text:
\section{Introduction}
Suslin and Voevodsky defined Suslin homology (also called singular
homology) $H_i^S(X,A)$ of a scheme of finite type over a field
$k$ with coefficients in an abelian group $A$ as
$\operatorname{Tor}_i(\operatorname{Cor}_k(\Delta^*,X), A)$. Here $\operatorname{Cor}_k(\Delta^i,X)$ is
the free abelian group generated by integral subschemes $Z$
of $\Delta^i\times X$ which are finite and surjective over $\Delta^i$,
and the differentials are given by alternating sums of pull-back maps
along face maps.
Suslin cohomology $H^i_S(X,A)$ is defined to be
$\operatorname{Ext}^i_{\text{\rm Ab}}(\operatorname{Cor}_k(\Delta^*,X),A)$. Suslin and Voevodsky showed
in \cite{susvoe98} that over a separably closed field in which $m$
is invertible, one has
\begin{equation}\label{susvoemain}
H^i_S(X,{{\mathbb Z}}/m)\cong H^i_{\text{\rm et}}(X,{{\mathbb Z}}/m)
\end{equation}
(see \cite{ichdejong} for the case of fields of characteristic $p$).

In the first half of this paper we study both the situation
that $m$ is a power of the characteristic of $k$, and that $k$
is not algebraically closed.
In the second half we focus on finite base fields and discuss a modified
version of Suslin homology, which is closely related to etale
cohomology on the one hand,  but is also expected to be finitely
generated. Moreover, its zeroth homology is ${{\mathbb Z}}^{\pi_o(X)}$ and
its first homology is expected to be an integral model of the abelianized
tame fundamental group.

We start by discussing the $p$-part of Suslin homology over an
algebraically closed field of characteristic $p$.
We show that assuming resolution of
singularities, the groups $H_i^S(X,{{\mathbb Z}}/p^r)$ are finite
abelian groups, and vanish outside the range $0\leq i \leq \dim X$.
Thus Suslin cohomology with finite coefficients
is etale cohomology away from the characteristic, but better behaved
than etale cohomology at the characteristic (for example,
$H^1_{\text{\rm et}}({\mathbb A}^1,{{\mathbb Z}}/p)$ is not finite).
Moreover, Suslin homology is a birational invariant in the following
strong sense:
If $X$ has a resolution of singularities $p:X'\to X$ which is an
isomorphism outside of the open subset $U$, then
$H_i^S(U,{{\mathbb Z}}/p^r)\cong H_i^S(X,{{\mathbb Z}}/p^r)$.

Next we examine the situation over non-algebraically closed fields.
We redefine Suslin homology and cohomology by imposing
Galois descent. Concretely, if $G_k$ is the absolute Galois
group of $k$, then we define Galois-Suslin homology to be
\begin{equation*}\label{galsushom}
H_i^{GS}(X,A) = H^{-i}R\Gamma(G_k,\operatorname{Cor}_{\bar k}(\Delta_{\bar k}^*,\bar X)),
\end{equation*}
and Galois-Suslin cohomology to be
\begin{equation*}\label{galsuscoh}
H^i_{GS}(X,A) = \operatorname{Ext}^i_{G_k}(\operatorname{Cor}_{\bar k}(\Delta_{\bar k}^*,\bar X), A).
\end{equation*}
Ideally one would like to define Galois-Suslin homology via Galois homology,
but we are not aware of such a theory. With rational coefficients,
the newly defined groups agree with the original groups. On the other hand,
with finite coefficients prime to the characteristic, the
proof of \eqref{susvoemain} in \cite{susvoe98} carries over to show that
$H^i_{GS}(X,{{\mathbb Z}}/m)\cong H^i_{\text{\rm et}}(X,{{\mathbb Z}}/m)$. As a corollary we obtain an
isomorphism between $H_0^{GS}(X,{{\mathbb Z}}/m)$ and the abelianized fundamental group
$\pi_1^{ab}(X)/m$ for any separated $X$ of finite type over a finite field.

The second half of the paper focuses on the case of a finite base field.
We work under the assumption of resolution of singularities in order
to see the picture of the properties which can expected.
The critical reader
can view our statements as theorems for schemes of dimension at most three,
and conjectures in general.
A theorem of Jannsen-Saito \cite{uweshuji} can
be generalized to show that Suslin homology and cohomology with finite
coefficients for any $X$ over a finite field is finite.
Rationally, $H_0^S(X,{{\mathbb Q}})\cong H^0_S(X,{{\mathbb Q}})\cong {{\mathbb Q}}^{\pi_0(X)}$. Most other
properties are equivalent to the following Conjecture $P_0$ considered in
\cite{ichkato}:
For $X$ smooth and proper over a finite field, $CH_0(X,i)$ is
torsion for $i\not=0$. This is a particular case of Parshin's conjecture
that $K_i(X)$ is torsion for $i\not=0$.
For example, Conjecture $P_0$ is equivalent to the vanishing of
$H_i^S(X,{{\mathbb Q}})$ for $i\not=0$ and all smooth $X$. For arbitrary $X$
of dimension $d$, Conjecture $P_0$ implies the
vanishing of $H_i^S(X,{{\mathbb Q}})$ outside of the range $0\leq i\leq d$
and its finite dimensionality in this range.
Combining the torsion
and rational case, we show that $H_i^S(X,{{\mathbb Z}})$ and $H^i_S(X,{{\mathbb Z}})$
are finitely generated for all $X$ if and only if Conjecture $P_0$ holds.

Over a finite field and with integral coefficients, it is more
natural to impose descent by the Weil group $G$ generated by
the Frobenius endomorphism instead of the Galois group
\cite{lichtenbaumweilI, ichweilI, ichduke, ichkato}.
We define arithmetic homology
$$H_i^{\text{\rm ar}}(X,A)=\operatorname{Tor}_i^G(\operatorname{Cor}_{\bar k}(\Delta_{\bar k}^*,\bar X),A)$$
and arithmetic cohomology
$$H^i_{\text{\rm ar}}(X,{{\mathbb Z}})=\operatorname{Ext}^i_G(\operatorname{Cor}_{\bar k}(\Delta_{\bar k}^*,\bar X),{{\mathbb Z}}).$$
We chose the notation to distinguish the groups from weight homology
and Weil-etale cohomology considered elsewhere.
We show that $H_0^{\text{\rm ar}}(X,{{\mathbb Z}})\cong H^0_{\text{\rm ar}}(X,{{\mathbb Z}})\cong {{\mathbb Z}}^{\pi_0(X)}$
and that arithmetic homology and cohomology lie in long exact
sequences with Galois-Suslin homology and cohomology, respectively.
They are finitely generated abelian groups if and only if
Conjecture $P_0$ holds.

The difference between arithmetic and Suslin homology
is measured by a third theory, which we call Kato-Suslin
homology, and which is defined as
$H_i^{KS}(X,A)=H_i(\operatorname{Cor}_{\bar k}(\Delta^*_{\bar k},\bar X)_G)$.
By definition there is a long exact sequence
$$ \cdots \to H_i^S(X,A) \to  H_{i+1}^{\text{\rm ar}}(X,A) \to
H_{i+1}^{KS}(X,A) \to H_{i-1}^{S}(X,A) \to \cdots .$$
It follows that $H_0^{KS}(X,A)={{\mathbb Z}}^{\pi_0(X)}$ for any $X$.
As a generalization of the integral version \cite{ichkato}
of Kato's conjecture \cite{kato}, we propose

\begin{conjecture} The groups $H_i^{KS}(X,A)$ vanish for all $i>0$
and all smooth $X$.
\end{conjecture}

Equivalently that there are short exact sequences
$$0\to H_{i+1}^S(\bar X,{{\mathbb Z}})_G \to H_i^S(X,{{\mathbb Z}}) \to H_i^S(\bar X,{{\mathbb Z}})^G\to 0$$
for all $i\geq 0$ and all smooth $X$.
We show that this conjecture, too, is equivalent to Conjecture $P_0$.
This leads us to conjecture on class field theory:

\begin{conjecture} For every $X$ separated and of finite type over ${{\mathbb F}}_q$,
there is a canonical injection
$$H_1^{\text{\rm ar}}(X,{{\mathbb Z}})\to \pi_1^t(X)^{ab}$$
with dense image.
\end{conjecture}

It might even be true that the relative group
$H_1^{\text{\rm ar}}(X,{{\mathbb Z}})^\circ := \ker (H_1^{\text{\rm ar}}(X,{{\mathbb Z}})\to {{\mathbb Z}}^{\pi_0(X)})$
is isomorphic to the geometric part of the abelianized fundamental group
defined in SGA 3X\S 6.
To support our conjecture, we note that the generalized Kato conjecture
above implies $H_0^S(X,{{\mathbb Z}})\cong H_1^{\text{\rm ar}}(X,{{\mathbb Z}})$ for smooth $X$, so that
in this case our conjecture becomes a theorem of Schmidt-Spiess \cite{ss}.
In addition, we show (independent of any conjectures)

\begin{proposition}
If $\frac{1}{l}\in {{\mathbb F}}_q$, then
$H_1^{\text{\rm ar}}(X,{{\mathbb Z}})^{\wedge l}\cong \pi_1^t(X)^{ab}(l)$ for arbitrary $X$.
\end{proposition}

In particular, the conjectured finite generation of $H_1^{\text{\rm ar}}(X,{{\mathbb Z}})$ implies
the conjecture away from the characteristic.
We also give a conditional result at the characteristic.

Notation: In this paper, scheme over a field $k$ means separated scheme
of finite type over $k$.

We thank Uwe Jannsen for interesting discussions
related to the subject of this paper, and Shuji Saito and Takeshi Saito
for helpful comments during a series of lectures I gave on the topic
of this paper at Tokyo University.

\section{Motivic homology}
Suslin homology $H_i^S(X,{{\mathbb Z}})$ of a scheme $X$ over a
field $k$ is defined as the homology of the global sections
$C_*^X(k)$ of the complex of etale sheaves
$C_*^X(-)=\operatorname{Cor}_k(-\times \Delta^*,X)$. Here
$\operatorname{Cor}_k(U,X)$ is the group
of universal relative cycles of $U\times Y/U$ \cite{susvoerel}.
If $U$ is smooth, then $\operatorname{Cor}_k(U,X)$
is the free abelian group generated by closed irreducible subschemes
of $U\times X$ which are finite and surjective over a connected
component of $U$.

More generally \cite{friedvoe},
motivic homology of weight $n$ are the extension groups in Voevodsky's category of geometrical mixed motives
$$ H_i(X,{{\mathbb Z}}(n))=\operatorname{Hom}_{DM^-_k}({{\mathbb Z}}(n)[i],M(X)),$$
and are isomorphic to
$$H_i(X,{{\mathbb Z}}(n))=\begin{cases}
H^{2n-i}_{(0)}({\mathbb A}^n,C_*^X)&n\geq 0\\
H_{i-2n-1}(C_*\big(\frac{c_0(X\times ({\mathbb A}^n-\{0\}))}
{c_0(X\times \{1\})}\big)(k)) &n< 0.
\end{cases}$$
Here cohomology is taken for the Nisnevich topology.
There is an obvious version with coefficients.
Motivic homology is a covariant functor on the category of schemes
of finite type over $k$, and has the following
additional properties, see \cite{friedvoe} (the final three properties
require resolution of singularities)

\begin{enumerate}
\item It is homotopy invariant.
\item It satisfies a projective bundle formula
$$ H_i(X\times {\mathbb P}^1,{{\mathbb Z}}(n))=
H_i(X,{{\mathbb Z}}(n))\oplus H_{i-2}(X,{{\mathbb Z}}(n-1)).$$
\item There is a Mayer-Vietoris long exact sequence for open covers.
\item  Given an abstract blow-up square
$$\begin{CD}
Z'@>>> X'\\
@VVV @VVV \\
Z@>>> X
\end{CD}$$
there is a long exact sequence
\begin{multline}\label{blowups}
\cdots\to H_{i+1}(X,{{\mathbb Z}}(n))\to
H_i(Z',{{\mathbb Z}}(n))\to \\
H_i(X',{{\mathbb Z}}(n))\oplus H_i(Z,{{\mathbb Z}}(n))\to H_i(X,{{\mathbb Z}}(n))\to \cdots
\end{multline}
\item If $X$ is proper, then motivic homology agrees with higher
Chow groups, $H_i(X,{{\mathbb Z}}(n))\cong CH_n(X,i-2n)$.
\item If $X$ is smooth of pure dimension $d$, then motivic homology
agrees with motivic cohomology with compact support,
$$H_i(X,{{\mathbb Z}}(n))\cong H^{2d-i}_c(X,{{\mathbb Z}}(d-n)).$$
In particular, if $Z$ is
a closed subscheme of a smooth scheme $X$ of pure dimension $d$,
then we have a long exact sequence
\begin{equation}\label{localizations}
\cdots \to H_i(U,{{\mathbb Z}}(n)) \to H_i(X,{{\mathbb Z}}(n)) \to
H^{2d-i}_c(Z,{{\mathbb Z}}(d-n))\to\cdots .
\end{equation}
\end{enumerate}

In order to remove the hypothesis on resolution of singularities,
it would be sufficient to find a proof of Theorem 5.5(2) of
Friedlander-Voevodsky \cite{friedvoe} that does
not require resolution of singularities.
For all arguments in this paper (except the $p$-part of the Kato
conjecture) the sequences \eqref{blowups} and \eqref{localizations}
and the existence of a smooth and
proper model for every function field are sufficient.

\subsection{Suslin cohomology}
Suslin cohomology is by definition the dual of Suslin homology,
i.e. for an abelian group $A$ it is defined as
$$ H^i_S(X,A):= \operatorname{Ext}^i_{\text{\rm Ab}}({C}_*^X(k),A).$$
We have $H^i_S(X,{{\mathbb Q}}/{{\mathbb Z}})\cong \operatorname{Hom}(H_i^S(X,{{\mathbb Z}}),{{\mathbb Q}}/{{\mathbb Z}})$, and
a short exact sequence of abelian groups gives a long exact sequence of
cohomology groups, in particular long exact sequences
\begin{equation}\label{coeffseq}
\cdots \to H^i_S(X,{{\mathbb Z}})\to
H^i_S(X,{{\mathbb Z}})\to H^i_S(X,{{\mathbb Z}}/m)\to H^{i+1}_S(X,{{\mathbb Z}})\to \cdots.
\end{equation}
and
$$\cdots \to H^{i-1}_S(X,{{\mathbb Q}}/{{\mathbb Z}})\to H^i_S(X,{{\mathbb Z}})\to
H^i_S(X,{{\mathbb Q}})\to H^i_S(X,{{\mathbb Q}}/{{\mathbb Z}})\to \cdots.$$
Consequently, $H^i_S(X,{{\mathbb Z}})_{{\mathbb Q}}\cong H^i_S(X,{{\mathbb Q}})$ if
Suslin-homology is finitely generated.
If $A$ is a ring, then $H^i_S(X,A)\cong \operatorname{Ext}_A^i({C}_*^X(k)\otimes A,A)$,
and we get a spectral sequence
\begin{equation}\label{extss}
E_2^{s,t}= \operatorname{Ext}^s_A(H_t^S(X,A),A)\Rightarrow H^{s+t}_S(X,A).
\end{equation}
In particular, there are perfect pairings
\begin{align*}
H_i^S(X,{{\mathbb Q}})\times H^i_S(X,{{\mathbb Q}}) &\to {{\mathbb Q}}\\
H_i^S(X,{{\mathbb Z}}/m)\times H^i_S(X,{{\mathbb Z}}/m) &\to {{\mathbb Z}}/m.
\end{align*}

\begin{lemma}\label{duall1}
There are natural pairings
$$ H^i_S(X,{{\mathbb Z}})/{\text{\rm tor}} \times H_i^S(X,{{\mathbb Z}})/{\text{\rm tor}} \to {{\mathbb Z}}$$
and
$$H^i_S(X,{{\mathbb Z}})_{\text{\rm tor}} \times H_{i-1}^S(X,{{\mathbb Z}})_{\text{\rm tor}}\to {{\mathbb Q}}/{{\mathbb Z}}.$$
\end{lemma}

\noindent{\it Proof. }
The spectral sequence \eqref{extss} gives a short exact sequence
\begin{equation}\label{cefff}
0\to \operatorname{Ext}^1(H_{i-1}^S(X,{{\mathbb Z}}),{{\mathbb Z}}) \to H^i_S(X,{{\mathbb Z}})\to
\operatorname{Hom}(H_i^S(X,{{\mathbb Z}}),{{\mathbb Z}})\to 0.
\end{equation}
The resulting map $H^i_S(X,{{\mathbb Z}})/{\text{\rm tor}} \twoheadrightarrow \operatorname{Hom}(H_i^S(X,{{\mathbb Z}}),{{\mathbb Z}})$
induces the first pairing. Since $\operatorname{Hom}(H_i^S(X,{{\mathbb Z}}),{{\mathbb Z}})$
is torsion free, we obtain the map
\begin{multline*}
H^i_S(X,{{\mathbb Z}})_{\text{\rm tor}} \hookrightarrow
\operatorname{Ext}^1(H_{i-1}^S(X,{{\mathbb Z}}),{{\mathbb Z}})\twoheadrightarrow \\
\operatorname{Ext}^1(H_{i-1}^S(X,{{\mathbb Z}})_{\text{\rm tor}} ,{{\mathbb Z}}) \stackrel{\sim}{\leftarrow}
\operatorname{Hom}(H_{i-1}^S(X,{{\mathbb Z}})_{\text{\rm tor}},{{\mathbb Q}}/{{\mathbb Z}})
\end{multline*}
for the second pairing.
\hfill $\Box$ \\

\subsection{Comparison to motivic cohomology}
Recall that in the category $DM^-_k$ of bounded above complexes
of homotopy invariant Nisnevich sheaves with transfers,
the motive $M(X)$ of $X$ is the complex of presheaves with
transfers ${C}_*^X$.
Since a field has no higher Nisnevich cohomology,
taking global sections over $k$ induces a canonical map
$$\operatorname{Hom}_{DM^-_k}(M(X),A) \to R\operatorname{Hom}_{{\text{\rm Ab}}}(C_*^X(k),R\Gamma(k,A)),$$
hence a natural map
\begin{equation}\label{canmap}
H^i_M(X,A)\to H^i_S(X,A).
\end{equation}
Even though the cohomology groups do
not depend on the base field, the map does. For example, if $L/k$
is an extension of degree $d$, then the diagram of groups isomorphic
to ${{\mathbb Z}}$,
$$\begin{CD}
H^0_M(\operatorname{Spec} k,{{\mathbb Z}})@= H^0_S(\operatorname{Spec} k,{{\mathbb Z}})\\
@| @VV\times d V\\
H^0_M(\operatorname{Spec} L,{{\mathbb Z}})@>>> H^0_S(\operatorname{Spec} L,{{\mathbb Z}})
\end{CD}$$
shows that the lower horizontal map is
multiplication by $d$.  We will see
below that conjecturally \eqref{canmap} is a map between finitely
generated groups which is rationally an isomorphism, and one
might ask if its Euler characteristic has any interpretation.

\section{The mod $p$ Suslin homology in characteristic $p$}
We examine the $p$-part of Suslin homology
in characteristic $p$. We assume
that $k$ is perfect and resolution of singularities exists
over $k$ in order to obtain stronger results.
We first give an auxiliary result on motivic cohomology
with compact support:

\begin{proposition}\label{theelemma}
Let $d=\dim X$.

a) We have $H^i_c(X,{{\mathbb Z}}/p^r(n))=0$ for $n>d$.

b) If $k$ is algebraically closed, then $H^i_c(X,{{\mathbb Z}}/p^r(d))$ is finite,
$H^i_c(X,{{\mathbb Q}}_p/{{\mathbb Z}}_p(d))$ is of cofinite type, and the groups vanish
unless $d\leq i\leq 2d$.
\end{proposition}

\noindent{\it Proof. }
By induction on the dimension and the localization sequence,
the statement for $X$ and a dense open subset of $X$
are equivalent. Hence replacing $X$ by a smooth subscheme and then
by a smooth and proper model, we can assume that $X$ is smooth
and proper. Then a) follows from \cite{ichmarcI}.  If $k$ is
algebraically closed, then
$$ H^i(X,{{\mathbb Z}}/p(d))\cong H^{i-d}(X_{Nis},\nu^d)\cong
H^{i-d}(X_{\text{\rm et}},\nu^d),$$
by \cite{ichmarcI} and \cite{katokuzumaki}. That the latter
group is finite and of cofinite type, respectively, can be derived
from \cite[Thm.1.11]{milnevalues}, and it vanishes outside of the given
range by reasons of cohomological dimension.
\hfill $\Box$ \\

\begin{theorem}\label{ppart}
Let $X$ be separated and of finite type over $k$.

a) The groups $H_i(X,{{\mathbb Z}}/p^r(n))$ vanish for all $n<0$.

b) If $k$ is algebraically closed, then the groups
$H_i^S(X,{{\mathbb Z}}/p^r)$ are finite,
the groups $H_i^S(X,{{\mathbb Q}}_p/{{\mathbb Z}}_p)$ are of cofinite type, and both vanish
unless $0\leq i\leq d$.
\end{theorem}

\noindent{\it Proof. }
If $X$ is smooth, then $H_i(X,{{\mathbb Z}}/p^r(n))\cong H^{2d-i}_c(X,{{\mathbb Z}}/p^r(d-n))$
and we conclude by the Proposition.
In general, we can assume by \eqref{blowups} and induction on the
number of irreducible components that $U$ is integral. Proceeding
by induction on the dimension, we choose a resolution of singularities
$X'$ of $X$, let $Z$ be the closed subscheme of $X$ where the map
$X'\to X$ is not an isomorphism, and let $Z'=Z\times_XX'$.
Then we conclude by the sequence \eqref{blowups}.
\hfill $\Box$ \\

\noindent {\bf Example. } If $X'$ is the blow up of a smooth scheme $X$ in a smooth
subscheme $Z$, then the strict transform $Z'=X'\times_XZ$ is a projective
bundle over $Z$, hence by the projective bundle formula
$H_i^S(Z,{{\mathbb Z}}/p^r)\cong H_i^S(Z',{{\mathbb Z}}/p^r)$ and
$H_i^S(X,{{\mathbb Z}}/p^r)\cong H_i^S(X',{{\mathbb Z}}/p^r)$. More generally, we have

\begin{proposition}\label{biratt}
Assume $X$ has a desingularization $p:X'\to X$ which is
an isomorphism outside of the dense open subset $U$. Then
$H_i^S(U,{{\mathbb Z}}/p^r)\cong  H_i^S(X,{{\mathbb Z}}/p^r)$.
In particular, the $p$-part of Suslin homology is a birational invariant.
\end{proposition}

The hypothesis is satisfied if $X$ is smooth, or if $U$ contains all
singular points of $X$ and
a resolution of singularities exists which is an isomorphism outside
of the singular points.

\medskip

\noindent{\it Proof. }
If $X$ is smooth, then this follows from Proposition \ref{theelemma}
and the localization sequence \eqref{localizations}.
In general, let $Z$ be the set of points where $p$ is not an isomorphism,
and consider the cartesian diagram
$$\begin{CD}
Z' @>>> U' @>>> X'\\
@VVV @VVV @VVV \\
Z@>>> U @>>> X.
\end{CD}$$
Comparing long exact sequence \eqref{blowups} of the left and outer
squares,
$$\begin{CD}
\to H_i^S(Z',{{\mathbb Z}}/p^r)@>>>  H_i^S(U',{{\mathbb Z}}/p^r)\oplus H_i^S(Z,{{\mathbb Z}}/p^r)@>>>
H_i^S(U,{{\mathbb Z}}/p^r)\to \\
@| @|@VVV \\
\to H_i^S(Z',{{\mathbb Z}}/p^r)@>>>  H_i^S(X',{{\mathbb Z}}/p^r)\oplus H_i^S(Z,{{\mathbb Z}}/p^r)@>>>
H_i^S(X,{{\mathbb Z}}/p^r)\to
\end{CD}$$
we see that
$H_i^S(U',{{\mathbb Z}}/p^r)\cong  H_i^S(X',{{\mathbb Z}}/p^r)$ implies
$H_i^S(U,{{\mathbb Z}}/p^r)\cong  H_i^S(X,{{\mathbb Z}}/p^r)$.
\hfill $\Box$ \\

\noindent {\bf Example. } If $X$ is a node, then the blow-up sequence gives
$H_i^S(X,{{\mathbb Z}}/p^r)\cong H_{i-1}^S(k,{{\mathbb Z}}/p^r)\oplus H_i^S(k,{{\mathbb Z}}/p^r)$,
which is ${{\mathbb Z}}/p^r$ for $i=0,1$ and vanishes otherwise.
Reid constructed a normal surface with a singular point
whose blow-up is a node, showing that
the $p$-part of Suslin homology is not a birational invariant for normal
schemes.

\begin{corollary}
The higher Chow groups $CH_0(X,i,{{\mathbb Z}}/p^r)$ and the logarithmic de Rham-Witt
cohomology groups $H^i(X_{\text{\rm et}},\nu_r^d)$, for $d=\dim X$,
are birational invariants.
\end{corollary}

\noindent{\it Proof. }
Suslin homology agrees with higher Chow groups for proper $X$,
and with motivic cohomology for smooth and proper $X$.
\hfill $\Box$ \\

Note that integrally $CH_0(X)$ is a birational invariant, but
the higher Chow groups $CH_0(X,i)$ are generally not.

Suslin and Voevodsky \cite[Thm.3.1]{susvoe98} show that for a smooth
compactification $\bar X$ of the smooth curve $X$,
$H_0^S(X,{{\mathbb Z}})$ is isomorphic to the relative Picard group $\operatorname{Pic}(\bar X,Y)$
and that all higher Suslin homology groups vanish.
Proposition \ref{biratt} implies that the
kernel and cokernel of $\operatorname{Pic}(\bar X,Y)\to \operatorname{Pic}(\bar X)$
are uniquely $p$-divisible.
Given $U$ with compactification $j:U\to X$, the
normalization $X^\sim$ of $X$ in $U$ is the affine bundle defined
by the integral closure of $\O_X$ in $j_*\O_U$. We call $X$ normal
in $U$ if $X^\sim\to X$ is an isomorphism.

\begin{proposition} If $X$ is normal in the curve $U$, then
$H_i^S(U,{{\mathbb Z}}/p)\cong H_i^S(X,{{\mathbb Z}}/p)$.
\end{proposition}

\noindent{\it Proof. }
This follows by applying the argument of Proposition \ref{biratt}
to $X'$ the normalization of $X$, $Z$ the closed subset
where $X'\to X$ is not an isomorphism, $Z'=X'\times_XZ$
and $U'=X'\times_XU$. Since $X$ is normal in $U$, we have
$Z\subseteq U$ and $Z'\subseteq U'$.
\hfill $\Box$ \\

\section{Galois properties}
Suslin homology is covariant, i.e. a separated map $f:X\to Y$ of
finite type induces
a map $f_*:\operatorname{Cor}_k(T,X)\to \operatorname{Cor}_k(T,Y)$ by sending a closed
irreducible subscheme $Z$ of $T\times X$, finite over $T$,
to the subscheme $[k(Z):k(f(Z))]\cdot f(Z)$ (note that $f(Z)$
is closed in $T\times Y$ and finite over $T$). On the other
hand, Suslin homology
is contravariant for finite flat maps $f:X\to Y$, because $f$
induces a map $f^*:\operatorname{Cor}_k(T,Y)\to \operatorname{Cor}_k(T,X)$ by composition with the
graph of $f$ in $\operatorname{Cor}_k(Y,X)$ (note that the graph is a universal
relative cycle in the sense of \cite{susvoerel}).
We examine the properties of Suslin homology under change of base-fields.

\begin{lemma}
Let $L/k$ be a finite extension of fields, $X$ a scheme over $k$
and $Y$ a scheme over $L$. Then
$ \operatorname{Cor}_L(Y,X_L)= \operatorname{Cor}_k(Y,X)$ and if $X$ is smooth, then
$ \operatorname{Cor}_L(X_L,Y)= \operatorname{Cor}_k(X,Y)$.
In particular, Suslin homology does not depend on the base field.
\end{lemma}

\noindent{\it Proof. }
The first statement follows because $Y\times_LX_L\cong Y\times_kX$. The
second statement follows because
the map $X_L\to X$ is finite and separated, hence a closed subscheme of
$X_L\times_LY \cong X\times_kY$ is finite and surjective over
$X_L$ if and only if it is finite and surjective over $X$.
\hfill $\Box$ \\

Given a scheme over $k$, the graph of the projection $X_L\to X$
in $X_L\times X$ gives elements $\Gamma_X\in \operatorname{Cor}_k(X,X_L)$
and $\Gamma_X^t\in \operatorname{Cor}_k(X_L,X)$.

\subsection{Covariance}
\begin{lemma}
a) If $X$ and $Y$ are separated schemes of finite type over $k$, then
the two maps
$$\operatorname{Cor}_L(X_L,Y_L)\to \operatorname{Cor}_k(X,Y)$$
induced by composition and precomposition, respectively,
with $\Gamma_Y^t$ and $\Gamma_X$ agree. Both maps send a generator
$Z\subseteq X_L\times_k Y\cong X\times_k Y_L$ to its image in $X\times Y$
with multiplicity $[k(Z):k(f(Z))]$, a divisor of $[L:k]$.

b) If $F/k$ is an infinite algebraic extension, then
$\lim_{L/k} \operatorname{Cor}_L(X_L,Y_L)=0$.
\end{lemma}

\noindent{\it Proof. }
The first part is easy. If $Z$ is of finite type over $k$,
then $k(Z)$ is a finitely generated field extension of $k$.
For every component $Z_i$ of $Z_F$, we
obtain a map $F\to F\otimes_k k(Z) \to k(Z_i)$, and
since $F$ is not finitely generated over $k$, neither is $k(Z_i)$.
Hence going up the tower of finite
extensions $L/k$ in $F$, the degree of $[k(W_L):k(Z)]$, for $W_L$ the
component of $Z_L$ corresponding to $Z_i$, goes to infinity.
\hfill $\Box$ \\

\subsection{Contravariance}

\begin{lemma}\label{contra}
a) If $X$ and $Y$ are schemes over $k$, then the two maps
$$\operatorname{Cor}_k(X,Y)\to \operatorname{Cor}_L(X_L,Y_L)$$
induced by composition and precomposition, respectively with $\Gamma_Y$ and
$\Gamma_X^t$ agree. Both maps send a generator
$Z\subseteq X\times Y$ to the cycle associated to
$Z_L\subseteq X\times_k Y_L\cong X_L\times_k Y$.
If $L/k$ is separable, this is a sum of the integral
subschemes lying over $Z$ with multiplicity one.
If $L/k$ is Galois with group $G$, then the maps induce an isomorphism
$$\operatorname{Cor}_k(X,Y)\cong  \operatorname{Cor}_L(X_L,Y_L)^G.$$

b) Varying $L$, $\operatorname{Cor}_L(X_L,Y_L)$ forms a etale sheaf on
$\operatorname{Spec} k$ with stalk
$M=\operatornamewithlimits{colim}_L \operatorname{Cor}_L(X_L,Y_L)\cong \operatorname{Cor}_{\bar k}(X_{\bar k},Y_{\bar k})$,
where $L$ runs through the finite extensions of $k$ in an
algebraic closure $\bar k$ of $k$. In particular,
$\operatorname{Cor}_L(X_L,Y_L)\cong M^{\operatorname{Gal}(\bar k/L)}$.
\end{lemma}

\noindent{\it Proof. }
Again, the first part is easy. If $L/k$ is separable,
$Z_L$ is finite and etale over $Z$,
hence $Z_L\cong \sum_i Z_i$, a finite sum of the integral
cycles lying over $Z$ with multiplicity one each.
If $L/k$ is moreover Galois, then $\operatorname{Cor}_k(X,Y)\cong \operatorname{Cor}_L(X_L,Y_L)^G$ and
$\operatorname{Cor}_{\bar k}(X_{\bar k},Y_{\bar k})\cong \operatornamewithlimits{colim} \operatorname{Cor}_L(X_L,Y_L)$
by EGA IV Thm. 8.10.5.
\hfill $\Box$ \\

The proposition suggests to work with the complex $C_*^X $ of
etale sheaves on $\operatorname{Spec} k$ given by
$${C}_*^X(L):=\operatorname{Cor}_L(\Delta^*_L,X_L)\cong \operatorname{Cor}_k(\Delta^*_L,X).$$

\begin{corollary}\label{descentss}
We have $H_i^S(\bar X,A)\cong \operatornamewithlimits{colim}_L H_i^S(X_L,A)$, and
there is a spectral sequence
$$E_2^{s,t}=  {\lim}^s H^t_S(X_L,A) \Rightarrow
H^{s+t}_S(\bar X,A).$$
The maps in the direct and inverse system are induced by contravariant
functoriality of Suslin homology for finite flat maps.
\end{corollary}

\noindent{\it Proof. }
This follows from the quasi-isomorphisms
$$ R\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k),{{\mathbb Z}})
\cong R\operatorname{Hom}_{\text{\rm Ab}}(\operatornamewithlimits{colim}_L {C}_*^X(L),{{\mathbb Z}})
\cong R\lim_L R\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(L),{{\mathbb Z}}).$$
\hfill $\Box$ \\

\subsection{Coinvariants}
If $G_k$ is the absolute Galois group of $k$, then
$\operatorname{Cor}_{\bar k}(\bar X,\bar Y)_{G_k}$ can be identified
with $\operatorname{Cor}_{k}(X,Y)$ by associating orbits of points
of $\bar X\times_{\bar k}\bar Y$ with their image in $X\times_k Y$.
However, this identification is neither compatible with covariant nor with
contravariant functoriality, and in particular not with the differentials
in the complex $C_*^X(k)$.
But the obstruction is torsion, and we can remedy this problem by
tensoring with ${{\mathbb Q}}$: Define an isomorphism
$$\tau:(\operatorname{Cor}_{\bar k}(\bar X,\bar Y)_{{\mathbb Q}})_{G_k}
\to \operatorname{Cor}_{k}(X,Y)_{{\mathbb Q}}.$$
as follows. A generator $1_{\bar Z}$ corresponding to the closed
irreducible subscheme $\bar Z\subseteq \bar X\times\bar Y$ is sent to
$\frac{1}{g_Z} 1_Z$, where $Z$ is the image of $\bar Z$ in $X\times Y$
and $g$ the number of irreducible components of $Z\times_k\bar k$,
i.e. $g_Z$ is the size of the Galois orbit of $\bar Z$.

\begin{lemma}\label{ratisom}
The isomorphism $\tau$ is functorial in both variables,
hence it induces an isomorphism of complexes
$$( C_*^X(\bar k)_{{\mathbb Q}} )_{G_k} \cong C_*^X(k)_{{\mathbb Q}}.$$
\end{lemma}

\noindent{\it Proof. }
This can be proved by direct verification.
We give an alternate proof. Consider the composition
$$\operatorname{Cor}_{k}(X,Y)\to \operatorname{Cor}_{\bar k}(\bar X,\bar Y)^{G_k}\to
\operatorname{Cor}_{\bar k}(\bar X,\bar Y)_{G_k}\xrightarrow{\tau} \operatorname{Cor}_{k}(X,Y)_{{\mathbb Q}}.$$
The middle map is induced by the identity, and is
multiplication by $g_Z$ on the component corresponding to $Z$.
All maps are isomorphisms upon tensoring with ${{\mathbb Q}}$. The first map, the
second map, and the composition are functorial, hence so is the $\tau$.
\hfill $\Box$ \\

\section{Etale theory}
Let $\bar k$ be the algebraic closure of $k$ with Galois group $G_k$,
and let $A$ be a continuous $G_k$-module.
Then ${C}_*^X(\bar k)\otimes A$
is a complex of continuous $G_k$-modules, and if $k$ has
finite cohomological dimension we define Galois-Suslin homology to be
$$H_i^{GS}(X,A)= H^{-i}R\Gamma(G_k, {C}_*^X(\bar k)\otimes A).$$
By construction, there is a spectral sequence
$$E^2_{s,t}=H^{-s}(G_k,H_t^S(\bar X,A))\Rightarrow
H_{s+t}^{GS}(X,A).$$
The case $X=\operatorname{Spec} k$ shows that Suslin homology does not agree with
etale Suslin homology, i.e. Suslin homology does not have Galois descent.
We define etale Suslin cohomology to be
\begin{equation}\label{suscohdef}
H^i_{GS}(X,A)= \operatorname{Ext}^i_{G_k}({C}_*^X(\bar k),A).
\end{equation}
This agrees with the old definition if $k$ is algebraically closed.
Let $\tau_*$ be the functor from $G_k$-modules to continuous
$G_k$-modules which sends $M$ to $\operatornamewithlimits{colim}_L M^{G_L}$, where
$L$ runs through the finite extensions of $k$.
It is easy to see that $R^i\tau_* M= \operatornamewithlimits{colim}_H H^i(H,M)$.

\begin{lemma}\label{suscohss}
We have  $H^i_{GS}(X,A)=H^i R\Gamma_{G_k}R\tau_*\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k), A)$.
In particular, there is a spectral sequence
\begin{equation}
E_2^{s,t}= H^s(G_k,R^t\tau_* \operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k), A))\Rightarrow
H^{s+t}_{GS}(X,A).
\end{equation}
\end{lemma}

\noindent{\it Proof. }
This is \cite[Ex. 0.8]{adt}. Since $C_*^X(\bar k)$ is a
complex of free ${{\mathbb Z}}$-modules, $\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k), -)$ is
exact and preserves injectives. Hence the derived functor of
$\tau_*\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k), -)$ is $R^t\tau_* $ applied
to $\operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(\bar k), -)$.
\hfill $\Box$ \\

\begin{proposition}\label{Qversion}
If $A$ is a ${{\mathbb Q}}$-vector space with trivial $G_k$-action, then
\begin{align*}
H_i^{GS}(X,A)&\cong H_i^S(X,A)\\
H^i_{GS}(X,A)&\cong H^i_S(X,A).
\end{align*}
\end{proposition}

\noindent{\it Proof. }
Since $-\otimes A$ is exact,
$H_i^S(X,A)= H_i(C_*^X(\bar k)^{G_k}\otimes A)$
as well as $H_i^{GS}(X,A)=H_i((C_*^X(\bar k)\otimes A )^{G_k})$
are isomorphic to the homology of the kernel of the map of complexes
$$C_*^X(\bar k)\otimes A \stackrel{\varphi-1}{\to}C_*^X(\bar k)\otimes A.$$
Since higher Galois cohomology is torsion, we have
$R^t\tau_*\operatorname{Hom}({C}_i^X(\bar k), A)=0$ for $t>0$, and
$H^s(G_k,\tau_* \operatorname{Hom}({C}_*^X(\bar k), A))=0$ for $s>0$. Hence
$H^i_{GS}(X,A)$ is isomorphic to the $i$th cohomology of
$$
\operatorname{Hom}_{G_k}({C}^X_*(\bar k),A)
\cong \operatorname{Hom}_{\text{\rm Ab}}({C}^X_*(\bar k)_{G_k},A)
\cong \operatorname{Hom}_{\text{\rm Ab}}({C}_*^X(k),A).
$$
The latter equality follows with Lemma \ref{ratisom}.
\hfill $\Box$ \\

\begin{theorem}\label{svduality}
If $m$ is invertible in $k$ and $A$ is a finitely generated
$m$-torsion $G_k$-module, then
$$ H^i_{GS}(X,A)\cong H^i_{{\text{\rm et}}}(X,A).$$
\end{theorem}

\noindent{\it Proof. }
This follows with the argument of Suslin-Voevodsky \cite{susvoe98}.
Indeed, let $f:(Sch/k)_h \to Et_k$ be the canonical map
from the large site with the h-topology of $k$ to the small etale site of $k$.
Clearly $f_*f^*{\cal F}\cong {\cal F}$, and the proof of Thm.4.5 in loc.cit. shows that
the cokernel of the injection $f^*f_* {\cal F} \to {\cal F}$ is uniquely $m$-divisible,
for any homotopy invariant presheaf with transfers
(like, for example, $C_i^X: U\mapsto \operatorname{Cor}_k(U\times \Delta^i,X)$). Hence
$$\operatorname{Ext}^i_h({\cal F}_h^\sim,f^*A)\cong \operatorname{Ext}^i_h(f^*f_*{\cal F}_h^\sim,f^*A)\cong
\operatorname{Ext}^i_{Et_k}(f_*{\cal F}^\sim_h,A)\cong \operatorname{Ext}^i_{G_k}({\cal F}(\bar k),A).$$
Then the argument of section 7 in loc.cit. together with
Theorem 6.7 can be descended from the algebraic closure of $k$ to $k$.
\hfill $\Box$ \\

\subsection{Duality results}
Duality results for the Galois cohomology of a field $k$
lead via theorem \ref{svduality} to duality results between
Galois-Suslin homology and cohomology over $k$.

\begin{theorem}\label{ffduality}
Let $k$ be a finite field, $A$ a finite $G_k$-module, and
$A^*=\operatorname{Hom}(A,{{\mathbb Q}}/{{\mathbb Z}})$.
Then there is a perfect pairing of finite groups
$$ H_{i-1}^{GS}(X,A) \times H^i_{GS}(X,A^*) \to {{\mathbb Q}}/{{\mathbb Z}}.$$
\end{theorem}

\noindent{\it Proof. }
According to \cite[Example 1.10]{adt} we have
$$\operatorname{Ext}^r_{G_k}(M,{{\mathbb Q}}/{{\mathbb Z}})\cong \operatorname{Ext}^{r+1}_{G_k}(M,{{\mathbb Z}})
\cong H^{1-r}(G_k,M)^*.$$
Hence
\begin{multline*}
\operatorname{Ext}^r_{G_k}(C_*^X(\bar k),\operatorname{Hom}(A,{{\mathbb Q}}/{{\mathbb Z}}))\cong
\operatorname{Ext}^r_{G_k}(C_*^X(\bar k)\otimes A,{{\mathbb Q}}/{{\mathbb Z}})\cong \\
H^{1-r}(G_k,C_*^X(\bar k)\otimes A)^*
\cong H_{r-1}^{GS}(X,A)^*.
\end{multline*}
\hfill $\Box$ \\

The case of non-torsion sheaves is discussed below.

\begin{theorem}
Let $k$ be a local field with finite residue field and separable
closure $k^s$. For a finite $G_k$-module $A$ let
$A^D=\operatorname{Hom}(A,(k^s)^\times)$.
Then we have isomorphisms
$$H^i_{GS}(X,A^D) \cong\operatorname{Hom}(H_{i-2}^{GS}(X,A) ,{{\mathbb Q}}/{{\mathbb Z}}).$$
\end{theorem}

\noindent{\it Proof. }
According to \cite[Thm.2.1]{adt} we have
$$\operatorname{Ext}^r_{G_k}(M,(k^s)^ \times)\cong H^{2-r}(G_k,M)^*$$
for every finite $G_k$-module $M$.
The rest of the proof is the same as above.
\hfill $\Box$ \\

\section{Finite base fields}
From now on we fix a finite field ${{\mathbb F}}_q$ with algebraic
closure $\bar {{\mathbb F}}_q$. To obtain the following results, we
assume resolution of singularities. This is
needed to use the sequences \eqref{blowups} and \eqref{localizations}
to reduce to the smooth
and projective case on the one hand, and the proof of
Jannsen-Saito \cite{uweshuji} of the Kato conjecture on the other hand
(however, Kerz and Saito announced a proof of the prime to $p$-part
of the Kato conjecture which does not require resolution of
singularities).
The critical reader is invited to view the following results as
conjectures which are theorems in dimension at most $3$.

We first present results on finite generation in the spirit
of \cite{uweshuji} and \cite{ichkato}.

\begin{theorem}\label{jsfingen}
For any $X/{{\mathbb F}}_q$ and any integer $m$,
the groups $H_i^S(X,{{\mathbb Z}}/m)$ and $H^i_S(X,{{\mathbb Z}}/m)$ are finitely generated.
\end{theorem}

\noindent{\it Proof. }
It suffices to consider the case of homology.
If $X$ is smooth and proper of dimension $d$, then
$H_i^S(X,{{\mathbb Z}}/m)\cong CH_0(X,i,{{\mathbb Z}}/m)\cong H^{2d-i}_c(X,{{\mathbb Z}}/m(d))$,
and the result follows from work of Jannsen-Saito \cite{uweshuji}.
The usual devisage then shows that $H^j_c(X,{{\mathbb Z}}/m(d))$ is finite
for all $X$ and $d\geq \dim X$, hence $H_i^S(X,{{\mathbb Z}}/m)$ is finite
for smooth $X$. Finally, one proceeds by induction on the dimension
of $X$ with the blow-up long-exact sequence to reduce to the case
$X$ smooth.
\hfill $\Box$ \\

\subsection{Rational Suslin-homology}
We have the following unconditional result:

\begin{theorem}\label{ratdegree0}
For every connected $X$, the map $H_0^S(X,{{\mathbb Q}})\to H_0^S({{\mathbb F}}_q,{{\mathbb Q}})\cong {{\mathbb Q}}$
is an isomorphism.
\end{theorem}

\noindent{\it Proof. }
By induction on the number of irreducible components and \eqref{blowups}
we can first assume that $X$ is
irreducible and then reduce to the situation where $X$ is smooth.
In this case, we use \eqref{localizations} and the following
Proposition to reduce to the smooth and proper case, where
$H_0^S(X,{{\mathbb Q}})=CH_0(X)_{{\mathbb Q}} \cong CH_0({{\mathbb F}}_q)_{{\mathbb Q}}$.
\hfill $\Box$ \\

\begin{proposition}
If $n>\dim X$, then $H^i_c(X,{{\mathbb Q}}(n))=0$
for $i\geq n+\dim X$.
\end{proposition}

\noindent{\it Proof. }
By induction on the dimension and the localization sequence
for motivic cohomology with compact support one sees that the
statement for $X$ and a dense open subscheme of $X$ are equivalent.
Hence we can assume that $X$ is smooth and proper of dimension $d$.
Comparing to higher Chow groups, one sees that this vanishes for
$i>d+n$ for dimension (of cycles) reasons.
For $i=d+n$, we obtain from the niveau spectral sequence a surjection
$$ \bigoplus_{X_{(0)}} H^{n-d}_M(k(x),{{\mathbb Q}}(n-d))
\twoheadrightarrow H^{d+n}_M(X,{{\mathbb Q}}(n)).$$
But the summands vanish for $n>d$ because higher Milnor $K$-theory of
finite fields is torsion.
\hfill $\Box$ \\

By definition, the groups $H_i(X,{{\mathbb Q}}(n))$ vanish for $i<n$.
We will consider the following conjecture $P_n$ of \cite{ichparshin}:

\medskip

\noindent{\bf Conjecture $P_n$:}
{\it For all smooth and projective schemes $X$ over the finite field ${{\mathbb F}}_q$,
the groups $H_i(X,{{\mathbb Q}}(n))$ vanish for $i\not=2n$.}

\medskip

This is a special case of Parshin's conjecture: If $X$ is smooth
and projective of dimension $d$, then
$$ H_i(X,{{\mathbb Q}}(n))\cong H^{2d-i}_M(X,{{\mathbb Q}}(d-n))\cong K_{i-2n}(X)^{(d-n)}$$
and, according to Parshin's conjecture, the latter $K$-group vanishes for
$i\not=2n$.
By the projective bundle formula, $P_n$ implies $P_{n-1}$.

\begin{proposition}\label{ratsituation}
a) Let $U$ be a curve. Then $H_i^S(U,{{\mathbb Q}})\cong H_i^S(X,{{\mathbb Q}})$
for any $X$ normal in $U$.

b) Assume conjecture $P_{-1}$. Then $H_i(X,{{\mathbb Q}}(n))=0$ for all $X$ and $n<0$,
and if $X$ has a desingularization $p:X'\to X$ which is an
isomorphism outside of the dense open subset $U$, then
$H_i^S(U,{{\mathbb Q}})\cong H_i^S(X,{{\mathbb Q}})$.
In particular, Suslin homology and higher Chow groups of
weight $0$ are birational invariant.

c) Under conjecture $P_0$, the groups $H_i^S(X,{{\mathbb Q}})$ are finite dimensional
and vanish unless $0\leq i\leq d$.

d) Conjecture $P_0$ is equivalent to the vanishing of $H_i^S(X,{{\mathbb Q}})$ for
all $i\not=0$ and all smooth $X$.
\end{proposition}

\noindent{\it Proof. }
The argument is the same as in Theorem \ref{ppart}.
To prove b), we have to show
that $H^i_c(X,{{\mathbb Q}}(n))=0$ for $n>d=\dim X$ under $P_{-1}$, and for
c) we have to show that $H^i_c(X,{{\mathbb Q}}(d))$ is finite dimensional and vanishes
unless $d\leq i\leq 2d$ under $P_0$.
By induction on the dimension and the localization sequence
we can assume that $X$ is smooth and projective. In this case, the
statement is Conjecture $P_{-1}$ and $P_0$, respectively,
plus the fact that $H_0^S(X,{{\mathbb Q}})\cong CH_0(X)_{{\mathbb Q}}$ is a finite dimensional
vector space.
The final statement follows from the exact sequence \eqref{localizations}
and the vanishing of $H^i_c(X,{{\mathbb Q}}(n))=0$ for $n>d=\dim X$ under $P_{-1}$.
\hfill $\Box$ \\

\begin{proposition}
Conjecture $P_0$ holds if and only if
the map $H^i_M(X,{{\mathbb Q}})\to H^i_S(X,{{\mathbb Q}})$ of \eqref{canmap}
is an isomorphism for all $X/{{\mathbb F}}_q$ and $i$.
\end{proposition}

\noindent{\it Proof. }
The second statement implies the first, because if the map is an
isomorphism, then $H^i_S(X,{{\mathbb Q}})=0$ for $i\not=0$ and $X$ smooth and proper,
and hence so is the dual $H_i^S(X,{{\mathbb Q}})$.
To show that $P_0$ implies the second statement, first note that
because the map is compatible with long exact blow-up sequences,
we can by induction on the dimension assume that $X$ is smooth
of dimension $d$. In this case, motivic cohomology vanishes above
degree $0$, and the same is true for Suslin cohomology in view
of Proposition \ref{ratsituation}d). To show that for connected $X$ the map
\eqref{canmap} is an isomorphism of ${{\mathbb Q}}$ in degree zero,
we consider the commutative diagram induced by the structure map
$$\begin{CD}
H^0_M({{\mathbb F}}_q,{{\mathbb Q}}) @>>> H^0_S({{\mathbb F}}_q,{{\mathbb Q}})\\
@VVV @VVV\\
H^0_M(X,{{\mathbb Q}}) @>>> H^0_S(X,{{\mathbb Q}})
\end{CD}$$
This reduces the problem to the case $X=\operatorname{Spec} {{\mathbb F}}_q$, where it can be
directly verified.
\hfill $\Box$ \\

\subsection{Integral coefficients}
Combining the torsion results \cite{uweshuji} with the rational results,
we obtain the following

\begin{proposition}
Conjecture $P_0$ is equivalent to the finite generation of
$H_i^S(X,{{\mathbb Z}})$ for all $X/{{\mathbb F}}_q$.
\end{proposition}

\noindent{\it Proof. }
If $X$ is smooth and proper, then according to the main theorem of
Jannsen-Saito \cite{uweshuji}, the groups
$H_i^S(X,{{\mathbb Q}}/{{\mathbb Z}})=CH_0(X,i,{{\mathbb Q}}/{{\mathbb Z}})$ are isomorphic to etale homology,
and hence finite for $i>0$ by the Weil-conjectures.
Hence finite generation of $H_i^S(X,{{\mathbb Z}})$ implies that $H_i^S(X,{{\mathbb Q}})=0$.

Conversely, we can by induction on the dimension assume that $X$ is smooth
and has a smooth and proper model. Expressing Suslin homology of
smooth schemes in terms of cohomology with compact support and again
using induction, it suffices to show that $H^i_M(X,{{\mathbb Z}}(n))$ is finitely
generated for smooth and proper $X$ and $n\geq \dim X$. Using the projective
bundle formula we can assume that $n=\dim X$, and then the statement follows
because $H^i_M(X,{{\mathbb Z}}(n))\cong CH_0(X,2n-i)$ is finitely generated according
to \cite[Thm 1.1]{ichkato}.
\hfill $\Box$ \\

Recall the pairings of Lemma \ref{duall1}. We call them
perfect if they identify one group with the
dual of the other group. In the torsion case, this implies that
the groups are finite, but in the free case this is not true:
For example, $\oplus_I {{\mathbb Z}}$ and $\prod_I {{\mathbb Z}}$ are in perfect duality.

\begin{proposition}\label{fingensushom}
Let $X$ be a separated scheme of finite type over a finite field.
Then the following statements are equivalent:
\begin{enumerate}
\item The groups $H_i^S(X,{{\mathbb Z}})$ are finitely generated for all $i$.
\item The groups $H^i_S(X,{{\mathbb Z}})$ are finitely generated for all $i$.
\item The groups $H^i_S(X,{{\mathbb Z}})$ are countable for all $i$.
\item The pairings of Lemma \ref{duall1} are perfect for all $i$.
\end{enumerate}
\end{proposition}

\noindent{\it Proof. }
a) $\Rightarrow$ b) $\Rightarrow$ c) are clear, and
c) $\Rightarrow$ a) follows from \cite[Prop.3F.12]{hatcher},
which states that if $A$ is not finitely generated, then either
$\operatorname{Hom}(A,{{\mathbb Z}})$ or $\operatorname{Ext}(A,{{\mathbb Z}})$ is uncountable.

Going through the proof of Lemma \ref{duall1} it is easy to see
that a) implies d). Conversely, if the pairing is perfect,
then ${}_{\text{\rm tor}} H_i^S(X,{{\mathbb Z}})$ is finite. Let $A=H^i_S(X,{{\mathbb Z}})/{\text{\rm tor}}$
and fix a prime $l$. Then $A/l$ is a quotient of
$H^i_S(X,{{\mathbb Z}})/l\subseteq H^i_S(X,{{\mathbb Z}}/l)$, and which is
finite by Theorem \ref{jsfingen}.
Choose lifts $b_i\in A$ of a basis of $A/l$ and
let $B$ be the finitely generated free abelian
subgroup of $A$ generated by the $b_i$. By construction, $A/B$ is
$l$-divisible, hence $H_i^S(X,{{\mathbb Z}})/{\text{\rm tor}}=\operatorname{Hom}(A,{{\mathbb Z}})\subseteq \operatorname{Hom}(B,{{\mathbb Z}})$
is finitely generated.
\hfill $\Box$ \\

\subsection{Algebraically closure of a finite field}
Suslin homology has properties similar to a Weil-cohomology theory.
Let $X_1$ be separated and of finite type over ${{\mathbb F}}_q$,
$X_n= X\times_{{{\mathbb F}}_q}{{\mathbb F}}_{q^n}$ and $X=X_1\times_{{{\mathbb F}}_q}\bar {{\mathbb F}}_q$.
From Corollary \ref{descentss}, we obtain a short exact sequence
$$ 0\to {\lim}^1 H^{t+1}_S(X_n,{{\mathbb Z}})\to H^t_S(X,{{\mathbb Z}})
\to\lim H^t_S(X_n,{{\mathbb Z}})\to 0.$$
The outer terms can be calculated with the $6$-term $\lim$-$\lim^1$-sequence
associated to \eqref{cefff}.
The theorem of Suslin and Voevodsky implies that
$$ \lim H^i_S(X,{{\mathbb Z}}/l^r)\cong H^i_{\text{\rm et}}(X,{{\mathbb Z}}_l)$$
for $l\not=p$. For $X$ is proper and $l=p$, we get the same
result from \cite{ichduality}
$$ H^i_S(X,{{\mathbb Z}}/p^r)\cong \operatorname{Hom}(CH_0(X,i,Z/p^r),{{\mathbb Z}}/p^r)\cong
H^i_{\text{\rm et}}(X,{{\mathbb Z}}/p^r).$$
We show that this is true integrally:

\begin{proposition}
Let $X$ be a smooth and proper curve over the algebraic closure
of a finite field $k$ of characteristic $p$. Then the non-vanishing cohomology
groups are
$$ H^i_S(X,{{\mathbb Z}})\cong
\begin{cases}
{{\mathbb Z}} & i=0\\
\lim_r \operatorname{Hom}_{GS}(\mu_{p^r}, \operatorname{Pic} X) \times
\prod_{l\not=p} T_l\operatorname{Pic} X (-1) & i=1\\
\prod_{l\not=p} {{\mathbb Z}}_l(-1) & i=2.
\end{cases} $$
The homomorphisms are maps of group schemes.
\end{proposition}

\noindent{\it Proof. }
By properness and smoothness we have
$$H_i^S(X,{{\mathbb Z}})\cong H^{2-i}_M(X,{{\mathbb Z}}(1))\cong
\begin{cases}
\operatorname{Pic} X & i=0;\\
k^\times & i=1;\\
0 & i\not=0,1.
\end{cases}$$
Now
$$\operatorname{Ext}^1(k^\times,{{\mathbb Z}})=
\operatorname{Hom}(\operatornamewithlimits{colim}_{p\not|m}\mu_m,{{\mathbb Q}}/{{\mathbb Z}})\cong \prod_{l\not=p}{{\mathbb Z}}_l(-1)$$
and since $\operatorname{Pic} X$ is finitely generated by torsion,
\begin{multline*}
\operatorname{Ext}^1(\operatorname{Pic} X,{{\mathbb Z}})\cong \operatorname{Hom}(\operatornamewithlimits{colim}_m {}_m\operatorname{Pic} X,{{\mathbb Q}}/{{\mathbb Z}})\cong \\
\lim \operatorname{Hom}_{GS}({}_m\operatorname{Pic} X,{{\mathbb Z}}/m)
\cong \lim_m \operatorname{Hom}_{GS}(\mu_m,{}_m\operatorname{Pic} X)
\end{multline*}
by the Weil-pairing.
\hfill $\Box$ \\

\begin{proposition}
Let $X$ be smooth, projective and connected over the algebraic
closure of a finite field. Assuming conjecture $P_0$, we have
$$ H^i_S(X,{{\mathbb Z}})\cong
\begin{cases}
{{\mathbb Z}} & i=0\\
\prod_l H^i_{\text{\rm et}}(X,{{\mathbb Z}}_l) & i\geq 1.
\end{cases} $$
In particular, the $l$-adic completion of $H^i_S(X,{{\mathbb Z}})$
is $l$-adic cohomology $H^i_{\text{\rm et}}(X,{{\mathbb Z}}_l)$ for all $l$.
\end{proposition}

\noindent{\it Proof. }
Let $d=\dim X$. By properness and smoothness we have
$$H_i^S(X,{{\mathbb Z}})\cong H^{2d-i}_M(X,{{\mathbb Z}}(d)).$$
Under hypothesis $P_0$, the groups $H_i^S(X,{{\mathbb Z}})$
are torsion for $i>0$, and $H_0^S(X,{{\mathbb Z}})=CH_0(X)$ is the product of
a finitely generated group and a torsion group. Hence for $i\geq 1$
we get by \eqref{cefff} that
\begin{multline*}
H^i_S(X,{{\mathbb Z}})\cong \operatorname{Ext}^1(H_{i-1}^S(X,{{\mathbb Z}}),{{\mathbb Z}})\cong
\operatorname{Hom}(H_{i-1}^S(X,{{\mathbb Z}})_{\text{\rm tor}},{{\mathbb Q}}/{{\mathbb Z}})\\ \cong
\operatorname{Hom}(H^{2d-i+1}_M(X,{{\mathbb Z}}(d))_{\text{\rm tor}},{{\mathbb Q}}/{{\mathbb Z}})
\cong
\operatorname{Hom}(H^{2d-i}_{\text{\rm et}}(X,{{\mathbb Q}}/{{\mathbb Z}}(d)),{{\mathbb Q}}/{{\mathbb Z}}) \\
\cong \operatorname{Hom}(\operatornamewithlimits{colim}_m H^{2d-i}_{\text{\rm et}}(X,{{\mathbb Z}}/m(d)),{{\mathbb Q}}/{{\mathbb Z}})
\cong
\lim_m \operatorname{Hom}(H^{2d-i}_{\text{\rm et}}(X,{{\mathbb Z}}/m(d)),{{\mathbb Z}}/m).
\end{multline*}
By Poincare-duality, the latter agrees with
$\lim H^i_{\text{\rm et}}(X,{{\mathbb Z}}/m) \cong \prod_l H^i_{\text{\rm et}}(X,{{\mathbb Z}}_l)$.
\hfill $\Box$ \\

\section{Arithmetic homology and cohomology}
We recall some definitions and results from \cite{ichweilI}.
Let $X$ be separated and of finite type over a finite field ${{\mathbb F}}_q$,
$\bar X=X\times_{{{\mathbb F}}_q}\bar{{\mathbb F}}_q$ and $G$ be the Weil-group of ${{\mathbb F}}_q$.
Let $\gamma: {\cal T}_G \to {\cal T}_{\hat G}$ be the functor
from the category of $G$-modules to the category of continuous
$\hat G=\operatorname{Gal}({{\mathbb F}}_q)$-modules which associated to $M$ the module
$\gamma_*=\operatornamewithlimits{colim}_m M^{mG}$, where the index set is ordered by divisibility.
It is easy to see that the forgetful functor is a left adjoint of $\gamma_*$,
hence $\gamma_*$ is left exact and preserves limits. The derived functors
$\gamma^i_*$ vanish for $i>1$, and
$\gamma^1_*M=R^1\gamma_*M=\operatornamewithlimits{colim} M_{mG}$, where the transition maps
are given by $M_{mG}\to M_{mnG}, x\mapsto \sum_{s\in mG/mnG}sx$.
Consequently, a complex $M^\cdot$ of $G$-modules
gives rise to an exact triangle of continuous $G_k$-modules
\begin{equation}\label{weiltriangle}
\gamma_* M^\cdot \to R\gamma_* M^\cdot \to \gamma^1_* M^\cdot[-1] .
\end{equation}
If $M=\gamma^*N$ is the restriction of a continuous $\hat G$-module, then
$\gamma_*M=N$ and $\gamma^1_*M=N\otimes {{\mathbb Q}}$. In particular, Weil-etale
cohomology and etale cohomology of torsion sheaves agree.
Note that the derived functors $\gamma_*$ restricted to the
category of $\hat G$-modules does not agree with the derived functors
of $\tau_*$ considered in Lemma \ref{suscohss}. Indeed,
$R^i\tau_*M = \operatornamewithlimits{colim}_L H^i(G_L,M)$ is the colimit of Galois cohomology
groups, whereas $R^i\gamma_*M = \operatornamewithlimits{colim}_m H^i(mG,M)$ is the colimit
of cohomology groups of the discrete group ${{\mathbb Z}}$.

\subsection{Homology}
We define arithmetic homology with coefficients in the $G$-module $A$ to be
$$H_i^{\text{\rm ar}}(X,A) := \operatorname{Tor}_i^G (C_*^X(\bar k),  A).$$
A concrete representative is the double complex
$$ C_*^X(\bar k)\otimes A \stackrel{1-\varphi}{\longrightarrow}
C_*^X(\bar k)\otimes A,$$
with the left and right term in homological degrees one and
zero, respectively, and with the Frobenius endomorphism $\varphi$ acting
diagonally. We obtain short exact sequences
\begin{equation}\label{poip}
0 \to H_i^S(\bar X,A)_G \to H_i^{\text{\rm ar}}(X,A)\to H_{i-1}^S(\bar X,A)^G\to 0.
\end{equation}

\begin{lemma}\label{pofinite}
The groups $H_i^{\text{\rm ar}}(X,{{\mathbb Z}}/m)$ are finite.
In particular, $H_i^{\text{\rm ar}}(X,{{\mathbb Z}})/m$ and ${}_mH_i^{\text{\rm ar}}(X,{{\mathbb Z}})$ are finite.
\end{lemma}

\noindent{\it Proof. }
The first statement follows from the short exact sequence \eqref{poip}.
Indeed, if $m$ is prime
to the characteristic, then we apply \eqref{susvoemain}
together with finite generation of etale cohomology, and if $m$
is a power of the characteristic, we apply Theorem \ref{ppart}
to obtain finiteness of the outer terms of \eqref{poip}.
The final statements follows from the long exact sequence
$$ \cdots \to H_i^{\text{\rm ar}}(X,{{\mathbb Z}})\stackrel{\times m}{\longrightarrow}
 H_i^{\text{\rm ar}}(X,{{\mathbb Z}})\to H_i^{\text{\rm ar}}(X,{{\mathbb Z}}/m)\to \cdots$$
\hfill $\Box$ \\

If $A$ is the restriction of a $\hat G$-module, then
\eqref{weiltriangle} applied to the complex of continuous
$\hat G$-modules ${C}_*^X(\bar k)\otimes A$, gives a long exact sequence
$$ \cdots \to H_i^{GS}(X,A) \to  H_{i+1}^{\text{\rm ar}}(X,A) \to
H_{i+1}^{GS}(X,A_{{\mathbb Q}}) \to H_{i-1}^{GS}(X,A) \to \cdots$$
With rational coefficients this sequence breaks up into
isomorphisms
\begin{equation}\label{ratweilsus}
H_i^{\text{\rm ar}}(X,{{\mathbb Q}})\cong H_i^S(X,{{\mathbb Q}})\oplus H_{i-1}^S(X,{{\mathbb Q}}).
\end{equation}

\subsection{Cohomology}
In analogy to \eqref{suscohdef}, we define arithmetic
cohomology with coefficients in the $G$-module $A$ to be
\begin{equation}\label{weilsusdef}
H^i_{\text{\rm ar}}(X,A)= \operatorname{Ext}^i_G(C_*^X(\bar k),A).
\end{equation}
Note the difference to the definition in \cite{lichtenbaumweilI},
which does not give well-behaved (i.e. finitely generated) groups
for schemes which are not smooth and proper.
A concrete representative is the double complex
$$ \operatorname{Hom}(C_*^X(\bar k),A)  \stackrel{1-\varphi}{\longrightarrow}
\operatorname{Hom}(C_*^X(\bar k),A),$$
where the left and right hand term are in cohomological degrees
zero and one, respectively.
There are short exact sequences
\begin{equation}\label{weilhsss}
0 \to H^{i-1}_S(\bar X,A)_G \to H^i_{\text{\rm ar}}(X,A)\to H^i_S(\bar X,A)^G\to 0.
\end{equation}

The proof of Lemma \ref{pofinite} also shows

\begin{lemma} The groups $H^i_{\text{\rm ar}}(X,{{\mathbb Z}}/m)$ are finite.
In particular, ${}_mH^i_{\text{\rm ar}}(X,{{\mathbb Z}})$ and $H^i_{\text{\rm ar}}(X,{{\mathbb Z}})/m$ are finite.
\end{lemma}

\begin{lemma}
For every $G$-module $A$, we have an isomorphism
$$H^i_{\text{\rm ar}}(X,A)\cong H^i_{GS}(X,R\gamma_*\gamma^*A).$$
\end{lemma}

\noindent{\it Proof. }
Since $M^G=(\gamma_*M)^{\hat G}$, Weil-Suslin cohomology is the Galois
cohomology of the
derived functor of $\gamma_*\operatorname{Hom}_{\text{\rm Ab}}(C_*^X(\bar k),-)$ on the
category of $G$-modules. By Lemma \ref{suscohss}, it suffices to
show that this derived functor agrees with
the derived functor of $\tau_*\operatorname{Hom}_{\text{\rm Ab}}(C_*^X(\bar k),\gamma_*-)$ on the
category of $G$-modules. But given a continuous $\hat G$-modules $M$ and a
$G$-module $N$, the inclusion
$$\tau_*\operatorname{Hom}_{\text{\rm Ab}}(M,\gamma_*N)\subseteq  \gamma_*\operatorname{Hom}_{\text{\rm Ab}}(\gamma^*M,N)$$
induced by the inclusion $\gamma_*N\subseteq N$
is an isomorphism. Indeed, if $f:M\to N$ is $H$-invariant and $m\in M$
is fixed by $H'$, then $f(m)$ is fixed by $H\cap H'$, hence $f$
factors through $\gamma_*N$.
\hfill $\Box$ \\

\begin{corollary}
If $A$ is a continuous $\hat G$-module, then there is a long exact sequence
$$ \cdots \to H^i_{GS}(X,A) \to  H^i_{\text{\rm ar}}(X,A) \to H^{i-1}_{GS}(X,A_{{\mathbb Q}})
\to H^{i+1}_{GS}(X,A) \to \cdots.$$
\end{corollary}

\noindent{\it Proof. }
This follows from the Lemma by applying the long exact
$\operatorname{Ext}^*_{\hat G}(C_*^X(\bar k),-)$-sequence to \eqref{weiltriangle}.
\hfill $\Box$ \\

\subsection{Finite generation and duality}

\begin{lemma}\label{duall}
There are natural pairings
$$ H^i_{\text{\rm ar}}(X,{{\mathbb Z}})/{\text{\rm tor}} \times H_i^{\text{\rm ar}}(X,{{\mathbb Z}})/{\text{\rm tor}} \to {{\mathbb Z}}$$
and
$$H^i_{\text{\rm ar}}(X,{{\mathbb Z}})_{\text{\rm tor}} \times H_{i-1}^{\text{\rm ar}}(X,{{\mathbb Z}})_{\text{\rm tor}}\to {{\mathbb Q}}/{{\mathbb Z}}.$$
\end{lemma}

\noindent{\it Proof. }
From the adjunction
$ \operatorname{Hom}_G(M,{{\mathbb Z}})\cong \operatorname{Hom}_{\text{\rm Ab}}(M_G,{{\mathbb Z}})$
and the fact that $L(-)_G = R(-)^G[-1]$,
we obtain by deriving a quasi-isomorphism
$$ R\operatorname{Hom}_G(C_*^X(\bar k),{{\mathbb Z}})\cong
R\operatorname{Hom}_{\text{\rm Ab}}(C_*^X(\bar k)\otimes^L_G{{\mathbb Z}},{{\mathbb Z}}) .$$
Now we obtain the pairing as in Lemma \ref{duall1}
using the resulting spectral sequence
$$ \operatorname{Ext}^s_{\text{\rm Ab}}(H_t^{\text{\rm ar}}(X,{{\mathbb Z}}),{{\mathbb Z}})\Rightarrow H^{s+t}_{\text{\rm ar}}(X,{{\mathbb Z}}).$$
\hfill $\Box$ \\

\begin{proposition}\label{fingenequi}
For a given separated scheme $X$ of finite type over ${{\mathbb F}}_q$,
the following statements are equivalent:
\begin{enumerate}
\item The groups $H_i^{\text{\rm ar}}(X,{{\mathbb Z}})$ are finitely generated.
\item The groups $H^i_{\text{\rm ar}}(X,{{\mathbb Z}})$ are finitely generated.
\item The groups $H^i_{\text{\rm ar}}(X,{{\mathbb Z}})$ are countable.
\item The pairings of Lemma \ref{duall} are perfect.
\end{enumerate}
\end{proposition}

\noindent{\it Proof. }
This is proved exactly as Proposition \ref{fingensushom}, with
Theorem \ref{jsfingen} replaced by Lemma \ref{pofinite}.
\hfill $\Box$ \\

We need a Weil-version of motivic cohomology
with compact support. We define $H^i_c(X_W,{{\mathbb Z}}(n))$ to be
the $i$th cohomology of $R\Gamma(G, R\Gamma_c(\bar X,{{\mathbb Z}}(n)))$,
where the inner term is a complex defining motivic cohomology
with compact support of $\bar X$. We use this notation to distinguish
it from arithmetic homology with compact support considered in
\cite{ichduke}, which is the cohomology of
$R\Gamma(G, R\Gamma_c(\bar X_{\text{\rm et}},{{\mathbb Z}}(n)))$. However, if $n\geq \dim X$,
which is the case of most importance for us, both theories agree.

Similar to \eqref{localizations} we obtain for a closed subscheme
$Z$ of a smooth scheme $X$ of pure dimension $d$ with open
complement $U$ a long exact sequence
\begin{equation}\label{localizationsw}
\cdots \to H_i^{\text{\rm ar}}(U,{{\mathbb Z}}) \to H_i^{\text{\rm ar}}(X,{{\mathbb Z}}) \to
H^{2d+1-i}_c(Z_W,{{\mathbb Z}}(d))\to\cdots .
\end{equation}
The shift by $1$ in degrees occurs because arithmetic homology is defined
using homology of $G$, whereas cohomology with compact support is defined
using cohomology of $G$.

\begin{proposition}
The following statements are equivalent:
\begin{enumerate}
\item Conjecture $P_0$.
\item The groups $H_i^{\text{\rm ar}}(X,{{\mathbb Z}})$ are finitely generated for all $X$.
\end{enumerate}
\end{proposition}

\noindent{\it Proof. }
a) $\Rightarrow$ b):
By induction on the dimension of $X$ and the blow-up square,
we can assume that $X$ is smooth of dimension $d$, where
$$H_i^{\text{\rm ar}}(X,{{\mathbb Z}})\cong H^{2d+1-i}_c(X_W,{{\mathbb Z}}(d)).$$
By localization for $H^*_c(X_W,{{\mathbb Z}}(d))$ and induction on the dimension
we can reduce the question to $X$ smooth and projective. In
this case ${{\mathbb Z}}(d)$ has etale hypercohomological descent
over an algebraically closed field by \cite{ichduality}, hence
$H^j_c(X_W,{{\mathbb Z}}(d))$ agrees with the Weil-etale cohomology $H^j_W(X,{{\mathbb Z}}(d))$
considered in \cite{ichweilI}.
These groups are the finitely generated for $i> 2d$
by \cite[Thm.7.3,7.5]{ichweilI}. By conjecture $P_0$, and the
isomorphism
$H^i_W(X,{{\mathbb Z}}(d))_{{\mathbb Q}}\cong CH_0(X,2d-i)_{{\mathbb Q}}\oplus CH_0(X,2d-i+1)_{{\mathbb Q}}$
of Thm.7.1c) loc.cit., these groups are torsion for $i<2d$, so that
the finite group $H^{i-1}(X_{\text{\rm et}},{{\mathbb Q}}/{{\mathbb Z}}(d))$ surjects onto
$H^i_W(X,{{\mathbb Z}}(d))$. Finally, $H^{2d}_W(X,{{\mathbb Z}}(d))$ is an extension
of the finitely generated group $CH_0(\bar X)^G$ by the finite group
$H^{2d-1}(\bar X_{\text{\rm et}},{{\mathbb Z}}(d))_G\cong H^{2d-2}(\bar X_{\text{\rm et}},{{\mathbb Q}}/{{\mathbb Z}}(d))_G$.

b) $\Rightarrow$ a) Consider the special case that $X$ is smooth
and projective. Then as above, $H_i^{\text{\rm ar}}(X,{{\mathbb Z}})\cong H^{2d+1-i}_W(X,{{\mathbb Z}}(d))$.
If this group is finitely generated, then we obtain from
the coefficient sequence that
$H^{2d+1-i}_W(X,{{\mathbb Z}}(d))\otimes {{\mathbb Z}}_l\cong \lim H^{2d+1-i}(X_{\text{\rm et}},{{\mathbb Z}}/l^r(d))$,
and the latter group is torsion for $i>1$ by the Weil-conjectures.
Now use \eqref{ratweilsus}.
\hfill $\Box$ \\

\begin{theorem}\label{degreezero}
For connected $X$, the map $H_0^{\text{\rm ar}}(X,{{\mathbb Z}})\to H_0^{\text{\rm ar}}({{\mathbb F}}_q,{{\mathbb Z}})\cong {{\mathbb Z}}$
is an isomorphism. In particular, we have $H_0^{\text{\rm ar}}(X,{{\mathbb Z}})\cong {{\mathbb Z}}^{\pi_0(X)}$.
\end{theorem}

\noindent{\it Proof. }
The proof is similar to the proof of Theorem \ref{ratdegree0}.
Again we use induction on the dimension and the blow-up sequence to reduce
to the situation where $X$ is irreducible and smooth.
In this case, we can use \eqref{localizationsw}
and the following Proposition to reduce to the smooth and proper case,
where we have
$H_0^{\text{\rm ar}}(X,{{\mathbb Z}})=CH_0(\bar X)_G \cong {{\mathbb Z}}$.
\hfill $\Box$ \\

\begin{proposition}\label{zerohigh}
If $n>\dim X$, then $H^i_c(X_W,{{\mathbb Z}}(n))=0$
for $i>n+\dim X$.
\end{proposition}

\noindent{\it Proof. }
By induction on the dimension and the localization sequence
for motivic cohomology with compact support one sees that the
statement for $X$ and a dense open subscheme of $X$ are equivalent.
Hence we can assume that $X$ is smooth and proper of dimension $d$.
In this case, $H^i_c(X_W,{{\mathbb Z}}(n))$ is an extension of $H^i_M(\bar X,{{\mathbb Z}}(n))^G$
by $H^{i-1}_M(\bar X,{{\mathbb Z}}(n))_G$. These groups vanish for
$i-1>d+n$ for dimension (of cycles) reasons.
For $i=d+n+1$, we have to show that $H^{d+n}_M(\bar X,{{\mathbb Z}}(n))_G$
vanishes. From the niveau spectral sequence for motivic cohomology
we obtain a surjection
$$ \bigoplus_{\bar X_{(0)}} H^{n-d}_M(k(x),{{\mathbb Z}}(n-d))
\twoheadrightarrow H^{d+n}_M(\bar X,{{\mathbb Z}}(n)).$$
The summands are isomorphic to $K_{n-d}^M(\bar{{\mathbb F}}_q)$.
If $n>d+1$, then they vanish because higher Milnor $K$-theory of
algebraically closed fields vanishes. If $n=d+1$, then the summands
are isomorphic to $(\bar {{\mathbb F}}_q)^\times$, whose coinvariants vanish.
\hfill $\Box$ \\

\section{A Kato type homology}
We construct a homology theory measuring the
difference between Suslin homology and arithmetic homology.
The cohomological theory can be defined analogously.
Kato-Suslin-homology with coefficients in the $G$-module $A$,
$H_i^{KS}(X,A)$ is defined as the $i$th homology of the complex
of coinvariants $(C_*^X(\bar k)\otimes A)_G$. If $A$ is
trivial as a $G$-module, then since
$(C_*^X(\bar k)\otimes A)^G \cong C_*^X(k)\otimes A$, we get the
short exact sequence of complexes
$$0\to C_*^X(k) \otimes A\to
C_*^X(\bar k)\otimes A \stackrel{1-\varphi}{\longrightarrow}
C_*^X(\bar k)\otimes A\to (C_*^X(\bar k)\otimes A)_\varphi \to 0$$
and hence a long exact sequence
$$ \cdots \to H_i^S(X,A) \to  H_{i+1}^{\text{\rm ar}}(X,A) \to
H_{i+1}^{KS}(X,A) \to H_{i-1}^{S}(X,A) \to \cdots .$$
By Theorem \ref{degreezero}
we have $H_0^{KS}(X,{{\mathbb Z}})\cong H_0^{\text{\rm ar}}(X,{{\mathbb Z}})\cong  {{\mathbb Z}}^{\pi_0(X)}$.
The following is a generalization of the integral
version \cite{ichkato} of Kato's conjecture \cite{kato}.

\begin{conjecture}\label{katoconj} (Generalized integral Kato-conjecture)
If $X$ is smooth, then $H_i^{KS}(X,{{\mathbb Z}})=0$ for $i>0$.
\end{conjecture}

Equivalently, the canonical map $H_i^S(X,{{\mathbb Z}})\cong H_{i+1}^{\text{\rm ar}}(X,{{\mathbb Z}})$
is an isomorphism for all smooth $X$ and all $i\geq 0$, i.e. there
are short exact sequences
$$0 \to H_{i+1}^S(\bar X,{{\mathbb Z}})_G \to H_i^S(X,{{\mathbb Z}})\to H_i^S(\bar X,{{\mathbb Z}})^G\to 0.$$

\begin{theorem}
Conjecture \ref{katoconj} is equivalent to conjecture $P_0$.
\end{theorem}

\noindent{\it Proof. }
If Conjecture \ref{katoconj} holds, then
$$ H_i^S(X,{{\mathbb Q}})\cong H_{i+1}^{\text{\rm ar}}(X,{{\mathbb Q}})\cong H_{i+1}^S(X,{{\mathbb Q}})\oplus
H_i^S(X,{{\mathbb Q}})$$
implies the vanishing of $H_i^S(X,{{\mathbb Q}})$ for $i>0$.

Conversely, we first claim that for smooth and proper $Z$,
the canonical map $H^i_c(Z,{{\mathbb Z}}(n))\to H^i_c(Z_W,{{\mathbb Z}}(n))$ is an isomorphism for
all $i$ if $n<\dim Z$, and for $i\leq 2n$ if $n=\dim Z$.
Indeed, if $n\geq \dim Z$ then the cohomology of ${{\mathbb Z}}(n)$ agrees with the
etale hypercohomology of ${{\mathbb Z}}(n)$, see \cite{ichduality},
hence satisfies Galois descent.
But according to (the proof of) Proposition \ref{ratsituation}b),
these groups are torsion groups, so that Galois descent $R\Gamma_{G_k}$
agrees with $R\Gamma_G$.

Using localization for cohomology with compact support and
induction on the dimension, we get next that
$H^i_c(Z,{{\mathbb Z}}(n))\cong H^i_c(Z_W,{{\mathbb Z}}(n))$ for all $i$ and all $Z$
with $n<\dim Z$. Now choose a smooth
and proper compactification $C$ of $X$.  Comparing the
exact sequences \eqref{localizations} and \eqref{localizationsw},
we see with the $5$-Lemma that the
isomorphism $H_i^S(C,{{\mathbb Z}})\cong H^{2d-i}_c(C,{{\mathbb Z}}(d))
\to H_{i+1}^{\text{\rm ar}}(C,{{\mathbb Z}})\cong H^{2d-i}_c(C_W,{{\mathbb Z}}(d))$ for $C$ implies the
same isomorphism for $X$ and $i\geq 0$.
\hfill $\Box$ \\

\section{Tamely ramified class field theory}
We propose the following conjecture relating Weil-Suslin homology
to class field theory:

\begin{conjecture}\label{CFT} (Tame reciprocity conjecture)
For any $X$ separated and of finite
type over a finite field, there is a canonical injection to
the tame abelianized fundamental group with dense image
$$ H_1^{\text{\rm ar}}(X,{{\mathbb Z}}) \to \pi_1^{t}(X)^{ab}.$$
\end{conjecture}

Note that the group $H_1^{\text{\rm ar}}(X,{{\mathbb Z}})$ is conjecturally finitely generated.
At this point, we do not have an explicit construction (associating
elements in the Galois groups to algebraic cycles) of the map.
One might even hope that
$H_1^{\text{\rm ar}}(X,{{\mathbb Z}})^\circ := \ker (H_1^{\text{\rm ar}}(X,{{\mathbb Z}})\to {{\mathbb Z}}^{\pi_0(X)}$
is finitely generated and isomorphic to the abelianized
geometric part of the tame fundamental group defined in SGA 3X\S 6.

Under Conjecture \ref{katoconj}, $H_0^S(X,{{\mathbb Z}})\cong H_1^{\text{\rm ar}}(X,{{\mathbb Z}})$
for smooth $X$,
and Conjecture \ref{CFT} is a theorem of Schmidt-Spiess \cite{ss}.

\begin{proposition}
a) We have $H_1^{\text{\rm ar}}(X,{{\mathbb Z}})^{\wedge l}\cong \pi_1^t(X)^{ab}(l)$.
In particular, the prime to $p$-part of Conjecture \ref{CFT} holds
if $H_1^{\text{\rm ar}}(X,{{\mathbb Z}})$ is finitely generated.

b) The analog statement holds for the $p$-part if
$X$ has a compactification $T$ which has a
desingularization which is an isomorphism outside of $X$.
\end{proposition}

\noindent{\it Proof. }
a) By Theorem \ref{degreezero}, $H_0^{\text{\rm ar}}(X,{{\mathbb Z}})$ contains
no divisible subgroup. Hence if $l\not=p$, we have by
Theorems \ref{svduality} and \ref{ffduality}
\begin{multline*}
H_1^{\text{\rm ar}}(X,{{\mathbb Z}})^{\wedge l} \cong \lim H_1^{\text{\rm ar}}(X,{{\mathbb Z}}/l^r)\cong
\lim H_0^{GS}(X,{{\mathbb Z}}/l^r) \\
\cong \lim H^1_{\text{\rm et}}(X,{{\mathbb Z}}/l^r)^*
\cong \pi_1^t(X)^{ab}(l).
\end{multline*}

b) Under the above hypothesis, we can use the duality result
of \cite{ichduality} for the proper scheme $T$ to get
with Proposition \ref{biratt}
\begin{multline*}
H_1^{\text{\rm ar}}(X,{{\mathbb Z}})\otimes {{\mathbb Z}}_p\cong  \lim H_0^{GS}(X,{{\mathbb Z}}/p^r)\cong
\lim H_0^{GS}(T,{{\mathbb Z}}/p^r)\\
\cong \lim H^1_{\text{\rm et}}(T,{{\mathbb Z}}/p^r)^*
\cong \pi_1(T)^{ab}(p)\cong \pi_1^t(X)^{ab}(p).
\end{multline*}
\hfill $\Box$ \\

redpajama_set_name: RedPajamaArXiv


OK, so this was really lunch. We went to a birthday party for one of Seth's coworkers and had some leftover pizza for dinner before we headed over. Dough Bakery had the Cuban as their sandwich special this week and we really wanted to try them. They were amazing. For those of you outside of Atlanta, Dough is a little piece of heaven. Seriously. They have a deli called Gutenfleishers inside the bakery where they make their own vegan sandwich meats. Yes. Freshly made, vegan deli meats. They have a wide variety too – everything from ham to spicy turkey to steak and several others in between. And it's all delicious. So, back to The Cuban – it's a pressed sandwich on fresh Cuban bread with Gutenfleischers HM (ham), Daiya Mozzarella, pickles, banana peppers mayo & mustard. Oh and you could choose between spicy mustard and agave Dijon for the mustard. Seth got potato salad for a side and I got the carrot kale salad. I don't know what kind of yum they use for the dressing on that kale, but just wow. They were having a breakdance contest and backspin promo – you got 5% off your order for each completed backspin. No one was dancing while we were there but we were ready for them if they did – we brought the video camera. Dough tweeted later that there were a couple of people brave enough to do backspins. We had a great time even without any breakdancing because we ran into several friends we've met at vegan meet ups, and also the president of GARP. (Psst – we're going to be on her radio show on Wednesday the 26th).
This place sounds amazing! I have to try it.


Bridges Evergreen Holdings ("Evergreen") is delighted to announce the launch of The Ethical Housing Company ("EHC"), an ambitious new property venture with a mission to create a portfolio of well-maintained, affordable homes for rent in Teesside.
The Ethical Housing Company (ethicalhousing.co.uk) will acquire suitable 1-3-bedroom properties in the region, focusing initially on Redcar, Cleveland and the neighbouring boroughs. Through its partner The Ethical Lettings Agency ("TELA"), it will then rent the properties out to people in housing need, including those on benefits, over the long term.
A lack of other suitable housing means vulnerable people often need to rely on the private rental sector for their accommodation. All too often, they face substandard living conditions, with landlords and agencies ill-equipped to support their needs – which leads to unstable tenancies and a risk of homelessness. There are now an estimated 4.5m people living in poverty in private rented accommodation in the UK, a total that has increased by c. 80% since 2007. In Redcar and Cleveland alone, there are an estimated 1,568 households assessed as needing affordable housing, 1,529 of which are currently in private rented accommodation, with an estimated 48% of current private rented households in receipt of Local Housing Allowance/Universal Credit to help pay rental costs.
Social entrepreneur Carla Keegans established TELA two years ago as a community interest company in her home town of Redcar. Carla, who has previously held senior roles in the local authority housing sector, continues to lead TELA, and will serve as the Director of EHC.
The Ethical Housing Company was launched officially in Redcar this week, with Tees Valley Mayor Ben Houchen also endorsing the initiative as a catalyst for change in the local area.
Evergreen is a permanent capital vehicle managed by specialist impact investor Bridges Fund Management ("Bridges"). Unlike standard private equity funds, Evergreen has no requirement to exit in a finite period – allowing it to provide financial, operational and impact management support to the Ethical Housing Company over the long-term. It has earmarked an initial £5m to support the launch, with an ambition to scale beyond this over time into other areas of the UK.
Bridges has extensive experience in property and housing. It has raised over £500m across four property investment funds, two of which have made substantial investments in lower-cost housing in Greater London. It has also now funded six social outcomes contracts commissioned by central and local Government with the aim of preventing or reducing homelessness. Bridges' Scott Greenhalgh will join both the EHC and TELA boards as a Non-Executive Director.
The Ethical Housing Company will exist to purchase and own decent affordable rented accommodation in the Teesside area, to meet local housing need. By taking a long-term approach, rents can be kept at affordable levels, thus providing a stable option for those in housing need. Working in partnership with TELA, the decent maintenance of properties and good tenancy management will also help to keep value in vulnerable housing markets and hence support local communities.
The Ethical Lettings Agency (TELA) is a Community Interest Company that lets and manages residential properties in the Redcar, Cleveland and wider Teesside area. It provides a range of expert services based on professional housing practice with a policy of simple and transparent fees. TELA strives to raise standards in the lettings industry to reflect the changing needs of customers and communities.
Bridges Fund Management (formerly Bridges Ventures) is a specialist fund manager focused exclusively on sustainable and impact investment, with offices in London, New York and San Francisco. Since 2002, the firm has raised over £900m to invest in SMEs, properties and social sector organisations that are helping to tackle some of society's biggest challenges, with a focus on four themes: health and wellbeing, education and skills, sustainable living and underserved markets.
Bridges Evergreen Holdings is a new kind of investment vehicle, designed to free ambitious mission-led organisations from the constraints of traditional fund structures.
It looks to back profit-with-purpose companies, public sector spin-outs, social sector organisations and employee-owned businesses.
It is structured as a holding company rather than a fund, so it can provide ongoing support over the long term, with no exit requirement.
In addition to financial capital, it also offers strategic, operational and impact management support plus access to the specialist networks of the Bridges team.


text:
Pseudomastax brevipennis är en insektsart som först beskrevs av Henri Saussure 1903.  Pseudomastax brevipennis ingår i släktet Pseudomastax och familjen Eumastacidae. Inga underarter finns listade i Catalogue of Life.

Källor

Hopprätvingar
brevipennis

redpajama_set_name: RedPajamaWikipedia


text:
Die Klass A 1956 war die 18. Saison der höchsten sowjetischen Fußballliga. Sie begann am 31. März und endete am 28. Oktober 1956.

Spartak Moskau gewann mit sechs Punkten Vorsprung vor Dynamo Moskau zum sechsten Mal die sowjetische Fußballmeisterschaft. Die letzten zwei Teams stiegen ab.

Abschlusstabelle
||Republik
 || RUS
 || RUS
 || RUS
 || UKR
 || RUS
 || MDA
 || UKR
 || GEO
 || RUS
 || RUS
 || RUS
 || RUS

Kreuztabelle

Weblinks
 Table of Honor auf soccerlibrary.free.fr
 Sowjet Union 1956 auf rsssf.com
 Meisterschaft 1956 auf footballfacts.ru (russisch)

Einzelnachweise

Fußballsaison 1956
1956


text:
Chambers Family Heritage
Isaac DAUDIN 1 was born in 1611 in Ile De Riveria, La Rochelle, Charente-Maritime, France. He died on 6 Apr 1671 in Montreal, Ile de Montreal, Quebec, Canada. He was buried on 7 Apr 1671 in Montreal, Ile de Montreal, Quebec, Canada. He married Anne JARNET about 1645 in Ile de Re, Larochelle, Aunis, Charente Maritime, France.
Anne JARNET [Parents] 1 was born about 1615 in Ile De Rhe, La Rochelle, Charente-Maritime, France. She died about 1647 in Quebec, Canada. She married Isaac DAUDIN about 1645 in Ile de Re, Larochelle, Aunis, Charente Maritime, France.
They had the following children:
F i Hélène DAUDIN
Jean GAUDET [Parents] was born in 1575 in Martaize, Loudun, Vienne, France. He died in 1671 in Port Royal, Acadia, Nova Scotia, Canada . He married Marie DAUSSY.
Marie DAUSSY [Parents] was born in 1580 in Martaize, Loudun, Vienne, France. She died in 1628 in Pouilly En Auxois, Chassemy, Aisne, France. She married Jean GAUDET.
F i Jeanette GOUDET
Andre JARNET [Parents] was born on 9 Aug 1642 in Royas,Vienne,Dauphine,France. He died on 13 Apr 1692 in Vercheres,,Quebec,Canada. He married Marguerite ANTHIAUME on 12 Jan 1676 in Montreal,Montreal,Quebec,Canada.
Marguerite ANTHIAUME was born in 1653 in St. Nicolas, Champs, Ile de France,France. She died on 4 Oct 1699 in Varennes,Chambly,Quebec,Canada . She married Andre JARNET on 12 Jan 1676 in Montreal,Montreal,Quebec,Canada.
F i Anne JARNET
Jean JARNET was born in 1600 in Royas,Vienne,Dauphine,France. He died in 1660 in Ste. Famile De Lisle D' Orleans, France. He married Perette SERMETTE.
Perette SERMETTE was born in 1615 in Crucilleux,Vienne,Dauphine,France. She died on 2 Nov 1675 in Salagnon,Vignieu,Dauphine,France. She married Jean JARNET.
M i Andre JARNET
Inconnu DAUSSY died .
He had the following children:
F i Marie DAUSSY
Jean GAUDET was born in 1553 in Martaize, Loudun, Vienne, France. He died .
M i Jean GAUDET
Roger LOWERY died . The cause of death was automobile accident.
F i Miss LOWERY
Charles FIEBERNITZ died . He married Helen MAULTHARP. The marriage ended in divorce.
Helen MAULTHARP died . She married Charles FIEBERNITZ. The marriage ended in divorce.
Other marriages:
WILSON, Mr.
M i Albert FIEBERNITZ
Mr. WILSON died . He married Helen MAULTHARP.
Helen MAULTHARP died . She married Mr. WILSON.
FIEBERNITZ, Charles
Francis Marion COOK [Parents] 1 was born on 11 May 1849 in Henry County, Georgia. He died on 10 Aug 1919 in Cullman County, Alabama. He was buried in New Hope Cemetery, Cullman County, Alabama. He married Julia COOK about 1871.
Julia COOK [Parents] 1 was born on 10 Aug 1848 in Georgia. She died on 20 Jul 1919 in Cullman County, Alabama. She was buried in New Hope Cemetery, Cullman County, Alabama. She married Francis Marion COOK about 1871.
Julia Cook's maiden name was Cook.
F i Susan Sarah COOK
Surname List | Name Index

redpajama_set_name: RedPajamaCommonCrawl


Archives|HAMILTON PASSES BEAT C.C.N.Y., 12-0; First Touchdown Made by Toss Across Goal Line After Line Plunges Fail. PENALTIES' MAR THE GAME Buff and Blue Tallies Again in the Closing Period With Skillful Air Attack.
HAMILTON PASSES BEAT C.C.N.Y., 12-0; First Touchdown Made by Toss Across Goal Line After Line Plunges Fail. PENALTIES' MAR THE GAME Buff and Blue Tallies Again in the Closing Period With Skillful Air Attack.


Today only, while supplies last, you can get this Hoover Vacuum Cleaner WindTunnel 3 High Performance Pet Bagless Corded Upright Vacuum for ONLY $95.99 (Retail $189.99).
-Includes pivoting dusting tool, pet Turbo tool, telescoping extension Wand and crevice tool.
-Box might come with empty compartments , but the item is not missing any parts.


As your best friend-type, Max has the most important role in the story. He appears kind and considerate towards people's feelings. When it comes down to it, he can be a bit sarcastic and playful (typically at Andy's suspense) and he can't take a compliment very easily. He appears quite confident in his delivery and tries his best to keep going even when the going gets tough, always being there for you whenever you need a friend.
- His shirt reads "bakuten" which means backflip. It's a tribute to the original inspiration for the character before he diverged.
- Max and Roxie are my longtime OCs. Whether it's canon in CD or not, Max is the youngest quintuplet. Roxie embodies all of their personalities.
- Max's pins are another tribute to the character he was initially inspired on before we changed him up.


In games, as in any creative discipline, it's relatively easy to find good creative products, but much harder to find truly great ones. What is it that separates great creative work — and the people who do it — from the rest? In Episode 17 of Vertical Slice, we're delving into the differences in mindset, focus, and process that differentiate the most vividly inventive creators. Let's see what we can learn from the best!
Chris finished Transistor! Six months after we first discussed it, this stylish action RPG remains a favorite of ours.
Derek is exploring a pastoral English mystery in Everybody's Gone to the Rapture. If you enjoy slower paced narrative games like Firewatch, this one will speak to you.
Designer Notes' interview with Amy Hennig provides a fascinating window into the thought processes of a great creative director.


Santa gets some instruction from his younger generation elf.
Warm Santa hugs for Christmas!
Wish one and all a merry time on Christmas with this bright and beautiful ecard.
Get 'em grooving with Santa through this fun ecard.
Fun With Santa On Christmas!
Spread lotsa holiday cheer around with this warm and bright Christmas ecard.
Spread lotsa holiday cheer around with this groovy Christmas ecard.
A song from the new album "Christmas Remixes"
It's time for merriment because Santa Claus is coming again! With Christmas Santa Claus cards and ecards, send your friends, family and loved ones sacks full of warm and happy wishes. Reach out to your loved ones with these cute and special ecards and wish them jingles of joy and lots of fun on this Christmas.


The courses in Futtsu Park are great for first-rate athletes, visually impaired runners, and wheelchair racers. The full course is 5.6 km, while the shorter course is 2.8 km. The road surface is extremely flat, which makes it good for such sports. The roads are uniform with no undulations, so they're good for building mental fortitude. You can build mental stamina by continuously running the same course.
On the map, Futtsu Park appears as a rectangle. It is easy for those with profound blindness and those with reduced vision to master the course. For example, you can feel after you've run a certain distance that it's time to turn. One corner has a bulge in it, so this feature makes it easy to know where you are.
The longest straight section is exactly one kilometer. You can learn to sense this distance by running it. Many people have honed their senses on this course. Some people count the number of steps. Say, the previous lap was 2480 steps, or 2500 steps. Some people have the ability to internally measure their own pace. They understand much better than sighted people where and how they are running.
If there were street lamps, you could even run at night, and lights would probably be good for preventing crime. If each corner had an audio signal, it would make the course even easier for visually impaired people to run. Worldwide, there are hardly any courses designed to be easy for visually impaired people to run. I think it would be good to make Futtsu Park the world's first such course.
Kyohei Yasuda Japan's Coach for Marathon Athletes at the London Paralympics was born in Chiba. In 1996, he achieved a gold medal as the guide runner for a visually impaired marathon runner at the Atlanta Paralympics. He followed this up with 6th place at the Sydney Paralympics in 2000, and 4th place at the Athens Paralympics in 2004. He has been Japan's coach for marathon athletes with a visual impairment at the Beijing Paralympics in 2008 and at the London Paralympics in 2012. Currently, he is making good use of his experience in efforts to coach amateur runners and in activities to strengthen and popularize the marathon for visually impaired people.
Distance: 2.3 km, 2.8 km, and 5.6 km road courses. Cross country course is located within the park.
Altitude difference: The road is almost level. The cross country course is moderately undulating.


text:
Q: Is it possible to backup my entire MacBook to Microsoft OneDrive? I have a lifetime of free 5TB cloud storage on Microsoft OneDrive via my university.
Is it possible to configure things or write a script so that the entire contents of my computer are continuously backed up to OneDrive?
Thank you!
Gabriel

A: A paid product that I've tried is Arq Backup. It supports backing up to OneDrive as well as other cloud products like Google Drive. However, it is not as fast and efficient as time machine or carbon copy cloner.

redpajama_set_name: RedPajamaStackExchange


We want you to have a consistent and personable experience when you contact us.
We publish regular reports on how we meet our performance standards, giving figures of complaints received and decisions made, together with general feedback on performance.
Previous Making your comment, compliment or complaint Next What if I need help?


On the Moos-Trail, every bike rider experiences a major adventure in the bike paradise of Zermatt – Matterhorn. The super soft tour from the hamlet of Furi to Zermatt is ideal for taking excellent advantage of gravity. Chainless too!
You can reach the Moos Trail by travelling through Zermatt to Winkelmatten on the Moostrasse to the hamlet of Furi (signalized from the train station square) or with the Matterhorn-Express up to Furi.
Before starting at the Restaurant Alm in Schweigmatten, the rider glances briefly into the valley towards Zermatt. You will soon be there!
Over 1.3 km with an average gradient of 7.5%, one measures 95 m of altitude on this Flow Country Trail. This is a perfect route to test whether it is possible to get down to the valley to Moos at the end of the trail, without even turning the pedals once. Occasionally, the rider catches a glance at a Stadel, the forest or a gondola, which rushes past just above his head.
The Moos Trail is equally suited for beginners as well as for experts and everyone in between.
Turnoff under the Restaurant Alm. Afterwards ride down the trail to Moos and back to the Furi street.
It is best to ride this trail several times after each other and test your sense of balance. On the third trip, a stop at the Gorner gorge is worthwhile.
Always wear gloves and a helmet.
Use marked bike trails and routes only, and observe all signposting. Please close gates after passing through.
Please be considerate to walkers and to plants and animals. As a rule, walkers have priority.
When planning, take into account participants' fitness level and ability: some routes lead above 3,000 metres.
All riding is at one's own risk.


text:
Patient Stories | Matthew
Cancer Survivor Feels Inspired to Give Back
Brain Tumor Survivorship Program
Contact / Refer
Matthew Grosser's diagnosis as a baby was devastating. He had something called an ependymoma, a brain tumor. Nurse practitioner Judy Correll's early memories of him are of his mother bringing him to the hospital in a stroller. Matt, now 22, is one of six boys in a family that has supported him through all of the trials of a cancer diagnosis and relapses. Matt prided himself on "not being different," although he says cancer has shaped who he is. He says he feels he needs to give back. He just graduated from the College of Mount St. Joseph, and now is pursuing a graduate degree in social work from the University of Louisville.
(Published September 2012)
Contact the Brain Tumor Center
Read more patient stories from the Brain Tumor Center


text:
Q: Parallel For Each Over Data Tables in a Data Set In VB.NET, can a parallel loop be performed on the tables inside a data set?
This has been answered.
My project is this:

*

*Read a CSV file into a data table in memory.

*Group the rows by sorting the data.

*Process the data in synchronous loop detecting each group by a change in key values.

There are many separate processes that read these CSV files which each have their own layout and processing specifics.
My goal is to process these "groups" of data in parallel without using Entity Framework.
Is the Data Set/Data Table approach the best?

A: Yes, you can. A DataSet and a DataTable are in memory collections. So if you can process them independently it's a good candidate for a parallel For Each.
Parallel.ForEach(
    dataSet.Tables.Cast(Of DataTable),
    Sub(table)
        ProcessTable(table)
    End Sub
)


text:
ABBA has a blissful, bleeding heart on 'Voyage'
1221 AB/Courtesy
By Dominic Marziali | Staff
Peach Tree Rascals impress with nostalgia at Great American Music Hall
Marissa Nadler's 'The Path of the Clouds' is veiled in mystique, reverie and truth
Ed Sheeran got married, but audiences are barely engaged on '='
Grade: 3.5/5.0
ABBA made its grand reentrance with two studio-polished, shimmering singles in September. They were bold and swooning from the start — a promise of old ABBA. In the lead single "I Still Have Faith in You," the group wonders, "Do I have it in me?/ I believe it is in there." It's as much about them like us. And then it's straight into the soaring piano glissando on "Don't Shut Me Down." Dancing queens, no doubt.
New ABBA doesn't quite exist, but it doesn't not exist either. The pop group's Agnetha Fältskog and Anni-Frid Lyngstad's voices have a bit of wear, and they're not current (though Voyage has been topping charts across the globe since its release). They've matured, and they get romanticism, grandeur; big things have had their moment. Their new tracks have moved on to the pressures of the climate, overtly on "Bumblebee," and freedom outright, flatly on "Ode to Freedom."
Björn Ulvaeus and Benny Andersson throw around hooks and arrangements as sharp as they ever did, now souped-up with synths and sounds of the future, without bending to trend. On "Keep an Eye on Dan," a near cousin to a video game's lobby music jacks up the tension before a love-lorn shootout breaks out with, "I know that this shouldn't be a traumatic event, but it is." ABBA understands its sound is of the past, yet timeless.
It's genuine to the ideals ABBA reaches for in Voyage that the group moves from gas-guzzling to tree-hugging. This is an album laden with nostalgia, but devoid of — for example — the acoustic strumming that characterized world hits such as "S.O.S." Instead, ABBA blends the minimalist ding of an idiophone with emotional piano and orchestral arrangements, all while lyrics spin elaborate narrative and melodrama.
The taps of an SOS aren't so far away, for good and bad. On the last track "Ode to Freedom," cellos, flutes and violins plead for harmony. Back on "Bumblebee," a track modern-day Elves might be caught belting out — well, there's the problem. It's an environmental anthem wisting for when Sauron wasn't on the horizon, for Kumbaya.
ABBA tries to balance present and past, and while Voyage is occasionally sharp enough to pull it off, the lyrics sometimes take on the sour taste of a bubble, like bread left in the fridge too long. Sobbing has never been done like ABBA since ABBA, but Voyage also finds the band falling victim to the same ills that plagued the parts of its discography that aged worst. "Chiquitita" had a flippancy to it, a carefree attitude that crops up in modified form on "Little Things," an ode to the material joys of Christmas morning.
ABBA holds dear to what it loves on Voyage, and in that way, ABBA sounds most stuck in the past — a refusal to learn, grow. Saving the bees is nice; reigning in corporations is better. Gifts are great, but "Little Things" doesn't have the personal oomph to realize the joys of giving.
At the same time: You can look at ABBA's album through an anti-capitalist lens, or take issue with their happy-ever-after melodrama. You can't, however, ignore this is not an album about such things. It's not a victory lap. It's a dare: "Do I have it in me," the super troupers wonder on the opener "I Still Have Faith in You." Confidence builds from the synth-folk barn-stomper "When You Danced With Me," up through the "Little Things" they love and straight into the bravado of "Don't Shut Me Down."
And when the group arrives at "Just a Notion," the question behind the album — why did they get back together? — gets answered. Destiny, with a roaring electric guitar and a bouncing bass line, brought them together. The band took a chance to make some swinging good music. The rest of the tracks are the extras, the fun they turn to with a wink and a nod to luck, love and everything else that got them in the studio 40 years later.
Contact Dominic Marziali at [email protected].
ABBA, Agnetha Fältskog, Anni-Frid Lyngstad, Benny Andersson, Björn Ulvaeus, Bumblebee, Don't Shut Me Down, I Still Have Faith in You, Little Things, Ode to Freedom
An ode to Stephen Sondheim's unexpected magic
Attitude of gratitude: Bears roll into season against rival Cardinal
'Benedetta' is visceral, whip-smart spiritual horror


Men vs. women: Learn how — and why — your gender may affect your dreams. Then, a nightmarish sleep disorder that causes violent thrashing ... could you be at risk? Plus, tips to prevent bedtime bloat. And, natural solutions for catching more ZZZs.


AXA Belgium is a lot more than you might think. We are part of the AXA Group, a world leader in financial services, with over 166.000 employees in 64 countries. Over the years, AXA Belgium - Insurance has waxed to become the market leader in this sector, serving over 3 million clients, individuals as well as large companies. Although we are based in Brussels, we have sites up and down the country.
The Procurement Department plays a strategic role at AXA. By keeping an eye on our expenditures and by making the right and sustainable choices with our suppliers and partners, we are further strengthening our position for the future. If you are passionate about purchasing, whether you are experienced in this respect or otherwise, you could be just the key player we are looking for to help us make this happen! Joining a dynamic team, you will be working on transformation projects aimed at building the AXA of tomorrow.
From your first day (and during the following weeks), you will be getting intensive induction training and all the support you need to enable you to swiftly find your feet.
A New Way of Working which a particular focus going out to flexible working hours and workspaces, as well as activity-based work spaces, to name a few elements.
A competitive Compensation & Benefits package (including a cafeteria scheme) including a range of personal benefits: group insurance, disability/hospitalisation/accident insurance, mobility-related benefits, reductions on bank and insurance products, child care during school holidays, laundry and ironing services, etc.
... and, above all, a pleasant working environment and friendly colleagues!