Differential Homological Algebra and General Relativity

In 1916, F.S. Macaulay developed specific localization techniques for dealing with “unmixed polynomial ideals” in commutative algebra, transform-ing them into what he called “inverse systems” of partial differential equations. In 1970, D.C. Spencer and coworkers studied the formal theory of such systems, using methods of homological algebra that were giving rise to “differential homological algebra”, replacing unmixed polynomial ideals by “pure differential modules”. The use of “differential 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 of arbitrary potential like functions. In 2012, we have been able to extend this result to arbitrary pure differential modules, introducing a “relative parametrization” where the potentials should satisfy compatible “differential constraints”. We recently noticed that General Relativity is just a way to parametrize the Cauchy stress equations by means of the formal adjoint of the Ricci operator in order to obtain a “minimum parametrization” by adding sufficiently many compatible differential constraints, exactly like the Lorenz condition in electromagnetism. In order to make this difficult paper rather self-contained, these unusual purely mathematical results are illustrated by many explicit examples, two of them dealing with contact transformations, and even strengthening the comments we recently provided on the mathematical foundations of General Relativity and Gauge Theory. They also bring additional doubts on the origin and existence of gravitational waves.


Introduction
The main purpose of this paper is to prove how apparently totally abstract mathematical tools, ranging among the most difficult ones existing in differential geometry and homological algebra today, can also become useful and enlighten many engineering or physical concepts (see the review Zbl 1079.93001 for the only application to control theory).
In the second section, we first sketch and then recall, with more specific references, the main (difficult) mathematical results on differential extension modules and differential double duality that are absolutely needed in order to understand the purity concept and, in particular, the so-called purity filtration of a differential module ([1] [2] [3] [4]). We also explain the unexpected link existing between involutivity and purity allowing to exhibit a relative parametrization of a pure differential module, even defined by a system of linear PD equations with coefficients in a non-constant differential field K. It is important 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.

Mathematical Tools
be the ring of differential operators with coefficients in a differential field K of characteristic zero, that is such that → → → → of right differential modules that can be transformed by a side-changing functor to an exact sequence of finitely generated left differential modules. This new presentation corresponds to the formal adjoint ( ) ad  of the linear differential operator  determined by the initial presentation but now with p unknowns and m equations, obtaining therefore a new finitely generated left differential module ad  , a result which is not evident at first sight (see [3] [13]). Using now a maximum free submodule equivalently, a maximum number of differentially linearly independent CC, and repeating this standard procedure while using the well known facts that the cokernel of this monomorphism is a torsion module and ). 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 classical control system when 1 n = ([3] [16]). This parametrization will be called an "absolute parametrization" as it only involves arbitrary "potential-like" functions (see [4] [8] [9] [15] [17] [18] [19] [20] for more details and examples, in particular the fact that the Einstein equations cannot be parametrized).
It is however essential to notice that such an approach is leading to a "vicious circle" because the only constructive way to check whether M is torsion-free or not is to use the differential double duality. For this reason, we briefly recall the five steps of the corresponding test allowing to know whether a given differential system or operator can be parametrized or not: • STEP 1: Write down the system in the form of a differential operator 1  .
If "no", we shall say that 1  cannot be parametrized.
The purpose of this paper is to extend such a result to a much more general situation, that is when M is not torsion-free, by using unexpected results first found by F.S. Macaulay in 1916 ([21]) through his study of "inverse systems" for "unmixed polynomial ideals".
where the codimension of Dm is n minus the dimension of the characteristic variety determined by m in the cor-responding system for one unknown, we may define the purity filtration as in cd M cd N r N M = = ∀ ⊂ and a torsion-free module is a 0-pure module.
Moreover, when ( ) K k cst K = = is a field of constants and 1 m = , a pure module is unmixed in the sense of Macaulay, that is defined by an ideal having an equidimensional primary decomposition. Example 2.1: As an elementary example with K k = =  , 1 m = , 2 n = , 2 p = , the differential module defined by 22 0 d y = , 12 0 d y = is not pure be- . We obtain therefore the purity filtration ( ) ( ) ( ) ( ) From Independently of the previous results, the following procedure, where one may have to change linearly the independent variables if necessary, is the heart towards the next effective definition of involution. It is intrinsic even though it must be checked in a particular coordinate system called δ -regular ([32] [33] [34]) and is quite simple for first order systems without zero order equations.
• Equations of class n: Solve the maximum number n q β of equations with respect to the jets of order q and class n. Then  α with d n ≤ when α is the smallest non-zero character in the case of an involutive symbol. Such a prolongation allows to compute in a unique way the principal ( pri ) jets from the parametric ( par ) other ones. This definition may also be applied to nonlinear systems as well.  can be given arbitrarily and may constitute the input variables in control theory, though it is not necessary to make such a choice. In this case, the intrinsic number 0 n q m α α β = = − > is called the n-character and is the system counterpart of the so-called "differential transcendence degree" in differential algebra and the "rank" in module theory. As we shall see in the next Section, the smallest non-zero character and the number of zero characters are intrinsic numbers that can most easily be known by bringing the system to involution and we have In the situation of the last remark, the following procedure will generalize for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the input do not appear ( [3], VI, Remark 1.14, p 802). For this, we just need to modify the Spencer form and we provide the procedure that must be followed in the case of a first order involutive system with no zero order equation, for example an involutive Spencer form.
• Look at the equations of class n solved with respect to 1 , , n n y y β  .
• Use integrations by parts like: equal to the number of zero characters, that is to the number of "full" classes in the Janet tabular of an involutive system.
with implicit summation on the multi-index, the highest value of µ with 0 a µ ≠ 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 ver, it is clear that D, as an algebra, is generated by to the concept of (explicit or formal) solutions. It is at this moment that we have to take into account the results of the previous section in order to understand that certain presentations will be much better than others, 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 filtration obtained from that of m D which is defined by the order of the jet coordinates q y in q D y . We have therefore the inductive limit Having in mind that K is a left D-module for the action → ∂ and that D is a bimodule over itself, we have only two possible constructions: DEFINITION 2.8: We define the system ( ) as the system of order q. We have the projective limit k k a f τµ µ = 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 definition has of course to do with the concept of a formal power series solution. However, for an arbitrary differential field K, the main novelty of this new approach is that such a definition has nothing to do with the concept of a formal power series solution (care) as illustrated in ( [39]). DEFINITION 2.9: We may define the right differential module 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: and thus recover exactly the Spencer operator though this is not evident at all. We also get ( ) and this intrinsic result can be most easily checked by using 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.   n-pure, its defining system is a finite dimensional vector space over K with a symbol of finite type, that is when 0 q g = is (trivially) involutive. Finally, when , we shall say that there is a "gap" in the purity filtration:   • STEP 1: Construct a free resolution of M, say: Suppress M in order to obtain the deleted sequence: in order to obtain the dual sequence heading backwards: The following nested chain of difficult propositions and theorems can be obtained, even in the non-commutative case, by combining the use of extension modules and bidualizing complexes in the framework of algebraic analysis. The main difficulty is to obtain first these results for the graded module provides right D-modules that can be transformed to left D-modules by means of the side changing functor and f m m M = ∀ ∈ , that is to say 0 f = because D is an integral domain. When 3 n = and the torsion-free module M is defined by the formally surjective div operator, the formal adjoint of div is grad − which defines a torsion module.
Also, when 1 n = as in classical control theory, a controllable system with coefficients in a differential field allows to define a torsion-free module M which is free in that case because a finitely generated module over a principal ideal do-  [2] and [3] [4] for more details). THEOREM 2.30: When M is r-pure, the characteristic ideal is thus unmixed, that is a finite intersection of prime ideals having the same codimension r and the characteristic set is equidimensional, that is the union of irreducible algebraic varieties having the same codimension r.
We shall now illustrate and apply this new procedure in the next two sections.

Motivating Examples
We shall discover that it is not evident to prove that it is an unmixed polynomial ideal and that the corresponding differential module is 1-pure.
The first result is provided by the existence of the primary decomposition obtained from the two existing factorizations ( [23]): Taking the respective radical ideals, we get the prime decomposition: The corresponding involutive system is: 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 ( ) 1 3 4 , , z y z z = while taking into account that 2 1 1 z y d y = = : , , y z z are torsion elements and we can eliminate ( ) 3 4 , z z in order to find the desired system that must be satisfied by y which is showing the inclu- The Janet tabular on the right allows at once to compute the characters 3 2 0 α = , Similarly, using certain parametric jet variables as new unknowns, we may set  1  4  2  3  3  4  3  3  3  3  1  3  2  3  3  4  2  2  1  2  2  1  2  4  3  1  1   class 3  0,  0,  0,  0  1 2 3  class 2 0, 0, 0, 0 1 2 class 1 0, 0 1 where we have separated the classes while using standard computer algebra notations this time instead of the jet notations used in the previous example. Contrary to what could be believed, this operator does not describe the Spencer sequence that could be obtained from the previous Janet sequence but we can use it exactly like a Janet sequence or exactly like a Spencer sequence. We obtain therefore a long strictly exact sequence of differential modules with only first order operators while replacing Dy by The differential module K d by the two PD equations of class 1 and is easily seen to be torsion-free with the two potentials ( ) 1 4 , z y z = .
Substituting into the PD equations of class 2 and 3, we obtain the generating differential constraints: From the Janet tabular we may construct at once the Janet sequence: Let us transform the initial second order involutive system for y into a first order involutive system for ( ) 1  2  3  4  5  1  2  3  4 , , , , z y z y z y z y z y = = = = = as follows:   if we take the residue or, equivalently, the residue of z does not belong to M. The differential module L defined by the above system is therefore 2-pure with a strict inclusion M L ⊂ and admits a free resolution with only 2 operators according to its Janet tabular.  3  3  2  3  2  3  2  1  2  1  3  2  3  3   2  1  1  3  2  3  1  3  2  3  2  2  1 The following injective absolute parametrization is well known and we let the reader find it by using differential double duality: J.-F. Pommaret  3  1  2  3  3  1  3 2  3  3  2  1 , , We obtain the Janet sequence with formally exact adjoint sequence: L ξ ω = . One obtains the system using jet notations: We let the reader prove that these three PD equations are differentially independent and we obtain the free resolution of M: x , we obtain the equivalent involutive system in δ -regular coordinates: The differential module which is therefore torsion-free and M is 2-pure in a coherent way. Substituting into the three upper equations, we obtain the desired relative parametrization by adding the differential constraint 3 0 d φ = . Coming back to the original coordinates, we obtain the relative parametrization: which is thus strikingly obtained from the previous contact parametrization by adding the only differential constraint 1 0 d φ = obtained by substituting it in the new system of Lie equations.

Applications
Before studying applications to mathematical physics, we shall start with an example describing in an explicit way the Janet and Spencer sequences used thereafter, both with their link, namely the relations existing between the dimensions of the respective Janet and Spencer bundles.
and construct the following successive commutative and exact diagrams followed by the corresponding dimensional diagrams that are used in order to construct effectively the respective Janet and Spencer differential sequences while comparing them.
In the present situation we notice that In this new situation, we now notice that  is thus no longer a monomorphism though we still have an isomorphism again. Finally, we may extend such a procedure to the conformal group of space-time by considering the system of infinitesimal conformal transformations of the Minkowski metric defined by the first order system ( ) 1 1 in such a way that we have the strict inclusions ( ) For this, we just need to introduce the metric density In that case, we shall refer to ( [5] or [14]) for the proof of the following technical results that will be used in this case (compare to [43]). Instead of the standard "upper dot" notation for derivative we shall identify the formal and the jet notations, setting thus This system is involutive and the corresponding generating CC for the second member ( ) 1 2 , ν ν is: ν is differentially dependent on 1 ν but 1 ν is also differentially dependent on 2 ν . Multiplying on the left by a test function θ and integrating by parts, the corresponding adjoint system of PD equations is: Multiplying now the first equation by the test function 1 ξ , the second equation by the test function 2 ξ , adding and integrating by parts, we get the canonical parametrization Dξ η = : ( ) 2  1  2  2  2  2  2  22  12  2   2  1  1  2  1  1  2  2  2  2  2  1  12  2  11  1   2  1 2 1 2 − ∂ + +∂ − ∂ + =   of the initial system with zero second member. This system is involutive and the kernel of this parametrization has differential rank equal to 1 because 1 ξ or 2 ξ can be given arbitrarily. x x which is again easily seen to be involutive by exchanging 1 x with 2 x .
With again a similar comment, setting now 1 We are now ready for understanding the meaning and usefulness of what we have called "relative parametrization" in ( [4]) by imposing the differential constraint In a different way, we may add the differential constraint The 4 generating CC only produce the desired system for ( ) 1 2 , η η as we wished.
We cannot impose the condition already found as it should give the identity 0 η = .
It is however also important to notice that the strictly exact long exact sequence: As an exercise, we finally invite the reader to study the situation met with the system ( ) Most physicists know the Maxwell equations in vacuum, eventually in dielectrics and magnets, but are largely unaware of the more delicate constitutive laws involved in field-matter couplings like piezzoelectricity, photoelasticity or streaming birefringence. In particular they do not know that the phenomenological laws of these phenomena have been given ... by Maxwell ( [7]). The situation is even more critical when they deal with invariance properties of Maxwell equations because of the previous comments ( [44]). Therefore, we shall first quickly recall what the use of adjoint operators and differential duality can bring when studying Maxwell equations as a first step before providing comments on the so-called gauge condition brought by the Danish physicist Ludwig Lorenz in 1867 and not by Hendrik Lorentz with name associated with the Lorentz transformations.
Though it is quite useful in actual practice, the following approach to Maxwell equations cannot be found in any textbook. Namely, avoiding any variational calculus based on given Minkowski constitutive laws F   between field F and induction  for dielectric or magnets, let us use differential duality and define the first set : , in a totally independent and intrinsic manner, using now contravariant tensor densities in place of covariant tensors. As T  T  T   T  T  T  T  T  T   T  T Using symbolic notations with an Euclidean metric instead of the Minkowski one because they are both locally constant while using the constitutive law but the differential module defined by the corresponding homogeneous system is not torsion-free. Proof: When , this second order system is formally integrable because it is homogeneous. However, even if we know a priori that necessarily it is not evident that such a condition is also sufficient, contrary to what is claimed in the literature. When 4 n = , using the Euclidean metric for simplicity, one can rewrite the system in the form: 3  3  3  4  3  2  1  3  44  33  22  11  3  4  3  2  1   2  2  2  3  4  3  2  1  2  44  33  22  11  2  4  3  2  1   1  1  1  3  4  3  2  1  1  44  33  22  11  1  4  3  2  1   3  2  1  4  4  4  4  34  24  14  33  22  11 1 2 3 4 1 2 3 4

d A d A d A d A d d A d A d A d A d A d A d A d A d d A d A d A d A d A d A d A d A d d A d A d A d A d A d A d A d A d A d A
Let us check that the second order symbol is involutive with three equations of class 4 and only one equation of class 3. Indeed, we have successively :   3  2  1  4  4  4  4  344  244  144  334  224  114  4   3  3  3  3  4  3  2  1  3  344  333  223  113  334  333  233  133  3   2  2  2  2  4  3  2  1  2  244  233  222  112  224  223  222  122  2   1  1  1  1  144 133 122 111    [17] for an explicit example).

d A d A d A d A d A d A d d A d A d A d A d A d A d A d A d d A d A d A d A d A d A d A d A d d A d A d A d A
We sum up all these results in the following tabular only depending on the tor essentially admits only one parametrization in dimension 2 n = which is minimum but the situation is quite different in dimension 3 n = . Indeed, the parametrization found by E. Beltrami in 1892 with 6 potentials ( [9]) is not minimal as the kernel of the Beltrami operator has differential rank 3 while the two other parametrizations respectively found by J.C. Maxwell in 1870 and by G.
Morera in 1892 are both minimal with only 3 potentials even though they are quite different because the first is cancelling 3 among the 6 potentials while the other is cancelling the 3 others. In particular, we point out the technical fact that it is quite difficult to prove that the Morera parametrization is providing an involutive system. These three tricky examples are proving that the possibility to exhibit different parametrizations of the stress equations that we have presented has surely nothing to do with the proper mathematical background of elasticity theory as it provides an explicit application of double differential duality in differential homological algebra. Also, the example presented in Section 3. A is proving that the existence of many different minimal parametrizations has surely nothing to do with the mathematical foundations of control theory. Similarly, we have just seen in the previous section that the so-called Lorenz condition has surely nothing to do with the mathematical foundations of EM. Such a comment will be now extended in a natural manner to GR.
With standard notations, denoting by * 2

S T Ω ∈
a perturbation of the non-degenerate metric ω , it is well known (see [8] [10] and [47]  , which is therefore useless because it contains 6 terms instead of 4 terms only, even though the corresponding operator is self-adjoint. Proof: For each given 1,2,3,4 s = the system under study is exactly the system used for studying the Lorenz condition in Proposition 4.B.1. Accordingly, nothing has to be changed in the proof of this proposition and we get an involutive second order system with 0 rs r d σ = as only CC in place of the conservation of current. Needless to say that this result has nothing to do with any concept of gauge theory as it is sometimes claimed ([8] [47]). Q.E.D.

Conclusion
In 1916, F.S. Macaulay used a new localization technique for studying unmixed polynomial ideals. In 2012, we have generalized this procedure in order to study pure differential modules, obtaining therefore a relative parametrization in place of the absolute parametrization already known for torsion-free modules and equivalent to controllability in the study of OD or PD control systems. Such a result is 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 minimum parametrizations obtained by adding, in a convenient but generally not intrinsic way, certain compatible differential constraints on the potentials. We have proved that this is exactly the kind of situation met in control theory, in EM with the Lorenz condition and in GR with gravitational waves. However, the systematic use of adjoint operators and differential duality is proving that the physical meaning of the potentials involved has absolutely nothing to do with the one usually adopted in these domains. Therefore, these results bring the need to revisit the mathematical foundations of Electromagnetism and Gravitation, thus of Gauge Theory and General Relativity, in particular Maxwell and Einstein equations, even if they seem apparently well established.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.