A restriction isomorphism for zero-cycles with coefficients in Milnor K-theory
Morten L\"uders
We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups over local and finite fields.
Zero_cycles_koefficients_milnor_k_theory.tex
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\title[Zero-cycles with coefficients in Milnor K-theory]{A restriction isomorphism for zero-cycles with coefficients in Milnor K-theory}
\author{Morten L\"uders}
\address{Fakult\"at f\"ur Mathematik, Universit\"at Regensburg, 93040 Regensburg, Germany}
\email{morten.lueders@mathematik.uni-regensburg.de}

\begin{document}
\begin{abstract}
We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups over local and finite fields.
\end{abstract}

\thanks{The author is supported by the DFG through CRC 1085 \textit{Higher Invariants} (Universit\"at Regensburg).}
\maketitle
\section{Introduction}
Let $\mathcal{O}_K$ be an excellent henselian discrete valuation ring with quotient field $K$ and residue field $k=\mathcal{O}_K/\pi\mathcal{O}_K$ and always assume that $1/n\in k$.
Let $X$ be a smooth and projective scheme over Spec$\mathcal{O}_K$ of fibre dimension $d$. Let $X_K$ denote the generic fibre and $X_0$ the reduced special fibre. By $X_{(p)}$ we denote the set of points $x\in X$ such that $\text{dim}(\overline{\{x\}})=p$, where $\overline{\{x\}}$ denotes the closure of $x$ in $X$.

We call the groups
$$\text{coker}(\oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x))$$
and 
$$H(\oplus_{x\in X^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X^{(d)}}K_{j-d}^Mk(x)\r \oplus_{x\in X^{(d+1)}}K_{j-d-1}^Mk(x))$$
as well as the higher Chow groups
$$CH^j(X_0,j-d)$$
and
$$CH^j(X,j-d)$$
Chow groups of zero-cycles with coefficients in Milnor K-theory. We will see in section \ref{Some identifications} that  the groups $\text{coker}(\oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x))$ and $CH^j(X_0,j-d)$ are isomorphic. The identification of $H(\oplus_{x\in X^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X^{(d)}}K_{j-d}^Mk(x)\r \oplus_{x\in X^{(d+1)}}K_{j-d-1}^Mk(x))$ and $CH^j(X,j-d)$ depends on the Gersten conjecture for a henselian DVR for higher Chow groups if we work integrally. However, which is sufficient for our purposes, they are isomorphic if considered with finite coefficients (see section \ref{Some identifications}).

In this article, we study the restriction homomorphism on higher Chow groups 
$$res^{CH}:CH^j(X,2j-i)\xrightarrow{} CH^j(X_0,2j-i)$$
for $i-j=d$. The restriction homomorphism $res^{CH}$ is defined to be the composition
$$CH^j(X,2j-i)\xrightarrow{}CH^j(X_K,2j-i)\xrightarrow{\cdot(-\pi)}CH^{j+1}(X_K,2j-i+1)\xrightarrow{\partial}CH^j(X_0,2j-i).$$
Here $\cdot(-\pi)$ is the product with a local parameter $-\pi\in CH^1(K,1)=K^{\times}$ defined in \cite[Sec. 5]{Bl86} and $\partial$ is the boundary map coming from the localization sequence for higher Chow groups (see \cite{Le01}). We call the composition $$CH^j(X_K,2j-i)\xrightarrow{\cdot(-\pi)}CH^{j+1}(X_K,2j-i+1)\xrightarrow{\partial}CH^j(X_0,2j-i)$$ a specialisation map and denote it by $sp^{CH}_\pi$. We note that $res^{CH}$ is independent of the choice of $\pi$ whereas $sp^{CH}_\pi$ depends on it. For more details see \cite[Sec. 3]{ADIKMP} or \cite{Lu16}. We denote higher Chow groups with coefficients in a ring $\Lambda$ by $CH^j(X,j-d)_{\Lambda}$. Our main result is the following:
\begin{theorem}\label{thmspchowmil}
Let $\Lambda=\Z/n\Z$. Then the restriction map
$$res^{CH}:CH^j(X,j-d)_{\Lambda}\xrightarrow{} CH^j(X_0,j-d)_{\Lambda}$$
is an isomorphism for all $j$.
\end{theorem}
This was conjectured in \cite[Conj. 10.3]{KEW16} by Kerz, Esnault and Wittenberg. More precisely, they conjecture that the corresponding restriction homomorphism for motivic cohomology
$res:H^{i,j}(X,\mathbb{Z}/n\mathbb{Z})\rightarrow H^{i,j}_{cdh}(X_0,\mathbb{Z}/n\mathbb{Z})$
is an isomorphism for $i-j=d$. 
We note that the case $j=d$ was first proved in \cite{SS10} assuming that $k$ is finite or separably closed and then generalised to arbitrary perfect residue fields in \cite{KEW16} using an idea of Bloch put forward in \cite{EWB16}. 

For $j=d+1$ and $k$ finite, theorem \ref{thmspchowmil} also follows from the Kato conjectures. In fact, Jannsen and Saito observe that for $j=d+1$ and $k$ finite, the \'etale cycle class map
$$\rho_X^{j,j-d}:CH^j(X,j-d)_{\Lambda} \r H^{d+j}_{\text{\'et}}(X,\Lambda(j))$$
fits into the exact sequence
\begin{equation*}
\begin{split}
...\r KH^{0}_{2+a}(X,\Z/n\Z)\r CH^{d+1}(X,a)_{\Lambda}\r H_{\text{\'et}}^{2d+2-a}(X,\Lambda(d+1))\\
\r KH^{0}_{1+a}(X,\Z/n\Z)\r CH^{d+1}(X,a-1)_{\Lambda}\r...,
\end{split}
\end{equation*}
where $KH^0_a(X,\Z/n\Z)$ denotes the homology of certain complexes $C^0_n(X)$ in degree $a$ defined by Kato. For more details see section \ref{sectionkatocomplexes}.

Theorem \ref{thmspchowmil} implies the following two well-known corollaries:
\begin{corollary}
Let $X_K$ be a scheme of dimension $d$ over a local field $K$ with good reduction. Then the groups
\begin{enumerate}
\item $CH^j(X_K,j-d)_{\Lambda}$ are finite for all $j\geq 0$.
\item $CH^j(X_K,j-d)_{\Lambda}=0$ for $j\geq d+3$.
\end{enumerate}
\end{corollary}

\begin{corollary}
Let $X$ be a smooth projective scheme over an excellent henselian discrete valuation ring $\O_K$ with finite or algebraically closed residue field. Then
$$\rho_X^{j,j-d}:CH^j(X,j-d)_{\Lambda} \r H^{d+j}_{\text{\'et}}(X,\Lambda(j))$$
is an isomorphism for all $j$.
\end{corollary}

In the last section, we turn to torsion questions for Chow groups of Zero-cycles with coefficients in Milnor K-theory. We show the following two propositions:
\begin{proposition}
Let $X$ be a smooth and proper scheme over a local field $K$. Let $n>0$ be a natural number prime to the characteristic of $K$. Then for all $j\geq 1$ the groups $$CH^{d+j}(X,j)[n]$$ are finite.
\end{proposition}

\begin{proposition}
Let $X$ be a smooth projective scheme of dimension $d$ over a finite field $k$ of characteristic $p>0$. Then the group
$$CH^{d+1}(X,1)$$
is finite. 
\end{proposition}

\textit{Acknowledgements.} I would like to thank Moritz Kerz for many helpful comments and discussions.

\section{Identification of Zero-cycles with coefficients in Milnor K-theory}\label{Some identifications}
We start by recalling some basic facts about Milnor K-theory.
\begin{definition}
Let $k$ be a field. We define the $n$-th Milnor K-group $K^M_n(k)$ to be the quotient of $(k^{\times})^{\otimes n}$ by the Steinberg group, i.e. the subgroup of $(k^{\times})^{\otimes n}$ generated by elements of the form $a_1\otimes...\otimes a_n$ satisfying $a_i+a_j=1$ for some $1\leq i<j\leq n$. Elements of $K^M_n(k)$ are called symbols and the image of $a_1\otimes...\otimes a_n$ in $K^M_n(k)$ is denoted by $\{a_1,...,a_n\}$.
\end{definition}

One can easily see that in $K^M_2(k)$ the following relations hold:
$$\{x,-x\}=0 \text{   and   } \{x,x\}=\{x,-1\}$$
This implies the following lemma:
\begin{lemma}\label{lemmagenmilnork}
Let $K$ be a discrete valuation ring with ring of integers $A$, local parameter $\pi$ and residue field $k$. Then $K^M_n(k)$ is generated by symbols of the form 
$$\{\pi,u_2,...,u_n\}$$
with $u_i\in A^{\times}$ for $2\leq i \leq n$. 
\end{lemma}
Keeping the notation of lemma \ref{lemmagenmilnork} and denoting the image of $u_i$ in $K^M_n(k)$ by $\bar u_i$, one can show that there exists a unique homomorphism
$$\partial:K^M_n(K)\r K^M_{n-1}(k)$$
satisfying
$$\partial(\{\pi,u_2,...,u_n\})= \{\bar u_2,...,\bar u_n\},$$
called the tame symbol, and a unique homomorphism
$$sp_\pi:K^M_n(K)\r K^M_n(k)$$
satisfying
$$sp_\pi(\{\pi^{i_1}u_1,\pi^{i_2}u_2,...,\pi^{i_n}u_n\})= \{\bar u_1,...,\bar u_n\},$$
called the specialisation map. Note that $sp_\pi$, unlike $\partial$, depends on the choice of a local parameter in $K$ and that $\partial(\{u_1,u_2,...,u_n\})=0$, if $u_i\in A^{\times}$ for all $1\leq i \leq n$.

We now return to the situation of the introduction: Let $\mathcal{O}_K$ be an excellent henselian discrete valuation ring with quotient field $K$ and residue field $k=\mathcal{O}_K/\pi\mathcal{O}_K$ and always assume that $1/n\in k$. 
Let $X$ be a smooth and projective scheme over Spec$\mathcal{O}_K$ of fibre dimension $d$. Let $X_K$ denote the generic fibre and $X_0$ the reduced special fibre. By $X_{(p)}$ we denote the set of points $x\in X$ such that $\text{dim}(\overline{\{x\}})=p$, where $\overline{\{x\}}$ denotes the closure of $x$ in $X$.

We use the following notation for Rost's Chow groups with coefficients in Milnor K-theory (see \cite[Sec. 5]{Ro96}):
$$C_p(X,m)=\bigoplus_{x\in X_{(p)}}(K_{m+p}^Mk(x))\otimes \mathbb{Z}/n\Z$$
$$Z_p(X,m)=\text{ker}[\partial:C_p(X,m)\rightarrow C_{p-1}(X,m)]$$
$$A_p(X,m)=H_p(C_*(X,m))$$
and similarly for $X_0$ (resp. $X_K$) replacing $X$ by $X_0$ (resp. $X_K$). Furthermore, let
$$C_p^g(X,m)=\bigoplus_{x\in X^g_{(p)}}(K_{m+p}^Mk(x))\otimes \mathbb{Z}/n\Z,$$
and
$$Z_p^g(X,m)=\text{ker}[\partial:C_p^g(X,m)\rightarrow C_{p-1}(X,m)]$$
be the corresponding groups supported on cycles in good position, i.e. the sum is taken over all $x\in X_{(p)}$ such that $\overline{\{x\}}$ is flat over $\O_K$.
Note that $C_k(X,-k)=Z_k(X)\otimes \Z/n\Z$, the group of $k$-cycles on $X$, i.e. the free abelian group generated by $k$-dimensional closed subschemes of $X$, tensored with $\Z/n\Z$.

Let now $\pi$ be a local parameter of $\O_K$. We define the restriction map
$$res_\pi: C_p(X,m)\rightarrow C_{p-1}(X_0,m+1)$$
to be the composition
$$res_\pi: C_p(X,m)\rightarrow C_{p-1}(X_K,m+1)\xrightarrow{\cdot\{-\pi\}} C_{p-1}(X_K,m+2)\xrightarrow{\partial}C_{p-1}(X_0,m+1),$$
where $C_p(X,m)\rightarrow C_{p-1}(X_K,m+1)$ is defined to be the identity on all elements supported on $X_{(p)}\setminus {X_0}_{(p)}$ and zero on ${X_0}_{(p)}$ and $\partial$ is the boundary map induced by the tame symbol. For the fact that this composition is compatible with the corresponding cycle complexes see \cite[Sec. 4]{Ro96}. For a detailed discussion of $res$ see \cite[Sec. 2]{Lu16}. We just recall that the restriction map $res_\pi$ depends on choice of $\pi$ but the induced map $res: A_p(X,m)\rightarrow A_{p-1}(X_0,m+1)$ is independent of such a choice. From now on, we will fix a $\pi\in\O_K$ and write $res$ instead of $res_\pi$. 

We now prove the identifications of the two versions of Chow groups of zero-cycles with coefficients in Milnor K-theory stated in the introduction.
\begin{proposition}\label{identificationwithZero-cycles with coefficients in Milnor K-theory}
For all $j\geq 0$, there are the following isomorphisms:
\begin{enumerate}
\item $CH^j(X_0,j-d)\cong \text{coker}(\oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x))$.
\item $CH^j(X,j-d)_\Lambda\cong A_1(X,j-d-1)$.

\end{enumerate}
\end{proposition}
\begin{proof}
In \cite[Sec. 10]{Bl86}, Bloch proves the existence of the spectral sequence
\begin{equation}\label{spseq1}
^{CH}E^{p,q}_1=\oplus_{x\in X_0^{(p)}}CH^{r-p}(\text{Spec}k(x),-p-q)\Rightarrow CH^r(X_0,-p-q).
\end{equation} 
Using the localization sequence for higher Chow groups for schemes over a regular noetherian base (see \cite{Le01}) and a limit argument, one also gets the existence of the spectral sequence 
\begin{equation}\label{spseq2}
^{CH}E^{p,q}_1=\oplus_{x\in X^{(p)}}CH^{r-p}(\text{Spec}k(x),-p-q)\Rightarrow CH^r(X,-p-q).
\end{equation} 

Setting $j=r$, and noting that $CH^r(k(x),r)\cong K^M_r(k(x))$, it follows from spectral sequence (\ref{spseq1}) that $CH^j(X_0,j-d)$ is isomorphic to the cokernel of 
$$\oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x)$$
since $E^{\bullet, \geq -j+1}_1=0$ as $CH^s(k(x),t)=0$ for $s>t$. 

Similarly, we get from spectral sequence (\ref{spseq2}) that $CH^j(X,j-d)_\Lambda$ is isomorphic to $A_1(X,j-d-1)$, noting that the map
$$\oplus_{x\in X^{(d)}}CH^{j-d}(k(x),j-d+1)_\Lambda\r \oplus_{x\in X^{(d+1)}}CH^{j-d-1}(k(x),j-d)_\Lambda$$
is surjective for all $j$ (see figure \ref{figure2}). This can be seen as follows: Let $x\in X^{(d+1)}$ and let $y$ be the generic point of a regular lift $Z$ of $x$ to $X$ which is flat over $\O_K$. Now by the Beilinson-Lichtenbaum conjecture (see theorem \ref{Beillichenbaum}), $CH^{j-d}(k(x),j-d+1)_\Lambda$ is isomorphic to $H_{}^{j-d-1}(k(y),\Lambda(j-d))$ and $CH^{j-d-1}(k(x),j-d)_\Lambda$ is isomorphic to $H_{}^{j-d-2}(k(x),\Lambda(j-d-1))$. The assertion now follows from the surjectivity of the map
$$\partial:H_{}^{j-d-1}(k(y),\Lambda(j-d))\r H_{}^{j-d-2}(k(x),\Lambda(j-d-1)).$$
Since $\O_{Z,x}$ is henselian, $H_{}^{j-d-2}(k(x),\Lambda(j-d-1))\cong H_{}^{j-d-2}(\O_{Z,x},\Lambda(j-d-1))$ by rigidity for \'etale cohomology. An element $\alpha\in H_{}^{j-d-2}(k(x),\Lambda(j-d-1))$ corresponding to an element $\alpha'\in H_{}^{j-d-2}(\O_{Z,x},\Lambda(j-d-1))$ lifts to an element $\alpha'\cup s\in H_{}^{j-d-1}(k(y),\Lambda(j-d))$, $s$ being a generator of the maximal ideal of $\O_{Z,x}$, with $\partial(\alpha'\cup s)=\alpha$ (see also \cite[Lem. 1.4 (2)]{Ka86}).
\end{proof}
\begin{remark}
\begin{enumerate}
\item For similar identifications for $X_0$ with motivic cohomology see also \cite{RS}.
\item In order to show the isomorphism 
\begin{equation*}
\begin{split} CH^j(X,j-d)\cong             \\ 
H(\oplus_{x\in X^{(d-1)}}K_{j-d+1}^Mk(x)\rightarrow \oplus_{x\in X^{(d)}}K_{j-d}^Mk(x)\r \oplus_{x\in X^{(d+1)}}K_{j-d-1}^Mk(x))
\end{split}
\end{equation*}
integrally, one would need to show the Gersten conjecture for a henselian DVR for higher Chow groups, i.e. the exactness of the seuqence
$$0\r CH^r(\text{Spec}A,q)\r CH^r(\text{Spec}K,q)\r CH^{r-1}(\text{Spec}k,q-1)\r 0$$
for a henselian discrete valuation ring $A$ with field of fractions $K$ and residue field $k$.
\item Let $A$ be as in $(2)$. If $k$ is of characteristic $p>0$, then the sequence 
$$0\r CH^r(A, \Z/p^r\Z, q)\r CH^r(K, \Z/p^r\Z, q)\r CH^{r-1}(k, \Z/p^r\Z, q-1)\r 0$$
is exact. This follows from the fact that in this case 
$CH^{r-1}(\text{Spec}k, \Z/p^r\Z, q-1)=0$ for $r\neq q$ by \cite[Thm. 1.1]{GL00} and that
$$CH^r(K, \Z/p^r\Z, r)\cong K^M_r(K)/p^r\r K^M_{r-1}(k)/p^r\cong CH^{r-1}(k, \Z/p^r\Z, r-1)$$
is surjective which implies that the long exact localization sequence
$$...\r CH^r(A, \Z/p^r\Z, q)\r CH^r(K, \Z/p^r\Z, q)\r CH^{r-1}(k, \Z/p^r\Z, q-1)\r ...$$
splits (see also \cite[Cor. 4.3]{Ge04}).
\item If $k$ is finite, then $CH^j(X_0,j-d)=0$ for $j>d+1$ since $K_2(k)=0$ in that case. If $K$ is a local field, then $CH^j(X_K,j-d)_\Lambda=0$ for $j>d+2$ since $K^M_n(K)$ is uniquely divisible for $n\geq 3$ (see \cite[VI. 7.1]{We13}). 
\end{enumerate}
\end{remark}

\begin{figure}
$$\begin{xy} 
  \xymatrix@-1.5em{
  &  & d & d+1 \\
 -j+1 & 0 &      0   &   0      \\
 -j & ... & \oplus_{x\in X^{(d)}}CH^{j-d}(k(x),j-d) \ar[r]^{}  &      \oplus_{x\in X^{(d+1)}}CH^{j-d-1}(k(x),j-d-1) \\
 -j-1 & ... & \oplus_{x\in X^{(d)}}CH^{j-d}(k(x),j-d+1) \ar[r]^{}  &   \oplus_{x\in X^{(d+1)}}CH^{j-d-1}(k(x),j-d)    \\
  & ... &      ...   &   ...           
  }
\end{xy} $$ 
\caption{Table of $^{CH}E^{p,q}_1$ for $X/\O_K$.}
\label{figure2}
\end{figure}


\section{Relation with Kato complexes}\label{sectionkatocomplexes}
In this section we recall some facts about the Kato conjectures which we will need in the following sections.

Let $X$ be an excellent scheme. In \cite{Ka86}, Kato defines the following complexes:
\begin{equation*}
\begin{split}
C^i_n(X): ...\r \oplus_{x\in X_a}H^{i+a+1}(k(x),\Z/n(i+a))\r ...\r \\
\oplus_{x\in X_1}H^{i+2}(k(x),\Z/n(i+1))\r \oplus_{x\in X_0}H^{i+1}(k(x),\Z/n(i))
\end{split}
\end{equation*}
Here the term $\oplus_{x\in X_a}H^{i+a+1}(k(x),\Z/n(i+a))$ is placed in degree $a$. We denote the homology of $C^i_n(X)$ in degree $a$ by $KH^i_a(X,\Z/n\Z)$. 

The complex $C^0_n(X)$ for a proper smooth scheme $X$ over a finite field or the ring of integers in a $1$-local field is the subject of the study of the Kato conjectures. The Kato conjectures say the following and have been fully proved by Jannsen, Kerz and Saito (see \cite{KeS12}):

\begin{conj}\label{Katoconjfinitefield}
Let $X$ be proper smooth scheme over a finite field. Then
$$KH^0_a(X,\Z/n\Z)=0 \quad \text{   for } a> 0.$$
\end{conj}
\begin{conj}\label{Katoconjlocalring}
Let $X$ be a regular scheme proper and flat over $\text{Spec}(\O_k)$, where $\O_k$ is the ring of integers in a local field. Then
$$KH^0_a(X,\Z/n\Z)=0 \quad\text{   for } a\geq 0.$$
\end{conj}

In \cite[Lem. 6.2]{JS}, Jannsen and Saito relate the complex $C^0_n(X)$ for a smooth scheme $X$ over a finite field to the \'etale cycle class map
$$\rho^{r,2r-s}_X:CH^r(X,2r-s)_{\Lambda}\r H_{\text{\'et}}^{s}(X,\Lambda(r))$$
for $r=d$. More presicely, they show that there is an exaxt sequence 
\begin{equation}\label{JSexactsequence}
\begin{split}
...\r KH^0_{q+2}(X,\Z/n\Z)\r CH^d(X,q)_{\Lambda}\r H_{\text{\'et}}^{2d-q}(X,\Lambda(d))\\
\r KH^0_{q+1}(X,\Z/n\Z)\r CH^d(X,q-1)_{\Lambda}\r...
\end{split}
\end{equation}
This sequence is a tool to deduce finiteness results for higher zero cycles with finite coefficients from the Kato conjectures. The proof of the exactness of (\ref{JSexactsequence}) uses the coniveau spectral sequence for the domain and target of $\rho^{r,q}_X$ and the following theorem of Voevodsky:
\begin{theorem}(Beilinson-Lichtenbaum conjecture, see \cite{Vo11})\label{Beillichenbaum}
Let $X$ be a smooth scheme over a field. Then the \'etale cycle map
$$\rho^{r,2r-s}_X:CH^r(X,2r-s)_{\Lambda}\r H_{\text{\'et}}^{s}(X,\Lambda(r))$$
is an isomorphism for $s\leq r$.
\end{theorem}

We recall the following proposition from \cite[Prop. 9.1]{KEW16}:

\begin{proposition}\label{katofinitefieldcase}
Let $X$ be a proper smooth scheme over a finite or algebraically closed field. Then the \'etale cycle map
$$\rho^{j,j-d+a}_X:CH^j(X,j-d+a)_{\Lambda}\r H_{\text{\'et}}^{j+d-a}(X,\Lambda(j))$$
is an isomorphism for all $j\geq d$ and all $a$ except possibly if $k$ is finite, $j=d$ and $a=-1$. In particular, the groups $CH^j(X,j-d+a)_{\Lambda}$ are finite if $j\geq d, a\geq 0$. 
\end{proposition}
\begin{proof}
We consider the spectral sequences
$$ ^{CH}E^{p,q}_1(X)=\oplus_{x\in X^{(p)}}CH^{j-p}(\text{Spec}k(x),-p-q)\Rightarrow CH^j(X,-p-q)$$
and
$$ ^{\text{\'et}}E_1^{p,q}(X,\Lambda(j))=\oplus_{x\in X^p}H^{q-p}(k(x),\Lambda(j-p))\Rightarrow H_{\text{\'et}}^{p+q}(X,\Lambda(j)).$$
The \'etale cycle class map $\rho^{r,q}_X$ induces a map of spectral sequences
$$\rho^{r,q}_X: ^{CH}E^{p,q}_1\r ^{\text{\'et}}E_1^{p,q+2j}$$
which by theorem \ref{Beillichenbaum} is an isomorphism for $q\leq -j$. By cohomological dimension, the difference between the two spectral sequences is given by
$$^{\text{\'et}}E_1^{\bullet,j+1}=C^{j-d}_n(X)$$
which is equal to the zero-complex by cohomological dimension under the assumption of $j>d$ or that $k$ is an algebraically closed field because of cohomological dimension. If $j=d$, then the complex $^{\text{\'et}}E_1^{\bullet,d+1}=C^{0}_n(X)$ is exact except for possibly the last term on the right due to conjecture \ref{Katoconjfinitefield}. 
\end{proof}

We now turn to the arithmetic case (see also \cite[Sec. 9]{KeS12}).
\begin{proposition}\label{katochetale}
Let $X$ be a proper smooth scheme over a henselian discrete valuation ring $\O_K$ with finite residue field $k$. Let $d$ be the relative dimension of $X$ over $\O_K$. Then for $j=d+1$, there is an exact sequence
\begin{equation}
\begin{split}
...\r KH^{0}_{2+a}(X,\Z/n\Z)\r CH^{d+1}(X,a)_{\Lambda}\r H_{\text{\'et}}^{2d+2-a}(X,\Lambda(d+1))\\
\r KH^{0}_{1+a}(X,\Z/n\Z)\r CH^{d+1}(X,a-1)_{\Lambda}\r...
\end{split}
\end{equation}
and for $j>d+1$, there are isomorphisms
$$CH^j(X,j-d+a)_{\Lambda}\r H_{\text{\'et}}^{j+d-a}(X,\Lambda(j)).$$
\end{proposition}
\begin{proof}
We keep the notation of the proof of proposition \ref{katofinitefieldcase}. Like there, we get a map of spectral sequences
$$\rho^{r,q}_X: ^{CH}E^{p,q}_1\r ^{\text{\'et}}E_1^{p,q+2j}$$
which by theorem \ref{Beillichenbaum} is an isomorphism for $q\leq -j$. The difference between the two spectral sequences is given by
$$^{\text{\'et}}E_1^{\bullet,j+1}=C^{j-d-1}_n(X)$$
if $j\geq d+1$ since all other rows vanish by cohomological dimension and $C^{j-d-1}_n(X)=0$ for $j\geq d+2$ again by cohomological dimension. This implies the proposition.
\end{proof}

\begin{remark}
For $X$ be a scheme over an excellent henselian discrete valuation ring with finite field and $j=d$, we are in the situation of \cite{SS10} which is more complex since there are two rows ($^{\text{\'et}}E_1^{\bullet,j+1}=C^{-1}_n(X)$ and $^{\text{\'et}}E_1^{\bullet,j+2}$) which might not be quasi-isomorphic to zero. In \cite{SS10} Saito and Sato show that $KH^{-1}_{a}(X,\Q_n/\Z_n)=0$ for $a=2,3$. 
\end{remark}

We keep the notation of proposition \ref{katochetale}. It follows from conjecture \ref{Katoconjlocalring} that 
$$res^{CH}:CH^{d+1}(X,a)_{\Lambda}\r CH^{d+1}(X_0,a)_{\Lambda}$$
is an isomorphism for all $a$. In the next section, we generalise this result for $a=1$ to arbitrary perfect residue fields. It remains an open problem if $res^{CH}:CH^{d+1}(X,a)_{\Lambda}\r CH^{d+1}(X_0,a)_{\Lambda}$ is an isomorphism for arbitrary perfect residue fields for all $a$. 

\section{Main theorem}
We keep the notation of the introduction. In this section we prove theorem \ref{thmspchowmil} of the introduction.
By the identifications of proposition \ref{identificationwithZero-cycles with coefficients in Milnor K-theory}, this comes down to studying the following commutative diagram:

$$\begin{xy} 
  \xymatrix{
  \oplus_{x\in X^{(d-1)}}K_{j-d+1}^Mk(x) \ar[r]^{sp} \ar[d]^{\partial}  &  \oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x) \ar[d]^{\partial}   \\
     \oplus_{x\in X^{(d)}}K_{j-d}^Mk(x) \ar[r]^{sp} \ar[d]^{\partial}   & \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x)  \\
     \oplus_{x\in X^{(d+1)}}K_{j-d-1}^Mk(x) &
  }
\end{xy} $$  

We first note the surjectivity of $res^{CH}$ resp. $res$. 
\begin{proposition}\label{surjchmiln}
The specialisation map
$$res:A_1(X,j-d-1)\xrightarrow{} A_0(X_0,j-d)$$
is surjective for all $j$. 
\end{proposition}
\begin{proof}
Let $\{\bar{u}_1,...,\bar{u}_{j-d}\}\in K_{j-d}^Mk(x)$ for some $x\in X_0^{(d)}$. Let $y\in X^{(d)}$ be the generic point of a lift $Z$ of $x$ which intersects $X_0$ transversally in $x$. Let $A$ be the stalk of $Z$ at $y$ and denote by $u_i\in A^{\times}$ a lift of $\bar{u}_i$ to the units of $A$. Then $res(\{u_1,...,u_{j-d}\})=\{\bar{u}_1,...,\bar{u}_{j-d}\}$ and $\partial(\{u_1,...,u_{j-d}\})=0$. 
\end{proof}

\begin{figure}
$$\begin{xy} 
  \xymatrix@-1.5em{
 j & ... & \oplus_{x\in X^{(1)}}H^{j-1}(k(x),\Lambda(j-1)) \ar[r]^{}  &      \oplus_{x\in X^{(2)}}H^{j-2}(k(x),\Lambda(j-2)) \\
 j-1 & ... & \oplus_{x\in X^{(1)}}H^{j-2}(k(x),\Lambda(j-1)) \ar[r]^{}  &      \oplus_{x\in X^{(2)}}H^{j-3}(k(x),\Lambda(j-2)) \\
  & ... &      ...   &   ...      \\
   2 & ...&    \oplus_{x\in X^{(1)}}H^1(k(x),\Lambda(j-1)) \ar[r]^{}    &  \oplus_{x\in X^{(2)}}H^0(k(x),\Lambda(j-2))  \\
   1 & ...&  \oplus_{x\in X^{(1)}}H^0(k(x),\Lambda(j-1)) & 0 \\
  0 & ... &      0   &   0      \\
   & 0 & 1 & 2
  }
\end{xy} $$ 
\caption{Table of $E^1_{p,q}(X,\Lambda)$ for $X/\O_K$ and $d=1$.}
\label{figure1}
\end{figure}

\begin{keylemma}\label{keylemmazerowithkoeffmilnork}
Let 
$$\alpha\in \text{ker}(Z_1^g(X,j-d-1)\r Z_0(X_0,j-d)),$$
then $\alpha\equiv 0 \in A_0(X,j-d-1)$. In particular, there is a well-defined map $$\phi: Z_0(X_0,j-d)\r A_1(X,j-d-1).$$
\end{keylemma}
\begin{proof}
We start with the case of relative dimension $d= 1$. Let $\Lambda:=\mathbb{Z}/n$ and $\Lambda(q):=\mu_n^{\otimes q}$. We consider the coniveau spectral sequence
$$E_1^{p,q}(X,\Lambda(j))=\oplus_{x\in X^p}H^{q-p}(k(x),\Lambda(j-p))\Rightarrow H_{\text{\'et}}^{p+q}(X,\Lambda(j))$$
for $X$ and $X_0$ respectively (see figure \ref{figure1} for $E^1_{p,q}(X,\Lambda)$ for $X/\O_K$ and $d=1$) and the norm residue isomorphism $K^M_n(k)/m\cong H^n(k,\mu_m^{\otimes n})$ to show that there are injective edge morphisms 
$$A_1(X,j-d-1)=E_2^{d,j}(X)\hookrightarrow H^{d+j}_{\text{\'et}}(X,\Lambda(j))$$ 
and 
$$A_0(X_0,j-d)=E_2^{d,j}(X_0)\hookrightarrow H^{d+j}_{\text{\'et}}(X_0,\Lambda(j)).$$ 
The injectivity in the second case, i.e. for $X_0$, is trivial since we just have two non-trivial columns for dimensional reasons. In the first case, we have three columns but $E_2^{2,j}(X,\Lambda(c))$ is equal to zero  for all $j$ since the map
$$\oplus_{x\in X^{(1)}}H^{j-2}(k(x),\Lambda(j-1)) \r     \oplus_{x\in X^{(2)}}H^{j-3}(k(x),\Lambda(j-2))$$
is surjective by the same arguments as in the proof of \ref{identificationwithZero-cycles with coefficients in Milnor K-theory}.
The restriction map induces a map between the respective spectral sequences for $X$ and $X_0$ and therefore a commutative diagram

$$\begin{xy} 
  \xymatrix{
  A_1(X,j-d-1) \ar[r]^{} \ar@{^{(}->}[d]_{} & A_0(X_0,j-d)  \ar@{^{(}->}[d]^{}  \\
     H^{d+j}_{\text{\'et}}(X,\Lambda(j)) \ar[r]^{\cong}  & H^{d+j}_{\text{\'et}}(X_0,\Lambda(j))  
  }
\end{xy} $$ 
whose lower horizontal morphism is an isomorphism by proper base change. It follows that $A_1(X,j-d-1)\rightarrow A_0(X_0,j-d)$ is injective. 

Let now $d>1$. Let 
$$\alpha\in \text{ker}(Z_1^g(X,j-d-1)\r Z_0(X_0,j-d)).$$  By definition, 
$$\alpha=\sum_{x\in X^g_{(1)}} \alpha_x\in  \text{ker}(res:\oplus_{x\in X^g_{(1)}}K_{j-d}^Mk(x) \r     \oplus_{x\in X_0^{(d)}}K_{j-d}^Mk(x))$$
and $\alpha\in \text{ker}(\partial:\oplus_{x\in X^{(d)}}K_{j-d}^Mk(x) \r\oplus_{x\in X^{(d+1)}}K_{j-d-1}^Mk(x))$ with $\alpha_x\in K_{j-d}^Mk(x)$.
We may assume that 
$$\alpha\in  \text{ker}(res:\oplus_{x\in X^g_{(1)}}K_{j-d}^Mk(x) \r  K_{j-d}^Mk(x))$$ for some $x\in X_0^{(d)}.$

Let $y\in X^g_{(1)}$ be the generic point of a lift $Z_1$ of $x$ which intersects $X_0$ transversally in $x$.
We may now assume that 
$$\alpha=(\alpha_y,\alpha_z)\in  \text{ker}(res:K_{j-d}^Mk(y)\oplus K_{j-d}^Mk(z) \r  K_{j-d}^Mk(x))$$ for some $z\in X^{(d)}.$ This follows from the fact that for every $z\in X^{(d)}$, we can lift $res(\alpha_z)$ to an element $\alpha_y\in K_{j-d}^Mk(y)$ such that $\partial(\alpha_y)=\partial(\alpha_z)$. This follows from the observation that if $\partial(\alpha_y)-\partial(\alpha_z)=\sum\{\bar{u}_1,...,\bar{u}_{j-d-1}\}\neq 0$, then choosing $u_i$ to be a lift of $\bar{u}_i$, for all $1\leq i\leq j-d-1$, which is a unit in the discrete valuation ring associated to $k(y)$, then we can modify $\alpha_y$ summandwise by $\{\pi,u_1,...,u_{j-d}\}$ since $res(\{\pi,u_1,...,u_{j-d}\})=0$ and $\partial(\{\pi,u_1,...,u_{j-d}\})=\{\bar{u}_1,...,\bar{u}_{j-d-1}\}$ until we get the claim.

We now apply an idea of Bloch to reduce our situation to the case that $Z_1=\overline{\{y\}}$ intersects $X_0$ transversally and that $Z_2=\overline{\{z\}}$ is regular (see \cite[App.]{EWB16}). Let $\tilde Z_2$ be the normalisation of $Z_2$. Since $\mathcal{O}_K$ is excellent, $\tilde Z_2\r Z_2$ is finite and projective. This implies that there is an imbedding $\tilde Z_2\hookrightarrow X':=X\times_{\text{Spec}\mathcal{O}_K}\mathbb{P}^N$ such that the following diagram commutes:
$$\begin{xy} 
  \xymatrix{
  \tilde Z_2 \ar[r]^{} \ar[d]_{} & X'=X\times_{\text{Spec}\mathcal{O}_K}\mathbb{P}^N  \ar[d]^{pr_X}  \\
     Z_2 \ar[r]^{} \ar[d]_{} & X \ar[d]^{}  \\
     \text{Spec}\mathcal{O}_K \ar[r]^{=}     &  \text{Spec}\mathcal{O}_K
  }
\end{xy} $$
Let $(\tilde Z_2\cap X_0')_{red}= x'$ for $x'$ an integral zero-dimensional subscheme of $X'_0$. Let $\tilde Z_1$ be a regular lift of $x'$ in $Z_1\times \mathbb{P}^N\subset X'$ which has intersection number $1$ with $X_0'$. We denote the generic points of $\tilde Z_1$ and $\tilde Z_2$ by $y'$ and $z'$ respectively. Note first now that $\alpha_z\in K_{j-d}^Mk(z')$. Then, taking into account ramification, we can lift $res(\alpha_z)\in K_{j-d}^Mk(x')$ to an element $\alpha_y'\in K_{j-d}^Mk(y')$ such that $(\alpha_y',\alpha_z)$ lies in the kernel of $res:\oplus_{x\in X'^{(d+N)}}K_{j-d+1}^Mk(x) \r \oplus_{x\in X_0'^{(d+N)}}K_{j-d}^Mk(x)$ as well as the kernel of $\partial:\oplus_{x\in X'^{(d+N)}}K_{j-d}^Mk(x) \r \oplus_{x\in X'^{(d+N-1)}}K_{j-d}^Mk(x)$ and such that, furthermore, we have that $pr_X((\alpha_y',\alpha_z))=(\alpha_y,\alpha_z)$.

It remains to show that $(\alpha_y',\alpha_z)$ is in the image of the boundary map $$\partial:\oplus_{x\in X'^{(d+N+1)}}K_{j-d+1}^Mk(x) \r     \oplus_{x\in X'^{(d+N)}}K_{j-d}^Mk(x).$$ 
Using a standard norm argument, we may assume that the residue field of $\text{Spec}\mathcal{O}_K$ is infinite. We proceed by an induction on the relative dimension of $X$ over $\text{Spec}\mathcal{O}_K$. By standard Bertini arguments (see f.e. \cite[Sec. 4]{KEW16} or \cite[Lem. 2.6]{Lu16}), we can find smooth closed subschemes $S_1,S_2\subset X'$ with the following properties:
\begin{enumerate}
         \item $S_1$ has fiber dimension one, $S_2$ has fiber dimension $d-1$.
         \item $S_1$ contains $\tilde Z_1$, $S_2$ contains $\tilde Z_2$.
         \item The intersection $S_1\cap S_2\cap X'_0$ consist of reduced points.
      \end{enumerate}
Let $\tilde Z_3$ denote the component of $S_1\cap S_2$ containing $x'$ and let $t'$ denote its generic point. 
Then again there is an $\alpha_{t'}\in K_{j-d}^Mk(t')$ such that
$res(\alpha_{t'})=res(\alpha_{y'})=-res(\alpha_{z})$ and $\partial(\alpha_{t'})=\partial(\alpha_{y'})=-\partial(\alpha_{z})$. 
Now by our induction assumption, both $(\alpha_y',\alpha_{t'})$ and $(\alpha_x,\alpha_{t'})$ map to zero in $A_1(X',j-d-1)$ and therefore so does $(\alpha_y',\alpha_z)$.
\end{proof}

\begin{proposition}\label{maintheoremintext}
The restriction map
$$res^{CH}:CH^j(X,j-d)_{\Lambda}\xrightarrow{} CH^j(X_0,j-d)_{\Lambda}$$
is an isomorphism. 
\end{proposition}
\begin{proof}
By the identification of proposition \ref{identificationwithZero-cycles with coefficients in Milnor K-theory}, it suffices to show that $res:A_1(X,j-d-1)\xrightarrow{} A_0(X_0,j-d)$ is an isomorphism.

We need to show that the map $\phi:Z_0(X_0,j-d)\r A_1(X,j-d-1)$ factorises through the group $A_0(X_0,j-d)$. In other words, we need to show that there is a $\overline{\phi}: A_0(X_0,j-d)\r A_1(X,j-d-1)$ such that the following diagram commutes:
$$\begin{xy} 
  \xymatrix{
    A_0(X_0,j-d) \ar[dr]^{\overline{\phi}}  &   \\
     Z_0(X_0,j-d) \ar[r]^{\phi} \ar[u]^{} & A_1(X,j-d-1)
  }
\end{xy} $$ 
Then $res\circ \overline{\phi}= id$ and since $\overline{\phi}$ is surjective the result follows.

Let $\alpha_0=(\alpha_0^1,...,\alpha_0^{j-d+1})\in C_1(X_0,j-d)=\oplus_{x\in X_0^{(d-1)}}K_{j-d+1}^Mk(x)$ be supported on some $x\in X_0^{(d-1)}$. As in the proof of \cite[Lem. 7.2]{SS10}, we can find a relative surface $Z\subset X$ containing $x$ which is regular at $x$  and such that $Z\cap X_0$ contains $\overline{\{x\}}$ with multiplicity $1$. Let $Z_0$ be the special fiber of $Z$ and denote by $\cup_{i\in I} Z_0^{(i)}\cup \overline{\{x\}}$ the union of the pairwise different irreducible components of $Z_0$. Here the irreducible components different from $\overline{\{x\}}$ are indexed by $I$. Let $z$ be the generic point of $Z$. Now as in the proof of \cite[Lem. 2.1]{Lu16}, we can for all $1\leq t\leq j-d+1$ find a lift $\alpha^t\in k(z)^{\times}$ of $\alpha_0^t$ which specialises to $\alpha_0^t$ in $k(x)^{\times}$ and to $1$ in $K(Z_0^{(i)})^{\times}$ for all $i\in I$. Let $\alpha=(\alpha^1,...,\alpha^{j-d+1})$. Then $\phi(\partial(\alpha_0))=\partial(\alpha)=0$ in $A_1(X,j-d-1)$ which implies the above factorisation.

The surjectivity of $\overline{\phi}$ follows from the surjectivity of $\phi$ which follows from key lemma \ref{keylemmazerowithkoeffmilnork}: 
Let $\alpha\in A_1(X,j-d-1)$. By arguments as in the last paragraph, one sees that $Z^g_1(X,j-d-1)$ generates $A_1(X,j-d-1)$. We may therefore asssume that $\alpha\in Z^g_1(X,j-d-1)$. Let $\alpha_0$ be the restriction of $\alpha$ to $Z_0(X-0,j-d)$ and $\alpha'$ be a lift of $\alpha_0$ to $Z^g_1(X,j-d-1)$. Then, by key lemma \ref{keylemmazerowithkoeffmilnork}, we have that $\alpha\equiv \alpha'\in A_1(X,j-d-1)$.
\end{proof}

\section{Applications}\label{applications}
We list some applications of proposition \ref{maintheoremintext}:

\begin{corollary}\label{cor1}
Let $X$ be a smooth projective scheme of relative dimension $d$ over an excellent henselian discrete valuation ring $\O_K$ with finite or algebraically closed residue field. Then
$$\rho_X^{j,j-d}:CH^j(X,j-d)_{\Lambda} \r H^{d+j}_{\text{\'et}}(X,\Lambda(j))$$
is an isomorphism for all $j$. 
\end{corollary}
\begin{proof}
Consider the diagram
$$\begin{xy} 
  \xymatrix{
  KH_2^{j-d-1}(X,\Z/n\Z) \ar[r]^{} & CH^j(X,j-d)_{\Lambda} \ar[r]^{\rho_X^{j,j-d}} \ar[d]_{\cong} & H^{d+j}_{\text{\'et}}(X,\Lambda(j)) \ar[d]^{\cong} \ar[r]^{} & KH_1^{j-d-1}(X,\Z/n\Z) \\
    & CH^j(X_0,j-d)_{\Lambda} \ar[r]^{\cong}  & H^{d+j}_{\text{\'et}}(X_0,\Lambda(j)). &
  }
\end{xy} $$ 
The left vertical isomorphism follows from proposition \ref{maintheoremintext}, the lower horizontal isomorphism from proposition \ref{katofinitefieldcase} and the right vertical isomorphism from proper base change. This implies that $\rho_X^{j,j-d}$ is an isomorphism for all $j$. 
\end{proof}

\begin{corollary}\label{cor2}
Let $X_K$ be a smooth projective scheme of dimension $d$ over a local field $K$ with good reduction. Then the groups
\begin{enumerate}
\item $CH^j(X_K,j-d)_{\Lambda}$ are finite for all $j\geq 0$.
\item $CH^j(X_K,j-d)_{\Lambda}=0$ for $j\geq d+3$.

\end{enumerate}
\end{corollary}
\begin{proof} Consider the localisation sequence
$$CH^{j-1}(X_0,j-d)_{\Lambda}\xrightarrow{i_*} CH^j(X,j-d)_{\Lambda}\xrightarrow{j^*} CH^j(X_K,j-d)_{\Lambda}\xrightarrow{}CH^{j-1}(X_0,j-d-1)_{\Lambda}.$$
By proposition \ref{maintheoremintext} and proposition \ref{katofinitefieldcase}, the groups $CH^j(X,j-d)_{\Lambda}$ are finite for all $j$ and vanish for $j\geq d+2$. By proposition \ref{katofinitefieldcase}, the groups $CH^{j-1}(X_0,j-d-1)_{\Lambda}$ are finite for all $j$ and vanish for $j\geq d+3$. This implies that $CH^j(X_K,j-d)_{\Lambda}$ is finite for all $j$, vanishes for $j\geq d+3$.
\end{proof}

\section{Torsion for zero-cycles with coefficients in Milnor K-theory}

In this section we prove finiteness theorems for torsion subgroups of some higher Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth (projective) schemes over $p$-adic local fields and finite fields. These theorems generealise results of \cite{CSS82} (see also \cite{CSS83}) and \cite{Sz00}. Many of the proofs are very similar.

\begin{notat} For an abelian groups $A$ we denote by $A[n]$ the kernel of the multiplication by $n$. For a prime $l$ we denote by $A\{l\}$ the $l$-primary torsion subgroups of $A$ and by $A_{\text{tors}}$ the entire torsion subgroup of $A$. 

For $X$ a scheme we denote by $\mathcal{H}^q(\mu_n^{\otimes m})$ the Zariski sheaf associated to the presheaf $U\mapsto H^q(U,\mu_n^{\otimes m})$.
\end{notat}

Let $X$ be a smooth variety over a field. Recall that by \cite{BO74} the Leray spectral sequence associated to the morphism of sites $X_{\text{\'et}}\r X_{\text{Zar}}\r \text{Spec}k$
\begin{equation}\label{spseqchangeofsites}
E_2^{p,q}=H^p(X,\mathcal{H}^q(\mu_n^{\otimes m}))\Rightarrow H^{p+q}_{\text{\'et}}(X,\mu_n^{\otimes m})
\end{equation}
and the coniveau spectral sequence
\begin{equation}\label{spseqchangeofsites2}
E_1^{p,q}=\oplus_{x\in X^p}H^{q-p}(k(x),\mu_n^{\otimes m-p})
\end{equation}
agree from $E_2$ onwards and that therefore in particular
$$H^{p}(X,\mathcal{H}^q(\mu_n^{\otimes m}))=0 \text{ for } p>q.$$

\begin{proposition}\label{propfinitenessCH2(C,1)[n]}
Let $S$ be a smooth surface over a field $k$. Let $n>0$ be a natural number prime to the characteristic of $k$. Then for all $j\geq 0$ we have the following diagramm:
$$\begin{xy} 
  \xymatrix{
   H^1(S,\mathcal{K}^M_{2+j}/n) \ar[d]^{\simeq}  \ar@{->>}[r]^{} &  CH^{2+j}(S,j)[n] & \\
     H^1(S,\mathcal{H}^{2+j}(\mu_n^{\otimes 2+j})) \ar[r]^{\simeq}  & E_\infty^{1,2+j} & \\
     & F^1H^{3+j} \ar@{->>}[u]^{} \ar@{^{(}->}[r]^{} & H^{3+j}_{\text{\'et}}(S,\mu_n^{\otimes 2+j})
  }
\end{xy} $$ 
In particular, if $k$ is either separably closed, local of dimension $1$ or finite, then the groups $$CH^{2+j}(S,j)[n]$$
are finite and the groups $CH^{2+j}(S,j)\{n\}$ are subquotients of $H^{3+j}_{\text{\'et}}(X,\Q_n/\Z_n(2+j))$.
\end{proposition}
\begin{proof}
By proposition \ref{identificationwithZero-cycles with coefficients in Milnor K-theory}, we know that $H^2(S,\mathcal{K}^M_{2+j})\cong CH^{2+j}(S,j)$ and therefore that
$$H^1(S,\mathcal{K}^M_{2+j}/n) \twoheadrightarrow CH^{2+j}(S,j)[n].$$
By the Bloch-Kato conjecture (see \cite{Vo11}), we have an isomorphism $$H^1(S,\mathcal{K}^M_{2+j}/n)\cong H^1(S,\mathcal{H}^{2+j}(\mu_n^{\otimes 2+j}))$$ in the Zariski topology. Since dim$(S)=2$, we have that $H^1(S,\mathcal{H}^{2+j}(\mu_n^{\otimes 2+j})) \cong E_2^{1,2+j}\cong E_\infty^{1,2+j}$ which is a quotient of $F^1H^{3+j}\subset H^{3+j}(S,\mu_n^{\otimes 2+j})$.
If $k$ is separabely closed, then $H^{3+j}_{\text{\'et}}(S,\mu_n^{\otimes 2+j})$ is finite by \cite[Ch. XVI, thm.  5.2]{SGA4}. This implies the finiteness for $k$ finite of local of dimension $1$ by the Hochschild-Serre spectral sequence.
\end{proof}

\begin{remark}
\begin{enumerate}
\item The case $j=0$ of proposition \ref{propfinitenessCH2(C,1)[n]} was shown in \cite{CSS82} and \cite{CSS83}. The case $j=1$ was shown in \cite{Sz00}.
\item
Let $X$ be a smooth projective scheme of dimension $d$ over a $p$-adic field $K$. In \cite[Sec. 1, Sec. 5]{AS07}, Asakura and Saito show that neither the groups $CH^d(X)_{\text{tors}}$ nor the groups $CH^{d+1}(X,1)_{\text{tors}}$ may be expected to be finite.
\item
It would be interesting to have a conjecture on the expected structure of $CH^j(X,j-d)^{\wedge p}$, the $p$-completion of $CH^j(X,j-d)$, for $X$ a variety over a $p$-adic field for $j>0$. For $j=0$ see \cite[Sec. 1]{Co95}.

\end{enumerate}
\end{remark}
For proper smooth schemes one can generalise the above proposition for $j\geq 1$ to arbitrary dimension:
\begin{proposition}\label{}
Let $X$ be a smooth and proper scheme over a local field $K$. Let $n>0$ be a natural number prime to the characteristic of $K$. Then for all $j\geq 1$ the groups $$CH^{d+j}(X,j)[n]$$ are finite.
\end{proposition}
\begin{proof}
For higher Chow groups we have the following exact sequence:
$$0\rightarrow CH^s(X_K,t)/n\rightarrow CH^s(X_K,t,\mathbb{Z}/n\mathbb{Z})\rightarrow CH^s(X_K,t-1)[n]\rightarrow 0$$
In order to study $CH^{d+j}(X_K,j)[n]$, one can therefore study the group $CH^{d+j}(X_K,j+1,\mathbb{Z}/n\mathbb{Z})$. 
By Levine's localization sequence for higher Chow groups, $CH^{d+j}(X_K,j+1,\mathbb{Z}/n\mathbb{Z})$ fits into the exact sequence 
$$CH^{d+j}(X,j+1,\mathbb{Z}/n\mathbb{Z})\rightarrow CH^{d+j}(X_K,j+1,\mathbb{Z}/n\mathbb{Z})\rightarrow CH^{d+j-1}(X_k,j,\mathbb{Z}/n\mathbb{Z}).$$
By proposition \ref{katofinitefieldcase}, proposition \ref{katochetale} and the Kato conjectures, the groups $CH^{d+j}(X,j+1,\mathbb{Z}/n\mathbb{Z})$ and $CH^{d+j-1}(X_k,j,\mathbb{Z}/n\mathbb{Z})$ are finite if $j\geq 1$. This implies that $CH^{d+j}(X_K,j)[n]$ is finite if $j\geq 1$.
\end{proof}

We now study $CH^{d+1}(X,1)$ for a smooth projective scheme $X$ of dimension $d$ over a finite field. $CH^d(X)$ is studied in unramified class field theory and treated for example in \cite{CSS83}, \cite{KaS83} and \cite[Sec. 8]{Sz10}.

We first note that for all $j\geq 0$, we have the following commutative diagram:
$$\begin{xy} 
  \xymatrix{
     H^{2d+j-1}(X,\Z/l^n\Z(d+j)) \ar[d]^{}  \ar@{->>}[r]^{} &  CH^{d+j}(X,j)[l^n] \ar[d]^{} \\
      H^{2d+j-1}_{\text{\'et}}(X,\Z/l^n\Z(d+j)) \ar[r]^{}  &  H^{2d+j}_{\text{\'et}}(X,\Z/l^m\Z(d+j))
  }
\end{xy} $$ 
Let us recall the construction (cf. \cite[Sec. 3]{Sz00}, where the commutativity of the above diagram is shown in detail, and \cite[Sec. 8]{Sz10}):
\begin{enumerate}
\item The upper horizontal map comes from the distinguished triangle
$$\Z(i)\xrightarrow{l^n} \Z(i)\r \Z/l^n\Z(i)\r \Z(i)[1].$$
\item The lower horizontal map comes from the exact sequence
$$0\r \mu^{\otimes d+j}_{l^m}\rightarrow{} \mu^{\otimes d+j}_{l^{mn}}\r \mu^{\otimes d+j}_{l^n}\r 0$$ 
on $X_{\text{\'et}}$ inducing the distinguished triangle
$$R\pi_*\mu^{\otimes d+j}_{l^m}\r R\pi_*\mu^{\otimes d+j}_{l^{mn}}\r R\pi_*\mu^{\otimes d+j}_{l^n} \r R\pi_*\mu^{\otimes d+j}_{l^m}[1].$$
in the $\text{D}(X_{\text{Zar}})$.
\item The two vertical maps are induced by the change of sites map $\pi:X_{\text{\'et}}\r X_{\text{Zar}}$, i.e. 
\begin{equation*} \begin{split}
H^{2d+j-1}(X,\Z/l^n\Z(d+j))\xrightarrow{\simeq} H^{2d+j-1}(X,\tau_{\leq d+j}R\pi_*\Z/l^n\Z(d+j))\\ \r  H^{2d+j-1}(X,R\pi_*\Z/l^n\Z(d+j))\xrightarrow{\simeq} H^{2d+j-1}_{\text{\'et}}(X,\Z/l^n\Z(d+j))
\end{split} \end{equation*} 
and 
$$H^{2d+j}(X,\Z(d+j))\r H^{2d+j}(X,\tau_{\leq d+j}R\pi_*\Z/l^n\Z(d+j))\r  H^{2d+j}_{\text{\'et}}(X,\Z/l^n\Z(d+j)).$$
\end{enumerate}

Taking the colimit over $n$ and the limit over $m$ in the above commutative diagram, we get the following commutative diagram:
$$\begin{xy} 
  \xymatrix{
     H^{2d+j-1}(X,\Q_l/\Z_l(d+j)) \ar[d]^{}  \ar@{->>}[r]^{} &  CH^{d+j}(X,j)\{l^n\} \ar[d]^{} \\
      H^{2d+j-1}_{\text{\'et}}(X,\Q_l/\Z_l(d+j)) \ar[r]^{}  &  H^{2d+j}_{\text{\'et}}(X,\Z_l(d+j))
  }
\end{xy} $$ 

\begin{proposition}\label{propositionfinitenesstorsionfinitefield}
Let $X$ be a smooth projective scheme of dimension $d$ over a finite field $k$ of characteristic $p>0$. Then the group
$$CH^{d+1}(X,1)$$
is finite. 
\end{proposition}
\begin{proof}
We first note that $CH^{d+1}(X,1)=CH^{d+1}(X,1)_{\text{tors}}$. It therefore suffices to show the finiteness of $CH^{d+1}(X,1)_{\text{tors}}$. Let $C$ be a smooth curve and $l$ be a prime number prime to $p$. In this case we have the following commutative diagram by the above discussion:
$$\begin{xy} 
  \xymatrix{
     H^{2}(C,\Q_l/\Z_l(2)) \ar@{^{(}->}[d]^{}  \ar@{->>}[r]^{} &  CH^{2}(C,1)\{l^n\} \ar[d]^{} \\
      H^{2}_{\text{\'et}}(C,\Q_l/\Z_l(2)) \ar@{^{(}->}[r]^{}  &  H^{3}_{\text{\'et}}(C,\Z_l(2))
  }
\end{xy} $$ 
The injectivity of the left vertical morphism follows from spectral sequence \ref{spseqchangeofsites}. The injectivity of the lower horizontal map follows from the vanishing of $H^2_{\text{\'et}}(C,\Q_l(2))$ which follows from the fact that the groups 
$$H^{i}_{\text{\'et}}(X,\Z_l(r))$$
are torsion for $i\neq 2r,2r+1$. This follows from the Weil conjectures (see \cite[Sec. 2]{CSS83}). In particular, the map $CH^{2}(C,1)\{l^n\}   \r  H^{3}(X,\Q_l/\Z_l(2))$ is injcetive. We now show that the map $CH^{d+1}(X,1)\{l^n\} \r  H^{2d+1}(X,\Q_l/\Z_l(d+1)$ is injective. Let $\alpha\in \text{ker}(CH^{d+1}(X,1)\{l^n\} \r  H^{2d+1}(X,\Q_l/\Z_l(d+1))$. By the Bertini theorem of Poonen (see \cite{Po08}), we can find a smooth curve $C$ containing the support of $\alpha$. Since $CH^{2}(C,1)$ is torsion, $\alpha\in CH^{2}(C,1)\{l^n\}$. The injectivity now follows from the following commutative diagram: 
$$\begin{xy} 
  \xymatrix{
      CH^{2}(C,1)\{l^n\} \ar[d]^{}  \ar@{^{(}->}[r]^{} &    H^{3}(C,\Q_l/\Z_l(2)) \ar[d]^{\simeq} \\
      CH^{d+1}(X,1)\{l^n\} \ar[r]^{}  &    H^{2d+1}(X,\Q_l/\Z_l(d+1))
  }
\end{xy} $$ 
The isomorphism on the right follows from the weak lefschetz theorem (see \cite[Thm. 7.1]{Mi80}). The finiteness of $CH^{d+1}(X,1)_{\text{tors}}$ now follows from the fact that for a smooth projective scheme $X$ over a finite field $k$, the groups 
$$H^{i}_{\text{\'et}}(X,\Q_l/\Z_l(r))$$
are finite for $i\neq 2r,2r+1$ and zero for almost all $l$ (see \cite[Thm. 2]{CSS82} and \cite[Sec. 2]{CSS83})
and the fact that $CH^{d+1}(X,1)$ does not contain any $p$-torsion. This follows from $0=H^{d}_{}(X,v_n(d+1))\twoheadrightarrow CH^{d+1}(X,1)[p^n]$ for dimension reasons.
\end{proof}

\begin{remark}
That $CH^{d+1}(X,1)$ is torsion is a special case of Parshin's conjecture saying that $K_i(X)\otimes \Q=0$ for a smooth projective scheme over a finite field if $i>0$. Proposition \ref{propositionfinitenesstorsionfinitefield}  says more: $CH^{d+1}(X,1)$ is not just torsion but finite.
\end{remark}
 
\bibliographystyle{siam}
\bibliography{Bibliografie}

\end{document}
Zero_cycles_koefficients_milnor_k_theory.bbl
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