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Differential Calculus on N-Graded Manifolds.

1. Introduction

This work addresses the differential calculus over N-graded commutative rings and on N-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 K be a commutative ring. A direct sum of K-modules

[mathematical expression not reproducible] (1)

is called the N-graded K-module. Its elements p are called homogeneous of degree [p] if p e

Definition 2. A K-ring A is called N-graded if it is an N-graded K-module [A.sup.*] (1) so that a product of homogeneous elements aa is a homogeneous element of degree [[alpha]] + [[alpha]']. In particular, it follows that [A.sup.0] is a K-ring, while [A.sup.k>0] and, accordingly, [A.sup.*] are [A.sup.0]-bimodules.

Any N-graded K-module P (1) admits the associated [Z.sub.2]-graded structure

[mathematical expression not reproducible]. (2)

Accordingly, an N-graded ring [A.sup.*] also is the Z2-graded one [A.sub.*] (Definition 18). The converse is not true. For instance, Clifford algebras are [Z.sub.2]-graded, but not N-graded rings. In general, an N-graded ring [A.sup.*] can admit different N- and [Z.sup.2]graded structures (Theorem 26).

Remark 3. Hereafter, we follow the notation [A.sup.*] (resp., [A.sup.*]) of a K-ring A endowed with an N-graded (resp., [Z.sub.2]-graded) structure. If there is no danger of confusion, the symbol [*] further stands both for N- and Z2-degree.

We further restrict our consideration to N-graded commutative rings.

Definition 4. An N-graded K-ring [A.sup.*] is said to be graded commutative if

[alpha][beta] = [(-1).sup.[[alpha]][[beta]]] [beta][alpha], [alpha], [beta] [member of] [A.sup.*]. (3)

In this case, [A.sup.0] is a commutative K-ring, and [A.sup.*] is an [A.sup.0]-ring.

Example 5. An N-graded commutative ring [A.sup.*] is commutative if [A.sup.2i+1] = 0. Conversely, any commutative ring A is an N-graded commutative one [A.sup.*], where [A.sup.>0] = 0.

An N-graded commutative ring [A.sup.*] possesses an associated [Z.sub.2]-graded commutative structure

[mathematical expression not reproducible], (4)

in accordance with Definition 19. The converse need not be true.

The differential calculus over N-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 [Z.sub.2]-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 [Z.sub.2]-graded commutative rings [5,8,9]. Since any N-graded commutative ring [A.sup.*] possesses the associated structure (4) of a [Z.sub.2]-graded commutative ring [A.sup.*] and the commutation relations (3) of its elements depend on their [Z.sub.2]-graded degree, the differential calculus over N-graded commutative rings is defined similarly to that over the [Z.sub.2]-graded ones (Section 4). Herewith, a linear N-graded differential operator, being an N-graded K-module homomorphism, is a [Z.sub.2]-graded homomorphism which obeys conditions (72). Consequently, it is a linear [Z.sub.2]-graded differential operator, too. However, the converse need not be true. Therefore, the differential calculus over N-graded commutative rings can possess properties which do not characterize the [Z.sub.2]-graded differential calculus. This is just the case of N-graded manifolds in comparison with the [Z.sub.2]-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 [Z.sub.2]-graded manifolds [5,8,9], define an N-graded manifold as a local-ringed space which is a sheaf in local N-graded commutative rings on a finite-dimensional real smooth manifold Z (Definition 47).

Since [Z.sub.2]-graded manifolds conventionally are sheaves in Grassmann algebras [8], we focus our consideration on local finitely generated N-graded commutative rings of the following type (Remark 8).

Definition 6. An N-graded commutative K-ring [[LAMBDA].sup.*] is called the Grassmann-graded K-ring if it is finitely generated in degree 1 (Definition 25) so that it is the exterior algebra of [[LAMBDA].sup.*] = [disjunction] [[LAMBDA].sup.1] of a K-module [[LAMBDA].sup.1] (Example 9).

A Grassmann-graded K-ring [[LAMBDA].sup.*] seen as a [[LAMBDA].sup.*]-graded commutative ring [[LAMBDA].sub.*] is a Grassmann algebra (Definition 23). A Grassmann algebra [[LAMBDA].sub.*], in turn, can admit different associated Grassmann-graded structures [[LAMBDA].sup.*]. However, since it is finitely generated in degree 1, all these structures mutually are isomorphic if K is a field by virtue of Theorem 26. Therefore, an N-graded manifold also is a conventional [Z.sub.2]-graded manifold. Conversely, any [Z.sub.2]-graded manifold is isomorphic to the N-graded one in accordance with Batchelor's Theorem 45. However, let us emphasize that though an N-graded manifold is [Z.sub.2]-graded and vice versa, the differential calculus on these graded manifolds is different.

The differential calculus on an N-graded manifold is the differential calculus over its structure N-graded commutative ring (Section 5). A key point is that derivations of the structure ring of graded functions on an N-graded manifold, unlike the [Z.sup.2]-graded one, are represented by sections of the smooth vector bundle [V.sub.E] (102) over its body manifold Z (Theorem 50). As a consequence, the Chevalley-Eilenberg differential calculus on an N-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 N-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 N-graded bundles (Definition 52) and graded jet manifolds (Definition 55). Our goal is the differential calculus [S.sup.*.sub.[infinity]] [F, Y] (123) of graded differential forms on an N-graded infinite order jet manifold ([mathematical expression not reproducible]). A key point is that this ring [S.sup.*.sub.[infinity]] [F,Y] is split into a bigraded variational bicomplex which provides Lagrangian theory in Grassmanngraded (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 A 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 1 [not equal to] 0. Nonzero

elements of a ring A constitute a multiplicative monoid. If it is a group, A is called the division ring. A field is a commutative division ring. A ring A is said to have no divisor of zero if an equality ab = 0, a, b [member of] A, implies either a = 0 or b = 0. For instance, this is a case of a division ring.

A subset J of an algebra A is said to be the left (resp., right) ideal if it is a subgroup of an additive group A and ab [member of] J (resp., ba [member of] 1) for all a [member of] A, be J. If J 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 J [subset] A, an additive factor group A/J is an algebra.

Definition 7. A ring A is called local if it has a unique maximal two-sided ideal. This ideal consists of all noninvertible elements of A.

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 [Z.sub.2]-graded commutative rings, too [8]. Grassmann-graded rings in Definition 6 and Grassmann algebras in Definition 23 are local.

Given an algebra A, an additive group P is said to be the left (resp., right) A-module if it is provided with a distributive multiplication A x P [right arrow] P by elements of A such that (ab)p = a(bp) (resp., p(ab) = (pa)b) for all a,b [member of] A and p [member of] P. If A is a ring, one additionally assumes that 1p = p = p1 for all p [member of] P. If P is both a left module over an algebra A and a right module over an algebra A', it is calledthe (A-A'-bimodule (the A-bimodule if A = A'). If A is a commutative algebra, an A-bimodule P is said to be commutative if ap = pa for all a [member of] A and p [member of] P. Any module over a commutative algebra A can be brought into a commutative bimodule. Therefore, unless otherwise stated (Section 3.1), any A-module over a commutative algebra is a commutative A-bimodule, which is called the A-module if there is no danger of confusion. A module over a field is called the vector space. If an algebra A is a commutative bimodule over a commutative ring K, it is said to be the K-algebra. Any algebra can be regarded as a Z-algebra.

The following are constructions of new modules over a commutative ring A from the old ones.

(i) A direct sum P [direct sum] P' of A-modules P and P' is an additive group P x P provided with an A-module structure

a(p,p') = (ap, ap'), p [member of] P, p [member of] P', a [member of] A. (5)

Let [{[P.sub.i]}.sub.i[member of]I] be a set of A-modules. Their direct sum [direct sum] [P.sub.i] consists of elements (..., [p.sub.i], ...) of the Cartesian product [PI][P.sub.i] such that [p.sub.i] [not equal to] 0 at most for a finite number of indices i [member of] I.

(ii) A tensor product P [[cross product].sub.A] Q of A-modules P and Q is an additive group which is generated by elements p [cross product] q, p [member of] P, q [member of] Q, obeying relations

[mathematical expression not reproducible]. (6)

It is endowed with an A-module structure

a(p [cross product] q) = (ap) [cross product]q = p[cross product] (qa) = (p[cross product]q) a. (7)

If a ring A is treated as an A-module, a tensor product A[[cross product].sub.A]Q is canonically isomorphic to Q via the assignment a[cross product]q [left and right arrow] aq, a [member of] A, q [member of] Q.

Example 9. Let Q be an A-module. Let us consider an N-graded module

[mathematical expression not reproducible]. (8)

This is an N-graded A-ring with respect to a tensor product [cross product]. It is called the tensor algebra of an A-module Q. Its quotient [disjucntion]Q with respect to an ideal generated by elements q [cross product] q +q [cross product] q, q,q [member of] Q, is an N-graded commutative algebra, called the exterior algebra of an A-module Q.

(i) Given a submodule Q of an A-module P, the quotient P/Q of an additive group P by its subgroup Q also is provided with an A-module structure. It is called the factor module.

(ii) A set [Hom.sub.A] (P, Q) of A-linear morphisms of an A-module P to an A-module Q naturally is an A-module. An A-module [p.sup.*] = HomA(P, A) is called the dual of an A-module P. There is a natural monomorphism P [right arrow] [P.sup.**].

A module P over a commutative ring A is called free if it admits a basis, that is, a linearly independent subset I c P such that each element of P has a unique expression as a linear combination of elements of I with a finite number of nonzero coefficients from a ring A. 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 P is projective if there exists a module Q such that P [direct sum] Q is a free module.

Theorem 10. If P is a projective module of finite rank, then its dual [P.sup.*] is so, and [P.sup.**] = P.

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 I is meant a set with an order relation < which satisfies the following conditions: (i) i < i, for all i [member of] I; (ii) if i < j and j < k, then i < k; (iii) for any i, j [member of] I, there exists k [member of] I such that i < k and j < k. It may happen that i [not equal to] j, but i < j and j < i.

A family of A-modules [{[P.sub.i]}.sub.i[member of]I], indexed by a directed set I, is called the direct system if, for any pair i < j, there is a morphism [r.sup.i.sub.j]: [P.sub.i] [right arrow] [P.sub.j] such that (i) [r.sup.i.sub.i] = Id [P.sub.i]; (ii) [r.sup.i.sub.j] [omicron] [r.sup.j.sub.k] = [r.sup.i.sub.k], i < j < k. A direct system of modules admits a direct limit.

Definition 11. This is an A-module PTO together with morphisms [r.sup.i.sub.[infinity]]: [P.sub.i] [right arrow] [P.sub.[infinity]] such that [r.sup.i.sub.[infinity]] = [r.sup.j.sub.[infinity]] [omicron] [r.sup.i.sub.j] for all i < j. A module [P.sub.[infinity]] consists of elements of a direct sum [direct sum] [P.sub.i] modulo the identification of elements of [P.sub.i] with their images in [P.sub.j] for all i < j.

Theorem 12. Direct limits commute with direct sums and tensor products of modules. Namely, let {[P.sub.i]} and {[Q.sub.i]} be two direct systems of A-modules which are indexed by the same directed set I, and let [P.sub.[infinity]] and [Q.sub.[infinity]] be their direct limits. Then direct limits of direct systems [[P.sub.i] [direct sum] [Q.sub.i]} and {[P.sub.i] [cross product] [Q.sub.i]} are [P.sub.[infinity]][Q.sub.[infinity]] and [P.sub.[infinity]] [cross product] [Q.sub.[infinity]], respectively.

Theorem 13. A morphism of a direct system [{[P.sub.i], [r.sup.i.sub.j]}.sub.I] to a direct system [{[Q.sub.i'], [[rho].sup.i'.sub.j']}.sub.I'] consists of an order preserving map f: I [right arrow] I' and A-module morphisms [[PHI].sub.i]: [P.sub.i] [right arrow] [Q.sub.f(i)] which obey compatibility conditions [[rho].sup.f(i).sub.f(j)] [omicron] [[PHI].sub.i] = [[PHI].sub.J] [omicron] [r.sup.i.sub.j]. If [P.sub.[infinity]] and [Q.sub.[infinity]] are direct limits of these direct systems, there exists a unique A-module morphism [[PHI].sub.[infinity]]: [P.sub.[infinity]] [right arrow] [Q.sub.[infinity]] such that [[rho].sup.f(i).sub.[infinity]] [omicron] [[PHI].sub.i] = [[PHI].sub.[infinity]] [omicron] [r.sup.i.sub.[infinity]].

Theorem 14. A construction of a direct limit morphism preserves monomorphisms and epimorphisms. If all [[PHI].sub.i]: [P.sub.i] [right arrow] [Q.sub.f(i)] are monomorphisms (resp., epimorphisms), so is [[PHI].sub.[infinity]]: [P.sub.[infinity]] [right arrow] [infinity].

In a case of inverse systems of modules, we restrict our consideration to inverse sequences

[mathematical expression not reproducible]. (9)

Its inverse limit is a module [P.sup.[infinity]] together with morphisms [[pi].sup.[infinity].sub.i]: [P.sup.[infinity]] [right arrow] [P.sup.i] so that [[pi].sup.[infinity].sub.i] = [[pi].sup.j.sub.i] [omicron] [[pi].sup.[infinity].sub.j] for all i < j. It consists of elements (..., [p.sup.i], ...), [p.sup.i] [member of] [P.sup.i], of the Cartesian product [PI][P.sup.i] such that [p.sup.i] = [[pi].sup.j.sub.i] ([p.sup.j]) for all i < j. A morphism of an inverse system [[P.sub.i], [[pi].sup.j.sub.i]} to an inverse system {[Q.sub.i], [[??].sup.i.sub.j]} consists of A-module morphisms: [[PHI].sub.i]: [P.sub.i] [right arrow] [Q.sub.i] which obey compatibility conditions

[[PHI].sub.j] [omicron] [[pi].sup.i.sub.j] = [[??].sup.i.sub.j] [omicron] [[PHI].sub.i]. (10)

If [P.sub.[infinity]] and [Q.sub.[infinity]] are inverse limits of these inverse systems, there exists a unique A-module morphism [mathematical expression not reproducible]. A construction of an inverse limits morphism preserves monomorphisms, but not epimorphisms.

Example 15. In particular, let {[P.sub.i], [[pi].sup.i.sub.j]} be an inverse system of A-modules and Q an A-module together with A-module morphisms [[PHI].sub.i]: Q [right arrow] [P.sub.i] which obey compatibility conditions [[PHI].sub.j] = [[pi].sup.i.sub.j] [[pi].sup.i.sub.j] [omicron] [[PHI].sub.i]. Then there exists a unique morphism [[PHI].sub.[infinity]]: Q [right arrow] [P.sub.[infinity]] such that [[PHI].sub.j] = [[pi].sup.[infinity].sub.j] [omicron] [[PHI].sub.[infinity]].

Example 16. Let {[P.sub.i], [[pi].sup.i.sub.j]} be an inverse system of A-modules and Q an A-module. Given a term [P.sub.r], let [[PHI].sub.r]: [P.sub.r] [right arrow] Q be an A-module morphism. It yields the pull-back morphisms

[mathematical expression not reproducible] (11)

which obviously obey the compatibility conditions (10). Then there exists a unique morphism [[PHI].sub.[infinity]]: [P.sub.[infinity]] [right arrow] Q such that [[PHI].sub.[infinity]] = [[PHI].sub.r] [omicron] [[pi].sup.[infinity].sub.r].

Example 17. Let {[P.sub.i]} be an inverse sequence of A-modules. Given an A-module Q, modules [Hom.sub.A] ([P.sub.i], Q) constitute a direct sequence whose direct limit is isomorphic to [Hom.sub.A]([P.sub.[infinity]]Q).

2.2. [Z.sub.2]-Graded Commutative Rings. A K-module Q is called [Z.sub.2]-graded if it is decomposed into a direct sum Q = [Q.sub.*] = [Q.sub.0] [direct sum] [Q.sub.1] of modules [Q.sub.0] and [Q.sub.1], called the even and odd parts of [Q.sub.*], respectively. A [Z.sub.2]-graded K-module is said to be free if it has a basis composed by graded-homogeneous elements.

A morphism [PHI]: [P.sub.*] [right arrow] [Q.sub.*] of [Z.sub.2]-graded K-modules is said to be an even (resp., odd) morphism if [PHI] preserves (resp., changes) the [Z.sub.2]-parity of all homogeneous elements. A morphism [PHI]: [P.sub.*] [right arrow] [Q.sub.*] of [Z.sub.2]-graded K-modules is called graded if it is represented by a sum of even and odd morphisms. A set [Hom.sub.K](P, Q) of these graded morphisms is a [Z.sub.2]-graded K-module.

Definition 18. A K-ring A is called [Z.sub.2]-graded if it is a [Z.sub.2]-graded K-module, [A.sub.*], and product of its homogeneous elements [alpha][alpha]' is a homogeneous element of degree ([a] + [a'])mod 2. In particular, [1] = 0. Its even part [A.sub.0] is a K-ring, and the odd one [A.sub.1] is an [A.sub.0]-bimodule.

Definition 19. A [Z.sub.2]-graded ring [A.sub.*] is called graded commutative if

aa = [(-1).sup.[a][a']] a' a, a,a' [member of] [A.sub.*]. (12)

Its even part [A.sub.0] belongs to the center [Z.sub.A] of a ring A.

Every N-graded commutative K-ring [A.sup.*] (Definition 4) possesses the associated [Z.sub.2]-graded commutative structure [A.sub.*] (4). For instance, the exterior algebra [disjunction] Q of a K-module Q in Example 9 is a [Z.sub.2]-graded commutative ring.

Definition 20. A [Z.sub.2]-graded commutative ring is called local if it contains a unique maximal [Z.sub.2]-graded ideal.

If K is a field, an exterior K-algebra exemplifies a local [Z.sub.2]-graded commutative ring. An ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is [Z.sub.2]-graded.

A Z2-graded commutative ring can admit different [Z.sub.2]-graded commutative structures [A.sub.*] in general (Example 21).

By automorphisms of a [Z.sub.2]-graded commutative ring [A.sub.*] are meant automorphisms of a K-ring A which are graded K-module morphisms of [A.sub.*]. Obviously, they are even, and they preserve a [Z.sub.2]-graded structure of A. However, there exist automorphisms [phi] of a K-ring A which do not possess this property in general. Then [A.sub.*] and [phi]([A.sub.*]) are isomorphic, but different [Z.sub.2]-graded commutative structures of a ring A. Moreover, it may happen that a K-ring A admits nonisomorphic [Z.sub.2]-graded commutative structures.

Example 21. Given a [Z.sub.2]-graded commutative ring [A.sub.*] and its odd element k, an automorphism

[phi]: [A.sub.0] [contains as member] a [right arrow] a, [A.sub.1] [contains as member] a [right arrow] a (1 + k), (13)

of a K-ring A does not preserve its original [Z.sub.2]-graded structure [A.sub.*].

Given a [Z.sub.2]-graded commutative ring [A.sub.*], a [Z.sub.2]-graded [A.sub.*]-module [Q.sub.*] is defined as an (A - A)-bimodule which is a [Z.sub.2]-graded K-module such that

[aq] = ([a] + [<j])mod 2, qa = [(-1).sup.[a][q]] aq, a [member of] [A.sub.*], q [member of] [Q.sub.*]. (14)

The following are constructions of new [Z.sub.2]-graded [A.sub.*]-modules from the old ones.

(i) A direct sum of [Z.sub.2]-graded modules and a [Z.sub.2]-graded factor module are defined just as those of modules over a commutative ring.

(ii) A tensor product [P.sub.*] [cross product] [Q.sub.*], of [Z.sub.2]-graded [A.sub.*]-modules [P.sub.*] and [Q.sub.*] is their tensor product as A-modules such that

[mathematical expression not reproducible]. (15)

In particular, the tensor algebra [cross product][P.sub.*] of a [Z.sub.2]-graded [A.sub.*]-module [P.sub.*] is defined just as that (8) of a module over a commutative ring. Its quotient [disjunction][P.sub.*] with respect to the ideal generated by elements

p [cross product] p' + [(-1).sup.[p][p']] p' [cross product] p, p, p' [member of] [P.sub.*] (16)

is the exterior algebra of a [Z.sub.2]-graded module [P.sub.*] with respect to the graded exterior product

p [conjunction] p' = -[(-1).sup.[p][p']] p' [conjunction] p. (17)

(iii) A graded morphism [phi]: [P.sub.*] [right arrow] Q, of [Z.sub.2]-graded [A.sub.*]-modules is their graded morphism as [Z.sub.2]-graded K-modules which obeys the relations

[PHI] (ap) = [(-1).sup.[[PHI][a]] a[PHI](p), p [member of] [P.sub.*], a [member of] [A.sub.*]. (18)

These morphisms form a [Z.sub.2]-graded [A.sub.*]-module [Hom.sub.A]([P.sub.*], [Q.sub.*]). A [Z.sub.2]-graded [A.sub.*]-module [P.sup.*] = [Hom.sub.A] ([P.sub.*], [A.sub.*]) is called the dual of a [Z.sub.2]-graded [A.sub.*]-module

In the sequel, we are concerned with Z2-graded manifolds (Section 5). They are sheaves in Grassmann algebras which are defined as follows.

Definition 22. A [Z.sub.2]-graded K-ring [[LAMBDA].sub.*] is said to be finitely generated in degree 1 if it is a free K-module of finite rank so that [[LAMBDA].sub.0] = K [cross product] [[LAMBDA].sup.2.sub.1].

It follows that a K-module A has a decomposition

[LAMBDA] = K [direct sum] R, R = [[LAMBDA].sub.1] [direct sum] [([[LAMBDA].sub.1]).sup.2]

where R is the ideal of nilpotents of a ring [LAMBDA]. A surjection [sigma]: [LAMBDA] [right arrow] K is called the body map.

Definition 23. A [Z.sub.2]-graded commutative K-ring [[LAMBDA].sub.*] is said to be the Grassmann algebra if it is finitely generated in degree 1 and is isomorphic to the exterior algebra [disjunction](R/[R.sup.2]) (Example 9) of a K-module R/[R.sup.2], where R is the ideal of nilpotents (19) of [[LAMBDA].sub.*].

An exterior algebra [disjunction]Q of a free K-module Q of finite rank is a Grassmann algebra. Conversely, a Grassmann algebra admits a structure of an exterior algebra [disjunction]Q by a choice of its minimal generating K-module Q [subset] [[LAMBDA].sub.1], and all these structures are mutually isomorphic if K is a field (Theorem 26). Automorphisms of a Grassmann algebra preserve its ideal R of nilpotents and the splitting (19), but need not the odd sector [[LAMBDA].sub.1] (Example 21).

A Grassmann algebra is local in accordance with Definition 20. Its ideal of nilpotents R is a unique maximal ideal which is graded in accordance with the decomposition (19).

Remark 24. Let A, be a Z2-graded commutative ring. A [Z.sub.2]-graded [A.sub.*]-algebra [G.sub.*] is called the Lie [A.sub.*]-superalgebra if its product [*, *], called the Lie superbracket, obeys the rules

[mathematical expression not reproducible]. (20)

Clearly, an even part [G.sub.0] of a Lie superalgebra [G.sub.*] is a Lie [A.sub.0]-algebra. Given an [A.sub.*]-superalgebra, a [Z.sub.2]-graded [A.sub.*]-module [P.sub.*] is called a [G.sub.*]-module if it is provided with an [A.sub.*]-bilinear map

[mathematical expression not reproducible]. (21)

2.3. N-Graded Commutative Rings. Let A = [A.sup.*] be an N-graded K-ring (Definition 2). Seen as a K-ring, it can admit different N-graded structures. All these structures are isomorphic in the following case [20].

Definition 25. An N-graded K-ring [A.sup.*] is called finitely generated in degree 1 if the following hold: (i) [A.sup.0] = K; (ii) [A.sup.1] is a free K-module of finite rank; (iii) [A.sup.*] is generated by [A.sup.1]; namely, if R is an ideal generated by [A.sup.1], then there is K-module isomorphism A/R = K, R/[R.sup.2] = [A.sup.1].

Theorem 26. Let K be afield, and let [A.sup.*] and [[LAMBDA].sup.*] be N-graded K-ringsfinitely generated in degree 1. If they are isomorphic as K-rings, there exists their graded isomorphism [PHI]: [A.sup.*] [right arrow] [LAMBDA]* so that [PHI]([A.sup.i]) = [[LAMBDA].sup.i] for all i [member of] N.

As was mentioned above, we restrict our consideration to N-graded commutative rings (Definition 4), unless they are the differential graded ones (Section 2.4). They also possess the associated [Z.sub.2]-graded commutative structure (4).

Definition 27. An N-graded commutative ring is called local if it contains a unique maximal N-graded ideal.

Certainly, if an N-graded ring [A.sup.*] is local, the associated [Z.sub.2]-graded ring [A.sub.*] is well. A Grassmann-graded ring over a field K is local. The ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is N-graded.

Given an N-graded commutative ring [A.sup.*], an N-graded [A.sup.*]-module [Q.sup.*] is defined as a graded ([A.sup.*] - [A.sup.*])-bimodule which is an N-graded K-module such that

qa = [(-1).sup.[a][q]] aq, [aq] = [a] + [q], (22) a [member of] [A.sup.*], q [member of] [Q.sup.*], (22)

and it also is a [Z.sub.2]-graded module. A direct sum, a tensor product of N-graded modules, and the exterior algebra [disjunction][Q.sup.*] of an N-graded module [Q.sup.*] are defined similarly to those of [Z.sub.2]-graded modules (Section 2.2), and they also are a direct sum, a tensor product, and an exterior algebra of associated [Z.sub.2]-graded modules, respectively.

A morphism [PHI]: [P.sup.*] [right arrow] [Q.sup.*] of N-graded [A.sup.*]-modules seen as K-modules is said to be homogeneous of degree [[PHI]] if [[PHI](p)] = [p] + [[PHI]] for all homogeneous elements p [member of] [P.sup.*] and the relations (18) hold. A morphism [PHI]: [P.sup.*] [right arrow] [Q.sup.*] of N-graded [A.sup.*]-modules as the K-ones is called the N-graded [A.sup.*]-module morphism if it is represented by a sum of homogeneous morphisms. Therefore, a set HomA([P.sup.*], [Q.sup.*]) of graded morphisms [P.sup.*] [right arrow] [Q.sup.*] is an N-graded [A.sup.*]-module. An N-graded [A.sup.*]-module [P.sup.*] = HomA([P.sup.*], [A.sup.*]) is called the dual of an N-graded [A.sup.*]-module [P.sup.*]. Certainly, an N-graded [A.sup.*]-module morphism of N-graded [A.sup.*]-modules is their Z2-graded [A.sup.*]-module morphism as associated [Z.sub.2]-graded modules, but the converse is true.

By automorphisms of an N-graded ring [A.sup.*] are meant automorphisms of a K-ring A which preserve its N-gradation [A.sup.*]. They also keep the associated [Z.sub.2]-structure [A.sub.*] of A. However, there exist automorphisms of a K-ring A which do not possess these properties in general.

Let [[LAMBDA].sup.*] be a Grassmann-graded K-ring (Definition 6). Its associated [Z.sub.2]-graded commutative ring is a Grassmann algebra [LAMBDA] (Definition 23). Conversely, any Grassmann algebra [LAMBDA] admits the associated structure of a Grassmann-graded ring [[LAMBDA].sup.*] by a choice of its minimal generating K-module [[LAMBDA].sup.1] [subset] [[LAMBDA].sub.1]. Given a generating basis {[c.sup.i]} for a K-module [[LAMBDA].sup.1], elements of a Grassmann-graded ring [[LAMBDA].sup.*] take a form

[mathematical expression not reproducible]. (23)

We agree to call {[c.sup.i]} the generating basis for the associated Grassmann algebra [[LAMBDA].sub.*] which brings it into a Grassmann-graded ring [[LAMBDA].sup.*].

Given a generating basis {[c.sup.i]} for a Grassmann-graded ring [[LAMBDA].sup.*], one can show that any K-ring automorphism is a composition of automorphisms

[c.sup.i] [right arrow] [c'.sup.i] = [[rho].sup.i.sub.j][c.sup.j] + [b.sup.i], (24)

where [rho] is an automorphism of a K-module [[LAMBDA].sup.1] and [b.sup.i] are odd elements of [[LAMBDA].sup.>2] and of morphisms

[c.sup.i] [right arrow] [c'.sub.i] = [c.sup.i] (1 + k), k [member of] [[LAMBDA].sub.1]. (25)

Automorphisms (24), where [b.sup.i] = 0, are automorphisms [c'.sup.i] = [[rho].sup.i.sub.j][c.sup.j] of a Grassmann-graded ring [[LAMBDA].sup.*]. If [b.sup.i] [not equal to] 0, the automorphism (24) preserves the associated [Z.sub.2]-graded structure [[LAMBDA].sub.*] of [LAMBDA] but does not keep its N-graded structure [[LAMBDA].sup.*]. It yields a different N-graded structure [[LAMBDA]'.sup.*], where {[c'.sup.i]} (24) is a basis for [[LAMBDA]'.sup.1] and the generating basis for [[LAMBDA]'.sup.*]. Automorphisms (25) preserve an even sector [[LAMBDA].sub.0] of [[LAMBDA].sup.*], but not the odd one [[LAMBDA].sub.1] (Example 21). However, it follows from Theorem 26 that different N- and [Z.sub.2]-graded structures of a Grassmann-graded ring are mutually isomorphic if K is a field. As a consequence, we come to the following.

Theorem 28. Given a Grassmann-graded ring [[LAMBDA].sup.*] over a field K, there exists a finite-dimensional vector space W over K so that [[LAMBDA].sup.*] is isomorphic to the exterior algebra [disjunction]W of W (Example 9) seen as a Grassmann-graded ring generated by W.

2.4. Differential N-Graded Rings. If an N-graded ring also is a cochain complex, we come to the following notion [5,17].

Definition 29. An N-graded K-ring [[OMEGA].sup.*] is called the differential graded ring (henceforth, DGR) if it is a cochain complex of K-modules

[mathematical expression not reproducible] (26)

with respect to a coboundary operator [delta] which obeys the graded Leibniz rule

[delta]([alpha][beta]) = [delta][alpha][beta] + [(-1).sup.[absolute value of ([alpha])] [alpha][delta][beta]. (27)

The cochain complex (26) is called the de Rham complex of a DGR ([[OMEGA].sup.*], [delta]). It also is said to be the differential graded calculus over a K-ring [[OMEGA].sup.0].

Given a DGR ([[OMEGA].sup.*], [delta]), one considers its minimal differential graded subring ([bar.[OMEGA]]*, [delta]) which contains [[OMEGA].sub.0]. Seen as a ([[OMEGA].sup.0] - [[OMEGA].sup.0])-ring, it is generated by elements [delta]a, a [member of] A, and consists of monomials [alpha] = [a.sub.0]S[a.sub.1] ... [delta][a.sub.k], [a.sub.i] [member of] [[OMEGA].sup.0], whose product obeys the juxtaposition rule

([a.sub.0][delta][a.sub.1]) ([b.sub.0][delta][b.sub.1]) = [a.sub.0]8 ([a.sub.1][b.sub.0]) [delta][b.sub.1] - [a.sub.0][a.sub.1][delta][b.sub.0][delta][b.sub.1] (28)

in accordance with equality (27). A complex ([[bar.[OMEGA]].sup.*], [delta]) is called the minimal differential graded calculus over [[OMEGA].sup.0]. Its cohomologyis said to be the de Rham cohomologyof ([[OMEGA].sup.*], [delta]).

One can associate a DGR to any Lie K-algebra G as follows [5, 21]. Let a K-ring Q be a G-module so that G acts

on Q on the left by endomorphisms

[mathematical expression not reproducible], (29)

(cf. Remark 24). For instance, Q = K and G : K [right arrow] 0. A K-multilinear skew-symmetric map

[mathematical expression not reproducible] (30)

is called the Q-valued fc-cochain on a Lie algebra G. These cochains form a G-module [C.sup.k][G; Q]. Let us put [C.sup.0][G;Q] = Q. We obtain the co chain complex

[mathematical expression not reproducible] (31)

with respect to the Chevalley-Eilenberg coboundary operators

[mathematical expression not reproducible], (32)

where the caret ^ denotes omission. Complex (31) is called the Chevalley-Eilenberg complex of a Lie algebra G with coefficients in a ring Q. 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, K throughout is a commutative ring without a divisor of zero. Let A be a commutative K-ring, and let P and Q be A-modules. A K-module [Hom.sub.K] (P, Q) of K-module homomorphisms O: P [right arrow] Q can be endowed with two different A-module structures

([alpha][PHI])(p) = a[PHI](p) ([PHI] * a) (p) = [PHI](ap) a [member of] A, p [member of] P. (33)

We refer to the second one as an [A.sup.*]-module structure. Let us put [[delta].sub.a][PHI] = a[PHI] - [PHI] * a, a [member of] A.

Definition 30. An element [DELTA] [member of] [Hom.sub.K] (P, Q) is called the linear s-order Q-valued differential operator on P if [mathematical expression not reproducible]. A set [Diff.sub.s] (P, Q) of these operators inherits the A- and [A.sup.*]-module structures (33).

In particular, linear zero-order differential operators obey a condition

[[delta].sub.a][DELTA] (p) = a[DELTA](p) - [DELTA] (ap) = 0, a [member of] A, p [member of] P, (34)

and, consequently, they coincide with A-module morphisms P [right arrow] Q. A linear first-order differential operator [DELTA] satisfies a relation

[mathematical expression not reproducible], (35)

Of course, an s-order differential operator is of (s + 1)-order. Therefore, there is a direct sequence

[Diff.sub.0] (P, Q) [Diff.sub.1] [right arrow] (P,Q) ... [right arrow] [Diff.sub.r] (P, Q) [right arrow] ... (36)

of linear Q-valued differential operators on an A-module P. Its direct limit is an A - [A.sup.*]-module [Diff.sub.[infinity]] (P, Q) of all linear Q-valued differential operators on P.

In particular, let P = A. Any linear zero-order Q-valued differential operator [DELTA] on A is defined by its value [DELTA](1). Then there is an A-module isomorphism [Diff.sub.0](A, Q) = Q via the association q [right arrow] [[DELTA].sub.q], where [DELTA]q is given by an equality [[DELTA].sub.q] (1) = q. A linear first-order differential operator [DELTA] on A fulfils a condition

[DELTA] (ab) = b[DELTA] (a) + a[DELTA] (b) - ba[DELTA] (1), a, b [member of] A. (37)

Definition 31. It is called a Q-valued derivation of A if [DELTA](1) = 0; that is, it obeys the Leibniz rule

[DELTA] (ab) = [DELTA] (a) b + b[DELTA]A (b), a, b [member of] A. (38)

If [pounds sterling] is a derivation of A, then a[pounds sterling] d is well for any a [member of] A. Hence, derivations of A constitute an A-module b(A, Q), called the derivation module of A.

If Q = A, the derivation module bA = b(A, A) of A also is a Lie algebra over a ring K with respect to a Lie bracket

[u, u'] = u [omicron] u' - u' [omicron] u, u, u' [member of] dA. (39)

3.2. Jets of Modules. A linear s-order differential operator on an A-module P is represented by a zero-order differential operator on a module of s-order jets of P (Theorem 33).

Given an A-module P, let A [[cross product].sub.K] P be a tensor product of K-modules A and P. We put

[[delta].sup.b] (a [cross product] p) = (ba) [cross product] p - a [cross product] (bp), p [member of] P, a, b [member of] A. (40)

Let us denote by [[mu].sup.k+1] a submodule of A [[cross product].sub.K] P generated by elements

[mathematical expression not reproducible]. (41)

Definition 32. A k-order jet module [J.sup.k] (P) of a module P is defined as the quotient of a K-module A [[cross product].sub.K] P by [[mu].sup.k+1]. We denote its elements c [[cross product].sub.k] p.

In particular, a first-order jet module [J.sup.1] (P) consists of elements c [[cross product].sub.1] p modulo the relations

[[delta].sup.a] [omicron] [[delta].sup.b] (1 [[cross product].sub.1] p) = ab [[cross product].sub.1] p - b [[cross product].sub.1] (ap) - a [[cross product].sub.1] (bp) + 1 [[cross product].sub.1] (abp) = 0. (42)

A K-module [J.sup.k] (P) is endowed with the A- and [A.sup.*]-module structures

b (a [[cross product].sub.k] p) = ba [[cross product].sub.k] p,

b * (a [[cross product].sub.k] p) = a [[cross product].sub.k] (bp).

There exists a module morphism

[J.sup.k] : P [contains as member] p [right arrow] 1 [[cross product].sub.k] p [member of] [J.sub.k] (P) (44)

of an A-module P to an [A.sup.*]-module [J.sup.k] (P) such that [J.sup.k] (P), seen as an A-module, is generated by elements [J.sup.k]p, p [member of] P. One can show the following [3, 4].

Theorem 33. Any linear k-order Q-valued differential operator [DELTA] on an A-module P uniquely factorizes as

[mathematical expression not reproducible] (45)

through the morphism [J.sup.k] (44) and some A-module homomorphism [f.sup.[DELTA]] : [J.sup.k] (P) [right arrow] Q. The correspondence [DELTA] [right arrow] [f.sup.[DELTA]] defines an A-module isomorphism

[Diff.sub.k] (P, Q) = [Hom.sub.A] ([J.sup.k] (P), Q). (46)

Due to monomorphisms [[mu].sup.r] [right arrow] [[mu].sup.s], r > s, there exist A-module epimorphisms of jet modules

[[pi].sup.i+1.sub.i]: [J.sup.i+1] (P) [right arrow] [J.sup.i] (P),

[[pi].sup.1.sub.0]: [J.sup.1] (P) [contains as member] a [[cross product].sub.1] p [right arrow] ap [member of] P. (47)

Thus, there is an inverse sequence

[mathematical expression not reproducible] (48)

of jet modules. Its inverse limit [J.sup.[infinity]] (P) is an A-module together with A-module morphisms

[[pi].sup.[infinity].sub.r]: [J.sup.[infinity]] (P) [right arrow] [J.sup.r] (P),

[[pi].sup.[infinity].sub.r<s] = [[pi].sup.s.sub.r] [omicron] [[pi].sup.[infinity].sub.s]. (49)

In particular, let us consider a module P together with the morphisms [J.sup.r] (44) which obey compatibility conditions [J.sup.r] (p) = [[pi].sup.r+k.sub.r] [omicron] [J.sup.r+k] (p), p [member of] P. Then it follows from Example 15 that there exists an A-module morphism

[J.sup.[infinity]]: P [contains as member] p [right arrow] (p, [J.sup.1]p, ..., [J.sup.r]p, ...) [member of] [J.sup.[infinity]] (P) (50)

so that [J.sup.r] (p) = [[pi].sup.[infinity].sub.r] [omicron] [J.sup.[infinity]] (p). The inverse sequence (48) yields a direct sequence

[mathematical expression not reproducible], (51)

where

[mathematical expression not reproducible] (52)

is the pull-back A-module morphism (11). Its direct limit is an A-module [Hom.sub.A] ([J.sup.[infinity]] (P), Q) (Example 17).

Theorem 34. One has the isomorphisms (46) of the direct systems (36) and (51) which leads to an A-module isomorphism

[Diff.sub.[infinity]] (P, Q) = [Hom.sub.A] ([J.sup.[infinity]] (P), Q) (53)

of their direct limits in accordance with Theorem 14.

Proof. Any element [[DELTA].sub.[infinity]] = [DELTA] [member of] [Diff.sub.[infinity]] (P, Q) factorizes as

[mathematical expression not reproducible] (54)

through the morphism [J.sup.[infinity]] (50) and an A-module homomorphism [f.sup.[DELTA].sub.[infinity]] = [f.sup.[summation]] [omicron] [[pi].sup.[infinity].sub.k] (Example 16) in accordance with the commutative diagram

[mathematical expression not reproducible] (55)

3.3. Chevalley-Eilenberg Differential Calculus over Commutative Rings. Since the derivation module [delta]A of a commutative K-ring A is a Lie K-algebra, one can associate to A the following DGR (59), called the Chevalley-Eilenberg differential calculus over A.

Given a Lie K-algebra [delta]A, let us consider the Chevalley-Eilenberg complex [C.sup.*] [[delta]A; A] (31) of [delta]A with coefficients in a ring A regarded as a [delta]A-module [5, 7]. This complex contains a subcomplex [O.sup.*] [[delta]A] of A-multilinear skew-symmetric maps

[mathematical expression not reproducible] (56)

with respect to the Chevalley-Eilenberg coboundary operator (32):

[mathematical expression not reproducible]. (57)

Indeed, it is readily justified that if [phi](56) is an A-multilinear map, d[phi] (57) is well. In particular,

(da) (u) = u (a), a [member of] [O.sup.0] [[delta]A] = A. (58)

It follows that d(1) = 0; that is, d is an [O.sup.1] [[delta]A]-valued derivation of A.

Let us define an N-graded A-module

[mathematical expression not reproducible]. (59)

It is provided with the structure of an N-graded A-ring with respect to a product

[mathematical expression not reproducible], (60)

where [??] denotes the sign of a permutation. This product obeys relations

[mathematical expression not reproducible]. (61)

By the first one, [O.sup.*] [[delta]A] is an N-graded commutative ring. Relation (61) shows that [O.sup.*] [[delta]A] is a DGR (Definition 29), called the Chevalley-Eilenberg differential calculus over a K-ring A.

Since [O.sup.1] [[delta]A] = [Hom.sub.A] ([delta]A, A) = [delta][A.sup.*] and, consequently, [delta]A [subset] [delta][A.sup.**] = [O.sup.1] [[[delta]A].sup.*], we have the interior product u][phi] = [phi](u), u [member of] [delta]A, [phi] [member of] [O.sup.1] [[delta]A]. It is extended as

(u [??] [phi]) ([u.sub.1], ..., [u.sub.k-1]) = k[phi] (u, [u.sub.1], ..., [u.sub.k-1]),

u [member of] [delta]A, [phi] [member of] [O.sup.*] [[delta]A], (62)

to a DGR ([O.sup.*] [[delta]A],d), and obeys a relation

u [??] ([phi] [conjunction] [sigma]) = u [??] [phi] [conjunction] [sigma] + [(-1).sup.[absolute value of ([phi])]] [phi] [conjunction] u [??] [sigma]. (63)

With the interior product (62), one defines a derivation

[L.sub.u] ([phi]) = d (u [??] [phi]) + u [??] d[phi], [phi] [member of] [O.sup.*] [[delta]A],

[L.sub.u] ([phi] [conjunction] [sigma]) = [L.sub.u] ([phi]) [conjunction] [sigma] + [phi] [conjunction] [L.sub.u][sigma], (64)

of an N-graded ring [O.sup.*] [[delta]A] for any u [member of] [delta]A. Then one can think of elements of [O.sup.*] [[delta]A] as being differential forms over A.

The minimal Chevalley-Eilenberg differential calculus [O.sup.*]A over a ring A consists of the monomials [a.sub.0]d[a.sub.1] [conjunction] ... [conjunction] d[a.sub.k], [a.sub.i] [member of] A. Its de Rham complex

[mathematical expression not reproducible] (65)

is called the de Rham complex of a K-ring A.

3.4. Differential Calculus over [C.sup.[infinity]] (X). Let X be a smooth manifold (Remark 35) and [C.sup.[infinity]] (X) an R-ring of real smooth functions on X. The differential calculus on a smooth manifold X is defined as that over a ring [C.sup.[infinity]] (X).

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 [C.sup.0.sub.X] of continuous functions (Example A.3), a sheaf [C.sup.[infinity].sub.X] of smooth real functions on X is defined. Its stalk [C.sup.[infinity].sub.x] at x [member of] X has a unique maximal ideal of germs of functions vanishing at x. Therefore, [C.sup.[infinity].sub.X] is a local-ringed space (Definition A.2). Though a sheaf [C.sup.[infinity].sub.X] exists on a topological space X, it fixes a unique smooth manifold structure on X as follows.

Theorem 36. Let X be a paracompact topological space and (X, R) a local-ringed space. Let X admit an open cover {U such that a sheaf R restricted to each Ui is isomorphic to a local-ringed space [mathematical expression not reproducible]. Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X, R) and (X, [C.sup.[infinity].sub.X]).

One can think of this result as being an equivalent definition of smooth real manifolds in terms of local-ringed spaces. A smooth manifold X also is algebraically reproduced as a certain subspace of the spectrum of a real ring [C.sup.[infinity]] (X) of smooth real functions on X [7,14].

Moreover, the well-known Serre-Swan theorem (Theorem 37) states the categorial equivalence between the vector bundles over a smooth manifold X and projective modules of finite rank over the ring [C.sup.[infinity]] (X) of smooth real functions on X. This theorem originally has been proved in the case of a compact manifold X, but it is generalized to an arbitrary smooth manifold [7, 22].

Theorem 37. Let X be a smooth manifold. A [C.sup.[infinity]](X)-module P is a projective module of finite rank iff it is isomorphic to the structure module Y(X) of global sections of some smooth vector bundle Y [right arrow] X over X.

In particular, the derivation module of a real ring [C.sup.[infinity]](X) coincides with a [C.sup.[infinity]] (X)-module [T.sub.1](X) of vector fields on X, that is, the structure module of sections of the tangent bundle TX of X. Hence, it is a projective [C.sup.[infinity]] (X)-module of finite rank. Its [C.sup.[infinity]](X)-dual [O.sup.1](X) = [T.sub.1][(X).sup.*] is the structure module [O.sup.1](X) of the cotangent bundle T*X of X which is a module of one-form on X and, conversely, [T.sub.1](X) = [O.sup.1] [(X).sup.*] (Theorem 10). It follows that the Chevalley-Eilenberg differential calculus over a real ring [C.sup.[infinity]](X) is exactly the DGR ([O.sup.*](X),d) of exterior forms on X, where the Chevalley-Eilenberg coboundary operator d (57) coincides with the exterior differential. Accordingly, the de Rham complex (65) of areal ring [C.sup.[infinity]](X) is the de Rham complex of a DGR [O.sup.*](X) of exterior forms on X. The cohomology of [O.sup.*](X) is called the de Rham cohomology [H.sup.*.sub.DR](X) of a manifold X.

Let Y [right arrow] X be a vector bundle and Y(X) its structure module. An r-order jet manifold [J.sup.r] Y of Y [right arrow] X (Appendix B) also is a smooth vector bundle, and its structure module [J.sup.r] Y(X) is exactly the r-order jet module J(Y(X)) of a [C.sup.[infinity]] (X)-module Y(X) (Definition 32) [3, 4].

In view of this fact and by virtue of Theorem 33, a liner k-order differential operator (Definition 30) on a projective [C.sup.[infinity]] (X)-module P of finite rank with values in a projective [C.sup.[infinity]] (X)-module Q of finite rank is represented by a linear bundle morphism [J.sup.k] Y [right arrow] E over X of a jet bundle [J.sup.k] Y [right arrow] X to a vector bundle E [right arrow] X, where Y [right arrow] X and E [right arrow] X are smooth vector bundles whose structure modules Y(X) and E(X) are isomorphic to P and Q, respectively, in accordance with Theorem 37.

This construction is generalized to a case of nonlinear differential operators [3,4,16].

Definition 38. Let Y [right arrow] X and E [right arrow] X be smooth fibre bundles. A bundle morphism [DELTA] : [J.sup.k]Y [right arrow] E over X is called the E-valued k-order differential operator on Y. This differential operator sends each section s of Y [right arrow] X to the section A [omicron] [J.sup.k]s of E [right arrow] X.

Jet manifolds [J.sup.k]Y of a fibre bundle Y [right arrow] X constitute the inverse sequence (B.7) whose inverse limit is an infinite order jet manifold [J.sup.[infinity]]Y (Definition B.1). Then any k-order E-valued differential operator [DELTA] on a fibre bundle Y (Definition 38) is defined by a continuous bundle map

[DELTA] [omicron] [[pi].sup.[infinity].sub.r] : [J.sup.[infinity]]Y [??] E. (66)

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 [O.sup.*.sub.r] = [O.sup.*]([J.sup.r]Y) of exterior forms on finite order jet manifolds

[mathematical expression not reproducible], (67)

where [[[pi].sup.r.sub.r-1].sup.*] are the pull-back monomorphisms. Its direct limit

[mathematical expression not reproducible] (68)

(Definition 11) consists of all exterior forms on finite order jet manifolds modulo the pull-back identification. In accordance with Theorem 13, [O.sup.*.sub.[infinity]] Y is a DGR which inherits operations of the exterior differential d and the exterior product [conjunction] of DGRs [O.sup.*.sub.r].

Theorem 39. The cohomology [H.sup.*]([O.sup.*.sub.[infinity]]Y) of the de Rham complex

[mathematical expression not reproducible] ... (69)

of a DGR [O.sup.*.sub.[infinity]] equals the de Rham cohomology [H.sup.*.sub.DR] (Y) of a fibre bundle Y.

Proof. The result follows from the fact that Y is a strong deformation retract of any jet manifold [J.sup.r]Y and, consequently, the de Rham cohomology of [J.sup.r]Y equals that of Y[23].

One can think of elements of Of Y as being differential forms on an infinite order jet manifold [J.sup.[infinity]]Y. A DGR [O.sup.*.sub.[infinity]]Y 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 Y [1,16, 24, 25].

4. Differential Calculus over N-Graded Commutative Rings

The differential calculus over N-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 [Z.sub.2]-graded commutative ring endowed with a fixed Ngraded structure (Remark 44).

Let K be a commutative ring without a divisor of zero and A an N-graded commutative K-ring. Let P and Q be Ngraded A-modules. An N-graded K-module [Hom.sub.K](E, Q) of N-graded K-module homomorphisms [PHI] : P [right arrow] Q can admit the two N-graded A-module structures

(a[PHI]) (p) = a[PHI] (p), ([PHI] * a) (p) = [PHI] (ap), a [member of] A, p [member of] P, (70)

called A- and [A.sup.*]-module structures, respectively. Let us put

[delta][PHI] = a[PHI] - [(-1).sup.[a][[PHI]]] [PHI] * a, a [member of] A. (71)

Definition 40. An element [DELTA] [member of] [Hom.sub.K](P, Q) is said to be the linear Q-valued N-graded differential operator of order s on P if

[mathematical expression not reproducible] (72)

for any tuple of s + 1 elements [a.sub.0], ..., [a.sub.s] of A. A set [Diff.sub.s](P, Q) of these operators inherits the N-graded A-module structures (70).

For instance, zero-order N-graded differential operators obey a condition

[[delta].sub.a][DELTA](p) = a[DELTA](p) - [(-1).sup.[a][[DELTA]]] [DELTA] (ap) = 0, a [member of] A, p [member of] P; (73)

that is, they coincide with graded A-module morphisms P [right arrow] Q. A first-order N-graded differential operator [DELTA] satisfies a condition

[mathematical expression not reproducible]. (74)

Graded differential operators on an N-graded A-module P form a direct system of N-graded (A - [A.sup.*])-modules.

[mathematical expression not reproducible]. (75)

Its limit [Diff.sub.[infinity]](P,Q) is a [Z.sub.2]-graded module of all Q-valued graded differential operators on P.

In particular, let P = A. Any zero-order Q-valued N-graded differential operator [DELTA] on A is defined by its value [DELTA](1). Then there is a graded A-module isomorphism [Diff.sub.0](A, Q) = Q via the association q [right arrow] [[DELTA].sub.q], where [[DELTA].sub.q] is given by the equality [[DELTA].sub.q](1) = q. A first-order Q-valued N-graded differential operator [DELTA] on A fulfils a condition

[DELTA](ab) = [DELTA](a)b + [(-1).sup.[a][[DELTA]]]a[DELTA](b) - [(-1).sup.([b] + [a])[DELTA]] ab[DELTA] (1), a b [member of] A.

Definition 41. It is called the Q-valued N-graded derivation of A if [DELTA](1) = 0, that is, if it obeys the graded Leibniz rule

[DELTA](ab) = [DELTA](a)b + [(-1).sup.[a][[DELTA]]] a[DELTA](b), a, b [member of] A. (77)

Any first-order N-graded differential operator on A falls into a sum

[DELTA](a) = [DELTA](1)a+[[DELTA](a) - [DELTA](1)a] (78)

of a zero-order graded differential operator [DELTA](1)a and an N-graded derivation [DELTA](a) - [DELTA](1)a. If [partial derivative] is an N-graded derivation of A, then a[partial derivative] is so for any a [member of] A. Hence, N-graded derivations of A constitute an N-graded A-module d(A, Q), called the graded derivation module.

If Q = A, the N-graded derivation module dA = d(A, A) also is a Lie K-superalgebra (Remark 24) with respect to the superbracket

[u,u']=u [omicron] u - [(-1).sup.[u][u']]u' [omicron] u, u, u' [member of] A. (79)

Example 42. Let [LAMBDA] be a Grassmann-graded ring provided with an odd generating basis {[c.sup.i]}. Its N-graded derivations are defined in full by their action on the generating elements [c.sup.i]. Let us consider odd derivations

[[partial derivative].sub.i]([c.sup.j]) = [[delta].sup.j.sub.i], [[partial derivative].sub.i] [omicron] [[partial derivative].sub.j] = -[[partial derivative].sub.j] [omicron] [[partial derivative].sub.i]. (80)

Then any N-graded derivation of A takes a form

u = [u.sup.i][[partial derivative].sub.i], [u.sub.i] [member of] A. (81)

Graded derivations (81) constitute the free N-graded Amodule dU of finite rank. It also is a finite-dimensional Lie superalgebra over K with respect to the superbracket (79). Any N-graded differential operator on a Grassmann-graded ring is a composition of graded derivations.

Remark 43. It should be emphasized that though an N-graded commutative ring is a particular noncommutative ring, N-graded differential operators in accordance with Definition 40 are not differential operators over a noncommutative ring [5-7]. For instance, N-graded derivations of A obey the graded Leibniz rule (77) which differs from the Leibniz rule

[partial derivative](ab) = [partial derivative](a)b + a[partial derivative](b) (82)

(cf. (38)) in the noncommutative differential calculus.

Since the graded derivation module dA of an N-graded commutative ring A is a Lie K-superalgebra, one can consider the Chevalley-Eilenberg complex

[mathematical expression not reproducible], (83)

where a K-ring A is seen as a dA-module [5,21]. Its cochains are A-modules

[C.sup.k] [dA; A] = [Hom.sub.K] ([??]dA,A) (84)

of K-linear N-graded morphisms of graded exterior products [??] dA of an N-graded K-module dA to A. Let us bring homogeneous elements of [[conjunction].sup.k] dA into a form

[[epsilon].sub.1] [conjunction] ... [[epsilon].sub.r] [conjunction] [[epsilon].sub.r+1] [conjunction] ... [conjunction] [[epsilon].sub.k], [[epsilon].sub.i] [member of] d[A.sub.0], [[epsilon].sub.j] [member of] d[A.sub.1]. (85)

Then a Chevalley-Eilenberg coboundary operator d of the complex (83) reads

[mathematical expression not reproducible], (86)

where the caret ^ denotes omission.

It is easily justified that complex (83) contains a subcomplex [O.sup.*][dA] of A-linear N-graded morphisms. The N-graded module [O.sup.*][dA] is provided with the graded exterior product

[mathematical expression not reproducible], (87)

where [u.sub.1], ..., [u.sub.r+s] are graded-homogeneous elements of dA and

[mathematical expression not reproducible]. (88)

The graded Chevalley-Eilenberg differential d (86) and the graded exterior product [conjunction] (87) bring [O.sup.*][dU] into a differential bigraded ring (henceforth, DBGR) whose elements obey relations

[mathematical expression not reproducible]. (89)

It is called the graded Chevalley-Eilenberg differential calculus over an N-graded commutative K-ring A. In particular, we have

[O.sup.1][dA] = [Hom.sub.A] (dA, A) = d[A.sup.*]. (90)

One can extend this duality relation to the graded interior product of u [member of] dA with any element [phi] [member of] [O.sup.*] [[delta]A] by the rules

[mathematical expression not reproducible],

As a consequence, any graded derivation u [member of] dA of A yields a graded derivation

[mathematical expression not reproducible], (92)

termed the graded Lie derivative of a differential bigraded ring [O.sup.*][dA].

The minimal graded Chevalley-Eilenberg differential calculus [O.sup.*] A [subset] [O.sup.*] [dA] over a N-graded commutative ring A consists of monomials [a.sub.0]d[a.sub.1] [conjunction] ... [conjunction] d[a.sub.k], [a.sub.i] [member of] A. The corresponding complex

[mathematical expression not reproducible] (93)

is called the bigraded de Rham complex of an N-graded commutative K-ring A.

Remark 44. Let us note that if A = [A.sup.0] is a commutative ring, N-graded differential operators and the graded Chevalley-Eilenberg differential calculus are reduced to the familiar commutative differential calculus. On the other hand, N-graded modules possess the associated structure of [Z.sub.2]-graded modules, and their N-graded homomorphisms are [Z.sub.2]-graded homomorphisms of this [Z.sub.2]-graded structure. Moreover, N-graded commutativity conditions in fact are the [Z.sub.2]-graded ones. Therefore, the differential calculus over N-graded rings is the [Z.sub.2]-graded differential calculus when the associated N-structure hold fixed. In particular, any Ngraded differential operator also is a [Z.sub.2]-graded differential operator, but the converse need not be true. For instance, the differential calculus over a Grassmann-graded K-ring [A.sup.*] (Example 42) is the differential calculus over an associated Grassmann algebra [A.sup.*]. Namely, the derivations (81) of [A.sup.*] also are derivations of a Grassmann algebra [A.sup.*], and any [Z.sub.2] graded differential operator on [A.sup.*] is a composition of these derivations.

5. N-Graded Manifolds

As was mentioned above, we define an N-graded manifold by analogy with smooth and [Z.sub.2]-graded manifolds as a localringed space which is a sheaf in local N-graded commutative rings on a finite-dimensional real smooth manifold X.

We start with the notion of a [Z.sub.2]-graded manifold [5,8,9]. It is defined as a local-ringed space (Z, A) (Definition A.2), where Z is an n-dimensional smooth manifold, and U = [U.sub.0] [direct sum] [U.sub.1] is a sheaf of real Grassmann algebras such that

(i) there is the exact sequence of sheaves

0 [right arrow] R [right arrow] U [??] [C.sup.[infinity].sub.Z] [right arrow] 0, R = [U.sub.1] + [([U.sub.1]).sup.2], (94)

where [C.sup.[infinity].sub.Z] is the sheaf of smooth real functions on Z;

(ii) R/[R.sup.2] is a locally free sheaf of [C.sup.[infinity].sub.Z]-modules of finite rank (with respect to pointwise operations), and the sheaf U is locally isomorphic to the exterior product [mathematical expression not reproducible].

A sheaf A is called the structure sheaf of a [Z.sub.2]-graded manifold (Z, A), and a manifold Z is said to be the body of (Z, A). Sections of the sheaf A are termed the graded functions on a [Z.sub.2]-graded manifold (Z, A). They make up a [Z.sub.2]-graded commutative [C.sup.[infinity]](Z)-ring A(Z) called the structure ring of (Z, A).

Given a [Z.sub.2]-graded manifold (Z, A), by the sheaf dU of graded derivations of A is meant a subsheaf of endomorphisms of the structure sheaf A such that any section u [member of] dU(U) of dU over an open subset U [subset] Z is a graded derivation of the real [Z.sub.2]-graded commutative ring A(G), that is, u [member of] d(U(U)). Conversely, one can show that, given open sets U' c U, there is a surjection of the graded derivation derivation modules d(U(G)) [right arrow] d(U(U')) [8]. It follows that any graded derivation of a local [Z.sub.2]-graded commutative ring A(G) also is a local section over U of a sheaf dU. Global sections of dU are called graded vector fields on a [Z.sub.2]-graded manifold (Z, A). They make up a graded derivation module dU(Z) of a real [Z.sub.2]-graded commutative ring A(Z). This module is a real Lie superalgebra (Remark 24) with respect to the superbracket (79).

Turn now to N-graded manifolds. By virtue of the well-known Batchelor theorem [5, 8], [Z.sub.2]-graded manifolds possess the following structure.

Theorem 45. Let (Z, A) be a [Z.sub.2]-graded manifold. There exists a vector bundle E [right arrow] Z with an m-dimensional typical fibre V such that the structure sheaf A of (Z, A) as a sheaf in real rings is isomorphic to the structure sheaf [U.sub.E] = [conjunction][E.sup.*.sub.Z] of germs of sections of the exterior bundle

[mathematical expression not reproducible], (95)

whose typical fibre is the Grassmann algebra A = [conjunction] [V.sup.*] 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 [Z.sub.2]-graded manifolds [7].

Theorem 46. Let Z be a smooth manifold. A [Z.sub.2]-graded commutative [C.sup.[infinity]](Z)-ring A is isomorphic to the structure ring of a [Z.sub.2]-graded manifold with a body Z iff it is the exterior algebra of some projective [C.sup.[infinity]] (Z)-module of finite rank.

In fact, the structure sheaf AE of a [Z.sub.2]-graded manifold (Z, A) in Theorem 45 is a sheaf in Grassmann-graded rings [[LAMBDA].sup.*], whose N-graded structure is fixed. Therefore, we come to the following notion of N-graded manifolds.

Definition 47. An N-graded manifold is defined to be a [Z.sub.2]-graded manifold whose Batchelor isomorphism (Z, [U.sub.E]) is fixed.

Thus, Theorem 45 states that a [Z.sub.2]-graded manifold is isomorphic to the N-graded one.

An N-graded manifold (Z, [U.sub.E]) is said to be modelled over a vector bundle E [right arrow] Z, and E is called its characteristic vector bundle. Its structure ring Ae is the structure module

[A.sub.E] = [U.sub.E] (Z) = [conjunction] [E.sup.*](Z) (96)

of sections of the exterior bundle [conjunction] [E.sup.*] (95). Automorphisms of an N-graded manifold (Z, [U.sub.E]) are restricted to those induced by automorphisms of its characteristic vector bundles E [right arrow] Z.

Accordingly, Serre-Swan Theorem 46 can be formulated for the N-graded ones.

Theorem 48. Let Z be a smooth manifold. An N-graded commutative [C.sup.[infinity]] (Z)-ring A is isomorphic to the structure ring of an N-graded manifold with a body Z iff it is the exterior algebra of some projective [C.sup.[infinity]] (Z)-module of finite rank.

Remark 49. One can treat a local-ringed space (Z, [U.sub.0] = [C.sup.[infinity].sub.Z]) as a trivial N-graded manifold whose characteristic vector bundle is E = Z x {0}. Its structure module is a ring [C.sup.[infinity]](Z) of smooth real functions on Z.

Given an N-graded manifold (Z, [U.sub.E]), every trivialization chart (U; [z.sup.A], [y.sup.a]) of its characteristic vector bundle E [right arrow] Z yields a splitting domain (U; [z.sup.A], [c.sup.a]) of (Z, [U.sub.E]). Graded functions on such a chart are A-valued functions

[mathematical expression not reproducible], (97)

where [mathematical expression not reproducible] are smooth functions on U and {[c.sup.a]} is the fibre basis for [E.sup.*]. One calls {[z.sup.A], [c.sup.a]} the local basis for an N-graded manifold (Z, [U.sub.E]). Transition functions [y'.sup.a] = [[rho].sup.a.sub.b]([z.sup.A])[y.sup.b] of bundle coordinates on E [right arrow] Z induce the corresponding transformation [c'.sup.a] = [[rho]'.sup.a.sub.b] ([z.sup.A])[c.sup.b] of the associated local basis for an N-graded manifold (Z, [U.sub.E]) and the according coordinate transformation law of graded functions (97).

The following is an essential peculiarity of an N-graded manifold (Z, [U.sub.E]) in comparison with the [Z.sub.2]-graded ones.

Theorem 50. Derivations of the structure module Ae of an N-graded manifold (Z, AE) are represented by sections of the vector bundle (102).

Proof. Due to the canonical splitting VE = ExE, the vertical tangent bundle VE of E - Z can be provided with fibre bases {[partial derivative]/[partial derivative][c.sup.a] }, which are the duals of bases {[c.sup.a]}. Then graded derivations of AE on a splitting domain (U;zA, ca) of (Z, AE) read

u = [u.sup.A][[partial derivative].sub.A] + [u.sup.a][partial derivative]/[partial derivative][c.sup.a], (98)

where [u.sup.[lambda]], [u.sup.a] are local graded functions on U. In particular,

[mathematical expression not reproducible]. (99)

The graded derivations (98) act on graded functions f [member of] [U.sub.E](U) (97) by a rule

u([f.sub.a...b][c.sup.a] ... [c.sup.b]) = [u.sup.A][[partial derivative].sub.A] ([f.sub.a...b]) [c.sup.a] ... [c.sup.b] + [u.sup.k][f.sub.a...b] [partial derivative]/[partial derivative][c.sup.k]] ([c.sup.a] ... [c.sup.b]). (100)

This rule implies the corresponding transformation law

[u'.sup.A] = [u.sup.A], [u'.sup.a] = [[rho].sup.a.sub.j][u.sup.j] + [u.sup.A]([[rho].sup.a.sub.j]) [c.sup.j] (101)

of graded derivations (98). It follows that they can be represented by sections of a vector bundle

[V.sub.E] = [conjunction] [E.sup.*] [??] TE [right arrow] Z. (102)

Thus, the graded derivation module dUE is isomorphic to the structure module [V.sub.E](Z) of global sections of the vector bundle [V.sub.E] [right arrow] Z (102).

Given the structure ring [U.sub.E] of graded functions on an N-graded manifold (Z, [U.sub.E]) and the real Lie superalgebra d[A.sub.E] of its graded derivations, let us consider the graded Chevalley-Eilenberg differential calculus

[mathematical expression not reproducible], (103)

over [S.sup.0][E; Z] = [A.sub.E]. Since a module [d.sub.AE] is isomorphic to the structure module of sections of a vector bundle [V.sub.E] [right arrow] Z, elements of [S.sup.*][E; Z] are represented by sections of the exterior bundle [conjunction] [[bar.V].sub.E] of the [A.sub.E]-dual

[mathematical expression not reproducible] (104)

of [V.sub.E]. With respect to the dual fibre bases {d[z.sup.A]} for [T.sup.*]Z and {d[c.sup.b]} for [E.sup.*], sections of [[bar.V].sub.E] (104) take a coordinate form

[mathematical expression not reproducible], (105)

The duality relation [S.sup.1][E; Z] = d[A.sup.*.sub.E](90) is given by a graded interior product

[mathematical expression not reproducible]. (106)

Elements of [S.sup.*] [E; Z] are called graded differential forms on an N-graded manifold (Z, [U.sub.E]). Seen as an [A.sub.E]-ring, the DBGR [S.sup.*] [E; Z] (103) on a splitting domain ([z.sup.A], [c.sup.a]) is locally generated by the graded one-form d[z.sup.A], d[c.sup.i] such that

d[z.sup.A] [conjunction] d[c.sup.i] = -d[c.sup.i] [conjunction] d[z.sup.A], d[c.sup.i] [conjunction] d[c.sup.j] = d[c.sup.j] [conjunction] d[c.sup.i]. (107)

Accordingly, the graded Chevalley-Eilenberg coboundary operator d (86), termed the graded exterior differential, reads

d[phi] = d[z.sup.A] [conjunction] [[partial derivative].sub.A][phi] + d[c.sup.a] [conjunction] [partial derivative]/[partial derivative][c.sup.a] [phi], (108)

where derivations [[partial derivative].sub.[lambda]], [partial derivative]/[partial derivative][c.sup.a] act on coefficients of graded exterior forms by formula (100), and they are graded commutative with graded forms d[z.sup.A] and d[c.sup.a]. Formulas (89), (90), and (92) hold.

Theorem 51. The DBGR [S.sup.*][E; Z] (103) is a minimal differential calculus over [A.sub.E]; that is, it is generated by elements df, f [member of] [A.sub.E].

Proof. Since d[A.sub.E] = [V.sub.E](Z), it is a projective [C.sup.[infinity]](Z)- and [A.sub.E]-module of finite rank, and so is its [A.sub.E]-dual [S.sup.1][E; Z] (Theorem 10). Hence, d[A.sub.E] is the Ag-dual of [S.sup.1][E; Z] and, consequently, [S.sup.1][E; Z] is generated by elements df, f [member of] [A.sub.E].

Cohomology of the DBGR S* [E; Z] (103) is called the de Rham cohomology of an N-graded manifold (Z, AE). It equals the de Rham cohomology of its body Z [9]. In particular, there exist both a cochain monomorphism [O.sup.*](Z) [right arrow] [S.sup.*][E;Z] and a body epimorphism [S.sup.*][E;Z] [right arrow] [O.sup.*](Z).

6. N-Graded Bundles and Jet Manifolds

A morphism of (both [Z.sub.2]- and N-) graded manifolds (Z, U) [right arrow] (Z', U) is defined as that of local-ringed spaces [phi] : Z [right arrow] Z', [??] : A' [right arrow] [[phi].sub.*] U, where [phi] is a manifold morphism and [??] is a sheaf morphism of A' to the direct image [[phi].sub.*] U of U onto Z' (Appendix A). This morphism of graded manifolds is said to be (i) a monomorphism if [phi] is an injection and [??] is an epimorphism and (ii) an epimorphism if is a surjection and [??] is a monomorphism.

An epimorphism of graded manifolds (Z, U) [right arrow] (Z', U'), where Z [right arrow] Z' is a fibre bundle is called the graded bundle [26, 27]. In this case, a sheaf monomorphism [degrees] induces a monomorphism of canonical presheaves [bar.U]' [right arrow] [bar.U], which associates to each open subset U [subset] Z the ring of sections of U' over [phi](U). Accordingly, there is a pull-back monomorphism of the structure rings U'(Z') [right arrow] U(Z) of graded functions on graded manifolds (Z', U') and (Z, U).

In particular, let (Y, U) be a graded manifold whose body is a fibre bundle Y [right arrow] X. Let us consider a trivial graded manifold (X, [U.sub.0] = [C.sup.[infinity].sub.X]) (Remark 49). Then we have a graded bundle

(Y, U) [right arrow] (X, [C.sup.[infinity].sub.X]). (109)

Let us denote it by (X, Y, U). Given a graded bundle (X, Y, U), the local basis for a graded manifold (Y, U) can take a form ([x.sup.[lambda]], [y.sup.i], [c.sup.a]), where ([x.sup.[lambda]], [y.sup.i]) are bundle coordinates of Y [right arrow] X.

Definition 52. One agrees to call the graded bundle (109) over a trivial graded manifold (X, [C.sup.[infinity].sub.X]) the graded bundle over a smooth manifold.

If Y [right arrow] X 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 Y [right arrow] X be a fibre bundle. Then a trivial graded manifold (Y, [C.sup.[infinity].sub.Y]) together with a real ring monomorphism [C.sup.[infinity]](X) [right arrow] [C.sup.[infinity]](Y) is a graded bundle (X, Y, [C.sup.[infinity].sub.Y]) of trivial graded manifolds (Y, [C.sup.[infinity].sub.Y]) [right arrow] (X, [C.sup.[infinity].sub.X]).

Remark 54. A graded manifold (X, U) itself can be treated as the graded bundle (X, X, U) (109) associated with the identity smooth bundle X [right arrow] X.

Let E [right arrow] Z and E' [right arrow] Z' be vector bundles and [PHI]: E [right arrow] E' their bundle morphism over a morphism : Z [right arrow] Z'. Then a section [s.sup.*] of the dual bundle [E'.sup.*] [right arrow] Z' yields the pull-back section [[PHI].sup.*][s.sup.*] of the dual bundle [E.sup.*] [right arrow] Z by a law

[v.sub.z]][[PHI].sup.*][s.sup.*] (z) = [PHI]([v.sub.z])][s.sup.*] ([phi](z)), [v.sub.z] [member of] [E.sub.z]. (110)

A bundle morphism ([PHI], [phi]) induces a morphism of N-graded manifolds

(Z, [U.sub.E]) [right arrow] (Z', [U.sub.E']). (111)

This is a pair ([phi], [??] = [[phi].sub.*] [omicron] [O.sup.*]) of a morphism [phi] of body manifolds and the composition [[phi].sub.*] [omicron] [[PHI].sup.*] of the pull-back [A.sub.E'] [contains as member] f [right arrow] [[PHI].sup.*] f [member of] [A.sub.E] of graded functions and the direct image [[phi].sub.*] of a sheaf [U.sub.E] onto Z'. Relative to local bases ([z.sup.A], [c.sup.a]) and ([z'.sup.A], [c'.sup.a]) for (Z, [U.sub.E]) and (Z', [U.sub.E']), the morphism (111) of N-graded manifolds reads z' = [phi](z), [??]([c'.sup.a]) = [[PHI].sup.a.sub.b](z)[c.sup.b].

The graded manifold morphism (111) is a monomorphism (resp., epimorphism) if [PHI] is a bundle injection (resp., surjection). In particular, the graded manifold morphism (111) is a graded bundle if O is a fibre bundle. Let [A.sub.E'] [right arrow] [A.sub.E] be the corresponding pull-back monomorphism of the structure rings. By virtue of Theorem 51 it yields a monomorphism of the DBGRs

[S.sup.*] [E'; Z'] [right arrow] [S.sup.*] [E; Z]. (112)

Let (Y, [U.sub.F]) be an N-graded manifold modelled over a vector bundle F [right arrow] Y. This is an N-graded bundle (X, Y, [U.sub.F]):

(Y, [U.sub.F]) [right arrow] (X, [C.sup.[infinity].sub.X]), (113)

modelled over a composite bundle

F [right arrow] Y [right arrow] X. (114)

The structure ring of graded functions on an N-graded manifold (Y, [U.sub.F]) is the graded commutative [C.sup.[infinity]](X)-ring [A.sub.F] = [conjunction] [F.sup.*](Y) (96). Let the composite bundle (114) be provided with adapted bundle coordinates ([x.sup.[lambda]], [y.sup.i], [q.sup.a]) possessing transition functions

[x'.sup.[lambda]] ([x.sup.[mu]]), [y'.sup.i] ([x.sup.[mu]], [y.sup.j]), [q'.sup.a] = [[rho].sup.a.sub.b] ([x.sup.[mu]], [y.sup.j])[q.sup.b]. (115)

Then the corresponding basis for an N-graded manifold (Y, [U.sub.F]) is ([x.sup.[lambda]], [y.sup.i], [c.sup.a]) together with transition functions [c'.sup.a] = [[rho].sup.a.sub.b] ([x.sup.[mu]], [j.sup.j])[c.sup.b]. We call it the local basis for an N-graded bundle (X,Y, [U.sub.F]).

As was shown above, the differential calculus on a fibre bundle Y [right arrow] X is formulated in terms of jet manifolds [J.sup.*]Y of Y. Being fibre bundles over X, they can be regarded as trivial graded bundles [mathematical expression not reproducible]. We describe their partners in a case of N-graded bundles as follows.

Let us note that, given an N-graded manifold (X, [U.sub.E]) and its structure ring [A.sub.E], one can define the jet module [J.sup.1][A.sub.E] of a [C.sup.[infinity]](X)-ring [A.sub.E] [5, 7]. It is a module of global sections of the jet bundle [J.sup.1]([conjunction][E.sup.*]). A problem is that [J.sup.1][A.sub.E] 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 N-graded manifolds [2, 9].

Let (X, [A.sub.E]) be an N-graded manifold modelled over a vector bundle E [right arrow] X. Let us consider a k-order jet manifold [J.sup.k]E of E. It is a vector bundle over X. Then let [mathematical expression not reproducible] be an N-graded manifold modelled over [J.sup.k]E [right arrow] X.

Definition 55. One calls [mathematical expression not reproducible] the graded jet manifold of an N-graded manifold (X, [A.sub.E]).

Given a splitting domain (U; [x.sup.[lambda]], [c.sup.a]) of an N-graded manifold (Z, [A.sub.E]), the adapted splitting domain of a graded jet manifold [mathematical expression not reproducible] reads

[mathematical expression not reproducible]. (116)

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 N-graded bundles over smooth manifolds as follows. Let (X, Y, [U.sub.F]) be the N-graded bundle (113) modelled over the composite bundle (114). It is readily observed that the jet manifold [mathematical expression not reproducible] be an N-graded manifold modelled over this vector bundle. Its local generating basis is ([x.sup.[lambda]], [y.sup.i.sub.[LAMBDA]], [c.sup.a.sub.[LAMBDA]]), 0 [less than or equal to] [absolute value of ([LAMBDA])] [less than or equal to] r.

Definition 56. One calls [mathematical expression not reproducible] the graded jet manifold of an N-graded bundle (X, Y, [U.sub.F]).

In particular, let Y [right arrow] X be a smooth bundle seen as a trivial graded bundle (X, Y, [C.sup.[infinity].sub.Y]) modelled over a composite bundle Y x {0} [right arrow] Y [right arrow] X. Then its graded jet manifold is a trivial graded bundle [mathematical expression not reproducible], that is, the jet manifold [J.sup.r]Y of Y. Thus, Definition 56 of graded jet manifolds of N-graded bundles is compatible with Definition 2 of jets of fibre bundles.

The affine bundles [J.sup.r+1]Y [right arrow] [J.sup.r]Y (B.7) and the corresponding fibre bundles [J.sup.r+1]F [right arrow] [J.sup.r]F also yield the N-graded bundles

[mathematical expression not reproducible], (117)

including the sheaf monomorphisms

[mathematical expression not reproducible], (118)

where [mathematical expression not reproducible] is the pull-back onto [J.sup.r+1]Y of the continuous fibre bundle [mathematical expression not reproducible]. The sheaf monomorphism (118) induces a monomorphism of canonical presheaves Aprp [mathematical expression not reproducible], which associates to each open subset U [subset] [J.sup.r+1] Y the ring of sections of [mathematical expression not reproducible]. Accordingly, there is a pull-back monomorphism of the structure rings

[mathematical expression not reproducible], (119)

of graded functions on graded manifolds [mathematical expression not reproducible]. As a consequence, we have the inverse sequence of graded manifolds

[mathematical expression not reproducible]. (120)

One can think of its inverse limit [mathematical expression not reproducible] as being the graded infinite order jet manifold whose body is an infinite order jet manifold [J.sup.[infinity]]Y and whose structure sheaf [mathematical expression not reproducible] is a sheaf of germs of graded functions on N-graded manifolds [mathematical expression not reproducible]. However [mathematical expression not reproducible] fails to be an N-graded manifold in a strict sense because the inverse limit [J.sup.[infinity]]Y of the sequence (B.7) is a Freche manifold, but not the smooth one.

By virtue of Theorem 51, the differential calculus [S.sup.*.sub.r] [F; Y] of graded differential forms on N-graded manifolds [mathematical expression not reproducible] is minimal. Therefore, the monomorphisms of structure rings (119) yield the pull-back monomorphisms (112) of DBGRs

[mathematical expression not reproducible]. (121)

As a consequence, we have a direct system of DBGRs

[mathematical expression not reproducible]. (122)

The DBGR [S.sup.*.sub.[infinity]][F;Y] associated with an N-graded bundle (X, Y, [U.sub.F]) is defined as the direct limit

[mathematical expression not reproducible] (123)

of the direct system (122). It includes all graded differential forms [mathematical expression not reproducible] modulo the monomorphisms (121). Its elements obey relations (89).

Monomorphisms [O.sup.r]([J.sup.r]Y) [right arrow] [S.sup.*] [F; Y] yield a monomorphism of direct system (67) to direct system (122) and, consequently, a monomorphism

[O.sup.*.sub.[infinity]]Y [right arrow] [S.sup.*.sub.[infinity]][F;Y] (124)

of their direct limits. Accordingly, body epimorphisms [S.sup.*.sub.r] [F; Y] [right arrow] [O.sup.*.sub.r]Y yield an epimorphism

[S.sup.*.sub.[infinity]] [F;Y] [right arrow] [O.sup.*.sub.[infinity]]Y. (125)

It is readily observed that the morphisms (124)-(125) are cochain morphisms between the de Rham complex (69) of [O.sup.*.sub.[infinity]]Y and the de Rham complex

[mathematical expression not reproducible]. (126)

of a DBGR [S.sup.*.sub.[infinity]] [F;Y]. Moreover, the corresponding homomorphisms of cohomology groups of these complexes are isomorphisms as follows [9].

Theorem 57. There is an isomorphism [H.sup.*]([S.sup.*.sub.[infinity]][F;Y]) = [H.sup.*.sub.DR](Y) of the cohomology [H.sup.*]([S.sup.*.sub.[infinity]][F; Y]) of the de Rham complex (126) to the de Rham cohomology [H.sup.*.sub.DR](Y) of Y.

As was mentioned above, the [S.sup.*.sub.[infinity]][F; Y] (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 X.

A sheaf on a topological space X is a topological fibre bundle [pi] : S [right arrow] X in modules over a commutative ring K, where a surjection [pi] is a local homeomorphism and fibres [S.sub.x], x [member of] X, called the stalks, are provided with the discrete topology. Global sections of a sheaf S constitute a K-module S(X), called the structure module of S.

A presheaf [S.sub.{U}] on a topological space X is defined if a module [S.sub.U] over a commutative ring K is assigned to every open subset U [subset] X ([S.sub.0] = 0) and if, for any pair of open subsets V [subset] U, there exists a restriction morphism

[mathematical expression not reproducible]. (A.1)

Every presheaf [S.sub.{U}] on a topological space X yields a sheaf on X whose stalk [S.sub.x] at a point x [member of] X is the direct limit of modules [S.sub.U], x [member of] U, with respect to restriction morphisms [r.sup.U.sub.V]. It means that, for each open neighborhood U of a point x, every element s [member of] [S.sub.U]. determines an element [s.sub.x] [member of] [S.sub.x], called the germ of s at x. Two elements s [member of] [S.sub.U] and s' [member of] [S.sub.V] belong to the same germ at x iff there exists an open neighborhood W [subset] U [intersection] V of v such that [r.sup.U.sub.W] s = [r.sup.V.sub.W]s'.

Example A.1. Let [C.sup.0.sub.{U}] be a presheaf of continuous real functions on a topological space X. Two such functions s and s' define the same germ [s.sub.x] if they coincide on an open neighborhood of x. Hence, we obtain a sheaf [C.sup.0.sub.x] of continuous functions on X.

Every sheaf S defines a presheaf S({U}) of modules S(U) of its local sections. It is called the canonical presheaf of S. The sheaf generated by the canonical presheaf of a sheaf S is S.

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 [S.sub.{U}] to a presheaf [S'.sub.{U}] on a topological space X is defined as a set of module morphisms [[gamma].sub.U] : [S.sub.U] [right arrow] [S'.sub.U] which commute with restriction morphisms. A morphism of presheaves yields a morphism of sheaves generated by these presheaves. This is a bundle morphism over X such that [[gamma].sub.x] : [S.sub.x] [right arrow] [S'.sub.x] is the direct limit of morphisms [[gamma].sub.U], x [member of] U. Conversely, any morphism of sheaves S [right arrow] S' on a topological space X yields a morphism of canonical presheaves of local sections of these sheaves. Let Hom(S[|.sub.U],S'[|.sub.U]) be a commutative group of sheaf morphisms S [|.sub.U] [right arrow] S'[|.sub.U] for any open subset U [subset] X. These groups are assembled into a presheaf and define the sheaf Hom(S, S') on X. There is a monomorphism

Hom (S,S') (U) [right arrow] Hom (s (U), S' (U)). (A.2)

A sheaf R on a topological space X is said to be the ringed space if its stalk [R.sub.x] at each point x [member of] X is a commutative ring [5, 7, 31]. A ringed space often is denoted by a pair (X, R) of a topological space X and a sheaf R of rings on X which are called the body and the structure sheaf of a ringed space, respectively.

Definition A.2. A ringed [C.sup.0.sub.X] 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 [C.sup.0.sub.x] of germs of continuous real functions on a topological space X (Example A.1) is a local-ringed space. Its stalk [C.sup.0.sub.x], x [member of] X, contains a unique maximal ideal of germs of functions vanishing at x.

Morphisms of local-ringed spaces are defined to be particular morphisms of sheaves on different topological spaces as follows. Let [phi] : X [right arrow] X' be a continuous map. Given a sheaf S on X, its direct image [[phi].sub.*] S on X' is generated by the presheaf of assignments X' [contains] U' [right arrow] S([[phi].sup.-1](U')) forany open subset U' [subset] X'. Conversely, given a sheaf S' on X', its inverse image [[phi].sup.*]S' on X is defined as the pull-back onto X of a continuous fibre bundle S' over X'; that is, [[phi].sup.*][S'.sub.x] = [S.sub.[phi](x)]. This sheaf is generated by the presheaf which associates to any open subset V [subset] X the direct limit of modules S'(U) over all open subsets U [subset] X' such that V [subset] [f.sup.-1](U).

By a morphism of ringed spaces (X, R) [right arrow] (X', R') is meant a pair ([phi], [THETA]) of a continuous map [phi] : X [right arrow] X' and a sheaf morphism [THETA]: R' [right arrow] [[phi].sub.*]R or, equivalently, a sheaf morphism [[phi].sup.*]R' [right arrow] R [31]. Restricted to each stalk, a sheaf morphism O is assumed to be a ring homomorphism. A morphism of ringed spaces is said to be (i) a monomorphism if [phi] is an injection and [THETA] is an epimorphism and (ii) an epimorphism if [phi] is a surjection, while [THETA] is a monomorphism.

Given a local-ringed space (X, R), a sheaf P on X is called the sheaf of R-modules if every stalk [P.sub.x], x [member of] [P.sub.x], x [member of] X, is an [R.sub.x]-module or, equivalently, if P(U) is an R(U)-module for any open subset U [subset] X. A sheaf of R-modules P is said to be locally free if there exists an open neighborhood U of every point x [member of] X such that P(U) is a free R(U)-module. If all these free modules are of the same finite rank, one says that P is of constant rank. The structure module of a locally free sheaf is called the locally free module.

B. Jet Manifolds

Let Y [right arrow] X be a smooth fibre bundle provided with bundle coordinates ([x.sup.[lambda]], [y.sup.i]). An r-order jet manifold [J.sup.r]Y of sections of a fibre bundle Y [right arrow] X is defined as the disjoint union of equivalence classes [j.sup.r.sub.x]s of sections s of Y [right arrow] X which are identified by r + 1 terms of their Taylor series at points of X [4,16,32]. The set [J.sup.r]Y is endowed with an atlas of the adapted coordinates

[mathematical expression not reproducible], (B.1)

[[gamma]'.sup.i.sub.[lambda]+[LAMBDA]] = [partial derivative][x.sup.[mu]]/[partial derivative]'[x.sup.[lambda]] [d.sub.[mu]][y'.sup.i.sub.[LAMBDA]], (B.2)

where the symbol [d.sub.[LAMBDA]] stands for the higher order total derivative

[mathematical expression not reproducible]. (B.3)

These derivatives act on exterior forms on [J.sup.r]Y and obey the relations

[mathematical expression not reproducible]. (B.4)

We use the compact notation [mathematical expression not reproducible].

The coordinates (B.1) bring a set [J.sup.r]Y into a smooth manifold. They are compatible with the natural surjections

[[pi].sup.r.sub.r-1]: [J.sup.r]Y [right arrow] [J.sup.r-1]Y. (B.5)

A glance at the transition functions (B.2) shows that they are affine bundles. It follows that Y is a strong deformation retract of any finite order jet manifold [J.sup.r]Y.

Given fibre bundles Y and Y' over X, every bundle morphism [THETA] : Y [right arrow] Y' over a diffeomorphism f of X admits the r-order jet prolongation to a morphism of r-order jet manifolds

[mathematical expression not reproducible]. (B.6)

Every section s of a fibre bundle Y [right arrow] X has the r-order jet prolongation to a section ([J.sup.r]s)(x) = [f.sup.r.sub.x] s of a jet bundle [J.sup.r]Y [right arrow] X.

The surjections (B.5) form the inverse sequence of finite order jet manifolds

[mathematical expression not reproducible]. (B.7)

Its inverse limit [J.sup.[infinity].sub.Y] is defined as a minimal set such that there exist surjections

[mathematical expression not reproducible]. (B.8)

obeying the relations [[pi].sup.[infinity].sub.r] = [[pi].sup.k.sub.r] [omicron] [[pi].sup.[omicron].sub.k] for all admissible k and r < k. It consists of those elements

(..., [z.sub.r], ..., [z.sub.k], ...), [z.sub.r] [member of] [J.sup.r]Y, [z.sub.k] [member of] [J.sup.k]Y, (B.9)

of the Cartesian product [[??].sub.k][J.sup.k]Y which satisfy the relations [z.sub.r] = [[pi].sup.k.sub.r]([z.sub.k]) for all k > r. One can think of elements of [J.sup.[infinity].Y] as being infinite order jets of sections of Y [right arrow] X identified by their Taylor series at points of X. A set [J.sup.[infinity].sub.Y] is provided with the inverse limit topology. This is the coarsest topology such that the surjections [[pi].sup.[infinity].sub.r] (B.8) are continuous. Its base consists of inverse images of open subsets of [J.sup.r]Y, r = 0, ..., under the maps [[pi].sup.[infinity].sub.r]. With the inverse limit topology, [J.sup.[infinity].sub.Y] is a paracompact Frechet manifold [16,24,25]. One can show that surjections are open maps admitting local sections; that is, [J.sup.[infinity]]Y [right arrow] [J.sup.r]Y are continuous bundles. A bundle coordinate atlas {[U.sub.Y], ([x.sup.[lambda]], [y.sup.i])} of Y [right arrow] X provides [J.sup.[infinity]]Y with a manifold coordinate atlas

[mathematical expression not reproducible]. (B.10)

Definition B.1. One calls [J.sup.[infinity].sub.Y] the infinite order jet manifold.

http://dx.doi.org/10.1155/2017/8271562

Disclosure

One of the authors G. Sardanashvily has passed away on September 1 this year. It is a greatloss to his grateful colleagues and students.

Competing Interests

The authors declare that they have no competing interests.

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G. Sardanashvily and W. Wachowski

Department of Theoretical Physics, Moscow State University, Moscow 119999, Russia

Correspondence should be addressed to W. Wachowski; vladvakh@gmail.com

Received 31 July 2016; Accepted 3 November 2016; Published 17 January 2017

Academic Editor: Mauro Nacinovich
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