The language of differential modules has been recently introduced in control theory as a way to understand in an intrinsic way the structural properties of systems of ordinary differential (OD) or partial differential (PD) equations (controllability, observability, identifiability, ...) [1] - [10] . A similar comment can be done for optimal control that is for variational calculus with differential constraints, and the author thanks Prof. Lars Andersson (Einstein Institute, Potsdam) for having suggested him to study the Lanczos potential within this new framework.

We start providing a few explicit examples in order to convince the reader that the corresponding computations are often becoming so tricky that nobody could achieve them or even imagine any underlying general algorithm, for example in the study of the mathematical foundations of control theory, elasticity theory or general relativity.

EXAMPLE 1.1: OD Control Theory

With one independent variable x, for example the time t in control theory or the curvilinear abcissa s in the study of a beam, and three unknowns. Setting formally for and so on, let us consider the system made by the two first order OD equations depending on a variable coefficient:

In control theory, if is a constant parameter, one could bring the system to any first order Kalman form and check that the corresponding control system is controllable if and only if, that is and (exercise), independently of the choice of 1 input and 2 outputs among the 3 control variables [11] . In addition to that, using the second OD equation in the form and substituting in the first, we get the only second order OD equation:

a result leading to a kind of “vicious circle” because the only way to test controllability is ... to bring this second order equation back to a first order system and there are a lot of possibilities. Again, in any case, the only critical values are and. Of course, one could dream about a direct approach providing the same result in an intrinsic way. Introducing the operator as the (formal) derivative with respect to x, we may rewrite the last equation in the form:

Replacing the operators and by the polynomials and, the two poly- nomials have a common root or and we find back the desired critical values

but such a result is not intrinsic at all. However, we notice that, for example.

Introducing, we get while that is, setting , we get now. Calling “torsion element” any scalar quantity made from the unknowns and their derivatives but satisfying at least one OD equation, we discover that such quantities do exist ... if and only if or (exercise). Of course, the existence of any torsion element breaks at once the controllability of the system but the converse is not evident at all, a result leading nevertheless to the feeling that a control system is controllable if and only if no torsion element can be found and such an idea can be extended “mutatis mutandis” to any system of PD equations [6] . However, this result could be useful if and only if there is a test for checking such a property of the system.

Now, using a variable parameter, not a word of the preceding approach is left but the concept of a torsion element still exists. We shall prove, at the end of the paper, that the condition becomes and that the computations needed are quite far from the previous ones. We ask the reader familiar with classical control theory to make his mind a few minutes (or hours!) to agree with us by trying to recover himself such a differential condition.

EXAMPLE 1.2: OD Optimal Control Theory

OD optimal control is the study of OD variational calculus with OD constraints described by OD control systems. However, while studying optimal control, the author of this paper has been surprised to discover that, in all cases, the OD constraints were defined by means of controllable control systems. It is only at the end of this paper that the importance of such an assumption will be explained. For the moment, we shall provide an example allowing to exhibit all the difficulties involved. For this, let be a solution of the following single input/single output (SISO) OD control system where a is a constant parameter:

Proceeding as before, the two polynomials replacing the respective operators are, and can only have the common root. Accordingly, the system is controllable if and only if for any choice of input and output. Now, let us introduce the so-called “cost function” and let us look at the extremum of the

integral under the previous OD constraint. It is well known that the proper way to study

such a problem is to introduce a Lagrange multiplier and to vary the new integral:

The corresponding Euler-Lagrange (EL) equations are:

to which we must add the OD constraint when varying. Summing the two EL equations, we get and two possibilities:

1) compatible with the constraint.

2).

Substituting, we get:

This system may not be formally integrable. Indeed, by substraction, we get and must consider the following two possibilities:

Summarising the results so far obtained, we discover that the Lagrange multiplier is known if and only if the system is controllable. Also, if, we may exhibit the parametrization and the cost function becomes. Equivalently, when the system is controllable it can be parametrized and the variational problem with constraint becomes a variational problem without any constraint which, some- times, does not provide EL equations. We finally understand that extending such a situation to PD variational calculus with PD constraints needs new techniques.

EXAMPLE 1.3: Elasticity Theory

In classical elasticity, the stress tensor density existing inside an elastic body is a symmetric 2-tensor density introduced by A. Cauchy in 1822. The corresponding Cauchy stress equations can be written as where the right member describes the local density of forces applied to the body, for example gravitation. With zero second member, we study the possibility to “parametrize” the system of PD equations, namely to express its general solution by means of a certain number of arbitrary functions or potentials, called stress functions. Of course, the problem is to know about the number of such functions and the order of the parametrizing operator. In what follows, the space has n local coordinates. For one may introduce the Euclidean metric while, for, one may consider the Minkowski metric. A few definitions used thereafter will be provided later on.

・ : There is no possible parametrization of.

・ : The stress equations become. Their second order parametrization has been provided by George Biddell Airy (1801-1892) in 1863 [12] . It can be simply recovered in the following manner:

We get the second order system:

which is involutive with one equation of class 2, 2 equations of class 1 and it is easy to check that the 2 corresponding first order CC are just the stress equations. As we have a system with constant coefficients, we may use localization [6] [13] in order to transform the 2 PD equations into the 2 linear equations and get

Setting, we finally get and obtain the previous parametrization by delocalizing, that is replacing now by.

・ : Things become quite more delicate when we try to parametrize the 3 PD equations:

Of course, localization could be used similarly by dealing with the 3 linear equations:

having rank 3 for 6 unknowns but, even if we succeed bringing all the fractions to the same denominator as before after easy but painful calculus, there is an additional difficulty which is well hidden. Indeed, coming back to the previous Example when, say, we should get

. Hence, setting, we only get a

parametrization of the first order OD equation leading to. Accordingly, localization does indeed provide a parametrization, ... if we already know there exists a possibility to parametrize the given system or if we are able to check that we have obtained such a parametrization by using involution, a way to supersede the use of Janet or Gröbner bases as was proved for the case [14] . Also, if we proceed along such a way, we should surely loose any geometric argument that could exist.

A direct computational approach has been provided by Eugenio Beltrami (1835-1900) in 1892 [15] , James Clerk Maxwell (1831-1879) in 1870 [16] and Giacinto Morera (1856-1909) in 1892 [17] by introducing the 6 stress functions through the parametrization obtained by considering:

and the additional 4 relations obtained by using a cyclic permutation of. The system:

is involutive with 3 equations of class 3, 3 equations of class 2 and no equation of class 1. The 3 CC are describing the stress equations which admit therefore a parametrization ... justifying the localization approach “a posteriori” but without any geometric framework [18] .

Surprisingly, the Maxwell parametrization is obtained by keeping while setting in order to obtain the system:

However, this system may not be involutive and no CC can be found “a priori” because the coordinate system is surely not d-regular. Indeed, effecting the linear change of coordinates, we obtain the involutive system:

and it is easy to check that the 3 CC obtained just amount to the desired 3 stress equations when coming back to the original system of coordinates. Again, if there is a geometrical background, this change of local coordinates is hidding it totally. Moreover, we notice that the stress functions kept in the procedure are just the ones on which is acting. The reason for such an apparently technical choice is related to very general deep arguments in the theory of differential modules that will only be explained at the end of the paper. The Morera parametrization is obtained similarly by keeping now while setting.

・ : As already explained, localization cannot be applied directly as we don't know if a parametrization may exist and in any case no analogy with the previous situations could be used. Moreover, no known differential geometric background could be used at first sight in order to provide a hint towards the solution. Now, if is the Minkowski metric and is the gravitational potential, then and a perturbation of may satisfy in vacuum the 10 second order Einstein equations for the 10 W:

by introducing the corresponding second order Einstein operator when [19] . Though it is well known that the corresponding second order Einstein operator is parametrizing the stress equations, the challenge of parametrizing Einstein equations has been proposed in 1970 by J. Wheeler for 1000 $ and solved negatively in 1995 by the author who only received 1 $. We shall see that, exactly as before and though it is quite striking, the key ingredient will be to use the linearized Riemann tensor considered as a second order operator [6] [20] . As an even more striking fact, we shall discover that the condition has only to do with Spencer cohomology for the symbol of the conformal Killing equations.

EXAMPLE 1.4: PD Control Theory

The aim of this last example is to prove that the possibility to exhibit two different parametrizations of the stress equations which has been presented in the previous example has surely nothing to do with the proper mathematical background of elasticity theory!

For this, let us consider the (trivially involutive) inhomogeneous PD equations with two independent variables, two unknown functions and a second member:

Multiplying on the left by a test function and integrating by parts, the corresponding inhomogeneous adjoint system of PD equations is:

Using crossed derivatives, we get and substituting, we get the two CC:

The corresponding generating CC for the second member is:

Therefore is differentially dependent on but is also differentially dependent on.

Multiplying the first equation by the test function, the second equation by the test function, adding and integrating by parts, we get the canonical parametrization:

of the initial system with zero second member. The system (up to sign) is involutive and the kernel of this parametrization has differential rank equal to 1.

Keeping while setting, we get the first minimal parametrization:

The system is again involutive (up to sign) and the parametrization is minimal because the kernel of this parametrization has differential rank equal to 0. With a similar comment, setting now while keeping, we get the second minimal parametrization:

EXAMPLE 1.5: PD Optimal Control Theory

Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to for simplicity. If is a point in the plane and is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the “small” deformation tensor with (independent) components . If we study a part of a deformed body, for example a thin elastic plane sheet, by means of a variational principle, we may introduce the local density of free energy and vary the total free energy with by introducing for in order

to obtain. Accordingly, the “decision” to define the stress tensor by

a symmetric matrix with is purely artificial within such a variational principle. Indeed, the usual Cauchy device (1828) assumes that each element of a boundary surface is acted on by a surface density of force with a linear dependence on the outward normal unit vector and does not make any assumption on the stress tensor. It is only by an equilibrium of forces and couples, namely the well known phenomenological static torsor equilibrium, that one can “prove” the symmetry of. However, even if we assume this symmetry, we now need the different summation.

An integration by parts and a change of sign produce the integral leading to the stress equations

already considered. This classical approach to elasticity theory, based on invariant theory with respect to the group of rigid motions, cannot therefore describe equilibrium of torsors by means of a variational principle where the proper torsor concept is totally lacking. It is however widely used through the technique of “finite elements” where it can also be applied to electromagnetism (EM) with similar quadratic (piezoelectricity) or cubic (photoelasticity) Lagrangian integrals. In this situation, the 4-potential A of EM is used in place of while the EM field is used in place of and the Maxwell equations are used in place of the Riemann CC for.

However, there exists another equivalent procedure dealing with a variational calculus with constraint. Indeed, as we shall see later on, the deformation tensor is not any symmetric tensor as it must satisfy compatibility conditions (CC), that is only when. In this case, introducing

the Lagrange multiplier, we have to vary the new integral for an

arbitrary. Setting, a double integration by parts now provides the parametrization of the stress equations by means of the Airy function and the formal adjoint of the Riemann CC, on the condition to observe that we have in fact as another way to under- stand the deep meaning of the factor “2” in the summation. The same variational calculus with constraint may thus also be used in order to “shortcut” the introduction of the EM potential.

Finally, using the constitutive relations of the material establishing an isomorphism, one can also introduce a local free energy in a variational problem having now for constraint the stress equations, with the same comment as above (see [6] , p. 915, for more details). The well known Minkowski constitutive relations can be similarly used for EM.

In arbitrary dimension, the above compatibility conditions are nothing else but the linearized Riemann tensor in Riemannian geometry, a crucial mathematical tool in the theory of general relativity and a good reason for studying the work of Cornelius Lanczos (1893-1974) as it can be found in [21] - [23] or in a few modern references [24] - [31] . The starting point of Lanczos has been to take EM as a model in order to introduce a

Lagrangian that should be quadratic in the Riemann tensor while con-

sidering it independently of its expression through the second order derivatives of a metric with inverse or the first order derivatives of the corresponding Christoffel symbols. According to the previous paragraph, the corresponding variational calculus must involve PD constraints made by the Bianchi identities and the new Lagrangian to vary must therefore contain as many Lagrange multipliers as the number of Bianchi identities (care!) that can be written under the form:

Meanwhile, Lanczos and followers have been looking for a kind of parametrization of the Bianchi identities, exactly like the Lagrange multiplier has been used as an Airy potential for the stress equations. However, we shall prove that the definition of a Riemann candidate and the answer to this question cannot be done without the knowledge of the Spencer cohomology. Moreover, we have pointed out the existence of well known couplings between elasticity and electromagnetism, namely piezoelectricity and photoelasticity, which are showing that, in the respective Lagrangians, the EM field is on equal footing with the deformation tensor and not with the Riemann tensor. This fact is showing the shift by one step that must be used in the physical inter-pretation of the differential sequences involved and cannot be avoided. Meanwhile, the ordinary derivatives can be used in place of the covariant derivatives when dealing with the linearized framework as the Christoffel symbols vanish when Euclidean or Minkowskian metrics are used.

The next tentative of Lanczos has been to extend his approach to the Weyl tensor:

The main problem is now that the Spencer cohomology of the symbols of the conformal Killing equations, in particular the 2-acyclicity, will be absolutely needed in order to study the Vessiot structure equations providing the Weyl tensor and its relation with the Riemann tensor. It will follow that the CC for the Weyl tensor are not first order contrary to the CC for the Riemann tensor made by the Bianchi identities, another reason for justifying the above shift by one step.

Finally, comparing the various parametrizations already obtained in the previous examples, it seems that the procedures are similar, even when dealing with systems having variable coefficients. The purpose of the paper is to prove that, in order to obtain a general algorithm, we shall need a lot of new tools involving at the same time commutative algebra, homological algebra, differential algebra and differential geometry that will be recalled in the next sections. Finally, like in any good crime story, it is only at the real end of the paper that we shall be able to revisit and compare all these examples in a unique framework.

2) MODULE THEORY

Before entering the heart of the next section dealing with extension modules, we need a few technical definitions and results from commutative algebra [13] [32] .

DEFINITION 2.1: A ring A is said to be unitary if it has a (unique) element such that and commutative if. A non-zero element is called a zero-divisor if one can find a non-zero such that and a ring is called an integral domain if it has no zero-divisor. From now on, all rings considered will be unitary integral domains as we shall deal mainly with rings of partial diffe- rential operators.

DEFINITION 2.2: A ring K is called a field if every non-zero element is a unit, that is one can find an element such that.

DEFINITION 2.3: A module M over a ring A or simply an A-module is a set of elements which is an abelian group for an addition with an action satisfying:

・

・

・

・

The set of modules over a ring A will be denoted by. A module over a field is called a vector space.

DEFINITION 2.4: A map between two A-modules is called a homomorphism over A if and. We successively define:

・

・

・

・

with an isomorphism induced by f.

DEFINITION 2.5: We say that a chain of modules and homomorphisms is a sequence if the composition of two successive such homomorphisms is zero. A sequence is said to be exact if the kernel of each map is equal to the image of the map preceding it. An injective homomorphism is called a monomorphism, a surjective homo- morphism is called an epimorphism and a bijective homomorphism is called an isomorphism. A short exact sequence is an exact sequence made by a monomorphism followed by an epimorphism.

PROPOSITION 2.6: If one has a short exact sequence:

then the following conditions are equivalent:

・ There exists an epimorphism such that (left inverse of f).

・ There exists a monomorphism such that (right inverse of g).

DEFINITION 2.7: In the above situation, we say that the short exact sequence splits. The relation provides an isomorphism with inverse. The short exact sequence cannot split over.

For the sake of clarity, as a few results will also be valid for modules over non-commutative rings, we shall denote by a bimodule M which is a left module for A with operation and a right module for B with operation. In the commutative case, lower indices are not needed. If and are two left A-modules, the set of A-linear maps will be denoted by or simply when there will be no confusion and there is a canonical isomorphism with inverse. When A is commutative, is again an A-module for the law. In the non-commutative case, things are much more complicate and we have:

LEMMA 2.8: Given and, then becomes a right module over B for the law. A similar result can be obtained with and, where now becomes a left module over B for the law.

THEOREM 2.9: If are A-modules, the sequence:

is exact if and only if the sequence:

is exact for any A-module N.

COROLLARY 2.10: The short exact sequence:

splits if and only if the short exact sequence:

is exact for any module N.

DEFINITION 2.11: If M is a module over a ring A, a system of generators of M over A is a family of elements of M such that any element of M can be written with only a finite number of nonzero. An A-module is called noetherian if every submodule of M (and thus M itself) is finitely generated.

One has the following standard technical result:

PROPOSITION 2.12: In a short exact sequence of modules, the central module is noetherian if and only if the two other modules are noetherian. As a byproduct, if A is a noetherian ring and M is a finitely generated module over A, then M is noetherian.

Accordingly, if M is generated by, there is an epimorphism . The kernel of this epimorphism is thus also finitely generated, say by and we therefore obtain the exact sequence that can be extended inductively to the left. Such a property will always be assumed in the sequel.

DEFINITION 2.13: In this case, we say that M is finitely presented.

We now turn to the definition and brief study of tensor products of modules over rings that will not be necessarily commutative unless stated explicitly.

Let be a right A-module and be a left A-module. We may introduce the free -module made by finite formal linear combinations of elements of with coefficients in.

DEFINITION 2.14: The tensor product of M and N over A is the -module obtained by quotienting the above -module by the submodule generated by the elements of the form:

and the image of will be denoted by.

It follows from the definition that we have the relations:

and there is a canonical isomorphism. When A is commutative, we may use left modules only and becomes a left A-module.

EXAMPLE 2.15: If and, we have because .

We present the technique of localization in order to introduce rings and modules of fractions.

Definition 2.16: A subset S of a ring A is said to be multiplicatively closed if and. By a left ring of fractions or left localization of a noncommutative ring A with respect to a multiplicatively closed subset S of A, we mean a ring denoted by with a monomorphism or simply a such that:

1) s is invertible in, with inverse or simply.

2) Each element of or fraction has the form for some.

We have to distinguish carefully from and we recover the standard notation of the com- mutative case when two fractions and can be reduced to the same denominator. The follow- ing proposition is essential for constructing localizations.

Proposition 2.17: If there exists a left localization of A with respect to S, then we must have . A set S satisfying this condition is called a left Ore set.

Proof: As must be a ring, the element in must be of the form for some . Accordingly, with.

Q.E.D.

Lemma 2.18: If S is a left Ore set in a ring A, then and two fractions can be brought to the same denominator.

Proof: From the left Ore condition, we can find and such that. More generally, we can find such that and we successively get:

so that the two fractions and can be brought to the same denominator.

Q.E.D.

Let us now define an equivalence relation on by saying that if one can find such that and. Such a relation is clearly reflexive and symmetric, thus we only need to prove that it is transitive. So let and. Then we can find such that and. Also we can find such that and. Now, from the Ore condition, one can find such that and thus , that is to say. As A is an integral domain, we have as wished. We finally define to be the quotient of by the above equivalence relation with. The sum will be defined to be and the product will be defined to be whenever.

A similar approach can be used in order to define and construct modules of fractions whenever S satisfies the two conditions of the last proposition. For this we need a preliminary lemma:

LEMMA 2.19: If S is a left Ore set in a ring A and M is a left module over A, the set:

is a submodule of M called the S-torsion submodule of M.

Proof: If, we may find such that. Now, we can find such that and we successively get. Also, , using the Ore condition for S, we can find such that and we get.

Q.E.D.

DEFINITION 2.20: By a left module of fractions or left localization of M with respect to S, we mean a left module over both with a homomorphism such that:

1) Each element of has the form for.

2).

In order to construct, we shall define an equivalence relation on by saying that if there exists such that and. The main property of localization is ex- pressed by the following theorem:

Theorem 2.21: If one has an exact sequence:

then one also has the exact sequence:

where.

As a link between tensor product and localization, we notice that the multiplication map given by induces an isomorphism of modules over when is considered as a right module over A with and M as a left module over A. In particular, when A is a commutative integral domain and, the field is called the field of fractions of A and we have the short exact sequence:

If now M is a left A-module, we may tensor this sequence by M on the right with but we do not get in general an exact sequence. The defect of exactness on the left is nothing else but the torsion submodule and we have the long exact sequence:

as we may describe the central map as follows:

As we saw in the Introduction, such a result allows to understand why controllability has to do with localization which is introduced implicitly through the transfer matrix in control theory. In particular, a module M is said to be a torsion module if and a torsion-free module if.

DEFINITION 2.22: A module in is called a free module if it has a basis, that is a system of generators linearly independent over A. When a module F is free, the number of generators in a basis, and thus in any basis, is called the rank of F over A and is denoted by. In particular, if F is free of finite rank r, then. More generally, a module P is said to be projective if there exists another (projective) module Q such that and any short exact sequence splits if it ends with a projective module (see [6] , p. 638-644) for a formal test).

If M is any module over a ring A and F is a maximum free submodule of M, then is a torsion module. Indeed, if, then one can find such that because, otherwise, should be free submodules of M with a strict inclusion. In that case, the rank of M is by definition the rank of F over A. When A is commutative, one has:

LEMMA 2.23:.

Proof: Taking the tensor product by K over A of the short exact sequence, we get an isomorphism because (exercise) and the lemma follows from the definition of the rank.

Q.E.D.

PROPOSITION 2.24: (additivity property of the rank) If is a short exact sequence of modules over a ring A, then we have.

Proof : Let us consider the following diagram with exact left/right columns and central row:

where is a maximum free submodule of and is a torsion module. Pulling back by g the image under of a basis of, we may obtain by linearity a map and we define where and are the canonical projections on each factor of the direct sum. We have and. Hence, the diagram is commutative and thus exact with trivially. Finally, if and are torsion modules, it is easy to check that T is a torsion module too and is thus a maximum free submodule of M.

Q.E.D.

DEFINITION 2.25: If is any morphism, the rank of f will be defined to be.

We provide a few additional properties of the rank that will be used in the sequel. For this we shall set and, for any morphism we shall denote by the corresponding morphism which is such that.

PROPOSITION 2.26: When A is a commutative integral domain and M is a finitely presented module over A, then.

Proof: Applying to the short exact sequence in the proof of the preceding lemma while taking into account, we get a monomorphism and obtain therefore. However, as with because M is finitely generated, we get too because. It

follows that and thus.

Now, if is a finite presentation of M, applying to this presentation, we get the ker/coker exact sequence:

Applying to this sequence while taking into account the isomorphisms, we get the ker/coker exact sequence:

Counting the ranks, we obtain:

and thus:

As both two numbers in this sum are non-negative, they must be zero and we finally get .

Q.E.D.

COROLLARY 2.27: Under the condition of the proposition, we have.

Proof: Introducing the exact sequence:

we have:. Applying and taking into account Theorem 2.9, we have the exact sequence:

and thus:. Using the preceding proposition, we get and

, that is to say.

Q.E.D.

3) HOMOLOGICAL ALGEBRA

We need a few definitions and results from homological algebra [13] [33] [34] and start recalling the well known Cramer’s rule for linear systems through the exactness of the ker/coker sequence for modules when is a linear map (homomorphism):

In the case of vector spaces over a field K, we successively have, , of compatibility conditions, and obtain by sub- straction:

In the case of modules, we may replace the dimension by the rank and obtain the same relations because of the additive property of the rank. We may also define cohomology theory as follows:

DEFINITION 3.1: If one has a sequence, that is if, then one may introduce the submodules and define the cohomology at M to be the quotient.

We now introduce the extension modules in an elementary manner, using the standard notation. Using a free resolution of an A-module M, that is to say a long exact sequence:

where are free modules, namely modules isomorphic to powers of A and. We may take out M and obtain the deleted sequence:

which is of course no longer exact. We may apply the functor and obtain the sequence:

in order to state:

DEFINITION 3.2: We set:

The extension modules have the following three main properties [6] [13] [33] [34] :

PROPOSITION 3.3: The extension modules do not depend on the resolution of M chosen.

PROPOSITION 3.4: If is a short exact sequence of A-modules, then we have the following connecting long exact sequence:

of extension modules. Moreover whenever P is a projective module.

PROPOSITION 3.5: is a torsion module,.

Proof: Having in mind that and, we obtain and

. However, we started from a resolution, that is an exact se-

quence in which. It follows that and thus , that is to say is a torsion module for,.

Q.E.D.

The next theorem and its corollary constitute the main results that will be used for applications through a classification of modules [1] [2] [6] - [8] [34] [35] :

THEOREM 3.6: The following long exact sequence:

is isomorphic to the ker/coker long exact sequence for the central morphism which is defined by.

Proof: Introducing, we may obtain two short exact sequences, a left one starting with K and a right one finishing with K as follows:

Using the two corresponding long exact connecting sequences, we get from the one starting with which is also providing the left exact column of the next diagram and the exact central row of this diagram from the one starting with. The Theorem is finally obtained by a chase proving that the full diagram is commutative and exact:

Q.E.D.

COROLLARY 3.7:.

Proof: As is a torsion module, we have therefore. Now, if, we may find such that and because A is an integral domain, that is and thus.

Q.E.D.

DEFINITION 3.8: A module M will be called torsion-free if and reflexive if.

Despite all these results, a major difficulty still remains. Indeed, we have as a left module over A but, using the bimodule structure of and Lemma 2.13, it follows that is a right module over A and thus is also a right module over A. However, as we shall see, all the differential modules used through applications will be left modules over the ring of differential operators and it will therefore not be possible to use dual sequences as we did without being able to “pass from left to right and vice-versa”. For this purpose we now need many delicate results from differential geometry, in particular a way to deal with the formal adjoint of an operator as we did many times in the Introduction.

4) SYSTEM THEORY

If E is a vector bundle over the base manifold X with projection and local coordinates projecting onto for and, identifying a map with its graph, a (local) section is such that on U and we write or simply. For any

change of local coordinates on E, the change of section is

such that. The new vector bundle obtained by changing the transi- tion matrix A to its inverse is called the dual vector bundle of E. Differentiating with respect to and using new coordinates in place of, we obtain. Introducing a multi-index with length and prolonging the procedure up to order q, we may construct in this way, by patching coordinates, a vector bundle over X, called the jet bundle of

order q with local coordinates with and. We have therefore epimor-

phisms. For a later use, we shall set and define the operator on sections by the local formula

. Moreover, a jet coordinate is said to be of class i if

. We finally introduce the Spencer operator with

.

DEFINITION 4.1: A system of PD equations of order q on E is a vector subbundle locally defined by a constant rank system of linear equations for the jets of order q of the form. Its first

prolongation will be defined by the equations

which may not provide a system of constant rank as can easily be seen for where the rank drops at.

The next definition of formal integrability (FI) will be crucial for our purpose.

DEFINITION 4.2: A system is said to be formally integrable if the are vector bundles (regularity condition) and no new equation of order can be obtained by prolonging the given PD equations more than r times, or, equivalently, we have induced epimorphisms allowing to compute “step by step” formal power series solutions.

A formal test has been first sketched by C. Riquier in 1910 [36] , then improved by M. Janet in 1920 [20] [37] and by E. Cartan in 1945 [38] , finally rediscovered in 1965, totally independently, by B. Buchberger who introduced Gröbner bases, using the name of his thesis advisor. However all these tentatives have been largely superseded and achieved in an intrinsic way, again totally independently of the previous approaches, by D.C. Spencer in 1965 [20] [39] [40] .

DEFINITION 4.3: The family of vector spaces over X defined by the purely linear equations for is called the symbol at order and only depends on.

The following procedure, where one may have to change linearly the independent variables if necessary, is the key towards the next definition which is intrinsic even though it must be checked in a particular coordinate system called d-regular (see [6] [20] and [39] for more details):

・ Equations of class n: Solve the maximum number of equations with respect to the jets of order q and class n. Then call multiplicative variables.

・ Equations of class i: Solve the maximum number of remaining equations with respect to the jets of order q and class i. Then call multiplicative variables and non-multiplicative variables.

・ Remaining equations of order: Call non-multiplicative variables.

DEFINITION 4.4: The above multiplicative and non-multiplicative variables can be visualized respectively by integers and dots in the corresponding Janet board. 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 variables. The following numbers are called characters:

For an involutive system, can be given arbitrarily.

For an involutive system of order q in the above solved form, we shall use to denote by the principal jet coordinates, namely the leading terms of the solved equations in the sense of involution. Accordingly, any formal derivative of a principal jet coordinate is again a principal jet coordinate. The remaining jet coordinates will be called parametric jet coordinates and denoted by. Now, the symbol of is the zero symbol and is thus trivially involutive at any order q. Accordingly, if we introduce the multiplicative variables for the parametric jets of order q and class i, the formal derivative or a parametric jet of strict order q and class i by one of its multiplicative variables is uniquely obtained and cannot be a principal jet of order which is coming from a uniquely defined principal jet of order q and class i.

PROPOSITION 4.5: Using the Janet board and the definition of involutivity, we get:

Let T be the tangent vector bundle of vector fields on X, be the cotangent vector bundle of 1-forms on X and be the vector bundle of s-forms on X with usual bases where we have set. Also, let be the vector bundle of symmetric q-covariant tensors. Moreover, if are two vector fields on X, we may define their bracket by the local formula

leading to the Jacobi identity

. We may finally introduce the exterior derivative

with in the Poincaré sequence:

In a purely algebraic setting, one has [20] [39] [40] :

PROPOSITION 4.6: There exists a map which restricts to and.

Proof: Let us introduce the family of s-forms and set. We obtain at once.

Q.E.D.

The kernel of each in the first case is equal to the image of the preceding but this may no longer be true in the restricted case and we set (see [39] , p. 85-88 for more details):

DEFINITION 4.7: We denote by and respectively

the coboundary space, cocycle space and cohomology space at of the restricted d-sequence which only depend on g_q and may not be vector bundles. The symbol g_q is said to be s-acyclic if , involutive if it is n-acyclic and finite type if becomes trivially involutive for r large enough. For a later use, we notice that a symbol is involutive and of finite type if and only if. Finally, is involutive if we set.

FI CRITERION 4.8: If is an epimorphism of vector bundles and is 2-acyclic (involutive), then is formally integrable (involutive).

EXAMPLE 4.9: The system defined by the three PD equations

is homogeneous and thus automatically formally integrable but is not involutive though finite type because. Elementary computations of ranks of matrices show that the d-map:

is a isomorphism and thus is 2-acyclic with, a crucial intrinsic property totally absent from any “old” work and quite more easy to handle than its Koszul dual.

The main use of involution is to construct differential sequences that are made up by successive compatibility conditions (CC) of order one. In particular, when is involutive, the differential operator of order q with space of solutions is said to be involutive and one has the canonical linear Janet sequence ( [31] , p. 144):

where each other operator is first order involutive and generates the CC of the preceding one with the Janet bundles. As the Janet sequence can be “cut at any place”, that is can also be constructed anew from any intermediate operator, 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 still believe (with provides the simplest example). Moreover, the fiber dimension of the Janet bundles can be computed at once inductively from the board of multiplicative and non-multiplicative variables that can be exhibited for by working out the board for 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 of dots in the column i just indicates the number of CC of class i for with. When is not involutive but formally integrable and the r-prolongation of its symbol becomes 2-acyclic, it is known that the generating CC are of order (see [39] , Example 6, p. 120 and previous Example).

DEFINITION 4.10: More generally, a differential sequence is said to be formally exact if each operator generates the CC of the operator preceding it.

EXAMPLE 4.11: ( [41] , §38, p 40, is providing the first intuition of formal integrability) The second order system is neither formally integrable nor involutive. Indeed, we get and, that is to say each first and second prolongation does bring a new second order PD equation. Considering the new system, the question is to decide whether this system is involutive or not. In such a simple situation, as there is no PD equation of class 3, the evident permutation of coordinates provides the following involutive second order system with one equation of class 3, 2 equations of class 2 and 1 equation of class 1:

We have and the corresponding CC system is easily seen to be the following involutive first order system:

The final CC system is the involutive first order system:

We get therefore the (formally exact) Janet sequence:

However, keeping only and while using the fact that commutes with, we get the formally exact sequence which is not a Janet sequence. We finally check that each is separately differentially dependent on because we have successively .

Finally, we may extend the restriction of the Spencer operator to:

in order to construct the first Spencer sequence which is another resolution of because the kernel of the first D is such that when q is large enough.

5) DIFFERENTIAL MODULES

Let K be a differential field, that is a field containing with n commuting derivations with such that, and. Using an implicit summation on multi-indices, we may introduce the (noncom- mutative) ring of differential operators with elements such that and. The highest value of with is called the order of the operator P and the ring D with multiplication is filtred by the order q of the operators. We have the filtration. Moreover, it is clear that D, as an algebra, is generated by and with if we identify an element with the vector field of differential geometry, but with now. It follows that is a bimodule over itself, being at the same time a left D-module by the composition and a right D-module by the composition with.

If we introduce differential indeterminates, we may extend to for. Therefore, setting and calling the differential module of equations, we obtain by residue the differential module or D-module, denoting the residue of by when there can be a confusion. Introducing the two free differential modules, we obtain equivalently the free presentation of order q when and. We shall moreover assume that provides a strict morphism, namely that the corresponding system is formally integrable. It follows that M can be endowed with a quotient filtration obtained from that of which is defined by the order of the jet coordinates y_q in. We have therefore the inductive limit with but it is important to notice that in this particular case. It also follows from Noetherian arguments and involution that though we have in general only. As, we may introduce the forgetful functor.

More generally, introducing the successive CC as in the preceding section while changing slightly the numbering of the respective operators, we may finally obtain the free resolution of M, namely the exact sequence. In actual practice, one must never forget that acts on the left on column vectors in the operator case and on the right on row vectors in the module case. Also, with a slight abuse of language, when is involutive as in Section 2 and thus is involutive, one should say that M has an involutive presentation of order q or that is involutive.

DEFINITION 5.1: Setting, we have and. Such a definition can be extended to any matrix of operators by using the transposed matrix of adjoint operators and we get:

from integration by part, where is a row vector of test functions and the usual contraction. We quote

the useful formulas (care about the

minus sign) and as in ( [32] , p. 610-612).

REMARK 5.2: As can be seen from the examples of the Introduction, when is involutive, then may not be involutive. In the differential framework, we may set. Comparing to similar concepts used in differential algebra, this number is just the maximum number of differentially independent equations to be found in the differential module I of equations. Indeed, pointing out that differential indeter- minates in differential algebra are nothing else than jet coordinates in differential geometry and using standard

notations, we have. In that case, the differential ideal I automatically generates a prime

differential ideal providing a differential extension with and differential transcendence degree, a result explaining the notations [39] . Now, from the dimension formulas of, we obtain at once and thus in a coherent way with any free presentation of M starting with. However, acts on the left in differential geometry but on the right in the theory of differential modules. For an operator of order zero, we recognize the fact that the rank of a matrix is equal to the rank of the transposed matrix.

PROPOSITION 5.3: If is a local diffeomorphisms on X, we may set and we have the identity:

and the adjoint of the well defined intrinsic operator is ( minus) the well defined intrinsic operator. Accordingly, if we have an operator, we obtain the formal adjoint operator.

Now, with operational notations, let us consider the two differential sequences:

where generates all the CC of. Then but may not gene- rate all the CC of. Passing to the module framework, we just recognize the definition of. Now, exactly like we defined the differential module M from, let us define the differential module N from. Then does not depend on the presentation of M.

Having in mind that D is a K-algebra, that K is a left D-module with the standard action and that D is a bimodule over itself, we have only two possible constructions leading to the following two definitions:

DEFINITION 5.4: We may define the inverse system of M and introduce as the inverse system of order q.

DEFINITION 5.5: We may define the right differential module.

The first definition is leading to the inverse systems introduced by Macaulay in [41] (see [43] for details). As for the second, we have (see [1] , p. 21, [6] , p. 483-495, [42] , [43] for details).

THEOREM 5.6: When M and N are left D-modules, then and are left D-modules. In particular is also a left D-module for the Spencer operator. Moreover, the structures of left D-modules existing therefore on and are now coherent with the adjoint isomor- phism for:

Proof: For any, let us define:

It is easy to check that in the operator sense and that is the standard bracket of vector fields. We have in particular with d in place of any:

For any with arbitrary and, we may then define:

and conclude similarly with:

Using K in place of N, we finally get that is we recognize exactly the Spencer operator with now and thus:

In fact, R is the projective limit of in a coherent way with jet theory [18] [19] .

The next result is entrelacing the two left structures that we have just provided through the formula defining the map whenever is given and . Using any, we get successively in L:

and thus or simply.

Q.E.D.

COROLLARY 5.7: If M and N are right D-modules, then is a left D-module. Moreover, if M is a left D-module and N is a right D-module, then is a right D-module.

Proof: If M and N are right D-modules, we just need to set and conclude as before. Similarly, if M is a left D-module and N is a right D-module, we just need to set.

Q.E.D.

REMARK 5.8: When and, , then cannot be endowed with any left or right differential structure. When and, then cannot be endowed with any left or right differential structure (see [1] , p. 24 for more details).

As is a bimodule, then is a right D-module according to Lemma 2.13 and the module N defined by the ker/coker sequence is thus a right module.

COROLLARY 5.9: We have the side changing procedure with inverse whenever.

Proof: According to the above Theorem, we just need to prove that has a natural right module structure over D. For this, if is a volume form with coefficient, we may set when. As D is generated by K and T, we just need to check that the above formula has an intrinsic meaning for any. In that case, we check at once:

by introducing the Lie derivative of with respect to, along the intrinsic formula where is the interior multiplication and d is the exterior derivative of exterior forms. According to well known properties of the Lie derivative, we get:

Q.E.D.

Collecting all the results so far obtained, if a differential operator is given in the framework of differential geometry, we may keep the same notation in the framework of differential modules which are left modules over the ring D of linear differential operators and apply duality, provided we use the notation and deal with right differential modules or use the notation and deal again with left differential modules by using the conversion procedure.

DEFINITION 5.10: If a differential operator is given, a direct problem is to find (generating) compatibility conditions (CC) as an operator such that. Conversely, given, the inverse problem will be to look for such that generates the CC of and we shall say that is parametrized by if such an operator is existing.

REMARK 5.11: Of course, solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem. However, though the direct problem always has a solution, the inverse problem may not have a solution at all and the case of the Einstein operator is one of the best non-trivial PD counterexamples (compare [10] to [34] ). It is rather striking to discover that, in the case of OD operators, it took almost 50 years to understand that the possibility to solve the inverse problem was equivalent to the controllability of the corresponding control system (compare [11] to [34] ).

As, any operator is the adjoint of a certain operator and we get:

FORMAL TEST 5.12: The double duality test needed in order to check whether or not and to find out a parametrization if has 5 steps which are drawn in the following diagram where generates the CC of and generates the CC of:

THEOREM 5.13: parametrized by.

REMARK 5.14: When an operator can be parametrized by an operator, we may ask whether or not can be parametrized again by an operator and so on. A good comparison can be made with hunting rifles as a few among them, called double rifles, are equipped with a double trigger mechanism, allowing to shoot again once one has already shot. In a mathematical manner, the question is to know whether the diffe- rential module defined by is torsion-free or reflexive. The main difficulty is that these intrinsic properties highly depend on the choice of the parametrizing operator. The simplest example is provided by the Poincaré sequence for n = 3 made by the successive operators. Indeed, any student knows that is parametrizing div and that grad is parametrizing curl. However, we may parametrize by

choosing with 2 potentials only instead of the usual 3 poten-

tials and cannot proceed ahead as before. Other important examples will be provided in the next section dealing with applications, in particular the one involving Einstein equations when. This comment points out the reason for using the extension modules.

It remains to study a delicate question on which all the examples of the Introduction were focussing. Indeed, if a parametrization of a given system of OD or PD equations is possible, that is, equivalently, if the corresponding differential module is torsion-free, it appears that different parametrizations may exist with quite different numbers of potentials needed. Accordingly, it should be useful to know about the possibility to have upper and lower bounds for these numbers when, particularly in elasticity theory, because when, an OD module M with being automatically isomorphic to a free module E, the number of potentials needed is equal to. We shall use the language of differential modules in order to improve and apply a few results already presented in ( [7] , Theorem 7+ Appendix).

THEOREM 5.15: Let be a finite free presentation of the differential module and assume we already know that by using the formal test. Accordingly, we have obtained the exact sequence of free differential modules where is the parametrizing operator. Then, there exists other parametrizations called minimal parametrizations and such that is a torsion module.

Proof: We first explain the reason for using the word “minimal”. Indeed, we have but also and thus as a way to get a lower bound for the number of potentials but not to get a differential geometric framework.

While applying the formal test in the operator language, is describing the (generating) CC of and we shall denote by the (generating) CC of as we did in Example 1.3. In the module framework, going on with left differential modules, when F is a free left module, we shall denote by the corresponding converted left differential module of the right differential module. The reader not familiar with duality may look at the situations met in electromagnetism and elasticity in ( [6] , p. 492-495). If and is the largest free differential submodule of L (in Example 1.3, D in Example 1.4), then is a torsion module and we have the following commutative and exact diagram:

where the central vertical monomorphism is obtained by pulling a basis of back to as we did in the diagram of Proposition 2.24. Coming back to the operators and, we get the following commutative and exact diagram allowing to define by composition:

We have:

and obtain by duality the following commutative and exact diagram where:

However, though the upper sequence is exact by definition because , the lower induced sequence may not be exact. With for simplicity, and the induced epimorphism, we obtain:

Accordingly, is a minimal parametrization of contrary to in general and we invite the reader to repeat the proof by using operators and their adjoints as in the formal test.

Q.E.D.

6) APPLICATIONS

EXAMPLE 6.1: OD Control Theory Revisited

The following result is well known and can be found in any textbook of algebra [6] [13] [32] :

PROPOSITION 6.2: If A is a principal ideal domain, that is if any ideal in A is generated by a single element, then any torsion-free module over A is free and thus projective.

As this is the case of the ring when, we obtain the following corollary of the preceding parametrizing Theorem, allowing to extend the Kalman test of controllability to PD systems with variable coefficients as we did in the Introduction (see [6] - [9] [11] for details).

COROLLARY 6.3: If is surjective, then is injective if and only if M is projective.

Proof: As is surjective, replacing M by P, we have the following short exact sequence:

As P is projective, this short exact sequence splits with [6] [13] [32] . Using Proposition 2.6, we can construct a right inverse operator of with now for the corresponding morphisms. Applying duality and Corollary 2.10, we get the short exact sequence:

It follows that is surjective and the adjoint operator is injective because.

Conversely, if is injective, there exists a left inverse of providing a right inverse of (care). We may thus use again Corollary 2.10 because and. Meanwhile, we have proved that, if and, it is always possible to find an injective parametrization but Example 1.4 is showing that this result is no longer true when.

Q.E.D.

Multiplying the control system of Example 1.1 by a test function and integrating by parts, the kernel of the operator thus obtained is defined by the OD equations:

The formal adjoint of the operator defining the control system is thus injective if and only if we have, a result absolutely not evident at first sight but explaining why we used the same notation for a test function and for a Lagrange multiplier.

EXAMPLE 6.4: Elasticity Theory Revisited

The Killing operator is a defined by with where is the displacement vector, is the Lie derivative of with respect to and is the infinitesimal deformation tensor of textbooks. It is a Lie operator because its solutions satisfy. The corresponding first order Killing system is not involutive because its symbol is finite type with first prolongation and thus. Accordingly, as is a flat constant metric, the second order CC are described by an operator coming from the linearization of the Riemann tensor obtained in a standard way by setting with a small parameter, dividing by t and taking the limit when. Finally, raising the index by means of the metric, the adjoint operator is defined by the intrinsic stress equations where is the covariant derivative and the Christoffel symbols ( [6] , p. 494, [44] , p. 236).