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\begin{document}

\title{On the relation of Voevodsky's algebraic cobordism to Quillen's $K$-theory}

\author{I.~Panin\footnote{Universit\"at Bielefeld, SFB 710, Bielefeld, Germany}
\footnote{Steklov Institute of Mathematics at St.~Petersburg, Russia}
\and K.~Pimenov\footnotemark[2]
\and O.~R{\"o}ndigs\footnote{Institut f\"ur Mathematik, Universit\"at Osnabr\"uck, Osnabr\"uck, Germany}}

\date{\today\thanks{The authors thank
the SFB-701 at the Universit\"at Bielefeld, the RTN-Network HPRN-CT-2002-00287, the
RFFI-grant 03-01-00633a, and
INTAS-03-51-3251 for their support.}}

\maketitle

\begin{abstract}
Quillen's algebraic $K$-theory is reconstructed via Voevodsky's
algebraic cobordism. More precisely,
for a ground field $k$
the algebraic cobordism $\Pro^1$-spectrum
$\mathrm{MGL}$ of Voevodsky
is considered as
a commutative ring $\Pro^1$-spectra.
Setting
$\mathrm{MGL}^i = \oplus_{2q-p = i} \ \mathrm{MGL}^{p,q}$
we regard the bigraded theory
$\mathrm{MGL}^{p,q}$
as just a graded theory.
There is a unique ring morphism
$\phi\colon \mathrm{MGL}^0(k) \to \mathbb Z$
which sends the class
$[X]_{\mathrm{MGL}}$
of a smooth projective $k$-variety $X$ to the Euler
characteristic
$\chi(X, \mathcal O_X)$
of the structure sheaf
$\mathcal O_X$.
%the functor
%$(X,U) \mapsto \mathrm{MGL}^{\ast}(X,U) \otimes_{\mathrm{MGL}^{0}(k)} \mathbb Z$
%is an ring cohomology theory on the category
%$\SmOp$ in the sense of
%\cite{PS}. More over
Our main result states that
there is a canonical grade preserving isomorphism of ring
cohomology theories on the category
$\SmOp/k$
$$
\varphi\colon \mathrm{MGL}^{\ast}(X,U) \otimes_{\mathrm{MGL}^{0}(k)} \mathbb Z \to
\mathrm{K}^{TT}_{- *}(X,U)= \mathrm{K}^{\prime}_{- *}(X-U),
$$
in the sense of
\cite{PSorcoh},
where
$K^{TT}_*$
is the Thomason-Trobaugh $K$-theory and
$\mathrm{K}^{\prime}_*$
is Quillen's
$\mathrm{K}^{\prime}$-theory.
In particular, the left hand side is a ring cohomology theory.
Moreover both theories are oriented in the sense of
\cite{PSorcoh}
and
$\varphi$
respects the orientations.
The result is an algebraic version of a theorem due to Conner and Floyd.
That theorem reconstructs complex $K$-theory via complex cobordism
\cite{CF}.
%If
%$char(k)=0$,
%then
%$\phi$ coincides with the composition
%$\mathrm{MGL}^0(k) \xra{r} \Omega \xra{\chi} \mathbb Z$
%where $r$ is the geometric realization map to the complex
%cobordism ring
%$\Omega$
%and
%$\chi$ is a unique ring homomorphism taking classes
%$\Pro^n(\mathbb C)$
%of complex projective spaces to the unite
%$1 \in \mathbb Z$.
\end{abstract}



\section{A motivic version of a theorem by Conner and Floyd}
\label{Introduction}
Our main result relates
Voevodsky's algebraic cobordism theory
$\mathrm{MGL}^{\ast,\ast}$
to Quillen's
$K^{\prime}$-theory.
We refer to
\cite[Appendix]{BGL}
for the basic terminology, notation, constructions, definitions,
results. Let $S$ be a Noetherian separated finite-dimensional scheme $S$.
One may think of $S$ being the spectrum of a field or the integers.
A {\em motivic space
over\/} $S$ is a functor
\[ A\colon \Sm/S^{op} \to \mathbf{sSet} \]
(see \cite[Appendix]{BGL}). The category of motivic spaces over $S$ is denoted
$\mathbf{M}(S)$. This definition of a motivic space is different from the one considered
by Morel and Voevodsky in \cite{MV} -- they consider only those simplicial presheaves
which are sheaves in the Nisnevich topology on $\Sm/S$. With our definition the
Thomason-Trobaugh $K$-theory functor obtained by using big vector bundles is a motivic space
on the nose. It is not a simplicial Nisnevich sheaf. This is why we prefer to work with the
above notion of ``space''.

We write
$\mathrm{H}^{\cm}_\bullet (S)$
for the pointed motivic homotopy category
and
$\mathrm{SH}^\cm(S)$
for
the stable motivic homotopy category over $S$ as constructed in
\cite[A.3.9, A.5.6]{BGL}.
By
\cite[A.3.11 resp.~A.5.6]{BGL}
there are canonical equivalences to
$\mathrm{H}_\bullet(S)$
of
\cite{MV}
resp.
$\mathrm{SH}(S)$
of
\cite{V1}.
Both
$\mathrm{H}^\cm_\bullet(S)$
and
$\mathrm{SH}^\cm(S)$
are equipped with
closed symmetric monoidal structures such that the
$\Pro^1$-suspension spectrum functor
is a strict symmetric monoidal functor
\[ \Sigma^\infty_{\Pro^1} \colon \mathrm{H}^\cm_\bullet(S)\to \mathrm{SH}^\cm(S). \]
Here
$\Pro^1$
is considered as a motivic space pointed by
$\infty \in \Pro^1$.
The symmetric monoidal structure $(\wedge,\mathbb{I}_S = \Sigma^\infty_{\Pro^1}S_+)$
on the homotopy category
$\mathrm{SH}^\cm(S)$ is constructed on the model category level by
employing the category $\MSS(S)$ of symmetric $\Pro^1$-spectra.
It satisfies the properties required by
Theorem 5.6 of Voevodsky congress talk
\cite{V1}. From now on we will usually omit the superscript $(-)^\cm$.

Given a $\Pro^1$-spectrum $E$ one has a cohomology theory on the category
of pointed spaces. Namely, for a pointed space $(A,a)$ set
$E^{p,q}(A,a)=Hom_{\mathrm{H}^{\cm}_\bullet (S)}(\Sigma^\infty_{\Pro^1}(A,a), \Sigma^{p,q}(E))$
and
$E^{\ast,\ast}(A,a)= \oplus_{p,q} E^{p,q}(A,a)$.
A cohomology theory on the category of non-pointed spaces is defined as follows.
For a non-pointed space
$A$ set
$E^{p,q}(A)=E^{p,q}(A_+,+)$
and
$E^{\ast,\ast}(A)= \oplus_{p,q} E^{p,q}(A)$.

Each
$X \in \Sm/S$
defines a motivic space constant in the siplicial direction
taking an $S$-smooth $U$ to
$\mathrm{Mor}_S(U,X)$. This motivic space is non-pointed.
So we regard $S$-smooth varieties as motivic spaces (non-pointed)
and set
$$
E^{p,q}(X)=E^{p,q}(X_+,+).
$$

Given a $\Pro^1$-spectrum $E$ we will reduce the double grading on the cohomology
theory
$E^{\ast,\ast}$
to a grading. Namely, set
$E^m = \oplus_{m = p-2q}E^{p,q}$
and
$E^{\ast}=\oplus_{m}E^m$.
{\it We often will write}
$E^{\ast}(k)$
{\it for}
$E^{\ast}(\mathrm{Spec}(k))$
{\it below in this text}.

A $\Pro^1$-ring spectrum is a monoid
$(E,\mu,e)$
in
$(\mathrm{SH(S)},\wedge, \mathbb{I}_S)$.
A commutative $\Pro^1$-ring
spectrum is a commutative monoid
$(E,\mu,e)$
in
$(\mathrm{SH(S)},\wedge, 1)$.

The cohomology theory $E^{\ast}$ defined by a
$\Pro^1$-ring spectrum is a ring cohomology theory.
The cohomology theory $E^{\ast}$ defined by a commutative
$\Pro^1$-ring spectrum is a ring cohomology theory,
however it is not necessary graded commutative.
The cohomology theory $E^{\ast}$ defined by an oriented commutative
$\Pro^1$-ring spectrum is a graded commutative ring cohomology theory.


Occasionally a $\Pro^1$-ring spectrum $(E,\mu,e)$ might have a model
$(E',\mu',e')$ which is a symmetric $\Pro^1$-ring spectrum, that is, a
symmetric $\Pro^1$-spectrum $E'$ equipped with a strict multiplication $\mu'\colon E'\wedge E'
\to E'$
which is strictly associative and strictly unital for the unit
$e'\colon \Sigma^\infty_{\Pro^1}(S_+) \to E'$. This is the case for the algebraic
cobordism $\Pro^1$-ring spectrum $\MGL$, as described below. Such a model for the
algebraic $K$-theory $\Pro^1$-ring
spectrum $\mathrm{BGL}$ is currently not known to us.

For the rest of the paper let $k$ be a field and $S=\mathrm{Spec}(k)$.
Usually $S$ will be replaced by $k$ in the notation.
%We will write
%$\mathrm{H}^{cm}_\bullet (k)$
%and
%$\mathrm{SH(k)}$
%for the
%$\mathrm{H}^{cm}_\bullet (S)$
%and
%$\mathrm{SH(S)}$
%respectively.
We work in this text with the algebraic cobordism
$\Pro^1$-spectrum
$\mathrm{MGL}$
and the algebraic $K$-theory
$\Pro^1$-spectrum
$\mathrm{BGL}$
as described in
\cite[Defn.~1.2.4]{BGL}
and
\cite[Sect.~2.1]{UniThm}
respectively.
The spectrum
$\mathrm{MGL}$
is a commutative ring
$\Pro^1$-spectrum
by that construction.
%which will be sketched below (see Defn.~\ref{textMGL}).
The spectrum
$\mathrm{BGL}$
is equipped with a structure of a commutative
$\Pro^1$-ring spectrum
as explained in
\cite[Thm.~2.1.1]{BGL}.
Let
$K^{TT}_*$
be Thomason-Trobaugh $K$-theory functor
\cite{TT}.
There is a canonical isomorphism
$$
Ad\colon K^{TT}_{- *} \to \mathrm{BGL}^{*,0}
$$
of ring cohomology theories
on the category $\SmOp/S$
in the sense of
\cite{PSorcoh}.
An invertible Bott element
$\beta \in \mathrm{BGL}^{2,1}(\Spec(k))$
is constructed in
\cite[Section 1.3]{BGL}.
%Now recall a notation.
%For a $\Pro^1$-spectrum $E$ and a pointed motivic space $A$ set
%\[ E^{i}(X)= \bigoplus_{p-2q=i} E^{p,q}(X) = \Hom_{\SH(S)}(A,S^{p,q}\wedge E) \]
%where $S^{p,q}$ is the sphere spectrum having twist $q\in \ZZ$ and simplicial dimension
%$p-q\in \ZZ$.
For every pointed motivic space $A$ the morphism
\begin{eqnarray}
\label{BGLandTT}
\mathrm{BGL}^{*,0}(A) \otimes \mathrm{BGL}^{0}(\Spec(k)) \to
\mathrm{BGL}^{\ast,\ast}(A)
\end{eqnarray}
given by
$a \otimes b \mapsto a \cup b$
is a ring isomorphism by \cite[Sect.~1.3]{BGL}. Furthermore
$\mathrm{BGL}^{0}(\Spec(k))= \ZZ[\beta, \beta^{-1}]$
is the ring of Laurent polynomials on the Bott element
$\beta$. To say the same in a different way,
\begin{eqnarray}
\label{BGLlaurantAndTT}
\mathrm{BGL}^{*,0}(A)[\beta, \beta^{-1}]\cong
\mathrm{BGL}^{\ast,\ast}(A).
\end{eqnarray}
The special case
$A = X/(X\minus Z)$
where $X$ is a smooth $k$-variety and
$Z \subset X$ is a closed subset
implies the following result
\cite[Cor.~1.3.6]{BGL}.

\begin{cor}
\label{LaurantPolinomial}
Let $X$ be a smooth $k$-scheme, $Z$ a closed subset of $X$
and $U = X\minus Z$ its open complement. Then
there are isomorphisms
\begin{eqnarray}
\label{BGLAndTTrings1}
K^{TT}_{- *, Z}(X)[\beta, \beta^{-1}] & \cong &\mathrm{BGL}^{\ast,\ast}(X/U) = \mathrm{BGL}^{\ast}(X/U)\\
\label{BGLAndTTrings2}
K^{TT}_{- *,Z}(X) & \cong &
 \mathrm{BGL}^{\ast,\ast}(X/U)/(\beta + 1)\mathrm{BGL}^{\ast,\ast}(X/U)
\end{eqnarray}
of ring cohomology theories on $\SmOp/k$ in the sense of
\cite{PSorcoh}.
\end{cor}

We refer to
\cite{UniThm}
for a construction of the commutative
ring
$\Pro^1$-spectrum
$\mathrm{MGL}$.
For the purposes of the present preprint we will need to know
only two properties of that spectrum. Those properties are:
Quillen universality and $\mathrm{MGL}$-cellularity
(see Subsection
\ref{AgeneralResult}
below).







\subsection{Oriented commutative ring spectra}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Following Adams and Morel we define an orientation of a commutative
$\Pro^1$-ring spectrum. However we prefer to use Thom classes instead of
Chern classes. Consider the pointed motivic space
$\Pro^{\infty}= \colim_{n\geq 0} \Pro^n$ having base point
$g_1 \colon S = \Pro^0 \hra \Pro^\infty$.

The tautological ``vector bundle''
$\mathcal{T}(1) = \mathcal{O}_{\Pro^\infty}(-1)$ is also known as the
Hopf bundle. It has zero section $z\colon \Pro^\infty \hra \mathcal{T}(1)$.
The fiber over the point
$g_1 \in \Pro^{\infty}$
is
$\mathbb{A}^1$.
For a vector bundle $V$ over a smooth
$S$-scheme $X$ with zero section $z\colon X \hra V$
consider
a Nisnevich sheaf associated with
the presheaf
$Y \mapsto V(Y)/\bigl(V\minus z(X)\bigr)(Y)$
on the Nisnevich site
$\Sm/S$.
The {\em Thom space\/} $\Th(V)$ of $V$ is defined as that
Nisnevich sheaf regarded as a presheaf.
In particular $\Th(V)$ is a pointed motivic space in the
sense of
\cite[Defn.~A.1.1]{BGL}.
Its Nisnevich sheafification coincides with Voevodsky's Thom space \cite[p. 422]{V1},
since
$\Th(V)$
is already a Nisnevich sheaf.
The Thom space of the Hopf bundle is then defined as the colimit
$\Th(\mathcal{T}(1)) = \colim_{n\geq 0} \Th\bigl(\mathcal{O}_{\Pro^n}(-1)\bigr)$.
Abbreviate $T =\Th(\Aff^1_S) = \Aff^1_S/(\Aff^1_S\minus \lbrace 0 \rbrace)$.

Let $E$ be a commutative ring $\Pro^1$-spectrum. The unit gives rise
to an element
$1\in E^{0,0}(\Spec(k)_+)$.
Applying the
$\Pro^1$-suspension
isomorphism to that element we get an element
$\Sigma_{\Pro^1}(1) \in E^{2,1}(\Pro^1/\lbrace \infty \rbrace))$.
The canonical covering of $\Pro^1$ defines
motivic weak equivalences
\[ \xymatrix{\Pro^1/\lbrace \infty \rbrace \ar[r]^-\sim & \Pro^1/\Aff^1 &
\Aff^1/\Aff^1\minus \lbrace 0 \rbrace = T \ar[l]_-\sim,} \]
which in turn define pull-back isomorphisms
$E(\Pro^1/\lbrace \infty \rbrace) \leftarrow E(\Aff^1/\Aff^1\minus \lbrace 0 \rbrace) \rightarrow  E(T)$ .
Denote
$\Sigma_{T}(1)$
the image of
$\Sigma_{\Pro^1}(1)$
in
$E^{2,1}(T)$.


\begin{defn}
\label{OrientationViaThom}
Let $E$ be a commutative ring $\Pro^1$-spectrum.
A {\em Thom  orientation of\/}
$E$ is an element
$th \in E^{2,1}(\Th(\mathcal T(1))$
%= =E^{2,1}(\mathbb{MGL}_1)$ %{\Pro^{\infty}}(\mathcal T(1))$
such that its restriction to the Thom space of the fibre over the distinguished
point coincides with the element
$\Sigma_{T}(1) \in E^{2,1}(T)$.
A {\em Chern orientation of\/}
$E$ is an element
$c \in E^{2,1}(\Pro^{\infty})$
such that
$c|_{\Pro^1}= - \Sigma_{\Pro^1}(1)$.
{\em An orientation} of $E$ is either a Thom orientation or a Chern orientation.
Two Thom orientations of $E$ coincide if respecting Thom elements coincides.
Two Chern orientations of $E$ coincide if respecting Chern elements coincides.
One says that a Thom orientation $th$ of $E$ coincides
with a Chern orientation $c$ of $E$ provided that
$c = z^*(th)$ or equivalently
the element
$th$ coincides with the one
$th(\mathcal O(-1))$
given by
(\ref{ThomClass})
below.
\end{defn}

% It should be regarded
% as the Thom class of the Hopf bundle $\mathcal{O}(-1)$ on $\Pro^\infty$.
%%% \begin{defn}
%%% \label{OrientationViaChern}
%%% Let $E$ be a commutative ring $\Pro^1$-spectrum.
%%% \end{defn}

\begin{rem}
\label{ThomAndChern}
The element $th$ should be regarded as a Thom class of the tautological
line bundle
$\mathcal{T}(1)= \mathcal O(-1)$
over
$\Pro^{\infty}$.
The element $c$ should be regarded as a Chern class of the tautological
line bundle
$\mathcal{T}(1)= \mathcal O(-1)$
over
$\Pro^{\infty}$.
\end{rem}

%\begin{rem}
%\label{ThomAndChern}
%Let $c$ be an orientation of $E$. Consider
%$th(\mathcal T(1)) \in E^{2,1}_{\Pro^{\infty}}(\mathcal T(1))$
%given by
%(\ref{ThomClass})
%and set \ $th= th(\mathcal T(1))$.
%It is straight forward to check that
%$th|_{Th(1)}= \Sigma_{\Pro^1}(1)$.
%Thus $th$ is an orientation of $E$.
%Clearly
%$c =z^*(th) \in E^{2,1}(\Pro^{\infty})$.
%Whence the two definition of orientations of $E$
%are equivalent.
%\end{rem}


\begin{example}
\label{OrientationsOfMGLandK}
The following orientations given right below are relevant for our work.
Here $\mathrm{MGL}$ denotes the $\Pro^1$-ring spectrum
representing algebraic cobordism obtained in
\cite[Defn 2.1.1]{UniThm}
and $\mathrm{BGL}$ denotes the $\Pro^1$-ring spectrum
representing algebraic $K$-theory
constructed in
\cite[Theorem 2.2.1]{BGL}.
\begin{itemize}
\item
Let
$u_1: \Sigma^{\infty}_{\Pro^1}(\Th(\mathcal{T}(1)))(-1) \to \mathrm{MGL}$
be the canonical map of
$\Pro^1$-spectra. Set
$th^{\mathrm{MGL}} =u_1 \in \mathrm{MGL}^{2,1}(\Th(\mathcal{T}(1)))$.
Since
$th^{\mathrm{MGL}}|_{\Th(\mathbf{1})}= \Sigma_{\Pro^1}(1)$
in
$\mathrm{MGL}^{2,1}(\Th(\mathbf{1}))$,
the class
$th^{\mathrm{MGL}}$
is an orientation of
$\mathrm{MGL}$.
\item
Set
$c =(- \beta) \cup ([\mathcal O]-[\mathcal O(1)])
\in \mathrm{BGL}^{2,1}(\Pro^{\infty})$.
The relation
(11)
from
\cite{BGL}
shows that
the class $c$ is an orientation of
$\mathrm{BGL}$.
%Consider
%$th(\mathcal O(-1)) \in \mathrm{BGL}^{2,1}_{\Pro^{\infty}}(\mathcal O(-1))$
%given by
%(\ref{ThomClass})
%and set
%$th^K = th(\mathcal O(-1))$.
%The class
%$th^K$
%is the same orientation of
%$\mathrm{BGL}$.
\end{itemize}
\end{example}

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

\section{Oriented cohomology theories}
\label{OrientedSpectraAndTheory}

Let $(E,c)$ be an oriented commutative $\Pro^1$-ring spectrum. In this Section we compute
the $E$-cohomology of infinite
Grassmannians and their products. The results
are the expected ones
\ref{CohomologyOfGr}.

The oriented $\Pro^1$-ring spectrum $(E,c)$ defines an oriented cohomology theory on
$\SmOp$ in the sense of
\cite[Defn.~3.1]{PSorcoh}
as follows.
The restriction of the functor $E^{\ast,\ast}$ to the category
$\Sm/S$ is a ring cohomology theory.
By
\cite[Th.~3.35]{PSorcoh}
it remains to construct
a Chern structure on
$E^{\ast,\ast}|_{\SmOp}$
in the sense of
\cite[Defn.3.2]{PSorcoh}.
Let
$H(k)$
be the homotopy category of spaces over $k$.
The functor isomorphism
$\Hom_{\mathrm{H(k)}}(- , \Pro^{\infty}) \to \mathrm{Pic}(-)$
%$\Hom_{\mathrm{H}_\bullet(S)}(- , \Pro^{\infty}) \to \mathrm{Pic}(-)$
on the category
$\Sm/S$
provided by
\cite[Thm.~4.3.8]{MV}
sends the class of the identity map
$\Pro^\infty \to {\Pro^{\infty}}$ to the class of the tautological line bundle
$\mathcal{O}(-1)$
over
$\Pro^{\infty}$.
For a line bundle $L$ over
$X\in \Sm/S$ let
$[L]$ be the class of $L$ in the group
$\mathrm{Pic}(X)$.
Let
$f_L\colon X \to \Pro^{\infty}$
be a morphism in
$\mathrm{H(k)}$ corresponding to
the class $[L]$
under the functor isomorphism above.
For a line bundle $L$ over $X\in \Sm/S$
set
$c(L)=f^\ast_L(c) \in E^{2,1}(X)$.
Clearly,
$c(\mathcal{O}(-1))=c$.
The assignment
$L/X \mapsto c(L)$
is a Chern structure on
$E^{\ast,\ast}|_{\SmOp}$
since
$c|_{\Pro^1}= - \Sigma_{\Pro^1}(1) \in E^{2,1}(\Pro^1,\infty)$.
With that Chern structure
$E^{\ast,\ast}|_{\SmOp}$
is an oriented ring cohomology theory
in the sense of
\cite{PSorcoh}.
In particular,
$(\mathrm{BGL},c^K)$
defines an oriented ring cohomology theory on
$\SmOp$.



Given this Chern structure, one
obtains a theory of Thom classes
$V/X \mapsto th(V) \in E^{2\mathrm{rank}(V),\mathrm{rank}(V)}\bigl(\Th_X(V)\bigr)$
on the cohomology theory
$E^{\ast,\ast}|_{\SmOp/S}$
in the sense of
\cite[Defn.~3.32]{PSorcoh} as follows.
There is a unique theory of Chern classes
$V \mapsto c_i(V) \in E^{2i,i}(X)$
such that for every line bundle $L$ on $X$ one has
$c_1(L)=c(L)$. For a rank $r$ vector bundle
$V$ over $X$ consider the vector bundle
$W:= {\mathbf{1}} \oplus V$
and the associated projective vector bundle
$\Pro(W)$
of lines in $W$.
Set
\begin{eqnarray}
\label{ThomBarClass}
\bar th(V)= c_r(p^\ast(V) \otimes \mathcal{O}_{\Pro(W)}(1)) \in E^{2r,r}(\Pro(W)).
\end{eqnarray}
It follows from
\cite[Cor.~3.18]{PSorcoh}
that the support extension map
\[E^{2r,r}\bigl(\Pro(W)/(\Pro(W)\smallsetminus \Pro(\mathbf{1}))\bigr)
\to E^{2r,r}\bigl(\Pro(W)\bigr)\]
is injective and
$\bar th(E) \in E^{2r,r}\bigl(\Pro(W)/(\Pro(W)\smallsetminus \Pro(\mathbf{1}))\bigr) $.
Set
\begin{eqnarray}
\label{ThomClass}
th(E)= j^\ast(\bar th(E)) \in E^{2r,r}\bigl(\Th_X(V)\bigr),
\end{eqnarray}
where
$j\colon \Th_X(V) \to \Pro(W)/ (\Pro(W) \smallsetminus \Pro({\bf 1}))$
is the canonical motivic weak equivalence of pointed motivic spaces
induced by the open embedding $V\hra \Pro(W)$.
The assignment $V/X$ to $th(V)$ is a theory of Thom classes
on $E^{\ast,\ast}|_{\SmOp}$
(see the proof of
\cite[Thm.~3.35]{PSorcoh}). So the Thom classes are natural,
multiplicative and satisfy the following Thom isomorphism property.

\begin{thm}
\label{ThomIsomorphism}
For a rank $r$ vector bundle
$p\colon V \to X$
on $X\in \Sm/S$ with zero section $z\colon X\hra V$, the map
\[\cup th(V)\colon  E^{\ast,\ast}(X) \to E^{\ast+2r,\ast+r}\bigl(V/(V\minus z(X))\bigr) \]
is an isomorphism of the two-sided
$E^{\ast,\ast}(X)$-modules,
where
$-\cup th(V)$
is written for the composition map
$\bigl(\cup th(V)\bigr) \circ p^\ast$.
\end{thm}

\begin{proof}
  See \cite[Defn.~3.32.(4)]{PSorcoh}.
\end{proof}



%\begin{rem}
%\label{ThomViaChern}
%If an element $th$ is an orientation of $E$ then set
%$c:=z^*(th) \in E^{2,1}(\Pro^{\infty})$
%as in Remark
%\ref{ThomAndChern}.
%Then $(E,c)$ is an oriented commutative ring
%$\Pro^1$-spectrum and
%the Thom class
%$th(\mathcal O(-1)) \in E^{2,1}_{\Pro^{\infty}}(\mathcal O(-1))$
%given by
%(\ref{ThomClass})
%coincides with the element $th$
%(see
%\cite[Thm.??]{PSorcoh}).
%In particular,
%$th(\mathcal O(-1))|_{Th({\bf 1})}= \Sigma_{\Pro^1}(1)$.
%\end{rem}

Analogous to
\cite[p. 422]{V1}
one obtains for
vector bundles $V \to X$ and $W\to Y$ in $\Sm/S$
a canonical map of pointed motivic spaces
$\Th(V) \wedge \Th(W) \to \Th(V \times_S W)$
which is a motivic weak equivalence as defined in \cite[Defn.~3.1.6]{BGL}.
In fact, the canonical map becomes an isomorphism
after Nisnevich (even Zariski) sheafification.
Taking
$Y=S$
and
$W= \mathbf{1}$ the trivial line bundle
yields a motivic weak equivalence
$\Th(V) \wedge T \to Th(V \oplus \mathbf{1})$.
The canonical covering of $\Pro^1$ defines
motivic weak equivalences
\[ \xymatrix{T = \Aff^1/\Aff^1\minus \lbrace 0 \rbrace \ar[r]^-\sim & \Pro^1/\Aff^1 &
 \Pro^1 \ar[l]_-\sim} \]
and the arrow
$T = \Aff^1/\Aff^1\minus \lbrace 0 \rbrace \to  \Pro^1/\Pro^1\minus \lbrace 0 \rbrace$
is an isomorphism.
Hence one may switch between $T$ and $\Pro^1$ as desired.


\begin{cor}
\label{ThomAndSuspension}
For
$W=V \oplus {\mathbf{1}}$
consider the composite motivic weak equivalence
%%% $\epsilon\colon \Th(V) \wedge \Pro^1 \to \Th(W)$
%%% be the composite motivic weak equivalence
$\epsilon\colon \Th(V) \wedge \Pro^1 \to \Th(V) \wedge \Pro^1/\Aff^1 \leftarrow \Th(V) \wedge T \to \Th(W)$
in
$\mathrm{H}_\bullet(S)$.
Then the diagram
$$
\begin{CD}
E^{*+2r,*+r}(Th(V)) @>{\Sigma_{\Pro^1}}>>  E^{*+2r+2,*+r+1}(Th(V) \wedge \Pro^1) \\
@A{id}AA                                                 @A{\epsilon^*}AA \\
E^{*+2r,*+r}(Th(V)) @>{\Sigma_{T}}>>  E^{*+2r+2,*+r+1}(Th(W)) \\
@A{\cup th(V)}AA                                         @A{\cup th(W)}AA    \\
E^{*,*}(X)        @>{id}>>     E^{*,*}(X).
\end{CD}
$$
commutes.
\end{cor}


\begin{thm}
\label{CohomologyOfGr}
Let
$c_i= c_i(\mathcal T(n)) \in E^{2i,i}(\Gr(n))$
be the $i$-th Chern class of the tautological bundle
$\mathcal T(n)$. Then
$$
E^{\ast,\ast}(\Gr(n))= E^{\ast,\ast}(k)[[c_1,c_2, \dots, c_n]]
$$
is the formal power series on the $c_i$'s.
The inclusion
$i\colon \Gr(n) \hra \Gr(n+1)$
satisfies
$i^*(c_m)=c_m$
for $m < n+1$ and
$i^*(c_{n+1})=0$.
\end{thm}





\subsection{A general result}
\label{AgeneralResult}
The main result of this Section is Theorem
\ref{AlmostlyMain}.
The complex cobordism spectrum, equipped with its natural orientation,
is a universal oriented ring cohomology theory by Quillen's universality
theorem \cite{Quillen:mu}.
A motivic version of this universality theorem
is proved in
\cite{UniThm}
(see
\cite{Vez}
for the original statement).
We consider
$\mathrm{MGL}$
with the commutative monoid structure described in
\cite[Defn 2.1.1]{UniThm}
and with the orientation
$th^{\mathrm{MGL}}$
described in
\ref{OrientationsOfMGLandK}.

By a cofibration we mean below in the text a cofibration
with respect to the closed model structure on the category 
$\mathbf{M}(S)$
(see \cite[Appendix]{BGL}).
Recall that for a
$\Pro^1$-spectrum $E$ and a cofibration
$Y \to X$
the group
$E^{p,q}(X,Y)$
is defined as the cohomology
$E^{p,q}(X/Y,Y/Y)$
of the pointed space
$(X/Y,+)$
(if
$Y$
is the empty set, then one should take the group
$E^{p,q}(X_+,+)$
for
$E^{p,q}(X,Y)$).

\begin{defn}[Universality Property]
\label{UniversalityDefn}
Let
$(\mathrm{U},\mathrm{u})$
be an oriented
commutative ring
$\Pro^1$-spectrum over a field $k$.
We say that
$(\mathrm{U},\mathrm{u})$
satisfies Quillen universality property, if
for each commutative ring
$\Pro^1$-spectrum $E$ over $k$
the assignment
$\varphi \mapsto \varphi(\mathrm{u}) \in \mathrm{U}^{2,1}(\Th(\mathcal T(1)))$
identifies the set of monoid morphisms
\begin{eqnarray}
\label{OrientationMap}
\varphi\colon \mathrm{U} \to \mathrm{E}
\end{eqnarray}
in the motivic stable homotopy category
$\mathrm{SH}^\cm(S)$
with the set of orientations of $\mathrm{E}$.
\end{defn}

%For a $\Pro^1$-spectrum $\mathrm{F}$, a pointed motivic space $A$ and
%an integer $i$ set
%$\mathrm{F}^{i}(A)= \bigoplus_{p-2q=i} \mathrm{F}^{p,q}(A)$ and $\mathrm{F}^\ast(A) = \bigoplus_{i\in \ZZ} \mathrm{F}^i(A)$.
Let
$(\mathrm{U},\mathrm{u})$
be an oriented commutative ring
$\Pro^1$-spectrum over $k$.
Let
$(\mathrm{E},\mathrm{th})$
oriented commutative ring
$\Pro^1$-spectrum over $k$.
Let
\begin{eqnarray}
\label{MainMOrphism}
\varphi\colon \mathrm{U} \to \mathrm{E}
\end{eqnarray}
be a monoid morphism in
$\mathrm{SH}^{cm}(k)$
such that $\varphi(u) =th$.
For every space $X$ over $k$
and a cofibration
$Y \to X$
and a unique morphism
$f\colon X/Y \to \mathrm{Spec}(k)$
one has a commutative
diagram of $U^0(k)$-module homomorphisms.
\[
\xymatrix{
\mathrm{U}^{\ast}(X,Y) \ar[r]^-{\varphi_{X,Y}} &    \mathrm{E}^{\ast}(X,Y)    \\
\mathrm{U}^{0}(k) \ar[u]^-{f^\ast} \ar[r]^{\varphi^0_S} &  \mathrm{E}^{0}(k)
\ar[u]_-{f^\ast}}\]
%We write $S$ for
%$\mathrm{Spec}(k)$
%here.
It is known that for each
oriented commutative ring
$\Pro^1$-spectrum $(F,v)$
and each space $A$ the ring
$F^0(A)$
is contained in the center of
$F^{\ast}(A)$.
The last commutative diagram
induces two homomorphisms
\begin{eqnarray}
\label{CFhomPeriodic}
\bar \varphi_{X,Y}\colon \mathrm{U}^{\ast}(X,Y) \otimes_{\mathrm{U}^{0}(k)}
\mathrm{E}^{0}(k) \to
\mathrm{E}^{\ast}(X,Y)
\end{eqnarray}

\begin{eqnarray}
\label{CFhomPeriodic}
\bar \varphi^0_{X,Y}\colon \mathrm{U}^{0}(X,Y) \otimes_{\mathrm{U}^{0}(k)}
\mathrm{E}^{0}(k) \to
\mathrm{E}^{0}(X,Y)
\end{eqnarray}
which are natural in a cofibration
$Y \to X$.

Since this moment choose
$(\mathrm{BGL}, \mathrm{th}^K)$
for
$(\mathrm{E},\mathrm{th})$
(see Example
\ref{OrientationsOfMGLandK}).
Set
$\bar {\mathrm{U}}^{\ast}(X,Y)=
\mathrm{U}^{\ast}(X,Y) \otimes_{\mathrm{U}^{0}(k)} \mathrm{BGL}^{0}(k)$,
$\bar {\mathrm{U}}^{0}(X,Y)=
\mathrm{U}^{0}(X,Y) \otimes_{\mathrm{U}^{0}(k)} \mathrm{BGL}^{0}(k)$.
%and write
%$\bar \varphi_X$
%for the above homomorphism.

\begin{defn}[Weakly MGL-Cellular]
\label{MGLWeaklyCellular}
A Quillen universal oriented commutative ring
$\Pro^1$-spectrum
$(\mathrm{U}, \mathrm{u})$
is called weakly $MGL$-cellular if there exists an integer $N$
such that
the map
$\bar \varphi^0_{U_n,\ast}$
is an isomorphism for
$n \geq N$ .
\end{defn}

\begin{rem}
By the Universality Theorem
\cite{Vez}
or
\cite{UniThm}
the
$\Pro^1$-spectrum
$\mathrm{MGL}$
is Quillen universal. That is why we choose to write
$\mathrm{MGL}$-cellular in the definition above.
The following theorem motivates two last Definitions.
\end{rem}

\begin{thm}
\label{AlmostlyMain}
Let
$(\mathrm{U},\mathrm{u})$
be an oriented commutative ring
$\Pro^1$-spectrum over a field $k$
satisfying the Quillen universality property.
Suppose
$(\mathrm{U},\mathrm{u})$
is weakly $MGL$-cellular.
Then
for each cofibration
$Y \to X$of small spaces over the field $k$
the homomorphism
$\bar \varphi_{X,Y}$
is an isomorphism.
\end{thm}

%\begin{rem}
%It will be checked in the next Subsection that the pair
%$(\mathrm{MGL},\mathrm{th}^{MGL})$
%is an oriented commutative ring
%$\Pro^1$-spectrum over $k$
%satisfying the Quillen universality property
%and that pair is weakly
%$MGL$-cellular as well.
%So the last Theorem shows the relation of the algebraic cobordisms
%of Voevodsky to Quillen's $K$-theory.
%\end{rem}

\begin{proof}
The proof consists of several steps.
Our first aim is to prove that homomorphisms
$\bar \varphi^0_{X,Y}$
are isomorphisms.
We beging with constructing a section of the natural
transformation
\[
\varphi^{0,0}\colon \mathrm{U}^{0,0} \to \mathrm{BGL}^{0,0}
\]
of functors on the category of cofibrations of small spaces.
To do this we begin with recalling
that for every oriented commutative
$\Pro^1$-ring spectrum
$(E,th)$
the ring cohomology theory
$E^{\ast,\ast}|_{\SmOp}$
is an oriented cohomology theory on the category
$\SmOp$
(see Section
\ref{OrientedSpectraAndTheory}).
Let
$\mathbb{F}_{E,th}$
be the induced commutative formal group law
over the ring
$E^0(k)$.
Let
$\Omega$
be the complex cobordism ring
and let
$l_{E,th}\colon \Omega \to E^0(k)$
be the unique ring homomorphism,
which takes the universal formal group
$\mathbb{F}_{\Omega}$
to
$\mathbb{F}_{E,th}$.
Set
\begin{eqnarray}
\label{ClassOfPro}
[\Pro^n]_{E}= l_{E,th}([\mathbb C \mathbb P^n]),
\end{eqnarray}
where
$[\mathbb C \mathbb P^n]$
is the class of the complex projective space
$\mathbb C \mathbb P^n$
in
$\Omega$.
Although the class
$[\Pro^n]_{E}$
depends on the orientation class $th$, we
use the notation
$[\Pro^n]_{E}$
instead. If
$(E^{\prime},th^{\prime})$
is another oriented commutative
$\Pro^1$-ring spectrum and
$\psi\colon E \to E^{\prime}$
is a monoid homomorphism in the
category
$\mathrm{SH}^\cm(S)$
which preserves orientation classes, then it
sends the formal group law
$\mathbb{F}_{E,th}$
to
$\mathbb{F}_{E^{\prime},th^{\prime}}$.
In particular
$
\psi([\Pro^n]_{E})=[\Pro^n]_{E^{\prime}}.
$
Applying this observation to the monoid homomorphism
$\varphi$
one obtains
\[
\varphi([\Pro^1]_{\mathrm{U}})= [\Pro^1]_{\mathrm{BGL}}.
\]
%\begin{clm}
%\label{ProLineAndBott}
%$[\Pro^1]_{\mathrm{BGL}}= - \beta^{-1}$.
%\end{clm}
To compute
$[\Pro^1]_{\mathrm{BGL}}$
recall that
the coefficient at
$XY$ in the formal group law
$\mathbb F_{\Omega}$
coincides with the class\
$-[\mathbb C\mathbb P^1]$
in
$\Omega$.
The formal group law
$\mathbb F_{\mathrm{BGL}}$
coincides with
$X+Y+\beta^{-1}XY$,
since
$c^{\mathrm{BGL}}(L)=([\bf 1]- [L^{\vee}])(- \beta)$.
Thus one gets
\[ [\Pro^1]_{\mathrm{BGL}}= - \beta^{-1} \]
{\em We are ready to construct a section}.
Consider the map
\begin{eqnarray}
\label{Section}
s\colon \Sigma_{\Pro^1}^{\infty}(\mathbb Z \times \mathrm{Gr}) \to \mathrm{U}
\end{eqnarray}
in the stable homotopy category category
$\mathrm{SH}^\cm(S)$
given by the element
\[
c^{\mathrm{U}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{U}}
\in \mathrm{U}^{0,0}(\mathbb Z \times \mathrm{Gr}).
\]
\begin{clm}
\label{PhiOfTheClass}
One has
$\varphi(c^{\mathrm{U}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{U}})=
\tau_{\infty}- \infty
\in \mathrm{BGL}^{0,0}(\mathbb Z \times \mathrm{Gr})$.
\end{clm}
In fact,
%\begin{eqnarray*}
$$\varphi(c^{\mathrm{U}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{U}})=
c^{\mathrm{BGL}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{BGL}}=
(\infty - \tau_{\infty}) \cup \beta \cup (- \beta^{-1})$$
$$
=
\tau_{\infty}- \infty.
$$
%\end{eqnarray*}
Claim \ref{PhiOfTheClass} shows that the composite map
\[ \varphi \circ s\colon \Sigma_{\Pro^1}^{\infty}(\mathbb Z \times \mathrm{Gr}) \to \mathrm{BGL} \]
coincides with the adjoint of the
motivic
weak equivalence
$i\colon \mathbb Z \times \mathrm{Gr} \to \K=\K_0$
from
\cite[Lemma 1.2.2]{BGL}.
Thus for every cofibration
$Y \to X$ of small
motivic spaces the map
\[
s_{X,Y}\colon \mathrm{BGL}^{0,0}(X/Y)= [X/Y, \K_0]=
[X/Y, \mathbb Z \times \mathrm{Gr}] \to [\Sigma_{\Pro^1}^{\infty}(X/Y), \mathrm{U}]=
\mathrm{U}^{0,0}(X/Y)
\]
is a {\it section} of the map
$\varphi^{0,0}_{X,Y}\colon \mathrm{U}^{0,0}(X,Y) \to \mathrm{BGL}^{0,0}(X,Y)$.
Moreover, the section
$s_{X,Y}$
is natural in the cofibration
$Y \to X$.

Next we extend the section $s$ to a section
$\bar s^0\colon \mathrm{BGL}^0 \to \overline{\mathrm{U}}^0$
of the natural transformation
$\bar \varphi^0\colon \overline{\mathrm{U}}^0 \to \mathrm{BGL}^0$
of functors on the category of cofibrations.
To achieve this, recall that
\[
\mathrm{BGL}^0=\mathrm{BGL}^{0,0}[\beta,\beta^{-1}]
\]
for the Bott element
$\beta \in \mathrm{BGL}^{2,1}(k)$ (see
(\ref{BGLlaurantAndTT})).
Thus
for every cofibration $Y \to X$
every element
$\alpha \in \mathrm{BGL}^0(X,Y)$
can be presented in a unique way in the form
$a \cup \beta^i$
with
$a \in \mathrm{BGL}^{0,0}(X,Y)$.
Define
\begin{eqnarray}
\label{Salgebraic}
\bar s^0_{X,Y}\colon \mathrm{BGL}^0 \to \overline{\mathrm{U}}^0
\end{eqnarray}
by
$\bar s_{X,Y}^0(a \cup \beta^i)= s_{X,Y}(a) \otimes \beta^i \in \overline{\mathrm{U}}^0(A)$,
where
$a \in \mathrm{BGL}^{0,0}(X,Y)$.
It is immediate that $s_A^0$ is natural in cofibration $Y \to X$.
The following computation proves the claim which is right below the row of computation
$$
\bar \varphi^0_A(\bar s^0(a \cup \beta^i))=
\bar \varphi^0_A(s(a) \otimes \beta^i)=
\varphi(s(a)) \cup \beta^i=
a \cup \beta^i.
$$
\begin{clm}
The map $\bar s^0_{X,Y}$
is a section of
$\bar \varphi^0_{X,Y}$.
\end{clm}

%In fact,
%$
%\bar \varphi^0_A(\bar s^0(a \cup \beta^i))=
%\bar \varphi^0_A(s(a) \otimes \beta^i)=
%\varphi(s(a)) \cup \beta^i=
%a \cup \beta^i.
%$

Now observe the following.
If for a cofibration $Y \to X$ the map
$\bar \varphi^0_{X,Y}$
is an isomorphism,
then
$\bar s^0_{X,Y}$
is an isomorphism inverse to
$\bar \varphi^0_{X,Y}$.
In particular, one has
$\bar s^0_{X,Y} \circ \bar \varphi^0_{X,Y}= \id$.

The homomorphism
$\bar \varphi^0_{X,Y}$
is an isomorphism for cofibrations of the form
$\ast \to U_n$
with
$n \geq N$,
since
$U$ is weakly
$\mathrm{MGL}$-cellular.
Taking
$\ast \to U_n$
as a cofibration
$Y \to X$
and the class
$[u_n] \in \mathrm{U}^{2n,n}(\mathrm{U}_n, \ast)$
of the canonical morphism
$u_n\colon \Sigma^{\infty}_{\Pro^1}\mathrm U_n(-n) \to \mathrm{U}$
we get the following relation:

\begin{eqnarray}
\label{KeyRelationForU}
(\bar s^0_{\mathrm{U}_n, \ast} \circ \varphi^0_{\mathrm{U}_n, \ast})([u_n])
= [u_n] \otimes 1 \in
\overline{\mathrm{U}}^0(\mathrm{U}_n, \ast).
\end{eqnarray}

Now we are ready to check that
$\bar \varphi^0_A$
is an isomorphism for all cofibrations
$Y \to X$
of small motivic spaces.
Recall that for a
cofibrations
$Y \to X$
of small motivic spaces
there is a canonical isomorphism of the form
\begin{eqnarray}
\label{IndLimit}
\mathrm{U}^{2i,i}(X,Y)= \colim_n [\Sigma^{2n,n}(X/Y, Y/Y),
\mathrm U_{i+n}]_{\mathrm{H}_\bullet(S)}
\end{eqnarray}
where
$\Sigma^{2n,n}= \Sigma^n_{\Pro^1}$
(if $Y$ is empty then one should replace the pair
$(X/Y, Y/Y)$ by the one $(X_+,+)$).
This isomorphism implies that for every element
$a \in \mathrm{U}^{2i,i}(X,Y)$
there exists an integer
$n \geq 0$ such that
$\Sigma^{2n,n}(a)= f^*([u_n])$
for an appropriate map
$f\colon \Sigma^{2n,n}(X/Y) \to \mathrm{U}_{i+n}$
in the homotopy category
$\mathrm{H}^{cm}_\bullet(S)$. Here
$\Sigma^{2n,n}(a)$
is the $n$-fold
$\Sigma_{\Pro^1}$-suspension
of $a$.

The surjectivity of
$\bar \varphi^0_{X,Y}$
is clear, since
$\bar s^0_{X,Y}$
is its section.
It remains to check the injectivity of
$\bar \varphi^0_{X,Y}$.
Take a homogeneous element
$\alpha \in \overline{\mathrm{U}}^{2i,i}(X,Y) \subseteq \overline{\mathrm{U}}^0(X,Y)$
such that
$\bar \varphi^0_{X,Y}(\alpha)=0$.
It has the form
$\alpha= a \otimes \beta^m$
for a homogeneous element
$a \in \mathrm{U}^{0,0}(X,Y)$.
Since the element
$\beta$
is invertible in
$\mathrm{BGL}^{\ast,\ast}(k)$, one concludes
$\varphi^0_{X,Y}(a)=0$.

Choose an integer $n \geq 0$ such that
$\Sigma^{2n,n}(a)= f^\ast([u_n])$
and write
$A$ for $X/Y$ to short the notation.
The map $\varphi$ of $\Pro^1$-spectra
respects the suspension isomorphisms.
Thus
$\varphi_{\Sigma^{2n,n}A}(\Sigma^{2n,n}(a))= \Sigma^{2n,n}(\varphi_A(a))=0$
and
$(\bar s^0_{\Sigma^{2n,n}A} \circ \varphi_{\Sigma^{2n,n}A})(\Sigma^{2n,n}(a))=0$
too.
The chain of relations in
$\overline{\mathrm{U}}^0(\Sigma^{2n,n}A)$
given by
\begin{eqnarray*}
  0 & = \bigl(\bar s_{\Sigma^{2n,n}A}^0 \circ \varphi_{\Sigma^{2n,n}A}\bigr)
  \bigl(\Sigma^{2n,n}(a)\bigr)= \bigl(\bar s_{\Sigma^{2n,n}A}^0 \circ
  \varphi_{\Sigma^{2n,n}A}\bigr)\bigl(f^*([u_n])\bigr) \\
   & = f^*\bigl((\bar s^0_{\mathrm{U}_{n+i}} \circ \varphi_{\mathrm{U}_{n+i}})
   ( [u_n] ) \bigr) =f^*([u_n] \otimes 1)= f^*([u_n]) \otimes 1 \\
& = \Sigma^{2n,n}(a) \otimes 1
\end{eqnarray*}
implies that
$\Sigma^{2n,n}(a \otimes 1)= \Sigma^{2n,n}(a) \otimes 1=0$.
Because the $n$-fold suspension map
\[
\Sigma^{2n,n}\colon \overline{\mathrm{U}}^0(X/Y, Y/Y) \to
\overline{\mathrm{U}}^0(\Sigma^{2n,n}(X/Y, Y/Y)) \]
is an isomorphism,
$a \otimes 1=0$
in
$\overline{\mathrm{U}}^0(X/Y)=\overline{\mathrm{U}}^0(X,Y)$.
This proves the injectivity and hence the bijectivity of
$\bar \varphi^0_{X,Y}$
for cofibrations of all small motivic spaces.

To prove that
$\bar \varphi_{X,Y}$
is an isomorphism
for cofibrations of all small motivic spaces
we will use
the fact that
$\bar \varphi_{X,Y}$
respects the
$\Pro^1$-suspension isomorphisms.
Set $A=X/Y$.


For every integer
$i \in \mathbb{Z}$
choose an integer
$n \geq 0$
with
$n \geq i$.
Then for a pointed motivic space $A$ one may form the suspension
$\mathbb{G}^{\wedge n}_m \wedge S^{n-i}_s \wedge A = S^{n,n}\wedge S^{n-i,0}\wedge A$
in the category of pointed motivic spaces, which supplies the commutative diagram
\[
\xymatrix{
\mathrm{BGL}^{i}(A) \ar[r]^-{\Sigma^{2n,n}}_-\cong &
\mathrm{BGL}^{i}(S^{2n,n} \wedge A) &
\mathrm{BGL}^0(S^{n,n}\wedge S^{n-i,0}\wedge A) \ar[l]_-{\Sigma^{i,0}}^-\cong \\
\mathrm{U}^{i}(A) \ar[r]^-{\Sigma^{2n,n}}_-\cong \ar[u]^-{\bar \varphi^i_A}&
\mathrm{U}^{i}(S^{2n,n} \wedge A) \ar[u]^-{\bar \varphi^i_{S^{2n,n}\wedge A}}&
\mathrm{U}^0(S^{n,n}\wedge S^{n-i,0}\wedge A) \ar[l]_-{\Sigma^{i,0}}^-\cong
\ar[u]_-{\bar \varphi^0_{S^{n,n}\wedge S^{n-i,0}\wedge A}}^-\cong} \]
with the suspension isomorphisms
$\Sigma^{2n,n} = \Sigma^n_{\Pro^1}$
and
$\Sigma^{i,0}$.
The map
$\bar \varphi^0_B$
is an isomorphism for $B$ a small pointed motivic space, hence so is
$\bar \varphi^{i}_A$.
We proved that the map
$\bar \varphi_{X,Y}$
is an isomorphism.
Theorem
\ref{AlmostlyMain}
is proven.

\end{proof}




\subsection{The $\mathrm{MGL}$-cellularity of the $\mathrm{MGL}$}
\begin{thm}
\label{Preliminary}
The oriented commutative ring
$\Pro^1$-spectrum
$(\mathrm{MGL}, th^\mathrm{MGL})$
from Example
\ref{OrientationsOfMGLandK}
is weakly
$\mathrm{MGL}$-cellular.


\end{thm}
\begin{proof}
We must check that the homomorphism
$\bar \varphi^0_{X,Y}$
is an isomorphism for
$({X,Y})$
being
$(Th(\mathcal{T}_n),\ast) = (\mathrm{MGL}_n,\ast)$.
We check that inspecting step by step
motivic spaces
$\mathrm{Spec}(k)$, $\Pro^{\infty}$, $\Gr(n)$
and the pair
$(Th(\mathcal{T}_n),\ast) = (\mathrm{MGL}_n,\ast)$.



The map
$\bar \varphi^0_{k}$
is an isomorphism,
since it is the identity map.
By the case $n=1$ of Theorem
~\ref{CohomologyOfGr}
one has
$\overline{\mathrm{MGL}}^{\ast}(\Pro^{\infty})=
\overline{\mathrm{MGL}}^{\ast}(k)[[c^{\mathrm{MGL}}]]$,
whence
\[ \overline{\mathrm{MGL}}^0(\Pro^{\infty})=
\overline{\mathrm{MGL}}^0(k)[[c^{\mathrm{MGL}}]]\]
(the formal power series on the first Chern class
$c^{\mathrm{MGL}}$
of the tautological line bundle
$\mathcal O(-1)$).
The same holds for
$\mathrm{BGL}$. Namely
\[ \mathrm{BGL}^0(\Pro^{\infty})= \mathrm{BGL}^0(k)[[c^{\mathrm{BGL}}]]. \]
By its definition the morphism
$\varphi$
takes the orientation class
$th^{\mathrm{MGL}}$
to the orientation class
$th^K$
and so it preserves the first Chern class.
Whence the map
$\bar \varphi^0_{\Pro^{\infty}}$
coincides with a map of formal power series
induced by the isomorphism
$\bar \varphi^0_{k}$
of the coefficients rings.
%The map
%$\bar \varphi^0_{\Spec(k)}$
%is an isomorphism.
Hence
$\bar \varphi^0_{\Pro^{\infty}}$
is an isomorphism as well.

Consider now
$X= \Gr(n)$. By Theorem
\ref{CohomologyOfGr}
its
$\mathrm{MGL}$-cohomology ring
is the ring of formal power series on
the Chern classes
of the tautological bundle
$\mathcal T_n$
over the coefficient ring
$\mathrm{MGL}^{\ast,\ast}(k)$.
The same holds for the
$\mathrm{BGL}$-cohomology ring.
As observed above, the map
$\varphi$
preserves the first Chern class,
thus it takes Chern classes to the Chern classes.
Whence
$\bar \varphi^0_{\Gr(n)}$
is an isomorphism as well.

Now consider
$(X,Y) = (\Th(\mathcal T_n), \ast)$.
The morphism
$\varphi$
respects Thom classes
(see
(\ref{ThomBarClass})
and
(\ref{ThomClass})).
The vertical arrows in the commutative diagram
\[ \xymatrix{
\overline{\mathrm{MGL}}^0((\Th(\mathcal{T}_n), \ast)) \ar[r]^-{\bar \varphi^0_{\Th(\mathcal{T}_n), \ast)}} &
\mathrm{BGL}^0(\Th(\mathcal{T}_n))    \\
\overline{\mathrm{MGL}}^0(\Gr(n))  \ar[r]^-{\bar \varphi^0_{\Gr(n)}}
\ar[u]^-{\overline{\mathrm{thom}}^\mathrm{MGL}} &  \mathrm{BGL}^0(G(n))
\ar[u]_-{{\mathrm{thom}}^{\mathrm{BGL}}}}\]
are isomorphisms induced by the
the Thom isomorphism \ref{ThomIsomorphism}.
The map
$\bar \varphi^0_{\Gr(n)}$
is an isomorphism by the preceding case,
whence
$\bar \varphi^0_{Th(\mathcal T_n), \ast)}$
is an isomorphism too.

\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Main Result}
Let $k$ be a field and $S= \mathrm{Spec}(k)$.
By Theorem
\cite[Theorem 2.2.1]{UniThm}
and Example
\ref{OrientationsOfMGLandK}
there exists a unique monoid morphism
\begin{eqnarray}
\label{MainMOrphism}
\varphi\colon \mathrm{MGL} \to \mathrm{BGL}
\end{eqnarray}
in
$\mathrm{SH}^\cm(S)$
such that $\varphi(th^\mathrm{MGL}) =th^K$.
For every cofibration $Y \to X$ of motivic spaces over $k$ a unique morphism
$f\colon X/Y \to S$
induces the homomorphism
\begin{eqnarray}
\label{CFhomPeriodic}
\bar \varphi_{X,Y}\colon \overline{\mathrm{MGL}}^{\ast}({X,Y})\colon = \mathrm{MGL}^{\ast}({X,Y}) \otimes_{\mathrm{MGL}^{0}(k)}
\mathrm{BGL}^{0}(k) \to
\mathrm{BGL}^{\ast}({X,Y})
\end{eqnarray}
which is natural in cofibration $Y \to X$.
Recall that a space $A$ is called small if
the covariant functor
$\Sigma^\infty_{\Pro^1} A$ represents on
$\mathrm{SH}^\cm(S)$ commutes with arbitrary coproducts.

\begin{thm}
\label{PreMainThm}
The homomorphism $\bar \varphi_{X,Y}$
is an isomorphism for all cofibrations
$Y \to X$ of small motivic spaces.
\end{thm}

In fact, the
$(\mathrm{MGL}, th^{\mathrm{MGL}})$
is Quillen universal
by
\cite{Vez}
or by Theorem
$2.2.1$ from
\cite{UniThm}
and weakly
$\mathrm{MGL}$-cellular
by Theorem
\ref{Preliminary}
above.
Theorem
\ref{AlmostlyMain}
completes the proof.

%Here a pointed motivic space $A$ is called {\em small\/} if
%the covariant functor $\Sigma^\infty_{\Pro^1} A$ represents on
%$\SH(S)$ commutes with arbitrary coproducts.
%From Theorem~\ref{PreMainThm} one may
%thus deduce the following more general result.

%\begin{cor}
%\label{cor:mainthm}
%Let $E$ be a $\Pro^1$-spectrum of finite type, i.e.,~a retract in $\SH(S)$
%of a $\Pro^1$-spectrum obtained by attaching finitely many cells. Then
%there is an isomorphism
%%\[ \mathrm{MGL}^{\ast}(E) \otimes_{\mathrm{MGL}^{0}(S_+)}
%\mathrm{BGL}^{0}(S_+) \to
%\mathrm{BGL}^{\ast}(E)\]
%which is natural in $E$.
%\end{cor}

%\begin{rem}
%\label{PhiOnPoint}
%For $X\in \Sm_k$ with structural morphism $f\colon X\to S=\Spec(k)$ set
%$[X] = f_\ast(1)$, where $f_\ast\colon \mathrm{MGL}^{0}(X_+) \to \mathrm{MGL}^0(S_+)$
%is the pushforward map defined in \cite{PSorcoh}.
%The ring homomorphism
%\[ \varphi_S\colon  \mathrm{MGL}^{0}(S_+) \to \mathrm{BGL}^{0}(S_+)=
%\mathbb{Z}[\beta, \beta^{-1}] \]
%sends the class
%$[X] \in \mathrm{MGL}^{0}(k)$
%of an irreducible smooth projective variety
%$X$ to the element
%$(-1)^{- dim X} \cdot \chi(X, \mathcal O_X) \cdot \beta^{- dim X}$,
%where
%$\chi(X, \mathcal O_X)$
%is the Euler characteristic of the structure sheaf
%$\mathcal O_X$.
%\end{rem}

\begin{rem}
\label{MorelHopkins}
There is an unpublished result due to Morel and Hopkins,
which states that there is a canonical isomorphism
of the form
$$
\mathrm{MGL}^{\ast,\ast}(X) \otimes_{\mathbb{L}} \mathbb{Z}[\beta, \beta^{-1}] \to
\mathrm{BGL}^{\ast,\ast}(X)
$$
where $\mathbb{L}$ denotes the Lazard ring carrying the universal formal group law.
If the canonical homomorphism
$\mathbb{L} \to \mathrm{MGL}^{0}(k)$
is an isomorphism, Theorem
\ref{PreMainThm} implies their result.
\end{rem}

Let $X$ be a smooth $k$-scheme and $Z\subseteq X$ a closed subset, with
open complement $U\subseteq X$.
Consider the motivic space
$X/U$
and take the quotients of both sides of the isomorphism
(\ref{CFhomPeriodic})
modulo the principal ideal
generated by the element
$1\otimes(\beta +1)$.
Corollary
\ref{LaurantPolinomial}
then implies the following isomorphism
\begin{eqnarray}
\label{CFhomomorphism}
\bar {\bar \varphi}_{X/U}\colon  \overline {\overline{\mathrm{MGL}}}^{\ast}({X,Y})= \mathrm{MGL}^{\ast}(X/U) \otimes_{\mathrm{MGL}^{0}(k)}
\mathbb{Z} \to K^{TT}_{- *,Z}(X)
\end{eqnarray}
where
$K^{TT}_{*,Z}(X)$
are the
Thomason-Trobaugh $K$-groups with supports.
This family of isomorphisms shows that the functor
\[ (X,X\minus Z)\mapsto \mathrm{MGL}^\ast(X/(X\minus Z))\otimes_{\mathrm{MGL}^0(k) }\ZZ =
\colon \overline{\overline{\mathrm{MGL}}}^{\ast}(X/(X \minus Z))\]
is a ring cohomology theory in the sense of
\cite{PSorcoh}.
This implies the first part of our main result.

\begin{thm}[Main Theorem]
\label{MainThm}
Let $X\in \Sm_k$ and $Z\subseteq X$ be a closed subset.
\begin{itemize}
\item
The family of isomorphisms
\begin{eqnarray}
\label{CFhomomorphism2}
\bar {\bar \varphi}_{X/(X-Z)}\colon \overline{\overline{\mathrm{MGL}}}^{\ast}(X/(X \minus Z)) \to
K^{TT}_{- *,Z}(X)
\end{eqnarray}
form an isomorphism
$\bar {\bar \varphi}$
of ring cohomology theories on
$\SmOp/k$.
\item
The $\bar {\bar \varphi}$ respects orientations provided that
$\mathrm{MGL}^{\ast}$
and
$K^{TT}_{- \ast}$ are considered
as oriented cohomology theories
in the sense of
\cite{PSorcoh}
with orientations given
by the Thom class
$th^{\mathrm{MGL}} \otimes 1$
from
\ref{OrientationsOfMGLandK}
and
the Chern structure
$L/X \mapsto [\mathcal O]-[L^{-1}]$.
In particular, the composition
\[ \xymatrix{
\mathrm{MGL}^{0}(k) \ar[r] & \mathrm{MGL}^0(k) \tensor \ZZ \ar[r] &  \mathbb Z \\
     a \ar@{|->}[r] & a\tensor 1 \quad b\tensor c \ar@{|->}[r] & \varphi(b)\cdot c }\]
sends the class
$[X] \in \mathrm{MGL}^{0}(X)$
of a smooth projective $k$-variety
$X$ to
the Euler
characteristic
$\chi(X, \mathcal O_X)$
of the structure sheaf
$\mathcal O_X$.
\end{itemize}
\end{thm}

\begin{proof}
The first part is already proven. To prove the second one consider
the orientations
$th^{\mathrm{MGL}}$
and
$th^K$
from
\ref{OrientationsOfMGLandK}.
Note that
by the very definition of
$\varphi$
it sends
$th^{\mathrm{MGL}}$
to
$th^K$.
Thus it respects the Chern structures on
$\mathrm{MGL}^{\ast}$
and
$\mathrm{BGL}^{\ast}$
described in Section
\ref{OrientedSpectraAndTheory}.

The quotient map
$\mathrm{BGL}^{\ast} \to K^{TT}_{- *}$
takes the Bott element
$\beta$ to $(-1)$.
Thus it takes the Chern structure on
$\mathrm{BGL}^{\ast}$
to the Chern structure on
$K^{TT}_{- \ast}$
given by
$L/X \mapsto [\mathcal O]-[L^{-1}] \in K_0(X)$.
This shows that
$\bar {\bar \varphi}\colon \overline{\overline{\mathrm{MGL}}}^{\ast} \to K^{TT}_{- \ast}$
respects the orientations described in the Theorem
\ref{MainThm}.

Let
$f \mapsto f_{\mathrm{MGL}}$
resp.~$f \mapsto f_K$
be the integrations on
$\mathrm{MGL}^{\ast}$
resp.~
$K^{TT}_{- *}$
given by these Chern structures via Theorem
\cite[Thm.~4.1.4]{PSpush}.
By Theorem
\cite[Thm.~1.1.10]{PSrr}
the composition
$\mathrm{MGL}^\ast \to \mathrm{BGL}^\ast \to K^{TT}_{- \ast}$
respects the integrations on
$\mathrm{MGL}^{\ast}$
and
$K^{TT}_{- \ast}$
since it preserves the Chern structures.
In particular, given a smooth projective
$S$-scheme $f\colon X \to \Spec(k)$,
the diagram
\[
\xymatrix{
\overline{\overline{\mathrm{MGL}}}^0(X) \ar[r]^-{\bar {\bar \varphi}} \ar[d]_-{f_\mathrm{MGL}} &
  K^{TT}_0(X) \ar[d]^-{f_K}    \\
\overline{\overline{\mathrm{MGL}}}^0(k)  \ar[r]^-{\bar {\bar \varphi}} &  K^{TT}_0(k)} \]
commutes
where
$f_{\mathrm{MGL}}$
and
$f_K$
are the push-forward maps for
$\mathrm{MGL}^{\ast}$
and
$K^{TT}_{- \ast}$
respectively.
The integration
$f \mapsto f_K$
on
$K^{TT}_{- \ast}$
respecting the Chern structure
$L \mapsto [\mathcal O]-[L^{-1}]$
coincides with the one given by
the higher direct images by Theorem
\cite[Thm.~1.1.11]{PSrr}.
The last one sends the class
$[V] \in K_0(X)$
of a vector bundle $V$ over a smooth projective
variety $X$ to the Euler characteristic
$\chi(X, \mathcal V)$
of the sheaf $\mathcal{V}$ of sections of
$V$.

Recall that for an oriented cohomology theory
$A$ with a Chern structure
$L \mapsto c(L)$
and for a smooth projective variety
$f\colon X \to \Spec(k)$
its class
$[X]_A \in A^{\mathrm{even}}(\Spec(k))$
is defined as
$f_A(1)$,
where
$f_A\colon A(X) \to A(\Spec(k))$
is the push-forward respecting the Chern structure
(see
\cite[Thm.~4.1.4]{PSpush}).
The $f_A$ depends on the Chern structure.
However we write just $f_A$ for the push-forward operator.
Taking the element
$1\in \mathrm{MGL}^{0,0}(X)$
and using the commutativity of the very last
diagram  we see that
\[\bar {\bar \varphi} ([X]_{\mathrm{MGL}} \otimes 1)= \chi(X, \mathcal{O}_X).\]
Whence the Theorem.

\end{proof}




\begin{thebibliography}{MMM}
\label{References}

\bibitem[Ad]{Ad}
\emph{J.~F.~Adams.} Stable homotopy and generalised cohomology.
University of Chicago Press, 1974.

\bibitem[CF]{CF}
\emph{P.~E.~Conner, E.~E.~Floyd.} The relation of cobordism to
$K$-theory. Lect. Notes Math., 28 (1966), Springer-Verlag, Berlin.


\bibitem[J]{J}
\emph{J.~F.~Jardine.}
Motivic symmetric spectra.
Doc. Math. 5 (2000), 445--553.

\bibitem[LM1]{LM1}
\emph{M.~Levine, F.~Morel.} Algebraic cobordism I. Preprint, 116 pages (2002).
http://www.math.uiuc.edu/K-theory/0547/2002

\bibitem[MV]{MV}
\emph{F.~Morel, V.~Voevodsky.} $\Aff^1$-homotopy theory of schemes.
Publ. IHES \ 90, (1999), 45--143 (2001).

\bibitem[PPR1]{BGL}
\emph{I.~Panin, K.~Pimenov, O.~R\"{o}ndigs.}
On Voevodsky's algebraic $K$-theory
spectrum $\mathrm{BGL}$. Preprint, 47 pages (2007). Available via
http://www.math.uiuc.edu/K-theory/0838/

\bibitem[PPR2]{UniThm}
\emph{I.~Panin, K.~Pimenov, O.~R\"{o}ndigs.}
A universality theorem for Voevodsky's algebraic cobordism spectrum $\mathrm{MGL}$.
Preprint, 11 pages (2007).
http://www.math.uiuc.edu/K-theory/????

\bibitem[PS1]{PSorcoh}
\emph{I.~Panin.}  ({\it After I.~Panin and A.~Smirnov})
Oriented cohomology theories on algebraic varieties,
Special issue in honor of H. Bass on his seventieth birthday. Part III.
$K$-Theory, 30 (2003), no. 3, 265--314.


\bibitem[PS2]{PSrr}
\emph{I.~Panin.} ({\it After I.~Panin and A.~Smirnov})
Riemann--Roch theorems for oriented cohomology,
in ``Axiomatic, enriched and motivic homotopy theory'' (ed. J.P.C.Greenless)
261--333, NATO Sci. Ser. II Math. Phys. Chem., 131, \\
Kluwer Acad. Publ., Dordrecht, 2004

\bibitem[PS3]{PSpush}
\emph{I.~Panin.} ({\it After I.~Panin and A.~Smirnov})
Push-forwards in oriented cohomology theories of algebraic varieties: II,
Preprint,
http://www.math.uiuc.edu/K-theory/0619/2003

\bibitem[PY]{PY}
\emph{I.~Panin, S.~Yagunov.} Rigidity for orientable functors. J.
Pure and Appl. Algebra, 172, (2002), 49--77.

\bibitem[Qu1]{Quillen:mu}
\emph{D.~Quillen}.
On the formal group laws of unoriented and complex cobordism theory.
Bull.~Am.~Math.~Soc. 75, 1293-1298 (1969)

\bibitem[Qu2]{Qu3}
\emph{D.~Quillen.} Higher algebraic $K$-theory: I. Lect.~Notes Math.
341 (1973), 85--147.


\bibitem[TT]{TT}
\emph{R.~Thomason, T.~Trobaugh.} \ Higher algebraic K-theory of schemes
and of derived categories.
In The Grothendieck festschift, 3, 247-436, Birkhauser, Boston, 1990.

\bibitem[Ve]{Vez}
\emph{G.~Vezzosi.}
Brown-Peterson spectra in stable $\Aff^1$-homotopy theory.
Rend. Semin. Mat. Univ. Padova 106, 47-64 (2001).

\bibitem[V1]{V1}
\emph{V.~Voevodsky.} $\mathbf{A}^1$-Homotopy theory.
Doc. Math., Extra Vol. ICM 1998(I), 417-442.


\end{thebibliography}



\end{document}


\begin{proof}

For a $\Pro^1$-spectrum $E$, a pointed motivic space $A$ and
an integer $i$ set
$E^{i}(A)= \bigoplus_{p-2q=i} E^{p,q}(A)$ and $E^\ast(A) = \bigoplus_{i\in \ZZ} E^i(A)$.
By Theorem~\ref{UniversalityThm} there exists a unique monoid morphism
\begin{eqnarray}
\label{MainMOrphism}
\varphi\colon \mathrm{MGL} \to \mathrm{BGL}
\end{eqnarray}
in
$\mathrm{SH}(S)$
such that $\varphi(th^\mathrm{MGL}) =th^K$.
For every smooth scheme $X$ over $S$ with structural morphism
$f\colon X\to S$ one has a commutative
diagram of ring homomorphisms.
\[
\xymatrix{
\mathrm{MGL}^{\ast}(X_+) \ar[r]^-{\varphi_X} &    \mathrm{BGL}^{\ast}(X_+)    \\
\mathrm{MGL}^{0}(S_+) \ar[u]^-{f^\ast} \ar[r]^{\varphi^0_S} &  \mathrm{BGL}^{0}(S_+)
\ar[u]_-{f^\ast}}\]

It
induces a ring homomorphism
\begin{eqnarray}
\label{CFhomPeriodic}
\bar \varphi_X\colon \mathrm{MGL}^{\ast}(X_+) \otimes_{\mathrm{MGL}^{0}(S_+)}
\mathrm{BGL}^{0}(S_+) \to
\mathrm{BGL}^{\ast}(X_+)
\end{eqnarray}
which is natural in $X$.
For the construction of the homomorphism~(\ref{CFhomPeriodic}) it suffices
to observe that $\mathrm{MGL}^{\ast,\ast}(E)$ has a natural
$\mathrm{MGL}^{\ast,\ast}(S_+)$-module structure for any $\Pro^1$-spectrum $E$,
and similarly with $\mathrm{MGL}$ replaced by $\mathrm{BGL}$.
In particular, this is the case for $E = \Sigma^\infty_{\Pro^1} A$,
where $A$ is a pointed motivic space over $S$.


\begin{thm}
\label{PreMainThm}
The ring homomorphism $\bar \varphi_A$
is an isomorphism for all small pointed motivic spaces $A$.
\end{thm}

Here a pointed motivic space $A$ is called {\em small\/} if
the covariant functor $\Sigma^\infty_{\Pro^1} A$ represents on
$\SH(S)$ commutes with arbitrary coproducts.
%From Theorem~\ref{PreMainThm} one may
%thus deduce the following more general result.

%\begin{cor}
%\label{cor:mainthm}
%Let $E$ be a $\Pro^1$-spectrum of finite type, i.e.,~a retract in $\SH(S)$
%of a $\Pro^1$-spectrum obtained by attaching finitely many cells. Then
%there is an isomorphism
%%\[ \mathrm{MGL}^{\ast}(E) \otimes_{\mathrm{MGL}^{0}(S_+)}
%\mathrm{BGL}^{0}(S_+) \to
%\mathrm{BGL}^{\ast}(E)\]
%which is natural in $E$.
%\end{cor}

\begin{rem}
\label{PhiOnPoint}
For $X\in \Sm_k$ with structural morphism $f\colon X\to S=\Spec(k)$ set
$[X] = f_\ast(1)$, where $f_\ast\colon \mathrm{MGL}^{0}(X_+) \to \mathrm{MGL}^0(S_+)$
is the pushforward map defined in \cite{PSorcoh}.
The ring homomorphism
\[ \varphi_S\colon  \mathrm{MGL}^{0}(S_+) \to \mathrm{BGL}^{0}(S_+)=
\mathbb{Z}[\beta, \beta^{-1}] \]
sends the class
$[X] \in \mathrm{MGL}^{0}(k)$
of an irreducible smooth projective variety
$X$ to the element
$(-1)^{- dim X} \cdot \chi(X, \mathcal O_X) \cdot \beta^{- dim X}$,
where
$\chi(X, \mathcal O_X)$
is the Euler characteristic of the structure sheaf
$\mathcal O_X$.
\end{rem}

\begin{rem}
\label{MorelHopkins}
There is an unpublished result due to Morel and Hopkins,
which states that there is a canonical isomorphism
of the form
$$
\mathrm{MGL}^{\ast,\ast}(X_+) \otimes_{\mathbb{L}} \mathbb{Z}[\beta, \beta^{-1}] \to
\mathrm{BGL}^{\ast,\ast}(X_+)
$$
where $\mathbb{L}$ denotes the Lazard ring carrying the universal formal group law.
If the canonical homomorphism
$\mathbb{L} \to \mathrm{MGL}^{0}(k)$
is an isomorphism, Theorem
\ref{PreMainThm} implies their result.
\end{rem}

Let $X$ be a smooth $k$-scheme and $Z\subseteq X$ a closed subset, with
open complement $U\subseteq X$.
Consider the pointed motivic space
$X/U$
and take the quotients of both sides of the isomorphism
(\ref{CFhomPeriodic})
modulo the principal ideal
generated by the element
$1\otimes(\beta +1)$.
Corollary
\ref{LaurantPolinomial}
then implies the following isomorphism
\begin{eqnarray}
\label{CFhomomorphism}
\bar {\bar \varphi}_{X/U}\colon  \mathrm{MGL}^{\ast}(X/U) \otimes_{\mathrm{MGL}^{0}(S_+)}
\mathbb{Z} \to K^{TT}_{- *,Z}(X)
\end{eqnarray}
where
$K^{TT}_{*,Z}(X)$
are the
Thomason-Trobaugh $K$-groups with supports. This family of isomorphisms shows that the
functor \[ (X,X\minus Z)\mapsto \mathrm{MGL}^\ast(X/(X\minus X))\otimes_{\mathrm{MGL}^0(S_+) }\ZZ \] is a
ring cohomology theory in the sense of \cite{PSorcoh}. This implies our main result.

\begin{thm}
\label{MainThm}
Let $X\in \Sm_k$ and $Z\subseteq X$ be a closed subset.
The family of isomorphisms
\begin{eqnarray}
\label{CFhomomorphism2}
\bar {\bar \varphi}_{X/(X-Z)}\colon \mathrm{MGL}^{\ast}(X/(X\minus Z))
\otimes_{\mathrm{MGL}^{0}(S_+)} \mathbb{Z} \to
K^{TT}_{- *,Z}(X)
\end{eqnarray}
form an isomorphism
$\bar {\bar \varphi}$
of ring cohomology theories on
$\SmOp/k$.
It respects orientations if
$\mathrm{MGL}^{\ast}$
and
$K^{TT}_{- \ast}$ are considered
as oriented cohomology theories
in the sense of
\cite{PSorcoh}
with orientations given
by the Thom class
$th^{\mathrm{MGL}}$
from
\ref{OrientationOfMGLandK}
and
the Chern structure
$L/X \mapsto [\mathcal O]-[L^{-1}]$.
In particular, the composition
\[ \xymatrix{
\mathrm{MGL}^{0}(S_+) \ar[r] & \mathrm{MGL}^0(S_+) \tensor \ZZ \ar[r] &  \mathbb Z \\
     a \ar@{|->}[r] & a\tensor 1 \quad b\tensor c \ar@{|->}[r] & \phi(b)\cdot c }\]
sends the class
$[X] \in \mathrm{MGL}^{0}(X_+)$
of a smooth projective $k$-variety
$X$ to
the Euler
characteristic
$\chi(X, \mathcal O_X)$
of the structure sheaf
$\mathcal O_X$.
\end{thm}

\begin{proof}[Proof of Theorem \ref{PreMainThm}]
%In the following we will leave out the extra basepoint $(-)_+$ in the notation.
For a pointed motivic space $A$ set
\begin{eqnarray*}
 \overline{\mathrm{MGL}}^{\ast}(A) & =
  \mathrm{MGL}^{\ast}(A) \otimes_{\mathrm{MGL}^0(S_+)} \mathrm{BGL}^0(S_+) \\
  \overline{\mathrm{MGL}}^0(A) & =
  \mathrm{MGL}^0(A) \otimes_{\mathrm{MGL}^0(S_+)} \mathrm{BGL}^0(S_+)
\end{eqnarray*}
Let
\begin{eqnarray}
\label{PhiAlg}
\bar \varphi^0_A\colon \overline{\mathrm{MGL}}^0(A) \to
\mathrm{BGL}^0(A)
\end{eqnarray}
be the restriction of
$\bar \varphi_A$
to
$\overline{\mathrm{MGL}}^0(A)$. The proof consists of several steps.
Our first aim is to prove the result for the homomorphisms
$\bar \varphi^0_A$.
We begin with the following statement:

\begin{lem}
\label{PtProGrTh}
The homomorphism $\bar \varphi^0_A$
is an isomorphism for $A$ being one of the pointed motivic spaces
$S_+$, $\Pro^{\infty}$, $\Gr(n)$ and $Th(\mathcal{T}_n) = \mathbb{MGL}_n$.
\end{lem}

\begin{proof}
The map
$\bar \varphi^0_{S_+}$
is an isomorphism,
since it is the identity map.
By the first case of Theorem~\ref{CohomologyOfGr}
$\overline{\mathrm{MGL}}^{\ast}(\Pro^{\infty})=
\overline{\mathrm{MGL}}^{\ast}(S_+)[[c^{\mathrm{MGL}}]]$,
whence
\[ \overline{\mathrm{MGL}}^0(\Pro^{\infty})=
\overline{\mathrm{MGL}}^0(S_+)[[c^{\mathrm{MGL}}]]\]
(the formal power series on the first Chern class
$c^{\mathrm{MGL}}$
of the tautological line bundle
$\mathcal O(-1)$).
The same holds for
$\mathrm{BGL}$. Namely
\[ \mathrm{BGL}^0(\Pro^{\infty})= \mathrm{BGL}^0(S_+)[[c^{\mathrm{BGL}}]]. \]
By its definition the morphism
$\varphi$
sends the orientation class
$th^{\mathrm{MGL}}$
to the orientation class
$th^K$
Thus it preserves the first Chern class.
Whence the map
$\bar \varphi^0_{\Pro^{\infty}}$
coincides with a map of formal power series
induced by the isomorphism
$\bar \varphi^0_{S_+}$
of the coefficients rings.
%The map
%$\bar \varphi^0_{\Spec(k)}$
%is an isomorphism.
Hence
$\bar \varphi^0_{\Pro^{\infty}}$
is an isomorphism as well.

Consider now
$X= \Gr(n)$. By Theorem
\ref{CohomologyOfGr}
its
$\mathrm{MGL}$-cohomology ring
is the ring of formal power series on
the Chern classes
of the tautological bundle
$\mathcal T_n$
over the coefficient ring
$\mathrm{MGL}^{\ast,\ast}(S_+)$.
The same holds for
$\mathrm{BGL}$.
As observed above, the map
$\varphi$
preserves the first Chern class,
thus it takes Chern classes to the Chern classes.
Whence
$\bar \varphi^0_{\Gr(n)}$
is an isomorphism as well.

Now consider
$A=\Th(\mathcal T_n)$.
The morphism
$\varphi$
respects Thom classes
(see
(\ref{ThomBarClass})
and
(\ref{ThomClass})).
The vertical arrows in the commutative diagram
\[ \xymatrix{
\overline{\mathrm{MGL}}^0(\Th(\mathcal{T}_n)) \ar[r]^-{\varphi^0_{\Th(\mathcal{T}_n)}} &
\mathrm{BGL}^0(\Th(\mathcal{T}_n))    \\
\overline{\mathrm{MGL}}^0(\Gr(n))  \ar[r]^-{\varphi^0_{\Gr(n)}}
\ar[u]^-{\overline{\mathrm{thom}}^\mathrm{MGL}} &  \mathrm{BGL}^0(G(n))
\ar[u]_-{\overline{\mathrm{thom}}^{\mathrm{BGL}}}}\]
are isomorphisms induced by the
the Thom isomorphism \ref{ThomIsomorphism}.
The map
$\varphi^0_{\Gr(n)}$
is an isomorphism by the preceding case,
whence
$\varphi^0_{Th(\mathcal T_n)}$
is an isomorphism too.
\end{proof}

Our next aim is to construct a section of the natural
transformation
\[
\varphi^{0,0}\colon \mathrm{MGL}^{0,0} \to \mathrm{BGL}^{0,0}
\]
of functors on the category of pointed motivic spaces. To do this we begin with recalling
that for every oriented commutative
$\Pro^1$-ring spectrum
$(E,th)$
the ring cohomology theory
$E^{\ast,\ast}|_{\SmOp}$
is an oriented cohomology theory on the category
$\SmOp$
(see Section
\ref{OrientedSpectraAndTheory}).
Let
$\mathbb{F}_{E,th}$
be the induced commutative formal group law
over the ring
$E^0(S_+)$.
Let
$\Omega$
be the complex cobordism ring
and let
$l_{E,th}\colon \Omega \to E^0(S_+)$
be the unique ring homomorphism,
which takes the universal formal group
$\mathbb{F}_{\Omega}$
to
$\mathbb{F}_{E,th}$.
Set
\begin{eqnarray}
\label{ClassOfPro}
[\Pro^n]_{E}= l_{E,th}([\mathbb C \mathbb P^n]),
\end{eqnarray}
where
$[\mathbb C \mathbb P^n]$
is the class of the complex projective space
$\mathbb C \mathbb P^n$
in
$\Omega$.
Although the class
$[\Pro^n]_{E}$
depends on the orientation class $th$, we
use the notation
$[\Pro^n]_{E}$
instead. If
$(E^{\prime},th^{\prime})$
is another oriented commutative
$\Pro^1$-ring spectrum and
$\psi\colon E \to E^{\prime}$
is a monoid homomorphism in the
category
$\mathrm{SH}(S)$
which preserves orientation classes, then it
sends the formal group law
$\mathbb{F}_{E,th}$
to
$\mathbb{F}_{E^{\prime},th^{\prime}}$.
In particular
$
\psi([\Pro^n]_{E})=[\Pro^n]_{E^{\prime}}.
$
Applying this observation to the monoid homomorphism
$\varphi$
one obtains
\[
\varphi([\Pro^1]_{\mathrm{MGL}})= [\Pro^1]_{\mathrm{BGL}}.
\]
%\begin{clm}
%\label{ProLineAndBott}
%$[\Pro^1]_{\mathrm{BGL}}= - \beta^{-1}$.
%\end{clm}
To compute
$[\Pro^1]_{\mathrm{BGL}}$
recall that
the coefficient at
$XY$ in the formal group law
$\mathbb F_{\Omega}$
coincides with the class\
$-[\mathbb C\mathbb P^1]$
in
$\Omega$.
The formal group law
$\mathbb F_{\mathrm{BGL}}$
coincides with
$X+Y+\beta^{-1}XY$,
since
$c^{\mathrm{BGL}}(L)=([\bf 1]- [L^{\vee}])(- \beta)$.
Thus one gets
\[ [\Pro^1]_{\mathrm{BGL}}= - \beta^{-1} \]
{\em We are ready to construct a section}.
Consider the map
\begin{eqnarray}
\label{Section}
s\colon \Sigma_{\Pro^1}^{\infty}(\mathbb Z \times \mathrm{Gr}) \to \mathrm{MGL}
\end{eqnarray}
in the stable homotopy category category
$\mathrm{SH}(S)$
given by the element
\[
c^{\mathrm{MGL}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{MGL}}
\in \mathrm{MGL}^{0,0}(\mathbb Z \times \mathrm{Gr}).
\]
\begin{clm}
\label{PhiOfTheClass}
One has
$\varphi(c^{\mathrm{MGL}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{MGL}})=
\tau_{\infty}- \infty
\in \mathrm{BGL}^{0,0}(\mathbb Z \times \mathrm{Gr})$.
\end{clm}
In fact,
\begin{eqnarray*}
\varphi(c^{\mathrm{MGL}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{MGL}}) & =
c^{\mathrm{BGL}}_1(\infty - \tau_{\infty}^{\vee}) \cup [\Pro^1]_{\mathrm{BGL}} \\
 & = (\infty - \tau_{\infty}) \cup \beta \cup (- \beta^{-1})=
\tau_{\infty}- \infty.
\end{eqnarray*}
Claim \ref{PhiOfTheClass} shows that the composite map
\[ \varphi \circ s\colon \Sigma_{\Pro^1}^{\infty}(\mathbb Z \times \mathrm{Gr}) \to \mathrm{BGL} \]
coincides with the adjoint of the
motivic
weak equivalence
$i\colon \mathbb Z \times \mathrm{Gr} \to \K=\K_0$
from
\cite[Lemma 1.2.2]{BGL}.
Thus for every pointed motivic space $A$ the map
\[
s_A\colon \mathrm{BGL}^{0,0}(A)= [A, \K_0]=
[A, \mathbb Z \times \mathrm{Gr}] \to [\Sigma_{\Pro^1}^{\infty}(A), \mathrm{MGL}]=
\mathrm{MGL}^{0,0}(A)
\]
is a {\it section} of the map
$\varphi^{0,0}\colon \mathrm{MGL}^{0,0}(A) \to \mathrm{BGL}^{0,0}(A)$.
Moreover, the section $s_A$ is natural in $A$.

Next we extend the section $s$ to a section
$\bar s^0\colon \mathrm{BGL}^0 \to \overline{\mathrm{MGL}}^0$
of the natural transformation
$\bar \varphi^0\colon \overline{\mathrm{MGL}}^0 \to \mathrm{BGL}^0$
of functors on the category of pointed motivic spaces. To achieve this, recall that
\[
\mathrm{BGL}^0=\mathrm{BGL}^{0,0}[\beta,\beta^{-1}]
\]
for the Bott element
$\beta \in \mathrm{BGL}^{2,1}(S_+)$ (see
(\ref{BGLlaurantAndTT})).
Thus every element
$\alpha \in \mathrm{BGL}^0(A)$
can be presented in a unique way in the form
$a \cup \beta^i$
with
$a \in \mathrm{BGL}^{0,0}(A)$.
Define
\begin{eqnarray}
\label{Salgebraic}
\bar s^0\colon \mathrm{BGL}^0 \to \overline{\mathrm{MGL}}^0
\end{eqnarray}
by
$\bar s_A^0(a \cup \beta^i)= s_A(a) \otimes \beta^i \in \overline{\mathrm{MGL}}^0(A)$,
where
$a \in \mathrm{BGL}^{0,0}(A)$. It is immediate that $s_A^0$ is natural in $A$.

\begin{clm}
The map $\bar s^0_A$
is a section of
$\bar \varphi^0_A$.
\end{clm}

In fact,
$
\bar \varphi^0_A(\bar s^0(a \cup \beta^i))=
\bar \varphi^0_A(s(a) \otimes \beta^i)=
\varphi(s(a)) \cup \beta^i=
a \cup \beta^i.
$
Now observe the following.
If for a pointed motivic space $A$ the map
$\bar \varphi^0_A$
is an isomorphism,
then
$\bar s^0_A$
is an isomorphism inverse to
$\bar \varphi^0_A$.
In particular, one has
$\bar s^0_A \circ \bar \varphi^0_A=\id$.
Taking
$A= \Th(\mathcal{T}_n)$
and the class
$[u_n] \in \mathrm{MGL}^{2n,n}(\Th(\mathcal{T}_n))$
of the canonical morphism
$u_n\colon \Sigma^{\infty}_{\Pro^1}\Th(\mathcal T_n)(-n) \to \mathrm{MGL}$
implies the following relation:

\begin{eqnarray}
\label{KeyRelationForU}
(\bar s^0_{\Th(\mathcal T_n)} \circ \varphi^0_{\Th(\mathcal T_n)})([u_n])= [u_n] \otimes 1 \in
\overline{\mathrm{MGL}}^0(\Th(\mathcal{T}_n)).
\end{eqnarray}

Now we are ready to check that
$\bar \varphi^0_A$
is an isomorphism for all small pointed motivic spaces $A$.
Recall that for a small pointed motivic space $A$
there is a canonical isomorphism of the form
\begin{eqnarray}
\label{IndLimit}
\mathrm{MGL}^{2i,i}(A)= \colim_n [\Sigma^{2n,n}(A),
\Th(\mathcal T_{i+n})]_{\mathrm{H}_\bullet(S)}
\end{eqnarray}
where $\Sigma^{2n,n}= \Sigma^n_{\Pro^1}$.
This isomorphism implies that for every element
$a \in \mathrm{MGL}^{2i,i}(A)$
there exists an integer
$n \geq 0$ such that
$\Sigma^{2n,n}(a)= f^*([u_n])$
for an appropriate map
$f\colon \Sigma^{2n,n}(A) \to \Th(\mathcal{T}_{i+n})$
in the homotopy category
$\mathrm{H}_\bullet(S)$. Here
$\Sigma^{2n,n}(a)$
is the $n$-fold
$\Sigma_{\Pro^1}$-suspension
of $a$.

The surjectivity of
$\bar \varphi^0_A$
is clear, since
$\bar s^0_A$
is its section.
It remains to check the injectivity of
$\bar \varphi^0_A$.
Take a homogeneous element
$\alpha \in \overline{\mathrm{MGL}}^{2i,i}(A) \subseteq \overline{\mathrm{MGL}}^0(A)$
such that
$\bar \varphi^0_A(\alpha)=0$.
It has the form
$\alpha= a \otimes \beta^m$
for a homogeneous element
$a \in \mathrm{MGL}^0(A)$.
Since the element
$\beta$
is invertible in
$\mathrm{BGL}^{\ast,\ast}(A)$, one concludes
$\varphi^0_A(a)=0$.

Choose an integer $n \geq 0$ such that
$\Sigma^{2n,n}(a)= f^\ast([u_n])$.
The map $\varphi$ of $\Pro^1$-spectra
respects the suspension isomorphisms.
Thus
$\varphi_A(\Sigma^{2n,n}(a))= \Sigma^{2n,n}(\varphi_A(a))=0$
and
$(\bar s^0_A \circ \varphi_A)(\Sigma^{2n,n}(a))=0$
too.
The chain of relations in
$\overline{\mathrm{MGL}}^0(\Sigma^{2n,n}A)$
given by
\begin{eqnarray*}
  0 & = \bigl(\bar s_{\Sigma^{2n,n}A}^0 \circ \varphi_{\Sigma^{2n,n}A}\bigr)
  \bigl(\Sigma^{2n,n}(a)\bigr)= \bigl(\bar s_{\Sigma^{2n,n}A}^0 \circ
  \varphi_{\Sigma^{2n,n}A}\bigr)\bigl(f^*([u_n])\bigr) \\
   & = f^*\bigl((\bar s^0_{\Th(\mathcal{T}(n+i))} \circ \varphi_{\Th(\mathcal{T}(n+i))})
   ( [u_n] ) \bigr) =f^*([u_n] \otimes 1)= f^*([u_n]) \otimes 1 \\
& = \Sigma^{2n,n}(a) \otimes 1
\end{eqnarray*}
implies that
$\Sigma^{2n,n}(a \otimes 1)= \Sigma^{2n,n}(a) \otimes 1=0$.
Because the $n$-fold suspension map
\[
\Sigma^{2n,n}\colon \overline{\mathrm{MGL}}^0(A) \to
\overline{\mathrm{MGL}}^0(\Sigma^{2n,n}(A)) \]
is an isomorphism,
$a \otimes 1=0$
in
$\overline{\mathrm{MGL}}^0(A)$.
This proves the injectivity and hence the bijectivity of
$\bar \varphi^0_A$ for small pointed motivic spaces $A$.
%Thus
%$\bar \varphi^0_X$
%is an isomorphism for small spaces $X$.

To prove that
$\bar \varphi_A$
is an isomorphism for small pointed motivic spaces we will use
the fact that
$\bar \varphi_A$
respects the $\Pro^1$-suspension isomorphisms.
Recall that for a $\Pro^1$-spectrum $E$ and a pointed motivic space $A$ set
\begin{eqnarray}
\label{NewGrading}
E^{i}(A)= \oplus_{p-2q=i} E^{p,q}(A).
\end{eqnarray}
The restriction of $\bar \varphi_A$ to $\overline{\mathrm{MGL}}^i(A)$ is denoted
\begin{eqnarray}
\label{PhiAlg2}
\bar \varphi^{i}_A\colon \overline{\mathrm{MGL}}^{i}(A) \to
\mathrm{BGL}^{i}(A)
\end{eqnarray}
For every integer
$i \in \mathbb{Z}$
choose an integer
$n \geq 0$
with
$n \geq i$.
Then for a pointed motivic space $A$ one may form the suspension
$\mathbb{G}^{\wedge n}_m \wedge S^{n-i}_s \wedge A = S^{n,n}\wedge S^{n-i,0}\wedge A$
in the category of pointed motivic spaces, which supplies the commutative diagram
\[
\xymatrix{
\mathrm{BGL}^{i}(A) \ar[r]^-{\Sigma^{2n,n}}_-\cong &
\mathrm{BGL}^{i}(S^{2n,n} \wedge A) &
\mathrm{BGL}^0(S^{n,n}\wedge S^{n-i,0}\wedge A) \ar[l]_-{\Sigma^{i,0}}^-\cong \\
\mathrm{MGL}^{i}(A) \ar[r]^-{\Sigma^{2n,n}}_-\cong \ar[u]^-{\bar \varphi^i_A}&
\mathrm{MGL}^{i}(S^{2n,n} \wedge A) \ar[u]^-{\bar \varphi^i_{S^{2n,n}\wedge A}}&
\mathrm{MGL}^0(S^{n,n}\wedge S^{n-i,0}\wedge A) \ar[l]_-{\Sigma^{i,0}}^-\cong
\ar[u]_-{\bar \varphi^0_{S^{n,n}\wedge S^{n-i,0}\wedge A}}^-\cong} \]
with the suspension isomorphisms
$\Sigma^{2n,n} = \Sigma^n_{\Pro^1}$
and
$\Sigma^{i,0}$.
The map
$\bar \varphi^0_A$
is an isomorphism for $A$ a small pointed motivic space, hence so is
$\bar \varphi^{i}_A$.
Theorem
\ref{PreMainThm}
is proven.
\end{proof}

%\begin{rem}
%It might be that the morphism
%$s$ from
%(\ref{Section})
%gives a section on the category of
%{\it small spectra}.
%Arguments proving the injectivity of
%$\bar \varphi^0$
%works well for the small spectra too.
%\end{rem}























