aa r X i v : . [ m a t h - ph ] A ug On Koszul-Tate resolutions and Sullivan models
Damjan Pištalo and Norbert Poncin
Abstract
We report on Koszul-Tate resolutions in Algebra, in Mathematical Physics, in Coho-mological Analysis of PDE-s, and in Homotopy Theory. Further, we define an abstractKoszul-Tate resolution in the frame of D -Geometry, i.e., geometry over differential oper-ators. We prove Comparison Theorems for these resolutions, thus providing a dictionarybetween the different fields. Eventually, we show that all these resolutions are of the new D -geometric type. MSC 2010 : 18G55, 16E45, 35A27, 32C38, 16S32
Keywords : Koszul-Tate resolution, relative Sullivan algebra, gauge theory, partial differentialequation, jet bundle, compatibility complex, model category, homotopy theory, D -module Contents DG D A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Koszul-Tate resolution implemented by a D -ideal . . . . . . . . . . . . . . . . . 21 D -Geometry 23 n Koszul-Tate resolutions D -geometric KTR . . . . . . . . . . . . . . . . . . . . . . . 256.2 Cofibrant replacement KTR seen as D -geometric KTR . . . . . . . . . . . . . . 266.3 Change of perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 KTR of a reducible theory seen as D -geometric KTR . . . . . . . . . . . . . . . 306.5 KTR of a reducible theory versus cofibrant replacement KTR . . . . . . . . . . 326.6 KTR of a reducible theory versus KTR in Cohomological Analysis . . . . . . . 336.7 KTR in Cohomological Analysis seen as D -geometric KTR . . . . . . . . . . . . 37 D -geometry 60 D -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638.4 Construction of non-split relative Sullivan D -algebras . . . . . . . . . . . . . . . 64 The present paper arose from our interest in a coordinate-free approach to the moduli spaceof solutions of a system of partial differential equations (
PDE -s) modulo symmetries and in n Koszul-Tate resolutions
PDE -s [Vin01] is a landmark in this field. It interprets the solution space as asmooth manifold inside an infinite jet space. Beilinson and Drinfeld [BD04] view this solutionspace as a D -scheme, where D denotes the ring of linear differential operators of a smoothscheme X .We are convinced that the best framework for the Batalin-Vilkovisky formalism ishomotopical algebraic D -geometry [BPP17a], a combination of D -Geometry and Homotopi-cal Algebraic Geometry [TV08]. This idea leads to derived D -stacks, i.e., sheaves from thecategory DG D A of differential graded commutative algebras over D to the category of simpli-cial sets. The definition of the sheaf condition uses an appropriate model structure on DG D A [BPP17b]. The corresponding cofibration-trivial-fibration factorization and cofibrant replace-ment functor provide a minimal relative Sullivan D -algebra, which turned out to be a goodcandidate for the Koszul-Tate resolution – the first step of the Batalin-Vilkovisky construction.In this paper, we report on a series of Koszul-Tate resolutions: on the Koszul resolution[CE48] of a regular sequence , the Koszul-Tate resolution [Tat57] of a quotient ring , the Koszul-Tate resolution in Gauge Field Theory [HT92] – in particular on the Koszul-Tate resolution ina regular first-order on-shell reducible gauge theory [BBH00] and the Koszul-Tate resolution in Cohomological Analysis of PDE -s [Ver02]. Each one of these resolutions builds on thechronologically preceding ones.Next we comment on relative Sullivan algebras. In [Qui69], Quillen introduces model cat-egories as a suitable framework for resolutions. A standard model structure on the category DG Q A of differential graded commutative algebras over the field Q of rational numbers, aswell as the small object argument, lead, for any morphism, to a relative Sullivan Q -algebrathat models this morphism [Hal83] – the relative Sullivan minimal model of the morphism.Although this relative Sullivan algebra is quasi-isomorphic to the target of the considered mor-phism and thus resolves this target differential graded algebra if it is concentrated in degreezero, and although it has the same structure as the Koszul-Tate resolution of a quotient ring,relative Sullivan minimal models and Koszul-Tate resolutions appeared independently in theirrespective fields. We observed (see above) this similarity in structure after having defined amodel structure on DG D A , in order to deal with the Batalin-Vilkovisky formalism. Sincethis projective model structure exists only if the underlying smooth scheme X (see above)is a smooth affine variety, our candidate-Koszul-Tate-resolution (see above), which we finallycalled the cofibrant replacement Koszul-Tate resolution [BPP17b], also exists only in this case.We noticed that its main structure can nevertheless be extended to the general situation ofan arbitrary smooth scheme, what leads then to a general abstract Koszul-Tate resolution,which we describe in this paper and which does always exist. Since we observed later onthat an equivalent structure is used in [BD04] under the name of semi-free differential graded D -algebra, we refer to the latter resolution as the D -geometric Koszul-Tate resolution.Beyond the surveys that we described in the two preceding paragraphs, we show in thepresent paper that all the Koszul-Tate resolutions that we reviewed are D -geometric Koszul- n Koszul-Tate resolutions D -geometry is the appropriate setting for the Batalin-Vilkovisky formalism.The comparisons of the various Koszul-Tate resolutions are a major difficulty in viewof the distinct languages used. Since we believe that such passages – between the Alge-bra and Physics worlds [Tat57], [HT92], [Bar10], the world of partial differential equations[Vin01], [Ver02], the world of Homotopy Theory [Qui69], [Sul77], [Hal83], and the world of D -Geometry [BD04] ([BPP17a], [BPP17b]) – are lacking in the literature, we give precise com-parison results for the Koszul-Tate resolutions that we considered, thus providing a kind ofdictionary between different fields.We assume that most readers are familiar with homotopy and model categories (if not, aconcise introduction can be found in the appendices of [BPP15a] and [BPP15b]), but we givea short introduction to regular first-order on-shell reducible field theories and provide in theappendix a smallest possible introduction to the jet bundle formalism in Field Theory andCohomological Analysis. Let Σ be an embedded p -dimensional submanifold of R n . This means that, for each x ∈ Σ ,there is an open neighborhood Ω ⊂ R n such that Σ ∩ Ω is described by a regular cartesianequation E ∈ C ∞ (Ω , R n − p ) . By ‘regular’ we mean that the equations E a ∈ C ∞ (Ω , R ) areindependent, i.e., that the rank ρ ( ∂ x E ) is equal to n − p , for all x ∈ Σ ∩ Ω . Assume forsimplicity that the first n − p columns of the Jacobian matrix are independent and use thedecomposition x = ( x ′ , x ′′ ) ∈ R n − p × R p . Then, locally, in the neighborhood of Σ , we have E = E ( x ′ , x ′′ ) ⇔ x ′ = x ′ ( E, x ′′ ) . It follows that, locally, in the new coordinates ( E, x ′′ ) , theequation of Σ is E = 0 .Adopt now the standpoint of Mathematical Physics and consider a submanifold Σ ⊂ R n that is globally described by the equations E a = 0 , for all a .One of the fundamental consequences of regularity is the structure of the ideal I (Σ) madeof those smooth functions C ∞ ( R n ) that vanish on Σ . It is clear that any linear combination F = P a F a E a , F a ∈ C ∞ ( R n ) , of the equations belongs to I (Σ) . Conversely, if F ∈ I (Σ) , weget, working in the coordinates ( E, x ′′ ) , F ( E, x ′′ ) = Z d t (cid:0) F ( tE, x ′′ ) (cid:1) d t = X a E a Z ( ∂ E a F ) ( tE, x ′′ ) d t =: X a F a E a . We are now prepared to recall the construction of the Koszul resolution of the functionalgebra C ∞ (Σ) of Σ : E a = 0 , ∀ a ∈ { , . . . , n − p } , (1)where the E a are the first coordinates of an appropriate coordinate system ( E, x ′′ ) of R n . n Koszul-Tate resolutions Definition 1.
The
Koszul complex of the regular surface (1) is the chain complex madeof the free Grassmann algebra
K = C ∞ ( R n ) ⊗ S [ φ a ∗ ] on n − p odd generators φ a ∗ – associated to the equations (1) – and of the Koszul differential δ K = X a E a ∂ φ a ∗ . (2) Remark 2.
Notice that the base ring for the tensor products ⊗ and S has not been specifiedand that these products are merely a sensible notation for graded commutative polynomialsin the generators with coefficients in C ∞ ( R n ) . Proposition 3.
The Koszul complex of Σ is a resolution of C ∞ (Σ) , i.e., the homology of (K , δ K ) is given by H (K) = C ∞ (Σ) and H k (K) = 0 , ∀ k > . (3) We refer to this resolution as the
Koszul resolution of C ∞ (Σ) . Indeed, in degree 0, the cycles are the functions in C ∞ ( R n ) and the boundaries are theelements of δ K { X b F b φ b ∗ } = { X a F a E a } = I (Σ) , so that H (K) = C ∞ (Σ) . The proof that the higher homology spaces vanish is technical andnot really instructive. It is based on the fact that the operator h = P a φ a ∗ ∂ E a is a homotopybetween the Euler vector field or number operator E a ∂ E a + φ a ∗ ∂ φ a ∗ and the zero chain map,so that any chain c reads c ( E, x ′′ , φ ∗ ) = c (0 , x ′′ ,
0) + Z d λλ (( δ K h + hδ K ) c )( λE, x ′′ , λφ ∗ ) . Let R be a commutative unital ring, M a module over R of finite rank r , and let d ∈ M ∗ := Hom R ( M, R ) be an R -linear map. Definition 4.
The
Koszul complex of the covector d is the graded R -module V R M endowed with the differential δ K given by the extension of d as a degree − derivation. Wedenote the Koszul chain complex of d by K [ d ] . More precisely, the Koszul differential is given by δ K ( m ∧ . . . ∧ m k ) = k X ℓ =1 ( − ℓ − ( dm ℓ ) m ∧ . . . b ℓ . . . ∧ m k . Assume now that M = R r is free with basis ( e a ) a . In this case, the linear map d is given by d = ( E , . . . , E r ) ( E a ∈ R ). It is well-known that the Koszul complex K [ E , . . . , E r ] coincideswith the tensor product K [ E ] ⊗ . . . ⊗ K [ E r ] of the Koszul complexes K [ E a ] , where E a is n Koszul-Tate resolutions R -linear map from Re a to R . Indeed, in both cases the underlying R -module is L rk =0 R ∁ kr ( ∁ kr is the binomial coefficient) and the differential is defined by δ K ( e a ∧ . . . ∧ e a k ) = k X ℓ =1 ( − ℓ − E a ℓ e a ∧ . . . b ℓ . . . ∧ e a k . The degree 0 Koszul homology module H ( K [ E , . . . , E r ]) is the quotient of the kernel R bythe image { δ K ( ρ ) = P a ρ a E a , ρ ∈ R r } , i.e., the quotient R/ ( E , . . . , E r ) (4)of the ring R by its ideal generated by the E a .The considered Koszul complex can of course be written K [ E , . . . , E r ] = R ⊗ S [ e , . . . , e r ] and δ K = X a E a ∂ e a , provided we view the e a as degree 1 generators: the definitions of the present subsection andthe just mentioned homology result coincide with those of the preceding subsection.As easily checked, the degree 1 homology module is (at least if R is a Q -algebra) thequotient of the cycles { ρ ∈ R r : P a ρ a E a = 0 } by the trivial cycles { ρ = Θ ( E , . . . , E r ) ˜ : Θ ∈ Sk( r, R ) } , where ‘tilde’ is the transpose and where Sk( r, R ) denotes the skew-symmetric r × r matriceswith entries in R .In the language of the preceding subsection this means that H ( K [ E , . . . , E r ]) is given bythe linear relations between the equations modulo the trivial relations.If all the ‘relations’ are trivial, as well as all the ‘higher relations’ pertaining to the higherhomology modules, the Koszul complex is a resolution of the quotient (4) of ‘on-shell functions’. Definition 5.
A sequence ( E , . . . , E r ) of elements of R is called regular , if E a is not a zerodivisor of R/ ( E , . . . , E a − ) , for all a ∈ { , . . . , r } , and if R/ ( E , . . . , E r ) = 0 . To illustrate the definition, we consider the case r = 2 . The existence of a relation ρ E + ρ E = 0 is equivalent to E [ ρ ] = 0 , with [ ρ ] ∈ R/ ( E ) . It follows from the regularity assumption that ρ = ρ E , so that E ( ρ + ρ E ) = 0 . Applying again the regularity, we obtain ρ = − ρ E and, finally, ρ ρ ! = − ρ ρ ! E E ! , so that the linear combination ρ E + ρ E = − ρ E E + ρ E E vanishes trivially. This factthat regularity implies that all relations are trivial, can be extended, first to higher r , andsecond to higher relations. Actually we have the following (well-known) mathematical variantof Proposition 3: n Koszul-Tate resolutions Proposition 6.
If the sequence ( E , . . . , E r ) of elements of R is regular, the Koszul complex K [ E , . . . , E r ] resolves the quotient ( ). More generally, if the ring R is local and the R -module M is a finitely generated, a sequence ( E , . . . , E r ) of elements of R is M -regular if and only ifthe Koszul complex K [ E , . . . , E r ] resolves the quotient M/ ( E , . . . , E r ) M .
Remark 7.
It follows that, if the sequence ( E , . . . , E r ) of elements of R is regular, then anyrelation P a ρ a E a = 0 is trivial , in the sense that ρ = Θ ( E , . . . , E r ) ˜ , with Θ ∈ Sk( r, R ) . In [Tat57], J. Tate starts from a Noetherian commutative unital ring R , defines the category DGRA of differential graded commutative unital R -algebras A as usual except that the R -module A in degree 0 is assumed to be just R · A , he calls such a differential graded R -algebra A free, if there exist homogeneous generators ( e , e , . . . ) such that A = R ⊗ S [ e , e , . . . ] andeach R -module A n of degree n > contains only a finite number of e i , and, finally, says that A ∈ DGRA is acyclic if H n ( A ) = 0 , for all n > (for most other authors acyclic means that onehas in addition H ( A ) = 0 ).The paper contains two main theorems. Theorem 8.
For any ideal I ⊂ R of any Noetherian commutative unital ring R , there existsa free resolution of R/I in DGRA .Sketch of Proof.
Note first that, for a commutative ring, Noetherian, i.e, the property thatany ascending chain of ideals stabilizes, is equivalent to the property that any ideal is finitelygenerated. Let now ( E , . . . , E r ) be the generators of the ideal I , set X = R ⊗ S [ e , . . . , e r ] , (5)with all generators e a in degree 1, and define the differential d on X by d = X a E a ∂ e a . (6)The homology module H ( X ) is the module R/I . The complex ( X , d ) is clearly the Koszulcomplex ( K [ E , . . . , E r ] , δ K ) of the sequence ( E , . . . , E r ) – see Subsection 2.2. However,since this sequence is here not assumed to be regular, the higher homology modules do notnecessarily vanish. If the module H ( X ) does not vanish, we choose F , . . . , F s ∈ ker d suchthat the homology classes [ F b ] generate H ( X ) , and we set X = R ⊗ S [ e , . . . , e r , f , . . . , f s ] , (7)with all generators f b in degree 2 and d defined by d = X a E a ∂ e a + X b F b ∂ f b . (8) n Koszul-Tate resolutions H ( X ) = H ( X ) = R/I . As for H ( X ) , note that ker d = ker d and that im d (resp., im d ) is made of the linear combinations of the type X b r b F b + X a ′ a ′′ r a ′ a ′′ ( E a ′ e a ′′ − E a ′′ e a ′ ) resp., X a ′ a ′′ r a ′ a ′′ ( E a ′ e a ′′ − E a ′′ e a ′ ) ! . Let now [ c ] ∈ H ( X ) . Since [ c ] ∈ H ( X ) , we have [ c ] = [ P b r b F b ] , so that c = X b r b F b + X a ′ a ′′ r a ′ a ′′ ( E a ′ e a ′′ − E a ′′ e a ′ ) and [ c ] = 0 . It suffices to iterate the procedure and to construct X k , such that H ( X k ) = R/I and H p ( X k ) = 0 , for all ≤ p ≤ k . Then, the inductive limit X of the direct system of freedifferential graded R -algebras X k is the resolving free differential graded R -algebra. (cid:3) Remark 9.
We refer to Tate’s extension X of the Koszul complex X associated to I =( E , . . . , E r ) as the Koszul-Tate resolution of R/ ( E , . . . , E r ) . Tate’s method allows to finda resolution, even if the sequence ( E , . . . , E r ) is not regular, i.e., if not all ‘relations’ and‘higher relations’ are trivial. The procedure starts from the Koszul complex (5)-(6), whosechain module is constructed from generators e a associated to the equations E a (see (5)) andwhose differential is the corresponding characteristic differential (6). Then, one associatesadditional generators f b to the non-trivial 1-cycles F b = P a F ab e a or non-trivial relations d F b = P a F ab E a = 0 (see (7)) and extends the differential accordingly (see (8)), in order tokill the non-trivial 1-cycles or relations. The procedure is now iterated, i.e., still new generatorsare added and new similar extensions of the differential are considered to kill the higher non-trivial relations. The Noetherian hypothesis allows to obtain finitely generated terms X p , p ≥ .The second theorem of [Tat57] is valid without the Noetherian property: Theorem 10.
Let I ⊂ R be an ideal of a commutative unital ring R . Assume that there exista commutative unital ring S , as well as ideals P ⊂ J ⊂ S , which are generated by regular S -sequences ( P , . . . , P s ) and ( J , . . . , J r ) , respectively, and which are such that S/P = R and J/P = I . Denote the classes in these quotients by ¯ • , set E a = ¯ J a , and set P b = P a s ab J a .Then the differential graded R -algebra Y = R ⊗ S [ e , . . . , e r , f , . . . , f s ] , (9) with all generators e a (resp., f b ) in degree 1 (resp., 2) and with differential d defined by d = X a E a ∂ e a + X b ( X a ¯ s ab e a ) ∂ f b , (10) is a free resolution of R/I . n Koszul-Tate resolutions E a of I as the equations of a shell in some ambient space, the ring R as thefunctions of this space, and the ideal I = ( E , . . . , E r ) as the functions that vanish on-shell.Moreover, a new concept of triviality will appear. Until now, a relation between the equations E a was considered as trivial, if the column of its coefficients could be obtained by applying askew-symmetric matrix to the column made of the E a (Remark 7). We will refer to a relationbetween the equations as weakly trivial , if all its coefficients vanish on-shell. It is clear thattrivial implies weakly trivial: Theorem 10.1 in [HT92] shows that in the context of Physics aweakly trivial relation is always trivial. Remark 11.
The assumptions of Theorem 10 imply that Y is a resolution of R/I , what inturn entails that there exist relations between the equations, which are not trivial and thusnot weakly trivial, i.e., at least one of their coefficients does not vanish on-shell. Further, theserelations are independent, in the sense that, if a linear combination between them vanisheson-shell, then all the coefficients vanish on-shell.The interest of this remark resides in the fact that, in the Mathematical Physics’ literature,this context – existence of non-weakly-trivial relations between the equations and only weaklytrivial relations between these relations – does not result as here from a Koszul-Tate resolution,which was constructed under certain assumptions, but this setting is essentially the startingpoint in the Physicists’ attempt to build a Koszul-Tate resolution of shell functions – which isitself the first step in the construction of the Batalin-Vilkovisky ( BV ) resolution (interestingideas and a survey on BV can be found in [Sta97]) of the functions of the shell modulo gaugesymmetries (see Appendix A, Section 7).Let us explain Remark 11.Since, for any b , the class ¯ • of the generator P b = P a s ab J a ∈ P vanishes, we have betweenthe equations the relation X a ¯ s ab E a = 0 . (11)The kernel ker d is made of the 1-chains c = P a r a e a that induce a relation P a r a E a = 0 ,and, as easily checked, the image im d is made of the boundaries dc of the 2-chains c , whichare (at least if R is a Q -algebra) of the type dc = X a X c ρ ac E c e a + X a X d ρ d ¯ s ad e a , (12)where ( ρ ac ) is a shew-symmetric r × r matrix and where ( ρ d ) is an s × matrix, both withentries in R . Remember that the new generators f d have been added to make homologicallynon-trivial 1-cycles trivial. The 1-cycle P a ¯ s ab e a (see (11)) is visibly homologically trivial dueto the adjunction of the generators f d , which are responsible for the second term in the RHS of Equation (12). In other words, it was not homologically trivial before the addition of these n Koszul-Tate resolutions ¯ s ab are not of the form P c ρ ac E c , with ( ρ ac ) skew-symmetric, or, still,the relation P a ¯ s ab E a = 0 is not trivial and so not weakly trivial (a (longer) direct proof of thisfact can be given). Remark 12.
Similarly, still other generators have to be added, if not all relations between thejust considered relations are homologically trivial. Since no additional generators were added,all relations between the relations (11) are homologically trivial, i.e., more precisely, for anyrelation P b σ b ¯ s ab = 0 , a ∈ { , . . . , r } , the corresponding 2-cycle P b σ b f b is homologicallytrivial.Concerning the statement that there exist only weakly trivial ‘relations’ between the re-lations (11), we now assume that P b σ b ¯ s ab ∈ I , for all a , and prove that σ b ∈ I , for all b .Since X b σ b ¯ s ab = X c z ca E c , we have the weakly trivial relation X a X c z ca E c E a = X b σ b X a ¯ s ab E a = 0 , which is, in view of [HT92, Theorem 10.1], trivial, what means that the matrix ( z ca ) can bechosen shew-symmetric. Hence, the sum c below is a 2-chain and dc = d ( 12 X a X c z ac e a e c + X b σ b f b ) = X a X c z ac E c e a + X a X b σ b ¯ s ab e a = − X a X b σ b ¯ s ab e a + X a X b σ b ¯ s ab e a = 0 . As Y is acyclic, the 2-cycle c is the boundary of a 3-chain c – made of terms in e a e c e g andof the term P a P b r ab e a f b . The terms of the first type induce in the boundary only terms in e a e c and the terms of the second type generate, in addition to terms in e a e c , the terms X b X a r ab E a f b . In view of freeness, we deduce that σ b = P a r ab E a ∈ I . Remark 13.
The differential in Theorem 10 is analogous to that of Theorem 8: in Theorem 8we dealt with the relations P a F ab E a = 0 , b ∈ { , . . . , s } , and added the term P b ( P a F ab e a ) ∂ f b to the differential, and in Theorem 10 the relations are P a ¯ s ab E a = 0 , b ∈ { , . . . , s } , and weadded the term P b ( P a ¯ s ab e a ) ∂ f b . Remark 14.
In the following, we use standard concepts, results, and notation of the theoryof
PDE -s in the jet bundle formalism [Vin01] (see Appendix A, Section 7). n Koszul-Tate resolutions In field theory , fields are sections φ ∈ Γ( π ) of a vector bundle π : E → X . Sincewe will consider gauge theories from the standpoint of Physics, we work systematically in atrivialization of E (fiber coordinates u = ( u , . . . , u r ) – we will sometimes write u a instead of u ) over a coordinate patch of X (coordinates x = ( x , . . . , x n ) – we may write x i instead of x ), or, we just assume that E = R n × R r . The dynamics of the considered field theory is givenby a functional S acting on compactly supported sections φ ∈ Γ( π ) , S [ φ ] = Z X L ( x i , u aα ) | j k − φ d x ∈ R , where j k − denotes the ( k − -jet and where the Lagrangian L is a function L ∈ F ( π k − ) ofthe ( k − -jet bundle of π (jet bundle coordinates ( x i , u aα ) ) such that L ( x i ,
0) = 0 (it sufficesto set e F ( x i , u aα ) := F ( x i , u aα ) − F ( x i , , for any function F ∈ F = F ( π ∞ ) of the infinite jetspace of π , to see that F = C ∞ ( X ) ⊕ e F , where the functions in e F vanish on the zero section).Equivalently, we may use the corresponding Euler-Lagrange equations δ u a L| j k φ = ( − D x ) α ∂ u aα L| j k φ = 0 , (13)where δ u a is the algebraicized Euler-Lagrange operator and D x i the total derivative withrespect to x i (see Appendix A, Section 7).The extended algebraicized Euler-Lagrange equations D αx δ u a L = 0 (14)define the constraint surface or shell Σ in the infinite jet space J ∞ ( π ) . The solutions φ of theoriginal Euler-Lagrange equations (13) are those compactly supported sections φ ∈ Γ( π ) thatsatisfy the condition ( j ∞ φ )( X ) ⊂ Σ (we mostly ignore local aspects). We denote by I (Σ) ⊂ F the ideal of those functions in F that vanish on-shell. If f ∈ I (Σ) , we write f ≈ .As for any system of linear equations, we may find linear relations between the consideredequations (14) (see ‘compatibility complex’ in Appendix A), i.e., relations of the type N aα D αx δ u a L ≡ , (15)with N aα ∈ F . It is easy to write such relations, if we use coefficients in I (Σ) . Indeed,for any functions n [ ab ] ∈ F (that are antisymmetric in a, b ), we have the linear relation n [ ab ] ∂ u b L ∂ u a L ≡ between the equations ∂ u a L = 0 . What we actually have in mind arenon-trivial linear relations, i.e., relations of the type (15), but with at least one coefficient N aα / ∈ I (Σ) . We refer to such relations as non-trivial Noether identities .A deep result [Noe18, Kos11], which is already present in elementary Mechanics, is the1:1 correspondence between, roughly speaking, ‘symmetries of the action’ (resp., ‘gauge sym-metries’) and conserved currents (resp., Noether identities). It motivates the definition of a gauge theory as a field theory (see above) with non-trivial Noether identities. n Koszul-Tate resolutions regularity conditions for first-order reducible gaugetheories can be formulated as follows: Assumption 1 . For any ℓ ∈ N , the LHS -s D αx δ u a L of the equations of Σ , up to order k + ℓ (i.e., since L ∈ F ( π k − ) , we consider derivatives D αx up to order ℓ ), can be separated into twopackages E a and E ∆ (of course, the ranges of ( α, a ) and of ( a , ∆) are the same) (we couldeven only ask that the D αx δ u a L and the ( E a , E ∆ ) be related by an invertible matrix, i.e., that D αx δ u a L = M α a a E a + M α ∆ a E ∆ , where the matrix M = ( M α a a , M α ∆ a ) , with row index ( α, a ) , is invertible; however, to simplify,we often ignore this matrix in the following, just as we ignore, as mentioned before, a numberof local aspects). Assumption 2 . The functions E a ∈ F ( π k + ℓ ) are independent. This is the actual reg-ularity condition (see Subsection 2.1). In other words, we assume that (locally – but weignore this restriction) the E a = E a ( x i , u aα ) can be chosen as the first fiber coordinates of anew coordinate system ( x i , E a , u ′′ aα ) in J k + ℓ ( π ) : ( x i , u ′ aα , u ′′ aα ) ↔ ( x i , E a , u ′′ aα ) . Assumption 3 . The functions E ∆ are linear consequences of the functions E a : E ∆ = F a ∆ E a , with F a ∆ ∈ F ( π k + ℓ ) . It follows that E ∆ = 0 , if E a = 0 : the E a (resp., E ∆ ) are the independent (resp., dependent) equations . Assumption 4 . The dependent equations E ∆ are total derivatives of a finite numberof dependent equations E δ = F b δ E b ( δ ∈ { , . . . , K } ), i.e., there is a finite number K of generators E δ by differentiation : E ∆ = D βx E δ . Assumption 5 . Note that the differences E ∆ − F a ∆ E a ≡ are non-trivial Noether identi-ties. We assume that, if E ∆ = D βx E δ , the derivative D βx of the Noether identity E δ − F b δ E b ≡ is the preceding Noether identity associated to E ∆ . If we write this requirement out, we findan invertibility condition for some matrix, which is called the first-order reducibility as-sumption ( IA ) of the considered gauge theory.The assumptions 1-5 are satisfied in many physically relevant examples, in particular inthe Klein-Gordon case and in electromagnetism.Consider now a regular first-order reducible gauge theory , i.e., a field theory, whichadmits non-trivial Noether identities (i.e., non-trivial gauge symmetries) and satisfies the as-sumptions 1-5. Proposition 15.
In a regular first-order reducible gauge theory, there exists an irreducible setof non-trivial Noether operators.
Indeed, consider the Noether identities E δ − F b δ E b ≡ and write them in the form R aδα D αx δ u a L ≡ δ ∈ { , . . . , K } ) . (16) n Koszul-Tate resolutions D αx δ u a L is given by the action of an invertible matrix M on thetuple made of the E a , E ∆ . We often assume for simplicity that this matrix is identity. Even ifwe take this matrix into account, we see easily that the Noether identities (16) are non-trivial .A compatibility operator (roughly, non-trivial linear total differential relations betweenthe equations – see Appendix A) can itself admit a compatibility operator (relations be-tween the relations). Similarly, Noether identities can be related by first-stage Noether iden-tities, which satisfy second-stage Noether identities... It is naturel to refer to the existence ofnon-trivial higher-stage Noether identities as the reducibility of the considered gauge theory.Since we deal in this text with a first-order reducible gauge theory, no non-trivial first-stageNoether identity should exist, i.e., any linear total differential operator ( S β . . . S Kβ ) D βx suchthat S δβ D βx ◦ R aδα D αx = 0 should be trivial, should vanish, or, still, all its coefficients shouldvanish. As mentioned, in the present approach to the Koszul-Tate resolution, ‘trivial’ (resp.,‘non-trivial’) means what has been called ‘weakly trivial’ (resp., ‘not weakly trivial’) in thepreceding subsection, i.e., it means that all the coefficients vanish (resp., at least one coefficientdoes not vanish) on Σ . Hence, we actually deal with first-order on-shell reducibility . Thismeans that S δβ D βx ◦ R aδα D αx ≈ must imply that S δβ ≈ ∀ δ ∈ { , . . . , K } ) . (17)It can be shown [Bar10] that this first-order on-shell reducibility condition really holds – inview of the above first-order reducibility assumption ( IA ).In view of (16) and (17), the linear total / horizontal differential operators R aδ = R aδα D αx arethe irreducible set of non-trivial Noether operators, which has been announced in Proposition15. Remark 16.
Observe that regularity does no longer mean, as in Subsection 2.1, that allthe equations E a are independent, but that some equations E a are independent. The otherequations E ∆ are dependent and they are generated via differentiation by a finite numberof dependent equations E δ . These dependent generators induce Noether identities, i.e., non-trivial relations between the equations. These relations are themselves on-shell independent,i.e., there are no non-trivial first-stage Noether identities. The latter situation is referred toas ‘ irreducibility ’ in [Bar10], whereas it is called ‘ first-order reducibility ’ in [HT92] . In this subsection, we report on a
Koszul-Tate resolution of the algebra C ∞ (Σ) = F /I (Σ) of functions of the shell Σ , in the case of a regular (on-shell) first-order reduciblegauge theory. We are thus in the situation (16) – (17), which has already been described inRemark 11, and we build a resolution that is similar to the one of Theorem 10. Since theirreducible non-trivial Noether operators R aδ , or, still, the Noether identities R aδα D αx δ u a L ≡ and their extensions D βx R aδα D αx δ u a L ≡ , (18) n Koszul-Tate resolutions P a ¯ s ab E a = 0 of Remark 11, we do, just asin Theorem 10, not only associate degree 1 generators φ α ∗ a to the equations D αx δ u a L = 0 of Σ ,but we assign further degree 2 generators C β ∗ δ to the (irreducible) relations (18) (no degree3 generators are needed). The candidate for a Koszul-Tate resolution of C ∞ (Σ) is then thechain complex, whose chains are the elements of the free Grassmann algebra KT =
F ⊗ S [ φ α ∗ a , C β ∗ δ ] (19)(see Equation (9)) and whose differential is defined by δ KT = D αx δ u a L ∂ φ α ∗ a + D βx ( R aδα φ α ∗ a ) ∂ C β ∗ δ (see Equation (10)).Just as the fiber coordinates u aα (in the following, we denote them by φ aα ) of the jet space of E are algebraizations of the derivatives ∂ αx φ a of the components of a section φ (field) ofthe vector bundle π : E → X , the generators φ α ∗ a and C β ∗ δ symbolize the total derivatives D αx φ ∗ a and D βx C ∗ δ of the components of sections φ ∗ and C ∗ (fermionic antifield and bosonicantifield) of the pullback bundles π ∗∞ F → J ∞ E and π ∗∞ F → J ∞ E of some vector bundles F → X and F → X . Hence, the φ α ∗ a and C β ∗ δ can be thought of as the fiber coordinates ofthe horizontal jet spaces of π ∗∞ F and π ∗∞ F , respectively.In the sequel, we thus put the antifields φ ∗ and C ∗ on an equal footing with the fields φ .More precisely, we extend the definition of the total derivatives by setting ¯ D x i = ∂ x i + φ aiα ∂ φ aα + φ iα ∗ a ∂ φ α ∗ a + C iβ ∗ δ ∂ C β ∗ δ , (20)so that they act on functions of the extended jet space , and we finally define the Koszul-Tatedifferential by δ KT = D αx δ u a L ∂ φ α ∗ a + ¯ D βx ( R aδα ¯ D αx φ ∗ a ) ∂ C β ∗ δ . (21)The homology of (KT , δ KT ) is actually concentrated in degree 0, where it coincides with C ∞ (Σ) . Indeed, the -cycles are the functions F and the -boundaries are the δ KT (cid:16)X F aα φ α ∗ a (cid:17) = X F aα D αx δ u a L ≈ . In view of the regularity assumption 2, the equations E a play the same role as in Subsection2.1, so that the ideal I (Σ) of those functions of F that vanish on Σ is made of the combinations P F a E a . Therefore, not only any -boundary belongs to I (Σ) , but, conversely, any functionof I (Σ) reads X F a E a = X F a ( M − ) a a α D αx δ u a L = δ KT (cid:16)X F a ( M − ) a a α φ α ∗ a (cid:17) and is therefore a -boundary. It follows that H (KT) = F /I (Σ) = C ∞ (Σ) . To show that thehomology vanishes in higher degrees, one needs the first-order reducibility assumption ( IA )[Bar10]. n Koszul-Tate resolutions irreducible set of non-trivial Noether operators R aδ is generating ,in the sense that any Noether operator ( N α . . . N rα ) D αx , i.e., any total differential operator suchthat N aα D αx δ u a L ≡ , uniquely reads N aα D αx = S δγ D γx ◦ R aδβ D βx + M [ a,bα,β ] D βx δ u b L D αx , (22)where the coefficients belong to F and satisfy S δγ and M [ a,bα,β ] = − M [ b,aβ,α ] . Hence, in a regularfirst-order reducible gauge theory, any Noether operator ( N . . . N r ) coincides on-shell witha composite ( S δ ◦ R δ . . . S δ ◦ R rδ ) of the irreducible set of Noether operators with some totaldifferential operators. This result is actually a quite straightforward corollary of the fact that H (KT) = 0 . The existence of non-trivial first- or higher-stage Noether identities is referred to as ‘re-ducibility’ in [Bar10] and as ‘higher order reducibility’ in [HT92]. The precise description ofhigher order reducibility and of the corresponding physical background [HT92] would lead farbeyond the scope of this text. Let us thus just mention that, from a mathematical standpoint,higher order reducibility is similar to Verbovetsky’s framework, which we describe in the nextsection, except that Verbovetsky considers regular off-shell reducibility.
Below we detail some ideas of [Ver02] adopting a slightly different standpoint.
Remark 17.
As in the preceding subsection, we will use – now without further reference– standard concepts, results, and notation of the cohomological analysis of
PDE -s [Vin01].For a summary of the needed knowledge, we refer the reader to Appendix A, Section 7. InSubsection 3.2, we also use some ideas of the D -geometric approach to PDE -s [BD04]. Somedetails can be found in Appendix B, Section 8, as well as in [BPP15a], [BPP15b], [BPP17a],[BPP17b].
In Subsection 2.4, we described – within the smooth geometric setting and for a fixedchoice of coordinates – the classical Koszul-Tate resolution used in Mathematical Physics. Thestarting point was made of field theoretic Euler-Lagrange equations, with Noether identitiesrelating them, and with precise regularity and first-order on-shell reducibility assumptions.In the present case, the context will be as well smooth geometry and, just as in MathematicalPhysics, we will work in local coordinates , although some aspects are developed in a coordinate-free manner. Our springboard will be any not necessarily linear
PDE , for which we formulate regularity and off-shell reducibility conditions . n Koszul-Tate resolutions π : E → X and ρ : F → X be smooth vector bundles of ranks r and r , respectively, over a smooth manifold of dimension n . Take a not necessarily linearformally integrable PDE Σ ⊂ J k ( π ) of order k , which is implemented by a not necessarilylinear differential operator D ∈ DO k ( π, ρ ) : Σ = ker ψ D , where ψ D ∈ FB ( J k ( π ) , F ) is therepresentative fiber bundle morphism of D . Recall (see Section 7) that DO k ( π, ρ ) ≃ FB ( J k ( π ) , F ) ≃ F k ( π, ρ ) := Γ( π ∗ k ( ρ )) ⊂ Γ( π ∗∞ ( ρ )) =: Γ( R ) =: R (in the sequel, we often denote a vector bundle over X by a Greek lower-case character,its pullback over J ∞ ( π ) by the corresponding Latin capital, and the module of sections ofthe latter by the same calligraphic letter). As usual, we denote by Σ ⊂ J ∞ ( π ) the infiniteprolongation of Σ ⊂ J k ( π ) : Σ = ker ψ ∞ D , where ψ ∞ D ∈ FB ( J ∞ ( π ) , J ∞ ( ρ )) is the infiniteprolongation of ψ D .We now describe the locality and regularity hypotheses used in [Ver02]. In fact, the authorassumes that Σ is contained in a small open subset U ⊂ J ∞ ( π ) , in which there exist coordinates ( x i , u aα ) . Also in the bundle ρ fiber coordinates are fixed (we will not need their denotation,only its index λ ∈ { , . . . , r } will be used). In addition to these triviality conditions, heformulates a regularity requirement for Σ . Just as for the classical Koszul-Tate resolution ofMathematical Physics, it is assumed that some equations of Σ can be chosen as first or lastcoordinates of a new system (of course, the equations of Σ read in the considered trivializations D αx ψ λD = 0 , for all α ∈ N n and λ ∈ { , . . . , r } ). More precisely, the neighborhood U of Σ isassumed to be a trivial bundle over Σ , in the sense that there is an isomorphism Φ : U → Σ × V ,where V is a star-shaped neighborhood of 0 in R ∞ , such that the coordinates v = ( v , v , . . . ) in V are precisely certain equations of Σ (not necessarily all of them): for any a ∈ N , thereis an α a ∈ N n and a λ a ∈ { , . . . , r } , such that v a = D α a x ψ λ a D . This means that the fibercoordinates v ( κ ) of a point κ ∈ Σ , which are obtained by projecting Φ( κ ) on the second factor V , vanish. In addition, the projection of Φ( κ ) , κ ∈ Σ , on the first factor Σ , is simply κ .Although in the following we systematically consider the open subset U ⊂ J ∞ ( π ) insteadof the whole jet space, we do not always insist on this restriction (and even write for simplicitysometimes J ∞ ( π ) instead of U ).The latter regularity condition has the same fundamental consequence as in Subsections2.1 and 2.4.2: if a function F ∈ F vanishes on Σ , it is a finite sum of the type F = X F α a ,λ a D α a x ψ λ a D , with F α a ,λ a ∈ F . In other words, a function F ∈ F belongs to the ideal I (Σ) if and only if itreads F = Ψ( ψ D ) , for some Ψ ∈ C Diff( R , F ) . In Subsection 2.4 , we assumed first-order on-shell reducibility, i.e., we assumed that thereare no on-shell first stage Noether identities. More precisely, there does exist a generatingirreducible set of Noether operators R aδ = R aδα D αx , or, still, a horizontal linear differential n Koszul-Tate resolutions ∆ = R . . . R r ... ...R K . . . R rK , i.e., an operator ∆ ∈ C Diff( π ∗∞ ( ρ ) , π ∗∞ ( ρ )) (in the considered special case of Subsec-tion 2.4, the bundle ρ coincides with the bundle π ). In this new notation, the relations R aδα D αx δ u a L ≡ , for all δ ∈ { , . . . , K } , read ∆ ( δ u • L ) ≡ . Note that the LHS of the al-gebraicized Euler-Lagrange equations δ u • L = 0 is the representative morphism ψ D of a notnecessarily linear differential operator D ∈ DO( π, ρ ) . The universal linearization of the lat-ter is a horizontal linear differential operator ℓ D ∈ C Diff( π ∗∞ ( π ) , π ∗∞ ( ρ )) . When linearizingthe identity ∆ ( ψ D ) ≡ , we get ∆ ◦ ℓ D = 0 . Since ∆ is generating, it does not vanishand, for any operator ∇ ∈ C Diff( π ∗∞ ( ρ ) , π ∗∞ ( ρ ′ )) , such that ∇ ( ψ D ) ≡ , there is an opera-tor (cid:3) ∈ C Diff( π ∗∞ ( ρ ) , π ∗∞ ( ρ ′ )) , such that ∇ ≈ (cid:3) ◦ ∆ , see Equation (22). Hence, roughlyspeaking, the restriction ∆ | Σ is an on-shell compatibility operator for ℓ D | Σ , and the men-tioned first-order on-shell reducibility means that there is no on-shell compatibility operatorfor ∆ | Σ , see Equation (17). We now come back to the context of [Ver02] . The restricted linearization ℓ D | Σ of theconsidered operator D admits a compatibility operator ∆ Σ ∈ C Diff( R | Σ , R | Σ ) . One of thefirst results in [Ver02] states that ∆ Σ can be extended to an operator ∆ ∈ C Diff( R , R ) , suchthat ∆ ( ψ D ) = 0 . Just as any other horizontal linear differential operator, the extension ∆ admits a formally exact compatibility complex. However, the latter is a priori neither finite,nor are its F -modules R i modules of sections of vector bundles of finite rank. One of the mainassumptions of [Ver02] is that there exists a finite formally exact compatibility complex −→ R −→ R −→ . . . ∆ k − −→ R k − −→ , (23)whose F -modules R i are all modules R i = Γ( R i ) = Γ( π ∗∞ ( ρ i )) , where the ρ i : F i → X are rank r i smooth vector bundles, and whose arrows are horizontal operators ∆ i ∈ C Diff( R i , R i +1 ) .This hypothesis is of course an off-shell reducibility condition . Formal exactness of (23) implies in particular that, when applying the horizontal infinitejet functor ¯ J ∞ to the complex (23), we get an exact sequence of F -modules: −→ ¯ J ∞ ( R ) ¯ ψ ∞ ∆1 −→ ¯ J ∞ ( R ) ¯ ψ ∞ ∆2 −→ . . . ¯ ψ ∞ ∆ k − −→ ¯ J ∞ ( R k − ) −→ . (24)Next we use the left exact contravariant Hom functor Hom F ( − , F ) , what leads to the exactsequence Hom F ( ¯ J ∞ ( R ) , F ) −◦ ¯ ψ ∞ ∆1 ←− Hom F ( ¯ J ∞ ( R ) , F ) −◦ ¯ ψ ∞ ∆2 ←− . . . −◦ ¯ ψ ∞ ∆ k − ←− Hom F ( ¯ J ∞ ( R k − ) , F ) ←− (25) n Koszul-Tate resolutions F -modules. The identification of representative morphisms with the corresponding differ-ential operators finally gives the exact sequence C Diff( R , F ) −◦ ∆ ←− C Diff( R , F ) −◦ ∆ ←− . . . −◦ ∆ k − ←− C Diff( R k − , F ) ←− (26)[Vin01, Section 5.5.5]. The completion −→ C Diff( R k − , F ) −◦ ∆ k − −→ C Diff( R k − , F ) −◦ ∆ k − −→ . . . −◦ ∆ −→ C Diff( R , F ) − ( ψ D ) −→ F −→ (27)of the latter sequence by − ( ψ D ) is a complex of F -modules for the natural grading given bythe subscripts of the R i . This complex, which is exact in all spots, except, maybe, in degrees0 and 1, is actually made of F [ D ] -modules (see Section 8). Indeed, in view of Equation (104),we have F [ D ] := F ⊗ D ≃ CD ( J ∞ ( π )) := C Diff( F , F ) , so that the F [ D ] -action is given by left composition (except for F ). Hence, the arrows of thiscomplex are F [ D ] -linear maps and the complex itself is a differential graded F [ D ] -module ( C Diff( R • , F ) , δ KT ) ∈ DG F [ D ] M , where δ KT is the direct sum of the maps in (27). The graded symmetric tensor algebra functor S F sends this module to the free differential graded F [ D ] -algebra ( KT , δ KT ) := ( S F C Diff( R • , F ) , δ KT ) ∈ DG F [ D ] A , (28)whose differential is a degree − graded derivation of the graded symmetric tensor product.The latter complex is the Koszul-Tate complex , in the sense of [Ver02], associated to theconsidered partial differential equation.The homology space H ( KT ) coincides with C ∞ (Σ) (in view of the above-mentioned fun-damental consequence of the regularity condition, the standard argument goes through) andthe higher homology spaces vanish (as suggested by the above sequences). To prove thisstatement, it suffices to show that the Koszul-Tate complex (28) coincides – as claimed –with the Koszul-Tate complex defined in [Ver02] and to use the corresponding result therein.The algebra of Koszul-Tate chains is defined in [Ver02] as the graded polynomial functionalgebra P ol( ¯ J ∞ ( R • )) . As usual, the polynomial functions P ol( ¯ J ∞ ( R • )) are the smoothfunctions F ( ¯ J ∞ ( R • )) that are polynomial along the fibers of the considered bundle – here ¯ J ∞ ( R • ) → J ∞ ( π ) . Just as the polynomial functions of a vector bundle G → X are defined by P ol( G ) := Γ( S G ∗ ) ≃ S C ∞ ( X ) Γ( G ∗ ) = S C ∞ ( X ) Hom C ∞ ( X ) (Γ( G ) , C ∞ ( X )) , the polynomial functions considered here are defined by P ol( ¯ J ∞ ( R • )) := S F Hom F ( ¯ J ∞ ( R • ) , F ) ≃ S F C Diff( R • , F ) . Hence, the Koszul-Tate chains of [Ver02] and those defined above do coincide. Moreover,the Koszul-Tate differential is defined in [Ver02] as an odd evolutionary vector field δ of n Koszul-Tate resolutions ¯ J ∞ ( R • ) . Such a graded derivation, when restricted as here to P ol( ¯ J ∞ ( R • )) , is completelydefined by its values on the polynomial functions that are linear along the fibers, i.e., on Hom F ( ¯ J ∞ ( R • ) , F ) ≃ C Diff( R • , F ) – and by its values on F . But on ∇ i ∈ C Diff( R i , F ) (resp., F ∈ F ), this evolutionary field is given by δ ( ∇ i ) = ∇ i ◦ ∆ i − , if i ≥ , and by δ ( ∇ ) = ∇ ( ψ D ) (resp., δ ( F ) = 0 ) [Ver02, Proposition 5]. Hence, the odd derivations δ and δ KT coincide, the Koszul-Tate complexes ( P ol( ¯ J ∞ ( R • )) , δ ) and ( KT , δ KT ) coincide, and so dotheir homologies. Remark 18.
In this section, we use the model structure of the category DG D A of differentialnon-negatively graded commutative unital algebras over the ring D of differential operators.We aim at providing an, as far as possible, self-contained exposition. For further detailson definitions, results, on D -modules, sheaves, model categories . . . , the reader may consultAppendix B, Section 8, as well as [BPP15a], [BPP15b], and [BPP17b], and in particular theappendices therein. Note that, whereas the frame for the preceding sections was algebra orsmooth geometry, the context of the mentioned papers and this section is algebraic geometry.We will work over a smooth scheme , since for an arbitrary, maybe singular, scheme X , thenotion of left D X -module is meaningless [BD04, Remark p. 56]. DG D A Let X be a smooth scheme and let O X (resp., D X ) be the sheaf of rings of functions(resp., differential operators) of X . Denote by qcCAlg ( O X ) (resp., qcCAlg ( D X ) ) the categoryof commutative unital O X -algebras (resp., commutative unital D X -algebras, i.e., commutativeunital O X -algebras, whose O X -module structure can be extended to a D X -module structure,such that vector fields θ ∈ D X act as derivations on the product) that are quasi-coherent as O X -modules. We will refer to the objects of this category as O X -algebras (resp., D X -algebras) (this convention differs from the one adopted in [BPP15a], [BPP15b], and[BPP17b]). The forgetful functor has a left adjoint [BD04] J ∞ : qcCAlg ( O X ) → qcCAlg ( D X ) : For , called the jet functor (see Appendix B, Subsection 8.2). Proposition 19.
Let π : E → X be an algebraic vector bundle of finite rank over a smoothscheme X and denote by O E the structure sheaf of the scheme E . If π ∗ stands for the directimage by π, we have O EX := π ∗ O E ∈ qcCAlg ( O X ) and thus J ∞ ( O EX ) ∈ qcCAlg ( D X ) . The D X -algebra J ∞ ( O EX ) (or its total section D X ( X ) -algebra J ∞ ( O EX )( X ) ) is the D -geometric counterpart of the function algebra O ( J ∞ E ) = F ( π ∞ ) = F of the infinite jet spaceof a smooth vector bundle π : E → X . Note that we prefer in this section the notation J ℓ E to the notation J ℓ ( π ) ( ≤ ℓ ≤ ∞ ). Proposition 19 is rather natural. A proof can be found inAppendix B, Subsection 8.1. n Koszul-Tate resolutions Remark 20.
In [BPP15a] and [BPP15b], as well as in [BPP17b], we proved in particularthat the category DG D A of differential non-negatively graded commutative unital algebras over D = D X ( X ) admits a cofibrantly generated model structure, if X is a smooth affine variety .This theorem results from the transfer of the model structure on the category DG D M of differ-ential non-negatively graded D -modules to the category DG D A . Actually, the categories underinvestigation are the category DG + qcMod ( D X ) of sheaves of differential non-negatively graded O X -quasi-coherent D X -modules and the category DG + qcCAlg ( D X ) of sheaves of differentialnon-negatively graded O X -quasi-coherent commutative unital D X -algebras, i.e., of commuta-tive monoids in the symmetric monoidal category DG + qcMod ( D X ) . The restriction to a smoothaffine variety (both assumptions are necessary) allows to show that the total section functoryields an equivalence of categories Γ( X, − ) : DG + qcCAlg ( D X ) ⇄ DG D A , (29)and similarly for DG + qcMod ( D X ) and DG D M . These equivalences allow in turn to avoid theproblem of the non-existence of a projective model structure on DG + qcMod ( D X ) for an arbitrarysmooth scheme [Gil06] and so the problem of the non-existence of a transferred structure on DG + qcCAlg ( D X ) .Before we describe the model structure of DG D A , we recall the Definition 21 ([BPP17b]) . A relative Sullivan D -algebra ( RS D A ) is a DG D A -morphism ( standard definition ) ( A, d A ) → ( A ⊗ S V, d )( the tensor product functor ⊗ and the graded symmetric tensor algebra functor S are takenover the ring O = O X ( X ) and the differential d is usually not the standard differential on atensor product ) that sends a ∈ A to a ⊗ O ∈ A ⊗ S V . Here V is a free non-negatively graded D -module V = M α ∈ J D · v α , which admits a homogeneous basis ( v α ) α ∈ J that is indexed by a well-ordered set J , and is suchthat dv α = d (1 A ⊗ v α ) ∈ A ⊗ S V <α , (30) for all α ∈ J . In the last requirement, we set V <α := L β<α D · v β . We refer to Property (30)by saying that d is lowering .A RS D A with the property α ≤ β ⇒ deg v α ≤ deg v β (31) ( resp., with the property d = d A ⊗ id + id ⊗ d S , where d S is a differential on S V ( in particularthe differential d S = 0 ) ; over ( A, d A ) = ( O ,
0) ) is called a minimal RS D A ( resp., a split RS D A ; a Sullivan D -algebra ( S D A ) ) and it is often simply denoted by ( A ⊗ S V, d ) ( resp., ( A ⊗ S V, d ) ; ( S V, d ) ) . n Koszul-Tate resolutions D -algebra is similar to the notion of relative Sullivan Q -algebra, which originates from Rational Homotopy Theory. Theorem 22.
The category DG D A of differential non-negatively graded commutative unitalalgebras over the ring D = D X ( X ) of total sections of the sheaf D X of differential operatorsof a smooth affine variety X , is a finitely ( and thus a cofibrantly ) generated model category ( in the sense of [GS06] and in the sense of [Hov07] ) . The weak equivalences are the DG D A -morphisms that induce an isomorphism in homology, the fibrations are the DG D A -morphismsthat are surjective in all positive degrees p > , and the cofibrations are exactly the retracts ofthe relative Sullivan D -algebras. Further, we describe in [BPP15a], [BPP15b], and [BPP17b] explicit functorial cofibration-fibration factorizations, as well as an explicit functorial cofibrant replacement functor. Thesedescriptions are too long to be recalled here.When remembering that the coproduct in DG D A is the tensor product, we get from [Hir05]: Proposition 23.
For any differential graded D -algebra A , the coslice category A ↓ DG D A carries a cofibrantly generated model structure given by the adjoint pair L ⊗ : DG D A ⇄ A ↓ DG D A : For , in the sense that its distinguished morphism classes are defined by For and itsgenerating cofibrations and generating trivial cofibrations are given by the functor L ⊗ , whichsends B in DG D A to A → A ⊗ B in A ↓ DG D A . D -ideal A partial differential equation (see Appendix A, Section 7) of order k acting on the sections φ of a smooth vector bundle π : E → X is a smooth fiber subbundle Σ ⊂ J k E , and (at leastif Σ is formally integrable) its infinite prolongation Σ ⊂ J ∞ E is a smooth manifold. If Σ is implemented by a differential operator D with representative morphism ψ , we have Σ =ker ψ and Σ = ker ψ ∞ , where ψ ∞ is the representative morphism of the infinite prolongation j ∞ ◦ D of D . In coordinates: the equation of Σ is ψ ( x i , u aα ) = 0 and the equation of Σ is ( D βx ψ )( x i , u aα ) = 0 , ∀ β . These equations are the algebraizations of the PDE -s ψ ( x i , ∂ αx φ a ) = 0 and d βx ( ψ ( x i , ∂ αx φ a )) = 0 , ∀ β . Since the latter differential equations have the same solutions, we can focus on Σ instead of Σ .Hence, a PDE Σ can be thought of as a manifold Σ , or, in view of the space-algebra duality,as the function algebra C ∞ (Σ) , which is (see above) the quotient of the algebra O ( J ∞ E ) bythe ideal I of all functions of O ( J ∞ E ) that vanish on Σ . A PDE acting on the sections of E can thus finally be interpreted as an ideal I ⊂ O ( J ∞ E ) . It follows that, in our present D -geometric context, where we considered an algebraic vector bundle π : E → X over a smoothaffine variety X , we think about a PDE acting on the sections of E , as a D -ideal (i.e., an O -ideal and a D -submodule) I ⊂ J , where J := J ∞ ( O EX )( X ) = Γ( X, J ∞ ( O EX )) ∈ D A n Koszul-Tate resolutions Q := Q ( π, I ) := J / I ∈ D A as the D -algebra of thecorresponding shell functions. Our goal is to resolve this D -algebra.The fundamental concepts of the jet bundle formalism are the Cartan distribution andthe Cartan connection, or, still, horizontal linear differential operators C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) between pullback bundles π ∗∞ ( η i ) : π ∗∞ F i → J ∞ E of smooth vector bundles η i : F i → X .Hence, jets lead to a systematic base change X J ∞ E . The remark is essential, in the sensethat both, the classical Koszul-Tate resolution of Mathematical Physics (constructed above inthe context of a regular first-order on-shell reducible gauge theory) and Verbovetsky’s Koszul-Tate resolution (induced by the compatibility complex of the linearization of a differentialequation), use the jet formalism to resolve shell functions, and thus enclose the base change • → X • → J ∞ E . This means that, in the dual function algebra setting, or, in the presentsituation, in the dual D -algebra setting, we pass from DG D A , i.e., from the coslice category O ( X ) ↓ DG D A ( O ( X ) := O = O X ( X ) is the base ring for the tensor product in DG D A and ( O , is the initial object in DG D A ) to the coslice category O ( J ∞ E ) ↓ DG D A .A first candidate for a resolution of Q = J / I ∈ D A is of course the cofibrant replace-ment of Q in DG D A given by the functorial ‘Cofibration – Trivial Fibration’ factorization of[BPP17b, Theorem 28], when applied to the unique DG D A -morphism O → Q . Indeed, thisdecomposition implements a functorial cofibrant replacement functor Q ([BPP17b, Theorem34]) with value Q ( Q ) = S V described in [BPP17b, Theorem 28]: O S V ∼ ։ Q , where (resp., ։ , ∼ → ) denotes a cofibration (resp., a fibration, a weak equivalence (here anisomorphism in homology)). Since Q is concentrated in degree 0 and has 0 differential, it isclear that H k ( S V ) vanishes, except in degree 0 where it coincides with Q , so that S V is indeeda resolution of Q .In the next section, we suggest a general and precise definition of a Koszul-Tate resolution.Although such a definition does not seem to exist in the literature, it is commonly acceptedthat a Koszul-Tate resolution of the quotient Q of a commutative ring k by an ideal I is a k -algebra that resolves Q = k/I .The natural idea – to get a resolving J -algebra for Q – is to replace S V by J ⊗ S V , and,more precisely, to consider the ‘Cofibration – Trivial Fibration’ decomposition J J ⊗ S V ∼ ։ Q (32)of the canonical DG D A -morphism J → Q [BPP17b, Theorem 28]. The differential graded D -algebra J ⊗ S V is a J -algebra that resolves Q = J / I , but it is of course not a cofibrantreplacement, since the left algebra in (32) is not the initial object O in DG D A (further, theconsidered factorization does not canonically induce a cofibrant replacement in DG D A , sinceit can be shown that the morphism O → J is not a cofibration). However, as emphasizedabove, the Koszul-Tate problem requires a passage from the category DG D A to the category J ↓ DG D A (under the D -geometric counterpart J of O ( J ∞ E ) ). It is easily checked that, inthe latter undercategory, J ⊗ S V is a cofibrant replacement of Q . n Koszul-Tate resolutions Definition 24 ([BPP17b]) . Let
J ∈ D A be a D -algebra and let I ⊂ J be a D -ideal. Thealgebra J ⊗ S V ∈ DG D A given by the ‘Cofibration – Trivial Fibration’ factorization of J → J / I is a J -algebra that resolves J / I . Moreover, the algebra J ⊗ S V ( in fact J J ⊗ S V ) isa cofibrant replacement of J / I ( in fact of J → J / I ) in the model category J ↓ DG D A . Werefer to J ⊗ S V as the cofibrant replacement Koszul-Tate resolution of J / I . D -Geometry In view of Subsection 4.2, a Koszul-Tate resolution of a DG D A -morphism J → Q , where
J ∈ D A , should be an algebra C ∈ DG D A , as well as a J -algebra. This suggests to combinethe D -action ⊲ and the J -action ⊳ in an action ⋄ of the ring J [ D ] := J ⊗ O D of linear differential operators with coefficients in J , by setting, for any j ∈ J , D ∈ D , and c ∈ C , ( j ⊗ D ) ⋄ c = (( j ⊗ O ) ◦ (1 J ⊗ D )) ⋄ c := j ⊳ ( D ⊲ c ) . The introduction of the ring J [ D ] is the more natural as the algebra J = J ∞ ( O EX )( X ) ∈D A is the D -geometric counterpart of the algebra O ( J ∞ E ) = F = F ( π ∞ ) (that we denotein Appendix A, to simplify, also by F ( π ) ), and as the J -module J [ D ] = J ⊗ O D ∈ D M isthe D -geometric analog of the F -module F ( π ) ⊗ C ∞ ( X ) D ( X ) ≃ CD ( F , F ) used in smoothgeometry (see Appendix A). Indeed, as stressed in Subsection 4.2, horizontal linear differentialoperators CD ( F , F ) are the fundamental ingredient of the Koszul-Tate resolutions in Mathe-matical Physics and in Cohomological Analysis of PDE -s. Therefore, the passage from DG D A to DG J [ D ] A corresponds to the necessary encryption of horizontal differential operators in the D -geometric approach to the Koszul-Tate resolution and to the Batalin-Vilkovisky formalism.We will use the following notation. For any monoidal category ( C , ⊗ , I ) and any monoid ( A , µ, η ) in C , we denote by Mod C ( A ) the category of (left) A -modules in C , i.e., of C -objects M together with a C -morphism ν : A ⊗ M → M , such that the usual associativity and unitalitydiagrams commute. If C is symmetric monoidal, the category CMon ( C ) is the category ofcommutative monoids in C . Finally, for any additive (or even Abelian) category E , we denoteby Ch + ( E ) the category of non-negatively graded chain complexes in E .If A ∈ D A ⊂ DG D A is a differential graded D -algebra concentrated in degree 0 and withzero differential, we have Mod DG D M ( A ) = Ch + ( Mod D M ( A )) = Ch + ( A [ D ] M ) = DG A [ D ] M , (33)since, as well-known [BD04], Mod D M ( A ) = Mod ( A [ D ])) =: A [ D ] M . It follows from (33) that DG A [ D ] A := CMon ( DG A [ D ] M ) = CMon ( Mod DG D M ( A )) ≃ A ↓ DG D A , (34) n Koszul-Tate resolutions DG D A -map A →B , with source
A ∈ D A , should be an object C in C ∈ A ↓ DG D A ≃ DG A [ D ] A = CMon ( DG A [ D ] M ) . Hence, in the general situation, over a smooth – not necessarily affine – scheme X , weconsider, in addition to the above mentioned category DG + qcCAlg ( D X ) = CMon ( DG + qcMod ( D X )) , also the category DG + qcCAlg ( A X [ D X ]) = CMon ( DG + qcMod ( A X [ D X ])) (35)of differential non-negatively graded O X -quasi-coherent commutative unital A X [ D X ] -algebras,where A X ∈ qcCAlg ( D X ) and A X [ D X ] = A X ⊗ O X D X . For simplicity, we refer to the objects of the category (35) as differential graded A [ D X ] -algebras (thus writing A instead of A X ). A few details on A X [ D X ] and DG + qcCAlg ( A X [ D X ]) can be found in Appendix B, Section 8 (we recommend to read Definition 58 andExample 59).Notice now that the cofibrant replacement Koszul-Tate resolution (see Definition 24) of a DG D A -map J → Q , J ∈ D A , is the DG D A -cofibration J J ⊗ S V , whose target resolves Q (see Equation (32)) and which is, in view of Theorem 22, a retract of a relative Sullivan D -algebra, and, in view of [BPP17b, Theorem 28], even just a minimal (non-split) relativeSullivan D -algebra (see Definition 21). This observation suggests the following two definitions,which generalize Definition 21 and the just recalled Definition 24, respectively, taking intoaccount the above-motivated passage to the category (35): Definition 25.
Let X be a smooth scheme and let A be a D X -algebra. A differential graded A [ D X ] -algebra C is said to be of Sullivan type , if it admits an increasing filtration C ⊂ C ⊂ . . . by differential graded D X -subalgebras, such that there is a differential graded D X -algebramorphism A → C ( we set C − := A ) and that C k ( k ≥ is isomorphic as differential graded D X -algebra to C k ≃ C k − ⊗ S V k , where V k is a locally projective graded D X -submodule of C k such that d C k V k ⊂ C k − . Definition 26.
Let X be a smooth scheme, let A be a D X -algebra, and let φ : A → B be adifferential graded D X -algebra morphism. A D -geometric Koszul-Tate resolution of φ isa differential graded A [ D X ] -algebra morphism ψ : C → B , which is a quasi-isomorphism in thecategory of differential graded A [ D X ] -modules, and whose source C is of Sullivan type. Remark 27.
Observe first that a quasi-isomorphism in the category of differential graded A [ D X ] -modules is a morphism that induces a bijection in homology, i.e., is an A -linear quasi-isomorphism in the category of differential graded D X -modules. Further, the differential on n Koszul-Tate resolutions C k − ⊗ S V k is d C k and, since d C k is a degree − graded derivation, it is completely defined bythe differential of the differential graded D X -subalgebra C k − and the restriction d C k | V k (notethat, for c ∈ C k − and v, w ∈ V k , for instance, we have c ⊗ ( v ⊙ w ) = ( c ⊗ O X ) ⋆ (1 ⊗ v ) ⋆ (1 ⊗ w ) ,where is the unit in C k − and ⋆ the multiplication in C k ).These definitions show that the confinement to the smooth affine case in Section 4 doesnot only allow to use the artefacts of the model categorical environment, i.e., to computethe cofibrant replacement Koszul-Tate resolution, but allows also to discover the fundamentalstructure of this Koszul-Tate resolution, and to extend this structure to the general case of anarbitrary smooth scheme X .The requirement that C be equipped with an increasing filtration by differential graded D X -subalgebras C k ( k ≥ ) and that there exists a differential graded D X -algebra morphism j : A → C , is equivalent to the condition that C be filtered by a sequence C ⊂ C ⊂ . . . of differential graded A [ D X ] -subalgebras. Indeed, since j : A → C , as well as the canonicalinclusions i k : C k − → C k ( k ≥ ), are differential graded D X -algebra morphisms, we havedifferential graded D X -algebra morphisms j k = i k ◦ . . . ◦ i ◦ j : A → C k that provide afiltering sequence C ⊂ C ⊂ . . . of differential graded A [ D X ] -subalgebras. Conversely, such asequence gives a differential graded D X -algebra morphism A ∋ a a ⊳ C ∈ C . Hence, a D -geometric Koszul-Tate resolution of a differential graded D X -algebra morphism φ : A → B is the same as an A -semi-free resolution of φ in the sense of [BD04]. It follows [BD04] thatthe next proposition holds. Proposition 28.
Let X be a smooth scheme and A a D X -algebra. Any differential graded D X -algebra morphism A → B admits a D -geometric Koszul-Tate resolution. This holds inparticular if A = J ∞ ( O EX ) ∈ qcCAlg ( D X ) is the D X -algebra ‘of functions of the infinite jetspace’ of an algebraic vector bundle π : E → X of finite rank over a smooth scheme X . In the following, we use the acronym
KTR for ‘Koszul-Tate resolution’. Our goal is to showthat all the
KTR -s that we considered so far are D -geometric KTR -s, as well as to compareseveral
KTR -s. D -geometric KTR Tate’s
KTR [Tat57, Theorem 1], which we described briefly in the proof of Theorem 8, ispurely algebraic, there is no underlying space X , and there are no differential operators D = D ( X ) . Of course, one could consider the special situation where the Noetherian commutativeunital ring R is an algebra over a commutative unital algebra O over some field, define lineardifferential operators D on O algebraically (the algebraic approach to differential operatorsis well-known, see, e.g., [GKP13a]), and compare Tate’s resolution – in this case – with the D -geometric KTR . We see however no advantage in running through the technicalities of n Koszul-Tate resolutions D -geometric KTR is exactlythe same as that of Tate’s resolution (ignore D and take A = R ). Remark 29. D -geometric KTR can be traced back to minimal models in Homotopy Theory[Hal83]. Let us start with a short historical note. Since the categories of topological spaces andsimplicial sets have equivalent homotopy categories, simplicial sets are purely combinatorialmodels for classical Homotopy Theory. Kan constructed in 1958 algebro-combinatorial models:simplicial groups. In 1969, Quillen proved that the homotopy categories of simply-connectedrational topological spaces and of connected differential graded Lie Q -algebras are equivalent.Similarly, in 1977, Sullivan showed that there exists a categorical equivalence between thehomotopy categories of simply-connected rational topological spaces with finite Betti numbersand of differential graded commutative Q -algebras (category DG Q A ) ( A • , d ) , whose cohomologyspaces satisfy H ( A • , d ) = Q , H ( A • , d ) = 0 , and H n ( A • , d ) is finite-dimensional for any n . This correspondence became really efficient due to the introduction of relative Sullivanminimal models of DG Q A -morphisms – which are specific relative Sullivan Q -algebras – . Suchmodels are (nowadays) obtained from the application of the small object argument to a mostnatural cofibrantly generated model structure on DG Q A . Hence, the cofibrant replacement KTR , which is a relative Sullivan minimal model, and its generalization, the D -geometric KTR , have no apparent link with Tate’s
KTR and with the
KTR -s in Mathematical Physicsand Cohomological Analysis, which are based on [Tat57]. Indeed, Tate’s paper is a work inHomological Algebra and it originates from the attempt to replace the Koszul resolution ofa regular sequence by a resolution that is valid even when the sequence is not regular. Theanalogy between these two types of
KTR -s, the Tate type and the Sullivan type, might thusseem astonishing. However, both, Tate and Sullivan (and his successors), just looked for agood ‘resolution’ of a commutative ring, and they used (in our opinion independently) the same‘naive’ technique – the addition of generators to kill cycles or obstructions to isomorphisms inhomology – . This justifies our decision to refer to relative Sullivan minimal models – minimalKoszul-Sullivan extensions in [Hal83] – as Koszul-Tate resolutions.It is now clear that the
KTR -s in Algebra, Mathematical Physics, Cohomological Analysis,Homotopy Theory, and D -Geometry, have all roughly the same structure. In some areasspecific assumptions reduce more or less strongly the size of the corresponding KTR . Thedifficulty is to switch between the different fields and respective languages (to establish a kindof dictionary) and to prove precise comparison results, such as, for instance, the result that,except for Tate’s
KTR , all the others are rigorously D -geometric ones. D -geometric KTR Proposition 30.
The cofibrant replacement Koszul-Tate resolution of a DG D A -map φ : J →Q , J ∈ D A , is a D -geometric Koszul-Tate resolution of φ . n Koszul-Tate resolutions D -geometric resolution is a generalization of the notion of cofibrantreplacement resolution to the case of an arbitrary smooth scheme, this proposition is ratherobvious. Here is its precise proof. Proof.
Let
J ⊗ S V be the cofibrant replacement resolution of a DG D A -map φ : J → Q , J ∈ D A . Since the underlying X is a smooth affine variety, we replace the sheaves in Section5 by their total sections. The construction in Section 9 of [BPP17b] – which leads to Theorem28 of [BPP17b] – directly implies that the minimal relative Sullivan D -algebra J → J ⊗ S V isof Sullivan type. Indeed, R := J ⊗ S V is obtained as the union of a sequence R ⊂ R ⊂ . . . of differential graded D -algebras, where R k ( k ≥ ) is defined by R k = R k − ⊗ S G k ( R − = J )and where G k is a free non-negatively graded D -module. Since the differential graded D -algebrastructure on R k − ⊗ S G k is obtained by means of Lemma 60 in Subsection 8.4, it is clear thatthe differential δ k of R k satisfies δ k G k ⊂ R k − . It now suffices to check that the DG D A -trivial-fibration q : J ⊗ S V ∼ ։ Q , which is also obtained by an iterated application of Lemma 60, isa DG J [ D ] A -map, i.e., that its source and target are objects in the latter category and that q is J -linear. In view of Example 59, the DG D A -morphisms j : J ∋ ι ι ⊗ O ∈ J ⊗ S V and φ : J ∋ ι [ ι ] ∈ Q endow the two target algebras J ⊗ S V (with multiplication ⋄ ) and Q (with multiplication ∗ ) with natural DG J [ D ] A -structures ι ⊳ T = ( ι ⊗ O ) ⋄ T and ι ⊳ Q = [ ι ] ∗ Q .
As for the J -linearity of q , we have q ( ι ⊳ T ) = q (( ι ⊗ O ) ⋄ T ) = q ( ι ⊗ O ) ∗ q ( T ) = φ ( ι ) ∗ q ( T ) = [ ι ] ∗ q ( T ) = ι ⊳ q ( T ) , as, by construction, q ( ι ⊗ O ) = φ ( ι ) . Depending on the author(s), the concept of D X -module is considered over a base space X that is a finite-dimensional smooth manifold [Cos11] or a finite-dimensional complex manifold[KS90], a smooth algebraic variety [HTT08] or a smooth scheme [BD04] over a fixed base fieldof characteristic zero.In [BPP17b], our base space is a smooth affine algebraic variety X . This enables us toreplace sheaves by their total sections (which are much easier to handle) – e.g., we substitute DG D A , with D = D X ( X ) , to DG + qcCAlg ( D X ) . However, all the results that we obtain in[BPP17b] after the passage to total sections, are also valid for other underlying spaces X .Indeed, the only instance (after the passage), where we still use the nature of X , is the resultthat the O -module, O = O X ( X ) , of linear differential operators D = D X ( X ) over a smoothaffine algebraic variety X is flat (and even projective [BPP17a]).For the KTR -s in Mathematical Physics and in Cohomological Analysis of
PDE -s, the space X is an n -dimensional smooth manifold , and even an open subset X ⊂ R n , so that D = D ( X ) is a free module over O = O ( X ) , hence a projective and a flat one. Moreover, the context n Koszul-Tate resolutions KTR -s – smooth geometry – is usually presented in terms of global sections andmorphisms between them [BPP17b, Subsection 11.3]. It follows that:
Remark 31.
In the contexts of the
KTR -s from Mathematical Physics and from CohomologicalAnalysis, total sections replace sheaves, D -modules can be used, and the results of [BPP17b]are valid. For instance, Lemma 60 holds, the cofibrant replacement KTR makes sense, and sodoes the total-sections-version of the D -geometric KTR .We can thus try to show that the
KTR of a regular first-order on-shell reducible gaugetheory is a D -geometric KTR . The Koszul-Tate complex of such a theory, see Subsection 2.4.2,can be rewritten as
KT =
F ⊗ S V , where F = F ( π ∞ ) and V = M α,a R · φ α ∗ a ⊕ M β,δ R · C β ∗ δ , (36)and where the tensor products are over R . The complex (KT , δ KT ) is thus a chain complex inthe category of F -modules.The algebra F can be endowed with a D -module structure. Since we work in fixed coordi-nates, any D ∈ D uniquely reads D = P | α |≤ k D α ( x ) ∂ αx , for some integer k ∈ N and functions D α ∈ O . As observed in Equation (138) (and, maybe, partially in Equation (104)), the actionof D on F ∈ F should be defined by D · F = C ( D ) F = X | α |≤ k D α ( x ) D αx F , where C denotes the horizontal lift. It is easily seen that this definition actually provides a D -module structure, since, for any composable linear differential operators ∆ ∈ Diff( η , η ) and ∆ ∈ Diff( η , η ) between vector bundles η i over X , the horizontal lifts C (∆ ) ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) and C (∆ ) ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) satisfy C (∆ ◦ ∆ ) = C (∆ ) ◦ C (∆ ) . This result holds [KV98] for any vector bundles π : E → X and η i : F i → X . For the trivialbundle π : R n × R r → R n that we fixed at the beginning of Subsection 2.4 and for the trivial linebundle η i : R n × R → R n , we get Diff( η i , η j ) = D and C Diff( π ∗∞ ( η i ) , π ∗∞ ( η j )) = C Diff( F , F ) ,i.e., we get the situation that we considered above.It is clear that this D -module structure of F and the O -algebra structure of F are com-patible in the sense that vector fields act as derivations. Hence, F is a D -algebra. Moreover,the ideal I (Σ) of those functions of F that vanish on Σ : D αx δ u a L = 0 , is an O -ideal and a D -submodule, hence a D -ideal. As for the submodule structure, note that if F ∈ I (Σ) and D ∈ D , one has ( D · F ) | Σ = ( C ( D ) F ) | Σ = C ( D ) Σ F | Σ = 0 , n Koszul-Tate resolutions C ∞ (Σ) = F /I (Σ) is a D -algebra for theaction D · [ F ] = [ D · F ] and the multiplication [ F ][ G ] = [ F G ] . It follows that the passage φ : F ∋ F [ F ] ∈ C ∞ (Σ) (37)to the quotient is a D -algebra map. Example 59 shows that the action F ⊳ [ G ] := [ F ][ G ] = [ F G ] endows C ∞ (Σ) is an F [ D ] -algebra structure. Remark 32.
In view of Equation (37) the algebra C ∞ (Σ) fits into the framework of Defini-tion 26 of a D -geometric KTR , as well as into the framework of Definition 24 of a cofibrantreplacement
KTR .In Subsection 8.2, we observed that the D -action on the fiber coordinates x ( k ) of an infinitejet space with base coordinate t satisfies the equations ∂ t · x ( k ) = D t x ( k ) = x ( k +1) . In Subsection 2.4.2, we viewed the degree 1 generators φ α ∗ a (resp., the degree 2 generators C β ∗ δ )as fiber coordinates of an infinite horizontal jet space with base coordinates ( x i , u aα ) and wenoticed that this interpretation comes along with the replacement of the total derivatives D x i by the extended total derivatives ¯ D x i . It is therefore natural to define the D -action on thefiber coordinates φ α ∗ a (resp., C β ∗ δ ) by ∂ x i · φ α ∗ a := ¯ D x i φ α ∗ a = φ iα ∗ a (resp., by ∂ x i · C β ∗ δ := ¯ D x i C β ∗ δ = C iβ ∗ δ ) . In particular, we obtain ∂ αx · φ ∗ a = ¯ D αx φ ∗ a = φ α ∗ a ( resp., ∂ βx · C ∗ δ = ¯ D βx C ∗ δ = C β ∗ δ ) . (38)Eventually, it is natural to replace the underlying module V of Equation (36) by the freenon-negatively graded D -module V = M a D · φ ∗ a ⊕ M δ D · C ∗ δ (39)over the components of the antifields φ ∗ and C ∗ . The F -module of Koszul-Tate chains thenreads KT =
F ⊗ R S R V = F ⊗ O S O V , (40)where the
RHS is also a graded D -algebra.Any element c of this graded D -algebra reads non-uniquely as a finite sum c = X F ( D a · φ ∗ a ) . . . (∆ δ · C ∗ δ ) , n Koszul-Tate resolutions F ∈ F and D a , ∆ δ ∈ D , and where we omitted the tensor products. The Koszul-Tatedifferential δ KT , which is well-defined on KT , acts as a graded derivation and is thus completelyknown, if it is known on the D a · φ ∗ a and the ∆ δ · C ∗ δ . For any D = D α ∂ αx , we have, in view ofthe definitions given above, δ KT ( D · φ ∗ a ) = D α δ KT ( ∂ αx · φ ∗ a ) = D α δ KT ( φ α ∗ a ) = D α D αx δ u a L = D · ( δ u a L ) = D · δ KT ( φ ∗ a ) . (41)Similarly, we get δ KT ( D · C ∗ δ ) = D α δ KT ( ∂ αx · C ∗ δ ) = D α δ KT ( C α ∗ δ ) = D α ¯ D αx ( R aδβ ¯ D βx φ ∗ a ) = D α ¯ D αx ( R aδβ φ β ∗ a ) . The extended total derivative ¯ D αx of R aδβ φ β ∗ a is a sum of terms of the type D α x R aδβ ¯ D α x φ β ∗ a = ( ∂ α x · R aδβ ) ( ∂ α x · φ β ∗ a ) , so that, in view of the definition of the D -action on the tensor product of F and S O V , we find ¯ D αx ( R aδβ φ β ∗ a ) = ∂ αx · ( R aδβ φ β ∗ a ) . Eventually, δ KT ( D · C ∗ δ ) = D · δ KT ( C ∗ δ ) . (42) Remark 33.
The equations (40), (41), and (42) show that KT is a graded D -algebra andthat (KT , δ KT ) is a chain complex in the category of D -modules. D -geometric KTR In the following, we apply Lemma 60 from Subsection 8.4, which allows to construct non-split relative Sullivan D -algebras ( RS D A -s), as well as DG D A -morphisms from such a Sullivanalgebra to another differential graded D -algebra.Let V := L a D · φ ∗ a . To endow the graded D -algebra C := F ⊗ O S O V (43)with a differential graded D -algebra structure d , we set, dφ ∗ a := δ u a L ∈ F , (44)extend d to V by D -linearity, and equip C with the differential d given by d ( F ( D · φ ∗ a ) (∆ · φ ∗ b )) := ( F d ( D · φ ∗ a ))(∆ · φ ∗ b ) − ( F d (∆ · φ ∗ b ))( D · φ ∗ a ) , where we omitted the tensor products and considered, to increase clarity, an element of degree2. Then the natural DG D A -morphism ı : ( F , ∋ F F ⊗ O ∈ ( C , d ) is a RS D A . Since δ KT is also a graded derivation that is D -linear (Equation (41)) and coincides with d on thegenerators φ ∗ a , the RS D A is actually the DG D A -morphism ı : ( F , ∋ F F ⊗ O ∈ ( C , δ KT ) . (45) n Koszul-Tate resolutions D -algebra C ∞ (Σ) = F /I (Σ) and the D A -morphism φ : F → C ∞ (Σ) (Equation (37)). To define a DG D A -morphism q : C → C ∞ (Σ) , (46)it suffices to set q ( φ ∗ a ) = 0 ∈ ( C ∞ (Σ)) ∩ − ( φ ( dφ ∗ a )) , (47)to extend q by D -linearity to V , and to define q in degree 0 by q ( F ) = φ ( F ) = [ F ] and indegree ≥ by q = 0 . As for Condition (47), note that φ ( dφ ∗ a ) = [ δ u a L ] = 0 , in view of thedefinition of Σ .An anew application of Lemma 60, where the role that was played above by ( F , (resp., V ) is now assumed by ( C , δ KT ) (resp., V := L δ D · C ∗ δ ), endows the graded D -algebra C := C ⊗ O S O V (48)with a differential graded D -algebra structure d that, similar to d above, is fully defined by d C ∗ δ = R aδα ( ∂ αx · φ ∗ a ) ∈ ( C ) ∩ δ − { } . (49)Indeed, in view of Equation (18), we have δ KT ( R aδα ( ∂ αx · φ ∗ a )) = R aδα D αx δ u a L ≡ . To compare the differential d with the differential δ KT , note that d is extended to V by D -linearity and that its value on c = F ( D · φ ∗ a ) (∆ · C ∗ δ ) ( ∇ · C ∗ ε ) , for instance, is d c = δ KT ( F ( D · φ ∗ a )) (∆ · C ∗ δ ) ( ∇ · C ∗ ε ) − ( F ( D · φ ∗ a ) d(∆ · C ∗ δ )) ( ∇ · C ∗ ε ) − ( F ( D · φ ∗ a ) d( ∇ · C ∗ ε )) (∆ · C ∗ δ ) . As δ KT is a graded derivation that is D -linear (Equation (42)) and coincides with d on thegenerators C ∗ δ , we get d = δ KT on C . Hence, the DG D A -morphism : ( C , δ KT ) ∋ c c ⊗ O ∈ ( C , δ KT ) (50)is a relative Sullivan D -algebra.Start now from the DG D A -morphism q , and define a DG D A -morphism q : C → C ∞ (Σ) (51)by setting q ( C ∗ δ ) = 0 ∈ ( C ∞ (Σ)) ∩ − ( q ( δ KT C ∗ δ )) , extending q by D -linearity to V , and by defining q in degree 0 by q ( F ) = [ F ] and in degree ≥ by q = 0 . n Koszul-Tate resolutions V = V ⊕ V as graded D -module, the graded D -algebras S O V = S O ( V ⊕ V ) and S O V ⊗ O S O V are isomorphic. Hence, the same holds for the graded D -algebras KT =
F ⊗ O S O V and C = F ⊗ O S O V ⊗ O S O V . It follows that ◦ ı : ( F , → (KT , δ KT ) is a DG D A -morphism and thus allows to endow (KT , δ KT ) with a DG F [ D ] A -structure – see Example 59. Theorem 34.
The Koszul-Tate resolution of the function algebra C ∞ (Σ) of the infinite pro-longation manifold Σ of the Euler-Lagrange equations of a regular first-order on-shell reduciblegauge theory is a D -geometric Koszul-Tate resolution ( in the smooth setting – see beginning ofSubsection 6.3 ) of the canonical D -algebra map F → C ∞ (Σ) , where F is the function algebraof the infinite jet space in which Σ is located and where C ∞ (Σ) is the quotient of F by theideal of those functions of F that vanish on Σ .Proof. Most of the proof is given in the preparation that precedes the theorem. For instance,it is clear from what has been said that KT ≃ C admits an increasing filtration C ⊂ C ⊂C ⊂ . . . by DG D -subalgebras, such that there is a DG D -algebra morphism F → C ( weset C := F ) and that C k ( k ≥ is isomorphic as DG D -algebra to C k ≃ C k − ⊗ O S O V k , where V k is a free graded D -submodule of C k such that δ KT V k ⊂ C k − : KT is of Sullivantype. We already mentioned that KT ≃ C and C ∞ (Σ) are DG F [ D ] -algebras. It now sufficesto show that the DG D A -morphism q := q : KT → C ∞ (Σ) is F -linear and induces an F - and D -linear bijection q ♯ of degree 0 between the graded module H • (KT) and the module C ∞ (Σ) concentrated in degree 0. First, q is F -linear, as, if F, G ∈ F , we obtain
F ⊳ q ( G ) = F ⊳ [ G ] = [ F G ] = q ( F G ) . Hence, the induced map q ♯ has the required properties, except, maybe, bijectivity. In degree ≥ , the homology H • (KT) vanishes, just as C ∞ (Σ) . In degree , the homology is given by C ∞ (Σ) = F /I (Σ) , where F (resp., I (Σ) ) are the 0-cycles (resp., -boundaries), and q ♯ [ F ] = q ( F ) = [ F ] is the identity. Recall first that, in the setting of a
KTR from Mathematical Physics, the concept ofcofibrant replacement Koszul-Tate resolution makes sense. Secondly, it is clear a priori thatthe general functorial cofibrant replacement KT resolution ( KT , δ KT ) is much larger thanthe KT resolution (KT , δ KT ) , which is subject to size-reducing irreducibility (i.e., first-orderreducibility) conditions and is far from being functorial.More precisely, the KT resolution (KT , δ KT ) is the DG F [ D ] A KT =
F ⊗ O S O V , where V is the free graded D -module with homogeneous basis [ { φ ∗ a , C ∗ δ } n Koszul-Tate resolutions , ), endowed with the degree − , F - and D -linear gradedderivation defined by δ KT ( φ ∗ a ) = δ u a L and δ KT ( C ∗ δ ) = R aδα ( ∂ αx · φ ∗ a ) . The results of [BPP17b], applied to the DG D A -map φ : ( F , → ( C ∞ (Σ) , , show that thecofibrant replacement KT resolution ( KT , δ KT ) is the DG F [ D ] A KT = F ⊗ O S O V , where V is the free graded D -module with homogeneous basis [ { I f , I σ n , , I σ n , , . . . , I kσ n , , . . . } , for all f ∈ C ∞ (Σ) and ‘numerous’ σ n (of degree n ≥ ), which are described in [BPP17b,Theorem 28] and in the proof that precedes this result (the degrees of the generators are , n + 1 , n + 1 , . . . , n + 1 , . . . ). Here δ KT is the degree − , F - and D -linear graded derivationdefined by δ KT ( I f ) = 0 and δ KT ( I kσ n , ) = σ n . When using the just mentioned description in [BPP17b, Theorem 28], one sees quite easilythat the injective map i , defined by i ( φ ∗ a ) = I δ ua L , ∈ V and i ( C ∗ δ ) = I (cid:16) R aδα (cid:16) ∂ αx · I δua L , (cid:17) , (cid:17) ∈ V , is a DG F [ D ] A -morphism i : (KT , δ KT ) → ( KT , δ KT ) . Proposition 35.
The Koszul-Tate resolution of the function algebra C ∞ (Σ) of the infinite pro-longation manifold Σ of the Euler-Lagrange equations of a regular first-order reducible gaugetheory is a differential graded F [ D ] -subalgebra of the cofibrant replacement Koszul-Tate res-olution ( in the smooth setting – see beginning of Subsection 6.3 ) of the quotient D -algebra C ∞ (Σ) . We compare the Koszul-Tate complex (KT , δ KT ) of a regular first-order on-shell reduciblefield theory, which is defined in coordinates, with the Koszul-Tate complex ( KT , δ KT ) of Sub-section 3.2, which is subject to regularity and higher-order off-shell reducibility conditions, andis – although fixed coordinates are considered – mostly defined in the coordinate-free languageof Cohomological Analysis of PDE -s. The difficulty is to pass from one setting to the other.Let us stress that in the following KT and KT refer to these two different complexes, and letus mention that this section might be easier to read after a revision of Section 3 and of partsof Appendix A, Section 7.In the contexts of KT and KT the underlying space is an open subset X ⊂ R n . We thushave O = C ∞ ( X ) and D = D ( X ) . n Koszul-Tate resolutions ( KT , δ KT ) is defined from a compatibility complex −→ R −→ R −→ . . . ∆ k − −→ R k − −→ made of F -modules R j := Γ( R j ) := Γ( π ∗∞ ( F j )) – here π ∞ : J ∞ E → X is the infinite jet spaceof π : E → X , a rank r smooth vector bundle over X , F is the function algebra of J ∞ E, and ρ j : F j → X is a rank r j smooth vector bundle over X – and of horizontal differentialoperators ∆ j : R j → R j +1 between them. The Koszul-Tate chains KT are the elements of thealgebra S F C Diff( R • , F ) , where R • := Γ( R • ) := Γ( π ∗∞ ( F • )) and where R • (resp., R • , F • ) is the direct sum of the R j (resp., R j , F j ). Since C : F ⊗ O Diff(Γ( F • ) , O ) → C Diff( R • , F ) is an F -module isomorphism (Equation (103)), we get KT ≃ S F ( F ⊗ O Diff(Γ( F • ) , O )) ≃ F ⊗ O S O Diff(Γ( F • ) , O ) . As already mentioned, we work in fixed coordinates. The coordinates of E are denotedby ( x i , u a ) and those of J ∞ E by ( x i , u aα ) . Similarly, we symbolize the coordinates of F • by ( x i , v λ ( j )) – where j ∈ { , . . . , k − } and λ ∈ { , . . . , r j } – , those of R • by ( x i , u aα , v λ ( j )) , andthose of ¯ J ∞ ( R • ) by ( x i , u aα , v λβ ( j )) . Hence, a linear differential operator D ∈ Diff(Γ( F • ) , O ) ,when applied to a section v ∈ Γ( F • ) , reads D v = X α ( D α ( x ) . . . D P j r j α ( x )) ∂ αx ...v λ ( j )( x i ) ... , so that it is natural to view it as an element of the free non-negatively graded D -module V := k − M j =1 r j M λ =1 D · v λ ( j ) (52)over formal generators of degree j , which we also denote by v λ ( j ) . Hence, we get the F -moduleisomorphism KT ≃ F ⊗ O S O V , (53)where the
RHS is also a graded D -algebra.The comparison of Equations (52) and (53) with Equations (39) and (40) shows that thealgebras KT and KT are defined similarly. More precisely: n Koszul-Tate resolutions Remark 36.
Whereas the complex KT contains the antifields φ ∗ and C ∗ – with components φ ∗ a and C ∗ δ that correspond to the considered equations δ u a L ( x i , u aα ) and the irreducible relations R aδα D αx δ u a L ( x i , u aα ) ≡ between them – , the complex KT contains antifields v (1) , v (2) , v (3) , ... – whose components v λ (1) , v λ (2) , v λ (3) , ... correspond to the equations ψ D ∈ R , i.e., the equations ψ λD (1)( x i , u aα ) ,the reducible relations ∆ ( ψ D ) = 0 between them, i.e., the relations (∆ ( ψ D )) λ (2)( x i , u aα ) ≡ , the relations ∆ ◦ ∆ = 0 between these relations, ... – .To further compare KT and KT , we must of course use here the same basic definitions asin Subsection 6.3. Hence, in analogy with (38), we set ∂ βx · v λ ( j ) := ¯ D βx v λ ( j ) = v λβ ( j ) , (54)where ¯ D x i = ∂ x i + u aiα ∂ u aα + v λiβ ( j ) ∂ v λβ ( j ) . (55)We are now prepared to compare the Koszul-Tate differentials δ KT and δ KT . As mentionedin Subsection 3.2, the differential δ KT is completely defined by its values on C Diff( R • , F ) ≃ Hom F ( ¯ J ∞ ( R • ) , F ) ≃ P ol ( ¯ J ∞ ( R • )) and its values on F . Here superscript 1 refers to functions that are linear in the fiber coordi-nates v λβ ( j ) . To simplify the notation and to nevertheless distinguish the sections v λ ( j )( x i , u aα ) of R • (resp., the sections v λβ ( j )( x i , u aα ) of ¯ J ∞ ( R • ) ) from the fiber coordinates v λ ( j ) of R • (resp., the fiber coordinates v λβ ( j ) of ¯ J ∞ ( R • ) ), we write ˜ v λ ( j ) (resp., ˜ v λβ ( j ) ) for sections. Inthe considered fixed coordinates, the preceding identifications read, i.e., such a differentialoperator ∇ and the corresponding linear jet space function F ∇ read (with obvious notation) ∇ v = X β ( . . . ∇ λβ ( j )( x i , u aα ) . . . ) D βx ... ˜ v λ ( j ) ... ≃ F ∇ ( x i , u aα , v λβ ( j )) = X β ( . . . ∇ λβ ( j )( x i , u aα ) . . . ) ...v λβ ( j ) ... . (56)Since δ KT vanishes on F , it is completely defined by its values on the v λβ ( j ) , exactly as δ KT is fully defined by its values on the φ α ∗ a and the C β ∗ δ . Note still, before proceeding, that,for horizontal linear differential operators C Diff( R j , R j +1 ) valued in a not necessarily rank 1 n Koszul-Tate resolutions ∇ λβ ( j ) is replaced by a matrix of coefficients ∇ µλβ ( j + 1 , j ) .Recall now from Subsection 3.2 that, if F ∈ F and ∇ j ∈ C Diff( R j , F ) , we have δ KT ( F ) = 0 , δ KT ( ∇ ) = ∇ ( ψ D ) , and δ KT ( ∇ j ) = ∇ j ◦ ∆ j − , ∀ j ≥ . (57)The equations (56) and (57) lead to the equation δ KT ( v λβ (1)) = δ KT ( D βx ˜ v λ (1)) = D βx ( ψ λD (1)) (58)– which is entirely similar to the definition δ KT ( φ α ∗ a ) = D αx ( δ u a L ) . (59)For j ∈ { , . . . , k − } , we find analogously δ KT ( v λβ ( j )) = δ KT ( D βx ˜ v λ ( j )) = D βx (cid:16) (∆ j − ˜ v ( j − λ ( j ) (cid:17) = D βx (cid:16) (∆ λµγ ( j, j − x i , u aα ) D γx ˜ v µ ( j − (cid:17) , in view of the above remark on matrix coefficients. When using again the identification (56),we finally get δ KT ( v λβ ( j )) = ¯ D βx (cid:16) (∆ λµγ ( j, j − x i , u aα ) v µγ ( j − (cid:17) = ¯ D βx (cid:16) F λ ∆ j − (cid:17) . For j = 2 , we thus find the equation δ KT ( v λβ (2)) = ¯ D βx (cid:16) ∆ λµγ (2 ,
1) ¯ D γx v µ (1) (cid:17) , (60)where we omitted the variables ( x i , u aα ) – which is fully analogous to the definition δ KT ( C β ∗ δ ) = ¯ D βx (cid:16) R µδγ ¯ D γx φ ∗ µ (cid:17) . (61)We conclude with the observation that the Koszul-Tate differential δ KT = X βλ ¯ D βx (cid:16) ψ λD (cid:17) ∂ v λβ (1) + k − X j =2 X βλ ¯ D βx (cid:16) F λ ∆ j − (cid:17) ∂ v λβ ( j ) is the evolutionary vector field, or symmetry of the Cartan distribution, that is obtained asthe prolongation δ X to the horizontal jet space ¯ J ∞ ( R • ) → J ∞ E of the vertical vector field X = X λ ψ λD ∂ v λ (1) + k − X j =2 X λ F λ ∆ j − ∂ v λ ( j ) of the bundle R • → J ∞ E with coefficients in F ( ¯ J ∞ ( R • )) , see Equation (120).Remark 36, Equations (52), (53), (39), and (40), as well as Equations (58), (60), (59), and(61), show that: Remark 37.
The
KTR in Cohomological Analysis [Ver02] is the natural extension of the
KTR of a first-order reducible field theory and it thus corresponds exactly to the
KTR of ahigher-order reducible field theory [HT92]. n Koszul-Tate resolutions D -geometric KTR It is clear that, since the
KTR in a first-order reducible theory is a D -geometric KTR (Theorem 34), the natural extension of this
KTR is D -geometric as well. In view of Remark37, we thus have the Theorem 38.
The Koszul-Tate resolution of C ∞ (Σ) from Cohomological Analysis of PDE -sis a D -geometric Koszul-Tate resolution of the D -algebra map F → C ∞ (Σ) ( in the smoothsetting – see Remark 31 ) , where F is the function algebra of the infinite jet space in which Σ is located.Proof. The proof is similar to the proof of Theorem 34.
The goal of the present section is to explain a number of concepts that are of importancein the Geometry of
PDE -s. Additional details can be found, for instance, in [KV98].
Consider a differential equation ( DE ) ψ ( t, φ ( t ) , d t φ, . . . , d kt φ ) ≡ , (62)with evident notation. When defining the k -jet of φ ( t ) by j kt φ = ( t, φ ( t ) , d t φ, . . . , d kt φ ) , we may rewrite this DE as ψ ( t, u, u , . . . , u k ) | j kt φ ≡ . (63)Here ( t, u, u , . . . , u k ) are independent variables of what is called the k -jet space. Roughlyspeaking, the (purely) algebraic equation ψ ( t, u, u , . . . , u k ) = 0 (64)defines a hypersurface Σ in the k -jet space (or, better, since t plays a distinguished role,a subbundle Σ of the k -jet bundle), and a solution of the considered DE is nothing but afunction φ ( t ) such that the graph of its k -jet is located on Σ . This is one of the key-aspectsof the jet bundle approach to partial differential equations ( PDE -s) – which will be formalizedin the following. Usually the k -jet is defined by j kt φ = ( φ ( t ) , d t φ, . . . , d kt φ ) , so that ‘graph’ is actually the proper denomina-tion. In view of our modified definition, ‘graph’ means in this text ‘image’. n Koszul-Tate resolutions π : E → X be a smooth vector bundle of rank rk( π ) = r over a smooth n -dimensionalmanifold. For k ∈ N , the k -jet j km φ at m ∈ X of a local smooth section φ ∈ Γ( π ) of π that isdefined around m , is the equivalence class of all local sections of π , such that in any trivializingchart ( x, u ) = ( x i , u a ) of π around m , the local coordinates of these sections coincide at x ( m ) ,together with their partial derivatives at x ( m ) up to order k (it actually suffices that theycoincide in one trivializing chart). We define the k -jet set J k ( π ) of π by J k ( π ) = { j km φ : m ∈ X, φ ∈ Γ( π ) } . The k -jet set is a smooth finite rank vector bundle π k : J k ( π ) → X – the k -jet bundle .Indeed, any trivializing chart ( x i , u a ) of π induces a trivializing chart ( x i , u aα ) of π k , definedby x i ( j km φ ) = x i ( m ) and u aα ( j km φ ) = ∂ αx φ a | x ( m ) , where α ∈ N n and | α | ≤ k . For k ≤ ℓ , there is a ‘truncation’ vector bundle (epi)morphism π kℓ : J ℓ ( π ) → J k ( π ) , so that ( J k ( π ) , π kℓ ) is an inverse system. The limit of this diagram is the ∞ -jet space π ∞ : J ∞ ( π ) → X together with the natural projections π k ∞ : J ∞ ( π ) → J k ( π ) .Coordinates ( x i , u aα ) of J ∞ ( π ) can be obtained from coordinates ( x i , u a ) of π , as above, bydefining an infinite number of coordinates u aα that correspond to the partial derivatives ∂ αx ofthe components φ a = u a ( φ ( x )) of the sections φ of π . We denote the algebra of smoothfunctions of J k ( π ) by F k = F k ( π ) . The canonical epimorphisms π kℓ induce inclusions F k ⊂F ℓ . The colimit of this direct system is the algebra F = S k F k (we will also write F ( π ) , F ∞ , or F ∞ ( π ) ) of smooth functions of J ∞ ( π ) . It follows that any smooth function of J ∞ ( π ) is a smooth function of some J k ( π ) . Note eventually that j k : Γ( π ) → Γ( π k ) and that j ∞ : Γ( π ) → Γ( π ∞ ) (in fact, we should, as above, consider the case k = ∞ separately, as alimit case; however, here and in the following, we refrain from presenting these details).We will use jet bundles to define differential operators between sections of vector bundles.Let π ′ : E ′ → X be a second vector bundle and take the pullback bundle π ∗ k ( π ′ ) , k ∈ N , seeFigure 1. Consider now the F k ( π ) -module of sections Γ( π ∗ k ( π ′ )) . If π ′ : X × R → X , the latter π ∗ k E ′ E ′ J k ( π ) X π ′ pπ ∗ k ( π ′ ) π k Figure 1: Pullback bundlecan be naturally identified with F k ( π ) . This justifies the notation F k ( π, π ′ ) := Γ( π ∗ k ( π ′ )) . Wedenote the composite of ψ ∈ F k ( π, π ′ ) ⊂ C ∞ ( J k ( π ) , π ∗ k E ′ ) n Koszul-Tate resolutions p ∈ C ∞ ( π ∗ k E ′ , E ′ ) also by ψ . Hence, ψ ∈ C ∞ ( J k ( π ) , E ′ ) , and, for any point j km φ ∈ J k ( π ) ,we have ψ ( j km φ ) ∈ E ′ m , i.e., ψ is a fiber bundle morphism ψ ∈ FB ( J k ( π ) , E ′ ) . We thus get anisomorphism of C ∞ ( X ) -modules: Γ( π ∗ k ( π ′ )) = F k ( π, π ′ ) ≃ FB ( J k ( π ) , E ′ ) . (65)Since, for every section φ ∈ Γ( π ) , the composite of j k φ ∈ Γ( π k ) ⊂ C ∞ ( X, J k ( π )) and ψ is a section ψ ◦ ( j k φ ) ∈ Γ( π ′ ) , we see that ψ ∈ F k ( π, π ′ ) implements a map D : Γ( π ) ∋ φ D ( φ ) = ψ ◦ ( j k φ ) ∈ Γ( π ′ ) , such that the value D ( φ ) | m only depends on j km φ . We therefore say that D is a not necessarilylinear differential operator of order k between π and π ′ . Definition 39.
A (not necessarily linear) differential operator D ∈ DO k ( π, π ′ ) of order k from π to π ′ is a map D : Γ( π ) → Γ( π ′ ) that factors through the k -jet bundle, i.e., that reads D = ψ D ◦ ( j k − ) , (66) for some section or fiber bundle morphism ψ D ∈ F k ( π, π ′ ) ≃ FB ( J k ( π ) , E ′ ) . This morphism,which is visibly unique, is the representative morphism of D .
In trivializations of π and π ′ over the same chart ( U, x ) of X , such a k -th order differentialoperator reads ψ bD ( x, ∂ αx φ a ) = ψ bD ( x, u aα ) | j kx φ , ( a ∈ { , . . . , rk( π ) } , b ∈ { , . . . , rk( π ′ ) } , | α | ≤ k ) . (67)If both ranks are 1 and we write ψ (resp., t ) instead of ψ D (resp., x = ( x , . . . , x n ) ), we recover ψ ( t, φ ( t ) , d t φ, . . . , d kt φ ) = ψ ( t, u, u , . . . , u k ) | j kt φ (68)(see beginning of 7.1).The composite of a differential operator D ∈ DO k ( π, π ′ ) and a differential operator D ′ ∈ DO ℓ ( π ′ , π ′′ ) is a differential operator D ′ ◦ D ∈ DO k + ℓ ( π, π ′′ ) . The set DO k ( π, π ′ ) is a C ∞ ( X ) -module. There is a canonical C ∞ ( X ) -module isomor-phism DO k ( π, π ′ ) ≃ F k ( π, π ′ ) ≃ FB ( J k ( π ) , E ′ ) . (69)The natural surjective morphisms π kℓ , k ≤ ℓ , give rise to inclusions DO k ( π, π ′ ) ⊂ DO ℓ ( π, π ′ ) ,thus leading to an increasing sequence of C ∞ ( X ) -modules. The colimit is the filtered C ∞ ( X ) -module DO( π, π ′ ) = [ i DO i ( π, π ′ ) (70)of all differential operators from π to π ′ . n Koszul-Tate resolutions r, r ′ ∈ R and φ, φ ′ ∈ Γ( π ) , we have D ( rφ + r ′ φ ′ ) = r D ( φ ) + r ′ D ( φ ′ ) , the differential operator D is said to be linear. We denote the C ∞ ( X ) -submodule made of thelinear differential operators of order k (resp., of all linear differential operators) from π to π ′ by Diff k ( π, π ′ ) ⊂ DO k ( π, π ′ ) ( resp., Diff( π, π ′ ) ⊂ DO( π, π ′ )) . In trivializations of π and π ′ over the same chart ( U, x ) of X , a linear differential operator D of order k reads ψ bD ( x, ∂ αx φ a ) = ψ bD ( x, u aα ) | j kx φ , ( a ∈ { , . . . , rk( π ) } , b ∈ { , . . . , rk( π ′ ) } , | α | ≤ k ) , (71)where the ψ bD are C ∞ ( x ( U )) -linear in the derivatives, i.e., ψ bD ( x, ∂ αx φ a ) = X α,a M bαa ( x ) ∂ αx φ a . In fact, a differential operator is a linear operator D ∈ Diff k ( π, π ′ ) if and only if itsrepresentative morphism is a vector bundle morphism ψ D ∈ VB ( J k ( π ) , E ′ ) (not only a fiberbundle morphism), i.e., a C ∞ ( X ) -linear map ψ D ∈ Hom C ∞ ( X ) (Γ( π k ) , Γ( π ′ )) (denoted by thesame symbol). This passage from the vector bundle map to the linear map between sectionsallows to replace D ( − ) = ψ D ◦ ( j k − ) , see (66), by D ( − ) = ( ψ D ◦ j k )( − ) . Therefore,
Proposition 40. A linear differential operator D ∈ Diff k ( π, π ′ ) is an R -linear map D :Γ( π ) → Γ( π ′ ) that factors through the k -jet bundle, i.e., that reads D = ψ D ◦ j k , (72) for some (and thus unique) vector bundle or C ∞ ( X ) -module morphism ψ D ∈ VB ( J k ( π ) , E ′ ) ≃ Hom C ∞ ( X ) (Γ( π k ) , Γ( π ′ )) . Hence the isomorphisms of C ∞ ( X ) -modules Diff k ( π, π ′ ) ≃ VB ( J k ( π ) , E ′ ) ≃ Hom C ∞ ( X ) (Γ( π k ) , Γ( π ′ )) , (73) and Diff( π, π ′ ) ≃ VB ( J ∞ ( π ) , E ′ ) ≃ Hom C ∞ ( X ) (Γ( π ∞ ) , Γ( π ′ )) . (74)We close the present section with the remark that, in the case π = π ′ = pr : X × R → X ,the differential operators Diff( π, π ′ ) act on functions C ∞ ( X ) , and that we then write D ( X ) instead of Diff(pr , pr ) ; in other words: Remark 41.
We denote by D ( X ) the associative unital R -algebra of linear differential oper-ators acting on functions C ∞ ( X ) of a smooth manifold X . n Koszul-Tate resolutions A second fundamental feature is that one prefers replacing the original system of
PDE -sby an enlarged system, its prolongation, which also takes into account the differential conse-quences of the original one. More precisely, if φ ( t ) satisfies the original DE (62), we have , forany ℓ ∈ N ,d rt ( ψ ( t, φ ( t ) , d t φ, . . . , d kt φ )) = ( ∂ t + u ∂ u + u ∂ u + . . . ) r ψ ( t, u, u , . . . , u k ) | j k + ℓt φ =: D rt ( ψ ( t, u, u , . . . , u k )) | j k + ℓt φ ≡ , ∀ r ≤ ℓ . (75)Let us stress that the ‘total derivative’ D t or ‘horizontal lift’ D t of d t is actually an infinitesum. The DE (62) and the system of DE -s (75), have clearly the same solutions, so we mayfocus just as well on (75). The corresponding system of algebraic equations ( D rt ψ )( t, u, u , . . . , u k , u k +1 , . . . , u k + r ) = 0 , ∀ r ≤ ℓ (76)defines a ‘surface’ Σ ℓ in the ( k + ℓ ) -jet space. A solution of the original DE (62) is now afunction φ such that the graph gr( j k + ℓ φ ) is a subset of Σ ℓ . The ‘surface’ Σ ℓ is referred to asthe ℓ -th prolongation of the considered DE or differential operator.To grasp the interest in differential consequences, consider for instance the integrationproblem ∂ x i F = f i ( i ∈ { , . . . , n } ) in R n – where notation is obvious – . The differential con-sequences of this (system of) PDE (-s) include the equations ∂ x j ∂ x i F = ∂ x j f i ( i, j ∈ { , . . . , n } ),hence, they include the compatibility conditions ∂ x j f i = ∂ x i f j .In the case k = ℓ = 1 , the equation of Σ ⊂ J (resp., of Σ ⊂ J ) is ψ ( t, u, u ) = 0 (resp., ψ ( t, u, u ) = 0 and ( D t ψ )( t, u, u , u ) = 0) , (see (76)). Hence, Σ is the set of points j t φ ∈ J such that j t φ ∈ Σ and ( ∂ t ψ + u ∂ u ψ + u ∂ u ψ ) | j t φ = ∂ t ψ | j t φ + d t φ | t ∂ u ψ | j t φ + d t φ | t ∂ u ψ | j t φ = 0 . The last requirement means that the tangent vector (1 , d t φ | t , d t φ | t ) at t of the curve ( t, φ ( t ) , d t φ ) ∈ J is an element of the vector space T j t φ Σ : ∂ t ψ | j t φ t + ∂ u ψ | j t φ u + ∂ u ψ | j t φ u = 0 that is tangent to Σ at j t φ . Thus, Σ = { j t φ ∈ J : gr( j φ ) is tangent to Σ at j t φ } . (77)Observe that the equations of Σ and Σ show that Σ ℓ is not necessarily a smooth manifoldand that π : Σ → Σ is not necessarily a smooth fiber bundle.We now define partial differential equations and their prolongations in a coordinate-freemanner. n Koszul-Tate resolutions Definition 42. A partial differential equation ( resp., a linear partial differential equation ) of order k ( k ≥ acting on sections φ ∈ Γ( π ) of a vector bundle π , is a smooth fiber ( resp.,vector ) subbundle π k : Σ → X of J k ( π ) . The ℓ -th prolongation of Σ ( 0 ≤ ℓ ≤ ∞ ) is thesubset Σ ℓ = { j k + ℓm φ ∈ J k + ℓ ( π ) : gr( j k φ ) is tangent up to order ℓ to Σ at j km φ } (78) of J k + ℓ ( π ) . A ( local ) solution of Σ is a ( local ) section φ of π such that gr( j k φ ) ⊂ Σ . Note that the definition of the prolongation means that the points j k + ℓm φ of Σ ℓ provide ℓ -thorder approximations gr( j k φ ) of possible solutions of Σ . Remark 43.
1. In the following we always assume that the considered equation Σ ⊂ J k ( π ) is formally integrable (see also Subsection 7.8), i.e., that • the prolongations Σ ℓ are smooth manifolds (0 ≤ ℓ ≤ ∞ ) , and • the maps π k + ℓ,k + ℓ +1 : Σ ℓ +1 → Σ ℓ (0 ≤ ℓ < ∞ ) are smooth fiber bundles.2. Let us stress as well that it follows from Definition 42 (see also introduction to the presentsubsection 7.2) that φ is a solution of Σ if gr( j k + ℓ φ ) ⊂ Σ ℓ , (79)for some ≤ ℓ ≤ ∞ , and that, conversely, we have (79) for every ℓ , if φ is a solution.A PDE (resp., a linear
PDE ) Σ of order k in π is implemented by a differentialoperator D ∈ DO k ( π, π ′ ) (resp., D ∈ Diff k ( π, π ′ ) ), if Σ = ker ψ D , where π ′ : E ′ → X is avector bundle and where ψ D ∈ FB ( J k ( π ) , E ′ ) (resp., ψ D ∈ VB ( J k ( π ) , E ′ ) ) is the representativemorphism of D .
In this case, the differential operator j ℓ ◦ D is of order k + ℓ and acts from π to π ′ ℓ . Its decomposition j ℓ ◦ D = ψ j ℓ ◦ D ◦ j k + ℓ (80)corresponds to Equation (75). In the sequel we write ψ ℓD : J k + ℓ ( π ) → J ℓ ( π ′ ) (81)for the representative morphism ψ j ℓ ◦ D of the ℓ -th prolongation j ℓ ◦ D of D . It is now clearthat Σ ℓ = ker ψ ℓD , (82)i.e., that the ℓ -th prolongation of the equation is given by the ℓ -th prolongation of the corre-sponding differential operator (see Equation (76)). n Koszul-Tate resolutions Jet spaces π k : J k ( π ) → X , ≤ k ≤ ∞ , come equipped with a natural geometric structure,their Cartan distribution C k = C k ( π ) , i.e., with an assignment C k : J k ( π ) ∋ κ k
7→ C kκ k ⊂ T κ k ( J k ( π )) (83)of a vector subspace C kκ k of the corresponding tangent space to any point of the jet space. Thissubspace can be defined in a coordinate-free manner, which will however not be detailed here.The next proposition gives the coordinate description of C kκ k . Proposition 44.
Let π : E → X be a vector bundle of rank r over a manifold of dimension n . For any k ≥ and any κ k ∈ J k ( π ) , the Cartan space C kκ k = C kκ k ( π ) is generated by thevectors D ≤ k − x i | κ k = ∂ x i + r X a =1 X | α |≤ k − u aiα ∂ u aα | κ k and ∂ u aα | κ k ,i ∈ { , . . . , n } , a ∈ { , . . . , r } , | α | = k , (84) where ( x i , u aα ) is a trivializing chart of J k ( π ) around π k ( κ k ) . In the limit case k = ∞ , theCartan space C ∞ κ ∞ is generated by the total derivatives D x i | κ ∞ , i ∈ { , . . . , n } . (85)The existence of the extra generators ∂ u aα ( a ∈ { , . . . , r } , | α | = k ) makes the Cartandistribution C k = C k ( π ) non-integrable . Indeed, take, to simplify, the case k = n = r = 1 and note that the bracket [ D ≤ t , ∂ u ] = [ ∂ t + u ∂ u , ∂ u ] = − ∂ u of local vector fields in C is not located in C . This problem disappears at the limit k = ∞ : the Cartan distribution C ∞ = C ∞ ( π ) is n -dimensional and integrable (indeed [ D x i , D x j ] = 0 ). Remark 45.
In the sequel, we deal with limits, e.g., infinite prolongations Σ ∞ . Whenever noconfusion arises, we omit the sub- and superscripts ∞ , thus writing Σ (resp., κ, C , . . . ) insteadof Σ ∞ (resp., κ ∞ , C ∞ , . . . ).Consider now a PDE Σ ⊂ J k ( π ) of order k on π (as mentioned before, we systematicallyassume that the considered PDE -s are formally integrable). We define the Cartan distribution C k (Σ ) of Σ by C k (Σ ) : Σ ∋ κ k
7→ C kκ k ∩ T κ k Σ ⊂ T κ k Σ , (86)and the Cartan distribution C (Σ) of Σ ⊂ J ∞ ( π ) by C (Σ) : Σ ∋ κ
7→ C κ ∩ T κ Σ ⊂ T κ Σ . (87)It can be shown that C κ = C κ ( π ) ⊂ T κ Σ , (88)so that C (Σ) = C ( π ) | Σ . (89)Moreover, just as C ( π ) , the Cartan distribution C (Σ) = C ( π ) | Σ is n -dimensional and integrable .Eventually: n Koszul-Tate resolutions Proposition 46.
The maximal dimensional ( n -dimensional ) integral manifolds of the Cartandistribution C ( π ) ( resp., C (Σ) ) are the graphs gr( j ∞ φ ) of the infinite jets of the local sections φ ∈ Γ loc ( π ) ( resp., the local solutions φ ∈ Γ loc ( π ) of Σ ) . Hence, the set of maximal dimensional integral manifolds in (Σ , C (Σ)) can be identifiedwith the set of solutions of Σ . Since all relevant information about the original PDE Σ is thusencrypted in the pair (Σ , C (Σ)) , the partial differential equation Σ is frequently identified withthe ‘diffiety’ (Σ , C (Σ)) . Diffieties , or, explicitly, differential varieties, are for partial differentialequations what algebraic varieties are for algebraic equations. Diffieties are (often infinite-dimensional) manifolds equipped with a Cartan distribution; they are locally equivalent toinfinite prolongations of differential equations. The Cartan distribution allows developing on adiffiety a specific differential calculus, called Secondary Calculus, whose objects are cohomologyclasses of differential complexes. Many characteristics of a diffiety, i.e., of the correspondingsystems of partial differential equations, can be expressed in terms of Secondary Calculus andvice versa.
Since C ( π ) : J ∞ ( π ) ∋ κ
7→ C κ ( π ) ⊂ T κ J ∞ ( π ) , where C κ ( π ) is the tangent space at κ to the graphs gr( j ∞ φ ) of the sections j ∞ φ that passthrough κ at m = π ∞ ( κ ) , the following statements are rather obvious: • T κ π ∞ : C κ ( π ) → T m X is a vector space isomorphism (it is easily seen that this derivativesends D x i | κ to ∂ x i | π ∞ ( κ ) ). • The F ( π ) -module C Θ( π ) := Γ( C ( π )) (resp., Θ v ( π ) ) of sections of the subbundle C ( π ) ⊂ T J ∞ ( π ) (resp., of π ∞ -vertical vector fields of J ∞ ( π ) ) is a submodule of the F ( π ) -module Θ( π ) of vector fields of J ∞ ( π ) . More precisely, we have Θ( π ) = C Θ( π ) ⊕ Θ v ( π ) . (90)This suggests the idea of connection, i.e., of a C ∞ ( X ) -linear lift (map with the obvious pro-jection property) C : Θ( X ) ∋ θ
7→ C θ ∈ C Θ( π ) . (91)Indeed, its suffices to set, for any κ ∈ J ∞ ( π ) with projection π ∞ ( κ ) = m , ( C θ ) κ := ( T κ π ∞ ) − θ m ∈ C κ ( π ) ⊂ T κ J ∞ ( π ) . (92)This connection C on J ∞ ( π ) is the Cartan connection induced by the Cartan distribution C ( π ) on J ∞ ( π ) . n Koszul-Tate resolutions ( x i , u aα ) of J ∞ ( π ) over U around m = π ∞ ( κ ) , the Cartanspace C κ ( π ) is generated by the D x i | κ , the horizontal vector fields H ∈ C Θ( π ) are locallygenerated over functions of J ∞ ( π ) by the total derivatives D x i : H | π − ∞ ( U ) = X j H j ( x i , u aα ) D x j . (93)Since T κ π ∞ ( D x j | κ ) = ∂ x j | m , a vector field θ | U = P j θ j ( x i ) ∂ x j is lifted to ( C θ ) | π − ∞ ( U ) = X j θ j ( x i ) D x j . (94)Let us also mention, for the sake of completeness, that a vector field T ∈ Θ( π ) ( resp., a verticalvector field V ∈ Θ v ( π ) ) locally reads T | π − ∞ ( U ) = X j T j ( x i , u aα ) ∂ x j + X bβ T bβ ( x i , u aα ) ∂ u bβ ( resp., V | π − ∞ ( U ) = X bβ V bβ ( x i , u aα ) ∂ u bβ ) . (95)We are now able to rewrite the definition of a horizontal lift C θ in a useful way. If θ ∈ Θ( X ) and F ∈ F ( π ) , and if φ is a local section in Γ( π ) that is defined around m ∈ X , we get ( j ∞ φ ) ∗ (( C θ ) F ) | m = (( C θ ) F ) | j ∞ m φ = (cid:0) ( C θ ) j ∞ m φ F (cid:1) | j ∞ m φ = (cid:0) (( T π ∞ ) − θ m ) F (cid:1) | j ∞ m φ = θ m ( F ◦ j ∞ φ ) | m = θ (( j ∞ φ ) ∗ F ) | m . Indeed, the isomorphism ( T π ∞ ) − sends a partial derivative to the corresponding totalderivative. Observe also that, although the function F ◦ j ∞ φ depends on φ , its derivative θ m ( F ◦ j ∞ φ ) | m depends only on j ∞ m φ . Hence, the
Proposition 47.
For any θ ∈ Θ( X ) , F ∈ F ( π ) , and φ ∈ Γ loc ( π ) , we have ( j ∞ φ ) ∗ (( C θ ) F ) = θ (( j ∞ φ ) ∗ F ) . (96)It is clear that we could define the Cartan connection (92) by means of (96), and thatEquation (96) is the generalization of Equation (75).We already explained that [ C Θ( π ) , C Θ( π )] ⊂ C Θ( π ) . Moreover, it immediately followsfrom (96) that C [ θ, θ ′ ] = [ C θ, C θ ′ ] . In other words, the integrable Cartan distribution of J ∞ ( π ) induces a flat Cartan connection on J ∞ ( π ) → X . Further, the increasing sequence C (Θ( X )) ⊂C Θ( π ) ⊂ Θ( π ) is a sequence of Lie subalgebras . Eventually, if Σ is the infinite prolongationof a PDE on π , we set C Θ(Σ) := Γ( C (Σ)) , where C (Σ) is the Cartan distribution of Σ . This F (Σ) -module is locally generated by the D x i | Σ . When restricting the lifts C θ to Σ , we get aconnection C : Θ( X ) → C Θ(Σ) , the Cartan connection on Σ , which is flat as well. Hence, theintegrable Cartan distribution of Σ induces a flat Cartan connection on Σ → X, which is therestriction of the connection on the infinite jet space. n Koszul-Tate resolutions Total differential operators (
TDO -s)
Ψ = X β Ψ β ( x i , u aα ) D βx (97)are of primary importance in Field Theory. The fundamental property is that TDO -s act notonly on F ( π ) , but also on F (Σ) . This is of course due to the fact that total derivatives restrictto (horizontal) vector fields of Σ (see Equation (88)), and is not true for ordinary differentialoperators T = X γ T γ ( x i , u aα ) . . . ◦ ∂ γ j x j ◦ . . . ◦ ∂ γ bβ u bβ ◦ . . . (98)of J ∞ ( π ) . An interesting subclass of TDO -s are the lifts C ∆ = X β ∆ β ( x i ) D βx (99)of linear differential operators ∆ = P β ∆ β ( x i ) ∂ βx acting on C ∞ ( X ) . These lifts can be definedexactly as the lifts of base vector fields in (96).Note first that differential operators act usually not only on functions C ∞ ( X ) (resp., on F ( π ) (functions of J ∞ ( π ) )), but act between sections Γ( η k ) (locally: R r k -valued functions on‘ X ’) of rank r k vector bundles η k : E k → X (resp., between sections F ( π, η k ) = Γ( π ∗∞ ( η k )) (locally: R r k -valued functions on ‘ J ∞ ( π ) ’) of the bullbacks π ∗∞ ( η k ) : π ∗∞ ( E k ) → J ∞ ( π ) ofthese bundles). Hence, the Definition 48.
Let π : E → X and η k : E k → X ( k ∈ { , } ) be vector bundles. The lift of a linear differential operator ∆ : Γ( η ) → Γ( η ) is the linear differential operator C ∆ : F ( π, η ) → F ( π, η ) ( of same order ) defined by ( j ∞ φ ) ∗ (( C ∆) S ) = ∆(( j ∞ φ ) ∗ S ) , (100) where S ∈ F ( π, η ) and φ ∈ Γ loc ( π ) . The difference with lifts C θ = X j θ j ( x i ) D x j ∈ C Θ( π ) of vector fields is that the horizontal or C -vector fields C Θ( π ) had been defined before the lifts C θ . Here, i.e., for lifts C ∆ of differential operators, we still need to find the proper definition of C -differential operators C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) . In view of (93), these C -differential operatorsshould locally be the TDO -s Ψ = X β Ψ β ( x i , u aα ) D βx , see (97). Since, for any F ∈ F ( π ) and any φ ∈ Γ( π ) , this model C -differential operator Ψ satisfies (Ψ F ) ◦ j ∞ φ = X β (Ψ β ◦ j ∞ φ ) (( D βx F ) ◦ j ∞ φ ) = X β (Ψ β ◦ j ∞ φ ) ∂ βx ( F ◦ j ∞ φ ) =: Ψ φ ( F ◦ j ∞ φ ) , n Koszul-Tate resolutions ( j ∞ φ ) ∗ (Ψ F ) = Ψ φ (( j ∞ φ ) ∗ F ) , where the RHS Ψ • (see its definition) is a not necessarily linear differential operator in φ ∈ Γ( π ) with values Ψ φ in linear differential operators on C ∞ ( X ) . This motivates the Definition 49.
A linear differential operator
Ψ : F ( π, η ) → F ( π, η ) is a C -differentialoperator Ψ ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) , if, for any φ ∈ Γ( π ) , there exists a linear differentialoperator Ψ φ : Γ( η ) → Γ( η ) , such that, for any S ∈ F ( π, η ) , the equality ( j ∞ φ ) ∗ (Ψ S ) = Ψ φ (( j ∞ φ ) ∗ S ) (101) holds. This definition captures correctly our intuition of C -differential operators. Since it is clearfrom its definition that the lift C of differential operators respects composition, we have, locally, X β Ψ β ( x i , u aα ) D βx = X β Ψ β ( x i , u aα ) C ( ∂ βx ) . It can be shown [KV98] that this result is global:
Proposition 50.
Any Ψ ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) reads Ψ = X β Ψ β C ∆ β , (102) where the sum is finite, where Ψ β ∈ F ( π ) , and where ∆ β ∈ Diff( η , η ) . In other words, C -differential operators are generated over F ( π ) by lifts. Moreover, just as
TDO -s, C -differential operators can be restricted to the infinite prolon-gation Σ of a PDE . More precisely [KV98],
Corollary 51.
For any C -differential operator Ψ : F ( π, η ) → F ( π, η ) and any infiniteprolongation Σ ⊂ J ∞ ( π ) , there is a linear differential operator Ψ Σ : F (Σ , η ) → F (Σ , η ) suchthat, for every s ∈ F ( π, η ) , we have Ψ Σ ( s | Σ ) = (Ψ s ) | Σ . Finally, we have the important
Corollary 52.
There is a canonical F ( π ) -module isomorphism C : F ( π ) ⊗ C ∞ ( X ) Diff( η , η ) → C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) (103) between the linear differential operators with coefficients in the jet space functions and thecorresponding C -differential operators. In particular, in the case of the trivial line bundle η = η , we get the isomorphism C : F ( π ) ⊗ C ∞ ( X ) D ( X ) → CD ( J ∞ ( π )) . (104) n Koszul-Tate resolutions Proof.
Observe first that the action of a differential operator F ⊗ ∆ , with F ∈ F ( π ) and ∆ ∈ D ( X ) , on a function f ∈ C ∞ ( X ) is naturally defined by ( F ⊗ ∆)( f ) = F ((∆ f ) ◦ π ∞ ) . The action ( F ⊗ ∆)( s ) , ∆ ∈ Diff( η , η ) and s ∈ Γ( η ) , is defined similarly: ( F ⊗ ∆)( s ) = F ((∆ s ) ◦ π ∞ ) . (105)The map C : F ( π ) ⊗ C ∞ ( X ) Diff( η , η ) ∋ F ⊗ ∆ F C ∆ ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) , (106)is obviously well-defined and F ( π ) -linear. To prove injectivity, assume that F ( C ∆)( S ) = 0 ,for all S ∈ Γ( π ∗∞ ( η )) , in particular, for all S = s ◦ π ∞ , s ∈ Γ( η ) . It follows from (100) that ( F ◦ j ∞ φ ) ∆ s = ( F ((∆ s ) ◦ π ∞ )) ◦ j ∞ φ = 0 , for all s, φ . Eventually, (105) allows to conclude that F ⊗ ∆ = 0 . As for surjectivity, recallthat any C -differential operator Ψ reads P β Ψ β C ∆ β , and note that P β Ψ β ⊗ ∆ β is a preimageof Ψ .Let us summarize in coordinate language what we achieved so far. Consider a PDE ψ b ( x i , ∂ αx φ a ) ≡ , ∀ b , whose LHS sends sections φ = ( φ a ( x )) a ∈ Γ( π ) to sections ψ = ( ψ b ( x )) b := ( ψ b ( x i , ∂ αx φ a )) b ∈ Γ( η ) . We take into account the linear differential consequences ∆ ψ b ( x i , ∂ αx φ a ) := X β M cβb ( x ) ∂ βx ψ b ( x i , ∂ αx φ a ) ≡ , ∀ c of this equation, where ∆ ∈ Diff( η , η ) . The latter condition can be rewritten in the form ( C ∆) ψ b ( x i , u aα ) | j ∞ x φ = X β M cβb ( x ) D βx ψ b ( x i , u aα ) | j ∞ x φ ≡ , ∀ c , thus leading to a C -differential operator C ∆ ∈ C Diff( π ∗∞ ( η ) , π ∗∞ ( η )) . Just as the value ψ b ( x i , ∂ αx φ a ) | m at m ∈ X (in fact we mean here the coordinates of m ; the same notational abuse will betolerated in the sequel) of the image of φ = ( φ a ( x )) a ∈ Γ( π ) by a differential operator in DO k ( π, η ) only depends on the values ∂ αx φ a | m of the coefficients of the ‘Taylor expansion’ of φ at m up to order k , the value X β N cβb ( x i , u aα ) D βx ψ b ( x i , u aα ) | κ n Koszul-Tate resolutions κ ∈ J ∞ ( π ) of the image of ψ = ( ψ b ( x i , u aα )) b ∈ Γ( π ∗∞ ( η )) by a C -differential operator in C Diff k ( π ∗∞ ( η ) , π ∗∞ ( η )) only depends on the values D βx ψ b ( x i , u aα ) | κ of the total or horizontalderivatives of ψ at κ up to order k . In fact, the C -differential calculus is similar to the ordinarydifferential calculus. For k ∈ N ∪ {∞} , the horizontal k -jet ¯ kκ S at κ ∈ J ∞ ( π ) of a localsection S ∈ Γ( π ∗∞ ( η )) that is defined around κ is the equivalence class of all such local sections,whose coordinate forms in a trivializing chart ( x i , u aα , v b ) around κ coincide at κ , together withtheir total derivatives at κ up to order k . Remark 53.
In the following, if π : E → X and ρ : F → X are two vector bundles, we set R := π ∗∞ ( ρ ) and R := Γ( R ) = Γ( π ∗∞ ( ρ )) .The set ¯ J k ( H ) = { ¯ kκ S : κ ∈ J ∞ ( π ) , S ∈ H } is a vector bundle H ,k : ¯ J k ( H ) → J ∞ ( π ) , called the horizontal k -jet bundle . A trivializingchart ( x i , u aα , v b ) of H induces a trivializing chart ( x i , u aα , v bβ ) of H ,k given by x i (¯ kκ S ) = x i ( κ ) , u aα (¯ kκ S ) = u aα ( κ ) , v bβ (¯ kκ S ) = D βx S b | κ . (107)As already suggested above here, the C -differential or horizontal differential operators Ψ ∈ C Diff k ( H , H ) are those Ψ ∈ Hom R ( H , H ) that factor through the horizontal k -jet bundle ¯ J k ( H ) , i.e., that read Ψ = ψ ◦ ¯ k , for some(and thus unique) vector bundle map ψ ∈ VB ( H ,k , H ) ≃ Hom F ( π ) (Γ( ¯ J k ( H )) , H ) . Actually, the whole theory of jet bundles can be transferred to horizontal jet bundles [Ver02].Indeed, it follows from what has been said that, in the coordinate setting, horizontal jet bundlesare just jet bundles with extra coordinates u aα in the base. The concept of symmetry is of fundamental importance in many fields of Science and de-serves special attention. The notion is quite straightforward – at least in elementary situations– . For instance, when thinking about an axial symmetry of a plane domain S , we get a bijec-tion p from the plane to itself, such that p ( S ) = S . Similarly, a symmetry of an equation Σ ⊂ J k ( π ) should be a fiber bundle automorphism (or, just a diffeomorphism) ψ of J k ( π ) such that ψ (Σ ) = Σ . (108) n Koszul-Tate resolutions J k ( π ) is the Cartan distribution C k (i.e., the infinitesi-mal object that encodes jet prolongations of sections), it seems natural to ask that a symmetryrespect the Cartan distribution (or, better, that its tangent map does).In the following, we focus on automorphisms of J k ( π ) that respect C k , thus omitting firstCondition (108). We refer to such automorphisms as Lie automorphisms of π k . In particular,we may ask whether it is possible to build a Lie automorphism of π k as a prolongation of anautomorphism of π . It is easily seen that, if
Ψ = ( ψ , ψ ) is a fiber bundle automorphism of π : E → X , we canprolong it to a fiber bundle automorphism j ℓ Ψ := ( ψ , j ℓ ψ ) of π ℓ : J ℓ ( π ) → X . It actuallysuffices to recall that ψφψ − ∈ Γ( π ) , for any φ ∈ Γ( π ) (as elsewhere in this text, we donot insist here on the possibility that φ might be defined only locally), and to consider thewell-defined fiber bundle automorphism j ℓ ψ : J ℓ ( π ) ∋ j ℓm φ j ℓψ ( m ) ( ψφψ − ) ∈ J ℓ ( π ) . It can easily be checked that the lift j ℓ Ψ is a Lie automorphism, i.e., that, for any κ ℓ ∈ J ℓ ( π ) ,the inclusion ( T κ ℓ j ℓ ψ )( C ℓκ ℓ ) ⊂ C ℓj ℓκℓ ψ (109)holds. Let us still mention that the prolongation j ℓ ψ : J ℓ ( π ) → J ℓ ( π ) of ψ : J ( π ) → J ( π ) isreally a lifting, in the sense that π ℓ ◦ j ℓ ψ = ψ ◦ π ℓ . Instead of considering finite automorphisms or diffeomorphisms, we can take an interestin infinitesimal ones, i.e, in vector fields. Note that a vector field Ξ ∈ Θ( π ) , i.e., a field of π : E → X (we avoid writing Θ( π ) , since this notation is used instead of the more precise Θ( π ∞ ) ), is a π - projectable vector field if and only if T π Ξ e = ξ π ( e ) , for all e ∈ E , i.e., if andonly if there is a vector field ξ ∈ Θ( X ) that is π -related to Ξ . It is well-known that this meansthat π intertwines the flows ψ Ξ t and ψ ξt , i.e., that π ◦ ψ Ξ t = ψ ξt ◦ π (assume for simplicity thatthe flows are globally defined). In other words, Ψ Ξ t = ( ψ ξt , ψ Ξ t ) is a 1-parameter group of fiberbundle isomorphisms of π : E → X , and it can thus be prolonged to a 1-parameter group ofLie isomorphisms j ℓ Ψ Ξ t = ( ψ ξt , j ℓ ψ Ξ t ) of π ℓ : J ℓ ( π ) → X . The latter implements a vector field j ℓ Ξ ∈ Θ( π ℓ ) – the ℓ -jet prolongation of the projectable vector field Ξ ∈ Θ( π ) – . In otherwords, the lift j ℓ Ξ is given by ( j ℓ Ξ) j ℓm φ = d t | t =0 j ℓψ ξt ( m ) ( ψ Ξ t φψ ξ − t ) . The flow of the prolongation j ℓ Ξ of Ξ is thus the prolongation j ℓ ψ Ξ t of the flow of Ξ , which ismade of Lie isomorphisms. The explicit coordinate computation of the lift of the projectablefield Ξ = X j A j ( x i ) ∂ x j + X b B b ( x i , u a ) ∂ u b = X j A j ( ∂ x j + u bj ∂ u b ) + X b ( B b − A j u bj ) ∂ u b (110) n Koszul-Tate resolutions j ℓ Ξ = X j A j D ≤ ℓ − x j + X b X | β |≤ ℓ − D βx ( B b − A j u bj ) ∂ u bβ (111)[Kru73]. Note that the first term (resp., second term) of the lift is obtained by extending thetotal derivatives D ≤ x j in (110) to D ≤ ℓ − x j (resp., by adding new terms whose coefficients are thecorresponding total derivatives of the coefficients in (110)).Hence, any fiber bundle automorphism of π (resp., any projectable vector field of π ) canbe prolonged to a fiber bundle automorphism of π ℓ (resp., a vector field of π ℓ ) that respects(whose flow respects) the Cartan distribution C ℓ . The result can be generalized to arbitrarydiffeomorphisms ψ : J ( π ) → J ( π ) (resp., vector fields Ξ ∈ Θ( π ) ). More precisely, anydiffeomorphism (resp., vector field) of π can be lifted to a diffeomorphism (resp., vector field) of π ℓ that (whose flow) respects the Cartan distribution. We refer to such distribution respectingdiffeomorphisms and vector fields as Lie transformations and
Lie fields , respectively (inthe case ℓ = 0 , any vector in T e E is tangent to a section, so C e = T e E , and Lie transformations(resp., Lie fields) are just diffeomorphisms (resp., vector fields)). The lift to π ℓ of an arbitraryvector field of π , i.e., of Ξ = X j A j ( x i , u a ) ∂ x j + X b B b ( x i , u a ) ∂ u b = X j A j ( ∂ x j + u bj ∂ u b ) + X b ( B b − A j u bj ) ∂ u b , (112)is locally given by the same formula (111) as before [Vit11] (any Lie transformation (resp., Liefield) of π k can be lifted to a Lie transformation (resp., Lie field) of any π k + ℓ ). Conversely, anyLie transformation (resp., any Lie field) of π ℓ is the lift of a diffeomorphism (resp., a vectorfield) of π , at least if rk( π ) > , [KV98], [Vit11]. In view of what has been said above, a symmetry of an equation Σ ⊂ J k ( π ) is a Lietransformation ψ of J k ( π ) such that ψ (Σ ) = Σ . As also mentioned before, we do in thistext usually not insist on possible local aspects. For instance, we could consider here localsymmetries of Σ ⊂ J k ( π ) , i.e., Lie transformations ψ of an open subset U ⊂ J k ( π ) such that ψ ( U ∩ Σ ) = U ∩ Σ . The notion of infinitesimal symmetry of an equation Σ ⊂ J k ( π ) isnow clear as well. It is a Lie field τ of J k ( π ) that is tangent to Σ , i.e., such that τ κ ∈ T κ Σ , for all κ ∈ Σ . Let us recall that we systematically assume that the considered equations are formallyintegrable (see Remark 43 and Subsection 7.8). Just as a Lie transformation (resp., a Liefield) of J k ( π ) lifts to a Lie transformation (resp., a Lie field) of any J k + ℓ ( π ) , a symmetry(resp., an infinitesimal symmetry) of Σ ⊂ J k ( π ) lifts to a symmetry (resp., an infinitesimalsymmetry) of any Σ ℓ ⊂ J k + ℓ ( π ) (the converse is true as well) [KV98, Prop. 3.23]. Hence, asymmetry (resp., an infinitesimal symmetry) of Σ induces a symmetry (resp., an infinitesimal n Koszul-Tate resolutions Σ := Σ ∞ . To avoid diffeomorphisms of infinite dimensional spaces, we considerin the following only infinitesimal symmetries and call them just symmetries . Further, we willstudy not only the symmetries of Σ that are implemented by symmetries of Σ (such inducedsymmetries are Lie fields, i.e., the derivatives of the diffeomorphisms obtained from their flowsrespect the Cartan distribution), but ‘all symmetries’ of Σ (such ‘higher symmetries’ willrespect the Cartan distribution in a generalized sense).Recall that a symmetry of Σ = Σ ∞ is a vector field T ∈ Θ( π ) of J ∞ ( π ) that is tangentto Σ and that is Lie. A higher symmetry of Σ (or simply a symmetry of Σ whenever noconfusion is possible) is a vector field T ∈ Θ( π ) that is tangent to Σ and respects the Cartandistribution C = C ( π ) of J ∞ ( π ) , not in the preceding sense that the derivatives of its flowrespect C , but in the sense that [ T, C Θ( π )] ⊂ C Θ( π ) , (113)where C Θ( π ) = Γ( C ( π )) is the space of Cartan fields. Just as above, where we omitted Condition (108), we will forget now temporarily thetangency condition, and study infinite jet space vector fields that satisfy the Cartan condition(113). These fields will be called in the following symmetries of C . In view of the Jacobiidentity, the space Θ C ( π ) of symmetries of C is a Lie R -subalgebra of Θ( π ) . Since C is integrable,Cartan fields C Θ( π ) are trivial symmetries of C , and, by definition, they thus form a Lieideal of Θ C ( π ) . The quotient sym( π ) := Θ C ( π ) / C Θ( π ) is the Lie algebra of proper symmetries of C . In view of the Cartan connection (90), wehave the direct sum decomposition Θ C ( π ) = C Θ( π ) ⊕ EΘ( π ) , (114)where EΘ( π ) = { T ∈ Θ v ( π ) : [ T, C Θ( π )] ⊂ C Θ( π ) } . (115)It follows that sym( π ) ≃ EΘ( π ) , (116)i.e., that any proper symmetry of C is naturally represented by a vertical symmetry, or, still,by an evolutionary vector field .Although it is not difficult, we will not explain here that for any V ∈ Θ v ( π ) the symmetryor the evolutionary condition is equivalent to [ V, C (Θ( X ))] = 0 . Since the lifts C (Θ( X )) are locally generated over C ∞ ( X ) by the total derivatives and sincethe local form of a vertical vector field is completely defined by its values on the coordinate n Koszul-Tate resolutions u aα , this condition reads locally [ V, D x i ] = 0 , or, still, [ V, D x i ]( u aα ) = 0 . Noticing that D x i u aα = u aiα , we finally obtain V aiα = V ( u aiα ) = V ( D x i u aα ) = D x i ( V ( u aα )) = D x i V aα . In other words, V ∈ Θ v ( π ) is a local symmetry or evolutionary field if and only if its coefficientssatisfy V aiα = D x i V aα . (117)This shows that evolutionary vector fields V ∈ EΘ( π ) are completely determined (locally, bytheir coefficients V a , i.e., globally,) by their restriction V | F ∈ Der v ( F , F ) .More precisely, there is a 1:1 correspondence between EΘ( π ) and Der v ( F , F ) . It is worthto further elaborate on this idea. Let X ∈ Der( F , F ) . Locally, this is a vector field X of J ( π ) with coefficients in functions of J ∞ ( π ) : X = X j A j ( x i , u aα ) ∂ x j + X b B b ( x i , u aα ) ∂ u b = X j A j ( ∂ x j + u bj ∂ u b ) + X b ( B b − A j u bj ) ∂ u b . (118)Such a field can be prolonged to a field of J ∞ ( π ) in the way specified by formula (111), exactlyas in the particular cases (110) and (112) – except that ℓ = ∞ here. The prolonged vectorfield (111) is the sum of a term in C Θ( π ) (horizontal fields are locally generated over F by thetotal derivatives) and a term in EΘ( π ) (see Equation (117)). In particular, if we start from X ∈
Der v ( F , F ) , i.e., locally, from X = X b B b ( x i , u aα ) ∂ u b , (119)we obtain the evolutionary vector field δ X = X b,β D βx B b ∂ u bβ ∈ EΘ( π ) . (120)Note that a local vertical derivation (119) is the same as a local section B = ( B b ( x i , u aα )) b of π ∗∞ ( π ) . The point is that this isomorphism Der v ( F , F ) ≃ Γ( π ∗∞ ( π )) = F ( π, π ) =: κ ( π ) (121)holds globally and that the local evolutionary fields (120), computed from the global X ∈
Der v ( F , F ) , can be glued to provide a global evolutionary field δ X ∈ EΘ( π ) .It is noteworthy that the δ : κ ( π ) ∋ X 7→ δ X ∈ EΘ( π ) (122)allows to push the F ( π ) -module structure of κ ( π ) forward to EΘ( π ) (this multiplication isdifferent (!) from that of vector fields of π ∞ by functions of π ∞ ) and to pull the Lie algebrastructure of EΘ( π ) back to κ ( π ) . n Koszul-Tate resolutions δ allows introducing a linearization of a not necessarilylinear differential operator D ∈ DO( π, π ′ ) ≃ ψ D ∈ F ( π, π ′ ) between two vector bundles π and π ′ . For any X ∈ κ ( π ) , one can extend the action on F ( π ) of δ X ∈ EΘ( π ) to an action on F ( π, π ′ ) . Locally, this claim is obvious – the point is that the extended action is actually aglobal one – . The operator ℓ D : κ ( π ) ∋ X 7→ ℓ D X := δ X ψ D ∈ F ( π, π ′ ) (123)is the universal linearization operator of D . In view of (120), we have ℓ D X = δ X ψ D = X b,β ∂ u bβ ψ D D βx X b . (124)In fact, the partial derivatives ∂ u bβ ( b ∈ { , . . . , rk( π ) } ) act on the components ψ aD ( a ∈{ , . . . , rk( π ′ ) } ) of ψ D . In other words, the coordinate expression of the linearization operator is ℓ D = X β (cid:16) ∂ u bβ ψ aD (cid:17) a,b D βx , (125)where a (resp., b ) refers to the row (resp., column). The linearization of any (not necessarilylinear) differential operator D ∈ DO( π, π ′ ) is a ( linear ) horizontal differential operator ℓ D ∈ C Diff( π ∗∞ ( π ) , π ∗∞ ( π ′ )) . (126)Observe also that the coefficients ∂ u bβ ψ D of the linearization of D ≃ ψ D or of ker ψ D = Σ arecoefficients of the equation of the tangent space of Σ . To upgrade an evolutionary vector field V ∈ EΘ( π ) of J ∞ ( π ) to a symmetry of Σ (a proper generalized symmetry of the equation Σ ), we must still add the requirement that V κ ∈ T κ J ∞ ( π ) be tangent to the prolongation Σ ⊂ J ∞ ( π ) when κ ∈ Σ : V κ ∈ T κ Σ , for all κ ∈ Σ . In other words, the considered evolutionary field is a symmetry of the equation Σ ifand only if it acts on functions F (Σ) of the infinite prolongation Σ of Σ . The space of allsymmetries of Σ is a Lie R -algebra that we denote by EΘ(Σ) .To finish this review of symmetries, we ask what classical and higher symmetries mean lo-cally, in coordinates, in the case the considered formally integrable equation Σ is implementedby a differential operator D ≃ ψ D , i.e., Σ = ker ψ D . Let first τ ∈ Θ( π k ) be a Lie field that is tangent to Σ . This Lie field is (if rk( π ) > ) thelift τ = j k Ξ of a vector field Ξ ∈ Θ( π ) . Further, the tangency property means locally that,for any κ k ∈ Σ , we have L j k Ξ ψ D | κ k ≃ h ( ψ D ( κ k + hτ κ k ) − ψ D ( κ k )) = 0 . (127) n Koszul-Tate resolutions Ξ transforms a solution into a solution up to terms oforder ≥ in the infinitesimal parameter).Consider now X ∈ κ ( π ) , as well as the corresponding proper symmetry δ X ∈ EΘ( π ) of C .When remembering that this field is a symmetry δ X ∈ EΘ(Σ) of Σ if and only if it acts on F (Σ) , we conclude rather easily that the Σ -symmetry condition for δ X is ( δ X ψ D ) | Σ = 0 , (128)or, still, ( ℓ D X ) | Σ = ℓ D | Σ X | Σ = 0 , (129)since ℓ D is a horizontal differential operator and can thus be restricted. In other words, if wedenote the restriction of the linearization ℓ D (resp., of the section X ) by ℓ Σ (resp., X Σ ), weget the Proposition 54.
Let Σ be a formally integrable PDE in π , implemented by a differentialoperator and with infinite prolongation Σ . An evolutionary vector field δ X generated by X ∈ κ ( π ) is a symmetry δ X ∈ EΘ(Σ) of Σ under the necessary and sufficient condition that X Σ ∈ ker ℓ Σ . (130) Remark 55.
We suggest to read this Subsection after Subsection 2.4.1.We finally explain the gauge theoretical concepts of symmetry of the Euler-Lagrange equa-tions, symmetry of the action, and gauge symmetry. As usual, we denote the coordinates ofthe considered trivial bundle π : E = R n × R r → X = R n by ( x i , u a ) and the Lagrangian ofthe theory by L ( x i , u aα ) .As mentioned above, a vector field X of J ( π ) with coefficients in functions of J ∞ ( π ) (see Equation (118)) can be prolonged to a field of J ∞ ( π ) in the way described by Equation(111) (with ℓ = ∞ ). This prolongation j ∞ X ∈ Θ( π ) is the sum of a horizontal vector field A j D x j ∈ C Θ( π ) and an evolutionary vector field δ X ∈ EΘ( π ) .In conformity with the symmetry conditions (127) and (128), we say that the generalizedvector field X ∈ Der( F , F ) is a symmetry of the Euler-Lagrange equations δ u a L| j k φ =0 , ∀ a , if δ X ( δ u a L ) ≈ , ∀ a . (131)As said before, the requirement means that the infinitesimal transformation induced by X transforms a solution into a solution up to terms of order ≥ in the infinitesimal parameter.As for the concept of symmetry of the action, remember first a well-known fact of La-grangian Mechanics. In Electromagnetism, the gauge the transformation F ′ = F − ∂ t θ, ~A ′ = ~A + ~ ∇ θ n Koszul-Tate resolutions F and ~A are the scalar and vector potentials, θ is a function of time and positions, and ~ ∇ is thegradient) modifies the generalized electromagnetic potential U = e ( F − ~v · ~A ) ( e is the chargeand ~v the velocity of the considered particle) and thus leads to different Lagrangians L and L ′ .However, it is easily seen that the latter differ by the total derivative L ′ − L = d t of a function of time and positions, and that the Euler-Lagrange equations associated to L and L ′ , hence,the dynamics, are therefore the same. This observation can be extended to the present fieldtheoretic context. Two Lagrangians L , L ′ ∈ e F implement the same Euler-Lagrange equationsif and only if they differ by a total divergence: δ u a L = δ u a L ′ , ∀ a ⇔ L ′ − L = D x i i , i ∈ e F . This indicates that two action functionals S L and S L ′ , which are defined by Lagrangians L and L ′ , coincide (on all compactly supported sections) if and only if the underlying Lagrangians L , L ′ differ by a total divergence. It is thus natural to identify the space of action function-als S L with the space of classes [ L ] of functions L ∈ e F considered up to total divergence.Alternatively, an action can be viewed as a class [ L d x ] , where d x = d x . . . d x n and where L d x ≃ L d x + D x i i d x . A symmetry of the action is now a generalized vector field X , such that δ X [ L d x ] = [0] . This definition only makes sense, if we define how the prolongation δ X acts on the differentialform d x and show that its action on [ L d x ] is well-defined. We confine ourselves here tomentioning that the symmetry condition finally reads δ X L = D x i i , where i ∈ F , i.e., just requires that δ X L be a total divergence. Moreover, any symmetry ofthe action is a symmetry of the Euler-Lagrange equations (but the converse is not true).Eventually, a gauge symmetry is a symmetry X ( f ) = A j ( x i , u aα ) ∂ x j + B b ( x i , u aα ) ∂ u b = A j ( ∂ x j + u bj ∂ u b ) + ( B b − A j u bj ) ∂ u b (132)of the action, whose coefficients A j = A j ( f ) = A jα D αx f and B b = B b ( f ) = B bβ D βx f are the values of some total differential operators on an arbitrary / a varying function f ∈ F .Symmetries of the action (resp., symmetries of the action obtained as value of a gaugesymmetry on a specific / a fixed function f ∈ F ) are often termed as global symmetries (resp., local symmetries ). Further, we call symmetry in characteristic form a symmetrygiven by a vertical generalized vector field X = C b ( x i , u aα ) ∂ u b ∈ Der v ( F , F ) . n Koszul-Tate resolutions X (see Equation (132)) provides a symmetry X = ( B b − A j u bj ) ∂ u b in characteristic form (note that X is a symmetry, since δ X = δ X ). Einstein qualified Noether’s results as a monument of mathematical thinking. The tightrelationship between symmetries and conserved quantities is part of each course in ClassicalMechanics. More precisely, Noether’s theorems claim that there exists a 1:1 correspondencebetween (equivalence classes of) symmetries of the action in characteristic form and (equiv-alence classes of) ‘conserved currents’, and that there exists a 1:1 correspondence betweengauge symmetries in characteristic form and Noether identities [Noe18], [Kos11].The latter correspondence is via formal adjoint operators. More precisely, if N aα D αx δ u a L ≡ is a Noether identity, we consider the total differential operator N with components N a = N aα D αx , and define the corresponding gauge symmetry in characteristic form X ( f ) = C a ( f ) ∂ u a as the adjoint N + of N , i.e., by C a ( f ) = N a + ( f ) = ( − D x ) α ( N aα f ) . The converse associa-tion is similar. It follows that non-trivial Noether identities correspond to non-trivial gaugesymmetries in characteristic form. An overdetermined system is a system of linear equations that are not independent, sothat the existence of a solution is subject to compatibility conditions .The simplest example of an overdetermined system is a system of linear equations LX = C , where L ∈ gl( p × n, R ) , X ∈ R n , and C ∈ R p , whose rank ρ ( L ) = p . This meansthat, between the ( LHS -s of the) equations, i.e., between the rows L i ⋆ of L , there do exist non-trivial linear relations. In the following, we assume for simplicity that there is exactly one suchrelation, L p ⋆ = P p − j =1 λ j L j ⋆ , with λ j ∈ R . This existence of non-trivial linear relationsbetween the equations is equivalent to the existence of a non-zero linear operator, in theconsidered case, the non-zero linear operator Λ = ( λ , . . . , λ p − , − ∈ gl(1 × p, R ) , suchthat Λ ◦ L = 0 . Hence, the existence of a solution X requires that C satisfies the compatibilitycondition C ∈ ker Λ , i.e., C p = P p − j =1 λ j C j . In this case, the original system reduces to L ′ X = C ′ , with self-explaining notation, and, in view of our assumption, we have ρ ( L ′ ) = p − .Of course, a homogeneous system always reduces. The most general solution then dependson n − ( p − ≥ parameters, so that C ∈ im L and the complex R n L −→ R p Λ −→ R is exact . n Koszul-Tate resolutions R n , which corresponds to the system of linear PDE -s d f = ω , where d : C ∞ ( R n ) → Ω ( R n ) is the de Rham differential. The non-triviallinear partial differential relations ∂ x j ∂ x i f − ∂ x i ∂ x j f = 0 (133) between the PDE -s can be equivalently written as d d = 0 , where the non-zero linearpartial differential operator d is the de Rham operator on 1-forms: C ∞ ( R n ) d −→ Ω ( R n ) d −→ Ω ( R n ) . The existence of a solution implies that the compatibility condition ω ∈ ker d holds. Sincethe complex is exact , we then have ω ∈ im d , i.e., the considered PDE admits a solution .More generally, let D ∈ Diff( π, π ′ ) be a linear differential operator between smooth sectionsof vector bundles π : E → X and π ′ : E ′ → X over a manifold X . The linear (homogeneous) PDE implemented by D ≃ ψ D is called overdetermined , if there exists a non-zero lineardifferential operator ∆ ∈ Diff( π ′ , π ′′ ) , such that Γ( π ) D −→ Γ( π ′ ) ∆ −→ Γ( π ′′ ) is a complex (of C ∞ ( X ) -modules). We then say that ∆ is a compatibility operator for D ,if the pair (∆ , π ′′ ) is universal in the obvious sense.Just as the original operator D can be overdetermined (non-trivial linear differential rela-tions between the corresponding equations – compatibility operator), a compatibility operator ∆ can itself be overdetermined (relations between the relations – new compatibility operator).This then leads to a compatibility complex of the original operator D : Γ( π ) D −→ Γ( π ′ ) ∆ −→ Γ( π ′′ ) ∆ −→ Γ( π ′′′ ) ∆ −→ . . . In fact, any D ∈ Diff k ( π, π ′ ) admits a compatibility complex in the abelian category Mod ( O ) of modules over O = C ∞ ( X ) , but not necessarily in the non-abelian category rC ∞ VB ( X ) offinite rank smooth vector bundles over X . Indeed, for any k ∈ N , the algebraicized k -prolongation ψ k D ∈ Hom O (Γ( π k + k ) , Γ( π ′ k )) of D admits a cokernel ψ ∈ Hom O (Γ( π ′ k ) , P ) in Mod ( O ) , which represents a differential operator ∆ ∈ Diff k ( π ′ , P ) . Since ψ is the cokernelof ψ k D , the operator ∆ satisfies ∆ ◦ D = ψ ◦ j k ◦ D = ψ ◦ ψ k D ◦ j k + k = 0 . (134)In fact ∆ is universal and is thus a compatibility operator of D . When turning the crankagain and again, we obtain a compatibility complex of D : Γ( π ) D −→ Γ( π ′ ) ∆ −→ P −→ P −→ . . . (135)Here we actually use the algebraic approach – in the frame of O -modules – to differential oper-ators, see for instance [KV98], [GKP13b], [GKP13a]. However, the O -modules P , P , . . . are n Koszul-Tate resolutions Γ( π ′′ ) , Γ( π ′′′ ) , . . . of sections of vector bundles of finite rank.In the following, we stay within the setting of algebraic differential operators and considera diagram of the type we just used to construct a compatibility operator (see Equations (134),(80), (75)): · · · −→ P i − i − −→ P i ∆ i −→ P i +1 −→ · · · j ki − ki + ℓ y j ki + ℓ y j ℓ y · · · −→ J ki − ki + ℓ ( P i − ) ψ ki + ℓ ∆ i − −→ J ki + ℓ ( P i ) ψ ℓ ∆ i −→ J ℓ ( P i +1 ) −→ · · · (136)Here P i − , P i , P i +1 are O -modules, ∆ i − ∈ Diff k i − ( P i − , P i ) , ∆ i ∈ Diff k i ( P i , P i +1 ) , ℓ ∈ N ,and J k ( P ) is the algebraic counterpart of Γ( J k ( P )) , where P → X is a vector bundle and J k ( P ) is the ordinary k -jet bundle (‘algebraic counterpart’ means that, in the geometric case P = Γ( P ) , we have J k ( P ) = Γ( J k ( P )) ).The bottom row of (136) is made of prolonged algebraicized operators, or, still, prolongedformal operators (acting on formal derivatives, i.e., on jet space coordinates). The study offormal operators is referred to as the formal theory . Note that the word ‘formal’ appearsnaturally here and refers to the algebraicized or jet space setting.It is clear (see above) that one of the main questions in the context of compatibilitycomplexes is exactness (exactness of the top row in (136)), i.e., ‘the question whether theconsidered equation admits a solution whenever the compatibility condition is satisfied’. Thequestion of exactness can of course also be considered in the (simpler) formal theory (exactnessof the bottom row).More precisely, a compatibility complex (top row) is called formally exact , if the corre-sponding formal complex (bottom row) is exact, for any ℓ ∈ N . In this case, the main task isto look for criteria for exactness of the original (top row) complex.We will not investigate the latter problem. On the other hand, it is important to knowthat [KV98], for any sufficiently large k ∈ N , the compatibility complex (135) is formallyexact, for any operator D . We actually have the Proposition 56.
Any linear differential operator D ∈ Diff( π, π ′ ) admits a formally exactcompatibility complex. The same is true for any horizontal linear differential operator D ∈C Diff( π ∗∞ ( η ) , π ∗∞ ( η ′ )) . We now briefly comment on formal integrability of a linear partial differential equation Σ or linear differential operator D .The first observation is that the category rC ∞ VB ( X ) is not Abelian. Indeed, kernels, likee.g., Σ ℓ = ker ψ ℓD , are not necessarily vector bundles over X . The reason is that, if ψ : E → E ′ n Koszul-Tate resolutions X , the rank ρ ( ψ m ) of the linear map ψ m : E m → E ′ m mayvary with m ∈ X . Then, the kernel ker ψ := ` m ∈ X ker ψ m is a bundle of vector spaces ofvarying dimension rk( E ) − ρ ( ψ m ) . However, if the rank ρ ( ψ ) is constant, it is easily seen thatthe kernel ker ψ is a vector bundle over X . Therefore, it is natural to ask that D ≃ ψ D be regular , i.e., that the rank ρ ( ψ ℓD ) be constant, for any ℓ ∈ N , or, still, that Σ ℓ = ker ψ ℓD be avector bundle over X , for any ℓ ∈ N .The second remark is that, if D is of order k , the prolongation Σ ℓ is the kernel in J k + ℓ ( E ) of the differential consequences ψ ℓD up to order ℓ of the equation ψ D = 0 . It follows that anysolution in J k + ℓ +1 ( E ) of the system ψ ℓ +1 D = 0 (differential consequences up to order ℓ + 1 )projects by π k + ℓ,k + ℓ +1 to a solution in J k + ℓ ( E ) of the system ψ ℓD = 0 (differential consequencesup to order ℓ ): π k + ℓ,k + ℓ +1 Σ ℓ +1 ⊂ Σ ℓ . On the other hand, any family j k + ℓm φ ( m ∈ X ) of solutions of ψ ℓD = 0 can be extended to afamily j k + ℓ +1 m φ ( m ∈ X ) of solutions of ψ ℓ +1 D = 0 . Of course, the best situation is when anysolution of ψ ℓD = 0 can be extended to a solution of ψ ℓ +1 D = 0 , i.e., when π k + ℓ,k + ℓ +1 Σ ℓ +1 = Σ ℓ . This shows that the existence of extended formal solutions , i.e., formal integrability, isa simplifying requirement.Actually we say that a linear differential operator D ≃ ψ D is formally integrable , if it isregular and if extended formal solutions do exist, i.e., more precisely, if Σ ℓ is a vector bundle,for all ℓ ∈ N , and the vector bundle map π k + ℓ,k + ℓ +1 : Σ ℓ +1 → Σ ℓ is surjective, for all ℓ ∈ N .In the present text, all partial differential equations Σ , even those that are not implementedby a differential operator, are assumed to be formally integrable in the sense of Remark43 [KV98]. D -geometry Remark 57.
This section should be read together with Section 4, where notation and moti-vation are explained.
Proposition 19, which states roughly speaking that the function algebra of the total spaceof a vector bundle can be viewed as an algebra over the function algebra of the base, is almostobvious. We nevertheless check the details carefully.Let π : E → X be an affine morphism of schemes (i.e., a locally ringed space morphism ( π, π ♯ ) : ( E, O E ) → ( X, O X ) such that there is an affine cover of X whose preimages by π areaffine), in particular a vector bundle. In the following, we consider the sheaf O E ∈ Sh ( E ) as n Koszul-Tate resolutions O EX := π ∗ O E ∈ Sh ( X ) , where π ∗ denotes the direct image of sheaves. It is known [Har97]that π ∗ induces an equivalence of the categories qcMod ( O E ) and qcMod ( O X ) ∩ Mod ( O EX ) , withself-explaining notation. It follows that O EX ∈ qcMod ( O X ) . Moreover, O EX is clearly a (sheafof) commutative unital ring(s). To see that O EX ∈ qcCAlg ( O X ) , recall first that such analgebra is a commutative monoid in qcMod ( O X ) , i.e., that it is an object in qcMod ( O X ) thatcarries an associative commutative unital multiplication, which is a morphism in qcMod ( O X ) (and similarly for the unit). It suffices to examine the O X -linearity of the multiplication (andof the unit – what is also simple). Start with noticing that, for any open V ⊂ X , f ∈ O X ( V ) and F ∈ O EX ( V ) = O E ( π − ( V )) , the action of f on F is defined via the ring morphism π ♯ : O X ( V ) → O E ( π − ( V )) by f · F := π ♯ ( f ) ⋆ F , where ⋆ is the ring multiplication. Hence, the multiplication ⋆ is O X ( V ) -bilinear, i.e., ⋆ : O EX ( V ) ⊗ O X ( V ) O EX ( V ) → O EX ( V ) is O X ( V ) -linear, and this presheaf morphism induces a sheaf morphism ⋆ : O EX ⊗ O X O EX → O EX in O X -modules. We give some information about the construction of the jet functor J ∞ : qcCAlg ( O X ) → qcCAlg ( D X ) : For as left adjoint of the forgetful functor. We assume that the smooth scheme X is a smoothaffine algebraic variety, so that we can substitute global sections to sheaves and thus avoidsheaf-theoretic subtleties – but the same proof goes through in the general case. We denoteby O (resp., D ) the algebra O X ( X ) (resp., D X ( X ) ).The functor J ∞ must be left adjoint to the forgetful functor For , i.e., for B ∈ O A := CAlg ( O ) and A ∈ D A := CAlg ( D ) , we must have Hom D A ( J ∞ B, A ) ≃ Hom O A ( B, For A ) , (137)functorially in A, B . The construction of J ∞ B is quite natural. We start from the D -module D ⊗ O B (in the tensor product we consider D as endowed with its right O -module structure),and consider the D -algebra S O ( D ⊗ O B ) over D ⊗ O B ( S is the symmetric tensor algebrafunctor). Since Equation (137) suggests the existence of an O -algebra morphism B → J ∞ B ,we define J ∞ B as the quotient of the D -algebra S O ( D ⊗ O B ) by a D -ideal such that thenatural inclusion i : B ∋ b ⊗ b ∈ S O ( D ⊗ O B ) becomes an O -algebra morphism Π ◦ i : B → J ∞ B when composed with the natural projection Π . Since an O -algebra morphism is an O -linear map (a condition that is automatically satisfiedhere) that respects the multiplications and the units, we must ensure that Π(1 ⊗ ( bb ′ )) = Π(1 ⊗ b ) ⊙ Π(1 ⊗ b ′ ) = Π((1 ⊗ b ) ⊙ (1 ⊗ b ′ )) and Π(1 ⊗ B ) = Π(1) , n Koszul-Tate resolutions (resp., B ) denotes the unit in O (resp., B ) and where ⊙ is the symmetric tensorproduct (we denote the product of two residue classes by the same symbol). Hence, weconsider the D -ideal K generated by the elements D · (cid:0) (1 ⊗ b ) ⊙ (1 ⊗ b ′ ) − ⊗ ( bb ′ ) (cid:1) ∈ S O ( D ⊗ O B ) and D · (1 ⊗ B − ∈ S O ( D ⊗ O B ) , where D · denotes the action of an arbitrary differential operator D ∈ D .It now suffices to show that J ∞ : O A ∋ B
7→ J ∞ B := S O ( D ⊗ O B ) /K ∈ D A possesses the adjointness property (137).If f : J ∞ B → A is a D -algebra morphism, the map ˜ f : B ∋ b f (Π(1 ⊗ b )) ∈ For A is obviously an O -algebra morphism.Conversely, let g : B → For A be an O -algebra morphism. The map ¯ g : D ⊗ O B ∋ D ⊗ b D · ( g ( b )) ∈ A is a well-defined D -module morphism. Since S O ( D ⊗ O B ) is the free D -algebra over the D -module D⊗ O B , the D -module morphism ¯ g can be uniquely extended to a D -algebra morphism ¯ g : S O ( D ⊗ O B ) → A . As ¯ g vanishes on K (note that ¯ g (1) = 1 A , where A is the unit in A ),it descends to the quotient J ∞ B . Hence, the searched D -algebra morphism ¯ g : J ∞ B → A .Consider the example of a trivial line bundle π : E = R ∋ ( t, x ) t ∈ X = R and set O = O X ( X ) := R [ t ] and B := O EX ( X ) = O E ( E ) := R [ t, x ] ∈ O A . It is easily seen that thesymmetric algebra S O ( D ⊗ O B ) coincides with the polynomial algebra R [ t, ∂ it ⊗ x j ] , where i, j ∈ N . When dividing the ideal K out, we obtain J ∞ ( B ) = R [ t, x, ∂ t ⊗ x, ∂ t ⊗ x, . . . ] ∈ D A . Indeed, the initial generator ∂ t ⊗ x (resp., ∂ t ⊗ B ), for instance, coincides in the quotientwith ∂ t ⊗ x = ∂ t · ((1 ⊗ x ) ⊙ (1 ⊗ x )) ( resp., ∂ t ⊗ B = ∂ t · . This generator is thus a polynomial in ∂ t ⊗ x and ⊗ x ≃ x (resp., is thus equal to 0, since ∂ t acts on the element of the D -module O ) and can therefore be omitted in the quotient.Hence, the announced result. When setting x ( k ) := ∂ kt ⊗ x , we get J ∞ ( B ) = R [ t, x, x (1) , x (2) , . . . ] ∈ D A , i.e., we obtain indeed the polynomial function algebra of the infinite jet space of π .Observe also that by definition ∂ t · x ( k ) = x ( k +1) , i.e., that ∂ t · x ( k ) = ( ∂ t + x (1) ∂ x + x (2) ∂ x (1) + . . . ) x ( k ) = D t x ( k ) , (138)where D t is the total derivative. Since the vector field ∂ t ∈ D acts on a function in J ∞ ( B ) asderivation, the action of a differential operator of the base on a function in J ∞ ( B ) coincideswith the natural action of the corresponding total differential operator. n Koszul-Tate resolutions D -algebra Let X be a smooth scheme and let A ∈ qcCAlg ( D X ) with multiplication ⋆ (let us recallthat D X is generated by the sheaf O X of functions and the sheaf Θ X of vector fields). Wedenote the action on a ∈ A by f ∈ O X (resp., θ ∈ Θ X ) by f · a (resp., ∇ θ a ).The qcCAlg ( D X ) -morphism ϕ : O X → A , which is defined by ϕ ( f ) = ϕ ( f · O X ) = f · ϕ (1 O X ) = f · A , is injective, since it is the composition of the injective qcCAlg ( D X ) -morphism O X ∋ f f ⊗ A ∈ O X ⊗ O X A and the bijective qcCAlg ( D X ) -morphism O X ⊗ O X A ∋ f ⊗ a f · a ∈ A .Hence, an element f ∈ O X is viewed as an element in A via the identification f ≃ f · A , and f · a = f · (1 A ⋆ a ) = ( f · A ) ⋆ a ≃ f ⋆ a . (139)The ring A [ D X ] of differential operators on X with coefficients in A is the D X -module A [ D X ] := A ⊗ O X D X , endowed with the associative unital R -algebra structure ◦ defined, for a, a ′ ∈ A , θ ∈ Θ X , and D ∈ D X , by ( a ⊗ O ) ◦ ( a ′ ⊗ D ) = ( a ⋆ a ′ ) ⊗ D (140)and (1 A ⊗ θ ) ◦ ( a ′ ⊗ D ) = ( ∇ θ a ′ ) ⊗ D + a ′ ⊗ ( θ ◦ D ) . (141)This multiplication is canonically extended to a first factor of the type a ⊗ ( f ◦ θ ◦ θ ′ ) = (( a ⋆ f ) ⊗ O ) ◦ (1 A ⊗ θ ) ◦ (1 A ⊗ θ ′ ) . It is straightforwardly checked that the usual relations like, e.g., θ ◦ θ ′ = θ ′ ◦ θ + [ θ, θ ′ ] , do notlead to any contradiction. Moreover, the embedding A ∋ a a ⊗ O ∈ A [ D X ] is an associative unital algebra morphism (i.e., A is a subalgebra of A [ D X ] ), whereas theembedding Θ X ∋ θ A ⊗ θ ∈ A [ D X ] is a Lie algebra morphism (i.e., Θ X is a Lie subalgebra of A [ D X ] ). These inclusions extend toan associative unital algebra morphism D X ∋ D A ⊗ D ∈ A [ D X ] . Let us now focus on the category DG + qcCAlg ( A [ D X ]) of differential non-negatively graded O X -quasi-coherent commutative unital A [ D X ] -algebras. As already mentioned in Equation(35): n Koszul-Tate resolutions Definition 58. A differential non-negatively graded A [ D X ] -algebra is an object of thecategory CMon ( DG + qcMod ( A [ D X ])) of commutative monoids in the category of differential non-negatively graded O X -quasi-coherent A [ D X ] -modules, i.e., it is a differential graded commu-tative A -algebra, as well as a differential graded D X -module A • ∈ DG + qcMod ( D X ) , such thatvector fields act as derivations on the A -action on A • and on the multiplication of A • . A mor-phism of differential graded A [ D X ] -algebras is a morphism of differential graded D X -modulesthat is A -linear and respects the multiplications and the units. The category of differentialgraded A [ D X ] -algebras and morphisms between them will be denoted by DG + qcCAlg ( A [ D X ]) . In other words, a differential graded A [ D X ] -algebra is a differential graded A -algebra,as well as a differential graded D X -algebra, such that the A -action and the D X -action arecompatible in the sense that vector fields Θ X ⊂ D X act on the A -action ⊳ as derivations. Example 59.
Let A be, as above, a D X -algebra. Any differential graded D X -algebra mor-phism f : A → B • allows to endow B • with a differential graded A [ D X ] -algebra structure, i.e.,to view B • as an object B • ∈ DG + qcCAlg ( A [ D X ]) . Indeed, it suffices to set a ⊳ b := f ( a ) ⋆ B b , with self-explaining notation. Verifications are straightforward. In particular, A can be in-terpreted as differential graded A [ D X ] -algebra with A -action ⊳ given by the A -multiplication ⋆ A . D -algebras For convenience, we recall Lemma 22 of [BPP17b], which is needed in the main part ofthis text.
Lemma 60.
Let ( T, d T ) ∈ DG D A , let ( g j ) j ∈ J be a family of symbols of degree n j ∈ N , and let V = L j ∈ J D · g j be the free non-negatively graded D -module with homogeneous basis ( g j ) j ∈ J .(i) To endow the graded D -algebra T ⊗ S V with a differential graded D -algebra structure d , it suffices to define dg j ∈ T n j − ∩ d − T { } , (142) to extend d as D -linear map to V , and to equip T ⊗ S V with the differential d given, for any t ∈ T p , v ∈ V n , . . . , v k ∈ V n k , by d ( t ⊗ v ⊙ . . . ⊙ v k ) = d T ( t ) ⊗ v ⊙ . . . ⊙ v k + ( − p k X ℓ =1 ( − n ℓ P j<ℓ n j ( t ∗ d ( v ℓ )) ⊗ v ⊙ . . . b ℓ . . . ⊙ v k , (143) where ∗ is the multiplication in T . If J is a well-ordered set, the natural map ( T, d T ) ∋ t t ⊗ O ∈ ( T ⊗ S V, d ) is a relative Sullivan D -algebra. n Koszul-Tate resolutions (ii) Moreover, if ( B, d B ) ∈ DG D A and p ∈ DG D A ( T, B ) , it suffices – to define a morphism q ∈ DG D A ( T ⊗ S V, B ) (where the differential graded D -algebra ( T ⊗ S V, d ) is constructed asdescribed in (i)) – to define q ( g j ) ∈ B n j ∩ d − B { p d ( g j ) } , (144) to extend q as D -linear map to V , and to define q on T ⊗ S V by q ( t ⊗ v ⊙ . . . ⊙ v k ) = p ( t ) ⋆ q ( v ) ⋆ . . . ⋆ q ( v k ) , (145) where ⋆ denotes the multiplication in B . Lemma 60 is natural. Indeed, the differential d (Equation (60)) is the unique differentialthat restricts to d T on T , maps V to T, and provides a differential graded D -algebra structureon the graded D -algebra T ⊗ S V . Similarly, the morphism q (Equation (145)) is the unique DG D A -morphism q : ( T ⊗ S V, d ) → ( B, d B ) that restricts to p : ( T, d T ) → ( B, d B ) on T . SinceLemma 60 allows to build relative Sullivan D -algebras, a similar construction might exist inRational Homotopy Theory (in any case, we found this canonical construction independently). The authors are grateful to Jim Stasheff for having read the first version of the presentpaper. His comments and suggestions allowed to significantly improve their text. They wouldalso like to thank Gennaro di Brino for a discussion on an algebraic geometric issue. Further,the authors acknowledge the systematic use of online encyclopedias.
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