac{{\rm
fil}_{D^{\prime}}H^{1}(K)}{{\rm fil}^{{\rm bk}}_{D^{\prime}}H^{1}(K)}\neq 0$
if $K$ is the function field of $X^{\prime}$. It follows that ${\rm
Ker}(\pi^{{\rm adiv}}_{1}(X^{\prime},D^{\prime})\twoheadrightarrow\pi^{{\rm
abk}}_{1}(X^{\prime},D^{\prime}))\neq 0$. Equivalently, ${\rm Ker}(\pi^{{\rm
adiv}}_{1}(X^{\prime},D^{\prime})_{0}\twoheadrightarrow\pi^{{\rm
abk}}_{1}(X^{\prime},D^{\prime})_{0})\neq 0$.

We now look at the the commutative diagram

(10.2) $\textstyle{{\rm
CH}_{0}(X^{\prime}|D^{\prime})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}_{X^{\prime}|D^{\prime}}}$$\textstyle{\pi^{{\rm
adiv}}_{1}(X^{\prime},D^{\prime})_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{4}_{\mathcal{M}}(X^{\prime}|D^{\prime},{\mathbb{Z}}(2))_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{\rm
cyc}}^{\prime}_{X^{\prime}|D^{\prime}}}$$\textstyle{\pi^{{\rm
abk}}_{1}(X^{\prime},D^{\prime})_{0}.}$

The top horizontal arrow is an isomorphism by [16, Theorems 1.2, 1.3] and the
bottom horizontal arrow is an isomorphism by Corollary 1.4. We conclude that
the left vertical arrow is surjective but not injective.

To complete the construction of a counterexample, what remains is to find a
pair $(X,C)$ such that $d({\mathfrak{f}})\subset\Omega^{1}_{{\mathfrak{f}}}$
is not zero. But this is an easy exercise. For instance, take
$X={\mathbb{P}}^{2}_{k}$ and $C\subset{\mathbb{P}}^{2}_{k}$ a coordinate
hyperplane. Then ${\mathfrak{f}}\cong k(t)$ is a purely transcendental
extension of degree one and $d(t)$ is a free generator of
$\Omega^{1}_{{\mathfrak{f}}}$. We remark that we had assumed above that $p\neq
2$, but this condition can be removed using the proof of [15, Theorem 6.3].

Let $(X^{\prime},D^{\prime})$ be as above and let $F=k(t)$ be a purely
transcendental extension of degree one. We let $\widetilde{X}=X^{\prime}_{F}$
and $\widetilde{D}=D^{\prime}_{F}$. Then we have a commutative diagram

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

where the horizontal arrows are the flat pull-back maps. It easily follows
from the proof of [32, Proposition 4.3] that the top horizontal arrow is
injective. It follows that the right vertical arrow is not injective. This
shows the failure of Nisnevich descent for the Chow groups with modulus over
infinite fields too.

_Acknowledgements._ Gupta was supported by the SFB 1085 _Higher Invariants_
(Universität Regensburg).

## References

  * [1] A. Abbes, T. Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math., 124, (2002), no. 5, 879–920.
  * [2] M. Artin, A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics, 306, (1972-1973), Séminaire de Géométrie Algébrique du Bois-Marie 1963–64 (SGA 4).
  * [3] F. Binda, A. Krishna, S. Saito, Bloch’s formula for 0-cycles with modulus and the higher dimensional class field theory, arXiv:200201856 [math.AG], (2020).
  * [4] F. Binda, S. Saito, Relative cycles with moduli and regulator maps, J. Math. Inst. Jussieu, 18, (2019), 1233–1293.
  * [5] S. Bloch, K. Kato, $p$-adic étale cohomology, Publ. Math. Inst. Hautes Études Sci., 63, (1986), 107–152.
  * [6] J. Colliot-Thèléne, J. Sansuc, C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J., 50, (1983), 763–801.
  * [7] C. Dahlhausen, Milnor $K$-theory of complete discrete valuation rings with finite residue fields, J. Pure Appl. Algebra, 222, (2018), no. 6, 1355–1371.
  * [8] A. J. de Jong et al., The Stacks Project, Available at http://stacks.math.columbia.edu. (2020).
  * [9] W. Fulton, Intersection theory, Second Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge. A Series of Modern Surveys in Mathematics, 2, Springer-Verlag, Berlin, (1998).
  * [10] T. Geisser, L. Hesselholt, On the $K$-theory of complete regular local ${\mathbb{F}}_{p}$-algebras, Topology, 45, (2006), 475–493.
  * [11] T. Geisser, M. Levine, The $K$-theory of fields in characteristic $p$, Invent. Math., 139, (2000), 459–493.
  * [12] M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mémo. Soc. Math. France, $2^{e}$ série, 21, (1985), 1–87.
  * [13] M. Gros, N. Suwa, La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique, Duke Math. J., 57, (1988), no. 2, 615–628.
  * [14] R. Gupta, A. Krishna, Relative $K$-theory via 0-cycles in finite characteristic, arXiv:1910.06630 [math.AG], (2020).
  * [15] R. Gupta, A. Krishna, Reciprocity for Kato-Saito idele class group with modulus, arXiv:2008.05719v3 [math.AG], (2021).
  * [16] R. Gupta, A. Krishna, Idele class groups with modulus, arXiv:2101.04609v1 [math.AG], (2021).
  * [17] R. Hartshorne, Algebraic Geometry, Graduate Text in Mathematics, 52, (1997), Springer-Verlag.
  * [18] L. Hesselholt, Topological Hochschild homology and de Rham-Witt complex of ${\mathbb{Z}}_{(p)}$-algebras, Contemp. Math., 346, (2004), 253–259.
  * [19] L . Hesselholt, I. Madsen, On the K-theory of local fields, Ann. of Math., (2) 158, 1 (2003), 1–113.
  * [20] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Ec. Norm. Supér., (4) 12, (1979), 501–661.
  * [21] L. Illusie, M. Raynaud, Les suites spectrales associées au complexe de de Rham-Witt , Publ. Math. Inst. Hautes Études Sci., 57, (1983), 73–212.
  * [22] U. Jannsen, S. Saito, Y. Zhao, Duality for relative logarithmic de Rham-Witt sheaves and wildly ramified class field theory over finite fields, Comp. Math., 154, (2008), 1306–1331.
  * [23] K. Kato, Milnor $K$-theory and Chow group of zero cycles, Applications of algebraic K-theory to algebraic geometry and number theory, Contemp. Math., 55, Amer. Math. Soc, Providence, RI, (1986), 241–255.
  * [24] K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, in Algebraic K-Theory and Algebraic Number Theory (Honolulu, Hawaii, 1987), Contemp. Math., 83, Amer. Math. Soc., Providence, (1989), 101–131.
  * [25] K. Kato, S. Saito, Unramified class field theory of arithmetic surfaces, Ann. of Math., 118, No. 2, (1983), 241–275.
  * [26] K. Kato, S. Saito, Global class field theory of arithmetic schemes, Applications of algebraic K-theory to algebraic geometry and number theory, Contemp. Math., 55, Amer. Math. Soc, Providence, RI, (1986), 255–331.
  * [27] M. Kerz, Milnor K-theory of local rings with finite residue fields, J. Alg. Geom., 19, (2010), 173–191.
  * [28] M. Kerz, Ideles in higher dimensions, Math. Res. Lett., 18, (2011), 699–713.
  * [29] M. Kerz, S. Saito, Chow group of 0-cycles with modulus and higher-dimensional class field theory, Duke Math. J., 165, (2016), 2811–2897.
  * [30] M. Kerz, Y. Zhao, Higher ideles and class field theory, Nagoya Math. J., 236, (2019), 214–250.
  * [31] M. Kolster, $K_{2}$ of noncommutative local rings, J. Algebra, 95, (1985), 173–200.
  * [32] A. Krishna, J. Park, On additive higher Chow groups of affine schemes, Doc. Math., 21, (2016), 49–89.
  * [33] A. Krishna, J. Park, de Rham-Witt sheaves via algebraic cycles, Comp. Math., (to appear), arXiv:1504.08181v7 [math.AG], (2021).
  * [34] G. Laumon, Semi-continuité du conducteur de Swan (d’aprés P. Deligne), in ‘The Euler-Poincaré characteristic’ edited by J.-L. Verdier, Soc. Math. France, Paris, Astérisque, 83, (1981), 173–219.
  * [35] M. Lüders, On the relative Gersten conjecture for Milnor $K$-theory in the smooth case, arXiv:2010.02622v1 [math.AG], (2020).
  * [36] M. Lüders, M. Morrow, Milnor $K$-theory of $p$-adic rings, arXiv:2101.01092v1 [math.KT], (2021).
  * [37] S. Matsuda, On the Swan conductors in positive characteristic, Amer. J. Math., 119, (1997), 705–739.
  * [38] J. Milne, Duality in the flat cohomology of a surface, Ann. Sci. Éc. Norm. Supér., (4), 9, (1976), 171–201.
  * [39] J. Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math., 108, (1986), 297–360.
  * [40] M. Morrow, K-theory and logarithmic Hodge-Witt sheaves of formal schemes in characteristic $p$, Ann. Sci. Éc. Norm. Supér., (4), 52, no. 6, (2019), 1537–1601.
  * [41] L. Ribes, P. Zalesskii, Profinite Groups, 2nd, Ergebnisse der Mathematik und iher Grenzgibiete, 40, Springer, (2010).
  * [42] K. Rülling, S. Saito, Higher Chow groups with modulus and relative Milnor $K$-theory, Trans. Amer. Math. Soc., 370, (2018), 987–1043.
  * [43] S. Saito, A global duality theorem for varieties over global fields, Algebraic $K$-Theory: Connections with Geometry and Topology, Springer, (1989), 425–444.
  * [44] Y. Zhao, Duality for relative logarithmic de Rham-Witt sheaves on semistable schemes over ${\mathbb{F}}_{q}[[t]]$, Doc. Math., 23, (2018), 1925–1967.