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\begin{document}

\Title{The oriented Chow ring}
\ShortTitle
\SubTitle
\Author{Jean Fasel}
\ShortAuthor
\EndTitle
\Abstract
We define a ring structure on the total oriented Chow group of any integral smooth scheme over a field of characteristic different from $2$.
\EndAbstract
\MSC 
14C15, 14C17, 14C99, 14F43
\EndMSC
\KEY
Oriented Chow groups, Chow groups, Grothendieck-Witt groups and Witt groups
\EndKEY
\Address
D-Math
ETH Zentrum
8092 Z\"urich
Switzerland
\Address
\Address
\Address
\EndAddress


\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Let $A$ be a commutative noetherian ring of Krull dimension $n$ and $P$ a projective $A$-module of rank $d$. One can ask the following question: does $P$ admit a free factor of rank one? Serre proved a long time ago that the answer is always positive when $d>n$. So in fact the first interesting case is when $P$ is projective of rank equal to the dimension of $A$. Suppose now that $X$ is an integral smooth scheme over a field $k$ of characteristic not $2$. To deal with the above question, Barge and Morel introduced the oriented Chow groups $\ch jX$ of $X$ (see \cite{BM}) and associated to each vector bundle $E$ of rank $n$ an Euler class $\che nE$ in $\ch nX$. Morel proved recently that if $X=\spec A$ we have $\che nP=0$ if and only if $P\simeq Q\oplus A$ given $n=2$ or $n\geq 4$ (see \cite{Mo} or \cite{JF} for the case $n=2$). It is therefore important to provide more tools, such as a ring structure, to compute the oriented Chow groups and the Euler classes. 

To define $\ch pX$ consider the fibre product of the complex in Milnor K-theory 

$$\xymatrix@C=1.1em{0\ar[r] & K^M_p(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} K^M_{p-1}(k(x_1))}\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} K^M_{p-n}(k(x_n))}\ar[r] & 0}$$
and the Gersten-Witt complex restricted to the fundamental ideals


$$\xymatrix@C=1.2em{0\ar[r] & I^p(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} I^{p-1}(\mathcal O_{X,{x_1}})}\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} I^{p-n}(\mathcal O_{X,x_n})}\ar[r] & 0}$$
over the quotient complex

$$\xymatrix@C=1.2em{0\ar[r] & I^p/I^{p+1}(k(X))\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} I^{p-n}/I^{p+1-n}(\mathcal O_{X,x_n})}\ar[r] & 0.}$$
The group $\ch pX$ is defined as the $p$-th cohomology group of the fibre product. Roughly speaking, an element of $\ch pX$ is the class of a sum of varieties of codimension $p$ with a quadratic form defined on each variety. We oviously have a map $\ch pX\to CH^p(X)$. 

Using the functoriality of the two complexes we see that the oriented Chow groups satisfy good functorial properties (see \cite{JF}). For example, we have a pull-back morphism $f^*:\ch jX\to \ch jY$ associated to each flat morphism $f:Y\to X$ and a push-forward morphism $g_*:\chs jYL\to \ch {j+r}X$ associated to each proper morphism $g:Y\to X$, where $r=\dim X-\dim Y$ and $L$ is a suitable line bundle over $Y$. Provided this functorial behaviour, it is possible to produce an oriented intersection theory. This is what we do in this paper using the usual strategy (see for example \cite{Fu} or \cite{Ro}). First we define an exterior product 
$$\ch jX\times \ch iY\to \ch {i+j}{X\times Y}$$
and then a Gysin-like homomorphism $i^!:\ch {d}{X}\to \ch {d}Y$ associated to a closed embedding $i:Y\to X$ of smooth schemes. The product is then defined as the composition 

$$\xymatrix{\ch jX\times \ch iX\ar[r] & \ch {i+j}{X\times X} \ar[r]^-{\triangle^!} & \ch {i+j}X}$$
where $\triangle:X\to X\times X$ is the diagonal embedding. To define the exterior product, we first note that Rost already defined an exterior product on the homology of the complex in Milnor K-theory (\cite{Ro}). Thus it is enough to define an exterior product on the homology of the Gersten-Witt complex and show that both exterior products coincide over the quotient complex. We use the usual product on derived Witt groups (\cite{GN}) and show that this product passes to homology using the Leibnitz rule proved by Balmer (see \cite{Ba1}). 

The definition of the Gysin-like map is done by following the ideas of Rost (\cite{Ro}). It uses the deformation to the normal cone to modifiy any closed embedding to a nicer closed embedding and uses also the long exact sequence associated to a triple $(Z,X,U)$ where $Z$ is a closed subset of $X$ and $U=X\setminus Z$.

The product that we obtain has the meaning of intersecting varieties with quadratic forms defined on them. It is therefore not a surprise that the natural map $\ch {tot}X\to CH^{tot}(X)$ turns out to be a ring homomorphism. There is however a surprise: the product that we obtain is neither commutative nor anticommutative. This comes from the fact that the product of triangulated Grothendieck-Witt groups $GW^i\times GW^j\to GW^{i+j}$ does not satisfy any commutativity property.  

The organization of this paper is as follows: In section 2, we recall some basic results on triangulated Witt groups. This includes the construction of the Gersten-Witt complex, and some results on products and consanguinity. In section 3, we construct the oriented Chow groups, recall some results and prove some basic facts. The definition of the exterior product takes place in section 4 and the definition of the oriented Gysin map in section 5. In this part, we also prove the functoriality of this map. Finally we put all the pieces together in section 6 and prove some basic results in section 7.      
   



\subsection{Conventions}
All schemes are smooth and integral over a field $k$ of characteristic different from $2$, or are localizations of such schemes. For any two schemes $X$ and $Y$ we will always denote by $X\times Y$ the fibre product $X\times_{\spec k} Y$. 



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\section{Preliminaries}

\subsection{Witt groups}
We recall here some basic facts on Witt groups of triangulated categories following the exposition of \cite{Ba1}. We suppose that for any triangulated category $\mathcal C$ and any objects $A,B$ of $\mathcal C$ the group $\homm{}AB$ is uniquely $2$-divisible. We also suppose that all triangulated categories are essentially small.

\begin{defin}
Let $\mathcal C$ be a triangulated category. A duality on $\mathcal C$ is a triple $(D,\delta,\varpi)$ where $\delta=\pm 1$, $D:\mathcal C\to \mathcal C$ is a $\delta$-exact contravariant functor and $\varpi:1\simeq D^2$ is an isomorphism of functors satisfying $D(\varpi_A)\circ \varpi_{DA}=id_{DA}$ and $T(\varpi_A)=\varpi_{TA}$ for all $A\in \mathcal C$. A triangulated category $\mathcal C$ with a duality $(D,\delta,\varpi)$ is written $(\mathcal C,D,\delta,\varpi)$.  
\end{defin}

\begin{exem}\label{dual}
Let $X$ be a regular scheme and $\mathcal P(X)$ the category of locally free coherent $\O_X$-modules. Let $D^b(\mathcal P(X))$ be the triangulated category of bounded complexes of objects of $\mathcal P(X)$. Then the usual duality $^\vee$ on $\mathcal P(X)$ defined by $P^\vee=\homm {\O_X}P{\O_X}$ induces a $1$-exact duality on $D^b(\mathcal P(X))$. We also denote this derived duality by $^\vee$. Moreover, the canonical isomorphism $ev:P\to P^{\vee\vee}$ for any locally free module $P$ induces a canonical isomorphism $\varpi:1\to ^{\vee\vee}$ in $D^b(\mathcal P(X))$. More generally, if $L$ is any invertible module over $X$, then the duality $\homm {\O_X}{\_}L$ on $\mathcal P(X)$ also induces a duality on $D^b(\mathcal P(X))$.
\end{exem}


\begin{defin}
Let $(\mathcal C,D,\delta,\varpi)$ be a triangulated category with duality. For any $i\in \Z$, define $(D^{(i)},\delta^{(i)},\varpi^{(i)})$ by  $D^{(i)}=T^i\circ D$, $\delta^{(i)}=(-1)^i\delta$ and $\varpi^{(i)}=\delta^i(-1)^{i(i+1)/2}\varpi$. It is easy to check that $(D^{(i)},\delta^{(i)},\varpi^{(i)})$ is a duality on $\mathcal C$. It is called the $i^{th}$-shifted duality of $(D,\delta,\varpi)$.
\end{defin}

\begin{defin}
Let $(\mathcal C,D,\delta,\varpi)$ be a triangulated category with duality, $A\in \mathcal C$ and $i\in\Z$. A morphism $\varphi:A\to D^{(i)}A$ is $i$-symmetric if the following diagram commutes:

$$\xymatrix{A\ar[r]^-\varphi\ar[d]_-{\varpi^{(i)}_A} & D^{(i)}A\ar@{=}[d] \\
(D^{(i)})^2(A)\ar[r]_-{D^{(i)}\varphi} & D^{(i)}A.}$$ The couple $(A,\varphi)$ is called an $i$-symmetric pair.
\end{defin}

\begin{defin}
We denote by $Symm^i(\mathcal C)$ the monoid of isometry classes of $i$-symmetric pairs.
\end{defin}

\begin{defin}
An $i$-symmetric form is an $i$-symmetric pair $(A,\varphi)$ where $\varphi$ is an isomorphism.
\end{defin}

\begin{thm}\label{symmetric_cone}
Let $(\mathcal C,D,\delta,\varpi)$ be a triangulated category with duality and let $(A,\phi)$ be an $i$-symmetric pair. Choose an exact triangle containing $\phi$

$$\xymatrix{A\ar[r]^-\phi & D^{(i)}A\ar[r]^-\alpha & C\ar[r]^-\beta & TA.}$$
Then there exists an $(i+1)$-symmetric isomorphism $\psi:C\to D^{(i+1)}C$ such that the following diagram commutes

$$\xymatrix@C=4.0em{A\ar[r]^-\phi\ar[d]_-{\varpi^{(i)}} & D^{(i)}A\ar[r]^-\alpha\ar@{=}[d] & C\ar[r]^-\beta\ar@{-->}[d]_-\psi & TA\ar[d]^-{T\varpi^{(i)}} \\
D^{(i)}(D^{(i)}A)\ar[r]_-{D^{(i)}\phi} & D^{(i)}A\ar[r]_-{\delta^{(i+1)}D^{(i+1)}\beta} & D^{(i+1)}C\ar[r]_-{D^{(i+1)}\alpha} & T(D^{(i)}(D^{(i)}A))}$$ 
where the rows are exact triangles and the second one is the dual of the first. Moreover, the $(i+1)$-symmetric form $(C,\psi)$ is unique up to isometry. It is denoted by $\co{A,\phi}$.

\end{thm}

\begin{proof}
See \cite{Ba2}, Theorem 1.6.
\end{proof}

\begin{exem}\label{hyperbol}
Let $P\in \mathcal C$. For any $i$, the morphism $0:P\to D^{(i)}P$ is symmetric and then $\co{P,0}$ is well defined.
\end{exem}

\begin{cor}
The above construction gives a well defined homomorphism of monoids $d^i:Symm^{(i)}(\mathcal C)\to Symm^{(i+1)}(\mathcal C)$ such that $d^{i+1}d^i=0$.
\end{cor}

\begin{defin}
Let $(\mathcal C,D,\delta,\varpi)$ be a triangulated category with duality. The Witt group $W^i(\mathcal C)$ is defined as $\ker{d^{i}}/\im {d^{i+1}}$. Remark that $\ker{d^{i}}$ is just the monoid of isometry classes of $i$-symmetric forms. 
\end{defin}

\begin{defin}\label{groth_witt}
Let $(\mathcal C,D,\delta,\varpi)$ be a triangulated category with duality. The Grothendieck-Witt group $GW^i(\mathcal C)$ is defined as the quotient of $\ker {d^i}$ by the submonoid generated by the elements $\co{A,\phi}-\co{A,0}$ where $A\in \mathcal C$ and $\phi$ is $(i-1)$-symmetric ($0$ is also seen as an $(i-1)$-symmetric morphism). 
\end{defin}

\begin{exem}
Let $(D^b(\mathcal P(X)),^\vee,1,\varpi)$ be the triangulated category with duality defined in the example \ref{dual}. Its Witt groups are the Witt groups $W^i(X)$ of the scheme $X$ as defined in \cite{Ba3}.
\end{exem}

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\subsection{Products}
Provided a pairing $\otimes:\mathcal C\times \mathcal D\to\mathcal M$ of triangulated categories with duality and assuming that this pairing satisfies some nice conditions, the authors of \cite{GN} define a pairing of Witt groups. We briefly recall some definitions (see 1.2 and 1.11 in \cite{GN}):

\begin{defin}
Let $\mathcal C, \mathcal D$ and $\mathcal M$ be triangulated categories. A product between $\mathcal C$ and $\mathcal D$ with codomain $\mathcal M$ is a covariant bi-functor 
$$\otimes:\mathcal C\times \mathcal D\to \mathcal M$$ 
exact in both variables and satisfying the following condition: the functorial isomorphisms $r_{A,B}:A\otimes TB\simeq T(A\otimes B)$ and $l_{A,B}:TA\otimes B\simeq T(A\otimes B)$ make the diagram

$$\xymatrix{TA\otimes TB\ar[r]^-{l_{A,TB}}\ar[d]_-{r_{TA,B}}  & T(A\otimes TB)\ar[d]^-{T(r_{A,B})}\\
T(TA\otimes B)\ar[r]_-{T(l_{A,B})} & T^2(A\otimes B)}$$
skew-commutative.
\end{defin}  

\begin{defin}\label{dual_pairing}
Let $\mathcal C, \mathcal D$ and $\mathcal M$ be triangulated categories with dualities. Where there is no possible confusion, we drop the subscripts for $D,\delta$ and $\varpi$. A dualizing pairing between $\mathcal C$ and $\mathcal D$ with codomain $\mathcal M$ is a product $\otimes$ with isomorphisms 

$$\eta_{A,B}:DA\otimes DB\simeq D(A\otimes B)$$
natural in $A$ and $B$ which make the following diagrams commute
\begin{enumerate}[1.]
\item $$\xymatrix{A\otimes B\ar[r]^-{\varpi_A\otimes\varpi_B}\ar[d]_-{\varpi_{A\otimes B}} & D^2A\otimes D^2B\ar[d]^-{\eta_{DA,DB}} \\
D^2(A\otimes B)\ar[r]_-{D(\eta_{A,B})} & D(DA\otimes DB)}$$

\item $$\xymatrix@C=3.5em{T(DTA\otimes DB)\ar[d]_-{\delta_\mathcal C\delta_\mathcal M T(\eta_{TA,B})} & DA\otimes DB\ar[l]_-{l_{DTA,DB}}\ar[d]_-{\eta_{A,B}}\ar[r]^-{r_{DA,DTB}} & T(DA\otimes TDB)\ar[d]^-{\delta_\mathcal L\delta_\mathcal M T(\eta_{A,TB})} \\
TD(TA\otimes B) & D(A\otimes B)\ar[l]^-{TD(l_{A,B})}\ar[r]_-{TD(r_{A,B})} & TD(A\otimes TB).}$$

\end{enumerate} 
\end{defin} 


\begin{thm}\label{prod_1}
Let $\mathcal C, \mathcal D$ and $\mathcal M$ be triangulated categories with duality.
Let $\otimes:\mathcal C\times \mathcal D\to \mathcal M$ be a dualizing pairing between $\mathcal C$ and $\mathcal D$ with codomain $\mathcal M$. Then $\otimes$ induces for all $i,j\in\Z$ a pairing

$$\star:W^i(\mathcal C)\times W^j(\mathcal D)\to W^{i+j}(\mathcal M).$$
\end{thm}

\begin{proof}
See \cite{GN}, thm 2.9.
\end{proof}

\begin{exem}\label{def_produit}
Let $(D^b(\mathcal P(X)),^\vee,1,\varpi)$ be the triangulated category with duality defined in example \ref{dual}. The usual tensor product induces a dualizing pairing of triangulated categories and then a product $W^i(X)\times W^j(X)\to W^{i+j}(X)$. Suppose that $L$ and $N$ are invertible modules over $X$. Then $\homm {\O_X}{\_}L$, $\homm {\O_X}{\_}N$ and $\homm {\O_X}{\_}{L\otimes N}$ give dualities $^\sharp$, $^\natural$ and $^\flat$ on $D^b(\mathcal P(X))$. The tensor product gives a dualizing pairing
$$\otimes:(D^b(\mathcal P(X)),^\sharp,1,\varpi)\times (D^b(\mathcal P(X)),^\natural,1,\varpi) \to (D^b(\mathcal P(X)),^\flat,1,\varpi).$$
\end{exem}



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\subsection{Supports}
We briefly recall the notion of triangulated category with supports following \cite{Ba1}.

\begin{defin}\label{supp_1}
Let $X$ be a topological space. A triangulated category defined over $X$ is a pair $(\mathcal C,\text{Supp})$ where $\mathcal C$ is a triangulated category and Supp assigns to each object $A\in \mathcal C$ a closed subset $\supp A$ of $X$ such that the following rules are satisfied:
\begin{itemize}
\item[(S1)]{$\supp A=\emptyset\iff A\simeq 0$.}
\item[(S2)]{$\supp{A\oplus B}=\supp A\cup \supp B$.}
\item[(S3)]{$\supp A=\supp{TA}$.}
\item[(S4)]{For every distinguished triangle
$$\xymatrix{A\ar[r] & B\ar[r] & C\ar[r] & TA}$$
we have $\supp C\subset \supp A\cup\supp B$.}

\end{itemize}
\end{defin}


When $\mathcal I$ is a saturated triangulated subcategory of $\mathcal C$ and $S$ is the multiplicative system of morphisms whose cone is in $\mathcal I$, then we can construct a support on the category $S^{-1}\mathcal C$. This is done in \cite{Ba4} when $\mathcal C$ has a tensor product. However we will only need some basic facts, so we prove the following lemma:

\begin{lem}\label{supp_2}
let $\mathcal C$ be a triangulated category defined over a topological space $X$. Let $\mathcal I$ be a triangulated subcategory of $\mathcal C$ and let $\supp{\mathcal I}=\cup_{A\in\mathcal I}\supp A$. Suppose that $\supp A\subset\supp{\mathcal I}$ implies $A\in \mathcal I$. Let $S$ be the multiplicative system in $\mathcal C$ of morphisms $f$ such that $\co f\in \mathcal I$ and let

$$\xymatrix{\mathcal I\ar[r] & \mathcal C\ar[r] & S^{-1}\mathcal C}$$
be the exact sequence of triangulated categories obtained by inverting $S$. Then $S^{-1}\mathcal C$ is a triangulated category defined over $X^\prime=X\setminus\supp{\mathcal I}$ (with the induced topology).
\end{lem}

\begin{proof}

It is easy to see that the rules (S1), (S2) and (S3) are satisfied. We only have to prove (S4).

First observe that if $s:A\to B$ is a morphism in $S$ and

$$\xymatrix{A\ar[r]^-s & B\ar[r] & C\ar[r] & TA}$$
is an exact triangle in $\mathcal C$ containing $s$, then $\sup SA=\sup SB$ (use (S4) for the category $\mathcal C$). This shows that $\sup SA=\sup S{A^\prime}$ if $A\simeq A^\prime$ in $S^{-1}\mathcal C$. Now observe that any exact triangle 

$$\xymatrix{A\ar[r]^-\alpha & B\ar[r] & C\ar[r] & TA}$$
in $S^{-1}\mathcal C$ is isomorphic to the localization of an exact triangle in $\mathcal C$. This shows that $\sup SC\subset \sup SA\cup\sup SB$.

\end{proof}

\begin{exem}\label{def_support} 
Let $D^b(\mathcal P(X))$ be the usual triangulated category. Define the support of an object $P\in D^b(\mathcal P(X))$ as the union of the support of all the cohomology groups of $P$, i.e

$$\supp P=\displaystyle{\bigcup_i \supp {H^i(P)}}.$$
Then it is easy to see that $(D^b(\mathcal P(X)),\text{Supp})$ is a triangulated category with support.
Denote by $D^b(\mathcal P(X))^{(k)}$ the full subcategory of $D^b(\mathcal P(X))$ of objects whose support is of codimension $\geq k$. Then $D^b(\mathcal P(X))^{(k)}$ is a triangulated category and the following sequence

$$D^b(\mathcal P(X))^{(k)}\to D^b(\mathcal P(X))\to D^b(\mathcal P(X))/D^b(\mathcal P(X))^{(k)}$$
satisfies the conditions of Lemma \ref{supp_2}. So $D^b(\mathcal P(X))/D^b(\mathcal P(X))^{(k)}$ is a triangulated category over $X^\prime=\{ x\in X\  \vert\ \codim x\leq k-1\}$.

\end{exem}

The following definitions are also due to Balmer (see \cite{Ba1}):

\begin{defin}
Let $(\mathcal C,\text{Supp})$ be a triangulated category over $X$ and assume that $\mathcal C$ has a structure of triangulated category with duality $(\mathcal C,D,\delta,\varpi)$. Then we say that $\mathcal C$ is a triangulated category with duality defined over $X$ if 
\begin{itemize} 
\item[(S5)]{$\supp A=\supp {DA}$ for every object $A$.}
\end{itemize}

\end{defin}

\begin{defin}
Let $(\mathcal C,\text{Supp}_\mathcal C)$, $(\mathcal D,\text{Supp}_\mathcal D)$ and $(\mathcal F,\text{Supp}_\mathcal F)$ be triangulated categories defined over $X$. Suppose that 

$$\otimes:\mathcal C\times \mathcal D\to \mathcal F$$
is a pairing of triangulated categories. The pairing $\otimes$ is defined over $X$ if 
\begin{itemize}
\item[(S6)]{$\sup {\mathcal F}{A\otimes B}=\sup {\mathcal C}A\cap \sup {\mathcal D}B.$}
\end{itemize}

\begin{exem}
The triangulated category $D^b(\mathcal P(X))$ with the support defined in the example \ref{def_support} and the pairing of example \ref{def_produit} satisfy the condition (S5) and (S6).
\end{exem}


\end{defin}

\begin{defin}
The degeneracy locus of a symmetric pair $(A,\alpha)$ is defined to be the support of the cone of $\alpha$:

$$\del\alpha=\supp {\co\alpha}.$$

\end{defin}


\begin{defin}
Let $(\mathcal C, \text{Supp})$ be a triangulated category with duality defined over $X$.
The consanguinity of two symmetric pairs $\alpha$ and $\beta$ is defined to be the following subset of $X$:

$$\cons \alpha\beta=(\supp \alpha\cap\del\beta)\cup (\del\alpha\cap \supp\beta).$$

\end{defin}

We are now ready to state the Leibnitz formula:

\begin{thm}[Leibnitz formula]\label{leibnitz}
Assume that we have a dualizing pairing $\otimes:\mathcal C\times\mathcal D\to\mathcal F$ of triangulated categories with dualities over $X$.
Let $\alpha$ and $\beta$ be two symmetric pairs such that $\del\alpha\cap\del\beta=\emptyset$. Then we have an isometry 

$$\delta_{\mathcal F}\cdot d(\alpha\star\beta)=\delta_{\mathcal C}\cdot d(\alpha)\star\beta+\delta_{\mathcal D}\cdot \alpha\star d(\beta)$$
where $\delta_\mathcal C,\delta_\mathcal D,\delta_\mathcal F$ are the signs involved in the dualities of $\mathcal C, \mathcal D$ and $\mathcal F$.

\end{thm}

\begin{proof}
See \cite{Ba1}, Theorem 5.2.
\end{proof}




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


\section{Oriented Chow groups}

Let $(D^b(\mathcal P(X)),^\vee,1,\varpi)$ be the triangulated category with the usual duality of example \ref{dual} and consider its full subcategory $D^b(\mathcal P(X))^{(i)}$ of objects with supports of codimension $\geq i$ (here we use the support defined in example \ref{def_support}). Then the duality on $D^b(\mathcal P(X))$ induces dualities on $D^b(\mathcal P(X))^{(i)}$ for any $i$ (\cite{Ba2}). It is also clear that $D^b(\mathcal P(X))^{(i+1)}\subset D^b(\mathcal P(X))^{(i)}$ for any $i$. 

\begin{defin}
For all $i\in \N$, denote by $D^b_i(X)$ the triangulated category $D^b(\mathcal P(X))^{(i)}/D^b(\mathcal P(X))^{(i+1)}$.
\end{defin}

Suppose that $(A,\alpha)$ is an $i$-symmetric form in $D^b_i(X)$. Then there exists an $i$-symmetric pair $(B,\beta)$ such that the localization of $(B,\beta)$ is $(A,\alpha)$ (by localization we mean the map $Symm^i(D^b(\mathcal P(X))^{(i)})\to Symm^i(D^b_i(X))$ induced by the functor $D^b(\mathcal P(X))^{(i)}\to D^b_i(X)$). Applying \ref{symmetric_cone}, we get an $(i+1)$-symmetric form $(C,\psi)$. By construction, $C\in D^b(\mathcal P(X))^{(i+1)}$. Localizing this form we get a form $(C,\psi)$ in $W^{i+1}(D_{i+1}^b(X))$. At first sight, this construction depends on some choices but in fact this is not the case (see \cite{Ba2}, Corollary 4.16). Hence we get a well defined homomorphism

$$d^i:W^i(D^b_i(X))\to W^{i+1}(D^b_{i+1}(X)).$$


\begin{thm}
Let $X$ be a regular scheme of dimension $n$. Then we have a complex

$$\xymatrix{0\ar[r] & W^0(D^b_0(X))\ar[r]^-{d^0} & W^1(D^b_1(X))\ar[r]^-{d^1} & \ldots \ar[r]^-{d^n} & W^n(D^b_n(X))\ar[r] & 0.}$$
\end{thm}

\begin{proof}
See \cite{BW}, Theorem 3.1 and paragraph 8. 
\end{proof}

Let $A$ be a regular local ring. We denote by $W^{fl}(A)$ the Witt group of finite length modules over $A$ (see \cite{QSS} for more informations about Witt groups of finite length modules). The following proposition holds:


\begin{prop}\label{isomfl}
We have isomorphisms 
$$W^i(D^b_i(X))\simeq \bigoplus_{x\in X^{(i)}} W^{fl}(\mathcal O_{X,x}).$$
\end{prop}

\begin{proof}
See \cite{BW}, Theorem 6.1 and Theorem 6.2.
\end{proof}

\begin{rem}\label{explicit}
Since we use the isomorphism of the above proposition, we briefly recall how to obtain a symmetric complex from a finite length module. For more details, see \cite{BW} or \cite{JF} (chapter 3). Choose a point $x\in X^{(i)}$, a finite length $\O_{X,x}$-module $M$ and a symmetric isomorphism $\phi:M\to \exten i{\O_{X,x}}M{\O_{X,x}}$. Let $P_\bullet$ be a resolution of $M$ by locally free coherent $\O_{X,x}$-modules. Then $P_\bullet$ can be chosen of the form

$$\xymatrix{0\ar[r] & P_i\ar[r] & \ldots\ar[r] & P_0\ar[r] & M\ar[r] & 0.}$$ 
Dualizing this complex and shifting $i$ times gives the following diagram

$$\xymatrix{0\ar[r] & P_i\ar[r]\ar@{-->}[d]_-{\exists} & \ldots\ar[r] & P_0\ar[r]\ar@{-->}[d]_-{\exists} & M\ar[r]\ar[d]_-\phi & 0 \\
0\ar[r] & P_0^\vee\ar[r] & \ldots\ar[r] & P_i^\vee\ar[r] & \exten i{\O_{X,x}}M{\O_{X,x}}\ar[r] & 0.}$$
Using $\phi$ we get a symmetric morphism $\varphi:P_\bullet\to (P_\bullet)^\vee$. Thus we have constructed an $i$-symmetric pair in the category $D^b(\mathcal P(\O_{X,x}))$ from the pair $(M,\phi)$. Since $D^b_i(X)\simeq \displaystyle{\coprod_{x\in X^{(i)}} D^b(\mathcal P(\mathcal O_{X,x}))}$ (\cite{BW}, Proposition 7.1), we can see the pair $(P_\bullet,\varphi)$ as a symmetric pair in $D^b_i(X)$. 

\end{rem}

\begin{defin}\label{defin_gersten}
The complex 

$$\xymatrix@C=1.2em{0\ar[r] & W^{fl}(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} W^{fl}(\mathcal O_{X,{x_1}})}\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} W^{fl}(\mathcal O_{X,x_n})}\ar[r] & 0}$$
is called the Gersten-Witt complex of $X$. We denote it by $C(X,W)$.
\end{defin}

This complex is obtained by using the usual duality $^\vee$ on the triangulated category $D^b(\mathcal P(X))$ (example \ref{dual}). For any invertible module $L$ over $X$, we have a duality derived from the functor $^\sharp=\homm {\O_{X,x}}{\_}L$ and we can apply the same construction to get a Gersten-Witt complex. 

\begin{defin}
Let $X$ be a regular scheme and $L$ an invertible $\O_X$-module. We denote by $C(X,W,L)$ the Gersten-Witt complex obtained from the duality $^\sharp$.
\end{defin}


\begin{thm}
Let $A$ be a regular semi-local $k$-algebra and $X=\spec A$. Then for any $i>0$ we have $H^i(C(X,W))=0$.
\end{thm}

\begin{proof}
See \cite{BGPW}, Theorem 6.1.

\end{proof}

Let $A$ be a regular local ring of dimension $n$. Denote by $F$ the residue field of $A$. Then any choice of a generator $\xi\in \ext nF$ gives an isomorphism $\alpha_\xi:W(F)\to W^{fl}(A)$. Recall that $I(F)$ is the fundamental ideal of $W(F)$. If $n\leq 0$, put $I^n(F)=W(F)$. 

\begin{defin}
For any $n\in \Z$ let $I_{fl}^n(A)$ be the image of $I^n(F)$ by $\alpha_\xi$.  
\end{defin}

\begin{rem}
It is easily seen that $I_{fl}^n(A)$ does not depend on the choice of the generator $\xi\in \ext nF$.
\end{rem}


\begin{prop}
The differential $d$ of the Gersten-Witt complex satisfies $d(I_{fl}^m(\mathcal O_{X,x}))\subset I_{fl}^{m-1}(\mathcal O_{X,y})$ for any $m\in \Z$, $x\in X^{(i)}$ and $y\in X^{(i-1)}$.    
\end{prop}

\begin{proof}
See \cite{Gi}, Theorem 6.4 or \cite{JF}, Theorem 9.2.4. 
\end{proof}

\begin{defin}
Let $L$ be an invertible $\O_X$-module. We denote by $C(X,I^d,L)$ the complex

$$\xymatrix@C=1.2em{0\ar[r] & I_{fl}^d(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} I_{fl}^{d-1}(\mathcal O_{X,{x_1}})}\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} I_{fl}^{d-n}(\mathcal O_{X,x_n})}\ar[r] & 0.}$$
\end{defin}

\begin{rem}\label{Izero}
In particular, we have $C(X,I^0,L)=C(X,W,L)$.
\end{rem}

\begin{thm}\label{exa_I}
Let $A$ be a regular local $k$-algebra. Then for any $i>0$ we have $H^i(C(X,I^d))=0$. 
\end{thm}

\begin{proof}
See \cite{Gi}, Corollary 7.7.
\end{proof}

Of course, there is an inclusion $C(X,I^{d+1},L)\to C(X,I^d,L)$ and therefore we get a quotient complex.

\begin{defin}
Denote by $C(X,\overline I^d)$ the complex $C(X,I^d,L)/C(X,I^{d+1},L)$.
\end{defin}

\begin{rem}
For any invertible module $L$ the complexes $C(X,I^d)/C(X,I^{d+1})$ and $C(X,I^d,L)/C(X,I^{d+1},L)$ are canonically isomorphic (see \cite{JF}, Corollary E.1.3), so we can drop the $L$ in $C(X,\overline I^d)$. 
\end{rem}

\begin{rem}
The complex $C(X,\overline I^d)$ is of the form

$$\xymatrix@C=1.2em{0\ar[r] & I_{fl}^d(k(X))/I_{fl}^{d+1}(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} I_{fl}^{d-1}(\mathcal O_{X,{x_1}})/I_{fl}^{d}(\mathcal O_{X,{x_1}})}\ar[r] & \ldots.}$$
\end{rem}


\begin{rem}
As a consequence of Theorem \ref{exa_I}, we immediately see that $H^i(C(X,\overline I^d))=0$ for $i>0$ if $X=\spec A$ where $A$ is a regular local $k$-algebra. 
\end{rem}

Let $F$ be a field and denote by $K_i^M(F)$ the $i$-th Milnor K-theory group of $F$. If $i<0$ it is convenient to put $K_i^M(F)=0$.


\begin{con} Let $X$ be a scheme. Then for any $d$ we have a complex 

$$\xymatrix@C=1.0em{0\ar[r] & K^M_d(k(X))\ar[r] & \displaystyle{\bigoplus_{x_1\in X^{(1)}} K^M_{d-1}(k(x_1))}\ar[r] & \ldots \ar[r] & \displaystyle{\bigoplus_{x_n\in X^{(n)}} K^M_{d-n}(k(x_n))}\ar[r] & 0.}$$
We denote it by $C(X,K^M_d)$.
\end{con}

\begin{proof}
See \cite{Ro}, paragraph 3 or \cite{JF} chapter 2.
\end{proof}

We also have the exactness of this complex when $X$ is the spectrum of a smooth semi-local $k$-algebra:

\begin{thm}\label{exa_K}
Let $A$ be a smooth local $k$-algebra. Then for all $i>0$ we have $H^i(C(X,K^M_d))=0$.
\end{thm}

\begin{proof}
See \cite{Ro}, Theorem 6.1.
\end{proof}

If $F$ is a field, recall that we have a homomorphism 

$$s:K^M_j(F)\to I^j(F)/I^{j+1}(F)$$
given by $s(\{a_1,\ldots,a_j\})=<1,-a_1>\otimes\ldots\otimes <1,-a_j>$. The following is true: 

\begin{lem}
The homomorphisms $s$ induce a morphism of complexes 
$$s:C(X,K^M_d)\to C(X,\overline I^d).$$
\end{lem}

\begin{proof}
See \cite{JF}, Proposition 10.2.5.
\end{proof}

\begin{defin}
Let $C(X,G^d,L)$ be the fibre product of $C(X,K^M_d)$ and $C(X,I^d,L)$ over $C(X,\overline I^d)$:

$$\xymatrix{C(X,G^d,L)\ar[r]\ar[d] & C(X,I^d,L)\ar[d]^-\pi \\
C(X,K^M_d)\ar[r]_-s & C(X,\overline I^d).}$$
\end{defin}


\begin{defin}
Let $X$ be a smooth scheme and $L$ an invertible $\O_X$-module. The $j$-th oriented Chow group $\chs jXL$ of $X$ twisted by $L$ is the group $H^j(C(X,G^j,L))$.
\end{defin}

\begin{rem}\label{groth_witt2}
Denote by $GW^j(D^b_j(X))$ the $j$-th Grothendieck-Witt group of the category $D^b_j(X)$ with the duality derived from $\homm {\O_X}{\_}L$ (see definition \ref{groth_witt}).
By definition, the complex $C(X,G^j,L)$ is 

$$\xymatrix@C=2em{\ldots\ar[r] & C(X,G^j,L)_{j-1}\ar[r] & GW^{j}(D^b_j(X))\ar[r]^-{d^j} & W^{j+1}(D^b_{j+1}(X))\ar[r] & \ldots}$$
Hence $\chs jXL$ is a quotient of $\ker{d^j}$.
  
\end{rem}

We also have the exactness of the complex $C(X,G^d,L)$ in the local case: 

\begin{thm}\label{exa_G}
Let $A$ be a smooth local $k$-algebra and $X=\spec A$. Then $H^i(C(X,G^j))=0$ for all $j$ and all $i>0$.
\end{thm}

\begin{proof}
As $C(X,G^j)$ is the fibre product of the complexes $C(X,K^M_j)$ and $C(X,I^j)$ over $C(X,\overline I^j)$, we have an exact sequence of complexes

$$\xymatrix{0\ar[r] &C(X,G^j)\ar[r] & C(X,I^j)\oplus C(X,K^M_j)\ar[r] & C(X,\overline I^j)\ar[r] & 0 }$$
inducing a long exact sequence in cohomology. It follows then from Theorems \ref{exa_I} and \ref{exa_K} that $H^i(C(X,G^j))=0$ if $i>1$. For $i=1$, we have an exact sequence

$$\xymatrix@C=1.2em{H^0(C(X,I^j))\oplus H^0(C(X,K^M_j))\ar[r] & H^0(C(X,\overline I^j))\ar[r] & H^1(C(X,G^j))\ar[r] & 0.}$$
The exact sequence of complexes 

$$\xymatrix{0\ar[r] & C(X,I^{j+1})\ar[r] & C(X,I^j)\ar[r] & C(X,\overline I^j)\ar[r] & 0}$$
shows that $H^0(C(X,I^j))$ maps onto $H^0(C(X,\overline I^j))$.
\end{proof}

\begin{defin}
Let $X$ be a smooth scheme and $L$ an invertible $\O_X$-module. We define the sheaf $\mathcal G^j_L$ on $X$ by $\mathcal G^j_L(U)=H^0(C(U,G^j,L))$.
\end{defin}
We have:

\begin{thm}
Let $X$ be a smooth scheme of dimension $n$. Then for any $i$ we have 
$$H^i_{Zar}(X,\mathcal G^j_L)\simeq H^i(C(X,G^j,L)).$$
\end{thm}

\begin{proof}
Define sheaves $\mathcal C_l$ by $\mathcal C_l(U)=C(U,G^j,L)_l$ for any $l\geq 0$. It is clear that the $\mathcal C_l$ are flasque sheaves. We have a complex of sheaves over $X$

$$\xymatrix{0\ar[r] & \mathcal G^j_L\ar[r] & \mathcal C_0\ar[r] & \mathcal C_1\ar[r] & \ldots\ar[r] & \mathcal C_n\ar[r] & 0.}$$
Theorem \ref{exa_G} shows that this complex is a flasque resolution of $\mathcal G^j_L$. Thus the theorem is proved. 

\end{proof}


Suppose that $f:X\to Y$ is a flat morphism. Since it preserves codimensions, it induces a morphism of complexes 

$$f^*:C(Y,G^j,L)\to C(X,G^j,f^*L)$$
for any $j\in\N$ and any line bundle $L$ over $Y$ (\cite{JF}, Corollary 10.4.2). Hence we have:

\begin{thm}\label{homotopy}
Let $f:X\to Y$ be a flat morphism and $L$ a line bundle over $Y$. Then, for any $i,j$ we have homomorphisms

$$f^*:H^i(C(Y,G^j,L))\to H^i(C(X,G^j,f^*L)).$$
In particular, if $E$ is a vector bundle over $Y$ and $\pi:E\to Y$ is the projection, we have isomorphisms

$$\pi^*:H^i(C(Y,G^j,L))\to H^i(C(E,G^j,\pi^*L)).$$
\end{thm}

\begin{proof}
We have a morphism of complexes $f^*:C(Y,G^j,L)\to C(X,G^j,f^*L)$ which gives the induced homomorphisms in cohomology. For the proof of homotopy invariance, see Corollary 11.3.2 in \cite{JF}.
\end{proof}

\begin{prop}\label{func_dom}
Let $f:X\to Y$ and $g:Y\to Z$ be flat morphisms. Then $(gf)^*=f^*g^*$.
\end{prop}

\begin{proof}
See \cite{JF}, Proposition 3.4.9.
\end{proof}



Suppose that $f:X\to Y$ is a finite morphism with $\dim Y-\dim X=~r$. Consider the morphism of locally ringed spaces $\overline f:(X,\O_X)\to (Y,f_*\O_X)$ induced by $f$. If $X$ is smooth, then $L=\overline f^*\exten r{\O_Y}{f_*\O_X}{\O_Y}$ is an invertible module over $Y$ (\cite{Gi1}, Corollary 6.6) and we get a morphism of complexes (of degree r)
 
$$f_*:C(X,G^{j-r},L\otimes f^*N)\to C(Y,G^j,N)$$
for any invertible module $N$ over $Y$ (\cite{JF}, Corollary 5.3.7). 

\begin{prop}\label{fin_morphism}
Let $f:X\to Y$ be a finite morphism between smooth schemes. Let $\dim Y-\dim X=r$ and $N$ be an invertible module over $Y$. Then the morphism of complexes $f_*$ induces a homomorphism

$$f_*:H^{i-r}(C(X,G^{j-r},L\otimes f^*N))\to H^i(C(Y,G^j,N)).$$
\end{prop}
In particular, we have (\cite{JF}, Remark 9.3.5):

\begin{prop}\label{finite_morphism}
Let $f:X\to Y$ be a closed immersion of codimension $r$ between smooth schemes. Then $f$ induces an isomorphism  

$$f_*:H^{i-r}(C(X,G^{j-r},L\otimes f^*N))\to H^i_Y(C(Y,G^j,N))$$
for any $i,j$ and any invertible module $N$ over $Y$.
 
\end{prop}

\begin{imrem}
If $f:X\to Y$ is a closed immersion, then $f_*$ will always be the map with support:

$$f_*:H^{i-r}(C(X,G^{j-r},L\otimes f^*N))\to H^i_Y(C(Y,G^j,N))$$
\end{imrem}
The transfer for finite morphisms is functorial (\cite{JF}, proposition 5.3.8):

\begin{prop}\label{func_finite}
Let $f:X\to Y$ and $g:Y\to Z$ be finite morphisms. Then $g_*f_*=(gf)_*$.
\end{prop} 

\begin{rem}\label{in_cartier}
Let $X$ be a smooth scheme and $D$ be a smooth effective Cartier divisor on $X$. Let $i:D\to X$ be the inclusion and $L(D)$ be the line bundle over $X$ associated to $D$. Then there is a canonical section $s\in L(D)$ (see \cite{Fu}, Appendix B.4.5) and therefore an exact sequence

$$\xymatrix{0\ar[r] & \mathcal O_X\ar[r]^-s & L(D)\ar[r] & i_*\mathcal O_D\ar[r] & 0.}$$
Applying $\homm {\mathcal O_X}{\_}{L(D)}$ and shifting, we obtain the following diagram

$$\xymatrix@C=0.9em{0\ar[r] & \mathcal O_X\ar[r]^-s\ar[d]_-\simeq & L(D)\ar[r]\ar[d]_-\simeq & i_*\mathcal O_D\ar[r]\ar@{-->}[d] & 0\\
0\ar[r] & \homm {\mathcal O_X}{L(D)}{L(D)}\ar[r]_-s & \homm {\mathcal O_X}{\mathcal O_X}{L(D)}\ar[r] & \exten 1{\mathcal O_X}{i_*\mathcal O_D}{L(D)}\ar[r] & 0}$$
which shows that $\exten 1{\mathcal O_X}{i_*\mathcal O_D}{\mathcal O_X}\otimes L(D)\simeq i_*\mathcal O_D$. 
Proposition \ref{finite_morphism} shows that we then have an isomorphism

$$i_*:H^{i-1}(C(D,G^{j-1},i^*L(D)))\to H^i_{D}(C(X,G^j)).$$
\end{rem}

\begin{lem}\label{fibre_product}
Let $g:X\to Y$ be a flat morphism and $f:Z\to Y$ a finite morphism. Consider the following fibre product

$$\xymatrix{V\ar[r]^-{f^\prime}\ar[d]_-{g^\prime} & X\ar[d]^-g\\
Z\ar[r]_-f & Y.}$$
Then $(f^\prime)_* (g^\prime)^*=g^*f_*$.
\end{lem}

\begin{proof}
See \cite{JF}, Corollary 12.2.8.
\end{proof}

\begin{rem}
Of course, in the above fibre product we suppose that $V$ is also smooth and integral. Such a strong assumption is not necessary in general, but this case is sufficient for our purposes.
\end{rem}


\begin{rem}
It is possible to define a map $f_*$ when the morphism $f$ is proper (see \cite{JF}) but we don't use this fact here.
\end{rem}




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

\section{The exterior product}

Let $X$ and $Y$ be two schemes. The fibre product $X\times Y$ comes equipped with two projections $p_1:X\times Y\to X$ and $p_2:X\times Y\to Y$. %

\begin{lem}
For any $i,j\in \N$, the pairing 

$$\boxtimes:D^b_i(X)\times D^b_j(Y)\to D^b_{i+j}(X\times Y)$$
given by $P\boxtimes Q=p_1^*P\otimes p_2^*Q$ is a dualizing pairing of triangulated categories with duality.
\end{lem}

\begin{proof}
Straight verification.
\end{proof}

\begin{cor}
For any $i,j\in \N$, the pairing 

$$\boxtimes:D^b_i(X)\times D^b_j(Y)\to D^b_{i+j}(X\times Y)$$
induces a pairing 

$$\star:W^i(D^b_i(X))\times W^j(D^b_j(Y))\to W^{i+j}(D^b_{i+j}(X\times Y)).$$
\end{cor}

\begin{proof}
Clear by Theorem \ref{prod_1}.
\end{proof}


\begin{cor}
Let $\psi\in W^j(D^b_j(Y))$. Then we have a homomorphism

$$\mu_\psi:W^j(D^b_i(X))\to W^{i+j}(D^b_{i+j}(X\times Y))$$
given by $\mu_\psi(\varphi)=\varphi\star\psi$.

\end{cor}

Recall that we have isomorphisms $W^i(D^b_i(X))\simeq \displaystyle{\bigoplus_{x\in X^{(i)}}W^{fl}(\mathcal O_{X,x})}$ (Proposition \ref{isomfl}).
\begin{defin}
Denote by $I^s(D^b_i(X))$ the preimage of $\displaystyle{\bigoplus_{x\in X^{(i)}}I_{fl}^s(\mathcal O_{X,x})}$ under the above isomorphism.
\end{defin}

\begin{prop}\label{prod_id}
For any $m,p\in \N$ the product 

$$\star:W^i(D^b_i(X))\times W^j(D^b_j(Y))\to W^{i+j}(D^b_{i+j}(X\times Y))$$
induces a product 

$$\star:I^m(D^b_i(X))\times I^n(D^b_j(Y))\to I^{m+n}(D^b_{i+j}(X\times Y)).$$

\end{prop}

\begin{proof}
Let $x\in X^{(i)}$ and $y\in Y^{(j)}$. It is clear that the product can be computed locally (use \cite{GN}, theorem 3.2). So we can suppose that $X=\spec{A}$ and $Y=\spec{B}$ where $A$ and $B$ are local in $x$ and $y$ respectively. Recall that we have the following diagram

$$\xymatrix{X\times Y\ar[r]^-{p_2}\ar[d]_-{p_1} & Y\ar[d] \\
X\ar[r] & \spec k.}$$
Let $P$ be an $A$-projective resolution of $k(x)$ and $Q$ be a $B$-projective resolution of $k(y)$. Consider a symmetric form $\rho:k(x)\to \ext i{k(x)}$ and a symmetric form $\mu:k(y)\to \exten jB{k(y)}B$. Then $p_1^*(\rho)$ is a symmetric isomorphism supported by the complex $P\otimes B$ and $p_2^*(\mu)$ is a symmetric isomorphism supported by the complex $A\otimes Q$. Clearly, the complex $P\otimes B\otimes A\otimes Q$ has its homology supported in degree $0$, and this homology is isomorphic to $k(x)\otimes_k k(y)$. Let $u$ be a point of $\spec {k(x)\otimes_k k(y)}$. Then the restriction of $p_1^*\rho\otimes p_2^*\mu$ to $u$ is a finite length module $M$ whose support is on $u$ with a symmetric form 
$$M\to \exten {i+j}{(A\otimes B)_u}M{(A\otimes B)_u}.$$
Taking its class in the Witt group, we obtain a $k(u)$-vector space $V$ with a symmetric form $\psi:V\to \exten {i+j}{(A\otimes B)_u}V{(A\otimes B)_u}$. Now choose a unit $a\in k(x)^\times$. Consider the image $a_u$ of $a$ under the homomorphism $k(x)\to k(u)$. The class of $p_1^*(a\rho)\otimes p_2^*(\mu)$ is the symmetric form 
$$a_u\psi:V\to \exten {i+j}{(A\otimes B)_u}V{(A\otimes B)_u}.$$
As the same property holds for any unit $b\in k(y)^\times$, we conclude that 

$$p_1^*(<1,-a_1>\otimes\ldots\otimes<1,-a_n>\rho)\otimes p_2^*(<1,-b_1>\otimes\ldots\otimes<1,-b_m>\mu)$$
is equal to $<1,-(a_1)_u>\otimes\ldots\otimes<1,-(b_m)_u>\psi$.   
  
\end{proof}
Recall that for any scheme $X$ we have a Gersten-Witt complex (Definition \ref{defin_gersten})

$$\xymatrix{C(X,W): & \ldots\ar[r] & W^r(D^b_r(X))\ar[r]^-{d^r_X} & W^{r+1}(D^b_{r+1}(X))\ar[r] & \ldots}$$
and a complex $C(X,I^d)$:

$$\xymatrix@C=1.7em{\ldots\ar[r] & \displaystyle{\bigoplus_{x_r\in X^{(r)}} I_{fl}^{d-r}(\mathcal O_{X,{x_r}})}\ar[r] & \displaystyle{\bigoplus_{x_{r+1}\in X^{(r+1)}} I_{fl}^{d-r-1}(\mathcal O_{X,{x_{r+1}}})}\ar[r] &\ldots .}$$
The above proposition gives:

\begin{cor}
The product 

$$\star:C(X,W)\times C(Y,W)\to C(X\times Y,W)$$
induces for any $r,s\in N$ a product 

$$\star:C(X,I^r)\times C(Y,I^s)\to C(X\times Y,I^{r+s}).$$
\end{cor}
Now we investigate the relations between $\star$ and the differentials of the complexes.


\begin{prop}
Let $\psi\in W^j(D^b_j(Y))$ be such that $d_j^Y(\psi)=0$. Then the following diagram commutes

$$\xymatrix{W^i(D^b_i(X))\ar[r]^-{d^i_X}\ar[d]_-{(-1)^j\mu_\psi} & W^{i+1}(D^b_{i+1}(X))\ar[d]^-{\mu_\psi} \\
W^{i+j}(D^b_{i+j}(X\times Y))\ar[r]_-{d^{i+j}_{X\times Y}} & W^{i+j+1}(D^b_{i+j+1}(X\times Y)).}$$  
\end{prop}

\begin{proof}
Let $\varphi\in W^i(D^b_i(X))$. Let $X_{i+1}$ be the set of points of $X$ of codimension $\geq i+1$, $Y_{j+1}$ the set of point of $Y$ of codimension $\geq j+1$ and $(X\times Y)_{i+j+1}$ the set of points of $X\times Y$ of codimension $\geq i+j+1$. By Lemma \ref{supp_2}, the triangulated categories $D^b_i(X),D^b_j(Y)$ and $D^b_{i+j}(X\times Y)$ are defined over the topological spaces $X\setminus X_{i+1}$, $Y\setminus Y_{j+1}$ and $(X\times Y)\setminus (X\times Y)_{i+j+1}$. Let $\alpha\in Symm^i(D^b(\mathcal P(X))^{(i)})$ and $\beta\in Symm^j(D^b(\mathcal P(Y))^{(j)})$ be symmetric pairs representing $\varphi$ and $\psi$. By definition, we have $\del\alpha\in D^b(\mathcal P(X))^{(i+1)}$, $\del\beta\in D^b(\mathcal P(Y))^{(j+1)}$ and $d\beta$ is neutral. It is easily seen that $\supp{dp_1^*\alpha}\cap \supp{dp_2^*\beta}=\emptyset$ in the topological space $(X\times Y)\setminus (X\times Y)_{i+j+1}$. Theorem \ref{leibnitz} implies that 

$$(-1)^{i+j}d(p_1^*\alpha\star p_2^*\beta)=(-1)^i dp_1^*\alpha\star p_2^*\beta +(-1)^jp_1^*\alpha\star dp_2^*\beta.$$
Using Theorem \ref{prod_1}, we see that we have in $W^{i+j}(D^b_{i+j}(X\times Y))$ the equality 

$$(-1)^jd_{X\times Y}^{i+j}(p_1^*\varphi\star p_2^*\psi)=p_1^* d^i_X(\varphi)\star p_2^*\psi.$$ 
\end{proof}

The following corollary is obvious.

\begin{cor}\label{commut_I}
Let $\psi\in I^m(D^b_j(Y))$ be such that $d_j^Y(\psi)=0$. Then the following diagram commutes

$$\xymatrix{I^p(D^b_i(X))\ar[r]^-{d^i_X}\ar[d]_-{(-1)^j\mu_\psi} & I^{p-1}(D^b_{i+1}(X))\ar[d]^-{\mu_\psi} \\
I^{p+m}(D^b_{i+j}(X\times Y))\ar[r]_-{d^{i+j}_{X\times Y}} & I^{p+m-1}(D^b_{i+j+1}(X\times Y)).}$$
\end{cor}


We now have to deal with the complex in Milnor K-theory. Let $C(X,K^M_r)$, $C(Y,K^M_s)$ and $C(X\times Y,K^M_{r+s})$ be the complexes in Milnor K-theory associated to $X,Y$ and $X\times Y$. In \cite{Ro}, the author defines a product 

$$\odot:C(X,K^M_r)^i\times C(Y,K^M_s)^j\to C(X\times Y,K^M_{r+s})^{i+j}$$ as follows:
Let $u\in (X\times Y)^{(i+j)}$, $x\in X^{(i)}$, $y\in Y^{(j)}$ be such that $x$ and $y$ are the projections of $u$. Let $\rho=\{a_1,\ldots,a_{r-i}\}\in K_{r-i}^M(k(x))$ and $\mu=\{b_1,\ldots,b_{s-j}\}\in K_{s-j}^M(k(y))$. Then  

$$(\rho\odot\mu)_u=l((k(x)\otimes_k k(y))_u)\{(a_1)_u,\ldots,(a_{d-i})_u,(b_1)_u,\ldots,(b_{s-j})_u\}$$
where the $(a_l)_u$ and $(b_t)_u$ are the images of the $a_l$ and $b_t$ under the inclusions $k(x)\to k(u)$ and $k(y)\to k(u)$, and $l((k(x)\otimes_k k(y))_u)$ is the length of the module $k(x)\otimes_k k(y)$ localized in $u$.

\begin{lem}

For any $\rho\in C(X,K_r^M)^i$ and $\mu\in C(Y,K_s^M)^j$ we have 
$$d(\rho\odot\mu)=d(\rho)\odot \mu+(-1)^j\rho\odot d(\mu).$$
\end{lem}

\begin{proof}
See \cite{Ro}, 14.4, p 391.
\end{proof}

\begin{cor}\label{commut_K}
Let $\mu\in C(Y,K_s^M)^j$ be such that $d\mu=0$. Then the following diagram commutes:

$$\xymatrix{C(X,K^M_r)^i\ar[r]^-{d_X^i}\ar[d]_-{\odot\mu} & C(X,K^M_r)^{i+1}\ar[d]^-{\odot\mu} \\
C(X\times Y,K^M_{r+s})^{i+j}\ar[r]_-{d_{X\times Y}^{i+j}}  & C(X\times Y,K^M_{r+s})^{i+j+1}.}$$
\end{cor}

\begin{proof}
Obvious.
\end{proof}

Now we compare the products $\star$ and $\odot$.

\begin{prop}\label{compar}
The following diagram commutes:

$$\xymatrix{C(X,K^M_r)^i\times C(Y,K^M_s)^j\ar[r]^-\odot\ar[d]_-{s_{(r-i)}\times s_{(s-j)}} & C(X\times Y,K^M_{r+s})^{i+j}\ar[d]^-{s_{(r+s-i-j)}} \\
C(X,\overline I^r)^i\times C(Y,\overline I^s)^j \ar[r]_-{\star} & C(X\times Y,\overline I^{r+s})^{i+j}.}$$
\end{prop} 

\begin{proof}

Let $\{a_1,\ldots,a_{r-i}\}\in K^M_{r-i}(k(x))$ and $\{b_1,\ldots,b_{s-j}\}\in K^M_{s-j}(k(y))$. Let $\rho^\prime$ be a symmetric isomorphism

$$\rho^\prime:k(x)\to \exten i{\mathcal O_{X,x}}{k(x)}{\mathcal O_{X,x}}$$
and $\mu^\prime$ a symmetric isomorphism

$$\mu^\prime:k(y)\to \exten j{\mathcal O_{Y,y}}{k(y)}{\mathcal O_{Y,y}}.$$
We then have $s_{(r-i)}(\{a_1,\ldots,a_{r-i}\})=<1,-a_1>\otimes\ldots \otimes<1,-a_{r-i}>\rho^\prime:=\rho$ and $s_{(s-j)}(\{b_1,\ldots,b_{s-j}\})=<1,-b_1>\otimes\ldots\otimes<1,-b_{s-j}>\mu^\prime:=\mu$.
Choose a point $u$ in $(X\times Y)^{(i+j)}$ lying over $x$ and $y$. The proof of Proposition \ref{prod_id} shows that 

$$(\rho\star\mu)_u=s_{(r+s-i-j)}(\{(a_1)_u,\ldots,(a_{r-i})_u,(b_1)_u,\ldots, (b_{s-j})_u\})\varphi$$
where $\varphi:M\to \exten {i+j}{\mathcal O_{X\times Y,u}}{M}{\mathcal O_{X\times Y,u}}$ is a symmetric isomorphism and $M$ is a $k(u)$-vector space. But $\text{dim}_{k(u)}M\equiv l((k(x)\otimes k(y))_u)\pmod 2$ where $l$ denotes the length. So we have in $C(X\times Y,\overline I^{r+s})^{i+j}$ the equality

$$(\rho\star\mu)_u=s_{(r+s-i-j)}(\{(a_1)_u,\ldots,(a_{r-i})_u,(b_1)_u,\ldots, (b_{s-j})_u\})l((k(x)\otimes k(y))_u).$$
The right hand term is equal to $s_{(r+s-i-j)}(\{a_1,\ldots,a_{r-i}\}\odot\{b_1,\ldots,b_{s-j}\})$ by definition.
\end{proof}

\begin{cor}
The products 

$$\star:C(X,I^r)\times C(Y,I^s)\to C(X\times Y,I^{r+s})$$
and 

$$\odot:C(X,K_r^M)\times C(Y,K_s^M)\to C(X\times Y,K_{r+s}^M)$$
give a product

$$\diamond:C(X,G^r)\times C(Y,G^s)\to C(X\times Y,G^{r+s}).$$

\end{cor}

\begin{cor}
Let $\mu\in C(Y,G^s)^j$ such that $d^i_Y\mu=0$. Then $\mu$ induces a product

$$\_\diamond\mu:H^i(C(X,G^r))\to H^{i+j}(C(X\times Y,G^{r+s})).$$
\end{cor}

\begin{proof}
This a direct consequence of Proposition \ref{compar}, Corollary \ref{commut_I} and Corollary \ref{commut_K}.
\end{proof}

Next we have to check that $\_\diamond\mu$ is well defined on the cohomolgy class of~$\mu$.

\begin{lem}
Let $\gamma\in C(Y,G^s)^{j-1}$ and $\mu=d_Y^{j-1}\gamma$. Then $\_\diamond\mu=0$.
\end{lem}

\begin{proof}
Suppose that $\alpha$ is such that $d^i_X\alpha=0$. By Corollaries \ref{commut_I} and \ref{commut_K} we have up to signs $d^{i+j-1}_{X\times Y}(\alpha\diamond\gamma)=\alpha\diamond d^{j-1}\gamma=\alpha\diamond\mu$. So $\alpha\diamond\mu$ is trivial in $H^{i+j}(C(X\times Y,G^{r+s}))$.
\end{proof}

Finally:
\begin{thm}\label{extprod}
Let $X$ and $Y$ be smooth schemes. Then for any $i,j,r,s\in \N$ the product

$$\diamond:C(X,G^r)\times C(Y,G^s)\to C(X\times Y,G^{r+s})$$
induces an exterior product

$$\times:H^i(C(X,G^r))\times H^j(C(Y,G^s))\to H^{i+j}(C(X\times Y,G^{r+s})).$$
\end{thm}
This exterior product can also be defined with complexes twisted by invertible modules.

\begin{thm}\label{extprodtwist}
Let $X$ and $Y$ be smooth schemes. Let $L$ and $N$ be invertible modules over $X$ and $Y$ respectively. For any $i,j,r,s\in \N$, the pairing

$$\diamond:C(X,G^r,L)\times C(Y,G^s,N)\to C(X\times Y,G^{r+s},p_1^*L\otimes p_2^*N)$$
induces an exterior product

$$\times:H^i(C(X,G^r,L))\times H^j(C(Y,G^s,N))\to H^{i+j}(C(X\times Y,G^{r+s},p_1^*L\otimes p_2^*N)).$$

\end{thm}

\begin{proof}
Left to the reader.
\end{proof}

If $i=d$ and $j=s$, we obtain the following corollary:

\begin{cor}
Let $X$ and $Y$ be smooth schemes. Then for any $i,j\in\N$ the product

$$\diamond:C(X,G^r)\times C(Y,G^s)\to C(X\times Y,G^{r+s})$$
gives an exterior product

$$\times:\ch iX\times\ch jY\to \ch {i+j}{X\times Y}.$$
\end{cor}

Next we prove some properties of this exterior product:

\begin{prop}\label{assoc_ext}
The exterior product $\times$ is associative. 
\end{prop}

\begin{proof}
It clearly suffices to prove that the exterior products $\star$ and $\odot$ are associative. For $\star$ this is clear because of the associativity of the tensor product (up to isomorphism). For the second, see (14.2) in \cite{Ro}.
\end{proof}

Now we deal with the commutativity. Let $X$ and $Y$ be smooth schemes and let $\tau:X\times Y\to Y\times X$ be the flip. We have:

\begin{lem}\label{anticommut_ext1}
Let $\mu\in H^i(C(X,K^M_r))$ and $\eta\in H^j(C(Y,K^M_s))$. Then we have $\tau^*(\eta\odot\mu)=(-1)^{(r-i)(s-j)}(\mu\odot\eta)$.
\end{lem} 

\begin{proof}
This is clear from the definition.
\end{proof}

\begin{lem}\label{anticommut_ext2}
Let $\mu\in H^i(C(X,I^r))$ and $\eta\in H^j(C(Y,I^s))$. Then we have $\tau^*(\eta\star\mu)=(-1)^{ij}(\mu\star\eta)$.
\end{lem}

\begin{proof}
It is clear by the skew-commutativity of the product of Witt groups (\cite{GN}, Theorem 2.9).
\end{proof}

\begin{rem}
Of course, the associativity and the anticommutativity of the exterior product are also true for the twisted product of Theorem \ref{extprodtwist}. 
\end{rem}


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

\section{Intersection with a smooth subscheme}

\subsection{The oriented Gysin map}
The goal of this section is to define for any closed embedding $i:Y\to X$ of smooth schemes an oriented Gysin map $i^!:H^r(C(X,G^j))\to H^r(C(Y,G^j))$.
In order to define such a map, we slightly adapt the ideas of Rost (\cite{Ro}, paragraph 11).

First we briefly recall the properties of the deformation to the normal cone. For more details, see \cite{Fu} (chapter 5) or \cite{Ro} (chapter 10). Let $Y$ be a closed subscheme of a smooth scheme $X$. Then there is a smooth scheme $D(X,Y)$, a closed imbedding $j:Y\times \A^1\hookrightarrow D(X,Y)$ and a flat morphism $\rho:D(X,Y)\to \A^1$ such that the following diagram commutes 

$$\xymatrix{Y\times \A^1\ar[r]^-j\ar[rd]_-{pr} & D(X,Y)\ar[d]^-\rho\\
 & \A^1}$$ 
and 
\begin{enumerate}[(1)]
\item $\rho^{-1}(\A^1-0)=X\times (\A^1-0)$ and the restriction of $j$ is the closed imbedding $i\times 1:Y\times (\A^1-0)\hookrightarrow X\times(\A^1-0)$.
\item $\rho^{-1}(0)=N_YX$, where $N_YX$ is the normal cone to $Y$ in $X$ and the restriczion of $j$ is the embedding as the zero section $s_0:Y\to N_YX$
\end{enumerate}

The scheme $D(X,Y)$ can be obtained as follows:
Consider the blow-up $M$ of $X\times \A^1$ along $Y\times 0$ and the blow-up $\tilde X$ of $X\times 0$ along $Y\times 0$. Then define $D(X,Y)$ to be $M\setminus\tilde X$. 

If $Y$ is smooth in a smooth scheme $X$, then it is locally of complete intersection and $N_YX$ is a vector bundle over $Y$ of rank $\dim X-\dim Y$. Moreover, $N_YX$ is Cartier divisor on $D(X,Y)$. We denote by $L(N)$ the associated line bundle. A straight computation shows that the restriction of $L(N)$ to $N_YX$ is trivial. 

Let $U=\A^1-0$ and consider the form

$$<1,-t>:\mathcal O_U^2\to \mathcal O_U^2$$
in $W^0(D^b(U))$. Now let $X$ be a smooth scheme and consider the projection $\eta:X\times U\to U$. Then $\eta^*(<1,-t>)\in W^0(D^b(X\times U))$ and we also denote it by $<1,-t>$. Since the support of this form is $X\times U$, the tensor product gives a functor 

$$<1,-t>\otimes\_: D^b_i(X\times U)\to D^b_i(X\times U).$$
Using the fact that $<1,-t>$ is symmetric, we see that this functor is duality preserving (see \cite{GN}, Definition 1.8 and Lemma 1.14) and therefore induces for any $i$ a homomorphism

$$<1,-t>\otimes\_:W^i(D^b_i(X\times U))\to W^i(D^b_i(X\times U)).$$
For some sign reasons that will be made clearer in Lemma \ref{prep_assoc3}, we will in fact consider for any $i$ the homomorphism

$$m_t:W^i(D^b_i(X\times U))\to W^i(D^b_i(X\times U))$$
defined by $m_t(\alpha)=(-1)^{i+1}<1,-t>\otimes\alpha$.

\begin{lem}
For any $i,j\in\N$ the homomorphism $m_t$ induces a homomorphism

$$I^j(D^b_i(X\times U))\to I^{j+1}(D^b_i(X\times U))$$ 
and the following diagram commutes

$$\xymatrix{I^j(D^b_i(X\times U))\ar[r]^-{d^i}\ar[d]_-{-m_t} & I^{j-1}(D^b_{i+1}(X\times U))\ar[d]^-{m_t} \\
I^{j+1}(D^b_i(X\times U))\ar[r]_-{d^i} & I^j(D^b_{i+1}(X\times U)).}$$
\end{lem}

\begin{proof}
The first assertion is clear. Now $<1,-t>$ is a global isomorphism and we can use Theorem 2.10 in \cite{GN} (or Theorem \ref{leibnitz} in the present paper) to see that $$d^i(<1,-t>\otimes \alpha)=<1,-t>\otimes d^i\alpha$$
for any $\alpha\in I^j(D^b_i(X\times U))$.
The first term is $(-1)^{i+1}d^i(m_t(\alpha))$ and the second one is $(-1)^{i+2}m_t(d^i\alpha)$.
\end{proof}

Now consider $t\in \mathcal O_{X\times U}^*$. For any $i$ and any $x\in X\times U$, we have a multiplication by $t$:

$$n_t:K_i^M(k(x))\to K^M_{i+1}(k(x))$$
defined by $n_t(\{a_1,\ldots,a_i\})=\{t,a_1,\ldots,a_i\}$.

\begin{lem}
For any $i,j\in N$ the following diagram commutes

$$\xymatrix{C(X,K_j^M)^i\ar[r]^-{d^i}\ar[d]_-{-n_t} &  C(X,K_{j}^M)^{i+1}\ar[d]^-{n_t}\\
C(X,K_{j+1}^M)^i\ar[r]_-{d^i} & C(X,K_{j+1}^M)^{i+1}.}$$
\end{lem}

\begin{proof}
See \cite{Ro}, Proposition 4.6.
\end{proof}

\begin{cord}\label{multit}
The homomorphisms $m_t$ and $n_t$ induce for any $i,j\in N$ a homomorphism

$$\{t\}:H^{i}(C(X\times U,G^j))\to H^i(C(X\times U,G^{j+1})).$$
We call this homomorphism \emph{multiplication by $t$}.
\end{cord}

\begin{proof}
It suffices to show that $m_t$ and $n_t$ give the same operation on $C(X,\overline I^j)$. It is straightforward.
\end{proof}

We will need the following lemma:

\begin{lem}\label{com_t}
Let $f:X\to Y$ be a flat morphism of smooth schemes. Then for any $i,j$ the following diagram commutes

$$\xymatrix{H^i(C(Y\times U,G^j))\ar[r]^-{\{t\}}\ar[d]_-{(f\times 1)^*} & H^i(C(Y\times U,G^{j+1}))\ar[d]^-{(f\times 1)^*}\\
H^i(C(X\times U,G^j))\ar[r]_-{\{t\}} & H^i(C(X\times U,G^{j+1})).}$$
\end{lem}

\begin{proof}
First observe that $(f\times 1)^*(<1,-t>)=<1,-t>$ by definition. Then for any $\alpha\in I^r(D^b_i(X))$ we have $(f\times 1)^*(m_t\alpha)=m_t((f\times 1)^*\alpha)$ (use \cite{GN}, Theorem 3.4). On the other hand, we have $(f\times 1)^*(n_t(\alpha))=n_t((f\times 1^*)\alpha)$ for any $\alpha\in K^M_r(k(y))$ (\cite{Ro}, Lemma 4.3). Putting this together, we get the conclusion.
\end{proof}

Let $Y\to X$ be a closed embedding of smooth schemes and consider the deformation to the normal cone space $D(X,Y)$. Then $N_YX$ is a Cartier divisor and its complement in $D(X,Y)$ is $X\times U$. We have a long exact sequence associated to this triple (\cite{JF}, Corollary 10.4.9):

$$\xymatrix@C=0.8em{ H^i(C(D(X,Y),G^{j+1}))\ar[r] & H^i(C(X\times U,G^{j+1}))\ar[r]^-\partial & H^{i+1}_{N_YX}(C(D(X,Y),G^{j+1}))}$$
Because the restriction to $N_YX$ of its associated line bundle is trivial, we also have an isomorphism (Remark \ref{in_cartier})

$$\kappa_*:H^i(C(N_YX,G^{j}))\to H^{i+1}_{N_YX}(C(D(X,Y),G^{j+1})).$$
Let $q:N_YX\to Y$ and $\pi:X\times U\to X$ be the projections and consider the following composition:

$$\xymatrix{H^i(C(X,G^{j}))\ar[d]_-{\pi^*}\ar@{-->}[rr] & & H^i(C(Y,G^{j}))\\
H^i(C(X\times U,G^j))\ar[r]_-{\{t\}} & H^i(C(X\times U,G^{j+1}))\ar[r]_-{(\kappa_*)^{-1}\partial} & H^i(C(N_YX,G^{j}))\ar[u]_-{(q^*)^{-1}}.}$$    
 


\begin{defin}\label{ori_gysin}
Let $Y$ be a smooth subscheme of a smooth scheme $X$ with inclusion $i:Y\to X$. We denote by $i^!:H^r(C(X,G^j))\to H^r(C(Y,G^j))$ and call {\it oriented Gysin map} the composition $(q^*)^{-1}(\kappa_*)^{-1}\partial\{t\}\pi^*$.
\end{defin} 


\begin{rem}\label{twist_ori_gysin}
Let $i:Y\to X$ be a closed immersion of smooth schemes and let $L$ be an invertible $\O_X$ module. Then we have a twisted version of the oriented Gysin map:

$$i^!:H^r(C(X,G^j,L))\to H^r(C(Y,G^j,i^*L)).$$
\end{rem}

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

\subsection{Functoriality}

The goal of this section is to prove that for any inclusions of smooth schemes $\xymatrix{Z\ar[r]^-i & Y\ar[r]^-j & X}$ we have $(ji)^!=i^!j^!$. The strategy is not new. We follow the exposition of the sections 11, 12 and 13 in \cite{Ro}.  First we prove some lemmas:

\begin{lem}\label{prep_assoc1}
Let $i:Y\to X$ be a closed immersion and $g:V\to X$ be a flat morphism. Consider the following fibre product

$$\xymatrix{W\ar[r]^-{i^\prime}\ar[d]_-{g^\prime} & V\ar[d]^-g\\
Y\ar[r]_-i & X.}$$
Then we have $(g^\prime)^*i^!=(i^\prime)^!g^*$.
\end{lem}

\begin{proof}
Let $D(X,Y)$ be the deformation to the normal cone associated to $i:Y\hookrightarrow X$ and $D(V,W)$ be the deformation associated to $i^\prime:W\hookrightarrow V$. Let $U=\A^1-0$. Because of the universal properties of blow-ups, we see that $g$ and $g^\prime$ give a morphism $D(g):D(V,W)\to D(X,Y)$ such that the following diagram commutes:

$$\xymatrix{D(V,W)\ar[d]_-{D(g)} & V\times U\ar[d]^-{g\times 1}\ar[l]_-{\iota^\prime}\\
D(X,Y) & X\times U\ar[l]_-{\iota}}$$
where $\iota$ and $\iota^\prime$ are the inclusions of the respective open subsets. We also get a morphism $N(g):N_WV\to N_YX$ such that these diagrams commute:

$$\xymatrix{N_WV\ar[r]^-{q^\prime}\ar[d]_-{N(g)} & W\ar[d]^-{g^\prime}\\ 
N_YX\ar[r]_-q & Y}\quad\quad \xymatrix{N_WV\ar[r]^{\kappa^\prime}\ar[d]_-{N(g)} & D(V,W)\ar[d]^-{D(g)}\\
N_YX\ar[r]_-\kappa & D(X,Y).} $$
Now use Propositions \ref{func_dom} and \ref{fibre_product}, Lemma \ref{com_t} and the diagram

$$\xymatrix{W\ar[d]_-{g^\prime} & N_WV\ar[r]^-{\kappa^\prime}\ar[d]^-{N(g)}\ar[l]_-{q^\prime} & D(V,W)\ar[d]^-{D(g)} & V\times U\ar[d]^-{g\times 1}\ar[l]_-{\iota^\prime}\ar[r]^-{\pi^\prime} & V\ar[d]^-g\\
Y & N_YX\ar[r]_-{\kappa}\ar[l]^-q & D(X,Y) & X\times U\ar[l]^-{\iota}\ar[r]_-{\pi} & X}$$
to conclude (observe that $D(g)$ and $N(g)$ are flat because of \cite{Ro}, Remark 10.1).

\end{proof}

\begin{lem}
Let $\xymatrix@C=1.2em{Z\ar[r]^-i & Y\ar[r]^-j & X}$ be inclusions of smooth schemes. Then we have inclusions $a:N_ZY\to N_ZX$, $c:i^*N_YX\to N_YX$ and isomorphisms $N_{(i^*N_YX)}(N_YX)\simeq N_ZY\oplus i^*N_YX\simeq N_{(N_ZY)}(N_ZX).$
\end{lem}

\begin{proof}
The first two assertions are trivial. The relation (2.1) in \cite{Ne} shows that we have canonical isomorphisms
$$N_{(i^*N_YX)}(N_YX)\simeq N_ZY\oplus i^*N_YX\simeq N_{(N_ZY)}(N_ZX).$$
 
\end{proof}


\begin{lem}\label{prep_assoc2}
Let $\xymatrix@C=1.2em{Z\ar[r]^-i & Y\ar[r]^-j & X}$ be inclusions of smooth schemes. Let $a:N_ZY\to N_ZX$, $c:i^*N_YX\to N_YX$ be the inclusions and 
and $q:N_YX\to Y$, $r:N_ZX\to Z$, $s_1:N_{(i^*N_YX)}(N_YX)\to i^*N_YX$, $s_2:N_{(N_ZY)}(N_ZX)\to N_ZY$ the projections. Then we have $(s_1)^*c^!q^*j^!=(s_2)^*a^!r^*(ji)^!$
\end{lem}

\begin{proof}
Consider the deformation to the normal cone spaces $D(Y,Z)$ and $D(X,Z)$. Using the universal property of blow-ups, we get a map $D(Y,Z)\to D(X,Z)$ such that the following diagram commutes

$$\xymatrix{N_ZY\ar[r]^-a\ar[d] & N_ZX\ar[d]\\
D(Y,Z)\ar[r] & D(X,Z)\\
Y\times U\ar[r]_-{j\times 1}\ar[u] & X\times U\ar[u]}$$
where the top vertical maps are inclusions of the exceptional fiber in the deformation to the normal space and the bottom vertical maps are inclusions of open subsets. It is easy to check that the map $D(Y,Z)\to D(X,Z)$ is a closed immersion. Let $D(X,Y,Z)$ be the deformation to the normal cone space associated to this closed immersion. Using again the universal property of blow-ups, we see that the above diagram gives a sequence

$$\xymatrix{D(N_ZX,N_ZY)\ar[r] & D(X,Y,Z) & D(X,Y)\times U\ar[l]}$$
where the first map is a closed immersion and the second one is an open immersion. Consider now the space $D(X,Y,Z)$. We have an open immersion  $D(X,Y,Z)\to D(X,Z)\times U$ and a closed immersion (as the special fiber) $N_{D(Y,Z)}D(X,Z)\to D(X,Y,Z)$. In fact, this exceptional fiber is isomorphic to $D(N_YX,i^*N_YX)$ (see \cite{Ne}, paragraph 3.2). So we get a diagram

$$\xymatrix{N_YX\ar[r]^-{\kappa} & D(X,Y) & X\times U\ar[l]_-\iota\ar[r]^-\pi & X\\
N_YX\times U\ar[r]^-{\kappa}\ar[u]^-{\pi}\ar[d]_-\iota & D(X,Y)\times U\ar[u]_-\pi\ar[d]^-\iota & X\times U\times U\ar[r]^-\pi\ar[l]_-\iota\ar[u]_-\pi\ar[d]^-\iota & X\times U\ar[u]_-\pi\ar[d]^-\iota\\
D(N_YX,i^*N_YX)\ar[r]^-\kappa & D(X,Y,Z) & D(X,Z)\times U\ar[l]_-\iota\ar[r]^-\pi & D(X,Z) \\
N_{(i^*N_YX)}N_YX\ar[u]^-\kappa\ar[r]_-\kappa & D(N_ZX,N_ZY)\ar[u]_-\kappa & N_ZX\times U\ar[u]_-{\kappa}\ar[r]_-\pi\ar[l]^-\iota & N_ZX\ar[u]_-{\kappa}}$$
where all the lines are deformations to the normal cone, the first, third and fourth columns are also deformations to the normal cone. This diagram is commutative (see \cite{Ne}, paragraph 3.2). The maps $\kappa$ denote inclusions of special fibers, $\iota$ denote the inclusions of the complement of these special fibers and $\pi$ denote the projections. The map $q^*j^!$ is obtained by composing the operations (in cohomology) of the top row and $s_1^*b^!$ is obtained by working with the left column. Similarly, $r^*(ji)^!$ and $s_2^*a^!$ are deduced from the right column and the bottom row. Now all the squares appearing in this diagram are commutative and give commutative diagrams in cohomology (Proposition \ref{func_dom}, Proposition \ref{func_finite} and Lemma \ref{fibre_product}). Using this and Lemma \ref{com_t}, we get the result. 


\end{proof}

\begin{lem}\label{prep_assoc3} 
Let $V,X$ and $W$ be smooth schemes. Consider the following commutative diagram

$$\xymatrix{W\ar[r]^-i\ar[rd]_-{p^\prime} & V\ar[d]^-p\\
 & X}$$
where $p,p^\prime$ are flat and $i$ is a closed immersion. Suppose that the composition $N_WV\to W\to X$ is of the same relative dimension as $\pi$. Then $i^!p^*=(p^\prime)^*$.
\end{lem}

\begin{proof}
Let $D(V,W)$ be the deformation to the normal cone associated to $i$ and $b:D(V,W)\to V\times\A^1$ be the blow-down map. We have a commutative diagram 

$$\xymatrix{W & N_VW\ar[r]^-\kappa\ar[l]_-q & D(V,W)\ar[r]^-b & V\times \A^1\ar[r]^{p\times 1} & X\times \A^1\\
 & & V\times U\ar[u]^-\iota\ar@{=}[r] & V\times U\ar[r]_{p\times 1}\ar[u]^-\iota\ar[d]_-\pi & X\times U\ar[d]^-{\pi^\prime}\ar[u]_-\iota\\
 & &  & V\ar[r]_-p & X.}$$
By definition, $i^!p^*=(q^*)^{-1}(\kappa_*)^{-1}\partial\{t\}\pi^*p^*$. Using Lemma \ref{func_dom}, we get $i^!p^*=(q^*)^{-1}(\kappa_*)^{-1}\partial\{t\}(p\times 1)^*(\pi^\prime)^*$. By \ref{com_t}, this gives
$$(q^*)^{-1}(\kappa_*)^{-1}\partial\{t\}(p\times 1)^*(\pi^\prime)^*=(q^*)^{-1}(\kappa_*)^{-1}\partial (p\times 1)^*\{t\}(\pi^\prime)^*.$$
Using Remark 10.1 in \cite{Ro}, we see that $f:=(p\times 1)b$ is flat. We have a commutative diagram

$$\xymatrix@C=1.1em{ H^i(C(X\times\A^1,G^j))\ar[r]\ar[d]_-{f^*} & H^i(C(X\times U,G^j))\ar[r]^-{\partial^\prime}\ar[d]_-{(p\times 1)^*} & H^{i+1}_X(C(X\times \A^1,G^j))\ar[r]\ar[d]_-{f^*} &\\
 H^i(C(D(V,W),G^j))\ar[r] & H^i(C(V\times U,G^j))\ar[r]_-\partial & H^{i+1}_{N_VW}(C(D(V,W),G^j))\ar[r] & }$$
where the first line is the localization long exact sequence associated to the triple $(X\times U,X\times \A^1,X\times 0)$ and the second line is the one associated to the triple $(V\times U,D(V,W),N_VW)$. Then

$$(q^*)^{-1}(\kappa_*)^{-1}\partial (p\times 1)^*\{t\}(\pi^\prime)^*=(q^*)^{-1}(\kappa_*)^{-1}f^*\partial^\prime\{t\}(\pi^\prime)^*.$$
Consider next the fibre product

$$\xymatrix{N_VW\ar[r]^-\kappa\ar[d]_-{p^\prime q} & D(V,W)\ar[d]^-f\\
X\ar[r]_-{i_0} & X\times A^1}$$
where $i_0:X\to X\times\A^1$ is the inclusion in $0$. Using Lemma \ref{fibre_product}, we finally find $i^!p^*=(p^\prime)^*(i_0)_*^{-1}\partial^\prime\{t\}(\pi^\prime)^*$. It remains to show that $(i_0)_*^{-1}\partial^\prime\{t\}(\pi^\prime)^*=Id$ to finish the proof. At the level of Milnor $K$-theory, this is Lemma 4.5 in \cite{Ro}. Thus we only have to prove this result at the level of the Witt groups. Let $\alpha\in W^i(D^b_i(X))$ be such that $d\alpha=0\in W^{i+1}(D^b_{i+1}(X))$. Now $\del{(\pi^\prime)^*\alpha}\cap\del{<1,-t>}$ is a closed subset of $X\times \A^1$ of codimension $\geq i+2$. Therefore we can use \ref{leibnitz} to compute

$$(-1)^{i}d(<1,-t>\otimes \alpha)=d(<1,-t>)\otimes \alpha +(-1)^i<1,-t>\otimes d\alpha .$$
As $\alpha$ is a cycle, we have $d\alpha=0$ and then 
$$(-1)^{i}d(<1,-t>\otimes \alpha)=d(<1,-t>)\otimes \alpha=-dt\otimes\alpha.$$ 
By definition of $m_t$, we find $d(m_t(\alpha))=dt\otimes\alpha$. The latter is precisely $(i_0)_*\alpha$ (see \cite{GH}, Lemma 2.8).

\end{proof}

Now we have all the tools to prove the following theorem:

\begin{thm}\label{func_gysin}
Let $\xymatrix@C=1.2em{Z\ar[r]^-i & Y\ar[r]^-j & X}$ be inclusions of smooth schemes. Then $(ji)^!=i^!j^!$.
\end{thm}

\begin{proof}
Let $q:N_YX\to Y$, $p:N_ZY\to Z$ and $r:N_ZX\to Z$ be the projections. Consider also the projections $s_1:N_{(i^*N_YX)}(N_YX)\to i^*N_YX$ and $s_2:N_{(N_ZY)}(N_ZX)\to N_ZY$. Denote by $a:N_ZY\to N_ZX$ and $c:i^*N_YX\to N_YX$ the inclusions. We also have a fibre product

$$\xymatrix{i^*N_YX\ar[r]^-c\ar[d]_-{q^\prime} & N_YX\ar[d]^-q \\
Z\ar[r]_-i & Y.}$$
Then

$$(s_1)^*(q^\prime)^*i^!j^!=(s_1)^*c^!q^*j^!=(s_2)^*a^!r^*(ji)^!=(s_2)^*p^*(ji)^!$$
where the first equality is due to Lemma \ref{prep_assoc1}, the second is due to Lemma \ref{prep_assoc2} and the third to Lemma \ref{prep_assoc3}. As $(s_2)^*p^*$ induces an isomorphism in cohomology and $q^\prime s_1=p s_2$, we get the result. 
\end{proof}

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

\section{The ring structure}

Let $X$ be a smooth scheme and let $\triangle:X\to X\times X$ be the diagonal inclusion. For any $i,j,r,s$ we have an exterior product (Theorem \ref{extprod}) $$H^i(C(X,G^r))\times H^j(C(X,G^s))\to H^{i+j}(C(X\times X,G^{r+s}))$$
and an oriented Gysin map (definition \ref{ori_gysin})

$$\triangle^!:H^{i+j}(C(X\times X,G^{r+s}))\to H^{i+j}(C(X,G^{r+s})).$$

\begin{defin}
We denote by $\cdot$ the composition $\triangle^!\circ \times$.
\end{defin}

\begin{rem}\label{twist_product}
If $X$ is a smooth scheme and $L,N$ are invertible $\O_X$-modules, then using Theorem \ref{extprodtwist} and Remark \ref{twist_ori_gysin} we see that there is a product 

$$\cdot:H^i(C(X,G^i,L))\times H^j(C(X,G^j,N))\to H^{i+j}(C(X,G^{i+j},L\otimes N)).$$
\end{rem}

\begin{rem}
In particular, we have for any $i,r\in \N$ a product 
$$\cdot:H^i(C(X,G^i))\times H^j(C(X,G^j))\to H^{i+j}(C(X,G^{i+j}))$$ which by definition is a product $\ch iX\times \ch jX\to \ch {i+j}X$.
\end{rem}

\begin{rem}
It is clear from our construction that we also can define a product 

$$\cdot: H^i(C(X,K^M_r))\times H^j(C(X,K^M_s))\to H^{i+j}(C(X,K^M_{r+s})).$$
This product coincide with the one defined by Rost (\cite{Ro}, chapter 14) and the natural projections $\pi:C(X,G^p)\to C(X,K^M_p)$ give a commutative diagram

$$\xymatrix{H^i(C(X,G^r))\times H^j(C(X,G^s))\ar[r]^-{\cdot}\ar[d]_-{\pi\times\pi} &  H^{i+j}(C(X,G^{r+s}))\ar[d]^-\pi  \\
H^i(C(X,K^M_r))\times H^j(C(X,K^M_s))\ar[r]_-{\cdot} & H^{i+j}(C(X,K^M_{r+s})).}$$
\end{rem}

\begin{rem}\label{prod_gersten}
Our technique provides also a product on the cohomology of the Gersten-Witt complex of a scheme. That is, we have a product

$$\cdot: H^i(C(X,W))\times H^j(C(X,W))\to H^{i+j}(C(X,W)).$$ 
\end{rem}

Now we prove the associativity of the product we have defined. 

\begin{prop}
The product $\cdot$ is associative.
\end{prop}

\begin{proof}
First note that the exterior product is associative (Proposition \ref{assoc_ext}). We consider the following fibre product diagram

$$\xymatrix{X\ar[r]^-\triangle\ar[d]_-\triangle & X\times X\ar[d]^-{1\times\triangle} \\
X\times X\ar[r]_-{\triangle\times 1} & X\times X\times X .}$$
We see that $((1\times\triangle)\triangle)^!=((\triangle\times 1)\triangle)^!$. Theorem \ref{func_gysin} shows that we have in fact $\triangle^!(1\times\triangle)^!=\triangle^!(\triangle\times 1)^!$. Since $(1\times\triangle)^!$ is clearly $1\times\triangle^!$ and $(\triangle\times 1)^!=\triangle^!\times 1$, the associativity is proved.
\end{proof}


\begin{rem}In general, the product is not commutative. This is due to the fact that $\times$ and $\star$ do not commute with the flip $\tau:X\times X\to X\times X$ (see \ref{anticommut_ext1} and \ref{anticommut_ext2}). Moreover, the product is not anticommutative because the signs in \ref{anticommut_ext1} and \ref{anticommut_ext2} are not compatible. However, let $\alpha\in \ch iX$ and $\beta\in \ch jX$. Then $\alpha\cdot\beta$ is an element of $\ch {i+j}X$ and is therefore represented by a sum $\sum (P_s,\psi_s)\in \ker{d^{i+j}}$ where  

$$d^{i+j}:GW^{i+j}(D^b_{i+j}(X))\to W^{i+j+1}(D^b_{i+j+1}(X))$$
(see Remark \ref{groth_witt2}). Using \ref{anticommut_ext1} and \ref{anticommut_ext2}, we see that $\beta\cdot\alpha=\sum (P_s,(-1)^{ij}\psi_s)$. For a more precise statement, the reader is referred to Theorem \ref{gw_prod}.

\end{rem}

Now remark that there is a canonical class $1_X$ in $\ch 0X$ given by the symmetric form $<1>$ in $GW(k(X))$.

\begin{prop}
The class $1_X$ is a left and right unit for the product $\cdot$.
\end{prop}

\begin{proof}
Let $p_2:X\times X\to X$ be the second projection and consider the following commutative diagram

$$\xymatrix{X\ar[r]^-\triangle\ar[rd]_-{Id} & X\times X\ar[d]^-{p_2} \\
 & X.}$$
By Lemma \ref{prep_assoc3}, we see that $\triangle^!(p_2)^*=(Id)^*=Id$. Consider $\mu\in H^i(C(X,G^j))$. It is clear that $1_X\times \mu=(p_2)^*(\mu)$ and then $1_X\cdot\mu=\mu$. Replacing $p_2$ by $p_1$ shows that $1_X$ is also a right unit.
\end{proof}

Hence we have:

\begin{thm}
Let $X$ be a smooth scheme and let $\ch *X$ be the total oriented Chow group of $X$. Then the product $\cdot$ turns $\ch *X$ into a graded associative ring with unit.
\end{thm}

Taking the twists into account, we get the following theorem:

\begin{thm}
Let $X$ be a smooth scheme and let $\displaystyle{\bigoplus_{L\in \text{Pic}(X)/2}\chs *XL}$ be the total twisted oriented Chow group of $X$. Then the product $\cdot$ turns this group into a graded associative ring with unit.
\end{thm}

\begin{defin}
Let $X$ be a smooth scheme. We call {\it oriented Chow ring} the ring $\ch *X$ and {\it twisted oriented Chow ring} the ring $\displaystyle{\bigoplus_{L\in \text{Pic}(X)/2}\chs *XL}$.  
\end{defin}

The following proposition is obvious:

\begin{prop}
Let $X$ be a smooth scheme. Then the natural homomorphism $\ch *X\to CH^*(X)$ is a ring homomorphism.
\end{prop}

\begin{rem}
The same methods show that the product of Remark \ref{prod_gersten} gives a graded associative anticommutative ring structure on the total cohomology group $H^*(C(X,W))$ of the Gersten-Witt complex associated to $X$. 
\end{rem}


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

\section{Basic properties}

We first show that the oriented Chow ring is a functorial construction. 
\begin{defin}
Let $X$ and $Y$ be smooth schemes and $f:X\to Y$ a morphism. Consider the graph morphism $\gamma_f:X\to X\times Y$. We define 

$$f^!:\ch *Y\to \ch *X$$
by $f^!(y)=\gamma_f^!(1_X\times y)$ for any $y\in \ch *Y$. 
 
\end{defin}

\begin{prop}
The map $f^!:\ch *Y\to \ch *X$ is a ring homomorphism.
\end{prop}

\begin{proof}
We only have to check that $f^!(y\cdot z)=f^!(y)\cdot f^!(z)$ for any $y,z\in \ch *Y$. Consider the following commutative diagram:

$$\xymatrix{X\ar[r]^-{\gamma_f}\ar[d]_-{\triangle_X} & X\times Y\ar[d]^-{\triangle_{X\times Y}} \\
X\times X\ar[r]_-{\gamma_f\times\gamma_f} & (X\times Y)\times (X\times Y). }$$
Theorem \ref{func_gysin} shows that $\gamma_f^!\triangle_{X\times Y}^!=\triangle_X^!(\gamma_f\times\gamma_f)^!$. Applying this to the cycle $1_X\times y\times 1_X\times z$, we obtain the result.
\end{proof}

\begin{rem}
The proposition shows that $\ch *{\_}$ is a functor from the category of smooth schemes to the category of rings. It is clear that the homomorphisms $\ch *X\to CH^*(X)$ give a natural transformation $\ch *{\_}\to CH^*(\_)$.
\end{rem}

In the case where $f:X\to Y$ is a flat morphism, we can identify $f^!$ more precisely.

\begin{prop}
Let $f:X\to Y$ be a flat morphism. Then $f^!=f^*$. 
\end{prop}

\begin{proof}
Consider the following commutative diagram:

$$\xymatrix{X\ar[r]^-{\gamma_f}\ar[rd]_-f & X\times Y\ar[d]^-p \\
 & Y}$$
where $p:X\times Y\to Y$ is the projection. Lemma \ref{prep_assoc3} implies that $\gamma_f^!p^*=f^*$. Since $p^*\beta=1_X\times \beta$ for any oriented cycle on $Y$, the result is proved.
\end{proof}
Let $Z\subset X$ be a closed subset of pure codimension $i$. As $D^b_Z(X)\subset D^b(X)^{(i)}$, we have a homomorphism $GW^i_Z(X)\to GW^i(D^b(X)^{(i)})$. Composing with the localization, we obtain a homomorphism $GW^i_Z(X)\to GW^i(D^b_i(X))$. As the composition $GW^i(D^b(X)^{(i)})\to GW^i(D^b_i(X))\to W^{i+1}(D^b(X)^{(i+1)})$ is zero (see \cite{Ba2}), we finally obtain a homomorphism (Remark \ref{groth_witt2}):

$$\alpha_Z:GW^i_Z(X)\to \ch iX.$$ 

\begin{rem}\label{commut_gw}
Let $f:X\to Y$ be a flat morphism and $Z\subset Y$ be a closed subset of pure codimension $i$. The definitions of $f^*$ for the Grothendieck-Witt group and the definition of $f^*$ for the oriented Chow groups show that the following diagram commutes (see \cite{JF}, Theorem 3.2.2 and Corollary 10.4.2):

$$\xymatrix{GW^i_Z(Y)\ar[r]^-{\alpha_Z}\ar[d]_-{f^*} & \ch iY\ar[d]^-{f^*}\\
GW^i_{f^{-1}Z}(X)\ar[r]_-{\alpha_{(f^{-1}Z)}} & \ch iX.}$$ 
\end{rem}
The next theorem shows that our intersection product is the expected one:

\begin{thm}\label{gw_prod}
Let $Z,T\subset X$ be closed subschemes of respective pure codimension $i$ and $j$. Suppose that $Z\cap T$ is of pure codimension $i+j$. Then the following diagram commutes

$$\xymatrix{GW^i_Z(X)\times GW^j_T(X)\ar[d]_-{\alpha_Z\times\alpha_T}\ar[r]^-\star & GW^{i+j}_{Z\cap T}(X)\ar[d]^-{\alpha_{Z\cap T}} \\
\ch iX\times\ch jX\ar[r]_-\cdot & \ch {i+j}X.}$$  
\end{thm}

\begin{proof}
Let $\gamma\in GW^i_Z(X)$ and $\delta\in GW^j_T(X)$. Consider the deformation to the normal cone space $D(X\times X,X)$ and the blow down map $b:D(X\times X,X)\to X\times X\times \A^1$. We have the following commutative diagram

$$\xymatrix{ X\ar@{=}[r] & X\ar[d]_-\triangle & & \\
N_X(X\times X)\ar[r]^-{b^\prime}\ar[u]_-q\ar[d]_-\kappa & X\times X\ar[d]^-{i_0} & & \\
D(X\times X,X)\ar[r]_-b & X\times X\times \A^1\ar[r]^-{\pi^\prime} & X\times X & (1)\\
X\times X\times U\ar@{=}[r]\ar[u]^-\iota & X\times X\times U\ar[u]_-\iota\ar[ru]_-\pi & & }$$
where $i_0$ is the inclusion in $0$, $q$ is the projection and the two bottom squares are fibre products. By definition, we have 

$$\alpha_Z(\gamma)\cdot\alpha_T(\delta)=(q^*)^{-1}(\kappa_*)^{-1}\partial\{t\}\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta)).$$
Let $F=b^{-1}(\pi^\prime)^{-1}(p_1^{-1}Z\cap p_2^{-1}T)$ in $D(X\times X,X)$. By commutativity of the above diagram and Remark \ref{commut_gw}, we have (note that $b^*$ is defined at the level of the Grothendieck-Witt groups, but not at the level of the oriented Chow groups):
$$\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta))=\alpha_{\iota^{-1}F}(\iota^*b^*(\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta))=\iota^*\alpha_F(b^*(\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)).$$ 
We have to compute $(\kappa_*)^{-1}\partial\{t\}\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta))$. By definition of $\partial$, we have to consider any element $\nu\in C(D(X\times X,X),G^{i+j+1})_{i+j}$ having the property that $\iota^*\nu=\{t\}\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta))$ and then compute $d_G(\nu)$ where

$$d_G:C(D(X\times X,X),G^{i+j+1})_{i+j}\to C(D(X\times X,X),G^{i+j+1})_{i+j+1}$$
is the differential of the complex $C(D(X\times X,X),G^{i+j+1})$. Consider the commutative diagram

$$\xymatrix{D(X\times X,X)\ar[r]^-b & X\times X\times \A^1\ar[r]^-{pr} & \A^1\\
X\times X\times U\ar[u]^-\iota\ar@{=}[r] & X\times X\times U\ar[u]^-\iota\ar[ru]_-\eta.}$$
This shows that $N_X(X\times X)$ is the principal Cartier divisor in $D(X\times X,X)$ defined by $f:=b^*pr^*(t)$. Diagram (1) gives  
$$F\cap N_X(X\times X)=\kappa^{-1}F=\kappa^{-1}b^{-1}(\pi^\prime)^{-1}(p_1^{-1}Z\cap p_2^{-1}T)=q^{-1}(Z\cap T).$$
As $Z\cap T$ is of codimension $i+j$ in $X$ and $q$ is flat, $q^{-1}(Z\cap T)$ is also of codimension $i+j$ in $N_X(X\times X)$ and hence is of codimension $i+j+1$ in $D(X\times X,X)$. Using this, it is not hard to see that $F$ is of codimension $i+j$ in $D(X\times X,X)$. 

Consider the form $b^*(\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)$. Its support is $F$. Localizing at the generic points of $F$, we obtain a form $\nu_0$ in $W^{i+j}(D^b_{i+j}(D(X\times X,X)))$. We also obtain an element $\nu_1$ in $\displaystyle{\bigoplus_{x\in F^{(0)}}K_0(k(x))}$. The above computation shows that $f$ is a unit in $k(x)$ for any generic point $x$ of $F$. We get an element 

$$\nu:=((-1)^{i+j+1}<1,-f>\otimes \nu_0,\{f\}\cdot\nu_1)\in C(D(X\times X,X),G^{i+j+1})_{i+j}$$
which satisfy $\iota^*\nu=\{t\}\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta))$. A straightforward computation (use \ref{leibnitz} again) shows that $d_G(\nu)=df\otimes b^*(\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)$ in the group $GW^{i+j+1}(D^b_{i+j+1}(D(X\times X,X)))$. But $df=b^*dt$ and 
$$b^*dt\otimes b^*(\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)=b^*(dt\otimes (\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta))$$
(\cite{GN}, Theorem 3.2). Since $dt\otimes (\pi^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)=(i_0)_*(p_1^*\gamma\otimes p_2^*\delta)$ (\cite{GH}, Lemma 2.8), we finally obtain 

$$(\kappa_*)^{-1}\partial \{t\}\pi^*(\alpha_Z(\gamma)\times\alpha_T(\delta))=\alpha_{F\cap N_X(X\times X)}((b^\prime)^*(p_1^*\gamma\otimes p_2^*\delta)).$$  
We have a commutative diagram

$$\xymatrix{N_X(X\times X)\ar[r]^-{b^\prime}\ar[d]_-q & X\times X\\
X.\ar[ru]_-\triangle &}$$  
Now $\triangle^{-1}(p_1^{-1}Z\cap p_2^{-1}T)=Z\cap T$ and using the diagram, we see that

$$\alpha_Z(\gamma)\cdot\alpha_T(\beta)=\alpha_{Z\cap T}(\triangle^*(p_1^*\gamma\otimes p_2^*\delta)).$$
Hence it only remains to show that $\triangle^*(p_1^*\gamma\otimes p_2^*\delta)=\gamma\star\delta$ to finish the proof. This is clear by \cite{GN}, Theorem 3.2.



\end{proof}

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