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add notes on the theory

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Nacho B. R. 2025-07-08 09:46:01 +02:00
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%%%%%%%%%%%%%%%%%% THEOREM STYLES AND NUMERATION
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%%%%%%%%%%%%%%%%%%
\begin{document}
%\author{Author 1, Author 2}
%\title[Short tittle] {tittle}
\title {Computation of $\mathcal A _e$-codimension}
%\address{Adress 1st author}
%\email{1stauthor@email.com}
%\address{Adress 2nd author}
%\email{2ndauthor@email.com}
%\thanks{Supported by }
%\subjclass[2000]{Primary XXXXX; Secondary YYYYY, ZZZZZ}
%\keywords{Some keywords}
%\begin{abstract} The abstract
%\end{abstract}
\maketitle
\section{Theoretical framework}
\subsection{The $\mathcal A_e$-tangent over the target}
Consider a polynomial map-germ $f\colon (\C^n,0)\to(\C^p,0),$ with coordinate functions $f_j(x)$, such that the map given by the first $n$ coordinates functions, \[\overline f\colon (\C^n,0)\to (\C^n,0)\] is finite. To avoid confusion, we write \[X=(\C^n,0)\quad \text{and}\quad \overline f\colon X\to (\C^n,0).\] Let \[g_1,\dots,g_m\in \overline f_*\cO_n\] be a minimal set of generators of $\overline f_*\cO_n$, which is the ring $\cO_X$, regarded as an $\cO_n$-algebra via composition with $\overline f$. From the fact that $X$ is Cohen-Macaulay and $\dim X=n$ it follows that there is an isomorphism
\[\bigoplus^m \cO_n\xrightarrow{\varphi}\overline f_*\cO_X,\]
given by
\[(H^k)_{k=1}^m\mapsto \sum_{k=1}^m H^k\cdot g_k,\] where $H^k\cdot g_k=(H^k\circ \overline f)\,g_k.$
\begin{rem}All our troubles have to do with the behaviour of the inverse of $\varphi$. Assuming $\overline f$ to be a polynomial mapping and the generators $g_k$ to be polynomials, the isomorphism $\varphi$ takes polynomial vectors to polynomial functions. However, the inverse $\varphi^{-1}$ need not take polynomials to polynomial vectors. A simple example is provided by $\overline f(x)=x+x^2$, with a single generator $g_1=1$. The function $h(x)=x$ cannot be expressed as a polynomial in $x+x^2$.
\end{rem}
Similarly, consider the the $\cO_n$-module\[\theta(f)=\bigoplus^p \overline f_*\cO_X\] and the isomorphism
\[\bigoplus^p\bigoplus^m \cO_n \xrightarrow{\psi}\overline f_*\cO_X,\]
given by
\[(H^k_j)_{\substack{1 \leq k \leq m \\ 1 \leq j \leq p}}\mapsto (\sum_{k=1}^m H^k_j\cdot g_k)_{1\leq j\leq p},\]
Finally consider the $\mathcal A_e$-tangent submodule
\[T\mathcal A_e(f)=\left\langle \pd{f}{x_i}\mid 1\leq i\leq n\right\rangle_{\cO_X}+\bigoplus^p f^*\cO_p,\]
where $f^*\cO_p$ stands for the subring of $\cO_n$ consisting of functions of the form $f^*H=H\circ f$, with $H\in \cO_p$ (Caution: composition is taken with $f$ and not with $\overline f$).
\begin{prop}\label{propGeneratorsOfTA}As an $\cO_n$-module, the space $T\mathcal A_e(f)$ is generated by the following collection of vectors \smallskip
\begin{itemize}
\item $\dfrac{\partial f}{\partial x_i}\,g_k$, with $1\leq i\leq n$, $1\leq k\leq m$
\medskip
\item $u_j^\alpha=(0,\dots,\overset{\stackrel{j}{\smile}}m_\alpha,\dots,0)$, with $1\leq j\leq p$, $\alpha=(\alpha_{n+1},\dots,\alpha_p)$ and $0\leq \alpha_\alpha\leq m$, and where
\[m_\alpha=f_{n+1}^{\alpha_{n+1}}\dots f_{p}^{\alpha_{p}}\]
\end{itemize}
\end{prop}
\begin{rem}\label{remAlpha}Question: is it enough to take $\alpha=(\alpha_{n+1},\dots,\alpha_p)$ with $0\leq \vert \alpha\vert \leq m$ instead?
\end{rem}
\subsection{The $\mathcal A_e$-codimension}
By definition, the $\mathcal A_e$-codimension of $f$ is
\[\codim_{\mathcal A_e}(f)=\dim_\C\frac{\theta(f)}{T\mathcal A_e(f)}\]
The difficulty in computing the $\mathcal A_e$-codimension resides in the fact that it is a subset of $\theta(f)$, thus it consists of vectors in $\cO_X$, but it is not a $\cO_X$ module. To solve this, we may move everything isomorphically to $\bigoplus^p\bigoplus^m\cO_n$, applying the isomorphism $\psi$, so that the calculation that we actually perform is
\[\codim_{\mathcal A_e}(f)=\dim_\C\frac{\bigoplus^p\bigoplus^m\cO_n}{\psi^{-1}(T\mathcal A_e(f))}\]
Since $\psi$ is an isomorphism of $\cO_n$-modules, the generators of $\psi^{-1}(T\mathcal A_e(f))$ are
\begin{itemize}
\item $\psi^{-1}\left(\dfrac{\partial f}{\partial x_i}\,g_k\right)$, with $1\leq i\leq n$, $1\leq k\leq m$
\medskip
\item $\psi^{-1}(u_j^\alpha)=(0,\dots,\varphi^{-1}(m_\alpha),\dots,0)$, with $1\leq j\leq p$, $\alpha=(\alpha_{n+1},\dots,\alpha_p)$ and $0\leq \alpha_\alpha\leq m$, and where
\[m_\alpha=f_{n+1}^{\alpha_{n+1}}\dots f_{p}^{\alpha_{p}}\]
\end{itemize}
\begin{rem}\label{remLessCalculations} Observe that, in order to compute $\psi^{-1}(u^\alpha_j)$, it is enough to compute $\varphi^{-1}(f_{\ell})$, for $n+1\leq \ell\leq p$
\end{rem}
\subsection{Computing $\mathcal A_e$-codimension by approximations}
The obstacle to compute $\mathcal A_e$-codimension is that we must compute the generators \[\psi^{-1}(\dfrac{\partial f}{\partial x_i}\,g_k) \quad\text{and}\quad \psi^{-1}(u_j^\alpha),\]which may even fail to have polynomial entries. This may be solved by introducing a decreasing filtration of $\bigoplus^p\bigoplus^m\cO_n$.
\begin{definition}\label{defFiltration}A \emph{complete, separated, origin supported filtration} of an $\cO_n$ module $M$ is a sequence
\[M_0\supseteq M_1\supseteq \dots \supseteq M_r\supset \dots\]
of $\cO_n$-modules, satisfying the following properties.
\begin{enumerate}
\item Completeness: $M_0=M$.
\smallskip
\item Separation: $\bigcap_{r=1}^\infty M_r=0$.
\smallskip
\item Origin supported: $\supp (M_r/M_{r+1})\subseteq \{0\}$, for all $r$.
\end{enumerate}
From now on, we just use the term filtration, and always assume them to be complete, separated and origin supported.
\end{definition}
\begin{rem}\label{remFiltrationConditions}
Completeness is easily achieved, by adding an extra initial term $M_0=M$ to any given filtration.
Separation may be though of as the property that ensures that the approximations that we will introduce later on actually converge to the desired result.
The origin supported condition is equivalent to the fact that $\supp (M/M_{r+1})\subseteq \{0\}$ (By induction, the stalks at $x\neq 0$ of $M_0$ and $M_r$ must agree). For the module $M=\bigoplus^p\bigoplus^m\cO_n$ that we study, this is in turn equivalent to the fact that, for all $r$, there exists $K$, such that $\bigoplus^p\bigoplus^m\mathfrak m_n^K\subseteq M_r$, where $\mathfrak m_n$ stands for the maximal ideal in $\cO_n$).
\end{rem}
\begin{definition}Given a filtration $M_\bullet$ of $\bigoplus^p\bigoplus^m\cO_n$, we define
\[d_r=\dim_\C\frac{\psi^{-1}(T\mathcal A_e(f))+M_r}{\psi^{-1}(T\mathcal A_e(f))+M_{r+1}}<\infty\]
(Observe that the fact that $d_r<\infty$ is a direct consequence of the origin supported condition for the filtration).
\end{definition}
Any such filtration gives a way to express the $\mathcal A_e$-codimension as an infinite sum, or as a ``degree $r$ aproximation'' plus a tail, that is:
\begin{enumerate}
\item$\codim_{\mathcal A_e}(f)=d_0+d_1+\dots$
\smallskip
\item $\codim_{\mathcal A_e}(f)=d_0+d_1+\dots+d_{r}+\dim_\C\frac{\psi^{-1}(T\mathcal A_e(f))+M_{r+1}}{\psi^{-1}(T\mathcal A_e(f))}$
\end{enumerate}
Both expressions are helpful in different situations and allow us to compute $\codim_{\mathcal A_e}(f)$ by a series of aproximations of $\psi^{-1}(T\mathcal A_e(f))$, defined as follows.
\begin{definition} Given a map $f$ as above and a filtration $M_\bullet$ of $\bigoplus^p\bigoplus^m\cO_n$. An \emph{approximating sequence} for $T\cA_e(f)$ (subject to $M_\bullet$) is a sequence $A_\bullet$ of $\cO_n$-submodules
$A_r\leq \bigoplus^p\bigoplus^m\cO_n$, called \emph{approximations}, such that
\[A_r\equiv \psi^{-1}(T\mathcal A_e(f))\mod M_{r+1}\]
or, equivalently,
\[\psi(A_r)\equiv T\mathcal A_e(f)\mod \psi(M_{r+1}).\]
\end{definition}
Then, clearly we have that
\[d_r=\dim_\C\frac{A_r+M_r}{A_r+M_{r+1}}.\]
\subsection{The computation method}\label{secComputationalMethod}
The notions of the previous sections gives two strategies to compute $\codim_{\mathcal A_e}(f)$ by computing the successive approximations $A_r$:
We start computing $A_r$ and $d_r$ for $r=0,1,2,\dots$ and, for each $r$,
\begin{enumerate}
\item If $d_r=0$, then find the biggest number $L$ such that \[d_{r-L}=\dots=d_{r-1}=d_r=0.\] If $L$ satisfies \textcolor{red}{something which can be decided directly from the expression of $f$ (This is Du Plessis' criterion, which we need to adapt slightly to our setup)}, then all $d_j$ are zero for $j\geq r$. In that case,
\[\codim_{\mathcal A_e}(f)=d_0+\dots+d_{r-L-1}.\]
\item If $\psi(A_r)=T\mathcal A_e(f)$, then
\[\codim_{\mathcal A_e}(f)=d_0+d_1+\dots+d_{r}+\dim_\C\frac{A_r+M_{r+1}}{A_r}\]
\end{enumerate}
\begin{rem}
On one hand, strategy (1) finishes if and only if $T\mathcal A_e(f)<\infty$. In particular, it is unable to decide whether of not $f$ is $\mathcal A$-finite. On the other hand, we are only able to use strategy (2) if at some point our approximation is exact. But, if we do, we are able to decide whether or not $f$ is $\mathcal A$-finite in finite time, since this happens if and only if $\dim_\C\frac{A_r+M_{r+1}}{A_r}<\infty$
\end{rem}
\begin{rem}\label{remDr}
Alternatively (but probably not equivalently in terms of computing efficiency) we may define
\[D_r=\dim_\C\frac{\bigoplus^p\bigoplus^m\cO_n}{\psi^{-1}(T\mathcal A_e(f))+M_{r+1}}=\dim_\C\frac{\bigoplus^p\bigoplus^m\cO_n}{A_r+M_{r+1}}.\]
Then we have an increasing sequence $D_0\leq D_1\leq\dots$ such that
\[\codim_{\mathcal A_e}(f)=\lim_{r\to\infty}D_r\]
and obviously $D_r=d_0+\dots d_r$, The two strategies above now look as follows:
We start computing $A_r$ and $D_r$ for $r=0,1,2,\cdots$, and for each $r$
\begin{enumerate}
\item If $D_r=D_{r-1}$, then find the biggest number $L$ such that \[D_{r-L}=\dots=D_{r-1}=D_r=0.\] There is a criterion by Du Plessis, computable in terms of $A_r$, which ensures that, if $L$ satisfies \textcolor{red}{something (We need to make this precise)}, then all remaining $D_j$ for $j\geq r$ are equal to $D_r$. Consequently,
\[\codim_{\mathcal A_e}(f)=D_r.\]
\item If $\psi(A_r)=T\mathcal A_e(f)$, then
\[\codim_{\mathcal A_e}(f)=D_r\]
\end{enumerate}
\end{rem}
\section{The algorithm} Assume that, given a map-germ $f$, for every integer $r$, we have
\begin{enumerate}
\item The module $M_r$ of a filtration $M_\bullet$ of $\bigoplus^p\bigoplus^m\cO_n$.
\item A procedure to compute, for any vector $v\in \theta(f)$ and every integer $r\geq 0$, a matrix $H$ such that
\[\psi(H)\equiv v\mod \psi(M_{r+1}).\]
(Although not strictly necessary, it isconvenient to ask for this procedure to return $H=0$ if $v\in \psi(M_{r+1})$, or at least if $v=0$).
\end{enumerate}
Then, the computational method of Section \ref{secComputationalMethod} can be implemented as follows:
{\sc Input:}
\begin{itemize}
\item The map $f$,
\item The generators $g_1,\dots,g_m$ of $\overline f_*\cO_X$.
\item (Optional) a degree bound $K$.
\end{itemize}
{\sc Output:}
\begin{itemize}
\item An integer $a$, which may be of the following forms:
\begin{itemize}
\item $a\geq 0$, in which case $\codim_{\mathcal A_e}(f)=a$
\item $a=-1$, in which case $\codim_{\mathcal A_e}(f)=\infty$
\item $a=-2$, in which case the degree bound $K$ has been surpassed during calculations (in a sense to be made precise later)
\end{itemize}
\end{itemize}
{\sc
Steps:}
\begin{enumerate}
\item {\sc Set}
\begin{itemize}
\item $a=-2$
\item $r=0$
\item $M=\bigoplus^p\bigoplus^m\cO_n$
\item {\sc Set} $v_1,\dots,v_q=$ the collection of generators of $T\mathcal A_e(f)$ as an $\cO_n$-module (see Proposition \ref{propGeneratorsOfTA})
\item $V_1,\dots,V_q=0$
\end{itemize}
\medskip
\item\label{ItemLoop}{\sc If $r$ is too big for $K$} \textcolor{red}{(To be made precise)}
\begin{itemize}
\item {\sc return $a$}
\end{itemize}
\medskip
\item\label{refItemElseFor} {\sc Else:}
\begin{enumerate}
\item {\sc For $i=1,\dots,q$}\label{refItemFor}
\begin{itemize}
\item {\sc Find a matrix $R_i$, such that}
\[\psi(R_i)\equiv v_i\mod \psi(M_{r+1}).\]
(See Remark \ref{remEfficiency})
\item $V_i=V_i+R_i$
\item $v_i=v_i-\psi(R_i)$
\end{itemize}
\medskip
\item $d_r=\dim_\C\dfrac{\langle V_1,\dots,V_q\rangle+M_r}{\langle V_1,\dots,V_q\rangle+M_{r+1}}$
\medskip
\item
{\sc If $v_1=\dots=v_q=0$, then}
\begin{itemize}
\item $T=\dim_\C\dfrac{\langle V_1,\dots,V_q\rangle+M_r}{\langle V_1,\dots,V_q\rangle+M_{r+1}}$
\smallskip
\item {\sc If $T=-1$}
\begin{itemize}
\item {\sc return $T$}
\end{itemize}
\smallskip
\item {\sc Else,}
\begin{itemize}
\item{\sc return $d_0+\dots+d_r+T$}
\end{itemize}
\end{itemize}
\item {\sc Else}
\begin{itemize}
\item {\sc If $d_r=0$ and we have enough consecutive $d_{r-L}=\dots=d_r=0$} \textcolor{red}{(To be made precise)},
\begin{itemize}
\item {\sc return $d_0+\dots+d_{r-L-1}$.}
\end{itemize}
\item {\sc Else,}
\begin{itemize}
\item $r=r+1$
\item {\sc go to step \ref{ItemLoop}}
\end{itemize}
\end{itemize}
\end{enumerate}
\end{enumerate}
\begin{rem}\label{remEfficiency} Step \ref{refItemFor} involves a huge amount of calculations. Remarks \ref{remAlpha}
and \ref{remLessCalculations} should be taken into account in order to reduce it. Without being very precise, in principle, it should be enough to compute at most the consecutive approximations of certain set of monomials, or of the products, $x_ig_j$ or something like that
\end{rem}
\begin{rem}\label{remEfficientQuotients}
If we replace $d_r$ by $D_r$ as in Remark \ref{remDr} and choose $M_r$ adequately, the calculations of $D_r$ required are calculations in a free module over a quotient ring, which could speed up calculations.
\end{rem}
\section{Filtration choices and their associated approximations}
The method explained above requires to
\begin{enumerate}
\item Chose a filtration $M_\bullet$ as in Definition \ref{defFiltration}
\item Computing aproximations $H\in M_r$ of residues $v\in \psi(M_{r})$ modulo the chosen filtration:
\end{enumerate}
We must take into account that at this point we do not know how to compute approximations modulo an arbitrary submodule $\psi(M_r)$, but only approximations of the form
\[\psi(H)\equiv v\mod \bigoplus^p \km_X^N.\]
Theoretically, this is not a problem, because the origin-supported condition together with the finiteness of $\overline f$ (see Remark \ref{remFiltrationConditions}) ensure that, for every $r$, there is some exponent $N$ such that
\[\bigoplus^p \km_X^N\subseteq M_r\] and clearly an approximation modulo $\bigoplus^p \km_X^N$ is a valid approximation modulo $M_r$ as well.
\begin{rem}Notice that we do not need to compute $N$ before proceeding to compute the approximations. We can fix a tentative $N$, compute the approximation modulo $\bigoplus^p \km_X^N$. Then compute its residue modulo $\psi(M_r)$. If the residue is zero, the desired approximation has been obtained. Otherwise, the residue of this approximation is approximated again with $N$ replaced by $N+1$.
\end{rem}
It is however likely that the choice of the filtration affects the efficiency of the algorithm. The idea of this section is to collect the different choices that we want to experiment with.
\subsection{The target-degree filtration}
For each $r$, take
\[M_r=\bigoplus^p\bigoplus^m \km_n^r\]
For this filtration, if we let \[\operatorname{ord}(\overline f)=\max\{\operatorname{ord}(f_1),\dots,\operatorname{ord}(f_n)\}\qquad \rho=\max\{\operatorname{ord}(g_1),\dots,\operatorname{ord}(g_n)\},\]
For each $r$, our tentative initial $N$ is
\[N= r\cdot \operatorname{ord}(\overline f)+\rho\]
\subsection{The source-degree filtration}
For each $r$, take
\[\mathfrak n_{k,r}=\{X^\alpha\mid \sum_{j=1}^n\alpha_j\cdot \operatorname{ord}(f_j)\operatorname{ord}(g_k)\geq r\}\]
The tentative initial $N$ is
\[N= r\]
\section{Procedures}
\begin{enumerate}
\item GeneratingSetOverC:
{\sc Input:} list of polynomials $H=h_1,\dots,h_n$
{\sc Output:}
\begin{itemize}
\item basis $Q=q_1,\dots,q_r$ of $\Span_\C\{h_1,\dots,h_n\}$.
\item $n\times r$ matrix $M$ with entries in $\C$ such that
\[H=MQ\]
\end{itemize}
Should consist just on calling \[{\sc coef (ideal(h1,\dots,h_n),x_1\cdot\dots\cdot x_n))}\] and applying Gauss reduction to the resulting matrix while keeping track of the steps to undo the process.
Remark: Ideally, we would provide a generating set over $\cO_n$, and a matrix $M$ with entries in $\cO_n$, but producing this may be too costly.
\bigskip
\item ListEntries:
{\sc Input:} list of vectors $V=v_1,\dots,v_n$ of size $p$
{\sc Output:}
\begin{itemize}
\item list of polinomials $h_1,\dots,h_m$ consisting of the entries of $v_i$ (in order of appearance)
\item the integer $p$
\end{itemize}
\bigskip
\item AproxModuloM:
{\sc Input:}
\begin{itemize}
\item polynomial $h$
\item module M
\end{itemize}
{\sc Output:}
\begin{itemize}
\item vector $H=H_1,\dots,H_m$ such that
\[h\equiv\varphi(H)\mod M\]
\end{itemize}
\end{enumerate}
\section{To do list}
\begin{enumerate}
\item Minimizing ammount of approximations: Much of the calculations are repeated, simply because they appear at different entries of vectors. It seems that, for every graded piece $M_r/M_{r+1}$ we should be pin down a set of monomials that need to be approximated, and use these to write down all the required vectors. Also, we may be able to exploit the expressions of the monomials in lower degree to speed up the current degree calculations.
Idea: replace $v_1,\dots,v_q$ by a basis of the $\C$-vector space spanned by all entries of $v_1,\dots,v_q$.
\item Replacement of generators: Figure out whether a generator $v$ can be replaced by a combination $\tilde v$ of the existing generators with target coefficients, in a way such that
\begin{itemize}
\item The submodule generated is the same using $v$ or $\tilde v$
\item The approximation of $\tilde v$ is exact
\end{itemize}
\end{enumerate}
%
%\section{\textcolor{red}{Old Stuff }}
%
%Consider a polynomial map-germ $f\colon (\C^n,0)\to(\C^p,0),$ with coordinate functions $f_i(x)$, such that the map given by the first $n$ coordinates functions, \[\overline f\colon (\C^n,0)\to (\C^n,0)\] is finite. To avoid confusion, we write \[X=(\C^n,0)\quad \text{and}\quad \overline f\colon X\to (\C^n,0).\] Let \[g_1,\dots,g_m\in \overline f_*\cO_n\] be a set of generators of $\overline f_*\cO_n$, which is the ring $\cO_X$, regarded as an $\cO_n$-algebra via composition with $\overline f$.
%Consider the $\cO_n$-module\[\theta(f)=\bigoplus^p \overline f_*\cO_X\cong \bigoplus^p\bigoplus^m \cO_n \]
%and the $\mathcal A_e$-tangent submodule
%\[T\mathcal A _e(f)\leq \theta(f).\]
%This is the smallest submodule of $\theta(f)$ containing all vectors \[h(x)\cdot\pd{f}{x_i},\quad h\in \cO_X, 1\leq i\leq n,\] and the submodule \[\bigoplus^p \C[f_1,\dots,f_p].\]
%\begin{prop}
%With the above notations, $T\mathcal A _e(f)$ is generated as an $\cO_n$-module by
%\begin{itemize}
%\item The vectors
%\[\pd{f}{x_i}\cdot g_\ell,\quad i=1,\dots,n,\ell=1,\dots,m\]
%\item The vectors
%\[(0,\dots,H,\dots,0),\]
%where $H$ is of the form $f_{n+1}^{\alpha_{n+1}}\dots f_{p}^{\alpha_{p}}$, with $0\leq \alpha_i\leq m-1$.
%\end{itemize}
%\end{prop}
%Let $\km$ be the maximal ideal in $\cO_X$ and write
%\[\mathcal M^{(j)}=\bigoplus^p \km^j\leq \theta(f)\]
%Consider a sequence of submodules $T\mathcal A _e(f)^{(j)}\subseteq \theta(f)$, such that
%\[T\mathcal A _e(f)^{(j)}\equiv T\mathcal A _e(f)\mod \cM^{(j+1)},\]
%and write
%\[d_j=\dim_\C\frac{T\mathcal A _e(f)^{(j)}+\cM^{(j)}}{T\mathcal A _e(f)^{(j)}+\cM^{(j+1)}}=\dim_\C\frac{T\mathcal A _e(f)+\cM^{(j)}}{T\mathcal A _e(f)+\cM^{(j+1)}}\]
%
%\begin{prop}[Case 1]Assume that there exists $k$ such that $T\mathcal A _e(f)^{(k)}=T\mathcal A _e(f)$. Then,
%\[\operatorname{cod}_{\mathcal A _e}(f)=d_0+d_1+d_2+\cdots+d_{k-1}+\dim_\C\frac{T\mathcal A _e(f)^{(k)}+\cM^{(k)}}{T\mathcal A _e(f)^{(k)}}\]
%\end{prop}
%
%\begin{prop}[Case 2]Assume that there exists $\ell$ as in Du Plesis result (\textcolor{red}{to be made precise}) and assume that there exists $k$ such that $d_{k+1}=\dots=d_{k+\ell}=0$. Then,
%\[\operatorname{cod}_{\mathcal A _e}(f)=d_1+d_2+\cdots+d_{k}.\]
%
%\end{prop}
%
%\section{to do list}
%\begin{enumerate}
%\item Replace $\cM^{(k)}$ by any decreasing sequence of $\cO_n$-modules
%\[\theta(f)=\cM^{(0)}\geq \cM^{(1)}\geq\dots\geq \cM^{(j)}\geq \cdots,\]
%such that
%\[\bigcap_j\cM^{(j)}=0\]
%(the idea is to compute aproximations modulo $\cM^{(j)}$, and then let $\cM^{(j+1)}$ be the smallest module containing the ``residues'', or something like that, in a way that, if the aproximation modulo $\cM^{(j)}$ is exact, then we may take $\cM^{(j+1)}=0$)
%\item Formalize the fact that we are going to compute things in $\bigoplus^p\bigoplus^m \cO_n$, and replace the corresponding modules by the isomorphic images of their aproximations modulo $\cM^{(j)}$ in $\bigoplus^p\bigoplus^m \cO_n$
%\end{enumerate}
\end{document}