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# Primary decomposition of modules:
a computational differential approach

Justin Chen  School of Mathematics, Georgia Institute of Technology, Atlanta,
Georgia<EMAIL_ADDRESS>and  Yairon Cid-Ruiz  Department of
Mathematics: Algebra and Geometry, Ghent University, Krijgslaan 281 – S25,
9000 Gent, Belgium<EMAIL_ADDRESS>
###### Abstract.

We study primary submodules and primary decompositions from a differential and
computational point of view. Our main theoretical contribution is a general
structure theory and a representation theorem for primary submodules of an
arbitrary finitely generated module over a polynomial ring. We characterize
primary submodules in terms of differential operators and punctual Quot
schemes. Moreover, we introduce and implement an algorithm that computes a
minimal differential primary decomposition for a module.

###### Key words and phrases:

primary decomposition, differential primary decomposition, primary submodule,
differential operators, Noetherian operators, punctual Quot scheme, Weyl
algebra, join of ideals.

###### 2010 Mathematics Subject Classification:

13N10, 13N99, 13E05, 14C05.

##  1\. Introduction

The existence of primary decompositions has long been known, since the
classical works of Lasker [lasker1905theorie] and Noether
[noether1921idealtheorie]: over a Noetherian commutative ring, every proper
submodule of a finitely generated module can be expressed as a finite
intersection of primary submodules. Accordingly, one can view primary
submodules as the basic building blocks for arbitrary modules. In this paper,
we study the central notions of primary submodules and primary decompositions
from a differential and computational point of view.

Let $\mathbb{k}$ be a field of characteristic zero and $R$ a polynomial ring
$R=\mathbb{k}[x_{1},\ldots,x_{n}]$. The main objective of this paper is to
characterize primary $R$-submodules with the use of _differential operators_
and _punctual Quot schemes_. We achieve this goal in Theorem 3.1 (see also
Section missing 3), which can be seen as an extension from ideals to
$R$-modules of the representation theorem given in [PRIM_IDEALS_DIFF_EQS].
Consequently, we introduce an algorithm that computes a minimal _differential
primary decomposition_ for modules. This algorithm, along with others (see
Section missing 4), have been implemented in the computer algebra system
Macaulay2 [MACAULAY2].

The program of characterizing ideal membership in a polynomial ring with
differential conditions was initiated by Gröbner [Groebner] in the 1930s, and
he successfully employed Macaulay’s theory of inverse systems to characterize
membership in an ideal primary to a rational maximal ideal. Nevertheless, a
complete differential characterization of primary submodules over a polynomial
ring was obtained in 1970 by analysts, in the form of the _Fundamental
Principle_ of Ehrenpreis [Ehrenpreis] and Palamodov [PALAMODOV]. Subsequent
algebraic approaches were given in [BRUMFIEL_DIFF_PRIM] and
[OBERST_NOETH_OPS]. More recently, the study of primary ideals and primary
submodules via differential operators has continued in e.g. [DAMIANO],
[NOETH_OPS], [PRIM_IDEALS_DIFF_EQS], [CHKL], [CCHKL] and [DIFF_PRIM_DEC].

Let $D_{n}$ denote the _Weyl algebra_
$D_{n}={\text{Diff}}_{R/\mathbb{k}}(R,R)=R\langle\partial_{x_{1}},\ldots,\partial_{x_{n}}\rangle$.
Let ${\mathfrak{p}}\in{\text{Spec}}(R)$ be a prime ideal, and $U\subseteq
R^{r}$ a ${\mathfrak{p}}$-primary $R$-submodule of a free $R$-module of rank
$r$. Following Palamodov’s terminology, we say that
$\delta_{1},\ldots,\delta_{m}\in
D_{n}^{r}\cong{\text{Diff}}_{R/\mathbb{k}}(R^{r},R)$ is a set of _Noetherian
operators_ representing $U$ if we have the equality

$U=\big{\\{}w\in R^{r}\mid\delta_{i}(w)\in{\mathfrak{p}}\text{ for all }1\leq
i\leq m\big{\\}}.$

In a similar fashion to [PRIM_IDEALS_DIFF_EQS], we parametrize primary
submodules via a number of different sets of objects, one of which yields a
set of Noetherian operators (see Theorem 3.1). We provide an algorithm that
computes a set of Noetherian operators for a submodule in §missing 4.1. In the
other direction, we give an algorithm that computes the submodule
corresponding to a set of Noetherian operators in §missing 4.1. The following
example displays some of the gadgets used in Theorem 3.1.

######  Example .

Let $R=\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]$ and
${\mathfrak{p}}=\left(x_{1}-x_{3},x_{2}-x_{4}\right)\in{\text{Spec}}(R)$. The
$R$-submodule

$U=\text{image}_{R}\small\begin{bmatrix}{x}_{1}-{x}_{3}&0&{x}_{2}-{x}_{4}&0\\\
-{x}_{2}+{x}_{4}&{x}_{1}-{x}_{3}&{x}_{2}{x}_{3}-{x}_{3}{x}_{4}&{x}_{2}^{2}-2\,{x}_{2}{x}_{4}+{x}_{4}^{2}\end{bmatrix}\subseteq
R^{2}$

is ${\mathfrak{p}}$-primary of multiplicity $3$ over ${\mathfrak{p}}$. Let
$\mathbb{F}=R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}=\mathbb{Q}(\overline{x_{1}},\overline{x_{2}})$
be the residue field of ${\mathfrak{p}}$, where
$\overline{x_{i}}\in\mathbb{F}$ denotes the class of $x_{i}\in R$. Under the
bijective correspondence (a) $\leftrightarrow$ (b) of Theorem 3.1, we obtain a
$\mathbb{F}[[y_{1},y_{2}]]$-submodule corresponding to $U$, namely

$\mathbb{V}=\text{image}_{\mathbb{F}[[y_{1},y_{2}]]}\small\begin{bmatrix}y_{1}&0&y_{2}\\\
-y_{2}&y_{1}&\overline{x_{1}}y_{2}\end{bmatrix}\subseteq\mathbb{F}[[y_{1},y_{2}]]^{2}.$

Since
$\dim_{\mathbb{F}}\big{(}\mathbb{F}[[y_{1},y_{2}]]^{2}/\mathbb{V}\big{)}=3$,
the submodule $\mathbb{V}$ corresponds to a point in the punctual Quot scheme
$\text{Quot}^{3}\big{(}\mathbb{F}[[y_{1},y_{2}]]^{2}\big{)}$. Employing the
correspondences (b) $\leftrightarrow$ (c) and (c) $\leftrightarrow$ (d) of
Theorem 3.1, we get the following set of Noetherian operators
$\delta_{1}=\small\begin{bmatrix}1\\\
0\end{bmatrix},\;\delta_{2}=\small\begin{bmatrix}0\\\
1\end{bmatrix},\;\delta_{3}=\small\begin{bmatrix}\partial_{x_{1}}-x_{1}\partial_{x_{2}}\\\
\partial_{x_{2}}\end{bmatrix}\in D_{4}^{2}$ for $U$. In other words, the
following equality holds

$U=\left\\{(w_{1},w_{2})\in
R^{2}\,\mid\,w_{1}\in{\mathfrak{p}},\;w_{2}\in{\mathfrak{p}}\,\text{ and
}\,\frac{\partial w_{1}}{\partial_{x_{1}}}-x_{1}\frac{\partial
w_{1}}{\partial_{x_{2}}}+\frac{\partial
w_{2}}{\partial_{x_{2}}}\in{\mathfrak{p}}\right\\}.$

Therefore, instead of describing $U$ via its generators, one could do so with
the Noetherian operators $\delta_{1},\delta_{2},\delta_{3}$, or with the point
in $\text{Quot}^{3}\big{(}\mathbb{F}[[y_{1},y_{2}]]^{2}\big{)}$ given by
$\mathbb{V}$.

Recently, the notion of a _differential primary decomposition_ for a module
was introduced in [DIFF_PRIM_DEC]. This notion is a natural generalization of
Noetherian operators for (not necessarily primary) modules. Let $U\subseteq
R^{r}$ be an $R$-submodule with associated primes
$\text{Ass}_{R}(R^{r}/U)=\\{{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{k}\\}\subseteq{\text{Spec}}(R)$.
We now wish to describe $U$ in the following way

$U=\big{\\{}w\in R^{r}\mid\delta(w)\in{\mathfrak{p}}_{i}\text{ for all
}\delta\in\mathfrak{A}_{i}\text{ and }1\leq i\leq k\big{\\}},$

where each $\mathfrak{A}_{i}\subseteq D_{n}^{r}$ is a finite set of
differential operators (for more details, see §missing 4.2). In
[DIFF_PRIM_DEC], it was shown that there exists a differential primary
decomposition for $U$ of size equal to the _arithmetic multiplicity_ of $U$
(see §missing 4.2, 4.2) and that this is the minimal possible size.

Building on Theorem 3.1 and results from [DIFF_PRIM_DEC], we introduce an
algorithm for computing minimal differential primary decompositions (see
§missing 4.2). This algorithm is an extension of [DIFF_PRIM_DEC, Algorithm
5.4] from ideals to modules. The following example shows that a minimal
differential primary decomposition need not be obtained by concatenating sets
of Noetherian operators for each primary component.

######  Example .

Let $R=\mathbb{Q}[x_{1},x_{2},x_{3}]$ and consider the $R$-submodule
$U=\text{image}_{R}\small\begin{bmatrix}{x}_{1}^{2}&{x}_{1}{x}_{2}&{x}_{2}^{2}\\\
{x}_{2}^{2}&{x}_{2}{x}_{3}&{x}_{3}^{2}\end{bmatrix}\subseteq R^{2}$. Following
the algorithm of Section missing 2, we can compute a primary decomposition
$U=U_{1}\cap U_{2}\cap U_{3}$, where

$\displaystyle
U_{1}=\text{image}_{R}\small\begin{bmatrix}{x}_{1}&{x}_{2}^{2}&0\\\
{x}_{3}&{x}_{3}^{2}&{x}_{2}^{2}-{x}_{1}{x}_{3}\end{bmatrix},$
$\displaystyle\quad
U_{2}=\text{image}_{R}\small\begin{bmatrix}{x}_{3}&{x}_{2}^{2}&0&{x}_{1}{x}_{2}&{x}_{1}^{2}\\\
0&0&{x}_{3}^{2}&{x}_{2}{x}_{3}&{x}_{2}^{2}\end{bmatrix},$ $\displaystyle
U_{3}=\text{image}_{R}$
$\displaystyle\small\begin{bmatrix}0&{x}_{1}^{2}&{x}_{2}^{2}&{x}_{1}{x}_{2}&0\\\
{x}_{1}&0&{x}_{3}^{2}&{x}_{2}{x}_{3}&{x}_{2}^{2}\end{bmatrix}.$

The $R$-submodules $U_{i}$ are primary with associated primes
${\mathfrak{p}}_{1}=\left(x_{2}^{2}-x_{1}x_{3}\right)$,
${\mathfrak{p}}_{2}=\left(x_{2},x_{3}\right)$,
${\mathfrak{p}}_{3}=\left(x_{1},x_{2}\right)$, respectively. The
multiplicities of $\\{U_{1},U_{2},U_{3}\\}$ over
$\\{{\mathfrak{p}}_{1},{\mathfrak{p}}_{2},{\mathfrak{p}}_{3}\\}$ are
$\\{1,5,5\\}$. By Theorem 3.1, we can describe $U_{1}$, $U_{2}$, $U_{3}$ by
sets of Noetherian operators of sizes $1$, $5$, $5$, respectively. These sets
of Noetherian operators give a differential primary decomposition for $U$ of
size $11$. However, this naive computation is not optimal, as
${\text{amult}}(U)=3$. Indeed, a minimal differential primary decomposition
for $U$ is given by

$U=\left\\{(w_{1},w_{2})\in
R^{2}\,\mid\,-x_{3}w_{1}+x_{1}w_{2}\in{\mathfrak{p}}_{1},\frac{\partial
w_{2}}{\partial_{x_{3}}}\in{\mathfrak{p}}_{2},\frac{\partial
w_{1}}{\partial_{x_{1}}}\in{\mathfrak{p}}_{3}\right\\},$

with only one differential operator needed per associated prime.

The basic outline of this paper is as follows. In Section missing 2, we review
classical primary decomposition, and present a general algorithm for modules.
In Section missing 3, we prove our representation theorem (Theorem 3.1) for
primary submodules of a free module, as well as an extension to arbitrary
finitely generated modules (Section missing 3). In Section missing 4, we
present several algorithms of a differential nature, including one for
computing minimal differential primary decompositions. In Section missing 5,
we present an intrinsic differential description of certain ideals that come
from the join construction. Finally, in Section missing 6, we illustrate the
various algorithms on examples with our Macaulay2 implementation.

##  2\. A general primary decomposition algorithm

We begin with a general algorithm for primary decomposition of modules,
inspired by the work of Eisenbud-Huneke-Vasconcelos [EHV], with a particular
focus on computational aspects. Although primary decomposition for ideals has
been well-studied in the literature, the case of modules is considerably less
prominent (which we hope to remedy with this exposition!); cf.
[rutman1992grobner] as well as [Indrees, idrees2014primary,
idrees2015algorithm] for some treatments.

We start with some reductions to the main case of interest. First, we reduce
to the case of polynomial rings: let $\mathbb{k}$ be an arbitrary field, and
$T$ a finitely generated $\mathbb{k}$-algebra. If
$R=\mathbb{k}[x_{1},\ldots,x_{n}]$ is a polynomial ring with a surjection
$\pi:R\twoheadrightarrow T$ (with corresponding closed embedding of spectra
$\pi^{*}:{\text{Spec}}(T)\hookrightarrow{\text{Spec}}(R)$), then for any
$T$-module $N$, one has
$\text{Ass}_{R}(N\big{|}_{R})=\pi^{*}(\text{Ass}_{T}(N))$, where
$N\big{|}_{R}$ denotes $N$ viewed as an $R$-module. In this way we may compute
associated primes and primary components of $N$ over $T$, by first computing
that of $N\big{|}_{R}$ over $R$, and then applying $\pi$.

Next, to simplify the notation on modules, note that if $M^{\prime}\subseteq
M$ is a submodule, then a primary decomposition of $M^{\prime}$ in $M$ can be
obtained by lifting a primary decomposition of $0$ in $M/M^{\prime}$. Thus we
may always take $M^{\prime}=0$ (a benefit afforded by working with general
modules), and state the problem as follows: for a finitely generated module
$M\neq 0$ over a polynomial ring $R$, find submodules
$Q_{1},\ldots,Q_{s}\subseteq M$ such that $\bigcap_{i=1}^{s}Q_{i}=0$ and
$|\text{Ass}_{R}(M/Q_{i})|=1$. The decomposition should moreover be _minimal_
, in the sense that $\bigcap_{j\neq i}Q_{j}\neq 0$ for all $i$, and also
$\text{Ass}_{R}(M/Q_{i})=\text{Ass}_{R}(M/Q_{j})\iff i=j$.

The primary decomposition algorithm described here proceeds in 2 steps: first,
find all associated primes of $M$, and second, determine valid $P_{i}$-primary
components $Q_{i}$ for each associated prime $P_{i}$ (note that by uniqueness
of associated primes from a primary decomposition, such a decomposition will
automatically be minimal). For the first step, following [EHV], we first
reduce the problem of computing all associated primes of a module, to
computing minimal primes of ideals:

######  Theorem 2.1 ([EHV, Theorem 1.1]).

For any $i\geq 0$, the associated primes of $M$ of codimension $i$ are
precisely the minimal primes of $\operatorname{ann}\text{Ext}_{R}^{i}(M,R)$ of
codimension $i$.

In view of this, we may compute $\text{Ass}_{R}(M)$ via oracles to compute (1)
a free resolution of a module $M$ (and thus any Ext modules
$\text{Ext}_{R}^{\bullet}(M,\cdot)$), and (2) minimal primes of any ideal
$I\subseteq R$, which we henceforth assume are given (in practice, both are
well-optimized in Macaulay2). Note that following the above procedure
iteratively will naturally produce a list of associated primes which are
weakly ordered by codimension (e.g. all codimension 1 primes appear before any
codimension 2 primes, etc.).

For the second step, namely producing valid primary components, we proceed
inductively. Order the associated primes $P_{1},\ldots,P_{s}$ of $M$ by a
linear extension of the partial order by inclusion, i.e. $P_{i}\subseteq
P_{j}\implies i\leq j$ (note that this is automatic if the associated primes
are weakly ordered by codimension, as in the previous paragraph). In
particular, as a base case $P_{1}$ is a minimal prime of $M$ (i.e. a minimal
prime of $\operatorname{ann}M$). Primary components to minimal primes are
uniquely determined, and can be obtained as follows:

######  Proposition .

Let $P\in{\text{Spec}}(R)$, and let $M\to M_{P}$ be the localization map.
Then:

  1. (1)

[EISEN_COMM, Theorem 3.10(d)] $\ker(M\to M_{P})$ equals the intersection of
all $P_{i}$-primary components of $0$ in $M$ for $P_{i}\in\text{Ass}_{R}(M)$,
$P_{i}\subseteq P$ (in particular, this intersection is uniquely determined by
$M$ and $P$).

  2. (2)

[EISEN_COMM, Proposition 3.13] Suppose $f\in R$ is such that for
$P_{i}\in\text{Ass}_{R}(M)$, one has $f\in P_{i}\iff P_{i}\not\subseteq P$.
Then $\ker(M\to M_{P})=\ker(M\to M_{f})=0:_{M}f^{\infty}$.

An element $f$ as in Section missing 2(2) can be obtained as follows: for each
associated prime $P_{i}$ not contained in $P$, choose a generator $g_{i}$ of
$P_{i}$ not contained in $P$; then take $f:=\prod_{P_{i}\not\subseteq
P}g_{i}$. Taking $P=P_{i}$ for some minimal prime $P_{i}$ of $M$ in Section
missing 2 shows that given an oracle to compute saturations, we may obtain
(the unique) primary components corresponding to minimal primes of $M$.

It then remains to compute a valid $P$-primary component $Q$, for an embedded
prime $P$. In this case such a $Q$ is not unique; indeed there are always
infinitely many valid choices for $Q$. The next proposition gives one class of
such choices:

######  Proposition  ([EHV], p. 27-28).

For $P\in\text{Ass}_{R}(M)$ and $j\geq 0$, fix generators
$P=(f_{1},\ldots,f_{m})$ for $P$, and set
$P^{[j]}:=(f_{1}^{j},\ldots,f_{m}^{j})$. Then for $j\gg 0$, the submodule
$Q_{[j]}:=\operatorname{hull}(P^{[j]}M,M)$ is a valid $P$-primary component of
$0$ in $M$.

Here $\operatorname{hull}(N,M)$ is the equidimensional hull of $N$ in $M$,
i.e. the intersection of all primary components of $N$ of maximal dimension.
There are a number of ways to compute $Q_{[j]}$: the first is via Section
missing 2, viz. $Q_{[j]}=\ker(M\to(M/P^{[j]}M)_{P})$. Another method is given
in [EHV, Theorem 1.1(2)], which is a general way to compute hulls via iterated
Ext modules, and yet another method is given in [EHV, Algorithm 1.2].

To find a stopping criterion for the exponent $j$ in Section missing 2, note
that inductively, we may assume that $P_{i}$-primary components for any
$P_{i}\subsetneq P$ have already been computed, and so their intersection
$\mathcal{V}:=\bigcap_{P_{i}\subsetneq P}Q_{i}$ is also known. By Section
missing 2, we also know $\mathcal{U}:=\bigcap_{P_{i}\subseteq P}Q_{i}$. Then
for any $j\geq 0$, the submodule $Q_{[j]}$ in Section missing 2 is a valid
$P$-primary component if and only if $Q_{[j]}\cap\mathcal{V}=\mathcal{U}$. We
may thus find a valid $P$-primary component as follows: initialize $j$ at some
starting value, and compute $Q_{[j]}$. If
$Q_{[j]}\cap\mathcal{V}\neq\mathcal{U}$, then increment $j$, and repeat until
a valid candidate $Q_{[j]}$ is found.

######  Remark .

A few remarks are in order concerning efficiency of the algorithm described
above:

  1. (1)

Both choices of starting value of $j$, and the function used to increment $j$,
are relevant considerations for efficiency of the algorithm. If the starting
value of $j$ is too small, or the increment function grows too slowly, then
invalid candidates may be computed many times. On the other hand, the
computation time for $Q_{[j]}$ tends to increase as $j$ increases, so it is
desirable not to take $j$ unnecessarily large. The current implementation in
Macaulay2 uses the increment $j\to\lceil\frac{3j}{2}\rceil$ (the starting
value is more complicated, depending on the degrees of generators of $P$ and
of $\operatorname{ann}M$).

  2. (2)

The use of bracket powers $P^{[j]}$ rather than ordinary powers $P^{j}$ in
Section missing 2 is also for efficiency: if either $j$ or the number of
generators of $P$ is large, then $P^{j}$ may take much longer to compute than
$P^{[j]}$.

  3. (3)

Of the 3 methods given above for computing embedded components $Q_{[j]}$,
usually the first method (namely as a kernel of a localization map) is the
most efficient, although this is not always the case (in some examples, the
second method can be drastically faster). Note that in the first method, it is
necessary to compute $\text{Ass}_{R}(M/P^{[j]}M)$, which is in general
strictly bigger than $\\{P\\}$.

  4. (4)

The necessity of realizing $\ker(M\to M_{P})$ as a saturation in Section
missing 2(2) stems from the fact that $M_{P}$ typically does not have a finite
presentation as an $R$-module. Although one could also express $\ker(M\to
M_{P})$ as a saturation $0:_{M}(P^{\prime})^{\infty}$, where $P^{\prime}$
equals the intersection of all associated primes of $M$ not contained in $P$,
it is almost always much more efficient to compute saturations by a single
element, than by general ideals.

  5. (5)

As a general rule, computation of associated primes is the most time-consuming
step in this procedure. Once the associated primes are known, the minimal
primary components tend to be computed very quickly, and the time for
computing the embedded components can vary based on the method chosen (cf.
point (3) above). In particular, this algorithm tends to perform well for
modules whose free resolutions can be cheaply computed (e.g. when the number
of variables is small).

In summary, the algorithm described here reduces general primary decomposition
for modules to the following tasks (some of which can be seen as special cases
of primary decomposition):

  1. (1)

computation of Ext modules (in fact, $\text{Ext}_{R}^{\bullet}(\cdot,R)$
suffices),

  2. (2)

computation of minimal primes of an ideal $I\subseteq R$,

  3. (3)

computation of colon modules (of which saturations and annihilators are
special cases),

  4. (4)

computation of intersections of submodules.

##  3\. A representation theorem for primary submodules

We now leave the classical picture, and adopt a differential point of view.
Our main theorem in this section parametrizes primary submodules of a free
module of finite rank in terms of _punctual Quot schemes_ , _vector spaces
closed under differentiation_ and _subbimodules of the Weyl-Noether module_.
This extends the main result of [PRIM_IDEALS_DIFF_EQS] to the case of modules.
As a simple corollary, we also obtain a representation theorem for primary
submodules of an arbitrary module.

######  Setup .

For the rest of this section we fix the following notation:

  * –

Let $\mathbb{k}$ be a field of characteristic zero, and
$R:=\mathbb{k}[x_{1},\ldots,x_{n}]$ a polynomial ring over $\mathbb{k}$.

  * –

For an integer $r\geq 0$, let $R^{r}$ be a free $R$-module of rank $r$.

  * –

Let ${\mathfrak{p}}\in{\text{Spec}}(R)$ be a prime ideal with codimension
$c:=\text{ht}({\mathfrak{p}})$.

  * –

The residue field of ${\mathfrak{p}}$ is denoted
$\mathbb{F}:=k({\mathfrak{p}})=\text{Quot}(R/{\mathfrak{p}})=R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}$.

  * –

A subset of variables
$\\{x_{i_{1}},\ldots,x_{i_{\ell}}\\}\subseteq\\{x_{1},\ldots,x_{n}\\}$ is
_independent_ modulo ${\mathfrak{p}}$ if their images in $R/{\mathfrak{p}}$
are algebraically independent over $\mathbb{k}$, or equivalently
$\mathbb{k}[x_{i_{1}},\ldots,x_{i_{\ell}}]\cap{\mathfrak{p}}=\\{0\\}$. After
possibly permuting the variables, we may assume that
$\\{x_{c+1},\ldots,x_{n}\\}$ is a _basis_ modulo ${\mathfrak{p}}$, i.e. a
maximal set of independent variables modulo ${\mathfrak{p}}$ (see [Mateusz,
Example 13.2]).

  * –

Let $\mathbb{L}:=\mathbb{k}(x_{c+1},\ldots,x_{n})$ denote the field of
rational functions in the basis variables (which is a purely transcendental
extension of $\mathbb{k}$), and $S$ be the polynomial ring
$S:=\mathbb{k}(x_{c+1},\ldots,x_{n})[x_{1},\ldots,x_{c}]$ (which is a
localization of $R$, as
$S\cong\mathbb{L}\otimes_{\mathbb{k}[x_{c+1},\ldots,x_{n}]}R$).

  * –

The _Weyl algebra_ and the _relative Weyl algebra_ are denoted by
$D_{n}:=R\big{<}\partial_{x_{1}},\ldots,\partial_{x_{n}}\big{>}$ and
$D_{n,c}:=R\big{<}\partial_{x_{1}},\ldots,\partial_{x_{c}}\big{>}\subseteq
D_{n}$, respectively.

  * –

The _multiplicity_ of a ${\mathfrak{p}}$-primary submodule $U\subseteq R^{r}$
is defined as
${\text{length}}_{R_{\mathfrak{p}}}\left(R_{\mathfrak{p}}^{r}/U_{\mathfrak{p}}\right)$.

  * –

For an integer $m\geq 0$, the _punctual Quot scheme_ is a parameter space
${\rm Quot}^{m}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\right)$ whose
$\mathbb{F}$-points parametrize all
$\mathbb{F}[[y_{1},\ldots,y_{c}]]$-submodules
$\mathbb{V}\subseteq\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}$ of colength $m$,
i.e. which satisfy
$\dim_{\mathbb{F}}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}/\mathbb{V}\right)=m$.

  * –

We say that $\delta_{1},\ldots,\delta_{m}\in
D_{n}^{r}\cong{\text{Diff}}_{R/\mathbb{k}}(R^{r},R)$ is a set of _Noetherian
operators_ for a ${\mathfrak{p}}$-primary submodule $U\subseteq R^{r}$ if the
following equality holds

(1) $U=\big{\\{}w\in R^{r}\mid\delta_{i}(w)\in{\mathfrak{p}}\text{ for all
}1\leq i\leq m\big{\\}}.$

We can now state our main result:

######  Theorem 3.1.

The following four sets of objects are in bijective correspondence:

  1. (a)

${\mathfrak{p}}$-primary $R$-submodules $U\subseteq R^{r}$ of multiplicity $m$
over ${\mathfrak{p}}$,

  2. (b)

$\mathbb{F}$-points in the punctual Quot scheme ${\rm
Quot}^{m}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\right)$,

  3. (c)

$m$-dimensional $\mathbb{F}$-subspaces of
$\,\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$ that are closed under differentiation,

  4. (d)

$m$-dimensional $\mathbb{F}$-subspaces of the Weyl-Noether module
$\mathbb{F}\otimes_{R}D_{n,c}^{r}$ that are $R$-bimodules.

Moreover, any basis of the $\mathbb{F}$-subspace in part (d) can be lifted to
Noetherian operators $\delta_{1},\ldots,\delta_{m}\in D_{n,c}^{r}$ for the
$R$-submodule $U$ in part (a).

We structure the proof of Theorem 3.1 as follows. The correspondence (a)
$\leftrightarrow$ (b) is detailed in Theorem 3.2. The map $\gamma$ defined in
(missing) 6 yields a bijection

(2)
$\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{${\mathfrak{p}}$-primary
$R$-submodules of $R^{r}$}\\\ \mbox{of multiplicity $m$ over
${\mathfrak{p}}$}\end{array}\right\\}\par&\longleftrightarrow&\left\\{\begin{array}[]{c}\mbox{points
in }{\rm Quot}^{m}(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r})\\\
\end{array}\right\\}\\\
U&\longrightarrow&\mathbb{V}=\gamma(U)+(y_{1},\ldots,y_{c})^{m}\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\\\
U=\gamma^{-1}(\mathbb{V})&\longleftarrow&\mathbb{V}.\end{array}$

The correspondence (b) $\leftrightarrow$ (c) is detailed in Theorem 3.3. We
regard the polynomial ring $\mathbb{F}[z_{1},\ldots,z_{c}]$ as an
$\mathbb{F}[[y_{1},\ldots,y_{c}]]$-module by letting $y_{i}$ act as
$\partial_{z_{i}}$, i.e. $y_{i}\cdot F:=\frac{\partial F}{\partial_{z_{i}}}$
for any $F\in\mathbb{F}[z_{1},\ldots,z_{c}]$. By Macaulay inverse systems (see
also (missing) 7, (missing) 8) we have a bijection

(3) $\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{points in }{\rm
Quot}^{m}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\right)\\\
\end{array}\right\\}\par&\longleftrightarrow&\left\\{\begin{array}[]{c}\mbox{$m$-dimensional
$\mathbb{F}$-subspaces of}\\\ \mbox{$\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$
closed under differentiation}\end{array}\right\\}\\\
\mathbb{V}&\longrightarrow&\mathbb{W}=\mathbb{V}^{\perp}\\\
\mathbb{V}=\mathbb{W}^{\perp}&\longleftarrow&\mathbb{W}.\\\ \end{array}$

Finally, the correspondence (c) $\leftrightarrow$ (d) is detailed in §missing
3.3. The map $\Omega$ defined in (missing) 9 yields a bijection

(4) $\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{$m$-dimensional
$\mathbb{F}$-subspaces of}\\\ \mbox{$\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$
closed under
differentiation}\end{array}\right\\}\par&\longleftrightarrow&\left\\{\begin{array}[]{c}\mbox{$m$-dimensional
$\mathbb{F}$-subspaces of}\\\ \mbox{$\mathbb{F}\otimes_{R}D_{n,c}^{r}$ that
are $R$-bimodules. }\end{array}\right\\}\\\
\mathbb{W}&\longrightarrow&\mathcal{E}=\Omega(W)\\\
\mathbb{W}=\Omega^{-1}(\mathcal{E})&\longleftarrow&\mathcal{E}.\\\
\end{array}$

Furthermore, by §missing 3.1, we can lift elements from
$\mathbb{F}\otimes_{R}D_{n,c}$ to $D_{n,c}$.

We now extend Theorem 3.1 to arbitrary modules. Let $M$ be a finitely
generated $R$-module which can be generated by $r$ elements, so that there is
a short exact sequence of $R$-modules

(5) $0\to K\to R^{r}\to M\to 0.$

Let $U\subseteq M$ be a ${\mathfrak{p}}$-primary $R$-submodule of $M$ of
multiplicity
$m=\text{length}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}}/U_{\mathfrak{p}})$ over
${\mathfrak{p}}$. There is a unique $R$-submodule $\widetilde{U}\subseteq
R^{r}$ containing $K$ such that $\widetilde{U}/K\cong U$, which we call the
_lift_ of $U$ to $R^{r}$. Since $R^{r}/\widetilde{U}\cong M/U$, it follows
that $\widetilde{U}$ is a ${\mathfrak{p}}$-primary submodule of $R^{r}$ with
the same multiplicity as $U$ over ${\mathfrak{p}}$. This convenient fact
allows us to lift ${\mathfrak{p}}$-primary submodules of $M$ to
${\mathfrak{p}}$-primary submodules of $R^{r}$. In terms of the syzygies
$K\subseteq R^{r}$ of $M$ in (missing) 5, we define the following objects:

  * –

Let $\mathbb{V}^{\prime}\subseteq\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}$ be the
$\mathbb{F}[[y_{1},\ldots,y_{c}]]$-submodule

$\mathbb{V}^{\prime}:=\gamma(K)+(y_{1},\ldots,y_{c})^{m}\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\subseteq\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}.$

  * –

Let $\mathbb{W}^{\prime}:=(\mathbb{V}^{\prime})^{\perp}$ be the corresponding
$m$-dimensional $\mathbb{F}$-subspace of $\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$
closed under differentiation.

  * –

Let $\mathcal{E}^{\prime}:=\Omega(W^{\prime})$ be the resulting
$m$-dimensional $\mathbb{F}$-subspace of $\mathbb{F}\otimes_{R}D_{n,c}^{r}$
which is an $R$-bimodule.

We can now state the extension of Theorem 3.1 to an arbitrary finitely
generated $R$-module.

######  Corollary .

With the above notation, the following four sets of objects are in bijective
correspondence:

  1. (a)

${\mathfrak{p}}$-primary $R$-submodules $U\subseteq M$ of multiplicity $m$
over ${\mathfrak{p}}$,

  2. (b)

$\mathbb{F}$-points $\mathbb{V}\subseteq\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}$
in the punctual Quot scheme ${\rm
Quot}^{m}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\right)$ with
$\mathbb{V}\supset\mathbb{V}^{\prime}$,

  3. (c)

$m$-dimensional $\mathbb{F}$-subspaces
$\mathbb{W}\subseteq\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$ that are closed under
differentiation with $\mathbb{W}\subseteq\mathbb{W}^{\prime}$,

  4. (d)

$m$-dimensional $\mathbb{F}$-subspaces
$\mathcal{E}\subseteq\mathbb{F}\otimes_{R}D_{n,c}^{r}$ of the Weyl-Noether
module that are $R$-bimodules with $\mathcal{E}\subseteq\mathcal{E}^{\prime}$.

Moreover, any basis of the $\mathbb{F}$-subspace in part (d) can be lifted to
Noetherian operators $\delta_{1},\ldots,\delta_{m}\in D_{n,c}^{r}$ for the
lift $\widetilde{U}\subseteq R^{r}$ of the $R$-submodule $U$ in part (a).

###  3.1. A basic recap on differential operators

In this subsection, we recall basic properties of differential operators to be
used in the proof of the main theorem (for further details, the reader is
referred to [EGAIV_IV, §16]). For $R$-modules $M$ and $N$, we regard
$\text{Hom}_{\mathbb{k}}(M,N)$ as an $(R\otimes_{\mathbb{k}}R)$-module, by
setting

$\left((s\otimes_{\mathbb{k}}t)\delta\right)(w)=s\delta(tw)\quad\text{ for all
}\delta\in\text{Hom}_{\mathbb{k}}(M,N),\;w\in M,\;s,t\in R.$

We use the bracket notation $[\delta,s](w):=\delta(sw)-s\delta(w)$ for
$\delta\in\text{Hom}_{\mathbb{k}}(M,N)$, $s\in R$ and $w\in M$.

Unless otherwise specified, whenever we consider an
$(R\otimes_{\mathbb{k}}R)$-module as an $R$-module, we do so by letting $R$
act via the left factor of $R\otimes_{\mathbb{k}}R$.

######  Definition .

Let $M$ and $N$ be $R$-modules. The $m$-th order $\mathbb{k}$-linear
differential operators, denoted
${\text{Diff}}_{R/\mathbb{k}}^{m}(M,N)\subseteq\text{Hom}_{\mathbb{k}}(M,N)$,
form an $(R\otimes_{\mathbb{k}}R)$-module that is defined inductively by

  1. (i)

${\text{Diff}}_{R/\mathbb{k}}^{0}(M,N):=\text{Hom}_{R}(M,N)$.

  2. (ii)

${\text{Diff}}_{R/\mathbb{k}}^{m}(M,N):=\big{\\{}\delta\in\text{Hom}_{\mathbb{k}}(M,N)\,\mid\,[\delta,s]\in{\text{Diff}}_{R/\mathbb{k}}^{m-1}(M,N)\text{
for all }s\in R\big{\\}}$.

The set of all $\mathbb{k}$-linear differential operators from $M$ to $N$ is
the $(R\otimes_{\mathbb{k}}R)$-module

${\text{Diff}}_{R/\mathbb{k}}(M,N):=\bigcup_{m=0}^{\infty}{\text{Diff}}_{R/\mathbb{k}}^{m}(M,N).$

Following the notation used in [NOETH_OPS, PRIM_IDEALS_DIFF_EQS,
DIFF_PRIM_DEC], subsets
$\mathcal{E}\subseteq{\text{Diff}}_{R/\mathbb{k}}(M,N)$ are viewed as systems
of differential equations, and their solution spaces over $\mathbb{k}$ are
defined as

${\rm Sol}(\mathcal{E}):=\big{\\{}w\in M\,\mid\,\delta(w)=0\text{ for all
}\delta\in\mathcal{E}\big{\\}}=\bigcap_{\delta\in\mathcal{E}}{\rm
Ker}(\delta).$

######  Example .

As $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ is a polynomial ring over a field
$\mathbb{k}$ of characteristic zero, ${\text{Diff}}_{R/\mathbb{k}}(R,R)$ is
the Weyl algebra
$D_{n}=R\langle\partial_{x_{1}},\ldots,\partial_{x_{n}}\rangle=\bigoplus_{\alpha\in\mathbb{N}^{n}}R\partial_{\mathbf{x}}^{\alpha}$.

We next describe differential operators via the module of principal parts.
Consider the multiplication map $\mu:R\otimes_{\mathbb{k}}R\to R$,
$s\otimes_{\mathbb{k}}t\mapsto st$, and define
$\Delta_{R/\mathbb{k}}:=\text{Ker}(\mu)$, which is an ideal in
$R\otimes_{\mathbb{k}}R$. One can alternatively define differential operators
as follows:

######  Proposition  ([AFFINE_HOPF_I, Proposition 2.2.3]).

Let $M,N$ be $R$-modules. Then ${\text{Diff}}_{R/\mathbb{k}}^{m}(M,N)$ is the
$(R\otimes_{\mathbb{k}}R)$-submodule of $\text{Hom}_{\mathbb{k}}(M,N)$
annihilated by $\Delta_{R/\mathbb{k}}^{m+1}$.

######  Definition .

Let $M$ be an $R$-module. The module of $m$-th principal parts of $M$ is
defined as

$P_{R/\mathbb{k}}^{m}(M):=\frac{R\otimes_{\mathbb{k}}M}{\Delta_{R/\mathbb{k}}^{m+1}\left(R\otimes_{\mathbb{k}}M\right)}.$

This is a module over $R\otimes_{\mathbb{k}}R$ and thus also over $R$. For
simplicity, set $P_{R/\mathbb{k}}^{m}:=P_{R/\mathbb{k}}^{m}(R)$.

For any $R$-module $M$, consider the universal map $d^{m}:M\to
P_{R/\mathbb{k}}^{m}(M)$, $w\mapsto\overline{1\otimes_{\mathbb{k}}w}$. The
next result is a fundamental characterization of differential operators.

######  Proposition  ([EGAIV_IV, Proposition 16.8.4], [AFFINE_HOPF_I, Theorem
2.2.6]).

Let $M$ and $N$ be $R$-modules and let $m\geq 0$. Then the following map is an
isomorphism of $R$-modules:

$\displaystyle{\left(d^{m}\right)}^{*}:\text{Hom}_{R}\left(P_{R/\mathbb{k}}^{m}(M),N\right)$
$\displaystyle\xrightarrow{\cong}{\text{Diff}}_{R/\mathbb{k}}^{m}(M,N),$
$\displaystyle\varphi\quad$ $\displaystyle\mapsto\quad\varphi\circ d^{m}.$

We recall an explicit description of differential operators on free modules.
Let $J\subseteq R$ be an ideal and consider the canonical map
$\pi:R\twoheadrightarrow R/J$. We wish to describe
${\text{Diff}}_{R/\mathbb{k}}^{m}(F,R/J)$ for a free $R$-module $F=R^{r}$.
Since
${\text{Diff}}_{R/\mathbb{k}}^{m}(F,R/J)\cong{{\text{Diff}}_{R/\mathbb{k}}^{m}(R,R/J)}^{r}$
(see e.g. [NOETH_OPS, Lemma 2.7]), it is enough to describe
${\text{Diff}}_{R/\mathbb{k}}^{m}(R,R/J)$. We have an induced map:

${\text{Diff}}_{R/\mathbb{K}}^{m}(\pi):{\text{Diff}}_{R/\mathbb{K}}^{m}(R,R)\to{\text{Diff}}_{R/\mathbb{K}}^{m}(R,R/J),\quad\delta\mapsto\overline{\delta}=\pi\circ\delta.$

######  Lemma  ([PRIM_IDEALS_DIFF_EQS, Lemma 2]).

With the above notation, the following statements hold:

  1. (i)

${\text{Diff}}_{R/\mathbb{k}}^{m}(R,R/J)=\bigoplus_{\lvert\alpha\rvert\leq
m}(R/J)\overline{\partial_{\mathbf{x}}^{\alpha}}$ where
$\overline{\partial_{\mathbf{x}}^{\alpha}}=\pi\circ\partial_{\mathbf{x}}^{\alpha}.$

  2. (ii)

${\text{Diff}}_{R/\mathbb{k}}^{m}(\pi)$ is surjective: a differential operator
$\epsilon=\sum_{\lvert\alpha\rvert\leq
m}\overline{s_{\alpha}}\overline{\partial_{\mathbf{x}}^{\alpha}}\in{\text{Diff}}_{R/\mathbb{K}}^{m}(R,R/J)$
with $s_{\alpha}\in R$ can be lifted to $\delta=\sum_{\lvert\alpha\rvert\leq
m}s_{\alpha}\partial_{\mathbf{x}}^{\alpha}\in{\text{Diff}}_{R/\mathbb{k}}^{m}(R,R)$.

The notation below will be useful for describing the
$(R\otimes_{\mathbb{k}}R)$-action:

######  Notation .

Let
$T:=R\otimes_{\mathbb{k}}R=\mathbb{k}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$
be a polynomial ring in $2n$ variables, where $x_{i}$ represents
$x_{i}\otimes_{\mathbb{k}}1$ and $y_{i}$ represents
$1\otimes_{\mathbb{k}}x_{i}-x_{i}\otimes_{\mathbb{k}}1$. The action of $T$ on
$\text{Hom}_{\mathbb{k}}(M,N)$ is defined as follows: for all
$\delta\in\text{Hom}_{\mathbb{k}}(M,N)$ and $w\in M$,

$(x_{i}\cdot\delta)(w)\,:=\,x_{i}\delta(w)\quad\text{ and
}\quad(y_{i}\cdot\delta)(w)\,:=\,\delta(x_{i}w)-x_{i}\delta(w)=\left[\delta,x_{i}\right](w)$

for all $1\leq i\leq n$.

######  Remark .

Viewing $P_{R/\mathbb{k}}^{m}$ and ${\text{Diff}}_{R/\mathbb{k}}^{m}(R,R/J)$
as $T$-modules yields the following useful descriptions:

  1. (i)

$P_{R/\mathbb{k}}^{m}\,=\,\bigoplus_{\alpha\in\mathbb{N}^{n},\lvert\alpha\rvert\leq
m}Ry_{1}^{\alpha_{1}}\cdots y_{n}^{\alpha_{n}}$.

  2. (ii)

Under the isomorphism
${\text{Diff}}_{R/\mathbb{k}}^{m}(R,R/J)\cong\text{Hom}_{R}(P_{R/\mathbb{k}}^{m},R/J)$
(cf. §missing 3.1), the dual basis element $\left(y_{1}^{\alpha_{1}}\cdots
y_{n}^{\alpha_{n}}\right)^{*}$ corresponds to the differential operator
$\frac{1}{\alpha_{1}!\cdots\alpha_{n}!}\overline{\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{n}}^{\alpha_{n}}}$.

###### Proof.

For explicit computations, see [DIFF_PRIM_DEC, §5]. ∎

The next proposition describes the differential operators of an arbitrary
finitely generated module.

######  Proposition .

Let $0\to K\to F\to M\to 0$ be a short exact sequence of $R$-modules where $F$
is $R$-free of finite rank. Let $N$ be an $R$-module. Then, for all $m\geq 0$,
we have

${\text{Diff}}_{R/\mathbb{k}}^{m}(M,N)\,=\,\big{\\{}\delta\in{\text{Diff}}_{R/\mathbb{k}}^{m}(F,N)\mid\delta(K)=0\big{\\}}.$

###### Proof.

By left exactness of ${\text{Diff}}_{R/\mathbb{k}}^{m}(\bullet,N)$, one has
${\text{Diff}}_{R/\mathbb{k}}^{m}(M,N)\hookrightarrow{\text{Diff}}_{R/\mathbb{k}}^{m}(F,N)$
(see e.g. [NOETH_OPS, Lemma 2.7]). The inclusion “$\subseteq$” is clear.
Conversely, if $\delta\in{\text{Diff}}_{R/\mathbb{k}}^{m}(F,N)$ and
$\delta(K)=0$, then there is a unique map
$\overline{\delta}\in\text{Hom}_{\mathbb{k}}(M,N)$ induced by $\delta$, and by
§missing 3.1, $\Delta_{R/\mathbb{k}}^{m+1}\cdot\delta=0$, which implies
$\Delta_{R/\mathbb{k}}^{m+1}\cdot\overline{\delta}=0$. Then
$\overline{\delta}\in{\text{Diff}}_{R/\mathbb{k}}^{m}(M,N)$ by §missing 3.1
again, and the desired result follows. ∎

###  3.2. Punctual Quot schemes and Macaulay inverse systems

We are now ready to begin describing the correspondences in our main result.
The purpose of this subsection is to parametrize ${\mathfrak{p}}$-primary
$R$-submodules of a free module $R^{r}$ via punctual Quot schemes and Macaulay
inverse systems.

Let $\mathbb{B}$ be the power series ring
$\mathbb{B}:=\mathbb{F}[[y_{1},\ldots,y_{c}]]$ over the residue field
$\mathbb{F}=R_{\mathfrak{p}}/{\mathfrak{p}}R_{\mathfrak{p}}$ and denote its
maximal ideal by ${\mathfrak{n}}:=(y_{1},\ldots,y_{c})\subseteq\mathbb{B}$.
The punctual Quot scheme ${\rm Quot}^{m}\left(\mathbb{B}^{r}\right)$
parametrizes all quotients

$\mathbb{B}^{r}\,=\,\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}\,\twoheadrightarrow\,T$

such that $T$ has finite length $m$, and consequently, $T$ is only supported
at the maximal ideal ${\mathfrak{n}}$. More precisely, the
$\mathbb{F}$-rational points of ${\rm Quot}^{m}\left(\mathbb{B}^{r}\right)$
correspond to $\mathbb{B}$-submodules $\mathbb{V}\subseteq\mathbb{B}^{r}$ such
that $\dim_{\mathbb{F}}(\mathbb{B}^{r}/\mathbb{V})=m$ (in this paper, _a point
in ${\rm Quot}^{m}\left(\mathbb{B}^{r}\right)$_ will always mean _an
$\mathbb{F}$-rational point in ${\rm Quot}^{m}\left(\mathbb{B}^{r}\right)$_).
This parameter space has been considered in several papers, e.g.
[baranovsky2000moduli], [ellingsrud1999irreducibility], [HENNI_GUIMARES]. When
$r=1$, ${\rm Quot}^{m}\left(\mathbb{B}^{r}\right)$ coincides with the
_punctual Hilbert scheme_ ${\rm
Hilb}^{m}\left(\mathbb{F}[[y_{1},\ldots,y_{c}]]\right)$ studied by Briançon
[Bri77] and Iarrobino [IARROBINO_HILB].

The following basic fact shows that any point in ${\rm
Quot}^{m}\left(\mathbb{B}^{r}\right)$ can be identified with a
$(\mathbb{B}/{\mathfrak{n}}^{m}\mathbb{B})$-submodule of
$\mathbb{B}^{r}/{\mathfrak{n}}^{m}\mathbb{B}^{r}$.

######  Proposition .

For any $\mathbb{B}$-submodule $\mathbb{V}\subseteq\mathbb{B}^{r}$ with
colength $m=\dim_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V}\right)$, one has
$\mathbb{V}\supseteq{\mathfrak{n}}^{m}\mathbb{B}^{r}$.

###### Proof.

Consider the associated graded module

$\text{gr}(\mathbb{B}^{r}/\mathbb{V})\;:=\;\bigoplus_{k=0}^{\infty}{\mathfrak{n}}^{k}\mathbb{B}^{r}/\left({\mathfrak{n}}^{k+1}\mathbb{B}^{r}+\left({\mathfrak{n}}^{k}\mathbb{B}^{r}\cap\mathbb{V}\right)\right)$

which satisfies
$\dim_{\mathbb{F}}\left(\text{gr}(\mathbb{B}^{r}/\mathbb{V})\right)=\dim_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V}\right)$.
If $\mathbb{V}\not\supseteq{\mathfrak{n}}^{m}\mathbb{B}^{r}$, then
$\left[\text{gr}(\mathbb{B}^{r}/\mathbb{V})\right]_{k}\neq 0$ for all $0\leq
k\leq m$, so
$m=\dim_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V}\right)=\dim_{\mathbb{F}}\left(\text{gr}(\mathbb{B}^{r}/\mathbb{V})\right)\geq
m+1$, a contradiction. ∎

Following the approach of [PRIM_IDEALS_DIFF_EQS], we have an injection

$\eta:R\hookrightarrow\mathbb{B}\,,\qquad\begin{matrix}x_{i}&\mapsto&y_{i}+\overline{x_{i}},&\\!\\!\\!\\!\\!\mbox{
for }1\leq i\leq c,\\\ x_{j}&\mapsto&\overline{x_{j}},&\quad\,\mbox{ for
}c+1\leq j\leq n,\end{matrix}$

where $\overline{x_{i}}$ denotes the class of $x_{i}$ in $\mathbb{F}$ for
$1\leq i\leq n$. We now define the induced injection

(6) $\gamma:R^{r}\hookrightarrow\mathbb{B}^{r},\quad(f_{1},\ldots,f_{r})\in
R^{r}\mapsto(\eta(f_{1}),\ldots,\eta(f_{r}))\in\mathbb{B}^{r}.$

The map $\gamma$ provides the first correspondence (a) $\leftrightarrow$ (b)
in our main theorem.

######  Theorem 3.2.

With the above notation, there is a bijection

$\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{${\mathfrak{p}}$-primary
$R$-submodules of $R^{r}$}\\\ \mbox{of multiplicity $m$ over
${\mathfrak{p}}$}\end{array}\right\\}\par&\longleftrightarrow&\left\\{\begin{array}[]{c}\mbox{points
in }{\rm Quot}^{m}(\mathbb{B}^{r})\\\ \end{array}\right\\}\\\
U&\longrightarrow&\mathbb{V}=\gamma(U)+{\mathfrak{n}}^{m}\mathbb{B}^{r}\\\
U=\gamma^{-1}(\mathbb{V})&\longleftarrow&\mathbb{V}.\end{array}$

###### Proof.

The canonical map $R^{r}\hookrightarrow S^{r}$, $U\subseteq R^{r}\mapsto
US^{r}\subseteq S^{r}$ gives a bijection between ${\mathfrak{p}}$-primary
$R$-submodules and ${\mathfrak{p}}S$-primary $S$-submodules (see e.g.
[MATSUMURA, Theorem 4.1]). By [PRIM_IDEALS_DIFF_EQS, Proposition 1], for any
$m\geq 0$ the map $\eta:R\hookrightarrow\mathbb{B}$ induces an isomorphism of
local rings
$S/{\mathfrak{p}}^{m}S\xrightarrow{\cong}\mathbb{B}/{\mathfrak{n}}^{m}\mathbb{B}$.
Accordingly, we obtain a commutative diagram

${R^{r}}$${S^{r}}$${\mathbb{B}^{r}}$${S^{r}/{\mathfrak{p}}^{m}S^{r}}$${\mathbb{B}^{r}/{\mathfrak{n}}^{m}\mathbb{B}^{r}.}$$\cong$$\gamma$

By §missing 3.2, any $\mathbb{B}$-submodule
$\mathbb{V}\subseteq\mathbb{B}^{r}$ with
$\dim_{\mathbb{F}}(\mathbb{B}^{r}/\mathbb{V})=m$ contains
${\mathfrak{n}}^{m}\mathbb{B}^{r}$. Similarly, any ${\mathfrak{p}}S$-primary
$S$-submodule $V\subseteq S^{r}$ with $\text{length}_{S}(S^{r}/V)=m$ contains
${\mathfrak{p}}^{m}S$. Therefore, the result follows because under the above
identifications the constancy of multiplicity equal to $m$ will not change. ∎

We next recall the well-known Macaulay inverse systems for modules. Consider
the injective hull $E:=E_{\mathbb{B}}(\mathbb{F})$ of the residue field
$\mathbb{F}\cong\mathbb{B}/{\mathfrak{n}}$ of $\mathbb{B}$. Since $\mathbb{B}$
is a formal power series ring, this can be identified with the set of inverse
polynomials:

$E\cong\mathbb{F}[y_{1}^{-1},\ldots,y_{c}^{-1}]$

(for more details see e.g. [Brodmann_Sharp_local_cohom, Lemma 11.2.3, Example
13.5.3] or [BRUNS_HERZOG, Theorem 3.5.8]). Let $\mathbb{E}$ be the polynomial
ring $\mathbb{E}:=\mathbb{F}[z_{1},\ldots,z_{c}]$, regarded as a
$\mathbb{B}$-module by letting $y_{i}$ act as $\partial_{z_{i}}$, i.e.
$y_{i}\cdot F:=\frac{\partial F}{\partial_{z_{i}}}$ for any $F\in\mathbb{E}$.
Since $\mathbb{F}$ has characteristic zero, there is an isomorphism of
$\mathbb{B}$-modules

$E\cong\mathbb{F}[y_{1}^{-1},\ldots,y_{c}^{-1}]\xrightarrow{\cong}\mathbb{F}[z_{1},\ldots,z_{c}]=\mathbb{E},\qquad\frac{1}{\mathbf{y}^{\alpha}}=\frac{1}{y_{1}^{\alpha_{1}}\cdots
y_{c}^{\alpha_{c}}}\mapsto\frac{\mathbf{z}^{\alpha}}{\alpha!}=\frac{z_{1}^{\alpha_{1}}\cdots
z_{c}^{\alpha_{c}}}{\alpha_{1}!\cdots\alpha_{c}!}.$

We describe Macaulay inverse systems via Matlis duality. Let
${\left(\bullet\right)}^{\vee}:=\text{Hom}_{\mathbb{B}}\left(\bullet,E\right)$
denote the Matlis dual functor (see e.g. [BRUNS_HERZOG, Theorem 3.2.13]). For
any $\mathbb{B}$-submodule $\mathbb{V}\subseteq\mathbb{B}^{r}$, there is a
natural identification

(7)
$\bigl{(}\mathbb{B}^{r}/\mathbb{V}\bigr{)}^{\vee}\cong\mathbb{V}^{\perp}:=\left\\{w\in\mathbb{E}^{r}\mid
v\cdot w=\sum_{i=1}^{r}v_{i}\cdot w_{i}=0\mbox{ for all
}v\in\mathbb{V}\right\\}\subseteq\mathbb{E}^{r}.$

On the other hand, any $\mathbb{B}$-submodule
$\mathbb{W}\subseteq\mathbb{E}^{r}$ is an $\mathbb{F}$-subspace of
$\mathbb{E}^{r}$ that is closed under differentiation, since $y_{i}$ acts as
the operator $\partial_{z_{i}}$. Thus we also obtain an identification

(8)
$\bigl{(}\mathbb{E}^{r}/\mathbb{W}\bigr{)}^{\vee}\cong\mathbb{W}^{\perp}:=\left\\{v\in\mathbb{B}^{r}\mid
v\cdot w=\sum_{i=1}^{r}v_{i}\cdot w_{i}=0\mbox{ for all
}w\in\mathbb{W}\right\\}\subseteq\mathbb{B}^{r}.$

The above discussion then recovers the following well-known result.

######  Theorem 3.3 (Macaulay inverse systems).

With the above notation, there is a bijection

$\begin{array}[]{ccc}\left\\{\begin{array}[]{c}\mbox{points in }{\rm
Quot}^{m}\left(\mathbb{B}^{r}\right)\\\
\end{array}\right\\}\par&\longleftrightarrow&\left\\{\begin{array}[]{c}\mbox{$m$-dimensional
$\mathbb{F}$-subspaces of}\\\ \mbox{$\mathbb{E}^{r}$ closed under
differentiation}\end{array}\right\\}\\\
\mathbb{V}&\longrightarrow&\mathbb{W}=\mathbb{V}^{\perp}\\\
\mathbb{V}=\mathbb{W}^{\perp}&\longleftarrow&\mathbb{W}.\\\ \end{array}$

###  3.3. The proof of the representation theorem

In this subsection, we complete the proof of Theorem 3.1. First, we recall
some notation and results from [PRIM_IDEALS_DIFF_EQS]. Every differential
operator $\delta\in D_{n,c}$ is a unique $\mathbb{k}$-linear combination of
standard monomials
$\,\mathbf{x}^{\alpha}\partial_{\mathbf{x}}^{\beta}=x_{1}^{\alpha_{1}}\cdots
x_{n}^{\alpha_{n}}\partial_{x_{1}}^{\beta_{1}}\cdots\partial_{x_{c}}^{\beta_{c}}$,
where $\alpha_{i},\beta_{i}\in\mathbb{N}$. Consider the _Weyl-Noether module_

$\mathbb{F}\otimes_{R}D_{n,c}^{r}:=\mathbb{F}\otimes_{R}{{\text{Diff}}_{R/\mathbb{k}[x_{c+1},\ldots,x_{n}]}(R,R)}^{r}\cong\mathbb{F}\otimes_{S}{{\text{Diff}}_{S/\mathbb{L}}(S,S)}^{r}$

(for the isomorphism on the right, see e.g. [NOETH_OPS, Lemma 2.7]). From
§missing 3.1 and the fact that $P_{S/\mathbb{L}}^{m}$ is a free $S$-module,
the Weyl-Noether module admits the following description

$\mathbb{F}\otimes_{R}D_{n,c}^{r}=\mathbb{F}\otimes_{S}\biggl{(}\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
m\end{subarray}}{{\text{Diff}}_{S/\mathbb{L}}^{m}(S,S)}^{r}\biggr{)}\cong\lim\limits_{\begin{subarray}{c}\longrightarrow\\\
m\end{subarray}}{\text{Diff}}_{S/\mathbb{L}}^{m}(S,\mathbb{F})^{r}={\text{Diff}}_{S/\mathbb{L}}(S,\mathbb{F})^{r}.$

Applying §missing 3.1 with $J={\mathfrak{p}}S$ gives
$\mathbb{F}\otimes_{R}D_{n,c}\cong{\text{Diff}}_{S/\mathbb{L}}(S,\mathbb{F})\cong\bigoplus_{\alpha\in\mathbb{N}^{c}}\mathbb{F}\overline{\partial_{\mathbf{x}}^{\alpha}}$.
We then get an isomorphism of $\mathbb{F}$-vector spaces:

$\omega:\mathbb{E}=\mathbb{F}[z_{1},\ldots,z_{c}]\rightarrow\mathbb{F}\otimes_{R}D_{n,c},\quad\mathbf{z}^{\alpha}\mapsto\overline{\partial_{\mathbf{x}}^{\alpha}}\;\text{
for all }\alpha\in\mathbb{N}^{c},$

which in turn induces an isomorphism

(9) $\Omega:\mathbb{E}^{r}\to\mathbb{F}\otimes_{R}D_{n,c}^{r}.$

Let $\mathfrak{A}$ be the polynomial ring
$\mathfrak{A}:=\mathbb{F}[y_{1},\ldots,y_{c}]$. Using §missing 3.1 over the
field $\mathbb{L}=\mathbb{k}(x_{c+1},\ldots,x_{n})$ with
$T_{S}=S\otimes_{\mathbb{L}}S=\mathbb{L}[x_{1},\ldots,x_{c},y_{1},\ldots,y_{c}]$
gives that
$\mathfrak{A}\cong\mathbb{F}\otimes_{\mathbb{L}}S\cong\mathbb{F}\otimes_{S}(S\otimes_{\mathbb{L}}S)\cong\mathbb{F}\otimes_{S}T_{S}$,
and that
$\mathbb{F}\otimes_{R}D_{n,c}^{r}\cong{\text{Diff}}_{S/\mathbb{L}}(S,\mathbb{F})^{r}\cong{\text{Diff}}_{S/\mathbb{L}}(S^{r},\mathbb{F})$
has a natural structure of $\mathfrak{A}$-module. We now identify the power
series ring $\mathbb{B}=\mathbb{F}[[y_{1},\ldots,y_{c}]]$ of §missing 3.2 with
the completion $\widehat{\mathfrak{A}_{\mathfrak{n}}}$ of $\mathfrak{A}$ with
respect to the maximal irrelevant ideal
${\mathfrak{n}}=(y_{1},\ldots,y_{c})\subseteq\mathfrak{A}$; by an abuse of
notation ${\mathfrak{n}}$ is seen interchangeably as an ideal in both
$\mathfrak{A}$ and $\mathbb{B}$. In this way, we get actions of $\mathfrak{A}$
on both $\mathbb{E}$ and $\mathbb{F}\otimes_{R}D_{n,c}$. Explicitly, for
$\alpha\in\mathbb{N}^{c}$ and $1\leq i\leq c$,

$y_{i}\cdot\mathbf{z}^{\alpha}:=\alpha_{i}z_{1}^{\alpha_{1}}\cdots
z_{i}^{\alpha_{i}-1}\cdots z_{c}^{\alpha_{c}}\quad\text{ and }\quad
y_{i}\cdot\overline{\partial_{\mathbf{x}}^{\alpha}}:=\left[\overline{\partial_{\mathbf{x}}^{\alpha}},x_{i}\right]=\alpha_{i}\overline{\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{i}}^{\alpha_{i}-1}\cdots\partial_{x_{c}}^{\alpha_{c}}}.$

Therefore the map $\Omega$ in (missing) 9 gives a bijection between
$\mathbb{F}$-vector subspaces of $\mathbb{E}^{r}$ closed under differentiation
and $\mathfrak{A}$-submodules of $\mathbb{F}\otimes_{R}D_{n,c}^{r}$. The
latter structure as an $\mathfrak{A}$-submodule is equivalent to being an
$R$-subbimodule of the Weyl-Noether module $\mathbb{F}\otimes_{R}D_{n,c}^{r}$.

###### Proof of Theorem 3.1.

The bijections (a) $\leftrightarrow$ (b) and (b) $\leftrightarrow$ (c) have
been described in Theorem 3.2 and Theorem 3.3, respectively. By the discussion
above, the map $\Omega$ in (missing) 9 provides the bijection (c)
$\leftrightarrow$ (d). Due to §missing 3.1, we can lift differential operators
from $\mathbb{F}\otimes_{R}D_{n,c}^{r}$ to $D_{n,c}^{r}$ (cf.
[PRIM_IDEALS_DIFF_EQS, Remarks 7, 8]).

To finish the proof of the theorem, it suffices to show that an
$\mathbb{F}$-basis of the subspace in (d) lifts to a set of Noetherian
operators for the submodule $U$ in (a). That is,

  1. (1)

let $U\subseteq R^{r}$ be a ${\mathfrak{p}}$-primary $R$-submodule of
multiplicity $m$ over ${\mathfrak{p}}$,

  2. (2)

by Theorem 3.2 let
$\mathbb{V}:=\gamma(U)+{\mathfrak{n}}^{m}\mathbb{B}^{r}\subseteq\mathbb{B}^{r}$
be the corresponding point in $\text{Quot}^{m}(\mathbb{B}^{r})$,

  3. (3)

by Theorem 3.3 let $\mathbb{W}:=\mathbb{V}^{\perp}\subseteq\mathbb{E}^{r}$ be
the corresponding $m$-dimensional $\mathbb{F}$-subspace of $\mathbb{E}^{r}$
closed under differentiation,

  4. (4)

let
$\mathcal{E}:=\Omega(\mathbb{W})\subseteq\mathbb{F}\otimes_{R}D_{n,c}^{r}\cong{\text{Diff}}_{S/\mathbb{L}}(S^{r},\mathbb{F})$
be the corresponding $R$-subbimodule of $\mathbb{F}\otimes_{R}D_{n,c}^{r}$,

then we claim $\text{Sol}(\mathcal{E})=U\otimes_{R}S$.

Similarly to [NOETH_OPS, Lemma 3.14] and [PRIM_IDEALS_DIFF_EQS, Proposition
3], the statements below hold:

  * •

${\text{Diff}}_{S/\mathbb{L}}^{m-1}(S^{r},\mathbb{F})\cong\text{Hom}_{S}\big{(}\text{P}_{S/\mathbb{L}}^{m-1}(S^{r}),\mathbb{F}\big{)}\cong\text{Hom}_{\mathbb{F}}\left(\mathfrak{A}^{r}/{\mathfrak{n}}^{m}\mathfrak{A}^{r},\mathbb{F}\right)$
by §missing 3.1 and Hom-tensor adjunction.

  * •

Since $\mathbb{V}\supseteq{\mathfrak{n}}^{m}\mathbb{B}^{r}$ (cf. §missing 3.2)
and
$\mathbb{B}^{r}/{\mathfrak{n}}^{m}\mathbb{B}^{r}\cong\mathfrak{A}^{r}/{\mathfrak{n}}^{m}\mathfrak{A}^{r}$,
we get
$\mathcal{F}\subseteq{\text{Diff}}_{S/\mathbb{L}}^{m-1}(S^{r},\mathbb{F})$
determined by
$\text{Hom}_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V},\mathbb{F}\right)$.
Here we have
$\mathcal{F}\cong\text{Hom}_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V},\mathbb{F}\right)\subseteq\text{Hom}_{\mathbb{F}}\left(\mathbb{B}^{r}/{\mathfrak{n}}^{m}\mathbb{B}^{r},\mathbb{F}\right)$.

  * •

$\text{Sol}(\mathcal{F})=U\otimes_{R}S$ (see [NOETH_OPS, Lemma 3.14(iv)] and
[PRIM_IDEALS_DIFF_EQS, Proposition 3(iii)]).

It thus suffices to show that $\mathcal{E}$ and $\mathcal{F}$ coincide as
$R$-subbimodules of $\mathbb{F}\otimes_{R}D_{n,c}^{r}$. By the perfect pairing
of [PRIM_IDEALS_DIFF_EQS, Proof of Theorem 6.1] or general duality results
(see e.g. [EISEN_COMM, Proposition 21.4]), there are isomorphisms

(10)
$\displaystyle\begin{split}\mathcal{F}&\cong\text{Hom}_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V},\mathbb{F}\right)\\\
&\cong\text{Hom}_{\mathbb{B}}\big{(}\mathbb{B}^{r}/\mathbb{V},\text{Hom}_{\mathbb{F}}(\mathbb{B}/{\mathfrak{n}}^{m},\mathbb{F})\big{)}\qquad(\text{by
Hom-tensor adjunction})\\\
&\cong\text{Hom}_{\mathbb{B}}\big{(}\mathbb{B}^{r}/\mathbb{V},\text{Hom}_{\mathbb{B}}(\mathbb{B}/{\mathfrak{n}}^{m},E)\big{)}\qquad(\text{by
\cite[cite]{[\@@bibref{}{EISEN_COMM}{}{}, Proposition 21.4]}})\\\
&\cong\text{Hom}_{\mathbb{B}}\left(\mathbb{B}^{r}/\mathbb{V},E\right)=\left(\mathbb{B}^{r}/\mathbb{V}\right)^{\vee}\qquad\quad(\text{by
Hom-tensor adjunction}).\end{split}$

Recall from §missing 3.2 that the isomorphism
$\left(\mathbb{B}^{r}/\mathbb{V}\right)^{\vee}\cong\mathbb{V}^{\perp}=\mathbb{W}$
is explicitly described by identifying the inverted monomial
$\frac{1}{\mathbf{y}^{\alpha}}=\frac{1}{y_{1}^{\alpha_{1}}\cdots
y_{c}^{\alpha_{c}}}$ with
$\frac{\mathbf{z}^{\alpha}}{\alpha!}=\frac{z_{1}^{\alpha_{1}}\cdots
z_{c}^{\alpha_{c}}}{\alpha_{1}!\cdots\alpha_{c}!}$ for all
$\alpha=(\alpha_{1},\ldots,\alpha_{c})\in\mathbb{N}^{c}$. On the other hand,
the isomorphism
$\left(\mathbb{B}^{r}/\mathbb{V}\right)^{\vee}\cong\mathcal{F}$ is explicitly
described by identifying the inverted monomial $\frac{1}{\mathbf{y}^{\alpha}}$
with the dual monomial $\left(\mathbf{y}^{\alpha}\right)^{*}$ and then with
$\frac{1}{\alpha!}\overline{\partial_{\mathbf{x}}^{\alpha}}$ (see §missing
3.1); notice that we have an explicit isomorphism

$\left(\mathbb{B}/{\mathfrak{n}}^{m}\right)^{\vee}=\text{Hom}_{\mathbb{B}}(\mathbb{B}/{\mathfrak{n}}^{m},E)=\left(0:_{\mathbb{F}[y_{1}^{-1},\ldots,y_{c}^{-1}]}{\mathfrak{n}}^{m}\right)\cong\text{Hom}_{\mathbb{F}}\left(\mathbb{B}/{\mathfrak{n}}^{m},\mathbb{F}\right),\qquad\frac{1}{\mathbf{y}^{\alpha}}\mapsto\left(\mathbf{y}^{\alpha}\right)^{*}.$

Thus $\mathcal{E}$ and $\mathcal{F}$ do indeed coincide as $R$-subbimodules of
$\mathbb{F}\otimes_{R}D_{n,c}^{r}$, as desired. ∎

Finally, we have the following consequence.

###### Proof of Section missing 3.

From the short exact sequence $0\to K\to R^{r}\to M\to 0$, an $R$-submodule
$U\subseteq M$ corresponds to a unique $R$-submodule $\widetilde{U}\subseteq
R^{r}$ such that $\widetilde{U}\supset K$, and since $M/U\cong
R^{r}/\widetilde{U}$, it follows that $\widetilde{U}$ is a
${\mathfrak{p}}$-primary submodule of $R^{r}$ of multiplicity $m$ over
${\mathfrak{p}}$.

Let $\widetilde{U}\subseteq R^{r}$ be a ${\mathfrak{p}}$-primary submodule of
multiplicity $m$ over ${\mathfrak{p}}$. By using the correspondences of
Theorem 3.1, we set
$\mathbb{V}=\gamma\big{(}\widetilde{U}\big{)}\mathbb{B}^{r}+{\mathfrak{n}}^{m}\mathbb{B}^{r}$,
$\mathbb{W}=\mathbb{V}^{\perp}$ and $\mathcal{E}=\Omega(\mathbb{W})$. In the
same way, let
$\mathbb{V}^{\prime}=\gamma(K)\mathbb{B}^{r}+{\mathfrak{n}}^{m}\mathbb{B}^{r}$,
$\mathbb{W}^{\prime}=(\mathbb{V}^{\prime})^{\perp}$ and
$\mathcal{E}^{\prime}=\Omega(\mathbb{W}^{\prime})$. Since the following four
conditions are equivalent

$\widetilde{U}\supset
K,\quad\mathbb{V}\supset\mathbb{V}^{\prime},\quad\mathbb{W}\subseteq\mathbb{W}^{\prime}\quad\text{and}\quad\mathcal{E}\subseteq\mathcal{E}^{\prime},$

the result follows directly from Theorem 3.1. ∎

##  4\. Differential algorithms

In this section, we present several algorithms, based on Section missing 3 and
the previous papers [PRIM_IDEALS_DIFF_EQS] and [DIFF_PRIM_DEC], that deal with
the task of representing modules via differential operators. Together, they
show that there currently exist powerful and increasingly versatile
differential tools to represent modules computationally. These algorithms are:

  1. (I)

§missing 4.1: compute a set of Noetherian operators for a primary submodule.

  2. (II)

§missing 4.1: compute the primary submodule determined by a set of
differential operators. This can be seen as the inverse process to §missing
4.1.

  3. (III)

§missing 4.2: compute a minimal differential primary decomposition for a
submodule.

###  4.1. Noetherian operators vs primary submodules

This subsection deals with the problem of representing a primary submodule via
Noetherian operators. We continue to use the notation of Section missing 3,
cf. Section missing 3. First, we give an algorithm to compute a set of
Noetherian operators for a primary submodule.

######  Algorithm  (Noetherian operators for a primary submodule).

Input: A ${\mathfrak{p}}$-primary submodule $U\subseteq R^{r}$ of $R^{r}$ of
multiplicity $m$ over ${\mathfrak{p}}$.
Output: A set of Noetherian operators $\delta_{1},\ldots,\delta_{m}\in
D_{n,c}^{r}$ that represents $U$ as in (missing) 1.

  1. (1)

Compute the $\mathbb{F}[[y_{1},\ldots,y_{c}]]$-module
$\mathbb{V}=\gamma(U)+(y_{1},\ldots,y_{c})^{m}\mathbb{F}[[y_{1},\ldots,y_{c}]]^{r}$
that corresponds to $U$ as in (missing) 2.

  2. (2)

Using linear algebra over $\mathbb{F}$, compute an $\mathbb{F}$-basis
$\\{B_{1},\ldots,B_{m}\\}\subseteq\mathbb{F}[z_{1},\ldots,z_{c}]^{r}$ for the
inverse system $\mathbb{W}=\mathbb{V}^{\perp}$ as in (missing) 3.

  3. (3)

Compute
$C_{1}:=\Omega(B_{1}),\ldots,C_{m}:=\Omega(B_{m})\in\mathbb{F}\otimes_{R}D_{n,c}^{r}$
as in (missing) 4.

  4. (4)

Return lifts of $C_{1},\ldots,C_{m}$ in $D_{n,c}^{r}$, as guaranteed by
§missing 3.1.

###### Proof of correctness of §missing 4.1.

The correctness of this algorithm follows from Theorem 3.1. ∎

In §missing 4.1 the output is a set of Noetherian operators in the relative
Weyl algebra $D_{n,c}^{r}$. We now consider the reverse process, starting from
operators in the whole Weyl algebra $D_{n}$. We start with some basic facts
regarding modules defined via differential operators.

######  Remark .

Let
$\delta_{1},\ldots,\delta_{m}\in{\text{Diff}}_{R/\mathbb{k}}(R^{r},R)=D_{n}^{r}$
be differential operators. Let $\mathcal{G}\subseteq D_{n}^{r}$ be the
$R$-bimodule generated by $\delta_{1},\ldots,\delta_{m}$. The following
statements hold:

  1. (i)

$\\{w\in R^{r}\mid\delta(w)\in{\mathfrak{p}}\text{ for all
}\delta\in\mathcal{G}\\}$ is a ${\mathfrak{p}}$-primary $R$-submodule of
$R^{r}$.

  2. (ii)

$\\{w\in R^{r}\mid\delta(w)\in{\mathfrak{p}}\text{ for all
}\delta\in\mathcal{G}\\}\subseteq\\{w\in
R^{r}\mid\delta_{i}(w)\in{\mathfrak{p}}\text{ for all }1\leq i\leq m\\}$, and
equality holds if and only if the right hand side is an $R$-submodule of
$R^{r}$.

###### Proof.

For more details, see [NOETH_OPS, §3] and specifically [NOETH_OPS, Proposition
3.5]. ∎

In view of §missing 4.1, it is desirable to treat the following “closure
operation”: given finitely many differential operators
$\delta_{1},\ldots,\delta_{m}\in D_{n}^{r}$, compute the corresponding
${\mathfrak{p}}$-primary $R$-submodule

$\\{w\in R^{r}\mid\delta(w)\in{\mathfrak{p}}\text{ for all
}\delta\in\mathcal{G}\\},$

where $\mathcal{G}\subseteq D_{n}^{r}$ is the $R$-bimodule generated by
$\delta_{1},\ldots,\delta_{m}$.

The subsequent arguments follow verbatim the techniques used in Section
missing 3. Here we use the whole sets of variables $y_{1},\ldots,y_{n}$ and
$z_{1},\ldots,z_{n}$ instead of just $y_{1},\ldots,y_{c}$ and
$z_{1},\ldots,z_{c}$, respectively. The only issue with taking whole sets of
variables is that Theorem 3.2 is no longer be valid (as [PRIM_IDEALS_DIFF_EQS,
Proposition 1] requires $\mathbb{F}/\mathbb{k}(x_{c+1},\ldots,x_{n})$ to be
algebraic), but we may circumvent this via §missing 4.1. We have a canonical
map

(11)
$\Phi:{\text{Diff}}_{R/\mathbb{k}}(R^{r},R)=D_{n}^{r}\rightarrow\mathbb{F}\otimes_{R}D_{n}^{r}.$

Following §missing 3.1, by §missing 3.1 (see §missing 3.1) we obtain the
isomorphism

(12)
$\mathbb{F}\otimes_{R}{\text{Diff}}_{R/\mathbb{k}}^{m}(R^{r},R)\cong\mathbb{F}\otimes_{R}\text{Hom}_{R}\big{(}P_{R/\mathbb{k}}^{m}(R^{r}),R\big{)}\cong\text{Hom}_{\mathbb{F}}\left(\frac{\mathfrak{A}^{r}}{{\mathfrak{n}}^{m+1}\mathfrak{A}^{r}},\mathbb{F}\right)$

where $\mathfrak{A}=\mathbb{F}[y_{1},\ldots,y_{n}]$ and
${\mathfrak{n}}=(y_{1},\ldots,y_{n})$. Let
$\mathbb{B}=\mathbb{F}[[y_{1},\ldots,y_{n}]]$ and
$\mathbb{E}=\mathbb{F}[z_{1},\ldots,z_{n}]$, and as before, consider
$\mathbb{E}$ as a $\mathbb{B}$-module by setting $y_{i}=\partial_{z_{i}}$ for
all $1\leq i\leq n$. As is (missing) 7 and (missing) 8, we can define
$\mathbb{V}^{\perp}$ and $\mathbb{W}^{\perp}$ for
$\mathbb{V}\subseteq\mathbb{B}^{r}=\mathbb{F}[[y_{1},\ldots,y_{n}]]^{r}$ and
$\mathbb{W}\subseteq\mathbb{E}^{r}=\mathbb{F}[z_{1},\ldots,z_{n}]^{r}$,
respectively. In this setting, Macaulay inverse systems (Theorem 3.3) are also
valid. Now, the equivalent map of $\gamma$ in (missing) 6 is given by

(13)
$\Gamma:R^{r}\hookrightarrow\mathbb{B}^{r}=\mathbb{F}[[y_{1},\ldots,y_{n}]]^{r},\quad(f_{1},\ldots,f_{r})\in
R^{r}\mapsto(\eta^{\prime}(f_{1}),\ldots,\eta^{\prime}(f_{r}))\in\mathbb{B}^{r}$

where $\eta^{\prime}:R\hookrightarrow\mathbb{B}$, $x_{i}\mapsto
y_{i}+\overline{x_{i}}$ for all $1\leq i\leq n$, and the equivalent of the map
$\Omega$ in (missing) 9 is given by

(14)
$\Psi:\mathbb{E}^{r}=\mathbb{F}[z_{1},\ldots,z_{n}]^{r}\rightarrow\mathbb{F}\otimes_{R}D_{n}^{r}$

and is induced by
$\mathbf{z}^{\alpha}\mapsto\overline{\partial_{\mathbf{x}}^{\alpha}}$ for all
$\alpha\in\mathbb{N}^{n}$ (in this setting, under the isomorphism (missing) 12
with $r=1$ and $|\alpha|\leq m$, the dual monomial
$\left(\mathbf{y}^{\alpha}\right)^{*}$ coincides with
$\overline{\partial_{\mathbf{x}}^{\alpha}}\in\mathbb{F}\otimes_{R}D_{n}$).

After the above discussion, we can present the following algorithm which can
be seen as an inverse process to §missing 4.1.

######  Algorithm  (Primary submodule that corresponds to a set of
differential operators).

Input: A prime ideal ${\mathfrak{p}}\in{\text{Spec}}(R)$ and a set of
differential operators $\delta_{1},\ldots,\delta_{m}\in D_{n}^{r}$.
Output: The ${\mathfrak{p}}$-primary $R$-submodule $\\{w\in
R^{r}\mid\delta(w)\in{\mathfrak{p}}\text{ for all }\delta\in\mathcal{G}\\}$ of
$R^{r}$, where $\mathcal{G}\subseteq D_{n}^{r}$ is the $R$-bimodule generated
by $\delta_{1},\ldots,\delta_{m}$.

  1. (1)

By using (missing) 11 and (missing) 14, compute
$A_{1}:=\Psi^{-1}(\Phi(\delta_{1})),\ldots,A_{m}:=\Psi^{-1}(\Phi(\delta_{m}))\in\mathbb{F}[z_{1},\ldots,z_{n}]^{r}$.

  2. (2)

Using linear algebra over $\mathbb{F}$, compute the inverse system

$\mathbb{V}:=\left(A_{1},\ldots,A_{m}\right)^{\perp}=\Big{\\{}v\in\mathbb{F}[[y_{1},\ldots,y_{n}]]^{r}\mid
v\cdot A_{i}=0\mbox{ for all }1\leq i\leq m\Big{\\}}.$

  3. (3)

Return the $R$-submodule $U:=\Gamma^{-1}(\mathbb{V})\subseteq R^{r}$ where
$\Gamma$ is the map in (missing) 13.

###### Proof of correctness of §missing 4.1.

Let $\mathcal{G}\subseteq D_{n}^{r}$ be the $R$-bimodule generated by
$\delta_{1},\ldots,\delta_{m}$, and let
$\mathcal{E}=\Phi(\mathcal{G})\subseteq\mathbb{F}\otimes_{R}D_{n}^{r}$. Then
$\mathbb{W}=\Psi^{-1}(\mathcal{E})\subseteq\mathbb{E}^{r}=\mathbb{F}[z_{1},\ldots,z_{n}]^{r}$
is an $\mathbb{F}$-vector subspace that is closed under differentiation (the
same arguments of (missing) 4). Also the equalities
$\mathbb{V}=\mathbb{W}^{\perp}\subseteq\mathbb{B}^{r}$ and
$\mathbb{W}=\mathbb{V}^{\perp}\subseteq\mathbb{E}^{r}$ hold. In a similar way
to the proof of Theorem 3.1 (see (missing) 10), we obtain
$\mathcal{E}\cong\text{Hom}_{\mathbb{F}}\left(\mathbb{B}^{r}/\mathbb{V},\mathbb{F}\right)$.
By the same general arguments of [NOETH_OPS, Lemma 3.14] and
[PRIM_IDEALS_DIFF_EQS, Proposition 3], it follows that $\\{w\in
R^{r}\mid\delta(w)\in{\mathfrak{p}}\text{ for all
}\delta\in\mathcal{G}\\}=\text{Sol}(\mathcal{E})=\Gamma^{-1}(\mathbb{V})=U$,
which is ${\mathfrak{p}}$-primary by §missing 4.1. ∎

###  4.2. Differential primary decomposition for modules

The main goal of this subsection is to provide an algorithm that computes a
_minimal differential primary decomposition_ for general modules. The concept
of a _differential primary decomposition_ was introduced in [DIFF_PRIM_DEC],
and an algorithm for the case of ideals was given in [DIFF_PRIM_DEC, Algorithm
5.4].

First, we extend the notation in Section missing 3, for multiple primes
${\mathfrak{p}}_{i}$. For a set of variables
$\mathcal{S}=\\{x_{i_{1}},\ldots,x_{i_{\ell}}\\}\subseteq\\{x_{1},\ldots,x_{n}\\}$,
consider the corresponding _relative Weyl algebra_ , which consists of
$\mathbb{K}[\mathcal{S}]$-linear differential operators on $R$:

$D_{n}(\mathcal{S}):={\text{Diff}}_{R/\mathbb{K}[\mathcal{S}]}(R,R)=R\big{<}\partial_{x_{i}}\mid
x_{i}\not\in\mathcal{S}\big{>}\subseteq
R\langle\partial_{x_{1}},\ldots,\partial_{x_{n}}\rangle=D_{n}.$

We now recall the definition of the main object of interest in this section.

######  Definition  (cf. [DIFF_PRIM_DEC, Definition 5.2, Definition 4.1]).

Let $U\subseteq R^{r}$ be an $R$-module, with associated primes
$\text{Ass}(R^{r}/U)=:\\{{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{k}\\}$. A
_differential primary decomposition_ of $U$ is a list of triples

$({\mathfrak{p}}_{1},\mathcal{S}_{1},\mathfrak{A}_{1}),\;({\mathfrak{p}}_{2},\mathcal{S}_{2},\mathfrak{A}_{2}),\;\ldots,\;({\mathfrak{p}}_{k},\mathcal{S}_{k},\mathfrak{A}_{k})$

where $\mathcal{S}_{i}$ is a basis modulo ${\mathfrak{p}}_{i}$ and
$\mathfrak{A}_{i}\subseteq D_{n}(\mathcal{S}_{i})^{r}$ is a finite set of
differential operators such that

$\qquad U_{\mathfrak{p}}\cap
R^{r}\,\,=\;\bigcap_{i:{\mathfrak{p}}_{i}\subseteq{\mathfrak{p}}}\\!\\!\big{\\{}w\in
R^{r}\mid\delta(w)\in{\mathfrak{p}}_{i}\text{ for all
}\delta\in\mathfrak{A}_{i}\big{\\}}\quad\text{for each
${\mathfrak{p}}\in\text{Ass}(R^{r}/U)$.}$

These conditions imply that $U\;=\;\big{\\{}w\in
R^{r}\mid\delta(w)\in{\mathfrak{p}}_{i}\text{ for all
}\delta\in\mathfrak{A}_{i}\text{ and }1\leq i\leq k\big{\\}}$. The _size_ of
the differential primary decomposition is defined to be
$\sum_{i=1}^{k}|\mathfrak{A}_{i}|$.

######  Definition  ([STV_DEGREE], [DIFF_PRIM_DEC, Definition 4.3]).

For an $R$-submodule $U\subseteq R^{r}$, its _arithmetic multiplicity_ is the
non-negative integer

${\text{amult}}(U)\,\,:=\sum_{{\mathfrak{p}}\in\text{Ass}(R^{r}/U)}\\!\\!\\!\text{length}_{R_{\mathfrak{p}}}\left(\text{H}_{{\mathfrak{p}}}^{0}\Big{(}R^{r}_{\mathfrak{p}}/U_{\mathfrak{p}}\Big{)}\right)\quad=\sum_{{\mathfrak{p}}\in\text{Ass}(R^{r}/U)}\\!\\!\\!\text{length}_{R_{\mathfrak{p}}}\left(\frac{\left(U_{\mathfrak{p}}:_{R^{r}_{\mathfrak{p}}}{({\mathfrak{p}}R_{\mathfrak{p}})}^{\infty}\right)}{U_{\mathfrak{p}}}\right).$

In [DIFF_PRIM_DEC], it was shown that the arithmetic multiplicity is the
minimal size for a differential primary decomposition. The following algorithm
computes a minimal differential primary decomposition for a module. We note
that this algorithm does not depend on computing a primary decomposition.

######  Algorithm  (Differential primary decomposition for modules).

Input: An $R$-submodule $U\subseteq R^{r}$.
Output: A differential primary decomposition for $U$ of minimal size ${\rm
amult}(U)$.

  1. (1)

Compute the set of associated primes
$\text{Ass}(R^{r}/U)=:\\{{\mathfrak{p}}_{1},\ldots,{\mathfrak{p}}_{k}\\}$,
e.g. via Theorem 2.1.

  2. (2)

For $i$ from $1$ to $k$ do:

    1. (2.1)

Compute a basis $\mathcal{S}_{i}$ modulo ${\mathfrak{p}}_{i}$, and let
$\mathbb{F}_{i}:=k({\mathfrak{p}}_{i})$ be the residue field of
${\mathfrak{p}}_{i}$.

    2. (2.2)

Compute $\mathcal{U}:=U_{{\mathfrak{p}}_{i}}\cap R^{r}$ – this is the
intersection of all primary components of $U$ whose associated prime is
contained in ${\mathfrak{p}}_{i}$, e.g. via Section missing 2.

    3. (2.3)

Compute $\mathcal{V}:=\mathcal{U}:_{R^{r}}{\mathfrak{p}}_{i}^{\infty}$ – this
is the intersection of all primary components of $U$ whose associated prime is
strictly contained in ${\mathfrak{p}}_{i}$, e.g. via Section missing 2.

    4. (2.4)

Find $m>0$ giving the isomorphism in [DIFF_PRIM_DEC, Proposition 4.5](i):

$\mathcal{V}/\mathcal{U}\xrightarrow{\cong}(\mathcal{V}+{{\mathfrak{p}}_{i}}^{m}R^{r})/(\mathcal{U}+{{\mathfrak{p}}_{i}}^{m}R^{r}).$

    5. (2.5)

By using Theorem 3.1 and §missing 4.1, compute the $\mathbb{F}_{i}$-vector
subspaces $\mathcal{E}$ and $\mathcal{H}$ of the Weyl-Noether module
$\mathbb{F}_{i}\otimes_{R}D_{n}(\mathcal{S}_{i})^{r}$. These are
$(R\otimes_{\mathbb{K}[\mathcal{S}_{i}]}R)$-modules that correspond to the
${\mathfrak{p}}_{i}$-primary submodules
$\mathcal{U}+{\mathfrak{p}}_{i}^{m}R^{r}$ and
$\mathcal{V}+{\mathfrak{p}}_{i}^{m}R^{r}$, respectively.

    6. (2.6)

Compute an $\mathbb{F}_{i}$-basis $\overline{\mathfrak{A}_{i}}$ of an
$\mathbb{F}_{i}$-vector space complement $\mathcal{G}$ of $\mathcal{H}$ in
$\mathcal{E}$, i.e. $\mathcal{E}=\mathcal{H}\oplus\mathcal{G}$.

    7. (2.7)

Lift the basis $\overline{\mathfrak{A}_{i}}$ to a subset
$\mathfrak{A}_{i}\subseteq D_{n}(\mathcal{S}_{i})^{r}$.

  3. (3)

Return the triples $({\mathfrak{p}}_{1},\mathcal{S}_{1},\mathfrak{A}_{1})$,
$\ldots$, $({\mathfrak{p}}_{k},\mathcal{S}_{k},\mathfrak{A}_{k})$.

###### Proof of correctness of §missing 4.2.

This algorithm is correct because it realizes the steps in the proof of
[DIFF_PRIM_DEC, Theorem 5.3 (i)], as generalized by Theorem 3.1 and §missing
4.1. (For the case of ideals, see [DIFF_PRIM_DEC, Algorithm 5.4].) ∎

######  Remark .

The reverse process of §missing 4.2 can also be achieved. If we are given a
differential primary decomposition
$({\mathfrak{p}}_{1},\mathcal{S}_{1},\mathfrak{A}_{1})$, $\ldots$,
$({\mathfrak{p}}_{k},\mathcal{S}_{k},\mathfrak{A}_{k})$ as the output of
§missing 4.2, then via §missing 4.1 we can compute the
${\mathfrak{p}}_{i}$-primary submodule $U_{i}\subseteq R^{r}$ that corresponds
to $\mathfrak{A}_{i}$, so that $U=\bigcap_{i=1}^{k}U_{i}$. In this way a
differential primary decomposition gives a primary decomposition as in Section
missing 2.

##  5\. A differential description of joins

In this short section, we provide an intrinsic differential description of the
ideal join of an ideal and a primary ideal with respect to the maximal
irrelevant ideal. This result was implicitly obtained in the proof of
[PRIM_IDEALS_DIFF_EQS, Theorem 7.1], but the statement was not explicitly
given as the objective was to get an extension of a result of Sullivant
[SULLIVANT_SYMB, Proposition 2.8].

Let $\mathbb{k}$ be a field of characteristic zero,
$R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a polynomial ring and
${\mathfrak{m}}=(x_{1},\ldots,x_{n})\subseteq R$ be the maximal irrelevant
ideal. If $J$ and $K$ are ideals in $R$, then their join is given by the new
ideal

$J\star
K\,\,\,:=\,\,\,\Big{(}J(\mathbf{v})\,+\,K(\mathbf{w})\,+\,\left(x_{i}-v_{i}-w_{i}\mid
1\leq i\leq n\right)\Big{)}\,\,\cap\,\,R,$

where $J(\mathbf{v})\subseteq\mathbb{k}[v_{1},\ldots,v_{n}]$ is the ideal $J$
with new variables $v_{i}$ substituted for $x_{i}$ and
$K(\mathbf{w})\subseteq\mathbb{k}[w_{1},\ldots,w_{n}]$ is the ideal $K$ with
$w_{i}$ substituted for $x_{i}$.

The next definitions are obtained by mimicking [PRIM_IDEALS_DIFF_EQS,
Definition 4].

######  Definition .

Let $M\subseteq R$ be an $\mathfrak{m}$-primary ideal. We compute a finite
dimensional $\mathbb{k}$-subspace
$\mathfrak{A}(M)\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$
of differential operators with constant coefficients by performing the
following steps:

  1. (i)

Interpret the variable $x_{i}$ as $\partial_{z_{i}}$ for $i=1,\ldots,n$.

  2. (ii)

Compute $M^{\perp}=\left\\{F\in\mathbb{k}[z_{1},\ldots,z_{n}]\mid f\cdot
F=0\mbox{ for all }f\in M\right\\}$.

  3. (iii)

Let
$\mathfrak{A}(M)\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$
be the image of $M^{\perp}$ under the map
$\mathbf{z}^{\alpha}\mapsto\partial_{\mathbf{x}}^{\alpha}$.

A $\mathbb{k}$-subspace
$V\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$ is said to be
_closed under the bracket operation_ if $[\delta,x_{i}]\in V$ for all
$\delta\in V$ and $1\leq i\leq n$. Recall that
$\left[\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{i}}^{\alpha_{i}}\cdots\partial_{x_{n}}^{\alpha_{n}},\,x_{i}\right]=\alpha_{i}\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{i}}^{\alpha_{i}-1}\cdots\partial_{x_{n}}^{\alpha_{n}}$
for all $\alpha\in\mathbb{N}^{n}$ and $1\leq i\leq n$.

######  Definition .

Let $V\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$ be a
finite dimensional $\mathbb{k}$-subspace closed under the bracket operation.
We compute an ${\mathfrak{m}}$-primary ideal $\mathfrak{B}(V)\subseteq R$ by
performing the following steps:

  1. (i)

Interpret the variable $x_{i}$ as $\partial_{z_{i}}$ for $i=1,\ldots,n$.

  2. (ii)

Let $W\subseteq\mathbb{k}[z_{1},\ldots,z_{n}]$ be the image of $V$ under the
map $\partial_{\mathbf{x}}^{\alpha}\mapsto\mathbf{z}^{\alpha}$.

  3. (iii)

Let $\mathfrak{B}(V)=W^{\perp}=\left\\{f\in R\mid f\cdot F=0\mbox{ for all
}F\in W\right\\}$.

The following result can be easily deduced from [PRIM_IDEALS_DIFF_EQS, Theorem
7.1].

######  Theorem 5.1.

Let $J\subseteq R$ be an ideal. Then, the following statements hold:

  1. (i)

If $M\subseteq R$ is an ${\mathfrak{m}}$-primary ideal, then $J\star
M\;=\;\big{\\{}f\in R\mid\delta\cdot f\in J\;\text{ for all
}\;\delta\in\mathfrak{A}(M)\big{\\}}.$

  2. (ii)

If $V\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$ is a
finite dimensional $\mathbb{k}$-subspace closed under the bracket operation,
then $\big{\\{}f\in R\mid\delta\cdot f\in J\;\text{ for all }\;\delta\in
V\big{\\}}\;=\;J\star\mathfrak{B}(V).$

###### Proof.

(i) This is the statement of [PRIM_IDEALS_DIFF_EQS, Theorem 7.1 (i)].

(ii) Let $V\subseteq\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$ be a
finite dimensional $\mathbb{k}$-subspace closed under the bracket operation,
and set $M$ to be the corresponding ${\mathfrak{m}}$-primary ideal
$M=\mathfrak{B}(V)$. Notice that $V=\mathfrak{A}(M)$ and $M=\mathfrak{B}(V)$,
as both $\mathfrak{A}(M)$ and $\mathfrak{B}(V)$ are computed via Macaulay
inverse systems. By [PRIM_IDEALS_DIFF_EQS, Theorem 7.1 (i)], we obtain
$\big{\\{}f\in R\mid\delta\cdot f\in J\;\text{ for all }\;\delta\in
V\big{\\}}\;=\;J\star M,$ and so the result follows. ∎

Of particular interest is the case when $J$ is a prime ideal. Let
${\mathfrak{p}}\in{\text{Spec}}(R)$ be a prime ideal. A
${\mathfrak{p}}$-primary ideal $Q\subseteq R$ is said to be _representable by
differential operators with constant coefficients_ if there exist
$\delta_{1},\ldots,\delta_{m}\in\mathbb{k}[\partial_{x_{1}},\ldots,\partial_{x_{n}}]$
such that $Q=\\{f\in R\mid\delta_{i}\cdot f\in{\mathfrak{p}}\text{ for all
}1\leq i\leq m\\}$. A ${\mathfrak{p}}$-primary $Q\subseteq R$ is said to be
_representable by the join construction_ if there exists an
${\mathfrak{m}}$-primary ideal $M\subseteq R$ such that $Q={\mathfrak{p}}\star
M$.

######  Corollary .

Let ${\mathfrak{p}}\in{\text{Spec}}(R)$ be a prime ideal and $Q\subseteq R$ be
a ${\mathfrak{p}}$-primary ideal. Then, $Q$ is representable by differential
operators with constant coefficients if and only if $Q$ is representable by
the join construction.

###### Proof.

It follows from Theorem 5.1. ∎

##  6\. Examples

We conclude by demonstrating the algorithms in Section missing 2 and Section
missing 4 on some examples. The main commands in our implementation are:

  * –

primaryDecomposition(Module): executes an implementation of the algorithm in
Section missing 2.

  * –

noetherianOperators: executes an implementation of §missing 4.1.

  * –

getModuleFromNoetherianOperators: executes an implementation of §missing 4.1.

  * –

differentialPrimaryDecomposition: executes an implementation of §missing 4.2.

######  Remark .

The algorithms above have been implemented in Macaulay2 [MACAULAY2].
primaryDecomposition is part of the default package PrimaryDecomposition, and
the algorithm for modules is available to use as of version $1.17$. The
differential algorithms will appear in the package NoetherianOperators [CCHKL]
starting from version $1.18$. For convenience, we have included these
functions in a separate ancillary file “modulesNoetherianOperators.m2”.

We start with a simple example that first appeared in [DIFF_PRIM_DEC, Example
4.4].

######  Example .

Let $R=\mathbb{Q}[x_{1},x_{2},x_{3}]$ and $U\subseteq R^{2}$ be the
$R$-submodule $U={\rm
image}_{R}\\!\small\begin{bmatrix}x_{1}^{2}&x_{1}x_{2}&x_{1}x_{3}\\\
x_{2}^{2}&x_{2}x_{3}&x_{3}^{2}\end{bmatrix}$. We compute a primary
decomposition and a minimal differential primary decomposition for $U$:

    Macaulay2, version 1.17.2.1
    i1 : load "modulesNoetherianOperators.m2";
    i2 : printPD = M -> apply(primaryDecomposition M, Q -> trim image(gens Q | relations Q));
    i3 : R = QQ[x_1,x_2,x_3];
    i4 : U = image matrix {{x_1^2,x_1*x_2,x_1*x_3}, {x_2^2,x_2*x_3,x_3^2}};
    i5 : M = R^2 / U;
    i6 : L1 = printPD M
    o6 = {image | 0 x_1 |, image | x_1 x_2^2 0            |, image | x_3 x_2^2 0     x_1x_2 x_1^2 |}
                | 1 0   |        | x_3 x_3^2 x_2^2-x_1x_3 |        | 0   0     x_3^2 x_2x_3 x_2^2 |
    o7 : all(L1, isPrimary_M) and U == intersect L1
    o7 = true
    i8 : L2 = differentialPrimaryDecomposition U
                                       2
    o8 = {{ideal x , {| 1 |}}, {ideal(x  - x x ), {| -x_3 |}}, {ideal (x , x ), {| 0    |}}}
                  1   | 0 |            2    1 3    | x_1  |             3   2    | dx_3 |
    o8 : List
    i9 : U == intersect apply(L2, getModuleFromNoetherianOperators)
    o9 = true
    i10 : amult U
    o10 = 3
    Ψ

Notice that ${\text{amult}}(U)=3$ is the size of the computed differential
primary decomposition.

######  Example .

Let $R=\mathbb{Q}[x_{1},x_{2},x_{3},x_{4}]$ and $U\subseteq R^{2}$ be the
$R$-submodule

$U\;=\;{\rm
image}_{R}\\!\small\begin{bmatrix}x_{1}x_{2}&x_{2}x_{3}&x_{3}x_{4}&x_{4}x_{1}\\\
x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&x_{4}^{2}\end{bmatrix}.$

    i11 : R = QQ[x1,x2,x3,x4];
    i12 : U = image matrix{{x1*x2,x2*x3,x3*x4,x4*x1}, {x1^2,x2^2,x3^2,x4^2}};
    i13 : M = R^2 / U;
    i14 : L1 = printPD M;
    i15 : all(L1, isPrimary_M) and U == intersect L1
    o15 = true
    i16 : netList transpose{associatedPrimes M, L1}

          +-----------------------------------+---------------------------------------------------------------------------+
    o16 = |ideal (x4, x2)                     |image | 0 x4 x2 |                                                          |
          |                                   |      | 1 0  0  |                                                          |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x3, x1)                     |image | 0 x3 x1 |                                                          |
          |                                   |      | 1 0  0  |                                                          |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x4, x3, x2)                 |image | x4 0  x2 0    x2x3 x3^6 |                                          |
          |                                   |      | 0  x4 x1 x3^2 x2^2 0    |                                          |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x4, x3, x1)                 |image | x1 x3 0  0    x3x4 x4^6 |                                          |
          |                                   |      | 0  x2 x1 x4^2 x3^2 0    |                                          |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x4, x2, x1)                 |image | x2 x4 0  x1x4 0    x1^6 |                                          |
          |                                   |      | 0  x3 x2 x4^2 x1^2 0    |                                          |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x3, x2, x1)                 |image | x3 x1 0  0    0         x2^6 |                                     |
          |                                   |      | 0  x4 x3 x2^2 x1^2-x2x4 0    |                                     |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x3 - x4, x2 - x4, x1 - x4)  |image | 1 x3-x4 x2-x4 x1-x4 |                                              |
          |                                   |      | 1 0     0     0     |                                              |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x3 + x4, x2 - x4, x1 + x4)  |image | -1 x3+x4 x2-x4 x1+x4 |                                             |
          |                                   |      | 1  0     0     0     |                                             |
          +-----------------------------------+---------------------------------------------------------------------------+
          |                           2     2 |                                                                           |
          |ideal (x2 + x4, x1 + x3, x3  + x4 )|image | x2+x4 x3  x1 x4 0     -x4 |                                        |
          |                                   |      | 0     -x4 x4 x3 x2+x4 x1  |                                        |
          +-----------------------------------+---------------------------------------------------------------------------+
          |ideal (x4, x3, x2, x1)             |image | x1x4 x3x4 x2x3 x1x2 x4^4 x3^3x4 x3^4 x2^3x3 x2^4 x1^3x2 x1^4 0    ||
          |                                   |      | x4^2 x3^2 x2^2 x1^2 0    0      0    0      0    0      0    x4^4 ||
          +-----------------------------------+---------------------------------------------------------------------------+
    i17 : L2 = differentialPrimaryDecomposition U;
    i18 : netList L2
          +-----------------------------------+---------------------------------------------------------------------------------+
    o18 = |ideal (x3, x1)                     |{| 1 |}                                                                          |
          |                                   | | 0 |                                                                           |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x4, x2)                     |{| 1 |}                                                                          |
          |                                   | | 0 |                                                                           |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x4, x3, x2)                 |{| -x1dx2 |, | -x1dx2^2 |, | -x1^2dx2^3-6x1dx2dx3 |, | -x1^2dx2^4-12x1dx2^2dx3 |}|
          |                                   | | 1      |  | 2dx2     |  | 3x1dx2^2+6dx3        |  | 4x1dx2^3+24dx2dx3       | |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x4, x3, x1)                 |{| -x2dx3 |, | -x2dx3^2 |, | -x2^2dx3^3-6x2dx3dx4 |, | -x2^2dx3^4-12x2dx3^2dx4 |}|
          |                                   | | 1      |  | 2dx3     |  | 3x2dx3^2+6dx4        |  | 4x2dx3^3+24dx3dx4       | |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x4, x2, x1)                 |{| -x3dx4 |, | -x3dx4^2 |, | -x3^2dx4^3-6x3dx1dx4 |, | -x3^2dx4^4-12x3dx1dx4^2 |}|
          |                                   | | 1      |  | 2dx4     |  | 3x3dx4^2+6dx1        |  | 4x3dx4^3+24dx1dx4       | |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x3, x2, x1)                 |{| -x4dx1 |, | -x4dx1^2 |, | -x4^2dx1^3-6x4dx1dx2 |, | -x4^2dx1^4-12x4dx1^2dx2 |}|
          |                                   | | 1      |  | 2dx1     |  | 3x4dx1^2+6dx2        |  | 4x4dx1^3+24dx1dx2       | |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x3 - x4, x2 - x4, x1 - x4)  |{| -1 |}                                                                         |
          |                                   | | 1  |                                                                          |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x3 + x4, x2 - x4, x1 + x4)  |{| 1 |}                                                                          |
          |                                   | | 1 |                                                                           |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |                           2     2 |                                                                                |
          |ideal (x2 + x4, x1 + x3, x3  + x4 )|{| -x3 |}                                                                        |
          |                                   | | x4  |                                                                         |
          +-----------------------------------+---------------------------------------------------------------------------------+
          |ideal (x4, x3, x2, x1)             |{| -2dx1dx2dx3^2-2dx1^2dx3dx4           |}                                       |
          |                                   | | 2dx1dx2^2dx3+dx1^2dx3^2+2dx1dx3dx4^2 |                                        |
          +-----------------------------------+---------------------------------------------------------------------------------+
    i19 : U == intersect apply(L2, getModuleFromNoetherianOperators)
    o19 = true
    i20 : amult U
    o20 = 22

Again, ${\text{amult}}(U)=22$ is the size of the computed differential primary
decomposition.

## Acknowledgments

We thank Rida Ait El Manssour, Marc Härkönen, Anton Leykin and Bernd Sturmfels
for several joyful discussions on the topics appearing in this paper. We are
especially grateful to Marc Härkönen for his improvements to some of the
Macaulay2 implementations, and to Bernd Sturmfels for his constant guidance
and support, throughout the duration of this project.

## References