Abstract
The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, over -graded commutative rings and on -graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and on -graded manifolds. We follow the notion of an -graded manifold as a local-ringed space whose body is a smooth manifold . A key point is that the graded derivation module of the structure ring of graded functions on an -graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body . Accordingly, the Chevalley–Eilenberg differential calculus on an -graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on -graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of -graded bundles.
1. Introduction
This work addresses the differential calculus over -graded commutative rings and on -graded manifolds defined as local-ringed spaces. This differential calculus provides formalism of differential operators and Lagrangian theory in Grassmann-graded (even and odd) variables [1, 2].
Definition 1. Let be a commutative ring. A direct sum of -modulesis called the -graded -module. Its elements are called homogeneous of degree if .
Definition 2. A -ring is called -graded if it is an -graded -module (1) so that a product of homogeneous elements is a homogeneous element of degree . In particular, it follows that is a -ring, while and, accordingly, are -bimodules.
Any -graded -module (1) admits the associated -graded structure Accordingly, an -graded ring also is the -graded one (Definition 18). The converse is not true. For instance, Clifford algebras are -graded, but not -graded rings. In general, an -graded ring can admit different - and -graded structures (Theorem 26).
Remark 3. Hereafter, we follow the notation (resp., ) of a -ring endowed with an -graded (resp., -graded) structure. If there is no danger of confusion, the symbol further stands both for - and -degree.
We further restrict our consideration to -graded commutative rings.
Definition 4. An -graded -ring is said to be graded commutative ifIn this case, is a commutative -ring, and is an -ring.
Example 5. An -graded commutative ring is commutative if . Conversely, any commutative ring is an -graded commutative one , where .
An -graded commutative ring possesses an associated -graded commutative structurein accordance with Definition 19. The converse need not be true.
The differential calculus over -graded commutative rings (Section 4) is a straightforward generalization of the conventional differential calculus over commutative rings, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus over rings (Section 3) [3–5]. However, this is not a particular case of the differential calculus over noncommutative rings. One can generalize a construction of the Chevalley–Eilenberg differential calculus to a case of an arbitrary ring [5–7]. However, an extension of the notion of a linear differential operator to noncommutative rings meets difficulties [5]. A key point is that multiplication in a noncommutative ring is not a zero-order differential operator.
One overcomes this difficulty in a case of -graded commutative rings by means of reformulating the notion of linear differential operators (Remark 43). As a result, the differential calculus technique has been extended to -graded commutative rings [5, 8, 9]. Since any -graded commutative ring possesses the associated structure (4) of a -graded commutative ring and the commutation relations (3) of its elements depend on their -graded degree, the differential calculus over -graded commutative rings is defined similarly to that over the -graded ones (Section 4). Herewith, a linear -graded differential operator, being an -graded -module homomorphism, is a -graded homomorphism which obeys conditions (72). Consequently, it is a linear -graded differential operator, too. However, the converse need not be true. Therefore, the differential calculus over -graded commutative rings can possess properties which do not characterize the -graded differential calculus. This is just the case of -graded manifolds in comparison with the -graded ones (Theorem 50).
There are different notions of graded manifolds [8, 10–13]. We follow the conventional definition of manifolds as local-ringed spaces and, by analogy with smooth manifolds [14, 15] and -graded manifolds [5, 8, 9], define an -graded manifold as a local-ringed space which is a sheaf in local -graded commutative rings on a finite-dimensional real smooth manifold (Definition 47).
Since -graded manifolds conventionally are sheaves in Grassmann algebras [8], we focus our consideration on local finitely generated -graded commutative rings of the following type (Remark 8).
Definition 6. An -graded commutative -ring is called the Grassmann-graded -ring if it is finitely generated in degree 1 (Definition 25) so that it is the exterior algebra of of a -module (Example 9).
A Grassmann-graded -ring seen as a -graded commutative ring is a Grassmann algebra (Definition 23). A Grassmann algebra , in turn, can admit different associated Grassmann-graded structures . However, since it is finitely generated in degree 1, all these structures mutually are isomorphic if is a field by virtue of Theorem 26. Therefore, an -graded manifold also is a conventional -graded manifold. Conversely, any -graded manifold is isomorphic to the -graded one in accordance with Batchelor’s Theorem 45. However, let us emphasize that though an -graded manifold is -graded and vice versa, the differential calculus on these graded manifolds is different.
The differential calculus on an -graded manifold is the differential calculus over its structure -graded commutative ring (Section 5). A key point is that derivations of the structure ring of graded functions on an -graded manifold, unlike the -graded one, are represented by sections of the smooth vector bundle (102) over its body manifold (Theorem 50). As a consequence, the Chevalley–Eilenberg differential calculus on an -graded manifold provides it with the de Rham complex (103) of graded differential forms.
Just this fact enables us to extend the differential calculus on -graded manifolds to nonlinear differential operators (Section 6). We follow conventional formalism of (nonlinear) differential operators on smooth fibre bundles in terms of their jet manifolds (Appendix B) [3, 4, 16]. We develop the technique of -graded bundles (Definition 52) and graded jet manifolds (Definition 55). Our goal is the differential calculus (123) of graded differential forms on an -graded infinite order jet manifold . A key point is that this ring is split into a bigraded variational bicomplex which provides Lagrangian theory in Grassmann-graded (even and odd) variables [1, 2, 9].
2. Algebraic Preliminary
This section summarizes the relevant basics on commutative rings [17–19] and graded commutative rings [5, 7, 8].
2.1. Commutative Rings
An algebra is defined to be an additive group which additionally is provided with distributive multiplication. All algebras throughout are associative, unless they are Lie algebras and Lie superalgebras. By a ring is meant a unital algebra with a unit element . Nonzero elements of a ring constitute a multiplicative monoid. If it is a group, is called the division ring. A field is a commutative division ring. A ring is said to have no divisor of zero if an equality , , implies either or . For instance, this is a case of a division ring.
A subset of an algebra is said to be the left (resp., right) ideal if it is a subgroup of an additive group and (resp., ) for all , . If is both a left and right ideal, it is called the two-sided ideal. For instance, any ideal of a commutative algebra is two-sided. A proper ideal of a ring is said to be maximal if it does not belong to another proper ideal. Given a two-sided ideal , an additive factor group is an algebra.
Definition 7. A ring is called local if it has a unique maximal two-sided ideal. This ideal consists of all noninvertible elements of .
Any division ring, for example, a field, is local. Its unique maximal ideal consists of the zero element. A homomorphism of local rings is assumed to send a maximal ideal to a maximal ideal.
Remark 8. Local rings conventionally are defined in commutative algebra [18, 19]. This notion has been extended to -graded commutative rings, too [8]. Grassmann-graded rings in Definition 6 and Grassmann algebras in Definition 23 are local.
Given an algebra , an additive group is said to be the left (resp., right) -module if it is provided with a distributive multiplication by elements of such that (resp., ) for all and . If is a ring, one additionally assumes that for all . If is both a left module over an algebra and a right module over an algebra , it is called the -bimodule (the -bimodule if ). If is a commutative algebra, an -bimodule is said to be commutative if for all and . Any module over a commutative algebra can be brought into a commutative bimodule. Therefore, unless otherwise stated (Section 3.1), any -module over a commutative algebra is a commutative -bimodule, which is called the -module if there is no danger of confusion. A module over a field is called the vector space. If an algebra is a commutative bimodule over a commutative ring , it is said to be the -algebra. Any algebra can be regarded as a -algebra.
The following are constructions of new modules over a commutative ring from the old ones.
(i) A direct sum of -modules and is an additive group provided with an -module structure Let be a set of -modules. Their direct sum consists of elements of the Cartesian product such that at most for a finite number of indices .
(ii) A tensor product of -modules and is an additive group which is generated by elements , , , obeying relations It is endowed with an -module structure If a ring is treated as an -module, a tensor product is canonically isomorphic to via the assignment , , .
Example 9. Let be an -module. Let us consider an -graded moduleThis is an -graded -ring with respect to a tensor product . It is called the tensor algebra of an -module . Its quotient with respect to an ideal generated by elements , , is an -graded commutative algebra, called the exterior algebra of an -module .
(i) Given a submodule of an -module , the quotient of an additive group by its subgroup also is provided with an -module structure. It is called the factor module.
(ii) A set of -linear morphisms of an -module to an -module naturally is an -module. An -module is called the dual of an -module . There is a natural monomorphism .
A module over a commutative ring is called free if it admits a basis, that is, a linearly independent subset such that each element of has a unique expression as a linear combination of elements of with a finite number of nonzero coefficients from a ring . Any module over a field is free. Every module is isomorphic to the quotient of a free module. A module is said to be of finite rank if it is the quotient of a free module with a finite basis. One says that a module is projective if there exists a module such that is a free module.
Theorem 10. If is a projective module of finite rank, then its dual is so, and .
The forthcoming constructions of direct and inverse limits of modules over commutative rings also are extended to a case of modules over graded commutative rings.
By a directed set is meant a set with an order relation which satisfies the following conditions: (i) , for all ; (ii) if and , then ; (iii) for any , there exists such that and . It may happen that , but and .
A family of -modules , indexed by a directed set , is called the direct system if, for any pair , there is a morphism such that (i) ; (ii) , . A direct system of modules admits a direct limit.
Definition 11. This is an -module together with morphisms such that for all . A module consists of elements of a direct sum modulo the identification of elements of with their images in for all .
Theorem 12. Direct limits commute with direct sums and tensor products of modules. Namely, let and be two direct systems of -modules which are indexed by the same directed set , and let and be their direct limits. Then direct limits of direct systems and are and , respectively.
Theorem 13. A morphism of a direct system to a direct system consists of an order preserving map and -module morphisms which obey compatibility conditions . If and are direct limits of these direct systems, there exists a unique -module morphism such that
Theorem 14. A construction of a direct limit morphism preserves monomorphisms and epimorphisms. If all are monomorphisms (resp., epimorphisms), so is .
In a case of inverse systems of modules, we restrict our consideration to inverse sequences Its inverse limit is a module together with morphisms so that for all . It consists of elements , , of the Cartesian product such that for all . A morphism of an inverse system to an inverse system consists of -module morphisms which obey compatibility conditionsIf and are inverse limits of these inverse systems, there exists a unique -module morphism such that . A construction of an inverse limits morphism preserves monomorphisms, but not epimorphisms.
Example 15. In particular, let be an inverse system of -modules and an -module together with -module morphisms which obey compatibility conditions . Then there exists a unique morphism such that .
Example 16. Let be an inverse system of -modules and an -module. Given a term , let be an -module morphism. It yields the pull-back morphismswhich obviously obey the compatibility conditions (10). Then there exists a unique morphism such that .
Example 17. Let be an inverse sequence of -modules. Given an -module , modules constitute a direct sequence whose direct limit is isomorphic to .
2.2. -Graded Commutative Rings
A -module is called -graded if it is decomposed into a direct sum of modules and , called the even and odd parts of , respectively. A -graded -module is said to be free if it has a basis composed by graded-homogeneous elements.
A morphism of -graded -modules is said to be an even (resp., odd) morphism if preserves (resp., changes) the -parity of all homogeneous elements. A morphism of -graded -modules is called graded if it is represented by a sum of even and odd morphisms. A set of these graded morphisms is a -graded -module.
Definition 18. A -ring is called -graded if it is a -graded -module , and a product of its homogeneous elements is a homogeneous element of degree . In particular, . Its even part is a -ring, and the odd one is an -bimodule.
Definition 19. A -graded ring is called graded commutative if Its even part belongs to the center of a ring .
Every -graded commutative -ring (Definition 4) possesses the associated -graded commutative structure (4). For instance, the exterior algebra of a -module in Example 9 is a -graded commutative ring.
Definition 20. A -graded commutative ring is called local if it contains a unique maximal -graded ideal.
If is a field, an exterior -algebra exemplifies a local -graded commutative ring. An ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is -graded.
A -graded commutative ring can admit different -graded commutative structures in general (Example 21).
By automorphisms of a -graded commutative ring are meant automorphisms of a -ring which are graded -module morphisms of . Obviously, they are even, and they preserve a -graded structure of . However, there exist automorphisms of a -ring which do not possess this property in general. Then and are isomorphic, but different -graded commutative structures of a ring . Moreover, it may happen that a -ring admits nonisomorphic -graded commutative structures.
Example 21. Given a -graded commutative ring and its odd element , an automorphismof a -ring does not preserve its original -graded structure .
Given a -graded commutative ring , a -graded -module is defined as an -bimodule which is a -graded -module such that
The following are constructions of new -graded -modules from the old ones.
(i) A direct sum of -graded modules and a -graded factor module are defined just as those of modules over a commutative ring.
(ii) A tensor product of -graded -modules and is their tensor product as -modules such that In particular, the tensor algebra of a -graded -module is defined just as that (8) of a module over a commutative ring. Its quotient with respect to the ideal generated by elements is the exterior algebra of a -graded module with respect to the graded exterior product
(iii) A graded morphism of -graded -modules is their graded morphism as -graded -modules which obeys the relationsThese morphisms form a -graded -module . A -graded -module is called the dual of a -graded -module .
In the sequel, we are concerned with -graded manifolds (Section 5). They are sheaves in Grassmann algebras which are defined as follows.
Definition 22. A -graded -ring is said to be finitely generated in degree 1 if it is a free -module of finite rank so that .
It follows that a -module has a decompositionwhere is the ideal of nilpotents of a ring . A surjection is called the body map.
Definition 23. A -graded commutative -ring is said to be the Grassmann algebra if it is finitely generated in degree 1 and is isomorphic to the exterior algebra (Example 9) of a -module , where is the ideal of nilpotents (19) of .
An exterior algebra of a free -module of finite rank is a Grassmann algebra. Conversely, a Grassmann algebra admits a structure of an exterior algebra by a choice of its minimal generating -module , and all these structures are mutually isomorphic if is a field (Theorem 26). Automorphisms of a Grassmann algebra preserve its ideal of nilpotents and the splitting (19), but need not the odd sector (Example 21).
A Grassmann algebra is local in accordance with Definition 20. Its ideal of nilpotents is a unique maximal ideal which is graded in accordance with the decomposition (19).
Remark 24. Let be a -graded commutative ring. A -graded -algebra is called the Lie -superalgebra if its product , called the Lie superbracket, obeys the rules Clearly, an even part of a Lie superalgebra is a Lie -algebra. Given an -superalgebra, a -graded -module is called a -module if it is provided with an -bilinear map
2.3. -Graded Commutative Rings
Let be an -graded -ring (Definition 2). Seen as a -ring, it can admit different -graded structures. All these structures are isomorphic in the following case [20].
Definition 25. An -graded -ring is called finitely generated in degree 1 if the following hold: (i) ; (ii) is a free -module of finite rank; (iii) is generated by ; namely, if is an ideal generated by , then there is -module isomorphism , .
Theorem 26. Let be a field, and let and be -graded -rings finitely generated in degree 1. If they are isomorphic as -rings, there exists their graded isomorphism so that for all .
As was mentioned above, we restrict our consideration to -graded commutative rings (Definition 4), unless they are the differential graded ones (Section 2.4). They also possess the associated -graded commutative structure (4).
Definition 27. An -graded commutative ring is called local if it contains a unique maximal -graded ideal.
Certainly, if an -graded ring is local, the associated -graded ring is well. A Grassmann-graded ring over a field is local. The ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is -graded.
Given an -graded commutative ring , an -graded -module is defined as a graded -bimodule which is an -graded -module such that and it also is a -graded module. A direct sum, a tensor product of -graded modules, and the exterior algebra of an -graded module are defined similarly to those of -graded modules (Section 2.2), and they also are a direct sum, a tensor product, and an exterior algebra of associated -graded modules, respectively.
A morphism of -graded -modules seen as -modules is said to be homogeneous of degree if for all homogeneous elements and the relations (18) hold. A morphism of -graded -modules as the -ones is called the -graded -module morphism if it is represented by a sum of homogeneous morphisms. Therefore, a set of graded morphisms is an -graded -module. An -graded -module is called the dual of an -graded -module . Certainly, an -graded -module morphism of -graded -modules is their -graded -module morphism as associated -graded modules, but the converse is true.
By automorphisms of an -graded ring are meant automorphisms of a -ring which preserve its -gradation . They also keep the associated -structure of . However, there exist automorphisms of a -ring which do not possess these properties in general.
Let be a Grassmann-graded -ring (Definition 6). Its associated -graded commutative ring is a Grassmann algebra (Definition 23). Conversely, any Grassmann algebra admits the associated structure of a Grassmann-graded ring by a choice of its minimal generating -module . Given a generating basis for a -module , elements of a Grassmann-graded ring take a formWe agree to call the generating basis for the associated Grassmann algebra which brings it into a Grassmann-graded ring .
Given a generating basis for a Grassmann-graded ring , one can show that any -ring automorphism is a composition of automorphismswhere is an automorphism of a -module and are odd elements of and of morphismsAutomorphisms (24), where , are automorphisms of a Grassmann-graded ring . If , the automorphism (24) preserves the associated -graded structure of but does not keep its -graded structure . It yields a different -graded structure , where (24) is a basis for and the generating basis for . Automorphisms (25) preserve an even sector of , but not the odd one (Example 21). However, it follows from Theorem 26 that different - and -graded structures of a Grassmann-graded ring are mutually isomorphic if is a field. As a consequence, we come to the following.
Theorem 28. Given a Grassmann-graded ring over a field , there exists a finite-dimensional vector space over so that is isomorphic to the exterior algebra of (Example 9) seen as a Grassmann-graded ring generated by .
2.4. Differential -Graded Rings
If an -graded ring also is a cochain complex, we come to the following notion [5, 17].
Definition 29. An -graded -ring is called the differential graded ring (henceforth, DGR) if it is a cochain complex of -moduleswith respect to a coboundary operator which obeys the graded Leibniz ruleThe cochain complex (26) is called the de Rham complex of a DGR . It also is said to be the differential graded calculus over a -ring .
Given a DGR , one considers its minimal differential graded subring which contains . Seen as a -ring, it is generated by elements , , and consists of monomials , , whose product obeys the juxtaposition rule in accordance with equality (27). A complex is called the minimal differential graded calculus over . Its cohomology is said to be the de Rham cohomology of .
One can associate a DGR to any Lie -algebra as follows [5, 21]. Let a -ring be a -module so that acts on on the left by endomorphisms (cf. Remark 24). For instance, and . A -multilinear skew-symmetric mapis called the -valued -cochain on a Lie algebra . These cochains form a -module . Let us put . We obtain the cochain complexwith respect to the Chevalley–Eilenberg coboundary operators where the caret denotes omission. Complex (31) is called the Chevalley–Eilenberg complex of a Lie algebra with coefficients in a ring . It is a DGR with respect to the exterior product of skew-symmetric maps (30).
A construction of the Chevalley–Eilenberg complex is extended to Lie superalgebras [5, 21].
3. Differential Calculus over Commutative Rings
Conventional technique of the differential calculus over commutative rings includes formalism of linear differential operators and the Chevalley–Eilenberg differential calculus [3–5].
3.1. Differential Operators on Modules over Commutative Rings
As was mentioned above, throughout is a commutative ring without a divisor of zero. Let be a commutative -ring, and let and be -modules. A -module of -module homomorphisms can be endowed with two different -module structuresWe refer to the second one as an -module structure. Let us put , .
Definition 30. An element is called the linear -order -valued differential operator on if for any tuple of elements of . A set of these operators inherits the - and -module structures (33).
In particular, linear zero-order differential operators obey a condition and, consequently, they coincide with -module morphisms . A linear first-order differential operator satisfies a relation Of course, an -order differential operator is of -order. Therefore, there is a direct sequenceof linear -valued differential operators on an -module . Its direct limit is an -module of all linear -valued differential operators on .
In particular, let . Any linear zero-order -valued differential operator on is defined by its value . Then there is an -module isomorphism via the association , where is given by an equality . A linear first-order differential operator on fulfils a condition
Definition 31. It is called a -valued derivation of if ; that is, it obeys the Leibniz rule
If is a derivation of , then is well for any . Hence, derivations of constitute an -module , called the derivation module of .
If , the derivation module of also is a Lie algebra over a ring with respect to a Lie bracket
3.2. Jets of Modules
A linear -order differential operator on an -module is represented by a zero-order differential operator on a module of -order jets of (Theorem 33).
Given an -module , let be a tensor product of -modules and . We put Let us denote by a submodule of generated by elements
Definition 32. A -order jet module of a module is defined as the quotient of a -module by . We denote its elements .
In particular, a first-order jet module consists of elements modulo the relations
A -module is endowed with the - and -module structures There exists a module morphismof an -module to an -module such that , seen as an -module, is generated by elements , . One can show the following [3, 4].
Theorem 33. Any linear -order -valued differential operator on an -module uniquely factorizes as through the morphism (44) and some -module homomorphism . The correspondence defines an -module isomorphism
Due to monomorphisms , , there exist -module epimorphisms of jet modules Thus, there is an inverse sequenceof jet modules. Its inverse limit is an -module together with -module morphisms
In particular, let us consider a module together with the morphisms (44) which obey compatibility conditions , . Then it follows from Example 15 that there exists an -module morphismso that . The inverse sequence (48) yields a direct sequencewhere is the pull-back -module morphism (11). Its direct limit is an -module (Example 17).
Theorem 34. One has the isomorphisms (46) of the direct systems (36) and (51) which leads to an -module isomorphismof their direct limits in accordance with Theorem 14.
Proof. Any element factorizes as through the morphism (50) and an -module homomorphism (Example 16) in accordance with the commutative diagram
3.3. Chevalley–Eilenberg Differential Calculus over Commutative Rings
Since the derivation module of a commutative -ring is a Lie -algebra, one can associate to the following DGR (59), called the Chevalley–Eilenberg differential calculus over .
Given a Lie -algebra , let us consider the Chevalley–Eilenberg complex (31) of with coefficients in a ring regarded as a -module [5, 7]. This complex contains a subcomplex of -multilinear skew-symmetric mapswith respect to the Chevalley–Eilenberg coboundary operator (32):Indeed, it is readily justified that if (56) is an -multilinear map, (57) is well. In particular, It follows that ; that is, is an -valued derivation of .
Let us define an -graded -moduleIt is provided with the structure of an -graded -ring with respect to a product where denotes the sign of a permutation. This product obeys relationsBy the first one, is an -graded commutative ring. Relation (61) shows that is a DGR (Definition 29), called the Chevalley–Eilenberg differential calculus over a -ring .
Since and, consequently, , we have the interior product , , . It is extended asto a DGR , and obeys a relation With the interior product (62), one defines a derivation of an -graded ring for any . Then one can think of elements of as being differential forms over .
The minimal Chevalley–Eilenberg differential calculus over a ring consists of the monomials , . Its de Rham complexis called the de Rham complex of a -ring .
3.4. Differential Calculus over
Let be a smooth manifold (Remark 35) and an -ring of real smooth functions on . The differential calculus on a smooth manifold is defined as that over a ring .
Remark 35. Throughout the work, smooth manifolds are finite-dimensional real manifolds. We follow the notion of a manifold without boundary. A smooth manifold customarily is assumed to be Hausdorff and second-countable topological space. Consequently, it is a locally compact countable at infinity space and a paracompact space, which admits the partition of unity by smooth real functions. Unless otherwise stated, manifolds are assumed to be connected.
Similarly to a sheaf of continuous functions (Example A.3), a sheaf of smooth real functions on is defined. Its stalk at has a unique maximal ideal of germs of functions vanishing at . Therefore, is a local-ringed space (Definition A.2). Though a sheaf exists on a topological space , it fixes a unique smooth manifold structure on as follows.
Theorem 36. Let be a paracompact topological space and a local-ringed space. Let admit an open cover such that a sheaf restricted to each is isomorphic to a local-ringed space . Then is an -dimensional smooth manifold together with a natural isomorphism of local-ringed spaces and .
One can think of this result as being an equivalent definition of smooth real manifolds in terms of local-ringed spaces. A smooth manifold also is algebraically reproduced as a certain subspace of the spectrum of a real ring of smooth real functions on [7, 14].
Moreover, the well-known Serre–Swan theorem (Theorem 37) states the categorial equivalence between the vector bundles over a smooth manifold and projective modules of finite rank over the ring of smooth real functions on . This theorem originally has been proved in the case of a compact manifold , but it is generalized to an arbitrary smooth manifold [7, 22].
Theorem 37. Let be a smooth manifold. A -module is a projective module of finite rank iff it is isomorphic to the structure module of global sections of some smooth vector bundle over .
In particular, the derivation module of a real ring coincides with a -module of vector fields on , that is, the structure module of sections of the tangent bundle of . Hence, it is a projective -module of finite rank. Its -dual is the structure module of the cotangent bundle of which is a module of one-form on and, conversely, (Theorem 10). It follows that the Chevalley–Eilenberg differential calculus over a real ring is exactly the DGR of exterior forms on , where the Chevalley–Eilenberg coboundary operator (57) coincides with the exterior differential. Accordingly, the de Rham complex (65) of a real ring is the de Rham complex of a DGR of exterior forms on . The cohomology of is called the de Rham cohomology of a manifold .
Let be a vector bundle and its structure module. An -order jet manifold of (Appendix B) also is a smooth vector bundle, and its structure module is exactly the -order jet module of a -module (Definition 32) [3, 4].
In view of this fact and by virtue of Theorem 33, a liner -order differential operator (Definition 30) on a projective -module of finite rank with values in a projective -module of finite rank is represented by a linear bundle morphism over of a jet bundle to a vector bundle , where and are smooth vector bundles whose structure modules and are isomorphic to and , respectively, in accordance with Theorem 37.
This construction is generalized to a case of nonlinear differential operators [3, 4, 16].
Definition 38. Let and be smooth fibre bundles. A bundle morphism over is called the -valued -order differential operator on . This differential operator sends each section of to the section of .
Jet manifolds of a fibre bundle constitute the inverse sequence (B.7) whose inverse limit is an infinite order jet manifold (Definition B.1). Then any -order -valued differential operator on a fibre bundle (Definition 38) is defined by a continuous bundle map For instance, differential operators in Lagrangian theory on fibre bundles, for example, Euler–Lagrange operators, are represented by certain exterior forms on finite order jet manifolds [4, 16].
The inverse sequence (B.7) of jet manifolds yields the direct sequence of DGRs of exterior forms on finite order jet manifoldswhere are the pull-back monomorphisms. Its direct limit (Definition 11) consists of all exterior forms on finite order jet manifolds modulo the pull-back identification. In accordance with Theorem 13, is a DGR which inherits operations of the exterior differential and the exterior product of DGRs .
Theorem 39. The cohomology of the de Rham complexof a DGR equals the de Rham cohomology of a fibre bundle .
Proof. The result follows from the fact that is a strong deformation retract of any jet manifold and, consequently, the de Rham cohomology of equals that of [23].
One can think of elements of as being differential forms on an infinite order jet manifold . A DGR is split into a variational bicomplex. Its cohomology provides the global first variational formula for Lagrangians and Euler–Lagrange operators of a Lagrangian formalism on a fibre bundle [1, 16, 24, 25].
4. Differential Calculus over -Graded Commutative Rings
The differential calculus over -graded commutative rings is defined similarly to that over commutative rings, but it differs from the differential calculus over noncommutative rings (Remark 43). It also provides the differential calculus over a -graded commutative ring endowed with a fixed -graded structure (Remark 44).
Let be a commutative ring without a divisor of zero and an -graded commutative -ring. Let and be -graded -modules. An -graded -module of -graded -module homomorphisms can admit the two -graded -module structurescalled - and -module structures, respectively. Let us put
Definition 40. An element is said to be the linear -valued -graded differential operator of order on iffor any tuple of elements of . A set of these operators inherits the -graded -module structures (70).
For instance, zero-order -graded differential operators obey a condition that is, they coincide with graded -module morphisms . A first-order -graded differential operator satisfies a condition
Graded differential operators on an -graded -module form a direct system of -graded -modules.Its limit is a -graded module of all -valued graded differential operators on .
In particular, let . Any zero-order -valued -graded differential operator on is defined by its value . Then there is a graded -module isomorphism via the association , where is given by the equality . A first-order -valued -graded differential operator on fulfils a condition
Definition 41. It is called the -valued -graded derivation of if , that is, if it obeys the graded Leibniz rule
Any first-order -graded differential operator on falls into a sum of a zero-order graded differential operator and an -graded derivation . If is an -graded derivation of , then is so for any . Hence, -graded derivations of constitute an -graded -module , called the graded derivation module.
If , the -graded derivation module also is a Lie -superalgebra (Remark 24) with respect to the superbracket
Example 42. Let be a Grassmann-graded ring provided with an odd generating basis . Its -graded derivations are defined in full by their action on the generating elements . Let us consider odd derivations Then any -graded derivation of takes a formGraded derivations (81) constitute the free -graded -module of finite rank. It also is a finite-dimensional Lie superalgebra over with respect to the superbracket (79). Any -graded differential operator on a Grassmann-graded ring is a composition of graded derivations.
Remark 43. It should be emphasized that though an -graded commutative ring is a particular noncommutative ring, -graded differential operators in accordance with Definition 40 are not differential operators over a noncommutative ring [5–7]. For instance, -graded derivations of obey the graded Leibniz rule (77) which differs from the Leibniz rule (cf. (38)) in the noncommutative differential calculus.
Since the graded derivation module of an -graded commutative ring is a Lie -superalgebra, one can consider the Chevalley–Eilenberg complexwhere a -ring is seen as a -module [5, 21]. Its cochains are -modules of -linear -graded morphisms of graded exterior products of an -graded -module to . Let us bring homogeneous elements of into a form Then a Chevalley–Eilenberg coboundary operator of the complex (83) reads where the caret denotes omission.
It is easily justified that complex (83) contains a subcomplex of -linear -graded morphisms. The -graded module is provided with the graded exterior product where are graded-homogeneous elements of and The graded Chevalley–Eilenberg differential (86) and the graded exterior product (87) bring into a differential bigraded ring (henceforth, DBGR) whose elements obey relationsIt is called the graded Chevalley–Eilenberg differential calculus over an -graded commutative -ring . In particular, we haveOne can extend this duality relation to the graded interior product of with any element by the rules As a consequence, any graded derivation of yields a graded derivation termed the graded Lie derivative of a differential bigraded ring .
The minimal graded Chevalley–Eilenberg differential calculus over a -graded commutative ring consists of monomials , . The corresponding complexis called the bigraded de Rham complex of an -graded commutative -ring .
Remark 44. Let us note that if is a commutative ring, -graded differential operators and the graded Chevalley–Eilenberg differential calculus are reduced to the familiar commutative differential calculus. On the other hand, -graded modules possess the associated structure of -graded modules, and their -graded homomorphisms are -graded homomorphisms of this -graded structure. Moreover, -graded commutativity conditions in fact are the -graded ones. Therefore, the differential calculus over -graded rings is the -graded differential calculus when the associated -structure hold fixed. In particular, any -graded differential operator also is a -graded differential operator, but the converse need not be true. For instance, the differential calculus over a Grassmann-graded -ring (Example 42) is the differential calculus over an associated Grassmann algebra . Namely, the derivations (81) of also are derivations of a Grassmann algebra , and any -graded differential operator on is a composition of these derivations.
5. -Graded Manifolds
As was mentioned above, we define an -graded manifold by analogy with smooth and -graded manifolds as a local-ringed space which is a sheaf in local -graded commutative rings on a finite-dimensional real smooth manifold .
We start with the notion of a -graded manifold [5, 8, 9]. It is defined as a local-ringed space (Definition A.2), where is an -dimensional smooth manifold, and is a sheaf of real Grassmann algebras such that(i)there is the exact sequence of sheaves where is the sheaf of smooth real functions on ;(ii) is a locally free sheaf of -modules of finite rank (with respect to pointwise operations), and the sheaf is locally isomorphic to the exterior product .
A sheaf is called the structure sheaf of a -graded manifold , and a manifold is said to be the body of . Sections of the sheaf are termed the graded functions on a -graded manifold . They make up a -graded commutative -ring called the structure ring of .
Given a -graded manifold , by the sheaf of graded derivations of is meant a subsheaf of endomorphisms of the structure sheaf such that any section of over an open subset is a graded derivation of the real -graded commutative ring , that is, . Conversely, one can show that, given open sets , there is a surjection of the graded derivation derivation modules [8]. It follows that any graded derivation of a local -graded commutative ring also is a local section over of a sheaf . Global sections of are called graded vector fields on a -graded manifold . They make up a graded derivation module of a real -graded commutative ring . This module is a real Lie superalgebra (Remark 24) with respect to the superbracket (79).
Turn now to -graded manifolds. By virtue of the well-known Batchelor theorem [5, 8], -graded manifolds possess the following structure.
Theorem 45. Let be a -graded manifold. There exists a vector bundle with an -dimensional typical fibre such that the structure sheaf of as a sheaf in real rings is isomorphic to the structure sheaf of germs of sections of the exterior bundlewhose typical fibre is the Grassmann algebra in Theorem 28.
It should be emphasized that Batchelor’s isomorphism in Theorem 45 fails to be canonical.
Combining Batchelor Theorem 45 and classical Serre–Swan Theorem 37, we come to the following Serre–Swan theorem for -graded manifolds [7].
Theorem 46. Let be a smooth manifold. A -graded commutative -ring is isomorphic to the structure ring of a -graded manifold with a body iff it is the exterior algebra of some projective -module of finite rank.
In fact, the structure sheaf of a -graded manifold in Theorem 45 is a sheaf in Grassmann-graded rings , whose -graded structure is fixed. Therefore, we come to the following notion of -graded manifolds.
Definition 47. An -graded manifold is defined to be a -graded manifold whose Batchelor isomorphism is fixed.
Thus, Theorem 45 states that a -graded manifold is isomorphic to the -graded one.
An -graded manifold is said to be modelled over a vector bundle , and is called its characteristic vector bundle. Its structure ring is the structure moduleof sections of the exterior bundle (95). Automorphisms of an -graded manifold are restricted to those induced by automorphisms of its characteristic vector bundles .
Accordingly, Serre–Swan Theorem 46 can be formulated for the -graded ones.
Theorem 48. Let be a smooth manifold. An -graded commutative -ring is isomorphic to the structure ring of an -graded manifold with a body iff it is the exterior algebra of some projective -module of finite rank.
Remark 49. One can treat a local-ringed space as a trivial -graded manifold whose characteristic vector bundle is . Its structure module is a ring of smooth real functions on .
Given an -graded manifold , every trivialization chart of its characteristic vector bundle yields a splitting domain of . Graded functions on such a chart are -valued functionswhere are smooth functions on and is the fibre basis for . One calls the local basis for an -graded manifold . Transition functions of bundle coordinates on induce the corresponding transformation of the associated local basis for an -graded manifold and the according coordinate transformation law of graded functions (97).
The following is an essential peculiarity of an -graded manifold in comparison with the -graded ones.
Theorem 50. Derivations of the structure module of an -graded manifold are represented by sections of the vector bundle (102).
Proof. Due to the canonical splitting , the vertical tangent bundle of can be provided with fibre bases , which are the duals of bases . Then graded derivations of on a splitting domain of readwhere are local graded functions on . In particular, The graded derivations (98) act on graded functions (97) by a ruleThis rule implies the corresponding transformation law of graded derivations (98). It follows that they can be represented by sections of a vector bundleThus, the graded derivation module is isomorphic to the structure module of global sections of the vector bundle (102).
Given the structure ring of graded functions on an -graded manifold and the real Lie superalgebra of its graded derivations, let us consider the graded Chevalley–Eilenberg differential calculus over . Since a module is isomorphic to the structure module of sections of a vector bundle , elements of are represented by sections of the exterior bundle of the -dual of . With respect to the dual fibre bases for and for , sections of (104) take a coordinate form The duality relation (90) is given by a graded interior product
Elements of are called graded differential forms on an -graded manifold . Seen as an -ring, the DBGR (103) on a splitting domain is locally generated by the graded one-form , such that Accordingly, the graded Chevalley–Eilenberg coboundary operator (86), termed the graded exterior differential, reads where derivations , act on coefficients of graded exterior forms by formula (100), and they are graded commutative with graded forms and . Formulas (89), (90), and (92) hold.
Theorem 51. The DBGR (103) is a minimal differential calculus over ; that is, it is generated by elements , .
Proof. Since , it is a projective - and -module of finite rank, and so is its -dual (Theorem 10). Hence, is the -dual of and, consequently, is generated by elements , .
Cohomology of the DBGR (103) is called the de Rham cohomology of an -graded manifold . It equals the de Rham cohomology of its body [9]. In particular, there exist both a cochain monomorphism and a body epimorphism .
6. -Graded Bundles and Jet Manifolds
A morphism of (both - and -) graded manifolds is defined as that of local-ringed spaces , , where is a manifold morphism and is a sheaf morphism of to the direct image of onto (Appendix A). This morphism of graded manifolds is said to be (i) a monomorphism if is an injection and is an epimorphism and (ii) an epimorphism if is a surjection and is a monomorphism.
An epimorphism of graded manifolds , where is a fibre bundle is called the graded bundle [26, 27]. In this case, a sheaf monomorphism induces a monomorphism of canonical presheaves , which associates to each open subset the ring of sections of over . Accordingly, there is a pull-back monomorphism of the structure rings of graded functions on graded manifolds and .
In particular, let be a graded manifold whose body is a fibre bundle . Let us consider a trivial graded manifold (Remark 49). Then we have a graded bundleLet us denote it by . Given a graded bundle , the local basis for a graded manifold can take a form , where are bundle coordinates of .
Definition 52. One agrees to call the graded bundle (109) over a trivial graded manifold the graded bundle over a smooth manifold.
If is a vector bundle, the graded bundle (109) is a particular case of graded fibre bundles in [26, 28] when their base is a trivial graded manifold.
Remark 53. Let be a fibre bundle. Then a trivial graded manifold together with a real ring monomorphism is a graded bundle of trivial graded manifolds .
Remark 54. A graded manifold itself can be treated as the graded bundle (109) associated with the identity smooth bundle .
Let and be vector bundles and their bundle morphism over a morphism . Then a section of the dual bundle yields the pull-back section of the dual bundle by a law A bundle morphism induces a morphism of -graded manifoldsThis is a pair of a morphism of body manifolds and the composition of the pull-back of graded functions and the direct image of a sheaf onto . Relative to local bases and for and , the morphism (111) of -graded manifolds reads , .
The graded manifold morphism (111) is a monomorphism (resp., epimorphism) if is a bundle injection (resp., surjection). In particular, the graded manifold morphism (111) is a graded bundle if is a fibre bundle. Let be the corresponding pull-back monomorphism of the structure rings. By virtue of Theorem 51 it yields a monomorphism of the DBGRs
Let be an -graded manifold modelled over a vector bundle . This is an -graded bundle :modelled over a composite bundleThe structure ring of graded functions on an -graded manifold is the graded commutative -ring (96). Let the composite bundle (114) be provided with adapted bundle coordinates possessing transition functions Then the corresponding basis for an -graded manifold is together with transition functions . We call it the local basis for an -graded bundle .
As was shown above, the differential calculus on a fibre bundle is formulated in terms of jet manifolds of . Being fibre bundles over , they can be regarded as trivial graded bundles . We describe their partners in a case of -graded bundles as follows.
Let us note that, given an -graded manifold and its structure ring , one can define the jet module of a -ring [5, 7]. It is a module of global sections of the jet bundle . A problem is that fails to be a structure ring of some graded manifold. By this reason, we have suggested a different construction of jets of graded manifolds (Definition 55), though it is applied only to -graded manifolds [2, 9].
Let be an -graded manifold modelled over a vector bundle . Let us consider a -order jet manifold of . It is a vector bundle over . Then let be an -graded manifold modelled over .
Definition 55. One calls the graded jet manifold of an -graded manifold .
Given a splitting domain of an -graded manifold , the adapted splitting domain of a graded jet manifold reads
As was mentioned above, a graded manifold is a particular graded bundle over its body (Remark 54). Then Definition 55 of graded jet manifolds is generalized to -graded bundles over smooth manifolds as follows. Let be the -graded bundle (113) modelled over the composite bundle (114). It is readily observed that the jet manifold of is a vector bundle coordinated by , . Let be an -graded manifold modelled over this vector bundle. Its local generating basis is , .
Definition 56. One calls the graded jet manifold of an -graded bundle .
In particular, let be a smooth bundle seen as a trivial graded bundle modelled over a composite bundle . Then its graded jet manifold is a trivial graded bundle , that is, the jet manifold of . Thus, Definition 56 of graded jet manifolds of -graded bundles is compatible with Definition 2 of jets of fibre bundles.
The affine bundles (B.7) and the corresponding fibre bundles also yield the -graded bundles including the sheaf monomorphismswhere is the pull-back onto of the continuous fibre bundle . The sheaf monomorphism (118) induces a monomorphism of canonical presheaves , which associates to each open subset the ring of sections of over . Accordingly, there is a pull-back monomorphism of the structure ringsof graded functions on graded manifolds and . As a consequence, we have the inverse sequence of graded manifolds One can think of its inverse limit as being the graded infinite order jet manifold whose body is an infinite order jet manifold and whose structure sheaf is a sheaf of germs of graded functions on -graded manifolds . However fails to be an -graded manifold in a strict sense because the inverse limit of the sequence (B.7) is a Fréche manifold, but not the smooth one.
By virtue of Theorem 51, the differential calculus of graded differential forms on -graded manifolds is minimal. Therefore, the monomorphisms of structure rings (119) yield the pull-back monomorphisms (112) of DBGRsAs a consequence, we have a direct system of DBGRsThe DBGR associated with an -graded bundle is defined as the direct limitof the direct system (122). It includes all graded differential forms on graded manifolds modulo the monomorphisms (121). Its elements obey relations (89).
Monomorphisms yield a monomorphism of direct system (67) to direct system (122) and, consequently, a monomorphismof their direct limits. Accordingly, body epimorphisms yield an epimorphismIt is readily observed that the morphisms (124)–(125) are cochain morphisms between the de Rham complex (69) of and the de Rham complexof a DBGR . Moreover, the corresponding homomorphisms of cohomology groups of these complexes are isomorphisms as follows [9].
Theorem 57. There is an isomorphism of the cohomology of the de Rham complex (126) to the de Rham cohomology of .
As was mentioned above, the (123) is split into a graded variational bicomplex which provides Lagrangian theory in Grassmann-graded (even and odd) variables [1, 2, 9].
Appendix
A. Local-Ringed Spaces
We follow the terminology of [29, 30]. Unless otherwise stated, all presheaves and sheaves are considered on the same topological space .
A sheaf on a topological space is a topological fibre bundle in modules over a commutative ring , where a surjection is a local homeomorphism and fibres , , called the stalks, are provided with the discrete topology. Global sections of a sheaf constitute a -module , called the structure module of .
A presheaf on a topological space is defined if a module over a commutative ring is assigned to every open subset and if, for any pair of open subsets , there exists a restriction morphism
Every presheaf on a topological space yields a sheaf on whose stalk at a point is the direct limit of modules , with respect to restriction morphisms . It means that, for each open neighborhood of a point , every element determines an element , called the germ of at . Two elements and belong to the same germ at iff there exists an open neighborhood of such that .
Example A.1. Let be a presheaf of continuous real functions on a topological space . Two such functions and define the same germ if they coincide on an open neighborhood of . Hence, we obtain a sheaf of continuous functions on .
Every sheaf defines a presheaf of modules of its local sections. It is called the canonical presheaf of . The sheaf generated by the canonical presheaf of a sheaf is .
A direct sum and a tensor product of presheaves (as families of modules) and sheaves (as fibre bundles in modules) are naturally defined. By virtue of Theorem 12, a direct sum (resp., a tensor product) of presheaves generates a direct sum (resp., a tensor product) of sheaves.
A morphism of a presheaf to a presheaf on a topological space is defined as a set of module morphisms which commute with restriction morphisms. A morphism of presheaves yields a morphism of sheaves generated by these presheaves. This is a bundle morphism over such that is the direct limit of morphisms , . Conversely, any morphism of sheaves on a topological space yields a morphism of canonical presheaves of local sections of these sheaves. Let be a commutative group of sheaf morphisms for any open subset . These groups are assembled into a presheaf and define the sheaf on . There is a monomorphism
A sheaf on a topological space is said to be the ringed space if its stalk at each point is a commutative ring [5, 7, 31]. A ringed space often is denoted by a pair of a topological space and a sheaf of rings on which are called the body and the structure sheaf of a ringed space, respectively.
Definition A.2. A ringed space is said to be the local-ringed space (the geometric space in the terminology of [31]) if it is a sheaf of local rings.
Example A.3. A sheaf of germs of continuous real functions on a topological space (Example A.1) is a local-ringed space. Its stalk , , contains a unique maximal ideal of germs of functions vanishing at .
Morphisms of local-ringed spaces are defined to be particular morphisms of sheaves on different topological spaces as follows. Let be a continuous map. Given a sheaf on , its direct image on is generated by the presheaf of assignments for any open subset . Conversely, given a sheaf on , its inverse image on is defined as the pull-back onto of a continuous fibre bundle over ; that is, . This sheaf is generated by the presheaf which associates to any open subset the direct limit of modules over all open subsets such that .
By a morphism of ringed spaces is meant a pair of a continuous map and a sheaf morphism or, equivalently, a sheaf morphism [31]. Restricted to each stalk, a sheaf morphism is assumed to be a ring homomorphism. A morphism of ringed spaces is said to be (i) a monomorphism if is an injection and is an epimorphism and (ii) an epimorphism if is a surjection, while is a monomorphism.
Given a local-ringed space , a sheaf on is called the sheaf of -modules if every stalk , , is an -module or, equivalently, if is an -module for any open subset . A sheaf of -modules is said to be locally free if there exists an open neighborhood of every point such that is a free -module. If all these free modules are of the same finite rank, one says that is of constant rank. The structure module of a locally free sheaf is called the locally free module.
B. Jet Manifolds
Let be a smooth fibre bundle provided with bundle coordinates . An -order jet manifold of sections of a fibre bundle is defined as the disjoint union of equivalence classes of sections of which are identified by terms of their Taylor series at points of [4, 16, 32]. The set is endowed with an atlas of the adapted coordinateswhere the symbol stands for the higher order total derivative These derivatives act on exterior forms on and obey the relations We use the compact notation , .
The coordinates (B.1) bring a set into a smooth manifold. They are compatible with the natural surjectionsA glance at the transition functions (B.2) shows that they are affine bundles. It follows that is a strong deformation retract of any finite order jet manifold .
Given fibre bundles and over , every bundle morphism over a diffeomorphism of admits the -order jet prolongation to a morphism of -order jet manifolds Every section of a fibre bundle has the -order jet prolongation to a section of a jet bundle .
The surjections (B.5) form the inverse sequence of finite order jet manifoldsIts inverse limit is defined as a minimal set such that there exist surjectionsobeying the relations for all admissible and . It consists of those elements of the Cartesian product which satisfy the relations for all . One can think of elements of as being infinite order jets of sections of identified by their Taylor series at points of . A set is provided with the inverse limit topology. This is the coarsest topology such that the surjections (B.8) are continuous. Its base consists of inverse images of open subsets of , , under the maps . With the inverse limit topology, is a paracompact Fréchet manifold [16, 24, 25]. One can show that surjections are open maps admitting local sections; that is, are continuous bundles. A bundle coordinate atlas of provides with a manifold coordinate atlas
Definition B.1. One calls the infinite order jet manifold.
Disclosure
One of the authors G. Sardanashvily has passed away on September 1 this year. It is a great loss to his grateful colleagues and students.
Competing Interests
The authors declare that they have no competing interests.