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{
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%\rightline{PDMI PREPRINT ---  {\hskip2cm}}
\vskip 4 truecm
\centerline{\atfont A PURITY THEOREM  }
\vskip 1 truecm
\centerline{\atfont  FOR FUNCTORS WITH TRANSFERS}

\vskip 2 truecm
\centerline{\aufont  K.~Zainoulline}
\vskip 1.5 truecm
\centerline{Russian Academy of Sciences}
\smallskip
\centerline{Steklov Mathematical Institute}
\smallskip
\centerline{St.~Petersburg Department}
\smallskip
\centerline{Fontanka 27, St.~Petersburg 191011, Russia}
\bigskip
\centerline{\abfont e-mail: kirill\@pdmi.ras.ru}
\smallskip
\centerline{\abfont URL: http://www.pdmi.ras.ru/\~{}kirill/index.html}
\bigskip
\centerline{April 5, 2000}
\vskip 1.5 truecm


\centerline{\abtfont  ABSTRACT} 
\medskip
{Let $R$ be a local regular ring of geometric type
and $K$ be its field of fractions.
Let $\f$ be a covariant functor from the category of $R$-algebras
to abelian groups satisfying some additional properties (continuity, existence
of well behaving transfer map).
We show that the following equality holds for the subgroups of the group $\f(K)$:
$$
\bigcap_{\p\in\spec R, ht\p=1}\im\{\f(R_\p)\ra\f(K)\}= \im\{\f(R)\ra \f(K)\},
$$
where all maps are induced by the canonical inclusions.
}
\vskip1cm

{\narrower\narrower\noindent\abfont
\advance \parindent by 3mm
\noindent
\par}
\vfill\nopagenumbers\eject
}

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\head{Agreements}\endhead

All rings are assumed to be commutative with unit.

Let $A$ and $S$ be any rings.
By an $A$-algebra $S$ we will mean the pair $(S, i)$, 
where $i: A \ra S$ is a ring homomorphism. 
Sometimes, we will write $S$ instead of the pair $(S,i)$ 
keeping in mind the homomorphism $i$.
We will denote an $A$-algebra $S$ by $A \xra{i} S$.

By a morphism $f : (S, i) \ra (S',i')$ between two $A$-algebras
we will mean such a ring homomorphism $f:S \ra S'$ that $f\circ i=i'$.

Let $\f$ be a covariant functor from the category of $A$-algebras 
to the category of abelian groups.
Let $A \xra{i} R$ be an $A$-algebra.
Since every $R$-algebra $T$ has the $A$-algebra structure  
induced by the homomorphism $i$  the functor $\f$ may be considered as
a functor given on the category of $R$-algebras.
To make a difference from $\f$ this functor is denoted by $\fr$ and is called by 
the restriction of the functor $\f$ to the category of $R$-algebras via the homomorphism $i$. 
So on objects the functor $\fr$ is defined as
$\fr(R \xra{t} T)=\f(A \xra{i} R \xra{t} T)$, for any $R$-algebra $R\xra{t}T$.


\head \S 1. The Purity Theorem \endhead

Let $A$ be a smooth algebra over an infinite field $k$.
Let $R=A_\q$ be the localization of the algebra $A$ at some prime ideal $\q\in \spec A$
and let $K$ denote its field of fractions.
So $R$ is a local regular ring of geometric type.
Such a ring will be the main object of our discussion.

\definition{Definition}
Let $\g$ be a covariant  functor 
from the category of $R$-algebras  to
the category of abelian groups.
We say that the purity holds for the functor $\g$ 
if   the following equality holds for the subgroups of $\g(K)$:
$$
\bigcap_{
\smallmatrix
\p\in\spec R, \cr
ht\p=1
\endmatrix
}\im\{\g(R_\p)\ra\g(K)\}= \im\{\g(R)\ra \g(K)\},
\tag{$*$}
$$
where all maps are induced by the canonical inclusions. 
\qed
\enddefinition

This paper is devoted to the proof of the following theorem:

\proclaim{Theorem I}{(\rm The Purity Theorem)}
Let $R$ be a local regular ring of geometric type got by localizing a smooth $k$-algebra $A$.
Let $\f$ be a functor from the category of $A$-algebras to the category of abelian groups 
and $\fr$ denote its restriction to the category of $R$-algebras 
via the canonical inclusion $A \hra{i_R} R$. 
Let $\f$ and $\fr$ satisfy properties {\bf C}, {\bf T} and {\bf E}
described at the end of this section. 

Then the purity holds for the functor $\fr$.
\endproclaim

In the constant case, i.e. when the functor $\f$ is given on the category of $k$-algebras, we have

\proclaim{Theorem II}
Let $R$ be a local regular ring of geometric type got by localizing a smooth $k$-algebra $A$.
Let $\f$ be a functor from the category of $k$-algebras 
to the category of abelian groups and 
$\fr$ denote its restriction to the category of $R$-algebras
via the canonical inclusion $k\hra{i_R} R$. 
Let $\f$ and $\fr$ satisfy properties {\bf C} and {\bf T}
described below. 

Then the purity holds for the functor $\fr$.
\endproclaim

Below we restate these theorems in a slightly different form.

Under the hypotheses of the theorems above the main result of paper \cite{Za}
states that the map
$\f(R) \xra{i_K^*} \f(K)$ induced by the canonical inclusion $R\hra{i_K} K$  is injective.
Thus we may identify the group $\f(R)$ with its image in $\f(K)$.
Since the localization $R_\p$ at height 1 prime $\p$ is
a local regular ring we may rewrite the purity condition
for the functor $\fr$ as
$$
\bigcap_{
\smallmatrix
\p\in\spec R, \cr
ht\p =1
\endsmallmatrix
} \f(R_\p)=\f(R),
\tag{$*'$}
$$
where the left and the right parts are considered as subgroups of $\f(K)$.

\proclaim{Theorem III} Under the hypotheses of Theorem I or II
the following complex is exact:
$$
0 \lra \f(R) \xlra{i_K^*} \f(K) \xlra{\oplus\can_\p} \bigoplus_{
\smallmatrix
\p\in\spec R, \cr
ht\p =1
\endsmallmatrix
} \f(K)/\f(R_\p),
$$
where $\can_\p:\f(K) \ra \f(K)/\f(R_\p)$ is the canonical map.
\endproclaim

\demo{Proof}
By property {\bf C} of the functor $\f$ we have the canonical isomorphism
$\varinjlim_{g\in R}\f(R_g) \cong \f(K)$.
Thus there exists such a localization $R_g$ that
an element $\a \in \f(K)$ comes from some element $\a_g \in \f(R_g)$. 
Since $R$ is smooth over $k$
there are only finite number of prime ideals of height 1 lying over $g$. This means
that the element $\a$ doesn't lie only in finite number of subgroups 
$\f(R_\p)$ in $\f(K)$.
Hence the homomorphism $\oplus \can_\p$ is well-defined.

The injectivity of the map $i_K^*$ was already mentioned.
Clearly the complex is exact at the $\f(K)$-term if and only if the purity
($*'$) holds for the functor $\fr$.
\qed
\enddemo

\demo{Example 1}
Consider a linear algebraic group $G$ over the base field $k$
which is rational as the $k$-variety.
Let $T$ be a torus over $k$ and let $\mu: G\ra T$ be a group morphism which
is surjective over a separable closure $\bar{k}$ of the field $k$, i.e.
the map $\mu(\bar{k}):G(\bar{k})\ra T(\bar{k})$ is surjective.

Let $\f$ be a functor from the category of $k$-algebras to the category
of abelian groups defined as follows:
$$
\f: S \mapsto T(S)/\mu(G(S))
$$
for any $k$-algebra $S$. Then the purity holds for the functor $\fr$
and by Theorem III we get

\proclaim{Theorem IV}
The following sequence of abelian groups is exact:
$$
0 \lra T(R)/\mu(G(R)) \xlra{i_K^*} T(K)/\mu(G(K)) \xlra{\oplus_\p \can_\p}
\bigoplus_{
\smallmatrix
\p\in\spec R, \cr
ht\p =1
\endsmallmatrix
}
T(K)/\mu(G(K))\cdot T(R_\p)
$$
\endproclaim

Indeed, by Theorem II we only have to check that the functor $\f$ and its restriction $\fr$
satisfies properties {\bf C} and {\bf T}.

The property {\bf C} follows immediately.
To check the property {\bf T} we consider the obvious norm map $N: T(S) \ra T(R)$
for any $R$-algebra $S$ finitely generated projective as the $R$-module.

To get the desired transfer map $\tr^{S}_R:\fr(S)\ra \fr(R)$ we have to verify
the following inclusion, usually called Norm Principle: $N(\mu(G(S)))\subset \mu(G(R))$.

To do that we consider the following commutative diagram:
$$
\CD
T(S) @>{N}>> T(R)/\mu(G(R)) @>{\partial }>> \hml(R,\ker \mu) \\
@V{f'}VV @V{f}VV @V{g}VV \\
T(S_K) @>>{N}> T(K)/\mu(G(K)) @>>{\partial_K }> \hml(K,{\ker \mu}_K)
\endCD
$$
where $S_K=S\otimes_R K$ denotes the scalar extension.

Observe now that $g$ has trivial kernel by the result of \cite{CO}.
Since the map $\partial $ has trivial kernel the map $f$ has trivial kernel as well.

Let now $\a\in \mu(G(S))\subset T(S)$ be an element. We have to show that
$N(\a)\in \mu(G(R))$.
By the commutativity of the left square of the diagram and the injectivity of $f$
it suffices to verify that $N(f'(\a))\in \mu(G(K))$.
This is indeed the case by the Norm Principle of Merkurjev \cite{Me}
because the group $G$ is rational.

Thus the norm map $N$ induces a transfer map $\tr^S_R$.
The properties {\bf T}.(a) , {\bf T}.(b1) and {\bf T}.(b2) of the transfer map follows immediately.
\qed 
\enddemo

\demo{Example 2}
Suppose the group $G$ from the previous example is a torus which is
not necessary a rational $k$-variety.
Then the purity holds for the functor $\fr$ from Example 1. 

Indeed, in the previous notation the following diagram commutes
$$
\CD
G(S) @>{N_G}>> G(R) \\
@V{\mu(S)}VV @VV{\mu(R)}V \\
T(S) @>>{N}> T(R)
\endCD
$$
Therefore the norm map $N$ induces a well-defined transfer map $\tr^S_R:\fr(S)\ra \fr(R)$.
Again the properties  {\bf T}.(a) , {\bf T}.(b1) and {\bf T}.(b2) 
of the transfer map hold and the continuity of the functor $\f$ holds as well. 
\qed
\enddemo

\demo{Example 3}
Let $\A$ be an Azumaya algebra over the local regular ring $R$. 
Let $\nrd: \A^* \ra R^*$ denote the reduced norm homomorphism.
For any $R$-algebra $S$ let $\A_S=\A\otimes_R S$ be the extended Azumaya algebra over $S$.

Let $\f$ be a functor from the category of $R$-algebras to the category
of abelian groups defined as follows:
$$
\f: S \mapsto S^*/\nrd (\A_S^*)
$$
for any $R$-algebra $S$. Then the purity holds for the functor $\fr$ and by Theorem III we get

\proclaim{Theorem V}
The following sequence of abelian groups is exact:
$$
0\lra R^*/\nrd(\A^*) \xlra{i_K^*} K^*/\nrd(\A_K^*) \xlra{\oplus_\p \can_\p} \
\bigoplus_{
\smallmatrix
\p\in\spec R, \cr
ht\p =1
\endsmallmatrix
} K^*/\nrd(\A_K^*)\cdot R_\p^*
$$
\endproclaim

Observe that this is the result of I.Panin and A.Suslin (Theorem I, \cite{PS}) proved
by completely different way.

Indeed, all necessary properties 
{\bf C}, {\bf T} and {\bf E} 
of the functor $\f$ and its restriction $\fr$
were checked in 3.2 of \cite{Za}.
\qed
\enddemo

\demo{Example 4} 
Under the notation of the previous example 
assume that there is the additional structure on our Azumaya algebra:

Let $(\A,\s)$ be an Azumaya algebra with unitary involution over the local regular ring
$R$ ($char\; k \neq 2$).
It means that there is a tower $\A/C/R$, where $C$ is the center of $\A$ and 
$C/R$ is an etale quadratic extension with restricted involution $\s$.
Therefore, $C_K/K$ is a separable quadratic extension of the corresponding fields of fractions.

Let $\U(\A_S)=\{a \in \A_S \mid aa^\s=1 \}$ be the unitary group of an algebra $(\A_S,\s)$ for
any $R$-algebra $S$. And let $\U(C_S)$ denote the unitary group of the center $C_S$
of the algebra $\A_S$. 

Let $\f$ be a functor from the category of $R$-algebras to the category
of abelian groups defined as follows:
$$
\f: S \mapsto \U(C_S)/\nrd (\U(\A_S))
$$
for any $R$-algebra $S$. Then the purity holds for the functor $\fr$ and by Theorem III we get

\proclaim{Theorem VI} The following sequence of abelian groups is exact:
$$
0\ra \U(C)/\nrd(\U(\A)) \xlra{i_K^*} \U(C_K)/\nrd(\U(\A_K)) \xlra{\oplus_\p \can_\p} \
\bigoplus_{
\smallmatrix
\p\in\spec R, \cr
ht\p =1
\endsmallmatrix
} \U(C_K)/\nrd(\U(\A_K))\cdot \U(C_\p)
$$
\endproclaim

Indeed, all necessary properties 
{\bf C}, {\bf T} and {\bf E} 
of the functor $\f$ and its restriction $\fr$
were checked in 3.4 of \cite{Za}.
\qed
\enddemo

Now we describe  the properties  
{\bf C}, {\bf T} and {\bf E}
of the functors $\f$ and $\fr$ mentioned above (compare with sect.~1 and~2, \cite{Za}):

\subhead C \endsubhead
(continuity)
For any $A$-algebra $S$ essentially smooth over $k$ and
for any multiplicative system $M$ in $S$ the canonical map
$\varinjlim_{g \in M} \f(S_g) \ra \f(M^{-1}S)$ is an isomorphism,
where $M^{-1}S$ is the localization of $S$ with respect to $M$.

\subhead T \endsubhead
(transfer)
For any $R$-algebra $T$ finitely generated projective as the $R$-module
there is given a homomorphism $\tr_R^T: \fr(T) \ra \fr(R)$ called transfer map such that

\itemitem{(a)}{\rm (additivity and normalization)} 
$\tr_R^{R\times T}(x)= \pr_R^*(x) + \tr_R^T(\pr_T^*(x))$
for every $x\in \f(R\times T)$, 
where $\pr_R^*$ and $\pr_T^*$ are induced by the canonical projections;

\itemitem{(b1)}{\rm (scalar extension and homotopy invariance)} 
for an $R[t]$-algebra $S$ finitely generated projective as the $R[t]$-module
(thus, $S/(t)$ and $S/(t-1)$ are finitely generated projective as the $R$-modules)
the following diagram commutes:
$$
\CD
\fr(S) @>{\can_0^*}>> \fr(S/(t)) \\
@V{\can_1^*}VV        @VV{\tr_0}V \\
\fr(S/(t-1)) @>>{\tr_1}> \fr(R)
\endCD
$$
where $\can_0^*$, $\can_1^*$ are induced by the canonical projections and
$\tr_0$, $\tr_1$ denote the corresponding transfer maps
$\tr_R^{S/(t)}$ and $\tr_R^{S/(t-1)}$.

\itemitem{(b2)} the following diagram 
induced by extension of scalars via the canonical inclusion $R \hra{} K$ commutes:
$$
\CD
\fr(T) @>>> \fr(T\otimes_R K) \\
@V{\tr^T_R}VV @VV{\tr^{T\otimes_R K}_K}V \\
\fr(R) @>>> \fr(K)
\endCD
$$

\subhead E \endsubhead
(extension property)
Given an $R$-algebra $R  \xra{i} S$ essentially smooth over $k$, 
given an $A$-algebra $A \xra{i_S} S$ and 
an augmentation $S \xra{\e} R$ of the inclusion $i$, 
such that the following diagram commutes:
$$
\CD
A @>{i_S}>> S \\
@| @VV{\e}V \\
A @>>{i_R}> R
\endCD
$$
and given a multiplicative system $M$ with respect to a finite set
$\{\m_i\}_{i\in \I}$ of maximal ideals in $S$
with the property $\e^{-1}(\m_R)\subset \cup_{i\in \I}\m_i$,
there exist:

\itemitem{(a)} A localization $S_g$ and a finite etale extension $e: S_g \ra \St$
for a certain $g\in M$;

\itemitem{(b)} An augmentation $\et: \St \ra R$ for the
inclusion $R \xra{i} S_g \xra{e} \St$ such that the following
diagram commutes: 
$$
\CD
S_g @>{e}>> \St \\
@V{\e}VV @VV{\et}V \\
R @= R
\endCD
$$

\itemitem{(c)} A natural transformation $\Phi: \f \ra \fr$
between two functors $\f$ and $\fr$ restricted to the
category of $\St$-algebras via the maps $A \xra{i_S} S_g \xra{e} \St$ and
$R \xra{i} S_g \xra{e} \St$
correspondingly, such that the morphism
$$
\CD
 \f(\St \xra{\et} R \ra T) @>{\Phi(\St \xra{\et}R \ra T)}>>  \fr(\St \xra{\et}R 
\ra T)
\endCD
$$
is the identity.
In particular, the following diagram commutes:
$$
\CD
\f(\St) @>{\Phi(\St)}>> \fr(\St) \\
@V{\et^*}VV   @VV{\et^*}V \\
\f(R) @= \fr(R)
\endCD
$$
where $R$ is considered as the $\St$-algebra via the augmentation $\et$.
\qed

\ 

The present paper is organized as follows:

Section 2 is devoted to the proof of the Purity Theorem.
This proof is based on a further development of the technique offered in the papers
of K.Zainoulline, \cite{Za}, and I.Panin, M.Ojanguren, \cite{PO}, \cite{PO$_1$}.

In section 3 we prove the Divisor Theorem
--- the key point of the proof of the Purity Theorem.
It was motivated by the work of V.Voevodsky \cite{Vo}.

We finish by proving a stronger version of Geometric Presentation Lemma (Lemma~5.2, \cite{PO}). 

\demo{Acknowledgements}
I~am grateful to Ivan Panin for the useful discussions and ideas led
to the proof of the Purity Theorem.
I thank INTAS project YSF-99-3966 for the financial support.
\enddemo

% ********************************************

% ********************************************

% ********************************************

\vfil\eject

\head \S 2. The Proof of the Purity Theorem \endhead

Let $R=A_\q$ be the localization of a smooth $k$-algebra $A$
at some prime ideal $\q\in \spec A$ and let $K$ be its field of fractions.

Let $\f$ be a covariant functor
from the category of $A$-algebras to abelian groups and
let $\fr$ denote its restriction to the category of $R$-algebras
via the canonical inclusion $A \hra{i_R} R$.
By the very assumption  functors $\f$ and $\fr$
satisfy properties {\bf C}, {\bf T} and {\bf E}.

Let $\f_{nr}(K)$ denote the left part of the equality ($*'$), sect.1, i.e
$$
\f_{nr}(K)=\bigcap_{
\smallmatrix
\p\in\spec A \cr ht \p=1, \p\subset \q
\endmatrix
} \f(A_\p) =\bigcap_{
\smallmatrix
\p\in\spec R, \cr 
ht \p =1
\endmatrix
} \f(R_\p)\subset \f(K).
$$

We have to prove that any element $\a\in \f_{nr}(K)$ 
lies in the image of the map $\f(R) \xra{i_K^*} \f(K)$
induced by the canonical inclusion $R\hra{i_K} K$.

To do this it is enought to prove the following two assertions:

\demo{1}
For any regular function $f\in \q$ there are two commutative diagrams:
$$
\text{(i)}
\CD
\f(A_f) @>{\phi}>> \fr(S_f) @>{i_C^*}>> \fr(C_f) \\
@. @V{\Psi}VV @VV{\Psi_K}V \\
 @. \fr(R) @>>{i_K^*}> \fr(K)
\endCD
\qquad\qquad
\text{(ii)}
\quad
\CD
\f(A_f) @>{i_C^* \circ \phi}>> \fr(C_f) \\
@V{i_K^*}VV @VV{\e_K^*}V \\
\f(K) @= \fr(K)
\endCD
$$
for some algebras $S_f, C_f$ and some morphisms $\phi, i_C^*, \Psi, \Psi_K, \e_K^*$;
\enddemo

\demo{2}
There exists such a regular function $f\in A$
and an element $\a_f\in \f(A_f)$ satisfying $i_K^*(\a_f)=\a$ that
in the case of $f\in \q$ the following relation holds:
$$
\text{(iii)} \qquad \Psi_K(i_C^* \circ \phi(\a_f))=\e_K^*(i_C^* \circ \phi(\a_f)),
$$
where all morphisms are taken from the corresponding diagrams (i) and (ii) above.
\qed
\enddemo

Indeed, let $f\in A$ be the regular function and $\a_f\in \f(A_f)$ be the element from
assertion 2.

If $f\notin \q$ then there is the commutative diagram 
where all maps are induced by the canonical inclusions:
$$
\CD
\f(A_f) @>{i_R^*}>> \f(R) \\
@| @VV{i_K^*}V \\
\f(A_f) @>>{i_K^*}> \f(K)
\endCD
$$
So $i_K^*(i_R^*(\a_f))=i_K^*(\a_f) =\a$ and we get the required.

In the case of $f\in \q$ we have the chain of equalities:
$$
i_K^*(\Psi\circ \phi(\a_f))\overset{\text{i}}\to=
\Psi_K(i_C^* \circ \phi(\a_f))\overset{\text{iii}}\to= 
\e_K^*(i_C^* \circ \phi(\a_f))\overset{\text{ii}}\to=
i_K^*(\a_f) =\a
$$
and we get the required as well.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subhead 1 \endsubhead
Let $f\in \q$ be a regular function. Our aim is to construct the diagrams (i) and (ii).

We start by building up the diagram (i). More precisely we do this in three steps 
(see items 1.1-1.3 below): 
First (1.1) we construct the $R$-algebra $S$ and the map $\phi:\f(A_f)\ra \fr(S_f)$.
Then (1.2) we produce the homomorphism $\Psi:\fr(S_f)\ra\fr(R)$ (1.2).
On the last step (1.3) we construct the $K$-algebra $C$, the
homomorphism $\Psi_K:\fr(C_f)\ra\fr(K)$ and show the commutativity
of the diagram (i).

We end up by producing the diagram (ii) (see item 1.4 below).

\subhead 1.1 \endsubhead 
To construct the $R$-algebra $S$ and the map $\f(A_f)\xra{\phi}\fr(S_f)$ 
we use the following result
(see Lemma~5.12, \cite{Qu},  or sect.7, \cite{PO}):

\proclaim{Lemma}
Let $A$ be a smooth finite type algebra of dimension $d$ over a field $k$,
let $f\in A$ be a regular function,
and let $\I$ be a finite subset of $\spec A$.
Then there exist functions $x_1,\ldots,x_d$ in $A$
algebraically independent over $k$
and such that  if $P=k[x_1,\ldots,x_{d-1}] \hra{q} A$, then
A/(f) is finite over $P$;
A is smooth over $P$ at points of $\I$; and
the inclusion $q$ factors as
$q: P \hra{} P[x_d] \xra{q_1} A$, where $q_1$ is finite.
\endproclaim

Apply this lemma to the given $k$-algebra $A$, the regular function $f\in \q$
and the subset $\I=\{\q \}$. 

Put $S=A\otimes_P R$ to be the tensor product from the canonical diagram:
$$
\CD
A @>{i_S}>> A\otimes_P R \\
@A{q}AA  @AA{i}A \\
P @>>{r}> R
\endCD
$$
where the map $r$ is the composition 
$r: P \hra{q} A \hra{i_R} R$ and 
$i_S: a \mapsto a\otimes 1$, $i: r \mapsto 1\otimes r$. 

Consider the homomorphism $\f(A_f) \xra{i_S^*} \f(S_f)$ 
induced by the map $A_f\xra{i_S}S_f$.

In the constant case, i.e. when the functor $\f$ is given 
on the category of $k$-algebras (see Theorem II), 
there is the identity transformation between
the restrictions of the functors $\f$ and $\fr$ 
to the category of $S$-algebras (see 1.3.1, \cite{Za}).
Thus we have $\f(S_f)=\fr(S_f)$ and we put $i_S^*$ 
to be the desired map $\phi$. 
\qed

In the general case (Theorem I)  the homomorphism $i_S^*$ takes
values in the group $\f(S_f)$ which is different from the group $\fr(S_f)$.
To avoid this problem we use property {\bf E} of the functor $\f$.
More precisely, following the steps 1-4 of the proof of Proposition, sect.2, [Za],
we construct the $3\times 5$ commutative diagram for the triple $(A,R,f)$:
$$
\CD
\f(A_f) @>{i_g^*\circ i_S^*}>> \f(S_{gf}) @>{e^*}>> \f(\St_f) @>{i_2^*}>> \f(\St_{hf})
@>{\Phi(\St_{hf})}>> \fr(\St_{hf}) \\ 
@AAA  @AAA  @AAA  @AAA  @AAA \\
\f(A) @>{i_g^*\circ i_S^*}>> \f(S_g) @>{e^*}>> \f(\St)
@>{i_2^*}>> \f(\St_h) @>{\Phi(\St_h)}>> \fr(\St_h) \\ 
@V{i_R^*}VV  @V{\e^*}VV  @V{\et^*}VV  @V{\et_h^*}VV  @VV{\et_h^*}V \\
\f(R) @= \f(R) @= \f(R) @= \f(R) @= \fr(R)
\endCD
$$
Then we put $S=\St_h$ to be the desired $R$-algebra
and the composition $\phi=\Phi(\St_{hf})\circ i_2^* \circ e^* \circ i_g^*\circ i_S^*$
to be  the desired map $\f(A_f)\ra \fr(S_f)$.
\qed

In the subsequent discussion we will use the following properties (see 1.2, \cite{Za})
of the constructed $R$-algebra $S$:

\item{S1.}
$S$ is finite over the polynomial ring $R[t]$ and
the quotient $S/(f)$ is finite over $R$ (we identify $f$ with $f\otimes 1$);

\item{S2.}
There is the augmentation map $\e: S\ra R$ for the $R$-algebra $S$ defined as
$\e(a\otimes r)=ar$;

\item{S3.}
$S$ is essentially smooth over the field $k$ and
$S/\m S$ is smooth over  the residue field $R/\m $
at the maximal ideal $\e^{-1}(\m)/\m S$,
where $\m$ denotes the maximal
ideal of the local ring $R$.

In particular, there is the  commutative diagram:
$$
\CD
A @>{i_S}>> S \\
@V{i_R}VV @VV{\e}V \\
R @= R
\endCD
$$
where $\e : S \ra R$ is the augmentation from S2 for the inclusion $i: R \ra S$.
And for the regular function $f\in S$ we have $f \notin \ker\e$ and $f \in \e^{-1}(\m)$.

\demo{Remark} From this point on 
we may assume that $S$ is a domain that satisfy properties S1-3.

Indeed, since $S$ is smooth over $k$ it is the product of smooth domains 
(irreducible components), i.e.,
$S=\prod_i S_i$.
Consider the domain $S_j$ such that $\e(S_j)\neq 0$.
Observe that for other domains $S_i$, $i\neq j$, we have $\e(S_i)=0$.
Replace the algebra $S$ by the domain $S_j$ and the map $i_S$ by the composition
$\pr_j\circ i_S$, where $\pr_j:S\ra S_j$ is the canonical projection. 
Clearly properties  S1-3 will still hold.
\qed
\enddemo

\subhead 1.2 \endsubhead
Now let us  produce the map $\Psi:\fr(S_f)\ra \fr(R)$.

Consider the $R$-algebra $S$ constructed above.
Since S is a domain and it satisfy properties S1-3 we are under the hypotheses of 
Geometric Presentation Lemma (sect.4).
Applying this Lemma we get a regular function  $t'\in \ker \e$,
where $\e$ is the augmentation map from S2,  such that:

\item{G1.} 
$S$ is finite over $R[t']$;

\item{G2.} 
There is an ideal $J$ coprime with $\ker\e$ and $\ker\e \cap J=(t')$;

\item{G3.} 
$(f)$ is coprime with $J$ and with $(t'-1)$.

\item{G4.}
Algebras $S/(t')$ and $S/(t'-1)$ are etale over $R$.

We construct the homomorphism $\Psi: \fr(S_f) \ra \fr(R)$ according to the following definition:

\definition{Definition} 
Let we have an $R$-algebra $S$, an augmentation map $\e: S\ra R$,  
regular functions $f \in S$ and $t'\in \ker \e$ that satisfy properties G1-3.
Under these conditions we define the homomorphism
$\Psi: \fr(S_f) \ra \fr(R)$ by putting
$\Psi(\beta)=\tr_1({\can_1'}^*(\beta)) - \tr_J({\can_J'}^*(\beta))$ for every $\beta\in\fr(S_f)$,
where $\can_1':S_f\ra S/(t'-1)$ and $\can_J':S_f \ra S/J$ denote the 
canonical maps;
$\tr_1=\tr^{S/(t'-1)}_R$ and  $\tr_J=\tr^{S/J}_R$ denote the transfer maps.


Observe that the canonical maps  $\can_1'$
and $\can_J'$ exist because of property G3 saying that the 
function $f$ is invertible in $S/(t'-1)$ and $S/J$.
The transfer maps $\tr_1$ and $\tr_J$ exist by property {\bf T} since 
the $R$-algebras
$S/(t'-1)$ and $S/J$ are finitely generated and projective as the $R$-modules. 
Indeed, since $S$ and $R[t']$ are both regular 
property G1 and Grothendieck's theorem (Corollary 18.17, \cite{Ei}) 
show that $S$ is finitely generated and projective as
the $R[t']$-module.
Extending the scalars via $R[t'] \xlra{t'\mapsto 0} R$ and 
$R[t'] \xlra{t'\mapsto 1} R$ 
we get that $S/(t')$ and  $S/(t'-1)$ are finitely generated projective as  the 
$R$-modules.
By property G2 we get the canonical decomposition 
$S/(t')\cong S/\ker\e \times S/J\cong R\times S/J$.
Thus $S/J$ is finitely generated projective as the $R$-module as well. 
\qed
\enddefinition

\demo{Remark}
Let us show that the following diagram commutes 
(it means that the homomorphism $\Psi$ can be considered
as a lifting of the augmentation map $\e^*$):

$$
\CD
\fr(S) @>>> \fr(S_f) \\
@V{\e^*}VV @VV{\Psi}V \\
\fr(R) @= \fr(R)
\endCD
$$

To do this we follow the proof of Specialization Lemma, sect.1.1, [Za]:

Let an element $\a_f \in \fr(S_f)$ comes from an element $\a \in \fr(S)$
via the map $\fr(S)\ra \fr(S_f)$ induced by the canonical inclusion.

By  property G3 there are two canonical  compositions:
$$
\can_J: S\ra S_f \xra{\can_J'} S/J\quad\text{and}\quad \can_1: S\ra S_f\xra{\can_1'} S/(t'-1).
$$
Thus we have
$$
\Psi(\a_f)=\tr_1(\can_1^*(\a)) - \tr_J(\can_J^*(\a)).
$$

By  property {\bf T}.(b1) we have $\tr_1(\can_1^*(\a))=\tr_0(\can_0^*(\a))$,
where $\tr_0=\tr^{S/(t')}_R$ denotes the transfer map 
and $S \xlra{\can_0} S/(t')$ is the canonical map.

By  property {\bf T}.(a) applied to the decomposition $S/(t')\cong S/\ker\e \times S/J$
we get $\tr_0(\can_0^*(\a))=\e^*(\a)+\tr_J(\can_J^*(\a))$.
Thus we have
$$
\Psi(\a_f)=\tr_0(\can_0^*(\a))-\tr_J(\can_J^*(\a))=\e^*(\a). \qed
$$
\enddemo

\subhead 1.3 \endsubhead 
To construct the $K$-algebra $C$ and the homomorphism $\Psi_K$
consider the scalar extension diagram
induced by the canonical inclusion $R\hra{i_K} K$:
$$
\CD
S @>{i_C}>> C=S\otimes_R K \\
@A{i}AA @AA{i}A \\
R @>>{i_K}> K
\endCD
$$
It is easy to see that the $K$-algebra $C$ and the regular function $f\in C$
satisfy properties  S1-3.
Indeed,  since $K[t\otimes 1]= R[t]\otimes_RK$ and 
$S\otimes_R K/(f\otimes 1)=S/(f)\otimes_R K$
the property  S1 holds for the corresponding functions 
$t=t\otimes 1$ and $f=f\otimes 1$ in $S\otimes_R K$.
Put the augmentation $\e_K: S\otimes_R K \ra K$ to be the multiplication 
$\e_K(s\otimes k)=\e(s)\cdot k$.
The property  S3 follows immediately since it is compatible with the scalar extension.

Consider all corresponding data got by the scalar extension:
the $K$-algebra $C$, the augmentation $\e_K$, the functions
$t'=t'\otimes 1$, $f=f\otimes 1 \in C$ and the ideals $\ker\e_K=\ker\e \otimes 1$, 
$J=J\otimes 1 \subset C$. Clearly it satisfy properties  G1-4.
Let $\Psi_K: \fr({C}_f) \ra \fr(K)$ be the homomorphism defined for these data
according to 1.2.

\proclaim{Lemma} The following diagram commutes:
$$
\CD
\fr(S_f) @>{i_C^*}>> \fr({C}_f) \\
@V{\Psi}VV @VV{\Psi_K}V \\
\fr(R) @>>{i_K^*}> \fr(K)
\endCD
$$
\endproclaim

\demo{Proof} It follows immediately from the definition of homomorphism $\Psi$ 
and from
two commutative diagrams, where the upper squares are got by applying the 
functor $\fr$ to the
corresponding scalar extension diagrams, the lower squares coincide
with those from the property {\bf T}.(b2) :
$$
\CD
\fr(S_f) @>{i_C^*}>> \fr({C}_f) \\
@V{{\can_1'}^*}VV @VV{{\can_1'}^*}V \\
\fr(S/(t'-1)) @>>> \fr(C/(t'-1)) \\
@V{\tr_1}VV @VV{\tr_1}V \\
\fr(R) @>>{i_K^*}> \fr(K)
\endCD
\qquad
\qquad
\CD
\fr(S_f) @>{i_C^*}>> \fr({C}_f) \\
@V{{\can_J'}^*}VV @VV{{\can_J'}^*}V \\
\fr(S/J) @>>> \fr(C/J) \\
@V{\tr_J}VV @VV{\tr_J}V \\
\fr(R) @>>{i_K^*}> \fr(K)
\endCD \qed
$$
\enddemo


Finally taking together the map $\phi: \f(A_f)\ra \fr(S_f)$ constructed
in  1.1 and the square diagram from the lemma above we get the desired diagram (i).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subhead 1.4 \endsubhead
To get the diagram (ii) in the constant case (see 1.1)
we apply the functor $\f$ to the following commutative diagram:
$$
\CD
A_f @>{i_C\circ i_S}>> C_f \\
@V{i_K}VV @VV{\e_K}V \\
K @= K
\endCD
$$
where $C=S\otimes_R K=A\otimes_P K$,
$i_C\circ i_S:a\mapsto a\otimes 1$
and  $\e_K: a\otimes k\mapsto ak$.

In the general case consider the $3\times 5$ diagram for the 
triple $(A, R, f)$ mentioned in 1.1.
By extending the scalars via the canonical inclusion $R\hra{i_K} K$
and denoting $T=S\otimes_R K$
we get the following commutative diagram:
$$
\CD
\f(A_f) @>{i_g^*\circ i_T^*}>> \f(T_{gf}) @>{e^*}>> \f(\Tt_f) @>{i_2^*}>> \f(\Tt_{hf})
@>{\Phi(\Tt_{hf})}>> \fr(\Tt_{hf}) \\ 
@AAA  @AAA  @AAA  @AAA  @AAA \\
\f(A) @>{i_g^*\circ i_T^*}>> \f(T_g) @>{e^*}>> \f(\Tt)
@>{i_2^*}>> \f(\Tt_h) @>{\Phi(\Tt_h)}>> \fr(\Tt_h) \\ 
@V{i_K^*}VV  @V{\e^*}VV  @V{\et^*}VV  @V{\et_h^*}VV  @VV{\et_h^*}V \\
\f(K) @= \f(K) @= \f(K) @= \f(K) @= \fr(K)
\endCD
$$

One has the obvious homomorphisms from the first row of this diagram to the
last row given by $i_K$ and the extended augmentations. 
So there is the $2\times 5$ commutative diagram:
$$
\CD
\f(A_f) @>{i_g^*\circ i_T^*}>> \f(T_{gf}) @>{e^*}>> \f(\Tt_f) @>{i_2^*}>> \f(\Tt_{hf})
@>{\Phi(\Tt_{hf})}>> \fr(\Tt_{hf}) \\ 
@V{i_K^*}VV  @V{\e^*}VV  @V{\et^*}VV  @V{\et_h^*}VV  @VV{\et_h^*}V \\
\f(K) @= \f(K) @= \f(K) @= \f(K) @= \fr(K)
\endCD
$$
Indeed, the middle squares commutes since the augmentations are compatible with each other.
The first square commutes by definition of $T$.
And the last square commutes by property {\bf E}.(c).

In our notation the $K$-algebra $C$  coincides with $\Tt_h$,
the composition $i_C^*\circ \phi$ coincides with the upper row 
$\Phi(\Tt_{hf})\circ i_2^*\circ e^*\circ i_g^* \circ i_T^*$
and the augmentation $\e_K$  coincides with $\et_h$. So the outer contour
of this diagram gives us the desired diagram (ii).


\subhead 2 \endsubhead
Let  $\a \in \f_{nr}(K)$ be the given element.
Our goal is to check the assertion 2. 

More precisely we have to construct 
a regular function $f\in A$ and an element 
$\a_f\in \f(A_f)$ with the property $i_K^*(\a_f)=\a$ 
such that relation (iii) holds.
To do this we use the following lemma:

\proclaim{2.1 Lemma} There exist $n$ regular functions $f_i\in A$, $i=1\ldots n$,
and $n$ elements $\a_{f_i}\in \f(A_{f_i})$ such that

\item{\rm (a)}
The elements $\a_{f_i}$
have the same image in $\f(K)$
under the maps induced by the canonical inclusions
$A_{f_i} \ra K$ and this image coincides with the element $\a$.

\item{\rm (b)}
The closed subset
$\spec A \setminus (\spec A_{f_1} \cup \ldots \cup \spec A_{f_n})$
is of codimension at least 2 in $\spec A$ \ 
(a family of functions satisfying this condition
will be called coprime in $A$ in codimension 1).

\item{\rm (c)}
The elements $\a_{f_i}$
have the same image in $\f(A_{f_1\ldots f_n})$
under the maps induced by the canonical inclusions
$A_{f_i} \ra A_{f_1\ldots f_n}$.
\endproclaim

\demo{Proof}
First we prove the local version of the lemma got by writing
$R$ instead of $A$ everywhere in the conditions.

By property {\bf C} of the functor $\f$ we have the canonical isomorphism
$\varinjlim_{f \in \m} \f(R_f) \xra{\cong} \f(K)$,
where $\m$ is the maximal ideal of the local ring $R$.
So there exists a regular function $g\in \m$ such that
the element $\a$ comes from an element $\a_g \in \f(R_g)$.

Let $(g)=\p_1^*\p_2^*\ldots \p_m^*$ be the factorization
of the principal ideal $(g)$
into the product of principle prime ideals of height 1
(here '$*$' means any natural number).
It is unique and exists because $R$ is a local regular ring and, thus, factorial \cite{AK}.

By property {\bf C} we have $m$ canonical isomorphisms
$\varinjlim_{f\in \m\setminus \p_i} \f(R_f) \xra{\cong} \f(R_{\p_i})$, $i=1\ldots m$.
So for every $i$ there exists
such a regular function $g_i \in \m\setminus\p_i$ that
the element $\a \in \f_{nr}(K) \subset \f(R_{\p_i})$
comes from an element $\a_{g_i} \in \f(R_{g_i})$.

By the construction the elements $\a_g, \a_{g_1}, \ldots ,\a_{g_m}$
have the same image in $\f(K)$ and this image
coincides with $\a$.

Since $(g)=\p_1^*\p_2^*\ldots\p_m^*$ and $g_i\notin \p_i$, $i=1\ldots m$,
there is no such a prime ideal $\p \in \spec R$ of height 1 that
$g, g_1, \ldots ,g_m \in \p$.
This implies that the functions $g, g_1, \ldots g_m$
are coprime in $R$ in codimension 1.

\proclaim{Sublemma}
Let $f$, $g\in \m$ be two regular functions.
Let $\a_f\in \f(R_f)$ and $\a_g \in \f(R_g)$ be two elements that have
the same image in $\f(K)$.

Then there exists such a regular function $h\in \m$  coprime
with $f$ and $g$ in $R$ in codimension 1 that
the elements $\a_f$ and $\a_g$ have the same image in $\f(R_{fgh})$.
\endproclaim

\demo{Proof of Sublemma}
By property {\bf C} for the fixed regular function $fg\in \m$
we have the canonical isomorphism
$\varinjlim_{h \in \m} \f(R_{fgh}) \xra{\cong} \f(K)$.

Since the element $\a_f-\a_g$ has the trivial image in $\f(K)$
there exists such a regular function $h'\in \m$ that
the element $\a_f-\a_g$ has the trivial image in $\f(R_{fgh'})$.

Consider the factorizations of the ideals $(fg)$ and $(h')$ into the
product of principle prime ideals of height 1.
Let $(d)$ be the greatest common divisor of $(fg)$ and $(h')$ in these factorizations.
Then we have $h'=hd$ for some regular function $h$.
Clearly $h$ is coprime with $fg$ in $R$ in codimension 1 and
$R_{fgh}=R_{fgh'}$.
\qed
\enddemo

Apply now Sublemma to the each pair of regular functions
$(g,g_i)$, $i=1\ldots m$.
We get $m$ regular functions $h_i$.

Put $f_1=g$, $f_2=g_1h_1$, $f_3=g_2h_2$, $\ldots$ , $f_{m+1}=g_mh_m$ and $n=m+1$.
Let $\a_{f_1},\a_{f_2},\ldots ,\a_{f_n}$
denote the corresponding images of the elements
$\a_g, \a_{g_1}, \ldots, \a_{g_m}$.
We claim that $f_i$, $i=1\ldots n$, are required functions and
$\a_{f_i}$ are required elements.

Indeed, (a) holds for the elements $\a_{f_i}$ by definition.
Since each function $h_i$ is coprime with $g$ in $R$ in codimension 1
the functions $g, g_1h_1, \ldots, g_mh_m$ are coprime in codimension 1 as well
and we get (b).
Item (c) holds because the elements $\a_g$ and $\a_{g_ih_i}$
have the same image in $\f(R_{gg_ih_i})$.
Thus we have proved the local version of the lemma.

Let us lift the functions $f_i$ and the elements $\a_{f_i}$
by the following way:

Since $\varinjlim_{f\notin \q}A_f =R$
there exists such a regular function $g\notin \q$ that
the functions $f_i\in R$ are regular on the localization $A_g$.
Thus replacing $A_g$ by $A$ we may assume $f_i\in A$.

The canonical isomorphism
$\varinjlim_{f\notin \q}\f(A_{ff_i})\xra{\cong}\f(R_{f_i})$
implies that for every $i$ there exists such a regular
function $h_i\notin \q$ that the element
$\a_{f_i}\in \f(R_{f_i})$ comes from $\f(A_{h_if_i})$.
Replacing $A_{h_1h_2\ldots h_n}$ by $A$ we may assume
$\a_{f_i}\in \f(A_{f_i})$.

So we get $n$ regular functions $f_i\in A$ and $n$ elements
$\a_{f_i} \in \f(A_{f_i})$ that still satisfy conditions (a) and (c).

To see (b) let the closed subset
$\spec A \setminus (\spec A_{f_1}\cup\ldots\cup\spec A_{f_n})$
contains a finite number of irreducible components
$\p_j\in \spec A$ of codimension 1.
Since (b) holds for the local version
of the lemma we may assume that $\p_j \subsetneq \q$ for all $j$.
Take such a regular function $h\notin \q$ that $h\in \p_j$ for all $j$.
Then replacing $A_h$ by $A$ we get the required.

And we finish the proof of Lemma.
\qed
\enddemo

Applying this lemma we get the set of
$n$ regular functions $f_1, \ldots, f_n \in A$ and
$n$ elements $\a_{f_i}\in \f(A_{f_i})$, $i=1\ldots n$, that satisfy (a), (b) and (c).

Put $f=f_1f_2\ldots f_n$.
According to (c) let $\a_f$ denote the image in $\f(A_f)$ of the elements $\a_{f_i}$ 
under the map induced by the canonical inclusions $A_{f_i} \ra A_f$.
By (a) we have $i_K^*(\a_f)=\a$.
We may assume that $f, f_1, f_2, \ldots , f_n \in \q$ otherwise the Purity Theorem
will follows immediately from the arguments concerning assertions 1 and 2 at the end of page 7.

We claim that the regular function $f\in \q$ and the element $\a_f\in \f(A_f)$ satisfy relation (iii).
In other words $f$ is the desired function and $\a_f$ is the desired element.

\subhead 2.2 \endsubhead 
Consider the diagrams (i) and (ii)  constructed (see item 1) for the regular function $f$.
Let us rewrite relation (iii)
in terms of the homomorphism $\Phi_{C_f}: \I(C_f) \ra Hom(\fr(C_f),\fr(K))$ defined in 1.3, sect.3.

Consider the data (see 1.3) used to construct the homomorphism $\Psi_K: \fr(C_f)\ra\fr(K)$:
the $K$-algebra $C$, the augmentation map $\e_K$, the regular functions
$f$ and $t'$, the ideals $\ker\e_K$ and $J$. Recall that it satisfy properties G1-4.

Since $f\notin \ker\e_K$ (see 1.1) and $C/\ker\e_K=K$ the ideal
$(f)$ is coprime with $I=\ker\e_K$. 
Combining this fact with properties G2 and G3 
we get that $(f)$ is coprime with ideals $I$, $J$, $(t')=IJ$ and $(t'-1)$.
By property G4 and G1 the algebras $C/I$, $C/J$, $C/(t')$ and $C/(t'-1)$
are etale over $R$.

According to Lemma 1.5, sect.3, the homomorphism $\Psi_K$
can be rewritten as follows:
$$
\Psi_K=\tr_1\circ {\can_1'}^* - \tr_J \circ {\can_J'}^*=\Phi_{C_f}((t'-1))-\Phi_{C_f}(J)=
$$
$$
=\Phi_{C_f}((t'-1))-\Phi_{C_f}((t'))+\Phi_{C_f}(I)=\Phi_{C_f}((\tfrac{t'-1}{t'})\cdot I),
$$
and for the augmentation map $\e_K^*:\fr(C_f)\ra\fr(K)$ we have
$\e_K^*=\Phi_{C_f}(I)$.
Thus in terms of the homomorphism $\Phi_{C_f}$
relation (iii) looks as follows:
$$
\Phi_{C_f}((\tfrac{t'-1}{t'})\cdot I)(i_C^*\circ \phi(\a_f))=\Phi_{C_f}(I)(i_C^*\circ \phi(\a_f)).
$$

Let $\C=\spec C$ denote the affine smooth curve over the field $K$ and
$\C^0=\spec C_f$ be its principal open subset.
Let $\Delta$ be the $K$-rational point corresponding to the augmentation $\e_K$
and $(\d)$ be the divisor of the rational function $\d=\tfrac{t'-1}{t'}$ on $\C^0$.
Then using the geometric notation for the homomorphism $\Phi$ 
we may rewrite relation (iii) as
$$
\Phi_{\C^0}(\Delta + (\d))(\a^0)=\Phi_{\C^0}(\Delta)(\a^0), \quad \a^0=i_C^*\circ \phi(\a_f).
\tag{$*$}
$$
So to prove assertion 2 we have to show that relation ($*$) holds.
To do this we use Cor.2.2, sect.3.


\subhead 2.3 \endsubhead 
Let us produce the data 
that satisfy the hypotheses of Theorem 2.1 of sect.3. They are given at the end of this section.

Recall that we have $n$ regular functions $f_1, f_2, \ldots ,f_n \in \q\subset A$ and
$n$ elements $\a_{f_i}\in \f(A_{f_i})$, $i=1\ldots n$, 
that satisfy conditions (a), (b) and (c) of Lemma 2.1.

For every $i=1\ldots n$ let $f_i$ denote the image in $C$ 
of the function $f_i$ under the composition  $i_2\circ e\circ i_g\circ i_S: A\ra C$
(see the upper row of the $3\times 5$ diagram from 1.1) or 
under the composition $i_C\circ i_S: A \ra C$ 
in the constant case.
Put $\C_i=\spec C_{f_i}$
and let $Z_i=\C\setminus \C_i$ denote the closed subset corresponding
to the principal ideal $(f_i)$ in $C$.

By condition (b) of Lemma 2.1
the closed subset $\spec A \setminus \cup_{i=1}^n \spec A_{f_i}$ 
is of codimension at least 2 in $\spec A$. 
Since the composition $A\ra C$ above is flat 
the closed subset $\C \setminus \cup_{i=1}^n \C_i $ 
is of codimension at least 2 in $\C$ as well. Thus it is empty. In other words we have
$\cap_{i=1}^n Z_i =\emptyset$.

Since $f=f_1f_2\ldots f_n$ we have $\C^0=\cap_{i=1}^n \C_i=\C\setminus \cup_{i=1}^n Z_i$.

For every $i=1\ldots n$ let $\a_i$ denote the image in $\fr(\C_i)$ 
of the  element $\a_{f_i}\in \f(A_{f_i})$ under
the map $\f(A_{f_i}) \xlra{i_C^*\circ \phi} \fr(C_{f_i})$ 
induced by the composition above.
Since the following diagram commutes for every $i$:
$$
\CD
\f(A_{f_i}) @>{i_C^*\circ \phi}>> \fr(C_{f_i}) \\
@VVV @VVV \\
\f(A_f) @>>{i_C^*\circ \phi}> \fr(C_f)
\endCD
$$
we get
$\a_i|_{\C^0}=i_C^*\circ \phi (\a_{f_i})|_{C_f}=
i_C^*\circ \phi (\a_{f_i}|_{A_f})=i_C^*\circ \phi (\a_f)=\a^0$.
\qed

 
Consider now the regular function $t'\in \ker\e_K\subset C$ arisen in 2.2.

Let $\hat{\C}=\C\amalg \C_\infty$ be the smooth projective model of the affine curve $\C$.
Let $\AK=\spec K[t']$ be the affine line and 
$\PK=\proj K[T_0,T_1]$ be the projective line with $T_0/T_1=t'$.
In other words we have $\PK=\AK\amalg \{\infty\}$, where $\infty=(1,0)$.

By property S1 the morphism $\C \xra{t'} \AK$ is finite. 
So there is a finite morphism $\hat{\C}\xra{\hat{t'}} \PK$ such that $\C_\infty=\hat{t'}^{-1}(\infty)$ and
the following diagram commutes, where the horizontal morphisms are the open immersions:
$$
\CD
\C @>>> \hat{\C} \\
@V{t'}VV @VV{\hat{t'}}V \\
\AK @>>> \PK
\endCD
$$

Consider the rational function $\hat{\d}=\tfrac{T_0-T_1}{T_0}$ on $\hat{\C}$.

Clearly $\hat{\d}$ is regular in a neighbourhood of $\C_\infty$ and $\hat{\d}|_{\C_\infty}\equiv \hat{\d}|_{(1:0)}=1$.
Obviously we have $\hat{\d}|_{\C}=\d$ and $(\d)=(\hat{\d})$.
As it was mentioned in 2.2 the element $f$ is coprime with $(t')$ and $(t'-1)$. 
Therefore we have $\supp\; (\d) \subset \C^0$.
Since the algebras $C/(t'-1)$ and $C/(t')$ are etale over $R$
the divisor $(\d)$ is unramified (see Definition 1.4, sect.3).
\qed

\ 

Summarizing the discussion above we have the following data:
\item{$\bullet$} the functor $\fr$ restricted to the category of $K$-algebras and
the smooth affine curve $\C$ over $K$;

\item{$\bullet$} the $n$ closed subsets $Z_i\subset \C$, $i=1\ldots n$, 
and $n$ elements $\a_i\in \fr(\C\setminus Z_i)$
such that $\bigcap_{i=1}^n Z_i =\emptyset$,  $\C\setminus \bigcup_{i=1}^n Z_i =\C^0$
and $\a_i|_{\C^0}=\a^0$;

\item{$\bullet$} the rational function $\bar{\d}$ 
on the smooth projective model $\hat{\C}=\C\amalg \C_\infty$ 
of the affine curve $\C$
such that
$\hat{\d}|_{\C_{\infty}}\equiv 1$ 
(the function $\hat{\d}$ is regular in a 
neighbourhood of $\C_{\infty}$), $(\hat{\d})=(\d)$ and 
the divisor $(\d)$ is unramified with $\supp\,(\d) \subset \C^0$.

Observe that we are under the hypotheses of Theorem 2.1, sect.3 now.
So applying Corollary 2.2, sect.3, we get relation ($*$). 
And we finish the proof of assertion 2 and the proof of the Purity Theorem. 
\qed



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\head \S 3. The Divisor Theorem  \endhead

\subhead 1.1 \endsubhead
Let $K$ be an infinite field.
Let $\g$ be a covariant functor from the category of $K$-algebras
to the category of abelian groups that satisfies property {\bf T}.
We also use the geometric notation for the functor $\g$
as a contravariant functor from the category of affine $K$-schemes to the 
category of abelian groups.

\subhead 1.2 \endsubhead
Let $\C$ be a smooth affine curve over the field $K$.
There is an isomorphism between the group $Div(\C)$ of divisors
of the curve $\C$ and the group $\I(C)$ of fractional ideals of the Dedekind 
domain $C=K[\C]$.
Let $I_D$ denote the fractional ideal corresponding to divisor $D\in Div(\C)$.
Let $I_D=\p_1^{n_1}\p_2^{n_2}\ldots\p_m^{n_m}$, $n_i\in \Z$, be the 
factorization of $I_D$
into the product of prime ideals $\p_i \in \spec C$.


\definition{1.3. Definition}
We define a homomorphism 
$$
\Phi_C: \I(C) \lra Hom(\g(C),\g(K))
$$
or using the geometric notation
$$
\Phi_{\C}: Div(\C) \lra Hom(\g(\C),\g(\spec K))
$$
by the following way:

For a divisor $D=n_1x_1+n_2x_2+\ldots n_mx_m$ and the corresponding
fractional ideal $I_D=\p_1^{n_1}\p_2^{n_2}\ldots \p_m^{n_m}$,
we put 
$\Phi_{\C}(D)=\Phi_C(I_D)=n_1\Psi_{\p_1}+n_2\Psi_{\p_2}+\ldots + 
n_m\Psi_{\p_m}$, where
$$
\Psi_{\p_i}: \g(C) \xlra{\can_i^*} \g(C/\p_i) \xlra{\tr} \g(K), \quad i=1\ldots 
m
$$ 
is the composition of the morphism induced by the canonical map $C \xra{\can} 
C/\p_i$
and the transfer map. The latter exists  since
the residue field $C/\p_i$ is finite over $K$. \qed
\enddefinition

\definition{1.4. Definition}
A divisor $D=n_1x_1+\ldots +n_mx_m$ of a smooth curve $\C$ is called unramified 
if
$n_i=\pm 1$ for all $i=1\ldots m$. Correspondingly a fractional ideal $I$ is 
called
unramified if $I=\p_1^{\pm 1}\p_2^{\pm 1}\ldots\p_m^{\pm 1}$.

It is easy to see that an effective divisor $D$ is unramified iff
the corresponding algebra $C/I_D$ is etale over $K$.
\qed
\enddefinition

\proclaim{1.5. Lemma} Let $D=x_1+\ldots +x_m$ be an unramified effective divisor 
of the curve $\C$. 
Then the homomorphism
$\Phi_{\C}(D)=\Phi_C(I_D)$ coincides with the composition 
$$
\Psi_{I_D}: \g(C) \xlra{\can_{I_D}^*} \g(C/I_D) \xlra{\tr} \g(K).
$$
\endproclaim

\demo{Proof} It is enough to check the case when $D=x_1+x_2$.

By Chinees Remainder Theorem we have $C/I_D=C/\p_1\times C/\p_2$.
By additivity of the transfer {\bf T}.(a) we get:
$$
\tr^{C/I_D}_K(\can_{I_D}^*(x))=\tr^{C/\p_1}_K(\can_1^*(x))+\tr^{C/\p_2}_K(\can_2
^*(x)),
$$
for all elements $x\in \g(C)$. Thus 
$\Psi_{I_D}=\Psi_{\p_1}+\Psi_{\p_2}=\Phi_C(I_D)$.
\qed
\enddemo

\subhead 1.6 \endsubhead
Let $\hat{\C}$ be the smooth projective model of the affine curve $\C$, i.e.,
$\hat{\C}=\C \amalg \{\infty\}$ is the smooth projective curve over $K$, 
where $\{\infty\}$ denotes a closed subset of $\hat{\C}$ finite over $K$.
Informally speaking the set $\{\infty \}$ consists from infinity points of the 
affine curve $\C$.
Observe that $\C$ is the open affine subset of $\hat{\C}$.


\subhead 2 \endsubhead
The goal of this section is to prove the following result:

\proclaim{2.1. Theorem {\rm (The Divisor Theorem)}} Let there are given a 
functor $\g$ and a smooth affine curve $\C$.
Thus we get the homomorphism $\Phi$ and the smooth projective model
$\hat{\C}=\C \amalg \{\infty\}$.

Let there are given $n$ closed subsets  $Z_1, Z_2, \ldots, Z_n$ of the affine 
curve $\C$
with $\bigcap_{i=1}^n Z_i =\emptyset$.
Denote $Z=\bigcup_{i=1}^n Z_i$, $\C^0=\C\setminus Z$ and
$\C_i=\C\setminus Z_i$, $i=1\ldots n$.

Let there are given $n$ elements $\a_i\in \g(\C_i)$, $i=1\ldots n$, and
element $\a^0 \in \g(\C^0)$
such that $\a_i|_{\C^0}=\a^0$ for all $i=1\ldots n$.

Let $\d$ be a rational function on the projective curve $\hat{\C}$ such that 
$\d|_{\{\infty\}}\equiv 1$ (we assume that the function $\d$ is regular in a 
neighbourhood of infinity) and 
the divisor $(\d)$ is unramified with $\supp\,(\d) \subset\C^0$.

Then $\Phi_{\C^0}((\d))(\a^0)=0$.
\endproclaim

\proclaim{2.2. Corollary}
Let we are in conditions of the theorem above and there are given a rational 
point $\D$ 
on the affine curve $\C^0$. 
Then $\Phi_{\C^0}(\D+(\d))(\a^0)=\Phi_{\C^0}(\D)(\a^0)$.
\endproclaim

\subhead 2.3 \endsubhead
First of all we introduce the equivalence relation on the set of divisors
of the projective curve $\hat{\C}$.
Let $\C^0$ be an open subset
of the affine curve $\C$.
Let 
$$
\Phi_{\C^0}: Div(\C^0) \lra Hom(\g(\C^0),\g(\spec K))
$$
be the corresponding homomorphism.
Clearly the following diagram commutes
$$
\CD
Div(\C^0) @>{\Phi_{\C_0}}>> Hom(\g(\C^0),\g(\spec K)) \\
@VVV @VVV \\
Div(\C) @>>{\Phi_{\C}}> Hom(\g(\C),\g(\spec K))
\endCD
\tag{$*$}
$$
where vertical arrows are induced by the open immersion $\C^0 \hra{} \C$.

\definition{2.4. Definition}
We say that divisors $D_0$ and $D_1$ of the projective curve $\hat{\C}$ are 
equivalent on $\C^0$
if $\supp D_0$, $\supp D_1 \subset \C^0$ and 
$\Phi_{\C^0}(D_0)=\Phi_{\C^0}(D_1)$.
\qed
\enddefinition


\proclaim{2.5. Lemma} Let $Z$ be a closed subset of $\C$.
Let $D_0$ and $D_1$ be two divisors of $\hat{\C}$
such that $\supp D_0$, $\supp D_1 \subset \C\setminus Z$ and
there exists a rational function $(\d)$ on $\hat{\C}$ with properties:

\item{$\bullet$} $D_0-D_1=(\d)$; 
$\qquad\bullet$ $\d|_{\{\infty\}}\cong 1$, $\d|_Z\cong 1$; 
$\qquad\bullet$ the principal divisor $(\d)$ is unramified.

Then divisors $D_0$ and $D_1$ are equivalent on $\C\setminus Z$.
\endproclaim

\demo{Proof} Since $\Phi_{\C^0}$ is a homomorphism and the diagram $(*)$ 
commutes
it is enough to prove that $\Phi_{\C^0}((\d))=0$ for some
open subset $\C^0$ of the curve $\hat{\C}$ such that $\supp\,(\d) \subset \C^0 
\subset \C\setminus Z$.

The rational function $(\d)$ defines a finite morphism $\d: \hat{\C} \ra \PK$.
Let $\C^0=\hat{\C}\setminus \d^{-1}(\{1\})$ be an open subset of $\hat{\C}$.
Since $\d|_{\{\infty\}}\equiv 1$ and $\d|_Z \equiv 1$ 
we get that $\C^0$ is the open subset in $\C\setminus Z$.
Consider the fibred product diagram
$$
\CD
\hat{\C} @<<< \C^0 @. \supset \qquad (\d)_0 \qquad \; @.  (\d)_1\\
@V{\d}V{\text{finite}}V @V{\d}V{\text{finite}}V @VVV @VVV\\
\PK @<{\text{open}}<{\text{immersion}}< \PK\setminus\{1\} @. \{0\} @. 
\{\infty\}\\
@. @V{\a}V{\cong}V @VVV @VVV\\
 @. \AK @. \{0\} @. \{1\}
\endCD
$$
where $\a$ is an isomorphism sending $\{0\}$ to $\{0\}$ and $\{\infty\}$ to
$\{1\}$. 

Look on the principal unramified divisor $(\d)$. By definition $(\d)=(\d)_0-
(\d)_1$,
where $(\d)_0$ and $(\d)_1$ are corresponding 
preimages of $\{0\}$ and $\{1\}$ under the finite morphism $\a\circ\d: \C^0 \ra 
\AK$.
Obviously we have $\supp\,(\d)$, $\supp\,(\d)_0$, $\supp\,(\d)_1 \subset \C^0$ 
and
$\Phi_{\C^0}((\d))=\Phi_{\C^0}((\d)_0)-\Phi_{\C^0}((\d)_1)$. 

Thus our aim is to prove  $\Phi_{\C^0}((\d)_0)=\Phi_{\C^0}((\d)_1)$ 
for  unramified effective divisors $(\d)_0=\sum_k x_k^0$ and $(\d)_1=\sum_k 
x_k^1$.

Turn now to the algebraic language.
Let $C=K[\C^0]$ be a coordinate ring of the smooth affine curve $\C^0$. 
The fact that $\C^0$ is finite over $\AK$ via the morphism $t=\a\circ\d$ means
that $C$ is finitely generated $K[t]$-module. Moreover, $C$ is projective 
$K[t]$-module
because $C$ is smooth over $K$. 

It is easy to see that divisors $(\d)_0$ and $(\d)_1$ of the curve $\C^0$
correspond to the principal ideals $(t)$ and $(t-1)$ of the ring $C$.
By the previous lemma
we get
$$
\Phi_{\C^0}((\d)_i)=\Phi_C((t-i))=\Psi_{(t-i)}: \g(C) \xlra{\can^*} \g(C/(t-i)) 
\xlra{\tr_i} \g(K), \quad i=0,1.
$$
And the equality $\Phi_{\C^0}((\d)_0)=\Phi_{\C^0}((\d)_1)$ follows from the 
axiom {\bf T}.(b1).
\qed
\enddemo

\demo{Proof of the Theorem} 
Apply the following procedure inductively $n$ times starting with the given 
rational function $\d=\d_0$.

\demo{Procedure} For the rational function $\d_i$, $i=0\ldots n-1$, 
we find a rational function $\d_{i+1}$ such that

\item{$\bullet$} $\d_{i+1}|_{\{\infty\}}\equiv 1$; 
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\, \bullet$  
$\d_{i+1}|_{Z\setminus Z_{i+1}}\equiv 1$ and $\d_{i+1}|_{Z_{i+1}} \equiv \d_i|_{Z_{i+1}}$;
\item{$\bullet$} $\supp\,(\d_{i+1}) \subset \C^0$ and $\supp\,(\d_{i+1}) \cap 
\supp\,(\d_i)=\emptyset$;
$\quad \bullet$  the divisor $(\d_{i+1})$ is unramified.
\enddemo

Then we have 
$$
\Phi_{\C^0}((\d_i))(\a^0)=\Phi_{\C_{i+1}}((\d_i))(\a_{i+1})=
\Phi_{\C_{i+1}}((\d_{i+1}))(\a_{i+1})=\Phi_{\C^0}((\d_{i+1}))(\a^0),
\tag{$**$}
$$
where the left and the right equalities hold
because of the diagram $(*)$ and the middle equality holds because 
of the previous lemma applied to divisors $(\d_i)$ and $(\d_{i+1})$
on $\C_{i+1}$. Indeed, we have unramified divisor $(\d_i)-
(\d_{i+1})=(\d_i/\d_{i+1})$ and 
$(\d_i/\d_{i+1})|_{Z_{i+1}\cup \{\infty\}}\equiv 1$ which means that
the divisor $(\d_i)$ is equivalent to $(\d_{i+1})$ on $\C_{i+1}$.


Thus we get $n+1$ rational functions $\d=\d_0, \d_1, \ldots, \d_n$ and $n$
equalities $(**)$.

Taking all these equalities together we get 
$\Phi_{\C^0}((\d))(\a^0)=\Phi_{\C^0}((\d_n))(\a^0)$,
where restriction $\d_n|_{Z\setminus Z_n \cup Z\setminus Z_{n-1} \cup \ldots 
\cup Z\setminus Z_1}\equiv 1$.
Since $\bigcap_{i=1}^n Z_i=\emptyset$ we have $\d_n|_Z\equiv 1$.

The previous lemma shows that divisors $(\d_n)$ and $(1)=0$ are equivalent on 
$\C^0=\C\setminus Z$.
Thus $\Phi_{\C^0}((\d_n))(\a^0)=\Phi_{\C^0}((1))(\a^0)=0$.
And we have finished.
\qed
\enddemo

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\head \S 4. The Geometric Presentation Lemma \endhead

In this section we prove a stronger version of Lemma 5.2, \cite{PO}:

\proclaim{Lemma}
Let $R$ be a local essentially smooth algebra over an infinite field $k$, 
$\m$ its maximal ideal and $S$ an essentially smooth $k$-algebra (S is a domain), 
which is finite over the polynomial algebra $R[t]$. 
Suppose that $\e: S\ra R$ is an $R$-augmentation and let $I=\ker\e$.
Assume that $S/\m S$ is smooth over  the residue field $R/\m$
at the maximal ideal $\e^{-1}(\m)/\m S$. 
Given regular function $f\in S$ such that $S/(f)$ is finite over $R$
we can find a $t'\in I$ such that

\item{\rm G1.} 
$S$ is finite over $R[t']$;

\item{\rm G2.} 
There is an ideal $J$ coprime with $I$ and $I\cap J=(t')$; 

\item{\rm G3.} {\rm (a)} 
$(f)$ is coprime with $J$; \ \  {\rm (b)} $(f)$ is coprime with $(t'-1)$;

\item{\rm G4.} {\rm (a)}
$S/(t')$ is etale over $R$; \ \  {\rm (b)} $S/(t'-1)$ is etale over $R$.
\endproclaim

\demo{Proof} The proof  bases up on the proof of Lemma 5.2, \cite{PO} 
The only difference is the property G4:
By construction from \cite{PO} we have some freedom in the choice of the element $t'$. 
Showing  the algebra $S/(t')$ is etale over $R$ for almost all elements $t'$, we 
get the required. 
\enddemo

\subhead 1\endsubhead
Replacing $t$ by $t- \e(t)$ we may assume that $t\in I$. We denote by "bar" the 
reduction
modulo $\m$. By assumption made on $S$ the quotient $\bar{S}=S/\m S$ is 
smooth
over the residue field $\bar{R}=R/\m$ at its maximal ideal $\bar{I}=\e^{-
1}(\m)/\m S$.
Choose an $\a\in R$ such that $\bar{\a}$ is a local parameter of the 
localization  $\bar{S}_{\bar{I}}$
of $\bar{S}$ at $\bar{I}$. By Chinees Remainder's theorem we may assume that 
$\bar{\a}$
doesn't vanish at zeros of $\bar{f}$ different from $\bar{I}$. By Bertini's 
theorem we may assume
that the algebra $\bar{S}/(\bar{\a})$ is etale over $\bar{R}$. Without changing 
$\bar{\a}$
we may replace $\a$ by $\a -\e(\a)$ and assume that $\a \in I$. Since $S$ is 
integral over $R[t]$
there exists a relation of integral dependence
$$
\a^n + p_1(t)\a^{n-1} +\ldots + p_n(t) =0.
$$
For any $r\in k^*$ and any $N$ larger than the degree of each $p_i(t)$, putting 
$t'=\a-rt^N$
we see from the equation above that $t$ is integral over $R[t']$. Hence S, which 
is integral
over $R[t]$, is integral over $R[t']$ and property G1 holds. 

\subhead 2 \endsubhead
Clearly $t'\in I$. To insure that $\bar{t}'$ is also a local
parameter of $\bar{S}_{\bar{I}}$ it suffices to take $N \geq 2$. By assumption 
$S$ and $R[t']$
are smooth and since $S$ is finite over $R[t']$, $S$ is finitely generated 
projective as the $R[t']$-module (see Corollary 18.17 of \cite{Ei})
and hence $S/(t')$ is free as the  $R$-module. Since $\bar{t}'$ is a local 
parameter of $\bar{S}_{\bar{I}}$,
$\bar{S}/(\bar{t}')$ is etale over $\bar{R}$ at the augmentation ideal $\bar{I}$ 
and so we can find
a $g\notin I+\m S$ such that $(S/(t'))_g$ is etale over $R$. By Sublemma 5.3 of 
\cite{PO} 
we get property G2.

\subhead 3 \endsubhead
Consider the following fibred product diagram:
$$
\CD
\spec \bar{S}[u]/(\bar{\a}-u\bar{t}^N) @<<<  \spec \bar{S}/(\bar{t}')\\
@V{\pi}VV @VVV \\
{\Bbb A}^1_k  @<<< \spec k
\endCD
$$
which maps $\spec k$ to the rational point $\{r\} \in {\Bbb A}^1_k$.

Since the fibre of $\pi$ at the rational point $\{r=0\} \in {\Bbb A}^1_k$ is 
etale over $\spec k$ (it coincides with $\spec \bar{S}/(\bar{\a})$)
the fibres of $\pi$ at almost all rational points of ${\Bbb A}^1_k$ are etale over $\spec k$.
It means that the algebra $\bar{S}/(\bar{t}')=\bar{S}/(\bar{\a}-r\bar{t}^N)$ is etale over $k$
for almost all $r\in k^*$.

Consider the scalar extension diagram induced by the canonical map $R\ra \bar{R}$
$$
\CD
S/(t') @>>> \bar{S}/(\bar{t}') \\
@AAA  @AAA \\
R @>>> \bar{R}
\endCD
$$
As it was mentioned in 2 the $S/(t')$ is free as the $R$-module. Therefore it is flat over $R$.
If $S/(t')$ is flat over $R$ and $\bar{S}/(\bar{t}')$ is etale over $\bar{R}$
then $S/(t')$ is etale over $R$. Thus the algebra $S/(t')$ is etale over $R$ for 
almost all $r\in k^*$,
i.e., property G4.(a) holds for almost all $r\in k^*$.

To check property G3.(a) we refer to the end of the proof of Lemma 5.2, 
\cite{PO}.
It turns out that G3.(a) holds for almost any choice of $r\in k^*$.

Hence we may choose the element $t'$ (the corresponding $r\in k^*$) such that 
properties G3.(a) and G4.(a) hold.

\subhead 4 \endsubhead
It remains to check G3.(b) and G4.(b). 
Since $S/(t')$ is already etale  over $R$
the algebra $S/(t'-\lambda)$ is etale over $R$ for almost 
all $\lambda \in k^*$.
Since $S/(f)$ is semilocal the element $(t'-\lambda)$ is invertible in $S/(f)$ 
for almost all $\lambda \in k^*$.
Hence we may choose the element $\lambda \in k^*$ such that the algebra 
$S/(t'-\lambda)$ is etale over $R$ and $(t'-\lambda)$ is coprime with $(f)$.

Without perturbing conditions G1, G2, G3.(a) and G4.(a) 
we may replace $t'$ by $\tfrac{1}{\lambda}t'$ and thus satisfy G3.(b) and 
G4.(b).
\qed

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{\it Espaces principaux homog\`{e}nes localement triviaux}
// Inst. Hautes \'{E}tudes Sci. Publ. Math. 1992. N 75. p.97-122.

\item{[Ei]} D.~Eisenbud,
{\it Commutative Algebra} //
GTM 150.
Springer-Verlag 1995. 797 p.

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{\it Algebraic Geometry}
// GTM 52,
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\item{[Me]} A.~Merkurjev,
{\it Norm principle for algebraic groups}
// St.~Petersburg Math. J.  Vol.7, N 2. 1996.

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{\it Etale Cohomology}
// Princeton Mathematical Series 33,
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\item{[PO]} I.~Panin, M.~Ojanguren, 
{\it A Purity Theorem for the Witt group}
// SFB 343, Bielefeld 1997, Preprint 117.

\item{[PO$_1$]} I.~Panin, M.~Ojanguren,
{\it Rationally trivial hermitian spaces are locally trivial} //
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\item{[PS]} I.~Panin, A.~Suslin
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// St.~Pet. Math. J. Vol.9. N 4. 1998.

\item{[Qu]} D.~Quillen
{\it Higher algebraic K-theory I.
Algebraic $K$-theory I}
// Lecture Notes in Math. Vol. 341.
Springer-Verlag, Berlin 1973. p.85-147.

\item{[Vo]} V.~Voevodsky,
{\it Homology of schemes II}
 in Cycles, Transfers and Motivic Homology Theories 
// Princeton University Press 1998, An.Math.Studies, 282p.

\item{[Za]} K.~Zainoulline, 
{\it On Grothendieck Conjecture about principal homogeneous spaces 
for some classical algebraic groups} 
// Algebra i Analiz 2000, Vol.12, N 1, p.150-184 (russian).
English transl. to appear  at St.Petersburg Math. J. 
(the draft english version is available at 
http://www.math.uiuc.edu/K-theory/0332/index.html)



\endRefs


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Content-type: text/plain; charset=US-ASCII
Content-transfer-encoding: 7BIT
Content-description: Text from file 'abstract.txt'

Author: Kirill Zainoulline
Title: A Purity Theorem for Functors with Transfers
Email: kirill@pdmi.ras.ru

1991 Mathematics Subject Classification:
Keywords and Phrases: purity problem, local regular ring,
Azumaya algebra, transfer

Abstract Text (5-10 lines, please try to avoid TeX commands, as this
will be displayed in html):

Let R be a local regular ring got by localizing a smooth k-algebra A and let K be it's
field of fractions. Let F be a covariant functor from the category of A-algebras to
abelian groups that satisfies some additional properties (continuity, existence of well
behaved transfer map). 

In this paper we show that the subgroup im{F(R) --> F(K)} of the group F(K) coincides
with the intersection of the subgroups im{F(Rp) --> F(K)}, where all maps are induced
by the canonical inclusions and Rp runs through localizations of the local ring R at all
prime ideals p of height 1. 

--Message-Boundary-30641--

