arXiv:1212.4590v1 [math.AP] 19 Dec 2012

RELATIVE PARAMETRIZATION OF
LINEAR MULTIDIMENSIONAL SYSTEMS
J.F. Pommaret
CERMICS, Ecole Nationale des Ponts et Chaussées,
6/8 Av. Blaise Pascal, 77455 Marne-la-Vallée Cedex 02, France
E-mail: jean-francois.pommaret@wanadoo.fr, pommaret@cermics.enpc.fr
URL: http://cermics.enpc.fr/∼pommaret/home.html
ABSTRACT :
In the last chapter of his book ”The Algebraic Theory of Modular Systems ” published in 1916,
F. S. Macaulay developped specific techniques for dealing with ”unmixed polynomial ideals ” by
introducing what he called ”inverse systems ”. The purpose of this paper is to extend such a
point of view to differential modules defined by linear multidimensional systems, that is by linear
systems of ordinary differential (OD) or partial differential (PD) equations of any order, with any
number of independent variables, any number of unknowns and even with variable coefficients.
The first and main idea is to replace unmixed polynomial ideals by pure differential modules.
The second idea is to notice that a module is 0-pure if and only if it is torsion-free and thus if and
only if it admits an ” absolute parametrization ” by means of arbitrary potential like functions, or,
equivalently, if it can be embedded into a free module by means of an ” absolute localization ”.
The third idea is to refer to a difficult theorem of algebraic analysis saying that an r-pure module
can be embedded into a module of projective dimension r, that is a module admitting a projective
resolution with exactly r operators.
The fourth and final idea is to establish a link between the use of extension modules for such a
purpose and specific formal properties of the underlying multidimensional system through the use
of involution and a ”relative localization ” leading to a ” relative parametrization ”.
The paper is written in a rather effective self-contained way and we provide many explicit examples
that should become test examples for a future use of computer algebra.
KEY WORDS :
Unmixed ideal, algebraic analysis, homological algebra, extension module, projective dimension,
torsion-free module, pure module, characteristic variety, formal integrability, involution, Spencer
operator, inverse system.
1) INTRODUCTION :
Let D = K[d1 , ..., dn ] = K[d] be the ring of differential operators with coefficients in a differential field K with n commuting derivations ∂1 , ..., ∂n and commutation relations di a = adi +∂i a, ∀a ∈
K. If y 1 , ..., y m are m differential indeterminates, we may identify Dy 1 + ... + Dy m = Dy with
Dm and consider the finitely presented left differential module M with presentation Dp → Dm →
M → 0 determined by a given linear multidimensional system with n independent variables,
m unknowns and p equations. Applying the functor homD (•, D), we get the exact sequence
0 → homD (M, D) → Dm → Dp −→ N −→ 0 of right differential modules that can be transformed
by a side-changing functor to an exact sequence of finitely generated left differential modules.
This new presentation corresponds to the formal adjoint ad(D) of the linear differential operator
D determined by the initial presentation but now with p unknowns and m equations, obtaining
therefore a new finitely generated left differential module N and we may consider homD (M, D)
as the module of equations of the compatibility conditions (CC) of ad(D), a result not evident
at all (Compare to [24]). Using now a maximum free submodule 0 −→ Dl −→ homD (M, D)

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and repeating this standard procedure while using the well known fact that ad(ad(D)) = D,
we obtain therefore an embedding 0 → homD (homD (M, D), D) → Dl of left differential modules for a certain integer 1 ≤ l < m because K is a field and thus D is a noetherian bimodule over itself, a result leading to l = rkD (homD (M, D)) = rkD (M ) < m as in ([19], p
178,201)(See section 3 for the definition of the differential rankrkD ). Now, the kernel of the map
ǫ : M → homD (homD (M, D), D) : m → ǫ(m)(f ) = f (m), ∀f ∈ homD (M, D) is the torsion submodule t(M ) ⊆ M and ǫ is injective if and only if M is torsion-free, that is t(M ) = 0. In that case,
we obtain by composition an embedding 0 → M → Dl of M into a free module that can also be
obtained by localization if we introduce the ring of fractions S −1 D = DS −1 when S = D − {0}.
This result is quite important for applications as it provides a (minimum) parametrization of the
linear differential operator D and amounts to the controllability of a classical control system when
n = 1 ([24], p 258). This parametrization will be called an ”absolute parametrization ” as it only
involves arbitrary ”potential-like ” functions (See [1], [9], [18], [19], [20], [24], [25] and [32] for more
details and examples, in particular that of Einstein equations).
The purpose of this paper is to extend suh a result to a much more general situation, that
is when M is not torsion-free, by using unexpected results first found by F.S. Macaulay in 1916
through his study of ”inverse systems ” for ”unmixed polynomial ideals ”.
For this we define the purity filtration :
0 = tn (M ) ⊆ tn−1 (M ) ⊆ ... ⊆ t1 (M ) ⊆ t0 (M ) = t(M ) ⊆ M
by introducing tr (M ) = {m ∈ M | cd(Dm) > r} where the codimension of Dm is n minus
the dimension of the characteristic variety determined by m in the corresponding system for one
unknown. The module M is said to be r-pure if tr (M ) = 0, tr−1 (M ) = M or, equivalently, if
cd(M ) = cd(N ) = r, ∀N ⊂ M and a torsion-free module is a 0-pure module. Moreover, when
K = k = cst(K) is a field of constants and m = 1, a pure module is unmixed in the sense of
Macaulay, that is defined by an ideal having an equidimensional primary decomposition.
Example 1.1 : As an elementary example with K = k = Q, m = 1, n = 2, p = 2, the differential
module defined by d22 y = 0, d12 y = 0 is not pure because z ′ = d2 y satisfies d2 z ′ = 0, d1 z ′ = 0 while
z” = d1 y only satisfies d2 z” = 0 and ((χ2 )2 , χ1 χ2 ) = (χ1 ) ∩ (χ1 , χ2 )2 . We obtain therefore the
purity filtration 0 = t2 (M ) ⊂ t1 (M ) ⊂ t0 (M ) = t(M ) = M with strict inclusions as 0 6= z ′ ∈ t1 (M )
while z” ∈ t0 (M ) but z” ∈
/ t1 (M ).
From the few (difficult) references ([1],[9],[15],[18],[27]) dealing with extension modules extr (M ) =
and purity in the framework of algebraic analysis, it is known that M is r-pure if and
only if there is an embedding 0 → M → extrD (extrD (M, D), D). Indeed, the case r = 0 is exactly
the one already considered because ext0D (M, D) = homD (M, D) and the ker/coker exact sequence:

extrD (M, D)

0 −→ ext1 (N ) −→ M −→ ext0 (ext0 (M )) −→ ext2 (N ) −→ 0
allows to test the torsion-free property of M in actual practice by using the double-duality formula
t(M ) = ext1 (N ) as in ([19]). Also, when r ≥ 1, a similar construction that we shall recall and
illustrate in section 4 provides a finitely generated module L with projective dimension pdD (L) = r,
that is a minimum resolution of L with only r operators, and an embedding 0 → M → L that
allows to exhibit a relative parametrization of D because now the parametrizing potential-like functions are no longer arbitrary but must only depend on arbitrary functions of n − r variables.
Example 1.2 : With K = k = Q, m = 2, n = 3, r = 1, the differential module M defined by
the involutive system Φ1 ≡ d3 y 1 = 0, Φ2 ≡ d3 y 2 = 0, Φ3 ≡ d2 y 1 − d1 y 2 = 0 is 1-pure and admits
the resolution 0 −→ D −→ D3 −→ D2 −→ M −→ 0. The differential module L defined by
the system d3 z = 0 is also 1-pure and admits the resolution 0 −→ D −→ D −→ L −→ 0. We
finally obtain the relative parametrization y 1 = d1 z, y 2 = d2 z providing the strict inclusion M ⊂ L.
In a simple way, this result can be considered as a measure of how far a module is from being
projective, recalling that a module P is projective if there exists another (projective) module Q
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and a free module F such that P ⊕ Q ≃ F .
We adapt the ”relative localization” technique used by Macaulay and combine it with the ”involution” technique used in the formal theory of systems of partial differential equations in order
to obtain an explicit procedure for determining L when M is given. Many examples will illustrate these new methods that avoid the previous abstract arguments based on ”double duality”.
In particular, original non-commutative examples will also be presented. However, we point out
the fact that the latter method can be adapted without any change to the case of systems with
variables coefficients as it only depends on the use of adjoint operators but the following example
will explain by itself the type of difficulty involved.
Example 1.3 : Starting now with K = Q(x1 , x2 ), m = 2, n = 3, r = 1, the new differential module
M defined by d3 y 1 = 0, d3 y 2 = 0, d2 y 1 − d1 y 2 + x2 y 2 = 0 is also 1-pure and the differential module
L is again defined by d3 z = 0 as in the previous example. However we obtain the totally different
relative parametrization y 1 = d12 z − x2 d2 z + z, y 2 = d22 z providing the strict inclusion M ⊂ L.
More generally, we may consider a constant parameter a ∈ k = Q and consider the new system
d3 y 1 = 0, d3 y 2 = 0, d2 y 1 − d1 y 2 + ax2 y 2 = 0 depending on a. For a = 0 we find back the case of
the previous example and we let the reader wonder why the situation only changes when a 6= 0.
The content of the paper is just following the introduction.
In section 2 we recall the definitions and results from the formal theory of systems of OD/PD
equations that will be crucially used in the sequel. We pay a particular emphasize to the definition
of involution and the way to introduce the Spencer operator in this framework. We also study the
possibility and difficulty to use computer algebra in this framework.
In section 3 we recall the basic tools needed from module theory and homological algebra in
a way adapted to our purpose, in particular the definition of the extension modules, and provide
a few of their properties which, though well known by specialists of algebraic analysis, cannot be
found easily in the literature. Meanwhile, we provide a few links with the preceding section which
are not so well known. Many explicit examples will illustrate the main concepts in the commutative
(constant field k) and the non-commutative (differential field K) framework.
In section 4 we shall recall the proof of the theorem already quoted showing how to embed an
r-pure module M into another module L with projrective dimension equal to r. We shall provide
for the first time explicit computations of this result in order to point out the difficulty encountered
in such a procedure as a motivation for avoiding it.
In section 5 we extend the work of Macaulay, showing why only pure modules can fit with
relative localization in a coherent way with what happens for torsion-free modules. Meanwhile, we
shall extend for the first time this work to the non-commutative framework, showing in particular
that the operator introduced by Macaulay ([11], §60) for studying inverse systems is nothing else
than the Spencer operator. Many explicit examples, including highly non-trivial ones provided by
Macaulay himself, will be fully treated in such a way that any engineer, even with a poor knowledge of homological algebra, will nevertheless become intuitively able to understand and apply
these new techniques without reading the previous sections, just comparing to the way the same
examples have been treated in section 4 by means of another approach.
2) TOOLS FROM SYSTEM THEORY :
If X is a manifold of dimension n with local coordinates (x) = (x1 , ...xn ), we denote as usual by
T = T (X) the tangent bundle of X, by T ∗ = T ∗ (X) the cotangent bundle, by ∧r T ∗ the bundle of
r-forms and by Sq T ∗ the bundle of q-symmetric tensors. More generally, let E be a vector bundle
over X, that is (roughly) a manifold with local coordinates (xi , y k ) for i = 1, ..., n and k = 1, ..., m
simply denoted by (x, y), projection π : E → X : (x, y) → (x) and changes of local coordinates
x̄ = ϕ(x), ȳ = A(x)y. If E and F are two vector bundles over X with respective local coordinates
(x, y) and (x, z), we denote by E×X F the fibered product of E and F over X as the new vector
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bundle over X with local coordinates (x, y, z). We denote by f : X → E : (x) → (x, y = f (x)) a
global section of E, that is a map such that π ◦ f = idX but local sections over an open set U ⊂ X
may also be considered when needed. Under a change of coordinates, a section transforms like
f¯(ϕ(x)) = A(x)f (x) and the derivatives transform like:
∂ f¯l
(ϕ(x))∂i ϕr (x) = (∂i Alk (x))f k (x) + Alk (x)∂i f k (x)
∂ x̄r
We may introduce new coordinates (xi , y k , yik ) transforming like:
ȳrl ∂i ϕr (x) = (∂i Alk (x))y k + Alk (x)yik
k
We shall denote by Jq (E) the q-jet bundle of E with local coordinates (xi , y k , yik , yij
, ...) = (x, yq )
k
k
k
called jet coordinates and sections fq : (x) → (x, f (x), fi (x), fij (x), ...) = (x, fq (x)) transforming
like the sections jq (f ) : (x) → (x, f k (x), ∂i f k (x), ∂ij f k (x), ...) = (x, jq (f )(x)) where both fq and
jq (f ) are over the section f of E. Of course Jq (E) is a vector bundle over X with projection πq
while Jq+r (E) is a vector bundle over Jq (E) with projection πqq+r , ∀r ≥ 0.

DEFINITION 2.1: A linear system of order q on E is a vector sub-bundle Rq ⊂ Jq (E) and a
solution of Rq is a section f of E such that jq (f ) is a section of Rq .
Let µ = (µ1 , ..., µn ) be a multi-index with length |µ| = µ1 + ... + µn , class i if µ1 = ... = µi−1 =
0, µi 6= 0 and µ + 1i = (µ1 , ..., µi−1 , µi + 1, µi+1 , ..., µn ). We set yq = {yµk |1 ≤ k ≤ m, 0 ≤ |µ| ≤ q}
with yµk = y k when |µ| = 0. If E is a vector bundle over X with local coordinates (xi , y k ) for
i = 1, ..., n and k = 1, ..., m, we denote by Jq (E) the q-jet bundle of E with local coordinates
simply denoted by (x, yq ) and sections fq : (x) → (x, f k (x), fik (x), fijk (x), ...) transforming like
the section jq (f ) : (x) → (x, f k (x), ∂i f k (x), ∂ij f k (x), ...) when f is an arbitrary section of E.
Then both fq ∈ Jq (E) and jq (f ) ∈ Jq (E) are over f ∈ E and the Spencer operator just allows
to distinguish them by introducing a kind of ”difference” through the operator D : Jq+1 (E) →
T ∗ ⊗ Jq (E) : fq+1 → j1 (fq ) − fq+1 with local components (∂i f k (x) − fik (x), ∂i fjk (x) − fijk (x), ...)
k
(x). In a symbolic way, when changes of
and more generally (Dfq+1 )kµ,i (x) = ∂i fµk (x) − fµ+1
i
coordinates are not involved, it is sometimes useful to write down the components of D in the
form di = ∂i − δi and the restriction of D to the kernel Sq+1 T ∗ ⊗ E of the canonical projection
πqq+1 : Jq+1 (E) → Jq (E) is minus the Spencer map δ = dxi ∧ δi : Sq+1 T ∗ ⊗ E → T ∗ ⊗ Sq T ∗ ⊗ E.
The kernel of D is made by sections such that fq+1 = j1 (fq ) = j2 (fq−1 ) = ... = jq+1 (f ). Finally, if Rq ⊂ Jq (E) is a system of order q on E locally defined by linear equations Φτ (x, yq ) ≡
aτk µ (x)yµk = 0 and local coordinates (x, z) for the parametric jets up to order q, the r-prolongation
Rq+r = ρr (Rq ) = Jr (Rq ) ∩ Jq+r (E) ⊂ Jr (Jq (E)) is locally defined when r = 1 by the link
+ ∂i aτk µ (x)yµk = 0 and has symbol
ear equations Φτ (x, yq ) = 0, di Φτ (x, yq+1 ) ≡ aτk µ (x)yµ+1
i
gq+r = Rq+r ∩ Sq+r T ∗ ⊗ E ⊂ Jq+r (E) if one looks at the top order terms. If fq+1 ∈ Rq+1 is
over fq ∈ Rq , differentiating the identity aτk µ (x)fµk (x) ≡ 0 with respect to xi and substracting the
k
k
(x)) ≡ 0
(x)+∂i aτk µ (x)fµk (x) ≡ 0, we obtain the identity aτk µ (x)(∂i fµk (x)−fµ+1
identity aτk µ (x)fµ+1
i
i
∗
and thus the restriction D : Rq+1 → T ⊗ Rq ([17],[18],[30]).
q+r+1
DEFINITION 2.2: Rq is said to be formally integrable when the restriction πq+r
: Rq+r+1 →
Rq+r is an epimorphism ∀r ≥ 0 or, equivalently, when all the equations of order q + r are obtained
by r prolongations only ∀r ≥ 0. In that case, Rq+1 ⊂ J1 (Rq ) is a canonical equivalent formally
integrable first order system on Rq with no zero order equations, called the Spencer form.

Finding an intrinsic test has been achieved by D.C. Spencer in 1970 ([30]) along coordinate
dependent lines sketched by M. Janet in 1920 ([7]) and W. Gröbner in 1940 ([4],[6]). The key
ingredient, missing in the old approach, is provided by the following definition.
Let T ∗ be the cotangent vector bundle of 1-forms on X and ∧s T ∗ be the vector bundle of s-forms
on X with usual bases {dxI = dxi1 ∧ ... ∧ dxis } where we have set I = (i1 < ... < is ). Moreover, introducing the exterior derivative d : ∧s T ∗ −→ ∧s+1 T ∗ : ω = ωI (x)dxI −→ dω = ∂i ωI (x)dxi ∧ dxI ,
we have d2 = d ◦ d = 0 and may introduce the Poincaré sequence:

4

d

d

d

d

∧0 T ∗ −→ ∧1 T ∗ −→ ∧2 T ∗ −→ ... −→ ∧n T ∗ −→ 0
PROPOSITION 2.3: There exists a map δ : ∧s T ∗ ⊗ Sq+1 T ∗ ⊗ E → ∧s+1 T ∗ ⊗ Sq T ∗ ⊗ E which
restricts to δ : ∧s T ∗ ⊗ gq+1 → ∧s+1 T ∗ ⊗ gq and δ 2 = δ ◦ δ = 0.
k
k
.
Proof: Let us introduce the family of s-forms ω = {ωµk = vµ,i
dxI } and set (δω)kµ = dxi ∧ ωµ+1
i
2
k
i
j
k
We obtain at once (δ ω)µ = dx ∧ dx ∧ ωµ+1i +1j = 0.
Q.E.D.

The kernel of each δ in the first case is equal to the image of the preceding δ but this may no
longer be true in the restricted case and we set:
s
DEFINITION 2.4: We denote by Hq+r
(gq ) the cohomology at ∧s T ∗ ⊗ gq+r of the restricted
1
s
δ-sequence which only depends on gq . The symbol gq is said to be s-acyclic if Hq+r
= ... = Hq+r
=
0, ∀r ≥ 0, involutive if it is n-acyclic and finite type if gq+r = 0 becomes trivially involutive for r
large enough.

DEFINITION 2.5: Rq is said to be involutive when it is formally integrable and its symbol gq
δ

δ

is involutive, that is to say all the sequences ... → ∧s T ∗ ⊗ gq+r → ... are exact ∀0 ≤ s ≤ n, ∀r ≥ 0.
Equivalently, the following procedure, where one may have to change linearly the independent
variables if necessary, is the heart towards the next effective definition of involution. It is intrinsic
even though it must be checked in a particular coordinate system called δ-regular ([17],[18],[29])
and is particularly simple for first order systems without zero order equations.
• Equations of class n: Solve the maximum number βqn of equations with respect to the jets of
order q and class n. Then call (x1 , ..., xn ) multiplicative variables.
− − − − − − − − − − − − − − −−
• Equations of class i ≥ 1: Solve the maximum number βqi of remaining equations with respect
to the jets of order q and class i. Then call (x1 , ..., xi ) multiplicative variables and (xi+1 , ..., xn )
non-multiplicative variables.
−−−−−−−−−−−−−−−−−
• Remaining equations equations of order ≤ q − 1: Call (x1 , ..., xn ) non-multiplicative variables.
In actual practice, we shall use a multiplicative board where the multiplicative ”variables” are represented by their index in upper left position while the non-multiplicative variables are represented
by dots in lower right position.
DEFINITION 2.6: A system of PD equations is said to be involutive if its first prolongation
can be achieved by prolonging its equations only with respect to the corresponding multiplicative
(q+n−i−1)!
−βqi for i = 1, ..., n and
variables. In that case, we may introduce the characters αiq = m (q−1)!((n−i)!
we have dim(gq+1 ) = α1q + ... + αnq . Moreover, one can exhibit the Hilbert polynomial dim(Rq+r )
in r with leading term (α/d!)rd with d ≤ n when α is the smallest non-zero character in the case
of an involutive symbol. Such a prolongation allows to compute in a unique way the principal
(pri) jets from the parametric (par) other ones. This definition may also be applied to nonlinear
systems as well.
REMARK 2.7: For an involutive system with β = βqn < m, then (y β+1 , ..., y m ) can be given
arbitrarily and may constitute the input variables in control theory, though it is not necessary
to make such a choice. In this case, the intrinsic number α = αnq = m − β > 0 is called the
n-character and is the system counterpart of the so-called ”differential transcendence degree” in
differential algebra. As we shall see in the next section, the smallest non-zero character and the
number of zero characters are intrinsic numbers that cannot be known without bringing the system

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to involution and we have α1q ≥ ... ≥ αnq ≥ 0.
EXAMPLE 2.8: ([11], §38, p 40 where one can find the first intuition of formal integrability) The primary ideal q = ((χ1 )2 , χ1 χ3 − χ2 ) provides the system y11 = 0, y13 − y2 = 0 which
is neither formally integrable nor involutive. Indeed, we get d3 y11 − d1 (y13 − y2 ) = y12 and
d3 y12 − d2 (y13 − y2 ) = y22 , that is to say each first and second prolongation does bring a new second
order PD equation. Considering the new system y22 = 0, y12 = 0, y13 −y2 = 0, y11 = 0, the question
is to decide whether this system is involutive or not. One could use Janet or Gröbner algorithm but
with no insight towards involution. In such a simple situation, as there is no PD equation of class 3,
two evident permutations of coordinates (1, 2, 3) → (3, 2, 1) or (1, 2, 3) → (2, 3, 1) both provide one
equation of class 3, 2 equations of class 2 and 1 equation of clas 1. It is then easy to check directly
that the first permutation brings the involutive system y33 = 0, y23 = 0, y22 = 0, y13 − y2 = 0 that
will be used in the sequel and we have α32 = 0, α22 = 0, α12 = 2.
EXAMPLE 2.9: With n = 4, m = 1, q = 1, K = Q(x1 , x2 , x3 , x4 ), let us consider the system R1 :
{y4 − x3 y2 − y = 0,

y3 − x4 y1 = 0

Again, the reader will check easily that the subsystem R1′ ⊂ R1 :

 u≡
v≡

w≡

y4 − x3 y1 − y = 0
y3 − x4 y1 = 0
y2 − y1 = 0

1 2
1 2
1 2

3 4
3 •
• •

(1)

namely the projection R1 of R2 to R1 , is formally integrable and even involutive with one equation of class 4, one equation of class 3 and one equation of class 2.
In the situation of the last remark, the following theorem will generalizing for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the
input do not appear ([27], VI,1.14, p 802). For this, we just need to modify the Spencer form and
we provide the procedure that must be followed in the case of a first order involutive system with
no zero order equation, for example an involutive Spencer form.
• Look at the equations of class n solved with respect to yn1 , ..., ynβ .
• Use integrations by part like:
yn1 − a(x)ynβ+1 = dn (y 1 − a(x)y β+1 ) + ∂n a(x)y β+1 = ȳn1 + ∂n a(x)y β+1
• Modify y 1 , ..., y β to ȳ 1 , ..., ȳ β in order to ”absorb” the various ynβ+1 , ..., ynm only appearing in the
equations of class n.
We have the following unexpected result providing what we shall call reduced Spencer form:
THEOREM 2.10: The new equations of class n only contain yiβ+1 , ..., yim with 0 ≤ i ≤ n − 1
while the equations of class 1, ..., n − 1 no longer contain y β+1 , ..., y m and their jets. Accordingly,
as we shall see in the next section, any torsion element, if it exists, only depends on ȳ 1 , ..., ȳ β .
Proof: The first assertion comes from the absorption procedure. Now, if y m or yim should appear in an equation of class ≤ n−1, prolonging this equation with respect to the non-multiplicative
m
variable xn should bring ynm or yin
and (here involution is essential) we should get a linear combination of equations of various classes prolonged with respect to x1 , ..., xn−1 only, but this is
impossible and we get the desired reduced form.
Q.E.D.
jq

Φ

When Rq is involutive, the linear differential operator D : E → Jq (E) → Jq (E)/Rq = F0 of order
q with space of solutions Θ ⊂ E is said to be involutive and one has the canonical linear Janet
sequence ([17], p 144):
D

D

D

D

n
2
1
Fn −→ 0
... −→
F1 −→
0 −→ Θ −→ T −→ F0 −→

6

where each other operator is first order involutive and generates the compatibility conditions (CC)
of the preceding one. As the Janet sequence can be cut at any place, the numbering of the Janet
bundles has nothing to do with that of the Poincaré sequence for the exterior derivative, contrary
to what many physicists believe. Moreover, the dimensions of the Janet bundles can be computed
at once inductively from the board of multiplicative and non-multiplicative variables that can be
exhibited for D by working out the board for D1 and so on. For this, the number of rows of this
new board is the number of dots appearing in the initial board while the number nb(i) of dots in
the column i just indicates the number of CC of class i for i = 1, ..., n with nb(i) < nb(j), ∀i < j.
It follows that the successive first order operators D1 , ..., Dn are automatically in reduced Spencer
form.
EXAMPLE 2.11: Coming back to Example 2.9 and changing slightly our usual notations, we
get for D1 the following first order involutive system of CC in reduced Spencer form:
 3
 φ ≡
φ2 ≡
 1
φ ≡

d4 v − d3 u + x4 d1 u − x3 d1 v − v = 0
d4 w − d2 u + d1 u − x3 d1 w − w = 0
d3 w − d2 v + d1 v − x4 d1 w = 0

1 2
1 2
1 2

3 4
3 4
3 •

as d4 u does not appear in φ2 and φ3 while u does not appear in φ1 .
We finally obtain for D2 the only CC:
ψ ≡ d4 φ1 − d3 φ2 − d1 φ3 + x4 d1 φ2 − x3 d1 φ1 − φ1 = 0
DEFINITION 2.12: The Janet sequence is said to be locally exact at Fr if any local section of
Fr killed by Dr+1 is the image by Dr of a local section of Fr−1 . It is called locally exact if it is
locally exact at each Fr for 0 ≤ r ≤ n. The Poincaré sequence is locally exact, that is a closed
form is locally an exact form but counterexamples may exist ([18], p 373).
j1

Equivalently, we have the involutive first Spencer operator D1 : C0 = Rq → J1 (Rq ) →
J1 (Rq )/Rq+1 ≃ T ∗ ⊗ Rq /δ(gq+1 ) = C1 of order one induced by D : Rq+1 → T ∗ ⊗ Rq . Introducing
the Spencer bundles Cr = ∧r T ∗ ⊗ Rq /δ(∧r−1 T ∗ ⊗ gq+1 ), the first order involutive (r + 1)-Spencer
operator Dr+1 : Cr → Cr+1 is induced by D : ∧r T ∗ ⊗ Rq+1 → ∧r+1 T ∗ ⊗ Rq : α ⊗ ξq+1 →
dα ⊗ ξq + (−1)r α ∧ Dξq+1 and we obtain the canonical linear Spencer sequence ([17], p 150):
jq

D

D

D

D

n
3
2
1
Cn −→ 0
... −→
C2 −→
0 −→ Θ −→ C0 −→
C1 −→

as the canonical Janet sequence for the first order involutive system Rq+1 ⊂ J1 (Rq ).
The canonical Janet sequence and the canonical Spencer sequence can be connected by a
commutative diagram where the Spencer sequence is induced by the locally exact central horizontal sequence which is at the same time the Janet sequence for jq and the Spencer sequence for
Jq+1 (E) ⊂ J1 (Jq (E)) ([17], p 153) but this result will not be used in this paper (See [5],[20],[22],[23]
for more details on Cosserat and Maxwell equations, see ([16]-[21]) and in particular ([22],[23]) for
applications to engineering and mathematical physics).
REMARK 2.13: We shall revisit Example 2.8 in order to explain the word ”canonical ” that
has been used in the previous definitions. For this, starting with the inhomogeneous system
y33 = u, y13 − y2 = v, we obtain easily the following inhomogeneous involutive system with its
corresponding board of multiplicative and non-multiplicative variables:
 4
Φ


 3
Φ
Φ2


 1
Φ

≡
≡
≡
≡

y33
y23
y22
y13 − y2

=
=
=
=

u
d1 u − d3 v
d11 u − d13 v − d2 v
v

1
1
1
1

2
2
2
•

3
•
•
•

Using prolongation with respect to the 4 non-multiplicative variables involved should bring 4 first
order CC for the right members and we could wait for 4 third order CC involving u and v.
7

Surprisingly, we need the only CC Ψ ≡ d33 v − d13 u + d2 u = 0 and obtain the differential sequence:
0 −→ Θ −→ 1 −→ 2 −→ 1 −→ 0
as a single CC has no CC for itself (See ([18],p365) for the effective general procedure).
Such a differential sequence is quite different from the canonical Janet sequence:
D

D

D

1
2
0 −→ Θ −→ 1 −→ 4 −→
4 −→
1 −→ 0

which is the only sequence that can provide the Spencer sequence as we already said and could not
be obtained by simply using Gröbner bases. This remark will become essential in mathematical
physics (foundations of continuum mechanics, gauge theory, general relativity) where only involutive operators must be used ([20],[22],[23]). We also check that the Euler-Poincaré characteristic,
namely the alternate sum of the circled dimensions of the vector bundles involved, does not depend
on the differential sequence used as we get 1 − 2 + 1 = 1 − 4 + 4 − 1 = 0 (See [18], p 378).
In the same spirit, using certain parametric jet variables as new unknowns, we may set z 1 = y, z 2 =
y1 , z 3 = y2 , z 4 = y3 in order to obtain the following involutive first order system with no zero order
equation:

 class 3
class 2

class 1

d3 z 1 − z 4 = 0, d3 z 2 − z 3 = 0, d3 z 3 = 0, d3 z 4 = 0
d2 z 1 − z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 3 = 0, d2 z 4 = 0
d1 z 1 − z 2 = 0, d1 z 4 − z 3 = 0

1 2
1 2
1 •

3
•
•

where we have separated the classes. Contrary to what could be believed, this operator does not
describe the Spencer sequence that could be obtained from the previous Janet sequence. Indeed, introducing the trivial vector bundle E with local coordinates (x1 , x2 , x3 , y), it follows that J1 (E) has
local coordinates (x1 , x2 , x3 , z 1 , z 2 , z 3 , z 4 ). Now, the involutive system R2 ⊂ J2 (E) ⊂ J1 (J1 (E))
with involutive symbol g2 ⊂ S2 T ∗ ⊗ E ⊂ T ∗ ⊗ T ∗ ⊗ E ⊂ T ∗ ⊗ J1 (E) projects onto J1 (E)
but dim(R2 ) = 6 because par(R2 ) = {y, y1 , y2 , y3 , y11 , y12 } while we have only 4 unknowns
(z 1 , z 2 , z 3 , z 4 ). Nevertheless, as R2 projects onto J1 (E), we may construct a canonical Janet
sequence for this new system where the successive Janet bundles involved will be the Spencer
bundles Cr = ∧r T ∗ ⊗ J1 (E)/δ(∧r−1 T ∗ ⊗ g2 ) with a shift by one step in the numbering of the
bundles as now C0 = J1 (E) and the successive operators are induced by the composition of the
inclusion R2 ⊂ J2 (E) with the Spencer operator D : J2 (E) −→ T ∗ ⊗ J1 (E) as in ([17],p144,150) or
([18],p356). In any case, it is essential to notice that, both in the canonical Spencer sequence and
in the canonical Janet sequence, any intermediate operator can be constructed explicitely without
knowing the previous ones.
EXAMPLE 2.14: With m = 1, n = 4, q = 2, one could treat similarly the involutive system:
y44 = 0, y34 = 0, y33 = 0, y24 − y13 = 0 with one equation of class 4, two equations of class 3 and
one equation of class 2.
EXAMPLE 2.15: Coming back to the involutive system of Example 2.9 with variable coefficients, we let the reader prove that the Janet sequence is:
D

D

D

1
2
0 −→ Θ −→ 1 −→ 3 −→
3 −→
1 −→ 0

EXAMPLE 2.16: Let us finally consider the following involutive system of PD equations with
two independent variables (x1 , x2 ) and three unknowns (y 1 , y 2 , y 3 ), where again a is an arbitrary
constant parameter and we have set for simplicity yik = di y k :
 2
 y2 + y12 + y23 − y13 − ay 3
y 1 − y12 − y23 − y13 − ay 3
 21
y1 − y12 − 2y13

=0
=0
=0

1 2
1 2
1 •

Then the corresponding Janet sequence is:

D

D

1
1 −→ 0
0 −→ Θ −→ 3 −→ 3 −→

8

Finally, setting ȳ 1 = y 1 − y 3 , ȳ 2 = y 2 + y 3 , we obtain the new first order involutive system:
 2
 ȳ2 − ȳ12 − ay 3
ȳ 1 − ȳ12 − ay 3
 21
ȳ1 − ȳ12 = 0

=0
=0

1
1
1

2
2
•

with two equations of class 2 and one equation of class 1 in which y 3 surprisingly no longer appears.
If χ1 , ..., χn are n algebraic indeterminates or, in a more intrinsic way, if χ = χi dxi ∈ T ∗ is
a covector and D : E −→ F : ξ −→ aτk µ (x)∂µ ξ k (x) is a linear involutive operator of order q,
we may introduce the characteristic matrix a(x, χ) = (aτk µ (x)χµ | µ = q) and the resulting map
σχ (D) : E −→ F is called the symbol of D at χ. Then there are two possibilities:
• If maxχ rk(σχ (D) < m ⇔ αnq > 0: the characteristic matrix fails to be injective for any covector.
• If maxχ rk(σχ (D) = m ⇔ αnq = 0: the characteristic matrix fails to be injective if and only if
all the determinants of the m × m submatrices vanish. However, one can prove that this algebraic
ideal a ∈ K[χ] is not intrinsically defined and must be replaced by its radical rad(a) made by all
polynomials having a power in a. This radical ideal is called the characteristic ideal of the operator.
DEFINITION 2.17: For each x ∈ X, the algebraic set defined by the characteristic ideal is
called the characteristic set of D at x and V = ∪x∈X Vx is called the characteristic set of D.
One has the following important theorem ([18],[29]) that will play an important part later on:
THEOREM 2.18: (Hilbert-Serre) The dimension d(V ) of the characteristic set, that is the maximum dimension of the irreducible components, is equal to the number of non-zero characters while
the codimension cd(V ) = n − d(V ) is equal to the number of zero characters, that is to the number
of ”full ” classes in the board of multiplicative variables of an involutive system.
EXAMPLE 2.19: Coming back to Remark 2.12, we obtain a = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 ) =⇒
rad(a) = (χ2 , χ3 ) and thus cd(V ) = 2. However, if we take only into account Example 2.8, we
should only get the radical ideal (χ3 ) and the wrong result cd(V ) = 1. The reason for using the
radical can be seen from the equivalent first order system that shoul provide b = ((χ3 )4 , ...) with homogeneous polynomials of degree 4 and thus b ⊂ a with a strict inclusion though rad(a) = rad(b).
A similar situation can be obtained with Examples 1.1 and 2.9.
3) TOOLS FROM MODULE THEORY :
We may roughly say that, if a reader familiar with Gröbner bases ([4],[6]) and computer algebra
looks at the previous section, he will feel embarassed because he will believe that ”intrinsicness
is always competing with complexity ” as can be seen from Examples 2.8 + 2.12. However, even
if he admits that it may be useful to have intrinsic and thus canonical procedures, then looking
at the existing literature on differential modules ([1],[9],12]), he will really feel to be on another
planet as the main difficulty involved in the theory of differentia modules is to understand why and
where formal integrability and involution become essential tools to apply quite before dealing with
the homological background of ”algebraic analysis ” involving extension modules. This is the main
reason for which the case of variable coefficients is rarely treated ”by itself ” always refering to Weyl
algebras for examples and the main difficulty we found when writing ([18], in particular Chapter
IV). The central concept, essential for applications but well hidden in the literature dealing with
filtred modules ([14],p 383) and totally absent from the use of Gröbner bases because it amounts
to formal integrability by duality, is that of a ”strict morphism ”. Accordingly, the purpose of this
section will be to explain why such a definition, which seems to be purely technical, will be so
important for studying extension modules and purity.
If P = aµ dµ ∈ D = K[d], the highest value of |µ| with aµ 6= 0 is called the order of the operator
P and the ring D with multiplication (P, Q) −→ P ◦ Q = P Q is filtred by the order q of the
operators. We have the filtration 0 ⊂ K = D0 ⊂ D1 ⊂ ... ⊂ Dq ⊂ ... ⊂ D∞ = D. Moreover, it
is clear that D, as an algebra, is generated by K = D0 and T = D1 /D0 with D1 = K ⊕ T if we
9

identify an element ξ = ξ i di ∈ T with the vector field ξ = ξ i (x)∂i of differential geometry, but
with ξ i ∈ K now. It follows that D = D DD is a bimodule over itself, being at the same time a left
D-module by the composition P −→ QP and a right D-module by the composition P −→ P Q.
We define the adjoint functor ad : D −→ Dop : P = aµ dµ −→ ad(P ) = (−1)|µ| dµ aµ and we have
ad(ad(P )) = P . It is easy to check that ad(P Q) = ad(Q)ad(P ), ∀P, Q ∈ D. Such a definition can
also be extended to any matrix of operators by using the transposed matrix of adjoint operators
(See [18],[19],[22] for more details and applications to control theory and mathematical physics).
Accordingly, if y = (y 1 , ..., y m ) are differential indeterminates, then D acts on y k by setting
k
and y0k = y k . We may therefore use he jet coordidi y = yik −→ dµ y k = yµk with di yµk = yµ+1
i
nates in a formal way as in the previous section. Therefore, if a system of OD/PD equations is
written in the form Φτ ≡ aτk µ yµk = 0 with coefficients a ∈ K, we may introduce the free differential
module Dy = Dy 1 + ... + Dy m ≃ Dm and consider the differential submodule I = DΦ ⊂ Dy
which is usually called the module of equations, both with the differential module M = Dy/DΦ
or D-module and we may set M = D M if we want to specify the ring of differential operators.
The work of Macaulay only covers the case m = 1 with K replaced by k ⊆ cst(K). Again, we
k
+ ∂i aτk µ yµk
may introduce the formal prolongation with respect to di by setting di Φτ ≡ aτk µ yµ+1
i
k
k
in order to induce maps di : M −→ M : ȳµ −→ ȳµ+1i by residue if we use to denote the residue
Dy −→ M : y k −→ ȳ k by a bar as in algebraic geometry. However, for simplicity, we shall not
write down the bar when the background will indicate clearly if we are in Dy or in M .
k

As a byproduct, the differential modules we shall consider will always be finitely generated
(k = 1, ..., m < ∞) and finitely presented (τ = 1, ..., p < ∞). Equivalently, introducing the
matrix of operators D = (aτk µ dµ ) with m columns and p rows, we may introduce the morphism
D

Dp −→ Dm : (Pτ ) −→ (Pτ Φτ ) : P −→ P Φ = P D over D by acting with D on the left of these
row vectors while acting with D on the right of these row vectors and the presentation of M is
defined by the exact cokernel sequence Dp −→ Dm −→ M −→ 0. It is essential to notice that
the presentation only depends on K, D and Φ or D, that is to say never refers to the concept of
(explicit or formal) solutions. It is at this moment that we have to take into account the results
of the previous section in order to understant that certain presentations will be much better than
others, in particular to establish a link with formal integrability and involution.
It follows from its definition that M can be endowed with a quotient filtration obtained from
that of Dm which is defined by the order of the jet coordinates yq in Dq y. We have therefore the
inductive limit 0 ⊆ M0 ⊆ M1 ⊆ ... ⊆ Mq ⊆ ... ⊆ M∞ = M with di Mq ⊆ Mq+1 and M = DMq for
q ≫ 0 with prolongations Dr Mq ⊆ Mq+r , ∀q, r ≥ 0.
DEFINITION 3.1: ([14],p 383) If M and N are two differential modules and f : M −→ N is
a morphism over D compatible with the filtration, that is if f (Mq ) ⊂ Nq with induced morphism
fq : Mq −→ Nq , then f is a strict morphism if fq (Mq ) = f (M ) ∩ Nq , ∀q ≥ 0.
Equivalently, chasing in the following diagram:
0
↓

0
↓
fq

Mq
↓

−→ Nq
↓

M

−→

f

N

0
↓
−→ coker(fq )
↓

−→ 0

−→

−→ 0

coker(f )

then f is strict if the induced morphism coker(fq ) −→ coker(f ) is a monomorphism ∀q ≥ 0.
DEFINITION 3.2: An exact sequence of morphisms finishing at M is said to be a resolution of
M . If the differential modules involved apart from M are free, we shall say that we have a free
resolution of M . Moreover, a sequence of strict morphisms is called a strict sequence.
LEMMA 3.3: If f is a strict morphism as in the last definition, there are exact sequences

10

0 −→ coker(fq ) −→ coker(fq+1 ), ∀q ≥ 0.
Proof: As f is compatible with the filtrations and Mq ⊆ Mq+1 , Nq ⊆ Nq+1 , we have an induced
morphism coker(fq ) −→ coker(fq+1 ). Now, as f is also strict, we have the following commutative
and exact diagram:
0 −→

coker(fq )
−→ coker(f )
↓
k
0 −→ coker(fq+1 ) −→ coker(f )
The lemma finally follows from an elementary chase.
Q.E.D.
Having in mind that K is a left D-module with the standard action (D, K) −→ K : (di , a) −→
∂i a and that D is a bimodule over itself, we have only two possible constructions:
DEFINITION 3.4: We define the system R = homK (M, K) = M ∗ and set Rq = homK (Mq , K) =
Mq∗ as the system of order q. We have the projective limit R = R∞ −→ ... −→ Rq −→ ... −→
R1 −→ R0 . It follows that fq ∈ Rq : yµk −→ fµk ∈ K with aτk µ fµk = 0 defines a section at order
q and we may set f∞ = f ∈ R for a section of R. For an arbitrary differential field K, such a
definition has nothing to do with the concept of a formal power series solution (care).
DEFINITION 3.5: We may define the right differential module homD (M, D).
PROPOSITION 3.6: When M is a left D-module, then R is also a left D-module.
Proof: As D is generated by K and T as we already said, let us define:
(af )(m) = af (m),

∀a ∈ K, ∀m ∈ M

(ξf )(m) = ξf (m) − f (ξm),

∀ξ = ai di ∈ T, ∀m ∈ M

In the operator sense, it is easy to check that di a = adi + ∂i a and that ξη − ηξ = [ξ, η] is the
k
and thus
standard bracket of vector fields. We finally get (di f )kµ = (di f )(yµk ) = ∂i fµk − fµ+1
i
recover exactly the Spencer operator of the previous section though this is not evident at all. We
k
k
k
=⇒ di dj = dj di , ∀i, j = 1, ..., n and thus
+ fµ+1
− ∂j fµ+1
also get (di dj f )kµ = ∂ij fµk − ∂i fµ+1
i +1j
i
j
di Rq+1 ⊆ Rq =⇒ di R ⊂ R induces a well defined operator R −→ T ∗ ⊗ R : f −→ dxi ⊗ di f . This
result has been discovered (up to sign) by Macaulay in 1916 ([11]). For more details on the Spencer
operator and its applications, the reader may look at ([22],[23]).
Q.E.D.
As D is a bimodule over itself, it follows from this proposition that that D∗ = homK (D, K)
is a left D-module. Moreover, using Baer’s criterion ([28]), it is known that D∗ is an injective
D-module as there is a canonical isomorphism:
M ∗ = homK (M, K) ≃ homD (M, D∗ )
where both sides are well defined ([2], Prop 11, p 18;)([28], p 37).
DEFINITION 3.7: With any differential module M we shall associate the graded module
G = gr(M ) over the polynomial ring gr(D) ≃ K[χ] by setting G = ⊕∞
q=0 Gq with Gq = Mq /Mq+1
and we get gq = G∗q where the symbol gq is defined by the short exact sequences:
0 −→ Mq−1 −→ Mq −→ Gq −→ 0

=⇒

0 −→ gq −→ Rq −→ Rq−1 −→ 0

We have the short exact sequences 0 −→ Dq−1 −→ Dq −→ Sq T −→ 0 leading to grq (D) ≃ Sq T
and we may set as usual T ∗ = homK (T, K) in a coherent way with differential geometry. Moreover
any compatible morphism f : M −→ N induces a morphism gr(f ) : gr(M ) −→ gr(N ).

11

EXAMPLE 3.8: If K = Q(x), m = 1, n = 1, let us consider the system yxxx − yx = 0 for which
we may exhibit the basis of sections {f = (1, 0, 0, 0, ...), f ′ = (0, 1, 0, 1, ...), f ” = (0, 0, 1, 0, ...)} as in
([11],§59,p 67) or ([21]). We obtain dx f = 0, dx f ′ = −f −f ”, dx f ” = −f ′ and check that all the sections can be generated by a single one, namely f ” which describes the power series of ch(x)−1. With
1
now m = 2, let us consider the module defined by the system yxx
= 0, yx2 = 0. Setting y = y 2 −xy 1 ,
2
1
1
1
1
we successively get yx = −xyx − y , yxx = −2yx, yxxx = 0 =⇒ y = x2 yxx − yx , y 2 = x2 yxx − xyx + y
and a differential isomorphism with the module defined by the new system yxxx = 0. All the sections of the second system are easily seen to be generated by the single section f = (0, 0, 1, ...), a
2
1
= 0, ..., f 2 (x) = x2 , fx2 = 0, ...
result leading to the only generating section f 1 (x) = x2 , fx1 = − 21 , fxx
of the initial system but these sections do not describe solutions because ∂x f 1 − fx1 = 1 6= 0 and
∂x f 2 − fx2 = x 6= 0. We do not know any reference in computer algebra dealing with sections (See
[21] for more details)
Coming back to the presentation of M under study, we notice that the morphism D involved
is not compatible unless we shift the index of the filtration by ord(D) = q. In that case, we obtain
im(D) = I ⊂ Dy and may set Iq+r = I ∩ Dq+r y but we have in general Drp D ⊆ Iq+r only, that
is the equations of order q+r may not be produced by r prolongations only. We have thus obtained:
PROPOSITION 3.9: The morphism induced by D is strict if and only if D is formally integrable. Accordingly, the module version of both the Janet sequence and the Spencer sequence are
strictly exact sequences.
Proof: Using q + r instead of q in Lemma 3.3 and applying homK (•, K), we obtain the epimorphisms Rq+r+1 −→ Rq+r −→ 0, ∀r ≥ 0.
Q.E.D.
The reader will find in ([18], IV,3) more details on the relations existing between G and M
which are needed in order to study the non-commutative situation, at least when K is a differential field as such a case is hard enough. We obtain in particular the Hilbert polynomial
α d
dimK (Mq+r ) = dimK (Rq+r ) = d!
r + ... where α is called the multiplicity of M and we use to set
cdD (M ) = cd(M ) = n − r, rkD (M ) = rk(M ) = α if cd(M ) = 0 and 0 otherwise.
EXAMPLE 3.10: Coming back to the Example 2.8 of Macaulay, we obtain from Remark 2.13
the free resolution 0 −→ D −→ D2 −→ D −→ M −→ 0 but only the morphism on the left is strict
as for the morphism on the right we know that its image is indeed I2 = {y33 , y23 , y22 , y13 − y2 } and
not just {y33 , y13 −y2 }. However, bringing the system to involution, we get the strict free resolution
0 −→ D −→ D4 −→ D4 −→ D −→ M −→ 0 as the module version of the Janet sequence and we
let the reader exhibit the module version of the corresponding Spencer sequence as an exercise.
If M is a differential module over the ring D = K[d] of differential operators and m ∈ M , then
the differential submodule Dm ⊂ M is defined by a system of OD or PD equations for one unknown
and we may look for its codimension cd(Dm). A similar comment can be done for any differential
submodule M ′ ⊂ M . Sometimes, a single element m ∈ M , called differentially primitive element,
may generate M if Dm = M .
EXAMPLE 3.11: With K = Q, let us consider the differential moduleM defined by the system
1
1
yxx
− y 1 = 0, yx2 = 0. Then, setting y = y 1 − y 2 , we get yx = yx1 , yxx = yxx
= y 1 , yxx − y = y 2 and
thus yxxx − yx = 0 provides another way to describe M by means of a single element as in ([18],
p435). We have the following commutative and exact diagram:
0 −→
0 −→

D2
↓
D

D2
↓
D

−→
−→

(P, Q)
−→
(P (d2 − 1), Qd)
↓
↓
(P d + Q) −→ ((P d + Q)(d3 − d))
12

−→ M
k
−→ M

−→ 0
−→ 0

1
If now we consider the differential module M defined by yxx
− ay 1 = 0, yx2 = 0 where a is a constant
parameter, we cannot find a differentially primitive element when K = Q if a = 0 but we can when
K = Q(x) for any value of a, as in Example 3.8.

We may check the following definition in a constructive way ([27]):
DEFINITION 3.12: tr (M ) = {m ∈ M | cd(Dm) > r} is the greatest differential submodule of
M having codimension > r.
PROPOSITION 3.13: cd(M ) = cd(V ) = r ⇐⇒ αqn−r 6= 0, αn−r+1
= ... = αnq = 0 ⇐⇒ tr (M ) 6=
q
M, tr−1 (M ) = ... = t0 (M ) = t(M ) = M and this intrinsic result can be most easily checked by
using the Spencer form of the system defining M .
We are now in a good position for defining and studying purity for differential modules.
DEFINITION 3.14: M is r-pure ⇐⇒ tr (M ) = 0, tr−1 (M ) = M ⇐⇒ cd(Dm) = r, ∀m ∈ M . In
particular, M is 0-pure if t(M ) = 0 and, if cd(M ) = r but M is not r-pure, we may call M/tr (M )
the pure part of M . It follows that tr−1 (M )/tr (M ) is equal to zero or is r-pure (See the picture in
[18], p 545). Finally, when tr−1 (M ) = tr (M ), we shall say that there is a gap in the purity filtration:
0 = tn (M ) ⊆ tn−1 (M ) ⊆ ... ⊆ t1 (M ) ⊆ t0 (M ) = t(M ) ⊆ M
PROPOSITION 3.15: tr (M ) does not depend on the presentation or on the filtration of M .
EXAMPLE 3.16: If K = Q and M is defined by the involutive system y33 = 0, y23 = 0, y13 = 0,
then z = y3 satifies d3 z = 0, d2 z = 0, d1 z = 0 and cd(Dz) = 3 while z ′ = y2 only satisfies d3 z ′ = 0
and cd(Dz ′ ) = 1. We have the purity filtration 0 = t3 (M ) ⊂ t2 (M ) = t1 (M ) ⊂ t0 (M ) = t(M ) = M
with one gap and two strict inclusions.
We now recall the definition of the extension modules extiD (M, D) that we shall simply denote
by exti (M ) and the way to use their dimension or codimension. We point out once more that
these numbers cannot be obtained without bringing the underlying systems to involution in order
to get informations on M from informations on G. We divide the procedure into four steps that
can be achieved by means of computer algebra ([27]):
• Construct a free resolution of M , say:
... −→ Fi −→ ... −→ F1 −→ F0 −→ M −→ 0
• Suppress M in order to obtain the deleted sequence:
... −→ Fi −→ ... −→ F1 −→ F0 −→ 0
• Apply homD (•, D) in order to obtain the dual sequence heading backwards:
... ←− homD (Fi , D) ←− ... ←− homD (F1 , D) ←− homD (F0 , D) ←− 0
• Define exti (M ) to be the cohomology at homD (Fi , D) in the dual sequence with ext0 (M ) =
homD (M, D).
The following nested chain of difficult propositions and theorems can be obtained, even in the
non-commutative case, by combining the use of extension modules and bidualizing complexes in the
framework of algebraic analysis. The main difficulty is to obtain first these results for the graded
module G = gr(M ) by using techniques from commutative algebra before extending them to the
filtred module M as in ([1],[9],[18],[19]).
THEOREM 3.17: The extension modules do not depend on the resolution of M used.

13

PROPOSITION 3.18: Applying homD (•, D) provides right D-modules that can be transformed
to left D-modules by means of the side changing functor and vice-versa. Namely, if ND is a right
D-module, then D N = ∧n T ⊗K ND is the converted left D-module while, if D N is a left D-module,
then ND = ∧n T ∗ ⊗K D N is the converted right D-module.
PROPOSITION 3.19: Instead of applying homD (•, D) and the side changing functor in the
module framework, we may use ad in the operator framework. Namely, to any operator D : E −→ F
we may associate the formal adjoint ad(D) : ∧n T ∗ ⊗ F ∗ −→ ∧n T ∗ ⊗ E ∗ with the useful though
striking relation rkD (ad(D)) = rkD (D).
PROPOSITION 3.20: exti (M ) is a torsion module ∀1 ≤ i ≤ n but ext0 (M ) = homD (M, D)
may not be a torsion module.
EXAMPLE 3.21: When M is a torsion module, we have homD (M, D) = 0 (exercise). When
n = 3 and the torsion-free module M is defined by the formally surjective div operator, the formal
adjoint of div is −grad which defines a torsion module. Also, when n = 1 as in classical control
theory, a controllable system allows to define a torsion-free module M which is free in that case
and homD (M, D) is thus also a free module.
THEOREM 3.22:

exti (M ) = 0, ∀i ≥ n + 1.

THEOREM 3.23:

cd(exti (M )) ≥ i.

PROPOSITION 3.24:
THEOREM 3.25:

exti (M ) = 0, ∀i < cd(M ).
cd(M ) ≥ r ⇔ exti (M ) = 0, ∀i < r.

PROPOSITION 3.26:

cd(M ) = r =⇒ cd(extr (M )) = r and extr (M )is r-pure.

PROPOSITION 3.27:

extr (extr (M )) is equal to 0 or is r-pure, ∀0 ≤ r ≤ n.

PROPOSITION 3.28: If we set t−1 (M ) = M , there are exact sequences ∀0 ≤ r ≤ n:
0 −→ tr (M ) −→ tr−1 (M ) −→ extr (extr (M ))
THEOREM 3.29: If cd(M ) = r, then M is r-pure if and only if there is a monomorphism
0 −→ M −→ extr (extr (M )) of left differential modules.
THEOREM 3.30: M is pure ⇐⇒ exts (exts (M )) = 0, ∀s 6= cd(M ).
The last two theorems are known to characterize purity but it is however evident that they are
not very useful in actual practice.
THEOREM 3.31: When M is r-pure, the characteristic ideal is thus unmixed, that is a finite
intersection of prime ideals having the same codimension r and the characteristic set is equidimensional, that is the union of irreducible algebraic varieties having the same codimension r.
REMARK 3.32: For the reader knowing more about commutative algebra, we add a few details about the localization used in the primary decomposition of a module which are not so well
known ([3],[18],[21],[31]). For simplicity, setting k = cst(K), we shall denote by A = k[χ] the
polynomial ring isomorphic to D = k[d] and consider a module M over A. We denote as usual
by spec(A) the set of proper prime ideals in A, by max(A) the subset of maximal ideals in A
and by ass(M ) = {p ∈ spec(A)|∃0 6= m ∈ M, p = annA (m)} the (finite) set {p1 , ..., pt } of
associated prime ideals, while we denote by {p1 , ...ps } the subset of minimum associated prime
ideals. It is well known that M 6= 0 =⇒ ass(M ) 6= ∅. We recall that an ideal q ⊂ A is pprimary if ab ∈ q, b ∈
/ q =⇒ a ∈ rad(q) = p ∈ spec(A). We say that a module Q is p-primary
if am = 0, 0 6= m ∈ Q =⇒ a ∈ p = rad(q) ∈ spec(A) when q = annA (Q) or, equivalently,

14

ass(Q) = {p}. Similarly, we say that a module P is p-prime if am = 0, 0 6= m ∈ P =⇒ a ∈
p ∈ spec(A) when p = annA (P ). It follows that any p-prime or p-primary module is r-pure with
n − r = trd(A/p), a result generalizing ([11],§4, p 43). Accordingly, a module M is r-pure if and
only if a = annA (M ) admits a primary decomposition a = q1 ∩...∩qs and rad(a) = p1 ∩...∩ps with
cd(A/pi ) = cd(M ) = r, ∀i = 1, ..., s. In that case, the monomorphism 0 −→ M −→ ⊕p∈ass(M) Mp
induces a monomorphism 0 −→ M −→ Q1 ⊕ ... ⊕ Qs called primary embedding where the primary
modules Qi are the images of the localization morphisms M −→ Mpi = S −1 M with S = A − p
inducing epimorphisms M −→ Qi −→ 0 for i = 1, ..., s. Macaulay was only considering the case
M = A/a with primary decomposition a = q1 ∩ ... ∩ qs .
EXAMPLE 3.33: With k = Q and n = 3, then a = rad(a) = (χ1 , χ2 χ3 ) = (χ1 , χ2 ) ∩ (χ1 , χ3 ) is
unmixed and M = A/a is 2-pure while a = rad(a) = (χ1 χ2 , χ1 χ3 ) = (χ1 )∩(χ2 , χ3 ) is mixed, though
an intersection of two minimum prime ideals and M = A/a is not 1-pure. On the contrary, if one
has the primary decomposition a = ((χ1 )2 , χ1 χ2 , χ1 χ3 , χ2 χ3 ) = (χ1 , χ2 ) ∩ (χ1 , χ3 ) ∩ (χ1 , χ2 , χ3 )2 =
p1 ∩ p2 ∩ m2 and M = A/a, then ass(M ) = {p1 , p2 , m} with pi ⊂ m for i = 1, 2, though
rad(a) = p1 ∩ p2 as before. In that case, there is an embedding 0 −→ M −→ Q1 ⊕ Q2 ⊕ Q3
where Qi = A/pi is the image of the localization morphism M −→ Mpi for i = 1, 2 because p1
is killed by χ3 ∈ A − p1 , p2 is killed by χ2 ∈ A − p2 and Q3 = A/m2 is m-primary because
rad(m2 ) = m ∈ max(A). We have also an embedding 0 −→ M −→ Mp1 ⊕ Mp2 ⊕ Mm but no
element of the multiplicative set A − m = {1 + a|a ∈ m} can kill any element of M and the image
of M into Mm is thus isomorphic to M which is not a primay module. It is important to notice
that the example of Macaulay q = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 − χ2 ) provides a p-primary module
A/q with p = (χ3 , χ2 ) even though the annihilating ideal of G = gr(M ) is the homogeneous
ideal a = ((χ3 )2 , χ2 χ3 , (χ2 )2 , χ1 χ3 ) = ((χ2 )2 , χ3 ) ∩ (χ1 , χ2 , χ3 )2 which is a mixed ideal because
ass(A/a) = {(χ2 , χ3 ), (χ1 , χ2 , χ3 )}. However, we get rad(a) = (χ2 , χ3 ) in a coherent way.
PROBLEM : Is it possible to have a test for checking whether a differential module is pure or
not without using the previous results ?
4) MOTIVATION :
As we already said in the introduction and in the previous section, a torsion-free module M
is 0-pure because in that case t0 (M ) = t(M ) = 0. Accordingly, M can be embedded into a free
module F and the inclusion, which may not be strict when n > 1, provides a parametrization by
means of a finite number of potential-like arbitrary functions in the classical language of elasticity (Airy function) or electromagnetism (EM 4-potential). As it is clear that such a situation is
only a very particular case of purity, it remains to wonder what can happen for an r-pure module
whenever r ≥ 1. One has the following result ([9], [18], compare to [1], p494):
THEOREM 4.1: If M is an r-pure differential module with r ≥ 1, there exists a differential
module L with pd(L) ≤ r and an embedding M ⊆ L.
Proof: First of all we notice that we have r > 0 and thus any element m ∈ M is surely a torsion element because cd(Dm) > 0, that is M = t(M ) is a torsion module with ext0 (M ) = homD (M, D) =
0 because D is an integral domain. Let now ... −→ Fr −→ ... −→ F1 −→ F0 −→ N −→ 0 be a free
resolution of the right differential module N = extr (M ). According to Proposition 3.26, we have
cd(N ) = r > 0 too and N is also a torsion module with ext0 (N ) = homD (N, D) = 0. Applying the
functor homD (•, D) to the previous sequence or , equivalently, constructing the adjoint sequence in
the operator framework while using the fact that exti (N ) = 0, ∀i < r according to Theorem 3.25,
we obtain the finite long exact sequence with exactly r morphisms because N is finitely presented
and extr (N ) 6= 0:
0 −→ homD (F0 , D) −→ ... −→ homD (Fr−1 , D) −→ homD (Fr , D) −→ L −→ 0
where the left differential module L is the cokernel of the last morphism on the right. As
homD (F, D) is free whenever F is free because of the bimodule structure of D = D DD , the

15

corresponding deleted complex is:
0 −→ homD (F0 , D) −→ ... −→ homD (Fr−1 , D) −→ homD (Fr , D) −→ 0
Applying again homD (•, D) and using the reflexivity of any free module F , that is the isomorphism
homD (homD (F, D), D) ≃ F , we obtain the dual sequence:
0 −→ Fr −→ ... −→ F1 −→ F0 −→ 0
and a similar procedure may be followed with operators as we shall see in the next illustrating
examples ([18],[27]). This sequence is exact everywhere but at Fr and at F0 where its cohomology
is just N by definition, that is to say extr (L) = N = extr (M ). Looking for the cohomology at
homD (Fr , D) in the sequence obtained by duality from the resolution of N with coboundry module
Br and cocycle module Zr , we obtain the following commutative and exact diagram:
0
0
↓
↓
0 −→
Br
=
Br
↓
↓
0 −→
Zr
−→ homD (Fr , D)
↓
↓
0 −→ extr (N ) −→
L
↓
↓
0
0

−→ 0

Finally, composing the bottom monomorphism with the monomorphism 0 −→ M −→ extr (N )
provided by Theorem 3.29, we get the desired embedding M ⊆ L. It must be noticed that such a
procedure can be followed equally well in the commutative and non-commutative framework, that
is when K is a field of constants or a true differential field.
Q.E.D.
EXAMPLE 4.2: With K = Q, m = 1, n = 4, q = 2, let us study the 2-pure differential module
M defined by the involutive system:
 4
Φ


 3
Φ
Φ2


 1
Φ

≡
≡
≡
≡

y44
y34
y33
y24 − y13

=0
=0
=0
=0

1
1
1
1

2
2
2
2

3
3
3
•

4
•
•
•

From the board of multiplicative variables we may construct at once the Janet sequence:
D

D

D

1
2
4 −→
1 −→ 0
0 −→ Θ −→ 1 −→ 4 −→

where D1 is defined by the involutive system:
 4
Ψ


 3
Ψ
 Ψ2

 1
Ψ

≡
≡
≡
≡

d4 Φ3 − d3 Φ4
d4 Φ2 − d3Φ3
d4 Φ1 − d2 Φ4 + d1 Φ3
d3 Φ1 − d2 Φ3 + d1 Φ2

=0
=0
=0
=0

1
1
1
1

2
2
2
2

3
3
3
3

4
4
4
•

and D2 by the (trivially) involutive system:


Ω

≡

d4 Ψ1 − d3 Ψ2 + d2 Ψ4 − d1 Ψ3

=0

1 2

We have therefore the resolution:
0 −→ D −→ D4 −→ D4 −→ D −→ M −→ 0

16

3 4

leading to pd(M ) ≤ 3 and the deleted complex is:
0 −→ D −→ D4 −→ D4 −→ D −→ 0
Applying homD (•, D) to this sequence, we get the sequence:
0 ←− D ←− D4 ←− D4 ←− D ←− 0
which can be described by the following adjoint sequence:
ad(D)

←−

0 ←− 1

4

ad(D1 )

←−

4

ad(D2 )

←−

1 ←− 0

which is not a Janet sequence. As M is a torsion module, using now Theorem 3.25 we get
ext0 (M ) = 0, ext1 (M ) = 0 and we check that N = ext2 (M ) 6= 0. For this, dualizing Ψ by λ
and Ω by θ, we have to look for the CC of the inhomogeneous system:
 1
Ψ −→ −d4 θ = λ1


 2
Ψ −→
d3 θ = λ2
3
Ψ
−→
d1 θ = λ3


 4
Ψ −→ −d2 θ = λ4
which are not already provided by
 1
Φ −→


 2
Φ −→
Φ3 −→


 4
Φ −→

the system:

−d4 λ2 − d3 λ1
−d4 λ3 − d1 λ1
−(d4 λ4 − d2 λ1 ) + (d3 λ3 − d1 λ2 )
d3 λ4 + d2 λ2

=0
=0
=0
=0

One can check that the torsion module N can be generated by {u = d2 λ3 + d1 λ4 , v = d3 λ3 − d1 λ2 }
satisfying the involutive system:
 4
φ


 3
φ
φ2


 1
φ

with the two CC:


ψ2
ψ1

≡
≡
≡
≡

d4 u − d1 v
d4 v
d3 u − d2 v
d3 v

=0
=0
=0
=0

≡ d4 φ2 − d3 φ4 + d2 φ3 − d1 φ1
≡ d4 φ1 − d3 φ3

1
1
1
1

2
2
2
2

3
3
3
3

=0
=0

4
4
•
•

1
1

2 3
2 3

4
4

Accordingly, we have the following strict free resolution of N :
0 −→ D2 −→ D4 −→ D2 −→ N −→ 0
with deleted complex:
0 −→ D2 −→ D4 −→ D2 −→ 0
Applying homD (•, D), we get the desired resolution of L, namely:
0 ←− L ←− D2 ←− D4 ←− D2 ←− 0
Dualizing ψ by z, we finally discover that L is defined by the involutive system:

−φ1



−φ2
−φ3



φ4

−→
−→
−→
−→

d4 z 1 − d1 z 2
d4 z 2
d3 z 1 − d2 z 2
d3 z 2

17

=0
=0
=0
=0

1
1
1
1

2
2
2
2

3
3
3
3

4
4
•
•

and is therefore 2-pure with pd(L) ≤ 2 and a strict inclusion M ⊂ L defined by y = z 1 .
.
REMARK 4.3: In this example, we discover that, if L were also r-pure, we should therefore
have an embedding 0 −→ L −→ extr (extr (L)) = extr (N ) and thus an isomorphism extr (N ) = L
leading to an isomorphism Zr = homD (Fr , D) and to Fr+1 = 0, as can be checked on this example
with r = 2. It has been a challenge for the author during many months to find the following
counter-example showing that, sometimes L may not even be a torsion module.
EXAMPLE 4.4: According to the proof of the theorem, N = extr (M ) does not depend on the
resolution of M used while L does indeed depend on the resolution of N used. Coming back to
the system studied in Example 2.8 and Remark 2.12 with r = 2, we may use the shortest finite
free resolution of M already presented, namely 0 −→ D −→ D2 −→ D −→ M −→ 0. Therefore,
taking the adjoint of the only CC found, we may define N by the system:

v −→ d33 θ
=0
−u −→ d13 θ + d2 θ = 0
and obtain the corresponding involutive system:
 4
φ


 3
φ
φ2


 1
φ

≡
≡
≡
≡

d33 θ
d23 θ
d22 θ
d13 θ + d2 θ

=0
=0
=0
=0

1
1
1
1

2
2
2
•

3
•
•
•

1
1
1
1

2
2
2
2

We obtain the first order involutive system of CC:
 4
ψ


 3
ψ
ψ2


 1
ψ

≡
≡
≡
≡

d3 φ3 − d2 φ4
d3 φ2 − d2 φ3
d3 φ1 − d1 φ4 − φ3
d2 φ1 − d1 φ3 − φ2

=0
=0
=0
=0

3
3
3
•

with the only CC :
ω ≡ d3 ψ 1 − d2 ψ 2 + d1 ψ 4 + ψ 3 = 0 .
We may therefore introduce in reverse order the corresponding adjoint
volved in the Janet sequence we have just constructed:
 4
ψ −→ −d1 λ = ν 4


 3
ψ −→
λ = ν3
2
ψ −→
d2 λ = ν 2


 1
ψ −→ −d3 λ = ν 1
 4
4
4
2

 φ3 −→ µ3 ≡ d2 ν +4 d1 ν 1

φ −→ µ ≡ −(d3 ν − d1 ν ) + (d2 ν 3 − ν 2 )
φ2 −→ µ2 ≡ −d3 ν 3 − ν 1


 1
φ −→ µ1 ≡ −d3 ν 2 − d2 ν 1

operators of the ones in-

=0
=0
=0
=0

This last operator is defining L but is not involutive. We have the two torsion elements:
µ5 ≡ d2 ν 3 − ν 2 ,

µ6 ≡ d1 ν 3 + ν 4

which are generating ext2 (N ) and are easily seen to satisfy the involutive system:
d3 µ6 − µ5 = 0,

d3 µ5 = 0,

d2 µ6 − d1 µ5 = 0,

d2 µ5 = 0

because d2 µ5 ≡ d2 µ3 + d3 µ4 + d1 µ1 = 0. Finally, using the first equation, we may eliminate µ5
and identify µ6 with y because we have indeed d33 µ6 = 0, d13 µ6 − d2 µ6 = 0 in order to obtain the
strict inclusion M ⊂ L. Equivalently, we may also eliminate ν 1 and ν 2 respectively from µ2 and
µ3 in order to obtain:
µ4 −→ d13 (d1 ν 3 + ν 4 ) − d2 (d1 ν 3 + ν 4 ) = 0,
18

µ1 −→ d33 (d1 ν 3 + ν 4 ) = 0

but we may notice that L is not 2-pure and thus a torsion module because ν 3 (similarly ν 4 ) is
not by itself a torsion element of L. Such a situation is well known in control theory with the
SISO (single input u, single output y) system ẏ − u̇ = 0 because u (similarly y) is not by itself a
torsion element but z = y − u is a torsion element because ż = 0 (See the pages 9 and 10 of the
introduction in [17] for more details on such a comment).
PROBLEM : Is it possible to find an analogue of the previous theorem or of the case r = 0,
where L should be also r-pure with a free resolution having exactly r morphisms ?.
5) ABSOLUTE AND RELATIVE LOCALIZATIONS :
Surprisingly, the positive answer to such a problem has been given by Macaulay in ([11]) for
differential modules defined by systems with constant coefficients and only one unknown. Our purpose in this section is to generalize this resul to arbitrary differential modules defined by systems
of PD equations with coefficients in a differential field.
Now we hope that, after reading the previous section, the reader is convinced that the use of
extension modules is a quite important though striking tool for studying linear multidimensional
systems. Of course, as for any new language, it is necessary to apply it on many explicit examples
before being familiar with it. However, it is evident that it should be even more important to have
a direct approach allowing to exhibit the purity filtration and, in particular, to recognize whether
a differential module is pure or not. The purpose of this section is to combine the module approach
with the system approach, while taking into account the specific properties of the Spencer form in
a way rather similar to the use of the Kalman form of a control system when testing controllability, namely to check that an ordinary differential module is 0-pure. For this, we shall divide the
procedure into a few successive constructive steps that will be illustrated on explicit examples.
• STEP 1: Whenever a system Rq ⊂ Jq (E) is given, there is no way to obtain informations on
the corresponding module without bringing this system to an involutive or at least formally integrable system by means of prolongations and projections as in the Example 2.8 of Macaulay where
(2)
only the projection R2 ⊂ R2 of R4 to R2 is involutive. Of course, an homogeneous system with
constant coefficients is automatically formally integrable and one only needs to use a finite number
of prolongations in order to obtain an involutive symbol, though it is known that 2-acyclicity is
sufficient to obtain first order generating CC ([17]). However, it is essential to notice that it is
only with an involutive system that we are sure that the CC system is first order both with the
following ones in the Janet sequence.
EXAMPLE 5.1: With K = Q, m = 1, n = 3, q = 2, the homogeneous second order systems
y33 = 0, y23 − y11 = 0, y22 = 0 or y33 − y11 = 0, y23 = 0, y22 − y11 = 0 both have a 2-acyclic symbol
g3 of dimension 1 at order 3 (exercise) and a trivially involutive symbol g4 = 0 at order 4, such a
result leading to only one CC of order 2 with cd(M ) = 3 in both cases. We let the reader treat the
system y3 = 0, y12 = 0 similarly and conclude (Hint: (χ3 , χ1 χ2 ) = (χ3 , χ1 ) ∩ (χ3 , χ2 ) is unmixed).
It is however not evident that the homogeneous system y11 = 0, y12 = 0, y13 = 0, y23 = 0 of Example 3.33 is involutive.
Finally, according to section 2 and 3, this first step provides the characters α1q ≥ ... ≥ αnq ≥ 0
and the smallest non-zero character α = αqn−r 6= 0 providing cd(M ) = r, a result leading at once
to tr (M ) ⊂ M with a strict inclusion while tr−1 (M ) = ... = t0 (M ) = t(M ) = M . Of course, if
α = αnq 6= 0, then M cannot be a torsion module and t(M ) ⊂ M with a strict inclusion. The
following example proves nevertheless that it is much more delicate to study systems with variable
coefficients.
EXAMPLE 5.2: With K = Q(x2 ), n = 3, m = 1, q = 1, let us consider the differential module M
defined by the trivially involutive system y3 − x2 y1 = 0. We have cd(M ) = 1 but we can only say
that cd(Dz) ≥ 1, ∀z ∈ M . If we set z = y2 , proceeding as in Remark 2.12, we get the involutive
system:

19


 y3
y2

y1

=
=
=

x2 z3 − (x2 )2 z1
z
z3 − x2 z1

1 2
1 2
1 •

3
•
•

The differential submodule Dz ⊂ M is defined by the second order involutive system:


z33 − 2x2 z13 + (x2 )2 z11
z23 − x2 z12 − 2z1

=0
=0

1 2
1 2

3
•

and we get cd(Dz) = 1 exactly. However, even on such a very elementary example, it is not
evident that t0 (M ) = t(M ) = M is 1-pure. We also understand that the decoupling system for
any autonomous element in engineering sciences, like in magnetohydrodynamics, cannot be studied
without these new techniques if we want intrinsic results. Finally, if we denote by I the left ideal of
D = Dy generated by y3 −x2 y1 , we notice the relation ann(G) = (χ3 −x2 χ1 ) = gr(I) = rad(gr(I)).
However, we have ann(gr(Dz)) = ((χ3 − x2 χ1 )2 , χ2 (χ3 − x2 χ1 )) with radical equal to the prime
ideal (χ3 − x2 χ1 ) as before. Hence, in this example, the strict inclusion Dz ⊂ M does not imply
gr((Dz) ⊂ gr(M ) = G because otherwise we should get ann(G) ⊆ ann(gr(Dz) and this is the
reason for which only the radical must be considered as it does not depend on the filtration.
• STEP 2: Once we have obtained cd(M ) = r, in order to check that M is r-pure, it remains
to prove that tr (M ) = 0 as we already know that tr−1 (M ) = M . For this, the second step will
be to use the specific properties of the Spencer form Rq+1 ⊂ J1 (Rq ). More generally, it is possible to use any equivalent involutive first order system of the form R1 ⊂ J1 (E) with no zero
order equations, that is with an induced epimorphism R1 −→ E −→ 0 and such that the corresponding differential module is isomorphic and thus identified to the initial module as in Remark
2.13 . We have now the characters α11 ≥ ... ≥ αn1 ≥ 0 and the smallest non-zero character is
is still α = α1n−r 6= 0 providing of course the same codimension cd(M ) = r as in the first step.
Accordingly, the number r of non-zero characters and the number r of full classes is the same
as in the previous step. However, it must be noticed that the filtration may be different and the
following example explains once more why only the radical of the characteristic ideal must be used.
EXAMPLE 5.3: K = Q, n = 2, m = 1, q = 2.For the involutive system y22 = 0, y12 = 0, the characteristic ideal is a = ((χ2 )2 , χ1 χ2 ) =⇒ rad(a) = (χ2 ) =⇒ r = 1. Setting z 1 = y, z 2 = y1 , z 3 = y2 ,
we get the equivalent first order system d2 z 3 = 0, d2 z 2 = 0, d2 z 1 − z 3 = 0, d1 z 1 − z 2 = 0, d1 z 3 = 0
and the polynomial ideal generated by the 3 × 3 minors of the characteristic matrix is a =
((χ2 )3 , (χ2 )2 χ1 , χ2 (χ1 )2 ). Hence the characteristic ideal is rad(a) = (χ2 ) and r = 1 too.
EXAMPLE 5.4: For Example 2.8 we may set z 1 = y, z 2 = y1 , z 3 = y2 , z 4 = y3 and obtain the
first order involutive system :

 d3 z 4 = 0, d3 z 3 = 0, d3 z 2 − z 3 = 0, d3 z 1 − z 4 = 0
d2 z 4 = 0, d2 z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 1 − z 3 = 0

d1 z 4 − z 3 = 0, d1 z 1 − z 2 = 0

1
1
1

2 3
2 •
• •

with no zero-order equation. We have α31 = 4 − 4 = 0, α21 = 4 − 4 = 0, α11 = 4 − 2 = 2 =⇒ r = 2
too. We let the reader treat Example 4.2 similarly and obtain α41 = 0, α31 = 0, α21 = 2 =⇒ r = 2.
It is at this moment that we discover that such systems have particular properties not held by
other systems, apart from the fact that a canonical sequence may be constructed exactly like the
Spencer sequence or the first order part of the Janet sequence.
Shrinking the board of multiplicative variables, we obtain from the definition of involutiveness:
PROPOSITION 5.5: For an involutive first order system with no zero order equations and
solved with respect to the principal (pri) first order jets expressed by means of the parametric

20

(par) other first order jets, the system obtained by looking only at the PD equations of class 1+
... + class i only contains d1 , ..., di and is still involutive ∀i = 1, ..., n, after adopting the ordering
di+1 , ..., dn , d1 , ..., di .
EXAMPLE 5.6: Looking at Example 2.9, we notice that the systems:



y3 − x4 y1
y2 − y1
y2 − y1

=0
=0
=0

4
4
3

1
1

2 3
2 •

4 1

2

are both involutive. Also, looking at Example 5.4, we notice that the system:


d2 z 4 = 0, d2 z 3 = 0, d2 z 2 − d1 z 3 = 0, d2 z 1 − z 3 = 0
d1 z 4 − z 3 = 0, d1 z 1 − z 2 = 0

3
3

1
1

2
•

is still involutive and we let the reader treat Example 4.2 similarly.
We shall denote the corresponding differential module by Mn−i and we notice that M = M0
is defined by more equations than Mn−i . Accordingly, we have an epimorphism (specialization)
Mn−i −→ M −→ 0 and similarly epimorphisms Mn−i −→ Mn−i−1 −→ 0. Finally, as cd(M ) = r,
we notice that the classes n − r + 1, ..., n are full and we find therefore (χn )m , ..., (χn−r+1 )m among
the m × m minors with lower powers of χ1 , ..., χn−r for the other minors because the numbers of
equations of the lower classes are decreasing and thus strictly smaller than m. The characteristic
ideal is thus (χn , ..., χn−r+1 ) if we set χ1 , ..., χn−r to zero, in a coherent way. Finally, choosing
i = n − r, we get an epimorphism Mr −→ M −→ 0. The background will always indicate clearly
the meaning of the lower index and cannot be confused with the filtration index of M .
• STEP 3: We are now in position to study Mr with more details as a system in n − r variables
([11], §77, p 86). Its defining system has β1n−r = β = m − α equations of strict class n − r, a
smaller number of equations of class n − r − 1, ... , and eventually an even smaller number of
equations of class 1. Studying this system for itself, we may look for t(Mr ) exactly following the
known techniques working for any differential module, in particular double duality as described
in section 4. However, if one could find any (relative) torsion element z ∈ Mr , we could project
it to an element z ∈ M and we have N = Dz ⊆ M where we do not put a residue bar on the
new z for simplicity. Introducing the respective annihilator ideal a and b of M and N , we should
have a ⊆ rad(a) ⊆ rad(b) as it is the only result not depending on the filtration of the modules.
However, we know that z must be killed by at least one operator involving only d1 , ..., dn−r , in
addition to the operators involving separately (dn )m + ..., ..., (dn−r−1 )m + ... and we should obtain
tr−1 (M ) = M but tr (M ) 6= 0, that is M should not be pure. Hence M is r-pure if and only if Mr
is torsion-free as a differential module over K[d1 , ..., dn−r ]. In such a case, the system defining Mr
can be parametrized by α arbitrary unknowns among {y 1 , ..., y m } by using a so-called minimum
parametrization in the sense of ([24]). In actual practice, as shown on all the examples, one can use
a relative localization with respect to (d1 , ..., dn−r ) only by keeping (dn−r+1 , ..., dn ) untouched and
replacing (d1 , ..., dn−r ) by (χ1 , ..., χn−r ) considered as (constant) ”parameters” in the language of
Macaulay ([11], §43, p 45 and §77, p 86 with r instead of n − r and a different ordering). Such
a method provides therefore a quite useful and simple test for checking purity by linking it to
involutivity. An important intermediate result is provided by the next proposition.
PROPOSITION 5.7: The partial localization ”kills” the equations of class 1 up to class n− r − 1
(care) and finally only depends on the equations of strict class n − r.
Proof: Instead of using the column n − r in the multiplicative board, we provide the proof when
r = 0 by using the column n. Working out as usual the first order CC, we only look at the p < m
CC of class n for the equations Φ1 , ..., Φp of class 1 up to class n − 1 if we order the Φ starting
from the lower class involved. These p CC will be of a very specific form with a square p × p
operator matrix for (Φ1 , ..., Φp ) with diagonal operators of the form dn + ... where the dots denote
21

operators involving only d1 , ..., dn−1 in a quasi linear way with coefficients in K, the remaining
of the matrix only depending on d1 , ..., dn−1 for the (Φp+1 , ..., Φm ) of strict class n. Therefore, if
we have K = k = cst(K), the absolute localization is simply done by setting di −→ χi and the
determinant of the square matrix is equal to (χn )p + ... where the dots denote a polynomial of
degree < p in χp with coefficients involving only χ1 , ..., χn−1 . It follows that each Φ1 , ..., Φp can be
expressed as a linear combination over k(χ1 , ..., χn ) of the Φ of strict class n. The result is similar
for the variable coefficient case by using the graded machinery but needs much more work. In any
case, setting the Φ of strict class n equal to zero, we should obtain a zero graded module for the
Φ1 , ..., Φp which must be eequal to zero too. It must finally be noticed that the first order CC used
are in reduced Spencer form as the dn of the Φ of strict class n do not appear in the CC we have
used and these Φ do not appear in the other CC too.
Q.E.D
EXAMPLE 5.8: Coming back to Example 2.11, we notice that the only CC is an identity in
(u, v, w) and we may forget about u in order to obtain the new system for (v, w):
 3
 φ ≡ d4 v − x3 d1 v − v = 0
φ2 ≡ d4 w − x3 d1 w − w = 0
 1
φ ≡ d3 w − x4 d1 w − d2 v + d1 v = 0

1
1
1

2 3
2 3
2 3

4
4
•

defining a new module M with cd(M ) = 1 as the class 4 is now full. The same CC as before can
be written again in the form:
(d4 − x3 d1 − 1)φ1 = (d3 − x4 d1 )φ2 − (d2 − d1 )φ3
that we can localize in (d1 , d2 , d3 ) with:
φ1 = 0 =⇒ (d3 − x4 d1 )w = (d2 − d1 )v =⇒ v = (d3 − x4 d1 )y, w = (d2 − d1 )y
We let the reader check that we have indeed:
 2
φ ≡ (d4 − x3 d1 − 1)(d3 − x4 d1 )y
φ3 ≡
(d4 − x3 d1 − 1)(d2 − d1 )y

=
=

(d3 − x4 d1 )u = 0 =⇒ u = 0
(d2 − d1 )u = 0 =⇒ u = 0

We finally obtain for M a relative parametrization with the only constraint u ≡ (d4 −x3 d1 −1)y = 0
in a coherent way with Example 2.11.
• STEP 4: In this section we come back to the commutative situation with a field k of constants
and generalize the results of Macaulay in the following theorem which is recapitulating the results
so far obtained.
THEOREM 5.9: One has the commutative and exact diagram:
0 −→

t(Mr ) −→ Mr
↓
↓
0 −→ tr (M ) −→ M
↓
↓
0
0

−→ k(χ1 , ..., χn−r ) ⊗
↓
−→ k(χ1 , ..., χn−r ) ⊗
↓
0

Mr
M

Proof: For simplifying the notations in this diagram of modules over D, we have identified Mr as a
module over k[d1 , ..., dn−r ] with Mr as a module over k[dn−r+1 , ..., dn , d1 , ..., dn−r ] = k[d1 , ..., dn ] =
D while the localization of M just tells that the coefficients are now in the field k(χ1 , ..., χn−r ),
exactly following Macaulay. Moreover the central column is exact according to the definitionof Mr
and the right column is exact because localization preserves the exactness of a sequence.
For exampe, with k = Q, n = 2, m = 2, q = 2, r = 1, the differential module M defined by
the involutive system y22 = 0, y12 = 0 may also be defined by the first order involutive system
z 1 = y, z 2 = y1 , z 3 = y2 =⇒ d2 z 3 = 0, d2 z 2 = 0, d2 z 1 − z 3 = 0, d1 z 3 = 0, d1 z 1 − z 2 = 0. Then M1
is defined by the first order system d1 z 3 = 0, d1 z 1 − z 2 = 0 with torsion module t(M1 ) generated

22

by z 3 and the tensor product of M by k(χ1 ) is defined by d2 z 3 = 0, d2 z 2 = 0, d2 z 1 = 0, d1 z 3 =
0, z 3 = 0, χ1 z 1 − z 2 = 0 after division by χ1 in a coherent way with Example 1.1.
Q.E.D.
Finally, when Mr is torsion-free as a differential module over k[d1 , ..., dn−r ], then tr (M ) = 0
and we get the following generalization of a result provided by Macalualy ([11], §41, p 43):
COROLLARY 5.10: The differential module M is r-pure if and only if cd(M ) = r and there is
a monomorphism 0 −→ M −→ k(χ1 , ..., χn−r ) ⊗ M .
• STEP 5: The final idea is to embed Mr into a free module over K[d1 , ..., dn−r ] in order to
parametrize the corresponding system and substitute into the equations of class n−r+1, ..., n. However, if we look at Example 1.2, we should find after the substitution Φ1 ≡ z13 = 0, Φ2 ≡ z23 = 0
with one CC d2 Φ1 − d1 Φ2 = 0, that is on one side a module L which is not 1-pure and, on the
other side a module L having a finite free resolution with 2 operators. However, we forgot that
M , being pure, may be identified with its embedding into its localization. Hence, we get in fact
χ1 z3 = 0, χ2 z3 = 0 and thus only z3 = 0 is providing a convenient parametrizing module L.
Our purpose is to explain and illustrate this procedure for finding such an L in the general situation. Again, the main idea will be provided by this example. Indeed, we obtain the
only CC Ψ ≡ d3 Φ3 − d2 Φ1 + d1 Φ2 = 0. Substituting the parametrization, we get of course
Φ3 = 0 ⇐⇒ χ1 y 2 = χ2 y 1 , that is, among the two unknowns y 1 , y 2 we are left with only one,
say y 1 and, similarly, among the two equations Φ1 , Φ2 we are left with only one, say Φ1 , because
χ1 Φ2 = χ2 Φ1 from the CC which is of course compatible with the localization and we choose
z3 = 0 as χ1 z3 = 0 =⇒ z3 = 0.
The general situation may be treated similarly. Indeed, according to the previous step, we
are only concerned with the equations of class n − r + 1, ... , class n while the localization has
only to do with the β equations of strict class n − r (care) allowing to express β unknowns as
linear combinations of the α remaining unknowns with coefficients in k(χ1 , ..., χn−r ). To each such
equation are associated exactly r dots and each dot of index n − r + i provides a reduction of the
respective equations of class n − r + i for i = 1, ..., r. It follows that we are left with α equations of
each such class. When we ”delocalize”, replacing χi by di , we have to take into account the need to
take out the denominators and may find a few ”simplifications” as in the example just considered..
Finally, the maximum number r − 1 (care again) of dots found for one equation is obtained for the
equations of strict class n − r + 1 = n − (r − 1) and we have thus exhibited a system defining a
module L which is r-pure and admits a free resolution with exactly (r − 1) + 1 = r operators. In
any case, the reader must not forget that the localization of a module is useful only if we already
know that this module is torsion-free by means of the double-duality formula t(M ) = ext1 (N )
given in the introduction.
EXAMPLE 5.11: Let M be defined by the involutive system:

d3 y 4 + d1 y 2 − d1 y 1




d3 y 3 − d2 y 4 + d1 y 2 − d1 y 1



d3 y 2 + d1 y 2
d3 y 1 − d1 y 4 + d1 y 2




d2 y 2 − d1 y 4 + d1 y 1



d2 y 1 − d1 y 3 + d1 y 1

=0
=0
=0
=0
=0
=0

1
1
1
1
1
1

2
2
2
2
2
2

3
3
3
3
•
•

with characters α31 = 0, α21 = α = 2, α11 = 4. It follows that cd(M ) = 1 as only the class 4 is full
and we obtain the following relative localization showing that M is 1-pure:
y 1 = χ1 y, y 2 = χ1 z, y 3 = (χ1 + χ2 )y, y 4 = χ1 y + χ2 z
Substituting in the four equations of class 3, we only obtain the two equations:

23



d3 z + χ1 z
d3 y + (χ1 − χ2 )z − χ1 y

=0
=0

after a division by χ1 , χ2 and χ1 + χ2 . The parametrizing module L is thus defined by the two
equations:

d3 z + d1 z
=0
d3 y + (d1 − d2 )z − d1 y = 0
which are differentially independent and we have the relative parametrization:
y 1 = d1 y,

y 2 = d1 z,

y 3 = (d1 + d2 )y,

y 4 = d1 y + d2 z

Finally, M ⊂ L =⇒ ass(M ) ⊂ ass(L) =⇒ ass(M ) = {(d3 + d1 ), (d3 − d1 )} =⇒ annD (M ) =
(d3 + d1 ) ∩ (d3 − d1 ), a striking result showing that M can be embedded into the direct sum of two
primary differential modules according to Remark 3.32 (See [18] for more details).
EXAMPLE 5.12: With k = Q, m = 1, n = 3, let us consider the polynomial map χ1 = u5 , χ2 =
u3 , χ3 = u4 as in ([11], p 53). We have the exact sequence 0 −→ p −→ k[χ] −→ k[u] ⊂ k(u)
showing that p = ((χ2 )2 (χ3 ) − (χ1 )2 , (χ2 )3 − χ1 χ3 , (χ3 )2 − χ1 χ2 ) is a prime ideal ([18], p 126). The
corresponding prime differential module M is defined by the involutive system:

y333 − y123




y

233 − y122


y223 − y11
y222 − y13




y

133 − y112


y33 − y12

=0
=0
=0
=0
=0
=0

1
1
1
1
1
•

2
2
2
2
•
•

3
•
•
•
•
•

and is 2-pure. The localized system is finite type over k(χ1 )[d2 , d3 ] with par = {y, y2 , y3 , y22 , y23 }.
One can prove that p2 is not a primary ideal even though rad(p2 ) = p.
EXAMPLE 5.13: Similarly but now with k = Q, m = 1, n = 4, let us consider the polynomial map χ1 = uv, χ2 = u, χ3 = uv 3 , χ4 = uv 4 as in ([11], p 47). We have the exact sequence
0 −→ p −→ k[χ] −→ k[u, v] ⊂ k(u, v) showing that p = (χ2 χ4 − χ1 χ3 , (χ1 )3 − (χ2 )2 χ3 , (χ3 )3 −
χ1 (χ4 )2 , (χ1 )2 χ4 − χ2 (χ3 )2 ) is a prime ideal. It is not evident at all that the corresponding prime
differential module M can be defined by the homogeneous involutive system (exercise):

y444 − y224 − y134 − y123




y

344 − y111



y

334 − y114 − y112


y333 − y124 − y122 − y113
y244 + y224 − y123




y

234 − y133 + y111



y

144 + y124 − y113


y44 + y24 − y13

=0
=0
=0
=0
=0
=0
=0
=0

1
1
1
1
1
1
1
•

2
2
2
2
2
2
•
•

3
3
3
3
•
•
•
•

4
•
•
•
•
•
•
•

and is thus also 2-pure. The localized system is finite type over k(χ1 , χ2 )[d3 , d4 ] with par =
{y, y3 , y4 , y33 } and we have for example χ2 y34 − χ1 y33 + (χ1 )3 y = 0 in a coherent way with the
comments of Macaulay in ([11], §78, p 88, formula (A) and §88,89, p 98).
6) CONCLUSION :
In 1916, F.S. Macaulay discovered a new localization technique for studying unmixed polynomial ideals. We have been able to generalize this procedure for studying pure differential modules,
obtaining in particular a kind of relative parametrization generalizing the absolute parametrization already known for torsion-free modules and equivalent to controllability in classical control
theory. In the language of multidimensional systems theory, which is more intuitive, instead of
24

using arbitrary potential-like functions for the parametrization, the idea is now to use potentiallike functions which must satisfy a kind of minimum differential constraint limiting, in some sense,
the number of independent variables appearing in these functions, in a way similar to the situation met in the Cartan-Khäler theorem of analysis. For such a purpose, we have exhibited new
links between purity and involutivity, providing also a new insight into the primary decomposition
of modules and ideals by means of tools from the formal theory of linear multidimensional systems.
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