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%notes ou corrections Baptiste
\newcommand{\bap}[1]{\textcolor{red}{#1}}

%notes ou corrections Pablo
\newcommand{\pab}[1]{\textcolor{blue}{#1}}

%general commands

\newcommand{\eg}{{\sl e.g.}}
\newcommand{\ie}{{\sl i.e.}\ }
\newcommand{\loccit}{{\sl loc. cit.}}

\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cV}{{\mathcal V}}
%%%%%\newcommand{\mN}{{\mathbf N}}
\newcommand{\bbA}{{\mathbb{A}}}
\newcommand{\bbN}{{\mathbb{N}}}
\newcommand{\bbP}{{\mathbb{P}}}
%%%%%\newcommand{\mZ}{{\mathbf Z}}
\newcommand{\bbZ}{{\mathbb{Z}}}
%%%%%\newcommand{\mP}{{\mathbf P}}

\DeclareMathOperator{\Wcoh}{W}%Groupes de Witt cohérents
\DeclareMathOperator{\Wper}{W}%idem
\newcommand{\Wtot}{\Wcoh^{\text{\rm tot}}}
\newcommand{\bord}{\partial} % Bord
%\newcommand{\Sec}{s} % Section
%\newcommand{\cool}{special relative cellular } % situation avec flèche du blow-up
%\newcommand{\Cool}{Special relative cellular }

\newcommand{\half}{\frac{1}{2}}
\newcommand{\smallbullet}{{\scriptscriptstyle\bullet}}
\newcommand{\quadtext}[1]{\quad\text{#1}\quad}
\newcommand{\qquadtext}[1]{\qquad\text{#1}\qquad}

\newcommand{\can}{\omega} % Fibré canonique

\newcommand{\Bl}{B} % Blow-up
\newcommand{\Exc}{E} % Fibre exceptionnelle

%\newcommand{\arr}[1]{\stackrel{#1}{\longrightarrow}}

\newcommand{\deframe}{$(d\!\times\!e)$-frame}
\newcommand{\multi}[1]{\underline{#1}} % Multiindice, partition
\newcommand{\maxi}{{max}} % index of the last \lambda_i in a \Lambda
\newcommand{\row}{\rho}% number of non-zero rows of \Lambda
\newcommand{\zero}{\zeta}% number of zeros rows at the end of \Lambda
\newcommand{\potimes}[1]{^{\otimes #1}}

\newcommand{\Taut}{{\mathcal T}} % Fibré tautologique
\newcommand{\Flag}{{\mathcal Fl}} % Fibré tautologique
\newcommand{\Des}{\Omega} % Désingularisée
\newcommand{\Det}{\Delta} % Déterminant de fibré tautologiques
\newcommand{\gen}{\phi} % Générateur
\newcommand{\ff}{f} % Application de la désingularisée vers la Grassmanienne
\newcommand{\tr}{{tr}} % iso G(d,e)-> G(e,d) ou transposition de diagrame de Young
%%%%%\newcommand{\Grad}{\Gamma}% indices for Wtot(Gr_X(d,e))
\newcommand{\Free}{{\mathbb{F}}}% free module with basis \gen_\lambda's
\newcommand{\iso}{\phi}% iso induit par les \gen_\Lambda
\newcommand{\subb}{\lhd}% for subbundles
\newcommand{\subbeq}{\subb}% for subbundles \subb or = %%%%% A supprimer
\newcommand{\nsubb}{\dot{\ntriangleleft}}% attention

\newcommand{\rect}[2]{[#1\times #2]}
\newcommand{\restr}[1]{_{|_{\scriptstyle #1}}}% to restrict a map
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\newcommand{\too}{\longrightarrow}
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\newcommand{\ourfrac}[2]{\genfrac{}{}{0pt}{}{\scriptstyle #1}{\scriptstyle #2}}
\newcommand{\Ourfrac}[2]{\ourfrac{\displaystyle #1}{\displaystyle #2}}

\newcommand{\twist}{t} % twist of a generator
\newcommand{\Const}{C} % constant part of the twist of a generator
\newcommand{\Twist}{T} % twist of a generator
%\DeclareMathOperator{\perim}{Perim}% the perimeter
%\newcommand{\ptzero}{1_0} % point de twist nul
%\newcommand{\ptone}{1_1} % point de twist un
\newcommand{\pt}{1} % point sans son twist
\newcommand{\vide}{\varnothing} % diagramme vide

%\newcommand{\comb}[1]{(#1)}% ca fait une notation de trop

\DeclareMathOperator{\Spec}{\mathrm{Spec}}    % Spec
\DeclareMathOperator{\Grass}{\mathrm{Gr}}    % Grassmannienne
\DeclareMathOperator{\Gal}{\mathbb{G}}    % Galmèsienne
\DeclareMathOperator{\rank}{\mathrm{rk}} % Rank d'un fibré
\DeclareMathOperator{\Pic}{\mathrm{Pic}} % Groupe de Picard
\DeclareMathOperator{\shift}{\mathrm{shift}} % Shift d'un élément
\DeclareMathOperator{\Id}{\mathrm{Id}} % Identity
\DeclareMathOperator{\im}{\mathrm{im}} % Image
\DeclareMathOperator{\id}{\mathrm{id}} % identity
\DeclareMathOperator{\coim}{\mathrm{coim}} % Coimage


%theorem styles

\swapnumbers %pour que le numéro apparaisse devant le théorème
\theoremstyle{plain}
\newtheorem{theo}{Theorem}[section]
\newtheorem*{maintheo*}{Main Theorem}
\newtheorem{lemm}[theo]{Lemma}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{prop}[theo]{Proposition}

\theoremstyle{definition}
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\newtheorem*{conv*}{Convention}


\gdef\LOCAL{298.files}
%title

\title{{\bf Witt groups of Grassmann varieties}}
\author{Paul Balmer and Baptiste Calm{\`e}s}

\address{Paul Balmer, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA}
%\email{balmer@math.ucla.edu}%%%%% Je propose de ne pas les mettre online, pour eviter le spam.
\urladdr{http://www.math.ucla.edu/$\sim$balmer}

\address{Baptiste Calm\`es, DPMMS, Wilberforce Road, Cambridge, CB3 0WB, England}%%%%% Veux-tu mettre ton adresse a Louvain, plutot?
%\email{b.calmes@dpmms.cam.ac.uk}
\urladdr{http://www.dpmms.cam.ac.uk/site2002/People/calmes\_b.html}

\thanks{The first author is supported by NSF grant 0654397.}

\date\today

\keywords{Witt group, Grassmann variety, triangulated category,
cellular}

\subjclass{19G99, 11E81, 18E30}

%------------------------------

\begin{document}
\bibliographystyle{amsplain}
%\special{papersize=21cm,29.7cm}

\begin{abstract}
We compute the total Witt groups of (split) Grassmann varieties,
over any regular base~$X$. The answer is a free module over the
total Witt ring of~$X$. We provide an explicit basis for this free
module, which is indexed by a special class of Young diagrams, that
we call \emph{even} Young diagrams.
\end{abstract}

\maketitle

\tableofcontents

%------------------------------

\section*{Introduction}

%------------------------------

At first glance, it might be surprising for the non-specialist that
more than thirty years after the definition of the Witt group of a
scheme, by Knebusch \cite{Knebusch77b}, the Witt group of such a
classical variety as a Grassmannian has not been computed yet. This
is especially striking since analogous results for ordinary
cohomologies, for $K$-theory and for Chow groups have been settled a
long time ago. The goal of this article is to explain how Witt
groups differ from these sister theories and to prove the following:

%
\begin{maintheo*}[See Thm.\,\ref{main_thm}]
Let~$X$ be a regular noetherian and separated scheme over
$\bbZ[\half]$, of finite Krull dimension.
Let $0 < d < n$ be integers and let $\Grass_X(d,n)$
be the Grassmannian of $d$-dimensional subbundles of the trivial
$n$-dimensional vector bundle $\cV=\cO_X^n$ over~$X$. (More generally, we treat
any vector bundle $\cV$ admitting a complete flag of subbundles.) 

Then the
total Witt group of~$\Grass_X(d,n)$ is a free graded module over the
total Witt group of~$X$ with an explicit basis indexed by so-called
``even'' Young diagrams. The basis element corresponding to an even
Young diagram is essentially the push-forward of the unit along the
inclusion of the corresponding Schubert variety. The cardinal of
this basis equals $2\cdot\frac{\displaystyle(d'+e')!}{\displaystyle d'!\cdot e'!}$
where $d'=\big[\frac{\displaystyle d}{\displaystyle 2}\big]$ and $e'=\big[\frac{\displaystyle n-d}{\displaystyle 2}\big]$.
\end{maintheo*}

Let us explain this statement. The \emph{total} Witt group refers to the sum of the Witt groups
$\Wper^i(X,L)$
%
$$\Wtot(X) = \bigoplus_{\stackrel{\scriptstyle i \,\in\, \bbZ/4}{\scriptstyle L \,\in\, \Pic(X)/2}} \Wcoh^i(X,L)$$
%
for all possible shifts $i\in\bbZ/4$ and all possible twists
$L\in\Pic(X)/2$ in the duality, see Appendix~\ref{TotalWitt_app}. For $X=\Spec(F)$,
the spectrum of a field, the total Witt group boils down to the
classical Witt group~$\Wper(F)$ but even in that case the above
Theorem is new and the total Witt group of $\Grass_{F}(d,n)$
involves non-trivial shifted and twisted Witt groups. The result has
a very round form when stated for total Witt groups but the
classical unshifted Witt groups $\Wper^0(X,L)$ can be isolated, as
well as the unshifted and untwisted Witt group
$\Wper(X)=\Wper^0(X,\cO_X)$. Indeed, the announced basis consists of
homogeneous elements and we describe below how to read their explicit
shifts and twists on the Young diagrams. For instance, it is worth
noting that there are no new interesting antisymmetric forms in the
Witt groups of $\Grass_X(d,n)$, that is, except for those extended
from~$X$, see Corollary~\ref{ranks_cor}.

To describe our basis explicitly, we need to introduce \emph{even}
Young diagrams. We first consider ordinary Young diagrams sitting in the
upper left corner of a rectangle with $d$ rows and $e$ columns,
which we call the \emph{frame} of the diagram. See
Figure~\ref{framed_fig}.

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/YoungFramed.pstex_t}%pdf_t}%
\end{center}
\caption{Young diagram $\Lambda$ in \deframe}
\label{framed_fig}%
\end{figure}

We say that such a framed Young diagram~$\Lambda$ is \emph{even} if
all the segments of the boundary of~$\Lambda$ which are strictly
inside the frame have even length. That is, we allow $\Lambda$ to
have odd-length segments on its boundary only where it touches the
outside frame. See Figure~\ref{even_fig} for examples. (In Figures~\ref{G44_fig}, \ref{G45_fig} and~\ref{G55_fig} we further give all even diagram in \deframe\ for $(d,e)=(4,4)$, $(4,5)$ and $(5,5)$, respectively.)


\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/YoungEven.pstex_t}%pdf_t}%
\end{center}
\caption{Five examples of even Young diagrams}
\label{even_fig}%
\end{figure}

We shall see that basis elements are in bijective correspondence
with even Young diagrams in \deframe, for $e:=n-d$. Moreover, as
explained in Section~\ref{gen-Witt_sec}, the Witt class
$\gen_{d,e}(\Lambda)$ corresponding to such a diagram $\Lambda$
lives in the Witt group $\Wper^i\big(\Grass_X(d,n),L\big)$ for the
shift $i=|\Lambda|\in\bbZ/4$ equal to the \emph{area} of the diagram and
for the twist $L\in\Pic(\Grass_X(d,n))/2$ equal to the class of
$t(\Lambda)\cdot\Det$ where $\Det\in\Pic(\Grass_X(d,n))$ is the
determinant of the tautological bundle and where the integer
$\twist(\Lambda)$ is half the \emph{perimeter} of the diagram~$\Lambda$
(see Figure~\ref{twist_fig} in Section~\ref{gen-Witt_sec}). More
generally, when $\cV$ is not free but admits a complete flag of
subbundles, the twist of $\gen_{d,e}(\Lambda)$ also involves a
multiple of the determinant of~$\cV$, in the direct summand of
$\Pic(\Grass_X(d,n))/2$ coming from $\Pic(X)/2$.

\smallbreak

Let us put our result in perspective.
In its modern form, see for instance Laksov~\cite{Laksov72},
the computation of the cohomology (or $K$-theory, or Chow groups) of
a cellular variety usually relies on four important ingredients, (a)-(d) below. Let
$Z\subset V$ be a closed subscheme with open complement~$U$ (think of
$U$ as an open cell of~$V$ which is an $\bbA^*$-bundle over some smaller scheme). These results are:
\begin{enumerate}
\item A long exact sequence of localization relating the cohomology of~$V$, the cohomology of $V$ with support in $Z$ and the cohomology of~$U$.
\item A dévissage theorem, which identifies the cohomology of $Z$ with the cohomology of $V$ with support in~$Z$.
\item Homotopy invariance, that is the fact that the cohomology of the total space of an $\mathbb{A}^*$-bundle is the same as the cohomology of the base, which helps identifying the cohomology of $U$ with something simpler.
\item A way of splitting the localization long exact sequence for particular cellular decompositions, usually proving the connecting homomorphism to vanish.
\end{enumerate}

Looking back, it is now less surprising that Witt groups of Grassmann
varieties could not be computed in the 70's, because most of these
tools have only
been constructed recently for Witt groups of schemes. With the notable exception of
projective space over a field, see Arason~\cite{Arason80}, most of the recent computations for
projective varieties have awaited those tools to be available.

The long exact sequence could only be established by defining first
``higher" or ``shifted" Witt groups, as it was done
in~\cite{Balmer00} and~\cite{Balmer01} in 2000 by the first author
using the framework of triangulated categories. This enabled
Walter~\cite{Walter03} to compute the Witt groups of projective
bundles by decomposing their derived categories. The homotopy
invariance was proved by Gille~\cite{Gille03b}. The dévissage
theorem is usually a consequence of the definition of push-forwards
and was obtained in the form we need here by Hornbostel and the
second author in \cite{Calmes06}, although several forms of
dévissage were previously considered by various authors. Finally,
point~(d) was investigated by the authors in~\cite{BalmerCalmes08}
where it is shown that, under some geometric assumptions, the
connecting homomorphism can be computed by geometric means, using
only push-forwards and pull-backs. This is the part where Witt
groups differ in the most significant way from the sister theories
mentioned above. Unlike the case of oriented cohomology theories in
the sense of Levine-Morel~\cite{Levine07} or Panin~\cite{Panin04},
the connecting homomorphism of point~(d) is in general {\it not}
zero for Witt groups.

We use all the above techniques here, in the case of Grassmann
varieties and their usual cellular decompositions.

\smallbreak

Let us now comment on the organization of the paper.
Sections~\ref{Grass_sec} and~\ref{Young_sec} contain preparatory
material on Grassmann varieties, desingularizations of Schubert varieties and even Young diagrams.

Our generators of $\Wtot\big(\Grass_X(d,n)\big)$ are defined in
Section~\ref{gen-Witt_sec} as push-forwards of the unit forms of
certain desingularized Schubert varieties. The reader should keep in mind that pushing the unit form is not always possible, due to the
presence of line bundles in the definition of the push-forward.
Indeed, for a proper morphism $f:X \to Y$ of equidimension $\dim(f)$, between regular noetherian schemes (think $\dim f=\dim X-\dim Y$), the push-forward along~$f$ is
defined between the following Witt groups:
%
\begin{equation}
\label{push-forward_eq}%
\Wcoh^{i+\dim f}(X,\can_{f} \otimes f^*L) \too \Wcoh^{i}(Y,L)\,,
\end{equation}
%
where the line bundle $\can_f$ is the so-called \emph{relative line
bundle}. So, when $\cO_X$ is not (up to squares) of the form
$\can_{f} \otimes f^*L$ for some line bundle $L$ over~$Y$, one {\it
cannot} push-forward the unit form of~$X$, which lives
in~$\Wcoh^0(X,\cO_X)$. This is why we start
Section~\ref{gen-Witt_sec} by discussing the ``parity'' of the
relevant canonical bundles~$\can_{f}$. Although somewhat heavy,
these computations are elementary and are all based on a repeated
application of the computation of the relative canonical bundle of a
Grassmann bundle (Prop.\,\ref{rel-can-Grass_prop}). The
condition for a Young diagram to be even implies the existence of
such a push-forward for the unit of the desingularized Schubert cell
into the Grassmannian. In fact, we could push-forward the unit forms
for more Schubert cells but these additional generators would be
redundant. The even Young diagrams are chosen so that the
corresponding forms are also linearly independent.
	
Then, in Section~\ref{cell-dec_sec}, we recall the classical
relative cellular structure of the Grassmann varieties and in
Section~\ref{push-pull-bord_sec} the long exact sequence associated
to it. In the final Section~\ref{main_sec}, we compute how our
candidate-generators behave under the morphisms in the long exact
sequence, especially under the connecting homomorphism, which is
most of the time not zero (Cor.\,\ref{bord-non-zero_coro}). The
proof of the main theorem (Thm.\,\ref{main_thm}) then follows by
induction on the rank of the vector bundle~$\cV$.

\smallbreak

This article is written in the language of
``functors of points", which means that we describe schemes in terms
of their points (which are here flags) and morphisms of schemes as
how they act on those points. This method is completely rigorous in this
case. The original source is~\cite[\S\,I.1]{DemazureGabriel70} and
we also refer the reader to~\cite[Part 2]{Karpenko00b} for general
considerations on this subject. This language is customary when
dealing with flag varieties, see for instance~\cite{Laksov72} in
which it is used for the computation of Chow groups of Grassmann
varieties.

\begin{conv*}
%\label{X_conv}%
Throughout the paper, $X$ stands for a regular noetherian separated scheme over~$\bbZ[\half]$, of finite Krull dimension.
\end{conv*}

%------------------------------

\section{Combinatorics of Grassmann and flag varieties}
\label{Grass_sec}%

%------------------------------

We recall elementary facts about Grassmann varieties and desingularizations of Schubert cells. We also provide the necessary
material about canonical bundles to treat the push-forward
homomorphisms for Witt groups in Section~\ref{gen-Witt_sec}.

\begin{defi}\label{subb_def}%
A \emph{subbundle} $\cP\subb \cV$ of a vector
bundle $\cV$ over a scheme~$X$ is an $\cO_X$-submodule which is
locally a direct summand, \ie $\cP$ and $\cV/\cP$ are vector bundles.
\end{defi}

\begin{defi}
\label{Grass_def}%
Let $\cV$ be a vector bundle of rank $n>0$ over a
scheme~$X$ and let $d$ be an integer $0\leqslant d\leqslant n$. We
denote by $\Grass_X (d,\cV)$ the Grassmann bundle over~$X$
parameterizing the subbundles of rank~$d$ of~$\cV$. In the language
of functors of points, it means that for any morphism $f:\Spec(R)\to
X$, the set $\Grass_X(d,\cV)(R)$ consists of the $R$-submodules
$P\subb \cV(R)=f^*(\cV)$ which are direct summands of rank~$d$.

The scheme $\Grass_X(d,\cV)$ comes equipped with a smooth structural
morphism $\pi:\Grass_X(d,\cV) \to X$ and a tautological bundle
$\Taut_d=\Taut_d^{\Grass_X(d,\cV)}$ of rank~$d$, whose determinant
we denote by~$\Det_d$.
\end{defi}

\begin{prop}%[Picard group]
\label{Pic-Grass_prop}%
The scheme $\Grass_X(d,\cV)$ is smooth over~$X$ of relative
dimension $d(n-d)$. For $0<d<n$, the Picard group of
$\Grass_X(d,\cV)$ is given by
%
$$
\begin{array}{ccc}
\Pic(X) \oplus \bbZ & \cong & \Pic(\Grass_X(d,\cV))
\\
(\ell,m) & \mapsto & \pi^*(\ell) + m\, \Det_d.
\end{array}
$$
%
In case $d=0$ or $d=n$, the morphism $\pi:\Grass_X(d,\cV)\to X$ is
the identity.
\end{prop}

\begin{proof}
The Picard group of a regular scheme coincides with its Chow group
$CH^1$, which is computed in \cite{Laksov72} for Grassmannians: see
Theorem 16 for the case where $X$ is a field, and §13 to work over a
regular base $X$. Using the Pl\"ucker embedding, one checks that the
generator in \loccit{} is indeed $\Det_d$.
\end{proof}

\begin{coro}
\label{Pic-Grass_coro}%
We have $\Pic\big(\Grass_X(d,\cV)\big)/2\ =\ \Pic(X)/2\ \oplus\ \bbZ/2\cdot\Det_d$.
\qed
\end{coro}

\begin{prop}%[Relative canonical bundle]
\label{rel-can-Grass_prop}%
The relative canonical bundle $\can_{\Grass_X(d,\cV)/X}$ of the projection
$\pi:\Grass_X (d,\cV) \to X$ is $\can_{\Grass_X(d,\cV)/X}=-d \det \cV+n \Det_d$ in $\Pic(\Grass_X (d,\cV))$. In particular,
if $\cV=\cO_X^n$ is trivial, $\can_{\Grass_X(d,\cV)/X}=n\,\Det_d$.
\end{prop}
\begin{proof}
The morphism $\pi$ is smooth, so $\can_{\Grass_X(d,\cV)/X}$ is the determinant (highest exterior power) of the relative cotangent bundle of $\pi$. This cotangent bundle is the tautological bundle tensored by the dual of the tautological
quotient bundle (see~\cite[Appendix B.5.8]{Fulton98}). Taking the determinant, we get the result.
\end{proof}

\bigbreak \centerline{*\ *\ *}\goodbreak\bigbreak

We now extend the previous results from Grassmannians to some flag varieties.

\begin{defi}
\label{Flag_def}%
Let $k\geqslant1$ and $(\multi{d},\multi{e})$
be a pair of $k$-tuples of non-negative integers $\multi{d}=(d_1,\ldots,d_k)$ and
$\multi{e}=(e_1,\ldots,e_k)$ satisfying
%
\begin{equation}\label{pre-de_eq}%
0< d_1 < \cdots <d_k \qquad\text{and}\qquad
e_1+d_1\leqslant \cdots\leqslant e_k+d_k\,.
\end{equation}
%
(The second condition holds in particular if we have $e_1\leqslant\cdots\leqslant e_k$.) Consider a flag
%
\begin{equation}
\label{partial_eq}%
\cV_{d_1+e_1} \subb \cdots \subb \cV_{d_i+e_i} \subb \cdots \subb \cV_{d_k+e_k}
\end{equation}
%
of vector bundles over~$X$, where $\subb$ indicates subbundles in the strong sense of Definition~\ref{subb_def} and where the rank is given by the index: $\rank_X(\cV_r)=r$.


We associate to
this data the scheme $\Flag_{X}(\multi{d},\multi{e},\cV_{\smallbullet})$ over~$X$,
which parameterizes the flags of vector bundles $\cP_{d_1}\subb
\cP_{d_2}\subb\cdots\subb \cP_{d_k}$ such that $\rank \cP_{d_j}=d_j$
and $\cP_{d_j} \subb \cV_{d_j + e_j}$. As a functor of points, this gives for any morphism $f:Y\to X$
%
\begin{equation}\label{Flag_eq}%
\Flag_{X}(\multi{d},\multi{e},\cV_\smallbullet)(Y) := \left\{
\begin{array}{c}
\xymatrix@R=1.5ex@C=1.5ex{ 0 \ar@{}[r]|-{\subb} & \cP_{d_1}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_1}
\ar@{}[r]|{\subb} & \cP_{d_2}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_2}
\ar@{}[r]|{\subb} & \cdots \ar@{}[r]|{\subb} & \cP_{d_k}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_k}
\\
0 \ar@{}[r]|-{\subb} & f^*\cV_{d_1 + e_1} \ar@{}[r]|-{\subb}
& f^*\cV_{d_2 + e_2} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb}
& f^*\cV_{d_k + e_k} }
\end{array}
\right\}\,,
\end{equation}
%
where all $\cP_{d_i}$ are vector bundles over~$Y$ of rank~$d_i$ such that all inclusions are subbundles in the sense of Definition~\ref{subb_def}. The integers along inclusions indicate codimensions.
%
Following general practice, we shall drop the mention of $f^*$ in the sequel. Moreover, to avoid cumbersome notations, unless the original flag~\eqref{partial_eq} varies, we drop the mention of $\cV_\smallbullet$ from the notation\,: $\Flag_X(\multi{d},\multi{e})=\Flag_{X}(\multi{d},\multi{e},\cV_\smallbullet)$.
\end{defi}

\begin{exam}
\label{k=1_exa}%
For $k=1$, the scheme $\Flag_X(\multi{d},\multi{e})$ is simply $\Grass_X(d_1,\cV_{d_1+e_1})$.
\end{exam}

\begin{rema}
\label{morph_rem}%
For any choice $J$ of $k'$ indices among $\{1,\ldots,k\}$, one can consider the pair of $k'$-tuples $(\multi{d}',\multi{e}')$ obtained from $(\multi{d},\multi{e})$ by keeping $d_i$ and $e_i$ only for indices~$i\in J$. There is a natural morphism $\Flag_X(\multi{d},\multi{e})\to\Flag_X(\multi{d}',\multi{e}')$ over~$X$, obtained by dropping the $\cP_{d_j}$ for the non-chosen indices~$j$.

Furthermore, for any vector bundle $\cV$ such that $\cV_{d_k+e_k}\subb\cV$\,, there is a natural morphism $\ff_{\multi{d},\multi{e},\cV}$ of schemes over~$X$ as follows\,:
%
\begin{equation}
\label{pre-ff_eq}%
\begin{array}{rccccc}
\ff_{\multi{d},\multi{e},\cV}\,: &\Flag_X(\multi{d},\multi{e}) & \too &
\Grass_X(d,\cV_{d_k+e_k}) & \hookrightarrow & \Grass_X(d,\cV)
\\
&(\cP_{d_1},\ldots, \cP_{d_k}) & \longmapsto & \cP_{d_k} & \longmapsto & \cP_{d_k}\,,
\end{array}
\end{equation}
%
where the first morphism is as above and the second is a closed immersion.
\end{rema}

\begin{defi}
\label{tavtologic_def}%
The scheme $\Flag_{X}(\multi{d},\multi{e})$
is equipped with \emph{tautological bundles} $\Taut_{d_i}$, $1 \leqslant i
\leqslant k$, of rank $d_i$, whose determinants we denote by
$\Det_{d_i}:=\det(\Taut_{d_i})$. The stalk of $\Taut_{d_i}$ at a point $(\cP_{d_1},
\ldots, \cP_{d_k})$ is $\cP_{d_i}$. In ambiguous cases, the full notation for
$\Taut_{d_i}$ would be $\Taut_{d_i}^{\Flag_X(\multi{d},\multi{e},\cV_\smallbullet)}$.
\end{defi}

\begin{rema}\label{zero-Det_rem}%
If $e_i=0$ then the vector bundles $\Taut_{d_i}=\cV_{d_i}$ and
$\Det_{d_i}=\det\cV_{d_i}$ are both extended from~$X$.
\end{rema}


\begin{lemm}
\label{rel_lem}%
Let $k\geqslant2$ and let $(\multi{d},\multi{e})$ be a pair of
$k$-tuples satisfying~\eqref{pre-de_eq}. Let $\cV_\smallbullet$ be a
flag as in~\eqref{partial_eq}. Define the $(k-1)$-tuples $\multi{d}\restr{k-1}$ and
$\multi{e}\restr{k-1}$ as the restrictions of $\multi{d}$ and
$\multi{e}$ to the first $k-1$ entries. Consider the scheme
%
$$
Y:=\Flag_X(\,\multi{d}\restr{k-1}\,,\,\multi{e}\restr{k-1}\,,\,\cV_\smallbullet\,)\,,
$$
%
which only ``uses'' the first $k-1$ bundles $\cV_{d_1+e_1}\subb\cdots\subb\cV_{d_{k-1}+e_{k-1}}$. Consider the pull-back to~$Y$ of the remaining bundle, still denoted $\cV_{d_{k}+e_{k}}$\,. Observe that $\Taut_{d_{k-1}}^Y\subb\cV_{d_{k}+e_{k}}$ and consider the quotient bundle
%
$$
\tilde\cV:=\cV_{d_k+e_k}/\Taut_{d_{k-1}}^Y
$$
%
over~$Y$. It has rank $d_k-d_{k-1}+e_k$. We then have a canonical isomorphism of schemes over~$Y$ (hence over~$X$)\,:
%
\begin{equation}
\label{rel-Flag_eq}%
\Flag_X(\multi{d},\multi{e},\cV_\smallbullet)\quad\cong\quad\Grass_Y\big(d_k-d_{k-1},\tilde\cV\big)\,.
\end{equation}
%
Under this identification, we have $\Taut^{\Flag(\multi{d},\multi{e},\cV_\smallbullet)}_{d_i}=\Taut^Y_{d_i}$ for all $1\leqslant i\leqslant k-1$ and
%%
\begin{equation}
\label{conv-Taut_eq}%
\Taut_{d_{k}}^{\Flag_{Y}(\multi{d},\multi{e},\cV_\smallbullet)}/\Taut_{d_{k-1}}^{\Flag_{Y}(\multi{d},\multi{e},\cV_\smallbullet)}
=
\Taut_{d_k-d_{k-1}}^{\Grass_{Y}(d_k-d_{k-1},\tilde\cV)}\,.
\end{equation}
\end{lemm}

\begin{proof}
This simply amounts to the bijective correspondence between a flag $\cP_{d_1}\subb\cdots\subb\cP_{d_{k-1}}\subb\cP_{d_{k}}$ satisfying $\cP_{d_i}\subb\cV_{d_i+e_i}$ for all $1\leqslant i\leqslant k$ and the following data\,:
\begin{enumerate}
\item[(a)]
the beginning of this flag $\cP_{d_1}\subb\cdots\subb\cP_{d_{k-1}}$ satisfying $\cP_{d_i}\subb\cV_{d_i+e_i}$ for all $1\leqslant i\leqslant k-1$,
\item[(b)]
the bundle $\cP_{d_{k}}$ such that $\cP_{d_{k-1}}\subb\cP_{d_{k}}\subb\cV_{d_{k}+e_{k}}$
\end{enumerate}
and to observe that (b) is equivalent to a subbundle $\tilde\cP\subb\cV_{d_k+e_k}/\cP_{d_{k-1}}$ of rank $d_k-d_{k-1}$, where $\tilde\cP:=\cP_{d_{k}}/\cP_{d_{k-1}}$. Details are left to the reader.
\end{proof}


\begin{conv}
\label{zero_conv}%
When using $k$-tuples $\multi{d}=(d_1,\ldots,d_k)$, it will unify several formulas to simply define $d_0=0$.
\end{conv}

%
\begin{prop}
\label{Pic-Flag_prop}%
Let $\multi{d}$ and~$\multi{e}$ be two $k$-tuples as in~\eqref{pre-de_eq} and $\cV_\smallbullet$ be a flag as in~\eqref{partial_eq}.
%
Then $\Flag_X(\multi{d},\multi{e})$ is smooth over~$X$ of relative
dimension~$\sum_{i=1}^{k}(d_{i}-d_{i-1})\,e_i$.
The Picard group of $\Flag_X(\multi{d},\multi{e})$ is generated by
$\Pic(X)$ and the ``new'' bundles $\Det_{d_i}$\,:
%
\begin{equation} \label{Pic-Flag_eq}%
\Pic(\Flag_X(\multi{d},\multi{e})) \cong {\Pic}(X)\ \oplus\!
\bigoplus_{\Ourfrac{1\leqslant i\leqslant k}{\text{s.t.\ }e_i\neq0}} \bbZ \,\Det_{d_i}\,.
\end{equation}
%
The relative canonical bundle
$\can_{\Flag_X(\multi{d},\multi{e})/X}$ is given by the
formula
%
\begin{equation}\label{can-abs-Flag_eq}%
\begin{array}{rl}
\can_{\Flag_X(\multi{d},\multi{e})/X} =
& \sum\limits_{i=1}^k(-d_i+d_{i-1})\,\det\cV_{d_i+e_i}+
\\
& + \sum\limits_{i=1}^{k-1}(d_{i}-d_{i-1}
+ e_{i}-e_{i+1})\,\Det_{d_{i}}
+ (d_k-d_{k-1} + e_k)\,\Det_{d_k}\,,
\end{array}
\end{equation}
%
where $\Det_{d_i}=\det\cV_{d_i}$ if $e_i=0$ by Remark~\ref{zero-Det_rem} and where we use Convention~\ref{zero_conv}. In particular, for $k=1$, this reads $\can_{\Flag_X(\multi{d},\multi{e})/X}=-d_1\,\det\cV_{d_1+e_1}+(d_1+e_1)\,\Det_{d_1}$.
\end{prop}

\begin{proof}
By induction on~$k$. The case $k=1$ is that of a Grassmannian over~$X$ (Example~\ref{k=1_exa}) so the result follows from Propositions~\ref{Pic-Grass_prop} and~\ref{rel-can-Grass_prop}.

Let now $k\geqslant 2$. Consider $Y=\Flag_{X}(\multi{d}|_{k-1},\multi{e}|_{k-1},\cV_\smallbullet)$ and the bundle $\tilde\cV=\cV_{d_k+e_k}/\Taut_{d_{k-1}}$ over~$Y$, as in Lemma~\ref{rel_lem}. 
Recall that $\rank_Y(\tilde\cV)=d_k-d_{k-1}+e_k$, which is always strictly positive ($d_k>d_{k-1}$) and which is bigger than or
equal to $d_k-d_{k-1}$ with equality if and only if $e_k=0$. 
Equation~\eqref{rel-Flag_eq} and Propositions~\ref{Pic-Grass_prop} and~\ref{rel-can-Grass_prop} immediately give smoothness, the formula for the relative dimension and that for the Picard group~\eqref{Pic-Flag_eq}. 
Finally, to prove~\eqref{can-abs-Flag_eq}, observe that
%
\begin{align*}
& \can_{\Flag_{X}(\multi{d},\multi{e})/Y}
 = \can_{\Grass_{Y}(d_k-d_{k-1},\tilde\cV)\,/\,Y}
\\
& =
(-d_k+d_{k-1})\det(\tilde\cV) + \rank(\tilde\cV)\,\Det_{d_k-d_{k-1}}^{\Grass_{Y}(d_k-d_{k-1},\tilde\cV)}
\\
& = (-d_k+d_{k-1})\det(\tilde\cV) + \rank(\tilde\cV)\,
\big(\Det_{d_k}^{\Flag_X(\multi{d},\multi{e})}-\Det_{d_{k-1}}^{\Flag_X(\multi{d},\multi{e})}\big)
\\
&
=(-d_k+d_{k-1})\det\cV_{d_k+e_k}-e_k\,\Det_{d_{k-1}}
+(d_k-d_{k-1}+e_k)\,\Det_{d_{k}}\,.
\end{align*}
%
The first equality uses~\eqref{rel-Flag_eq}, the second comes from
Proposition~\ref{rel-can-Grass_prop} and the third from~\eqref{conv-Taut_eq}. The last equality is a direct
computation (in which we drop the mention of $\Flag_X(\multi{d},\multi{e})$ for readability). By induction hypothesis, we get $\can_{Y/X}$ from Equation~\eqref{can-abs-Flag_eq} applied for $k-1$, that is, for the flag variety~$Y$. Since $\can_{\Flag_{X}(\multi{d},\multi{e})/X}=\can_{\Flag_{X}(\multi{d},\multi{e})/Y}+\can_{Y/X}$ over~$\Flag_X(\multi{d},\multi{e})$, we get~\eqref{can-abs-Flag_eq} for~$k$ by adding the above.
\end{proof}

\begin{coro}
\label{can-Flag_cor}%
Let $\multi{d}$ and~$\multi{e}$ be two $k$-tuples as in~\eqref{pre-de_eq} and $\cV_\smallbullet$ be a flag as in~\eqref{partial_eq}. Let $\cV$ be a vector bundle of rank~$d+e$ such that $\cV_{d_k+e_k}\subb\cV$\,. The relative canonical bundle for the morphism $\ff_{\multi{d},\multi{e},\cV}:\Flag_X(\multi{d},\multi{e})\to\Grass_X(d,\cV)$ of~\eqref{pre-ff_eq} is given by
%
\begin{equation}
\label{can-Flag_eq}%
\begin{array}{rl}
\can_{\Flag_X(\multi{d},\multi{e})/\Grass_X(d,\cV)} & =\
\sum\limits_{i=1}^{k}(-d_i+d_{i-1})\,\det\cV_{d_i+e_i} + d_k\,\det\cV\ +
\\
+ \sum\limits_{i=1}^{k-1} & (d_{i}-d_{i-1}
+e_{i}-e_{i+1})\,\Det_{d_{i}} + (-d_{k-1} + e_k - e)\,\Det_{d_k}\,,
\end{array}
\end{equation}
%
where $\Det_{d_i}=\det\cV_{d_i}$ if $e_i=0$ by Remark~\ref{zero-Det_rem} and where we use Convention~\ref{zero_conv}. For $k=1$, this reads $\can_{\Flag_X(\multi{d},\multi{e})/\Grass_X(d,\cV)} =d_1\,\det(\cV/\cV_{d_1+e_1})+(e_1-e)\,\Det_{d_1}$.
\end{coro}

\begin{proof}
Subtract
$(\ff_{\multi{d},\multi{e},\cV})^*\,\can_{\Grass_X(d,\cV)/X}=-d_k\,\det\cV+(d_k+e)\,\Det_{d_k}$
(Prop.~\ref{rel-can-Grass_prop}) from the bundle
$\can_{\Flag_X(\multi{d},\multi{e})/X}$ given
in~\eqref{can-abs-Flag_eq}.
\end{proof}

\begin{rema}
When $\cV_{\smallbullet}=\cO_X^\smallbullet$, all the formulas are simpler, since all the $\det \cV_i$ are zero. This applies in particular when $X=\Spec(R)$ for a local ring $R$ .
\end{rema}


%------------------------------

\section{Even Young diagrams}
\label{Young_sec}%

%------------------------------

We introduce \emph{even} Young diagrams that will parameterize the basis of the total Witt group of the Grassmann variety, to be constructed in Section~\ref{gen-Witt_sec}.

\begin{defi}
\label{Young_def}%
Let $d,e\geqslant1$. We shall call \emph{Young
diagram in \deframe}, or simply \emph{$(d,e)$-diagram},
any $d$-tuple $\Lambda=(\Lambda_1,\Lambda_2,\ldots,\Lambda_d)$ of
integers such that:
%
$$
e \geqslant \Lambda_1 \geqslant \Lambda_2 \geqslant \ldots \geqslant \Lambda_d \geqslant 0.
$$
%
See Figure~\ref{framed_fig} in the Introduction. The \emph{area} of $\Lambda$ is
$|\Lambda|=\Lambda_1+\Lambda_2+\ldots+\Lambda_d$. These
$(d,e)$-diagrams are just ordinary Young diagrams displayed in the
upper left corner of a rectangle with $d$ rows and $e$ columns,
possibly leaving empty rows below and empty columns to the right of
the Young diagram. So, an ordinary Young diagram with $\rho$
rows and $\gamma$ columns defines a $(d,e)$-diagram
for any $d\geqslant \rho$ and $e\geqslant \gamma$.
\end{defi}

\begin{exam}
\label{empty-full_exa}%
We denote by $\varnothing$ the empty diagram $\varnothing=(0,\ldots,0)$. We denote by $\rect{d}{e}$ the full $d\times e$-rectangle $\rect{d}{e}=(e,\ldots,e)\in\bbN^d$.
\end{exam}

\begin{defi}
\label{de_def}%
Let $d,e\geqslant1$ and let $\Lambda$ be a Young diagram in
\deframe. The decreasing sequence
$\Lambda_1\geqslant\Lambda_2\geqslant\ldots\geqslant\Lambda_d$ can be written in a
unique way as a series of equalities and strict inequalities:
%
\begin{equation}
\label{dk_eq}%
\underbrace{\Lambda_1=\cdots=\Lambda_{d_1}}_{d_1\text{
terms}}\ >\
\underbrace{\Lambda_{d_1+1}=\ldots=\Lambda_{d_2}}_{d_2-d_1\text{
terms}}\ >\ \cdots\ >\
\underbrace{\Lambda_{d_{k-1}+1}=\cdots=\Lambda_{d_k}}_{d_k-d_{k-1}\text{
terms}}=\Lambda_d\,.
\end{equation}
%
Note that $d_k=d$. The integers $k\geqslant1$ and $0<d_1<\ldots<d_k$
depend on $\Lambda$. If we need to stress this we shall write
$k=k(\Lambda)$ and $d_i=d_i(\Lambda)$ for $1\leqslant i\leqslant
k(\Lambda)$.

For fixed $d$ and $e$, there is a bijection
(pictured on Figure~\ref{partition_fig}) between the Young diagrams
$\Lambda$ in
\deframe\ and pairs of $k$-tuples of integers
%
\begin{equation}
\label{de_eq}%
\begin{array}{lcl}
\multi{d}=(d_1,\ldots,d_k) & \text{such that} & 0< d_1 < \cdots <d_k
= d
\\
\multi{e}=(e_1,\ldots,e_k) & \text{such that} & 0\leqslant e_1 <
\cdots < e_k \leqslant e
\end{array}
\end{equation}
%
with $1\leqslant k\leqslant d$. The integers $k=k(\Lambda)$ and
$d_i=d_i(\Lambda)$ are the above ones and we set
$e_i=e_i(\Lambda):=e-\Lambda_{d_i}$ for all $i=1,\ldots,k$. The converse
construction is obvious.
\end{defi}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/YoungNombres.pstex_t}%pdf_t}%
\end{center}
\caption{Two examples of the two $k$-tuples $(d_1,\ldots,d_k)$ and
$(e_1,\ldots,e_k)$ corresponding to a Young diagram $\Lambda$ in \deframe.}
\label{partition_fig}%
\end{figure}

\begin{defi}\label{re-Flag_def}%
Let $d,e\geqslant1$. Fix a complete flag of vector bundles over~$X$
%
\begin{equation}
\label{complete_eq}%
0=\cV_0 \subb \cV_1 \subb \cdots \subb \cV_i \subb \cdots \subb \cV_{d+e}=:\cV\,.
\end{equation}
%
Note that we baptize $\cV$ the bundle of dimension $d+e$, to lighten notation.

Let $\Lambda$ be a Young diagram in
\deframe. By Definition~\ref{de_def}, this amounts to a pair $(\multi{d},\multi{e})$ of $k$-tuples of integers satisfying~\eqref{de_eq}, and hence satisfying~\eqref{pre-de_eq}. We can now apply Definition~\ref{Flag_def} to $\multi{d}$ and $\multi{e}$ and the flag~\eqref{partial_eq} taken from~\eqref{complete_eq} above\,:
%
\begin{equation}
\label{Flag-Lambda_eq}%
\Flag_X(d,e,\cV_\smallbullet;\Lambda):=\Flag_{X}(\multi{d},\multi{e},\cV_\smallbullet)
= \left\{
\begin{array}{c}
\xymatrix@R=1.5ex@C=1.5ex{ 0 \ar@{}[r]|-{\subb} & \cP_{d_1}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_1}
\ar@{}[r]|{\subb} & \cP_{d_2}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_2}
\ar@{}[r]|{\subb} & \cdots \ar@{}[r]|{\subb} & \cP_{d_k}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle e_k}
\\
0 \ar@{}[r]|-{\subb} & \cV_{d_1 + e_1} \ar@{}[r]|-{\subb} & \cV_{d_2
+ e_2} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d_k +
e_k} }
\end{array}
\right\}\,.
\end{equation}
%
As usual, instead of $\Flag_X(d,e,\cV_\smallbullet;\Lambda)$, we might simply write $\Flag(\Lambda)$ or anything ``in between'' depending on what is obvious from the context.

As in~\eqref{pre-ff_eq}, there is a natural morphism
$\ff_{\Lambda}$ from $\Flag_X(d,e;\Lambda)$ to $\Grass(d,\cV)$
%
\begin{equation}
\label{ff_eq}%
\begin{array}{rccc}
\ff_\Lambda=\ff_{d,e;\Lambda}:=\ff_{\multi{d},\multi{e},\cV}\,:
&\Flag_X(d,e;\Lambda) & \too & \Grass(d,\cV)
\\
&(\cP_{d_1},\ldots, \cP_{d_k}) & \longmapsto & \cP_{d_k}\,.
\end{array}
\end{equation}
%

When $X=\Spec(F)$ is a field, one can understand the image of
$\ff_\Lambda$ as the subset of those subspaces $\cP_d\subb \cV$ whose
intersection with each $\cV_{d_i+e_i}$ is of dimension at
least~$d_i$. This is the classical Schubert cell associated to the
diagram~$\Lambda$. It is pretty clear that $\ff_{\Lambda}$ is a
birational morphism. The advantage of $\Flag_X(d,e;\Lambda)$ over the Schubert
cell is that $\Flag_X(d,e;\Lambda)$ is not singular by Proposition~\ref{Pic-Flag_prop}.
\end{defi}

\begin{exam}
Following up on Example~\ref{k=1_exa}, when
$\Lambda=\varnothing$ is the empty diagram, that is for $k=1$ and
$e_1=e$, we have $\Flag_X(\varnothing)=\Grass_X(d,\cV)$ and $\ff_\varnothing$ is the identity. At the other
end, for $\Lambda=\rect{d}{e}$ the whole $(d\times e)$-rectangle,
that is for $k=1$ and $e_1=0$, we have
$\Flag_X(d,e;\Lambda)=\Grass_X(d,\cV_d)=X$ and $\ff_\Lambda$ is a closed immersion.
\end{exam}

\begin{defi}
\label{zerow_def}%
Let $\Lambda$ be a Young diagram in \deframe. We define $\row(\Lambda)\in\{0,\ldots,d\}$ to be the number of non-zero rows of $\Lambda$. Complementarily, we
define $\zero(\Lambda)=d-\row(\Lambda)$ to be the number of zero rows at the end of $\Lambda$, that is
%
$$
\begin{array}{lcll}
\row(\Lambda)=d & \text{and} & \zero(\Lambda)=0
 & \text{if }\Lambda_d>0\,,
\\
\row(\Lambda)=d_{k-1} & \text{and} & \zero(\Lambda)=d-d_{k-1}
& \text{if }\Lambda_d=0\,.
\end{array}
$$
%
For the empty diagram, we have $\row(\varnothing)=0$ and $\zero(\varnothing)=d$.
\end{defi}

We are going to use a certain class of $(d,e)$-diagrams, that we
call the \emph{even} $(d,e)$-diagrams. Defining them by a picture is
very easy. The condition to be even is that any segment of the
$(d,e)$-diagram which does not belong to the outer \deframe\ must
have even length. See Figure~\ref{even_fig}. The formal definition
is the following.

\begin{defi}
\label{even_def}%
Let $\Lambda$ be a Young diagram in \deframe\ and
let $\multi{d}$ and $\multi{e}$ be the associated $k$-tuples as in
Definition~\ref{de_def}. We say that $\Lambda$ is \emph{even} if all
the following conditions are satisfied:
\begin{enumerate}
\item[(i)] $d_{i+1}-d_i$ is even for all $i=1,\ldots,k-2$ (for
$k\geqslant 3$ otherwise no condition),
\item[(ii)] $e_{i+1}-e_i$ is even, for all
$i=1,\ldots,k-1$ (for $k\geqslant 2$),
\item[(iii)] when $0<e_1<e$ we also require $d_1$ to be even,
\item[(iv)] when $0<e_k<e$ we also require $d_k-d_{k-1}$ to be even.
\end{enumerate}
%
\end{defi}

\begin{exam}
For any $d,e\geqslant1$, both the empty diagram $\varnothing$ and
the full-rectangle $\rect{d}{e}$ are even $(d,e)$-diagrams (see Ex.\,\ref{empty-full_exa}). Indeed,
in both cases, $k=1$ and $\multi{d}=(d)$, whereas
$\multi{e}(\varnothing)=(e)$ and $\multi{e}(\rect{d}{e})=(0)$; so there is
no condition to check.

When $d=1$ or $e=1$, these are the only even Young diagram in \deframe.

For more examples, the reader can find all even Young diagrams in
the cases $(d,e)=(4,4)$, $(4,5)$ and $(5,5)$ in
Figures~\ref{G44_fig}, \ref{G45_fig} and~\ref{G55_fig}, at the end
of the paper.
\end{exam}

\begin{rema}
Definition~\ref{even_def} depends on $d$ and $e$ as well as on
the Young diagram~$\Lambda$. For an even $(d,e)$-diagram $\Lambda$ to remain even in a bigger frame, we might have one or two more conditions to check, namely~(iii) or~(iv) in Definition~\ref{even_def}, in the case where $\Lambda$ was
touching the right border or the bottom border of its \deframe.
\end{rema}

\begin{rema}
For each even $(d,e)$-diagram we will construct a basis element in
the total Witt group $\Wtot\big(\Grass_X(d,\cV)\big)$ of the
Grassmannian. The proof that these Witt classes actually form a
basis will proceed by induction on~$d+e=\rank(\cV)$, using the long
exact sequence of localization associated to a natural ``cellular''
decomposition of the Grassmannians. In that proof, we shall need the
description in terms of Young diagrams of the various Witt group
homomorphisms appearing in that long exact sequence. As we shall
see, these are the ones of the next propositions. This explains why
the following constructions are relevant here.
\end{rema}

\begin{prop}
\label{exact-Lambda_prop} %
Let $d,e\geqslant 2$.
\begin{enumerate}
\item There is a bijection
%
$$
\xymatrix@R=.3em{ \left\{\Ourfrac{\text{even Young
}(d,e-1)\text{-diagrams}}{\Lambda'\text{ such that
}\zero(\Lambda')\text{ is even}}\right\} \ar@{<->}[r]^-{\simeq} &
\left\{\Ourfrac{\text{even Young
}(d,e)\text{-diagrams}}{\Lambda\text{ such that
}\Lambda_d>0}\right\}
\\
\qquad\Lambda'\qquad\ar@{|->}[r] & \quad
(\Lambda'_1+1,\ldots,\Lambda'_d+1)\quad
\\
(\Lambda_1-1,\ldots,\Lambda_d-1) & \qquad\Lambda \qquad\ar@{|->}[l]}
$$
%
\medbreak
\item There is a bijection
%
$$
\xymatrix@R=.3em{ \left\{\Ourfrac{\text{even Young
}(d,e)\text{-diagrams}}{\Lambda\text{ such that
}\Lambda_d=0}\right\} \ar@{<->}[r]^-{\simeq} &
\left\{\Ourfrac{\text{even Young
}(d-1,e)\text{-diagrams}}{\Lambda''\text{ such that
}\Lambda''_{d-1}\text{ is even}}\right\}
\\
\qquad\Lambda\qquad\ar@{|->}[r] & \qquad\Lambda\restr{d-1,e}\quad
\\
(\Lambda_1'',\ldots,\Lambda_{d-1}'',0) & \qquad\Lambda''
\qquad\ar@{|->}[l]}
$$
%
\medbreak
\item There is a bijection
%
$$
\xymatrix@R=.3em{ \left\{\Ourfrac{\text{even Young
}(d-1,e)\text{-diagrams}}{\Lambda''\text{ such that
}\Lambda''_{d-1}\text{ is odd}}\right\} \ar@{<->}[r]^-{\simeq} &
\left\{\Ourfrac{\text{even Young
}(d,e-1)\text{-diagrams}}{\Lambda'\text{ such that
}\zero(\Lambda')\text{ is odd}}\right\}
\\
\qquad\Lambda''\qquad\ar@{|->}[r] &
(\Lambda_1''-1,\ldots,\Lambda_{d-1}''-1,0)
\\
(1+\Lambda_1',\ldots,1+\Lambda'_{d-1}) & \qquad\Lambda'
\qquad\ar@{|->}[l]}
$$
%
\end{enumerate}
\end{prop}

\begin{proof}
The proof essentially consists in checking that the announced
constructions are well-defined and that they preserve even diagrams.
Checking that they are mutually inverse constructions is
straightforward. The notation $\Lambda\restr{d-1,e}$ is the obvious
one\,: we view a diagram with empty last row in a smaller frame. All
this is most easily performed and followed on pictures. For
instance, the maps from left to right are pictured in the upper
parts of Figures~\ref{push_fig}, \ref{pull_fig} and~\ref{bord_fig}
below.
\end{proof}

%\eject

%
\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/Push.pstex_t}%pdf_t}%
\end{center}
\caption{Morphism $\bar\iota$ (and later $\iota_*$) on various $(d,e-1)$-diagrams $\Lambda'$.} \label{push_fig}%
\end{figure}

\vfill

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/Pull.pstex_t}%pdf_t}%
\end{center}
\caption{Morphism $\bar\kappa$ (and later $\kappa$) on various $(d,e)$-diagrams $\Lambda$.} \label{pull_fig}%
\end{figure}

\vfill

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/Bord.pstex_t}%pdf_t}%
\end{center}
\caption{Morphism $\bar\bord$ (and later $\bord$) on various
$(d-1,e)$-diagrams $\Lambda''$.} \label{bord_fig}%
\end{figure}

%\eject

\begin{coro}
\label{exact-Lambda_cor}%
Let $R$ be a ring, for instance $R=\bbZ$ or
later $R=\Wtot(X)$. For each $d,e\in\bbN$ let $\Free(d,e)$ be the
free $R$-module on the set of even $(d,e)$-diagrams. Then for any
$d,e\geqslant 1$, the following cyclic sequence is exact everywhere
%
\begin{equation}
\label{exact-Lambda_eq}%
\vcenter{\xymatrix{& \Free(d,e)
\ar@/^1em/[rd]^-{\bar\kappa}
\\
\Free(d,e-1) \ar@/^1em/[ru]^-{\bar\iota} && \Free(d-1,e)
\ar@/^2em/[ll]_-{\bar\bord} }}
\end{equation}
where the homomorphisms $\bar\iota$, $\bar\kappa$ and $\bar\bord$
are defined on the basis elements by the constructions of
Proposition~\ref{exact-Lambda_prop}, whenever they make sense, and
by zero otherwise. See Figures~\ref{push_fig},
\ref{pull_fig} and~\ref{bord_fig}. More precisely, we have:

\smallbreak

\noindent
The homomorphism $\bar\iota:\Free(d,e-1)\to \Free(d,e)$ is defined
on the basis by
\begin{equation}
\label{bar-iota_eq}%
\bar\iota(\Lambda')= \left\{ \begin{array}{ll}
(\Lambda'_1+1,\ldots,\Lambda'_d+1) & \text{if }
\zero(\Lambda')\text{ is even}
\\
0 & \text{if } \zero(\Lambda')\text{ is odd.}
\end{array}\right.
\end{equation}
%
The homomorphism $\bar\kappa:\Free(d,e)\to \Free(d-1,e)$ is defined
on the basis by
%
\begin{equation}
\label{bar-kappa_eq}%
\bar\kappa(\Lambda)= \left\{ \begin{array}{ll}
\Lambda\restr{d-1,e} & \text{ if } \Lambda_d=0
\\
0 & \text{ if }\Lambda_d>0\,.
\end{array}\right.
\end{equation}
%
The homomorphism $\bar\bord:\Free(d-1,e)\to \Free(d,e-1)$ is defined
on the basis by
\begin{equation}
\label{bar-bord_eq}%
\bar\bord(\Lambda'')=\left\{
\begin{array}{ll}
(\Lambda_1''-1,\ldots,\Lambda_{d-1}''-1,0) & \text{if
}\Lambda_{d-1}'' \text{ is odd}
\\
0 & \text{if } \Lambda_{d-1}''\text{ is even.}
\end{array}\right.
\end{equation}
\end{coro}

\begin{proof}
Indeed, the sequence~\eqref{exact-Lambda_eq} is even ``split" exact
as follows:
%
$$
\xymatrix@C=.1em{ &{}\bigoplus\limits_{\ourfrac{\Lambda\text{
s.t.}}{\Lambda_d=0}}R\cdot\Lambda\quad \oplus \
\bigoplus\limits_{\ourfrac{\Lambda\text{ s.t.}}{\Lambda_d>0}}
R\cdot\Lambda
\ar@/^3.5em/[rd]^(.75){\displaystyle\bar\kappa=\begin{pmatrix}0&0\\
\text{iso}&0\end{pmatrix}}
\\
{}\bigoplus\limits_{\ourfrac{\Lambda'\text{
s.t.}}{\zero(\Lambda')\text{ even}}} \kern-1em R\cdot\Lambda'\quad
\kern-0.5em \oplus \kern-0.5em \bigoplus\limits_{\ourfrac{\Lambda'\text{
s.t.}}{\zero(\Lambda'_d)\text{ odd}}} \kern-1em R\cdot\Lambda'
\kern-3em
\ar@/^3em/[ru]^(.25){\displaystyle\bar\iota=\begin{pmatrix}0&0\\
\text{iso}&0\end{pmatrix}} &&
\kern-4em{}\bigoplus\limits_{\ourfrac{\Lambda''\text{
s.t.}}{\Lambda''_{d-1}\text{ is odd}}} \!R\cdot\Lambda''\quad \kern-0.5em \oplus
\kern-0.5em \bigoplus\limits_{\ourfrac{\Lambda''\text{
s.t.}}{\Lambda''_{d-1}\text{ is even}}} \!R\cdot\Lambda''
\ar@/^5em/[ll]_(.53){\displaystyle\bar\bord=\begin{pmatrix}0&0\\
\text{iso}&0\end{pmatrix}} }
$$
%
where ``iso" indicates an isomorphism induced by a bijection of the
corresponding basis elements, according to
Proposition~\ref{exact-Lambda_prop}.
\end{proof}



%------------------------------

\section{Generators of the total Witt group}
\label{gen-Witt_sec}%

%------------------------------

For this section, let $\Lambda$ be a Young diagram in \deframe\ and recall the $k$-tuples $\multi{d}$ and $\multi{e}$ associated to $\Lambda$ in Definition~\ref{de_def}.

\begin{rema}
\label{cond-push_rem}%
Our goal is to construct classes in the total Witt group of $\Grass_X(d,\cV)$ by pushing-forward the unit form $1\in\Wcoh^0(\Flag_X(\Lambda))$ along the morphism $\ff_\Lambda:\Flag_X(\Lambda)\to \Grass_X(d,\cV)$ of~\eqref{ff_eq}. As recalled in~\eqref{push-forward_eq}, this push-forward only exists conditionally, namely only when the class of $\can_{\Flag_X(\Lambda)/\Grass_X(d,\cV)}$ in $\Pic(\Flag_X(\Lambda))/2$ belongs to the image of
%
$$
(\ff_\Lambda)^*\,:\
\Pic(\Grass_X(d,\cV))/2\too\Pic(\Flag_X(\Lambda))/2\,.
$$
%
This is true if and only if the following conditions are satisfied:
\begin{enumerate}
\item $d_i-d_{i-1}+e_{i+1}-e_{i}$ is even for every $i=2,\ldots,k-1$
(for $k\geqslant3$)
\item when $0<e_1<e$, require moreover $d_1+e_2-e_1$ even.
\end{enumerate}

We shall be more precise in Proposition~\ref{cond-even_prop} below but the reader can verify our claim using~\eqref{can-Flag_eq} in Corollary~\ref{can-Flag_cor}. In that expression, every term coming from~$X$ will also come from $\Grass_X(d,\cV)$ since $\ff_\Lambda$ is a morphism over~$X$. This also applies to $\Det_{d_1}=\det\cV_{d_1}$ in case $e_1=0$. Finally, the term in $\Det_d$ also comes from $\Grass_X(d,\cV)$ since $(\ff_\Lambda)^*\big(\Det_d^{\Grass_X(d,\cV)}\big)=\Det_d^{\Flag_X(\Lambda)}$, as can be checked on the tautological bundles.

Conditions~(a) and (b) hold in particular when $\Lambda$ is \emph{even} in the sense of Definition~\ref{even_def}. Indeed, for such $\Lambda$ not only the sum
$d_i-d_{i-1}+e_{i+1}-e_{i}$ is even but actually each term
$d_i-d_{i-1}$ and $e_{i+1}-e_{i}$ is. Compare Figure~\ref{conditions_fig}.

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/coins.pstex_t}%pdf_t}%
\end{center}
\caption{Framed Young diagram satisfying Conditions~(a) and~(b) of
Remark~\ref{cond-push_rem} but which is not even (at all).}
%%%%%Here $\Lambda=(3,3,3,2,1,1,1)$ in a $7\times 5$-frame; that is, $k=3$, $\multi{d}=(3,4,7)$ and $\multi{e}=(2,3,4)$.}
\label{conditions_fig}%
\end{figure}

When (a) and (b) hold (\eg\ for $\Lambda$ even), there exists a line bundle $L$ on $\Grass_X(d,\cV)$ such that $\can_{\ff_\Lambda}+\ff_\Lambda^*(L)=0$ in $\Pic(\Flag_X(\Lambda))\big/2$. 
Therefore, the push-forward~\eqref{push-forward_eq} applied to $f=\ff_\Lambda$ and $i=-\dim(\ff_\Lambda)$ yields a homomorphism\,:
%
$$
\Wcoh^{0}\big(\Flag(\Lambda),0\big) =
\Wcoh^{0}\big(\Flag(\Lambda),\can_{\ff_\Lambda}+\ff_\Lambda^*(L))\big)
\tooo{2}^{(\ff_\Lambda)_*}\Wcoh^{|\Lambda|}\big(\Grass_X(d,\cV),L\big)\,.
$$
%
(Use that $\dim(\ff_\Lambda)=\dim\Flag_X(\Lambda)-\dim\Grass_X(d,\cV)=-|\Lambda|$ by Propositions~\ref{Pic-Grass_prop} and~\ref{Pic-Flag_prop}.)
Consequently, we can produce a Witt class $(\ff_\Lambda)_*(1)$ over $\Grass_X(d,\cV)$. This is what we are going to do below for $\Lambda$ even, making the class of $L$ in $\Pic(\Grass_X(d,\cV))/2$ more explicit in terms of the diagram~$\Lambda$.
\end{rema}

\begin{rema}
\label{perim_rem}%
The perimeter of a Young diagram $\Lambda$ is an even integer.
Indeed, from the lower-left corner of $\Lambda$ to its upper-right
corner, there are two paths which follow the boundary (the upper
path and the lower path) and they have the same length, namely the
lattice distance between these two corners.
\end{rema}

\begin{defi}
\label{twist_def}%
Let $\Lambda$ be a Young diagram. We define $\twist(\Lambda)\in\bbZ/2$ to be the class of half the perimeter of~$\Lambda$. From the above remark, $\twist(\Lambda)$ is also the class of the (lattice) distance from the lower-left corner of $\Lambda$ to its upper-right corner. That is\,:
%
$$
\twist(\Lambda)=[\Lambda_1+\row(\Lambda)]\in\bbZ/2
$$
%
where $\row(\Lambda)$ is the number of non-zero rows of $\Lambda$ (Def.\,\ref{zerow_def}). Note that this Definition does not depend on an ambient frame.
\end{defi}

\begin{rema}
On an even Young diagram~$\Lambda$ in \deframe, there is another way to read $\twist(\Lambda)\in\bbZ/2$ on the diagram. Add the (parity of) the length of the segments where $\Lambda$ touches the right and the bottom of the frame. See Figure~\ref{twist_fig}. This is justified and generalized in Proposition~\ref{twist_prop}.
\end{rema}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/twist.pstex_t}%pdf_t}%
\end{center}
\caption{Class $\twist(\Lambda)\in\bbZ/2$ for different $\Lambda$.}
\label{twist_fig}%
\end{figure}

\begin{prop}
\label{twist_prop}%
Let $\Lambda$ be an even Young diagram in \deframe\ and let $\multi{d},\multi{e}$ be the associated $k$-tuples (Def.\,\ref{de_def}). Then $\twist(\Lambda)=[d_i+(e-e_j)]\in\bbZ/2$ for any $i,j\in\{1,\ldots,k\}$ such that $e_i<e$ (only $i=k$ should be avoided when $e_k=e$).
\end{prop}

\begin{proof}
Measure the half-perimeter of $\Lambda$ as the length of the lower boundary of~$\Lambda$, from the lower-left corner of $\Lambda$ to its upper-right corner (see Rem.\,\ref{perim_rem}). 
Since $\Lambda$ is even, all segments on this lower half-perimeter which are not on the outside frame have even length. 
So, the only two segments to contribute to the lower half-perimeter with possible odd length, are on the outside \deframe, \ie\,:
\begin{itemize}
\item the vertical segment most to the right, which has length $d_1$, and
\item the lowest horizontal segment, which has length $e-e_k$ when $\Lambda$ touches the lower part of the \deframe\ (otherwise $e_k=e$ and this length is even).
\end{itemize}
In any case, this shows that $\twist(\Lambda)=[d_1+(e-e_k)]\in\bbZ/2$, that is, the announced formula for $i=1$ and $j=k$. 
The other formulas follow from this one since by Definition~\ref{even_def} the successive differences $d_i-d_{i-1}$ and $e_{j+1}-e_j$ are even for all $i=2,\ldots,k-1$, for all $j=1,\ldots,k-1$, and also for $i=k$ when $e_k<e$.
\end{proof}

%We shall need the following observation\,:
%
%%
%\begin{lemm}
%\label{odd_lem}
%Let $\Lambda$ be an even Young diagram in \deframe. Then one of
%$\twist(\Lambda)$ or $|\Lambda|$ has to be even.
%\end{lemm}
%
%\begin{proof}
%Since $|\varnothing|=0$ is even, we can assume
%$\Lambda\neq\varnothing$. Let $1\leqslant j\leqslant d$ be the
%biggest index such that $\Lambda_j>0$. Let
%$\bar\Lambda=[j\times \Lambda_1]$ be the smallest rectangular Young
%diagram which contains $\Lambda$. Since $\bar\Lambda$ is obtained
%from $\Lambda$ by adding $2\times2$-squares ($\Lambda$ is even), we
%have $|\bar\Lambda|\equiv|\Lambda|$ and
%$\twist(\bar\Lambda)\equiv\twist(\Lambda)$ modulo~2. So, it suffices
%to prove that at least one of $|\bar\Lambda|=j\cdot \Lambda_1$ and
%$\twist(\bar\Lambda)=j+\Lambda_1$ is even, which is obvious.
%\end{proof}
%
%

\begin{defi}
\label{Twist_def}%
Let $\Lambda$ be an even Young diagram in
\deframe. We define \emph{the twist $\Twist(\Lambda)$ of
$\Lambda$} as the following class in $\Pic(\Grass_X(d,\cV))/2=\Pic(X)/2\ \oplus\ \bbZ/2\Det_d$ (see Cor.\,\ref{Pic-Grass_coro})\,:
%
$$
\Twist(\Lambda)=\Twist(\Lambda,d,e)=\row(\Lambda)\cdot\det \cV+\twist(\Lambda)\cdot\Det_d\,,
$$
%
where we recall that $\row(\Lambda)$ is the number of non-zero rows of~$\Lambda$ (Def.\,\ref{zerow_def}).
\end{defi}

\begin{rema}
The important part of $\Twist(\Lambda)$ is of course
$\twist(\Lambda)\,\Det_d$, which is not coming from the base~$X$.
For instance, when $\cV$ is trivial, the other term disappears; this
holds in particular when $X=\Spec(R)$ for a local ring $R$ 
\end{rema}

\begin{prop}
\label{cond-even_prop}%
Let $\Lambda$ be an even Young diagram in \deframe. Then
%
$$
\can_{\ff_\Lambda}+\ff_\Lambda^*(\Twist(\Lambda))=0
\quad\text{in}\quad
\Pic\big(\Flag_X(\Lambda)\big)\big/2\,.
$$
%
\end{prop}

\begin{proof}
Suppose $k(\Lambda)\geqslant2$. Remove from~\eqref{can-Flag_eq} all even coefficients coming from the fact that $\Lambda$ is even and use Proposition~\ref{twist_prop} for $i=k-1$ and $j=k$. This gives in $\Pic\big(\Flag_X(\Lambda)\big)\big/2$\,:
%
$$
\can_{\ff_\Lambda} =
d_1\det\cV_{d_1+e_1}
 +(d+d_{k-1})\,\det\cV_{d+e_k} + d\,\det\cV
 + d_1\,\Det_{d_1} + \twist(\Lambda)\,\Det_d\,.
$$
%
Now observe that $d_1\det\cV_{d_1+e_1}+d_1\,\Det_{d_1}=0$ in $\Pic\big(\Flag_X(\Lambda)\big)\big/2$. Indeed, either $e_1>0$ hence $d_1$ is even, or $e_1=0$ hence $\Det_{d_1}=\det\cV_{d_1}$ by Remark~\ref{zero-Det_rem}. So, we can simplify the above equation in $\Pic\big(\Flag_X(\Lambda)\big)\big/2$\,:
%
$$
\can_{\ff_\Lambda} =
(d+d_{k-1})\,\det\cV_{d+e_k} + d\,\det\cV + \twist(\Lambda)\,\Det_d\,.
$$
%
Now, if $e_k<e$ then $d_k-d_{k-1}$ is even and $\row(\Lambda)=d$\,; on the other hand, if $e_k=e$, then $\row(\Lambda)=d_{k-1}$. In any case, the above expression becomes $\row(\Lambda)\,\det\cV+\twist(\Lambda)\,\Det_d$, which is $\Twist(\Lambda)$ by Definition~\ref{Twist_def}.

Similarly, the case $k(\Lambda)=1$ is an easy consequence of
Corollary~\ref{can-Flag_cor}.
\end{proof}

%
\begin{prop}
\label{even-push_prop}%
Let $\Lambda$ be an even Young diagram in \deframe. Then the
push-forward for Witt groups induced by
$\ff_\Lambda:\Flag_X(\Lambda)\too\Grass_X(d,\cV)$ yields a
homomorphism:
%
$$
\Wcoh^{0}\big(\Flag_X(\Lambda),0\big) =
\Wcoh^{0}\big(\Flag_X(\Lambda)\,,\,\can_{\ff_\Lambda}+\ff_\Lambda^*(\Twist(\Lambda))\big)
\tooo{2}^{(\ff_\Lambda)_*}\Wcoh^{|\Lambda|}\big(\Grass_X(d,\cV),\Twist(\Lambda)\big)
$$
%
where the first identification comes from
Proposition~\ref{cond-even_prop} (see Appendix~\ref{TotalWitt_app}).
\end{prop}

\begin{proof}
Apply push-forward~\eqref{push-forward_eq}, for $i=-\dim(\ff_\Lambda)=|\Lambda|$ and $L=\Twist(\Lambda)$, as explained in Remark~\ref{cond-push_rem}.
\end{proof}

%
\begin{defi}
\label{gen-Witt_def}%
Let $\Lambda$ be an even Young diagram in \deframe. By
Proposition~\ref{even-push_prop}, we can push-forward along $\ff_\Lambda:\Flag_X(\Lambda)\to\Grass_X(d,\cV)$ the unit form $1\in\Wcoh^0(\Flag_X(\Lambda),0)$ to our announced Witt class
%
\begin{equation}
\label{elements_eq}%
\gen_{d,e}(\Lambda):=(\ff_{\Lambda})_*(1)\in
\Wcoh^{|\Lambda|}\big(\Grass_X(d,\cV)\,,\,\Twist(\Lambda)\big)\,.
\end{equation}
%
\end{defi}

\begin{exam}
\label{empty_exa}%
Let $\Lambda=\varnothing$ be the empty Young diagram in \deframe,
which is even, as we know. Then
$\gen_{d,e}(\varnothing)=1\in\Wcoh^0(\Grass_X(d,\cV),0)$ is the unit
form on the Grassmannian. Indeed,
$\ff_\varnothing:\Flag_X(d,e;\varnothing)\to \Grass_X(d,\cV)$ is the identity.
\end{exam}

\begin{defi}
\label{iso_def}%
Let $d,e\geqslant 1$. Consider the free $\Wtot(X)$-module
%
$$
\Free_X(d,e):=\bigoplus_{\Lambda\text{ even}}\Wtot(X)\cdot \Lambda
$$
%
with (formal) basis given by the even Young diagrams $\Lambda$ in \deframe. We have an obvious $\bbZ/2$-grading on $\Free_X(d,e)$ given by $\twist(\Lambda)\in\bbZ/2=\bbZ/2\,\Det_d$ (Def.\,\ref{twist_def})\,:
%
$$
\Free_X(d,e)=\bigoplus_{t\in\bbZ/2}\ \bigoplus_{\ourfrac{\Lambda\text{ even}}{\twist(\Lambda)=t\,\Det_d}}\Wtot(X)\cdot \Lambda\,.
$$
%
Now since $\Wtot(X)$ is a graded ring over $\bbZ/4\oplus\Pic(X)/2$ (see Appendix), we can extend this grading to $\Free_X(d,e)$ by letting $\Lambda$ have degree $|\Lambda|+\row(\Lambda)\det\cV$ in $\bbZ/4\oplus\Pic(X)/2$. 
Combining both observations, $\Free_X(d,e)$ becomes a graded abelian group over $\bbZ/4\oplus\Pic(X)/2\oplus\bbZ/2$. The latter is isomorphic to $\bbZ/4\oplus\Pic(\Grass_X(d,\cV))/2$ by Corollary~\ref{Pic-Grass_coro}. Explicitly, $\Lambda$ has degree
%
\begin{equation}
\label{twist_eq}%
\begin{array}{cccccccccc}
|\Lambda| & + & \row(\Lambda)\det\cV & + & \twist(\Lambda)\,\Det_d & =
& |\Lambda| & + & \Twist(\Lambda) \kern3em\text{in}
\\
\bbZ/4 & \oplus & \Pic(X)/2 & \oplus & \bbZ/2\,\Det_d & \cong & \bbZ/4 & \oplus & \Pic(\Grass_X(d,\cV))/2\,.
\end{array}
\end{equation}
%
We $\Wtot(X)$-linearly extend the $\gen_{d,e}(\Lambda)$ of Definition~\ref{gen-Witt_def} into a homomorphism
%
\begin{equation}
\label{iso_eq}%
\begin{array}{rccc}
\iso_{d,e}: & \Free_X(d,e) & \too & \Wtot\big(\Grass_X(d,\cV)\big)
\\
& \Lambda & \longmapsto & \gen_{d,e}(\Lambda)\,,\!
\end{array}
\end{equation}
%
which is homogeneous (of degree zero) with respect to the above grading, by~\eqref{elements_eq}.
\end{defi}

\begin{prop}
\label{deg_prop}%
Let $d,e\geq2$ and let $\cV^1:=\cV_{d+e-1}$ in the complete flag~\eqref{complete_eq}. With $\Free_X(d,e)$ graded as above over
$\bbZ/4\oplus\Pic(X)/2\oplus\bbZ/2\,\Det_d$, the homomorphisms
$\bar\iota$, $\bar\kappa$ and $\bar\bord$ of Corollary~\ref{exact-Lambda_cor}
are not always homogeneous (when $\cV/\cV^1$ is not trivial in
$\Pic(X)/2$). More precisely, they behave as follows\,:
%
\begin{enumerate}
%
\item
$\bar\iota$ is indeed homogeneous of degree $\big(\,d
\,,\,d\,(\cV/\cV^1)\,,\,1 \,\big)$.
%
\item
$\bar\kappa$ sends degree $(s,\ell,t)$ into degree
$\big(\,s\,,\,\ell+t\cdot(\cV/\cV^1) \,,\, t\,\big)$.
%
\item
$\bar\bord$ sends degree $(s,\ell,t)$ into degree
$\big(\,s-d+1\,,\,\ell+(t-d)\,(\cV/\cV^1) \,,\, t-1\,\big)$.
%
\end{enumerate}
\end{prop}

\begin{proof}
This is a direct inspection of the behavior of
$\deg(\gen_{d,e}(\Lambda))$ given in~\eqref{twist_eq} under the
combinatorial constructions of Corollary~\ref{exact-Lambda_cor}. We
leave this to the reader.
\end{proof}

\begin{coro}
\label{deg_cor}%
When $\cV/\cV^1$ is trivial (\eg\ when $\cV_\smallbullet=\cO_X^\smallbullet$, \eg\ when $X=\Spec(R)$ with $R$ a field or a local ring) then $\bar\iota$, $\bar\kappa$ and $\bar\bord$ are homogeneous.
\qed
\end{coro}

\begin{rema}
\label{ind_rem}%
Our main result is that $\iso_{d,e}$ is an isomorphism for all
$d,e\geqslant1$. We shall prove it by induction on $d+e$. The case $d=1$ or $e=1$ is that of the projective bundle $\bbP(\cV)$. In that case, Walter~\cite{Walter03} has proved
%
\begin{equation}
\label{Pn_eq}%
\Wtot(\bbP(\cV))=\Wtot(X)\cdot 1_{\bbP^m}\oplus\Wtot(X)\cdot\xi
\end{equation}
%
where $\xi\in\Wcoh^{d+e-1}(\bbP(\cV),\can_\pi)$ is such that
$\pi_*(\xi)=1\in\Wcoh^0(X)$ and $\pi:\bbP(\cV)\to X$ is the
structure morphism. We use the assumption that $\cV$ has a complete
flag to get the above closed formula,
see~\cite[Proposition~8.1]{Walter03}. One can check that $\xi$ is
equal to $\gen(\Lambda)$, up to a unit, where $\Lambda=\rect{d}{e}$
is the only non-empty even diagram in \deframe\ in our case.
Formula~\eqref{Pn_eq} was also proved in the case $\cV=\cO_X^{d+e}$
in~\cite{Balmer05}, using methods closer to the present geometric
philosophy. We could therefore assume the starting point of the
induction (\ie the case $d=1$ or $e=1$). Nevertheless, our proof of
the induction step can be adapted to give this initial step as well.
So, to be self-contained, we indicate how to do this in
Proposition~\ref{Pn_prop}.
\end{rema}


%------------------------------

\section{Cellular decomposition}
\label{cell-dec_sec}%

%------------------------------

We describe the usual relative cellular decomposition of
Grassmannians. Fix $d,e\geqslant2$ for the whole section.

\begin{nota}
Fix a complete flag $\cV_\smallbullet$ of vector bundles on~$X$
%
$$
0=\cV_0\subb\cV_1\subb\cdots\subb \cV_{d+e}=\cV\,,
$$
%
as in~\eqref{complete_eq}. We set $\cV^1=\cV_{d+e-1}$ to be the codimension one subbundle of~$\cV$. We have an obvious closed immersion $\Grass_X(d,\cV^1)\hookrightarrow\Grass_X(d,\cV)$, of codimension~$d$, whose open complement we denote by~$U_X(d,\cV_\smallbullet)$.
\end{nota}

\begin{nota}
\label{nsubb_not}%
Let $\cP_d\subb\cV$ be a subbundle of rank~$d$. We write $\cP_d\nsubb\cV^1$ to express that $\cP_d$ is not a subbundle of $\cV^1$ but moreover satisfies the equivalent conditions\,:
%
\begin{enumerate}
\item The natural map from
$\cP_d/(\cP_d\cap\cV^1)=(\cP_d+\cV^1)/\cV^1$ into $\cV/\cV^1$ is an
isomorphism.
\item $\cP_d\cap\cV^1$ is a subbundle of $\cP_d$ (in the strong
sense of Definition~\ref{subb_def}).
\end{enumerate}
%
Over a field, this amounts to $\cP_d\not\subset\cV^1$ but this is not sufficient in general.
\end{nota}

\begin{defi}
\label{main'_def}%
Using notations of Section~\ref{Grass_sec}, we have a commutative diagram
%
\begin{equation}
\label{main'_eq}%
\vcenter{\xymatrix@C=2em{ \Grass_X(d,\cV^1) \ar@{^{(}->}[r]^(.57){\iota}
\ar@{}[r]|(.4){|} & \Grass_X(d,\cV)
& U_X(d,\cV_\smallbullet)
 \ar@{_{(}->}[l]_-{\upsilon} \ar@{}[l]|-{\circ}
 \ar[d]^{\alpha}\ar@{_{(}->}[dl]_-{\tilde\upsilon}\ar@{}[dl]|-{\circ}
\\
\Flag_{X}((d-1,d),(e,e-1)) \ar[u]^{\tilde{\pi}}
\ar@{^{(}->}[r]^(.52){\tilde{\iota}} \ar@{}[r]|(.47){|} &
\Flag_{X}((d-1,d),(e,e)) \ar[u]^{\pi} \ar[r]^-{\tilde{\alpha}} &
\Flag_{X}(d-1,\cV^1)\,.
}}\kern-.7em
\end{equation}
%
which looks as follows on points\,:
%
\begin{equation}
\label{main-pts_eq}%
\vcenter{\xymatrix{ \{\cP_d \subb \cV^1 \} \ar@{^{(}->}[r]^{\iota}
\ar@{}[r]|(.4){|} & \{\cP_d \subb \cV \} & \{\cP_d \nsubb \cV^1 \}
\ar@{_{(}->}[l]_{\upsilon}\ar@{}[l]|-{\circ}
\ar@{_{(}->}[dl]_-{\tilde\upsilon}\ar@{}[dl]|-{\circ}
\ar[d]^{\alpha}
\\
\{\cP_{d-1} \subb \cP_{d} \subb \cV^1\} \ar[u]^-{\tilde{\pi}}
\ar@{^{(}->}[r]^(.37){\tilde{\iota}} \ar@{}[r]|(.4){|} & \{\cP_{d-1}
\subb \cP_d \subb \cV \;|\; \cP_{d-1} \subb \cV^1 \} \ar[u]^{\pi}
\ar[r]^-{\tilde{\alpha}} & \{ \cP_{d-1} \subb \cV^1 \}\,.}}\kern-1em
\end{equation}
%
Here $\iota$, $\tilde\iota$, $\pi$, $\tilde\pi$ and $\tilde\alpha$ are the obvious morphisms\,; the morphism $\tilde\upsilon$ maps $\cP_d$ to the flag $\cP_{d-1}\subb\cP_d$ with $\cP_{d-1}:=\cP_d \cap \cV^1$ (see Not.\,\ref{nsubb_not})\,; finally $\alpha$ is defined as $\tilde\alpha\circ\tilde\upsilon$.
\end{defi}

\begin{prop}
\label{blowup_prop}%
In Diagram~\eqref{main'_eq}, the scheme $\Flag_X((d-1,d),(e,e))$ is the blow-up of $\Grass_X(d,\cV)$ along $\Grass_X(d,\cV^1)$ with exceptional fiber $\Flag_X((d-1,d),(e,e-1))$.
\end{prop}

\begin{proof}
We can reduce to the case where $X$ is affine and $\cV$ is free and then to $X=\Spec(\bbZ)$ by compatibility of blow-ups with pull-back diagrams. We omit $X$ in the notation in the rest of the proof. Now, we first
show that there is a morphism from the announced blow-up $B$ to
$\Flag((d-1,d),(e,e))$ and then that this morphism is an
isomorphism. The inverse image of $\Grass(d,\cV)$ in
$\Flag((d-1,d),(e,e))$ is $\Flag((d-1,d),(e,e-1))=\bbP_Y
(\cV^1/\cP_{d-1})$ where $Y=\Grass(d-1,\cV^1)$, by construction. It
includes in $\Flag((d-1,d),(e,e))=\bbP_Y (\cV/\cP_{d-1})$, so it is
locally $\bbP^{e-1} \subset \bbP^e$ hence an effective Cartier
divisor (\ie a closed subscheme locally given by a principal
ideal). By the universal property of the blow-up construction (see
\cite[§ 8.1.2, Corollary 1.16]{Liu02}) we thus obtain a map $f: B
\to \Flag((d-1,d),(e,e))$, such that $\pi \circ f =\pi'$, where
$\pi':B \to \Grass(d,\cV)$ is the structural morphism of the
blow-up. In particular, $f$ is a finite (check over
$\Grass(d,\cV^1)$ and $U(d,\cV)$) birational morphism to a normal
variety, so it is an isomorphism: it is locally a finite morphism
$\Spec(S) \to \Spec(R)$ where $R$ is integrally closed.
\end{proof}

%
\begin{defi}
\label{cell_def}%
Let $\Bl_X(d,\cV_\smallbullet)=\Flag_X((d-1,d),(e,e))$ be the blow-up of
$\Grass_X(d,\cV)$ along $\Grass_X(d, \cV^1)$ and let
$\Exc_X(d,\cV_\smallbullet)=\Flag_X((d-1,d),(e,e-1))$ be the exceptional fiber.
By~\eqref{main'_eq}, $\Grass_X(d,\cV)$ admits a decomposition like
in~\cite[Hypothesis 1.2]{BalmerCalmes08}, namely
%
\begin{equation}
\label{main_eq}%
\vcenter{\xymatrix@C=5em{ \Grass_X(d,\cV^1)
\ar@{^{(}->}[r]^(.55){\iota} \ar@{}[r]|(.4){|} & \Grass_X(d,\cV) &
U_X(d,\cV_\smallbullet) \ar@{_{(}->}[l]_-{\upsilon}\ar@{}[l]|-{\circ}
\ar@{_{(}->}[dl]_-{\tilde\upsilon}\ar@{}[dl]|-{\circ}
\ar[d]^{\alpha}
\\
\Exc_X(d,\cV_\smallbullet) \ar[u]^{\tilde{\pi}}
\ar@{^{(}->}[r]^(.45){\tilde{\iota}} \ar@{}[r]|(.4){|} &
\Bl_X(d,\cV_\smallbullet) \ar[u]^{\pi} \ar[r]^-{\tilde{\alpha}} &
\Grass_X(d-1,\cV^1)\,.\! }}
\end{equation}
%
It is easy to see that $\alpha$ is an $\bbA^e$-bundle because $U_X(d,\cV_{\smallbullet})$ is canonically isomorphic to $\bbP_Y(\cV/\Taut^Y_{d-1})\setminus \bbP_Y(\cV^1/\Taut^Y_{d-1})$ where $Y=\Grass_X(d-1,\cV^1)$, with $\alpha$ corresponding to the structural morphism to~$Y$.
\end{defi}

%
\begin{rema}
\label{Pic-can_rem}%
We can compute the relevant Picard groups and canonical bundles, via the methods of Section~\ref{Grass_sec}.
%
Let us start with Picard groups, using~\eqref{Pic-Flag_eq}. Since
$\Pic(X)$ is a direct summand of the Picard group of all schemes
in~\eqref{main_eq}, we focus on the relative Picard groups
$\Pic(-)/\Pic(X)$. In short ``$\Pic(-)/\Pic(X)$ of~\eqref{main_eq}''
is
%
\begin{equation}
\label{Picards_eq}%
\vcenter{\xymatrix@C=4em{
\bbZ \Det_d \ar[d]_-{\tilde\pi^*=\left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)}
& \bbZ \Det_d \ar[l]_-{\iota^*=1} \ar[d]_-{\pi^*=\left(\begin{smallmatrix}0\\1\end{smallmatrix}\right)} \ar[r]^-{\upsilon^*=1}
& \bbZ\upsilon^*(\Det_d)=\bbZ \alpha^*(\Det_{d-1})\kern-1em
\\
\bbZ \Det_{d-1} \oplus \bbZ \Det_d
& \bbZ \Det_{d-1} \oplus \bbZ \Det_d
 \ar[l]^{\tilde\iota^*=\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)} \ar[ru]|-{\tilde\upsilon^*=\left(\begin{smallmatrix}1& 1\end{smallmatrix}\right)}
& \bbZ \Det_{d-1} \ar[l]^-{\tilde\alpha^*=\left(\begin{smallmatrix}1\\0\end{smallmatrix}\right)} \ar[u]_{\alpha^*=1}
}}
\end{equation}
%
(In the case $X=\Spec(R)$ for a local ring $R$, the Picard groups are
exactly as above.) Here we used that the closed subscheme
$\Grass_X(d,\cV^1)$ is of codimension $d\geqslant2$ in
$\Grass_X(d,\cV)$ to see that
$\upsilon^*:\Pic(\Grass_X(d,\cV))\cong\Pic(U_X(d,\cV_\smallbullet))$.
We also used that $e\geqslant2$, otherwise $e-1=0$ and
$\Det_{d-1}\in\Pic(X)$ by Remark~\ref{zero-Det_rem}. (When $d=1$,
resp.\ $e=1$, we loose all components $\bbZ \Det_{d-1}$, resp.\ all
components $\bbZ \Det_d$, in the previous diagram.) Alternatively,
~\eqref{Picards_eq} follows from the computation of the Picard
groups provided in~\cite[Proposition~A.6]{BalmerCalmes08}. Finally,
the maps into the upper corner of Diagram~\eqref{Picards_eq} are
obtained from
%
\begin{equation}
\label{alpha-upsilon-Pic_eq}%
\upsilon^*(\Det_d) - \alpha^*(\Det_{d-1}) =  \cV/\cV^1\,,
\end{equation}
%
which follows from Condition~(a) in Notation~\ref{nsubb_not}.

We shall use push-forward along some morphisms of~\eqref{main_eq}.
The relevant relative canonical bundles are given by
Corollary~\eqref{can-Flag_cor}\,:
%
\begin{eqnarray}
\can_\iota & = & -\Det_d +d\,(\cV/\cV^1) \label{rel-can-subGrass_eq}%
\\
\can_\pi & = & (d-1) \Det_{d-1} + (1-d) \Det_d + (d-1)\,\cV/\cV^1 \label{rel-can-Bl_eq}%
\\
\can_{\tilde\iota} & = & \cV/\cV^1 + \Det_{d-1} - \Det_d \label{rel-can-Exc-div_eq}%
\\
\can_{\tilde{\pi}} & = & d\, \Det_{d-1} + (1-d) \Det_d\,.\!
\label{rel-can-Exc_eq}
\end{eqnarray}
%
Again, when the fixed complete flag $\cV_\smallbullet=\cO_X^\smallbullet$ is trivial, the ``noise'' $\cV/\cV^1$ vanishes.
\end{rema}

We end this Section with two geometric lemmas which will be useful
in the proof of the main theorem.

%
\begin{lemm}
\label{double-PB_lem}%
Let $d,e\geqslant2$ and let $\Lambda$ be an even Young
$(d,e)$-diagram with empty last row (\ie $\Lambda_k=0$, \ie $\zero(\Lambda)>0$). Hence $\Lambda\restr{d-1,e}$ is an even
$(d-1,e)$-diagram. Then the base-change to $U_X(d,\cV_\smallbullet)$ of the
morphisms $\ff_{d,e;\Lambda}$ and
$\ff_{d-1,e;\Lambda\smallrestr{d-1,e}}$ coincide, that is, we have
two cartesian squares:
%
\begin{equation}
\label{double-PB_eq}%
\vcenter{ \xymatrix{ \Grass_X(d,\cV) \ar@{}[dr]|{\Box} & U_X(d,\cV_\smallbullet)
\ar@{_{(}->}[l]_-{\upsilon} \ar[r]^-{\alpha} \ar@{}[dr]|{\Box} &
\Grass_X(d-1,\cV^1)
\\
\Flag_X(d,e;\Lambda) \ar[u]^{\ff_{d,e;\Lambda}}
& U' \ar@{_{(}->}[l] \ar[r]^-{\alpha'} \ar[u]
& \Flag_{X}(d-1,e;\Lambda\restr{d-1,e})\,.\! \ar[u]_{\ff_{d-1,e;\Lambda\smallrestr{d-1,e}}}
}}
\end{equation}
\end{lemm}

\begin{proof}
Let us check this on points. Let $\multi{d}$ and $\multi{e}$ be the
$k$-tuples associated to $\Lambda$ as usual (Def.\,\ref{de_def}). We
need to distinguish two cases, namely $d_{k}-d_{k-1}>1$ and $d_k-d_{k-1}=1$.

When $d_k>d_{k-1}+1$, that is, when there is more than one zero line
at the end of $\Lambda$ (\ie $\zero(\Lambda)>1$), we then have
$k(\Lambda\restr{d-1,e})=k(\Lambda)=k$ and the $k$-tuples
$\multi{d}(\Lambda\restr{d-1,e})$ and
$\multi{e}(\Lambda\restr{d-1,e})$ are almost the same as $\multi{d}$
and $\multi{e}$ except for the last entry of
$\multi{d}(\Lambda\restr{d-1,e})$ which becomes~$d-1$.
Diagram~\eqref{double-PB_eq} then looks as follows on points (as
usual the $\cP_i$ and $\cP'_j$ are ``variables'' whereas the $\cV_i$
belong to the fixed complete flag):
%
$$\xy 0;<1ex,0ex>:
%ensemble haut gauche
(7,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_d \ar@{}[r]|-{\subb} & \cV \}
}
};
%ensemble haut milieu
(34,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_d \ar@{}[r]|-{\nsubb} & \cV^1 \}
}
};
%ensemble haut droite
(63,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP'_{d-1} \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble bas gauche
(0,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} &
\cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.0ex/{\labelstyle e_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.5ex/{\labelstyle e} \\
\cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV
}
}*\frm{\{}*\frm{\}};
%ensemble bas milieu et son rajout
(28,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} &
\cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.0ex/{\labelstyle e_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.5ex/{\labelstyle e} \\
\cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV
}
}*\frm{\{}*\frm{\}};
(45.3,7)*\xybox{
\xymatrix@R=1.5ex{
\cV^1 \\
\ar@{}[u]|{\rotatebox{90}{$\nsubb$}}
}
};
%ensemble bas droite
(56,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} & \cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.0ex/{\labelstyle e_{k-1}} \ar@{}[r]|-{\subb} & \cP'_{d-1}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.5ex/{\labelstyle e} \\
\cdots \ar@{}[r]|-{\subb} &
\cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV^1
}
}*\frm{\{}*\frm{\}};
%fleches de la ligne du haut
{\ar@{_{(}->} (30,15)*{};(15,15)*{}};
{\ar (43,15)*{};(59,15)*{} ^-{\alpha}};
%%fleches verticales
{\ar (10,5)*{};(10,13)*{}^{\ff_{d,e;\Lambda}}};
{\ar (38,5)*{};(38,13)*{}};
{\ar (65,5)*{};(65,13)*{}|{\ff_{d-1,e;\Lambda\smallrestr{d-1,e}}}};
%%fleches de la ligne du bas
{\ar@{_{(}->} (24,0)*{};(20.5,0)*{}};
{\ar (48.5,0)*{};(53,0)*{}^-{\alpha'}};
%carrés de diagrames cartésiens
(23,9)*{\labelstyle \Box};
(51,9)*{\labelstyle \Box};
\endxy$$
%
where the morphisms $\alpha$ sends $\cP_d$ to $\cP'_{d-1}:=\cP_d \cap
\cV^1$ and similarly for~$\alpha'$ .

On the other hand, when $d_k=d_{k-1}+1$, that is, when $\Lambda$ has
only one zero line (\ie $\zero(\Lambda)=1$), then we have
$k(\Lambda\restr{d-1,e})=k(\Lambda)-1=k-1$ and the $(k-1)$-tuples
$\multi{d}(\Lambda\restr{d-1,e})$ and
$\multi{e}(\Lambda\restr{d-1,e})$ are respectively $\multi{d}$ and
$\multi{e}$ truncated from their last entry.
Diagram~\eqref{double-PB_eq} then looks as follows on points:
%
$$
\xy 0;<1ex,0ex>:
%ensemble haut gauche
(6,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_d \ar@{}[r]|-{\subb} & \cV \}
}
};
%ensemble haut milieu
(37,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_d \ar@{}[r]|-{\nsubb} & \cV^1 \}
}
};
%ensemble haut droite
(66,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP'_{d-1} \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble bas gauche
(0,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} &
\cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.0ex/{\labelstyle e_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.5ex/{\labelstyle e} \\
\cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV
}
}*\frm{\{}*\frm{\}};
%ensemble bas milieu et son rajout
(31,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} &
\cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.0ex/{\labelstyle e_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}}_*!/_.5ex/{\labelstyle e} \\
\cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV
}
}*\frm{\{}*\frm{\}};
(48.3,7)*\xybox{
\xymatrix@R=1.5ex{
\cV^1 \\
\ar@{}[u]|{\rotatebox{90}{$\nsubb$}}
}
};
%ensemble bas droite
(62,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cdots \ar@{}[r]|-{\subb} & \cP_{d_{k-1}}
\ar@{}[d]|{\rotatebox{270}{$\subbeq$}}^*!/^.5ex/{\labelstyle
e_{k-1}} \\
\cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}}
}
}*\frm{\{}*\frm{\}};
%fleches de la ligne du haut
{\ar@{_{(}->} (33,15)*{};(14,15)*{}};
{\ar (45,15)*{};(62,15)*{} ^-{\alpha}};
%%fleches verticales
{\ar (9,5)*{};(9,13)*{}^{\ff_{d,e;\Lambda}}};
{\ar (40,5)*{};(40,13)*{}};
{\ar (69,5)*{};(69,13)*{}|{\ff_{d-1,e;\Lambda\smallrestr{d-1,e}}}};
%%fleches de la ligne du bas
{\ar@{_{(}->} (27,0)*{};(21,0)*{}};
{\ar (52,0)*{};(58.5,0)*{}^-{\alpha'}};
%carrés de diagrames cartésiens
(24,9)*{\labelstyle \Box};
(54,9)*{\labelstyle \Box};
\endxy
$$
%
where $\alpha$ still sends $\cP_d$ to $\cP'_{d-1}:=\cP_d \cap \cV^1$
and where $\ff_{d-1,e;\Lambda\smallrestr{d-1,e}}$ sends a flag
to~$\cP_{d_{k-1}}$. Note that in this case, $\alpha'$ drops the last
subspace~$\cP_d$ in the flag.

\smallbreak

In both cases, it is easy to check that the two squares are
cartesian.
\end{proof}

%
\begin{lemm}
\label{PB_lem}%
Let $d,e\geqslant2$ and let $\Lambda''$ be an even $(d-1,e)$-diagram such
that $\Lambda''_{d-1}$ is odd. Hence we can consider the even
$(d,e-1)$-diagram
$\Lambda'=(\Lambda''_1-1,\ldots,\Lambda''_{d-1}-1,0)$. Then, there
exists a commutative diagram:
%
\begin{equation}
\label{PB_eq}%
\vcenter{\xymatrix{ \Grass_X(d,\cV^1)
& \Exc_X(d,\cV_\smallbullet) \ar[r]^-{\tilde\alpha\,\tilde\iota} \ar[l]_-{\tilde\pi}
& \Grass_X(d-1,\cV^1)
\\
\Flag_X(d,e-1;\Lambda') \ar[u]^{\ff_{d,e-1;\Lambda'}}
& F' \ar[u]^-{f'} \ar[r] \ar[l]^-{\pi'} \ar@{}[ru]|{\Box}
& \Flag_X(d-1,e;\Lambda'') \ar[u]_{\ff_{d-1,e;\Lambda''}}
}}
\end{equation}
%
where $\Exc_X(d,\cV_\smallbullet)$ is the exceptional fiber of
Diagram~\eqref{main_eq} and where the right-hand square is
cartesian. Moreover, either $\pi'$ is an isomorphism or the scheme
$F'$ (with the morphism~$\pi'$) identifies with the blow-up of
$\Flag_X(d,e-1;\Lambda')$ along a closed regular subscheme of odd
codimension.
\end{lemm}

\begin{proof}
Let $k=k(\Lambda'')$,
$\multi{d}=\multi{d}(\Lambda'')$ and
$\multi{e}=\multi{e}(\Lambda'')$ as usual (Def.\,\ref{de_def}). We
need to distinguish two cases, namely $\Lambda''_{d-1}>1$ and $\Lambda''_{d-1}=1$.

Suppose first that $\Lambda''_{d-1}>1$. Then $k(\Lambda')=k+1$ and
$\multi{d}(\Lambda')$ and $\multi{e}(\Lambda')$ are just
$\multi{d}(\Lambda'')$ and $\multi{e}(\Lambda'')$ with one more
entry at the end, namely $d$ and $e-1$ respectively. We can
describe the pull-back in Diagram~\eqref{PB_eq} as follows (on
points):
%
$$\xy 0;<1ex,0ex>:
%ensemble haut gauche
(4,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_{d-1} \ar@{}[r]|-{\subb} & \cP_d \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble haut droite
(45,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_{d-1} \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble bas gauche
(0,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d-1} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d-1+e_{k}} \ar@{}[r]|-{\subb} & \cV^1
}
}*\frm{\{}*\frm{\}};
%ensemble bas droite
(40,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d-1} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d-1+e_{k}}
}
}*\frm{\{}*\frm{\}};
%%fleches verticales
{\ar (9,5)*{};(9,13)*{}^{f'}};
{\ar (48,5)*{};(48,13)*{}};
%fleche de la ligne du haut
{\ar (18,15)*{};(41.5,15)*{}};
%fleche de la ligne du bas
{\ar (24.5,0)*{};(34.5,0)*{}};
%carré de diagrame cartésien
(30,9)*{\labelstyle \Box};
\endxy$$

%
This pull-back $F'$ is $\Flag_X(d,e-1;\Lambda')$. So, take
$\pi'=\id$. The morphism $f'$ composed with $\tilde\pi$ sends a flag
$\cP_{d_1}\subb\cdots\subb\cP_{d-1}\subb\cP_d$ to $\cP_d$ and so
does~$\ff_{d,e-1;\Lambda'}$.

\smallbreak

Suppose now that $\Lambda''_{d-1}=1$. Then $k(\Lambda')=k$ and
$\multi{e}(\Lambda')=\multi{e}(\Lambda'')$, whereas
$\multi{d}(\Lambda')$ is obtained from $\multi{d}(\Lambda'')$ by
replacing its last entry by $d$. We can describe the pull-back in
Diagram~\eqref{PB_eq} as follows:
%
$$\xy 0;<1ex,0ex>:
%ensemble haut gauche
(11,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_{d-1} \ar@{}[r]|-{\subb} & \cP_d \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble haut droite
(54,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_{d-1} \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble bas gauche
(0,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d-1} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV_{d+e-2} \ar@{}[r]|-{\subb} & \cV^1
}
}*\frm{\{}*\frm{\}};
%ensemble bas droite
(50,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d-1} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d+e-2}
}
}*\frm{\{}*\frm{\}};
%%fleches verticales
{\ar (17,5)*{};(17,13)*{}^{f'}};
{\ar (57,5)*{};(57,13)*{}};
%fleche de la ligne du haut
{\ar (25,15)*{};(50,15)*{}};
%fleche de la ligne du bas
{\ar (36.5,0)*{};(44.5,0)*{}};
%carré de diagrame cartésien
(38,9)*{\labelstyle \Box};
\endxy$$%
where $f'$ is the obvious morphism. The left-hand square of~\eqref{PB_eq} is defined by\,:
%
$$\xy 0;<1ex,0ex>:
%ensemble haut gauche
(8,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_d \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble haut droite
(51,15)*\xybox{
\xymatrix@C=0.6ex{
\{ \cP_{d-1} \ar@{}[r]|-{\subb} & \cP_d \ar@{}[r]|-{\subb} & \cV^1 \}
}
};
%ensemble bas gauche
(0,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d_{k-1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cP_d \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV^1}
}*\frm{\{}*\frm{\}};
%ensemble bas droite
(40,0)*!(0,-2.5)\xybox{
\xymatrix@R=1.5ex@C=0.5ex{
\cP_{d_{1}} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cP_{d_{k-1}} \ar@{}[r]|-{\subb} & \cP_{d-1} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \ar@{}[r]|-{\subb} & \cP_{d} \ar@{}[d]|{\rotatebox{270}{$\subbeq$}} \\
\cV_{d_1+e_1} \ar@{}[r]|-{\subb} & \cdots \ar@{}[r]|-{\subb} & \cV_{d_{k-1}+e_{k-1}} \ar@{}[r]|-{\subb} & \cV_{d+e-2} \ar@{}[r]|-{\subb} & \cV^1
}
}*\frm{\{}*\frm{\}};
%%fleches verticales
{\ar (11,5)*{};(11,13)*{}};
{\ar (57,5)*{};(57,13)*{}^{f'}};
%fleche de la ligne du haut
{\ar (47,15)*{};(17,15)*{}_-{\tilde\pi}};
%fleche de la ligne du bas
{\ar (34.5,0)*{};(27.5,0)*{}_-{\pi'}};
%carré de diagrame cartésien
%%%%(33,9)*{\labelstyle \Box};
\endxy$$
%
The morphism $\pi':F'\to\Flag_X(d,e-1;\Lambda')$ simply drops $\cP_{d-1}$ in this case.

Let $\cV^2:=\cV_{d+e-2}$ and
$Y:=\Flag_X((d_1,\ldots,d_{k-1}),(e_1,\ldots,e_{k-1}),\cV_\smallbullet)$.
We have $\Flag_X(d,e-1;\Lambda')=\Grass_Y(d-d_{k-1},\cV^1/\Taut_{d_{k-1}})$ by Lemma~\ref{rel_lem}.
As in Definition~\ref{cell_def}, we consider the blow-up
$\Bl_Y(d-d_{k-1}\,,\,\cV^2/\Taut_{d_k-1}\subb\cV^1/\Taut_{d_k-1})$
of $\Grass_Y(d-d_{k-1},\cV^1/\Taut_{d_{k-1}})$ along the closed
regular immersion of~$\Grass_Y(d-d_{k-1},\cV^2/\Taut_{d_{k-1}})$. By
Proposition~\ref{blowup_prop}, this blow-up coincides with~$F'$ and
the morphism~$\pi$ of~\eqref{main-pts_eq} here becomes the above
morphism~$\pi'$. In other words, the following diagram commutes:
%
$$
\xymatrix{
\Flag_X(d,e-1;\Lambda') \ar@{=}[d]_{}
& F' \ar@{=}[d] \ar[l]_-{\pi'}
\\
\Grass_Y(d-d_{k-1},\cV^1/\Taut_{d_{k-1}})
& \Bl_Y(d-d_{k-1}, \,\cV^2/\Taut_{d_k-1}\subb\cV^1/\Taut_{d_{k-1}})\,.\! \ar[l]_-{\text{``}\pi\text{''}}
}
$$
%
The closed immersion $\Grass_Y(d-d_{k-1},\cV^2/\Taut_{d_{k-1}})\hookrightarrow \Grass_Y(d-d_{k-1},\cV^1/\Taut_{d_{k-1}})$ is of odd codimension equal to~$d-d_{k-1}$.
\end{proof}


%------------------------------

\section{Push-forward, pull-back and connecting homomorphism}
\label{push-pull-bord_sec}%

%------------------------------

We describe the long exact sequence on Witt groups associated to the
geometric decomposition $\Grass_X(d,\cV)=\Grass_X(d,\cV^1)\cup
U_X(d,\cV_\smallbullet)$, considered in Section~\ref{cell-dec_sec}. Recall the
important Diagram~\eqref{main_eq}:
%
\begin{equation}
\tag{\ref{main_eq}} \vcenter{\xymatrix@C=5em{ \Grass_X(d,\cV^1)
\ar@{^{(}->}[r]^(.55){\iota} \ar@{}[r]|(.4){|} & \Grass_X(d,\cV) &
U_X(d,\cV_\smallbullet) \ar@{_{(}->}[l]_-{\upsilon} \ar@{_{(}->}[dl]_-{\tilde\upsilon} \ar@{}[l]|-{\circ}
\ar[d]^{\alpha}
\\
\Exc_X(d,\cV_\smallbullet) \ar[u]^{\tilde{\pi}}
\ar@{^{(}->}[r]^(.45){\tilde{\iota}} \ar@{}[r]|(.4){|} &
\Bl_X(d,\cV_\smallbullet) \ar[u]^{\pi} \ar[r]^-{\tilde{\alpha}} &
\Grass_X(d-1,\cV^1)\,.\! }}
\end{equation}
%

\begin{nota}
\label{kappa_nota}%
On Witt groups,
$\alpha^*:\Wtot(\Grass_X(d-1,\cV^1))\isoto\Wtot(U_X(d,\cV_\smallbullet))$ is an
isomorphism by homotopy invariance and we define
%
$$
\kappa:=(\alpha^*)^{-1}\upsilon^*:\Wtot(\Grass_X(d,\cV))\too\Wtot(\Grass_X(d-1,\cV^1))\,.
$$
%
\end{nota}

\begin{prop}
\label{LES_prop}%
Let $L\in\Pic(\Grass_X(d,\cV))/2$. Then define
$L':=\can_{\iota}+\iota^*(L)$ in $\Pic(\Grass_X(d,\cV^1))/2$ and
$L'':=(\alpha^*)^{-1}(\upsilon^*(L))$ in
$\Pic(\Grass_X(d-1,\cV^1))/2$. There is a long exact sequence
%
$$
\xymatrix@C=3ex{ \cdots
\Wcoh^{i-d}(\Grass_X(d,\cV^1),L') \ar[r]^-{\iota_*} &
\Wcoh^{i}(\Grass_X(d,\cV),L) \ar[r]^-{\kappa} &
\Wcoh^i(\Grass_X(d-1,\cV^1),L'') \ar`d[l]`[dll]^{\bord}[dll]
\\
\Wcoh^{i-d+1}(\Grass_X(d,\cV^1),L') \ar[r]^-{\iota_*} &
\Wcoh^{i+1}(\Grass_X(d,\cV),L) \ar[r]^-{\kappa} &
\Wcoh^{i+1}(\Grass_X(d-1,\cV^1),L'')\cdots
\\
}$$
%
\end{prop}

\begin{proof}
This is the long exact sequence of localization~\cite{Balmer00}
associated to the codimension~$d$ closed immersion
$\Grass_X(d,\cV^1) \subset \Grass_X(d,\cV)$ in which we used
dévissage to replace $\Wcoh^i_{\Grass(d,\cV^1)}(\Grass_X(d,\cV),L)$
by $\Wcoh^{i-d}(\Grass_X(d,\cV^1),L')$ and homotopy invariance to replace
$\Wcoh^i(U_X(d,\cV_\smallbullet),\upsilon^*L)$ by
$\Wcoh^{i}(\Grass_X(d-1,\cV^1),L'')$. See~\cite[Sequence~(11)]{BalmerCalmes08}.
\end{proof}

\begin{coro}
\label{LES_cor}%
For $d,e\geqslant2$, there is a 3-term exact sequence of total Witt
groups
%
\begin{equation}
\label{LES-tot_eq}%
\xymatrix@C=3ex{ \Wtot(\Grass_X(d,\cV^1)) \ar[r]^-{\iota_*} &
\Wtot(\Grass_X(d,\cV)) \ar[r]^-{\kappa} & \Wtot(\Grass_X(d-1,\cV^1))
\ar`d[l]+/d5ex/`[ll]_{\bord}[ll]
\\
}
\end{equation}
%
\end{coro}

\begin{proof}
Add the long exact sequences of Proposition~\ref{LES_prop} over
$L\in\Pic(\Grass_X(d,\cV))/2=\Pic(X)/2\oplus\bbZ/2\,\Det_d$ and $i
\in \bbZ/4$. All three schemes have the same Picard groups
($d,e\geqslant 2$) by~\eqref{Picards_eq}. So when $L$ runs among all
possible values of $\Pic(\Grass_X(d,\cV))/2$, so does $L'$ in
$\Pic(\Grass_X(d,\cV^1))/2$ and $L''$ in
$\Pic(\Grass_X(d-1,\cV^1))/2$.
\end{proof}

\begin{rema}
\label{index_rem}%
When $d=1$, one can define the sum of the above proof but for each
$L''\in\Pic(\Grass_X(d-1,\cV^1)/2$, and each $i\in\bbZ/4$, the Witt
group $\Wcoh^i(\Grass_X(d-1,\cV^1),L'')$ will appear twice. This is
explained in Remark~\ref{Pic-can_rem}\,: the kernel of the
homomorphism
$\Pic(\Grass_X(d,\cV))/2\too\Pic(\Grass_X(d-1,\cV^1))/2$ is
$\bbZ/2\cdot\Det_d$\,. In this case, the third ``total" Witt group
of~\eqref{LES-tot_eq} must be replaced by two copies of it.

Similarly, when $e=1$, the total Witt group of $\Grass_X(d,\cV^1)$
will appear twice in the first entry of~\eqref{LES-tot_eq}. See more
in Remark~\ref{Pn_rem}.
\end{rema}

\begin{rema}
\label{twist_rem}%
All three groups in~\eqref{LES-tot_eq} are graded over
%
$$
\bbZ/4\oplus\Pic(X)/2\oplus\bbZ/2\,\Det_d\,.
$$
%
The homomorphisms $\iota_*$, $\kappa$ and $\bord$ are not always
homogeneous when the line bundle $\cV/\cV^1$ is not trivial (in
$\Pic(X)/2$). Indeed, the same happens as in Proposition~\ref{deg_prop}\,:
%
\begin{enumerate}
%
\item
$\iota_*$ is indeed homogeneous of degree $\big(\,d
\,,\,-d\cdot\cV/\cV^1\,,\,1 \,\big)$.
%
\item
$\kappa$ sends degree $(s,\ell,t)$ into degree
$\big(\,s\,,\,\ell+t\cdot(\cV/\cV^1) \,,\, t\,\big)$.
%
\item
$\bord$ sends degree $(s,\ell,t)$ into degree
$\big(\,s-d+1\,,\,\ell+(t-d)\,(\cV/\cV^1) \,,\, t-1\,\big)$.
%
\end{enumerate}
%
Note that if we restrict attention to the significant part of the twist (the part $\bbZ/2\,\Det_d$ not coming from $X$), $\kappa$ leaves it unchanged, and $\iota_*$ and $\bord$ change it.
\end{rema}

Finally, we shall also use the following general fact about
push-forwards along blow-up morphisms.

%
\begin{prop}
\label{push-1_prop}%
Let $X$ be a noetherian scheme (quasi-compact and quasi-separated is
enough) and let $Z\hookrightarrow X$ be a regular immersion of pure
codimension~$d$. Let $\pi:\Bl\to X$ be the blow-up of $X$ along~$Z$.
Then\,:
%
\begin{enumerate}
\item There is a natural isomorphism $R\pi_*(\cO_\Bl)\cong \cO_X$ in
the derived category of~$X$.
\item Assume further that $X$ is regular and that $\can_{\Bl/X}$ is
a square (which happens exactly when $d$ is odd
by~\cite[Proposition A.11 (iii)]{BalmerCalmes08}). Then the push-forward
$\pi_*:\Wper^0(\Bl)\to\Wper^0(X)$ maps the unit class $1_\Bl$ to the
unit class $1_X$. (This is even true on the level of forms, without
taking Witt classes.)
\end{enumerate}
\end{prop}

\begin{proof}
(a) can be found in SGA~6, see~\cite[Lemme~VII.3.5, p.\,441]{SGA6}
or the more recent account in~\cite[Lemme~2.3\,(a)]{Thomason93}.

(b) follows from~(a). Indeed, when $d=1$, we have $\Bl=X$ and there
is nothing to prove. When $d\geqslant3$ then line bundles over~$X$,
and homomorphisms between them, are determined by their restriction
to the open complement~$U=X\smallsetminus Z$ of~$Z$ since $Z$ is of
codimension at least~2. By the base-change formula for push-forward,
and since $\pi\restr{\pi^{-1}(U)}:\pi^{-1}(U)\to U$ is an
isomorphism, the result follows.
\end{proof}

%
\begin{rema}
\label{desing_rem}%
The main application of the previous result will appear at a slightly technical moment below. Yet, it also has a reassuring application, namely that our
candidate-generators $\gen_{d,e}(\Lambda)$ do not really depend on
the chosen desingularization $\Flag(\Lambda)$ of the Schubert subvariety
corresponding to the Young diagram~$\Lambda$. Indeed the unit will
remain the unit when pushed-forward between two such
desingularizations, if one is obtained from the other by blow-up.
(The condition on the relative bundle being even is automatically
satisfied if they both have even relative bundle with respect to the
Grassmannian.)
\end{rema}


%------------------------------

\section{Main result}
\label{main_sec}%

%------------------------------

We are now ready to state and prove the main result of the paper.

\begin{theo} \label{main_thm}%
Let $d,e\geqslant1$. Let $X$ be a regular scheme over~$\bbZ[\half]$. Let $\cV$ be a vector bundle of rank $d+e$
which admits a complete flag~\eqref{complete_eq} of subbundles. (For
instance, when $\cV$ is free.) Then, the elements
$\gen_{d,e}(\Lambda)$ of Definition~\ref{gen-Witt_def}, for $\Lambda$ even, form a basis of the graded $\Wtot(X)$-module
$\Wtot(\Grass_X(d,\cV))$.
\end{theo}

\begin{proof}
The proof occupies most of this Section. We fix a complete flag
%
$$
0=\cV_0\subb\cV_1\subb\cdots\subb\cV_{d+e}=\cV\,.
$$
%
We set as before $\cV^1:=\cV_{d+e-1}$.

The idea of the proof is very simple; it is an induction on
$\rank(\cV)=d+e$. The case $d=1$ or $e=1$ is slightly particular and
treated on its own (see Remark~\ref{ind_rem} or
Remark~\ref{Pn_rem}), so we assume $d,e\geqslant2$ and the result
proved for vector bundles of rank smaller than $d+e$.

The induction step is immediate by the 5-Lemma once we have
established the commutativity of the following
(horizontally-periodic) diagram:
%
\begin{equation}
\label{compare_eq}%
\kern-.3em\vcenter{\xymatrix@C=0.9em{ \Free_X(d,e-1)
\ar[r]^-{\bar\iota} \ar[d]_{\iso_{d,e-1}}^\simeq
& \Free_X(d,e) \ar[r]^-{\bar\kappa} \ar[d]_{\iso_{d,e}}
& \Free_X(d-1,e) \ar[r]^-{\bar\bord} \ar[d]_{\iso_{d-1,e}}^\simeq
& \Free_X(d,e-1) \ar[d]_{\iso_{d,e-1}}^\simeq
\\
\Wtot(\Grass_X(d,\cV^1)) \ar[r]^-{\iota_*} & \Wtot(\Grass_X(d,\cV))
\ar[r]^-{\kappa} & \Wtot(\Grass_X(d-1,\cV^1)) \ar[r]^-{\bord} &
\Wtot(\Grass_X(d,\cV^1)) }}
\end{equation}
%
whose first row is the exact sequence of
Corollary~\ref{exact-Lambda_cor} and whose second row is the exact
sequence~\eqref{LES-tot_eq}.

\goodbreak

To check that Diagram~\eqref{compare_eq} commutes amounts to
control how the morphisms $\iota_*$, $\kappa$ and $\bord$ behave
on the elements $\gen_{d,e}(\Lambda)$ constructed in
Def.\,\ref{gen-Witt_def}. Recall that the $\gen_{d,e}(\Lambda)$ are
only defined when $\Lambda$ is even. Note that Proposition~\ref{deg_prop} and Remark~\ref{twist_rem} imply that commutativity of~\eqref{compare_eq} makes sense as far as degrees as concerned. So, rephrasing the commutativity of~\eqref{compare_eq}, the core of the proof is the following:

\begin{prop}
\label{push-pull-bord_prop}%
Let $d,e\geqslant2$. Keep notations as above.
\begin{enumerate}
\item
Let $\Lambda'$ be an even Young diagram in $(d\!\times\!(e-1))$-frame.
The push-forward $\iota_*$ satisfies the following (see
Figure~\ref{push_fig}):
%
$$
\iota_*(\gen_{d,e-1}(\Lambda'))=\iso_{d,e}(\bar\iota(\Lambda'))\,.
$$
%
\item
Let $\Lambda$ be an even Young diagram in \deframe. The ``restriction'' morphism
$\kappa$ satisfies the following (see Figure~\ref{pull_fig}):
%
$$
\kappa(\gen_{d,e}(\Lambda))= \iso_{d-1,e}(\bar\kappa(\Lambda))\,.
$$
%
\item
Let $\Lambda''$ be an even Young diagram $((d-1)\!\times\!e)$-frame. The
connecting homomorphism $\bord$ satisfies the following (see
Figure~\ref{bord_fig}):
%
$$
\bord(\gen_{d-1,e}(\Lambda''))=
\iso_{d,e-1}(\bar\bord(\Lambda''))\,.
$$
%
\end{enumerate}
\end{prop}

\begin{proof}%[Proof of Proposition \ref{push-pull-bord_prop}]
Recall from Corollary~\ref{exact-Lambda_cor} that the definitions of
$\bar\iota$, $\bar\kappa$ and $\bar\bord$ each involve two cases.
The general organization of the proof is to start by proving the
equations of~(a), (b) and~(c) in the first cases (the non-zero
ones). The remaining cases will be easy to deduce from these.

\medbreak

\noindent\textit{Interesting case of~(a).} Let $\Lambda'$ be an even
$(d,e-1)$-diagram such that $\zero(\Lambda')$ is even. Let
$\multi{d}$ and $\multi{e}$ be the corresponding $k$-tuples
(Def.\,\ref{de_def}). By assumption we can consider the even
$(d,e)$-diagram
$\bar\iota(\Lambda'):=(\Lambda'_1+1,\ldots,\Lambda'_d+1)$. Observe
that it has the same associated $k$-tuples (with respect to the
\deframe). From~\eqref{Flag-Lambda_eq}, it is then easy to see that
$\Flag_{X}(d,e-1;\Lambda')=\Flag_{X}(d,e;\bar\iota(\Lambda'))$ and that the diagram
%
$$\xymatrix{
\Grass_X(d,\cV^1) \ar[r]^-{\iota} & \Grass_X(d,\cV)
\\
\Flag_{X}(d,e-1;\Lambda') \ar[u]^{\ff_{d,e-1;\Lambda'}} \ar@{=}[r] &
\Flag_{X}(d,e;\bar\iota(\Lambda')) \ar[u]_{\ff_{d,e,\bar\iota(\Lambda')}} }$$
%
commutes. In that case, (a) follows by composition of push-forwards.

\medbreak

\noindent\textit{Interesting case of~(b).}
Let $\Lambda$ be an even $(d,e)$-diagram such that
$\Lambda_d=0$. Let $\multi{d}$ and $\multi{e}$ be the corresponding
$k$-tuples (Def.\,\ref{de_def}). By assumption we can consider the
even $(d-1,e)$-diagram $\bar\kappa(\Lambda):=\Lambda\restr{d-1,e}$. We then have two cartesian squares:
%
$$
\xymatrix{ \Grass_X(d,\cV) \ar@{}[dr]|{\Box} & U_X(d,\cV_\smallbullet)
\ar@{_{(}->}[l]_-{\upsilon} \ar[r]^-{\alpha} \ar@{}[dr]|{\Box} &
\Grass_X(d-1,\cV^1)
\\
\Flag_X(d,e;\Lambda) \ar[u]^{\ff_{d,e;\Lambda}}
& U' \ar@{_{(}->}[l] \ar[r] \ar[u]
& \Flag_{X}(d-1,e;\bar\kappa(\Lambda))
\ar[u]_{\ff_{d-1,e;\bar\kappa(\Lambda)}} }
$$
%
by Lemma~\ref{double-PB_lem}. Replacing $\kappa$ by its definition
(see~\ref{kappa_nota}) and using that $\alpha^*$ is an isomorphism,
the equality in~(b) claims that
%
$$
\qquad\upsilon^*(\gen_{d,e}(\Lambda))=\alpha^*(\gen_{d-1,e}(\Lambda\restr{d-1,e}))\quad\quadtext{in}
\Wtot(U_X(d,\cV_\smallbullet))\,.
$$
%
This follows by base change on the above two cartesian squares
(see~\cite{Calmes06}, the horizontal morphisms are smooth, so flat,
and the vertical maps are proper, including $U'\to U_X(d,\cV_\smallbullet)$).
Both sides of the above equation are equal in $\Wtot(U_X(d,\cV_\smallbullet))$ to
the push-forward of the same $1_{U'}\in\Wcoh^0(U')$ along $U'\to
U_X(d,\cV_\smallbullet)$.

\medbreak

\noindent\textit{Interesting case of~(c).}
Let $\Lambda''$ be an even $(d-1,e)$-diagram such that
$\Lambda''_{d-1}$ is odd. Let $\multi{d}$ and $\multi{e}$ be the
corresponding $k$-tuples (Def.\,\ref{de_def}). By assumption, we can
consider the even $(d,e-1)$-diagram
%
$$
\bar\bord(\Lambda''):=(\Lambda''_1-1,\ldots,\Lambda''_{d-1}-1,0)\,.
$$

The twist $\twist(\Lambda'')$ is given by
$\twist(\Lambda'')=[d+1]\in\bbZ/2$. Using Diagram~\eqref{Picards_eq}
and Equation~\eqref{rel-can-Bl_eq}, one can easily check that this
is the twist for which the connecting homomorphism~$\bord$ can be
computed by first pulling back to $\Exc_X(d,\cV_\smallbullet)$ and
then pushing-forward to $\Grass_X(d,\cV^1)$, see~\cite[Theorem
2.6]{BalmerCalmes08}. By Lemma~\ref{PB_lem}, we have the right-hand
cartesian square in the following commutative diagram:
%
$$
\xymatrix{
\Grass_X(d,\cV^1)
& \Exc_X(d,\cV_\smallbullet) \ar[l]_-{\tilde\pi} \ar[r]^-{\tilde\alpha\tilde\iota} \ar@{}[dr]|{\Box} &
\Grass_X(d-1,\cV^1)
\\
\Flag_X(d,e-1;\bar\bord(\Lambda'')) \ar[u]^{\ff_{d,e-1;\bar\bord(\Lambda'')}}
& F' \ar[u]\ar[l]_-{\pi'}\ar[r]
& \Flag_X(d-1,e;\Lambda'')\,. \ar[u]_{\ff_{d-1,e;\Lambda''}} }$$
%
We can now compute $\bord(\gen_{d-1,e}(\Lambda''))$ by base-change
on the right-hand square and composition of push-forwards, starting
with the unit form on~$\Flag_X(d-1,e;\Lambda'')$. The key point is
to check that $\pi'_*$ preserves this unit form. By
Lemma~\ref{PB_lem}, we know that $\pi'$ is either an isomorphism or
a blow-up along a closed regular immersion of odd codimension. In
both cases $\pi'_*(1)=1$ by Proposition~\ref{push-1_prop} and we get
the result.

\medbreak

\noindent\textit{Remaining cases.} At this stage, we have proved the
equations of~(a), (b) and~(c) in the first cases (the non-zero
ones). In the remaining cases, both sides of these equations are
zero, as easily follows from the above cases and diagram chase
in~\eqref{compare_eq}.

For instance, let us prove that both sides of~(b) vanish for an even
$(d,e)$-diagram $\Lambda$ such that $\Lambda_d>0$. Since
$\bar\kappa(\Lambda)=0$ it suffices to show
$\kappa(\gen_{d,e}(\Lambda))=0$. By
Corollary~\ref{exact-Lambda_cor}, there exists an even
$(d,e-1)$-diagram $\Lambda'$ such that $\zero(\Lambda')$ is even and
$\bar\iota(\Lambda')=\Lambda$. We have established equation~(a) for
such $\Lambda'$. Applying $\kappa$ to this equation gives
$\kappa\iota_*(\gen_{d,e-1}(\Lambda'))=\kappa(\gen_{d,e}(\Lambda))$
hence the result since $\kappa\,\iota_*=0$. The other cases
($\Lambda'$ with $\zero(\Lambda')$ odd and $\Lambda''$ with
$\Lambda''_{d-1}$ even) can be proved similarly.
\end{proof}


\begin{rema}
\label{Pn_rem}%
Following up on Remark~\ref{ind_rem}, we now unfold the case $d=1$
(resp. $e=1$) of our constructions from Sections~\ref{cell-dec_sec}
and~\ref{push-pull-bord_sec}. The difference comes from the
apparition of $\Grass_X(0,\cV^1)$ (resp. $\Grass_X(d,\cV_d)$), which
is simply $X$, in which case the Picard groups do not look like
in~\eqref{Picards_eq}. This involves an artificial total Witt group
of $\Grass_X(0,\cV^1)$ (resp. $\Grass_X(d,\cV_d)$) in the
localization long exact sequence~\eqref{LES-tot_eq}, not summing on
its ``own'' twists but on the ``ambient'' twists
of~$\Grass_X(d,\cV)$, as explained in Remark~\ref{index_rem}.
Proposition~\ref{push-pull-bord_prop} then becomes the following
(only the cases involving $X$ are covered, the other ones having
already been done).
\end{rema}


\begin{prop}
\label{Pn_prop}%
Let $d=1$ (resp. $e=1$) and let $\pt_i$, $i\in \bbZ/2$ be two copies
of the unit form on $X$, one for each twist in $\bbZ/2\,\Det_d$ of
the ambient $\Grass_X(d,\cV)$. As before, $\rect{d}{e}$ is the full
diagram and $\vide$ is the empty one. When $e=1$, the push-forward satisfies the following:
\begin{equation}
\iota_*(\pt_{d+1})=0 \ \text{and}\ \iota_*(\pt_d)=\gen_{d,1}(\rect{d}{1}).
\end{equation}
When $d=1$, the morphism $\kappa$ satisfies the following:
\begin{equation}
\kappa(\gen_{1,e}(\vide))=\pt_0
\end{equation}
When $d=1$ (resp. $e=1$), the connecting homomorphism $\bord$ satisfies the following:
\begin{equation}
\bord(\pt_1)=\gen_{1,e}(\vide) \ \text{and}\ \bord(\pt_0)=0 \quad \text{(resp. } \bord(\rect{(d-1)}{1})=\pt_{d+1}\text{)\,.}
\end{equation}
\end{prop}
Since the proof works as the one of the previous proposition, we leave the details to the reader. See Figure~\ref{specialCases_fig}.
%%%%%
%This proof becomes very
%similar to the one in~\cite[\S\,7]{Balmer05} and we chose to
%simplify things by invoking~\cite{Balmer05}, as we did in
%Proposition~\ref{Pn_prop}. A motivated and careful reader can
%nevertheless follow our technique to get that special case as well.

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/specialCases.pstex_t}%pdf_t}%
\end{center}
\caption{Image of the generating elements by the morphisms $\iota_*$, $\kappa$ and $\bord$ in three special cases when $d=1$, $e=1$ and $d=e=1$. No arrow means mapped to zero.}
\label{specialCases_fig}%
\end{figure}

The induction step in the proof of Theorem~\ref{main_thm} then also works in the case $d=1$ or $e=1$, using $\Free_X(0,e)=\Free_X(d,0)=\Wtot(X).\pt_0 \oplus \Wtot(X).\pt_1$. Its proof is therefore completed.
\end{proof}

\begin{rema}
Of course every $\gen_{d,e}(\Lambda)$ has a particular shift and twist. When $X$ is local, and more generally when $\Wtot(X)=\Wcoh^0(X)$, one obtains a basis of $\Wcoh^i(\Grass_X(d,\cV),L)$, for $i\in\bbZ/4$
and $L\in\Pic/2$ fixed, by selecting the generators $\gen_{d,e}(\Lambda)$ with shift $i$ and twist~$L$.
\end{rema}

\begin{coro}
\label{ranks_cor}%
Let $d'$ (resp. $e'$) be the integral part of $d/2$ (resp. $e/2$)
and consider the binomial coefficient
$\binom{a+b}{a}=\frac{(a+b)!}{a!b!}$. As modules over $\Wtot(X)$,
the total Witt group of $\Grass_X(d,\cV)$ is free of rank $2 \binom{d'}{e'}$.
If we assume moreover that $\Wtot(X)=\Wcoh^0(X)$, for instance for $X$ local, then
%
\begin{itemize}
\item the classical Witt group of symmetric forms $\Wcoh^0(\Grass_X(d,\cV),0)$ has rank $\binom{d'+e'}{e'}$,
\item the classical Witt group of antisymmetric forms $\Wcoh^2(\Grass_X(d,\cV),0)$ is zero.
\item the Witt groups $\Wcoh^1(\Grass_X(d,\cV),1)$ and $\Wcoh^3(\Grass_X(d,\cV),1)$ are zero.
\end{itemize}
\end{coro}

\begin{proof}
Let ``$4$-block'' mean ``$2\times 2$ square''. Every even Young diagram $\Lambda$ is
either
\begin{enumerate}
\item \label{blocks}%
a union of $4$-blocks and $\gen(\Lambda)$ is in $\Wcoh^0(\Grass_X(d,\cV),0)$,
\item \label{row}%
a single row plus $4$-blocks ($e$ even) and $\gen(\Lambda)$ is in $\Wcoh^e(\Grass_X(d,\cV),1)$,
\item \label{col}%
a single column plus $4$-blocks ($d$ even) and $\gen(\Lambda)$ is in $\Wcoh^d(\Grass_X(d,\cV),1)$,
\item \label{col-row}%
a single row and a single column plus $4$-blocks ($d$ and $e$ odd) and $\gen(\Lambda)$ is in $\Wcoh^{d+e-1}(\Grass_X(d,\cV),0)$.
\end{enumerate}
All possibilities (a)-(d) are exclusive and can be enumerated easily
by counting the diagrams of $4$-blocks, which amounts to counting
the usual Young diagrams in $(d'\!\times\!e')$-frame. We get
$\binom{d'+e'}{e'}$ elements in case~(a), and the other results depend
on the parities of $d$ and $e$ but are also very easy to figure out
in each case.
\end{proof}

\begin{coro} \label{bord-non-zero_coro}%
The connecting homomorphism $\bord$ is zero (and thus the long exact
sequence $\eqref{LES-tot_eq}$ for $\Grass_X(d,\cV)$ decomposes as
split short exact sequences as for Chow groups) if and only if both
$d$ and $e$ are even.
\end{coro}

\begin{proof}
Looking back at the proof of the main theorem, and
at~\eqref{bar-bord_eq} (or Figure~\ref{bord_fig}), we see that
$\bord$ is zero if and only if there is no even $(d-1,e)$-diagram
$\Lambda''$ such that $\Lambda''_{d-1}$ is odd. This implies that
$e$ is even (otherwise $\Lambda''=\rect{(d-1)}{e}$ would be an even
diagram) and that $d-1$ is odd (otherwise $\Lambda''=(1,\ldots,1)$
would be an even diagram). Conversely, assume $e$ even and the
existence of an even $(d-1,e)$-diagram $\Lambda''$ such that
$\Lambda''_{d-1}$ is odd. Then $e_k$ is odd (since
$e=\Lambda''_{d-1}+e_k$ is even), hence all $e_i$ are odd since
$\Lambda''$ is an even diagram. In particular, $e_1$ is odd, hence
$e_1>0$ and therefore $d_1$ is even. This implies that
$d-1=d_k=(d_k-d_{k-1})+\ldots+(d_2-d_1)+d_1$ is even, \ie $d$ is
odd, as was to be shown.
\end{proof}

\begin{nota}
\label{Gal_not}%
For $d,e\geqslant1$, we denote by $\Gal_X(d,e)$ the split
Grassmannian
%
$$
\Gal_X(d,e)=\Grass_X(d,\cO_X^{d+e})\,.
$$
%
\end{nota}

\begin{exam}
Figure~\ref{G33_fig} shows how the different generators map to each
other by $\iota_*$, $\kappa$ and $\bord$ in the long exact
sequence~\eqref{LES-tot_eq} for~$\Gal(3,3)$.
\end{exam}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/G33.pstex_t}%pdf_t}%
\end{center}
\caption{Total long exact sequence on generators (no arrow means mapped to zero)}
\label{G33_fig}%
\end{figure}

\begin{exam}
Figures~\ref{G44_fig}, \ref{G45_fig} and~\ref{G55_fig} give the even Young diagrams in \deframe\ and the corresponding shifts in $\bbZ/4$ and twists in~$\bbZ/2\,\Det_d$ for the Grassmannians $\Gal(4,4)$, $\Gal(4,5)$ and $\Gal(5,5)$.
\end{exam}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/G44.pstex_t}%pdf_t}%
\end{center}
\caption{Diagrams of generators for $\Gal(4,4)$: first row in shift
$0$ and twist $0$, second row in shift $0$ and twist~$1$.}
\label{G44_fig}%
\end{figure}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/G45.pstex_t}%pdf_t}%
\end{center}
\caption{Diagrams of generators for $\Gal(4,5)$: first row in shift $0$ and twist $0$, second row in shift $0$ and twist~$1$.}
\label{G45_fig}%
\end{figure}

\begin{figure}[!ht]
\begin{center}
\input{\LOCAL/G55.pstex_t}%pdf_t}%
\end{center}
\caption{Diagrams of generators for $\Gal(5,5)$: first row in shift $0$ and twist $0$, second row in shift $1$ and twist~$1$.}
\label{G55_fig}%
\end{figure}

\bigbreak \centerline{*\ *\ *}\goodbreak\bigbreak

We conclude with a few comments.

\begin{rema}
We could have considered a larger set of elements
$\gen_{d,e}(\Lambda)$ using Remark~\ref{cond-push_rem} instead of
assuming $\Lambda$ even. This larger set is also stable by applying
$\iota_*$, $\kappa$ and $\bord$. Some of these extra elements are
then easily seen to be zero from the exact sequence, but not all of
them.
\end{rema}

\begin{rema}
Part of the ring structure on the total Witt group can be computed
at each induction step using the projection formula. Unfortunately,
this is not enough for the whole computation. Despite the results
for the Grothendieck and the Chow rings using basis of
Schubert cells, it is unclear to the authors what kind of
Littlewood-Richardson type rule one should expect.

Note however that all generators $\gen_{d,e}(\Lambda)$ are nilpotent
except for $\Lambda=\varnothing$. This can be checked using homotopy
invariance and a discussion on the supports or simply the fact that
these Witt classes are generically trivial.
\end{rema}

\begin{rema}
Although we don't need it here, it is possible to show that, for $\cV=\cO_X^{d+e}$, the
isomorphism $\Gal(d,e)=\Grass(d,\cV)\simeq
\Grass(e,\cV^\vee)=\Gal(e,d)$ sends $\gen_{d,e}(\Lambda)$ to
$\gen_{e,d}(\Lambda^\vee)$ where $\Lambda^\vee$ is the dual
partition.
\end{rema}

\appendix

\section{Total Witt group} \label{TotalWitt_app}%

For $i\in\bbZ$ and for a line bundle~$L$ over~$X$, the Witt group $\Wcoh^i(X,L)$ is the triangular Witt group of the bounded derived category of vector bundles over~$X$ with respect to the duality derived from $\mathcal{H}\textit{om}_{\cO_X}(-,L)$ and shifted $i$ times. See~\cite{Balmer00, Balmer01}. These groups are periodic in two elementary ways. First they are $4$-periodic in shift, \ie there is an isomorphism
%
$$
\Wcoh^i(X,L) \simeq \Wcoh^{i+4}(X,L)
$$
%
by \cite[Proposition 2.14]{Balmer00}. This isomorphism is completely canonical and easily goes through all other constructions, so we do not discuss it any further.

We then have a product
%
$$
\Wcoh^i(X,L) \times \Wcoh^j(X,M) \to \Wcoh^{i+j}(X,L \otimes M)
$$
%
by \cite{Gille03} that yields, for any line bundle $M$, a periodicity isomorphism
%
\begin{equation}
\label{Per_eq}%
\Wcoh^i(X,L) \simeq \Wcoh^i(X, L \otimes M^{\otimes 2})
\end{equation}
%
given by multiplication with the Witt class $[M \isoto M^{\vee}
\otimes M^{\otimes 2}]\in\Wcoh^0(X,M^{\otimes 2})$, where $M^{\vee}$
denotes the dual of~$M$.

From these two periodicity isomorphisms, it is clear that all the
information about the Witt groups of a scheme can be concentrated in
a \emph{total Witt group}, summing Witt groups indexed by elements in
$\Pic/2$ and integers in $\bbZ/4$. However, this total group is not
canonical since it involves the choice of a line bundle $L$ for
every class in $\Pic/2$. Furthermore, if we want to turn this total
Witt group into a ring, using the above product, we need to choose
isomorphisms between $L\otimes M$ and the line bundle representing
$L+M$ in $\Pic/2$, including the choice of ``square roots'' (for the
periodicity modulo 2), and so on. All this data should further
satisfy some compatibilities, of the highest sex appeal. Unfolding
these technicalities would simultaneously increase the size of the
article and reduce its readability, beyond the taste of any
potential reader. Therefore we make the following choice\,:

\begin{conv}
\label{no-noise_conv}%
We treat all periodicity isomorphisms as identities\,:
%
$$
\Wcoh^i(X,L) = \Wcoh^{i+4}(X,L)
\quad\text{and}\quad
\Wcoh^i(X,L)=\Wcoh^i(X,L\otimes M\potimes{2})\,.
$$
%
\end{conv}

We see two ways of providing a formal ground for this convention, at least in our discussion of the total Witt group of the Grassmann varieties over~$X$.

The first possibility is to assume the above choices for the base scheme~$X$ and then use the explicit isomorphism $\Pic(\Grass_X(d,\cV)) \cong \Pic(X) \oplus \bbZ\cdot\Det_d$ to extend the choices made for~$X$ to $\Grass(d,\cV)$, by using the tensor powers of $\Det_d$ in a natural way.

The second possibility can be applied beyond Grassmann varieties. It starts with the observation that two different choices of the above conglomerate of line bundles and compatibility isomorphisms would only differ ``by multiplication by a global unit''. The fact that some collection of Witt classes forms a basis of the graded Witt ring is stable by multiplication of these basis elements by units. As long as the proof of such a fact never uses addition in $\Wtot$ but only the 5-Lemma, then it is insensible to the ambiguity of the choices. Indeed, it is easily checked that the 5-Lemma holds if its 4 squares only commute ``up to units''.

Deciding which method is more pleasant is left to the insomniac reader.


%------------------------------

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
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}
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\end{document}

%------------------------------
%------------------------------

