Differential Homological Algebra and General Relativity
aa r X i v : . [ m a t h . G M ] J u l DIFFERENTIAL HOMOLOGICAL ALGEBRAAND GENERAL RELATIVITY
J.-F. PommaretCERMICS, Ecole des Ponts [email protected]://cermics.enpc.fr/ ∼ pommaret/home.html ABSTRACT :In 1916, F.S. Macaulay developed specific localization techniques for dealing with ”unmixed poly-nomial ideals” in commutative algebra, transforming them into what he called ”inverse systems”of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theoryof such systems, using methods of homological algebra that were giving rise to ”differential homo-logical algebra”, replacing unmixed polynomial ideals by ”pure differential modules”. The use of”extension modules” and ”differential double duality” is essential for such a purpose. In particular,0-pure differential modules are torsion-free and admit an ”absolute parametrization” by means ofarbitrary potential like functions. In 2012, we have been able to extend this result to arbitrarypure modules, introducing a ”relative parametrization” where the potentials should satisfy com-patible ”differential constraints”. We recently discovered that General Relativity is just a way toparametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator inorder to obtain a ”minimum parametrization” by adding sufficiently many compatible differentialconstraints, exactly like the Lorenz condition in electromagnetism. These unusual purely mathe-matical results are illustrated by many explicit examples and even strengthen the comments werecently provided on the mathematical foundations of General Relativity and Gauge Theory.
KEY WORDS :Unmixed polynomial ideal; Extension module; Torsion-free module; pure differential module;purity filtration; inverse system; involution; control theory; electromagnetism; General relativity.1 ) INTRODUCTION :Te main purpose of this paper is to prove how apparently totally abstract mathematical tools,ranging among the most difficult ones of differential geometry and homological algebra, can alsobecome useful and enlight many engineering or physical concepts (See the review Zbl 1079.93001for the only application to control theory).In the second section, we first sketch and then recall the main (difficult) mathematical results on differential extension modules and differential double duality that are absolutely needed in orderto understand the purity concept and, in particular, the so-called purity filtration of a differentialmodule ([1],[2],[24],[29]). We also explain the unexpected link existing between involutivity and purity allowing to exhibit a relative parametrization of a pure differential module, even definedby a system of linear PD equations with coefficients in a non-constant differential field K . It isimportant to notice that the reduced Spencer form which is used for such a purpose generalizes the Kalman form existing for an OD classical control system and we shall illustrate this fact.The third section will present for the first time a few explicit motivating academic examples inorder to illustrate the above mathematical results, in particular the unexpected striking situationsmet in the study of contact and unimodular contact structures.In the fourth section, we finally provide examples of applications, studying the mathematicalfoundations of OD/PD control theory ([24],[25]), electromagnetism (EM) ([31],[37]) and generalrelativity (GR) ([30],[33],[35]). Most of these examples can be now used as test examples for certaincomputer algebra packages recently developped for such a purpose ([43-44]).
2) MATEMATICAL TOOLS :Let D = K [ d , ..., d n ] = K [ d ] be the ring of differential operators with coefficients in a differen-tial field K of characteristic zero, that is such that Q ⊂ K , with n commuting derivations ∂ , ..., ∂ n and commutation relations d i a = ad i + ∂ i a, ∀ a ∈ K . If y , ..., y m are m differential indeterminates,we may identify Dy + ... + Dy m = Dy with D m and consider the finitely presented left differentialmodule M = D M with presentation D p → D m → M → n independent variables, m unknowns and p equations. Applying the functor hom D ( • , D ), we get the exact sequence 0 → hom D ( M, D ) → D m → D p −→ N D −→ right dif-ferential modules that can be transformed by a side-changing functor to an exact sequence of finitelygenerated 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 unknownsand m equations, obtaining therefore a new finitely generated left differential module N = D N andwe may consider hom D ( M, D ) as the module of equations of the compatibility conditions (CC) of ad ( D ), a result not evident at first sight (See [24],[38]). Using now a maximum free submodule0 −→ D l −→ hom D ( M, D ) and repeating this standard procedure while using the well known factthat ad ( ad ( D )) = D , we obtain therefore an embedding 0 → hom D ( hom D ( M, D ) , D ) → D l of leftdifferential modules for a certain integer 1 ≤ l < m because K is a field and thus D is a noetherianbimodule over itself, a result leading to l = rk D ( hom D ( M, D )) = rk D ( M ) < m as in ([22], p341,[23], [25] p 179)(See section 3 for the definition of the differential rank rk D ). Now, the kernelof the map ǫ : M → hom D ( hom D ( M, D ) , D ) : m → ǫ ( m )( f ) = f ( m ) , ∀ f ∈ hom D ( M, D ) is thetorsion 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 → D l of M into a free module(that can also be obtained by localization if we introduce the ring of fractions S − D = DS − when S = D − { } ). This result is quite important for applications as it provides a (minimal)parametrization of the linear differential operator D and amounts to the controllability of a classicalcontrol system when n = 1 ([12],[24]). This parametrization will be called an ” absolute parametriza-tion ” as it only involves arbitrary ” potential-like ” functions (See [23],[26],[29],[30],[33],[36],[39],[41]for more details and examples, in particular that of Einstein equations).The purpose of this paper is to extend such a result to a much more general situation, thatis when M is not torsion-free , by using unexpected results first found by F.S. Macaulay in 1916([15]) through his study of ” inverse systems ” for ” unmixed polynomial ideals ”.Introducing t r ( M ) = { m ∈ M | cd ( Dm ) > r } where the codimension of Dm is n minus thedimension of the characteristic variety determined by m in the corresponding system for one un-2nown, we may define the purity filtration as in ([1],[24],[29]):0 = t n ( M ) ⊆ t n − ( M ) ⊆ ... ⊆ t ( M ) ⊆ t ( M ) = t ( M ) ⊆ M The module M is said to be r - pure if t r ( M ) = 0 , t r − ( 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 afield of constants and m = 1, a pure module is unmixed in the sense of Macaulay, that is definedby an ideal having an equidimensional primary decomposition. Example 2.1 : As an elementary example with K = k = Q , m = 1 , n = 2 , p = 2, the differentialmodule defined by d y = 0 , d y = 0 is not pure because z ′ = d y satisfies d z ′ = 0 , d z ′ = 0 while z ” = d y only satisfies d z ” = 0 and (( χ ) , χ χ ) = ( χ ) ∩ ( χ , χ ) . We obtain therefore thepurity filtration 0 = t ( M ) ⊂ t ( M ) ⊂ t ( M ) = t ( M ) = M with strict inclusions as 0 = z ′ ∈ t ( M )while z ” ∈ t ( M ) but z ” / ∈ t ( M ).From the few (difficult) references ([[1],[2],[3],[10],[13],[14],[16],[17],[18],[24],[25],[29],[45],[46])dealing with the extension modules ext r ( M ) = ext rD ( M, D ) and purity in the framework of al-gebraic analysis, it is known that M is r -pure if and only if there is an embedding 0 → M → ext rD ( ext rD ( M, D ) , D ). Indeed, the case r = 0 is exactly the one already considered because ext D ( M, D ) = hom D ( M, D ) and the ker/coker exact sequence ([25],[29]):0 −→ ext ( N ) −→ M −→ ext ( ext ( M )) −→ ext ( N ) −→ M in actual practice by using the double-duality formula t ( M ) = ext ( N ) as in ([24],[25]).Independently of the previous results, the following procedure, where one may have to changelinearly the independent variables if necessary , is the heart towards the next effective definition ofinvolution. It is intrinsic even though it must be checked in a particular coordinate system called δ - regular ([19],[20],[32]) and is quite simple for first order systems without zero order equations. • Equations of class n : Solve the maximum number β nq of equations with respect to the jets oforder q and class n . Then call ( x , ..., x n ) multiplicative variables . • Equations of class i ≥
1: Solve the maximum number β iq of remaining equations with respectto the jets of order q and class i . Then call ( x , ..., x i ) multiplicative variables and ( x i +1 , ..., x n ) non-multiplicative variables . • Remaining equations equations of order ≤ q −
1: Call ( x , ..., x n ) non-multiplicative variables .In actual practice, we shall use a Janet tabular where the multiplicative ”variables” are representedby their index in upper left position while the non-multiplicative variables are represented by dotsin lower right position ([11],[19],[24]) (Compare to ([47]).
DEFINITION 2.2 : A system of PD equations is said to be involutive if its first prolongationcan be achieved by prolonging its equations only with respect to the corresponding multiplicativevariables. In that case, we may introduce the
Cartan characters α iq = m ( q + n − i − q − n − i )! − β iq for i = 1 , ..., n and we have dim ( g q ) = P α q = α q + ... + α nq and dim ( g q +1 ) = P iα iq = 1 α q + ... + nα nq .Moreover, one can exhibit the Hilbert polynomial dim ( R q + r ) in r with leading term ( α/d !) r d with d ≤ n when α is the smallest non-zero character in the case of an involutive symbol. Such aprolongation 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 2.3 : For an involutive system with β = β nq < m , then ( y β +1 , ..., y m ) can be givenarbitrarily and may constitute the input variables in control theory, though it is not necessary tomake such a choice. In this case , the intrinsic number α = α nq = m − β > n - character and is the system counterpart of the so-called ” differential transcendence degree ” in differential al-gebra and the ” rank ” in module theory. As we shall see in the next Section, the smallest non-zerocharacter and the number of zero characters are intrinsic numbers that can most easily be knownby bringing the system to involution and we have α q ≥ ... ≥ α nq ≥ we just need to modify the Spencerform and we provide the procedure that must be followed in the case of a first order involutivesystem with no zero order equation, for example an involutive Spencer form. • Look at the equations of class n solved with respect to y n , ..., y βn . • Use integrations by parts like: y n − a ( x ) y β +1 n = d n ( y − a ( x ) y β +1 ) + ∂ n a ( x ) y β +1 = ¯ y n + ∂ n a ( x ) y β +1 • Modify y , ..., y β to ¯ y , ..., ¯ y β in order to ” absorb ” the various y β +1 n , ..., y mn only appearing in theequations of class n .We have the following unexpected result providing what we shall call a reduced Spencer form : THEOREM 2.4 : The new equations of class n contain y , ..., y β and their jets but only contain y β +1 i , ..., y mi with 0 ≤ i ≤ n − , ..., n − 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 ¯ y , ..., ¯ y β .If χ , ..., χ n are n algebraic indeterminates or, in a more intrinsic way, if χ = χ i dx i ∈ T ∗ is acovector and D : E −→ F : ξ −→ a τµk ( x ) ∂ µ ξ k ( x ) is a linear involutive operator of order q , we mayintroduce the characteristic matrix a ( x, χ ) = ( a τµk ( x ) χ µ , | µ | = µ + ... + µ n = q ) and the resultingmap σ χ ( D ) : E −→ F is called the symbol of D at χ . Then there are two possibilities: • If max χ rk ( σ χ ( D ) < m ⇔ α nq >
0: the characteristic matrix fails to be injective for any covector. • If max χ rk ( σ χ ( D ) = m ⇔ α nq = 0: the characteristic matrix fails to be injective if and only ifall the determinants of the m × m submatrices vanish. However, one can prove that this algebraicideal a ∈ K [ χ ] is not intrinsically defined and must be replaced by its radical rad ( a ) made by allpolynomials having a power in a . This radical ideal is called the characteristic ideal of the operator. DEFINITION 2.5 : For each x ∈ X , the algebraic set defined by the characteristic ideal is calledthe characteristic set of D at x and V = ∪ x ∈ X V x is called the characteristic set of D while we keepthe word ” variety ” for an irreducible algebraic set defined by a prime ideal.One has the following important theorem ([24], [38]) that will play an important part later on: THEOREM 2.6 : (Hilbert-Serre) The dimension d ( V ) of the characteristic set, that is the maxi-mum dimension of the irreducible components, is equal to the number of non-zero characters whilethe codimension cd ( V ) = n − d ( V ) is equal to the number of zero characters, that is to the numberof ” full ” classes in the Janet tabular of an involutive system.If P = a µ d µ ∈ D = K [ d ] with implicit summation on the multi-index, the highest valueof | µ | with a µ = 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 ⊂ K = D ⊂ D ⊂ ... ⊂ D q ⊂ ... ⊂ D ∞ = D . Moreover, it is clear that D , as an al-gebra, is generated by K = D and T = D /D with D = K ⊕ T if we identify an element ξ = ξ i d i ∈ T with the vector field ξ = ξ i ( x ) ∂ i of differential geometry, but with ξ i ∈ K now. Itfollows that D = D D D is a bimodule over itself, being at the same time a left D -module by thecomposition P −→ QP and a right D -module by the composition P −→ P Q . We define the adjoint map ad : D −→ D op : P = a µ d µ −→ ad ( P ) = ( − | µ | d µ a µ and we have ad ( ad ( P )) = P . It is easyto check that ad ( P Q ) = ad ( Q ) ad ( P ) , ∀ P, Q ∈ D . Such a definition can also be extended to anymatrix of operators by using the transposed matrix of adjoint operators (See [24-26],[30],[40-42]for more details and applications to control theory and mathematical physics).Accordingly, if y = ( y , ..., y m ) are differential indeterminates, then D acts on y k by setting d µ y k = y kµ with d i y kµ = y kµ +1 i and y k = y k . We may therefore use the jet coordinates in a formal4ay as in the previous section. Therefore, if a system of OD/PD equations is written in the form:Φ τ ≡ a τµk y kµ = 0with coefficients a τµk ∈ K , we may introduce the free differential module Dy = Dy + ... + Dy m ≃ D m and consider the differential submodule I = D Φ ⊂ Dy which is usually called the module ofequations , both with the differential module M = Dy/D
Φ 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 may introduce the formal prolongation withrespect to d i by setting: d i Φ τ ≡ a τµk y kµ +1 i + ( ∂ i a τµk ) y kµ in order to induce maps d i : M −→ M : ¯ y kµ −→ ¯ y kµ +1 i if we use to denote the residue Dy −→ M : y k −→ ¯ y k by a bar as in algebraic geometry. However, for simplicity, we shall not write down thebar when the background will indicate clearly if we are in Dy or in M .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 = ( a τµk d µ ) with m columns and p rows, we may introduce the morphism D p D −→ D m : ( P τ ) −→ ( P τ Φ τ ) : P −→ P Φ = P D over D by acting with D on the left of theserow vectors while acting with D on the right of these row vectors and the presentation of M isdefined by the exact cokernel sequence D p −→ D m −→ M −→
0. It is essential to notice thatthe 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 resultsof the previous section in order to understant that certain presentations will be much better thanothers, in particular to establish a link with formal integrability and involution. DEFINITION 2.7 : It follows from its definition that M can be endowed with a quotient filtra-tion obtained from that of D m which is defined by the order of the jet coordinates y q in D q y .We have therefore the inductive limit ⊆ M ⊆ M ⊆ ... ⊆ M q ⊆ ... ⊆ M ∞ = M with d i M q ⊆ M q +1 and M = DM q for q ≫ D r M q ⊆ M q + r , ∀ q, r ≥
0. Weshall set gr ( M q ) = G q = M q /M q − and gr ( M ) = G = ⊕ q G q .Having in mind that K is a left D -module for the action ( D, K ) −→ K : ( d i , a ) −→ ∂ i a andthat D is a bimodule over itself, we have only two possible constructions : DEFINITION 2.8 : We define the system R = hom K ( M, K ) = M ∗ and set R q = hom K ( M q , K ) = M ∗ q as the system of order q . We have the projective limit R = R ∞ −→ ... −→ R q −→ ... −→ R −→ R . It follows that f q ∈ R q : y kµ −→ f kµ ∈ K with a τµk f kµ = 0 defines a section at order q and we may set f ∞ = f ∈ R for a section of R . For a ground field of constants k , this definitionhas of course to do with the concept of a formal power series solution. However, for an arbitrarydifferential field K , the main novelty of this new approach is that such a definition has nothing todo with the concept of a formal power series solution ( care ) as illustrated in ([27]). DEFINITION 2.9 : We may define the right differential module hom D ( M, D ). PROPOSITION 2.10 : 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 = a i d i ∈ T, ∀ m ∈ M In the operator sense, it is easy to check that d i a = ad i + ∂ i a and that ξη − ηξ = [ ξ, η ] is thestandard bracket of vector fields. We finally get ( d i f ) kµ = ( d i f )( y kµ ) = ∂ i f kµ − f kµ +1 i and thus recover exactly the Spencer operator though this is not evident at all . We also get ( d i d j f ) kµ = ∂ ij f kµ − i f kµ +1 j − ∂ j f kµ +1 i + f kµ +1 i +1 j = ⇒ d i d j = d j d i , ∀ i, j = 1 , ..., n and thus d i R q +1 ⊆ R q = ⇒ d i R ⊂ R induces a well defined operator R −→ T ∗ ⊗ R : f −→ dx i ⊗ d i f . This result has been discovered (upto sign) by Macaulay in 1916 ([15]). For more details on the Spencer operator and its applications,the reader may look at ([21],[22],[28],[48]). Q.E.D. DEFINITION 2.11 : t r ( M ) is the greatest differential submodule of M having codimension > r . PROPOSITION 2.12 : cd ( M ) = cd ( V ) = r ⇐⇒ α n − rq = 0 , α n − r +1 q = ... = α nq = 0 ⇐⇒ t r ( M ) = M, t r − ( M ) = ... = t ( M ) = t ( M ) = M and this intrinsic result can be most easily checked byusing the standard or reduced Spencer form of the system defining M .We are now in a good position for defining and studying purity for differential modules. DEFINITION 2.13 : M is r - pure ⇐⇒ t r ( M ) = 0 , t r − ( M ) = M ⇐⇒ cd ( Dm ) = r, ∀ m ∈ M .More generally, M is pure if it is r -pure for a certain 0 ≤ r ≤ n and M is pure if it is r -pure for acertain 0 ≤ r ≤ n . 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/t r ( M ) the pure part of M . It follows that t r − ( M ) /t r ( M ) is equal to zero or is r -pure (See the picture in [20], p 545). When M = t n − ( M ) is n -pure, its defining system is a finitedimensional vector space over K with a symbol of finite type, that is when g q = 0 is (trivially)involutive. Finally, when t r − ( M ) = t r ( M ), we shall say that there is a gap in the purity filtration:0 = t n ( M ) ⊆ t n − ( M ) ⊆ ... ⊆ t ( M ) ⊆ t ( M ) = t ( M ) ⊆ M PROPOSITION 2.14 : t r ( M ) does not depend on the presentation or on the filtration of M . EXAMPLE 2.15 : If K = Q and M is defined by the involutive system y = 0 , y = 0 , y = 0,then z = y satifies d z = 0 , d z = 0 , d z = 0 and cd ( Dz ) = 3 while z ′ = y only satisfies d z ′ = 0and cd ( Dz ′ ) = 1. We have the purity filtration 0 = t ( M ) ⊂ t ( M ) = t ( M ) ⊂ t ( M ) = t ( M ) = M with one gap and two strict inclusions.We now recall the definition of the extension modules ext iD ( M, D ) that we shall simply denoteby ext i ( M ) and the way to use their dimension or codimension. We point out once more that thesenumbers can be most easily obtained by bringing the underlying systems to involution in order toget informations on M from informations on G . We divide the procedure into four steps that canbe achieved by means of computer algebra ([43],[44]): • Construct a free resolution of M , say: ... −→ F i −→ ... −→ F −→ F −→ M −→ • Suppress M in order to obtain the deleted sequence : ... −→ F i −→ ... −→ F −→ F −→ • Apply hom D ( • , D ) in order to obtain the dual sequence heading backwards: ... ←− hom D ( F i , D ) ←− ... ←− hom D ( F , D ) ←− hom D ( F , D ) ←− • Define ext i ( M ) to be the cohomology at hom D ( F i , D ) in the dual sequence in such a way that ext ( M ) = hom D ( M, D ).The following nested chain of difficult propositions and theorems can be obtained, even in thenon-commutative case , by combining the use of extension modules and bidualizing complexes in theframework of algebraic analysis. The main difficulty is to obtain first these results for the gradedmodule G = gr ( M ) by using techniques from commutative algebra before extending them to the filtred module M as in ([1],[2],[10],[13],[14],[16],[24],[29],[40],[48]).6 HEOREM 2.16 : The extension modules do not depend on the resolution of M used. PROPOSITION 2.17 : Applying hom D ( • , D ) provides right D -modules that can be transformedto left D -modules by means of the side changing functor and vice-versa. Namely, if N D is a right D -module, then D N = ∧ n T ⊗ K N D is the converted left D -module while, if N = D N is a left D -module, then N D = ∧ n T ∗ ⊗ K N is the converted right D -module. PROPOSITION 2.18 : Instead of using hom D ( • , D ) and the side changing functor in the mod-ule 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 thoughstriking relation rk D ( ad ( D )) = rk D ( D ). PROPOSITION 2.19 : ext i ( M ) is a torsion module ∀ ≤ i ≤ n but ext ( M ) = hom D ( M, D )may not be a torsion module.
EXAMPLE 2.20 : When M is a torsion module, we have hom D ( M, D ) = 0 (exercise). When n = 3 and the torsion-free module M is defined by the formally surjective div operator, the formaladjoint of div is − grad which defines a torsion module. Also, when n = 1 as in classical controltheory, a controllable system with coefficients in a differential field allows to define a torsion-freemodule M which is free in that case because a finitely generated module over a principal idealdomain is free if and only if it is torsion-free and hom D ( M, D ) is thus also a free module.
THEOREM 2.21 : ext i ( M ) = 0 , ∀ i < cd ( M ) and ∀ i ≥ n + 1. THEOREM 2.22 : cd ( ext i ( M )) ≥ i . THEOREM 2.23 : cd ( M ) ≥ r ⇔ ext i ( M ) = 0 , ∀ i < r . PROPOSITION 2.24 : cd ( M ) = r = ⇒ cd ( ext r ( M )) = r and ext r ( M )is r -pure. PROPOSITION 2.25 : ext r ( ext r ( M )) is equal to 0 or is r -pure, ∀ ≤ r ≤ n . PROPOSITION 2.26 : If we set t − ( M ) = M , there are exact sequences ∀ ≤ r ≤ n :0 −→ t r ( M ) −→ t r − ( M ) −→ ext r ( ext r ( M )) THEOREM 2.27 : If cd ( M ) = r , then M is r -pure if and only if there is a monomorphism0 −→ M −→ ext r ( ext r ( M )) of left differential modules. THEOREM 3.28 : M is pure ⇐⇒ ext s ( ext s ( M )) = 0 , ∀ s = cd ( M ). COROLLARY 2.29 : If M is r -pure with r ≥
1, then it can be embedded into a differentialmodule L having a free resolution with only r operators.The previous theorems are known to characterize purity but it is however evident that they arenot very useful in actual practice. For more details on these two results which are absolutely out ofthe scope of this paper, see ([2], p 490-491) and ([24], p 547). Proposition 3.24 and Theorem 3.25come from the Cohen-Macaulay property of M , namely cd ( M ) = g ( M ) = inf { i | ext i ( M ) = 0 } where g ( M ) is called the grade of M (See [2] and [24],[29] for more details). THEOREM 2.30 : When M is r -pure, the characteristic ideal is thus unmixed , that is a finiteintersection of prime ideals having the same codimension r and the characteristic set is equidimen-sional , that is the union of irreducible algebraic varieties having the same codimension r .In 2013 we have provided a new effective test for checking purity while using the involutivityof the Spencer form with four steps as follows ([29]):7 STEP 1: Compute the involutive Spencer form of the system and the number r of full classes. • STEP 2: Select only the equations of class 1 to d ( M ) = n − r of this Spencer form which aremaking an involutive system over K [ d , ..., d ( n − r ) ]. • STEP 3: Using differential biduality for such a system, check if it defines a torsion-free module M ( n − r ) and work out a parametrization. • STEP 4: Substitute the above parametrization in the remaning equations of class n − r + 1 , ..., n of the Spencer form in order to get a system of PD equations which provides the parametrizingmodule L in such a way that M ⊆ L and L has a resolution with r operators. THEOREM 2.31 : As purity is an intrinsic property, we may work with an involutive Spencerform and M is r -pure if the classes n − r + 1 , ..., n are full and the module M ( n − r ) defined by theequations of class 1 + ... + class ( n − r ) is torsion-free. Hence M is 0-pure if it is torsion-free.We shall now illustrate and apply this new procedure in the next two sections.
3) MOTIVATING EXAMPLES : EXAMPLE 3.1 : With n = 3 , m = 1 and K = Q , let us consider the following polynomial ideal: a = (( χ ) , χ χ − χ χ , ( χ ) − χ χ ) ⊂ K [ χ , χ , χ ] = K [ χ ]We shall discover that it is not evident to prove that it is an unmixed polynomial ideal and thatthe corresponding differential module is 1-pure.The first result is provided by the existence of the primary decomposition obtained from the twoexisting factorizations: a = (( χ ) , χ − χ ) ∩ ( χ , χ ) = q ′ ∩ q ”Taking the respective radical ideals, we get the prime decomposition : rad ( a ) = ( χ , χ − χ ) ∩ ( χ , χ ) = p ′ ∩ p ” = rad ( q ′ ) ∩ rad ( q ”)The corresponding involutive system is: y = 0 y − y = 0 y − y = 0 1 2 31 2 • • with characters ( α = 1 − , α = 2 − , α = 3 − dim ( g ) = P α = 3.Setting ( z = y, z = y , z = y , z = y ), we obtain the involutive first order Spencer form: z = 0 , z − z = 0 , z − z = 0 , z − z = 0 z − z = 0 , z − z = 0 , z − z = 0 , z − z = 0 z − z = 0 1 2 31 2 • • • with new characters α = 4 − , α = 4 − , α = 4 − dim ( g ) = P α = 3.Both class 3 and class 2 are full while class 1 is defining a torsion-free module M (1) over K [ d ]by means of a trivially involutive system of class 1. Hence the differential module M is such that cd ( M ) = 2 and is 1-pure because it is 1-pure in this presentation.Suppressing the bar for the various residues, we are ready to exhibit the relative parametrization defining the parametrization module L because we may choose the 3 potentials ( z = y, z , z )while taking into account that z = y = d y : 8 z = 0 z − z = 0 z − z = 0 z − z = 0 z − z = 0 z − z = 0 1 2 31 2 31 2 31 2 • • • Both ( y, z , z ) are torsion elements and we can eliminate ( z , z ) in order to find the desired sys-tem that must be satisfied by y which is showing the inclusion M ⊂ L but we have indeed M = L because z = y , z = y . It follows that M admits a free resolution with only 2 operators, a resultfollowing at once from the last Janet tabular, contrary to the previous one .The reader may treat similarly the example a = ( χ , χ ) ∩ ( χ , χ ) and look at ([27]) for details.(Hint: use the involutive system y + y = 0 , y + y = 0 , y + y = 0 , y − y = 0). EXAMPLE 3.2 : With n = 3 , m = 1 , q = 2 , K = Q , D = K [ d , d , d ], let us consider the dif-ferential module M defined by the second order system P y ≡ y = 0 , Qy ≡ y − y = 0 firstconsidered by Macaulay in 1916 ([15],[39]). We shall prove that M is 2-pure through the inclusion0 → M → ext ( ext ( M )) directly and by finding out a relative parametrization, a result highlynot evident at first sight.First of all, in order to find out the codimension cd ( M ) = 2, we have to consider the equivalentinvolutive system: Φ ≡ y = u Φ ≡ y = d u − d v Φ ≡ y = d u − d v − d v Φ ≡ y − y = v • • • • The Janet tabular on the rigt allows at once to compute the characters α = 0 , α = 0 , α = α =3 − → D → D → D → D p → M → rad ( a ) = rad (( χ ) , χ χ , ( χ ) , χ χ ) = ( χ , χ ) = p ⇒ dim ( V ) = 1.As the classes 3 and 2 are full, it follows that d ( M ) = d ( Dy ) = 1 ⇒ cd ( M ) = n − y the canonical residue ¯ y of y after identifyoing D with Dy . We have constructedexplicitly in ([29]) a finite length resolution of N = ext ( M ) by pointing out that N does notdepend on the resolution of M used and one can refer to the single compatibility condition (CC) P ◦ Qy − Q ◦ P y = 0 for the initial system in the exact sequence made by second order operators:0 −→ D D −→ D D −→ D p −→ M −→ hom D ( • , D ) and the respective adjointoperators, we may define the torsion left differential module N by the long exact sequence:0 ←− N q ←− D ad ( D ) ←− D ad ( D ) −→ D ←− rk D ( M ) = 0 ⇒ rk D ( N ) = 1 − M = ext ( N ) = ext ( ext ( M )).Similarly, using certain parametric jet variables as new unknowns, we may set z = y, z = y , z = y , z = y in order to obtain the following involutive first order system with no zero order equation: class d z − z = 0 , d z − z = 0 , d z = 0 , d z = 0 class d z − z = 0 , d z − d z = 0 , d z = 0 , d z = 0 class d z − z = 0 , d z − z = 0 1 2 31 2 • • • where we have separated the classes while using standard computer algebra notations this timeinstead of the jet notations used in the previous example. Contrary to what could be believed, this9perator does not describe the Spencer sequence that could be obtained from the previous Janetsequence but we can use it exactly like a Janet sequence or exactly like a Spencer sequence. Weobtain therefore a long strictly exact sequence of differential modules with only first order operators while replacing Dy by Dz = Dz + Dz + Dz + Dz as follows:0 → D → D → D → D → M → − − M is defined over K [ d ] by the two PD equations of class 1 and is easilyseen to be torsion-free with the two potentials ( z = y, z ). Substituting into the PD equations ofclass 2 and 3, we obtain the generating differential constraints: d z − z = 0 d z = 0 d z − d z = 0 d z = 0 1 2 31 2 31 2 • • They define the parametrization module L and the inclusion M ⊆ L is obtained by eliminating z but we have indeed M = L because z = d y . EXAMPLE 3.3 : We have provided in ([29], Example 4.2) a case leading to a strict inclusion M ⊂ L that we revisit now totally in this new framework. With K = Q , m = 1 , n = 4 , q = 2, letus study the 2-pure differential module M defined by the involutive second order system: y = 0 y = 0 y = 0 y − y = 0 1 2 3 41 2 3 • • • • From the Janet tabular we may construct at once the Janet sequence:0 −→ Θ −→ D −→ D −→ D −→ −→ D is defined by the involutive system: d ( y ) − d ( y ) = 0 d ( y ) − d ( y ) = 0 d ( y − y ) − d ( y ) + d ( y ) = 0 d ( y − y ) − d ( y ) + d ( y ) = 0 1 2 3 41 2 3 41 2 3 41 2 3 • and so on. We have therefore a free resolution of M with 3 operators:0 −→ D −→ D −→ D −→ D −→ M −→ pd ( M ) ≤ rad (( χ ) , χ χ , ( χ ) , χ χ − χ χ ) = ( χ , χ ) = p ⇒ dim ( V ) = 1.Let us transform the initial second order involutive system for y into a first order involutive systemfor ( z = y, z = y , z = y , z = y , z = y ) as follows: d z − z = 0 , d z − d z = 0 , d z − d z = 0 , d z = 0 , d z = 0 d z − z = 0 , d z − d z = 0 , d z − d z = 0 , d z = 0 , d z = 0 d z − z = 0 , d z − d z = 0 , d z − d z = 0 d z − z = 0 1 2 3 41 2 3 • • • • • • with five equations of full class 4,five equations of full class 3, three equations of class 2 andfinally one equation of class 1. The equations of classes 2 and 1 are providing an involutivesystem over Q [ d , d ] defining a torsion-free module M (2) that can be parametrized by setting z = y, z = d y, z = d y, z = d z, z = d z with only 2 arbitrary potentials ( y, z ). Substitutingin the other equations of classes 3 and 4, we finally discover that L is defined by the involutive10ystem describing the relative parametrization: d y − d z = 0 d z = 0 d y − d z = 0 d z = 0 1 2 3 41 2 3 41 2 3 • • We have the strict inclusion M ⊂ L obtained by eliminating z because now z / ∈ Dy if we take theresidue or, equivalently, the residue of z does not belong to M . The differential module L definedby the above system is therefore 2-pure with a strict inclusion M ⊂ L and admits a free resolutionwith only 2 operators according to its Janet tabular. EXAMPLE 3.4 : (
Contact structure ) With n = m = 3 and K = Q ( x , x , x ) let us introduce theso-called contact α = dx − x dx and consider the first order system of infinitesimal Lieequations obtained by eliminating the contact factor ρ from the equations L ( ξ ) α = ρα . We let thereader check that he will obtain only the two equations Φ = 0 , Φ = 0 which is nevertheless neitherformally integrable nor even involutive. Using crossed derivatives one obtains the involutive system: Φ ≡ ∂ ξ + ∂ ξ + 2 x ∂ ξ = 0Φ ≡ ∂ ξ − x ∂ ξ = 0Φ ≡ ∂ ξ − x ∂ ξ + x ∂ ξ − ( x ) ∂ ξ − ξ = 0 1 2 31 2 31 2 • with the unique CC Ψ ≡ ∂ Φ − ∂ Φ − x ∂ Φ + Φ = 0. The following injective absoluteparametrization is well known and we let the reader find it by using differential double duality: φ − x ∂ φ = ξ , − ∂ φ = ξ , ∂ φ + x ∂ φ = ξ ⇒ ξ − x ξ = φ We obtain the Janet sequence0 → D − −→ D −→ D −→ → φ ξ Φ Ψwith formally exact adjoint sequence:0 ← ad ( D − ) ←− ad ( D ) ←− ad ( D ) ←− → θ ν µ λ and the resolution of the trivially torsion-free module M ≃ D :0 −→ D −→ D −→ D −→ M −→ EXAMPLE 3.5 : (
Unimodular contact structure ) With n = m = 3 and K = Q ( x , x , x ) letus introduce the 1-form ω = dx − x dx used as a geometric object and consider the first ordersystem of infinitesimal Lie equations from the equations L ( ξ ) ω = 0. One obtains the system usingjet notations: ξ − x ξ = 0 , ξ − x ξ − ξ = 0 , ξ − x ξ = 0We let the reader prove that these three PD equations are differentially independent and we obtainthe free resolution of M : 0 −→ D D −→ D −→ M −→ ←− N ←− D ad ( D ) ←− D ←− rk D ( M = rk D ( N ) = 3 − M and N are torsion modules with N = ext ( M ) ⇒ M = ext ( N ) = ext ( ext ( M )) and M is surely 1-pure. However, this sys-tem is not formally integrable, as it can be checked directly through crossed derivatives or bynoticing that L ( ξ ) dω = 0 with dω = dx ∧ dx and L ( ξ )( ω ∧ dω ) = 0 with ω ∧ dω = dx ∧ dx ∧ dx .Hence, we have to add the 3 first order equations: ξ + ξ , ξ = 0 , ξ = 0 ⇒ ξ = 0Exchanging x and x , we obtain the equivalent involutive system in δ -regular coordinates: ξ = 0 ξ = 0 ξ = 0 ξ + ξ = 0 ξ + x ξ − ξ = 0 ξ − x ξ = 0 1 2 31 2 31 2 31 2 • • • • The differential module M (2) over K [ d , d ] is defined by the three bottom equations. Setting now φ = ξ − x ξ , we deduce from the last bottom equation that ξ = − d φ and thus ξ = φ − x d φ .Finally, substituting in the equation before the last, we get ξ = d φ . We have thus obtainedan injective parametrization of M (2) which is therefore torsion-free and M is 2-pure in a coherentway. Substituting into the three upper equations, we obtain the desired relative parametrizationby adding the differential constraint d φ = 0. Coming back to the original coordinates, we obtainthe relative parametrization: φ − x d φ = ξ , − d φ = ξ , d φ = ξ with d φ = 0which is thus strikingly obtained from the previous contact parametrization by adding the onlydifferential constraint d φ = 0.
4) APPLICATIONS
Before studying applications to mathematical physics, we shall start with an example describingin an explicit way the Janet and Spencer sequences used thereafter, both with their link, namelythe relations existing between the dimensions of the respective Janet and Spencer bundles.
EXAMPLE 4.1 : When n = m = 2 , q = 2, ω is the Euclidean metric of X = R with Christoffelsymbols γ and metric density ˜ ω = ω/ p det ( ω ), we consider the two involutive systems of linearinfinitesimal Lie equations R ⊂ ˜ R ⊂ J ( T ) respectively defined by {L ( ξ ) ω = 0 , L ( ξ ) γ = 0 } and {L ( ξ )˜ ω = 0 , L ( ξ ) γ = 0 } . We have g = ˜ g = 0 and construct the following successive commutativeand exact diagrams followed by the corresponding dimensional diagrams that are used in order toconstruct effectively the respective Janet and Spencer differential sequences while comparing them.0 0 0 ց ↓ ↓ → S T ∗ ⊗ T → F ” → ↓ ↓ ց ↓ → R → J ( T ) → F → ↓ ↓ ↓ → R → J ( T ) → F ′ → ↓ ↓ ↓ ց ↓ ↓ → → → ↓ ↓ ց ↓ → → → → ↓ ↓ ↓ → → → → ↓ ↓ ↓ R = ρ ( R ) = J ( R ) ∩ J ( T ) ⊂ J ( J ( T )) and thus F ≃ J ( F ′ ) with F ” ≃ T ∗ ⊗ F ′ ≃ S T ∗ ⊗ T by counting the dimensions because we have surely F ⊂ J ( F ′ ) with g = 0. 0 0 0 ↓ ↓ ↓ → S T ∗ ⊗ T → T ∗ ⊗ F → F → ↓ ↓ ↓ k → R → J ( T ) → J ( F ) → F → ↓ ↓ ↓ ↓ → R → J ( T ) → F → ↓ ↓ ↓ ↓ ↓ ↓ → → → → ↓ ↓ ↓ k → → → → → k ↓ ↓ ↓ → → → → ↓ ↓ ↓ SP EN CER ↓ ↓ ↓ → Θ j q −→ C D −→ C D −→ C → ↓ ↓ ↓ → T j q −→ C ( T ) D −→ C ( T ) D −→ C ( T ) → k ↓ Φ ↓ Φ ↓ Φ −→ Θ → T D −→ F D −→ F D −→ F → ↓ ↓ ↓ JAN ET −→ ˜Θ j −→ D −→ D −→ −→ −→ Θ j −→ D −→ D −→ −→ SP EN CER ↓ ↓ ↓ −→ j −→ D −→ D −→ −→ HY BRID k ↓ Φ ↓ Φ ↓ Φ −→ Θ −→ D −→ D −→ D −→ −→ JAN ET −→ ˜Θ −→ D −→ D −→ D −→ −→
013 0 ↓ ↓ → T ∗ ⊗ ( ˜ C /C ) = ˜ C /C → ↓ ↓ ↓ → S T ∗ ⊗ T → T ∗ ⊗ F → F → k ↓ ↓ → S T ∗ ⊗ T → T ∗ ⊗ ˜ F → ˜ F → ↓ ↓ ↓ ↓ ↓ → → → ↓ ↓ ↓ → → → → k ↓ ↓ → → → → ↓ ↓ ↓ C r /C r ≃ ker ( F r → ˜ F r ) ≃ ∧ r T ∗ ⊗ ( ˜ C /C ) ⇒ dim ( ˜ C r ) − dim ( C r ) = dim ( F r ) − dim ( ˜ F r ) ⇒ dim ( C r ) + dim ( F r ) = dim ( ˜ C r ) + dim ( ˜ F r )In this new situation, we now notice that ˜ R ( ρ ( ˜ R ) = J ( ˜ R ) ∩ J ( T ) ⊂ J ( J ( T )) andthe induced morphism ˜ F → J ( ˜ F ′ ) is thus no longer a monomorphism though we still have anisomorphism ˜ F ” ≃ S T ∗ ⊗ T because ˜ g = 0 again. Finally, we may extend such a procedure tothe conformal group of space-time by considering the system of infinitesimal conformal transfor-mations of the Minkowski metric defined by the first order system ˆ R ⊂ J ( T ) in such a way thatwe have the strict inclusions R ⊂ ˜ R ⊂ ˆ R ⊂ J ( T ) with dim (ˆ g ) = n = 4. For this, we justneed to introduce the metric density ˆ ω = ω ( | det ( ω ) | ) − n and consider the system L ( ξ )ˆ ω = 0 ([38]):ˆ ω rj ∂ i ξ r + ˆ ω ir ∂ j ξ r − n ˆ ω ij ∂ r ξ r + ξ r ∂ r ˆ ω ij = 0 A) CONTROL THEORY : EXAMPLE 4.A.1 : (
OD control theory ) In classical control theory we have n = 1 and the onlyindependent variable is the time, simply denoted by x but we may choose any ground differentialfield like K = Q ( x ). In that case, we shall refer to ([22] or [25]) for the proof of the followingtechnical results that will be used in this case (Compare to [49]). Instead of the standard ”up-per dot” notation for derivative we shall identify the formal and the jet notations, setting thus d x y = dy = y x . With m = 2, let us consider the elementary Single Input/Single Output (SISO)second order system y xx − y x + a ( x ) y = 0 with a variable coefficient a ∈ K . The correspondingformally surjective operator is ∂ xx η − ∂ x η + a ( x ) η = ζ . Treating such a system by using classi-cal methods is not so easy when a is not constant as it cannot be possible to transform it to thestandard Kalman form. On the contrary, multiplying by a test function (or Lagrange multiplier) λ and integrating by parts, we obtain the adjoint system/operator: (cid:26) y −→ λ xx = µ y −→ λ x + aλ = µ • This system has a trivially involutive zero symbol but is not even formally integrable and we haveto consider : λ xx = µ λ x + aλ = µ ( ∂ x a − a ) λ = µ x − µ − aµ ••
14e have thus two possibilities: • We have a x − a = 0 and the adjoint system has the only zero solution, that is the adjointoperator is injective. In this case N = 0 and thus t ( M ) = ext ( N ) = 0 that is M is torsion-free.However, as n = 1 it follows that D = K [ d ] is a principal ideal ring which is therefore free and thusprojective ([Kunz,Rot]), that is M is torsion-free if and only if N = 0 and the system is controllable. • The Riccati equation a x − a = 0 is satisfied, for example if a = − /x and we get the CC µ x − µ − aµ = 0. Multiplying by a test function ξ and integrating by parts, we get the adjointoperator: (cid:26) µ −→ − ξ = η µ −→ − ξ x − aξ = η with only one first order generating CC, namely ∂ x η − η + aη = 0. It follows that N = 0 ⇒ ext ( N ) = 0 is a torsion module generated by the residue of z = y x − y + ay . We obtain indeeda torsion element as we can check at once that z x − az = 0 and wish good luck for control peopleto recover this result even on such an elementary example because the Kalman criterion is onlyworking for systems with constant coeficients (Compare [25] and [49]). EXAMPLE 4.A.2 : (
PD control theory ) With n = 2, let us consider the (trivially involutive)inhomogeneous single first order PD equations with two independent variables ( x , x ), two un-known functions ( η , η ) and a second member ζ : ∂ η − ∂ η + x η = ζ ⇔ D η = ζ The ring of differential operators is D = K [ d , d ] with K = Q ( x , x ). Multiplying on the left bya test function λ and integrating by parts, the corresponding adjoint operator is described by: (cid:26) η → − ∂ λ = µ η → ∂ λ + x λ = µ ⇔ ad ( D ) λ = µ Using crossed derivatives, this operator is injective because λ = ∂ µ + ∂ µ + x µ and we evenobtain a lift λ −→ µ −→ λ . Substituting, we get the two CC: (cid:26) ∂ µ + ∂ µ + x ∂ µ + 2 µ = ν ∂ µ + ∂ µ + 2 x ∂ µ + x ∂ µ + ( x ) µ − µ = ν • This system is involutive and the corresponding generating CC for the second member ( ν , ν ) is: ∂ ν − ∂ ν − x ν = 0Therefore ν is differentially dependent on ν but ν is also differentially dependent on ν . Multi-plying on the left by a test function θ and integrating by parts, the corresponding adjoint systemof PD equations is: (cid:26) ν → ∂ θ − x θ = ξ ν → − ∂ θ = ξ ⇔ ad ( D − ) θ = ξ Multiplying now the first equation by the test function ξ , the second equation by the test function ξ , adding and integrating by parts, we get the canonical parametrization D ξ = η : (cid:26) µ → ∂ ξ + ∂ ξ − x ∂ ξ − ξ = η µ → ∂ ξ − x ∂ ξ + ξ + ∂ ξ − x ∂ ξ + ( x ) ξ = η • of the initial system with zero second member. This system is involutive and the kernel of thisparametrization has differential rank equal to 1 because ξ or ξ can be given arbitrarily.Keeping now ξ = ξ while setting ξ = 0, we get the first second order minimal parametrization ξ → ( η , η ): (cid:26) ∂ ξ = η ∂ ξ − x ∂ ξ + ξ = η • ξ = 0while keeping ξ = ξ ′ , we get the second second order minimal parametrization ξ ′ → ( η , η ): (cid:26) ∂ ξ ′ − x ∂ ξ ′ + ( x ) ξ ′ = η ∂ ξ ′ − x ∂ ξ ′ − ξ ′ = η which is again easily seen to be involutive by exchanging x with x .With again a similar comment, setting now ξ = ∂ φ, ξ = − ∂ φ in the canonical parametrization,we obtain the third different second order minimal parametrization : (cid:26) x ∂ φ + 2 ∂ φ = η x ∂ φ − ( x ) ∂ φ + ∂ φ = η • We are now ready for understanding the meaning and usefulness of what we have called ” relativeparametrization ” in ([29]) by imposing the differential constraint ∂ ξ + ∂ ξ = 0 which is com-patible as we obtain indeed the new first order relative parametrization : ∂ ξ + ∂ ξ = 0 − x ∂ ξ − ξ = η − x ∂ ξ + ( x ) ξ + ξ = η • with 2 equations of class 2 (thus with class 2 full) and only 1 equtaion of class 1.In a different way, we may add the differential constraint ∂ ξ + ∂ ξ = 0 but we have to checkthat it is compatible with the previous parametrization. For this, we have to consider the followingsecond order system which is easily see to be involutive with 2 second order equations of (full)class 2, (only) 2 second order equations of class 1 and 1 equation of order 1: ∂ ξ + ∂ ξ = 0 ∂ ξ + ∂ ξ − x ∂ ξ − ξ = η ∂ ξ + ∂ ξ = 0 ∂ ξ − x ∂ ξ + ξ + ∂ ξ − x ∂ ξ + ( x ) ξ = η ∂ ξ + ∂ ξ = 0 1 21 21 • •• • The 4 generating CC only produce the desired system for ( η , η ) as we wished.We cannot impose the condition D − θ = ξ already found as it should give the identity 0 = η .It is however also important to notice that the strictly exact long exact sequence:0 −→ D D −→ D D −→ D D − −→ D −→ ζ −→ η −→ ζ , namely: ζ −→ ( − ∂ ζ + x ζ = η , − ∂ η = η ) −→ ∂ η − ∂ η + x η = ζ We have thus an isomorphism D ≃ D ⊕ M in the resolution 0 −→ D D −→ D p −→ M −→ r -extension modules ext r ( ) = 0 , ∀ r ≥ ∂ η − ∂ η + a ( x ) η whenever a ∈ K ( Hint : The controllability condition is now ∂ a = 0). The comparisonwith the previous OD case needs no comment. B) ELECTROMAGNETISM :Most physicists know the Maxwell equations in vacuum, eventually in dielectrics and magnets, butare largely unaware of the more delicate constitutive laws involved in field-matter couplings likepiezzoelectricity, photoelasticity or streaming birefringence. In particular they do not know thatthe phenomenological laws of these phenomena have been given ... by Maxwell ([37]). The situationis even more critical when they deal with invariance properties of Maxwell equations because of theprevious comments ([5]). Therefore, we shall first quickly recall what the use of adjoint operators16nd differential duality can bring when studying Maxwell equations as a first step before providingcomments on the so-called gauge condition brought by the danish physicist Ludwig Lorenz in 1867and not by Hendrik Lorentz with name associated with the Lorentz transformations.Theough it is quite useful in actual practice, the following approach to Maxwell equations cannotbe found in any textbook. Namely, avoiding any variational calculus based on given Minkowskiconstitutive laws
F ∼ F between field F and induction F for dielectric or magnets, let us usedifferential duality and define the first set M of Maxwell equations by d : ∧ T ∗ → ∧ T ∗ whilethe second set M will be defined by ad ( d ) : ∧ T ∗ ⊗ ∧ T → ∧ T ∗ ⊗ T with d : T ∗ → ∧ T ∗ , ina totally independent and intrinsic manner, using now contravariant tensor densities in place ofcovariant tensors . As we have already proved since a long time in ([20-22],[24],[37-38]), the keyresult is that these two sets of Maxwell equations are invariant by any diffeomorphism , contraryto what is generally believed ([5]). We recapitulate below this procedure in the form of a (locallyexact) differential sequence and its (locally exact) formal adjoint sequences where the left dottedarrow is the standard composition of operators: potential d −→ f ield → induction ad ( d ) −→ currentA −→ F → F −→ J which is responsible for EM waves, though it is equivalent to the composition: pseudopotential ad ( d ) −→ induction → f ield d −→ ∧ T ∗ The main difference is that we need to set J = 0 in the first approach because of M while weget automatically such a vanishing assumption in the second approach because of M , avoidingtherefore the Lorenz condition as in ([35], Remark 5.5). potential = ( A i ) d −→ ( ∂ i A j − ∂ j A i = F ij ) = f ield M −→ ( ∂ i F jk + ∂ j F ki + ∂ k F ij = 0) current = ( ∂ i F ij = J j ) M ←− ( F ij ) = induction ad ( d ) ←− pseudopotentialA FT ∗ d −→ ∧ T ∗ d = M −→ ∧ T ∗ ... ↓ ... ∧ T ∗ ⊗ T ad ( d )= M ←− ∧ T ∗ ⊗ ∧ T ad ( d ) ←− ∧ T ∗ ⊗ ∧ T l l l∧ T ∗ d ←− ∧ T ∗ d ←− T ∗ J F
Using symbolic notations with an euclidian metric instead of the Minkowski one because theyare both locally constant while using the constitutive law F = F for simplicity in vacuum while rais-ing or lowering the indices by means of the metric, we have the parametrization d i A j − d j A i = F ij and obtain by composition in the left upper square: d i ( d i A j − d j A i ) = d i d i A j − d j ( d i A i ) = J j ⇒ d j ( d i d i A j − d j d i A i ) = d j J j = 0with implicit summations on i and j . We may consider the composit homogeneous second ordersystem d ii A j − d ij A i = 0 which is automatically formally integrable and is easily seen (exercise) tobe involutive. The character α n is obtained by considering d nn A j − d jn A n for the equation giving J j . For J n we get d nn A n − d nn A n = 0 and thus α n = n − ( n −
1) = 1 a result showing thatthe corresponding differential module has rank 1 and there is thus only one CC, namely d j J j = 0with implicit summation on j . We prove that we may add the Lorenz condition d i A i = 0 tobring the rank to zero. Indeed, we have now the inhomogeneous system d i d i A j = J j and thedifferential constraint thus brought is compatible with the conservation of current. The corre-sponding homogeneous system obtained by adding the Lorenz constraint has second order symbolobtained by considering both d i d i A j = 0 and d ij A i = 0 or d ij A j = 0. We obtain therefore d nn A j = 0 , d nn A n = 0 showing that we have now α n = 0 and a torsion differential module. As amore important and effective result that does not seem to be known, we have:17 ROPOSITION 4.B.1 : When n = 4, the system: Ψ j ≡ d A j + ... + d A j = J j d Φ − Ψ ≡ d A + d A + d A − d A − d A − d A = −J d Φ ≡ d A + ... + d A = 0 d Φ ≡ d A + ... + d A = 0 d Φ ≡ d A + ... + d A = 0Φ ≡ d A + ... + d A = 0 1 2 3 41 2 3 • • • • • • •• • • • is involutive with four equations of class 4, two equations of class 3, one equation of class 2 andone equation of class 1. The 11 resulting CC only provide the conservation of current. Proof : Using the corresponding
Janet tabular on the let, one can check at once that the 4 CCbrought by the only first order equation Φ = 0 do not bring anything new, as they amount tocrossed derivatives, and that we are only left with the upper dots on the right side . However, for i = 1 , ,
3, we have d ( d i Φ) = d i ( d Φ − Ψ ) + d i Ψ and we are thus only left with a single CC,getting successively: d ( d Φ − Ψ ) ≡ d A + d A + d A − d A − d A − d A − d Ψ ≡ − d A − d A − d A − d A − d Ψ ≡ − d A − d A − d A − d A − d Ψ ≡ − d A − d A − d A − d A d ( d Φ) ≡ d A + d A + d A + d A d ( d Φ) ≡ d A + d A + d A + d A d ( d Φ) ≡ d A + d A + d A + d A Summing these 7 equations, we are left with the identity − ( d Φ + ... + d Φ) + d j Ψ j = d j J j = 0.It is important to notice that no other procedure can prove that we have an involutive symbolin δ -regular coordinates and this is the only way to compute effectively all the four characters(0 < < <
15) with 6 + 11 + 15 = 32 = (4 × − (4 + 4) for the dimension of the symbolof order 2, a result not evident at first sight. Accordingly, the so-called Lorenz gauge conditionis only a pure ” artifact ” amounting to a relative minimum parametrization with no importantphysical meaning as it can be avoided by using only the EM field F ([31],[34],[35]).Q.E.D.Such a new approach to a classical result is nevertheless bringing a totally unsatisfactory conse-quence. Using the well known correspondence between electromagnetism(EM) and elasticity (EL)used for all engineering computations with finite elements: EM potential ↔ EL displacement, EM f ield ↔ EL strain, EM induction ↔ EL stress where EL means elasticity, and instead of the left upper square in the diagram, ... we have toconsider the right upper square .We finally prove that the use of the linear and nonlinear Spencer operators drastically changes theprevious standard procedure in a way that could not even be imagined with classical methods.For such a purpose, we make a few comments on the implicit summation appearing in differentialduality. For example, we have, up to a divergence: X ,rk X k,r = X ,rk ( ∂ r ξ k − ξ kr ) = − ∂ r ( X ,rk ) ξ k − X ,rk ξ kr + ... In the conformal situation, we have ξ = ξ = ... = ξ nn = n ξ rr and obtain therefore, as factor of thefirs jets: X , ξ + X , ξ + ... + X ,nn ξ nn = ( X ,rr ) 1 n ξ rr = ( X ,rr ) ξ Going to the next order, we get as in ([26]), up to a divergence: X ,r ∂ r ξ = − ( ∂ r X ,r ) ξ + ... Cosserat equation forthe dilatation , namely the so-called virial equation that we provided in 2016 ([34], p 35) : ∂ r X ,r + X ,rr = 0generalizing the well known Cosserat equations for the rotations provided in 1909 ([6], p 137): ∂ r X ij,r + X i,j − X j,i = 0As for EM, substituting ∂ i ξ rrj − ∂ j ξ rri in the dual sum F i Cauchy stress operator essentially admits only one parametrization in dimension n = 2 which isminimum but the situation is quite different in dimension n = 3. Indeed, the parametrization foundby E. Beltrami in 1892 with 6 potentials ([33]) is not minimal as the kernel of the Beltrami operatorhas differential rank 3 while the two other parametrizations respectively found by J.C. Maxwellin 1870 and by G. Morera in 1892 are both minimal with only 3 potentials even though they arequite different because the first is cancelling 3 among the 6 potentials while the other is cancellingthe 3 others. In particular, we point out the technical fact that it is quite difficult to prove thatthe Morera parametrization is providing an involutive system. These three tricky examples areproving that the possibility to exhibit different parametrizations of the stress equations that wehave presented has surely nothing to do with the proper mathematical background of elasticitytheory as it provides an explicit application of double differential duality in differential homologicalalgebra. Also, the example presented in Section 3.A is proving that the existence of many different19inimal parametrizations has surely nothing to do with the mathematical foundations of controltheory. Similarly, we have just seen in the previous section that the so-called Lorenz conditionhas surely nothing to do with the mathematical foundations of EM. Such a comment will be nowextended in a natural manner to GR.With tandard notations, denoting by Ω ∈ S T ∗ a perturbation of the non-degenerate metric ω ,it is well known (See [30] and [35] for more details) that the linearization of the Ricci tensor R = ( R ij ) ∈ S T ∗ over the Minkowski metric, considered as a second order operator Ω → R , maybe written with four terms as:2 R ij = ω rs ( d ij Ω rs + d rs Ω ij − d ri Ω sj − d sj Ω ri ) = 2 R ji Multiplying by test functions ( λ ij ) ∈ ∧ T ∗ ⊗ S T and integrating by parts on space-time, we obtainthe following four terms describing the so-called gravitational waves equations :( ✷ λ rs + ω rs d ij λ ij − ω sj d ij λ ri − ω ri d ij λ sj )Ω rs = σ rs Ω rs where ✷ is the standard Dalembertian. Accordingly, we have: d r σ rs = ω ij d rij λ rs + ω rs d rij λ ij − ω sj d rij λ ri − ω ri d rij λ sj = 0The basic idea used in GR has been to simplify these equations by adding the differential con-straints d r λ rs = 0 in order to find only ✷ λ rs = σ rs , exactly like in the Lorenz condition for EM.Before going ahead, it is important to notice that when n = 2, the Lagrange multiplier λ is justthe Airy function φ and, using an integration by parts, we have the identity: φ ( d Ω + d Ω − d Ω ) = d φ Ω − d φ Ω + d Ω + div ( )providing the Airy parametrization of the Cauchy stress equations: σ = d φ, σ = σ = − d φ, σ = d φ where the Airy function has nothing to do with the perturbation of the metric .However, even if its clear that the constraints are compatible with the Cauchy equations, we dobelieve that the following result is not known as it does not contain any reference to the usual Einstein tensor E ij = R ij − ω ij tr ( R ) where tr ( R ) = ω rs R rs , which is therefore useless becauseit contains 6 terms instead of 4 terms only. PROPOSITION 4.C.1 : The system made by ✷ λ rs = σ rs and d r λ rs = 0 is a relative minimuminvolutive parametrization of the Cauchy equations describing the formal adjoint of the Killingoperator, that is Cauchy = ad ( Killing ) as operators. Proof : For each given s = 1 , , , exactly the system used for studyingthe Lorenz condition in Proposition 4.B.1. Accordingly, nothing has to be changed in the proof ofthis proposition and we get an involutive second order sysem with d r σ rs = 0 as only CC in placeof the conservation of current. Needless to say that this result has nothing to do with any conceptof gauge theory as it is sometimes claimed ([8],[30]). Q.E.D. 5) CONCLUSION :In 1916, F.S. Macaulay used a new localization technique for studying unmixed polynomial ideals.In 2013, we have generalized this procedure in order to study pure differential modules, obtain-ing therefore a relative parametrization in place of the absolute parametrization already knownfor torsion-free modules and equivalent to controllability in the study of OD or PD control sys-tems, a result showing that controllability does not depend on the choice of the control variables ,despite what engineers still believe. Meanwhile, we have pointed out the existence of minimumparametrizations obtained by adding in a convenient but generally not intrinsic way certain com-patible differential constraints on the potentials. We have proved that this is exactly the kind ofsituation met in control theory, in EM with the Lorenz condition and in GR with gravitationalwaves. However, the systematic use of adjoint operators and differential duality is proving that20he physical meaning of the potentials involved has absolutely nothing to do with the one usuallyadopted in these domains. Therefore, these results bring the need to revisit the mathematicalfoundations of Electromagnetism and Gravitation, thus of General Relativity and Gauge Theory,in particular Maxwell and Einstein equations, even if they seem apparently well established.