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\def\Wn{{\W^{\,n}}}
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\def\Wum{{\W^{\,1}}}
\def\Wus{{\W_{\operatorname{us}}}}
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\newcount\bala\bala=1
\newcount\balc\balc=2
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\newcount\hart\hart=4
\newcount\kell\kell=5
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%--------------------------------------------------

\topmatter
\title
Triangular Witt groups.\break
Part II\,: From usual to derived
\endtitle
\author
Paul Balmer
\endauthor
\address
Department of Mathematics, Rutgers University, PISCATAWAY, NJ 08854,
USA
\endaddress
\address e-mail address: balmer\@ math.rutgers.edu
\endaddress
\date
June 1999
\enddate
\abstract 
We establish that the {\it derived} Witt group is isomorphic to the
{\it usual} Witt group when 2 is invertible. This key result opens the
Ali Baba's cave of triangular Witt groups, linking the abstract
results of Part I to classical questions for the usual Witt group.
For commercial purposes, we survey the future applications of triangular
Witt groups in the introduction.
We also establish a connection between odd-indexed Witt groups and
{\it formations}. Finally, we prove that over a commutative local ring in
which 2 is a unit, the shifted derived Witt groups are all zero but
the usual one.
\endabstract
\subjclass 11E81, 18E30, 19G12\endsubjclass
\keywords Usual and derived Witt groups, surgery on symmetric complexes
\endkeywords
\endtopmatter


\document

\rightheadtext{From usual to derived Witt groups}% simplifies
                                                 % the top text
                                                 % on right pages


\centerline{\bigsmc Introduction}
\espaceavantpar

The {\it usual}\ \,Witt group is the one defined by Knebusch (see
[\the\kneb]) for algebraic varieties. We present the obvious
generalization of his definition to exact categories (see \S\,1)
without any feeling of paternity. The {\it derived}\ \, Witt group is
the Witt group of the derived category (see \S\,2), following the
general definition for triangulated categories introduced in
[\the\bala] and [\the\balc].  \medbreak

Let $\Cal E$ be an exact category with a duality $*\,:\Cal E\to\Cal
E$, \ie $*$ is a contravariant exact functor and $*^2\simeq\Id$. This
includes the case of schemes (even with twisted dualities) and
therefore rings and fields but also the case of abelian categories
with duality, like the category of finite length modules over a
regular local ring, for instance.  Given such an exact category, its
derived bounded category $\Db(\Cal E)$ is equipped with an induced
duality $\#\,:\Db(\Cal E)\to\Db(\Cal E)$ which is contravariant,
satisfies $\#^2\simeq\Id$ and is exact, this last notion of exactness
requiring some care.
%
Our main objective here is to establish that the natural homomorphism
from the usual Witt group to the derived Witt group\,:
$$\omega_{\Cal E}\,:\W(\Cal E)\too\W\big(\Db(\Cal E)\big)$$
is an isomorphism when $\half\in\Cal E$. This is theorem~4.3. The
homomorphism $\omega_{\Cal E}$ is described in 2.8.

Even if it might be of some metaphysical interest to know that a construction
depending on an exact category is actually an invariant of its derived
category, the reader will rapidly note that symmetric forms over
complexes are much ``heavier'' (to keep this text within the boundaries of
politeness) than the classical symmetric forms. Technical and
calligraphic obstacles arise, like the size of the complexes and of the
morphisms, like homotopies, like cone and cylinder constructions, like
(fractions of) quasi-isomorphisms and are moreover of considerable
unaesthetic weight.

Therefore, if the main question remains of course\,:
\vskip2mm

\centerline{\sl What shall we use derived Witt groups for\,?}
\vskip1mm

\noindent a collateral question is\,:
\vskip2mm

\centerline{\sl How can we avoid the technical toughness of the
  symmetric forms over complexes\,?}
\vskip2mm

The answer to the second question is obvious\,: we try to formalize
everything in the framework of triangulated categories. This was
successfully used in [\the\balc] to establish a long exact sequence for
localization. If, in some cases, the triangular approach appears to be too
ethereal, one can also have recourse to the more detailed formalism of
bi-Waldhausen categories.

\goodbreak
\medbreak
Let us advertise derived Witt groups in answering the first question\,:
usefulness. The following ideas are not developed herein. They will appear 
in forthcoming articles in collaboration with Charles Walter.


First of all, we already mentioned the localization exact sequence
constructed in part I of this series. Making all due allowance, this
gives for Witt groups the analogue of Thomason's result in K-theory,
namely a long exact sequence relating the Witt groups of a scheme, the
Witt groups of an open subscheme and the Witt groups on those
complexes of the scheme which are acyclic on the open subscheme (\cf
[\the\thom]). This is discussed in remark 4.9 below.

Secondly, among the triangulated categories with duality associated to
a given scheme, there is also the derived category of quasi-coherent
sheaves with coherent homology, considered by Hartshorne in
[\the\hart] and equipped with Grothendieck's duality.  This will lead
to the definition of what could be considered as the {\it Witt group
  of coherent modules}, by analogy with G-theory (or K$'$-theory).
This will also allow us to define, in this very flexible framework, a
direct image homomorphism (a {\it transfer}) for a large collection of
morphisms of schemes, introducing twists and shifts, all things which
make full sense in the derived categories as presented in [\the\hart].
As usual, over regular separated schemes, this will coincide with the
already existing Witt groups (of locally free modules).

Thirdly, out of the long exact sequence, we shall elaborate a spectral
sequence for a filtration of triangulated categories. Applying this to
schemes, we shall establish a Gersten-type spectral sequence,
analogous, once again, to the existing one in K-theory. This has very
easy new and powerful consequences on the Gersten complex over schemes
of low Krull dimension (until dimension 4) and opens the way to deal
with those problems in higher dimension. Among ``those problems'', we
do include the Gersten conjecture.

Fourthly, finally and obviously, the result\,: ``{\it derived Witt
  group equals usual Witt group}'' creates a new way for the
computation of some Witt groups through results on the derived
categories. Let me simply mention in this direction the possibility of
computing the Witt group of the projective space over a field using
well-known identifications of its derived category. This will re-prove
the famous and pretty hard theorem of Arason \big($\W(\Bbb
P^n_k)\simeq\W(k)$\big) and, more importantly, will  allow
similar computations for those projective schemes for which these
determinations of the derived categories have already been performed.
\medbreak

To put it in a nutshell, we install a machinery that allows a serious
treatment of the Witt groups within the well-known and powerful
strategies of K-theory. But, of course, one has to check first that
this new Witt group has something to do with the one we all love. This
is the goal of the present article.
\medbreak

Let us overview briefly its content. After the
presentations made in paragraphs 1 and 2, paragraphs 3 and 4 are
devoted to the proof of the main result $\omega_{\Cal
  E}\,:\Wus\isoto\Wder$.  Surjectivity of $\omega_{\Cal E}$ depends
upon the possibility of reducing complexes (a non-split version of
``surgery on complexes'' as topologists would say) and is presented in
\S\,3. Injectivity depends upon a general triangular result ({\it
  stably neutral forms are neutral}\,) established in part I of this
series and upon a simple reformulation of the classical sub-lagrangian
reduction (slogan\,: ``$\orth L/L$'') displayed in lemma~4.1.

Paragraph 5 is dedicated to shifted or higher Witt groups, whose
definitions are recalled, explained and detailed in remark 5.1. We
also mention a slight amelioration of the techniques of reduction used
in paragraph~3. This shows how a link could be made between
odd-indexed Witt groups and groups of {\it formations} as introduced
by Ranicki (see [\the\rani]) or Pardon (see [\the\pard]). In
proposition 5.3, we give generators of the odd-indexed Witt groups but
relations are not considered here. A complete treatment of this
question though important is not of crucial interest in this first
presentation. So we choose to focus on the usual Witt group for the
obvious reasons.

Finally, we also establish the reassuring result\,: {\it Over a
  commutative local ring (e.g.~a field) in which 2 is a unit, the only
one of the four shifted Witt groups which is not trivial is the usual one}
(theorem~5.6).




{\parindent=0pt

\espaceavantpar
\centerline{\bigsmc Table of content}
\smallbreak
1. The usual Witt group.
\espaceaprestruc

2. The derived Witt group.
\espaceaprestruc

3. Reducing complexes.
\espaceaprestruc

4. The isomorphism between usual and derived.
\espaceaprestruc

5. Some remarks on the shifted or higher Witt groups.
}



\goodbreak

\paragraph{The usual Witt group}

\truc{References} Standard references for exact categories are
[\the\kell], [\the\neem], [\the\quil] or appendix A of [\the\thom].

\defi
Let $\Cal E$ be an exact category and let $*:\Cal E\to \Cal E$
be a contravariant exact functor such that there exists an isomorphism
$$\pi:\Id\isoto*\circ*$$
satisfying $\dis{(\pi_M)}\circ\pi_{\dis
  M}=\Id_{\dis M}$ for all object $M$ of $\Cal E$. The triple $(\Cal
E,*,\pi)$ is called an {\it exact category with duality}.

\truc{Additive definitions} Let $(\Cal E,*,\pi)$ be an exact category
with duality. This is in particular an additive categories with
duality (see [\the\queb]). In this context, there are classical notions
that we recall here briefly. A {\it symmetric space} is a pair
$(P,\varphi)$ where $\varphi:P\isoto\dis P$ is a symmetric
isomorphism, meaning that $\dis\varphi\circ\pi_P=\varphi$. It is
common to refer to $\varphi$ as {\it the form} or sometimes {\it the
  symmetric form} over $P$.

Let $(P,\varphi)$ and $(Q,\psi)$ be symmetric spaces. The {\it
  orthogonal sum} of these spaces is the symmetric space $(P\oplus
Q,\pmatrix\varphi&0\\0&\psi\endpmatrix)$ and is denoted by
$(P,\varphi)\bot(Q,\psi)$.
%
An {\it isometry} between $(P,\varphi)$ and $(Q,\psi)$ is an isomorphism
$h:P\isoto Q$ such that $\dis h\,\psi\,h=\varphi$.
Isometry is of course an equivalence relation on symmetric spaces and
preserves orthogonal sum.

Assume that $\Cal E$ is essentially small. We define the {\it Witt
  monoid} of $\Cal E$ to be the set of isometry classes of symmetric
spaces, with orthogonal sum. We denote it by
$$(\MW(\Cal E),\bot).$$
``M'' stands for ``Monoid''.

\defi
Let $(\Cal E,*,\pi)$ be an exact category with duality.
A symmetric space $(P,\varphi)$ is called {\it metabolic} if it
possesses a {\it lagrangian}, that is a pair $(L,\alpha)$ where
$\alpha:L\rightarrowtail P$ is an admissible monomorphism such that
the following sequence is exact\,:
$$\xymatrix{L\ar@{ >->}[r]^-{\alpha}
&P\ar@{->>}[r]^-{\dis\alpha\varphi}
&\dis L.}$$

It is easy to check that a space isometric to a metabolic one is
metabolic and that the sum of two metabolic spaces is again
metabolic. In other words, we can consider the sub-monoid of $\MW(\Cal
E)$ consisting of isometry classes of metabolic spaces, which we
denote by
$$\NW(\Cal E).$$
``N'' stands for ``Neutral''.


\rem
Let $N\subset M$ be an inclusion of abelian monoids. Recall that the
quotient monoid $M/N$ is the set of equivalence classes of elements of
M under the following relation:
$x\sim y$ if and only if there exists $n_1,n_2\in N$ such that $x+n_1=y+n_2.$
The monoid $M/N$ is a group as soon as for any element $x\in M$ there exists an
element $y\in M$ such that $x+y\in N$.

It is easy to see that if $(P,\varphi)$ is a symmetric space
then the space $(P,\varphi)\bot(P,-\varphi)$ is metabolic.

\defi
Let $(\Cal E,*,\pi)$ be an essentially small exact category with
duality. The {\it Witt group} of $\Cal E$ is defined to be the
quotient
$$\W(\Cal E)={\MW(\Cal E)\over\NW(\Cal E)}$$
(see remark 1.5). It is
an abelian group. The class of symmetric space $(P,\varphi)$ in
$\W(\Cal E)$ is denoted by $[P,\varphi]$ and we say that two symmetric
spaces are {\it Witt-equivalent} if their classes in $\W(\Cal E)$
coincide. We call it the {\it usual} Witt group and denote it by
$\Wus(\Cal E)$ when we want to distinguish it from the derived one.

\exo
Define the natural notion of {\it morphism of exact categories
with duality} and check that the Witt group is a covariant functor.
Repeat this exercise for all the notions introduced hereafter.

\exa
Let $X$ be a scheme. Denote by $\Cal E(X)$ the category of $\Cal
O_X$-modules which are locally free of finite rank. It is an exact
category with the exact structure inherited from the
abelian category of (quasi-coherent) $\Cal O_X$-modules. Explicitly, a
sequence is exact if it is locally a (necessarily split)
short exact sequence of $\Cal O_{X,x}$-modules for all $x\in X$. The functor
$$\Cal H\hbox{\it om\,}_{\Cal O_X}(-,\Cal O_X)\,:\ \Cal E(X)\to \Cal E(X)$$
is a duality. By definition, the {\it usual
Witt group} of the scheme $X$ is $\Wus(X)=\W\big(\Cal E(X)\big)$.



\paragraph{The derived Witt group}


\truc{Once for all} Let $K$ denote a triangulated category and $T$ be
its {\it translation automorphism}. For the basic notions of
triangulated categories, the reference is [\the\weib, chapter 10] or
[\the\verd]. For the derived category of an exact category, the
reference is [\the\neem] or [\the\kell]. For triangulated categories
like we use them, see also \S\,0 of [\the\balc].

\defi
An additive contravariant functor $\#:K\to K$ is said to be
{\it exact} if $T\circ \#\cong\#\circ T^{-1}$ and if for any exact
triangle
$$\xytriangle A;B;C;u;v;w;$$
the following triangle is exact:
$$\magnspxytriangle 15;\di C;\di B;\di A;T(\di C).;\di v;\di u;T(\di w);$$
%
Suppose, moreover, that there exists an isomorphism of functors
$$\varpi:\Id\isoto\#\circ\#$$
such that\kern5mm$\varpi_{T(M)}=T(\varpi_M)
\link{.5}{and}
\di{(\varpi_M)}\circ\varpi_{\di M}=\Id_{\di M}
$\kern5mm for any object $M$ of $K$. Then the triple $(K,\#,\varpi)$
is called a {\it triangulated category with duality}.

\defi
Symmetric spaces, orthogonal sums and isometries are defined in the
same way as in 1.3 above. So, we get the Witt monoid $\MW(K)$. The
question is now to define an analogue to metabolic spaces. To avoid
confusion, we call them {\it neutral}.

A symmetric space $(P,\varphi)$ in a triangulated category with duality
$(K,\#,\varpi)$ is said to be {\it neutral} if it possesses a {\it
lagrangian}. In this context, a {\it lagrangian} is a triple
$(L,\alpha,z)$ where the morphisms $\alpha:L\to P$ and
$z:T\inv(\di L)\to L$ satisfy those two conditions\,:
\roster
\item the following triangle is exact\,:
$$\spxytriangle T\inv(\di L);L;P;\di L;z;\alpha;\di\alpha\,\varphi;$$
\item $T\inv(\di z)=\varpi_L\circ z$, which we shall abbreviate by\,: $s$
is symmetric for $T\inv\circ\#$.
\endroster
%
This is, from the psychological point of view, the analogue of
definition 1.4, namely the existence of a symmetric exact triangle
instead of a symmetric exact sequence.

As before, we define $\NW(K)$ to be the sub-monoid of $\MW(K)$
consisting in neutral symmetric spaces and we define the {\it Witt
  group} of $K$ to be
$$\W(K)={\MW(K)\over\NW(K)}.$$

\rem Actually, we have at our disposal a collection of Witt groups,
$\W^n(K)$ for $n\in\Z$, called the {\it shifted Witt groups} defined
in [\the\balc]. The above one is only $\W^0(K)$. We focus on this group when
$K$ is the derived category of some exact category (see 2.8 below).
Nevertheless, we shall reconsider shifted Witt groups in \S\,5.

\exo
Let $(L,\alpha,z)$ be a lagrangian of a symmetric space
$(P,\varphi)$. Show that $z=0$ if and only if $\alpha$ is a
monomorphism if and only if $P\simeq L\oplus\di L$ and
$\varphi\simeq\pmatrix 0&1\cr1&\beta\endpmatrix$ for some morphism
$\beta\,:\di L\to L$ such that $\di\beta=\beta$. Such a space is
called {\it split neutral} (or {\it split metabolic}). When $\beta=0$
it is called {\it hyperbolic}.

\truc{The homotopy category} 
Let $(\Cal E,*,\pi)$ be an exact category
with duality in the sense of definition 1.2. Let $K=\Kb(\Cal E)$ be
the homotopy category of $\Cal E$, whose objects are bounded chain
complexes of objects of $\Cal E$ and whose morphisms are chain
complexes morphisms up to homotopy. The triangulated structure of $K$
is given by the cone construction (\cf [\the\weib, paragraph 1.5]),
that is\,: a triangle
$$\xytriangle A;B;C;u;v;w;$$
is said to be {\it exact} if it is
isomorphic to the triangle of the mapping cone construction over $u$,
that is the following triangle over $u$, where $C(u)$ or $\cone(u)$ stands for
{\it cone of $u$}:
$$\xymatrix@C=25pt{&&&\hbox{degree $n$}\ar@{..>}[d]
\cr
\kern1.2cm A=\kern5mm{\cdots}\ar[r]\ar[d]_{\D u}
&A_{n+1}\ar[rr]^-{d_{n+1}}\ar[d]_{u_{n+1}}
&&A_n\ar[rr]^-{d_{n}}\ar[d]_{u_n}
&&A_{n-1}\ar[r]^-{d_{n-1}}\ar[d]_{u_{n-1}}
&{\cdots}
\cr
\kern1.2cm B=\kern5mm{\cdots}\ar[r]\ar[d]_{\D v}
&B_{n+1}\ar[rr]^-{d'_{n+1}}\ar[d]_{\pmatrix 0\\1\endpmatrix}
&&B_{n}\ar[rr]^-{d'_{n}}
\ar[d]_{\pmatrix 0\\1\endpmatrix}
&&B_{n-1}\ar[r]^-{d'_{n-1}}\ar[d]^{\pmatrix 0\\1\endpmatrix}
&{\cdots}
\cr
\kern1.2cm C(u):={\cdots}\ar[r]\ar[d]_{\D w}
&A_n\oplus B_{n+1}
\ar[rr]^-{\sppmatrix2 -d_{n}&0\\-u_{n}&d'_{n+1}\spendpmatrix2}
\ar[d]_{\pmatrix-1&0\endpmatrix}
&&A_{n-1}\oplus B_{n}
\ar[rr]^-{\sppmatrix2 -d_{n-1}&0\\-u_{n-1}&d'_n\spendpmatrix2}
\ar[d]_{\pmatrix-1&0\endpmatrix}
&&A_{n-2}\oplus B_{n-1}
\ar[r]
\ar[d]_{\pmatrix-1&0\endpmatrix}
&{\cdots}
\cr
\kern1.2cm T(A):=\kern1.5mm{\cdots}\ar[r]
&A_n\ar[rr]_-{-d_n}
&&A_{n-1}\ar[rr]_-{-d_{n-1}}
&&A_{n-2}\ar[r]
&{\cdots}
}
$$

Moreover $K$ is naturally endowed with a duality. If we denote a complex by
$(E_i,d_i)_{i\in\Z}$, then its dual would be the
complex
$$\di{\big((E_i,d_i)_{i\in\Z}\big)}:=
\big(\dis{(E_{-i})},\dis{(d_{-i+1})}\big)_{i\in\Z}.$$
On morphisms, with the same notations, $\#$ is given by 
$$\di{\big((f_i)_{i\in\Z}\big)}:=\big(\dis{(f_{-i})}\big)_{i\in\Z}.$$
This functor is well defined (up to homotopy), is exact and is a duality. The 
identification $\varpi:\Id_K\isoto \#\circ\#$ is given by
$$\varpi_E=\big((\varpi_E)_i\big)_{i\in\Z}:=\big(\pi_{E_i}\big)_{i\in\Z}$$
for any complex $E=(E_i)_{i\in\Z}$ in $K$.

The homotopy category is not satisfactory because the functor $\Cal
E\to\Kb(\Cal E)$, sending everything in degree 0, does not send an
exact sequence to an exact triangle (exercise). To pass over this
problem, one usually inverts quasi-isomorphisms. That's our next step.

\truc{Semi-saturated exact categories} Recall from [\the\neem,
def.\,1.11] that an exact category $\Cal E$ is called {\it
  semi-saturated} if every weakly split epimorphism is an admissible
epimorphism. In other words, if\break $p\,:E\to F$ is a morphism in
$\Cal E$ for which there exists another morphism $\sigma\,:F\to E$
such that $p\circ\sigma=\Id_F$, then one requires $p$ to be an
admissible epimorphism.

For example, a saturated exact category (= idempotent complete =
pseudo-abelian = Karoubian = every idempotent is the projection on a
direct summand) is semi-saturated. For example, if $X$ is a scheme,
the category $\Cal E(X)$ defined in 1.8 is saturated and hence
semi-saturated. So, for us, semi-saturation is a weak condition.

A semi-saturated exact category $\Cal E$ admits a fully faithful
embedding into an abelian category $\Cal A$ such that\,:
\roster
\item $\Cal E\subset\Cal A$ is closed under extension;

\item the exact functor $\Cal E\to\Cal A$ reflects exact sequences;
  
\item every morphism in $\Cal E$ which is an epimorphism in $\Cal A$
  is an admissible epimorphism in $\Cal E$.
\endroster
In our example $\Cal E=\Cal E(X)$, we already have $\Cal A=$ the category of (quasi-coherent) $\Cal O_X$-modules.

\truc{The derived category}
Let $\Cal E$ be a semi-saturated exact category.
Define $\Db(\Cal E)$ to be the localization of $\Kb(\Cal E)$
with respect to quasi-iso\-morphisms. Recall, to be sure, that a {\it
quasi-isomorphism} is a morphism  of complexes in $\Kb(\Cal E)$ whose
mapping cone is
acyclic. A bounded complex
$$\cdots\to0\to0\to P_m\ \too^{d_m}\ 
P_{m-1}\ \too^{d_{m-1}}P_{m-2}\too^{d_{m-2}}\cdots\too^{d_{n+2}} 
P_{n+1}\too^{d_{n+1}} P_n\to0\to0\cdots$$
is said to be {\it acyclic} if $d_m$ is an admissible monomorphism and
if the shorter complex
$$\kern2mm\cdots\to0\to0\to0\to P_{m-1}/P_m\too^{\bar d_{m-1}}
P_{m-2}\too^{d_{m-2}}\cdots\too^{d_{n+2}} P_{n+1}\too^{d_{n+1}}
P_n\to0\to0\cdots$$
is acyclic, where $\D P_m\into^{d_m}P_{m-1}\onto^p
P_{m-1}/P_m$ is a short exact sequence and $\bar
d_{m-1}:P_{m-1}/P_m\to P_{m-2}$ is induced by $d_{m-1}$, meaning that
$\bar d_{m-1}\,p=d_{m-1}$. To start this inductive definition, we say
that the zero complex is acyclic. Since we are on bounded complexes,
this is the same as saying that one can reduce the complex from the
other side and, in turn, is the same as saying (general definition)
that every differential $d_i$ factors as
$d_i=\alpha_{i-1}\circ\beta_i$ for short exact sequence\,:
$$Q_i\into^{\alpha_i}P_i\onto^{\beta_i}Q_{i-1}\kern1cm i\in\Z.$$

It is easy to see that the structure of triangulated category with
duality obtained on $\Kb(\Cal E)$ ``localizes'' to $\Db(\Cal E)$. We
still denote by $\#$ the duality on $\Db(\Cal E)$, which could be
called the {\it derived functor} of $*:\Cal E\to\Cal E$ even if it is
a weak acceptation of that concept. The identification $\varpi$
localizes as well. We obtain a triangulated category with duality\,:
$$\big(\Db(\Cal E),\#,\varpi\big).$$

Applying definition 2.3 to this triangulated category with duality, we
obtain what we call {\it the derived Witt group}\,:
$$\Wder(\Cal E,*,\pi):=\W\big(\Db(\Cal E),\#,\varpi\big).$$

Now, consider the functor
$$c_0:\Cal E\to\Db(\Cal E)$$
which sends an object to the complex
concentrated in degree 0 and does the same with the morphisms. It is
easy to check (and well known) that it sends exact sequences to exact
triangles. Namely, if
$$E\into_\alpha F\onto_\beta G$$
is an exact sequence in $\Cal E$ then
there exists a morphism $w:c_0(G)\to T\big(c_0(E)\big)$ in $\Db(\Cal
E)$ such that the following triangle is exact\,:
$$\spxytriangle
c_0(E);c_0(F);c_0(G);T\big(c_0(E)\big).;c_0(\alpha);c_0(\beta);w;$$
It is an easy exercise to verify that if the starting exact sequence is
the one characterizing a lagrangian (in the sense of definition 1.4),
then the triangle obtained with this construction satisfies the
properties of definition 2.3 (with $z=-T\inv(w)$).  Thus, we have the
following

\prop Let $(\Cal E,*,\pi)$ be a semi-saturated exact category with
duality and let $\big(\Db(\Cal E),\#,\varpi\big)$ be the associated
derived category with duality. Then the functor $c_0:\Cal E\to\Db(\Cal
E)$ induces a well defined and natural homomorphism of groups\,:
$$\omega_{\Cal E}\,: \Wus(\Cal E)\to\Wder(\Cal E).$$
\endprop

\exo Let $(\Cal E,*,\pi)$ be a semi-saturated exact category with
duality such that $\half\in\Cal E$ (see 3.1).

\roster
\item
Let $(P,\varphi)$ be a neutral
form in $\Db(\Cal E)$ and let $(L,\alpha,z)$ be a lagrangian of this
form. Show that the symmetric space $(P,\varphi)$ is uniquely
determined up to isometry by $L$ and $z:T\inv(\di L)\to L$ such that
$T\inv(\di z)=z$. {\it Solution: theorem 1.6 of [\the\balc]}.

\item
Let $(L,z)$ be such a pair. Suppose that $z$ is
moreover a good old morphism of complexes (no fractions) and that $\dis
z_{-i-1}=z_i$ for all $i\in\Z$. Prove that the cone of $z$ possesses a
symmetric form $\chi$ given in each degree by 
$$\cone(z)_i=\dis{(L_{-i})}\oplus L_i\tooo3^{\D\chi_i=\left(\matrix
    0&-1\cr-1&0\endmatrix\right)}
L_{i}\oplus\dis{(L_{-i})}=\dis{\big(\cone(z)_{-i}\big)}
=\big(\di{\cone(z)}\big)_i$$
for $i\in\Z$, which satisfies $w=\di v\chi$ where $v:L\to\cone(z)$ and
$w:\cone(z)\to\di L$ are the morphisms of the cone construction
described in 2.6\,:
$$\spxytriangle T\inv(\di L);L;\cone(z);\di L.;z;v;w;$$
Show that such
a form is {\it not} hyperbolic, in general, even if each $\chi_i$ looks
pretty hyperbolic.

\item Suppose that $z$ is only symmetric up to homotopy (instead of
  $\dis z_{-i-1}=z_i$). Write the entire exact triangle over $z$
  obtained from the cone construction together with the symmetric form
  on $\cone(z)$ satisfying condition (1) of definition 2.3 The form
  might be more complicated when $z$ is only symmetric in $\Db(\Cal
  E)$.
\endroster
  
\exa Let $X$ be a scheme. The exact category with duality $\Cal E(X)$
introduced in 1.8 has a derived category $\Db\big(\Cal E(X)\big)$
which Witt group shall be called the {\it derived Witt group of the
  scheme $X$}\,:
$$\Wder(X):=\W\big(\Db\big(\Cal E(X)\big)\big).$$

\rem This notion is already introduced in [\the\bala, \S\,2] in which
the notations are slightly different but do not affect $\W^0$ (see remark 
5.1 below). In [\loccit, thm.~4.29], we establish that $\omega$ is an
isomorphism when $\Cal E=\Cal P(R)$ is the category of finitely
generated projective $R$-modules over a commutative (noetherian) ring
$R$ containing $\half$. This is an additive version (this exact
category $\Cal P(R)$ is split) of the more general theorem stated
here. The result of [\loccit] does not apply to non-affine schemes.







\paragraph{Reducing complexes}

\defi Let $\Cal E$ be an additive category. We say that $\half\in\Cal
E$ if any morphism $f$ in $\Cal E$ can be written in an unique way as
$f=g+g$. In this case, we abbreviate $g=\half f$.

\thm
%
Let $\Cal E$ be a semi-saturated exact category with duality such that
$\half\in\Cal E$. Then the homomorphism $\omega_{\Cal E}\,: \Wus(\Cal
E)\to\Wder(\Cal E)$ is surjective.

\proof First, it is easy to see that any symmetric space in $\Db(\Cal
E)$ is represented by a symmetric quasi-isomorphism (see 3.6). Then,
we prove that it is Witt-equivalent to a form over a shorter complex
(see 3.7 and main lemma 3.9). Then an easy induction gives the
conclusion. The proof of theorem 3.2 will occupy the end of
paragraph~3.



\defi
Let $P=(P_i)_{i\in\Z}$ be a bounded complex. We say that $P$ is
{\it supported in $[m,n]$} where $m,n\in\Z$ if $P_i=0$ when $i>m$ or $n>i$\,:
$$P=\kern1cm
\cdots\to0\to0\to P_m\too^{d_m}
P_{m-1}\too^{d_{m-1}}\cdots\too^{d_{n+2}} 
P_{n+1}\too^{d_{n+1}} P_n\to0\to0\cdots$$

\kern-2mm

\defi
A morphism of complexes $s:P\to\di P$ is said to be {\it strongly
symmetric} if $s_{-i}^*=s_i$ for all $i\in\Z$.



\truc{First step}
Let $x\in\W\big(\Db(\Cal E)\big)$. Then, using only isometries and $\half$,
it is easy to see that $x$ is equal to a class $[P,s]$ in 
$\W\big(\Db(\Cal E)\big)$ where $P$ and $s$ are as follows\,:
\roster
\item $P$ is a (bounded) complex and $s:P\to\di P$ is a  morphism of complexes
(no fractions)
\item $s$ is a quasi-isomorphism;
\item the form $s$ is strongly symmetric.
\endroster
The proof will consist in an induction on the length of the support of $P$.
The following lemma is the easy part.
  
\lem Let $(P,s)$ be as in 3.6. Suppose that $P$ is
supported in $[m,-n]$, with $m>n\geq0$. Then $(P,s)$ is isomorphic 
in $\Db(\Cal E)$ to a symmetric space $(Q,t)$ such that
\roster
\item $(Q,t)$ is as in 3.6;
\item $Q$ is supported in $[n,-n]$.
\endroster

\proof The proof for $m=n+1$ gives the general case. Assume that
$n\geq 1$ and consider $(P,s)$\,:
$$\hskip-1.2cm\xymatrix{
\kern1.2cm P=\ar[d]_s
\cdots\,0\ar[r]
&P_{n+1}\ar[r]^{d_{n+1}}\ar[d]
&P_n\ar[r]^{d_{n}}\ar[d]_{s_n}
&{\cdots}\ar[r]^-{d_{-n+1}}
&P_{-n}\ar[r]\ar[d]^{\dis{s_n}}
&0\ar[r]
&0\,\cdots\\
\kern1.2cm \di P=
\cdots\,0\ar[r]
&0\ar[r]
&\dis{P_{-n}}\ar[r]_{\dis{d_{-n+1}}}
&{\cdots}\ar[r]_-{\dis{d_n}}
&\dis{P_{n}}\ar[r]_{\dis{d_{n+1}}}
&\dis{P_{n+1}}\ar[r]
&0\ \cdots
}
$$
The assumption that $s$ is a quasi-isomorphism means that the mapping
cone $C(s)$ is acyclic. A direct computation and the definition of
acyclic complexes in an exact category give that $d_{n+1}$ is an
admissible monomorphism. Choose an exact sequence
$$P_{n+1}\into_{d_{n+1}}P_n\onto_{q_n}Q_n.$$
Since $d_n\,d_{n+1}=0$ and $s_n\,d_{n+1}=0$, there exist morphisms
$e_n:Q_n\to P_{n-1}$ and $t_n:Q_n\to \dis{P_{-n}}$ such that the following
diagram commutes\,:
$$\xymatrix@C=15pt @R=15pt{
P_n\ar@{->>}[rd]_{\D q_n}\ar[rrr]^{\D d_n}\ar[ddd]_{\D s_n}
&&&P_{n-1}\ar[ddd]^{\D s_{n-1}}\cr
&Q_n\ar@{-->}[rru]_{\D\exists\,e_n}\ar@{-->}[ldd]^{\D\exists\,t_n}\cr
\cr
\dis{P_{-n}}\ar[rrr]_{\D \dis{d_{-n+1}}}
&&&\dis{P_{-n+1}}.
}
$$
The relation $s_{n-1}\,e_n=\dis{d_{-n+1}}\,t_n$ follows from the commutativity
of the rest of the diagram and the fact that $q$ is an epimorphism.
Now, it is easy to show that
$$\hskip-1.2cm\xymatrix{
\kern1.2cm Q:=\ar[d]_{\D t}
\cdots\,0\ar[r]
&Q_n\ar[r]^-{e_n}\ar[d]_{t_n}
&P_{n-1}\ar[r]^-{d_{n-1}}\ar[d]_{s_{n-1}}
&{\cdots}\ar[r]^-{d_{-n+2}}
&P_{-n+1}\ar[r]^-{d_{-n+1}}\ar[d]^{\dis{s_{n-1}}}
&P_{-n}\ar[r]\ar[d]^{\dis{t_n}}
&0\,\cdots\\
\kern1.2cm \di Q=
\cdots\,0\ar[r]
&\dis{P_{-n}}\ar[r]_-{\dis{d_{-n+1}}}
&\dis{P_{-n+1}}\ar[r]_-{\dis{d_{-n+2}}}
&{\cdots}\ar[r]_-{\dis{d_{n-1}}}
&\dis{P_{n-1}}\ar[r]_-{\dis{e_n}}
&\dis{Q_{n}}\ar[r]
&0\,\cdots
}
$$
is a symmetric morphism. Actually, the complexes $P$ and $Q$ are
quasi-isomorphic through\,:
$$\hskip-1.2cm\xymatrix{
\kern1.2cm P=\ar[d]_q
\cdots\,0\ar[r]
&P_{n+1}\ar[r]^-{d_{n+1}}\ar[d]
&P_n\ar[r]^-{d_{n}}\ar[d]_{q_n}
&P_{n-1}\ar[r]^-{d_{n-1}}\ddouble
&{\cdots}\ar[r]^-{d_{-n+1}}
&P_{-n}\ar[r]\ddouble
&0\,\cdots\\
\kern1.2cm Q=
\cdots\,0\ar[r]
&0\ar[r]
&Q_n\ar[r]_-{e_n}
&P_{n-1}\ar[r]_-{d_{n-1}}
&{\cdots}\ar[r]_-{d_{-n+1}}
&P_{-n}\ar[r]
&0\,\cdots
}
$$
We leave to the reader to check that this is a quasi-isomorphism
directly from the definition.

An immediate computation gives $\di q\,t\,q=s$. So, first, $t$ is a
quasi-isomorphism (and $(Q,t)$ is as in 3.6) and secondly the spaces
$(P,s)$ and $(Q,t)$ are isometric. This ends the proof of the lemma
when $n\geq1$.

Suppose $n=0$. Then the form $(P,s)$ is simply
$$\xymatrix{
{\cdots}0\ar[r]
&P_1\ar[r]^d\ar[d]
&P_0\ar[r]\ar[d]^{s=\dis s}
&0\ar[r]\ar[d]
&0{\cdots}\\
{\cdots}0\ar[r]
&0\ar[r]
&\dis{P_0}\ar[r]_{\dis d}
&\dis{P_1}\ar[r]
&0{\cdots}
}$$
and the proof contains a little change.
As before, since $s$ is a quasi-isomorphism, its cone is acyclic. The
$d$ is an admissible monomorphism that we can complete in an exact
sequence as before\,:
$$P_1\into_dP_0\onto_pQ.$$
Since $s\,d=0$ there exists a unique
morphism $\tilde t:Q\to\dis{P_0}$ such that $\tilde t\,p=s$. But now
$\dis d\,\tilde t\,p=\dis d\,s=0$ and then (since $p$ is an
epimorphism) $\dis d\,\tilde t=0$. On the other side, $*$ being
exact, the following sequence is exact\,:
$$\dis Q\into_{\dis p}\dis{P_0}\onto_{\dis d}\dis{P_1}.$$
The relation
$\dis d\,\tilde t=0$ induces the existence of a unique morphism
$t:Q\to\dis Q$ such that $\tilde t=\dis p\,t$.  Observe that $\dis
p\,t\,p=\tilde t\,p=s$.  Dualizing this last equation, we get $\dis
p\,\dis t\,p= \dis s=s$ by hypothesis. This implies that $\dis t=t$ by
uniqueness of $\tilde t$ and $t$.

Now it is clear that $(P,s)$ is quasi-isomorphic to $c_0(Q,t)$ through
the following quasi-isomorphism $q:P\to c_0(Q)$\,:
$$\xymatrix{
{\cdots}\,0\ar[r]
&P_1\ar[r]^d\ar[d]
&P_0\ar[r]\ar[d]^p
&0\ar[r]\ar[d]
&0\,{\cdots}\\
{\cdots}\,0\ar[r]
&0\ar[r]
&Q\ar[r]
&0\ar[r]
&0\,{\cdots}
}$$
Since $\di q\,c_0(t)\,q=s$, $c_0(t)$ is a quasi-isomorphism (and then 
$t\,:Q\to\dis Q$ is an isomorphism).
\qed

\truc{Main lemma}
Let $(P,s)$ be a symmetric space as in 3.6 and suppose that $P$ is
supported in $[n,-n]$ with $n\geq1$. Then there exists a symmetric
space $(Q,t)$ such that
\roster
\item $(Q,t)$ is as in 3.6;
\item $Q$ is supported in $[n,-(n-1)]$;
\item $[P,s]=-[Q,t]$ in $\W\big(\Db(\Cal E)\big)$.
\endroster

\proof
Suppose $n\geq2$.

Let $(P,s)$ be
$$\hskip-1.2cm\xymatrix{
\kern1.2cm P=\ar[d]_{\D s}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{d_{n}}\ar[d]_{s_n}
&P_{n-1}\ar[r]^-{d_{n-1}}\ar[d]_{s_{n-1}}
&{\cdots}\ar[r]^-{d_{-n+1}}
&P_{-n}\ar[r]\ar[d]^{s_{-n}}
&0\,{\cdots}\\
\kern1.2cm \di P=
{\cdots}\,0\ar[r]
&\dis{P_{-n}}\ar[r]_-{\dis{d_{-n+1}}}
&\dis{P_{-n+1}}\ar[r]_-{\dis{d_{-n+2}}}
&{\cdots}\ar[r]_-{\dis{d_{n}}}
&\dis{P_{n}}\ar[r]
&0\ {\cdots}
}
$$
For readability purpose, we omit the subscripts for the differentials $d_i$ as
well as for the morphisms $s_i$. They are forced by the subscripts of
the objects they are linking. Those who want to restore these
subscripts are recalled that, by assumption, $\dis{s_{-i}}=s_i$ for
all $i\in\Z$ and that we make an extensive use of this fact.

Define $(Q,t)$ to be
$$\hskip-1.2cm\xymatrix@C=15pt{&\hbox{(degree\ $n$)}\ar@{..>}[d]
&&&&&\hbox{(degree\ $-n$)}\ar@{..>}[d]
\cr
\kern1.2cm Q:=\ar[d]_{\D t}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix s\\ d\endpmatrix}\ar[d]
&\dis{P_{-n}}\oplus P_{n-1}
\ar[r]^-{\pmatrix 0&d\endpmatrix}
\ar[d]_{\pmatrix \dis d&-s\endpmatrix}
&P_{n-2}\ar@{.>}[r]^-{d}\ar[d]_{-s}\ar@{}[rd]|{{\cdots}}
&P_{-n+2}\ar[r]^-{d}\ar[d]_{-s}
&P_{-n+1}\ar[r]\ar[d]^{\pmatrix d\\-s\endpmatrix}
&0\ar[r]\ar[d]
&0\,{\cdots}\cr
\kern1.2cm \di Q=
{\cdots}\,0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]_-{\dis{d}}
&\dis{P_{-n+2}}\ar@{.>}[r]_-{\dis{d}}
&\dis{P_{n-2}}\ar[r]_-{\pmatrix0\\ \dis d\endpmatrix}
&P_{-n}\oplus\dis{P_{n-1}}
\ar[r]_-{\pmatrix s&\dis d\endpmatrix}
&\dis{P_n}\ar[r]
&0\ {\cdots}
}
$$
We are going to prove that $(P,s)\bot(Q,t)$ is neutral in $\Db(\Cal
E)$, which is enough since $(Q,t)$ obviously satisfies conditions (1) and (2)
of the lemma. Actually, to see that $t$ is a quasi-isomorphism, simply
compute the mapping cone of $t$ and observe that it is isomorphic (not
only quasi-isomorphic or homotopically equivalent but even 
isomorphic as a complex) to the mapping cone of $s$. This easy part
is left to the reader. Up to small sign tricks, the isomorphisms
are the obvious ones.

The rest of the proof is a little bit technical but involves no
particular difficulties. The reader can very well check every step,
the only knowledge required (and already used) being the construction
of the mapping cone.

First of all, we introduce a new complex $M$ and a morphism
$z:T\inv\di M\to M$. Namely, let
$$\hskip-1.2cm\xymatrix@C=15pt{
&\hbox{(degree $n$)}\morphism\dotted\tip\notip[d]
&&&&&\hbox{(degree $-n$)}\morphism\dotted\tip\notip[d]
\\
\kern1.2cm T\inv\di M=\ar[d]_{\D z}
{\cdots}\,0\ar[r]
&0\ar[r]\ar[d]
&P_n\ar[r]^-{-d}\ar[d]_{0}
&P_{n-1}\ar[r]\ar[d]_{0}
&{\cdots}\ar[r]^-{-d}
&P_{-n+2}\ar[r]^-{-d}\ar[d]^0
&P_{-n+1}\ar[r]\ar[d]^{s_{-n}\, d_{-n+1}}
&0\,{\cdots}\\
\kern1.6cm M:=\kern4mm{\cdots}\,0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]_-{\dis{d}}
&\dis{P_{-n+2}}\ar[r]
&{\cdots}\ar[r]_-{\dis{d}}
&\dis{P_{n-1}}\ar[r]_-{\dis d}
&\dis{P_{n}}\ar[r]
&0\ {\cdots}
}
$$
Observe that $T\inv(\di z)$ is the same map except that the first zero
map is replaced by $\dis{d_{-n+1}}\circ\dis{s_{-n}}$ and that the last
map (from $P_{-n+1}$ to $\dis{P_n}$) is zero. Actually the two chain
morphisms $z$ and $T\inv\di z$ are homotopic through the homotopy (we use
the strong symmetry of $s$)\,:
$$\hskip-1.2cm\xymatrix{
\kern1.2cm T\inv\di M=
{\cdots}\,0\ar[r]
&0\ar[r]
&P_n\ar[r]^-{-d}\ar[dl]
&P_{n-1}\ar[r]\ar[dl]_{s}
&{\cdots}\ar[r]^-{-d}\ar@{}[dl]|{{\cdots}}
&P_{-n+2}\ar[r]^-{-d}\ar@{}[dl]|{{\cdots}}
&P_{-n+1}\ar[r]\ar[dl]_{s}
&0\,{\cdots}\\
\kern2.2cm M={\cdots}\,0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]_-{\dis{d}}
&\dis{P_{-n+2}}\ar[r]
&{\cdots}\ar[r]_-{\dis{d}}
&\dis{P_{n-1}}\ar[r]_-{\dis d}
&\dis{P_{n}}\ar[r]
&0\ {\cdots}
}
$$
This means that $z=T\inv(\di z)$ in $\Kb(\Cal E)$ and {\sl a fortiori}\ 
in $\Db(\Cal E)$.
Now, consider the exact triangle obtained by the cone construction\,:
$$\spxytriangle T\inv\di M;M;Z;\di M.;z;z_1;z_2;$$
Explicitly, the complex $Z$ and the morphisms $z_1,\, z_2$ are\,:
$$\hskip-2.5cm\xymatrix@C=18pt{
\kern1.3cm T\inv\di M=\kern3mm{\cdots}\,0\ar[r]\ar[d]_{\D z} 
&0\ar[r]\ar[d]
&P_n\ar[r]^-{-d}\ar[d]_{0}
&P_{n-1}\ar[d]_{0}
\ar@{.>}[r]^-{\cdots\ -d\ \cdots}
&P_{-n+2}\ar[r]^-{-d}\ar[d]^0
&P_{-n+1}\ar[r]\ar[d]^{s\,d}
&0\,{\cdots}\\
\kern1.8cm M=\kern7mm{\cdots}\,0\ar[r]\ar[d]_{\D z_1}
&0\ar[r]\ar[d]
&\dis{P_{-n+1}}\ar[r]^-{\dis{d}}
\ar[d]_{\pmatrix 0\\1\endpmatrix}
&\dis{P_{-n+2}}\ar[d]^{\pmatrix 0\\1\endpmatrix}
\ar@{.>}[r]^-{\cdots\ \dis{d}\ \cdots}
&\dis{P_{n-1}}\ar[r]^-{\dis d}\ar[d]^{\pmatrix 0\\1\endpmatrix}
&\dis{P_{n}}\ar[r]\ar[d]^1
&0\ {\cdots} 
\\ 
\kern1.2cm Z:=C(z)={\cdots}\,0\ar[r]\ar[d]_{\D z_2} 
&P_n\ar[r]^-{\pmatrix d\\0\endpmatrix}\ar[d]_{-1} 
&P_{n-1}\oplus\dis{P_{-n+1}} 
\ar[r]^-{{{\pmatrix d&0\\0&\dis d\spendpmatrix2}\atop\ }} 
\ar[d]_{\pmatrix-1&0\endpmatrix} 
&P_{n-2}\oplus\dis{P_{-n+2}}  
\ar[d]_{\pmatrix-1&0\endpmatrix} 
\ar@{.>}[r]^-{{{\cdots\pmatrix d&0\\0&\dis d\spendpmatrix2\cdots}\atop\ }} 
&P_{-n+1}\oplus\dis{P_{n-1}} 
\ar[r]^-{{{\sppmatrix2-sd&\dis d\spendpmatrix2}\atop\ }} 
\ar[d]^{\pmatrix-1&0\endpmatrix} 
&\dis{P_n}\ar[r]\ar[d] 
&0\,{\cdots}\\ 
\kern1.7cm \di M=\kern6mm 
{\cdots}\,0\ar[r] 
&P_n\ar[r]_-{d} 
&P_{n-1}\ar[r]_-d 
&P_{n-2}\ar@{.>}[r]_-{\cdots\ d\ \cdots} 
&P_{-n+1}\ar[r] 
&0\ar[r] 
&0\,{\cdots}
}
$$
There is a symmetric form $\chi$ on $Z$ defined by
$$\hskip-1.2cm\xymatrix{
\kern1.2cm Z=\ar[d]_{\D\chi}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix d\\0\endpmatrix}\ar[d]_{-1}
&P_{n-1}\oplus\dis{P_{-n+1}}
\ar[r]^-{\pmatrix d&0\\0&\dis d\spendpmatrix2}
\ar[d]_{\pmatrix s&-1\\-1&0\endpmatrix}
&{\cdots}\ar[r]^-{\pmatrix d&0\\0&\dis d\spendpmatrix2}
&P_{-n+1}\oplus\dis{P_{n-1}}
\ar[r]^-{\sppmatrix2-sd&\dis d\spendpmatrix2}
\ar[d]^{\pmatrix s&-1\\-1&0\endpmatrix}
&\dis{P_n}\ar[r]\ar[d]^{-1}
&0\,{\cdots}\\
\kern1.2cm \di Z=
{\cdots}\,0\ar[r]
&P_n\ar[r]_-{\sppmatrix2-\dis d\,s\\ d \spendpmatrix1}
&\dis{P_{-n+1}}\oplus P_{n-1}
\ar[r]_-{\sppmatrix1 \dis d&0\\0&d\endpmatrix}
&{\cdots}\ar[r]_-{\sppmatrix1 \dis d&0\\0&d\endpmatrix}
&\dis{P_{n-1}}\oplus P_{-n+1}
\ar[r]_-{\pmatrix \dis d&0\endpmatrix}
&\dis{P_n}\ar[r]
&0\,{\cdots}
}
$$
One can verify that $\chi\circ z_1=\di z_2$, which means that the
triangle
$$\spxytriangle T\inv\di M;M;Z;\di M.;z;z_1;\di{z_1}\,\chi;$$
is exact. By definition 2.3, the space $(Z,\chi)$ is neutral (actually,
this one is already neutral in $\Kb(\Cal E)$). By the way, this gives
a solution to exercise 2.10, part (3).

It is now sufficient to prove that, in $\Db(\Cal E)$, the two following
spaces are isometric\,:
$$(Z,\chi)\simeq(P,s)\bot(Q,t).$$
Note that $Z\not\simeq P\oplus Q$ in $\Kb(\Cal E)$, but only in
$\Db(\Cal E)$, as we are going to see. The strategy is the
following. We are going to establish a certain number of equalities
true in $\Kb(\Cal E)$, labelled from (1) to (5). Then, we are going
to use the fact that $s$ and $t$ are quasi-isomorphisms to deduce new
equalities in $\Db(\Cal E)$. The final statement being 
$(Z,\chi)\simeq(P,s)\bot(Q,t)$ as announced.

{\sl Ex nihilo}, let's consider the following morphism $a:T\inv(\di
Q)\to P$, that we immediately present in the triangle of its mapping
cone\,:
$$\spxytriangle T\inv(\di Q);P;A;\di Q;a;a_1;a_2;$$
\vskip-5mm
$$\hskip-3.5cm\xymatrix@C=13pt{&\hbox{degree $n$}\ar@{..>}[d]
&&&&&\hbox{degree $-n$}\ar@{..>}[d]\cr
\kern1.2cm T\inv(\di Q)=\kern1mm
{\cdots}\,0\ar[r]\ar[d]\ar[d]_{\D a}
&0\ar[r]\ar[d]
&0\ar[r]\ar[d]
&\dis{P_{-n+1}}\ar[r]^-{-\dis{d}}\ar[d]_0
&{\cdots}\ar[r]^-{-\dis{d}}
&\dis{P_{n-2}}\ar[r]^-{\pmatrix0\\-\dis d\endpmatrix}\ar[d]^0
&P_{-n}\oplus\dis{P_{n-1}}
\ar[r]^-{{{\sppmatrix2 -s&-\dis d\spendpmatrix2}\atop\ }}
\ar[d]^{\pmatrix 1&0\endpmatrix}
&\dis{P_n}\ar[r]\ar[d]
&0\ {\cdots}
\\
\kern1.8cm P=\kern7mm{\cdots}\,0\ar[r]\ar[d]_{\D a_1}
&P_n\ar[r]^-{d}
\ar[d]_1
&P_{n-1}\ar[r]^-{d}
\ar[d]^{\pmatrix 0\\1\endpmatrix}
&P_{n-2}\ar[r]^-{d}
\ar[d]^{\pmatrix 0\\1\endpmatrix}
&{\cdots}\ar[r]^-{d}
&P_{-n+1}\ar[r]^-{d}\ar[d]^{\pmatrix 0\\0\\1\endpmatrix}
&P_{-n}\ar[r]\ar[d]^{\pmatrix 0\\1\endpmatrix}
&0\ar[r]\ar[d]
&0\ {\cdots}
\\
\kern1.2cm A:=C(a)=\ar[d]_{\D a_2}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix 0\\ d\endpmatrix}\ar[d]
&\dis P_{-n+1}\oplus P_{n-1}
\ar[r]^(.45){\sppmatrix2 \dis d&0\\0& d\endpmatrix}
\ar[d]_{\pmatrix-1&0\endpmatrix}
&\dis P_{-n+2}\oplus P_{n-2}
\ar[r]_(.8){{\ \atop{\sppmatrix2 \dis d&0\\0& d\endpmatrix}\,\cdots}}
\ar[d]_{\pmatrix-1&0\endpmatrix}
&{\cdots}\ar[r]^(.15){\pmatrix0&0\\ \dis d&0\\0& d\endpmatrix}
&P_{-n}\oplus\dis P_{n-1}\oplus P_{-n+1}
\ar[r]^(.55){\sppmatrix2 s&\dis d&0\\-1&0&d\spendpmatrix1}
\ar[d]^{\pmatrix-1&0&0\\0&-1&0\endpmatrix}
&\dis P_n\oplus P_{-n}\ar[r]\ar[d]^{\pmatrix-1&0\endpmatrix}
&0\ar[r]\ar[d]
&0\,{\cdots}
\\
\kern1.8cm \di Q=\kern4mm{\cdots}\,0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]_-{\dis{d}}
&\dis{P_{-n+2}}\ar[r]_(.7){\dis{d}\,\cdots}
&{\cdots}\ar[r]_-{\pmatrix0\\ \dis d\endpmatrix}
&P_{-n}\oplus\dis{P_{n-1}}
\ar[r]_-{\pmatrix s&\dis d\endpmatrix}
&\dis{P_n}\ar[r]
&0\ar[r]
&0\ {\cdots}
}
$$
%
We claim that there exists an homotopy equivalence $\rho:A\isoto Z$
and we give explicitly $\rho$ and $\rho\inv$\,:
$$\hskip-2.5cm\xymatrix{
\kern1.2cm A=\ar[dd]_{\D\rho}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix 0\\ d\endpmatrix}\ar[dd]_{-1}
&\dis{P_{-n+1}}\oplus P_{n-1}
\ar[r]^-{\pmatrix \dis d&0\\0& d\endpmatrix}
\ar[dd]^{\pmatrix0&-1\\1&0\endpmatrix\ \cdots}
&{\cdots}\ar[r]^-{\pmatrix0&0\\ \dis d&0\\0& d\endpmatrix}
&P_{-n}\oplus\dis{P_{n-1}}\oplus P_{-n+1}
\ar[r]^-{{\ \atop{\sppmatrix2 s&\dis d&0\\-1&0&d\spendpmatrix2}}}
\ar[dd]^{\pmatrix0&0&-1\\0&1&0\endpmatrix}
&\dis{P_n}\oplus P_{-n}\ar[r]\ar[dd]^{\pmatrix1&s\endpmatrix}
&0\,{\cdots}\\
\\
\kern1.2cm Z=\ar[dd]_{\D\rho\inv}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix d\\0\endpmatrix}\ar[dd]_{-1}
&P_{n-1}\oplus\dis{P_{-n+1}}
\ar[r]
\ar[dd]^{\pmatrix 0&1\\-1&0\endpmatrix\ \cdots}
&{\cdots}\ar[r]^-{\pmatrix d&0\\0&\dis d\endpmatrix}
&P_{-n+1}\oplus\dis{P_{n-1}}
\ar[r]^-{\pmatrix-sd&\dis d\endpmatrix}
\ar[dd]^{\pmatrix -d&0\\0&1\\-1&0\endpmatrix}
&\dis{P_n}\ar[r]\ar[dd]^{\pmatrix1\\0\endpmatrix}
&0\,{\cdots}\\
\\
\kern1.2cm A=
{\cdots}\,0\ar[r]
&P_n\ar[r]_-{\pmatrix 0\\ d\endpmatrix}
&\dis{P_{-n+1}}\oplus P_{n-1}
\ar[r]_-{\pmatrix \dis d&0\\0& d\endpmatrix}
&{\cdots}\ar[r]_-{\pmatrix0&0\\ \dis d&0\\0& d\endpmatrix}
&P_{-n}\oplus\dis{P_{n-1}}\oplus P_{-n+1}
\ar[r]_-{{\ \atop{\sppmatrix2 s&\dis d&0\\-1&0&d\spendpmatrix1}}}
&\dis{P_n}\oplus P_{-n}\ar[r]
&0\,{\cdots}
}
$$
One has $\rho\,\rho\inv=\Id_Z$ and it is easy to find the homotopy
that insures $\rho\inv\rho\sim\Id$.
%
Using this isomorphism and the (cone) exact triangle of $a$, we get an
exact triangle\,:
$$(\Delta)\kern1cm\spxytriangle T\inv(\di Q);P;Z;\di Q.;a;i;\pi;\kern1cm$$
where we have baptized $i:=\rho\,a_1$ and $\pi:=a_2\,\rho\inv$. Explicitly,
they are
$$\kern-2.5cm\xymatrix{
\kern1.2cm P={\cdots}\,0\ar[r]\ar[d]_{i}
&P_n\ar[r]^-{d}
\ar[d]_{-1}
&P_{n-1}\ar[r]^-{d}
\ar[d]^{\pmatrix -1\\0\endpmatrix}
&{\cdots}\ar[r]^-{d}
&P_{-n+2}\ar[r]^-{d}\ar[d]_{\pmatrix -1\\0\endpmatrix}
&P_{-n+1}\ar[r]^-{d}\ar[d]^{\pmatrix -1\\0\endpmatrix}
&P_{-n}\ar[r]\ar[d]^s
&0\ {\cdots}
\\
\kern1.2cm Z=\ar[d]_{\pi}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix d\\0\endpmatrix}\ar[d]
&P_{n-1}\oplus \dis{P_{-n+1}}
\ar[r]^-{\pmatrix d&0\\0&\dis d\endpmatrix}
\ar[d]_{\pmatrix0&-1\endpmatrix}
&{\cdots}\ar[r]
&P_{-n+2}\oplus \dis{P_{n-2}}
\ar[r]^-{\pmatrix d&0\\0&\dis d\spendpmatrix2}
\ar[d]_{\pmatrix 0&-1\endpmatrix}
&P_{-n+1}\oplus \dis{P_{n-1}}
\ar[r]^-{{{\sppmatrix2-sd&\dis d\spendpmatrix2}\atop\ }}
\ar[d]_{\pmatrix d&0\\0&-1\endpmatrix}
&\dis{P_n}\ar[r]\ar[d]^{-1}
&0\,{\cdots}
\\
\kern1.2cm \di Q=
{\cdots}\,0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]_-{\dis{d}}
&{\cdots}\ar[r]
&\dis{P_{n-2}}\ar[r]_-{\pmatrix0\\ \dis d\spendpmatrix2}
&P_{-n}\oplus\dis{P_{n-1}}
\ar[r]_-{\pmatrix s&\dis d\spendpmatrix2}
&\dis{P_n}\ar[r]
&0\ {\cdots}
}
$$
We consider now a very last morphism; let
$j:=\chi\inv\,\di\pi\,:Q\to Z$ be the following\,:
$$\kern-2.5cm\xymatrix{
\kern1.2cm Q=\ar[d]_{\D j}
{\cdots}\,0\ar[r]
&P_n\ar[r]^-{\pmatrix s\\d\endpmatrix}\ar[d]_1
&\dis{P_{-n}}\oplus P_{n-1}
\ar[r]^-{\pmatrix 0&d\endpmatrix}
\ar[d]_{\pmatrix0&1\\-\dis d&s\endpmatrix}
&P_{n-2}\ar[r]^-{d}\ar[d]_{\pmatrix1\\s\endpmatrix}
&{\cdots}\ar[r]
&P_{-n+1}\ar[r]\ar[d]^{\pmatrix1\\s\endpmatrix}
&0\ar[r]\ar[d]
&0\,{\cdots}
\\
\kern1.2cm Z=
{\cdots}\,0\ar[r]
&P_n\ar[r]_-{\pmatrix d\\0\endpmatrix}
&P_{n-1}\oplus \dis{P_{-n+1}}
\ar[r]_-{\pmatrix d&0\\0&\dis d\spendpmatrix2}
&P_{n-2}\oplus \dis{P_{-n+2}}
\ar[r]_-{\pmatrix d&0\\0&\dis d\spendpmatrix2}
&{\cdots}\ar[r]
&P_{-n+1}\oplus \dis{P_{n-1}}
\ar[r]_-{\sppmatrix2-sd&\dis d\spendpmatrix2}
&\dis{P_n}\ar[r]
&0\,{\cdots}
}
$$
%
We give now three equalities of morphisms of complexes and leave their
verification to the reader\,:
$$\eqalign{\di i\circ\chi\circ i&\lab31=s,\cr
\di{j}\circ\chi\circ j&\lab32=t,\cr
\pi\circ j&\lab33=t.}$$
%
Now, only up to homotopy, we have\,:
$$\kern2pt \di i\circ\chi\circ j\lab34=0.$$
This is easy and we
are pretty sure that our reader would have found the
homotopy\,:$\xymatrix@C=15pt{Q\ar@{.>}[r]^-{[+1]}&\di P}$, which is zero
  except in degree $n-1$ where it is $\pmatrix 1&0\endpmatrix$.
%
The second (and last) equality that we have to establish in $\Kb(\Cal
E)$ is
$$\kern23pt s\circ a\lab35=0.$$
Set the homotopy
$\xymatrix@C=15pt{\epsilon\,:T\inv(\di Q)\ar@{.>}[r]^-{[+1]}&\di P}$
to be\,:
$$\hskip-3.2cm\xymatrix{
\kern1.2cm T\inv(\di Q)=\ar@{.>}[d]_\epsilon^{[+1]}
{\cdots}\,0\ar[r]\ar[d]
&0\ar[r]
&0\ar[r]
&\dis{P_{-n+1}}\ar[r]^-{-\dis{d}}\dlto_1
&{\cdots}\ar[r]
&\dis{P_{n-2}}\ar[r]^-{\sppmatrix20\\-\dis d\spendpmatrix2}\dlto_1
&P_{-n}\oplus\dis{P_{n-1}}
\ar[r]^-{{{\sppmatrix2 -s&-\dis d\spendpmatrix2}\atop\ }}
\dlto|{\pmatrix 0&1\endpmatrix}
&\dis{P_n}\ar[r]\dlto_1
&0\ {\cdots}
\\
\kern1.2cm \di P=
{\cdots}\,0\ar[r]
&\dis{P_{-n}}\ar[r]_-{\dis{d_{-n+1}}}
&\dis{P_{-n+1}}\ar[r]_-{\dis{d_{-n+2}}}
&{\cdots}\ar[r]_-{\dis{d_{n-2}}}
&\dis{P_{n-2}}\ar[r]_-{\dis{d_{n-1}}}
&\dis{P_{n-1}}\ar[r]_-{\dis{d_n}}
&\dis{P_{n}}\ar[r]
&0\ {\cdots}
}
$$
Then, a direct verification gives
$$s\,a+\dis d\epsilon+\epsilon\bar d=0$$
where $\dis d$ (respectively $\bar
d$) denotes the differential in $\di P$ (respectively in $T\inv(\di Q)$).

Now, jump in the derived category $\Db(\Cal E)$ and invert happily $s$
and $t$ which are known to be quasi-isomorphisms. We obtain
$$a=0$$
directly from (5). Therefore the triangle $(\Delta)$ is split, meaning
that
$$P\into^iZ\onto^\pi\di Q$$
is a split short exact sequence.
Using (3), we have the following commutative diagram\,:
$$\xymatrix{
P\rrto^-{\pmatrix1\\0\endpmatrix}\ddouble
&&P\oplus Q\rrto^-{\pmatrix0&1\endpmatrix}
\ar[d]_{\pmatrix i&j\endpmatrix}
&&Q\ar[d]^t_{\simeq}
\\
P\rrto_-i
&&Z\rrto_-\pi
&&\di Q.
}
$$
Since both lines are split exact, the
morphism $h:=\pmatrix i&j\endpmatrix\,:P\oplus Q\to Z$ is an
isomorphism. Since the beginning, we know that $Z$ is equipped with a
neutral form $\chi$. Using $h$, we have that
$$\big(P\oplus Q,\di h\,\chi\,h\big)$$
is neutral. To conclude, compute $\di h\,\chi\,h$ using (1), (2) and
(4)\,: 
$$\di h\,\chi\,h=
\left(\matrix\di i\\ \di j\endmatrix\right)
\circ\chi\circ
\left(\matrix i&j\endmatrix\right)=
\left(\matrix \di i\,\chi\,i&\di i\,\chi\,j\\
\di j\,\chi\,i&\di j\,\chi\,j\endmatrix\right)
=\left(\matrix s&0\\0&t\endmatrix\right).$$
\vskip3mm

This gives the result when $n\geq2$. For $n=1$, some cone constructions
may have 3 factors in degree 0 (when $n$ was $\geq2$, $n-1$ and $-n+1$
were distinct integers. This is no more the case when
$n=1$). For instance, $(Q,t)$ should be defined as\,:
$$\hskip-1.2cm\xymatrix@C=15pt{&&&\hbox{(degree\ 0)}\ar@{..>}[d]
\cr
\kern1.2cm Q:=\ar[d]_{\D t}
{\cdots}\,0\ar[r]
&P_1\ar[rr]^-{\pmatrix s\\ d\endpmatrix}\ar[d]
&&\dis{P_{-1}}\oplus P_{0}
\ar[rr]
\ar[d]^{\pmatrix 0&d\cr\dis d&-s\endpmatrix}
&&0\ar[r]\ar[d]
&0\,{\cdots}\cr
\kern1.2cm \di Q=
{\cdots}\,0\ar[r]
&0\ar[rr]
&&P_{-1}\oplus\dis{P_{0}}\ar[rr]_-{\pmatrix s&\dis{d}\endpmatrix}
&&\dis{P_1}\ar[r]
&0\ {\cdots}
}
$$
Nevertheless, the argument is exactly the same and the reader
may find the exact triangles (over $z$ and over $a$) as well as
relations (1) to (5) directly. Then, the end of the proof is the
same (see 3.12 if needed). Our main lemma is now proven.
\qed

\truc{Last step}
The end of the proof of theorem 3.2 is now obvious. Using
alternatively lemma 3.11 and lemma 3.9 (for $m=n+1$), we reduce any
element of $\W\big(\Db(\Cal E)\big)$ to the class of a form (as in
3.6) over a complex concentrated in degree zero. Such a
quasi-isomorphism is necessarily a symmetric isomorphism in $\Cal E$.
\qed

\truc{Back entrance to proof 3.10} The idea of that proof is very
simple. Actually, the complexes $Q$, $M$, and other morphisms $z$,
$a$, that we take out of our pocket in proof 3.10 are produced by a
general argument. Consider a complex like the $P$ that we have at the
beginning of 3.10. Then the following map\,:
$$\hskip-1.2cm\xymatrix{
\kern5mm L:=\ar[d]_{\nu_1}
&{\cdots}\,0\ar[r]
&0\ar[r]\ar[d]
&{\cdots}\ar[r]
&0\ar[r]\ar[d]
&P_{-n}\ar[r]\ar[d]^1
&0\,{\cdots}\cr
P=
&{\cdots}\,0\ar[r]
&P_n\ar[r]^-{d_{n}}
&{\cdots}\ar[r]^-{d_{-n+2}}
&P_{-n+1}\ar[r]^-{d_{-n+1}}
&P_{-n}\ar[r]
&0\,{\cdots}
}
$$
is a sub-lagrangian of the symmetric space $(P,s)$ (in the sense of
[\the\balc, paragraph~3]). This makes use, of course, of the fact that
$n\geq1$ and that $s$ is a real morphism of complexes.

Then, we can follow the sub-lagrangian construction described in
[\loccit] and especially construct the diagram $\Omega$
of 3.1 [\loccit]. With those notations, $P$ is $P$, their $\varphi$ is
our $s$ and the morphism $\eta_0$ is easily found. Our $z$ (used in
3.10) is the $w=\eta_0\,\nu_0$ of [\loccit]. Our morphism $a$ is
$\nu_1\circ T\inv\eta_2$ in the homotopy category, which we prove to
be zero in the derived category. The morphism $t$ is guessed to match
in the diagram $\bar\Omega$ that appears in [\loccit, 3.5].

However, we cannot use formally the results of [\loccit] for two
reasons. First, they depend upon the assumption that the considered
triangulated category is ``noetherian'' and this is not necessary
here. Second reason, the constructions in [\loccit] are not explicit
and have to be written here into technical details to be sure that
$Q$ is really a shorter complex.  So, this conceptual complement is
only added to explain and strengthen the proof.

As we shall see in \S\,5, it is useful to understand this reduction of
complexes because it applies to shifted dualities too.






\paragraph{The isomorphism between usual and derived}

\lem 
Let $(\Cal E,*)$ be a semi-saturated exact category with duality such 
that $\half\in\Cal E$. Let $\Cal A$ be an abelian category as in 2.7.
Consider a complex $P$ in $\Cal E$ of the form\,:
$$P=\kern5mm\cdots0\too P_n\too^{\alpha_n}\cdots\too^{\alpha_2}
P_1\too^{\alpha_1}P_0\tooo{.6}^{\dis\alpha_1\circ\psi}
\dis{P_1}\too^{\dis\alpha_2}\cdots\too^{\dis\alpha_n}\dis{P_n}\too0\cdots$$
where $(P_0,\psi)$ is a symmetric space in $\Cal E$. Suppose that the
complex $P$, seen in $\Cal A$, is acyclic except in degree zero.
Denote by $\H_0(P)$ the homology of $P$ in degree zero. Denote by
$\Psi:P\isoto\di P$ the obvious form, that is identity in each degree
but $\psi$ in degree zero.  Then, all the following hold\,: \roster
\item $\H_0(P)$ is in $\Cal E$;
\item $\di P$ has also homology concentrated in degree zero and 
$\H_0(\di P)\cong\dis{\big(\H_0(P)\big)}$;
\item $\H_0(\Psi)$ induces a symmetric form on $\H_0(P)$;
\item most important\,: $[\H_0(P),\H_0(\Psi)]=-[P_0,\psi]$ in $\Wus(\Cal E)$;
\item If $P'$ is of the same form as $P$ and is also acyclic except in
  degree 0 and if $h:P\to P'$ is an isomorphism in $\Db (\Cal E)$,
  then $\H_0(h)$ is an isometry between the symmetric spaces
  $\big(\H_0(P),\H_0(\Psi)\big)$ and $\big(\H_0(P'),\H_0(\Psi')\big)$.
\endroster

\proof Suppose $n\geq 2$. In the abelian category $\Cal A$, the
morphism $\dis\alpha_n$ is an epimorphism because $\H_{-n}(P)=0$. By
2.7, part (3), it is an admissible epimorphism in $\Cal E$. Therefore
$\alpha_n$ is an admissible monomorphism and one can easily get rid of
$P_n$ and shorten the complex, keeping the same homology in degree
zero and the same structure. So we are reduced to the case $n=1$\,:
$$\kern-1.2cm\xymatrix{
\kern1.2cm P=\ar[d]_{\D\Psi}
{\cdots}\,0\ar[r]
&P_1\ar[r]^-{\alpha}\ar[d]_1
&P_0\ar[r]^{\dis\alpha\,\psi}\ar[d]_{\psi}
&\dis P_1\ar[r]\ar[d]_1
&0\,{\cdots}
\\
\kern1.2cm \di P=
{\cdots}\,0\ar[r]
&P_1\ar[r]_-{\psi\,\alpha}
&\dis P_0\ar[r]_-{\dis\alpha}
&\dis P_1\ar[r]
&0\,{\cdots}
}
$$
But, as before, $\dis\alpha$ is an admissible epimorphism and
$\alpha$ is an admissible monomorphism. The pair $(P_1,\alpha)$ is
called a sub-lagrangian of the symmetric space $(P_0,\psi)$ in the
{\bf usual sense}. Therefore, one can apply the usual sub-lagrangian
construction which is nothing but taking the homology in degree zero
of the above complex. This is classical and is not included here (see
[\the\kneb, \S\,2 prop.\,4 p.\,128 and \S\,4 thm.\,3
p.140 and its proof]). For instance, the orthogonal $\orth P_1$ of
$P_1$ is by definition $\ker(\dis\alpha\,\psi)$ which is in $\Cal E$.
The homology is then $\orth P_1/P_1$ and is in $\Cal E$ because the
map $P_1\too\orth P_1$ is an admissible monomorphism (still using the
semi-saturation). The form induced by $\psi$ on $\orth P_1/P_1$ is clearly
$\H_0(\Psi)$ and one has property (4) as usual.

The rest of the proof is left as an easy exercise.
\qed

\thm Let $\Cal E$ be a semi-saturated exact category with duality such that
$\half\in\Cal E$. Endow $\Db(\Cal E)$ with the induced duality as in
paragraph 2. Then the natural homomorphism\,:
$$\omega_{\Cal E}\,:\Wus(\Cal E)\too\W\big(\Db(\Cal E)\big)$$
sending everything in degree zero is an isomorphism.
\endthm

\proof In view of theorem 3.2, we only have to prove injectivity of
$\omega_{\Cal E}$. Let $x\in\ker(\omega_{\Cal E})$. Then
$x=[Q,\varphi]$ for some symmetric space $(Q,\varphi)$ in $\Cal E$. By
hypothesis, $\omega_{\Cal E}(x)=[c_0(Q),c_0(\varphi)]=0$ in
$\W\big(\Db(\Cal E)\big)$ using the notations of 2.8. This means that
$\big(c_0(Q),c_0(\varphi)\big)$ is stably neutral in $\big(\Db(\Cal
E),\#\big)$.

The key part of the proof is theorem 2.5 of [\the\balc] which says
that a stably neutral space is neutral in any triangulated category
with duality containing $\half$. That is
$\big(c_0(Q),c_0(\varphi)\big)$ is isomorphic to some cone form. In
other words, applying definition 2.3, we obtain easily that there
exists a complex $L$ in $\Cal E$, a morphism $z:T\inv(\di L)\too L$
%
$$\xymatrix@C=18pt{
&&\hbox{\rm degree\ $n$}\ar@{..>}[d]&&&\hbox{\rm degree\ $-n$}\ar@{..>}[d]\\
\kern1.2cm T\inv\di L=\ar[d]_{\D z}
{\cdots}\,0\ar[r]
&0\ar[r]\ar[d]
&\dis{L_{-n-1}}\ar[r]^-{-\dis d}\ar[d]_{z_{n}}
&\dis{L_{-n}}\ar[r]\ar[d]_{z_{n-1}}
&{\cdots}\ar[r]^-{-\dis d}
&\dis{L_{n-1}}\ar[r]^-{-\dis d}\ar[d]^{z_{-n}}
&\dis{L_{n}}\ar[r]\ar[d]^{z_{-n-1}}
&0\,{\cdots}\\
\kern1.6cm L=\kern5mm{\cdots}\,0\ar[r]
&0\ar[r]
&L_n\ar[r]_-{d}
&L_{n-1}\ar[r]
&{\cdots}\ar[r]_-{d}
&L_{-n}\ar[r]_-{d}
&L_{-n-1}\ar[r]
&0\ {\cdots}
}$$
such that\,:
\roster
\item $z$ is strongly symmetric, \ie $\dis{(z_{-i-1})}=z_i$ for\ all
  $i\in\Z$.
\item Let $Z=\cone(z)$. There is an isometry in $\Db(\Cal E)$ between
  $(Z,\chi)$ and $\big(c_0(Q),c_0(\varphi)\big)$ where
  $\chi=\di\chi:Z\isoto\di Z$ is the neutral form given by\,:
%
$$\hskip-2cm\xymatrix{
&&\hbox{\rm degree\ $n$}\ar@{..>}[d]&&\hbox{\rm degree\ $-n$}\ar@{..>}[d]\\
\kern1.5cm Z:=C(z)\ar[d]_{\D\chi}
{\cdots}\,0\ar[r]
&\dis{L_{-n-1}}\ar[r]^-{\pmatrix \dis d\\-z\endpmatrix}\ar[d]_{-1}
&\dis{L_{-n}}\oplus{L_{n}}
\ar[r]^-{\pmatrix \dis d&0\\-z&d\endpmatrix}
\ar[d]_{\pmatrix 0&-1\\-1&0\endpmatrix}
&{\cdots}\ar[r]^-{\pmatrix \dis d&0\\-z&d\endpmatrix}
&\dis{L_{n}}\oplus{L_{-n}}
\ar[r]^-{\sppmatrix2-z& d\endpmatrix}
\ar[d]^{\pmatrix 0&-1\\-1&0\endpmatrix}
&{L_{-n-1}}\ar[r]\ar[d]^{-1}
&0\,{\cdots}\\
\kern1.7cm \di Z=
\kern4mm{\cdots}\,0\ar[r]
&\dis{L_{-n-1}}\ar[r]_-{\sppmatrix2-\dis z\\ \dis d\spendpmatrix2}
&{L_{n}}\oplus\dis{L_{-n}}
\ar[r]_-{\pmatrix d&-\dis z\\0&\dis d\spendpmatrix2}
&{\cdots}\ar[r]_-{\pmatrix d&-\dis z\\0&\dis d\spendpmatrix2}
&{L_{-n}}\oplus\dis{L_{n}}
\ar[r]_-{\pmatrix\dis d&-\dis z\spendpmatrix2}
&{L_{-n-1}}\ar[r]
&0\,{\cdots}
}
$$
\endroster
Remarks\,: the integer $n$ does not mean anything and can be taken
large enough; the strong symmetry of $z$ comes from exercise
2.10, which is recommended to the confused reader (see also [\the\balc,
thm.\,1.6, part (2)] if necessary).

Note that in degree zero, $\chi$ is minus the hyperbolic space over
$\dis L_0\oplus L_0$. Using the isomorphism $\chi$ to modify $Z$ in
strictly negative degrees only, one obtains the following complex
$P$\,:
$$\xymatrix@C=25pt{&&&&\hbox{degree 0}\ar@{..>}[d]\cr
P:=\ {\cdots}\,\ar[rr]^-{\alpha_2:=\pmatrix \dis d&0\\-z&d\endpmatrix}
&&\dis{L_{-1}}\oplus L_1\ar[rr]^-{\alpha_1:=\pmatrix \dis d&0\\-z&d\endpmatrix}
&&\dis{L_0}\oplus L_0\ar[r]^-{\dis\alpha_1\,\psi}
&\dis{L_1}\oplus L_{-1}\ar[r]^-{\dis\alpha_2}
&\dis{L_2}\oplus L_{-2}\ar[r]^-{\dis\alpha_3}
&{\cdots} }$$
%
where $\psi=\pmatrix0&-1\cr-1&0\endpmatrix$ is a hyperbolic form over
$P_0:=\dis{L_0}\oplus L_0$. We check now that we are in the situation
of the above lemma. The form $\Psi:P\isoto\di P$ induced on $P$ from
$\chi$ on $Z$ is identity in all degree but degree zero where it is
$\psi=\pmatrix 0&-1\\-1&0\endpmatrix$. Moreover, this new complex $P$
is still isomorphic in $\Db(\Cal E)$ to $c_0\big(Q,\varphi\big)$.
Therefore $P$ has homology only in degree zero and this homology is
isomorphic to $Q$. The lemma 4.1 says that the form
$(Q,\varphi)\simeq\big(\H_0(P),\H_0(\Psi)\big)$ is Witt-equivalent to
$\big(L_0\oplus\dis{L_0},\pmatrix0&1\\1&0\endpmatrix\big)$, which is
Witt-trivial.
\qed

\cor Let $\Cal E$ be a semi-saturated exact category with duality such that
$\half\in\Cal E$. Then the inverse of the natural isomorphism
$\omega_{\Cal E}$ is given algorithmically by the explicit
constructions of 3.6, 3.8 and 3.10.
\endcor

\truc{Proof and remark} Since $\omega_{\Cal E}$ is an isomorphism, the
inverse consists simply in finding any pre-image.
\qed

The procedure in 3.6, 3.8 and 3.10 was the following: choose a
representation of the form $(P,s)$ where $s$ is a real quasi-isomorphism
and is strongly symmetric; then reduce $(P,s)$ on the right and on the
left alternatively using the explicit descriptions of $(Q,t)$ that can be
found in those proofs.

In [\the\bala, \S\,4], it was quite a pain to construct explicitly an
inverse to $\omega$ when the considered derived category was simply
the homotopy category (in the split case, there was no need to invert
quasi-isomorphisms). Even if the result of [\loccit] is a closed
formula instead of our present algorithmic reduction, it is quite
impressive how the proof of injectivity is shorter in the more general
context. The result ``{\it stably neutral implies neutral}\,'' [\the\balc,
thm 2.5], though very abstract, appears very useful here.



\thm
%
Let $X$ be a scheme such that $\half\in{\Cal O}_X$ (obvious
sense). There is a natural isomorphism\,:
$$\omega_X:\Wus(X)\too\Wder(X).$$
The same holds for skew-symmetric forms as well.
\endthm

\proof Let $\big(\Cal E(X),*,\can\big)$ be the exact category of 1.8 where
$\dis{(-)}=\Cal H\hbox{\it om\,}_{\Cal O_X}(-,\Cal O_X)$ and $\can\,:\Id\isoto
*^2$ is the canonical (usual) identification. It suffices then to apply
theorem 4.3 to the following exact categories\,:
$\big(\Cal E(X),*,\can\big)$ and $\big(\Cal E(X),*,-\can\big)$.
\qed

\rem
%
We give in [\the\balc, thm 5.1] a general localization exact sequence
for triangulated categories with duality, under the hypotheses of
``noetherianity''and of presence of $\half$.  This applies of course
to localization of schemes. Consider $U\subset X$ to be an open
subscheme of a noetherian scheme $X$. Denote by $\Db(X)$ the derived
bounded category of the exact $\Cal E(X)$ considered several times up
to here (see 1.8, 2.11). We can apply our localization theorem as soon
as $\Db(U)$ is a localization of $\Db(X)$. This is the case when $X$
is regular and separated (more general conditions are presented in
[\the\thom]). In that case, we have a long exact sequence\,:
$$\cdots\W^{n-1}(U)\too\W^n(J)\too\W^n(X)\too\W^n(U)\too\W^{n+1}(J)\too\cdots$$
where $J$ is the full subcategory of $\Db(X)$ on those complexes which
are acyclic over $U$.

We recall from [\the\balc, proposition 1.14] that the shifted (or
higher) Witt groups are 4-periodic. Theorem 4.7 gives a description of
half of them, namely the even-indexed Witt groups\,: $\W^0(X)$ is the
same as the usual Witt group and $\W^2(X)$ (which is the same as
$\W^0$ skew-symmetric) is the usual Witt group of skew-symmetric
forms. We discuss odd-indexed Witt groups in next paragraph.

For more general $X$, we shall have the ideal formulation by using the
derived category of all coherent $\Cal O_X$-modules with
Grothendieck's duality (see [\the\hart] for instance). This will appear later.



\paragraph{Some remarks on the shifted or higher Witt groups}

\truc{The four Witt groups\,: Explaning definitions and notations}
%
In [\the\balc] (respectively in [\the\bala]), we associated to a given
triangulated category with duality $(K,\#,\varpi)$ a collection of
Witt groups $\W^n$, $n\in\Z$, (respectively $\W_n^\epsilon$, $n\in\Z$,
$\epsilon=\pm1$). They should be remembered as being simply the Witt
groups for the shifted dualities $T^n\circ\#$. To be less simple, here
is a series of considerations that we have to make on those {\it
  shifted} Witt groups.
%
\roster
\smallbreak
\item
%
  The functor $T^n\circ\#$ does satisfy $(T^n\circ\#)^2\simeq\Id$
  because $T\circ\#\cong\#\circ T\inv$ and $\#^2\simeq\Id$. But it is
  not always {\it exact} in the sense of Definition 2.2 because the
  functor $T\,:K\to K$ is only skew-exact (\ie $T$ sends an exact
  triangle to a skew-exact triangle, that is a triangle which is exact when
  changing the sign of all the morphisms). Therefore, we had to deal
  carefully with the notion of exactness and introduced $\delta$-exact
  functors for $\delta=\pm1$ (for $T^n\circ\#$, $\delta=(-1)^n$). This
  affects also the notion of neutral forms. This is easy to
  understand\,: in the definition of neutral symmetric spaces, we
  required the existence of a symmetric exact triangle, but exactness
  involves some sign. We do not want to re-write all this here and we
  refer to [\the\balc, \S\,1].
%
\smallbreak
\item
%
  Those Witt groups are called sometimes {\it shifted} Witt groups
  because they use shifted dualities. Their interest is that they fit
  into a long exact sequence of localization (see thm.~5.1 in
  [\the\balc]) already mentioned in remark 4.9. This explains why we
  also call them {\it higher (and lower)} Witt groups. This
  cohomological use justifies also the notation $\W^n$. To have this
  nice property, we make the following definition of the translation
  of a triangulated category with $\delta$-duality\,:
  $T(K,\#,\varpi)=(K,T\circ\#,-\delta\cdot\varpi)$. In other words, we
  introduce a sign when we translate a duality but we introduce no
  sign when we translate a skew-duality. If we start with a
  triangulated category with (+1)-duality $(K,\#,\varpi)$, we have then\,:
$$T^n(K,\#,\varpi)\equal{{\text{def}}}%
\big(\,K\,,\,T^n\circ\#\,,\,(-1)^{n(n+1)\over2}\cdot\varpi\,\big)$$
for all $n\in\Z$.  We define then the {\it shifted Witt groups} very
simply by\,:
$$\W^n(K,\#,\varpi):=\W\big(T^n(K,\#,\varpi)\big).$$
%
\smallbreak
\item
%
  In [\the\bala], the groups $\W^n$ were written $\W_n^\epsilon$. This
  notation had the handicap of being inadequate for the harmony of the
  localization sequence (not yet established at that time).
  Nevertheless, it was more expressive because the ``$n$'' referred to
  the shift $T^n\circ\#$ and the $\epsilon=\pm1$ referred to
  $\epsilon$-symmetric forms, keeping the identification
  $\varpi\,:\Id\isoto\#^2$ unaltered. The dictionary is the following
  one\,: $\W^n=\W_n^\epsilon$ where $\epsilon=(-1)^{n(n+1)\over2}$
  (see above). We suggest to stick to the present notation $\W^n(K)$.
%
\smallbreak
\item
%
  There is an isomorphism of triangulated categories with
  $(-1)^n$-duality
  $$(K,T^n\circ\#,\varpi)\isoto(K,T^{n+2}\circ\#,\varpi)$$
  induced by
  $T\,:K\to K$ which gave in the old notations the isomorphism
  $\W^\epsilon_{n}(K)\isoto\W^\epsilon_{n+2}(K)$ and allowed us to
  reduce ourselves to four Witt groups\,: $\W^{\ 
    \pm}_{0\text{\,or\,}1}(K)$. In the new notations, note that $T$
  does {\bf not} give an isomorphism between $T^n(K,\#,\varpi)$ and
  $T^{n+2}(K,\#,\varpi)$ because of this sign trick. We only have
  $T^n(K,\#,\varpi)\cong T^{n+2}(K,\#,\hbox{\bf - \kern-5pt-}\varpi)$
  and therefore the 4-periodicity as\,:
  $$\W^n(K)\simeq\W^{n+4}(K)$$
  (see [\the\balc, prop.~1.14]). We also
  keep only four Witt groups\,: $\W^0$, $\W^1$, $\W^2$ and $\W^3$, for
  instance. The group $\W^2$ is the old $\W^-_0$, that is the group of
  skew-symmetric spaces for the unshifted duality.
%
\smallbreak
\item
%
  As a consequence of (4) above, in the case of the derived category
  of an exact category with duality $(\Cal E,*,\pi)$, theorem 4.3
  actually identifies $\W\big(\Db(\Cal E)\big)$ as being $\Wus(\Cal
  E)$ but also $\W^2\big(\Db(\Cal E)\big)$ as being
  $\W^-_{\text{us}}(\Cal E)=\Wus(\Cal E,*,-\pi)$ the usual Witt group
  of skew-symmetric forms. A direct group isomorphism\,:
  $$\W^-_{\text{us}}(\Cal E)\too\W^2\big(\Db(\Cal E),\#,\varpi\big)$$
  is given
  by the functor $c_1\,:\Cal E\to\Db(\Cal E)$ sending everything in
  degree 1.
%
\endroster




\prop Let $(\Cal E,*,\pi)$ be a semi-saturated exact category with duality such
that $\half\in\Cal E$. Endow $\Db(\Cal E)$ with the induced duality as
in paragraph 2. Then every symmetric space for the skew-duality $T\circ\#$
is Witt-equivalent to a space of the form $(P,\varphi)$ where $P$ is a
complex supported in $[1,0]$\,:
$$P=\ \cdots\,0\too0\too P_1\too P_0\too0\too0\,\cdots$$
and where $\varphi$ is a morphism of complexes (no fractions) from $P$ 
to $T(\di P)$ which is a quasi-isomorphism such that 
$\varphi_0=\dis\varphi_1\circ\pi_{P_0}$.

\proof
We shall omit this proof because we are not going to use this
result hereafter. It is an exercise to adapt the proofs of \S\,2 to
this shifted context. Remark 3.12 definitely helps.
\qed

\rem
Consider a symmetric form as described above\,:
$$\xymatrix{ \kern1cm P=\ \cdots\ar[r]\ar[d]_{\D\varphi} &0\ar[r]
  &P_1\ar[r]^-{d}\ar[d]_{\varphi_1} &P_0\ar[r]\ar[d]^{\dis\varphi_1}
  &0\ar[r] &{\cdots} \cr \kern8mm T(\di P)=\ \cdots\ar[r] &0\ar[r]
  &\dis P_0\ar[r]_-{-\dis d} &\dis P_1\ar[r] &0\ar[r] &{\cdots} }$$
Saying that $\varphi$ is a quasi-isomorphism is nothing but asking for
the exactness of the following complex (the mapping cone with all
signs changed)\,:
$$\xymatrix@C=15pt{{\cdots}\,0\ar[r] &0\ar[r] &P_1\ar[rr]^-{\pmatrix
    d\\\varphi_1\endpmatrix} &&P_0\oplus\dis P_0\ar[rr]^-{\pmatrix
    \dis\varphi_1&\dis d\endpmatrix} &&\dis P_1\ar[r] &0\ar[r]
  &0\,{\cdots} }$$
Now, we have a symmetric space $\big(P_0\oplus\dis
P_0,\pmatrix 0&1\cr1&0\endpmatrix\big)$ with two lagrangians $P_0$ and
$P_1$. This is the link with the notion of {\it formation} used by
Ranicki [\the\rani], Pardon [\the\pard] and others. Over a ring (or in
the split exact case), see in particular [\the\rani, prop. 2.3].

As explained in the introduction, we choose to focus our attention on
the un-shifted Witt group $\W^0$ and to identify it with the usual one
because this is the classical invariant largely used and studied. This
actually identifies half of the derived Witt groups, the even-indexed
ones as explained in remark 5.1, point (5). It is nevertheless of big
interest (mainly for computation of the 12-term localization sequence)
to have a simpler description of the odd-indexed Witt groups too.

\espaceavantpar
\centerline{*\ *\ *}

\truc{Notation} If $R$ is a commutative ring and a notation is defined
over schemes, say $N(X)$, then $N(R)$ will denote
$N\big(\Spec(R)\big)$.


\thm
Let $R$ be a commutative local ring in which 2 is invertible.
Then, among the derived Witt groups of $R$, we have $\W^1(R)=0$,
$\W^2(R)=0$ and $\W^3(R)=0$. That is there is only one non-trivial
Witt group, namely $\W^0(R)\cong\Wus(R)$. This holds in particular for
fields of characteristic not 2.
\endthm

\proof We denote by $\Db(R)\equal{\text{def}}\Db\big(\Cal
E(R)\big)=\Kb(\Cal E(R)\big)$ the derived category of finitely
generated projective (free) $R$-modules, which reduces to the homotopy
category because a quasi-isomorphism of bounded complexes of
projective modules is a homotopy equivalence. We also denote by $\#$
the duality induced by $\Hom_R(-,R)$ and by $\varpi$ the isomorphism
of functors $\Id\isoto\#^2$, defined to be the canonical
identification in each degree.

The computation of $\W^0(R)$ (respectively
$\W^2(R)\simeq\W^0\big(\Db(R),\#,-\varpi\big)$) to be the usual Witt
group of symmetric (respectively skew-symmetric) forms is now clear
from theorem 4.3 and remark 5.1, point (5). For the classical proof that $\W^{-}(R)=0$ see
[\the\miln, cor.~I.3.5, p.~7].

We give the proof that $W\big(\Db(R),T\circ\#,\varpi\big)=0$. The same for
$-\varpi$ is left to the reader.

Let $\frak m$ be the maximal ideal of $R$.

Recall that a complex of free $R$-modules of finite rank is
homotopically equivalent to a so called {\it minimal} complex, that is
a complex such that all the entries of the matrix of the differentials
are in $\frak m$.  To see this (exercise) it suffices to use
elementary operations and to have in mind that $R\setminus\frak m$ is
the set of units of $R$. Now, if a complex is minimal, its dual is
also minimal. Moreover, a homotopy equivalence between two minimal
complexes is necessarily an isomorphism. This is easy to check. In
fact, it suffices to note that a $n\times n$-matrix of the form
$1_n+A$ where $A$ has all entries in $\frak m$ is invertible (because
its determinant belongs to $1+\frak m$ and is therefore invertible). In
other words, an element of $W(\Db(R),T\circ\#,\varpi)$ can be
represented by an isomorphism\,:
$$\xymatrix{&&&&\hbox{degree 0}\ar@{..>}[d]\cr \kern1cm
  P:=\cdots\ar[r]\ar[d]_{\D\varphi}
  &P_3\ar[r]^-{d_3}\ar[d]_{\varphi_{3}}^\simeq
  &P_2\ar[r]^-{d_2}\ar[d]_{\varphi_{2}}^\simeq
  &P_1\ar[r]^-{d_1}\ar[d]_{\varphi_{1}}^\simeq
  &P_{0}\ar[r]^-{d_{0}}\ar[d]_{\dis\varphi_{1}}^\simeq
  &P_{-1}\ar[r]^-{d_{-1}}\ar[d]_{\dis\varphi_{2}}^\simeq
  &P_{-2}\ar[r]\ar[d]_{\dis\varphi_{3}}^\simeq &{\cdots} \cr
  \kern8mm T(\di P)=\cdots\ar[r] &\dis P_{-2}\ar[r]_-{-\dis d_{-1}}
  &\dis P_{-1}\ar[r]_-{-\dis d_{0}} &\dis P_0\ar[r]_-{-\dis d_{1}}
  &\dis P_{1}\ar[r]_-{-\dis d_{2}} &\dis P_{2}\ar[r]_-{-\dis d_{3}}
  &\dis P_{3}\ar[r] &{\cdots} }$$
with $P$ a bounded minimal complex
of finitely generated free $R$-modules and with $\varphi_i$ being an
isomorphism for all $i\in\Z$. As usual, we managed $\varphi$ to be
strongly symmetric since the beginning, which means here
$\varphi_i=\dis{(\varphi_{-i+1})}$ for all $i\in\Z$. Using then $\varphi_i$ to
replace $P_i$ by $\dis P_{-i+1}$ for all $i>0$, the symmetric space
$(P,\varphi)$ is isomorphic to the following\,:
$$\xymatrix{&&&&{\hbox{degree 0}}\ar@{..>}[d]\cr \kern1cm
  Q:=\cdots\ar[r]\ar[d]_{\D\chi} &\dis P_{-2}\ar[r]^-{-\dis
    d_{-1}}\ar[d]_{1} &\dis P_{-1}\ar[r]^-{-\dis d_0}\ar[d]_{1} &\dis
  P_0\ar[r]^-{z_0}\ar[d]_{1} &P_{0}\ar[r]^-{d_{0}}\ar[d]^{1}
  &P_{-1}\ar[r]^-{d_{-1}}\ar[d]^{1} &P_{-2}\ar[r]^-{d_{-2}}\ar[d]^{1}
  &{\cdots} \cr \kern8mm T(\di Q)=\cdots\ar[r] &\dis
  P_{-2}\ar[r]_-{-\dis d_{-1}} &\dis P_{-1}\ar[r]_-{-\dis d_0} &\dis
  P_0\ar[r]_-{-\dis z_0}\ar@{}[ur]|{\text{Watch\,!}}
  &P_{0}\ar[r]_-{d_{0}} &P_{-1}\ar[r]_-{d_{-1}}
  &P_{-2}\ar[r]_-{d_{-2}} &{\cdots} }$$
where
$z_0:=d_1\circ(\varphi_1)\inv\,:\dis P_0\to P_0$. Note that $\dis
z_0=-z_0$ from the first diagram. The form $\chi$ is the one obtained
from $\varphi$ through this isometry. But this last space $(Q,\chi)$
 is neutral by considering the cone construction on the following 
skew-symmetric morphism for $T\inv(T\,\#)=\#$\,:
$$\xymatrix{&&&{\hbox{degree 0}}\ar@{..>}[d]\cr \kern8mm \di
  L=\cdots\ar[r]\ar[d]_{\D z} &\dis P_{-2}\ar[r]^-{\dis d_{-1}}\ar[d]
  &\dis P_{-1}\ar[r]^-{\dis d_0}\ar[d] &\dis P_0\ar[r]\ar[d]_{-z_0}
  &0\ar[r]\ar[d] &0\ar[r]\ar[d] &{\cdots} \cr \kern1cm L:=\cdots\ar[r]
  &0\ar[r] &0\ar[r] &P_{0}\ar[r]_-{d_{0}} &P_{-1}\ar[r]_-{d_{-1}}
  &P_{-2}\ar[r]_-{d_{-2}} &{\cdots} }$$
A direct computation gives
that $(Q,\chi)=\cone(\di L,z)$ in the notations of [\the\balc,
definition 1.10].
\qed

\rem
This result should be compared with [\the\baks, lemma 4.2, p.~215].

\vfill\eject

\vfill\ \vskip1mm
\centerline{\bigsmc Bibliography}
\espaceaprespar


\bible Paul BALMER; Derived Witt groups of a scheme; Journal of pure
and applied algebra, to appear approx.\ 140 (1999);

\bible Paul BALMER; TWG I\,: the 12-term localization exact sequence;
preprint (1999) available on the preprint server\,: http://www.mathematik.uni-bielefeld.de/LAG/;

\bible A.~BAK, W.~SCHARLAU; Grothendieck and Witt groups of orders and finite groups; Inventiones mathematicae 23, pp.~207-240 (1974);

\bible Robin HARTSHORNE; Residues and duality; Lecture
Notes in Mathematics 20, Springer (1966);

\bible Bernhard KELLER; Derived categories and their uses; In:
Handbook of algebra, volume I, M.\ Hazewinke (ed.), Elsevier,
Amsterdam (1996), pp.~671-701;

\bible Manfred KNEBUSCH; Symmetric bilinear forms over algebraic varieties;
In: Conference on quadratic forms, Kingston 1976, Queen's papers on
pure and applied mathematics, number 46, pp.~103-283, (1977);

\bible J.\ MILNOR, D.\ HUSEMOLLER; Symmetric bilinear forms;
Ergebnisse der Mathematik 73, Springer (1973);

\bible Amnon NEEMAN; The derived category of an exact category;
J. of algebra 135, pp.~388-394 (1990);

\bible William PARDON; The exact sequence of a localization for Witt
groups; In\,: Algebraic K-theory, Evanston 1976, Lecture
Notes in Mathematics 551, pp.~336-379, Springer (1977);

\bible H.G.\ QUEBBEMANN, R.\ SCHARLAU, W.\ SCHARLAU, M.\ SCHULTE;
Quadratische Formen in additiven Kategorien; In: Colloque sur les
formes quadratiques, Montpellier 1975, Bulletin de la Socit\'et\'e
Math\'ematique de France, M\'emoire 48, pp.~93-101, (1976);

\bible Daniel QUILLEN; Higher K-theory;  Lecture
Notes in Math.~341, Springer (1966), pp.~85-147;

\bible Andrew RANICKI; The algebraic theory of surgery; Proceedings of
the London Mathematical Society (3) 40, pp.~87-192, (1980);

\bible R.\,W.\ THOMASON, T.\ TROBAUGH;Higher algebraic K-theory of
schemes and of derived categories; In\,: The Grothendieck Festschrift,
volume III, P.\ Cartier (ed.), Progress in mathematics volume 88,
pp.~247-435, Birk\"auser (1990);

\bible Jean-Louis VERDIER; Cat\'egories d\'eriv\'ees des cat\'egories
ab\'eliennes; Ast\'erisque 239, Soci\'et\'e math\'ematique de France
(1996);

\bible Charles WEIBEL; An introduction to homological algebra;
Cambridge University Press (1994);


\espaceavantpar \centerline{\bf Acknowledgements} \espaceaprespar I
would like to thank Marco Schlichting for having been the first one to
go through the main proof of reduction of complexes {\sl and not from
  the back door\,!} I am also indebted to Charles Walter and Andrew
Ranicki for reading the first drafts of this article, for making
valuable comments and for tempering my style when it happened to be
led astray by excessive enthusiasm.

\enddocument



