$f^{*}\colon C^{\rm t}(X)\to C^{\rm t}(X^{\prime}).$

###### Proof.

We can write $k^{\prime}$ as a union of $k_{i}:=k^{1/{p^{i}}}$. We choose an
integral normal compactification $\overline{X}$ of $X$ and let $u\colon
X\hookrightarrow\overline{X}$ be the inclusion. We now apply Proposition 4.15
and note that $X_{i}=X_{k_{i}}$ for every $i\geq 0$. Similarly,
$X^{\prime}=X_{k^{\prime}}$ because of the smoothness of $X$ over $k$. The
latter also implies that each $f_{ij}\colon X_{j}\to X_{i}$ is finite and
flat, as we observed in the proof of Corollary 4.19. ∎

###  4.5. Continuity of tame class group

Let $k$ be an imperfect field of characteristic $p>0$. Let $k^{\heartsuit}$
denote a fixed perfect closure of $k$. For
$X\in{\operatorname{\mathbf{Sch}}}_{k}$, we let $X^{\heartsuit}=X\times_{{\rm
Spec\,}(k)}{\rm Spec\,}(k^{\heartsuit})$. Note that $\pi\colon
X^{\heartsuit}\xrightarrow{\cong}\lim X_{\bullet}$, where the pro-scheme
$X_{\bullet}=\\{X_{i}\\}_{i\geq 0}$ is defined by taking $k_{i}=k^{1/{p^{i}}}$
and $X_{i}=X\times_{{\rm Spec\,}(k)}{\rm Spec\,}(k_{i})$. For every $j\geq i$,
the projection map $f_{ij}\colon X_{j}\to X_{i}$ is a finite flat radicial
morphism of degree $p^{j-i}$. We let $g_{i}\colon X_{i}\to X,\ f_{i}\colon
X^{\heartsuit}\to X_{i}$ and $f\colon X^{\heartsuit}\to X$ be the projection
maps. If $X\in{\mathbf{Sm}}_{k}$ is integral, then $X_{i}$ is an integral
smooth $k_{i}$-scheme for each $i$ and $X^{\heartsuit}$ is an integral smooth
$k^{\heartsuit}$-scheme. In particular, we have a homomorphism
${\underset{i\geq 0}{\varinjlim}}\ C^{\rm t}(X_{i})\to C^{\rm
t}(X^{\heartsuit})$ by Corollary 4.20. We denote this by $\pi^{*}$.

######  Lemma 4.21.

Assume that $X$ is a smooth and integral $k$-scheme. Then the pull-back map

$\pi^{*}\colon\ {\underset{i\geq 0}{\varinjlim}}\ C^{\rm t}(X_{i})\to C^{\rm
t}(X^{\heartsuit})$

is an isomorphism.

###### Proof.

Suppose that $y\in X^{\heartsuit}_{(0)}$ and $y_{j}=f_{j}(y)$ for any $j\geq
0$. Then there will exist an integer $j\gg 0$ and a closed subscheme
$Y_{j}\subset X_{j}$ such that $Y_{j}\times_{X_{j}}X^{\heartsuit}={\rm
Spec\,}(k(y))$. The support of such $Y_{j}$ must be the closed point $y_{j}$
of $X_{j}$. Furthermore, the fppf descent implies that $Y_{j}$ must be
reduced. In other words, $Y_{j}={\rm Spec\,}(k(y_{j}))$. The same argument
implies that ${\rm Spec\,}(k(y_{j}))\times_{X_{j}}X_{l}={\rm
Spec\,}(k(y_{l}))$ and ${\rm
Spec\,}(k(y_{j}))\times_{X_{j}}X^{\heartsuit}={\rm Spec\,}(k(y))$ for every
$l\geq j$. It follows from this and the definition of $f^{*}_{l}$ (see § 4.4)
that $f^{*}_{l}(a_{l})=a_{l}$ for every $l\geq j$ and $a_{l}\in
K^{M}_{1}(k(y_{l}))$. Since Milnor $K$-theory is a continuous functor on
fields, it follows that every $a\in K^{M}_{1}(k(y))$ has the property that
there exists $l\geq j$ and $a_{l}\in K^{M}_{1}(k(y_{l}))$ such that
$f^{*}_{l}(a_{l})=a$. This shows that $\pi^{*}$ is surjective.

To prove that $\pi^{*}$ is injective, we suppose that there exists
$\alpha\in\varinjlim_{j}\ C^{\rm t}(X_{j})$ such that $\pi^{*}(\alpha)=0$. We
can then find an integer $j\geq 0$, a 0-dimensional reduced closed subscheme
$S_{j}\subset X_{j}$ such that $\alpha$ is represented by an element of
$K_{1}(S_{j})$. We let $S^{\prime}$ be the set theoretic inverse image of
$S_{j}$ in $X^{\heartsuit}$ (with the reduced closed subscheme structure) so
that $f^{*}_{j}(\alpha)\in K_{1}(S^{\prime})$. As explained in the proof of
surjectivity of $\pi^{*}$, we can find $l\gg j$ and a reduced closed subscheme
$S_{l}\subset X_{l}$ such that $S_{l}\times_{X_{l}}X^{\heartsuit}=S^{\prime}$.
We can replace $\alpha$ by $f^{*}_{jl}(\alpha)\in K_{1}(S_{l})$ (note that
$S^{\prime}\to S_{l}\to S_{j}$ are bijections). We can also replace $k$ by any
$k_{l}$ and $\alpha$ by $f^{*}_{jl}(\alpha)$ with $l\gg j$. We can thus assume
that there exists a 0-dimensional reduced closed subscheme $S_{0}\subset X$
such that $\alpha$ is represented by an element of $K_{1}(S_{0})$.
Furthermore, $S^{\prime}=S_{0}\times_{X}X^{\heartsuit}$. In particular, if we
let $S_{j}=f_{j}(S^{\prime})$, then $S_{j}=S_{0}\times_{X}X_{j}$ and
$S^{\prime}=S_{j}\times_{X_{j}}X^{\heartsuit}$ for every $j\geq 0$.

We fix an integral compactification $u\colon X\hookrightarrow\overline{X}$ and
let $u^{\prime}\colon X^{\heartsuit}\hookrightarrow\overline{X}^{\heartsuit}$
be the induced open immersion. Since
$X^{\heartsuit}\in{\mathbf{Sm}}_{k^{\prime}}$, we see that $u^{\prime}$ has
factorization $X^{\heartsuit}\hookrightarrow(\overline{X}^{\heartsuit})_{\rm
red}\hookrightarrow\overline{X}^{\heartsuit}$. We let
$\overline{X}_{j}:=\overline{X}_{k_{j}}$. Since $f^{*}(\alpha)$ lies in the
image of $\partial^{1}_{X^{\heartsuit}}\colon F^{1}_{2}(X^{\heartsuit})\to
F^{0}_{1}(X^{\heartsuit})$, we can find integral curves
$C_{1},\ldots,C_{r}\subset X^{\heartsuit}$ satisfying the following. Let
$\overline{C}_{i}$ be the closure of $C_{i}$ in $\overline{X}^{\heartsuit}$
with the reduced (hence integral) closed subscheme structure and let
$\nu^{\prime}_{i}\colon\overline{Y}_{i}:=(\overline{C}_{i})_{n}\to\overline{X}^{\heartsuit}$
be the canonical map. Let
$T^{\prime}_{i}=(\nu^{\prime}_{i})^{-1}(\overline{X}^{\heartsuit}\setminus
X^{\heartsuit})$ and
$A^{\prime}_{i}={\mathcal{O}}_{\overline{Y}_{i},T^{\prime}_{i}}$. Let
$J^{\prime}_{i}$ be the Jacobson radical of $A^{\prime}_{i}$. Then, there
exist elements $\beta^{\prime}_{i}\in
K^{M}_{2}(A^{\prime}_{i},J^{\prime}_{i})\subset K^{M}_{2}(k^{\prime}(C_{i}))$
(in view of Lemma 4.5) such that $f^{*}(\alpha)=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}\partial(\beta^{\prime}_{i})\in K_{1}(S^{\prime})$.

We can now find an integer $j\gg 0$ and closed subschemes
$\overline{Z}^{j}_{i}\subset\overline{X}_{j}$ such that

(4.15)
$\overline{C}_{i}\cong\overline{Z}^{j}_{i}\times_{\overline{X}_{j}}\overline{X}^{\heartsuit}\
\forall\ 1\leq i\leq r.$

In particular, $\overline{C}_{i}\to\overline{Z}^{j}_{i}$ is faithfully flat.
By the fppf descent, each $\overline{Z}^{j}_{i}$ must be an integral curve in
$\overline{X}_{j}$. By the same argument, it follows that
$\overline{Z}^{l}_{i}:=\overline{Z}^{j}_{i}\times_{\overline{X}_{j}}\overline{X}_{l}$
is an integral curve in $\overline{X}_{l}$ for every $l\geq j$ and $1\leq
i\leq r$. Moreover, $\overline{C}_{i}\xrightarrow{\cong}{\underset{l\geq
j}{\varprojlim}}\ \overline{Z}^{l}_{i}$. We let
$Z^{l}_{i}=\overline{Z}^{l}_{i}\bigcap X_{l}$. Since
$\overline{Z}^{l}_{i}\subset(\overline{X}_{l})_{\rm
red}\subset((\overline{X}_{j})_{\rm red})_{l}$ for each $l$ and $i$ and
similarly $\overline{C}_{i}\subset(\overline{X}^{\heartsuit})_{\rm
red}\subset((\overline{X}_{j})_{\rm red})^{\heartsuit}$, we can pass from
$\overline{X}$ to $(\overline{X}_{j})_{\rm red}$ which allows us to assume
that $j=0$. Note that $(\overline{X}_{j})_{\rm red}$ is an integral
compactification of $X_{j}$ for every $j\geq 0$.

Let $\overline{Y}^{j}_{i}$ (resp. $Y^{j}_{i}$) be the normalization of
$\overline{Z}^{j}_{i}$ (resp. $Z^{j}_{i}$) and let
$\nu^{j}_{i}\colon\overline{Y}^{j}_{i}\to\overline{Z}^{j}_{i}$ be the
normalization maps. Since the normalization is a functor in the category of
integral schemes with dominant morphisms and this functor commutes with
projective limits of integral schemes with affine transition morphisms, it
follows that there is a commutative diagram

(4.16)
$\textstyle{\overline{Y}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{l}}$$\scriptstyle{\nu^{\prime}_{i}}$$\textstyle{\overline{Y}^{l}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{jl}}$$\scriptstyle{\nu^{l}_{i}}$$\textstyle{\overline{Y}^{j}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{j}_{i}}$$\textstyle{\overline{C}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{l}}$$\textstyle{\overline{Z}^{l}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{jl}}$$\textstyle{\overline{Z}^{j}_{i}}$

for every $l\geq j\geq 0$ and
$\overline{Y}_{i}\xrightarrow{\cong}{\underset{j\geq 0}{\varprojlim}}\
\overline{Y}^{j}_{i}$. If we let
$T^{j}_{i}=(\nu^{j}_{i})^{-1}(\overline{Z}^{j}_{i}\setminus
Z^{j}_{i})=(\nu^{j}_{i})^{-1}(\overline{X}_{j}\setminus X_{j})$ and
$A^{j}_{i}={\mathcal{O}}_{\overline{Y}^{j}_{i},T^{j}_{i}}$, then we moreover
have $T^{l}_{j}=(\phi_{jl})^{-1}(T^{j}_{i})$ and
$T^{\prime}_{i}=(\phi_{j})^{-1}(T^{j}_{i})$ for every $l\geq j\geq 0$. This
implies that ${\underset{j\geq 0}{\varinjlim}}\
A^{j}_{i}\xrightarrow{\cong}A^{\prime}_{i}$ for each $1\leq i\leq r$.

Since the transition maps of $\\{A^{j}_{i}\\}_{j\geq 0}$ are faithfully flat
and $J^{\prime}_{i}$ is finitely generated, we can find $j\gg 0$ such that
$J^{\prime}_{i}=J^{j}_{i}A^{\prime}_{i}$ for each $i$. In particular,
$J^{\prime}_{i}=\varinjlim_{j}J^{j}_{i}$. By the continuity of relative Milnor
$K$-theory, it follows that
$K^{M}_{2}(A^{\prime}_{i},J^{\prime}_{i})=\varinjlim_{j}K^{M}_{2}({A}^{j}_{i},{J}^{j}_{i})$
for every $i$. By passing from $X$ to $X_{j}$ with $j\gg 0$, we can therefore
assume that there exist

(4.17) $\beta^{0}_{i}\in K^{M}_{2}(A^{0}_{i},J^{0}_{i})\subset
K^{M}_{2}(k(Z^{0}_{i}))\ \mbox{such \ that}\
\beta^{\prime}_{i}=f^{*}(\beta^{0}_{i})\ \forall\ i.$

We now observe that the map $\phi_{0}\colon C_{i}\to Z^{0}_{i}$ is flat by
(4.15). Hence, for every open subscheme $U\subset Z^{0}_{i}$ and
$U^{\prime}=(\phi_{0})^{-1}(U)$, there is a commutative diagram of Quillen
$K$-theory localization sequences

$\textstyle{K_{2}(Z^{0}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}_{0}}$$\textstyle{K_{2}(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\phi^{*}_{0}}$$\textstyle{K_{1}(Z^{0}_{i}\setminus
U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}_{0}}$$\textstyle{K_{2}(C_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{2}(U^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{K_{1}(C_{i}\setminus
U^{\prime}),}$

where the vertical arrows are flat pull-back maps. Taking the limit over $U$
and noting that ${\rm Spec\,}(k^{\prime}(C_{i}))=C_{i}\times_{Z^{0}_{i}}{\rm
Spec\,}(k(Z^{0}_{i}))$, we get a commutative diagram

(4.18)
$\textstyle{K^{M}_{2}(k(Z^{0}_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\phi^{*}_{0}}$$\textstyle{F^{0}_{1}(Z^{0}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{*}_{0}}$$\textstyle{F^{0}_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{K^{M}_{2}(k^{\prime}(C_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{F^{0}_{1}(C_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{0}_{1}(X^{\heartsuit}),}$

where we have identified the Milnor $K_{i}$ and their boundary maps for fields
with Quillen $K_{i}$ and their boundary maps for $i\leq 2$. We conclude from
this that

$f^{*}\left(\alpha-\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}(\partial(\beta^{0}_{i}))\right)=\phi^{*}_{0}\left(\alpha-\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}(\partial(\beta^{0}_{i}))\right)=0\ \mbox{in}\
K_{1}(S^{\prime}).$

Using the continuity property of $K_{1}(-)$ again, it follows that
$g^{*}_{j}(\alpha-\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}(\partial(\beta^{0}_{i})))=\phi^{*}_{0j}((\alpha-\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}(\partial(\beta^{0}_{i})))=0$ in $K_{1}(S_{j})$ for
all $j\gg 0$. Equivalently, $g^{*}_{j}(\alpha)=\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\sum}}}g^{*}_{j}(\partial(\beta^{0}_{i}))$ in
$F^{0}_{1}(X_{j})$ for all $j\gg 0$. Since we have shown in (4.17) that
$\beta^{0}_{i}\in K^{M}_{2}(A^{0}_{i},J^{0}_{i})\subset
K^{M}_{2}(k(Z^{0}_{i}))$ for every $i$, it follows from Proposition 4.15 that
$g^{*}_{j}(\alpha)$ dies in $C^{\rm t}(X_{j})$. We have thus shown that
$\pi^{*}$ is injective. This concludes the proof. ∎

######  Remark 4.22.

If $k$ is a local field of characteristic $p>0$, then Lemma 4.21 allows one to
endow $C^{\rm t}(X^{\heartsuit})$ with the direct limit topology. It is then
clear that the map $f^{*}\colon C^{\rm t}(X)\to C^{\rm t}(X^{\heartsuit})$ is
continuous. Henceforth, we shall always assume $C^{\rm t}(X^{\heartsuit})$ to
be a topological abelian group in this manner whenever $k$ is a local field.

######  Corollary 4.23.

Keep the notations and assumptions of Lemma 4.21. Then the pull-back map

$\pi^{*}\colon C^{\rm t}(X)[\tfrac{1}{p}]\to C^{\rm
t}(X^{\heartsuit})[\tfrac{1}{p}]$

is an isomorphism.

###### Proof.

In view of Lemma 4.21, it suffices to show that if ${k^{\prime}}/k$ is a
finite purely inseparable extension and $X^{\prime}=X\times_{{\rm
Spec\,}(k)}{\rm Spec\,}(k^{\prime})$, then the pull-back map $\pi^{*}\colon
C^{\rm t}(X)[\tfrac{1}{p}]\to C^{\rm t}(X^{\prime})[\tfrac{1}{p}]$ is an
isomorphism if $\pi\colon X^{\prime}\to X$ is the projection. In this case,
the surjectivity follows from Lemma 4.24 and the injectivity from Corollary
4.19. ∎

######  Lemma 4.24.

Let ${k^{\prime}}/k$ be a purely inseparable field extension and let $m$ be an
integer prime to $p$. Then the pull-back maps

$\pi^{*}\colon{K^{M}_{n}(k)}[\tfrac{1}{p}]\to{K^{M}_{n}(k^{\prime})}[\tfrac{1}{p}]\
\mbox{and}\ \pi^{*}\colon{K^{M}_{n}(k)}/m\to{K^{M}_{n}(k^{\prime})}/m$

are isomorphisms for all $n\geq 0$.

###### Proof.

By a limit argument, we can easily reduce to the case when ${k^{\prime}}/k$ is
a finite extension. In this case, one knows that the composite map
${K^{M}_{n}(k)}\to{K^{M}_{n}(k^{\prime})}\xrightarrow{N_{{k^{\prime}}/k}}{K^{M}_{n}(k)}$
is multiplication by $p^{r}$ for some $r\geq 0$ by [7, § 5.4]. This implies
that $\pi^{*}$ are injective. For surjectivity, we can use the surjection
$K^{M}_{1}(k^{\prime})\otimes\cdots\otimes
K^{M}_{1}(k^{\prime})\twoheadrightarrow K^{M}_{n}(k^{\prime})$ (where the
tensor product is taken $n$ times) to reduce to showing the surjectivity for
$n=1$. On the other hand, it is easy to see that
${K^{M}_{1}(k^{\prime})}/{K^{M}_{1}(k)}$ is annihilated by $p^{r}$ for some
$r\geq 0$. In particular, it is uniquely $m$-divisible. ∎

##  5\. The idele class group

In this section, we shall define the idele class group of integral normal
schemes following Wiesend’s approach and present the tame class group as a
canonical quotient of the idele class group. We fix a local field $k$
throughout this section. Let $X\in{\operatorname{\mathbf{Sch}}}_{k}$ be an
integral normal $k$-scheme.

###  5.1. Wiesend style idele class groups

For $C\in{\mathcal{C}}(X)$ and $\overline{C}_{n}$ the unique compactification
of $C_{n}$, we let
$K^{M}_{2}(k(C_{n})_{\infty})={\underset{x\in\Delta(C_{n})}{\bigoplus}}K^{M}_{2}(k(C_{n})_{x})$,
where recall that $k(C_{n})_{x}$ is the completion of $k(C_{n})$ with respect
to the discrete valuation defined by $x$. We let
$U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})={\underset{x\in\Delta(C_{n})}{\bigoplus}}U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{x})$
(see § 3.3). We make the following definitions.

######  Definition 5.1.

  1. (1)

We let

$I(X)=\left({\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}K^{M}_{2}(k(C_{n})_{\infty})\right)\bigoplus\left({\underset{x\in
X_{(0)}}{\bigoplus}}K^{M}_{1}(k(x))\right).$

  2. (2)

If $\overline{X}$ is a complete integral normal $k$-scheme with an effective
Cartier divisor $D$ such that $X=\overline{X}\setminus D$, we let

$I(\overline{X},D)=\left({\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}\frac{K^{M}_{2}(k(C_{n})_{\infty})}{{\rm
fil}_{D}K^{M}_{2}(k(C_{n})_{\infty})}\right)\bigoplus\left({\underset{x\in
X_{(0)}}{\bigoplus}}K^{M}_{1}(k(x))\right),$

where ${\rm
fil}_{D}K^{M}_{2}(k(C_{n})_{\infty})={\underset{x\in\Delta(C_{n})}{\bigoplus}}K_{2}^{M}(\widehat{{\mathcal{O}}_{{C},x}},\widehat{I_{D,x}})$.

  3. (3)

We let

$I^{{\rm
t},W}(X)=\left({\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}\frac{K^{M}_{2}(k(C_{n})_{\infty})}{U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})}\right)\bigoplus\left({\underset{x\in
X_{(0)}}{\bigoplus}}K^{M}_{1}(k(x))\right).$

We shall call $I(X)$ the ‘idele group’ of $X$. We endow $I(X)$ with the
structure of a topological abelian group with the direct sum topology, where
each $K^{M}_{1}(k(x))$ is endowed with its canonical topology and each
$K^{M}_{2}(k(C_{n})_{y})$ is endowed with Kato’s topology if ${\rm char}(k)>0$
and discrete topology if ${\rm char}(k)=0$. We call $I(\overline{X},D)$ the
‘idele group with modulus’ of $(\overline{X},D)$ and $I^{{\rm t},W}(X)$ the
‘tame idele group’ of $X$. These are endowed with the quotient topologies.

For $C\in{\mathcal{C}}(X)$, the product of inclusions into the completions
along the points of $\Delta(C_{n})$ induces $\iota_{C}\colon
K^{M}_{2}(k(C_{n}))\to K^{M}_{2}(k(C_{n})_{\infty})$. We also have the
boundary map $\partial_{C}\colon K^{M}_{2}(k(C_{n}))\to F^{0}_{1}(C)$ (see §
4.2). We thus get a homomorphism
$K^{M}_{2}(k(C_{n}))\xrightarrow{(\iota_{C},\partial_{C})}K^{M}_{2}(k(C_{n})_{\infty})\bigoplus
F^{0}_{1}(C)\hookrightarrow K^{M}_{2}(k(C_{n})_{\infty})\bigoplus
F^{0}_{1}(X)$. We let
$F^{1,W}_{2}(X)={\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}K^{M}_{2}(k(C))$.

######  Definition 5.2.

The idele class group of $X$ is defined to be the cokernel of the homomorphism

$\sum_{C\in{\mathcal{C}}(X)}(\iota_{C},\partial_{C})\colon F^{1,W}_{2}(X)\to
I(X).$

It is a topological abelian group with respect to the quotient topology. We
denote this group by $C(X)$. We let

$C(\overline{X},D)={\rm Coker}\left(F^{1,W}_{2}(X)\to
I(\overline{X},D)\right)=\frac{C(X)}{{\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}{{\rm
fil}_{D}K^{M}_{2}(k(C_{n})_{\infty})}}$

and

$C^{{\rm t},W}(X)={\rm Coker}\left(F^{1,W}_{2}(X)\to I^{{\rm
t},W}(X)\right)=\frac{C(X)}{{\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}{U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})}}.$

These are topological abelian groups endowed with the quotient topologies.
There are canonical continuous surjections

(5.1) $C(X)\twoheadrightarrow C(\overline{X},D)\twoheadrightarrow C^{{\rm
t},W}(X).$

These groups are the local field analogues of the various idele class groups
of varieties over finite fields due to Wiesend (see [61, § 7])

###  5.2. Covariance of idele class group

We keep the above notations. We have the following covariant functoriality
property of the idele class group for a proper morphism between integral
normal schemes.

######  Proposition 5.3.

Let $f\colon X^{\prime}\to X$ be a proper morphism between integral normal
$k$-schemes. Then there is a push-forward homomorphism

$f_{*}\colon C(X^{\prime})\to C(X).$

###### Proof.

For any $x^{\prime}\in X^{\prime}_{(0)}$ with $x=f(x^{\prime})$, there is a
norm map $N_{{k(x^{\prime})}/{k(x)}}\colon K^{M}_{1}(k(x^{\prime}))\to
K^{M}_{1}(k(x))$. Taking the sum, we get a push-forward map $f_{*}\colon
F^{0}_{1}(X^{\prime})\to F^{0}_{1}(X)$, which is clearly continuous. We now
let $C^{\prime}\in{\mathcal{C}}(X^{\prime})$. If $f(C^{\prime})=x\in X_{(0)}$,
we let $f_{*}\colon
K^{M}_{2}(k(C^{\prime}_{n})_{\infty})\xrightarrow{\partial}{\underset{x^{\prime}\in\Delta(C^{\prime}_{n})}{\bigoplus}}K^{M}_{1}(k(x^{\prime}))\xrightarrow{\sum_{x^{\prime}}N_{{k(x^{\prime})}/{k(x)}}}K^{M}_{1}(k(x))$
be the composition of the boundary and the norm maps. If $f(C^{\prime})=C$ is
an integral curve, then the map $f\colon C^{\prime}\to C$ must be finite. In
particular, it induces a Cartesian square

(5.2)
$\textstyle{C^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\overline{C}^{\prime}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{C_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{C}_{n},}$

where the vertical arrows are finite flat and the horizontal arrows are open
immersions. Since $f^{-1}(\Delta(C_{n}))=\Delta(C^{\prime}_{n})$, we get the
norm map $N_{{C^{\prime}}/C}\colon K^{M}_{2}(k(C^{\prime}_{n})_{\infty})\to
K^{M}_{2}(k(C_{n})_{\infty})$. We have thus defined a homomorphism
$f_{*}\colon I(X^{\prime})\to I(X)$.

To show that $f_{*}$ descends to the level of the idele class groups, let
$C^{\prime}\in{\mathcal{C}}(X^{\prime})$. To show that the composition

(5.3) $K^{M}_{2}(k(C^{\prime}_{n}))\to
K^{M}_{2}(k(C^{\prime}_{n})_{\infty})\bigoplus F^{0}_{1}(C^{\prime}_{n})\to
K^{M}_{2}(k(C^{\prime}_{n})_{\infty})\bigoplus F^{0}_{1}(C^{\prime})\to C(X)$

is zero, we can replace $X^{\prime}$ by $C^{\prime}_{n}$. We can therefore
assume that $X^{\prime}$ is an integral normal curve over $k$.

Suppose first that $f(X^{\prime})=x\in X_{(0)}$ so that there is a
factorization $X^{\prime}\xrightarrow{f^{\prime}}{\rm
Spec\,}(k(x))\hookrightarrow X$ of $f$. Since the composition
$K^{M}_{2}(k(X^{\prime}))\to K^{M}_{2}(k(X^{\prime})_{\infty})\bigoplus
F^{0}_{1}(X^{\prime})\xrightarrow{f^{\prime}_{*}}K^{M}_{1}(k(x))$ is zero by
Suslin’s reciprocity law [94], it follows that $f_{*}$ descends to a
homomorphism $f_{*}\colon C(X^{\prime})\to K^{M}_{1}(k(x))\to C(X)$.

We now assume that $f(X^{\prime})$ is an integral curve $C$. Then $f\colon
X^{\prime}\to C$ is a finite map. In particular, we have a Cartesian square
such as (5.2) (where $C^{\prime}=X^{\prime}$). In order to show that the
composite map in (5.3) is zero, we can replace $X$ by $C_{n}$. We can
therefore assume that $f:X^{\prime}\to X$ is a finite morphism between
integral normal curves over $k$.

We now consider the diagrams

(5.4)
$\textstyle{K^{M}_{2}(k(\overline{X}^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{M}_{2}(k(\overline{X}^{\prime})_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{M}_{2}(k(\overline{X}^{\prime}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{F^{0}_{1}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{M}_{2}(k(\overline{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{M}_{2}(k(\overline{X})_{\infty})}$$\textstyle{K^{M}_{2}(k(\overline{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{F^{0}_{1}(X),}$

where the vertical arrows are the norm maps, left horizontal arrows are
induced by completions and the right horizontal arrows are the residue maps.
To finish the proof of the lemma, it remains to show that the two squares
commute.

Now, the right square commutes by [39, Lem. 3.4]. To show that the left square
commutes, we can replace Milnor $K_{2}$ by the Quillen $K_{2}$-functor on
fields and replace the norm map on Milnor $K$-theory by the transfer map on
Quillen $K$-theory by [52, § 2.4, Cor. 2]. We now let $x\in\Delta(X)$ and
$S=f^{-1}(\\{x\\})\subset\Delta(X^{\prime})$. Let
$A={\mathcal{O}}_{\overline{X},x}$ and
$A^{\prime}={\mathcal{O}}_{\overline{X}^{\prime},S}$. Then $A\hookrightarrow
A^{\prime}$ is a finite map of semilocal Dedekind domains. Let $J$ (resp.
$J^{\prime}$) denote the Jacobson radical of $A$ (resp. $A^{\prime}$). Let
$\widehat{A}$ (resp. $\widehat{A^{\prime}}$) denote the $J$-adic (resp.
$J^{\prime}$-adic) completion of $A$ (resp. $A^{\prime}$). Since
$\sqrt{JA^{\prime}}=J^{\prime}$, we see that $\widehat{A^{\prime}}$ coincides
with the $JA^{\prime}$-adic completion of $A^{\prime}$. Since the latter is
isomorphic to $\widehat{A}\otimes_{A}A^{\prime}$, we conclude that
$\widehat{A^{\prime}}\cong\widehat{A}\otimes_{A}A^{\prime}$. In particular,
$\widehat{A}\to\widehat{A^{\prime}}$ is a finite map of regular semilocal
rings of dimension one.

Let $F$ (resp. $F^{\prime}$) denote the quotient field of $A$ (resp. quotient
field of $A^{\prime}$). We let $\widehat{F}$ (resp. $\widehat{F^{\prime}}$)
denote the quotient field of $\widehat{A}$ (resp. ring of total quotients of
$\widehat{A^{\prime}}$). We then get

$\widehat{F}\otimes_{F}F^{\prime}=\widehat{F}\otimes_{F}F\otimes_{A}A^{\prime}\cong\widehat{F}\otimes_{A}A^{\prime}\cong\widehat{F}\otimes_{\widehat{A}}\widehat{A}\otimes_{A}A^{\prime}\cong\widehat{F}\otimes_{\widehat{A}}\widehat{A^{\prime}}\cong\widehat{F^{\prime}}.$

This shows that

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{F^{\prime}}}$

is a co-Cartesian square of fields, where the vertical inclusions are finite.
It follows from [80, § 7, Prop. 2.11] that the induced square of Quillen
$K$-theory, where the vertical arrows are finite push-forward and horizontal
arrows are flat pull-back, commutes. This concludes the proof. ∎

######  Remark 5.4.

The properness assumption in Proposition 5.3 can be removed. In order to do
this, one needs to consider one more case while defining $f_{*}$ on
generators. Without properness, it may happen that we have a dominant morphism
of curves $f\colon C^{\prime}\to C$ and a point
$x^{\prime}\in\Delta(C^{\prime}_{n})$ such that
$f(x^{\prime})=x\notin\Delta(C_{n})$. In this case, we define $f_{*}\colon
K^{M}_{2}(k(C^{\prime}_{n})_{x^{\prime}})\to K^{M}_{1}(k(x))$ as the
composition of the boundary and the norm maps. With this modification, one can
show following the above proof that $f_{*}$ descends to the level of the idele
class groups. We leave it as an exercise.

######  Remark 5.5.

We do not know if the norm map between Milnor $K_{2}$ of discrete valuation
fields is continuous with respect to Kato’s topology. If it were, it would
follow that the map $f_{*}$ in Proposition 5.3 is continuous.

###  5.3. Relation between $C^{\rm t}(X)$ and $C^{{\rm t},W}(X)$

We now compare the two notions of idele class groups with modulus and the tame
class groups we defined above. Let $j\colon X\hookrightarrow\overline{X}$ be
an open immersion into a complete integral normal $k$-scheme such that the
complementary closed subset is the support of an effective Cartier divisor
$D$. It is easy to check that the identify map of $F^{0}_{1}(X)$ induces
canonical continuous homomorphisms

(5.5) $\delta_{\overline{X}|D}\colon SK_{1}(\overline{X}|D)\to
C(\overline{X},D);\ \ \ \delta_{X}\colon C^{\rm t}(X)\to C^{{\rm t},W}(X).$

######  Lemma 5.6.

The maps $\delta_{\overline{X}|D}$ and $\delta_{X}$ are isomorphisms of
abelian groups. These are isomorphisms of topological abelian groups if ${\rm
char}(k)=0$.

###### Proof.

We shall prove that $\delta_{X}$ is an isomorphism as the proof for
$\delta_{\overline{X}|D}$ is identical modulo some obvious modifications. We
first show that $\delta_{X}$ is injective. Suppose that
$C_{1},\ldots,C_{r}\in{\mathcal{C}}(X)$ are distinct curves and $\alpha_{i}\in
K^{M}_{2}(k(C_{i}))$ such that
$\sum_{i}\iota_{C_{i}}(\alpha_{i})+\sum_{i}\partial_{C_{i}}(\alpha_{i})=\beta\in
F^{0}_{1}(X)$. Then we must have $\iota_{C_{i}}(\alpha_{i})\in
U^{\prime}_{1}K^{M}_{2}(k((C_{i})_{n})_{\infty})\subset
K^{M}_{2}(k(C_{i})_{\infty})$ for every $i$. Using Lemma 4.4 and Corollary
4.3, this implies that $\alpha_{i}\in{\rm
fil}_{\Delta((C_{i})_{n})}K^{M}_{2}(k(C_{i}))$ for every $i$. It follows from
this that $\alpha:=\sum_{i}\alpha_{i}\in F^{1}_{2}(X)$ and
$\partial^{1}_{X}(\alpha)=\beta$. We have thus shown that $\delta_{X}$ is
injective.

To show that $\delta_{X}$ is surjective, it suffices to show that the natural
map
$K^{M}_{2}(k(C_{n}))\to\frac{K^{M}_{2}(k(C_{n})_{\infty})}{U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})}$
is surjective for all $C\in{\mathcal{C}}(X)$. But this is a direct consequence
of the weak approximation theorem for Milnor $K$-theory [58, Lem. 6.3] once we
note that for all $x\in\Delta(C_{n})$, the subgroup
$U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{x})\subset K^{M}_{2}(k(C_{n})_{x})$ is open
in latter’s canonical topology.

Finally to show that $\delta_{X}$ is an isomorphism of topological abelian
groups if ${\rm char}(k)=0$, it suffices to show that $\delta_{X}$ is open in
this case. To show the latter, note that the source and the target of
$\delta_{X}$ have quotient topologies. We now recall an easy to prove general
fact that if $f\colon G\twoheadrightarrow H$ is a surjective homomorphism of
topological groups (with $H$ having the quotient topology), then $f$ is an
open map. Using this, it suffices to show that the inclusion of $F^{0}_{1}(X)$
in $I^{{\rm t},W}(X)$ is an open mapping. But this is clear because the other
summand of $I^{{\rm t},W}(X)$ has discrete topology. ∎

######  Remark 5.7.

Recall that in the definition of the topology on $C^{{\rm t},W}(X)$, we have
used Kato’s topology on each $K^{M}_{2}(k(C_{n})_{x})$ (where
$x\in\Delta(C_{n})$). However, the latter group also has another topology,
namely, the canonical topology which is generated by the open subgroups
$\\{U^{\prime}_{m}K^{M}_{2}(k(C_{n})_{x})\\}_{m\geq 0}$. Using the direct sum
of the canonical topologies of $K^{M}_{2}(k(C_{n})_{x})$, we can endow
$C^{{\rm t},W}(X)$ with another topology. Let us call it the ‘canonical
topology’. We can define the canonical topology on $C(\overline{X},D)$ in a
similar manner. The proof of Lemma 5.6 then shows that $\delta_{X}$ is an
isomorphism of topological abelian groups in any characteristic if we endow
$C^{{\rm t},W}(X)$ with its canonical topology. The same is true for
$\delta_{\overline{X}|D}$ too.

###  5.4. Relation between $C^{\rm t}(X)$ and $C(X)$

The following result provides a comparison between the idele and tame class
groups of normal schemes. This is the analogue of the first isomorphism of
Corollary 3.11 for class groups.

######  Lemma 5.8.

Let $X\in{\operatorname{\mathbf{Sch}}}_{k}$ be an integral normal scheme. Let
$m\in k^{\times}$ be a positive integer. Then the canonical surjection
$\kappa_{X}\colon C(X)\twoheadrightarrow C^{{\rm t},W}(X)$ induces an
isomorphism of topological abelian groups

$\kappa_{X}\colon{C(X)}/m\xrightarrow{\cong}{C^{{\rm t},W}(X)}/m.$

In particular, there is a canonical isomorphism of abelian groups
${C(X)}/m\xrightarrow{\cong}{C^{\rm t}(X)}/m$.

###### Proof.

In view of Lemma 5.6, we only need to prove the first part of the lemma. Now,
we note that ${C^{{\rm t},W}(X)}/m$ is a topological quotient of ${C(X)}/m$.
This implies that $\kappa_{X}$ is a continuous open map (see the proof of
Lemma 5.6). Hence, $\kappa_{X}$ will be an isomorphism of topological abelian
groups with ${{\mathbb{Z}}}/m$-coefficients if it is bijective.

Now, we see from Definition 5.2 that there is an exact sequence

${\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})\to
C(X)\to C^{{\rm t},W}(X)\to 0.$

We can therefore apply Lemma 5.9 to conclude the proof. ∎

######  Lemma 5.9.

Let $A$ be a Henselian local Noetherian ring containing an infinite field and
let ${\mathfrak{m}}$ be the maximal ideal of $A$. Let $I\subset{\mathfrak{m}}$
be an ${\mathfrak{m}}$-primary ideal and let $m$ be an integer prime to the
residue characteristic of $A$. Then ${K^{M}_{n}(A,I)}/m=0$ for all integers
$n\geq 0$.

###### Proof.

We can assume $n\geq 1$ because the lemma is trivial otherwise. We have a
short exact sequence

(5.6) $0\to K^{M}_{n}(A,I)\to K^{M}_{n}(A,{\mathfrak{m}})\to
K^{M}_{n}(A/I,{\mathfrak{m}}/I)\to 0.$

Since ${\mathfrak{m}}=\sqrt{I}$, it follows from [40, Lem. 2.6] that
$K^{M}_{n}(A/I,{\mathfrak{m}}/I)$ is uniquely $m$-divisible if the residue
characteristic of $A$ is positive. The same also holds if the residue
characteristic of $A$ is zero as a consequence of [36, Theorem 2.12]. Indeed,
since $A/I$ is ${\mathfrak{m}}$-adic complete local ring with residue field
$A/{\mathfrak{m}}$, the quotient map $A/I\to A/{\mathfrak{m}}$ admits a
splitting by Cohen structure theorem. Since a local ring containing an
infinite field is weakly $n$-stable for all $n\geq 1$, we can now apply [36,
Theorem 2.12] to conclude that $K^{M}_{n}(A/I,{\mathfrak{m}}/I)$ is uniquely
$m$-divisible. We can therefore assume that $I={\mathfrak{m}}$.

Now, there is a surjection
$K^{M}_{1}(A,{\mathfrak{m}})\otimes(A^{\times}\otimes\cdots\otimes
A^{\times})\twoheadrightarrow K^{M}_{n}(A,{\mathfrak{m}})$ by [55, Lem.
1.3.1], where the tensor product of $A^{\times}$ is taken $(n-1)$ times. It
suffices therefore to show that ${K^{M}_{1}(A,{\mathfrak{m}})}/m=0$. Since
$K^{M}_{1}(A,{\mathfrak{m}})\cong(1+{\mathfrak{m}})\subset A^{\times}$, the
desired statement follows easily from Hensel’s lemma, which is applicable
since $m\in A^{\times}$. We skip its proof. ∎

######  Corollary 5.10.

Let $\overline{X}\in{\operatorname{\mathbf{Sch}}}_{k}$ be a complete and
integral normal scheme. Let $D\subset\overline{X}$ be an effective Cartier
divisor with complement $X$. Let $m\in k^{\times}$ be a positive integer. Then
the canonical surjection $C(X)\twoheadrightarrow C(\overline{X},D)$ induces an
isomorphism of topological abelian groups

${C(X)}/m\xrightarrow{\cong}{C(\overline{X},D)}/m.$

In particular, there is a canonical isomorphism of abelian groups
${C(X)}/m\xrightarrow{\cong}{SK_{1}(\overline{X}|D)}/m$.

###### Proof.

Combine (5.1), Lemma 5.6 and Lemma 5.8. ∎

Let $j\colon X\hookrightarrow\overline{X}$ be an open immersion of integral
normal $k$-schemes such that $\overline{X}$ is complete. Let
$Z=\overline{X}\setminus X$ with the reduced closed subscheme structure. We
have the canonical map $\vartheta_{\overline{X}}\colon I^{{\rm t},W}(X)\to
I(\overline{X})$ which is identity on $F^{0}_{1}(X)$ and the boundary map
$K^{M}_{2}(k(C_{n})_{\infty})\to F^{0}_{1}(Z)$ (which kills
$U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{\infty})$) for $C\in{\mathcal{C}}(X)$. It
is easy to check that the diagram

(5.7)
$\textstyle{F^{1,W}_{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{C}\partial_{C}}$$\textstyle{F^{0}_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F^{1,W}_{2}(\overline{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sum_{C}\partial_{C}}$$\textstyle{F^{0}_{1}(\overline{X})}$

is commutative. This shows that $\vartheta_{\overline{X}}$ descends to a
homomorphism $\vartheta_{\overline{X}}\colon C^{{\rm t},W}(X)\to
C(\overline{X})=SK_{1}(\overline{X})$. The composition $C^{\rm t}(X)\to
C^{{\rm t},W}(X)\to SK_{1}(\overline{X})$ coincides with the one in (4.4).

###  5.5. Relation between $C^{\rm t}(X)$ and $C(\overline{X})$

We shall now prove a theorem relating $C^{\rm t}(X)$ with $C(\overline{X})$
when $\overline{X}$ is a smooth projective compactification of $X$. Before we
do this, we shall give a more refined presentation of $C(\overline{X})$ than
the one given in its definition. We shall use Bloch’s higher Chow groups for
this purpose. In this subsection, we let $k$ be any field and do not assume it
to be local until Theorem 5.13. We note here that Definition 4.7 of $C^{\rm
t}(X)$ makes sense over any field.

We let $\overline{X}$ be an integral smooth projective $k$-scheme of dimension
$d\geq 1$ and let $j\colon X\hookrightarrow\overline{X}$ be an open immersion
with the complementary closed subset $Z$ (having the reduced closed subscheme
structure). We let
$\square^{n}_{k}=({\mathbb{P}}^{1}_{k}\setminus\\{1\\})^{n}$ and
$\overline{\square}^{n}_{k}=({\mathbb{P}}^{1}_{k})^{n}$. For any closed
subscheme $Y\subset\overline{X}\times\overline{\square}^{n}_{k}$, we let
$g^{i}_{Y}\colon Y\to\overline{\square}^{1}_{k}$ be the projection to the
$i$-th factor of $\overline{\square}^{n}_{k}$ and let $f_{Y}\colon
Y\to\overline{X}$ be the projection to $\overline{X}$. We recall the following
from [63, § 1]. Let $z^{d+1}(\overline{X},\bullet)$ denote the cubical cycle
complex of $\overline{X}$ à la Bloch of codimension $(d+1)$-cycles. The
differential of this complex at level $n$ is given by
$\partial=\stackrel{{\scriptstyle
n}}{{\underset{i=1}{\sum}}}(-1)^{i}(\partial^{i}_{\infty}-\partial^{i}_{0})$.
The homology of this complex (modulo the subcomplex of degenerate cycles) at
level $n$ is the higher Chow group ${\rm CH}^{d+1}(\overline{X},n)$.

Let $(z^{d+1}_{Z}(\overline{X},\bullet),\partial_{Z})$ be the subcomplex of
$(z^{d+1}(\overline{X},\bullet),\partial)$ such that
$z^{d+1}_{Z}(\overline{X},n)$ is the free abelian group generated by
admissible integral cycles which intersect $Z\times F$ properly for each face
$F$ of $\square^{n}_{k}$. Since there can be no degenerate 0-cycles and
degenerate cycles are preserved under the boundary maps, one checks that the
homology of $z^{d+1}_{Z}(\overline{X},\bullet)$ (modulo the subcomplex of
degenerate cycles) at level one is the the cokernel of the map
$\partial_{Z}\colon z^{d+1}_{Z}(\overline{X},2)\to
z^{d+1}_{Z}(\overline{X},1)$, where $z^{d+1}_{Z}(\overline{X},1)$ is the free
abelian group generated by closed points of
$\overline{X}\times\square^{1}_{k}$ which neither lie in
$Z\times\square^{1}_{k}$ nor in any proper face of
$\overline{X}\times\square^{1}_{k}$ and $z^{d+1}_{Z}(\overline{X},2)$ is the
free abelian group generated by integral curves in
$\overline{X}\times\square^{2}_{k}$ which intersect all faces of
$X\times\square^{2}_{k}$ and $Z\times\square^{2}_{k}$ properly. In particular,
they do not meet any proper face of $Z\times\square^{2}_{k}$. By Bloch’s
moving lemma [11] (see [63, Thm. 1.10] for cubical version), the canonical map

(5.8)
$\frac{z^{d+1}_{Z}(\overline{X},1)}{\partial_{Z}(z^{d+1}_{Z}(\overline{X},2))}\to{\rm
CH}^{d+1}(\overline{X},1)$

is an isomorphism.

Recall from § 4.2 that $C(\overline{X})$ is the cokernel of the map
$\partial^{1}_{\overline{X}}\colon
F^{1}_{2}(\overline{X})={\underset{C\in{\mathcal{C}}(\overline{X})}{\bigoplus}}K^{M}_{2}(k(C))\to
F^{0}_{1}(\overline{X})$. It follows from the Gersten complex for the Milnor
$K$-theory of regular local rings (e.g., see Lemma 4.2) that for any
$C\in{\mathcal{C}}({X})$, one has
$\partial^{1}_{\overline{C}_{n}}(K^{M}_{2}({\mathcal{O}}_{\overline{C}_{n},\Delta(C_{n})}))\subset
F^{0}_{1}(C_{n})$. In particular,
$\partial^{1}_{\overline{X}}(K^{M}_{2}({\mathcal{O}}_{\overline{C}_{n},\Delta(C_{n})}))\subset
F^{0}_{1}(X)$. We write
$K^{M}_{2}({\mathcal{O}}_{\overline{C}_{n},\Delta(C_{n})})$ as ${\rm fil}_{\rm
nr}K^{M}_{2}(k(C))$ for $C\in{\mathcal{C}}({X})$. We let $F^{1,\rm
nr}_{2}({X})={\underset{C\in{\mathcal{C}}({X})}{\bigoplus}}{\rm fil}_{\rm
nr}K^{M}_{2}(k(C))$ and let $C^{\rm nr}({X})$ be the cokernel of the map
$\partial^{1}_{X}\colon F^{1,\rm nr}_{2}({X})\to F^{0}_{1}(X)$. Our key result
is the following.

######  Lemma 5.11.

The canonical map $C^{\rm nr}({X})\to C(\overline{X})$ is an isomorphism.

###### Proof.

For any closed point $x\in z^{d+1}(\overline{X},1)$, we let $y$ be the image
of $x$ in $\overline{X}$ and let
$\Psi_{1}([x])=N_{{k(x)}/{k(y)}}(g^{1}_{x})\in K^{M}_{1}(k(y))\subset
F^{0}_{1}(\overline{X})$. For an integral curve $Y\in
z^{d+1}(\overline{X},2)$, we let $\Psi_{2}([Y])=0$ if $f_{Y}\colon
Y\to\overline{X}$ is constant. If the scheme-theoretic image $f_{Y}(Y)$ is an
integral curve $\overline{Y^{\prime}}$, we let
$Y^{\prime}=\overline{Y^{\prime}}\cap X$. In this case, we let
$\Psi_{2}([Y])=N_{{k(Y)}/{k(Y^{\prime})}}(\\{g^{1}_{Y},g^{2}_{Y}\\})\in
K^{M}_{2}(k(Y^{\prime}))\subset F^{1}_{2}(\overline{X})$. By [67, Thm. 2.5],
$\Psi_{1}$ and $\Psi_{2}$ define a morphism of complexes

(5.9)
$\Psi\colon\left(z^{d+1}(\overline{X},2)\xrightarrow{\partial}z^{d+1}(\overline{X},1)\right)\to\left(F^{1}_{2}(\overline{X})\xrightarrow{\partial^{1}_{\overline{X}}}F^{0}_{1}(\overline{X})\right).$

such that the induced map $\Psi\colon{\rm CH}^{d+1}(\overline{X},1)={\rm
Coker}(\partial)\to{\rm Coker}(\partial^{1}_{\overline{X}})=C(\overline{X})$
is an isomorphism.

Since a 0-cycle in $z^{d+1}_{Z}(\overline{X},1)$ is supported in
$X\times\square^{1}_{k}$, we see that
$\Psi_{1}(z^{d+1}_{Z}(\overline{X},1))\subset F^{0}_{1}(X)$. If $Y$ is an
integral 1-cycle in $z^{d+1}_{Z}(\overline{X},2)$, then $Y$ does not lie in
$Z\times\square^{2}_{k}$ and does not meet any proper face of
$X\times\square^{2}_{k}$ inside $Z\times\square^{2}_{k}$. This implies that
the image of $\\{g^{1}_{Y},g^{2}_{Y}\\}$ dies in $K^{M}_{1}(k(y))$ for every
$y\in\nu^{-1}(Z)$, where $\overline{Y}$ is the closure of $Y$ in
$\overline{X}\times\overline{\square}^{2}_{k}$ and
$\nu\colon\overline{Y}_{n}\to\overline{X}\times\overline{\square}^{2}_{k}\to\overline{X}$
is the composite (finite) map. It follows from [39, Lem. 3.4] that the image
of $\Psi_{2}([Y])=N_{{k(Y)}/{k(Y^{\prime})}}(\\{g^{1}_{Y},g^{2}_{Y}\\})$ dies
in $K^{M}_{1}(k(y^{\prime}))$ for every $y^{\prime}\in\Delta(Y^{\prime}_{n})$.
An application of Lemma 4.2 shows that $\Psi_{2}([Y])\in{\rm fil}_{\rm
nr}K^{M}_{2}(k(Y^{\prime}))$. We conclude that
$\Psi_{2}(z^{d+1}_{Z}(\overline{X},2))\subset F^{1,\rm nr}_{2}({X})$. It
follows that (5.9) induces a commutative diagram of complexes

(5.10)
$\textstyle{\left(z^{d+1}_{Z}(\overline{X},2)\xrightarrow{\partial_{Z}}z^{d+1}_{Z}(\overline{X},1)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\left(F^{1,\rm
nr}_{2}({X})\xrightarrow{\partial^{1}_{X}}F^{0}_{1}({X})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(z^{d+1}(\overline{X},2)\xrightarrow{\partial}z^{d+1}(\overline{X},1)\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{\left(F^{1}_{2}(\overline{X})\xrightarrow{\partial^{1}_{\overline{X}}}F^{0}_{1}(\overline{X})\right),}$

where the vertical arrows are the canonical inclusions.

Taking the homology, we get a commutative diagram

(5.11) $\textstyle{{\rm
Coker}(\partial_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{\rm
Coker}(\partial^{1}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Coker}(\partial)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{\rm
Coker}(\partial^{1}_{\overline{X}}).}$

The left vertical arrow in (5.11) is an isomorphism by (5.8) and the bottom
horizontal arrow is an isomorphism by (5.9). It follows that the top
horizontal arrow is injective. To prove that this arrow is surjective, we fix
$x\in X_{(0)}$. Taking the graph of a rational function defines a homomorphism
$\Gamma_{x}\colon K^{M}_{1}(k(x))\to{\rm CH}^{1}(k(x),1)$ (see [102, § 2]).
Bloch’s push-forward map on the cycle complexes corresponding to inclusion
${\rm Spec\,}(k(x))\hookrightarrow\overline{X}$ induces a morphism
$z^{1}(k(x),\bullet)\to z^{d+1}(\overline{X},\bullet)$ which clearly factors
through $z^{d+1}_{Z}(\overline{X},\bullet)$. We thus have homomorphisms
$K^{M}_{1}(k(x))\xrightarrow{\cong}{\rm CH}^{1}(k(x),1)\to{\rm
Coker}(\partial_{Z})\xrightarrow{\Psi}{\rm Coker}(\partial^{1}_{X})$ whose
composition is identity on $K^{M}_{1}(k(x))$. This shows that the top
horizontal arrow in (5.11) is surjective. We have thus shown that all arrows
in (5.11) are isomorphisms. This concludes the proof. ∎

######  Corollary 5.12.

There is an exact sequence

${\underset{C\in{\mathcal{C}}(X),x\in\Delta(C_{n})}{\bigoplus}}{K^{M}_{2}(k(x))}\to{C^{\rm
t}(X)}\to{C(\overline{X})}\to 0.$

###### Proof.

This is a direct consequence of Definition 4.7 and Lemma 5.11. ∎

We can now prove the following analogue of the second isomorphism of Corollary
3.11 for the class groups.

######  Theorem 5.13.

Let $k$ be a local field of characteristic $p>0$. Let $\overline{X}$ be an
integral smooth projective $k$-scheme of dimension $d\geq 1$ and let $j\colon
X\hookrightarrow\overline{X}$ be an open immersion. Then
$\vartheta_{\overline{X}}\colon C^{\rm t}(X)\to C(\overline{X})$ is
surjective. The map

$\vartheta_{\overline{X}}\colon{C^{\rm
t}(X)}/{p^{n}}\to{C(\overline{X})}/{p^{n}}$

is an isomorphism of topological abelian groups for every integer $n\geq 1$.

###### Proof.

The first part of the theorem follows directly from Corollary 5.12. We now
show the second part. In view of Proposition 4.13, we only have to show the
group isomorphism. By Corollary 5.12, we have an exact sequence

${\underset{C\in{\mathcal{C}}(X),x\in\Delta(C_{n})}{\bigoplus}}{K^{M}_{2}(k(x))}/{p^{n}}\to{C^{\rm
t}(X)}/{p^{n}}\to{C(\overline{X})}/{p^{n}}\to 0.$

It suffices therefore to show that ${K^{M}_{2}(k(x))}/{p^{n}}=0$ if $x$ is a
closed point of a curve over $k$. To this end, we use the main result of [73],
which says that $K^{M}_{2}(k(x))$ is the direct sum of a (uniquely) divisible
group and the group of roots of unity in $k(x)$. Since the latter group is
cyclic of order prime to $p$ (e.g., see [79, Prop. II.5.7]), we are done. ∎

Scholium: For $m\geq 0$, we let $C^{\rm t}_{m}(X)$ be the cokernel of the
residue map

$\partial^{m}_{X}\colon{\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}{\rm
fil}_{\Delta(C_{n})}K^{M}_{m+1}(k(C))\to{\underset{x\in
X_{(0)}}{\bigoplus}}K^{M}_{m}(k(x)).$

This is the generalized version of the tame class group (see [111, Defn.
1.1]). Note that $C^{\rm t}_{1}(X)=C^{\rm t}(X)$ and $C^{\rm t}_{0}(X)$ is
canonically isomorphic to the Suslin homology $H^{S}_{0}(X)$, introduced by
Suslin-Voevodsky [96] (see [88, Thm. 5.1] for this isomorphism). Then the
proof of Lemma 5.11 actually shows the following. The only modifications one
needs to make are:

  1. (1)

Use [2, Thm. 1.1] to conclude that the lower horizontal arrow in (5.11) is an
isomorphism for any $m\geq 0$.

  2. (2)

Use Lemma 4.6 instead of Lemma 4.2 in the second paragraph and note that every
regular curve over a perfect field is smooth.

######  Theorem 5.14.

Let $m\geq 0$ be an integer. Then one has an exact sequence

${\underset{C\in{\mathcal{C}}(X),x\in\Delta(C_{n})}{\bigoplus}}{K^{M}_{m+1}(k(x))}\to{C^{\rm
t}_{m}(X)}\to{\rm CH}^{d+m}(\overline{X},m)\to 0$

for $m=0,1$. This is exact for every value of $m$ if $k$ is perfect.

######  Corollary 5.15.

Let $k$ be any perfect field of characteristic $p>0$. Then the canonical map

${H^{S}_{0}(X)}/{p^{n}}\to{{\rm CH}^{d}(\overline{X})}/{p^{n}}$

is an isomorphism

###### Proof.

Use Theorem 5.14 and the fact that the Milnor $K$-theory of a perfect field of
characteristic $p>0$ is $p^{n}$-divisible for every $n\geq 1$. ∎

######  Remark 5.16.

If $k$ is a finite field, one can easily show using the diagram (13.11) (with
$(2d+1,d+1)$ replaced by $(2d,d)$), Beilinson-Lichtenbaum conjecture (now a
theorem) and the theorem of Kerz-Saito on Kato’s conjecture (see [59]) that
${H^{S}_{0}(X)}/{m}\xrightarrow{\cong}{\pi^{\rm ab,{\rm t}}_{1}(X)}/{m}$ for
every $m\in k^{\times}$. In combination with Corollary 5.15 and the class
field theory of Kato-Saito [54], this gives a very short proof of the finite
coefficient version of the class field theory of Schmidt-Spiess [89].

###  5.6. Norm zero Idele class groups

Let $k$ be a local field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ an
integral normal scheme. To define the norm map on the idele class group with
modulus, we let $\overline{X}$ be a complete and integral normal $k$-scheme
with an effective Cartier divisor $D$ whose complement is $X$.

For every $x\in X_{(0)}$, there is a continuous norm map $N_{{k(x)}/k}\colon
K^{M}_{1}(k(x))\to K^{M}_{1}(k)$. Taking the sum over all closed points of
$X$, we get a homomorphism $N_{X}\colon F^{0}_{1}(X)\to k^{\times}$, which is
continuous by Lemma 3.1. If $C\in{\mathcal{C}}(X)$ and $x\in\Delta(C_{n})$, we
have continuous homomorphisms
$K^{M}_{2}(k(C_{n})_{x})\xrightarrow{\partial_{C_{n},x}}K^{M}_{1}(k(x))\xrightarrow{N_{{k(x)}/k}}K^{M}_{1}(k)$.
Note that the continuity of the residue map $\partial_{C_{n},x}$ strongly uses
Kato’s topology on $K^{M}_{2}(k(C_{n})_{x})$. Taking the direct sum, we get a
continuous homomorphism $N_{X}\colon I(X)\to k^{\times}$.

If $C\in{\mathcal{C}}(X)$, then the composite map
$K^{M}_{2}(k(C_{n}))\xrightarrow{(\iota_{C},\partial_{C})}I(X)\xrightarrow{N_{X}}k^{\times}$
is the same as the composite map
$K^{M}_{2}(k(C_{n}))\xrightarrow{\partial_{\overline{C}_{n}}}{\underset{x\in(\overline{C}_{n})_{(0)}}{\bigoplus}}K^{M}_{1}(k(x))\xrightarrow{N_{\overline{C}_{n}}}K^{M}_{1}(k)$.
But this composite map is zero by Suslin’s reciprocity law for Milnor
$K$-theory [94]. It follows that there is a continuous homomorphism
$N_{X}\colon C(X)\to k^{\times}$. Since the map $K^{M}_{2}(k(C_{n})_{x})\to
K^{M}_{1}(k)$ factors through the residue map $\partial_{C_{n},x}$ which is
trivial on $U^{\prime}_{1}K^{M}_{2}(k(C_{n})_{x})$, it follows that $N_{X}$
actually factors through $C^{{\rm t},W}(X)$. Hence, we get continuous norm
homomorphisms

(5.12) $C(X)\xrightarrow{N_{X}}k^{\times};\ \
C(\overline{X},D)\xrightarrow{N_{\overline{X}|D}}k^{\times};\ \ C^{\rm
t}(X)\xrightarrow{\cong}C^{{\rm t},W}(X)\xrightarrow{N_{X}}k^{\times}.$

We let $C(X)_{0}\subset C(X),\ C(\overline{X},D)_{0}\subset
C(\overline{X},D),\ C^{\rm t}(X)_{0}\subset C^{\rm t}(X)$ and $C^{{\rm
t},W}(X)_{0}\subset C^{{\rm t},W}(X)$ be the kernels of the above norm maps.
We shall call them the norm zero subgroups. Note that these subgroups are
closed in their ambient class groups because the canonical topology of
$k^{\times}$ is Hausdorff.

##  6\. Idele class groups via cycles with modulus

In this section, we shall present a relation of the idele class group with
modulus and the tame class group with the higher Chow groups with modulus.
This will provide a cycle theoretic description of these class groups. As a
byproduct, we shall prove an analogue of Bloch’s formula for higher Chow
groups with modulus for curves. We fix an arbitrary infinite field $k$
throughout this section.

###  6.1. Higher Chow group with modulus and class groups

Let $\overline{\square}^{1}_{k}={\rm Proj}(k[Y_{0},Y_{1}])$ and let
${\mathbb{A}}^{n}_{k}\cong{\rm Spec\,}(k[y_{1},\ldots,y_{n}])$ be the open
subset of $\overline{\square}_{k}^{n}$, where $(y_{1},\ldots,y_{n})$ denotes
the coordinate system of $\overline{\square}_{k}^{n}$ with
$y_{j}={Y^{j}_{1}}/{Y^{j}_{0}}$. Given
$Y\in{\operatorname{\mathbf{Sch}}}_{k}$, a rational map $f\colon
Y\dashedrightarrow\overline{\square}_{k}^{n}$ in
${\operatorname{\mathbf{Sch}}}_{k}$ and a point $y\in Y$ in the domain of
definition of $f$, we shall let $f_{i}(y)=(y_{i}\circ f)(y)$, where
$y_{i}\colon\overline{\square}_{k}^{n}\to\overline{\square}^{1}_{k}$ is the
$i$-th projection. For any $1\leq i\leq n$ and
$t\in\overline{\square}^{1}_{k}(k)$, we let $F^{t}_{n,i}$ denote the closed
subscheme of $\overline{\square}_{k}^{n}$ given by $\\{y_{i}=t\\}$. We let
$F^{t}_{n}=\stackrel{{\scriptstyle n}}{{\underset{i=1}{\sum}}}\ F^{t}_{n,i}$.

Let $\overline{X}$ be an integral normal $k$-scheme of dimension $d$ and
$D\subset\overline{X}$ an effective Cartier divisor with complement $X$. We
let $D_{0}=D_{\rm red}$ and $D_{+}=D+D_{0}$. Recall that the cycle complex
with modulus $z^{q}(\overline{X}|D,\bullet)$ is the subcomplex of Bloch’s
cycle complex $z^{q}(\overline{X},\bullet)$, where each integral cycle $Z\in
z^{q}(\overline{X},n)$ is required to additionally satisfy the following
(modulus) condition. If $\overline{Z}$ is the closure of $Z$ in
$\overline{X}\times\overline{\square}_{k}^{n}$ and
$\nu:\overline{Z}_{n}\to\overline{X}\times\overline{\square}_{k}^{n}$ is the
canonical map from the normalization of $\overline{Z}$, then the inequality

(6.1)
$\nu^{*}(D\times\overline{\square}_{k}^{n})\leq\nu^{*}(\overline{X}\times
F^{1}_{n})$

holds in the group of Weil divisors on $\overline{Z}_{n}$. The higher Chow
group with modulus ${\rm CH}^{q}(\overline{X}|D,i)$ is the $i$-th homology of
the complex $z^{q}(\overline{X}|D,\bullet)$ (modulo the subcomplex of
degenerate cycles). By definition, there is a canonical homomorphism ${\rm
CH}^{q}(\overline{X}|D,i)\to{\rm CH}^{q}(\overline{X},i)$. If $\overline{X}$
is complete of dimension $d$, there is a push-forward map ${\rm
CH}^{d+1}(\overline{X},1)\to{\rm CH}^{1}({\rm Spec\,}(k),1)\cong k^{\times}$.
Composing the two maps, we get $N_{\overline{X}|D}\colon{\rm
CH}^{d+1}(\overline{X}|D,1)\to k^{\times}$. We let ${\rm
CH}^{d+1}(\overline{X}|D,1)_{0}={\rm Ker}(N_{\overline{X}|D})$.

Our goal in this subsection is to prove the following result.

######  Theorem 6.1.

Assume that $\overline{X}$ is complete. Then we have the following.

  1. (1)

There is a canonical isomorphism of pro-abelian groups

${\operatorname{\rm cyc}}_{\overline{X}|D}\colon\\{{\rm
CH}^{d+1}(\overline{X}|nD,1)\\}_{n\geq
1}\xrightarrow{\cong}\\{SK_{1}(\overline{X}|nD)\\}_{n\geq 1}.$

  2. (2)

There is a canonical isomorphism of abelian groups

${\operatorname{\rm cyc}}_{\overline{X}|D}\colon{\rm
CH}^{d+1}(\overline{X}|D,1)\otimes
R\xrightarrow{\cong}SK_{1}(\overline{X}|D)\otimes R,$

where $R=[\tfrac{1}{p}]$ if ${\rm char}(k)=p>0$ and ${{\mathbb{Z}}}/m$ with
$m\neq 0$ if ${\rm char}(k)=0$.

We begin by constructing a cycle class map ${\operatorname{\rm
cyc}}_{\overline{X}|D}\colon{\rm CH}^{d+1}(\overline{X}|D_{+},1)\to
SK_{1}(\overline{X}|D)$. This does not require $\overline{X}$ to be complete.
The map $\Psi_{1}$ of (5.9) restricts a cycle class homomorphism
$\Psi_{1}\colon z^{d+1}(\overline{X}|D_{0},1)\to F^{0}_{1}(X)$.

######  Lemma 6.2.

The map $\Psi_{1}\colon z^{d+1}(\overline{X}|D_{0},1)\to F^{0}_{1}(X)$
descends to a homomorphism

${\operatorname{\rm cyc}}_{\overline{X}|D}\colon{\rm
CH}^{d+1}(\overline{X}|D_{+},1)\to SK_{1}(\overline{X}|D).$

###### Proof.

Using (5.9), it suffices to show that if $Z\in z^{d+1}(\overline{X}|D_{+},2)$
is an integral cycle, then $\Psi_{2}([Z])$ is zero if
$\pi_{\overline{X}}(\overline{Z})$ is a closed point and $\Psi_{2}([Z])\in{\rm
fil}_{\nu^{*}(D)}K^{M}_{2}(k(C))$ if
$\pi_{\overline{X}}(\overline{Z})=\overline{C}$ is an integral curve in
$\overline{X}$, where $\overline{Z}$ is the closure of $Z$ in
$\overline{\square}^{2}_{\overline{X}}$, $C=\overline{C}\cap X$,
$\pi_{\overline{X}}\colon\overline{\square}^{2}_{\overline{X}}\to\overline{X}$
is the projection and $\nu\colon\overline{C}_{n}\to\overline{X}$ is the
normalization morphism for $\overline{C}$.

To show the above, we fix an integral cycle $Z\in
z^{d+1}(\overline{X}|D_{+},2)$. If $\pi_{\overline{X}}(\overline{Z})$ is a
closed point, then the modulus condition implies that
$\overline{Z}\cap\overline{\square}^{2}_{D}=\emptyset$. In this case, we have
already seen in the proof of Lemma 5.11 that $\Psi_{2}([Z])=0$. Suppose now
that $\pi_{\overline{X}}(\overline{Z})$ is not a closed point. Then
$\pi_{\overline{X}}\colon\overline{Z}\to\overline{X}$ must be a finite
morphism. We let $\overline{C}\subset\overline{X}$ be the scheme theoretic
image of this finite morphism and let $C=\overline{C}\cap X$. Then the modulus
condition for $Z$ implies that $C\subset\overline{C}$ is dense open. As in §
5.5, let $g^{i}_{Z}\colon\overline{Z}\to\overline{\square}_{k}$ be the
projection to the $i$-th factor of $\overline{\square}^{2}$ and let
$g=\\{g^{1}_{Z},g^{2}_{Z}\\}\in K^{M}_{2}(k(Z))$. Note that neither
$g^{1}_{Z}$ nor $g^{2}_{Z}$ can be identically zero because $Z$ intersects the
faces of $\square^{2}_{\overline{X}}$ properly. Hence, $g$ is a well-defined
element.

We have a commutative diagram

(6.2)
$\textstyle{\overline{Z}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{\phi^{\prime}}$$\textstyle{\overline{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{C}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\overline{C}.}$

We let
$\widetilde{\nu}=\pi_{\overline{X}}\circ\nu^{\prime}\colon\overline{Z}_{n}\to\overline{X}$
and let $\widetilde{\nu}^{*}(D)=m_{1}[x_{1}]+\cdots+m_{r}[x_{r}]$. We set
$D_{Z}=(m_{1}+1)[x_{1}]+\cdots+(m_{r}+1)[x_{r}]$. We let
$R={\mathcal{O}}_{\overline{Z}_{n},D_{Z}}$ and let $F$ be the quotient field
of $R$. We let ${\mathfrak{m}}_{i}$ denote the maximal ideal of $R$
corresponding to the point $x_{i}$ for $1\leq i\leq r$. Then we have
$\widetilde{\nu}^{*}(D)\leq D_{Z}\leq\widetilde{\nu}^{*}(D_{+})$. Since $Z\in
z^{d+1}(\overline{X}|D_{+},2)$, these inequalities and the modulus condition
for $Z$ together imply that ${\rm ord}_{x_{i}}(g^{1}_{Z}-1)+{\rm
ord}_{x_{i}}(g^{2}_{Z}-1)\geq m_{i}+1$ for each $1\leq i\leq r$. By [37, Lem.
3.3] (with $R$ replaced by $R_{{\mathfrak{m}}_{i}}$), this implies that
$g\in(1+{\mathfrak{m}}^{m_{i}+1}_{i})K^{M}_{1}(F)\subset K^{M}_{1}(F)\cdot
K^{M}_{1}(F)\subset K^{M}_{2}(F)$ for every $1\leq i\leq r$. If we now apply
[41, Lem. 4.3] (with $A=R_{{\mathfrak{m}}_{i}}$), it follows that $g\in
U^{\prime}_{m_{i}}K^{M}_{2}(F)$ (see § 3.3) for every $1\leq i\leq r$. In
other words, $g\in\ \stackrel{{\scriptstyle
r}}{{\underset{i=1}{\bigcap}}}U^{\prime}_{m_{i}}K^{M}_{2}(F)$. Using Lemma
4.5, this implies that $g\in{\rm
fil}_{\widetilde{\nu}^{*}(D)}K^{M}_{2}(F)={\rm
fil}_{\widetilde{\nu}^{*}(D)}K^{M}_{2}(k(Z))$. We now conclude from [82, Lem.
6.19] and Lemma 4.5 that $\Psi_{2}([Z])=N_{{k(Z)}/{k(C)}}(g)\in{\rm
fil}_{\nu^{*}(D)}K^{M}_{2}(k(C))$. This finishes the proof of the lemma. ∎

To construct an inverse of ${\operatorname{\rm cyc}}_{\overline{X}|D}$, we let
$x\in X_{(0)}$. We have seen in the proof of Lemma 5.11 that the graph
function defines an isomorphism $K^{M}_{1}(k(x))\xrightarrow{\cong}{\rm
CH}^{1}(k(x),1)$. Since $x\in X_{(0)}$, we have a push-forward homomorphism
${\rm CH}^{1}(k(x),1)\to{\rm CH}^{d+1}(\overline{X}|D,1)$ (see [65, § 2]) and
it is clear that the composite $\phi_{x}\colon K^{M}_{1}(k(x))\to{\rm
CH}^{d+1}(\overline{X}|D,1)$ is simply the above graph homomorphism. Summing
over $(\phi_{x})_{x\in X_{(0)}}$, we get a homomorphism
$\phi_{\overline{X}|D}\colon F^{0}_{1}(X)\to{\rm CH}^{d+1}(\overline{X}|D,1)$.

######  Lemma 6.3.

If $\overline{X}$ is complete, then $\phi_{\overline{X}|D}$ induces a
homomorphism

$\phi_{\overline{X}|D}\colon SK_{1}(\overline{X}|D)\to{\rm
CH}^{d+1}(\overline{X}|D,1).$

###### Proof.

To show that $\phi_{\overline{X}|D}$ factors through $SK_{1}(\overline{X}|D)$,
we fix a curve $C\in{\mathcal{C}}(X)$ and let $\beta\in{\rm
fil}_{\nu^{*}(D)}K^{M}_{2}(k(C))$, where
$\nu\colon\overline{C}_{n}\to\overline{X}$ is the canonical map. Using the
push-forward map $\nu_{*}\colon{\rm
CH}^{2}(\overline{C}_{n}|\nu^{*}(D),1)\to{\rm CH}^{d+1}(\overline{X}|D,1)$, it
suffices to show that $\phi_{\overline{C}_{n}|\nu^{*}(D)}$ factors through
$SK_{1}(\overline{C}_{n}|\nu^{*}(D))$. We can therefore assume that
$\overline{X}$ is a complete integral regular curve over $k$. We let
$R={\mathcal{O}}_{\overline{X},D}$ and let $I\subset R$ be the ideal defining
the divisor $D$. By Lemma 4.5, we have ${\rm
fil}_{D}K^{M}_{2}(k(X))=K^{M}_{2}(R,I)$ and [38, Lem. 3.2] implies that the
latter group is generated by Steinberg symbols of the form $\\{1+a,b\\}$,
where $a\in I$ and $b\in R^{\times}$. We can therefore assume that
$\beta=\\{1+a,b\\}$ with $a\in I$ and $b\in R^{\times}$.

Let $\overline{Z}\subset\overline{\square}^{2}_{\overline{X}}$ be the closure
of the graph of the rational map
$\beta^{\prime}:=(1+a,b)=(f_{1},f_{2})\colon\overline{X}\dashedrightarrow\overline{\square}^{2}_{k}$.
Since $\overline{X}$ is a complete integral regular curve over $k$, the
rational function $f_{i}$ defines a regular morphism
$f_{i}\colon\overline{X}\to\overline{\square}^{1}_{k}$ for $i=1,2$.
Furthermore, $\overline{Z}$ is the graph
$\Gamma_{\beta^{\prime}}=\\{(x,f_{1}(x),f_{2}(x))|x\in\overline{X}\\}\subset\overline{\square}^{2}_{\overline{X}}$.
It follows that the projection map
$\pi_{\overline{X}}\colon\overline{Z}\to\overline{X}$ is an isomorphism. In
particular, $\overline{Z}$ is regular.

We let $Z=\overline{Z}\cap\square^{2}_{\overline{X}}$. It is easy to check
(e.g., see the proof of [67, Lem. 2.8]) that $Z\in z^{2}(\overline{X},2)$. Let
$A={\mathcal{O}}_{\overline{Z},E}$, where $E=\pi^{*}_{\overline{X}}(D)\cong
D=m_{1}[x_{1}]+\cdots+m_{r}[x_{r}]$, say. Since $a\in I$ and $b\in R^{\times}$
and the composition
$\overline{Z}\to\overline{\square}^{2}_{\overline{X}}\to\overline{X}$ induces
an isomorphism of semilocal rings $R\xrightarrow{\cong}A$, it follows that
${\rm ord}_{x_{i}}(f_{1}-1)\geq m_{i}$ and ${\rm ord}_{x_{i}}(f_{2}-1)\geq 0$,
and therefore, ${\rm ord}_{x_{i}}(f_{1}-1)+{\rm ord}_{x_{i}}(f_{2}-1)\geq
m_{i}$ for every $1\leq i\leq r$. In other words, the cycle $Z$ satisfies the
modulus condition. We conclude that $Z\in z^{2}(\overline{X}|D,2)$. It remains
to show that
$\partial([Z])=\phi_{\overline{X}|D}\circ\partial^{1}_{X}(\beta)$.

It is easy to see from the description of $\overline{Z}$ as the graph of the
morphism $(f_{1},f_{2})\colon\overline{X}\to\overline{\square}^{2}_{k}$ that

(6.3)
$\begin{array}[]{lll}\partial([Z])&:=&(\partial^{0}_{1}(Z)-\partial^{\infty}_{1}([Z]))+(\partial^{\infty}_{2}([Z])-\partial^{0}_{2}(Z))\\\
&=&([\Gamma_{f^{-1}_{1}(0)}]-[\Gamma_{f^{-1}_{1}(\infty)}])+([\Gamma_{f^{-1}_{2}(\infty)}]-[\Gamma_{f^{-1}_{2}(0)}])\in
z^{2}(\overline{X},1),\end{array}$

where
$\Gamma_{f^{-1}_{j}(t)}\subset\square^{1}_{X}\subset\overline{\square}^{1}_{\overline{X}}$
is the graph of the induced morphism $f_{i}\colon
f^{-1}_{j}(t)\to\overline{\square}^{1}_{k}$ from the scheme theoretic inverse
image of $\\{t\\}$ under the $j$-th projection with $i\neq j$ and
$t\in\\{0,\infty\\}$. We let $S^{i}_{t}$ be the support of the 0-cycle
$[f^{-1}_{i}(t)]\in z^{1}({X},0)$ for $i\in\\{1,2\\}$ and
$t\in\\{0,\infty\\}$. Since $Z$ intersects the faces of $\square^{2}_{X}$
properly, we can see that $S^{i}_{t}\cap S^{j}_{t^{\prime}}=\emptyset$
whenever $i\neq j$ or $t\neq t^{\prime}$.

On the other hand, recall that for any $i$, $t$ and any
$x\in\overline{X}_{(0)}$, one has that
$\partial_{x}(\beta)=\gamma_{x}((-1)^{v_{x}(f_{1})v_{x}(f_{2})}{(f_{1})^{v_{x}(f_{2})}}/{(f_{2})^{v_{x}(f_{1})}})$,
where $\partial_{x}\colon K^{M}_{2}(k(X))\to K^{M}_{1}(k(x))$ is the boundary
map of the Gersten complex for $K^{M}_{2}({\mathcal{O}}_{\overline{X},x})$ and
$\gamma_{x}\colon{\mathcal{O}}_{\overline{X},x}\twoheadrightarrow k(x)$ is the
quotient map. Note also that $\partial_{x}(\beta)=0$ if $x$ does not lie in
any $S^{i}_{t}$. Putting all of these together, we conclude that the image of
$\beta$ under the composite map
$K^{M}_{2}(k(X))\xrightarrow{\partial_{x}}K^{M}_{1}(k(x))\xrightarrow{\phi_{x}}{\rm
CH}^{2}(\overline{X}|D,1)$ is same as the $x$-component of the cycle class
$[\Gamma_{f^{-1}_{i}(t)}]$ when $x\in S^{i}_{t}$. Taking the sum over
$i\in\\{1,2\\}$ and $t\in\\{0,\infty\\}$, we conclude that
$\partial([Z])=\phi_{\overline{X}|D}\circ\partial^{1}_{X}(\beta)$. This
concludes the proof of the lemma. ∎

Proof of Theorem 6.1: It is easy to check that the composite map
$SK_{1}(\overline{X}|D_{+})\xrightarrow{\phi_{\overline{X}|D_{+}}}{\rm
CH}^{d+1}(\overline{X}|D_{+},1)\xrightarrow{{\operatorname{\rm
cyc}}_{\overline{X}|D}}SK_{1}(\overline{X}|D)$ is the canonical surjection
$SK_{1}(\overline{X}|D_{+})\twoheadrightarrow SK_{1}(\overline{X}|D)$. This
already shows that ${\operatorname{\rm
cyc}}_{\overline{X}|D}\circ\phi_{\overline{X}|D}$ is identity at the level of
pro-abelian groups. We now show that $\phi_{\overline{X}|D}$ is surjective,
which will complete the proof of part (1) of Theorem 6.1.

We let $[x]\in z^{d+1}(\overline{X}|D,1)$ be a 0-cycle represented by a closed
point $x$ and let $y=f(x)\in X$, where $f\colon{\rm
Spec\,}(k(x))\to\overline{X}$ is the restriction of the projection
$\overline{\square}^{1}_{\overline{X}}\to\overline{X}$. From the construction
of $\phi_{\overline{X}|D}$, we have a commutative diagram

(6.4)
$\textstyle{K^{M}_{1}(k(x))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{x}}$$\scriptstyle{f_{*}}$$\textstyle{{\rm
CH}^{1}(k(x),1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{SK_{1}(\overline{X}|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{\overline{X}|D}}$$\textstyle{{\rm
CH}^{d+1}(\overline{X}|D,1),}$

where the left vertical arrow is the composition
$K^{M}_{1}(k(x))\xrightarrow{N_{{k(x)}/{k(y)}}}K^{M}_{1}(k(y))\hookrightarrow
F^{0}_{1}(X)\twoheadrightarrow SK_{1}(\overline{X}|D)$. Since the sum of the
right vertical arrows ${\underset{x\in X_{(0)}}{\bigoplus}}{\rm
CH}^{1}(k(x),1)\to{\rm CH}^{d+1}(\overline{X}|D,1)$ is surjective as $x$ runs
through the closed points of ${\square}^{1}_{X}$ not meeting latter’s faces
and each $\phi_{x}$ is an isomorphism (e.g., see [102]), we conclude that
$\phi_{\overline{X}|D}$ is surjective.

We now prove part (2) of Theorem 6.1. After what we have shown above in the
proof of part (1), it suffices to show that the canonical surjection ${\rm
CH}^{d+1}(\overline{X}|D_{+},1)\otimes R\twoheadrightarrow{\rm
CH}^{d+1}(\overline{X}|D,1)\otimes R$ is an isomorphism when $R$ is as in part
(2). But this is already shown in [75, Thm. 1.3(2)]. ∎

Combining Lemma 4.10 and Theorem 6.1, we obtain the following cycle theoretic
interpretation of the tame class group of $X$ if it is not necessarily
complete or regular. We let ${\operatorname{\rm cyc}}^{\rm
t}_{\overline{X}|D}=\psi_{\overline{X}|D}\circ{\operatorname{\rm
cyc}}_{\overline{X}|D}$ (see (4.4)).

######  Corollary 6.4.

Let $R=[\tfrac{1}{p}]$ if ${\rm char}(k)=p>0$ and ${{\mathbb{Z}}}/m$ with
$m\neq 0$ if ${\rm char}(k)=0$. Then the cycle class map

${\operatorname{\rm cyc}}^{\rm t}_{\overline{X}|D}\colon{\rm
CH}^{d+1}(\overline{X}|D,1)\otimes R\to C^{\rm t}(X)\otimes R$

is an isomorphism. This isomorphism preserves the norm zero subgroups.

###  6.2. Bloch’s formula for higher Chow group with modulus on curves

As another consequence of Theorem 6.1 and the Thomason-Trobaugh descent
spectral sequence for relative $K$-theory, we obtain the following analogue of
Bloch’s formula for higher Chow groups of 0-cycles with modulus for curves, in
pro-setting. Note that no analogue of Bloch’s formula is yet known for higher
Chow groups of 0-cycles with modulus, even in dimension one.

For a Noetherian scheme $Z$ and a closed subscheme $W\subset Z$, we let
${\mathcal{K}}^{M}_{i,(Z,W)}={\rm
Ker}({\mathcal{K}}^{M}_{i,Z}\twoheadrightarrow{\mathcal{K}}^{M}_{i,W})$, where
${\mathcal{K}}^{M}_{i,Z}$ is the sheaf (in a given topology on $Z$) whose
stalks are the corresponding Milnor $K$-groups of the local rings of $Z$. We
let $K^{W}(Z)$ be the homotopy fiber of the pull-back morphism between the
(non-connective) spectra $K(Z)\to K(Z\setminus W)$ and let $K^{W}_{i}(Z)$ be
the $i$-th homotopy group of this homotopy fiber.

Assume that $X$ is regular of dimension $d$ and $x\in X_{(0)}$. We then have a
canonical isomorphism
$K^{M}_{n-d}(k(x))\xrightarrow{\cong}H^{d}_{x}(\overline{X},{\mathcal{K}}^{M}_{n,(\overline{X},D)})$
for every $n\geq 0$ (e.g., see [39, Lem. 3.7]), where the group on the right
is Zariski cohomology with support in $\\{x\\}$. By composing with the ‘forget
support’ map, we get $K^{M}_{n-d}(k(x))\to H^{d}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{n,(\overline{X},D)})$. We thus get a
homomorphism $z^{d+1}(\overline{X}|D,1)\xrightarrow{{\operatorname{\rm
cyc}}_{\overline{X}|D}}F^{0}_{1}(X)\to H^{d}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{d+1,(\overline{X},D)})$. We denote this
also by ${\operatorname{\rm cyc}}_{\overline{X}|D}$.

######  Proposition 6.5.

Assume that $\text{\rm dim}(X)=1$. Then the cycle class map
$z^{2}(\overline{X}|D,1)\to H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})$ induces an
isomorphism of pro-abelian groups

${\operatorname{\rm cyc}}_{\overline{X}|D}\colon\\{{\rm
CH}^{2}(\overline{X}|nD,1)\\}_{n\geq 1}\xrightarrow{\cong}\\{H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},nD)})\\}_{n\geq 1}.$

###### Proof.

By virtue of Theorem 6.1, we only need to show that there is a canonical
isomorphism

(6.5) $\gamma_{\overline{X}|D}\colon
SK_{1}(\overline{X}|D)\xrightarrow{\cong}H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})$

whose composition with the cycle class map $z^{2}(\overline{X}|D,1)\to
SK_{1}(\overline{X}|D)$ is the same as the one given in the statement of the
proposition. To this end, we first use the Thomason-Trobaugh localization
sequence for relative $K$-theory (see [101, Thm. 7.4]) to get an exact
sequence of relative $K$-groups

$K_{2}(\overline{X}\setminus S,D)\to K^{S}_{1}(\overline{X})\to
K_{1}(\overline{X},D)\to K_{1}(\overline{X}\setminus S,D)$

for every closed subset $S\subset\overline{X}$ contained in $X$.

Taking the filtered colimit over $S$ and using the continuity property of
$K$-theory, we get an exact sequence

(6.6) $K_{2}(A,I)\to F^{0}_{1}(X)\to K_{1}(\overline{X},D)\to K_{1}(A,I),$

where we let $A={\mathcal{O}}_{\overline{X},D}$ and $I={\rm
Ker}(A\twoheadrightarrow{\mathcal{O}}(D))$. One knows by the construction of
the localization sequence that the boundary map $K_{2}(A,I)\to F^{0}_{1}(X)$
coincides with the composition $K_{2}(A,I)\twoheadrightarrow
K^{M}_{2}(A,I)\xrightarrow{\partial^{1}_{X}}F^{0}_{1}(X)$, where the first
arrow is surjective by Lemma 4.1. We thus get an exact sequence

(6.7) $0\to SK_{1}(\overline{X}|D)\to K_{1}(\overline{X},D)\to K_{1}(A,I).$

On the other hand, the Thomason-Trobaugh descent spectral sequence (see [101,
Thm. 10.3]) gives rise to a short exact sequence

(6.8) $0\to H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})\to
K_{1}(\overline{X},D)\to H^{0}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{1,(\overline{X},D)})\to 0.$

Note that we used the isomorphisms
${\mathcal{K}}^{M}_{1,(\overline{X},D)}\cong{\mathcal{K}}_{1,(\overline{X},D)}$
and $H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}_{2,(\overline{X},D)})\xrightarrow{\cong}H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})$ that follow from
Lemma 4.1. Moreover, the functoriality of this spectral sequence gives rise to
a commutative diagram

$\textstyle{K_{1}(\overline{X},D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A,I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{H^{0}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{1,(\overline{X},D)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}_{\rm
zar}(Y,{\mathcal{K}}^{M}_{1,(Y,D)}),}$

where we let $Y={\rm Spec\,}(A)$.

It is an elementary verification that the right vertical arrow in the above
diagram is an isomorphism. Since the bottom horizontal arrow is injective, we
see that (6.7) is the same as the exact sequence

$0\to SK_{1}(\overline{X}|D)\to K_{1}(\overline{X},D)\to H^{0}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{1,(\overline{X},D)})\to 0.$

Comparing this with (6.8), we deduce (6.5). It is also clear from the proof
that the composition $z^{2}(\overline{X}|D,1)\to SK_{1}(\overline{X}|D)\to
H^{1}_{\rm zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})$ is the
same as the one stated in the proposition. ∎

######  Corollary 6.6.

If $\text{\rm dim}(X)=1$, then there is a canonical isomorphism

$C^{\rm t}(X)\xrightarrow{\cong}H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D_{0})}).$

###### Proof.

This follows from (6.5) once we observe that the map
$SK_{1}(\overline{X}|D_{0})\to C^{\rm t}(X)$ (see (4.4)) is an isomorphism if
$\text{\rm dim}(X)=1$. ∎

######  Corollary 6.7.

Assume that $\text{\rm dim}(X)=1$. Let $R=[\tfrac{1}{p}]$ if ${\rm
char}(k)=p>0$ and ${{\mathbb{Z}}}/m$ with $m\neq 0$ if ${\rm char}(k)=0$. Then
the cycle class map

${\operatorname{\rm cyc}}_{\overline{X}|D}\colon{\rm
CH}^{2}(\overline{X}|D,1)\otimes R\to H^{1}_{\rm
zar}(\overline{X},{\mathcal{K}}^{M}_{2,(\overline{X},D)})\otimes R$

is an isomorphism.

###### Proof.

Combine Theorem 6.1 and (6.5). ∎

##  7\. The reciprocity maps

In this section, we shall define the reciprocity homomorphisms from various
class groups of integral normal schemes over a local field to the
corresponding étale fundamental groups. We shall also establish some formal
properties of these homomorphisms. We shall use the following result from the
local class field theory as a key ingredient. For topological abelian groups
$G$ and $H$, we let ${\rm Hom}_{{\rm cf}}(G,H)={\rm
Hom}_{{\mathbf{Tab}}}(G,H)_{\rm tor}$.

######  Proposition 7.1.

Let $K$ be a complete discrete valuation field whose residue field
${\mathfrak{f}}$ is a local field. Then we have the following.

  1. (1)

There is a commutative diagram of continuous homomorphisms (called the
reciprocity homomorphisms)

(7.1)
$\textstyle{K^{M}_{2}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{K}}$$\scriptstyle{\partial}$$\textstyle{G_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{K^{M}_{1}({\mathfrak{f}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{{\mathfrak{f}}}}$$\textstyle{G_{{\mathfrak{f}}},}$

where the left vertical arrow is the residue map and the right vertical arrow
is the canonical map.

  2. (2)

The cup product pairing

(7.2) $\langle,\rangle\colon K^{M}_{2}(K)\otimes
H^{1}(K)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

induces isomorphisms

(7.3) $\rho^{\vee}_{K}\colon
H^{1}(K)\xrightarrow{\cong}\left\\{\begin{array}[]{ll}{\rm Hom}_{\rm
cf}(K^{M}_{2}(K),{{\mathbb{Q}}}/{{\mathbb{Z}}})&\mbox{if ${\rm char}(K)=0$}\\\
{\rm Hom}_{\mathbf{Tab}}(K^{M}_{2}(K),{{\mathbb{Q}}}/{{\mathbb{Z}}})&\mbox{if
${\rm char}(K)>0$}.\end{array}\right.$

  3. (3)

These isomorphisms induce the following.

$\rho^{\vee}_{K}\colon{\rm fil}_{0}H^{1}(K)\xrightarrow{\cong}{\rm Hom}_{\rm
cf}\left(\frac{K^{M}_{2}(K)}{U^{\prime}_{0}K^{M}_{2}(K)},{{\mathbb{Q}}}/{{\mathbb{Z}}}\right)\
\mbox{if}\ {\rm char}(K)=0.$ $\rho^{\vee}_{K}\colon{\rm
fil}_{m}H^{1}(K)\xrightarrow{\cong}{\rm
Hom}_{\mathbf{Tab}}\left(\frac{K^{M}_{2}(K)}{U^{\prime}_{m}K^{M}_{2}(K)},{{\mathbb{Q}}}/{{\mathbb{Z}}}\right)\
\forall\ m\geq 0\ \mbox{if}\ {\rm char}(K)>0.$

###### Proof.

The pairing (7.2) can be found in [52, § 3.2, p. 661]. The reciprocity map for
${\mathfrak{f}}$ is classical. The existence of $\rho_{K}$ follows from (7.2),
and its continuity is a direct consequence of [51, Thm. 2, p. 304] (see also
[52, § 3.5]). The commutativity of (7.1) is shown in [52, § 3.2, Lem. 6(2)].
The isomorphisms of (2) follow from Kato’s class field theory for
2-dimensional local fields (see [52, § 3.5 (ii), Rem. 3.5.4]). The first
isomorphism and the $m=0$ case of the second isomorphism in (3) follow by
combining the isomorphisms of (2) and diagram (7.1) with the classically known
fact that $\rho_{\mathfrak{f}}$ induces an isomorphism
$H^{1}({\mathfrak{f}})\xrightarrow{\rho^{\vee}_{\mathfrak{f}}}(K^{M}_{1}({\mathfrak{f}}))^{\vee}$
(e.g., see [85, Thm. 3.1]). The $m\geq 1$ case of the second isomorphism in
(3) follows from [70, Prop. 3.2.7] and [39, Thm. 6.3]. Recall that ${\rm
fil}_{m}H^{1}(K)=H^{1}(K)$ for $m\geq 1$ if ${\rm char}(K)=0$. ∎

###  7.1. Reciprocity for idele class group

We now let $k$ be a local field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ an
integral normal scheme with function field $K$. For $x\in X$, let
$\iota^{x}\colon{\rm Spec\,}(k(x))\to X$ be the inclusion. If $\text{\rm
dim}(X)=0$, Proposition 7.1 already provides the desired reciprocity
homomorphism. We therefore assume that $\text{\rm dim}(X)\geq 1$.

By taking sum of the composite maps
$K^{M}_{1}(k(x))\xrightarrow{\rho_{k(x)}}G_{k(x)}\to\pi^{\rm ab}_{1}(X)$ over
$x\in X_{(0)}$, we get a homomorphism $\rho^{1}_{X}\colon
F^{0}_{1}(X)\to\pi^{\rm ab}_{1}(X)$. For $C\in{\mathcal{C}}(X)$, Proposition
7.1 yields a continuous reciprocity homomorphism $\rho^{\eta}_{\infty}\colon
K^{M}_{2}(k(\eta)_{\infty})\to\pi^{\rm ab}_{1}({\rm
Spec\,}(k(\eta)_{\infty}))$, where $\eta\in C$ is the generic point. Composing
this with $\pi^{\rm ab}_{1}({\rm Spec\,}(k(\eta)_{\infty}))\to\pi^{\rm
ab}_{1}(k(\eta))\to\pi^{\rm ab}_{1}(X)$ and summing over
$C\in{\mathcal{C}}(X)$, we get a homomorphism
$\rho^{2}_{X}\colon{\underset{C\in{\mathcal{C}}(X)}{\bigoplus}}K^{M}_{2}(k(C_{n})_{\infty})\to\pi^{\rm
ab}_{1}(X)$. It follows from Lemma 3.1 that $\rho^{1}_{X}$ and $\rho^{2}_{X}$
are continuous. We let $\rho_{X}=(\rho^{2}_{X}+\rho^{1}_{X})\colon
I(X)\to\pi^{\rm ab}_{1}(X)$. This is clearly continuous. Our goal is to show
that $\rho_{X}$ factors through $C(X)$.

######  Lemma 7.2.

$\rho_{X}$ factors through a continuous homomorphism

$\rho_{X}\colon C(X)\to\pi^{\rm ab}_{1}(X).$

###### Proof.

The factorization will automatically imply continuity because $C(X)$ is a
topological quotient of $I(X)$. Now, it is clear from the definition of the
reciprocity homomorphism that the diagram

(7.4)
$\textstyle{I(C_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{C_{n}}}$$\scriptstyle{\nu_{*}}$$\textstyle{\pi^{\rm
ab}_{1}(k(C_{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{*}}$$\textstyle{I(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\pi^{\rm
ab}_{1}(X).}$

is commutative for every $C\in{\mathcal{C}}(X)$ if $\nu\colon C_{n}\to X$
denotes the canonical map. Using this, we are reduced to proving the lemma
when $\text{\rm dim}(X)=1$.

Let $\overline{X}$ be the unique normal compactification of $X$. We let
$I^{\prime}(\overline{X})={\underset{x\in\overline{X}_{(0)}}{\prod^{\prime}}}K^{M}_{2}(K_{x})$
be the restricted product with respect to the closed subgroups
$U^{\prime}_{0}K^{M}_{2}(K_{x})$. We endow
${\underset{x\in\overline{X}_{(0)}}{\prod}}K^{M}_{2}(K_{x})$ with the product
topology and $I^{\prime}(\overline{X})$ with the induced subspace topology. We
let
$I^{\prime}(X)={\underset{x\in{X}_{(0)}}{\prod^{\prime}}}K^{M}_{2}(K_{x})$. We
consider the diagram

(7.5)
$\textstyle{K^{M}_{2}(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\iota_{X},\partial_{X})}$$\scriptstyle{\alpha}$$\textstyle{I^{\prime}(\overline{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$$\textstyle{G_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\textstyle{\pi^{\rm
ab}_{1}(X),}$

where the right vertical arrow is the canonical surjection and $\alpha$ is
induced by taking product of completions. It is easy to check that this arrow
is well-defined. Our task is to show that
$\rho_{X}\circ(\iota_{X},\partial_{X})=0$.

To show the above, we first define $\beta$ and $\gamma$ such that (7.5)
commutes. To define $\beta$, note that we can write
$I^{\prime}(\overline{X})=K^{M}_{2}(K_{\infty})\bigoplus I^{\prime}(X)$. We
set $\beta$ to be the identity map on the first summand and
$\beta(a_{x})=\partial_{X,x}(a_{x})$ if $a_{x}\in K^{M}_{2}(K_{x})$ with $x\in
X_{(0)}$. This is well-defined because ${\rm
Ker}(\partial_{X,x})=U^{\prime}_{0}K^{M}_{2}(K_{x})$.

To define $\gamma$, let $\chi\in H^{1}(K)$ and $(a_{x})\in
I^{\prime}(\overline{X})$. We can find a dense open subscheme $U\subset X$
such that $\chi$ is unramified along $U$. That is, $\chi\in
H^{1}_{\text{\'{e}t}}(U,{{\mathbb{Q}}}/{{\mathbb{Z}}})\subset H^{1}(K)$.
Equivalently, $\chi$ factors through $\pi^{\rm ab}_{1}(U)$. It follows
therefore from Proposition 7.1(3) that $\sum_{x\in
X_{(0)}}\langle\chi_{x},a_{x}\rangle$ is a finite sum. In other words, the cup
product pairing of (7.2) induces a pairing

(7.6) $\langle,\rangle\colon H^{1}(K)\otimes
I^{\prime}(\overline{X})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

Equivalently, we get a homomorphism $\gamma\colon I^{\prime}(\overline{X})\to
G_{K}$ which is compatible with the reciprocity homomorphisms of the
completions of $K$. It is clear that $\gamma$ is continuous. Furthermore, the
commutative diagram (7.1) implies that the diagram (7.5) is commutative. Since
the composite horizontal arrow on the top in (7.5) is zero by [85, Thm. 1.1,
2.1], it follows that $\rho_{X}\circ(\iota_{X},\partial_{X})$ is zero. This
implies that $\rho_{X}$ factors through $C(X)$, as desired. ∎

###  7.2. Reciprocity for tame class group

Let $k$ be a local field and $X\in{\operatorname{\mathbf{Sch}}}_{k}$ an
integral normal scheme with function field $K$. We shall now construct the
reciprocity homomorphisms for the tame class groups of $X$. Since there is a
canonical continuous homomorphism $\delta_{X}\colon C^{\rm t}(X)\to C^{{\rm
t},W}(X)$, it suffices to construct the reciprocity homomorphism for the
latter group.

######  Proposition 7.3.

$\rho_{X}$ induces a continuous homomorphism $\rho^{\rm t}_{X}\colon C^{{\rm
t},W}(X)\to\pi^{\rm ab,{\rm t}}_{1}(X)$ such that we have a commutative
diagram

(7.7)
$\textstyle{C(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{X}}$$\scriptstyle{\kappa_{X}}$$\textstyle{\pi^{\rm
ab}_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{X}}$$\textstyle{C^{{\rm
t},W}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{{\rm
t}}_{X}}$$\textstyle{\pi^{\rm ab,{\rm t}}_{1}(X).}$

###### Proof.

Using the definition of $C^{{\rm t},W}(X)$ and Corollary 2.4, it suffices to
prove the proposition when $\text{\rm dim}(X)=1$. We assume this to be the
case and let $\overline{X}$ be the unique normal compactification of $X$. Let
$\eta$ denote the generic point of $X$. Using Proposition 3.4, Definition 5.2
and Lemma 7.2, it suffices to show that if $x\in\Delta(X)$, then the image of
$U^{\prime}_{1}K^{M}_{2}(K_{x})$ under the reciprocity map $\rho_{K_{x}}\colon
K^{M}_{2}(K_{x})\to G_{K_{x}}$ lies in the (closed) subgroup
$G^{(1)}_{K_{x}}$. If ${\rm char}(k)=0$, then $U^{\prime}_{1}K^{M}_{2}(K_{x})$
is a divisible group by Lemma 5.9. Hence, its image under $\rho_{K_{x}}$ must
be zero as $G_{K_{x}}$ is a profinite group. If ${\rm char}(k)>0$, then using
the Pontryagin duality and isomorphism (2.3), the assertion is equivalent to
showing that $\chi\circ\rho_{K_{x}}(a)=0$ for every $\chi\in{\rm
fil}_{1}H^{1}(K_{x})$ and $a\in U^{\prime}_{1}K^{M}_{2}(K_{x})$. But this
follows from Proposition 7.1(3). ∎

######  Corollary 7.4.

For an integral normal $k$-scheme $X$, there is a commutative diagram

(7.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C^{\rm
t}(X)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\rm
t}_{X,0}}$$\textstyle{C^{\rm
t}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{X}}$$\scriptstyle{\rho^{\rm
t}_{X}}$$\textstyle{k^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{k}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab,{\rm
t}}_{1}(X)_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi^{\rm
ab,{\rm
t}}_{1}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{X}}$$\textstyle{G_{k}.}$

###### Proof.

The only thing that may perhaps need an explanation is the commutativity of
the right square. But this follows from the construction of various maps in
the square and [52, § 3.2, Cor. 1]. ∎

######  Corollary 7.5.

Let $f\colon X^{\prime}\to X$ be a morphism of regular $k$-schemes. Then we
have a commutative diagram

(7.9) $\textstyle{C^{\rm
t}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\rm
t}_{X^{\prime}}}$$\scriptstyle{f_{*}}$$\textstyle{\pi^{\rm ab,{\rm
t}}_{1}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{C^{\rm
t}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\rm
t}_{X}}$$\textstyle{\pi^{\rm ab,{\rm t}}_{1}(X).}$

If $f$ is proper, the same holds for the reciprocity maps for idele class
groups.

###### Proof.

The vertical and horizontal arrows exist by Corollary 2.4, Proposition 4.13
(use Proposition 5.3 for idele class groups) and Proposition 7.3. The
commutativity of the diagram follows from the explicit construction of these
maps using [52, § 3.2, Cor. 1]. ∎

######  Remark 7.6.

Corollary 7.5 remains valid without the regularity assumption if we replace
$\pi^{\rm ab,{\rm t}}_{1}(-)$ by $\pi^{\rm ab,{\rm ctm}}_{1}(-)$.

The following result shows that over a local field of characteristic $p>0$,
there is no difference between the $p$-adic tame class field theory and the
$p$-adic unramified class field theory of smooth varieties.

######  Corollary 7.7.

Let $j\colon X\to\overline{X}$ be an open immersion between $k$-schemes such
that $\overline{X}$ is smooth and projective over $k$. Assume that ${\rm
char}(k)=p>0$. Then we have a commutative diagram of continuous homomorphisms

(7.10) $\textstyle{{C^{\rm
t}(X)}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho^{\rm
t}_{X}}$$\scriptstyle{j_{*}}$$\textstyle{{\pi^{\rm ab,{\rm
t}}_{1}(X)}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\textstyle{{C(\overline{X})}/{p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\overline{X}}}$$\textstyle{{\pi^{\rm
ab}_{1}(\overline{X})}/{p^{n}}}$

for every integer $n\geq 1$ such that the vertical arrows are isomorphisms.

###### Proof.

Combine Corollary 3.11, Proposition 4.13 and Theorem 5.13. ∎

###  7.3. Reciprocity for Idele class and Chow groups with modulus

Let $k$ be a local field and $\overline{X}$ a complete integral normal
$k$-scheme with an effective Cartier divisor $D\subset\overline{X}$ whose
complement is $X$. We assume that $X$ is regular. Let $K$ denote the function
field of $X$. We let ${\rm fil}^{c}_{D}H^{1}(K)$ be the subgroup of characters
$\chi$ of $G_{K}$ satisfying the property that $\chi\in
H^{1}_{\text{\'{e}t}}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}})$ and for every
$C\in{\mathcal{C}}(X)$ with canonical map
$\nu\colon\overline{C}_{n}\to\overline{X}$, the character $\nu^{*}(\chi)$ lies
in ${\rm fil}_{\nu^{*}(D)}H^{1}(k(C_{n}))$ (see § 2.4) under the map
$\nu^{*}\colon H^{1}_{\text{\'{e}t}}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}})\to
H^{1}_{\text{\'{e}t}}(C_{n},{{\mathbb{Q}}}/{{\mathbb{Z}}})$. We let $\pi^{\rm
ab}_{1}(\overline{X},D)=({\rm fil}^{c}_{D}(H^{1}(K)))^{\vee}$ with respect to
the discrete topology of ${\rm fil}^{c}_{D}H^{1}(K)$. This notion of étale
fundamental group with modulus was introduced by Deligne and Laumon [68] (see
[40, Definition 5.1]). It is clear that there is a canonical surjection
$\pi^{\rm ab}_{1}(X)\twoheadrightarrow\pi^{\rm ab}_{1}(\overline{X},D)$.

######  Proposition 7.8.

$\rho_{X}$ induces a continuous homomorphism

$\rho_{\overline{X}|D}\colon SK_{1}(\overline{X}|D)\to\pi^{\rm
ab}_{1}(\overline{X},D).$

In particular, there is a homomorphism

$\widetilde{\rho}_{\overline{X}|D}\colon{\rm
CH}^{d+1}(\overline{X}|D_{+},1)\to\pi^{\rm ab}_{1}(\overline{X},D)$

if $\text{\rm dim}(X)=d$.

###### Proof.

Using (5.5), we can replace $SK_{1}(\overline{X}|D)$ by $C(\overline{X},D)$.
The proof of the proposition is now identical to that of Proposition 7.3,
where we apply Proposition 7.1(3) for arbitrary values of $m\geq 1$ instead of
$m=1$. For the existence of $\widetilde{\rho}_{\overline{X}|D}$, use the map
${\rm CH}^{d+1}(\overline{X}|D_{+},1)\to SK_{1}(\overline{X}|D)$ in the proof
of Theorem 6.1 (where recall that $D_{+}=D+D_{\rm red}$). ∎

##  8\. Tame class group and motivic cohomology

The theory of motivic cohomology with compact support for schemes (due to
Voevodsky) will be a key player in our proofs of the main results of this
paper. Since we shall be working with schemes over local fields which may not
necessarily be perfect, we need to properly recall the results in the theory
of motives over imperfect fields that we shall use in this paper. This is the
purpose of this section. We shall conclude the section by establishing an
isomorphism between the tame class group with finite coefficients (prime to
the characteristic of the ground field) and certain motivic cohomology group
with compact support.

Throughout this section, we fix a field $k$ of characteristic exponent $p\geq
1$. We let $\Lambda$ be a commutative ring which we assume to be
${\mathbb{Z}}$ if $p=1$ or any ${\mathbb{Z}}[\tfrac{1}{p}]$-algebra if $p\geq
2$. We shall be more specific about this algebra as and when required. We let
${\operatorname{\mathbf{Reg}}}_{k}$ denote the category of regular
$k$-schemes.

###  8.1. Review of Voevodsky’s motives

It was shown by Cisinski-Déglise [17] that given a Noetherian $k$-scheme $X$,
there is a monoidal triangulated category of mixed motives
${\mathbf{DM}}(X,\Lambda)$ which coincides with the original construction of
Voevodsky [105] when $X$ is the spectrum of a perfect field and of Suslin [95]
when $X$ is the spectrum of an arbitrary field.

One defines $\underline{{\mathbf{DM}}}_{\rm cdh}(X,\Lambda)$ in the same
manner by replacing the Nisnevich topology on ${\mathbf{Sm}}_{X}$ by the cdh
topology on ${\operatorname{\mathbf{Sch}}}_{X}$. We let ${\mathbf{DM}}_{\rm
cdh}(X)$ be the full localizing triangulated subcategory of
$\underline{{\mathbf{DM}}}_{\rm cdh}(X,\Lambda)$ generated by motives of the
form $M_{X}(Y)(n)$ for $Y\in{\mathbf{Sm}}_{X}$ and $n\in{\mathbb{Z}}$. By the
main result of [15], the assignment $X\mapsto{\mathbf{DM}}_{\rm
cdh}(X,\Lambda)$ satisfies Grothendieck’s six functor formalism (see [17,
A.5]) in full generality in the category of Noetherian $k$-schemes of finite
Krull dimensions. In particular, it satisfies localization, absolute purity
and continuity. The constant Nisnevich (resp. cdh) sheaf with transfer
associated to the ring $\Lambda$ is the identity object for the monoidal
structure of ${\mathbf{DM}}(X,\Lambda)$ (resp. ${\mathbf{DM}}_{\rm
cdh}(X,\Lambda)$). We shall denote both of these by $\Lambda_{X}$. We recall
the following result from [15, Cor. 5.9].

######  Theorem 8.1.

The change of topology functor $a^{*}_{\rm
cdh}\colon{\mathbf{DM}}(X,\Lambda)\to{\mathbf{DM}}_{\rm cdh}(X,\Lambda)$ is an
equivalence of monoidal triangulated categories if
$X\in{\operatorname{\mathbf{Reg}}}_{k}$.

Using the continuity property and [15, Lem. 3.17, Prop. 7.1] (see also [17,
Prop. 9.1.14]), we have the following result in which we follow the notations
of § 4.5.

######  Proposition 8.2.

Assume that $k$ is an imperfect field and
$X\in{\operatorname{\mathbf{Sch}}}_{k}$. Let $i\geq 0$ be an integer. Let
$f_{i}\colon X^{\heartsuit}\to X_{i},\ f\colon X^{\heartsuit}\to X$ and
$g_{i}\colon X_{i}\to X$ be the projection maps. Then we have the following.

  1. (1)

Each of the compositions

${\mathbf{DM}}_{\rm cdh}(X,\Lambda)\xrightarrow{g^{*}_{i}}{\mathbf{DM}}_{\rm
cdh}(X_{i},\Lambda)\xrightarrow{(g_{i})_{*}}{\mathbf{DM}}_{\rm
cdh}(X,\Lambda)\ \mbox{and}$ ${\mathbf{DM}}_{\rm
cdh}(X_{i},\Lambda)\xrightarrow{(g_{i})_{*}}{\mathbf{DM}}_{\rm
cdh}(X,\Lambda)\xrightarrow{g^{*}_{i}}{\mathbf{DM}}_{\rm cdh}(X_{i},\Lambda)$

is multiplication by $p^{i}$.

  2. (2)

There is an equivalence of monoidal triangulated categories

$2-{{\underset{i\geq 0}{\varinjlim}}}\ {\mathbf{DM}}_{\rm
cdh}(X_{i},\Lambda)\xrightarrow{\simeq}{\mathbf{DM}}_{\rm
cdh}(X^{\heartsuit},\Lambda),$

where the left hand side is a colimit of 2-categories. In particular, the
pull-back functors

${\mathbf{DM}}_{\rm cdh}(X,\Lambda)\xrightarrow{g^{*}_{i}}{\mathbf{DM}}_{\rm
cdh}(X_{i},\Lambda)\xrightarrow{f^{*}_{i}}{\mathbf{DM}}_{\rm
cdh}(X^{\heartsuit},\Lambda)$

are equivalences of monoidal triangulated categories.

For a closed immersion $\iota\colon W\hookrightarrow X$ with structure maps
$f\colon X\to{\rm Spec\,}(k)$, and $h=f\circ\iota$, recall the following
definitions of motives in ${\mathbf{DM}}_{\rm
cdh}(k,\Lambda)\simeq{\mathbf{DM}}(k,\Lambda)$ (e.g., see [15, § 8.5]).

(8.1) $M_{k}(X)=f_{\sharp}\Lambda_{X}\cong f_{!}f^{!}\Lambda_{k},\
M^{c}_{k}(X)=f_{*}f^{!}\Lambda_{k}\ \mbox{and}\
M^{W}_{k}(X)=h_{!}\iota^{*}f^{!}\Lambda_{k},$

where all functors are considered on the premotivic category
$X\mapsto{\mathbf{DM}}_{\rm cdh}(X,\Lambda)$. Recall that
${\mathbb{Z}}_{k}(1)={{\mathbb{Z}}_{\rm
tr}({\mathbb{P}}^{1}_{k})}/{{\mathbb{Z}}_{\rm tr}({\rm Spec\,}(k))}$ as a
presheaf with transfer via the identification ${\rm Spec\,}(k)\cong
k(\infty)$. In particular, $\Lambda_{k}(1)\cong
M_{k}({{\mathbb{P}}^{1}_{k}}/{{\rm Spec\,}(k)})\in{\mathbf{DM}}(k,\Lambda)$.
We shall drop highlighting the base field $k$ in the notations of motives once
it is fixed in a context.

######  Definition 8.3.

We let

(8.2) $H_{i}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(\Lambda_{k}(j)[i],M_{k}(X));$ $H^{i}(X,\Lambda(j))={\rm
Hom}_{{\mathbf{DM}}_{\rm cdh}(k,\Lambda)}(M_{k}(X),\Lambda_{k}(j)[i]);$
$H^{i}_{c}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(M^{c}_{k}(X),\Lambda_{k}(j)[i]);$
$H^{i}_{W}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(M^{W}_{k}(X),\Lambda_{k}(j)[i]).$

The above groups are respectively called the (motivic) homology, cohomology,
cohomology with compact support and cohomology with support in $W$ of $X$. If
$X={\rm Spec\,}(A)$ is affine, we shall often use the notation $A$ instead of
$X$ while using these groups.

######  Lemma 8.4.

Given a closed immersion $\iota\colon W\hookrightarrow X$ with the inclusion
$u\colon U\hookrightarrow X$ of the complement, there are long exact sequences

$(1)\hskip 28.45274pt\cdots\to
H^{i}_{W}(X,\Lambda(j))\xrightarrow{\iota_{*}}H^{i}(X,\Lambda(j))\xrightarrow{u^{*}}H^{i}(U,\Lambda(j))\xrightarrow{\partial}H^{i+1}_{W}(X,\Lambda(j))\to\cdots.$
$(2)\hskip 28.45274pt\cdots\to
H^{i}_{c}(U,\Lambda(j))\xrightarrow{u_{*}}H^{i}_{c}(X,\Lambda(j))\xrightarrow{\iota^{*}}H^{i}_{c}(W,\Lambda(j))\xrightarrow{\partial}H^{i+1}_{c}(U,\Lambda(j))\to\cdots.$

###### Proof.

We let $g=f\circ u$ and $h=f\circ\iota$, where $f\colon X\to{\rm Spec\,}(k)$
is the structure map. We then have a distinguished triangle

$\iota_{*}\iota^{!}f^{!}\Lambda_{k}\to f^{!}\Lambda_{k}\to
u_{*}u^{*}f^{!}\Lambda_{k}$

in ${\mathbf{DM}}_{\rm cdh}(X,\Lambda)$ by [15, Thm. 5.11]. Using the
isomorphism $u^{*}\cong u^{!}$ and subsequently applying $f_{*}$, we get a
distinguished triangle

$M^{c}_{k}(W)\to M^{c}_{k}(X)\to M^{c}_{k}(U)$

in ${\mathbf{DM}}_{\rm cdh}(k,\Lambda)$. This yields the second exact sequence
in a standard way.

For the first exact sequence, we use the distinguished triangle

$u_{!}u^{*}f^{!}\Lambda_{k}\to
f^{!}\Lambda_{k}\to\iota_{*}\iota^{*}f^{!}\Lambda_{k}$

in ${\mathbf{DM}}_{\rm cdh}(X,\Lambda)$ by [15, Thm. 5.11]. We again proceed
as in the previous case to get a distinguished triangle

$M_{k}(U)\to M_{k}(X)\to M^{W}_{k}(X)$

in ${\mathbf{DM}}_{\rm cdh}(k,\Lambda)$. This yields the desired exact
sequence of cohomology groups. ∎

The following description of motivic cohomology groups will be used in § 9 to
compare them with étale cohomology.

######  Lemma 8.5.

In the notations of Lemma 8.4, there are canonical isomorphisms

$H^{i}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(\Lambda_{k},f_{*}\Lambda_{X}(j)[i]);$
$H^{i}_{c}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(\Lambda_{k},f_{!}\Lambda_{X}(j)[i]);$
$H^{i}_{W}(X,\Lambda(j))={\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(X,\Lambda)}(\iota_{*}\Lambda_{W},\Lambda_{X}(j)[i]).$

###### Proof.

This is well known to experts and we give a brief sketch. We shall use the
notations and distinguished triangles in the proof of Lemma 8.4. Part (1) is a
direct consequence of the adjointness of $(f_{\sharp},f^{*})$ and
$(f^{*},f_{*})$ using the canonical isomorphism
$f^{*}\Lambda_{k}\xrightarrow{\cong}\Lambda_{X}$. Part (2) follows by using
the distinguished triangles

$M_{k}(Z)\to M_{k}(\overline{X})\to M^{c}_{k}(X);\ \ f_{!}\Lambda_{X}\to
g_{*}\Lambda_{\overline{X}}\to h_{*}\Lambda_{Z}$

in ${\mathbf{DM}}_{\rm cdh}(k,\Lambda)$ and applying (1) to $\overline{X}$ and
$Z$, where $j\colon X\hookrightarrow\overline{X}$ is a compactification of $X$
and $\iota\colon Z\hookrightarrow\overline{X}$ is the complementary reduced
closed subscheme. Part (3) follows by using the distinguished triangles

$M_{k}(U)\to M_{k}(X)\to M^{W}_{k}(X);\ \ h_{*}\iota^{!}\Lambda_{X}\to
f_{*}\Lambda_{X}\to g_{*}\Lambda_{U}$

in ${\mathbf{DM}}_{\rm cdh}(k,\Lambda)$, the adjoint pair
$(\iota_{*},\iota^{!})$ and the isomorphism
$h^{*}\Lambda_{k}\xrightarrow{\cong}\Lambda_{Z}$. ∎

###  8.2. Motivic cohomology via stable homotopy category

We let ${\mathbf{SH}}(X)$ be the localization of the monoidal stable homotopy
category of smooth schemes over $X$ at $p$ (see [24, § 2.1] for its
definition). By its definition, every morphism in ${\mathbf{SH}}(X)$ is a
${\mathbb{Z}}[\tfrac{1}{p}]$-module. One defines
$\underline{{\mathbf{SH}}}_{\rm cdh}(X)$ in the same manner by replacing the
Nisnevich topology on ${\mathbf{Sm}}_{X}$ by the cdh topology on
${\operatorname{\mathbf{Sch}}}_{X}$. We let ${\mathbf{SH}}_{{\rm cdh}}(X)$ be
the full triangulated subcategory of $\underline{{\mathbf{SH}}}_{\rm cdh}(X)$
generated by the objects $\Sigma^{\infty}_{T}Y_{+}$ for
$Y\in{\mathbf{Sm}}_{X}$.

For $X\in{\operatorname{\mathbf{Sch}}}_{k}$, let
$\Sigma^{i,j}=\Sigma^{i-j}_{s}\circ\Sigma^{j}_{t}$, where $\Sigma_{s}$ and
$\Sigma_{t}$ are the simplicial and the Tate circles in ${\mathbf{SH}}(X)$,
respectively (e.g., see [66, § 2.1]). We let ${\mathbb{S}}_{X}$ be the sphere
spectrum (the unit object) of ${\mathbf{SH}}(X)$. Recall from [21, § 2.2] that
${\mathbf{SH}}(X)$ is a spectral category, i.e., it is a category enriched in
spectra. In particular, there is an $S^{1}$-spectrum ${\rm
Maps}_{{\mathbf{SH}}(X)}(A,B)$ for any $A,B\in{\mathbf{SH}}(X)$ such that one
has $\pi_{n}({\rm Maps}_{{\mathbf{SH}}(X)}(A,B))\cong{\rm
Hom}_{{\mathbf{SH}}(X)}(\Sigma^{n}_{s}A,B)$. The same is true for
${{\mathbf{SH}}}_{\rm cdh}(X)$. Recall that the canonical morphism of sites
$\epsilon_{X}\colon({\operatorname{\mathbf{Sch}}}_{X})_{\rm
cdh}\to({\mathbf{Sm}}_{k})_{\rm nis}$ induces a push-forward triangulated
functor $a^{\rm cdh}_{*}\colon{\mathbf{SH}}_{\rm cdh}(X)\to{\mathbf{SH}}(X)$
which is a right adjoint of the pull-back functor $a^{*}_{\rm cdh}$.
Furthermore, $a^{\rm cdh}_{*}$ is the localization of ${\mathbf{SH}}(X)$ with
respect to cdh-covers. In particular, $a^{\rm cdh}_{*}\colon{\mathbf{SH}}_{\rm
cdh}(X)\to{\mathbf{SH}}(X)$ is a full embedding of triangulated categories
(e.g., see [23, Prop. 2.13]).

By [15, § 2], forgetting the transfer structure and then taking the
$T$-suspension defines a functor
$\phi_{X}\colon{{\mathbf{DM}}}(X,\Lambda)\to{{\mathbf{SH}}}(X)$ which has a
strictly monoidal left adjoint $\psi_{X}$. The latter is uniquely determined
via the left Kan extension by the property that
$\psi_{X}(\Sigma^{\infty}_{T}Y_{+})=f_{\sharp}(\Lambda_{Y})$ for any $f\colon
Y\to X$ in ${\mathbf{Sm}}_{X}$. We let ${\rm
H}\Lambda_{X}:=\phi_{X}(\Lambda_{X})$. If $f\colon X\to{\rm Spec\,}(k)$ is the
structure map, then the exchange transformation
$f^{*}\phi_{k}\to\phi_{X}f^{*}$ induces a canonical map ${\rm
H}\Lambda_{X/k}:=f^{*}{\rm H}\Lambda_{k}\to{\rm H}\Lambda_{X}$. We similarly
define ${\rm H}\Lambda^{{\rm cdh}}_{X}$ and ${\rm H}\Lambda^{{\rm
cdh}}_{X/k}\in{\mathbf{SH}}(X)$ using the adjoint pair
$(\psi_{X},\phi_{X})\colon\underline{{\mathbf{DM}}}_{{\rm
cdh}}(X,\Lambda)\to\underline{{\mathbf{SH}}}_{{\rm cdh}}(X)$ which has the
property that $\psi_{X}(\Sigma^{\infty}_{T}Y_{+})=f_{\sharp}(\Lambda_{Y})$ for
any $f\colon Y\to X$ in ${\operatorname{\mathbf{Sch}}}_{X}$. Note that ${\rm
H}\Lambda_{{(-)}/k}$ and ${\rm H}\Lambda^{{\rm cdh}}_{{(-)}/k}$ are absolute
ring spectra functors on ${\operatorname{\mathbf{Sch}}}_{k}$ (see [77,
Defn.1.1] for the definition of absolute ring spectra). It is clear from the
above description of $\phi_{X}$ that $a^{*}_{\rm
cdh}\circ\phi_{X}=\phi_{X}\circ a^{*}_{\rm cdh}$ on the subcategory of compact
objects in ${\mathbf{DM}}(X,\Lambda)$.

######  Lemma 8.6.

The map ${\rm H}\Lambda_{X/k}\to{\rm H}\Lambda_{X}$ is an isomorphism if
$X\in{\operatorname{\mathbf{Reg}}}_{k}$. In particular, ${\rm H}\Lambda_{(-)}$
is an absolute ring spectrum functor on ${\operatorname{\mathbf{Reg}}}_{k}$.
Moreover, there is a canonical isomorphism $a^{*}_{\rm cdh}({\rm
H}\Lambda_{X/k})\cong{\rm H}\Lambda^{{\rm cdh}}_{X/k}$ for
$X\in{\operatorname{\mathbf{Sch}}}_{k}$.

###### Proof.

The first claim follows from [15, Prop. 3.10]. To prove the second claim, it
suffices to show the isomorphism $a^{*}_{\rm cdh}({\rm H}\Lambda_{k})\cong{\rm
H}\Lambda^{{\rm cdh}}_{k}$. But this is well known, e.g., follows directly
from the isomorphism $a^{*}_{\rm
cdh}\circ\phi_{k}(\Lambda_{k})\cong\phi_{k}\circ a^{*}_{\rm
cdh}(\Lambda_{k}):={\rm H}\Lambda^{\rm cdh}_{k}$, observed above. ∎

######  Lemma 8.7.

The canonical map ${\rm H}\Lambda_{k}\to a^{\rm cdh}_{*}{\rm H}\Lambda^{\rm
cdh}_{k}$ is an isomorphism.

###### Proof.

By [23, Prop. 2.7], it suffices to show that the map

${\rm Hom}_{{\mathbf{SH}}(k)}(\Sigma^{\infty}_{T}X_{+},\Sigma^{i,j}{\rm
H}\Lambda_{k})\to{\rm
Hom}_{{\mathbf{SH}}(k)}(\Sigma^{\infty}_{T}X_{+},\Sigma^{i,j}a^{\rm
cdh}_{*}{\rm H}\Lambda^{\rm cdh}_{k})$

is an isomorphism for every $X\in{\mathbf{Sm}}_{k}$. To show this, we note
that

$\begin{array}[]{lll}{\rm
Hom}_{{\mathbf{SH}}(k)}(\Sigma^{\infty}_{T}X_{+},\Sigma^{i,j}{\rm
H}\Lambda_{k})&{\cong}^{1}&{\rm
Hom}_{{\mathbf{DM}}(k,\Lambda)}(f_{\sharp}\Lambda_{X},\Lambda_{k}(j)[i])\\\
&{\cong}^{2}&{\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(f_{\sharp}\Lambda^{\rm cdh}_{X},\Lambda^{\rm
cdh}_{k}(j)[i])\\\ &{\cong}^{3}&{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(a^{*}_{\rm cdh}\Sigma^{\infty}_{T}X_{+},\Sigma^{i,j}{\rm
H}\Lambda^{\rm cdh}_{k})\\\ &{\cong}^{4}&{\rm
Hom}_{{\mathbf{SH}}(k)}(\Sigma^{\infty}_{T}X_{+},\Sigma^{i,j}a^{\rm
cdh}_{*}{\rm H}\Lambda^{\rm cdh}_{k}),\end{array}$

where ${\cong}^{1}$ and ${\cong}^{3}$ follow from the adjointness of
$(\psi_{k},\phi_{k})$, ${\cong}^{2}$ follows from [15, Cor. 5.9] and
${\cong}^{4}$ follows from the adjointness of $(a^{*}_{\rm cdh},a^{\rm
cdh}_{*})$. This concludes the proof. ∎

For a closed immersion $\iota\colon W\hookrightarrow X$ with structure maps
$f\colon X\to{\rm Spec\,}(k)$, and $h=f\circ\iota$, we let

(8.3) $E^{i}(X,\Lambda(j))={\rm
Hom}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}{\rm H}\Lambda_{X/k});$
$E^{i}_{c}(X,\Lambda(j))={\rm
Hom}_{{\mathbf{SH}}(k)}({\mathbb{S}}_{k},\Sigma^{i,j}f_{!}{\rm
H}\Lambda_{X/k});$ $E^{i}_{W}(X,\Lambda(j))={\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{*}{\mathbb{S}}_{W},\Sigma^{i,j}{\rm
H}\Lambda_{X/k}).$

######  Lemma 8.8.

We have the following canonical isomorphisms.

  1. (1)

$H^{i}(X,\Lambda(j))\xrightarrow{\cong}E^{i}(X,\Lambda(j))$.

  2. (2)

$H^{i}_{c}(X,\Lambda(j))\xrightarrow{\cong}E^{i}_{c}(X,\Lambda(j))$ if
$X\in{\operatorname{\mathbf{Reg}}}_{k}$.

  3. (3)

$H^{i}_{W}(X,\Lambda(j))=E^{i}_{W}(X,\Lambda(j))$ if
$X,W\in{\operatorname{\mathbf{Reg}}}_{k}$.

###### Proof.

To prove (1), we use [66, Thm. 2.14] (see also the proof of Prop. 2.13 of op.
cit.), which says that there are canonical weak equivalences

(8.4) $\begin{array}[]{lll}{\rm
Maps}_{{\mathbf{SH}}({X})}({\mathbb{S}}_{X},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})&\xrightarrow{\simeq}&{\rm Maps}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{\sharp}{\mathbb{S}}_{X},\Sigma^{i,j}a^{*}_{\rm cdh}{\rm
H}\Lambda_{k})\\\ &\xrightarrow{\simeq}&{\rm Maps}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{\sharp}{\mathbb{S}}_{X},\Sigma^{i,j}{\rm H}\Lambda^{{\rm
cdh}}_{k})\\\ &\xrightarrow{\simeq}&{\rm Maps}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{\sharp}{\mathbb{S}}_{X},\phi_{k}(\Lambda_{k}(j)[i])),\\\
\end{array}$

where the second weak equivalence follows from Lemma 8.6. The induced maps
between the homotopy groups yield a canonical isomorphism

(8.5) ${\rm Hom}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})\xrightarrow{\simeq}{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{\sharp}{\mathbb{S}}_{X},\phi_{k}(\Lambda_{k}(j)[i])).$

On the other hand, via the equivalences
${\mathbf{DM}}(k,\Lambda)\xrightarrow{\simeq}{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)\xrightarrow{\simeq}\underline{{\mathbf{DM}}}_{\rm
cdh}(k,\Lambda)$ (see [15, Prop. 8.1]), there are isomorphisms

(8.6) $\begin{array}[]{lll}{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{\sharp}{\mathbb{S}}_{X},\phi_{k}(\Lambda_{k}(j)[i]))&\xrightarrow{\simeq}&{\rm
Hom}_{\underline{{\mathbf{SH}}}_{\rm
cdh}(k)}(\Sigma^{\infty}_{T}X_{+},\phi_{k}(\Lambda_{k}(j)[i]))\\\
&\xrightarrow{\simeq}&{\rm Hom}_{\underline{{\mathbf{DM}}}_{\rm
cdh}(k,\Lambda)}(\psi_{k}\Sigma^{\infty}_{T}X_{+},\Lambda_{k}(j)[i])\\\
&\xrightarrow{\simeq}&{\rm Hom}_{{{\mathbf{DM}}}_{\rm
cdh}(k,\Lambda)}(f_{\sharp}\Lambda_{X},\Lambda_{k}(j)[i])\\\
&\xrightarrow{\simeq}&{\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(M_{k}(X),\Lambda_{k}(j)[i])\\\
&\xrightarrow{\simeq}&H^{i}(X,\Lambda(j)).\end{array}$

Combining (8.5) and (8.6), we get (1).

To prove (2), we let $u\colon X\hookrightarrow\overline{X}$ be an open
immersion such that $\overline{X}$ is proper over $k$ with the structure map
$g\colon\overline{X}\to{\rm Spec\,}(k)$. We then get

(8.7) $\begin{array}[]{lll}{\rm
Hom}_{{\mathbf{SH}}(k)}({\mathbb{S}}_{k},\Sigma^{i,j}f_{!}{\rm
H}\Lambda_{X/k})&\xrightarrow{\simeq}&{\rm
Hom}_{{\mathbf{SH}}(k)}({\mathbb{S}}_{k},\Sigma^{i,j}g_{!}u_{!}{\rm
H}\Lambda_{X/k})\\\ &\xrightarrow{\simeq}&{\rm
Hom}_{{\mathbf{SH}}(k)}({\mathbb{S}}_{k},\Sigma^{i,j}g_{*}u_{!}{\rm
H}\Lambda_{X/k})\\\ &\xrightarrow{\simeq}&{\rm
Hom}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}u_{!}{\rm
H}\Lambda_{X/k}).\end{array}$

Since $a^{\rm cdh}_{*}$ is a full embedding, (8.4) implies that

(8.8) $\begin{array}[]{lll}{\rm
Maps}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{{\overline{X}}/k})&\xrightarrow{\simeq}&{\rm
Maps}_{{\mathbf{SH}}_{\rm
cdh}(k)}(g_{\sharp}{\mathbb{S}}_{\overline{X}},\Sigma^{i,j}a^{*}_{\rm cdh}{\rm
H}\Lambda_{k})\\\ &\xrightarrow{\simeq}&{\rm Maps}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}g_{\sharp}{\mathbb{S}}_{\overline{X}},\Sigma^{i,j}a^{\rm
cdh}_{*}a^{*}_{\rm cdh}{\rm H}\Lambda_{k})\\\ &\xrightarrow{\simeq}&{\rm
Maps}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}g_{\sharp}{\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{k}),\\\ \end{array}$

where the third isomorphism follows from Lemmas 8.6 and 8.7.

To proceed further, we let $Z=\overline{X}\setminus X$, endowed with the
reduced closed subscheme structure. Let $v\colon Z\hookrightarrow\overline{X}$
be the inclusion and $h=g\circ v$. Using the canonical isomorphism
$t_{\sharp}{\mathbb{S}}_{Y}\cong t_{*}t^{!}{\mathbb{S}}_{k}$ (for
$Y\in\\{\overline{X},Z\\}$ and $t\in\\{g,h\\}$) and the distinguished triangle
[6, § 4.5.3]

$v_{*}v^{!}g^{!}{\mathbb{S}}_{k}\to g^{!}{\mathbb{S}}_{k}\to
u_{*}u^{!}g^{!}{\mathbb{S}}_{k}$

in ${\mathbf{SH}}_{\rm cdh}(\overline{X})$, we obtain a distinguished triangle

(8.9) $h_{\sharp}{\mathbb{S}}_{Z}\to g_{\sharp}{\mathbb{S}}_{\overline{X}}\to
f_{*}f^{!}{\mathbb{S}}_{k}$

in ${\mathbf{SH}}_{\rm cdh}(k)$. In particular, a distinguished triangle

(8.10) $a^{\rm cdh}_{*}h_{\sharp}{\mathbb{S}}_{Z}\to a^{\rm
cdh}_{*}g_{\sharp}{\mathbb{S}}_{\overline{X}}\to a^{\rm
cdh}_{*}f_{*}f^{!}{\mathbb{S}}_{k}$

in ${\mathbf{SH}}(k)$.

We also have the distinguished triangle in ${\mathbf{SH}}(\overline{X})$ by
[6, § 4.5.3]

$u_{!}{\rm H}\Lambda_{X/k}\to{\rm H}\Lambda_{\overline{X}/k}\to v_{*}{\rm
H}\Lambda_{Z/k}.$

We now consider the induced commutative diagram of $S^{1}$-spectra

(8.11) $\textstyle{{\rm
Maps}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}v_{*}{\rm
H}\Lambda_{Z/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}g_{\sharp}{\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}h_{\sharp}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm H}\Lambda_{k}),}$

whose vertical arrows are given by (8.8). In particular, they are weak
equivalences. Taking the homotopy fibers of the two rows, we get a canonical
weak equivalence of $S^{1}$-spectra

${\rm
Maps}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}u_{!}{\rm
H}\Lambda_{X/k})\xrightarrow{\simeq}{\rm Maps}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}f_{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda_{k}).$

The induced homomorphisms between the homotopy groups yield a canonical
isomorphism

(8.12) ${\rm
Hom}_{{\mathbf{SH}}({\overline{X}})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}u_{!}{\rm
H}\Lambda_{X/k})\xrightarrow{\cong}{\rm Hom}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}f_{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda_{k}).$

We now note that since $X\in{\operatorname{\mathbf{Reg}}}_{k}$, the premotivic
categories $X\mapsto{\mathbf{SH}}_{\rm cdh}(X)$ and
$X\mapsto{\mathbf{DM}}_{\rm cdh}(X)$ satisfy the isomorphisms

$\psi_{k}f_{*}f^{!}{\mathbb{S}}_{k}{\cong}f_{*}\psi_{X}f^{!}{\mathbb{S}}_{k}\cong
f_{*}f^{!}\psi_{k}{\mathbb{S}}_{k}\cong f_{*}f^{!}\Lambda_{k},$

where the first three isomorphisms follow from (the cdh analogue of) [15, Cor.
3.2]. We thus get

(8.13) $\begin{array}[]{lll}{\rm Hom}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}f_{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda_{k})&\cong&{\rm
Hom}_{{\mathbf{SH}}(k)}(a^{\rm
cdh}_{*}f_{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}a^{\rm cdh}_{*}a^{*}_{\rm
cdh}{\rm H}\Lambda_{k})\\\ &\cong&{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(f_{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda^{\rm
cdh}_{k})\\\ &\cong&{\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k)}(\psi_{k}f_{*}f^{!}{\mathbb{S}}_{k},\Lambda_{k}(j)[i])\\\ &\cong&{\rm
Hom}_{{\mathbf{DM}}_{\rm cdh}(k)}(f_{*}f^{!}\Lambda_{k},\Lambda_{k}(j)[i])\\\
&\cong&H^{i}_{c}(X,\Lambda(j)),\\\ \end{array}$

where the first isomorphism follows Lemma 8.7 and the second isomorphism
follows from the fully faithfulness of $a^{\rm cdh}_{*}$. Combining (8.13)
with (8.7) and (8.12), we get (2).

To prove (3), we let $w\colon U\hookrightarrow X$ be the inclusion of the
complement. We let $g\colon U\to{\rm Spec\,}(k)$ and $h\colon W\to{\rm
Spec\,}(k)$ be the structure maps. We then have a canonical weak equivalence

(8.14) ${\rm Maps}_{{\mathbf{SH}}(U)}({\mathbb{S}}_{U},\Sigma^{i,j}{\rm
H}\Lambda_{U/k})\xrightarrow{\simeq}{\rm
Maps}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}w_{*}{\rm
H}\Lambda_{U/k}).$

We also have isomorphisms

(8.15) $\begin{array}[]{lll}{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{*}{\mathbb{S}}_{W},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})&\xrightarrow{\cong}&{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{!}{\mathbb{S}}_{W},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})\\\ &\xrightarrow{\cong}&{\rm
Hom}_{{\mathbf{SH}}(W)}({\mathbb{S}}_{W},\Sigma^{i,j}\iota^{!}{\rm
H}\Lambda_{X/k})\\\ &\xrightarrow{\cong}&{\rm
Hom}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}\iota_{*}\iota^{!}{\rm
H}\Lambda_{X/k}).\end{array}$

We next recall that there is a distinguished triangle

(8.16) $\iota_{*}\iota^{!}{\rm H}\Lambda_{X/k}\to{\rm H}\Lambda_{X/k}\to
w_{*}{\rm H}\Lambda_{U/k}$

in ${\mathbf{SH}}(X)$ by [6]. Using the canonical isomorphism
$t_{\sharp}{\mathbb{S}}_{Y}\cong t_{*}t^{!}{\mathbb{S}}_{k}$ (for
$Y\in\\{X,U\\}$ and $t\in\\{f,g\\}$) and the distinguished triangle

$w_{!}w^{!}f^{!}{\mathbb{S}}_{k}\to
f^{!}{\mathbb{S}}_{k}\to\iota_{!}\iota^{*}f^{!}{\mathbb{S}}_{k}$

in ${\mathbf{SH}}_{\rm cdh}(X)$, we also obtain a distinguished triangle

(8.17) $g_{\sharp}{\mathbb{S}}_{U}\to f_{\sharp}{\mathbb{S}}_{{X}}\to
h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k}$

in ${\mathbf{SH}}_{\rm cdh}(k)$.

Using (8.14), (8.16), (8.17) and comparing (8.4) for $X$ and $U$, we get a
weak equivalence

${\rm
Maps}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}\iota_{*}\iota^{!}{\rm
H}\Lambda_{X/k})\xrightarrow{\simeq}{\rm Maps}_{{\mathbf{SH}}_{\rm
cdh}(k)}(h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda^{{\rm
cdh}}_{k}).$

The induced maps between the homotopy groups yields a canonical isomorphism

(8.18) ${\rm
Hom}_{{\mathbf{SH}}(X)}({\mathbb{S}}_{X},\Sigma^{i,j}\iota_{*}\iota^{!}{\rm
H}\Lambda_{X/k})\xrightarrow{\cong}{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda^{{\rm
cdh}}_{k}).$

On the other hand, using the regularity of $X$ and $Z$, (the cdh analogue of)
[15, Cor. 3.2] implies that

$\psi_{k}h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k}\cong
h_{!}\iota^{*}f^{!}\psi_{k}{\mathbb{S}}_{k}\cong
h_{!}\iota^{*}f^{!}\Lambda_{k}.$

In particular,

$\begin{array}[]{lll}{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k},\Sigma^{i,j}{\rm H}\Lambda^{{\rm
cdh}}_{k})&\xrightarrow{\cong}&{\rm Hom}_{{\mathbf{SH}}_{\rm
cdh}(k)}(h_{!}\iota^{*}f^{!}{\mathbb{S}}_{k},\phi_{k}(\Lambda_{k}(j)[i]))\\\
&\xrightarrow{\cong}&{\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(\psi_{k}h_{!}\iota^{*}f^{!}\Lambda_{k},\Lambda_{k}(j)[i])\\\
&\xrightarrow{\cong}&{\rm Hom}_{{\mathbf{DM}}_{\rm
cdh}(k,\Lambda)}(h_{!}\iota^{*}f^{!}\Lambda_{k},\Lambda_{k}(j)[i])\\\
&\xrightarrow{\cong}&H^{i}_{W}(X,\Lambda(j)).\end{array}$

We now use (8.15) and (8.18) to conclude the proof of (3), and of the lemma. ∎

###  8.3. Gysin homomorphisms

In this subsection, we shall construct very general Gysin homomorphisms for
the cohomology theory $E^{*}(-,\Lambda(\bullet))$.

Let $\iota\colon Z\hookrightarrow X$ be a regular closed immersion of pure
codimension $n$ in ${\operatorname{\mathbf{Sch}}}_{k}$, i.e., $Z$ is locally
defined by a regular sequence of length $n$ on $X$. Let $W\subset Z$ be a
closed subset. Since ${\rm H}\Lambda_{{(-)}/{k}}$ is an oriented absolute
spectrum, it follows from [77, Defn. 2.8] (see also [21, Thm. 3.2.21]) that
there is a fundamental class $\eta^{X}_{Z}\in E^{2n}_{Z}(X,\Lambda(n))$ and a
cup product pairing $\bigcup\colon E^{i}_{W}(Z,\Lambda(j))\times
E^{2n}_{Z}(X,\Lambda(n))\to E^{2n+i}_{W}(X,\Lambda(n+j))$ (see [77, Defn.
1.6]) which together give rise to the refined Gysin homomorphism

(8.19) $\iota_{*}\colon E^{i}_{W}(Z,\Lambda(j))\to
E^{2n+i}_{W}(X,\Lambda(n+j));\ \ \alpha\mapsto\alpha\cup\eta^{X}_{Z}.$

This map is linear with respect to the action of the (relative) motivic
cohomology ring of $X$. Furthermore, for a flat morphism $f\colon
X^{\prime}\to X$ of $k$-schemes with $Z^{\prime}=Z\times_{X}X^{\prime}$, one
has $\eta^{X^{\prime}}_{Z^{\prime}}=f^{*}(\eta^{X}_{Z})$ (see [77, Prop.
2.10]). In particular, the refined Gysin map commutes with the flat pull-back
map $f^{*}$ on the cohomology of $Z$ and $X$ with support in $W$.

Recall that when $k$ is a perfect field and $X$ is a regular $k$-scheme, then
there is a natural duality isomorphism $\Phi_{X}\colon
M^{c}(X)^{*}(d)[2d]\xrightarrow{\cong}M(X)$ (see [56, Thm. 5.3.18(iii)]).
Furthermore, if $\iota\colon Z\hookrightarrow X$ is a closed immersion of
regular $k$-schemes of pure codimension $n$, then one has a purity isomorphism
$M_{Z}(X)\cong M(Z)(n)[2n]$ (e.g., see [17, (2.4.39.1)]).

When $Z$ is moreover complete (e.g., a closed point of $X$), the duality and
purity isomorphisms yield the Gysin homomorphism $\iota_{*}\colon
H^{i}(Z,\Lambda(j))\to H^{2n+i}_{c}(X,\Lambda(n+j))$ which is dual to the
push-forward homomorphism $\iota_{*}\colon H_{d-n-i}(Z,\Lambda(d-n-j))\to
H_{d-n-i}(X,\Lambda(d-n-j))$. In other words, there is a commutative diagram

(8.20)
$\textstyle{H^{i}(Z,\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{Z}}$$\scriptstyle{\cong}$$\scriptstyle{\iota_{*}}$$\textstyle{H_{2d-2n-i}(Z,\Lambda(d-n-j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{H^{2n+i}_{c}(X,\Lambda(n+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi_{X}}$$\scriptstyle{\cong}$$\textstyle{H_{2d-2n-i}(X,\Lambda(d-n-j)).}$

If $k$ is not perfect, we shall need Gysin homomorphism even when $Z$ is not
regular. The purity theorem does not hold in this case. However, the notions
of fundamental class and refined Gysin map of (8.19) allow us to show the
existence of the Gysin homomorphism for $E$-theory without assuming purity.
This is the context of the following result.

######  Lemma 8.9.

Let $\iota\colon Z\hookrightarrow X$ be a regular closed immersion of pure
codimension $n$ in ${\operatorname{\mathbf{Sch}}}_{k}$. Assume that $Z$ is
complete. Then there is a natural Gysin homomorphism

$\iota_{*}\colon E^{i}(Z,\Lambda(j))\to E^{2n+i}_{c}(X,\Lambda(n+j)).$

###### Proof.

By the construction of the refined Gysin homomorphism, it suffices to show
that the forget support map $E^{i}_{Z}(X,\Lambda(j))\to E^{i}(X,\Lambda(j))$
factors through $E^{i}_{c}(Z,\Lambda(j))$. To show this, we choose a
compactification $\overline{X}$ of $X$ and let $j\colon
X\hookrightarrow\overline{X}$ be the inclusion. Let $\iota^{\prime}\colon
W=\overline{X}\setminus X\hookrightarrow\overline{X}$ be the inclusion of the
complementary closed subset, endowed with the reduced subscheme structure. We
let $\tau=j\circ\iota\colon Z\hookrightarrow\overline{X}$.

Now, we note that

$\begin{array}[]{lll}{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{*}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})&\cong&{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{X/k})\\\ &\cong&{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{!}{\mathbb{S}}_{Z},j^{*}\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\\\ &\cong&{\rm
Hom}_{{\mathbf{SH}}(X)}(\iota_{!}{\mathbb{S}}_{Z},j^{!}\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\\\ &\cong&{\rm
Hom}_{{\mathbf{SH}}(\overline{X})}(j_{!}\iota_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\\\ &\cong&{\rm
Hom}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\\\ &\cong&{\rm
Hom}_{{\mathbf{SH}}(\overline{X})}(\tau_{*}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k}),\end{array}$

where the last isomorphism uses $\tau_{!}\cong\tau_{*}$, thanks to the
completeness of $Z$. It follows that there is a canonical (excision)
isomorphism

(8.21)
$E^{i}_{Z}(\overline{X},\Lambda(j))\xrightarrow{\cong}E^{i}_{Z}(X,\Lambda(j)).$

Note also that the map $E^{i}_{Z}(\overline{X},\Lambda(j))\to
E^{i}(X,\Lambda(j))$ factors through $E^{i}_{Z}(\overline{X},\Lambda(j))\to
E^{i}(\overline{X},\Lambda(j))$. Hence, we are reduced to showing that the map
$E^{i}_{Z}(\overline{X},\Lambda(j))\to E^{i}(\overline{X},\Lambda(j))$ factors
through $E^{i}_{c}({X},\Lambda(j))$.

Next, we claim that the composite

(8.22) ${\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\hskip 85.35826pt$ $\hskip 199.16928pt\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k})$

is null-homotopic, where the first arrow is induced by the unit of adjunction
${\mathbb{S}}_{\overline{X}}\to\tau_{*}{\mathbb{S}}_{Z}\cong\tau_{!}{\mathbb{S}}_{Z}$.

To prove the claim, we note that there is a commutative diagram

(8.23) $\textstyle{{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k}).}$

It suffices therefore to show that ${\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k})$ is contractible.

Finally, we observe that there are canonical weak equivalences of spectra

$\begin{array}[]{lll}{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k})&\cong&{\rm
Maps}_{{\mathbf{SH}}(Z)}({\mathbb{S}}_{Z},\tau^{!}\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k})\\\ &\cong&{\rm
Maps}_{{\mathbf{SH}}(Z)}({\mathbb{S}}_{Z},\beta_{*}\alpha^{!}\Sigma^{i,j}{\rm
H}\Lambda_{W/k}),\end{array}$

where $\alpha\colon W\times_{\overline{X}}Z\to W$ and $\beta\colon
W\times_{\overline{X}}Z\to Z$ are the canonical maps. The claim now follows
because $W\times_{\overline{X}}Z=\emptyset$.

Using the claim and the distinguished triangle of spectra (e.g., see [21,
Prop. 2.2.10])

(8.24) ${\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},j_{!}\Sigma^{i,j}{\rm
H}\Lambda_{X/k})\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\hskip 85.35826pt$ $\hskip 227.62204pt\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\iota^{\prime}_{*}\Sigma^{i,j}{\rm
H}\Lambda_{W/k}),$

we see that the map

${\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},\Sigma^{i,j}{\rm
H}\Lambda_{\overline{X}/k})$

factors through

(8.25) ${\rm
Maps}_{{\mathbf{SH}}(\overline{X})}(\tau_{!}{\mathbb{S}}_{Z},\Sigma^{i,j}{\rm
H}\Lambda_{{\overline{X}}/k})\to{\rm
Maps}_{{\mathbf{SH}}(\overline{X})}({\mathbb{S}}_{\overline{X}},j_{!}\Sigma^{i,j}{\rm
H}\Lambda_{X/k}).$

Considering the induced maps on $\pi_{0}$, we get the factorization

(8.26) $E^{i}_{Z}(\overline{X},\Lambda(j))\to E^{i}_{c}(X,\Lambda(j))\to
E^{i}(\overline{X},\Lambda(j)),$

as desired. This concludes the proof. ∎

###  8.4. A functorial property of Gysin map

One knows that the Gysin homomorphism commutes with the pull-back homomorphism
on motivic cohomology in a transverse square. But we need to know the relation
between these two homomorphisms when the square is not Cartesian. We consider
a special case of this situation in this subsection.

We assume that $k$ is a perfect field. We first assume that
$X\in{\mathbf{Sm}}_{k}$ is an integral scheme of dimension one. We let
$Z\subset X$ be a closed subscheme such that $Z_{\rm red}$ is a closed point
$x$ of $X$. Let $S={\rm Spec\,}(k(x))$ and let
$S\xrightarrow{h}Z\xrightarrow{g}X$ be the inclusions. Write $f=gh$ and
$e=\ell({\mathcal{O}}(Z))$. Assume that $g$ is a regular closed immersion.
Since $X$ is regular, $f$ is also a regular closed immersion. We thus have a
diagram

(8.27)
$\textstyle{E^{i}(Z,\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{*}}$$\scriptstyle{h^{*}}$$\scriptstyle{\cong}$$\textstyle{E^{2+i}_{S}(X,\Lambda(1+j))}$$\textstyle{E^{i}(S,\Lambda(j)).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$

######  Lemma 8.10.

Let $m$ be an integer prime to $p$ and $\Lambda={{\mathbb{Z}}}/m$. Then we
have the identity $g_{*}=e(f_{*}\circ h^{*})$.

###### Proof.

Suppose that we know the equality $\eta^{X}_{Z}=e\eta^{X}_{S}$. Then, for any
$i,j$ and $\alpha\in E^{i}(Z,\Lambda(j))$, we get

$g_{*}(\alpha)=\alpha\cup\eta^{X}_{Z}=h^{*}(\alpha)\cup\eta^{X}_{Z}=h^{*}(\alpha)\cup
e\eta^{X}_{S}=ef_{*}(h^{*}(\alpha)),$

where the second identity follows from the construction of the cup product in
[77, Defn. 1.6, p. 509] and the isomorphism (of the unit of adjunction)
${\mathbb{S}}_{Z}\xrightarrow{\cong}h_{*}{\mathbb{S}}_{S}$ in
${\mathbf{SH}}(Z)$. It remains to show that $\eta^{X}_{Z}=e\eta^{X}_{S}$. By
definition of the fundamental classes (see [77, Defn. 2.1]), this is
equivalent to showing that
$c_{1}({\mathcal{L}}_{Z})=e(c_{1}({\mathcal{L}}_{S}))$, where
$c_{1}({\mathcal{L}}_{D})$ denotes the ‘Chern class with support’ of the
divisor $D\subset X$ in $E^{2}_{D}(X,\Lambda(1))$. We are now done by [77,
Prop. 1.43] because ${\rm H}\Lambda_{X/k}\cong{\rm H}\Lambda_{X}$ is an
oriented absolute ring spectrum with additive group law (e.g., see [15, Rem.
3.7]). ∎

Suppose now that $k$ is an imperfect field and $X\in{\mathbf{Sm}}_{k}$ is an
integral scheme of dimension $d\geq 1$. We let $x\in X$ be a closed point and
let $Z={\rm Spec\,}(k(x))\times_{X}X^{\heartsuit}$. We let $y$ be the unique
point in $X^{\heartsuit}$ lying over $x$ so that ${\rm Spec\,}(k(y))=Z_{\rm
red}$. Let $e=\ell({\mathcal{O}}(Z))$. Let $u\colon{\rm
Spec\,}(k(x))\hookrightarrow X$ and $v\colon{\rm Spec\,}(k(y))\hookrightarrow
X^{\heartsuit}$ be the inclusions. Let $f\colon X^{\heartsuit}\to X$ and
$g\colon{\rm Spec\,}(k(y))\to{\rm Spec\,}(k(x))$ be the projections. We have
the refined Gysin maps $u_{*}\colon E^{i}(k(x),\Lambda(j))\to
E^{2d+i}_{x}(X,\Lambda(d+j))$ and $v_{*}\colon E^{i}(k(y),\Lambda(j))\to
E^{2d+i}_{y}(X^{\heartsuit},\Lambda(d+j))$.

######  Lemma 8.11.

Let $m$ be an integer prime to $p$ and $\Lambda={{\mathbb{Z}}}/m$. Then
$e\in\Lambda^{\times}$ and one has a commutative diagram

(8.28)
$\textstyle{E^{i}(k(x),\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{*}}$$\scriptstyle{eg^{*}}$$\textstyle{E^{2d+i}_{x}(X,\Lambda(d+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}}$$\textstyle{E^{i}(k(y),\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v_{*}}$$\textstyle{E^{2d+i}_{y}(X^{\heartsuit},\Lambda(d+j)).}$

###### Proof.

By the Bertini theorem of Altman-Kleiman [3, Thm. 1], we can find an integral
curve $C\subset X$ which passes through $x$ and is smooth over $k$. We now
consider the diagram

(8.29)
$\textstyle{E^{i}(k(x),\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{eg^{*}}$$\textstyle{E^{2+i}_{x}(C,\Lambda(1+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{2d+i}_{x}(X,\Lambda(d+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{i}(k(y),\Lambda(j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{2+i}_{y}(C^{\heartsuit},\Lambda(1+j))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{2d+i}_{y}(X^{\heartsuit},\Lambda(d+j)),}$

where the middle and the right vertical arrows are the pull-back maps, and the
horizontal arrows are the refined Gysin maps of (8.19). We note that
$C^{\heartsuit}=C\times_{X}X^{\heartsuit}$. It follows from [77, Prop. 1.8,
(1) and (3)] that $u_{*}$ (resp. $v_{*}$) is the composition of the two upper
(resp. lower) horizontal arrows, and the right square is commutative. It
suffices therefore to show that the left square is commutative. This allows us
to assume that $d=1$.

We now let ${\rm