A flatness property for filtered $\mathcal D$-modules
F.J. Castro-Jim\'enez and M. Granger
Let $\cM$ be a coherent module over the ring $\cD_X$ of linear differential operators on an analytic manifold $X$ and let $Z_1,\cdots , Z_k$ be $k$ germs of transverse hypersurfaces at a point $x\in X$. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of $\cM$ at $x$, give rise to a multifiltration $U_\bullet(\cM)$ of $\cM_x$ as in Sabbah's paper \cite{Sabbah} and to an analytic standard fan in a way similar to \cite{assi-castro-granger-jpaa-2}. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in \cite{Sabbah}, for which the use of \cite{Sab-Cas} is not possible.
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\author{F.J. Castro-Jim\'enez and M. Granger}
\title{A flatness property for filtered $\mathcal D$-modules}

\date{}
\begin{document}
\maketitle
%\tableofcontents

\begin{abstract} Let $\cM$ be a coherent module over the ring $\cD_X$ of linear differential operators on an
analytic manifold $X$ and let  $Z_1,\cdots , Z_k$ be $k$ germs of transverse hypersurfaces at a point $x\in X$.  The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of  the germ of $\cM$
at $x$, give rise to a multifiltration $U_\bullet(\cM)$  of $\cM_x$ as in Sabbah's paper   \cite{Sabbah}
and to an analytic standard fan   in a way similar to
\cite{assi-castro-granger-jpaa-2}. We prove here that this standard
fan is adapted to the multifiltration, in the sense of C. Sabbah. This
result completes the proof of the existence of an adapted fan in
\cite{Sabbah}, for which the use of   \cite{Sab-Cas} is not possible.
\end{abstract}


\section{Introduction} Let us consider a coherent module $\cM$ over the ring $\cD_X$ of differential operators on an
analytic manifold $X$. For any smooth hypersurface $Z$ of $X$,
Malgrange and Kashiwara defined a filtration along $Z$ for $\cD_X$
and the notion of a good filtration for $\cM$. Given a set of
transverse smooth hypersurfaces $Z_1, \ldots, Z_k$, Sabbah
considered in \cite{Sabbah} multifiltrations of $\cM$ indexed by
$k$-uples of relative integers. To be precise he dealt with
linear combinations over ${\bf Q}^+$ of the filtrations $V^{(j)}$
along each hypersurface $Z_j$ and with refinements $V^{\Gamma}$ of
the original multifiltration associated with each rational
polyedral simplicial cone in the positive quadrant of $({\bf
Q}^k)^{\star}$. The original multifiltration is the one which
corresponds to the case $\Gamma ={\bf N}^k$.

The aim of this paper is to clarify the flatness properties which
appear in \cite{Sabbah}, namely to prove the existence of a fan
$\cE$, such that for any cone $\Gamma $ in this fan, the Rees
module of the filtration $V^{\Gamma}$ is flat over the affine ring
$A_{\Gamma}$ of the toric blowing-up of ${\CC}^k$ associated with
the fan $\cE$. Such a fan is called an adapted fan in
\cite{Sabbah}. The reason for this clarification is that  the proof in  \cite{Sabbah}
depends on  the appendix \cite{Sab-Cas} in which the main tool is
a  division theorem in Rees rings of differential operators which is not correct
as stated. Indeed, the infinite process that its proof suggests, would require
monomials of unbounded degrees in the differential variables of the ring of operators.

One of the main consequences of the existence of an adapted fan as
developed in \cite{Sabbah} is that we thus complete the proof of
the existence of non trivial functional equations of
Bernstein-Sato type for a $k$--uple of functions, following the
argument of Sabbah in \cite{Sabbah}. It should be emphasized here
that this problem involves a  $\cD _{X\times \CC ^k}$-module
naturally associated with a $k$-uple of functions on $X$, and that
in this situation a multifiltration, along the transverse
hypersurfaces $t_j=0$ due to the factor $\CC ^k$, appears in a
natural way.

We must note here that the proof of the existence of
Bernstein-Sato equations has already been completed in Bahloul's
paper \cite{bahloul}, by a different method which avoids the
reference to a flatness property involved in the notion of an
adapted fan. Bahloul uses instead the analytic standard fan as
defined in \cite{assi-castro-granger-jpaa-2}, and described also
in \cite{bahloul-takayama}. It is therefore not completely
surprising that the adapted fan wanted in \cite{Sabbah} turns out in
fact to be the analytic standard  fan. This emphasizes the
interest of Bahloul's proof which also has the advantage of being
constructive. This constructiveness is valuable in an
algebraic setting also. In both proofs, Bahloul's  and
Sabbah's , the latter which is completed by this paper, the main step
is the proof
of the goodness of the so-called saturated filtration. The details
are given in section \ref{bfunctions}.

The starting point of the proof of our main theorem (Theorem
\ref{mainresult}) is a criterion of flatness by M. Herrmann and U.
Orbanz (see \cite{Herrmann-Orbanz}), for graded modules over
graded rings, where the grading is indexed by an arbitrary
commutative group. In the statement of this result there is no
reference to any finiteness property. In our case the indexing
group will be $\ZZ ^k$, and the ring a conical subring of the ring
of Laurent polynomials in $k$ variables with the obvious
multigrading by monomials. The main ingredient of the proof of
Theorem \ref{mainresult} is then the existence of a simultaneous
$L$--standard basis of a submodule $\cN\subset \cD^r$ for  all $L$
in a cone of the analytic standard  fan of $\cN$ (see
\cite{assi-castro-granger-jpaa-2}).



The plan of the paper is as follows. In section \ref{multif_rees},
we recall with more details what $V$-multifiltrations, and
Rees modules are, and the refinements of these multifiltrations with
respect to a rational polyhedral simplicial cone. We end this
section by defining the fiber at zero of these Rees modules seen
as modules  over the ring of an affine toric blowing up. We aim to
prove that this is the fiber of a flat deformation if the cone is
taken in the analytic standard fan.

In section 3  we recall the notion of an analytic standard fan as
developed in  \cite{assi-castro-granger-jpaa-2}, and we sketch the
easy generalisation which we need from the cyclic case treated in
\cite{assi-castro-granger-jpaa-2} to the general case.

In section 4 we finally prove that the analytic standard fan is an
adapted fan in the sense of \cite{Sabbah}.

\vspace{.6cm}

\noindent Acknowledgements.- During the preparation of this paper
both authors have been partially supported by Acci\'{o}n
Integrada-Picasso HF2004-0117. First author has been  also
partially supported by MTM2004-01165. Both authors are grateful to
the D\'epartement de Math\'ematiques et Applications of the
\'Ecole Normale Sup\'erieure (Paris) where the first was invited
during the preparation of the final part of this work. They also
thank Jenifer Granger for her proofreading.



\section{Multifiltrations, Rees  rings and Rees  modules}\label{multif_rees}

Let us denote by $\cO=\CC\{x_1,\ldots,x_n\}$ the complex
convergent power series  ring and by $\cD$ the ring of germs of
linear differential operators with holomorphic coefficients (i.e.
$\cD = \cO[\partial]= \cO[\partial_1,\ldots,\partial_n]$ where
$\partial_i$ is the partial derivative with respect to $x_i$ and
the product in $\cD$ is defined by the Leibnitz's rule:
$\partial_i a=a\partial_i +
\partial_i(a)$, for each $a\in \cO$).

An operator $P$ in $\cD$ can be written as
$$P=\sum_{\beta\in \NN^n}
p_\beta(x)\partial^\beta = \sum_{\alpha,\beta\in \NN^n}
p_{\alpha\beta}x^\alpha\partial^\beta$$ where the first sum is
finite, $\alpha = (\alpha_1,\ldots,\alpha_n),
\beta=(\beta_1,\ldots,\beta_n)$, $x^\alpha=x_1^{\alpha_1}\cdots
x_n^{\alpha_n}$, $\partial^\beta=\partial_1^{\beta_1}\cdots
\partial_n^{\beta_n}$, $p_\beta(x) \in \cO$ and $p_{\alpha\beta}\in \CC$.

For each $i=1,\ldots, n$, let us remember that the $V$-filtration
on $\cD$, with respect to the hypersurface $x_i=0$, was
defined by Malgrange and Kashiwara as:
$$ V^{(i)}_\ell(\cD)= V^{(i)}_\ell = \{P=\sum_{\alpha,\beta\in \NN^n}
p_{\alpha\beta}x^\alpha\partial^\beta \in \cD \, \vert \,
\beta_i-\alpha_i \leq \ell \} $$ for each $\ell\in\ZZ$. The family
$(V^{(i)}_\ell)_{\ell \in \ZZ}$ is an increasing exhaustive
filtration on the ring $\cD$. For $i=1$, the associated graded
ring $$\gr^{V^{(1)}}(\cD) = \bigoplus_\ell
\frac{V^{(1)}_\ell}{V^{(1)}_{\ell -1}}$$ is isomorphic to the ring
$\CC\{x_2,\ldots,x_n\}[x_1,\partial_1,\partial_2,\ldots,\partial_n]$,
graded by the so-called $V^{(1)}$--graduation, where the
homogeneous elements of $V^{(1)}$--degree $\ell$ are
$$\sum_{\alpha, \beta \in \NN^n; \, \beta_1-\alpha_1=\ell} p_{\alpha
\beta} x^\alpha
\partial^\beta .$$ We have similar descriptions for $i=2,\ldots,k$.

We can also consider a $V^{(i)}$-filtration on the free module
$\cD^r$ just by defining $V^{(i)}_\ell (\cD^r)= (V_\ell^{(i)}
\cD)^r$ and more generally, for any vector
$\un{m}=(m_1,\ldots,m_r)\in \ZZ^r$, one can define the shifted
filtration $V^{(i)}[\un{m}]_\ell (\cD^r) = \oplus_{j=1}^{r}
V^{(i)}_{\ell-m_j}\cD$. Such a filtered module is called a
$V^{(i)}$-filtered free module. All the $\cD$-modules considered
will be left modules.

\begin{definicion}\label{goodfiltration}
Let $\cM$ be a $\cD$-module.  We say that a filtration $U^{(i)}(\cM)$ indexed by ${\bf Z}$ is
a good $V^{(i)}$-filtration if there exists a presentation $\cM
=\frac{\cD ^r}{\cN}$ as a quotient by a left submodule $\cN$ of $\cD ^r$, and a weight vector
$\un{m}$ such that $U^{(i)}_\ell (\cM)=\pi (V^{(i)}[\un{m}]_\ell (\cD^r)) $ where $\pi $ is
the projection $\cD ^r \to \cM$.
\end{definicion}

\subsection{$V$-multifiltrations and $V^\Gamma$-multifiltrations}\label{Vfiltrations}
Let us fix an integer  $k$ such that $1\leq k \leq n$.

For each $s=(s_1,\ldots,s_k)\in \ZZ^k$, we shall denote
$V_s(\cD)= \bigcap_{i=1}^k V^{(i)}_{s_i}(\cD)$. The family
$\{V_s(\cD)\}_{s\in \ZZ^k}$ defines a multi-filtration on the ring
$\cD$. To simplify we shall say that $V_\bullet(\cD)$ is a
$k$--filtration on  $\cD$ or even, if no confusion is possible, a
filtration on $\cD$.


Let us consider a rational simplicial cone $\Gamma$ in the first
quadrant of the dual space $(\QQ^k)^*= Hom_\QQ(\QQ^{k},\QQ)$. We
denote by $\check{\Gamma}$ the dual cone of $\Gamma$, i.e.
$$\check{\Gamma} = \{a\in \QQ^{k}\,\vert \, \gamma(a) \geq 0,
\forall \gamma \in \Gamma\}.$$
We associate
with such a cone $\Gamma$ the affine variety, denoted by $S_\Gamma$,
with coordinate ring equal to $\CC[\check{\Gamma}\cap \ZZ^{k}]$.
We will denote $A_\Gamma =\CC[\check{\Gamma}\cap \ZZ^k]$ and
$A=\CC[\NN^k]$. We denote by $\cL(\Gamma)$ the set of primitive
elements in the 1-skeleton of $\Gamma$.

The multifiltration $V^\Gamma$ on $\cD$ is defined as follows: For
each $s\in \ZZ^k$ we define $$V^\Gamma_s(\cD) = \sum_{\sigma\in
\ZZ^{k}\,\vert\, L(\sigma)\leq L(s); \forall L\in \cL(\Gamma)}
V_\sigma(\cD).$$ Notice that the sum is indexed by $\sigma \in
s-\check{\Gamma}$ and that we have the inclusion $V_s(\cD) \subset
V_s^\Gamma(\cD)$. The family $V^\Gamma_\bullet(\cD)$ is a
multifiltration of the ring $\cD$, indexed by $s\in \ZZ ^k$. This
means that~:
$$V^\Gamma_{s}(\cD).V^\Gamma_{s'}(\cD)\subset V^\Gamma_{s+s'}(\cD)
\mbox{ and } \bigcup V^\Gamma_s(\cD)=\cD.$$

We may define, in a similar way to the case of one filtration, the notion of a free multi-filtered
module and that of a good multifiltration of a  finitely generated $\cD$-module   $\cM$.
For that purpose we chose   a shift
multivector $\un {n}=(n^{(1)},\cdots  ,n^{(r)})\in ( \ZZ ^k)^r$ called also a shift matrix, with columns
$n^{(i)}\in \ZZ ^k$, and a presentation $\cM =\frac{\cD ^r}{\cN}$ of $\cM$.

\begin{definicion}
The multifiltered free module associated with $\un{n}$ is the module  $\cD ^r$
endowed with the multifiltration indexed by  $s\in \ZZ^k$, and defined as:
$$V[\un {n}] _s(\cD^r)= \bigoplus _{i=1}^r  V_{s- n ^{(i)}}(\cD  )$$
\end{definicion}

\begin{definicion} A good multifiltration  of $\cM$, is a filtration indexed by $s\in \ZZ^k$, of the type
$$U _s(\cM)= \pi (V[\un {n}]_s(\cD ^r))=\frac{V[\un{n}]_s(\cD ^r) +
\cN }{\cN}$$
for some presentation $\pi :\cD ^r \to \cM$ of $\cM.$
\end{definicion}

In the obvious sense these multifiltrations  are compatible with
the multifiltered structure on the ring $\cD$. Remark that the
$i$-th generator $e_i$ of  $\cD ^r$ is then of multidegree $n
^{(i)}\in \ZZ^k$. We may also endow $\cN $ with the induced
filtration $U_s(\cN)=V[\un{n}]_s(\cD ^r)\cap \cN$, so that we also
have $$U _s(\cM)=\frac{V[\un{n}]_s(\cD ^r) }{U_s(\cN)}.$$


We may observe that the multifiltration on the free module $\cD ^r$ is
defined as the intersection of the $V^{(i)}$--filtration with
respect to the row vectors of $\un{n}$,  $\un{n} _i=(n^{(1)}_i
,\cdots , n^{(r)}_i)\in {\bf Z}^r$, that is
$$V[\un {n}] _s(\cD^r)=V ^{(1)}[\un{n} _1]_{s_1}(\cD ^r)\cap \cdots
 \cap V ^{(k)}[\un{n} _k]_{s_k}(\cD ^r),$$
but the analogue for $\cM$, with respect to the good $V^{(i)}$-filtrations as defined
in definition \ref{goodfiltration} is not true since the inclusion:
$$U_s(\cM) = \frac{V ^{(1)}[\un{n} _1]_{s_1}(\cD ^r)\cap \cdots
 \cap V ^{(k)}[\un{n} _k]_{s_k}(\cD ^r) + \cN}{ \cN}  \subset
\bigcap _{i=1}^k \frac{ V ^{(i)}[\un{n} _i]_{s_i}(\cD ^r) +  \cN}{ \cN}
=\bigcap _{i=1}^k U^{(i)}_{s_i}(\cM)$$
may be strict.

For each good filtration there is an associated $\Gamma
$-filtration compatible with the multifiltration
$V^\Gamma_\bullet(\cD)$ on $\cD $~: \\  For each $s \in \ZZ^k$ let
us consider $$U^\Gamma_s(\cM )= \sum_{\sigma\in \ZZ^{k}\,\vert\,
L(\sigma)\leq L(s); \forall L\in \cL(\Gamma)} U_\sigma (\cM).$$
The multifiltration $U^\Gamma_\bullet(\cM)$ is a good filtration
with respect to $V^\Gamma_\bullet(\cD)$.

The goodness means here that we can verify, with the presentation
of $\cM $ as above, that $U^\Gamma_s(\cM)=\pi (V[\un {n}]^\Gamma
_s(\cD^r)) $, is still the quotient of a filtration on the free
module $\cD^r$ which is a direct sum of convenient shifts of the
filtration $V_\bullet^\Gamma $ on the ring $\cD$. More precisely
the involved filtration $V[\un {n}]_\bullet^\Gamma $ is defined by
$$V[\un {n}]^\Gamma _s(\cD^r)=
\sum_{\sigma\in \ZZ^{k}\,\vert\, L(\sigma)\leq L(s); \forall L\in
\cL(\Gamma)}V[\un {n}] _{\sigma }(\cD^r) (\,=  \bigoplus _{i=1}^r
( V^\Gamma _{s- n ^{(i)}}(\cD  ))$$ and it remains to be remarked that:
$$U_s^{\Gamma }(\cM) =\sum_{\sigma\in \ZZ^{k}\,\vert\, L(\sigma)\leq L(s); \forall L\in
\cL(\Gamma)}U _{\sigma }(\cM)=\sum_{\sigma\in \ZZ^{k}\,\vert\,
L(\sigma)\leq L(s); \forall L\in \cL(\Gamma)}\frac{V[\un
{n}]_{\sigma }(\cD ^r) + \cN }{\cN} =\frac{V[\un {n}]^\Gamma
_s(\cD^r) + \cN }{\cN}$$ so that the goodness of the filtration $U
_{\bullet }(\cM )$ implies the goodness of $U _\bullet ^{\Gamma}
(\cM )$.


\subsection{Rees  rings and Rees  modules. }\label{Reesrings}


\subsubsection {Definition of Rees rings $\cR_V(\cD)$, $\cR^\Gamma(\cD)$ and of related  Rees
modules}\label{R_VD} The Rees  ring associated with the
$V$-multifiltration on $\cD$ is defined by: $$ \cR_V(\cD) =
\bigoplus_{s\in \ZZ^k} V_s(\cD)u^s$$ where $u=(u_1,\ldots,u_k)$
are variables and the product in the Rees  ring is induced by the
natural product in the Laurent polynomial ring $\cD[u^{\pm 1}] =
\cD[u_1,u_1^{-1},\ldots,u_k,u_k^{-1}]$.

By definition the Rees  ring $\cR_V(\cD)$ is a graded
$\CC$-algebra with values group $\ZZ^k$, whose homogeneous
elements are $Pu^s$ for $P\in V_s(\cD)$ and $s\in \ZZ^k$.


Similarly given a $\cD$-module $\cM= \cD
^r/\cN$ and a shift matrix $\un{n}$, we define from the good filtrations
$V[\un{n}]_s(\cD ^r)$, and $U_s(\cM) $, the Rees modules
$$\cR_{V[\un{n}]}(\cD^r) =
\oplus_{s\in \ZZ^k} V[\un{n}]_s(\cD^r)u^s$$
$$\cR_U(\cM)= \bigoplus_{s\in \ZZ^k} U_s(\cM)u^s.$$
Both have a  natural structure of mutigraded left $\cR_V(\cD)$-module
and we shall consider $\cR_U(\cM)$ as a sub-group of $\cM[u^{\pm
1}]$, the Laurent polynomials with coefficients in $\cM$.

% As a $\cR_V(\cD)$--module
%the Rees  module  $\cR_{V[\un{n}]}(\cD^r)$ is isomorphic to the
%free module $(\cR_V(\cD))^r$. As a multigraded module over the
%multigraded ring $\cR_V(\cD) $, its graded piece indexed by $s\in
%\ZZ ^k$ is
%$$V[\un {n}] _s(\cD ^r)u^s= (\bigoplus _{i=1}^r  V _{s- n ^{(i)}}(\cD  ))u^s
%= \bigoplus _{i=1}^r ( \cap _{j=1}^kV _{s_j- n ^{(i)}_j}(\cD
%))u^s$$


As $\cN $ is a left submodule of  $\cD ^r$, we can also consider
on $\cN $ the induced $V$-filtration:
$$V_s(\cN ):=\cN \cap V_s[\un{n}](\cD ^r)$$ for each $s\in \ZZ^k$ as defined before (see \ref{Vfiltrations}).
The abelian group $$\cR_V(\cN ):=\bigoplus_{s\in \ZZ^k} V_s(\cN
)u^s$$ is in fact an homogeneous submodule of $\cR_{V[\un{n}]}(\cD
^r)$. The $\cR_{V}(\cD)$--module $\cR_U(\cM)$ is naturally
isomorphic to the quotient $\cR_{V[\un{n}]}(\cD ^r)/\cR_V(\cN)$.
\begin{definicion}
We call the module $\cR_U(\cM)$ the Rees module associated with
the filtration $U_\bullet(\cM)$.
\end{definicion}




Let us now consider the filtration
$V^\Gamma_\bullet(\cD)$ as in \ref{Vfiltrations}. Then the Rees
ring associated with this filtration is defined in a similar way as
$$\cR^\Gamma(\cD) =
\bigoplus_{s\in \ZZ^ k} V_s^\Gamma(\cD) u^s \subset \cD[u^{\pm
1}]$$ the product being induced by the one of $\cD[u^{\pm 1}]$.

If $\cM=\frac{\cD ^r}{\cN}$ is a finitely
generated $\cD$-module and $U_\bullet (\cM)$ is a good filtration
on $\cM$ with respect to $V_\bullet(\cD)$ then, for any cone
$\Gamma$ as in \ref{Vfiltrations}, the abelian group
$$\cR^\Gamma(\cM) = \bigoplus_{s\in \ZZ^k} U^\Gamma_s(\cM) u^s
\subset \cM[u^{\pm 1}]$$ is a left graded
$\cR^{\Gamma}(\cD)$-module.
\begin{definicion}
We call the module $\cR^\Gamma(\cM)$
the Rees module associated with the filtration $U_\bullet(\cM)$
and the cone $\Gamma$.
\end{definicion}


The fact that a filtration is good in the sense of previous
section \ref{Vfiltrations} is then equivalent to the fact that the
Rees module of $\cM$ is finitely generated over $\cR_V(\cD)$.
Similarly the goodness of $U^\Gamma_\bullet(\cM)$ is equivalent to
the fact that the Rees module $\cR ^{\Gamma}{(\cM)}$ as defined
above, is finitely generated over $\cR^{\Gamma}{(\cD)}$. Since
$\cR^{\Gamma}{\cD}$ is a Noetherian ring, this implies in
particular an Artin type property: If $\cM' \subset \cM$ is a
submodule of a $\cD$-module $\cM$ endowed with a good filtration
$U_\bullet(\cM)$ then the induced filtration on $\cM'$, namely
$U^\Gamma_\bullet(\cM)\cap \cM'$ is good.

The ring $\cR^{\Gamma}{(\cD)}$ contains as a subring the
$\CC$--algebra $A_\Gamma = \CC[\check{\Gamma}\cap \ZZ^k]\subset
\CC [u,u^{-1}]$, all these rings being included in
$\cD[u,u^{-1}]$.


\begin{teorema}\label{forme_faible} There is a fan $\cE$ in $(\QQ^k)^*_+$ such that for each
cone $\Gamma\in\cE$ the Rees module $\cR^\Gamma(\cM)$ is flat over
$A_\Gamma$.
\end{teorema}

We will prove a more precise form of this  result we in
section \ref{flatness}, see theorem \ref{mainresult}, after having
recalled in section \ref{analytic_standard_fan} the notion of an
analytic standard fan.


\subsubsection{The fiber at $0$}\label{fibres} Let us denote by $\mathfrak m$ the ideal of $\cR_{V}(\cD)$
generated by $(u_1,\ldots,u_k)$.  This is a two sided ideal since
it is generated by central elements. The fiber at zero of
$\cR_{V}(\cD)$ (resp. of $\cR_{V[\un{n}]}(\cD ^r)$)  is by
definition the quotient ring (resp. the quotient module),
$$\frac{\cR_{V}(\cD)}{\mathfrak m} \,\,(\mbox{resp. }
\frac{\cR_{V}[\un{n}](\cD ^r)}{\mathfrak m.\cR_{V}[\un{n}](\cD
^r)}).$$ More generally the fiber at zero of
$\cR_{U}(\cM)=\frac{\cR_{V}[\un{n}](\cD ^r)}{\cR_{V}(\cN)}$ is the
quotient $$\frac{\cR_U(\cM)}{{\mathfrak m }{\cR_U(\cM)}}$$ which
is naturally isomorphic to the quotient
$$\frac{\cR_{V[\un{n}]}(\cD^r)}{\cR_{V}(\cN )+\mathfrak m.\cR_{V[\un{n}]}(\cD^r )}.$$

Notice that the fiber at 0 of the module $\cR_U(\cM)$ can  be zero
for a non-zero $\cR_U(\cM)$, as shown in the following example.
Let us denote by $I$ the principal ideal of $\cD$ generated by the
differential operator $P=1+x_1^2\partial_1$ and let us suppose
$k\geq 1$. Then $\cR_{V}(I)$ is the principal ideal of
$\cR_{V}(\cD)$ generated by $Pu_1^0=1+(x_1^2\partial
_{1}u_1^{-1})u_1$ and the fiber at zero of $\cR_{V}(\cD/I)$ is
then zero since $\cR_{V}(I )+\mathfrak m.\cR_{V}(\cD)$ contains
$P-(x_1^2\partial_{1}u_1^{-1})u_1=1$, so that it is equal to
$\cR_{V}(\cD).$

Similarly for any $k$-dimensional cone $\Gamma$ the $\CC$--algebra
$A_\Gamma = \CC[\check{\Gamma}\cap \ZZ^k]$ has a maximal ideal
$$\mathfrak m^{\Gamma }= \CC[\check{\Gamma}\cap \ZZ^k\setminus\{ 0\}]$$
because $\check{\Gamma}$ is strictly convex. Then we define the
fiber at zero of the Rees module $\cR^\Gamma(\cM)$ as
$\frac{\cR^\Gamma(\cM)}{\mathfrak m^{\Gamma }\cR^\Gamma(\cM)}.$ As
is explained in \cite{Sabbah}, this fiber is a module over the
ring $\frac{\cR^\Gamma(\cD)}{\mathfrak m^{\Gamma
}\cR^\Gamma(\cD)}\simeq gr ^{\Gamma}\cD$.


\vspace{1cm}

\subsubsection{Description of  Rees rings $\cR_V(\cD)$ and $\cR^\Gamma(\cD)$}\label{RgammaD}

It is useful to describe the Rees  ring $\cR_{V}=\cR_V(\cD)$
(resp. $\cR^\Gamma(\cD)$) as subrings   of the  ring of relative
differential operators $\cD_{\CC^n\times \CC^k/\CC^k}$ (resp
$\cD_{\CC^n\times S_{\Gamma}/S_{\Gamma}}$). This is nothing but an
explicit version of the interpretation of a Rees ring as a ring of
relative differential operators on the deformation of $Y\times
0_{\Gamma}$ to its normal cone in $\CC^n\times S_{\Gamma},$ see
\cite {Sabbah}. We denote by $S_\Gamma$ the algebraic variety
associated to the ring $A_\Gamma$.

Let us define first
$$\cA = \CC\{X'U,X''\}[X',U,\Delta]=
\CC\{X_1U_1,\ldots,X_kU_k,X_{k+1},\ldots,X_n\}[X',U,\Delta]$$
where $U=(U_1,\ldots,U_k)$, $X'=(X_1,\ldots,X_k)$,
$X''=(X_{k+1},\ldots,X_{n})$, $X=(X_1,\ldots,X_n)$,
$\Delta=(\Delta_1,\ldots,\Delta_n)$ are new variables satisfying
the following relations, for $i=1,\ldots,n$:
$$\Delta_i X_i = X_i \Delta_i + 1$$ the other relations being
trivial.

The ring $\cA$ is graded with $\ZZ^k$ as a values group (we will say
that $\cA$ is a $\ZZ^k$--graded ring) , namely we have $$\cA =
\bigoplus_{s\in \ZZ^k} \cA_s$$ where $\cA_s$ is the set $$ \cA_s =
\{\sum_{\alpha,\beta\in \NN^n; \sigma \in \NN^k} f_{\alpha\beta
\sigma} X^\alpha \Delta^\beta U^\sigma\in \cA \,\vert \,
f_{\alpha\beta\sigma}\in \CC\{X'U,X''\}, \sigma_i+\beta_i-\alpha_i
= s_i; i=1,\ldots,k\}.$$


\begin{propo}
There is an isomorphism  of $\ZZ^k$--graded rings
$$i=i_{V} : \cR_{V}(\cD) \rightarrow \cA$$ defined by:

\begin{itemize}
 \item $i(u_j)=U_j$ for $j=1,\ldots,k$.
\item $i(x_ju_j^{-1}) = X_j$ for $j=1,\ldots,k$ \; \mbox{ and}
\quad
$i(x_j)=X_j$ for $j=k+1,\ldots,n$.
 \item $i(\partial_ju_j)=
\Delta_j$ for $j=1,\ldots,k$
 \; \;  \mbox{ and} \quad $i(\partial_j)=
{\Delta_j}$ for $j=k+1,\ldots,n$.
\end{itemize}
\end{propo}

\begin{prueba}{.}
It is clear that $i$ is injective and that, by the formula
$i(x^{\alpha}\partial ^{\beta}u^s)=X^{\alpha}\Delta ^{\beta}(\Pi
_{i=1}^kU_i^{s_i+\alpha_i-\beta_i}),$
  $i(V_s(\cD)u^s)= \cA_s$ for all $s\in \ZZ^k$ .
\end{prueba}



Let us give now the same description for the ring $\cR^\Gamma(\cD)$. We shall
do it in the only case of interest for us,
when the cone $\Gamma$ is basic, which means that if it is defined by $k$
independent linear forms $\{L_1,\ldots,L_k\}$ generating the
lattice $\ZZ^k$. For the sake of simplicity
let us suppose $k=n$.

In this situation we are going to make a change
of variables in order to write $A_\Gamma $  as a polynomial ring.

For any $i$ we can write $L_i=(\ell_{i1},\ldots,\ell_{ik})$ and we
can suppose by a suitable ordering of the linear forms $L_i$  that
the determinant of the matrix $\cL=(\ell_{ij})$ equals  1.

Let us write $\cL'$ for the inverse matrix of $\cL$, then the
columns $\{C _1,\ldots,C _k\}$ of $\cL'$ form a basis of the dual
cone $\check{\Gamma}$.

 The family $u^{C _1},\ldots,u^{C _k}$ generates the
$\CC$--algebra $A_\Gamma = \CC[\check{\Gamma}\cap \ZZ^k]$. Here we
write $u^{C_j}=u_1^{c_{1j}}\cdots u_k^{c_{kj}}$ where the
$c_{ij}$'s are the entries of the column $C_j$.

We will write $W_i=U^{C_i}$ for $i=1,\ldots,k$. Similarly, we have
$U_j = W^{C_j(\cL)}$ for $j=1,\ldots,k$ where $C_j(\cL)$ is the
$j-th$ column of the matrix $\cL$. In section \ref{flatness} we
will only use the column $C_j=C_j(\cL')$. \\
Let us remark here
that if $k<n$ then we can still define $W$, the relationship
between $W$ and $U$ being exactly the same.

\begin{propo}
The ring $\cR^\Gamma(\cD)$ is isomorphic to the subring of
$\cA'=\CC\{X,W\}[\Delta]$, which contains all polynomials and in
which only convergent power series with respect to the
family of monomials $X_iU_i=X_iW^{C_i(\cL)}$ for $i=1,\cdots k$,
and $X_{k+1},\cdots , X_n$, in the general case $k\leq n$  are allowed.
\end{propo}
\begin{prueba}{.}
A monomial $X^\alpha \Delta^\beta U^{s-\beta+\alpha}$ can be
written in terms of $X,\Delta,W$ as $X^\alpha \Delta^\beta
W^{\cL(s-\beta+\alpha)}.$ All that  remains to be done is to enumerate
the monomials derived from
 the power series variables $x_1,\cdots , x_n$
\end{prueba}

 The ring $A_{\Gamma}$ is identified to the subring
$\CC[W](=\CC[W_1,\ldots,W_k]) \subset \cR^\Gamma(\cD)$ and this inclusion is flat. We
shall prove this statement in \ref{RGammaDplat}.


The fiber of $\cR^\Gamma(\cD)$ at the origin is by definition
$\cR^\Gamma(\cD)\otimes_{\CC[W]} \frac{\CC[W]}{(W)}$ which is
isomorphic to a Weyl algebra, namely the Weyl algebra
$\CC[X,\Delta]$. This Weyl algebra is endowed with a
$\ZZ^{k}$--graduation by $weight(X_i)=-\epsilon_i$ and
$weight(\Delta_i)=\epsilon_i$ for $i=1,\ldots,k$ where
$\epsilon_i$ is the vector in $\ZZ^{k}$ whose $j-th$ coordinate is
$\delta_{ij}$.




\subsection{Multifiltrations and Bernstein-Sato functional
equations}\label{bfunctions}

Let us detail the functional equation problem raised in the
introduction. It has already been remarked that it gives rise to a
situation where a multifiltration along transverse hypersurfaces
comes out in a natural way.  These equations are of the type
$$P(\lambda)f^{\lambda }=b(\lambda )f_1^{\lambda  _1+1}\cdots
f_k^{\lambda  _k+1}$$ where $\lambda =(\lambda _1,\cdots , \lambda
_k)$ is a $k$-uple of indeterminates, and $f=(f_1,\cdots , f_k)$
is a $k$-uple of analytic functions on $X$. They  are naturally
written in the module $\cO _X\big[ \lambda _1,\cdots , \lambda
_k,\frac{1}{f_1 \cdots f_k} \big]f^{\lambda }$, endowed with a
structure of a $\cD _X[\lambda ]$-module, which can be extended in
a way discovered by B. Malgrange to a $\cD _{X\times \CC
^k}$-module structure with an action of the $2k$ variables $t_j,
\partial _{t_j}$such that $\lambda _j=-\partial _{t_j}t_j.$ The module
$\cM$ that we have then to consider is the module generated over
the ring $\cD _{X\times \CC ^k}$ by $f^{\lambda }$, with its
naturally defined multifiltration $V_{\bullet}(\cD _{X\times \CC
^k})\cdot f^{\lambda }$ along the hypersurfaces $t_1=0, \cdots ,
t_k=0$. The proof of Sabbah in \cite{Sabbah} can be sketched as
follows: let us define for any linear  form $L\in (\QQ
^k)^{\star}$ with positive rational coefficients $\ell _j \geq 0$,
a filtration of $U^L_{\bullet}(\cM)$, associated with the linear
combination $\sum \ell _jV^{(j)}$ of the basic $V$- filtrations
(see notations at \ref{multif_rees}). Precisely we set (see also
the notation \ref{filtrationVL} below):
$$
V_{\bullet}^{L}(\cD _{X\times \CC ^k})=\{ P=\sum_{\mu,\nu \in
\NN^k} P_{\mu\nu}(x,\partial _x)t^{\mu} \partial _t^{\nu} \,, \;
\sum \ell _i(\nu _i-\mu _i)\leq \bullet \}$$
$$ U^L_{\bullet}(\cM)=V_{\bullet}^{L}(\cD _{X\times \CC ^k})\cdot
f^{\lambda }$$
 Then the existence of a functional equation, in
which $b(\lambda )$ is a product of affine forms $L(\lambda )+c$
comes out from the following three steps:

1) There are Bernstein-Sato polynomials $b_L(\lambda )$, relative
to each $L$, with functional  equations
$$b_L(\lambda )f^{\lambda }\in V_{<0}^L(\cD _{X\times \CC
^k})\cdot f^{\lambda }.$$

2) The saturated filtration
$$\ov{U_s}(\cM)=\bigcap _{L\in (\QQ ^k)^{\star}}U_{L(s)}^L(\cM)$$
can be defined by using only a {\em finite number} of fixed linear
forms.

3) The saturated filtration is good which is equivalent to the
existence  of a $k$-uple of integers $\kappa $, such that
$$\forall s\in \NN ^k,  \;\; U_s(\cM)\subset \ov{U_s}(\cM)\subset U_{s+\kappa }(\cM).$$


It is step 2)  which, in the proof in \cite{Sabbah}, makes an essential
use of  the notion of an adapted
fan, whose existence is proved in this paper, see theorem
\ref{forme_faible}. In  Bahloul's paper
\cite{bahloul}, this step is made by constructive methods which do not use the
flatness property.

\section{The analytic standard fan}\label{analytic_standard_fan}

In this section we will summarize the main results of
\cite{assi-castro-granger-jpaa-2}. Since these results are only
given for a module $\cD/I$  over $\cD$ which is a quotient by an
ideal we will then show briefly how to adapt them to a module of
the type $\frac{\cD ^r}{\cN}$. Let us remark that the fan used in
this paper is obtained by a restriction of the fan in
\cite{assi-castro-granger-jpaa-2} to a subset of linear forms of
the set $\cU$ defined below, namely the linear combinations of the
filtrations $V^{(i)}$.


Let $\cU$ be the set of linear forms $\Lambda: \RR^{2n}
\rightarrow \RR$, $\Lambda(\alpha,\beta)=\sum_{i=1}^ n e_i
\alpha_i + \sum_{i=1}^ n f_i \beta_i$ with $e_i+f_i \geq 0$ and
$e_i \leq 0$ for $i=1,\ldots,n$. If $$P=\sum_{\alpha \beta}
p_{\alpha \beta} x^\alpha
\partial^\beta$$ is an element in $\cD$ we define $\ord^\Lambda(P)$ --the
$\Lambda$-order of $P$-- to be the maximal value of
$\Lambda(\alpha,\beta)$ for $\alpha,\beta$ such that $p_{\alpha
\beta} \not= 0$.

The $\Lambda$-filtration $F_{\Lambda,\bullet} (\cD)$ is defined by
$$F_{\Lambda,\ell} = F_{\Lambda,\ell}(\cD) = \{P \in \cD \,\vert \, \ord^\Lambda(P)\leq
\ell\}$$ for any $\ell \in \RR$. We will write
$F_{\Lambda,<\ell}(\cD):= \{P \in \cD \,\vert \, \ord^\Lambda(P) <
\ell\}$. If $e_i=0, f_i=1$ for $i=1,\ldots,n$ the corresponding
$\Lambda$--filtration is nothing but the usual filtration by the
order of differential operators. We shall denote it simply
$F_\bullet(\cD)$.

Let us recall that we have fixed $k\leq n$. For each linear form
$L\in (\QQ^k)^*_+$ (i.e. the coefficients of $L$ are non-negative)
we denote by $\widetilde{L}$ the linear form on $\RR^{2n}$ defined
by $\widetilde{L}(\alpha,\beta)=L(\beta)-L(\alpha)$ where
$L(\alpha_1,\ldots,\alpha_n)=L(\alpha_1,\ldots,\alpha_k)$.

\begin{notacion}\label{filtrationVL}
We define the filtration $V_\bullet ^L$ on $\cD$, indexed by the
set of values $L(\ZZ^k)$, as:
$$V^L_{\ell}(\cD)=F_{\widetilde{L},\ell}(\cD)=\{ P\in \cD \,\vert \, \ord^{\widetilde{L}}(P)
\leq  \ell\}$$
\end{notacion}

% With the notation as in \ref{Vfiltrations} for a given linear form
% $L\in (\QQ^k)_+^*$, the multifiltration $V_\bullet^{\{L\}}(\cD)$,
% for $\{L\}$ being the degenerated cone $\QQ_+L$,  is directly
% related to this filtration
% $V^L_\bullet(\cD)=F_{\widetilde{L},\bullet}(\cD).$ Precisely, for
% any $s\in \ZZ^k$ the link is the following:
% $$V_s^{\{L\}}(\cD) = V^L_{L(s)}(\cD).$$

Let us fix $L\in (\QQ^k)^*_+$.  The graded ring  associated to the
filtration $V^L=F_{\widetilde{L}}$ on $\cD$,
is by definition
$$\gr^{\widetilde{L}}(\cD) = \bigoplus_{\ell \in \widetilde{L}(\ZZ^{2n})}
\frac{F_{\widetilde{L},\ell}}{F_{\widetilde{L},<\ell}}.$$ If no
confusion is possible we shall write simply $\ord^L$ and
$\gr^L(\cD)$ instead of $\ord^{\widetilde{L}}$ and
$\gr^{\widetilde{L}}(\cD)$.

The graded ring $\gr^L(\cD)$ is a ring of differential operators
and its structure is the following: suppose the coefficients of
the form $L$ are $(e_1,\ldots, e_k)$ and suppose we also have
ordered the variables to have $e_i>0$ for $1 \leq i \leq \ell$ for
some $\ell \leq k$.

Then the graded ring $\gr^L(\cD)$ is isomorphic to the ring
$$\CC\{x_{\ell+1},\ldots,x_n\}[x_1,\ldots, x_{\ell},
\partial_1,\ldots,\partial_{n}].
$$
In this ring the graduation is induced by the weights
$\weight(x_i)=-e_i, \,\weight(\partial_i)=e_i$ for $1\leq i \leq
\ell$ and $\weight(x_j)=\weight(\partial_j)=0$ otherwise. There is
only a finite number of types of these  rings, one for each
partition of $\{1,\ldots,k\}$ into two sets.

\begin{notacion} \label{sigmaL} For each $P\in \cD$ and for each $d \in L(\ZZ^{k})$ with
$\ord^L(P) \leq d$ we denote by $\sigma^L_d(P)$ --the $L$--symbol
of $P$ of order $d$-- the class of $P$ in $F_{\wL,d}/F_{\wL,<d}$.
The principal symbol of $P$ is by definition
$\sigma^L(P)=\sigma^L_{d}(P)$ if $d=\ord^L(P)$. For $P,Q \in \cD$
we have $\sigma^L(PQ) = \sigma^L(P)\sigma^L(Q)$. For any left
ideal $I$ in $\cD$ we denote by $\gr^L(I)$ the graded ideal of
$\gr^L(\cD)$ generated by the set $\{\sigma^L(P)\,\vert\, P \in
I\}$. We set $\sigma^L(0)=0$.
\end{notacion}

We denote by $\cD[t]$ the $\CC$-algebra $\cO[\partial,t] =
\CC\{x_1,\ldots,x_n\}[\partial_1,\ldots,\partial_n,t]$ with
relations ($t$ being a new variable)
\begin{itemize}
\item $[t,a]=[t,\partial_i]=[a,b]=[\partial_i,\partial_j]=0$,
\item $[\partial_i,a] = \frac{\partial a}{\partial x_i}t$,
\end{itemize}
for $a,b\in \cO$ and $i=1,\ldots,n$.

The ring $\cD[t]$ is isomorphic to the Rees ring associated with
the order filtration $F_\bullet$ on $\cD$. Since this Rees ring is
by definition $$\cR_F(\cD) = \bigoplus_{\ell\in \ZZ}
F_\ell(\cD)v^\ell \subset \cD[v]=\cD\otimes_\CC \CC[v]$$ for a new
variable $v$, we can define an isomorphism of graded rings $\iota
: \cD[t] \rightarrow \cR_F(\cD)$ by $\iota(a)=a=av^0,
\iota(\partial_j) =
\partial_j v,$ and $\iota(t)=1v=v$.

The natural graded structure of $\cR_F(\cD)$ can be translated on
$\cD[t]$. An homogeneous element of degree $d\in \ZZ$ in $\cD[t]$
is nothing but an expression $$\sum_{\ell + \vert \beta \vert = d}
a_{\ell\,\beta}\partial^\beta t^\ell$$ for some
$a_{\ell\,\beta}\in \cO.$

For $P=\sum_\beta p_\beta(x) \partial ^\beta \in \cD$, the element
$Pv^{\ord(P)}\in \cR_F(\cD)$ is called the homogenization of $P$,
where $\ord(P)$ is the usual order of $P$. It is useful to see
$Pv^{\ord(P)}$ as an element of $\cD[t]$. The homogenization of
$P$ is then denoted by $h(P)$ and we have
$$h(P)= \sum_\beta p_\beta(x)\partial^\beta t^{d-|\beta|}$$ for
$d=\ord(P)$ the usual order of $P$.

For each linear form $L\in (\QQ^k)^*_+$ we can define in a natural
way a filtration $V^L_\bullet(\cD[t])$ on $\cD[t]$, the $L$--order
--denoted $\ord^L(R)$-- of an element $$R=\sum_{\ell\alpha \beta }
r_{\ell \alpha \beta } x^\alpha\partial^\beta t^\ell$$ being the
maximal value of $L(\beta)-L(\alpha)$ for $r_{\ell\alpha\beta}
\not= 0$. The associated graded ring is denoted by
$\gr^L(\cD[t])$. We have a natural ring isomorphism  from
$\gr^L(\cD[t])$ onto $\gr^L(\cD)[t]$, where with the notations as
before we have $[\partial_i,x_j]=\delta_{ij}t$, $\delta_{ij}$
being the Kronecker symbol.

For each left ideal $I$ in $\cD$ we denote by $h(I)$ the left
homogeneous  ideal of $\cD[t]$ generated by $\{h(P)\,\vert \, P\in
I\}$. As in the case of $\cD$, we denote by $\gr^L(h(I))$ the
homogeneous  ideal --in $\gr^L(\cD[t])$-- generated by the set of
principal symbols $\sigma^L(G)$ of the elements $G$ in $h(I)$.


The main result of \cite{assi-castro-granger-jpaa-2} is the
following

\begin{teorema} Let $I$ be a non-zero left ideal of $\cD$ and let $h(I)$ be the associated
homogenized ideal in $\cD[t]$. Then there exists a partition $\cE$
of $\cU$ into convex rational polyhedral cones such that for any
$\Gamma \in \cE$ the ideals $\gr^\Lambda(h(I))$ and
$\gr^\Lambda(I)$ do not depend on $\Lambda\in \Gamma$.
\end{teorema}

Let us sketch the generalization of this result to the case  of a
$\cD $-module $\cM =\frac{\cD ^r}{\cN}$ given by a general
presentation, and endowed with a good multifiltration
$$U_s(\cM ) =
\frac{V_s[\un{n}](\cD ^r)}{V_s(\cN)}=\frac{V_s[\un{n}](\cD ^r) +
\cN }{\cN}$$ associated with a multivector shift $\un{n}\in
(\ZZ^k)^r$.

Here we have to restrict the set  $\cU$ to the set of linear forms
$\wL$, indexed by $L$ in $(\QQ^k)_+^*$, as defined above:
$$\wL(\alpha , \beta )= L(\beta)-L(\alpha).$$

Let us recall  the notation  \ref{filtrationVL} $V_{L(s)}^{L}(\cD)
= F_{\wL,L(s)}(\cD)$, which we shall extend to any free
multifiltered module.

Precisely, given a shift multivector
$\un{n}=(n^{(1)},\ldots,n^{(r)})\in (\ZZ^k)^r$ we define a family, indexed by $L\in
(\QQ^k)_+^*$, of $L({\bf Z}^k)$-filtrations on the module $\cM$  as follows:

$$V_{\ell}^{L}[L(\un{n})](\cD ^r) =
\bigoplus _{i=1}^r V_{\ell-\ell_i}^{L}(\cD)$$ where
$\ell_i=L(n^{(i)})$ for $i=1,\ldots,r$ and
$L(\un{n})=(\ell_1,\ldots,\ell_r)$.

$$U_{\ell}^{L}(\cM) =\frac{V_{\ell}^{L}[L(\un{n})](\cD ^r)}
{ V_{\ell}^{L}(\cN)}= \frac{V_{\ell}^{L}[L(\un{n})](\cD ^r) + \cN
}{\cN }.$$

We again have the notion of a graded associated module
$$\gr^{L}(\cM) = \bigoplus_{\ell \in L(\ZZ^{k})}
\frac{ U_{\ell}^{L}(\cM)}{ U_{<\ell}^{L}(\cM) } =\frac{\gr^{L}(\cD
^r)}{\gr^{L}(\cN)}.$$

We can endow the module $\cD[t]^r$ with a filtration of a filtered
$\cD  [t]$-module by $$V_{\ell}^{L}[L(\un{n})](\cD [t] ^r) =
\bigoplus _{i=1}^r V_{\ell-\ell_i}^{L}(\cD [t])$$ with the same
notations as before,  and the homogeneization of $(P_1, \ldots
,P_r)\in \cD ^r $ is just $$(t^{d-d_1}h(P_1), \ldots
,t^{d-d_r}h(P_r))$$ where $d_i$ is the usual order of the operator
$P_i$ and $d=\max d_i$.

We can now state the straightforward generalisation of the main
theorem in  \cite{assi-castro-granger-jpaa-2}.

\begin{teorema} Let $\cN$ be a non-zero left submodule  of $\cD ^r$
endowed with a good filtration,
and let $h(\cN)$ be the associated homogenized submodule in
$\cD[t]^r$. Then there exists a partition $\cE$ of $(\QQ^k)_+^*$
into convex rational polyhedral cones such that for any $\Gamma
\in \cE$, the submodules $\gr^{L}(h(\cN))$ and $\gr^{L}(\cN)$ do
not depend on $L\in \Gamma$. \end{teorema}

The proof of the theorem is identical to the one in
\cite{assi-castro-granger-jpaa-2}, since the main ingredient,
which is the division theorem, can be adapted to the module case.
See for example \cite{granger-oaku}, where an even more
general situation (with shifts for the variables $t$) is treated. In
particular we still have on every cone of the partition $\cE$ the
notion of a reduced Gr\"obner basis of $h(\cN )$, valid for any
linear form $L$ in the cone.


\section{Analytic standard  fan and flatness}\label{flatness}


After  Herrmann-Orbanz \cite{Herrmann-Orbanz} we have the
following criterion for flatness of a module $M$ over a
commutative ring $R,$ both being graded by a commutative group $G$:
$$M=\bigoplus_{g\in G} M_g , \quad R = \bigoplus_{g\in G} R_g $$
\begin{teorema}
The (not necessarily of finite type) graded $R$--module $M$ is
$R$-flat if and only if for any graded ideal $H\subset R$, we
have:
$$Tor_1^R(M,R/H)=0.$$
\end{teorema}



We will apply this criterion to  the Rees module $$\cR ^{\Gamma
}({\cM}) = \bigoplus _{s\in \ZZ ^k}U^{\Gamma }_s(\cM) u^s$$ graded
by the group $\ZZ^k$, seen as a module over the graded ring $\cR
^{\Gamma }({\cD})$, and therefore also as a module over the graded
subring $R=\CC[W]= \CC[W_1,\cdots, W_k]$ (see \ref{Reesrings}).


Note that for any $a\in \NN^k$, $\CC \cdot W^a$, is a homogeneous
subspace of $R$ generated by a non zero element of $\ZZ^k$-degree
$\cL' a$, since with the notation of section \ref{RgammaD} $W^a =
u^{\cL'a}$. Therefore the graded ideals of $R$
we have to take into consideration are the monomial ideals of the
polynomial ring $R$.


%$\dagger $
%
%\noindent Comment: Je laisse ceci ici pour memoire;
%
%\underline{\bf
%\large Question.} Is it true that we have the implication:
%$$ker(\hat{\kappa })=\{ 0\}\Longrightarrow \cR ^{\Gamma }{\cM}\mbox{ is }
%\CC[\underline{v}]-\mbox{flat}?$$
%
%$\dagger \dagger $

We now deal with the case of $\cM=\frac{\cD ^r}{\cN}$. If we endow
$\cN$ and $\cM$ with the induced and quotient filtration, from a
$V_{\bullet}[\un{n} ]$--filtration  on the free module $\cD ^r$,
we have the exact sequence:
$$ ({\cal S}) \qquad 0 \longrightarrow \cR ^{\Gamma }({\cN })  \longrightarrow \cR
^{\Gamma }({\cD ^r})
\longrightarrow \cR ^{\Gamma }({\cM}) \longrightarrow 0.$$

\begin{lema}\label{RGammaDplat}
The ring $\cR ^{\Gamma }({\cD})$ is flat over $\CC[W]$.
\end{lema}

\begin{prueba}{.}
By the theorem of Herrmann-Orbanz, it is sufficient to prove that
$$Tor_1^{\CC[W]}(\cR^\Gamma(\cD),\CC[W]/H)=0$$ for any homogeneous
ideal $H$. If we consider the tensor product by $\cR ^{\Gamma
}(\cD )$ of the exact sequence
$$0 \longrightarrow  H  \longrightarrow \CC[W]
\longrightarrow \CC[W]/H \longrightarrow 0,$$ we see that this
vanishing result is equivalent to the injectivity of the mapping
$$H  \otimes \cR ^{\Gamma }(\cD ) \longrightarrow \cR ^{\Gamma
}(\cD).$$ \indent Since the ideal $H$ is multihomogeneous, it is a
monomial ideal and we may consider an element \break
$\mathfrak{Q}=W^{a_1}\otimes Q_1+\ldots + W^{a_r}\otimes Q_r$ of
the kernel of this map. Let $Q_i=\sum q_{i,\alpha ,\beta ,
\ell}X^{\alpha } \Delta ^{\beta}W^{\ell }$ be the development of
$Q_i$.

The fact that $\mathfrak{Q}$ is in the kernel can be expressed by the
equation
$$(\star )\qquad \sum _{i=1}^r W^{a_i}Q_i=0 .$$

Let us consider the partition of $\bigcup (a_i+\NN ^k)\subset  \NN ^k$ given
by~:
$$\Delta _1=a_1+\NN ^k\; , \quad \Delta _i=(a_i+\NN ^k)\setminus
(\Delta _1\cup \ldots \cup \Delta _{i-1}), \mbox{ for } i=2,\cdots ,r$$
We may write $Q_i=\sum _{p=1} ^i Q_{i,p}$, with~:
$$Q_{i,p}=\sum _{\ell+a_i\in\Delta _p} q_{i,\alpha ,\beta , \ell}X^{\alpha }
\Delta ^{\beta}W^{\ell },$$ operators having disjoint $(W)$-Newton
diagrams. We set $a_i+v_{i,p}=a_p+w_{i,p}$, with
$v_{i,p},w_{i,p}\in \NN ^k$ minimal for the componentwise partial
ordering, and then we may write $Q_{i,p}=W^{v_{i,p}}R_{i,p}$ with
$R_{i,p}\in \cD$, and we may also remark that $W^{a_i}Q_{i,p}$ is
the part of the operator $W^{a_i}Q_i$, whose Newton diagram  is
contained in $\Delta _p$ so that the equation $(\star )$ splits
into:
$$(\star )_p \qquad \sum _{i=p}^r W^{a_i}Q_{i,p}= W^{a_p}\sum _{i=p}^r
W^{w_{i,p}}R_{i,p}=0 \,\mbox{ for }\, p=1, \ldots ,r .$$ This
finally implies the desired result by the following calculation~:
\begin{align*}
\mathfrak{Q}=& \sum _{i=1}^rW^{a_i}\otimes
( \sum _{p=1}^i W^{v_{i,p}}R_{i,p})=\sum _{p=1}^r \sum _{i=p}^r W^{a_i+v_{i,p}}
\otimes R_{i,p} \\ =& \sum _{p=1}^r \sum _{i=p}^r W^{a_p+w_{i,p}}
\otimes R_{i,p} = \sum _{p=1}^r W^{a_p} \otimes (\sum _{i=p}^r
W^{w_{i,p}} R_{i,p})=0
\end{align*}
\end{prueba}

By the theorem of Herrmann-Orbanz, the flatness of
$\cR^\Gamma(\cM)$ -- if $\Gamma$ is included in a cone of the
analytic fan of $h(\cN)$ -- is a consequence of the equality
$$Tor_1^{\CC[W]}(\cR^\Gamma(\cM),\CC[W]/H)=0$$ for any homogeneous, that
is monomial, ideal $H\subset \CC[W]$.

Because of the above  lemma, and of the exact sequence $({\cal
S})$, this $Tor$ module appears as the kernel of the map:
$$\CC[W]/H\otimes_{\CC[W]} \cR ^{\Gamma }({\cN}) =
\frac{\cR ^{\Gamma }({\cN})}{H\cdot \cR ^{\Gamma }({\cN }) }
\longrightarrow  \frac{\cR ^{\Gamma }({\cD}^r)}{H\cdot \cR
^{\Gamma }({\cD}^r)}$$ and we will prove that it is injective.


The injectivity means that $H\cdot \cR ^{\Gamma }({\cN})= H\cdot
\cR ^{\Gamma }({\cD}^r) \cap \cR ^{\Gamma }({\cN})$ for any
monomial ideal $H = \sum _{i=1}^\ell \CC [W]W^{a_i}.$ The
inclusion $H\cdot \cR ^{\Gamma }({\cN})\subset H\cdot \cR ^{\Gamma
}({\cD}^r) \cap \cR ^{\Gamma }({\cN})$ is obvious.

\begin{teorema} \label{mainresult} Let $\Gamma$ be a cone included in a cone
of the analytic standard fan of $h(\cN)$. Then $\cR^\Gamma(\cM)$ is
flat over $A_\Gamma=\CC[W]$. \end{teorema}

\begin{prueba}{.} We have to prove the inclusion $H\cdot \cR
^{\Gamma }({\cN}) \supset H\cdot  \cR ^{\Gamma }({\cD^r}) \cap \cR
^{\Gamma }({\cN})$ for any monomial ideal $H$ in $\CC[W]$. It is
enough to treat the case  where $H$ is generated by a set of
variables $\{W_{j_1},\ldots,W_{j_p}\}$ for some
$J=\{j_1,\ldots,j_p\}\subset \{1,\ldots,k\} $. We denote by $W_J$
the ideal generated by $\{W_{j_1},\ldots,W_{j_p}\}$ and we call it
a coordinate ideal. This is due to the two following lemmata.

\begin{lema}
Let $H_1\subset H_2 \subset \CC[W]$ be two ideals and $M$ a
$\CC[W]$--module  such that
$$Tor_1^{\CC[W]}(M,\CC[W]/H_2)=Tor_1^{\CC[W]}(M,H_2/H_1)=0$$
then $Tor_1^{\CC[W]}(M,\CC[W]/H_1)=0$.
\end{lema}
\begin{lema} Let $H$ be a monomial ideal in
$\CC[W]=\CC[W_1,\ldots,W_k]$. Then there exists a sequence of
monomial ideals $$H=H_0 \subset H_1 \subset \cdots \subset
H_r=\CC[W]$$ such that for any $i$ the quotient  $H_{i+1}/H_i$ is
isomorphic to $\CC[W]/W_{J(i)}$ for some $J(i)\subset
\{1,\ldots,k\}$.
\end{lema}

\begin{prueba}{.}  By noetherianity of $\CC[W]$ it is sufficient to
construct $H_1=H_0+\CC[W]m$ where $m$ is a monomial in $\CC[W]$
such that $m\not\in H_0$ and $(H_0:m)$ is some coordinate ideal
$W_J$. In the Artinian case just take an element in the socle and
in general use an induction on the dimension of $\CC[W]/H.$
\end{prueba}



So, in order to prove theorem \ref{mainresult} it is sufficient to
treat the case $H$ generated by $\{W_1,\ldots,W_p\}$ for any
$p\leq k$. We have (see \ref{RgammaD}) $W_i = U^{C_i}$ for
$i=1,\ldots,k$ where $C_i\in \ZZ^k$ is the $i$-th column of the
matrix $\cL'$. Since the considered submodules of $\cR ^{\Gamma
}(\cD ^r ) $ are $V_{\bullet}[\un{n} ]$--homogeneous, it is sufficient
to prove that any homogeneous element $Qu^s \in H\cdot
\cR^\Gamma(\cD^r) \cap \cR^\Gamma(\cN)$ belongs to $H\cdot
\cR^\Gamma(\cN)$. As $H\cdot \cR^\Gamma(\cD ^r)$ is the submodule
$\sum u^{C_i}\cR^\Gamma(\cD ^r)$, the vector-operator $Q$, which
belongs to $\cN$, can be written  $ Q= \sum_{i=1}^p Q_i$ for some
$Q_i \in V^\Gamma_{s-C_i}(\cD ^r)$.

For any $i,j=1,\ldots,p$ we have (see notations \ref{sigmaL}):
$$\ord^{L_i}(Q_j) \leq
L_i(s)-\delta_{ij}.$$
%$$\ord^{L_i}(\sum_{j}Q_j) = \ord^{L_i}(\sum_{j\not = i} Q_j) \leq
%L_i(s).$$
$$\sigma^{L_i}_{L_i(s)}(\sum_j Q_j) = \sigma^{L_i}_{L_i(s)}
(\sum_{j\neq i} Q_j).$$


There exists a family $\ell,\ell_1,\ldots,\ell_p$ of non negative
integer numbers such that $$t^\ell h(Q) = \sum_i t^{\ell_i}
h(Q_i).$$

Let us consider $H_1,\ldots,H_r$ a simultaneous reduced
$L$-standard basis of $h(\cN)$ for any $L$ in $\Gamma$. The
existence of such a simultaneous $L$-standard basis is a
consequence of  the inclusion of $\Gamma$ in a cone of the
analytic standard fan of $h(\cN)$ (see the proof of \cite[Th.
20]{assi-castro-granger-jpaa-2}). By the analytic division theorem
in $\cD[t]$ (see \cite[Th. 7]{assi-castro-granger-jpaa-2}, and
\cite[Th. 4.1.]{granger-oaku}) we can write $$ t^\ell h(Q) =
\sum_{m=1}^r A_m H_m $$ for some $F$--homogeneous elements $A_j
\in \cD[t]$.

For an element $B=\sum_{\alpha \beta} b_{\alpha \beta i} x^\alpha
\partial^\beta e_i$ in $\cD ^r$ we denote by $\cN_V(B)$ its
$V_{\bullet}[\un{n} ]$-Newton diagram. By definition this is the
subset of $\ZZ ^k$ defined by~:
$$\cN_V(B)= \{
(\beta_1-\alpha_1+n^{(i)}_1,\ldots,\beta_k-\alpha_k+n^{(i)}_k)\in
\ZZ^k\,\vert\, b_{\alpha \beta i}\neq 0\}.$$ If no confusion is
posible we shall denote $\beta-\alpha+n^{(i)}=
(\beta_1-\alpha_1+n^{(i)}_1,\ldots,\beta_k-\alpha_k+n^{(i)}_k)$.
We have an analogous definition for the $V_{\bullet}[\un{n}
]$-Newton diagram  of an element in $\cD[t]$.

We have $$\cN_V(t^\ell h(Q))= \cN_V(Q) \subset \bigcup_{i=1}^p
((s-C_i)-\check{\Gamma}).$$ As $H_m$ is $L$-homogeneous for all
$L\in \Gamma$, and as a consequence of the division theorem in
$\cD[t]$, we have $\cN(A_m)+V_{\bullet}[\un{n} ](\exp(H_m))
\subset \bigcup_{i=1}^p ((s-C_i)-\check{\Gamma})$ for
$m=1,\ldots,r$, where $V_{\bullet}[\un{n} ](\alpha,\beta ,i)$ if
by definition  $\beta-\alpha +n^{(i)}$ and $\exp$ stands for the
privileged exponent with respect to a previously fixed well
ordering on $\NN^{2n+1}\times \{ 1\cdots r\}$ (see \cite[pp.
174-175]{granger-oaku} for more details).

As a consequence of the division theorem we also have
$$\sigma^{L_1}_{L_1(s)}(t^\ell h(Q)) = \sum_{j=2}^p
\sigma^{L_1}_{L_1(s)} (t^{\ell_j} h(Q_j)) =
\sum_{m=1}^r\sigma^{L_1}_{L_1(s)} (A_mH_m) = \sum_{m=1}^r
a_m(x,\partial,t) \sigma^{L_1}_{d_{1,m}}(H_m)
$$ where $d_{1,m} = \ord^{L_1}(H_m)$ and $a_m(x,\partial,t)$ is
$L_1$-homogeneous. As the $V_{\bullet}[\un{n} ]$-Newton diagram
of $\sigma^{L_1}_{L_1(s)}(t^\ell h(Q)) = \sum_{j=2}^p
\sigma^{L_1}_{L_1(s)} (t^{\ell_j} h(Q_j))$ is included in
$\bigcup_{j=2}^p ((s-C_j)-\check{\Gamma})$ we have, for any
$m=1,\ldots,r$ the inclusion
$$\cN_V(a_m(x,\partial,t)) +
V_{\bullet}[\un{n} ](\exp(H_m)) \subset \bigcup_{j=2}^p
((s-C_j)-\check{\Gamma}).$$

Let us write $$R_1= t^\ell h(Q) - \sum_{m=1}^r
a_m(x,\partial,t)H_m, \,\, {\mbox{ and }}\,\, Q_1'=(R_1)_{| t=1}.$$ We have :\\
$$\ord^{L_1} (R_1) \leq L_1(s)-1,$$ $$ \ord^{L_j}(R_1) \leq
L_j(s), j=2,\ldots,p,$$ and we can decompose each
$a_m(x,\partial,t)$ into a sum
$$a_m(x,\partial,t) = \sum_{j=2}^p A_{mj}$$ such that
$\cN_V(A_{mj}) + V_{\bullet}[\un{n} ](\exp(H_m)) \subset
((s-C_j)-\check{\Gamma})$ for $m=1,\ldots,r$ and $j=2,\ldots,p.$

Let us write $R_j = \sum_{m=1}^r A_{mj}H_m\in h(\cN)$ and $Q_j'=
(R_j)_{|t=1} \in \cN$ for $j=2,\ldots,p.$

We set $Q_1' = Q - \sum_{j=2} Q_j' \in \cN$. We have $Q_1' \in
V^{\Gamma }_{s-C_1}(\cN)$ by the properties of the orders $
\ord^{L_j}(R_1)$, and for $i=2,\cdots , k$ $Q_i' \in V^{\Gamma
}_{s-C_i}(\cN)$, by the properties of the  $V_{\bullet}[\un{n}
]$-Newton diagram of the $A_{mj}$. Therefore we can write
$$Qu^s=(R_1)_{|t=1}=Q_1'u^{s-C_1}u^{C_1} + \cdots + Q_p'u^{s-C_p}u^{C_p} \in H
\cR^\Gamma(\cN ).$$ That ends the proof.
\end{prueba}



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\end{document}

















%tout ce qu'on a ecrit avant sur le sujet suit ici:
Let $P=\sum _{i=1}^s\underline{v}^{\alpha ^i}Q_i$, with $Q_i\in
\cR ^{\Gamma }{\cD} $ be an element  of $H. \cR ^{\Gamma
}{\cD}\cap \cR ^{\Gamma }{I}$.

By the division theorem and taking into account the fact that the
remainder of $P\in \cR ^{\Gamma }{I}$ must be in $ker(\hat{\kappa
})=\{ 0\} $ we have the following standard division:
$$P=\sum _{j=1}^r\widehat{A_j}\widetilde{P_j}$$
We can also perform the division with respect to
$\underline{v}^{\alpha ^1},\cdots , \underline{v}^{\alpha ^s}$ of
the $\widehat{A_j}$:
$$\widehat{A_j}=\sum _{i=1}^s\underline{v}^{\alpha ^i}\widehat{R_{i,j}}+\widehat{B_j}$$
and since it is division by monomial in $\underline{v}$, each of
the operators $\widehat{R_{i,j}}$ and $\widehat{B_j}$ has its
Newton diagramm included in the same terms $E_i\times \NN ^k$ of
the partition of $\NN ^{2n}\times \NN^k$ determined by the
standard basis. We have the equalities
$$P=\sum _j(\sum_i\underline{v}^{\alpha ^i}\widehat{R_{i,j}})\widetilde{P_j}+
\sum _j\widehat{B_j}\widetilde{P_j}= \sum
_{i=1}^s\underline{v}^{\alpha ^i}Q_i $$ which we rewrite :
$$(1) \quad \sum _j\widehat{B_j}\widetilde{P_j}=\sum _{i=1}^s\underline{v}^{\alpha ^i}\widehat{Q'_i} $$
with $\widehat{Q'_i} =Q_i
-(\sum_j\widehat{R_{i,j}}\widetilde{P_j})$. The element in (1) is
an element of $H \widehat {\cR ^{\Gamma }{\cD}}$ and as such its
leading monomial is a multiple of some $\underline{v}^{\alpha
^i}$. Its leading exponent is then an element  $(\gamma ,
\alpha+\alpha ^i)$ of $(0,\alpha _i)+\NN ^{2n+k}$ with $\gamma \in
\NN ^{2n}$ and $ \alpha \in \NN ^k$. On the other hand $\sum
_j\widehat{B_j}\widetilde{P_j}$ is the standard division of the
common element in (1), therefore its exponent would be some
$exp(\widehat{B_j}\widetilde{P_j}).$ Since the exponent of
$\widetilde{P_j}$ is by construction of the standard basis in $\NN
^{2n}\times \{ 0\}$, we would have $exp(\widehat{B_j})\in \NN
^{2n}\times (\alpha ^i + \NN ^k)$ which contradicts the
construction of $\widehat{B_j}$.

Consequently the two members of the equation (1) are zero which
proves that :
$$(2) \quad P=\sum _i(\sum_j\underline{v}
^{\alpha ^i}\widehat{R_{i,j}})\widetilde{P_j}$$ and the fact that
$P$ is in $H.\cR ^{\Gamma }{I}\widehat{\cR ^{\Gamma }{\cD}}$

In fact this proves that $\widehat{\cR ^{\Gamma
}{\cM}}=\frac{\widehat{\cR ^{\Gamma }{\cD}}}{\cR ^{\Gamma
}{I}.\widehat{\cR ^{\Gamma }{\cD}}}$ is flat over $\CC
[\underline{v}]$, since $P$ could as well be a formal element in
the above argument. It is not enough for the result about $\cR
^{\Gamma }{\cD}$ itself since the $\widehat{R_{i,j}}$ are formal
operators.

{\bf Proof of the flatness of $\cR ^{\Gamma }{\cM}$ over
$\CC[\ov{v}]$, in the case $k=2$}
%How to get rid of the last condition??
We truncate the coefficient of the right member of $(2)$ up to
degree $N$ for a given $N$, which means that we set:
$\widehat{R_{i,j}}=R_{i,j}^{(N)}+\widehat{S_{i,j}^{(N)}}$,  with
$R_{i,j}^{(N)}$ obtained by truncation of $\widehat{R_{i,j}}$ at
order $N-|\alpha _i|$, which gives $$P=\sum _i(\sum_j\underline{v}
^{\alpha ^i}R_{i,j}^N)\widetilde{P_j}+\sum _i(\sum_j\underline{v}
^{\alpha ^i}\widehat{S_{i,j}^{(N)}})\widetilde{P_j}.$$ We have
$$\sum _i(\sum_j\underline{v} ^{\alpha
^i}R_{i,j}^N)\widetilde{P_j}\in H.\cR ^{\Gamma }{I},$$ and by
difference
$$P_1:=\sum _i(\sum_j\underline{v}
^{\alpha ^i}\widehat{S_{i,j}^{(N)}})\widetilde{P_j}\in H.\cR
^{\Gamma }{\cD}\cap \cR ^{\Gamma }{I}\cap (v_1,\cdots ,
v_k)^{N+1}H\cR ^{\Gamma }{I}\widehat{\cR ^{\Gamma }{\cD}}.$$
Knowing that $P_1$ is in favt polynomial in $\un{v}$ and by
inspection of the Newton diagramm, we may write: $P_1\in
(v_1,\cdots , v_k)^{N+1}H\cR ^{\Gamma }{\cD}. $ Finally we get the
inclusion:
\[\fbox{$H.\cR ^{\Gamma }{\cD}\cap
\cR ^{\Gamma }{I}\subset H.\cR ^{\Gamma }{I}+\cR ^{\Gamma }{I}\cap
(v_1,\cdots , v_k)^{N+1}H\cR ^{\Gamma }{\cD}$}\]

The ideal $H.\cR ^{\Gamma }{\cD}$ of $\cR ^{\Gamma }{\cD}$ is
generated by central element. We can therefore apply Artin Rees
lemma as well as if the ring $\cR ^{\Gamma }{\cD}$ was
commutative. There exists $N_0$ such that $\cR ^{\Gamma }{I}\cap
(v_1,\cdots , v_k)^{N+1}H\cR ^{\Gamma }{\cD} \subset \cR ^{\Gamma
}{I}\cap (v_1,\cdots , v_k)^{N+1}\cR ^{\Gamma }{\cD} =(v_1,\cdots
, v_k)^{N-N_0}(\cR ^{\Gamma }{I}\cap (v_1,\cdots , v_k)^{N_0+1}\cR
^{\Gamma }{\cD})$ which gives by choosing $N=N_0+1$:
$$\cR ^{\Gamma }{I}\cap (v_1,\cdots , v_k)^{N+1}H\cR ^{\Gamma }{\cD}
\subset (v_1,\cdots , v_k)\cR ^{\Gamma }{I}$$ and then $H.\cR
^{\Gamma }{\cD}\cap \cR ^{\Gamma }{I}\subset (v_1,\cdots , v_k)\cR
^{\Gamma }{I}.$

This complete the proof for the case $H=(v_1,\cdots ,
v_k)\CC[\un{v}]$.

If $H_1\subset H_2$ are now two ideal such that $H_2/H_1$ is
generated by one element and isomorphic to
$\CC[\un{v}]/(v_1,\cdots , v_k)$, the Tor sequence associated to
$0\to H_2/H_1\to \CC[\un{v}]/H_1\to \CC[\un{v}]/H_2\to 0$ and the
result above prove the implication~:
$$Tor _1^{\CC[\un{v}]}(\cR ^{\Gamma }{\cD}/\cR ^{\Gamma
}{I},\CC[\un{v}]/H_2)=0
\Longrightarrow Tor _1^{\CC[\un{v}]}(\cR ^{\Gamma }{\cD}/\cR ^{\Gamma }{I},\CC[\un{v}]/H_1)=0 .$$
In the case $k=2$, this reduces the proof to the case of a
principal ideal $\un{v}^{\alpha}$.

The proof goes as follows , independently now of the hypothesis
$k=2$.
% But I have to adapt it to the case when the module
% has more than one generators. I think its OK
Let $P\in \un{v}^{\alpha}\cR ^{\Gamma }{\cD}\cap \cR ^{\Gamma
}{I}$. Since the ring and the ideal are $\ZZ ^k$-graded, we may
assume that $P\in \un{v}^{\alpha}\cR ^{\Gamma }_s {\cD}$, for some
$s\in \ZZ ^k$, and $\un{v}^{\alpha}$, is in fact some
$\un{u}^{\sigma }$ for $\sigma $ a non zero element of the dual
cone $\check{\Gamma }$.

We can write $P=Q\un{u}^s$, with $Q\in {}^{\Gamma}U_s(\cD)\cap I:=
{}^{\Gamma}U_s I$, and furthermore the fact that $P$ is divisible
by $\un{v}^{\alpha}=\un{u}^{\sigma}$ can be written:
$$P\in \un{u}^{s}(\sum _{s\in s'+\sigma +\check{\Gamma }}U_{s'})$$
which means that in fact $Q\un{u}^{s-\sigma}\in
({}^{\Gamma}U_{s-\sigma}(\cD)\cap I)\un{u}^{s-\sigma}:= \cR
^{\Gamma}_{s-\sigma} I$. Finally we have
$$P=\un{v}^{\alpha}Q\un{u}^{s-\sigma}\in \un{v}^{\alpha}\cR
^{\Gamma}_ I$$ as required.




 [[GARBAGE Let $M$ be a left $\cD$-module (most of the
time $M$ will be $\cD/I$ for some left ideal $I\subset \cD$ or
even a quotient $\cD^r/N$ for some $N\subset \cD^r$).

Suppose we have chosen  a good $V_\bullet (\cD)$--multifiltration
$U_\bullet(M)$. Then for each cone $\Gamma$ one consider a new
multifiltration $U^\Gamma_\bullet (M)$ defined by $$U^\Gamma_s(M)=
\sum_{\sigma \in s+\check{\Gamma}} U_\sigma(M).$$ In particular we
have, by definition,
$$FV^\Gamma_s(\cD)= \sum_{\sigma \in s+\check{\Gamma}} FV_\sigma(\cD).$$

If $M=\cD/I$ then we can consider over $I$ the induced filtration
$FV_s^\Gamma(I):=FV_s^\Gamma(\cD) \cap I$. An then the associated
Rees module $\cR_{FV}^\Gamma(I)$ is a left ideal of
$\cR_{FV}^\Gamma(I)$ so that we have a natural isomorphism
$$\cR^\Gamma_U(M) =
\frac{\cR_{FV}^\Gamma(\cD)}{\cR^\Gamma_{FV}(I)}.$$

\begin{lema}
The following conditions are satisfied: \\ i)
$\CC[\check{\Gamma}\cap \ZZ^{k+1}] \otimes_{\CC[\NN^{k+1}]}
\cR_{FV}(\cD) \simeq \cR^\Gamma_{FV}(\cD)$. (Double check the case
$FV$). \\ ii) The kernel of the natural morphism
$$\CC[\check{\Gamma}\cap \ZZ^{k+1}] \otimes_{\CC[\NN^{k+1}]}
\cR_{U}(M) \simeq \cR^\Gamma_{U}(M)$$ equals the torsion over
$\CC[\check{\Gamma}\cap \ZZ^{k+1}]$.
\end{lema}
]]
\section*{Plan de travail et objectives}

a) {\bf Generalit\'{e}s sur les multifiltrations et les anneaux de
Rees. }

a.1) Description de $\cR_V(\cD)$ et $\cR_{FV}(\cD)$.

a.2) Description de $\cR^\Gamma(\cD)$.

a.3) Description de $\gr^\Gamma(\cD)$.

a.4) Description de $\widehat{\cR^\Gamma(\cD)}$.

\begin{lema}
S'il existe un \'{e}ventail $\{\Gamma\}$ tel que pour tout $\Gamma$ on
ait $\cR^\Gamma(M)$ est plat sur $\CC[\check{\Gamma}\cap \ZZ^{k}]$
alors le meme r\'{e}sultat est vrai pour toute face $\Gamma' $ de
$\Gamma$ \\ ((il faut consid\'{e}rer le morphism
$$\CC[\check{\Gamma}'\cap
\ZZ^{k}]\otimes_{\CC[\check{\Gamma}\cap \ZZ^{k}]} \cR^\Gamma(M)
\rightarrow \cR^{\Gamma'} (M)
$$
\end{lema}

[[Remarque: On doit peut-etre \'{e}noncer et montrer un resultat
analogue pour les $\cR^\Gamma_V$-filtrations.]]




\vspace{.5cm}

b) {\bf Completion et th\'{e}or\`{e}me de division adapt\'{e} \`{a}
$\gr^\Gamma(I)$ pour un id\'{e}al \`{a} gauche $I$ de $\cD$.} Plus
g\'{e}n\'{e}ralement th\'{e}or\`{e}me de division adapt\'{e} \`{a} $\gr^\Gamma(N)$ pour un
submodule $N$ de $\cD^r$.

b.1) Avec les notations canoniques, on voudrait montrer qu'il
existe un morphisme surjective

$$ \widehat{\cR^\Gamma(\cD)}^{E\times \NN^{k}}
\stackrel{\widehat{\kappa^\Gamma}}{\longrightarrow}
\frac{\widehat{\cR^\Gamma(\cD)}}{\cR^\Gamma(I)
\widehat{\cR^\Gamma(\cD)}}.$$

\vspace{.5cm}

c) {\bf Platitude}. On voudrait d\'{e}montrer le

\begin{teorema}
Si $Ker(\widehat{\kappa^\Gamma})=0$  alors $\cR^\Gamma(M)$ est
plat sur $\CC[\check{\Gamma}\cap \ZZ^{k+1}]$.
\end{teorema}


Vu qu'on ne travaille pas avec des modules de type fini sur les
alg\`{e}bres affines $S_\Gamma$ il faut sans doute utiliser le crit\`{e}re
de platitude de Hermann-Orbanz (voir aussi Matsumura).

En fait, il est probable qu'on ait besoin du r\'{e}sult suivant
\vspace{.5cm}

\begin{lema}
Si $\Omega$ est un sous-cone de $\Gamma$ et
$\Ker(\widehat{\kappa^\Omega}) \not= 0$ alors le morphisme naturel
$$\gr^\Omega(M) \rightarrow \gr^\Gamma(M)$$ a un noyau non nul.
\end{lema}

\vspace{.5cm}

d) {\bf Eventails adapt\'{e}s et applications.}

C'est ici qu'on demontre que la fiber diminue...


\subsection*{Quelques r\'{e}sultats utiles}

A) We shall denote the ring $\CC[\check{\Gamma}\cap \ZZ^{k}]$ by
$A_\Gamma$.

\begin{propo}
Si $\Gamma$ est un cone de l'\'{e}ventail de Gr\"obner alors
$\cR^\Gamma(M)$ est plat sur $A_\Gamma$.
\end{propo}

\begin{prueba}{.}
We shall prove that the morphism $$\widehat{\cR^\Gamma(\cD)^{E'}}
\stackrel{\widehat{\kappa^\Gamma}}{\longrightarrow }
\widehat{\cR^\Gamma(M)}$$ is bijective. [[Note.- Cela montre que
$\widehat{\cR^\Gamma(M)}$ est plat sur $A_\Gamma$ car
$\widehat{\cR^\Gamma(\cD)^{E'}}$ l'est.]]

Let $Pu^s\in V^\Gamma(\cD)$ ... (?) in the kernel of the morphism
[[ {\it the morphism associates $Pu^s$ with its remainder after
division} ]]

Let $\cF=\{Q_1,\ldots,Q_r\}$ be a family in $I$ such that for any
linear form $L \in interior(\Gamma)$, the family $\cF$ is a
$L$--standard basis of $I$ with
$\{\sigma^L(Q_1),\ldots,\sigma^L(Q_r)\}$ multi-homogeneous with
multi-weight $s^1,\ldots,s^r$ respectively. Write
$P_i=\sigma^{\Gamma}_{s^i}(Q_i) \in \gr^\Gamma_{s^i}(I).$ Then the
family $\cG=\{P_1,\ldots,P_r\}$ is an standard basis of
$\gr^\Gamma(I)$ (with respect to some well ordering to be detailed
later). Let us consider, to this end, a non zero multihomogeneous
element $P\in \gr^\Gamma_s(I)$. We have $P=[\overline{P}]$ (i.e.
$P$ is the class of $\overline{P}$) for some $\overline{P}$ in
$(FV)^\Gamma_s(I) = I \cap (FV)^\Gamma_s(\cD)$. [[To be
controlled: For each linear form $L$ in the interior of $\Gamma$
one has $P=\sigma^L(\overline{P})$]]. As each $P_i$ is
$L$-homogeneous we can divide $P$ by $\cG$ ...

...

 In the $\ell-th$ step of the division we get

$$\overline{P} = \sum_j \widetilde{A_j^{\ell}} \overline{P_j} +
\overline{R^{\ell}}$$

where the the partial remainder $\overline{R^{\ell}}$ belongs to
the  ideal $(W)^{\ell+1}$. So, we get

$${P}u^s = \sum_j {A_j^{\ell}}u^{s-s^j} {P_j}u^{s_j} +
{R^{\ell}}u^s$$ where ${R^{\ell}}u^s \in (FV)^\Gamma_s u^s \cap
(W)^{\ell+1}$ [[To be controlled: Je n'aime pas trop le m\'{e}lange
des $u$ avec les  $W$]]. So, by taking the inverse limit when
$\ell $ goes to $+\infty$ we get a zero remainder and this proves
that the kernel of $\widehat{\kappa^\Gamma}$ is zero. [[This proof
should be double checked]].
\end{prueba}


\vspace{.5cm}

B) {\bf Torsion and flatness}


\begin{lema} [[Reste \`{a} pr\'{e}ciser s'il s'agit d'une filtration
$U_\bullet(M)$ relative \`{a} $V_\bullet$ ou \`{a} $(FV)_\bullet$]]  If
$\cR_U(M)$ is flat over $\CC[u]$ then
$\frac{\cR_U(M)}{u_1\cR_U(M)}$ has no torsion over
$\CC[u_2,\ldots,u_k]$.
\end{lema}

\begin{prueba}{.}
\end{prueba}




zzz

...

%\subsection{Description of $\cR^\Gamma(\cD)$}
% To simplify we shall consider a good
%$V_\bullet(\cD)$--multifiltration $U_\bullet(M)$ over $M$ and
%$k=n$. [[Comment: So, we do not have $u_0$ here and the rings
%$\cA, \cA', \cA''$ are slightly different]] Here we have $x_i=
%X_iU_i$ and $\partial_i = \Delta_i/U_i$ for $i=1,\ldots,n$.

Let $\Gamma$ be a simplicial, rational, convex cone in the first
quadrant of  $(\QQ^k)^*$. We can suppose $\Gamma$ is defined by
$r$ linearly independent linear forms $L_1,\ldots,L_r$; here $r$
is in fact the dimension of the vector space generated by
$\Gamma$. To simplify let us consider first the case $k=n$.
[[Check what follows]] We denote by $L$ the square matrix whose
rows are $L_1,\ldots,L_k$ and we denote by $L_j'$ the $j-th$
column of $L$. We can suppose the determinant of $L$ to be equal
to 1. Call $\Lambda$ the inverse matrix of $L$ and its columns by
$\Lambda_1,\ldots,\Lambda_k$. Then we shall consider the morphisms
defined by $U_j=W^{L_j'}$ and $W_j=U^{\Lambda_j}$.


So, we have $$\cR^\Gamma(\cD)= \oplus \cR^\Gamma_s(\cD)=
\bigoplus_{s\in \ZZ^k} V^\Gamma_s(\cD)u^s \subset \cD\otimes_\CC
\CC[u^{\pm 1}]
$$
where $$V^\Gamma_s(\cD) = \sum_{\sigma\in \ZZ^k\,\vert\,
L_i(\sigma) \leq L_i(s), i=1,\ldots,k} V_\sigma(\cD).$$

Remember we consider the injective morphism $i$ from XXX in order
to associate to each monomial $$x^\alpha \partial^\beta u^s$$ in
$V^\Gamma_s(\cD)$ the monomial $$X^\alpha \Delta^\beta
U^{s-\beta+\alpha}$$ in the ring $\cA=\CC\{XU\}[X,\Delta, U^{\pm
1}]$ and then to this last monomial we associate the monomial
$$X^\alpha \Delta^\beta W^{\Lambda(s-\beta+\alpha)^t}$$ in the ring
$\cA'=\CC\{X,W\}[\Delta]$.

So, a monomial in $\cR^\Gamma_s(\cD)$ can we viewed as a monomial
$X^\alpha \Delta^\beta W^a$ such that $|\Lambda_i(a)-\alpha_i|$
are bounded for all $i=1,\ldots,k$.

[[Remark: To be controlled. In particular double check the use of
the matrix $\Lambda$ instead of the matrix $L$]]


Remark that if $\Gamma'$ is a face of $\Gamma$, for example
$\Gamma'=\langle L_2,\ldots,L_k\rangle$ then $A_{\Gamma'} = ...$.
((to be completed).

\begin{lema}
If $\cR^\Gamma(M)$ is flat over $A_\Gamma$ then
$\frac{\cR^\Gamma(M)}{W_1\cR^\Gamma(M)}$ is flat over
$\CC[W_2,\ldots,W_k]$.
\end{lema}

....

C) {\bf Examples a suivre}

C.1) We suppose $k=n=2$. Let us consider $I=\cD P$ with
$P=\partial_1+\partial_2$. Then
$$\cR_V(I)=\cR_V(\cD)(\partial_1+\partial_2)u_1u_2) =
\cR(\cD)((\frac{\\Delta_1}{U_1} + \frac{\\Delta_2}{U_2})U_1U_2) =
$$ $$ \Delta_1U_2+\Delta_2U_1.$$

Make the blow-up of the origin in $\CC^2$ and consider the chart
$U_1=W_1W_2, U_2=W_2$ (that corresponds to the cone
$\Gamma=<(1,0),(1,1)>$). Then we get the principal ideal generated
by $\Delta_1W_2+\Delta_2 W_1W_2=W_2(\Delta_1+\Delta_2W_1)$. We
have to consider the strict transform [[please, justify this]],
and so the ideal to be considered is generated by
$\Delta_1+\Delta_2W_1$. The quotient
$$\cR^\Gamma(M)= \frac{\cR^\Gamma(\cD)}{(\Delta_1+W_1\Delta_2)}$$
and it is flat over $\CC[W_1,W_2]$ since we can divide in
$\cR^\Gamma(\cD)$ (i.e. it is nor necessary to go to the
completion to perform the division as one can consider $\Delta_1$
as privileged exponent). So we get an isomorphism $$
\cR^\Gamma(\cD)^{E\times \NN^2}
\stackrel{\kappa^\Gamma}{\rightarrow}
\frac{\cR^\Gamma(\cD)}{(\Delta_1+W_1\Delta_2)}$$




\subsection{$\cR_V(I)$ and $\gr^V(I)$} The graded ring
associated to the $V$--filtration is
$$\gr^V(\cD)=\bigoplus_{k\in \ZZ} \frac{V_k}{V_{k-1}}$$ which
is in fact isomorphic to the ring of linear differential operators
$\CC\{x_1,\ldots,x_{n-1}\}[x_n,\partial_1,\ldots,\partial_n]$.

As $u$ is a central element in $\cR_V(\cD)$, the quotient
$\cR_V(\cD)/\cR_V(\cD)u$ is a graded ring, which is naturally
isomorphic to the graded ring $\gr^V(\cD)$.

[[Add the case (multi)]]


\subsection{Description of $\widehat{\cR^\Gamma(\cD)}$}

We shall describe the elements of $\widehat{\cR^\Gamma(\cD)}$. To
simplify we first consider the case $\Gamma=\NN^k$ and will omit
the superscript $\Gamma$ in the Rees ring and modules to be
considered.

We will write simply $\cR$ instead of $\cR_V(\cD)$ and $V_s$
instead of $\cV_s(\cD)$. In this case we have
$$\cR = \oplus \cR_{s\in \ZZ^k} u^s = V_s u^s.$$

So, we will rewrite $$x^\alpha \partial^\beta u^s$$ in $V_s$ as
the monomial $$X^\alpha \Delta^\beta U^{s-\beta+\alpha}$$ in the
ring $\cA=\CC\{XU\}[X,\Delta, U]$. We can identify $\cR$ to the
ring $\cA$. There is no $W$ here as we are considering
$\Gamma=\NN^k$.

The elements $U_1,\ldots,U_k$ are central in $\cA$ and we can
consider the completion of $\cA$ with respect to the $(U)$--adic
filtration. We have $$\widehat{\cA} = lim.proj._\ell
\frac{\cA}{(U)^\ell} = lim.proj._\ell
\frac{\CC[U,X,\Delta]}{(U)^{\ell+1}} = \CC[X,\Delta][[U]]$$ which
is isomorphic to the ring of formal power series in $k$ variables
$U=(U_1,\ldots,U_k)$ with coefficients in the Weyl algebra
$\CC[X,\Delta]$.

An element $P$  in $\widehat{\cR}$ can be written as $$P =
\sum_{s\in \NN^k} p_s U^s$$ where $p_s$ is an element in the Weyl
algebra $\CC[X,\Delta]$. We can also write $p_s= \sum_{\alpha
\beta} p_{s\alpha\beta} X^\alpha \Delta^\beta$ so that, the
element $P$ in $\cR$ can be written as a (infinite formal sum)

$$
\sum_{s\in \NN^k}(\sum_{\alpha \beta } p_{s \alpha \beta }X^\alpha
\Delta^\beta) U^s =  \sum_{s \alpha \beta } p_{s \alpha \beta }
X^\alpha \Delta^\beta U^s
$$ and in this last sum for each fixed $s$ there are only a finite
number of non zero $p_{s \alpha \beta}$.

It is  useful to get the expression of $P$ in terms of the
original variables $x,\partial, u$. To this end, we shall write
the monomial $$X^\alpha \Delta^\beta U^s$$ as the monomial
$$x^\alpha \partial^\beta u ^{s+\beta-\alpha}.$$ Now, we can write
$$P = \sum_{s\alpha \beta} p_{s\alpha \beta } x^\alpha
\partial^\beta u^{s-\alpha+\beta}
$$ and for a fixed $s$ there are only a finite number of non zero
$p_{s \alpha \beta}.$

It is also useful to write $s'=s-\alpha+\beta$ and suppose we fix
$s'$. Then the coefficient of $u^{s'}$ in $P$ is
$$\sum_{\alpha\beta}p_{s'+\alpha-\beta,\alpha\beta} x^\alpha
\partial^\beta$$ where $s'+\alpha-\beta\gg 0$ and for a fixed $s'+\alpha-\beta$ there are only a
finite number of non zero $p_{s'+\alpha-\beta,\alpha\beta}$.

If $\alpha\gg \beta$ (componentwise order) we write $$x^\alpha
\partial^\beta = x^{\alpha-\beta} x^{\beta} \partial^{\beta}.$$ If
$\beta \gg \alpha$ we write $$x^\alpha
\partial^\beta = x^{\alpha} \partial^{\alpha} \partial^{\beta-\alpha}.$$

More generally, suppose we have $\alpha_i \geq \beta_i$ for
$i=i_1,\ldots,i_p$ (for some $0\leq p \leq k$) and $\alpha_j <
\beta_j$ for $j\not\in \{i_1,\ldots,i_p\}$. Then, after  renaming
the variables, we can consider $i_1=1,\ldots,i_p=p$ and each
monomial
$$x^\alpha \partial^\beta$$ can be written as $$x_1^{\alpha_1-\beta_1}\cdots
x_p^{\alpha_p-\beta_p} x^\bullet \partial^\bullet
\partial_{p+1}^{\beta_{p+1}-\alpha_{p+1}}\cdots \partial_{k}^{\beta_{k}-\alpha_{k}}$$
where $\bullet_j = \min\{\alpha_j,\beta_j\}$ for $j=1,\ldots,k$.

Now, each monomial $x^\gamma \partial^\gamma$ is a polynomial in
$\CC[x\partial] = \CC[x_1\partial_1,\ldots,x_k\partial_k]$ and we
have the following commuting formula $$x^{\alpha'}p(x\partial) =
p(x\partial - \alpha')x^{\alpha'}$$ for each $\alpha'$ and for
each polynomial $p(x\partial) \in \CC[x\partial]$.

So, the coefficient of $u^{s'}$ in $P$ as before can be written as
a sum of objects with the form $$p_{\alpha \beta}(x\partial)
x_1^{\alpha_1-\beta_1}\cdots x_p^{\alpha_p-\beta_p}
\partial_{p+1}^{\beta_{p+1}-\alpha_{p+1}}\cdots \partial_{k}^{\beta_{k}-\alpha_{k}}  $$
where $s'-\beta+\alpha \gg 0$. So, $\beta-\alpha \ll s'$ and for a
fixed $s'$ only a finite number of monomial of type
$$\partial_{p+1}^{\beta_{p+1}-\alpha_{p+1}}\cdots
\partial_{k}^{\beta_{k}-\alpha_{k}} $$ appears.

Finally, we have proven that the coefficient os $u^{s'}$ in $P$
belongs to the ring $\cB=\CC[x\partial][[x]][\partial]$. This ring
is endowed with a $V$-filtration, defined in the natural way, and
the coefficient of $u^{s'}$ belongs in fact to $V_{s'}(\cB)$.







\subsection{Description of $\gr^V(\cD)$ and $gr^\Gamma(\cD)$}

\section{Completion. Adapted division theorem in
$\widehat{\cR^\Gamma(\cD^r)}$}

\subsection{Completion} With the notation of the previous sections
let us consider a submodule $N \subset \cD^r$ and $M=
\frac{\cD^r}{N}$. Let us denote by $\Gamma \subset (\QQ^k)^*_+$ a
basic simplicial cone and by $\CC[v]=\CC[v_1,\ldots,v_n]$ the
polynomial ring $\CC[\check{\Gamma} \cap \ZZ^k]$. We denote by
$\mathfrak m$ the maximal ideal $(v_1,\ldots,v_k)$ in $\CC[v]$. We
will also denote by $\mathfrak m$ the two sided ideal generated by
$v_1,\ldots,v_k$ in $\cR_\Gamma(\cD)$.

We denote by $\widehat{\cR_\Gamma(\cD)}$ the completion of the
ring $\cR_\Gamma(\cD)$ with respect to the two sided ideal
$\mathfrak m$, i.e. we have $$\widehat{\cR_\Gamma(\cD)} =
{lim.proj.}_{\ell} \frac{\cR_\Gamma(\cD)}{\cR_\Gamma(\cD)
{\mathfrak m}^\ell}$$

(a suivre)

\subsection{Adapted divisions}\label{adapted divisions}



\section{Flatness.}


\section{Adapted fans and applications}



\section{Division theorem on $\cR_V(\cD)$ and $\cR_{FV}(\cD)$}

Let us denote by $\cO=\CC\{x_1,\ldots,x_n\}$ the complex
convergent power series  ring and by $\cD$ the ring of germs of
linear differential operators with holomorphic coefficients (i.e.
$\cD=\cO[\partial_1,\ldots,\partial_n]$ where $\partial_i$ is the
partial derivative with respect to $x_i$ and the product in $\cD$
is defined by the Leibnitz's rule: $\partial_i a=a\partial_i +
\partial_i(a)$, for each $a\in \cO$).

An operator $P$ in $\cD$ can be written as
$$P=\sum_{\alpha,\beta\in \NN^n}
p_{\alpha\beta}x^\alpha\partial^\beta$$ where $\alpha =
(\alpha_1,\ldots,\alpha_n), \beta=(\beta_1,\ldots,\beta_n)$,
$x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$,
$\partial^\beta=\partial_1^{\beta_1}\cdots
\partial_n^{\beta_n}$ and $p_{\alpha\beta}$ is a complex
number.

Let us denote by $Y\subset \CC^n$ the hyperplane defined by
$x_n=0$. Remember that the $V$-filtration on $\cD$, with respect
to $Y$, has been defined by Malgrange and Kashiwara as: $$
V_k(\cD)=\{P=\sum_{\alpha,\beta\in \NN^n}
p_{\alpha\beta}x^\alpha\partial^\beta \, \vert \, \beta_n-\alpha_n
\leq k\} $$ for each $k\in\ZZ$.

\subsection{Rees  rings} The Rees  ring associated to the
$V$-filtration on $\cD$ is defined by: $$ \cR_V(\cD) =
\bigoplus_{k\in \ZZ} V_k(\cD)u^k.$$ The product in the Rees  ring
is induced by the natural product in the Laurent polynomial ring
$\cD[u,u^{-1}]$.

By definition the Rees  ring $\cR_V(\cD)$ is a graded
$\CC$-algebra, whose homogeneous elements are $Pu^k$ for $P\in
V_k(\cD)$ and $k\in \ZZ$.

Let $I$ be a left ideal in $\cD$.

We can first consider on $I$ the induced $V$-filtration:
$$V_k(I):=I\cap V_k(\cD).$$  The abelian group
$$\cR_V(I):=\bigoplus_{k\in \ZZ} V_k(I)u^k$$ is in fact an
homogeneous ideal of $\cR_V(\cD)$.

On the other hand, we can consider on the $\cD$-module $\cD/I$ the
induced $V$--filtration: $$V_k(\cD/I) =
\frac{V_k(\cD)}{V_k(I)}=\frac{V_k(\cD)+I}{I}$$ for $k\in \ZZ$. The
associated Rees  module is, by definition,
$$\cR_V(\cD/I)= \bigoplus_{k\in \ZZ} V_k(\cD/I)u^k.$$ We consider
this abelian group as a sub-group of $(\cD/I)[u,u^{-1}]$, the
Laurent polynomials with coefficients in $\cD/I$. In fact,
$\cR_V(\cD/I)$ as a natural structure of left $\cR_V(\cD)$-module,
which is naturally isomorphic to the quotient
$\cR_V(\cD)/\cR_V(I)$.

\vspace{1cm}

{\fbox{\parbox{60ex}{Describe an algorithm computing a generating
system of the ideal $\cR_V(I)$, taking as input a generating
system of $I$}}}


\subsection{$\cR_V(I)$ and $\gr^V(I)$} The graded ring
associated to the $V$--filtration is
$$\gr^V(\cD)=\bigoplus_{k\in \ZZ} \frac{V_k}{V_{k-1}}$$ which
is in fact isomorphic to the ring of linear differential operators
$\CC\{x_1,\ldots,x_{n-1}\}[x_n,\partial_1,\ldots,\partial_n]$.

As $u$ is a central element in $\cR_V(\cD)$, the quotient
$\cR_V(\cD)/\cR_V(\cD)u$ is a graded ring, which is naturally
isomorphic to the graded ring $\gr^V(\cD)$.


\section{Division theorem on $\cR_{\Gamma}(\cD)$}


\section{Flatening theorems in $\cD$-module theory: the relative case}

Let us consider a smooth complex analytic map $f: X \too S$, where
$X$ and $S$ are complex-analytic spaces. We will denote by
$\cD_{X\vert S}$ the sheaf of relative linear differential
operators. We fix a point $x\in X$ and let $s=f(x)\in S$. The
fiber $f^{-1}(s)$ of $f$ at $s$ will be noted $X_s$.

Let us consider a $\cD_{X|S}$-coherent module $\cN$. We will prove
the existence of a local flattner of $\cN$, considered as an
$\cO_S$-module, at the neighborhood of $s\in S$.

Let $(Z,s)\subset (S,s)$ be any germ of hypersurface, defined by a
local equation $e\in \cO_{S,s}$. We denote by $\cN^*$ the quotient
$\cN/\cK$ where $\cK$ is the kernel of the localization map $\cN
\too \cN_e$. In other words, $\cN^*$ is the local cohomology of
$\cN$ with respect to $Z$ in the neighborhood of $s$.

\begin{teorema}{\rm [ancien A.0.1]}
There exists a germ of closed analytic space $(Z,s)\subset (S,s)$
verifying  the following condition: For any analytic map
$\phi:(S',s')\too (S,s)$, the module
$(\cN\otimes_{\cO_S}\cO_{S'})_{s'}$ is $\cO_{S',s'}$-flat if and
only if the map $\phi$ factors through $(Z,s)$. Moreover, if the
ideal of $\phi^{-1}(Z)$ is principal in $\cO_{S',s'}$, then  the
natural surjective map $$
\cN\otimes_{\cO_{S,s}}\left(\frac{\cO_{S',s'}}{\mathfrak m}\right)
\too
(\cN\otimes_{\cO_{S,s}}\cO_{S',s'})^*\otimes_{\cO_{S',s'}}\left(\frac{\cO_{S',s'}}{{\mathfrak
m}}\right)$$ has a non trivial kernel, where ${\mathfrak m}$ is
the maximal ideal of $s'$ in $S'$.
\end{teorema}

\subsection{Lemme 2.1.1} For $i=1,\ldots,k\leq n$, let us consider
the hypersurface  $Y_i\subset X=\CC^n$ defined by $x_i=0$. Let us
denote by ${}^iV_\bullet(\cD)$ the $V$-filtration with respect to
$Y_i$. Finally, let consider, for each $s\in \ZZ^k$,  $$ V_s(\cD)
= \bigcup_{i=1}^k {}^iV_{s_i}(\cD)$$ where $s=(s_1,\ldots,s_k)$.

The associated Rees  ring is, by definition, $\cR_V(\cD) =
\bigoplus_{s\in \ZZ^k} V_s(\cD)u^s$ where $u=(u_1,\ldots,u_k)$ and
$u^s=u_1^{s_1}\cdots u_k^{s_k}$.

We will denote $M=\ZZ^k$.

For any convex rational simplicial cone $\Gamma \subset (Q^k)^*_+$
we consider the affine scheme
$S_\Gamma=Spec(\CC[\check{\Gamma}\cap M])$. Of course we have
$\CC[\NN^k] \subset \CC[\check{\Gamma}\cap M]$. We will denote
$A_\Gamma =\CC[\check{\Gamma}\cap M]$ and $A=\CC[\NN^k]$.

We denote by $\cL(\Gamma)$ the set of primitive elements in the
1-skeleton of $\Gamma$.

Let $\cM$ be a coherent $\cD_X$-module and $U_\bullet(\cM)$ a good
filtration, with respect to $V_\bullet(\cD)$.

For each $\sigma \in \ZZ^k$ let consider $${}^\Gamma
U_\sigma(\cM)= \sum_{\sigma\in \ZZ^{k+1}\,\vert\, L(s)\leq
L(\sigma); \forall L\in \cL(\Gamma)} U_s(\cM).$$ The
$k$-filtration ${}^\Gamma U_\bullet(\cM)$ is a good filtration
with respect to ${}^\Gamma V_\bullet(\cD)$. We will denote simply
$\cR_\Gamma(\cM)$ and $\cR_\Gamma(\cD)$ for the Rees  module and
the Rees   ring associated to ${}^\Gamma U_\bullet$ and ${}^\Gamma
V_\bullet$.

\begin{lema}{\rm [ancien 2.1.1]} \begin{enumerate} \item There exists a natural isomorphism
$$A_\Gamma  \otimes_{A} \cR_V(\cD) \too \cR_\Gamma(\cD).$$ \item
There exists a natural surjective map $$ \alpha_\Gamma (\cM):
A_\Gamma \otimes_{A} \cR_U(\cM) \too \cR_\Gamma(\cM)$$ whose
kernel equals the $A_\Gamma$--torsion of $A_\Gamma  \otimes_{A}
\cR_V(\cM).$
\end{enumerate}
\end{lema}

\begin{prueba}{.} The inclusion $U_\sigma \subset {}^\Gamma U_\sigma$
defines the natural map in the second part of the statement (and
the same is true for $V$ and ${}^\Gamma V$ in the first part).

The localization of $\cR_\Gamma(\cM)$ and $\cR_U(\cM)$ with
respect to $u_1\cdots u_k$  are both naturally isomorphic to
$\cM[u,u^{-\bf 1 }]$ where ${-\bf 1}=(-1,\ldots,-1)$. So, the
kernel $\cK$ of the map $\alpha_\Gamma(\cM)$ is zero after
localization by $u_1\cdots u_k$, that means that $\cK$ is exactly
the $A_\Gamma$ torsion of $A_\Gamma\otimes_A \cR_U(\cM)$ because
$\cR_\Gamma(\cM)$ has no $A_\Gamma$-torsion.

For the first part, $\cR_V(\cD)$ is $A$-flat and then $A_\Gamma
\otimes_{A} \cR_V(\cD)$ is $A_\Gamma$-flat, so it has no
$A\Gamma$-torsion and then the natural surjective map is an
isomorphism.
\end{prueba}



\subsection{Return to $\cR_{FV}^\Gamma$}

xxxx If $I$ is a left ideal in $\cD$ we may consider the exact
sequence
$$ 0 \rightarrow \cR_{FV}^\Gamma(I) \rightarrow
\cR_{FV}^\Gamma(\cD) \rightarrow \cR_{FV}^\Gamma(M) \rightarrow
0$$ where $M=\frac{\cD}{I}$. We tensor this exact sequence by
$\otimes _{\CC[W]} \frac{\CC[W]}{(W)}$ to obtain $$
\cR_{FV}^\Gamma(I) \otimes_{\CC[W]} \frac{\CC[W]}{(W)} \rightarrow
\gr^\Gamma(\cD) \rightarrow \gr^\Gamma(M) \rightarrow 0$$

The image of $\cR_{FV}^\Gamma(I) \otimes_{\CC[W]}
\frac{\CC[W]}{(W)}$ in $\gr^\Gamma(\cD)$ is $\gr^F(I)$.

Let $P$ be an element in $\gr^\Gamma(I)$. We can see $P$ as the
image of $\widetilde{P}\otimes 1$ for some $\widetilde{P}$ in
$\cR_{FV}^\Gamma(I)$.

Let   $\{P_1,\ldots,P_\ell\}$ be an  standard basis of
$\gr^\Gamma(I)$ with respect to the $(X,\Delta)$-- filtration and
denote $E$ the ideal of $\NN*^{2n}$ generated by the privileged
exponents of the $P_i$. We choose $\widetilde{P_i}$ in
$\cR^\Gamma(I)$ such that $P_i$ is the image of $\widetilde{P_i}$
by the morphism as before.

The objective is to provide us with a surjective morphism
$$ \widehat{\cR^\Gamma(\cD)}^{E\times \NN^{k+1}}
\stackrel{\hat{\kappa}}{\longrightarrow}
\frac{\widehat{\cR^\Gamma(\cD)}}{\cR^\Gamma(I)
\widehat{\cR^\Gamma(\cD)}}.$$ This morphism is defined by using a
division theorem in $\widehat{\cR^\Gamma(\cD)}$. This completion
is nothing but the inverse limit of the systems defined by the
quotients  $\frac{\cR^\Gamma(\cD)}{(W)^\ell}.$


\begin{lema} [see Atiyah-MacDonald (10.2)]
There is a natural exact sequence $$0 \rightarrow
\widehat{\cR^\Gamma(I)} \rightarrow \widehat{\cR^\Gamma(\cD)}
\rightarrow \widehat{\cR^\Gamma(M)} \rightarrow 0$$
\end{lema}

\begin{lema} [see Atiyah-MacDonald (10.3)]
There is a natural isomorphism $$\widehat{\cR^\Gamma(M)} \simeq
\widehat{\cR^\Gamma(\cD)}\otimes_{\cR^\Gamma(\cD)} \cR^\Gamma(M)$$
\end{lema}

\section{Flatening theorems in $\cD$-module theory: Rees modules}

\subsection{Commutative analytic flatner} Let us consider a
morphism $\pi : X \rightarrow Y$ where $X,Y$ are analytic spaces.
Let $\xi$ be a point in $X$ and $\eta= \pi(\xi)$. Denote
$A=\cO_{Y,\eta}$ and $B=\cO_{X,\xi}$. We are interested in finding
condition such that $\pi$ is flat at $\xi$ (i.e. such that the
ring morphism $\pi^* : A \rightarrow B$ is flat).

With the notation as in XXX we shall consider the $A$-module
surjective morphism $$A\{t\}^E \stackrel{\kappa}{\rightarrow }
B.$$

\begin{teorema}
$A \rightarrow B$ is flat if and only if $\kappa$ is bijective.
\end{teorema}

\begin{prueba}{.}
If $\kappa$ is bijective then $B$ is $A$--flat as $A\{t\}^E$ is
(because of the monotony of $E$). Suppose now $B$ is $A$--flat.

\begin{lema}
Let $(A,{\mathfrak m})$ be a noetherian local ring and $A
\rightarrow B_i$ a family of noetherian local $A$-algebras.
Suppose we have, for each $i$, a $B_i$--module $N_i$ finitely
generated over $B_i$. Write $N=\oplus_i N_i$ and consider
$K\subset N$ an $A$-submodule of $N$. Write $N' = N/K$. Then if
the morphism $$ N\otimes_A \kk  \rightarrow N'\otimes_A\kk$$ is an
isomorphism and $N'$ is $A$-flat, then $K=0$ and $N$ is $A$-flat.
\end{lema}

[[Ce lemme peut se lire aussi de cette fa\c{c}on: Si $N \rightarrow
N'$ est un morphisme surjectif et $N'$ n'est pas $A$-plat, alors
ce morphisme a un noyau no  nul]]
\end{prueba}







\section{Slopes and flatness}


Let us consider a few examples [[to be completed]]. \\ 1) Suppose
here $k=n=2$ and $\Gamma$ generated by two vectors $\{a_1,a_2\}$
with integer positive components $a_i=(a_{i1},a_{i2})$. Then the
dual cone is $\check{\Gamma}= \{(u_1,u_2)\in \QQ^2\,\vert\,
a_{i1}u_1+a_{i2}u_2 \geq 0; i=1,2\}$ and in this case $S_\Gamma =
... $ (please, check this last equality ... We have to consider
the points with integer components inside the generating rectangle
associated with the vectors $a_1,a_2$.) For example if the cone
$\Gamma$ is generated by the vectors $(1,0),(1,1)$ then $S_\Gamma
= \CC[u_1u_2^{-1},u_2]$. \\ 2) Let $L$ be a linear form in the
first quadrant of $(\QQ^k)^*$. We can consider the multifiltration
$V_\bullet^{\{L\}}(\cD)$ defined by
$$V_s^{\{L\}}(\cD) = \sum_{L(\sigma)\leq L(s)} V_\sigma(\cD).$$ An
operator $P=\sum_{\alpha \beta } p_{\alpha \beta} x^\alpha
\partial^\beta \in \cD$ is in $V_s^{\{L\}}(\cD)$ if and only if
$L(\beta-\alpha) \leq L(s)$ whenever $p_{\alpha \beta } \not=0$.
We have $V_s^{\{L\}}(\cD) = V_{s'}^{\{L\}}(\cD)$ if $L(s)=L(s')$.

For any pair $(\sigma, s)\in \ZZ^k \times \ZZ^k$ we have $L(\sigma) \leq
L(s)$ for any $L\in {\Gamma}$ if and only if $\sigma \in
s-\check{\Gamma}$. An homogeneous element of degree $s\in \ZZ^k$ in
$\cR^\Gamma(\cR)$ has the form
$$ Pu^s = (\sum_{P_\sigma \in V_\sigma(\cD); \sigma\in s-\check{\Gamma}}
P_\sigma) u^s$$ and then such an element can be written  as  $$
Pu^s = \left( (\sum_{\alpha \beta; \sigma\in s-\check{\Gamma};
\sigma-\beta+\alpha \gg 0} p_{\alpha \beta}^\sigma x^\alpha
\partial^\beta)u^{\sigma}\right)u^{s-\sigma}$$ for some $p_{\alpha
\beta}^\sigma \in \CC$. Here $\gg$ stands for the componentwise
partial order in $\ZZ^k$.




\section{Introduction} Let us consider a coherent module $\cM$ over the ring $\cD_X$ of differential operators on an
analytic manifold $X$. For any smooth hypersurface $Z$ of $X$,
Malgrange and Kashiwara defined a filtration along $Z$ for $\cD_X$
and the notion of a good filtration for $\cM$. Given a set of
transverse smooth hypersurfaces $Z_1, \ldots, Z_k$, Sabbah
considered in \cite{Sabbah} multifiltrations of $\cM$ indexed by
$k$-uples of relative integers. More precisely he deals with
linear combinations over ${\bf Q}^+$ of the filtrations $V_j$
along each hypersurface $Z_j$ and with refinements
$(V^{\Gamma}_{\un{s}})$ of the original multifiltration associated
with each rational polyedral simplicial cone in the positive
quadrant of $({\bf Q}^k)^{\star}$. The original multifiltration is
the one which corresponds to the case $\Gamma ={\bf N}^k$.

The aim of this paper is to clarify flatness properties which
appear in \cite{Sabbah}, namely the existence of a fan $\cE$, such
that for each cone $\Gamma $ in this fan, the Rees module of the
filtration $V^{\Gamma}$ is flat over the affine ring $A_{\Gamma}$
of the toric blowing-up of ${\CC}^k$ associated with the fan
$\cE$. Such a fan is called an adapted fan in \cite{Sabbah}. The
reason for this clearing up is that the appendix \cite{Sab-Cas} of
\cite{Sabbah} relies on division theorems which are not correct
because they would contain infinite processes involving the
differential variables.

One of the main consequences of the existence of an adapted fan as
developed in \cite{Sabbah} is that we thus complete the proof of
the existence of non trivial functional equations of
Bernstein-Sato type, following the argument of Sabbah in
\cite{Sabbah}, in a way detailed below. We must note here that
yhis proof has already been completed in Bahloul's paper
\cite{bahloul}, by a different method which avoids the reference
to a flatness property involved in the notion of an adapted fan.
Bahloul uses instead the analytic Gr\"obner fan as defined in
\cite{assi-castro-granger-jpaa-2}, and described also in
\cite{bahloul-takayama}, in a way that we will describe below.  It
is therefore not completely surprising that the adapted fan wanted
in \cite{Sabbah} is finally nothing else but the Gr\"obner fan.
This emphasizes the interest of Bahloul's proof which has also the
advantage of being constructive. This constructiveness is valuable
also in an algebraic setting. The main ingredient for our proof is
a criterion of flatness by M.Herrmann and U.Orbanz (see
\cite{Herrmann-Orbanz}), for graded modules over graded rings,
where the grading is indexed by an arbitrary commutative group. In
the statement of this result there is no reference to any
finiteness property. In our case the indexing group will be $\ZZ
^k$, and the ring a conical subring of the ring of Laurent
polynomials in $k$ variables with the obvious multigrading by
monomials.

Let us detail the functional equation problem. It should be
remarked that it gives rise to a situation where a multifiltration
along transverse hypersurfaces comes out in a natural way.  These
equations are of the type
$$P(\un{s})\un{f}^{\un{s}}=b(\un{s})f_1^{s_1+1}\cdots
f_k^{s_k+1},$$ where $\un{s}=(s_1,\cdots , s_k)$ is a $k$-uple of
indeterminates, and $f_1,\cdots , f_k$ are analytic functions on
$X$. They  are naturally written in the module $\cO _X\big[
s_1,\cdots , s_k,\frac{1}{f_1 \cdots f_k} \big]\un{f}^{\un{s}}$,
endowed with a structure of a $\cD _X[\un{s}]$-module, which can
be extended in a way discovered by B. Malgrange to a $\cD
_{X\times \CC ^k}$-module structure with action of the $2k$
variables $t_j, \partial _{t_j}$such that $s_j=-\partial
_{t_j}t_j.$ The module $\cM$ that we have then to consider is the
module generated over the ring $\cD _{X\times \CC ^k}$ by
$\un{f}^{\un{s}}$, with its naturally defined $k$-multifiltration
$V_{\bullet}(\cD _{X\times \CC ^k})\cdot \un{f}^{\un{s}}$ along
the hypersurfaces $t_1=0, \cdots , t_k=0$. The proof of Sabbah in
\cite{Sabbah} can be sketched as follows: let us define for any
linear  form $L\in (\QQ ^k)^{\star}$ with positive coefficients
$\alpha _j \geq 0$, a filtration $V^L_{\bullet}(\cM)$, associated
with the linear combination $\sum \alpha _jV_j$ of the basic $V$-
filtrations. Then the existence of a functional equation, in which
$b(\un{s})$ is a product of affine forms $L(\un{s})+c$ comes out
from the following three steps:

1) There are Bernstein-Sato polynomial $b_L(\un{s})$, relative to
each $L$, with functional  equations
$$b_L(\un{s})\un{f}^{\un{s}}\in V_{L(\un{s})-1}^L(\cD _{X\times \CC
^k})\cdot \un{f}^{\un{s}}.$$

2) The saturated filtration
$$\ov{V_s}(\cM)=\bigcap _{L\in (\QQ ^k)^{\star}}V_s^L(\cM)$$
can be defined by using only a {\em finite number} of fixed linear
forms.

3) The saturated filtration is good which means the existence of a
$k$-uple of integers $\kappa $, such that.
$$\forall s\in \NN ^k,  \;\; V_s^L(\cM)\subset \ov{V_s}(\cM)\subset V_{s+\kappa }^L(\cM)$$


The step 2) is the step where is made in \cite{Sabbah} an
essential use of the flatness property involved in the notion of
an adapted fan. This step is the one which Bahloul proves in
\cite{bahloul}, by constructive methods which do not use this
notion.

The plan of the paper is as follows.

In section 2, we recall with more details what are
V-multifiltrations, and Rees modules, and the refinements of these
filtration with respect to a rational diagram simplicial cone. We
end this section by defining the fiber at zero of these Rees
module over the affine ring of a toric blowing up. We aim to prove
that this is the fiber of a flat deformation if the cone is taken
in the Gr\"obner fan.

In section 3 three we recall the notion of an analytic standard
fan as developed in  \cite{assi-castro-granger-jpaa-2}, and we
sketch the easy generalisation which we need from the monogeneous
case treated in \cite{assi-castro-granger-jpaa-2} to the general
case.

In section 4 we finally prove that the Gr\"obner fan is an adapted
fan in the sense of \cite{Sabbah}

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