text:
\section{Main results}\label{Introduction}
Let $R$ be a commutative unital ring. Recall that an $R$-group scheme $\bG$ is called reductive, (respectively, semi-simple or simple),
if it is affine and smooth as an $R$-scheme and if, moreover,
for each algebraically closed field $\Omega$ and for each ring homomorphism $R\to\Omega$ the scalar extension $\bG_\Omega$ is
a connected reductive (respectively, semi-simple or simple)
algebraic group over $\Omega$.
The class of reductive group schemes contains
the class of semi-simple group schemes which in turn contains the class of simple group schemes.
This notion of a reductive $R$-group scheme
coincides with~\cite[Exp.~XIX, Definition~2.7]{SGA3}.
This notion of a simple $R$-group scheme coincides with the notion of a simple semi-simple $R$-group
scheme from Demazure and Grothendieck \cite[Exp. XIX, Definition 2.7 and Exp. XXIV, 5.3]{SGA3}.
Here is our first main result based on results of \cite{Pan2} and \cite{Pan3} and significantly extending the corresponding results of
\cite{Pan2} and \cite{Pan3}.
\begin{thm}
\label{Aus_Buksbaum_3}
Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be the fraction field of $R$.
Let
$\mu: \bG \to \bT$
be an $R$-group scheme morphism between reductive $R$-group schemes, which is smooth as a scheme morphism.
Suppose $T$ is an $R$-torus.
Then the map
$\bT(R)/\mu(\bG(R)) \to \bT(K)/\mu(\bG(K))$
is injective and the sequence
\begin{equation}
\label{Aus_Buks_sequence_3}
\{1\} \to \bT(R)/\mu(\bG(R)) \to
\bT(K)/\mu(\bG(K)) \xrightarrow{\sum r_{\mathfrak p}} \bigoplus_{\mathfrak p}
\bT(K)/[\bT(R_{\mathfrak p})\cdot \mu(\bG(K))] \to \{1\}
\end{equation}
is exact, where $\mathfrak p$ runs over all height one prime ideals of $R$
and each
$r_{\mathfrak p}$ is the natural map (the projection to the factor group).
\end{thm}
Let us comment on the first assertion of the theorem. Let $\bH$ be the kernel of $\mu$. It turns out that $\bH$ is
a quasi-reductive $R$-group scheme (see Definition \ref{pre-reductive}).
There is a sequence of group sheaves
$1\to \bH\to \bG\to \bT\to 1$,
which is exact in the \'{e}tale topology on $Spec R$.
Theorem \ref{MainThm1} yields now the injectivity of the map
$\bT(R)/\mu(\bG(R)) \to \bT(K)/\mu(\bG(K))$.

\begin{thm}\label{s_simple_groups}
Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be the fraction field of $R$.
Let $\bG_1$ and $\bG_2$ be two semi-simple $R$-group schemes.
Suppose the generic fibres $\bG_{1,K}$ and $\bG_{2,K}$ are isomorphic as algebraic $K$-groups. Then
the $R$-group schemes $\bG_1$ and $\bG_2$ are isomorphic.
\end{thm}
To prove Theorem \ref{s_simple_groups}
we need to work with automorphism group scheme of a semi-simple $R$-group scheme. The latter group scheme
is not geometrically connected in general.
So, Theorem \ref{s_simple_groups} can not be derived from \cite{FP} and \cite{Pan3}.

We state right below a theorem, which asserts that an extended version of Grothendieck--Serre conjecture holds for rings
$R$ as above. This latter theorem is proved in this paper. Theorem \ref{s_simple_groups} and the first assertion of
Theorem \ref{Aus_Buks_sequence_3} are derived from it.
To state the mentioned theorem it is convenient to give the following.
\begin{defn}[quasi-reductive]\label{pre-reductive}
Assume that $S$ is a Noetherian commutative ring. An $S$-group scheme $\bH$ is called {\rm quasi-reductive}
if there is a finite \'{e}tale $S$-group scheme $\bC$ and a smooth $S$-group scheme morphism
$\lambda: \bH \to \bC$ such that its kernel is a reductive $S$-group scheme and $\lambda$ is surjecive
locally in the \'{e}tale topology on $S$.
\end{defn}
Clearly, reductive $S$-group schemes are quasi-reductive.
Quasi-reductive $S$-group schemes are {\it affine and smooth} as $S$-schemes.
There are two types of quasi-reductive $S$-group schemes,
which we are focusing on in the present paper. The first one is
the automorphism group scheme of a semi-simple $S$-group scheme. The second one is obtained as follows:
take a reductive $S$-group scheme $\bG$, an $S$-torus $\bT$ and a smooth $S$-group morphism
$\mu: \bG \to \bT$. Then one can check that the kernel $\bH$ of $\mu$ is quasi-reductive.
It is an extension of a finite \'{e}tale $S$-group scheme $\bC$ of multiplicative type via a reductive
$S$-group scheme $\bG_0$.

Assume that $U$ is a regular scheme, $\bH$ is a quasi-reductive $U$-group scheme.
Recall that a $U$-scheme $\cH$ with an action of $\bH$ is called \emph{a principal $\bH$-bundle over $U$},
if $\cH$ is faithfully flat and quasi-compact over $U$ and the action is simple transitive,
that is, the natural morphism $\bH\times_U\cH\to\cH\times_U\cH$ is an isomorphism, see~\cite[Section~6]{Gr4}.
Since $\bH$ is $S$-smooth, such a bundle is trivial locally in \'{e}tale topology but in general not in Zariski topology.
Grothendieck and Serre conjectured that for a reductive $U$-group scheme $\bH$ a principal $\bH$-bundle $\cH$
over $U$  is trivial locally in Zariski topology, if it is trivial generically.
A {\it survey paper} on the topic is \cite{P2}.

The conjecture is true, if $\Gamma(U,\cO_U)$ contains a field (see \cite{FP} and \cite{Pan3}).
It is proved in \cite{Ni} that the conjecture is true in general for discrete valuation rings.
This result is extended in \cite{PSt} to the case of semi-local Dedekind integral domains
assuming that $\bG$ is simple simply connected and isotropic in a certain precise sense.
In \cite{NG} results of \cite{Ni} and \cite{PSt} are extended further.
It is proved there that the conjecture is true in general for the case of semi-local Dedekind integral domains.
The following result is a further extension of the main theorem of \cite{Pan3}.
\begin{thm}\label{MainThm1}
Let $R$ be a regular semi-local integral domain containing a field. Let $K$ be the fraction field of $R$.
Let $\bH$ be a quasi-reductive group scheme over $R$. Then the map
\[
  \textrm{H}^1_{\text{\'et}}(R,\bH)\to \textrm{H}^1_{\text{\'et}}(K,\bH),
\]
\noindent
induced by the inclusion of $R$ into $K$ has a trivial kernel.
In other words, under the above assumptions on $R$ and $\bH$, each principal $\bH$-bundle over $R$ having a $K$-rational point is trivial.
\end{thm}

\begin{cor}\label{CorOfMainThm1}
Under the hypothesis of Theorem~\ref{MainThm1}, the map
\[
  \textrm{H}^1_{\text{\'et}}(R,\bH)\to \textrm{H}^1_{\text{\'et}}(K,\bH),
\]
\noindent
induced by the inclusion of $R$ into $K$, is injective. Equivalently, if $\cH_1$ and $\cH_2$ are two principal $\bH$-bundles isomorphic over
$\Spec K$,
then they are isomorphic.
\end{cor}

\begin{proof}
Let $\cH_1$ and $\cH_2$ be two principal $\bH$-bundles isomorphic over
$\spec K$.
Let $\Iso(\cH_1,\cH_2)$ be the scheme of isomorphisms of principal $\bH$-bundles.
This scheme is a principal $\Aut\cH_1$-bundle.
By Theorem~\ref{MainThm1} it is trivial, and we see that $\cH_1\cong\cH_2$.
\end{proof}

Theorems  \ref{MainThm1} and \ref{s_simple_groups} are proved in Section \ref{2_theorems}.
Theorem \ref{Aus_Buksbaum_3} is proved in Section \ref{SectAus_Buksbaum_3}.

\section{Proof of Theorems \ref{MainThm1} and \ref{s_simple_groups}}\label{2_theorems}
We begin with the following general
\begin{lem}\label{f_flat_morphism}
Let $X$ be a regular semi-local integral domain. Let $\pi: X'\to X$ be a finite morphism.
Let $\eta\in X$ be the generic point of $X$. Then
sections of $\pi$
over $X$ are in the bijection with sections of $\pi$ over $\eta$.
\end{lem}

\begin{proof}
It suffices to check that each section $s: \eta\to X'$ of $\pi$
can be extended to a section of $\pi$ over $X$.

Decompose $\pi$ as a composition
$X' \xra{i} \Aff^n_X \xra{p} X$, where $p$ is the projection and $i$ is a closed embedding.
Let $s: \eta\to X'$ be a section of $\pi$. Since $X$ is a regular semi-local and $\pi$ is projective
there is a closed codimension two
subset $Z$ in $X$ and a section $\phi: X-Z \to X'$ of $\pi$ that extends $s$.
Since $\Gamma(X, \mathcal O_X)=\Gamma(X-Z, \mathcal O_X)$ the section $\phi$
is extended to a section $\tilde \phi$ of $\pi$. The lemma is proved.
\end{proof}

\begin{cor}\label{f_gr_scheme}
Let $X$, $\eta\in X$ be as in the previous lemma and $\bE$ be a finite \'{e}tale group $X$-scheme.
Then
the $\eta$-points of $\bE$ coincides with the $X$-points of $\bE$.
\end{cor}

\begin{cor}\label{inj_on_H1}
Under the hypothesis of Corollary \ref{f_gr_scheme}
the kernel of the pointed set map
$H^1_{\text{\'et}}(X,\bE)\to H^1_{\text{\'et}}(\eta,\bE)$
is trivial.
\end{cor}

\begin{proof}
Let $\cal E$ be a principal $\bE$-bundle over $X$. The standard descent arguments shows that
the $X$-scheme $\cal E$ is finite and \'{e}tale. Thus, ${\cal E}(X)=\cal E(\eta)$. This proves the corollary.
\end{proof}

\begin{proof}[Proof of Theorem \ref{MainThm1}]
Since $\bH$ is quasi-reductive $R$-group scheme, there is a finite \'{e}tale $R$-group scheme $\bC$ and a smooth $R$-group scheme morphism
$\lambda: \bH \to \bC$ such that its kernel $\bG$ is a reductive $R$-group scheme and $\lambda$ is
surjecive locally in the \'{e}tale topology on $S$.
The sequence of the \'{e}tale sheaves
$1\to \bG\to \bH\to \bC\to 1$ is exact. Thus, it induces a commutative diagram of pointed set maps with exact rows
$$
\SelectTips{cm}{}
\xymatrix @C=2pc @R=10pt {
\bC(R) \ar[r]^{\partial} \ar[dd]_{\alpha} & {H_\text{{\'et}}^1(R,\bG)} \ar[r] \ar[dd]_{\beta} & {H_\text{{\'et}}^1(R,\bH)} \ar[r] \ar[dd]_{\gamma} & {H_\text{{\'et}}^1(R,\bC)} \ar[dd]_{\delta} \\
{} & {} & {} & {} & {} \\
\bC(K) \ar[r]^{\partial} & {H_\text{{\'et}}^1(K,\bG)} \ar[r] & {H_\text{{\'et}}^1(K,\bH)} \ar[r] & {H_\text{{\'et}}^1(K,\bC)} \\
}
$$
The map $\alpha$ is bijective by Corollary \ref{f_gr_scheme}, the map $\delta$ has the trivial kernel by Corollary
\ref{inj_on_H1}, the map $\beta$ is injective by
\cite[Corollary 1.2]{Pan3}
Now a simple diagram chase shows that $ker(\gamma)=\ast$. This proves the theorem.
\end{proof}

\begin{rem}\label{Ch_G_P}
The statement of \cite[Lemma]{Ch-G-P} and its proof are non-acurate both, since
the authors do not assume
the injectivity of the map
$H_\text{{\'et}}^1(R,\bG^0)\to H_\text{{\'et}}^1(K,\bG^0_R)$.
\end{rem}

\begin{proof}[Proof of Theorem \ref{s_simple_groups}]
The $R$-group scheme $\underline {\text {Aut}}:=\underline {\text {Aut}}_{R-gr-sch}(\bG_1)$
is quasi-reductive by
\cite{D-G}.
The $R$-scheme
$\underline {\text {Iso}}:=\underline {\text {Iso}}_{R-gr-sch}({\bf G_1},{\bf G_2})$
is a principal $\underline {\text {Aut}}$-bundle.
An isomorphism $\phi: \bG_{1,K} \to \bG_{2,K}$ of algebraic $K$-groups gives a section of
$\underline {\text {Iso}}$ over $K$. So, $\underline {\text {Iso}}_K$
is a trivial principal $\underline {\text {Aut}}_K$-bundle.
Hence $\underline {\text {Iso}}$ is a trivial
principal $\underline {\text {Aut}}$-bundle by Theorem \ref{MainThm1}.
Thus, it has a section over $R$. So, there is an $R$-group scheme isomorphism
$\bG_1 \cong \bG_2$.
\end{proof}

\section{Proof of the first assertion of Theorem \ref{Aus_Buksbaum_3}}
\label{One_Lemma}
\begin{lem}\label{Kernel_G_to_T}
Let $X$ be a regular irreducible affine scheme.
Let $\bG$ be a reductive $X$-group scheme and $\bT$ be an $X$-torus.
Let $\mu: \bG \to \bT$
be an $X$-group schemes morphism, which is smooth as a scheme morphism.
Then the kernel of $\mu$ is
a quasi-reductive $X$-group scheme.
\end{lem}

\begin{proof}
Consider the coradical $Corad(\bG)$ of $\bG$ together with the canonical
$X$-group morphism $\alpha: \bG\to Corad(\bG)$. By the universal property of the $X$-group morphism $can$
there is a unique $X$-group morphism $\bar \mu: Corad(\bG)\to T$ such that
$\mu=\bar \mu\circ \alpha$. Since $\mu$ is surjective locally for the \'{e}tale topology, hence so is $\bar \mu$.
Let $\text{ker}(\bar \mu)$ be the kernel of $\bar \mu$ and let $\bH:=\alpha^{-1}(\text{ker}(\bar \mu))$
be the scheme theoretic pre-image of $\text{ker}(\bar \mu)$. Clearly, $\bH$ is a closed $X$-subgroup scheme of $\bG$, which
is the kernel of $\mu$. We must check that $\bH$ is a quasi-reductive.

The $X$-group scheme $\text{ker}(\bar \mu)$ is of multiplicative type. Hence there is a
finite $X$-group scheme $\bM$ of  multiplicative type and a faithfully flat $X$-group scheme morphism
$can: \text{ker}(\bar \mu)\to \bM$, which has the following property:
for any finite $X$-group scheme $\bM'$ of  multiplicative type
and an $X$-group morphism $\phi: \text{ker}(\bar \mu)\to \bM'$ there is a unique $X$-group morphism
$\psi: \bM\to \bM'$ with $\psi \circ can=\phi$. It is known that the kernel of $can$ is an $X$-torus.
Call it $\bT^0$. Since $\mu$ is smooth, hence so is $\bar \mu$. Thus, the $X$-group scheme
$\text{ker}(\bar \mu)$ is an $X$-smooth scheme. This yields that $M$ is \'{e}tale over $X$.

Let $\beta=\alpha|_{\bH}: \bH\to \text{ker}(\bar \mu)$
and let $\bG^0:=\beta^{-1}(\bT^0)$
be the scheme theoretic pre-image of $\bT^0$.
Clearly, $\bG^0$ is a closed $X$-subgroup scheme of $\bH$, which
is the kernel of the morphism $can\circ \beta: \bH\to \bM$.
Let $\gamma=\beta|_{\bG^0}: \bG^0\to \bT^0$.

The $X$-group scheme $\bM$ is finite and \'{e}tale.
The morphism $can$ is smooth. The morphism $\beta$ is smooth as
a base change of the smooth morphism $\alpha$. Thus,
$\lambda:=can\circ \beta$ is smooth.
It is also surjective locally in the \'{e}tale topology on $X$, because $can$ and $\beta$ have this property.
By the construction $\bG^0=\text{ker}(\lambda)$.
So, to prove that $\bH$ is quasi-reductive
it remains to check the reductivity of $\bG^0$.

The $X$-group scheme $\bG^0$ is affine as a closed $X$-subgroup scheme of the reductive $X$-group scheme $\bG$.
Prove now that $\bG^0$ is smooth over $X$. Indeed,
the morphism $\gamma$ is smooth as a base change of the smooth morphism $\alpha$.
The $X$-scheme $\bT^0$ is smooth, since it is an $X$-torus.
Thus, the $X$-scheme $\bG^0$ is smooth.

Write $X$ as $Spec S$ for a regular integral domain $S$.
It remains to verify that for
each algebraically closed field $\Omega$ and for each ring homomorphism $S\to\Omega$ the scalar extension $\bG^0_\Omega$ is
a connected reductive
algebraic group over $\Omega$.
Firstly, recall that $\text{ker}(\alpha)$ is a semi-simple $S$-group scheme. It is the
$S$-group scheme $\bG^{ss}$ under the notation of \cite{D-G}. Clearly,
$\text{ker}(\gamma)=\text{ker}(\alpha)$. Thus, $\text{ker}(\gamma)=\bG^{ss}$
is a semi-simple $S$-group scheme. Since the morphism $\gamma$ is smooth for
each algebraically closed field $\Omega$ and for each ring homomorphism $S\to\Omega$
we have an exact sequence of smooth algebraic groups over $\Omega$
$$1\to \bG^{ss}_\Omega \to \bG^0_\Omega \to \bT^0_\Omega \to 1.$$
The groups $\bT^0_\Omega$, $\bG^{ss}_\Omega$ are connected. Hence the group $\bG^0_\Omega$
is connected too. We know already that it is affine.

Finally, check that its unipotent radical $\mathbf U$ of $\bG^0_\Omega$ is trivial.
Since there is no non-trivial $\Omega$-group morphisms
$\mathbf U\to \bT^0_\Omega$, we conclude that $\mathbf U\subset \bG^{ss}_\Omega$. Since $\bG^{ss}_\Omega$
is semi-simple one has $\mathbf U=\{1\}$. This completes the proof of the reductivity of the $R$-group scheme $\bG^0$.
Thus, the $R$-group scheme $\bH$ is quasi-reductive. This proves the lemma.
\end{proof}

\begin{proof}[Proof of the first assertion of Theorem \ref{Aus_Buksbaum_3}]
Let $\bH$ be the kernel of $\mu$.
Since $\mu$ is smooth, the group scheme sequence
$$1 \to \bH \to \bG \to \bT \to 1$$
gives rise to an short exact sequence of group sheaves in the \'{e}tale topology.
In turn that sequence of sheaves induces a long exact sequence of pointed sets.
So, the boundary map
$\partial: \bT(R) \to \textrm{H}^1_{\text{\'et}}(R,\bH)$
fits in a commutative diagram
$$
\CD
\bT(R)/\mu (\bG(R)) @>>> \textrm{H}^1_{\text{\'et}}(R,\bH) \\
@VVV @VVV \\
\bT(K)/\mu (\bG(K)) @>>> \textrm{H}^1_{\text{\'et}}(K,\bH). \\
\endCD
$$
Clearly, the horizontal arrows have trivial kernels.
The right vertical arrow has trivial kernel by Lemma \ref{Kernel_G_to_T} and Theorem \ref{MainThm1}.
Thus the left vertical arrow has trivial kernel too. Since it is a group homomorphism,
it is injective.
\end{proof}

\section{Norms}
\label{SectNorms}
We recall here a construction from \cite{P1}.
Let $k\subset K\subset L$ be field extensions and assume that $L$
is finite separable over $K$. Let $K^{sep}$ be a separable closure
of $K$ and $\sigma_i:K\to K^{sep},\enspace 1\leq i\leq n$
the different embeddings of $K$ into $L$. Let $C$ be a $k$-smooth
commutative algebraic group scheme defined over $k$. One can define
a norm map
$${\mathcal N}_{L/K}:C(L)\to C(K)$$
by
${\mathcal N}_{L/K}(\alpha)=\prod_i C(\sigma_i)(\alpha) \in C(K^{sep})^{{\mathcal G}(K)} =C(K)\ .
$
In \cite{P1} following Suslin and Voevodsky
\cite[Sect.6]{SV} we
generalize this construction to finite flat ring extensions.
\smallskip
Let $p:X\to Y$ be a finite flat morphism of affine schemes.
Suppose that its rank is constant, equal to $d$. Denote by
$S^d(X/Y)$ the $d$-th symmetric power of $X$ over $Y$.

\smallskip
Let $k$ be a field. Let $\mathcal O$ be the semi-local ring of finitely many {\bf closed} points
on a smooth affine irreducible $k$-variety $X$.
Let $C$ be an affine smooth commutative $\mathcal O$-group scheme,
Let $p:X\to Y$ be a finite flat morphism of affine $\mathcal O$-schemes
and $f:X\to C$ any $\mathcal O$-morphism.
In \cite{P1} {\it the norm} $N_{X/Y}(f)$ of $f$ {\it is defined as the composite map}
\begin{equation}
\label{NormMap}
Y \xra{N_{X/Y}}
S^d(X/Y) \to S^d_{\mathcal O}(X) \xra{S^d_{\mathcal O}(f)} S^d_{\mathcal O}(C)\xra{\times} C
\end{equation}
Here we write $"\times"$ for the group law on $C$.
The norm maps $N_{X/Y}$ satisfy the following conditions
\begin{itemize}
\item[(i')]
Base change: for any map $f:Y'\to Y$ of affine schemes, putting
$X'=X\times_Y Y'$ we have a commutative diagram
$$
\begin{CD}
C(X)                  @>{(id \times f)^{*}}>>            C(X^{\prime})      \\
@V{N_{X/Y}}VV @VV{N_{X^{\prime}/Y^{\prime}}}V    \\
C(Y)                  @>{f^{*}}>>            C(Y^{\prime})
\end{CD}
$$
\item[(ii')]
multiplicativity: if $X=X_1 \amalg X_2$ then the diagram commutes
$$
\begin{CD}
C(X)                  @>{(id \times f)^{*}}>>            C(X_1) \times C(X_2)      \\
@V{N_{X/Y}}VV                              @VV{N_{X_1/Y}N_{X_2/Y}}V    \\
C(Y)                  @>{id}>>            C(Y)
\end{CD}
$$
\item[(iii')]
normalization: if $X=Y$ and the map $X \to Y$ is the identity then $N_{X/Y}=id_{C(X)}$.
\end{itemize}

\section{Unramified elements}
\label{SectUnramifiedElements}
Let $k$ be a field, $\mathcal O$ be the semi-local ring of finitely many closed points
on a $k$-smooth irreducible affine $k$-variety $X$.
Let $K$ be the fraction field of $\mathcal O$, that is $K=k(X)$.
Let
$$\mu: \bG \to \bT$$
be a smooth $\mathcal O$-morphism of reductive
$\mathcal O$-group schemes, with a torus $\bT$.
We work in this section with {\it the category of commutative Noetherian $\mathcal O$-algebras}.
For a commutative $\mathcal O$-algebra $S$ set
\begin{equation}
\label{MainFunctor}
{\cal F}(S)=\bT(S)/\mu(\bG(S)).
\end{equation}
For an element $\alpha \in \bT(S)$ we will write $\bar \alpha$ for its
image in ${\cal F}(S)$. {\it In this section we will write  ${\cal F}$
for the functor }
(\ref{MainFunctor}).
The following result is a particular case of the first part of Theorem \ref{Aus_Buksbaum_3} (those part is proved in Section \ref{One_Lemma}).
\begin{thm}
\label{NisnevichCor}
Let $S$ be an $\mathcal O$-algebra which is discrete valuation ring with fraction field $L$.
Then the map
${\cal F}(S) \to {\cal F}(L)$
is injective.
\end{thm}

\begin{lem}
\label{BoundaryInj}
Let $\mu: \bG \to \bT$ be the above morphism of our reductive group schemes.
Then for an $\mathcal O$-algebra $L$, where $L$ is a field, the boundary map
$\partial: \bT(L)/{\mu (\bG(L))} \to \textrm{H}^1_{\text{\'et}}(L,\bH)$
is injective.
\end{lem}

\begin{proof}
Repeat literally the proof of \cite[Lemma 6.2]{Pan2}.
\end{proof}

Let $k$, $\mathcal O$ and $K$ be as above in this Section.
Let $\mathcal K$ be a field containing $K$ and
$x: \mathcal K^* \to \mathbb Z$
be a discrete valuation vanishing on $K$.
Let $A_x$ be the valuation ring of $x$. Clearly,
$\mathcal O \subset A_x$.
Let
$\hat A_x$ and $\hat {\mathcal K}_x$
be the completions of $A_x$ and $\mathcal K$ with respect to $x$.
Let
$i:\mathcal K \hookrightarrow \hat {\mathcal K}_x$
be the inclusion. By Theorem
\ref{NisnevichCor}
the map
${\cal F}(\hat A_x)\to {\cal F}(\hat{\mathcal K}_x)$
is injective. We will identify
${\cal F}(\hat A_x)$
with its image under this map. Set
$$
{\cal F}_x(\mathcal K)=i_*^{-1}({\cal F}(\hat A_x)).
$$

The inclusion
$A_x\hookrightarrow \mathcal K$
induces a map
$
{\cal F}(A_x) \to {\cal F}(\mathcal K)
$
which is injective by Theorem
\ref{NisnevichCor}.
So both groups
${\cal F}(A_x)$ and ${\cal F}_x(\mathcal K)$
are subgroups of
${\cal F}(\mathcal K)$.
The following lemma shows that
${\cal F}_x(\mathcal K)$
coincides with the subgroup of $
{\cal F}(\mathcal K)$
consisting of all elements {\it unramified} at $x$.

\begin{lem}
\label{TwoUnramified}
${\cal F}(A_x)={\cal F}_x(\mathcal K)$.
\end{lem}

\begin{proof}
Repeat literally the proof of \cite[Lemma 6.3]{Pan2}.
\end{proof}
Let $S$ be an $\mathcal O$-algebra which is an integral domain and suppose $S$ is a regular ring.
Let $L$ be its fraction field.
For each height $1$ prime ideals $\mathfrak p$ in $S$ the group map
${\cal F}(S_{\mathfrak p})\to {\cal F}(L)$ is injective by the first part of Theorem \ref{Aus_Buksbaum_3}.
Define the {\it subgroup of $S$-unramified elements of $\mathcal F (L)$} as
\begin{equation}
\label{DefnUnramified}
\mathcal F_{nr,S}(L)=
\bigcap_{\mathfrak p \in Spec(S)^{(1)}} \mathcal F(S_{\mathfrak p}) \subseteq {\cal F}(L),
\end{equation}
where $Spec(S)^{(1)}$ is the set of height $1$ prime ideals in $S$.
Obviously the image of $\mathcal F(S)$ in $\mathcal F(L)$ is contained in
$\mathcal F_{nr,S}(L)$.
For
each height one prime $\mathfrak p$ in $S$
we construct {\it specialization maps}
$s_{\mathfrak p}: {\cal F}_{nr, S}(L) \to {\cal F} (l(\mathfrak p))$,
where $L$ is the field of fractions of $S$ and
$l(\mathfrak p)$
is the residue field of
$S$ at the prime $\mathfrak p$.

\begin{defn}
\label{SpecializationDef}
Let
$Ev_{\mathfrak p}: \bT(S_{\mathfrak p}) \to \bT(l(\mathfrak p))$
and
$ev_{\mathfrak p}: {\cal F}(S_{\mathfrak p}) \to {\cal F}(K(\mathfrak p))$
be the maps induced by the canonical $S$-algebra homomorphism
$S_{\mathfrak p} \to l(\mathfrak p)$.
Define a homomorphism
$s_{\mathfrak p}: {\cal F}_{nr, S}(L) \to {\cal F} (l(\mathfrak p))$
by
$s_{\mathfrak p}(\alpha)= ev_{\mathfrak p}(\tilde \alpha)$,
where
$\tilde \alpha$
is a lift of $\alpha$ to
${\cal F}(S_{\mathfrak p})$.
Theorem
\ref{NisnevichCor}
shows that the map $s_{\mathfrak p}$ is well-defined.
It is called the specialization map. The map $ev_{\mathfrak p}$ is called the evaluation
map at the prime $\mathfrak p$.

Obviously for
$\alpha \in \bT(S_\mathfrak p)$
one has
$s_{\mathfrak p}(\bar \alpha)=\overline {Ev_{\mathfrak p}(\alpha)}
\in {\cal F}(l(\mathfrak p))$.
\end{defn}

Let $k$, $\mathcal O$ and $K$ be as above in this Section.
The following two results are proved using literally the same arguments as in the proof of \cite[Thm. 6.5]{Pan2} and \cite[Cor. 6.6]{Pan2} respectively.
\begin{thm}[Homotopy invariance]
\label{HomInvNonram}
Let $K(t)$ be the rational function field in one variable over the field $K$.
Define ${\cal F}_{nr,K[t]}(K(t))$ by the formulae
(\ref{DefnUnramified}).
Then one has an equality
$$
{\cal F}(K)={\cal F}_{nr,K[t]}(K(t)).
$$
\end{thm}

\begin{cor}
\label{TwoSpecializations}
Let
$$
s_0, s_1: {\cal F}_{nr, K[t]}(K(t)) \to {\cal F}(K)
$$
be the specialization maps at zero and at one
(at the primes (t) and (t-1)). Then $s_0=s_1$.
\end{cor}

\begin{lem}
\label{KeyUnramifiedness}
Let $B \subset A$ be a finite extension of $K$-smooth algebras, which are integral domains
and each has dimension one.
Let $0 \neq f \in A$ and
let $h \in B\cap fA$ be such that the induced map
$B/hB\to A/fA$ is an isomorphism.
Suppose
$hA=fA\cap J^{\prime\prime}$
for an ideal
$J^{\prime\prime} \subseteq A$
co-prime to the principal ideal $fA$.

Let $E$ and $F$ be the field of fractions of $B$ and $A$ respectively.
Let $\alpha \in \bT(A_f)$ be such that
$\bar \alpha \in {\cal F}(F)$
is $A$-unramified. Then, for
$\beta= N_{F/E}(\alpha)$,
the class
$\bar \beta \in {\cal F}(E)$
is $B$-unramified.
\end{lem}

\begin{proof}
Repeat literally the proof of \cite[Lemma 6.7]{Pan2}.
\end{proof}

\section{Few recollections}
\label{First Application}
Let $X$ be an affine $k$-smooth irreducible $k$-variety, and let $x_1,x_2,\dots,x_n$ be closed points in $X$.
Let $\mathcal O$ be the semi-local ring $\mathcal O_{X,\{x_1,x_2,\dots,x_n\}}$.
Let $U=Spec(\mathcal O)$ and $can: U\hra X$ be the canonical embedding.
Let $\bG$ be a reductive
$X$-group scheme
and let
$\bG_U= can^*(\bG)$
be the pull-back of $\bG$ to $U$.
Let $\bT$ be an $X$-torus
and let
$\bT_U= can^*(\bT)$
be the pull-back of $\bG$ to $U$.
Let $\mu: \bG \to \bT$
be an $X$-group scheme morphism which is smooth as an $X$-scheme morphism.
Let $\mu_U=can^*(\mu)$.
The following result is \cite[Theorem 4.1]{Pan2}.

\begin{thm}
\label{equating3}
Given a non-zero function $\emph{f}\in k[X]$ vanishing at each point $x_i$,
there is a diagram of the form
\begin{equation}
\label{DeformationDiagram0}
    \xymatrix{
\Aff^1 \times U\ar[drr]_{\pr_U}&&\mathcal X^{\prime} \ar[d]^{}
\ar[ll]_{\sigma}\ar[d]_{q_U}
\ar[rr]^{q_X}&&X &\\
&&U \ar[urr]_{\can}\ar@/_0.8pc/[u]_{\Delta^{\prime}} &\\
    }
\end{equation}
with an irreducible affine scheme $\mathcal X^{\prime}$, a smooth morphism $q_U$, a finite surjective $U$-morphism $\sigma$ and an essentially smooth morphism $q_X$,
and a function $f^{\prime} \in q^*_X(\emph{f} \ )k[\mathcal X^{\prime}]$,
which enjoys the following properties:
\begin{itemize}
\item[\rm{(a)}]
if
$\mathcal Z^{\prime}$ is the closed subscheme of $\mathcal X^{\prime}$ defined by the ideal
$(f^{\prime})$, then the morphism
$\sigma|_{\mathcal Z^{\prime}}: \mathcal Z^{\prime} \to \Aff^1\times U$
is a closed embedding and the morphism
$q_U|_{\mathcal Z^{\prime}}: \mathcal Z^{\prime} \to U$ is finite;
\item[\rm{(a')}] $q_U\circ \Delta^{\prime}=id_U$ and $q_X\circ \Delta^{\prime}=can$ and $\sigma\circ \Delta^{\prime}=i_0$, \\
where $i_0$ is the zero section of the projection $\pr_U$;
\item[\rm{(b)}] $\sigma$
is \'{e}tale in a neighborhood of
$\mathcal Z^{\prime}\cup \Delta^{\prime}(U)$;
\item[\rm{(c)}]
$\sigma^{-1}(\sigma(\mathcal Z^{\prime}))=\mathcal Z^{\prime}\coprod \mathcal Z^{\prime\prime}$
scheme theoretically
for some closed subscheme $\mathcal Z^{\prime\prime}$
\\ and
$\mathcal Z^{\prime\prime} \cap \Delta^{\prime}(U)=\emptyset$;
\item[\rm{(d)}]
$\mathcal D_0:=\sigma^{-1}(\{0\} \times U)=\Delta^{\prime}(U)\coprod \mathcal D^{\prime}_0$
scheme theoretically
for some closed subscheme $\mathcal D^{\prime}_0$
and $\mathcal D^{\prime}_0 \cap \mathcal Z^{\prime}=\emptyset$;
\item[\rm{(e)}]
for $\mathcal D_1:=\sigma^{-1}(\{1\} \times U)$ one has
$\mathcal D_1 \cap \mathcal Z^{\prime}=\emptyset$.
\item[\rm{(f)}]
there is a monic polynomial
$h \in \mathcal O[t]$
with
$(h)=Ker[\mathcal O[t] \xrightarrow{\sigma^*} k[\mathcal X^{\prime}] \xrightarrow{-} k[\mathcal X^{\prime}]/(f^{\prime})]$, \\
where the map bar takes any $g\in k[\mathcal X^{\prime}]$ to ${\bar g}\in k[\mathcal X^{\prime}]/(f^{\prime})$;\\
\item[\rm{(g)}] there are $\mathcal X^{\prime}$-group scheme isomorphisms
$\Phi:  q^*_U(\bG_U)\to q^*_X(\bG)$,
$\Psi:  q^*_U(\bT_U)\to q^*_X(\bT)$
with
$(\Delta^{\prime})^*(\Phi)= id_{\bG_U}$,
$(\Delta^{\prime})^*(\Psi)= id_{\bT_U}$
and
$q^*_X(\mu) \circ \Phi=\Psi \circ q^*_U(\mu_U)$.
\end{itemize}
\end{thm}

\begin{rem}
The triple $(q_U: \mathcal X^{\prime} \to U, f^{\prime}, \Delta^{\prime})$ is
{\it a nice triple} over $U$, since $\sigma$ is
a finite surjective $U$-morphism.
See \cite[Defn.3.1]{PSV} for the definition of a nice triple.

The morphism
$q_X$ is not equal to $can\circ q_U$, since
$f^{\prime} \in q^*_X(\textrm{f})k[\mathcal X^{\prime}]$
and the morphism
$q_U|_{\mathcal Z^{\prime}}: \mathcal Z^{\prime}=\{f^{\prime}=0\} \to U$ is finite.

\end{rem}
To formulate a consequence of the theorem
\ref{equating3} (see Corollary \ref{ElementaryNisSquareNew_1}),
note that using the items (b) and (c) of Theorem
\ref{equating3}
one can find an element
$g \in I(\mathcal Z^{\prime\prime})$
such that \\
(1) $(f^{\prime})+(g)=\Gamma(\mathcal X^{\prime}, \mathcal O_{\mathcal X^{\prime}})$, \\
(2) $Ker((\Delta^{\prime})^*)+(g)=\Gamma(\mathcal X^{\prime}, \mathcal O_{\mathcal X^{\prime}})$, \\
(3) $\sigma_g=\sigma|_{\mathcal X^{\prime}_g}: \mathcal X^{\prime}_g \to \Aff^1_U$ is \'{e}tale.\\

Here is the corollary. It is proved in \cite[Cor. 7.2]{Pan0}.
\begin{cor}
\label{ElementaryNisSquareNew_1}
The function $f^{\prime}$ from Theorem \ref{equating3}, the polynomial $h$ from the item $(\textrm{f} \ )$
of that Theorem, the morphism $\sigma: \mathcal X^{\prime} \to \Aff^1_U$
and the function
$g \in \Gamma(\mathcal X,\mathcal O_{\mathcal X} )$
defined just above
enjoy the following properties:
\begin{itemize}
\item[\rm{(i)}]
the morphism
$\sigma_g= \sigma|_{\mathcal X^{\prime}_g}: \mathcal X^{\prime}_g \to \Aff^1\times U $
is \'{e}tale,
\item[\rm{(ii)}]
data
$ (\mathcal O[t],\sigma^*_g: \mathcal O[t] \to \Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})_g, h ) $
satisfies the hypotheses of
\cite[Prop.2.6]{C-TO},
i.e.
$\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}} )_g$
is a finitely generated
$\mathcal O[t]$-algebra, the element $(\sigma_g)^*(h)$
is not a zero-divisor in
$\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}} )_g$
and
$\mathcal O[t]/(h)=\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})_g/h\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})_g$ \ ,
\item[\rm{(iii)}]
$(\Delta(U) \cup \mathcal Z') \subset \mathcal X^{\prime}_g$ \ and $\sigma_g \circ \Delta=i_0: U\to \Aff^1\times U$,
\item[\rm{(iv)}]
$\mathcal X^{\prime}_{gh} \subseteq \mathcal X^{\prime}_{gf^{\prime}}\subseteq \mathcal X^{\prime}_{f^{\prime}}\subseteq \mathcal X^{\prime}_{q^*_X(\emph{f})}$ \ ,
\item[\rm{(v)}]
$\mathcal O[t]/(h)=\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})/(f^{\prime})$,
$h\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})=(f^{\prime})\cap I(\mathcal Z^{\prime\prime})$
and
$(f^{\prime}) +I(\mathcal Z^{\prime\prime})=\Gamma(\mathcal X^{\prime},\mathcal O_{\mathcal X^{\prime}})$.
\end{itemize}
\end{cor}

\section{Purity}\label{purity}
Let $S$ be a regular ring, $\bG$ a reductive $S$-group scheme, $\bT$ an $S$-torus,
$\mu: \bG\to \bT$ an $S$-group scheme morphisms which is smooth as a scheme morphism.
Suppose $S$ is an integral domain. Let $L$ be its field of fractions. For each $S$-algebra $S'$
write ${\cal F}(S')$ for the group $\bT(S')/\mu(\bG(S'))$. For any $a\in \bT(S')$ write $\bar a$
for the class of $a$ in ${\cal F}(S')$. Let $\mathfrak p$ be a height one prime ideal in $S$,
then by Theorem \ref{Aus_Buksbaum_3} the group ${\cal F}(S_{\mathfrak p})$ is a subgroup in ${\cal F}(L)$.

Recall some notion. For an element $a\in \bT(L)$ and a height one prime $\mathfrak p\subset S$
we say that $\bar a\in {\cal F}(L)$ is {\it unramified at} $\mathfrak p$,
if $\bar a$ is in ${\cal F}(S_{\mathfrak p})$.
We say that the element $\bar a\in {\cal F}(L)$ is $S$-unramified if for any height one prime ideal $\mathfrak p$ in $S$
the element $\bar a$ is in ${\cal F}(S_{\mathfrak p})$. Clearly, the image of ${\cal F}(S)$ in ${\cal F}(L)$
is in $\cap \ {\cal F}(S_{\mathfrak p})$, where the intersection is taken over all height one primes of $S$.
We say that {\it purity holds for the ring} $S$ if
$$Im[{\cal F}(S)\to {\cal F}(L)]=\cap \ {\cal F}(S_{\mathfrak p}).$$
Equivalently, purity holds for $S$ if each $S$-unramified element of ${\cal F}(L)$ comes from ${\cal F}(S)$.
Clearly, the sequence
$\{1\} \to {\cal F}(S_{\mathfrak p}) \to
{\cal F}(L) \xrightarrow{r_{\mathfrak p}}
\bT(L)/[\bT(R_{\mathfrak p})\cdot \mu(\bG(L))] \to \{1\}
$
is exact, where $r_{\mathfrak p}$ is the factorization map.
Thus, an element $a\in \bT(L)$ its class $\bar a$ in ${\cal F}(L)$ is unramified at $\mathfrak p$,
if and only if $r_{\mathfrak p}(\bar a)=0$. Hence
purity holds for $S$ if and only if the sequence
${\cal F}(S)\to {\cal F}(L)\xrightarrow{\sum r_{\mathfrak p}} \oplus_{\mathfrak p} \bT(L)/[\bT(R_{\mathfrak p})\cdot \mu(\bG(L))]$
is exact. Our aim is to prove the following assertion:\\
$(\ast)$ Purity holds for the ring $R$, group schemes $\bG$,$\bT$ and the morphism $\mu$ as in Theorem \ref{Aus_Buksbaum_3}.\\\\
The proof is subdivided in few steps.\\
{\it Claim 1.}
Let $X$ be a $k$-smooth irreducible affine $k$-variety.
Let $\bG$ be a reductive $X$-group scheme, $\bT$ be an $X$-torus and
$\mu: \bG \to \bT$ be an $X$-group scheme morphism which is smooth as an $X$-scheme morphism.
Suppose the $k$-algebra $R$ is the semi-local ring of finitely many closed points
on  $X$. Then purity holds for $R$.

To prove this Claim we just repeat literally the proof of
\cite[Theorem 1.1]{Pan2} replacing references to
\cite[Corollary 4.3, (ii),(v)]{Pan2}
with the one to the items (ii) and (v) of
Corollary \ref{ElementaryNisSquareNew_1}.
Replacing also references to \cite[Lemma 6.7]{Pan2}
with the one to
Lemma \ref{KeyUnramifiedness}.
Replacing also references to \cite[Theorem 6.5]{Pan2}
with the one to Theorem \ref{HomInvNonram}.
Replacing also references to \cite[Corollary 6.6]{Pan2}
with the one to Corollary \ref{TwoSpecializations}.
Replacing also references to \cite[Theorem 4.1]{Pan2}
with the one to Theorem \ref{equating3}.
Replacing also references to \cite[Definition 6.4]{Pan2}
with the one to the remark at the end of Definition \ref{SpecializationDef}.
The Claim 1 is proved.\\\\
{\it Claim 2.}
Let $X$ be a $k$-smooth irreducible affine $k$-variety and $\xi_1,...,\xi_n$ be points of the scheme $Spec(k[X])$
such that for each pair $r,s$ the point $\xi_r$ is not in the closure $\overline {\{\xi_s\}}$ of $\xi_s$.
Let $R$ be the semi-local ring $\cO_{X,\xi_1,...,\xi_n}$ of scheme points $\xi_1,...,\xi_n$ of $Spec(k[X])$.
Let $\bG$ be a reductive $X$-group scheme, $\bT$ be an $X$-torus and
$\mu: \bG \to \bT$ be an $X$-group scheme morphism which is smooth as an $X$-scheme morphism.
Then purity holds for $R$.

To prove this Claim take an element $a \in \bT(k(X))$ such that $\bar a$ is unramified at each irreducible divisor $D$ containing
at least one of the points $\xi_r$. We have to prove that the element $\bar a\in {\cal F}(K)$ is in the image of ${\cal F}(R)$.
Clearly, there is a non-zero $f\in k[X]$ such that $a \in \bT(k[X_f])$. Write down the divisor $div(f)\in Div(X)$
in the form $div(f)=\Sigma m_iD_i + \Sigma n_jD'_j$ such that for each index $i$ there is an index $r$ with $\xi_r\in D_i$
and for any index $j$ and any index $r$ the point $\xi_r$ does not belong to $D'_j$.
There is an element $g\in k[X]$
such that for any index $j$ one has $D'_j$ is contained in the closed subset $\{g=0\}$ and $g$ does not belong to any of $\xi_r$'r.
Replacing $X$ with $X_g$ we see that $a \in \bT(k[X_f])$, $div(f)=\Sigma m_iD_i$ and $\bar a$ is unramified at each irreducible divisor $D_i$.
Hence $\bar a$ is unramified at each height one prime ideal of $k[X]$.
Our assumption on points $\xi_r$'s yield the following: one can choose
closed points $x_r\in \overline {\{\xi_s\}}$ such that for each $r\neq s$ the point $x_r$ is not in $\overline {\{\xi_s\}}$.
Particularly, for each $r\neq s$ one has $x_r\neq x_s$. The element $\bar a$ is unramified at each height one prime ideal of $k[X]$.
Thus, by Claim 1 the element $\bar a$ is in the image of ${\cal F}(\cO_{X,x_1,...,x_n})$.
So, the element $\bar a$ is in the image of ${\cal F}(\cO_{X,\xi_1,...,\xi_n})={\cal F}(R)$.
The Claim 2 is proved.\\\\
{\it Claim 3.}
The assertion $(\ast)$ is true.\\
In the rest of the section we prove Claim 3.
Clearly, we may assume that $k$ is a prime field and hence $k$ is perfect.
It follows from Popescu's theorem [Pop, Swa, Spi] that $R$
is a filtered inductive limit of smooth $k$-algebras $R_{\alpha}$. Modifying the inductive system
$R_{\alpha}$
if necessary, we can assume that each $R_{\alpha}$ is integral.
For each maximal ideal $\mathfrak m_i$ in $R$ ($i = 1,...,n$) set
$\mathfrak p_i = \phi^{-1}_{\alpha}(\mathfrak m_i)$.
The homomorphism $\phi_{\alpha}: R_{\alpha}\to R$ induces a homomorphism of semi-local rings
$\phi^{\prime}_{\alpha}: (R_{\alpha})_{\mathfrak p_1,...,\mathfrak p_n} \to R$.
Since this moment we will write $A_{\alpha}$ for $(R_{\alpha})_{\mathfrak p_1,...,\mathfrak p_n}$
and $A$ for $R$ (to keep consistency of notation).
Thus, $A$
is a filtered inductive limit of regular semi-local $k$-algebras $A_{\alpha}$.

There exist an index $\alpha$, a reductive
group scheme $\bG_{\alpha}$, a torus $\bT_{\alpha}$ over $A_{\alpha}$ and
an $A_{\alpha}$-group scheme morphism $\mu_{\alpha}: \bG_{\alpha} \to \bT_{\alpha}$
which is smooth as an $A_{\alpha}$-scheme morphism
such that
$\bG=\bG_{\alpha}\times_{Spec(A_{\alpha})} Spec(A)$,
$\bT=\bT_{\alpha}\times_{Spec(A_{\alpha})} Spec(A)$,
$\mu=\mu_{\alpha}\times_{Spec(A_{\alpha})} Spec(A)$.
Replacing the index system with a co-final one consisting of indexes $\beta\geq \alpha$,
we may and will suppose that the reductive
group scheme
$\bG$, the torus $\bT$ and the group scheme morphism $\mu: \bG\to \bT$
comes from $A_{\alpha}$, and $\mu$ is is smooth as an $A_{\alpha}$-scheme morphism.
These observations and Claim 2 yield the following intermediate result \\
$(\ast\ast)$ for these
$\bG$, $\bT$ and $\mu: \bG\to \bT$ over $A_{\alpha}$ purity holds
for each ring $A_{\beta}$ with $\beta\geq \alpha$.

Let now $K$ be the field of fractions of $A$ and, for each
$\beta\geq \alpha$,  let $K_\beta$ be the field of fractions of $A_\beta$.
For each index $\beta\geq \alpha$ let
$\mathfrak a_{\beta}$
be the kernel of the map
$\phi^{\prime}_{\beta}: A_{\beta} \to A$
and
$B_{\beta}=(A_{\beta})_{\mathfrak a_{\beta}}$.
Clearly, for each
$\beta\geq \alpha$,  $K_{\beta}$
is the field of fractions of
$B_{\beta}$.
The composition map
$A_{\beta} \to A \to K$
factors through $B_{\beta}$.
Since $A$ is a filtering direct limit of the $A_{\beta}$'s
we see that $K$ is a filtering direct limit of the $B_{\beta}$'s.
We will write
$\psi_{\beta}$
for the canonical morphism
$B_{\beta} \to K$.

\begin{lem}
\label{WeakGrothendieck}
For each index $\alpha$ the group map $W(B_{\alpha})\to W(K_{\alpha})$ is injective.
\end{lem}

\begin{proof}
Just apply the first part of Theorem \ref{Aus_Buksbaum_3} to the $k$-algebra $B_{\alpha}$.
\end{proof}

\begin{lem}
\label{LiftToFiniteLevel}
Let $a\in W(K)$ be an $A$-unramified element.
Then there exists an index $\alpha$ and an element
$b_{\alpha} \in W(B_{\beta})$
such that
$\psi_{\alpha}(b_{\alpha})=b$
and the class
$b_{\alpha} \in W(K_{\beta})$
is $A_{\alpha}$-unramified.
\end{lem}

\begin{proof}
Repeat literally the proof of \cite[Lemma 9.0.9]{P1}. It works
for the semi-local case as well.
\end{proof}
We complete the proof of Claim 3 as follows.
Let $a\in W(K)$ be an $A$-unramified element.
We have to check that it comes from $W(A)$.
By Lemma \ref{LiftToFiniteLevel}
there exists an index $\alpha$ and an element
$b_{\alpha} \in W(B_{\alpha})$
such that
$\psi_{\alpha}(b_{\alpha})=b$
and the class
$b_{\alpha} \in W(K_{\beta})$
is $A_{\alpha}$-unramified.
For this index $\alpha$ consider a commutative diagram of $k$-algebras
$$
\xymatrix{
    A_{\alpha}  \ar[d]\ar[rr]^-{\varphi_{\alpha}}\ar[d]_-{} && A \ar[d]^-{} &   \\
     B_{\alpha} \ar[d]\ar[rr]^-{\psi_{\alpha}}  &&  K \\
     K_{\alpha}.
    }
$$
The class
$\bar b_{\beta} \in {\cal F}(K_{\beta})$
is $A_{\beta}$-unramified.
Hence by the statement $(\ast\ast)$ there exists an element
$a_{\beta} \in \bT(A_{\beta})$
such that
$\bar b_{\beta}=\bar a_{\beta}$ in ${\cal F}(K_{\beta})$.
By Lemma
\ref{WeakGrothendieck}
one has an equality
$\bar b_{\beta}=\bar a_{\beta}$ in
${\cal F}(B_{\beta})$.
Hence $\bar b \in {\cal F}(K)$ coincides with
the image of the element $\phi_{\beta} (\bar a_{\beta})$
in ${\cal F}(K)$.
The Claim 3 is proved.
Thus, the sequence
\eqref{Aus_Buks_sequence_3} is exact at its middle term.

\section{Proof of Theorem \ref{Aus_Buksbaum_3}}
\label{SectAus_Buksbaum_3}
\begin{proof}[Proof of Theorem \ref{Aus_Buksbaum_3}]
The proof of the first assertion of Theorem \ref{Aus_Buksbaum_3} is given in Section \ref{One_Lemma}.
The exactness of the sequence
\eqref{Aus_Buks_sequence_3} at its middle term is proved in Section \ref{purity}.

Prove now the surjectivity of the map $\sum r_{\mathfrak p}$.
Clearly, it is sufficient to prove
the surjectivity of the map
$\mathbf T(K) \xrightarrow{\sum r'_{\mathfrak p}} \bigoplus_{\mathfrak p} \mathbf T(K)/\mathbf T(R_{\mathfrak p})$,
where $\mathfrak p$ runs over
the height $1$ primes of $R$ and $r'_{\mathfrak p}$ is the factorisation map.
We follow arguments from
\cite[Section 9]{Pan3}.

We prefer to switch to the scheme terminology.
Set $X:=Spec(R)$.
Consider a finite \'{e}tale Galois morphism $\pi: \tilde X\to X$ with irreducible $\tilde X$
such that the torus $\bT$ splits over $\tilde X$. Let $Gal:=Aut(\tilde X/X)$ be its Galois group.
Since the torus $\bT$ splits over $\tilde X$ we have a short exact sequence of $Gal$-modules
$$0\to \bT(\tilde \cO)\to \bT(\tilde K)\to \oplus_{y} \bT(\tilde K)/\bT(\tilde \cO_{X,y})\to 0,$$
where $\tilde \cO=\Gamma(\tilde X, \mathcal O_{\tilde X})$, $\tilde K$ is the fraction field of $\tilde \cO$,
$y$ runs over the set $X^{(1)}$ of all
codimension $1$ points of $X$
and for any $y\in X^{(1)}$ the ring
$\tilde \cO_{X,y}$ is the semi-local ring $\cO_{\tilde X, \tilde y}$ of the finite set $\tilde y=\pi^{-1}(y)$
on the scheme $\tilde X$. Write $\cO$ for $R$ to be consistent with the above notation.

The above short exact sequence of $Gal$-modules gives rise to a long exact sequence of $Gal$-cohomology groups of the form
$$0\to \bT(\cO)\xrightarrow{in} \bT(K)\to \oplus_{y} [\bT(\tilde K)/\bT(\tilde \cO_{X,y})]^{Gal} \to H^1(Gal,\bT(\tilde \cO))\xrightarrow{H^1(in)} H^1(Gal,\bT(\tilde K)).$$
We claim that the map $H^1(in)$ is a monomorphism. Indeed, the group $H^1(Gal,\bT(\tilde \cO))$ is a subgroup of the group $H^1_{et}(X, \bT)$
and the group $H^1(Gal,\bT(\tilde K))$ is a subgroup of the group $H^1_{et}(Spec~K, \bT_K)$. By Theorem \ref{MainThm1}
the group map $H^1_{et}(X, \bT)\to H^1_{et}(Spec~K, \bT_K)$ is a monomorphism. Thus, $H^1(in)$ is a monomorphism also.
So, we have a short exact sequence of the form
$0\to \bT(\cO)\xrightarrow{in} \bT(K)\to \oplus_{y} [\bT(\tilde K)/\bT(\tilde \cO_{X,y})]^{Gal}\to 0$.

There is also the complex $0\to \bT(\cO)\xrightarrow{in} \bT(K)\to \oplus_{y} \bT(K)/\bT(\cO_{X,y})$. Set
$\alpha=id_{\bT(\cO)}$, $\beta=id_{\bT(K)}$ and let $\gamma=\oplus_{y} \gamma_y $, where
$\gamma_y: \bT(K)/\bT(\cO_{X,y})\to [\bT(\tilde K)/\bT(\tilde \cO_{X,y})]^{Gal}$ is induced by the inclusion $K\subset \tilde K$.
The maps $\alpha$, $\beta$ and $\gamma$ form a morphism between this complex and the above short exact sequence. We claim that this morphism is an isomorphism.
This claim completes the proof of the theorem.

To prove this claim it is sufficient to prove that $\gamma$ is an isomophism. Since the map
$\bT(K)\to \oplus_{y} [\bT(\tilde K)/\bT(\tilde \cO_{X,y})]^{Gal}$
is an epimorphism, hence so is the map $\gamma$.
It remains to prove that $\gamma$ is a monomorphism.
To do this it is sufficient to check that for any point $y\in X^{(1)}$ the map
$\bT(K)/\bT(\cO_{X,y})\to \bT(\tilde K)/\bT(\tilde \cO_{X,y})$
is a monomorphism. We will write $\epsilon_y$ for the latter map.
We prove below that $ker(\epsilon_y)$ is a torsion group and
the group $\bT(K)/\bT(\cO_{X,y})$ has no torsion. These two claims show that the map
$\epsilon_y$ is injective indeed.

To prove that $ker(\epsilon_y)$ is a torsion group recall that there are norm maps
$N_{\tilde \cO_{X,y}/\cO_{X,y}}: \bT(\tilde \cO_{X,y})\to \bT(\cO_{X,y})$ and
$N_{\tilde K/K}: \bT(\tilde K)\to \bT(K)$ (see Section \ref{SectNorms}). These maps induce a homomorphism
$$N_y: \bT(\tilde K)/\bT(\tilde \cO_{X,y})\to \bT(K)/\bT(\cO_{X,y})$$
such that $N_y\circ \epsilon_y$ is the multiplication by the degree $d$ of $\tilde K$ over $K$.
Thus, $ker(\epsilon_y)$ is killed by the multiplication by $d$.

Show now that the group $\bT(K)/\bT(\cO_{X,y})$ has no torsion. Take an element $a_K\in \bT(K)$ and suppose that
its class in $\bT(K)/\bT(\cO_{X,y})$ is a torsion element. Let $\tilde a_K$ be the image of $a_K$ in $\bT(\tilde K)$.
Since $\bT$ splits over $\tilde K$ we see that $\bT(\tilde K)/\bT(\tilde \cO_{X,y})$ is torsion free. Thus,
the class of $\tilde a_K$ in $\bT(\tilde K)/\bT(\tilde \cO_{X,y})$ vanishes. So, there is a unique element $\tilde a$ in $\bT(\tilde \cO_{X,y})$
whose image in $\bT(\tilde K)$ is $\tilde a_K$.
Moreover, $\tilde a$ is a $Gal$-invariant element in $\bT(\tilde \cO_{X,y})$, because $\tilde a_K$ comes from $\bT(K)$.
Since $\bT(\tilde \cO_{X,y})^{Gal}=\bT(\cO_{X,y})$, there is a unique element $a\in \bT(\cO_{X,y})$
whose image in $\bT(\tilde \cO_{X,y})$ is $\tilde a$. Clearly, the image of $a$ into $\bT(K)$
is the element $a_K$. Thus, the class of $a_K$ in $\bT(K)/\bT(\cO_{X,y})$ vanishes.
So, the group $\bT(K)/\bT(\cO_{X,y})$ is torsion free.
The injectivity of $\epsilon_y$ is proved.
The surjectivity of the map $\sum r_{\mathfrak p}$ is proved.
Theorem \ref{Aus_Buksbaum_3} is proved.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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Is Inflation really back?
May 18, 2021 KAA Blog, News
The consumer price index (CPI), rose by 0.8%in April. A level not seen since 2008. Calculating this at annual rates, that is, if this rate continues for a year inflation for that year will be 9.2%! A year not withstanding, even if this 0.8% increase in the CPI was just a one month event, it would have been a cause for concern; coming on the heels of another 3 exceptionally high rates during the first quarters makes it an issue. Inflation at annual rate calculates at 6.0% during the first 4 months of this year and 4.2% over the last 12 months. This is also the largest 12-month increase since September of 2008.
Lehigh Valley business sentiment index zoomed up by 13.1% in April
May 6, 2021 KAA Blog, News
The Lehigh Valley Business Sentiment Index, the BSI, recorded a 13.1% jump in one quarter in April 2021, its highest growth in 10 years. In response to the COVID-19 restrictions, the index plunged to an all-time low of 43.9, a drop of 30.4% in April 2020. Initially, the recovery was slow; by the last quarter of 2020, the index rose to only 48.5, still 25% below its pre-pandemic level. This year, the picture is different; with the rapidly rising vaccination rates, most businesses have started to, at least, think about going back to normal or something resembling that.
The BSI rose 5.5% in January and another 13.1% in April, reaching 57.9, which is 29.1% above its April 2020. This is good news; however, for comparison basis we have to use pre-pandemic references, thus for the April data we are going to use as reference April 2019. And compared to April 2019, the index is still a good 10% short.
The LV BSI is based on the Greater Lehigh Valley Chamber of Commerce-Kamran Afshar survey of Valley businesses done on a quarterly bases since 1998.
https://www.kaa-inc.com/wp-content/uploads/2020/05/bsi-4-21_B9EH8gDk.mp4
Lehigh Valley business sentiment index rose by only 3.1% in January
February 26, 2021 KAA Blog, News
The Lehigh Valley's business sentiment index, the BSI, made a smaller than expected recovery in January, rising by only 3.1%, a drop from last October's 5.7% and last July's 6.4% increases, after its historical 31.6% fall in April. Even after these three rises, BSI for January 2021 has only reached 50.9, well below its January 2020 level of 64.2. However, it has now exceeded its average level during the Great Recession. The diminishing rate of growth of the index is a cause for concern as it indicates a lack of enthusiasm among Valley businesses.
In our model, the largest gain was in the index of plans for future hirings, which rose by 7.3% above its last October's level, while still 5% below its January 2020 level. It should be noted that this index peaked in October 2018 and was already down by 10.3% in January 2020. The drop in the index was a dramatic 17.8%, according to our April 2020 survey. And while it is now only 5% below its January 2020 level, it is a significant 14.7% below its October 2018 peak in the Valley.
One of the indicators that we calculate for our indices is their momentum. This indicator is the composition of the mass, strength, and direction of the index's movement. And this indicator, which was at around 100 back in 2019 for future hiring, has dropped close to 60 as of January 2021. All of these indicate that despite its rise in January, the index of future hirings has a higher tendency to drop than to rise over the next 6 months.
The only index in our model which did not rise in January was the index of future purchases index, which dropped slightly in January, pushing it down to 14.3% below its previous January level.
This index's momentum is now in around negative 80, a massive drop from positive 160 back in 2019. The index of future purchases is also more likely to drop in the next 6 months than to rise.
The hiring index over the last 6 months, which recorded a slight increase in January, is still 20% below its January 2020 level. This index's momentum also dropped from an average of positive 50 in 2019 to an average of negative 40 by January 2021.
The percent of local companies that reported net layoffs was around 5% during 2018 & 19, while the percentage of those with net hiring hovered close to 30%. These ratios have changed dramatically; in January 2021, 21.5% of the participants in our survey had net layoffs, while only 12.5% of them reported net hiring. To put it simply, layoffs have increased by fourfold, while hiring dropped in half.
The index which has lost more than any other in our model over the last 12 months is the index of actual purchases, which fell to 42.6% below its January 2020 level. And its momentum is down from more than positive 200 in 2019 to around negative 300 in January 2021.
As was expected, the accommodation-food & leisure sector had the largest layoffs by far over the last 6 months, followed by construction and manufacturing.
The Finance-insurance sector led the plans for hiring over the next 6 months in January, followed by manufacturing and retail. And for the first time in almost a year, accommodation-food & leisure sector shows positive plans for future hirings.
Purchasing plans index dropped a slight 2.8% in January 2021, while actual purchases over the last 6 months rose by 4.9%. However, despite rising over the last two quarters, the index is still 42.6% below its January 2020 level.
The Valley has recovered/replaced 87% of all the jobs it lost in March and April of last year. And the finance and manufacturing followed by the accommodation-food & leisure sectors have the largest plans for expanding employment in the next 6 months.
And while all sectors except retail and real estate reported lower revenues over the last 6 months, the picture for the next 6 months is much more positive. And except for the accommodation-food & leisure and education sectors, none of the others expect their revenues to shrink in the next 6 months.
Our local business survey still shows a heightened level of concern with the future of the economy. However, we also observed positive signs among the local businesses regarding their hiring and purchasing plans for the next 6 months.
Lehigh Valley's Inflation rate was 2.2% for 2020
Prepared by KADAC @ DeSales University
Director: Kamran Afshar, Ph.D.
Student Assistant: Brandon Reese
Lehigh Valley's inflation rate in 2020 was 2.2%, according to the Kamran Afshar Data Analytics Center (KADAC) @ DeSales University.
During the 1980s and 1990s, the inflation rate in the Valley was often less than the national average. Between 2000 and 2007, the Valley's inflation rate started to exceed that of the national average. Data collection was suspended between 2007 and 2017.
Data collection was restarted in 2018, with an updated basket of goods and services for the Valley. Lehigh Valley's inflation rate was 2.2% in 2020, which is 0.9 percentage points higher than the 1.4% inflation at the national average. According to Lehigh Valley's CPI, between December 2018 and December 2020, local prices have increased by 5.9%, compared to 3.7% at the national average.
Food prices rose by 5.5% in 2019 in the Valley, significantly higher than the national average of 1.8% for that year. In 2020, local food prices rose by 4.4%, a little higher than the national average of 3.9%. Fruits and vegetables, a sub-category of food, increased by 18.1% during 2019 and 2020; this is significantly higher than the national average of 1.8% during the same period. Another sub-category, food away from home, rose by 5.6% in 2019 and 7.0% in 2020, compared to 3.1% and 3.9%, respectively, at the national average.
In 2019 shelter costs in Lehigh Valley rose at a higher 5.3% in the Valley compared to the 3.2% at the national average. In 2020, the equation was reversed; the Valley's shelter inflation rate was a negative 1.4% compared to a positive 1.8% at the national average. A 2.9% decrease in the sub-category of rent and rental costs in the Valley compared to a 1.9% increase at the national average was the most prominent cause of this rate divergence. Over the last two years, shelter costs in the Valley have increased by 3.9% compared to 5.1% at the national average.
Medical care costs in the Valley, rose at a faster pace than the national average, which is consistent with historical data.
Transportation costs rose by 0.8% in 2019 in the Valley compared to 1.9% at the national average. In 2020, Valley's transportation costs rose 1.3%, compared to a drop of 2.4% at the national average. The main reason for this difference in 2020 was that motor fuel costs nationwide dropped more than those in the Valley.
One of the interesting flip-flops among the inflation rates that we observed in 2020 was the inflation rate of apparel, which historically has been negative or, at best, very low. Apparel costs in the Valley rose by 9.8% in 2020, after dropping 6.5% in 2019. Nationwide, however, apparel costs dropped in both years, resulting in a 5.1% drop compared to the Valley's 2.7% rise for 2019 and 2020.
The rate of inflation rose by 5.8% during 2019 and 2020 in the Valley. During the same period, inflation at the national average was 3.7%.
CPI, or the consumer price index, reflects the change in the cost of an average urban basket of goods and services purchased by consumers. Higher inflation means prices are rising at a higher pace. And the price of almost all the items in the consumer basket of goods is based on their supply and demand. Since Lehigh Valley does not appear to be subject to any negative supply chain issues that push its prices up, the Valley's higher inflation rate is most likely the result of relatively higher consumers in the Valley than the national average.
Lehigh Valley business sentiment index rose 6.4% in October
October 30, 2020 KAA Blog, News
The Lehigh Valley's business sentiment index the BSI made another small recovery in October, rising by 6.4%, similar to its 5.7% increase in July, after its historical 31.6% fall in April. Even after two increases, the October index rose to only 49.4, well below its January level of 64.2. And still below its average level during the Great Recession. The index's improvement is underwhelming, too small, and reeks of a lack of enthusiasm from the Valley's businesses participating in the survey.
Our October survey of Valley businesses was done before the release of the advanced estimate for the US economic growth, which shows a 33.1% increase in the 3rd quarter's GDP, after an unexpected 31.4% drop in the 2nd quarter as the results of the Covid-19 and all the shutdowns. Despite the large percentage gain, since it was from a smaller base, the GDP rose to only 91.3% of its previous level, leaving us $670 billion short of its last year's level.
In our Lehigh Valley model, the largest gain was for the new hires' index in the last 6 months, which rose by 10.4% since July. Unfortunately, the index is still 26.6% below its last year's level, even with that increase. Plans for future hiring did not show any measurable changes since July. Its index is still 8.5% below its last year's level, and worse, the momentum for both hiring indices is negative and lower than any time even during the Great Recession.
The percent of local companies that reported net layoffs were around 5% during 2018 & 19, while the percentage of those with net hiring hovered close to 30%. These ratios have changed dramatically; in October, 27.1% of the participants in our survey had net layoffs, while only 15.5% of them reported net hiring. To simplify it, layoffs have increased by 5 fold, while hiring dropped in half.
As was expected, accommodation-food & leisure sector had the largest layoffs by far over the last 6 months. That dubious distinction still follows the sector as it also has led in layoff plans for the next 6 months.
Retail and other services sectors had the largest net hiring in the last 6 months, while manufacturing is the lead in hiring plans for the next 6 months.
Purchasing plans indices also rose in October; actual purchases recorded a 6.2% increase while plans for future purchases rose by 9.0%. However, to put these in context, actual purchases are still 41.3% below where they were last year despite the October increase. Plans for future purchases is in better shape and is only 12.3% below its last year's level.
Accommodation-food & leisure sector also had the largest cutbacks in purchasing over the last 6 months. And they also lead the plans for cutbacks among other sectors over the next 6 months. The manufacturing sector leads the purchasing plans for the next 6 months.
It was a different story when it came to revenues; not everybody was in the same boat. While Real Estate companies reported very large increases in their revenues over the last 6 months, other sectors reported significant losses. The only exception was the professional and scientific sector, which reported a small increase in their revenues.
Healthcare leads the expected rise in revenues over the next 6 months, while accommodation-food & leisure sector still expects large drops in their revenues.
The economy reclaimed a good 91% of its losses during the 2nd quarter, however, it is still well below its last year's level. The most common question is; was the 3rd quarter growth the start of a V-shaped recovery or the end of the first up-tick inside a W-shaped recovery. Which translates to, is this the start of continuous recovery, or are we in for a second drop in the GDP.
Our local business survey clearly shows a heightened level of concern with the future of the economy. We did not observe any of the common signs when enthusiasm about the future counters fears of the present and usually results in sustained growth, something that was part and parcel of all previous recoveries.
This time around, businesses are not setting-up for recovery, it looks like they are more concerned with further risk reduction.
The results of the 2nd KADAC 2020 General Election poll
This is the second of two DeSales University – WFMZ polls to be released before the general election on November 3rd. This poll is to be released on Monday October 26, 2020. The survey data was collected between October 11 and 24 and includes the responses of likely voters residing in Pennsylvania's 7th congressional district. This area includes all of Lehigh and Northampton counties, as well as portions of Monroe County. The survey and subsequent analysis was conducted by the Kamran Afshar Data Analytics Center (KADAC) at DeSales University.
The results reported here are based on live telephone interviews of 448 likely voters (279 cell phones and 169 landlines). Respondents were selected through a randomized database of registered voters in the district. A weighting measurement has been used to make sure the survey is representative of the relative number of registered voters in each of the three counties in the 7th district. The margin of error for the survey is+/- 4.4% at a 95% level of confidence.
Our sample includes more Democrats (48%) than Republicans (37%), which resembles the breakdown of actual voter registration data in the district. However, only 30% of these voters identify as liberal while 35% reported that they are conservative. President Trump's approval rating dropped to 36% from 40% in our first poll, mirroring the drop in his approval of his response to the COVID-19 pandemic, which dropped slightly from 35% to 32%. Governor Tom Wolf's scores on both accounts remain high in the 7th district. 55% of likely voters approve of the job he is doing as governor, and 58% approve of how he has dealt with the pandemic specifically.
Turning to the election on November 3rd, voters signal stronger support for Democratic candidates. These findings are bolstered by evidence of higher engagement from Democrats during the early voting period In Pennsylvania, where 70% of the already returned ballots have come from registered Democrats. According to the U.S. Elections Project, run by Michael McDonald at the University of Florida, nearly 97,000 votes have already been cast in the three counties we cover as of October 23rd.
In one of the two federal elections on the ballot, we have seen movement towards former Vice President Joe Biden since our last poll, giving him a lead of 56%-37% over President Donald Trump, an increase of 4 points since our previous release two weeks ago. We have also seen some movement towards President Trump solidifying his Republican base, as 81% of Republicans in this poll indicate they will vote for the president, up from 76% previously. But he still trails Biden in partisan support, as 91% of Democrats say they will vote for him. In the race to represent the 7th district in the House of Representatives, the current representative, Democrat Susan Wild leads her Republican challenger Lisa Scheller 54%-35%, which is nearly identical to the 54%-36% spread in our last survey.
We have also seen a slight uptick in voters' reported willingness to take a government approved vaccine in 2020 since the last poll. 45% of respondents said they would be willing to take such a vaccine. But we have also seen an increase in the opposition to take a vaccine, with 38% of likely voters reporting that they would not take it. In our first poll, 41% indicated they would take it and 36% said they would not.
To determine the sample of likely voters, we only include registered voters in the 7th district of Pennsylvania who described themselves as either "very likely" or "somewhat likely" to vote in the upcoming general elections. The survey was designed by faculty at DeSales University in Center Valley, PA in conjunction with WFMZ. The poll was funded entirely by KADAC and WFMZ.
The presentation:
Poll-2-2020Download
The results of the first KADAC poll in 2020
This is the first of two DeSales University – WFMZ polls before the general election on November 3rd. This poll was released on Tuesday October 13, 2020. The survey was conducted between October 2nd and 10th and includes the responses of likely voters residing Pennsylvania's 7th congressional district, which includes all of Lehigh and Northampton counties, as well as parts of Monroe County and was conducted by the Kamran Afshar Data Analytics Center (KADAC) at DeSales University.
The results reported here are based on live telephone interviews of 466 likely voters (294 cell phones and 172 landlines.) Respondents were selected through a randomized database of registered voters in the district. A weighting measurement has been used to make the survey is representative of the relative number of registered voters in each of the three counties in the 7th district. The margin of error for the survey is +/- 4.3% at a 95% level of confidence.
According to the results of those who are likely to vote is as follows:
Participants in our sample identify themselves as slightly more conservative than liberal, while more Democratic than Republican. Our data shows a correlation between educational attainment and enthusiasm to vote in this general election.
Among likely voters participating in our survey, 40% approve of President Trump's handling of his job, while 54% disapprove. Another 6% neither approve nor disapprove. 55% approve of Tom Wolf's performance as governor, while 38% disapprove, and 7% neither approve or disapprove. Additionally, only 35% of respondents approve of President Trump's response to the COVID-19 pandemic, while 59% approve of Governor Wolf's response.
Former Vice President Joe Biden holds a 15-point lead (54%-39%) over President Donald Trump among likely voters. 86% percent of those who voted for Donald Trump in 2016, are planning to vote for him again in 2020. 96% of those who voted for Hillary Clinton 2016 are planning to vote for Joe Biden in 2020. Among those who voted for someone else in 2016, 20% are planning to vote for Trump and 45% for Biden. Among those who didn't vote in 2016, 36% are planning to vote for Trump and 43% for Biden.
Representative Susan Wild has a lead of 18 points (54%-36%) over Lisa Scheller, with particularly heavy support among higher educated likely voters. 85% percent of Democrats 19% of Republicans are planning to vote for Wild, while 71% of Republicans and 7% of Democrats are planning to vote for Scheller.
Registered voters in the 7th district are almost evenly divided among those who will not get a government approved vaccine before the end of this year and those who would, with 41% of respondents indicating they would take such a vaccine and 36% saying they would not. And even taking or not taking the vaccine was also highly political; 37% of Democrats vs. 50% of the republican stated that they take the vaccine, if available before the end of this year. Independents resembled the Democrats more in this case as only 36% of them stated that they will take the vaccine by the year's end.
Quarterly Economic Review of Lehigh Valley Businesses: 2020 Q3 LV Discourse @ WDIY
August 20, 2020 KAA Blog, News
https://www.wdiy.org/post/quarterly-economic-review-lehigh-valley-businesses-2020-q3-lv-discourse#stream/0
How are the Lehigh Valley's businesses doing? Dr. Kamran Afshar welcomes guest Tony Iannelli, CEO and President of the Greater Lehigh Valley Chamber of Commerce, to discuss the results of the latest quarterly economic survey of Lehigh Valley businesses.
Lehigh Valley business sentiment index rose 5.7% in July
August 7, 2020 KAA Blog, News
The Lehigh Valley's business sentiment index the BSI made a small recovery in July, rising by 5.7% after its historical 31.6% fall in April. The index is still below its average level during the Great Recession. The small July recovery could be either the start of a directional change or just a bounce.
The July survey revealed some interesting decision-making dilemmas facing businesses; for example, the largest losses and gains in July were among the purchasing indices; the index of plans for future purchases rose sharply, however, since it was from an all-time low, it is still 19% below its January level. On the other hand, the index of actual purchases over the last 6 months, continued to drop, falling to half of its January 2020 level.
Businesses, clearly are not buying, nor planning to raise their purchases close to where it was before Corona.
The percent of local companies which reported net layoffs, was 5% during 2018 & 19, while the percentage of those with net hiring hovered close to 30%. In January 2020, before the impact of COVID, we observed a significant change; percent of businesses with net layoffs almost doubled, while the percentage of those with net hiring dropped. By April, the percent of companies with net layoffs had tripled, while the percent of those with net hiring was almost halved. This negative trend became even worse in July.
Our employment indices also moved in opposite directions, the index of actual hiring over the last 6 months, continue to fall through July to all-time low. On the other hand, the index of plans for future hiring recorded a small increase indicating that businesses have started to plan for future hiring.
The index of actual revenues over the last 6 months fell to less than half of where it was in January, while expected future revenues, rose sharply in July, it is still well below its January level. Some business optimism is coming back, but in trickles.
All of the indices in our model are still well below where they were in January.
Most economists still believe that the 3rd quarter would witness the start of the recovery. How fast or slow that recovery will be, has very little to do with how fast or slow states open businesses or if children attend school in person. All of it, however, depends on how safe people feel in going out of their homes, going shopping, or even thinking of maybe visiting a restaurant! That kind of feeling does not come from wishful thinking and cannot be manufactured; for it to sustain, it has to be real. Our economic recovery is directly related to the rate at which we can bring this pandemic under control.
Lehigh Valley's inflation was 3.6% in 2019
June 30, 2020 KAA Blog, News
Lehigh Valley's inflation rate in 2019 was 3.6%, according to the Kamran Afshar Data Analytics Center (KADAC) @ DeSales University Consumer Price Index.
Lehigh Valley's inflation rate rose by 3.6% in 2019, higher than the 2.3% at the national average. The higher inflation in the Valley is mostly due to the sharper rise in food and shelter costs compared to the national average.
Food prices increased by 5.5% in the Valley, compared to 1.8% nationwide in 2019. In both the Valley as well as the national average, the increase in the overall food prices was mostly due to the sharper rises in the sub-category of meats, poultry, fish, and eggs. In the Lehigh Valley.
The cost of Shelter in Lehigh Valley also rose at a higher rate of 5.3% compared to 3.2% at the national average. This was mostly influenced by a 5.8% increase in rent and rental costs in the Valley compared to 3.3% at the national average.
On the lower side, Apparel costs dropped by 6.5% in the Valley compared to a drop of 1.2% at the national level. Lehigh Valley transportation costs rose at a slower 0.8% rate compared to 2.0% at the national level.
This is the first in a series of continuing reports on the Lehigh Valley CPI, from KADAC @ DeSales University. Future updates will be provided on a semi-annual basis.

redpajama_set_name: RedPajamaCommonCrawl


Zombies are among us; one need look no further than the Fox News network for proof of that, and it's only a matter of time before those nefarious necrotic beasties take their act nation wide. With his quasi-satirical ZOMBIE SURVIVAL GUIDE, Max Brooks prepared us for potential zombie warfare in the guise of a lovingly and meticulously crafted survival manual that read much like those hilariously misguided Boy Scout handbooks they gave out back when I was a kid (and probably hadn't updated since the late 1950's). While Brook's book ended up in most store's humor sections, its author wasn't entirely kidding, and, as his comparatively stone-faced follow-up WORLD WAR Z proves, to Max Brooks zombies are not a laughing matter.
Set in the not too distant future, World War Z is a chronicle of the "great panic"; an all-out zombie war that claimed a sizeable chunk of the Earth's population. The book is broken up into interviews with various "witnesses", ranging from a Canadian Special Forces soldier to an impoverished South African, who each offer their perspective on the undead plague, their particular exposure to it, and how they survived. Brooks gives each character a unique voice, lending the "documents" a sense of authenticity that makes this "Oral History of the Zombie War" read like the sort of report it's modeled after (the 911 Commission springs to mind). That's not to say that the book is boring; not by a long shot. Brooks' post-apocalyptic world is as dark and intriguing as the people who populate it, and, as an obvious devotee to the genre, the author serves up heaping portions of the requisite gore one would expect from such a tale.
World War Z is a bold experiment in horror literature, combining sociopolitical satire with a well-trodden genre theme that serves as a thinly veiled representation of any number of pandemic possibilities. This is a book that will entertain, for certain, but also disturbs on many levels, and, despite the occasional laugh, I seriously doubt Brooks will have to worry about this one getting lost in the humor section.


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Ende Enat Part 29– Kana TV Dubbing Amharic Drama
Added 1 week ago by GurshaPlus in Ende Enat Drama
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Fyffe House
Rich in human history, the story of this area began with the arrival of the ancestors of today's Māori people. Hunting the southern right whale, Europeans arrived over 600 years later. Today this house, along with the stories it tells of a developing town and the three families that lived here, are all that remains of the 'Waiopuka Fishery', Robert Fyfe's whaling station that was established in 1842.
The earliest part of the House is the single storied wing which sits on whale vertebrae foundations. It was originally built by Robert Fyfe for his cooper, who made barrels in which the whale oil was transported and stored. Robert Fyfe drowned in 1854 and his cousin George Fyffe (spelled with a double f, and after whom the House was named) managed the whaling station and associated farming activities. George built the rest of the House in two stages, the southern end in 1857 and it was completed in the early 1860s when he married his wife Catherine.
George died in 1867, having fallen off the jetty near his house. Catherine died the following year and, since there were no children, all of their property was sold. Fyffe House was bought by Joseph Wilkinson Goodall (1835-1911), who transferred to Kaikoura as the district constable in 1865. The Goodalls made few changes to Fyffe House itself, apart from enclosing part of the verandah to form an office. It seems likely that it was the Goodalls who first established a flower garden in the space between the two wings of the building, and who wallpapered the interior for the first time.
The land surrounding the house was subdivided into five sections after Goodall's death in 1911. The section on which Fyffe House stands passed through several hands until George Low bequeathed the house to the New Zealand Historic Places Trust (now Heritage New Zealand Pouhere Taonga). After his death in 1980, the NZHPT repaired the house and some years later repainted the exterior in its original pink. Fyffe House remains Kaikoura's oldest building, and has strong connections to Māori and non-Māori.
62 Avoca Street, Kaikoura
64-3-319 5835
fyffe@heritage.org.nz
www.heritage.org.nz/places/places-to-visit/marlborough-region/fyffe-house
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Sharwanand has come out with another light hearted entertainer 'Mahanubhavudu' which deals with issues and problems faced by an OCD disorder person. The movie has been marketed as a light hearted film and so does it live upto the expectations. Maruthi has kept it simple and straight. Though he has taken his own sweet time to establish his characters, his writing at parts (especially the comedy) looks to have won many hearts. Sharwanand is a star in the making and he very well knows how much to put into a role and how to present himself. He looks sharp and sleak as the OCD person.
Anand (played by Sharwa) is a OCD person who works as a software professional. He is super strict with cleanliness and gets irate upon coming across unhygenic things or behaviour. He falls for his office colleague, Meghana (played by Mehreen) who also likes hydenic men. Somehow things takes a turn after the arrival of Ramaraju, Meghana's father. Fate leads Anand to Meghana's village and the rest of the story is about how Anand drives his love towards marriage. Though the movie looks quite familiar and common, it is the clever characterization of Anand as a person with OCD that makes the movie worth watching.
Maruthi's similar movie 'Bhale Bhale Mogadivoy' had the issues of person suffering from amnesia. Though similarities reflect in the two, this movie has its moments to bank on for entertainment. While Sharwa is all natural and totally lovable in this role, Mehreen looks fabulous and her chemistry with Sharwa has worked in their favour. Vennela Kishore provides the needed comical support in the second half of the movie. Thaman's music and BGM stands out as one of the highlight in the movie while Nizar Shafi's cinematography is commendable.
Verdict: A light hearted movie, keep your expectations low and walk in with a mood to only get entertained. You will be more than satisfied. Easily the best pick for this festival. Go for it!
Positive reports are pouring in from both audience as well as critics. Especially Thaman's music and Sharwanand's performance are being applauded.
As the morning shows come to an end, lot of celebrities and critics are posting Tweet reviews of 'Mahanubhavudu' most of which are quite positive.
Mahanubhavudu is being termed as a neat comedy film which once again has a superb performance from Sharwanand. Critics are giving decent reviews and ratings for the film which makes a decent watch for this festival weekend.
Public and Critics gave unanimous thumbs up for the film through their Tweets.
Sharwanand always has a way out when he releases his movies with the biggies in a race. His movies provide the needed entertainment and relaxation of watching a light hearted movie. After 'Express Raja' and 'Shatamanam Bhavati', looks like this time Sharwa is all ready as the OCD disorder guy in 'Mahanubhavudu.' The movie tells the issues and troubles faced by a person with OCD disorder and what happens to his love in the process. The movie features Mehreen Kaur as the female lead and has been directed by Maruthi. UV Creations bank rolled the project.
Sharwanand is by far one of the finest actors in Telugu industry at the moment. Continuing his streak of light hearted roles, Sharwa looks completely stylish and uber-cool in this rom-com. We can expect to see a new avatar from Sharwanand in this.
Maruthi is known for his quirky concepts and ideas that he puts into each of his movies. Ever since 'Bhale Bhale Mogadivoy' he has upped his game and is trying to stand out with his work. The unique disorder itself stands as an attractive point to go and give this movie a watch.
Be whatever movie Thaman picks, he tries his best to stand out. And with the title song receiving great response, the album has already been on loop for many. Looks like the breezy entertainer will be even better with his songs and picturisation.
The movie is all set for a grand release on September 29 amidst the heavy Dussera competiton. Watch out this space for complete Mahanubhavudu reviews. We will also update you with the public response to Mahanubhavudu.


If you or a loved one has been misdiagnosed because your doctor failed to detect your cancer, solicitors at Imperium Law can help you make a claim.
We represent cancer claim patients who have suffered delayed treatment because of misdiagnosis, and family members who have lost loved ones because their cancer was detected too late.
Statistics from Cancer Research UK show that each year bowel cancer accounts for around 10% of all UK deaths.
Any delay in detecting bowel cancer can be devastating, and in some cases failure to promptly diagnosis the disease can be fatal.
A change in bowel movements.
Diarrhoea, incontinence, constipation and passing mucus.
In addition you may experience abdominal pain, bloating, anaemia, lethargy and vomiting. These symptoms can be attributed by your GP to other conditions and so it is important that your doctor takes a detailed history and investigates your symptoms thoroughly. Failure to do this can be grounds for a claim.
Early detection and treatment is therefore essential, however sadly not all cases are detected, and left untreated bowel cancer can be terminal.
We strongly believe in representing our clients on a 'No Win-No Fee' basis because that way every person can pursue their rights.
Everyone deserves compensation for the pain, suffering and distress caused by negligent treatment from a medical professional.
By representing our clients on a 'No Win-No Fee' basis, we ensure there is need to worry about the financial costs of making a claim and highlighting medical failings.
You only pay a fee if we win your case and you are awarded a payout.
As well as getting a payout to help with your or a loved one's recovery, or to compensate you for the death of a loved one, there is another reason to make a claim.
By claiming you can help improve treatment for all patients, because pointing out medical failings can help prevent others from experiencing a similar fate.
Statistics show that if detected at its earliest stage, the 5 year bowel cancer survival rate is 90%. Any delay in the diagnosis of the disease can therefore have very serious consequences.
Whilst bowel screening has vastly improved over the last two decades mistakes happen.
If you are told your screening results are negative and your doctor fails to make an accurate diagnosis, cancer can be left unchecked and can spread through the bowel wall into other parts of the body.
If this describes your situation or that of a loved one, call a solicitor at Imperium Law on 0800 633 5730. We can review your claim and advise if you could be entitled to a payout.
To make a successful claim it is necessary to show that you were misdiagnosed, and that the delay directly caused your condition and eventual medical outcome to worsen.
The standard of care provided fell below the standard of care expected of a qualified GP or clinician.
The delay directly caused an adverse outcome.
Compensation amounts vary according the physical effects, and depend on the stage you were at when you should have been diagnosed and the stage you were at when you were finally diagnosed. This can be mapped by an oncologist and the information used to help prove your claim and work out how much compensation you are entitled to receive.
If you would like to make a claim call us on 0800 633 5730 and we can review your case.