arXiv:2104.03385v1 [math.AC] 7 Apr 2021
A COMPUTATIONAL DIFFERENTIAL APPROACH
JUSTIN CHEN AND YAIRON CID-RUIZ
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.
1. INTRODUCTION
The existence of primary decompositions has long been known, since the classical works of Lasker [25] and Noether [28]: 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.
Letkbe a field of characteristic zero and R a polynomial ringR=k[x1, . . . ,xn]. The main objective of this paper is to characterize primaryR-submodules with the use ofdifferential operatorsand punctual Quot schemes. We achieve this goal in Theorem 3.2 (see also Corollary 3.3), which can be seen as an extension from ideals toR-modules of the representation theorem given in [8]. Consequently, we introduce an algorithm that computes a minimal differential primary decomposition for modules. This algorithm, along with others (seeSection 4), have been implemented in the computer algebra systemMacaulay2[16].
The program of characterizing ideal membership in a polynomial ring with differential conditions was initiated by Gr¨obner [17] 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 dif- ferential characterization of primary submodules over a polynomial ring was obtained in 1970 by analysts, in the form of theFundamental Principleof Ehrenpreis [12] and Palamodov [30]. Subsequent algebraic approaches were given in [4] and [29]. More recently, the study of primary ideals and primary submodules via differential operators has continued in e.g. [11], [9], [8], [7], [6] and [10].
LetDn denote the Weyl algebra Dn=DiffR/k(R,R) =Rh∂x1, . . . ,∂xni. Let p∈Spec(R)be a prime ideal, andU⊆Rrap-primaryR-submodule of a freeR-module of rankr. Following Palamodov’s termi- nology, we say thatδ1, . . . ,δm∈Drn=∼ DiffR/k(Rr,R)is a set of Noetherian operatorsrepresenting Uif we have the equality
U=
w∈Rr|δi(w)∈pfor all 16i6m .
2010Mathematics Subject Classification. 13N10, 13N99, 13E05, 14C05.
Key words and phrases. primary decomposition, differential primary decomposition, primary submodule, differential opera- tors, Noetherian operators, punctual Quot scheme, Weyl algebra, join of ideals.
1
In a similar fashion to [8], we parametrize primary submodules via a number of different sets of objects, one of which yields a set of Noetherian operators (seeTheorem 3.2). We provide an algorithm that com- putes a set of Noetherian operators for a submodule inAlgorithm 4.1. In the other direction, we give an algorithm that computes the submodule corresponding to a set of Noetherian operators inAlgorithm 4.3.
The following example displays some of the gadgets used inTheorem 3.2.
Example 1.1. LetR=Q[x1,x2,x3,x4]andp = (x1−x3,x2−x4)∈Spec(R). TheR-submodule U=imageR
x1−x3 0 x2−x4 0
−x2+x4 x1−x3 x2x3−x3x4 x22−2x2x4+x24
⊆R2
isp-primary of multiplicity 3 overp. LetF=Rp/pRp=Q(x1,x2)be the residue field ofp, wherexi∈F denotes the class ofxi∈R. Under the bijective correspondence (a) ↔ (b)ofTheorem 3.2, we obtain a F[[y1,y2]]-submodule corresponding toU, namely
V=imageF[[y1,y2]]
y1 0 y2
−y2 y1 x1y2
⊆F[[y1,y2]]2. Since dimF F[[y1,y2]]2/V
=3, the submodule V corresponds to a point in the punctual Quot scheme Quot3 F[[y1,y2]]2
. Employing the correspondences (b)↔(c)and(c)↔(d)ofTheorem 3.2, we get the following set of Noetherian operatorsδ1=
1 0
, δ2= 0
1
, δ3=
∂x1−x1∂x2
∂x2
∈D24forU. In other words, the following equality holds
U=
(w1,w2)∈R2 | w1∈p, w2∈p and ∂w1
∂x1
−x1
∂w1
∂x2
+∂w2
∂x2
∈p
.
Therefore, instead of describing U via its generators, one could do so with the Noetherian operators δ1,δ2,δ3, or with the point in Quot3 F[[y1,y2]]2
given byV.
Recently, the notion of adifferential primary decompositionfor a module was introduced in [10]. This notion is a natural generalization of Noetherian operators for (not necessarily primary) modules. LetU⊆Rr be anR-submodule with associated primes AssR(Rr/U) ={p1, . . . ,pk}⊆Spec(R). We now wish to describe Uin the following way
U=
w∈Rr|δ(w)∈pifor allδ∈Aiand 16i6k ,
where eachAi⊆Drnis a finite set of differential operators (for more details, see§4.2). In [10], it was shown that there exists a differential primary decomposition forUof size equal to thearithmetic multiplicityofU (seeDefinition 4.4,4.5) and that this is the minimal possible size.
Building onTheorem 3.2and results from [10], we introduce an algorithm for computing minimal differ- ential primary decompositions (seeAlgorithm 4.6). This algorithm is an extension of [10, 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 1.2. LetR=Q[x1,x2,x3]and consider theR-submoduleU=imageR
x21 x1x2 x22 x22 x2x3 x23
⊆R2. Fol- lowing the algorithm ofSection 2, we can compute a primary decomposition U=U1∩U2∩U3, where
U1=imageR
x1 x22 0 x3 x23 x22−x1x3
, U2=imageR
x3 x22 0 x1x2 x21 0 0 x23 x2x3 x22
, U3=imageR
0 x21 x22 x1x2 0 x1 0 x23 x2x3 x22
. TheR-submodules Uiare primary with associated primesp1= x22−x1x3
, p2= (x2,x3),p3= (x1,x2), respectively. The multiplicities of{U1,U2,U3}over {p1,p2,p3}are{1, 5, 5}. ByTheorem 3.2, we can de- scribeU1,U2,U3 by sets of Noetherian operators of sizes 1, 5, 5, respectively. These sets of Noetherian operators give a differential primary decomposition forUof size 11. However, this naive computation is not optimal, as amult(U) =3. Indeed, a minimal differential primary decomposition forUis given by
U=
(w1,w2)∈R2 | −x3w1+x1w2∈p1,∂w2
∂x3
∈p2,∂w1
∂x1
∈p3
, with only one differential operator needed per associated prime.
The basic outline of this paper is as follows. In Section 2, we review classical primary decomposi- tion, and present a general algorithm for modules. In Section 3, we prove our representation theorem (Theorem 3.2) for primary submodules of a free module, as well as an extension to arbitrary finitely gener- ated modules (Corollary 3.3). InSection 4, we present several algorithms of a differential nature, including one for computing minimal differential primary decompositions. InSection 5, we present an intrinsic differ- ential description of certain ideals that come from the join construction. Finally, inSection 6, we illustrate the various algorithms on examples with ourMacaulay2implementation.
2. AGENERAL PRIMARY DECOMPOSITION ALGORITHM
We begin with a general algorithm for primary decomposition of modules, inspired by the work of Eisenbud-Huneke-Vasconcelos [14], 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. [31] as well as [22–24] for some treatments.
We start with some reductions to the main case of interest. First, we reduce to the case of polynomial rings: letkbe an arbitrary field, and T a finitely generatedk-algebra. IfR=k[x1, . . . ,xn]is a polynomial ring with a surjectionπ:R։T(with corresponding closed embedding of spectraπ∗:Spec(T)֒→Spec(R)), then for anyT-module N, one has AssR(NR) =π∗(AssT(N)), whereNR denotesNviewed as an R- module. In this way we may compute associated primes and primary components of Nover T, by first computing that ofNRoverR, and then applyingπ.
Next, to simplify the notation on modules, note that ifM′⊆Mis a submodule, then a primary decompo- sition ofM′inMcan be obtained by lifting a primary decomposition of 0 inM/M′. Thus we may always takeM′=0 (a benefit afforded by working with general modules), and state the problem as follows: for a finitely generated moduleM6=0 over a polynomial ringR, find submodulesQ1, . . . ,Qs⊆Msuch that Ts
i=1Qi=0 and|AssR(M/Qi)|=1. The decomposition should moreover beminimal, in the sense that T
j6=iQj6=0 for alli, and also AssR(M/Qi) =AssR(M/Qj) ⇐⇒ i=j.
The primary decomposition algorithm described here proceeds in 2 steps: first, find all associated primes ofM, and second, determine validPi-primary componentsQifor each associated primePi(note that by uniqueness of associated primes from a primary decomposition, such a decomposition will automatically
be minimal). For the first step, following [14], we first reduce the problem of computing all associated primes of a module, to computing minimal primes of ideals:
Theorem 2.1([14, Theorem 1.1]). For anyi>0, the associated primes ofMof codimensioniare precisely the minimal primes ofann ExtiR(M,R)of codimensioni.
In view of this, we may compute AssR(M)via oracles to compute (1) a free resolution of a moduleM (and thus any Ext modules Ext•R(M,·)), and (2) minimal primes of any idealI⊆R, which we henceforth assume are given (in practice, both are well-optimized inMacaulay2). Note that following the above proce- dure 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 primesP1, . . . ,PsofMby a linear extension of the partial order by inclusion, i.e. Pi⊆Pj =⇒ i6j (note that this is automatic if the associated primes are weakly ordered by codimension, as in the previous paragraph). In particular, as a base caseP1 is a minimal prime of M(i.e. a minimal prime of annM). Primary components to minimal primes are uniquely determined, and can be obtained as follows:
Proposition 2.2. LetP∈Spec(R), and letM→MPbe the localization map. Then:
(1) [13, Theorem 3.10(d)] ker(M→MP)equals the intersection of allPi-primary components of0in MforPi∈AssR(M),Pi⊆P(in particular, this intersection is uniquely determined byMandP).
(2) [13, Proposition 3.13]Supposef∈Ris such that forPi∈AssR(M), one hasf∈Pi ⇐⇒ Pi6⊆P.
Thenker(M→MP) =ker(M→Mf) =0:Mf∞.
An element f as in Proposition 2.2(2) can be obtained as follows: for each associated prime Pi not contained inP, choose a generatorgiofPi not contained inP; then takef:=Q
Pi6⊆Pgi. TakingP=Pi
for some minimal primePiofMinProposition 2.2shows that given an oracle to compute saturations, we may obtain (the unique) primary components corresponding to minimal primes ofM.
It then remains to compute a validP-primary componentQ, for an embedded primeP. In this case such aQis not unique; indeed there are always infinitely many valid choices forQ. The next proposition gives one class of such choices:
Proposition 2.3([14], p. 27-28). ForP∈AssR(M)andj>0, fix generatorsP= (f1, . . . ,fm)forP, and set P[j]:= (fj1, . . . ,fjm). Then forj≫0, the submoduleQ[j]:=hull(P[j]M,M)is a validP-primary component of0inM.
Here hull(N,M)is the equidimensional hull ofNinM, i.e. the intersection of all primary components ofNof maximal dimension. There are a number of ways to computeQ[j]: the first is viaProposition 2.2, viz.Q[j]=ker(M→(M/P[j]M)P). Another method is given in [14, Theorem 1.1(2)], which is a general way to compute hulls via iterated Ext modules, and yet another method is given in [14, Algorithm 1.2].
To find a stopping criterion for the exponentjinProposition 2.3, note that inductively, we may assume that Pi-primary components for anyPi(P have already been computed, and so their intersection V:=
T
Pi(PQi is also known. ByProposition 2.2, we also know U:=T
Pi⊆PQi. Then for any j>0, the submodule Q[j] inProposition 2.3is a validP-primary component if and only ifQ[j]∩V=U. We may thus find a validP-primary component as follows: initializejat some starting value, and computeQ[j]. If Q[j]∩V6=U, then incrementj, and repeat until a valid candidateQ[j]is found.
Remark 2.4. A few remarks are in order concerning efficiency of the algorithm described above:
(1) Both choices of starting value ofj, and the function used to incrementj, 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 forQ[j]tends to increase asjincreases, so it is desirable not to takejunneces- sarily large. The current implementation in Macaulay2uses the incrementj→ ⌈3j2⌉(the starting value is more complicated, depending on the degrees of generators ofPand of annM).
(2) The use of bracket powers P[j] rather than ordinary powers Pj inProposition 2.3 is also for ef- ficiency: if either j or the number of generators of P is large, then Pj may take much longer to compute thanP[j].
(3) Of the 3 methods given above for computing embedded componentsQ[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 AssR(M/P[j]M), which is in general strictly bigger than{P}.
(4) The necessity of realizing ker(M→MP) as a saturation in Proposition 2.2(2) stems from the fact that MP typically does not have a finite presentation as an R-module. Although one could also express ker(M→MP) as a saturation 0:M(P′)∞, where P′ equals the intersection of all associated primes of Mnot contained in P, it is almost always much more efficient to compute saturations by a single element, than by general ideals.
(5) As a general rule, computation of associated primes is the most time-consuming step in this proce- dure. 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) computation of Ext modules (in fact, Ext•R(·,R)suffices), (2) computation of minimal primes of an idealI⊆R,
(3) computation of colon modules (of which saturations and annihilators are special cases), (4) computation of intersections of submodules.
3. AREPRESENTATION 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 ofpunctual Quot schemes, vector spaces closed under differentiationandsubbimodules of the Weyl-Noether module. This extends the main result of [8] to the case of modules. As a simple corollary, we also obtain a representation theorem for primary submodules of an arbitrary module.
Setup 3.1. For the rest of this section we fix the following notation:
– Letkbe a field of characteristic zero, andR:=k[x1, . . . ,xn]a polynomial ring overk.
– For an integerr>0, letRrbe a freeR-module of rankr.
– Letp∈Spec(R)be a prime ideal with codimensionc:=ht(p).
– The residue field ofpis denotedF:=k(p) =Quot(R/p) =Rp/pRp.
– A subset of variables {xi1, . . . ,xiℓ}⊆{x1, . . . ,xn} is independent modulo p if their images in R/p are algebraically independent overk, or equivalentlyk[xi1, . . . ,xiℓ]∩p ={0}. After possibly permuting the variables, we may assume that {xc+1, . . . ,xn} is a basis modulo p, i.e. a maximal set of independent variables modulop(see [27, Example 13.2]).
– LetL:=k(xc+1, . . . ,xn) denote the field of rational functions in the basis variables (which is a purely transcendental extension ofk), andSbe the polynomial ringS:=k(xc+1, . . . ,xn)[x1, . . . ,xc](which is a localization ofR, asS=∼L⊗k[xc+1,...,xn]R).
– The Weyl algebra and the relative Weyl algebra are denoted by Dn :=R
∂x1, . . . ,∂xn
and Dn,c :=
R
∂x1, . . . ,∂xc
⊆Dn, respectively.
– Themultiplicityof ap-primary submoduleU⊆Rris defined as lengthRp Rrp/Up
.
– For an integerm>0, thepunctual Quot schemeis a parameter space Quotm(F[[y1, . . . ,yc]]r)whoseF- points parametrize all F[[y1, . . . ,yc]]-submodules V⊆F[[y1, . . . ,yc]]r of colengthm, i.e. which satisfy dimF(F[[y1, . . . ,yc]]r/V) =m.
– We say thatδ1, . . . ,δm∈Drn=∼DiffR/k(Rr,R)is a set ofNoetherian operatorsfor ap-primary submodule U⊆Rrif the following equality holds
(1) U=
w∈Rr|δi(w)∈pfor all 16i6m .
We can now state our main result:
Theorem 3.2. The following four sets of objects are in bijective correspondence:
(a) p-primaryR-submodulesU⊆Rrof multiplicitymoverp, (b) F-points in the punctual Quot schemeQuotm(F[[y1, . . . ,yc]]r),
(c) m-dimensionalF-subspaces of F[z1, . . . ,zc]rthat are closed under differentiation,
(d) m-dimensionalF-subspaces of the Weyl-Noether moduleF⊗RDrn,cthat areR-bimodules.
Moreover, any basis of theF-subspace in part(d)can be lifted to Noetherian operatorsδ1, . . . ,δm∈Drn,c for theR-submoduleUin part(a).
We structure the proof ofTheorem 3.2as follows. The correspondence(a)↔(b)is detailed inTheorem 3.14.
The mapγdefined in(6)yields a bijection
(2)
p-primaryR-submodules ofRr of multiplicitymoverp
←→
points in Quotm(F[[y1, . . . ,yc]]r) U −→ V=γ(U) + (y1, . . . ,yc)mF[[y1, . . . ,yc]]r
U=γ−1(V) ←− V.
The correspondence(b)↔(c)is detailed inTheorem 3.15. We regard the polynomial ringF[z1, . . . ,zc] as anF[[y1, . . . ,yc]]-module by lettingyiact as∂zi, i.e.yi·F:=∂∂F
zi for anyF∈F[z1, . . . ,zc]. By Macaulay
inverse systems (see also(7),(8)) we have a bijection
(3)
points in Quotm(F[[y1, . . . ,yc]]r)
←→
m-dimensionalF-subspaces of F[z1, . . . ,zc]rclosed under differentiation
V −→ W=V⊥
V=W⊥ ←− W.
Finally, the correspondence(c)↔(d)is detailed in§3.3. The mapΩdefined in(9)yields a bijection
(4)
m-dimensionalF-subspaces of F[z1, . . . ,zc]rclosed under differentiation
←→
m-dimensionalF-subspaces of F⊗RDrn,cthat areR-bimodules.
W −→ E=Ω(W)
W=Ω−1(E) ←− E.
Furthermore, byLemma 3.9, we can lift elements fromF⊗RDn,ctoDn,c.
We now extendTheorem 3.2to arbitrary modules. LetMbe a finitely generatedR-module which can be generated byrelements, so that there is a short exact sequence ofR-modules
(5) 0→K→Rr→M→0.
LetU⊆Mbe a p-primaryR-submodule ofMof multiplicitym=lengthRp(Mp/Up)overp. There is a unique R-submodule Ue ⊆Rrcontaining Ksuch that U/Ke =∼ U, which we call the liftofUtoRr. Since Rr/Ue =∼ M/U, it follows thatUe is ap-primary submodule ofRrwith the same multiplicity as Uover p.
This convenient fact allows us to liftp-primary submodules ofMtop-primary submodules ofRr. In terms of the syzygiesK⊆RrofMin(5), we define the following objects:
– LetV′⊆F[[y1, . . . ,yc]]rbe theF[[y1, . . . ,yc]]-submodule
V′:=γ(K) + (y1, . . . ,yc)mF[[y1, . . . ,yc]]r⊆F[[y1, . . . ,yc]]r.
– LetW′:= (V′)⊥be the corresponding m-dimensionalF-subspace ofF[z1, . . . ,zc]rclosed under differ- entiation.
– LetE′:=Ω(W′)be the resultingm-dimensionalF-subspace ofF⊗RDrn,cwhich is anR-bimodule.
We can now state the extension ofTheorem 3.2to an arbitrary finitely generatedR-module.
Corollary 3.3. With the above notation, the following four sets of objects are in bijective correspondence:
(a) p-primaryR-submodulesU⊆Mof multiplicitymoverp,
(b) F-pointsV⊆F[[y1, . . . ,yc]]rin the punctual Quot schemeQuotm(F[[y1, . . . ,yc]]r)withV⊃V′, (c) m-dimensionalF-subspacesW⊆F[z1, . . . ,zc]rthat are closed under differentiation withW⊆W′, (d) m-dimensional F-subspaces E⊆F⊗RDrn,c of the Weyl-Noether module that are R-bimodules with
E⊆E′.
Moreover, any basis of theF-subspace in part(d)can be lifted to Noetherian operatorsδ1, . . . ,δm∈Drn,c for the liftUe ⊆Rrof theR-submoduleUin 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 [18, §16]).
ForR-modulesMandN, we regard Homk(M,N)as an(R⊗kR)-module, by setting ((s⊗kt)δ) (w) =sδ(tw) for allδ∈Homk(M,N),w∈M, s,t∈R.
We use the bracket notation[δ,s](w) :=δ(sw) −sδ(w)forδ∈Homk(M,N),s∈Randw∈M.
Unless otherwise specified, whenever we consider an(R⊗kR)-module as an R-module, we do so by lettingRact via the left factor ofR⊗kR.
Definition 3.4. Let M and N be R-modules. The m-th order k-linear differential operators, denoted DiffmR/k(M,N)⊆Homk(M,N), form an(R⊗kR)-module that is defined inductively by
(i) Diff0R/k(M,N) :=HomR(M,N).
(ii) DiffmR/k(M,N) :=
δ∈Homk(M,N) | [δ,s]∈Diffm−1R/k (M,N)for alls∈R . The set of allk-linear differential operators fromMtoNis the(R⊗kR)-module
DiffR/k(M,N) :=
[∞ m=0
DiffmR/k(M,N).
Following the notation used in [8–10], subsetsE⊆DiffR/k(M,N)are viewed as systems of differential equations, and their solution spaces overkare defined as
Sol(E) :=
w∈M | δ(w) =0 for allδ∈E = \
δ∈E
Ker(δ).
Example 3.5. AsR=k[x1, . . . ,xn]is a polynomial ring over a fieldkof characteristic zero, DiffR/k(R,R) is the Weyl algebraDn=Rh∂x1, . . . ,∂xni=L
α∈NnR∂αx.
We next describe differential operators via the module of principal parts. Consider the multiplication map µ:R⊗kR→R, s⊗kt7→st, and define ∆R/k:=Ker(µ), which is an ideal in R⊗kR. One can alternatively define differential operators as follows:
Proposition 3.6([20, Proposition 2.2.3]). LetM,Nbe R-modules. ThenDiffmR/k(M,N)is the(R⊗kR)- submodule ofHomk(M,N)annihilated by∆m+1R/k .
Definition 3.7. LetMbe anR-module. Themodule ofm-th principal partsofMis defined as PR/mk(M) := R⊗kM
∆m+1R/k (R⊗kM).
This is a module overR⊗kRand thus also overR. For simplicity, setPR/km :=PmR/k(R).
For anyR-moduleM, consider the universal mapdm:M→PmR/k(M),w7→1⊗kw. The next result is a fundamental characterization of differential operators.
Proposition 3.8([18, Proposition 16.8.4], [20, Theorem 2.2.6]). LetMandNbeR-modules and letm>0.
Then the following map is an isomorphism ofR-modules:
(dm)∗:HomR
PR/km (M),N =∼
−→DiffmR/k(M,N), ϕ 7→ ϕ◦dm.
We recall an explicit description of differential operators on free modules. LetJ⊆Rbe an ideal and con- sider the canonical mapπ:R։R/J. We wish to describe DiffmR/k(F,R/J)for a freeR-moduleF=Rr. Since
DiffmR/k(F,R/J)=∼DiffmR/k(R,R/J)r(see e.g. [9, Lemma 2.7]), it is enough to describe DiffmR/k(R,R/J). We have an induced map:
DiffmR/K(π) :DiffmR/K(R,R)→DiffmR/K(R,R/J), δ7→δ=π◦δ.
Lemma 3.9([8, Lemma 2]). With the above notation, the following statements hold:
(i) DiffmR/k(R,R/J) =L
|α|6m(R/J)∂αx where ∂αx =π◦∂αx. (ii) DiffmR/k(π)is surjective: a differential operator ǫ=P
|α|6msα∂αx ∈DiffmR/K(R,R/J)with sα∈R can be lifted toδ=P
|α|6msα∂αx ∈DiffmR/k(R,R).
The notation below will be useful for describing the(R⊗kR)-action:
Notation 3.10. LetT :=R⊗kR=k[x1, . . . ,xn,y1, . . . ,yn] be a polynomial ring in 2n variables, where xi representsxi⊗k1 and yi represents 1⊗kxi−xi⊗k1. The action ofT on Homk(M,N) is defined as follows: for allδ∈Homk(M,N)andw∈M,
(xi·δ)(w) := xiδ(w) and (yi·δ)(w) :=δ(xiw) −xiδ(w) = [δ,xi] (w) for all 16i6n.
Remark 3.11. ViewingPmR/kand DiffmR/k(R,R/J)asT-modules yields the following useful descriptions:
(i) PmR/k = L
α∈Nn,|α|6mRyα11· · ·yαnn.
(ii) Under the isomorphism DiffmR/k(R,R/J)=∼ HomR(PR/km ,R/J) (cf. Proposition 3.8), the dual basis element yα11· · ·yαnn∗
corresponds to the differential operatorα 1
1!···αn!∂αx11· · ·∂αxnn.
Proof. For explicit computations, see [10, §5].
The next proposition describes the differential operators of an arbitrary finitely generated module.
Proposition 3.12. Let0→K→F→M→0be a short exact sequence ofR-modules whereFisR-free of finite rank. LetNbe anR-module. Then, for allm>0, we have
DiffmR/k(M,N) =
δ∈DiffmR/k(F,N)|δ(K) =0 .
Proof. By left exactness of DiffmR/k(•,N), one has DiffmR/k(M,N)֒→DiffmR/k(F,N)(see e.g. [9, Lemma 2.7]). The inclusion “⊆” is clear. Conversely, ifδ∈DiffmR/k(F,N)andδ(K) =0, then there is a unique map δ∈Homk(M,N)induced byδ, and byProposition 3.6,∆m+1R/k ·δ=0, which implies∆m+1R/k ·δ=0. Then δ∈DiffmR/k(M,N)byProposition 3.6again, 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 p-primary R- submodules of a free moduleRrvia punctual Quot schemes and Macaulay inverse systems.
LetB be the power series ring B:=F[[y1, . . . ,yc]] over the residue field F= Rp/pRp and denote its maximal ideal byn := (y1, . . . ,yc)⊆B. The punctual Quot scheme Quotm(Br)parametrizes all quotients
Br = F[[y1, . . . ,yc]]r։ T
such thatT has finite lengthm, and consequently, T is only supported at the maximal idealn. More pre- cisely, theF-rational points of Quotm(Br)correspond toB-submodulesV⊆Brsuch that dimF(Br/V) =m
(in this paper,a point inQuotm(Br)will always meananF-rational point inQuotm(Br)). This parameter space has been considered in several papers, e.g. [1], [15], [19]. Whenr=1, Quotm(Br) coincides with thepunctual Hilbert schemeHilbm(F[[y1, . . . ,yc]])studied by Brianc¸on [2] and Iarrobino [21].
The following basic fact shows that any point in Quotm(Br)can be identified with a(B/nmB)-submodule ofBr/nmBr.
Proposition 3.13. For anyB-submoduleV⊆Brwith colengthm=dimF(Br/V), one hasV⊇nmBr. Proof. Consider the associated graded module
gr(Br/V) :=
M∞ k=0
nkBr/ nk+1Br+ nkBr∩V
which satisfies dimF(gr(Br/V)) =dimF(Br/V). IfV6⊇nmBr, then[gr(Br/V)]k6=0 for all 06k6m, so m=dimF(Br/V) =dimF(gr(Br/V))>m+1, a contradiction.
Following the approach of [8], we have an injection
η:R֒→B, xi 7→ yi+xi, for 16i6c, xj 7→ xj, forc+16j6n, wherexidenotes the class ofxiinFfor 16i6n. We now define the induced injection (6) γ:Rr֒→Br, (f1, . . . ,fr)∈Rr7→(η(f1), . . . ,η(fr))∈Br.
The mapγprovides the first correspondence(a)↔(b)in our main theorem.
Theorem 3.14. With the above notation, there is a bijection p-primaryR-submodules ofRr
of multiplicitymoverp
←→
points inQuotm(Br) U −→ V=γ(U) + nmBr
U=γ−1(V) ←− V.
Proof. The canonical mapRr֒→Sr,U⊆Rr7→USr⊆Srgives a bijection betweenp-primaryR-submodules andpS-primaryS-submodules (see e.g. [26, Theorem 4.1]). By [8, Proposition 1], for anym>0 the map η:R֒→Binduces an isomorphism of local ringsS/pmS−→=∼ B/nmB. Accordingly, we obtain a commuta- tive diagram
Rr
Sr Br
Sr/pmSr =∼ Br/nmBr. γ
By Proposition 3.13, any B-submodule V⊆Br with dimF(Br/V) =m contains nmBr. Similarly, any pS-primary S-submodule V⊆Sr with lengthS(Sr/V) =mcontains pmS. Therefore, the result follows because under the above identifications the constancy of multiplicity equal tomwill not change.
We next recall the well-known Macaulay inverse systems for modules. Consider the injective hullE:=
EB(F)of the residue fieldF=∼B/nofB. SinceBis a formal power series ring, this can be identified with the set of inverse polynomials:
E=∼F[y−11 , . . . ,y−1c ]
(for more details see e.g. [3, Lemma 11.2.3, Example 13.5.3] or [5, Theorem 3.5.8]). LetEbe the polyno- mial ringE:=F[z1, . . . ,zc], regarded as aB-module by lettingyiact as∂zi, i.e. yi·F:= ∂∂F
zi for anyF∈E.
SinceFhas characteristic zero, there is an isomorphism ofB-modules E=∼ F[y−11 , . . . ,y−1c ]−=→∼ F[z1, . . . ,zc] =E, 1
yα = 1 yα11· · ·yαcc
7→ zα
α! =zα11· · ·zαcc α1!· · ·αc!.
We describe Macaulay inverse systems via Matlis duality. Let(•)∨:=HomB(•,E)denote the Matlis dual functor (see e.g. [5, Theorem 3.2.13]). For anyB-submoduleV⊆Br, there is a natural identification (7) Br/V∨=∼V⊥:=
w∈Er|v·w= Xr i=1
vi·wi=0 for allv∈V
⊆Er.
On the other hand, anyB-submoduleW⊆Eris anF-subspace ofErthat is closed under differentiation, sinceyiacts as the operator∂zi. Thus we also obtain an identification
(8) Er/W∨=∼W⊥:=
v∈Br|v·w= Xr i=1
vi·wi=0 for allw∈W
⊆Br. The above discussion then recovers the following well-known result.
Theorem 3.15(Macaulay inverse systems). With the above notation, there is a bijection
points inQuotm(Br)
←→
m-dimensionalF-subspaces of Erclosed under differentiation
V −→ W=V⊥
V=W⊥ ←− W.
3.3. The proof of the representation theorem. In this subsection, we complete the proof ofTheorem 3.2.
First, we recall some notation and results from [8]. Every differential operatorδ∈Dn,cis a uniquek-linear combination of standard monomials xα∂βx =xα11· · ·xαnn∂βx11· · ·∂βxcc, whereαi,βi∈N. Consider theWeyl- Noether module
F⊗RDrn,c:=F⊗RDiffR/k[xc+1,...,xn](R,R)r=∼F⊗SDiffS/L(S,S)r
(for the isomorphism on the right, see e.g. [9, Lemma 2.7]). FromProposition 3.8and the fact thatPmS/Lis a freeS-module, the Weyl-Noether module admits the following description
F⊗RDrn,c=F⊗S
−→lim
m
DiffmS/L(S,S)r
=∼ lim−→
m
DiffmS/L(S,F)r=DiffS/L(S,F)r. Applying Lemma 3.9withJ= pSgivesF⊗RDn,c=∼DiffS/L(S,F)=∼ L
α∈NcF∂αx. We then get an iso- morphism ofF-vector spaces:
ω:E=F[z1, . . . ,zc]→F⊗RDn,c, zα7→∂αx for allα∈Nc,
which in turn induces an isomorphism
(9) Ω:Er→F⊗RDrn,c.
LetAbe the polynomial ring A :=F[y1, . . . ,yc]. UsingNotation 3.10over the fieldL=k(xc+1, . . . ,xn) withTS =S⊗LS=L[x1, . . . ,xc,y1, . . . ,yc]gives that A=∼ F⊗LS=∼ F⊗S(S⊗LS)=∼ F⊗STS, and that F⊗RDrn,c=∼ DiffS/L(S,F)r=∼ DiffS/L(Sr,F)has a natural structure of A-module. We now identify the power series ringB =F[[y1, . . . ,yc]] of §3.2with the completion Acn of Awith respect to the maximal irrelevant idealn = (y1, . . . ,yc)⊆A; by an abuse of notationnis seen interchangeably as an ideal in both AandB. In this way, we get actions ofAon bothEandF⊗RDn,c. Explicitly, forα∈Nc and 16i6c,
yi·zα:=αizα11· · ·zαii−1· · ·zαcc and yi·∂αx :=
∂αx,xi
=αi∂αx11· · ·∂αxii−1· · ·∂αxcc.
Therefore the mapΩin(9)gives a bijection betweenF-vector subspaces ofErclosed under differentiation and A-submodules of F⊗RDrn,c. The latter structure as an A-submodule is equivalent to being an R- subbimodule of the Weyl-Noether moduleF⊗RDrn,c.
Proof ofTheorem 3.2. The bijections (a)↔ (b)and (b)↔ (c) have been described inTheorem 3.14and Theorem 3.15, respectively. By the discussion above, the mapΩ in(9)provides the bijection(c)↔ (d).
Due toLemma 3.9, we can lift differential operators fromF⊗RDrn,ctoDrn,c(cf. [8, Remarks 7, 8]).
To finish the proof of the theorem, it suffices to show that anF-basis of the subspace in(d)lifts to a set of Noetherian operators for the submoduleUin(a). That is,
(1) letU⊆Rrbe ap-primaryR-submodule of multiplicitymoverp,
(2) byTheorem 3.14letV:=γ(U) + nmBr⊆Brbe the corresponding point in Quotm(Br),
(3) byTheorem 3.15letW:=V⊥⊆Erbe the correspondingm-dimensionalF-subspace ofErclosed under differentiation,
(4) letE:=Ω(W)⊆F⊗RDrn,c=∼ DiffS/L(Sr,F)be the correspondingR-subbimodule ofF⊗RDrn,c, then we claim Sol(E) =U⊗RS.
Similarly to [9, Lemma 3.14] and [8, Proposition 3], the statements below hold:
• Diffm−1S/L (Sr,F)=∼HomS Pm−1S/L (Sr),F=∼HomF(Ar/nmAr,F)byProposition 3.8and Hom-tensor adjunction.
• Since V⊇nmBr(cf. Proposition 3.13) andBr/nmBr= A∼ r/nmAr, we getF⊆Diffm−1S/L (Sr,F) determined by HomF(Br/V,F). Here we haveF ∼=HomF(Br/V,F)⊆HomF(Br/nmBr,F).
• Sol(F) =U⊗RS(see [9, Lemma 3.14(iv)] and [8, Proposition 3(iii)]).
It thus suffices to show thatEandFcoincide asR-subbimodules ofF⊗RDrn,c. By the perfect pairing of [8, Proof of Theorem 6.1] or general duality results (see e.g. [13, Proposition 21.4]), there are isomorphisms
F ∼=HomF(Br/V,F)
=∼HomB Br/V, HomF(B/nm,F)
(by Hom-tensor adjunction)
=∼HomB Br/V, HomB(B/nm,E)
(by [13, Proposition 21.4])
=∼HomB(Br/V,E) = (Br/V)∨ (by Hom-tensor adjunction).
(10)
Recall from§3.2that the isomorphism (Br/V)∨ =∼ V⊥=Wis explicitly described by identifying the in- verted monomial y1α = yα11
1 ···yαcc with zα!α = z
α1 1 ···zαcc
α1!···αc! for allα= (α1, . . . ,αc)∈Nc. On the other hand, the isomorphism(Br/V)∨=∼ Fis explicitly described by identifying the inverted monomial y1α with the dual monomial(yα)∗and then with α!1 ∂αx (seeRemark 3.11); notice that we have an explicit isomorphism
(B/nm)∨=HomB(B/nm,E) = 0:F[y−1
1 ,...,y−1c ]nm
=∼ HomF(B/nm,F), 1
yα 7→(yα)∗. ThusEandFdo indeed coincide asR-subbimodules ofF⊗RDrn,c, as desired.
Finally, we have the following consequence.
Proof ofCorollary 3.3. From the short exact sequence 0→K→Rr→M→0, anR-submoduleU⊆M corresponds to a uniqueR-submoduleUe ⊆Rrsuch thatUe ⊃K, and sinceM/U=∼Rr/U, it follows thate Ue is ap-primary submodule ofRrof multiplicitymoverp.
Let Ue ⊆Rr be a p-primary submodule of multiplicity m over p. By using the correspondences of Theorem 3.2, we setV=γ Ue
Br+ nmBr,W=V⊥and E=Ω(W). In the same way, letV′=γ(K)Br+ nmBr,W′= (V′)⊥andE′=Ω(W′). Since the following four conditions are equivalent
Ue ⊃K, V⊃V′, W⊆W′ and E⊆E′,
the result follows directly fromTheorem 3.2.
4. DIFFERENTIAL ALGORITHMS
In this section, we present several algorithms, based onSection 3and the previous papers [8] and [10], 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:
(I) Algorithm 4.1: compute a set of Noetherian operators for a primary submodule.
(II) Algorithm 4.3: compute the primary submodule determined by a set of differential operators. This can be seen as the inverse process toAlgorithm 4.1.
(III) Algorithm 4.6: compute a minimal differential primary decomposition for a submodule.
4.1. Noetherian operators vs primary submodules. This subsection deals with the problem of repre- senting a primary submodule via Noetherian operators. We continue to use the notation ofSection 3, cf.
Setup 3.1. First, we give an algorithm to compute a set of Noetherian operators for a primary submodule.
Algorithm 4.1(Noetherian operators for a primary submodule).
INPUT: Ap-primary submoduleU⊆RrofRrof multiplicitymoverp.
OUTPUT: A set of Noetherian operatorsδ1, . . . ,δm∈Drn,cthat representsUas in(1).
(1) Compute theF[[y1, . . . ,yc]]-moduleV=γ(U) + (y1, . . . ,yc)mF[[y1, . . . ,yc]]rthat corresponds toUas in(2).
(2) Using linear algebra overF, compute anF-basis {B1, . . . ,Bm}⊆F[z1, . . . ,zc]r for the inverse system W=V⊥as in(3).
(3) ComputeC1:=Ω(B1), . . . ,Cm:=Ω(Bm)∈F⊗RDrn,cas in(4).
(4) Return lifts ofC1, . . . ,CminDrn,c, as guaranteed byLemma 3.9.
Proof of correctness ofAlgorithm 4.1. The correctness of this algorithm follows fromTheorem 3.2.
InAlgorithm 4.1the output is a set of Noetherian operators in the relative Weyl algebraDrn,c. We now consider the reverse process, starting from operators in the whole Weyl algebra Dn. We start with some basic facts regarding modules defined via differential operators.
Remark 4.2. Let δ1, . . . ,δm ∈DiffR/k(Rr,R) =Drn be differential operators. Let G⊆Drn be the R- bimodule generated byδ1, . . . ,δm. The following statements hold:
(i) {w∈Rr|δ(w)∈pfor allδ∈G}is ap-primaryR-submodule ofRr.
(ii) {w∈Rr|δ(w)∈pfor allδ∈G}⊆{w∈Rr|δi(w)∈pfor all 16i6m}, and equality holds if and only if the right hand side is anR-submodule ofRr.
Proof. For more details, see [9, §3] and specifically [9, Proposition 3.5].
In view of Remark 4.2, it is desirable to treat the following “closure operation”: given finitely many differential operatorsδ1, . . . ,δm∈Drn, compute the correspondingp-primaryR-submodule
{w∈Rr|δ(w)∈pfor allδ∈G}, whereG⊆Drnis theR-bimodule generated byδ1, . . . ,δm.
The subsequent arguments follow verbatim the techniques used inSection 3. Here we use the whole sets of variablesy1, . . . ,yn andz1, . . . ,zninstead of justy1, . . . ,ycand z1, . . . ,zc, respectively. The only issue with taking whole sets of variables is thatTheorem 3.14is no longer be valid (as [8, Proposition 1] requires F/k(xc+1, . . . ,xn)to be algebraic), but we may circumvent this viaRemark 4.2. We have a canonical map (11) Φ:DiffR/k(Rr,R) =Drn→F⊗RDrn.
FollowingNotation 3.10, byProposition 3.8(seeRemark 3.11) we obtain the isomorphism (12) F⊗RDiffmR/k(Rr,R)=∼F⊗RHomR PR/km (Rr),R=∼ HomF
Ar nm+1Ar,F
whereA =F[y1, . . . ,yn]andn = (y1, . . . ,yn). LetB=F[[y1, . . . ,yn]]andE=F[z1, . . . ,zn], and as before, considerEas aB-module by settingyi=∂zi for all 16i6n. As is(7)and(8), we can defineV⊥andW⊥ forV⊆Br=F[[y1, . . . ,yn]]randW⊆Er=F[z1, . . . ,zn]r, respectively. In this setting, Macaulay inverse systems (Theorem 3.15) are also valid. Now, the equivalent map ofγin(6)is given by
(13) Γ :Rr֒→Br=F[[y1, . . . ,yn]]r, (f1, . . . ,fr)∈Rr7→(η′(f1), . . . ,η′(fr))∈Br
whereη′:R֒→B,xi7→yi+xifor all 16i6n, and the equivalent of the mapΩin(9)is given by (14) Ψ:Er=F[z1, . . . ,zn]r→F⊗RDrn
and is induced by zα7→∂αx for allα∈Nn (in this setting, under the isomorphism (12) withr=1 and
|α|6m, the dual monomial(yα)∗coincides with∂αx ∈F⊗RDn).
After the above discussion, we can present the following algorithm which can be seen as an inverse process toAlgorithm 4.1.
Algorithm 4.3(Primary submodule that corresponds to a set of differential operators).
INPUT: A prime idealp∈Spec(R)and a set of differential operatorsδ1, . . . ,δm∈Drn.
