# Differential Calculus onN-Graded Manifolds

Authors: G. Sardanashvily; W. Wachowski
Journal: Journal of Mathematics
 (2017)

---

## Abstract

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and on N-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 Z2-graded manifolds. We follow the notion of an N-graded manifold as a local-ringed space whose body is a smooth manifold Z. A key point is that the graded derivation module of the structure ring of graded functions on an N-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its body Z. Accordingly, the Chevalley–Eilenberg differential calculus on an N-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus on N-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds of N-graded bundles.


---

## Body

## 1. Introduction

This work addresses the differential calculus overN-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.
LetK be a commutative ring. A direct sum of K-modules(1)P=P∗=⨁i∈NPiis called the N-graded K-module. Its elements p are called homogeneous of degree [p] if p∈P[p].Definition 2.
AK-ring A is called N-graded if it is an N-graded K-module A∗ (1) so that a product of homogeneous elements αα′ is a homogeneous element of degree [α]+[α′]. In particular, it follows that A0 is a K-ring, while Ak>0 and, accordingly, A∗ are A0-bimodules.AnyN-graded K-module P (1) admits the associated Z2-graded structure (2)P=P0⊕P1,P0=⨁i∈NP2i,P1=⨁i∈NP2i+1. Accordingly, an N-graded ring A∗ also is the Z2-graded one A∗ (Definition 18). The converse is not true. For instance, Clifford algebras are Z2-graded, but not N-graded rings. In general, an N-graded ring A∗ can admit different N- and Z2-graded structures (Theorem 26).Remark 3.
Hereafter, we follow the notationA∗ (resp., A∗) of a K-ring A endowed with an N-graded (resp., Z2-graded) structure. If there is no danger of confusion, the symbol [·] further stands both for N- and Z2-degree.
We further restrict our consideration toN-graded commutative rings.Definition 4.
AnN-graded K-ring A∗ is said to be graded commutative if(3)αβ=-1αββα,α,β∈A∗.In this case, A0 is a commutative K-ring, and A∗ is an A0-ring.Example 5.
AnN-graded commutative ring A∗ is commutative if A2i+1=0. Conversely, any commutative ring A is an N-graded commutative one A∗, where A>0=0.AnN-graded commutative ring A∗ possesses an associated Z2-graded commutative structure(4)A0=⨁kA2k,A1=⨁kA2k+1,k∈N,αβ=-1αββα,α,β∈A∗,in accordance with Definition 19. The converse need not be true.The differential calculus overN-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 ofZ2-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 Z2-graded commutative rings [5, 8, 9]. Since any N-graded commutative ring A∗ possesses the associated structure (4) of a Z2-graded commutative ring A∗ and the commutation relations (3) of its elements depend on their Z2-graded degree, the differential calculus over N-graded commutative rings is defined similarly to that over the Z2-graded ones (Section 4). Herewith, a linear N-graded differential operator, being an N-graded K-module homomorphism, is a Z2-graded homomorphism which obeys conditions (72). Consequently, it is a linear Z2-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 Z2-graded differential calculus. This is just the case of N-graded manifolds in comparison with the Z2-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 Z2-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).SinceZ2-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.
AnN-graded commutative K-ring Λ∗ 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 Λ∗=⋀Λ1 of a K-module Λ1 (Example 9).A Grassmann-gradedK-ring Λ∗ seen as a Z2-graded commutative ring Λ∗ is a Grassmann algebra (Definition 23). A Grassmann algebra Λ∗, in turn, can admit different associated Grassmann-graded structures Λ∗. 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 Z2-graded manifold. Conversely, any Z2-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 Z2-graded andvice versa, the differential calculus on these graded manifolds is different.The differential calculus on anN-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 Z2-graded one, are represented by sections of the smooth vector bundle VE (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 onN-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∞∗[F,Y] (123) of graded differential forms on an N-graded infinite order jet manifold (J∞Y,AJ∞F). A key point is that this ring S∞∗[F,Y] 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 algebraA 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≠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∈A, implies either a=0 or b=0. For instance, this is a case of a division ring.A subsetI 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∈I (resp., ba∈I) for all a∈A, b∈I. If I 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 I⊂A, an additive factor group A/I is an algebra.Definition 7.
A ringA 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 Z2-graded commutative rings, too [8]. Grassmann-graded rings in Definition 6 and Grassmann algebras in Definition 23 are local.Given an algebraA, an additive group P is said to be the left (resp., right) A-module if it is provided with a distributive multiplication A×P→P by elements of A such that (ab)p=a(bp) (resp., p(ab)=(pa)b) for all a,b∈A and p∈P. If A is a ring, one additionally assumes that 1p=p=p1 for all p∈P. If P is both a left module over an algebra A and a right module over an algebra A′, it is called the (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∈A and p∈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 ringA from the old ones.(i) A direct sumP⊕P′ of A-modules P and P′ is an additive group P×P provided with an A-module structure (5)ap,p′=ap,ap′,p∈P,p′∈P′,a∈A. Let {Pi}i∈I be a set of A-modules. Their direct sum ⨁Pi consists of elements (…,pi,…) of the Cartesian product ∏‍Pi such that pi≠0 at most for a finite number of indices i∈I.(ii) A tensor productP⊗AQ of A-modules P and Q is an additive group which is generated by elements p⊗q, p∈P, q∈Q, obeying relations (6)p+p′⊗q=p⊗q+p′⊗q,p⊗q+q′=p⊗q+p⊗q′,pa⊗q=p⊗aq,p∈P,q∈Q,a∈A. It is endowed with an A-module structure (7)ap⊗q=ap⊗q=p⊗qa=p⊗qa. If a ring A is treated as an A-module, a tensor product A⊗AQ is canonically isomorphic to Q via the assignment a⊗q↔aq, a∈A, q∈Q.Example 9.
LetQ be an A-module. Let us consider an N-graded module(8)⨂Q=A⊕Q⊕⋯⊕⨂AkQ⊕⋯.This is an N-graded A-ring with respect to a tensor product ⨂. It is called the tensor algebra of an A-module Q. Its quotient ⋀Q with respect to an ideal generated by elements q⊗q′+q′⊗q, q,q′∈Q, is an N-graded commutative algebra, called the exterior algebra of an A-module Q.(i) Given a submoduleQ 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 setHomA(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∗=HomA(P,A) is called the dual of an A-module P. There is a natural monomorphism P→P∗∗.A moduleP over a commutative ring A is called free if it admits a basis, that is, a linearly independent subset I⊂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⊕Q is a free module.Theorem 10.
IfP is a projective module of finite rank, then its dual P∗ is so, and P∗∗=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 setI is meant a set with an order relation < which satisfies the following conditions: (i) i<i, for all i∈I; (ii) if i<j and j<k, then i<k; (iii) for any i,j∈I, there exists k∈I such that i<k and j<k. It may happen that i≠j, but i<j and j<i.A family ofA-modules {Pi}i∈I, indexed by a directed set I, is called the direct system if, for any pair i<j, there is a morphism rji:Pi→Pj such that (i) rii=IdPi; (ii) rji∘rkj=rki, i<j<k. A direct system of modules admits a direct limit.Definition 11.
This is anA-module P∞ together with morphisms r∞i:Pi→P∞ such that r∞i=r∞j∘rji for all i<j. A module P∞ consists of elements of a direct sum ⨁Pi modulo the identification of elements of Pi with their images in Pj for all i<j.Theorem 12.
Direct limits commute with direct sums and tensor products of modules. Namely, let{Pi} and {Qi} be two direct systems of A-modules which are indexed by the same directed set I, and let P∞ and Q∞ be their direct limits. Then direct limits of direct systems {Pi⊕Qi} and {Pi⊗Qi} are P∞Q∞ and P∞⊗Q∞, respectively.Theorem 13.
A morphism of a direct system{Pi,rji}I to a direct system Qi′,ρj′i′I′ consists of an order preserving map f:I→I′ and A-module morphisms Φi:Pi→Qf(i) which obey compatibility conditions ρf(j)f(i)∘Φi=Φj∘rji. If P∞ and Q∞ are direct limits of these direct systems, there exists a unique A-module morphism Φ∞:P∞→Q∞ such that ρ∞f(i)∘Φi=Φ∞∘r∞i.Theorem 14.
A construction of a direct limit morphism preserves monomorphisms and epimorphisms. If allΦi:Pi→Qf(i) are monomorphisms (resp., epimorphisms), so is Φ∞:P∞→Q∞.In a case of inverse systems of modules, we restrict our consideration to inverse sequences(9)P0⟵P1⟵⋯Pi←πii+1⋯. Its inverse limit is a module P∞ together with morphisms πi∞:P∞→Pi so that πi∞=πij∘πj∞ for all i<j. It consists of elements (…,pi,…), pi∈Pi, of the Cartesian product ∏‍Pi such that pi=πij(pj) for all i<j. A morphism of an inverse system {Pi,πji} to an inverse system {Qi,ϱji} consists of A-module morphisms Φi:Pi→Qi which obey compatibility conditions(10)Φj∘πji=ϱji∘Φi.If P∞ and Q∞ are inverse limits of these inverse systems, there exists a unique A-module morphism Φ∞:P∞→Q∞ such that Φj∘πj∞=ϱj∞∘Φ∞. A construction of an inverse limits morphism preserves monomorphisms, but not epimorphisms.Example 15.
In particular, let{Pi,πji} be an inverse system of A-modules and Q an A-module together with A-module morphisms Φi:Q→Pi which obey compatibility conditions Φj=πji∘Φi. Then there exists a unique morphism Φ∞:Q→P∞ such that Φj=πj∞∘Φ∞.Example 16.
LetPi,πji be an inverse system of A-modules and Q an A-module. Given a term Pr, let Φr:Pr→Q be an A-module morphism. It yields the pull-back morphisms(11)πrr+k∗Φr=Φr∘πrr+k:Pr+k⟶Qwhich obviously obey the compatibility conditions (10). Then there exists a unique morphism Φ∞:P∞→Q such that Φ∞=Φr∘πr∞.Example 17.
Let{Pi} be an inverse sequence of A-modules. Given an A-module Q, modules HomA(Pi,Q) constitute a direct sequence whose direct limit is isomorphic to HomA(P∞,Q).


### 2.2.Z2-Graded Commutative Rings

AK-module Q is called Z2-graded if it is decomposed into a direct sum Q=Q∗=Q0⊕Q1 of modules Q0 and Q1, called the even and odd parts of Q∗, respectively. A Z2-graded K-module is said to be free if it has a basis composed by graded-homogeneous elements.A morphismΦ:P∗→Q∗ of Z2-graded K-modules is said to be an even (resp., odd) morphism if Φ preserves (resp., changes) the Z2-parity of all homogeneous elements. A morphism Φ:P∗→Q∗ of Z2-graded K-modules is called graded if it is represented by a sum of even and odd morphisms. A set HomK(P,Q) of these graded morphisms is a Z2-graded K-module.Definition 18.
AK-ring A is called Z2-graded if it is a Z2-graded K-module A∗, and a product of its homogeneous elements αα′ is a homogeneous element of degree a+a′mod2. In particular, [1]=0. Its even part A0 is a K-ring, and the odd one A1 is an A0-bimodule.Definition 19.
AZ2-graded ring A∗ is called graded commutative if (12)aa′=-1aa′a′a,a,a′∈A∗. Its even part A0 belongs to the center ZA of a ring A.EveryN-graded commutative K-ring A∗ (Definition 4) possesses the associated Z2-graded commutative structure A∗ (4). For instance, the exterior algebra ⋀Q of a K-module Q in Example 9 is a Z2-graded commutative ring.Definition 20.
AZ2-graded commutative ring is called local if it contains a unique maximal Z2-graded ideal.IfK is a field, an exterior K-algebra exemplifies a local Z2-graded commutative ring. An ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is Z2-graded.AZ2-graded commutative ring can admit different Z2-graded commutative structures A∗ in general (Example 21).By automorphisms of aZ2-graded commutative ring A∗ are meant automorphisms of a K-ring A which are graded K-module morphisms of A∗. Obviously, they are even, and they preserve a Z2-graded structure of A. However, there exist automorphisms ϕ of a K-ring A which do not possess this property in general. Then A∗ and ϕ(A∗) are isomorphic, but different Z2-graded commutative structures of a ring A. Moreover, it may happen that a K-ring A admits nonisomorphic Z2-graded commutative structures.Example 21.
Given aZ2-graded commutative ring A∗ and its odd element κ, an automorphism(13)ϕ:A0∋a⟶a,A1∋a⟶a1+κ,of a K-ring A does not preserve its original Z2-graded structure A∗.Given aZ2-graded commutative ring A∗, a Z2-graded A∗-module Q∗ is defined as an (A-A)-bimodule which is a Z2-graded K-module such that (14)aq=a+qmod2,qa=-1aqaq,a∈A∗,q∈Q∗.The following are constructions of newZ2-graded A∗-modules from the old ones.(i) A direct sum ofZ2-graded modules and a Z2-graded factor module are defined just as those of modules over a commutative ring.(ii) A tensor productP∗⊗Q∗ of Z2-graded A∗-modules P∗ and Q∗ is their tensor product as A-modules such that (15)p⊗q=p+qmod2,p∈P∗,q∈Q∗,ap⊗q=-1papa⊗q=-1pap⊗aq,a∈A∗. In particular, the tensor algebra ⨂P∗ of a Z2-graded A∗-module P∗ is defined just as that (8) of a module over a commutative ring. Its quotient ⋀P∗ with respect to the ideal generated by elements (16)p⊗p′+-1pp′p′⊗p,p,p′∈P∗ is the exterior algebra of a Z2-graded module P∗ with respect to the graded exterior product(17)p∧p′=--1pp′p′∧p.(iii) A graded morphismΦ:P∗→Q∗ of Z2-graded A∗-modules is their graded morphism as Z2-graded K-modules which obeys the relations(18)Φap=-1ΦaaΦp,p∈P∗,a∈A∗.These morphisms form a Z2-graded A∗-module HomA(P∗,Q∗). A Z2-graded A∗-module P∗=HomA(P∗,A∗) is called the dual of a Z2-graded A∗-module P∗.In the sequel, we are concerned withZ2-graded manifolds (Section 5). They are sheaves in Grassmann algebras which are defined as follows.Definition 22.
AZ2-graded K-ring Λ∗ is said to be finitely generated in degree 1 if it is a free K-module of finite rank so that Λ0=K⊕Λ12.It follows that aK-module Λ has a decomposition(19)Λ=K⊕R,R=Λ1⊕Λ12,where R is the ideal of nilpotents of a ring Λ. A surjection σ:Λ→K is called the body map.Definition 23.
AZ2-graded commutative K-ring Λ∗ is said to be the Grassmann algebra if it is finitely generated in degree 1 and is isomorphic to the exterior algebra ⋀(R/R2) (Example 9) of a K-module R/R2, where R is the ideal of nilpotents (19) of Λ∗.An exterior algebra⋀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 ⋀Q by a choice of its minimal generating K-module Q⊂Λ1, 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 Λ1 (Example 21).A Grassmann algebra is local in accordance with Definition20. Its ideal of nilpotents R is a unique maximal ideal which is graded in accordance with the decomposition (19).Remark 24.
LetA∗ be a Z2-graded commutative ring. A Z2-graded A∗-algebra G∗ is called the Lie A∗-superalgebra if its product [·,·], called the Lie superbracket, obeys the rules (20)ε,ε′=--1εε′ε′,ε,-1εε′′ε,ε′,ε′′+-1ε′εε′,ε′′,ε+-1ε′′ε′ε′′,ε,ε′=0. Clearly, an even part G0 of a Lie superalgebra G∗ is a Lie A0-algebra. Given an A∗-superalgebra, a Z2-graded A∗-module P∗ is called a G∗-module if it is provided with an A∗-bilinear map (21)G∗×P∗∋ε,p⟶εp∈P∗,εp=ε+pmod2,ε,ε′p=ε∘ε′--1εε′ε′∘εp.


### 2.3.N-Graded Commutative Rings

LetA=A∗ 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.
AnN-graded K-ring A∗ is called finitely generated in degree 1 if the following hold: (i) A0=K; (ii) A1 is a free K-module of finite rank; (iii) A∗ is generated by A1; namely, if R is an ideal generated by A1, then there is K-module isomorphism A/R=K, R/R2=A1.Theorem 26.
LetK be a field, and let A∗ and Λ∗ be N-graded K-rings finitely generated in degree 1. If they are isomorphic as K-rings, there exists their graded isomorphism Φ:A∗→Λ∗ so that Φ(Ai)=Λi for all i∈N.As was mentioned above, we restrict our consideration toN-graded commutative rings (Definition 4), unless they are the differential graded ones (Section 2.4). They also possess the associated Z2-graded commutative structure (4).Definition 27.
AnN-graded commutative ring is called local if it contains a unique maximal N-graded ideal.Certainly, if anN-graded ring A∗ is local, the associated Z2-graded ring A∗ 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 anN-graded commutative ring A∗, an N-graded A∗-module Q∗ is defined as a graded (A∗-A∗)-bimodule which is an N-graded K-module such that (22)qa=-1aqaq,aq=a+q,a∈A∗,q∈Q∗, and it also is a Z2-graded module. A direct sum, a tensor product of N-graded modules, and the exterior algebra ⋀Q∗ of an N-graded module Q∗ are defined similarly to those of Z2-graded modules (Section 2.2), and they also are a direct sum, a tensor product, and an exterior algebra of associated Z2-graded modules, respectively.A morphismΦ:P∗→Q∗ of N-graded A∗-modules seen as K-modules is said to be homogeneous of degree [Φ] if [Φ(p)]=[p]+[Φ] for all homogeneous elements p∈P∗ and the relations (18) hold. A morphism Φ:P∗→Q∗ of N-graded A∗-modules as the K-ones is called the N-graded A∗-module morphism if it is represented by a sum of homogeneous morphisms. Therefore, a set HomA(P∗,Q∗) of graded morphisms P∗→Q∗ is an N-graded A∗-module. An N-graded A∗-module P∗=HomA(P∗,A∗) is called the dual of an N-graded A∗-module P∗. Certainly, an N-graded A∗-module morphism of N-graded A∗-modules is their Z2-graded A∗-module morphism as associated Z2-graded modules, but the converse is true.By automorphisms of anN-graded ring A∗ are meant automorphisms of a K-ring A which preserve its N-gradation A∗. They also keep the associated Z2-structure A∗ of A. However, there exist automorphisms of a K-ring A which do not possess these properties in general.LetΛ∗ be a Grassmann-graded K-ring (Definition 6). Its associated Z2-graded commutative ring is a Grassmann algebra Λ (Definition 23). Conversely, any Grassmann algebra Λ admits the associated structure of a Grassmann-graded ring Λ∗ by a choice of its minimal generating K-module Λ1⊂Λ1. Given a generating basis {ci} for a K-module Λ1, elements of a Grassmann-graded ring Λ∗ take a form(23)a=∑k=0,1,…∑ai1⋯ikci1⋯cik.We agree to call {ci} the generating basis for the associated Grassmann algebra Λ∗ which brings it into a Grassmann-graded ring Λ∗.Given a generating basis{ci} for a Grassmann-graded ring Λ∗, one can show that any K-ring automorphism is a composition of automorphisms(24)ci⟶c′i=ρjicj+bi,where ρ is an automorphism of a K-module Λ1 and bi are odd elements of Λ>2 and of morphisms(25)ci⟶c′i=ci1+κ,κ∈Λ1.Automorphisms (24), where bi=0, are automorphisms c′i=ρjicj of a Grassmann-graded ring Λ∗. If bi≠0, the automorphism (24) preserves the associated Z2-graded structure Λ∗ of Λ but does not keep its N-graded structure Λ∗. It yields a different N-graded structure Λ′∗, where {c′i} (24) is a basis for Λ′1 and the generating basis for Λ′∗. Automorphisms (25) preserve an even sector Λ0 of Λ∗, but not the odd one Λ1 (Example 21). However, it follows from Theorem 26 that different N- and Z2-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 ringΛ∗ over a field K, there exists a finite-dimensional vector space W over K so that Λ∗ is isomorphic to the exterior algebra ⋀W of W (Example 9) seen as a Grassmann-graded ring generated by W.


### 2.4. DifferentialN-Graded Rings

If anN-graded ring also is a cochain complex, we come to the following notion [5, 17].Definition 29.
AnN-graded K-ring Ω∗ is called the differential graded ring (henceforth, DGR) if it is a cochain complex of K-modules(26)0⟶K⟶Ω0⟶δΩ1⟶δ⋯Ωk⟶δ⋯with respect to a coboundary operator δ which obeys the graded Leibniz rule(27)δαβ=δαβ+-1ααδβ.
The cochain complex (26) is called the de Rham complex of a DGR (Ω∗,δ). It also is said to be the differential graded calculus over a K-ring Ω0.Given a DGR(Ω∗,δ), one considers its minimal differential graded subring (Ω¯∗,δ) which contains Ω0. Seen as a (Ω0-Ω0)-ring, it is generated by elements δa, a∈A, and consists of monomials α=a0δa1⋯δak, ai∈Ω0, whose product obeys the juxtaposition rule (28)a0δa1b0δb1=a0δa1b0δb1-a0a1δb0δb1 in accordance with equality (27). A complex (Ω¯∗,δ) is called the minimal differential graded calculus over Ω0. Its cohomology is said to be the de Rham cohomology of (Ω∗,δ).One can associate a DGR to any LieK-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 (29)G×Q∋ε,q⟶εq∈Q,ε,ε′q=ε∘ε′-ε′∘εq,ε,ε′∈G, (cf. Remark 24). For instance, Q=K and G:K→0. A K-multilinear skew-symmetric map(30)ck:⨉kG⟶Qis called the Q-valued k-cochain on a Lie algebra G. These cochains form a G-module Ck[G;Q]. Let us put C0[G;Q]=Q. We obtain the cochain complex(31)0⟶K⟶Q⟶δC1G;Q⟶δ⋯CkG;Q⟶δ⋯with respect to the Chevalley–Eilenberg coboundary operators(32)δckε0,…,εk=∑i=0k-1iεickε0,…,ε^i,…,εk+∑1≤i<j≤k-1i+jckεi,εj,ε0,…,ε^i,…,ε^j,…,εk, 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].


## 2.1. Commutative Rings

An algebraA 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≠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∈A, implies either a=0 or b=0. For instance, this is a case of a division ring.A subsetI 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∈I (resp., ba∈I) for all a∈A, b∈I. If I 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 I⊂A, an additive factor group A/I is an algebra.Definition 7.
A ringA 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 Z2-graded commutative rings, too [8]. Grassmann-graded rings in Definition 6 and Grassmann algebras in Definition 23 are local.Given an algebraA, an additive group P is said to be the left (resp., right) A-module if it is provided with a distributive multiplication A×P→P by elements of A such that (ab)p=a(bp) (resp., p(ab)=(pa)b) for all a,b∈A and p∈P. If A is a ring, one additionally assumes that 1p=p=p1 for all p∈P. If P is both a left module over an algebra A and a right module over an algebra A′, it is called the (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∈A and p∈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 ringA from the old ones.(i) A direct sumP⊕P′ of A-modules P and P′ is an additive group P×P provided with an A-module structure (5)ap,p′=ap,ap′,p∈P,p′∈P′,a∈A. Let {Pi}i∈I be a set of A-modules. Their direct sum ⨁Pi consists of elements (…,pi,…) of the Cartesian product ∏‍Pi such that pi≠0 at most for a finite number of indices i∈I.(ii) A tensor productP⊗AQ of A-modules P and Q is an additive group which is generated by elements p⊗q, p∈P, q∈Q, obeying relations (6)p+p′⊗q=p⊗q+p′⊗q,p⊗q+q′=p⊗q+p⊗q′,pa⊗q=p⊗aq,p∈P,q∈Q,a∈A. It is endowed with an A-module structure (7)ap⊗q=ap⊗q=p⊗qa=p⊗qa. If a ring A is treated as an A-module, a tensor product A⊗AQ is canonically isomorphic to Q via the assignment a⊗q↔aq, a∈A, q∈Q.Example 9.
LetQ be an A-module. Let us consider an N-graded module(8)⨂Q=A⊕Q⊕⋯⊕⨂AkQ⊕⋯.This is an N-graded A-ring with respect to a tensor product ⨂. It is called the tensor algebra of an A-module Q. Its quotient ⋀Q with respect to an ideal generated by elements q⊗q′+q′⊗q, q,q′∈Q, is an N-graded commutative algebra, called the exterior algebra of an A-module Q.(i) Given a submoduleQ 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 setHomA(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∗=HomA(P,A) is called the dual of an A-module P. There is a natural monomorphism P→P∗∗.A moduleP over a commutative ring A is called free if it admits a basis, that is, a linearly independent subset I⊂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⊕Q is a free module.Theorem 10.
IfP is a projective module of finite rank, then its dual P∗ is so, and P∗∗=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 setI is meant a set with an order relation < which satisfies the following conditions: (i) i<i, for all i∈I; (ii) if i<j and j<k, then i<k; (iii) for any i,j∈I, there exists k∈I such that i<k and j<k. It may happen that i≠j, but i<j and j<i.A family ofA-modules {Pi}i∈I, indexed by a directed set I, is called the direct system if, for any pair i<j, there is a morphism rji:Pi→Pj such that (i) rii=IdPi; (ii) rji∘rkj=rki, i<j<k. A direct system of modules admits a direct limit.Definition 11.
This is anA-module P∞ together with morphisms r∞i:Pi→P∞ such that r∞i=r∞j∘rji for all i<j. A module P∞ consists of elements of a direct sum ⨁Pi modulo the identification of elements of Pi with their images in Pj for all i<j.Theorem 12.
Direct limits commute with direct sums and tensor products of modules. Namely, let{Pi} and {Qi} be two direct systems of A-modules which are indexed by the same directed set I, and let P∞ and Q∞ be their direct limits. Then direct limits of direct systems {Pi⊕Qi} and {Pi⊗Qi} are P∞Q∞ and P∞⊗Q∞, respectively.Theorem 13.
A morphism of a direct system{Pi,rji}I to a direct system Qi′,ρj′i′I′ consists of an order preserving map f:I→I′ and A-module morphisms Φi:Pi→Qf(i) which obey compatibility conditions ρf(j)f(i)∘Φi=Φj∘rji. If P∞ and Q∞ are direct limits of these direct systems, there exists a unique A-module morphism Φ∞:P∞→Q∞ such that ρ∞f(i)∘Φi=Φ∞∘r∞i.Theorem 14.
A construction of a direct limit morphism preserves monomorphisms and epimorphisms. If allΦi:Pi→Qf(i) are monomorphisms (resp., epimorphisms), so is Φ∞:P∞→Q∞.In a case of inverse systems of modules, we restrict our consideration to inverse sequences(9)P0⟵P1⟵⋯Pi←πii+1⋯. Its inverse limit is a module P∞ together with morphisms πi∞:P∞→Pi so that πi∞=πij∘πj∞ for all i<j. It consists of elements (…,pi,…), pi∈Pi, of the Cartesian product ∏‍Pi such that pi=πij(pj) for all i<j. A morphism of an inverse system {Pi,πji} to an inverse system {Qi,ϱji} consists of A-module morphisms Φi:Pi→Qi which obey compatibility conditions(10)Φj∘πji=ϱji∘Φi.If P∞ and Q∞ are inverse limits of these inverse systems, there exists a unique A-module morphism Φ∞:P∞→Q∞ such that Φj∘πj∞=ϱj∞∘Φ∞. A construction of an inverse limits morphism preserves monomorphisms, but not epimorphisms.Example 15.
In particular, let{Pi,πji} be an inverse system of A-modules and Q an A-module together with A-module morphisms Φi:Q→Pi which obey compatibility conditions Φj=πji∘Φi. Then there exists a unique morphism Φ∞:Q→P∞ such that Φj=πj∞∘Φ∞.Example 16.
LetPi,πji be an inverse system of A-modules and Q an A-module. Given a term Pr, let Φr:Pr→Q be an A-module morphism. It yields the pull-back morphisms(11)πrr+k∗Φr=Φr∘πrr+k:Pr+k⟶Qwhich obviously obey the compatibility conditions (10). Then there exists a unique morphism Φ∞:P∞→Q such that Φ∞=Φr∘πr∞.Example 17.
Let{Pi} be an inverse sequence of A-modules. Given an A-module Q, modules HomA(Pi,Q) constitute a direct sequence whose direct limit is isomorphic to HomA(P∞,Q).


## 2.2.Z2-Graded Commutative Rings

AK-module Q is called Z2-graded if it is decomposed into a direct sum Q=Q∗=Q0⊕Q1 of modules Q0 and Q1, called the even and odd parts of Q∗, respectively. A Z2-graded K-module is said to be free if it has a basis composed by graded-homogeneous elements.A morphismΦ:P∗→Q∗ of Z2-graded K-modules is said to be an even (resp., odd) morphism if Φ preserves (resp., changes) the Z2-parity of all homogeneous elements. A morphism Φ:P∗→Q∗ of Z2-graded K-modules is called graded if it is represented by a sum of even and odd morphisms. A set HomK(P,Q) of these graded morphisms is a Z2-graded K-module.Definition 18.
AK-ring A is called Z2-graded if it is a Z2-graded K-module A∗, and a product of its homogeneous elements αα′ is a homogeneous element of degree a+a′mod2. In particular, [1]=0. Its even part A0 is a K-ring, and the odd one A1 is an A0-bimodule.Definition 19.
AZ2-graded ring A∗ is called graded commutative if (12)aa′=-1aa′a′a,a,a′∈A∗. Its even part A0 belongs to the center ZA of a ring A.EveryN-graded commutative K-ring A∗ (Definition 4) possesses the associated Z2-graded commutative structure A∗ (4). For instance, the exterior algebra ⋀Q of a K-module Q in Example 9 is a Z2-graded commutative ring.Definition 20.
AZ2-graded commutative ring is called local if it contains a unique maximal Z2-graded ideal.IfK is a field, an exterior K-algebra exemplifies a local Z2-graded commutative ring. An ideal of its nilpotents is a unique maximal ideal of its noninvertible elements which also is Z2-graded.AZ2-graded commutative ring can admit different Z2-graded commutative structures A∗ in general (Example 21).By automorphisms of aZ2-graded commutative ring A∗ are meant automorphisms of a K-ring A which are graded K-module morphisms of A∗. Obviously, they are even, and they preserve a Z2-graded structure of A. However, there exist automorphisms ϕ of a K-ring A which do not possess this property in general. Then A∗ and ϕ(A∗) are isomorphic, but different Z2-graded commutative structures of a ring A. Moreover, it may happen that a K-ring A admits nonisomorphic Z2-graded commutative structures.Example 21.
Given aZ2-graded commutative ring A∗ and its odd element κ, an automorphism(13)ϕ:A0∋a⟶a,A1∋a⟶a1+κ,of a K-ring A does not preserve its original Z2-graded structure A∗.Given aZ2-graded commutative ring A∗, a Z2-graded A∗-module Q∗ is defined as an (A-A)-bimodule which is a Z2-graded K-module such that (14)aq=a+qmod2,qa=-1aqaq,a∈A∗,q∈Q∗.The following are constructions of newZ2-graded A∗-modules from the old ones.(i) A direct sum ofZ2-graded modules and a Z2-graded factor module are defined just as those of modules over a commutative ring.(ii) A tensor productP∗⊗Q∗ of Z2-graded A∗-modules P∗ and Q∗ is their tensor product as A-modules such that (15)p⊗q=p+qmod2,p∈P∗,q∈Q∗,ap⊗q=-1papa⊗q=-1pap⊗aq,a∈A∗. In particular, the tensor algebra ⨂P∗ of a Z2-graded A∗-module P∗ is defined just as that (8) of a module over a commutative ring. Its quotient ⋀P∗ with respect to the ideal generated by elements (16)p⊗p′+-1pp′p′⊗p,p,p′∈P∗ is the exterior algebra of a Z2-graded module P∗ with respect to the graded exterior product(17)p∧p′=--1pp′p′∧p.(iii) A graded morphismΦ:P∗→Q∗ of Z2-graded A∗-modules is their graded morphism as Z2-graded K-modules which obeys the relations(18)Φap=-1ΦaaΦp,p∈P∗,a∈A∗.These morphisms form a Z2-graded A∗-module HomA(P∗,Q∗). A Z2-graded A∗-module P∗=HomA(P∗,A∗) is called the dual of a Z2-graded A∗-module P∗.In the sequel, we are concerned withZ2-graded manifolds (Section 5). They are sheaves in Grassmann algebras which are defined as follows.Definition 22.
AZ2-graded K-ring Λ∗ is said to be finitely generated in degree 1 if it is a free K-module of finite rank so that Λ0=K⊕Λ12.It follows that aK-module Λ has a decomposition(19)Λ=K⊕R,R=Λ1⊕Λ12,where R is the ideal of nilpotents of a ring Λ. A surjection σ:Λ→K is called the body map.Definition 23.
AZ2-graded commutative K-ring Λ∗ is said to be the Grassmann algebra if it is finitely generated in degree 1 and is isomorphic to the exterior algebra ⋀(R/R2) (Example 9) of a K-module R/R2, where R is the ideal of nilpotents (19) of Λ∗.An exterior algebra⋀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 ⋀Q by a choice of its minimal generating K-module Q⊂Λ1, 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 Λ1 (Example 21).A Grassmann algebra is local in accordance with Definition20. Its ideal of nilpotents R is a unique maximal ideal which is graded in accordance with the decomposition (19).Remark 24.
LetA∗ be a Z2-graded commutative ring. A Z2-graded A∗-algebra G∗ is called the Lie A∗-superalgebra if its product [·,·], called the Lie superbracket, obeys the rules (20)ε,ε′=--1εε′ε′,ε,-1εε′′ε,ε′,ε′′+-1ε′εε′,ε′′,ε+-1ε′′ε′ε′′,ε,ε′=0. Clearly, an even part G0 of a Lie superalgebra G∗ is a Lie A0-algebra. Given an A∗-superalgebra, a Z2-graded A∗-module P∗ is called a G∗-module if it is provided with an A∗-bilinear map (21)G∗×P∗∋ε,p⟶εp∈P∗,εp=ε+pmod2,ε,ε′p=ε∘ε′--1εε′ε′∘εp.


## 2.3.N-Graded Commutative Rings

LetA=A∗ 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.
AnN-graded K-ring A∗ is called finitely generated in degree 1 if the following hold: (i) A0=K; (ii) A1 is a free K-module of finite rank; (iii) A∗ is generated by A1; namely, if R is an ideal generated by A1, then there is K-module isomorphism A/R=K, R/R2=A1.Theorem 26.
LetK be a field, and let A∗ and Λ∗ be N-graded K-rings finitely generated in degree 1. If they are isomorphic as K-rings, there exists their graded isomorphism Φ:A∗→Λ∗ so that Φ(Ai)=Λi for all i∈N.As was mentioned above, we restrict our consideration toN-graded commutative rings (Definition 4), unless they are the differential graded ones (Section 2.4). They also possess the associated Z2-graded commutative structure (4).Definition 27.
AnN-graded commutative ring is called local if it contains a unique maximal N-graded ideal.Certainly, if anN-graded ring A∗ is local, the associated Z2-graded ring A∗ 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 anN-graded commutative ring A∗, an N-graded A∗-module Q∗ is defined as a graded (A∗-A∗)-bimodule which is an N-graded K-module such that (22)qa=-1aqaq,aq=a+q,a∈A∗,q∈Q∗, and it also is a Z2-graded module. A direct sum, a tensor product of N-graded modules, and the exterior algebra ⋀Q∗ of an N-graded module Q∗ are defined similarly to those of Z2-graded modules (Section 2.2), and they also are a direct sum, a tensor product, and an exterior algebra of associated Z2-graded modules, respectively.A morphismΦ:P∗→Q∗ of N-graded A∗-modules seen as K-modules is said to be homogeneous of degree [Φ] if [Φ(p)]=[p]+[Φ] for all homogeneous elements p∈P∗ and the relations (18) hold. A morphism Φ:P∗→Q∗ of N-graded A∗-modules as the K-ones is called the N-graded A∗-module morphism if it is represented by a sum of homogeneous morphisms. Therefore, a set HomA(P∗,Q∗) of graded morphisms P∗→Q∗ is an N-graded A∗-module. An N-graded A∗-module P∗=HomA(P∗,A∗) is called the dual of an N-graded A∗-module P∗. Certainly, an N-graded A∗-module morphism of N-graded A∗-modules is their Z2-graded A∗-module morphism as associated Z2-graded modules, but the converse is true.By automorphisms of anN-graded ring A∗ are meant automorphisms of a K-ring A which preserve its N-gradation A∗. They also keep the associated Z2-structure A∗ of A. However, there exist automorphisms of a K-ring A which do not possess these properties in general.LetΛ∗ be a Grassmann-graded K-ring (Definition 6). Its associated Z2-graded commutative ring is a Grassmann algebra Λ (Definition 23). Conversely, any Grassmann algebra Λ admits the associated structure of a Grassmann-graded ring Λ∗ by a choice of its minimal generating K-module Λ1⊂Λ1. Given a generating basis {ci} for a K-module Λ1, elements of a Grassmann-graded ring Λ∗ take a form(23)a=∑k=0,1,…∑ai1⋯ikci1⋯cik.We agree to call {ci} the generating basis for the associated Grassmann algebra Λ∗ which brings it into a Grassmann-graded ring Λ∗.Given a generating basis{ci} for a Grassmann-graded ring Λ∗, one can show that any K-ring automorphism is a composition of automorphisms(24)ci⟶c′i=ρjicj+bi,where ρ is an automorphism of a K-module Λ1 and bi are odd elements of Λ>2 and of morphisms(25)ci⟶c′i=ci1+κ,κ∈Λ1.Automorphisms (24), where bi=0, are automorphisms c′i=ρjicj of a Grassmann-graded ring Λ∗. If bi≠0, the automorphism (24) preserves the associated Z2-graded structure Λ∗ of Λ but does not keep its N-graded structure Λ∗. It yields a different N-graded structure Λ′∗, where {c′i} (24) is a basis for Λ′1 and the generating basis for Λ′∗. Automorphisms (25) preserve an even sector Λ0 of Λ∗, but not the odd one Λ1 (Example 21). However, it follows from Theorem 26 that different N- and Z2-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 ringΛ∗ over a field K, there exists a finite-dimensional vector space W over K so that Λ∗ is isomorphic to the exterior algebra ⋀W of W (Example 9) seen as a Grassmann-graded ring generated by W.


## 2.4. DifferentialN-Graded Rings

If anN-graded ring also is a cochain complex, we come to the following notion [5, 17].Definition 29.
AnN-graded K-ring Ω∗ is called the differential graded ring (henceforth, DGR) if it is a cochain complex of K-modules(26)0⟶K⟶Ω0⟶δΩ1⟶δ⋯Ωk⟶δ⋯with respect to a coboundary operator δ which obeys the graded Leibniz rule(27)δαβ=δαβ+-1ααδβ.
The cochain complex (26) is called the de Rham complex of a DGR (Ω∗,δ). It also is said to be the differential graded calculus over a K-ring Ω0.Given a DGR(Ω∗,δ), one considers its minimal differential graded subring (Ω¯∗,δ) which contains Ω0. Seen as a (Ω0-Ω0)-ring, it is generated by elements δa, a∈A, and consists of monomials α=a0δa1⋯δak, ai∈Ω0, whose product obeys the juxtaposition rule (28)a0δa1b0δb1=a0δa1b0δb1-a0a1δb0δb1 in accordance with equality (27). A complex (Ω¯∗,δ) is called the minimal differential graded calculus over Ω0. Its cohomology is said to be the de Rham cohomology of (Ω∗,δ).One can associate a DGR to any LieK-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 (29)G×Q∋ε,q⟶εq∈Q,ε,ε′q=ε∘ε′-ε′∘εq,ε,ε′∈G, (cf. Remark 24). For instance, Q=K and G:K→0. A K-multilinear skew-symmetric map(30)ck:⨉kG⟶Qis called the Q-valued k-cochain on a Lie algebra G. These cochains form a G-module Ck[G;Q]. Let us put C0[G;Q]=Q. We obtain the cochain complex(31)0⟶K⟶Q⟶δC1G;Q⟶δ⋯CkG;Q⟶δ⋯with respect to the Chevalley–Eilenberg coboundary operators(32)δckε0,…,εk=∑i=0k-1iεickε0,…,ε^i,…,εk+∑1≤i<j≤k-1i+jckεi,εj,ε0,…,ε^i,…,ε^j,…,εk, 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 HomK(P,Q) of K-module homomorphisms Φ:P→Q can be endowed with two different A-module structures(33)aΦp=aΦp,Φ•ap=Φap,a∈A,p∈P.We refer to the second one as an A•-module structure. Let us put δaΦ=aΦ-Φ•a, a∈A.Definition 30.
An elementΔ∈HomK(P,Q) is called the linear s-order Q-valued differential operator on P if (δa0∘⋯∘δas)Δ=0 for any tuple of s+1 elements a0,…,as of A. A set Diffs(P,Q) of these operators inherits the A- and A•-module structures (33).In particular, linear zero-order differential operators obey a condition(34)δaΔp=aΔp-Δap=0,a∈A,p∈P, and, consequently, they coincide with A-module morphisms P→Q. A linear first-order differential operator Δ satisfies a relation (35)δb∘δaΔp=baΔp-bΔap-aΔbp+Δabp=0,a,b∈A. Of course, an s-order differential operator is of (s+1)-order. Therefore, there is a direct sequence(36)Diff0P,Q⟶Diff1P,Q⋯⟶DiffrP,Q⟶⋯of linear Q-valued differential operators on an A-module P. Its direct limit is an A-A•-module Diff∞(P,Q) of all linear Q-valued differential operators on P.In particular, letP=A. Any linear zero-order Q-valued differential operator Δ on A is defined by its value Δ(1). Then there is an A-module isomorphism Diff0(A,Q)=Q via the association q→Δq, where Δq is given by an equality Δq(1)=q. A linear first-order differential operator Δ on A fulfils a condition (37)Δab=bΔa+aΔb-baΔ1,a,b∈A.Definition 31.
It is called aQ-valued derivation of A if Δ(1)=0; that is, it obeys the Leibniz rule(38)Δab=Δab+aΔb,a,b∈A.If∂ is a derivation of A, then a∂ is well for any a∈A. Hence, derivations of A constitute an A-module d(A,Q), called the derivation module of A.IfQ=A, the derivation module dA=d(A,A) of A also is a Lie algebra over a ring K with respect to a Lie bracket(39)u,u′=u∘u′-u′∘u,u,u′∈dA.


### 3.2. Jets of Modules

A linears-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 anA-module P, let A⊗KP be a tensor product of K-modules A and P. We put (40)δba⊗p=ba⊗p-a⊗bp,p∈P,a,b∈A. Let us denote by μk+1 a submodule of A⊗KP generated by elements (41)δb0∘⋯∘δbka⊗p=aδb0∘⋯∘δbk1⊗p.Definition 32.
Ak-order jet module Jk(P) of a module P is defined as the quotient of a K-module A⊗KP by μk+1. We denote its elements c⊗kp.In particular, a first-order jet moduleJ1(P) consists of elements c⊗1p modulo the relations (42)δa∘δb1⊗1p=ab⊗1p-b⊗1ap-a⊗1bp+1⊗1abp=0.AK-module Jk(P) is endowed with the A- and A•-module structures (43)ba⊗kp=ba⊗kp,b•a⊗kp=a⊗kbp. There exists a module morphism(44)Jk:P∋p⟶1⊗kp∈JkPof an A-module P to an A•-module Jk(P) such that Jk(P), seen as an A-module, is generated by elements Jkp, p∈P. One can show the following [3, 4].Theorem 33.
Any lineark-order Q-valued differential operator Δ on an A-module P uniquely factorizes as (45)Δ:P⟶JkJkP⟶fΔQ through the morphism Jk (44) and some A-module homomorphism fΔ:Jk(P)→Q. The correspondence Δ→fΔ defines an A-module isomorphism(46)DiffkP,Q=HomAJkP,Q.Due to monomorphismsμr→μs, r>s, there exist A-module epimorphisms of jet modules (47)πii+1:Ji+1P⟶JiP,π01:J1P∋a⊗1p⟶ap∈P. Thus, there is an inverse sequence(48)P⟵π01J1P⋯⟵πr-1rJrP⟵⋯of jet modules. Its inverse limit J∞(P) is an A-module together with A-module morphisms (49)πr∞:J∞P→JrP,πr<s∞=πrs∘πs∞.In particular, let us consider a moduleP together with the morphisms Jr (44) which obey compatibility conditions Jr(p)=πrr+k∘Jr+k(p), p∈P. Then it follows from Example 15 that there exists an A-module morphism(50)J∞:P∋p⟶p,J1p,…,Jrp,…∈J∞Pso that Jr(p)=πr∞∘J∞(p). The inverse sequence (48) yields a direct sequence(51)HomAP,Q⟶π01∗HomAJ1P,Q⋯⟶πr-1r∗HomAJrP,Q⋯,where (52)πr-1r∗:HomAJr-1P,Q∋Δ⟶Δ∘πr-1r∈HomAJrP,Q is the pull-back A-module morphism (11). Its direct limit is an A-module HomA(J∞(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(53)Diff∞P,Q=HomAJ∞P,Qof their direct limits in accordance with Theorem 14.Proof.
Any elementΔ∞=Δ∈Diff∞(P,Q) factorizes as (54)Δ∞:P⟶J∞J∞P⟶f∞ΔQ through the morphism J∞ (50) and an A-module homomorphism f∞Δ=fΔ∘πk∞ (Example 16) in accordance with the commutative diagram(55)


### 3.3. Chevalley–Eilenberg Differential Calculus over Commutative Rings

Since the derivation moduledA 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 LieK-algebra dA, let us consider the Chevalley–Eilenberg complex C∗[dA;A] (31) of dA with coefficients in a ring A regarded as a dA-module [5, 7]. This complex contains a subcomplex O∗dA of A-multilinear skew-symmetric maps(56)OkdA=HomA⨉kdA,A∋ϕ:⨉kdA⟶Awith respect to the Chevalley–Eilenberg coboundary operator (32):(57)dϕu0,…,uk=∑i=0k-1iuiϕu0,…,ui^,…,uk+∑i<j-1i+jϕui,uj,u0,…,u^i,…,u^j,…,uk.Indeed, it is readily justified that if ϕ (56) is an A-multilinear map, dϕ (57) is well. In particular, (58)dau=ua,a∈O0dA=A. It follows that d(1)=0; that is, d is an O1[dA]-valued derivation of A.Let us define anN-graded A-module(59)O∗dA=⨁i∈NOidA.It is provided with the structure of an N-graded A-ring with respect to a product(60)ϕ∧ϕ′u1,…,ur+s=∑i1<⋯<ir;j1<⋯<jssgn1⋯r+si1⋯irj1⋯jsϕui1,…,uirϕ′uj1,…,ujs,ϕ∈OrdA,ϕ′∈OsdA,uk∈dA, where sgn…… denotes the sign of a permutation. This product obeys relations(61)ϕ∧ϕ′=-1ϕϕ′ϕ′∧ϕ,dϕ∧ϕ′=dϕ∧ϕ′+-1ϕϕ∧dϕ′,ϕ,ϕ′∈O∗dA.By the first one, O∗[dA] is an N-graded commutative ring. Relation (61) shows that O∗[dA] is a DGR (Definition 29), called the Chevalley–Eilenberg differential calculus over a K-ring A.SinceO1[dA]=HomA(dA,A)=dA∗ and, consequently, dA⊂dA∗∗=O1dA∗, we have the interior product u⌋ϕ=ϕ(u), u∈dA, ϕ∈O1[dA]. It is extended as(62)uϕu1,…,uk-1=kϕu,u1,…,uk-1,u∈dA,ϕ∈O∗dA,to a DGR (O∗[dA],d), and obeys a relation (63)uϕ∧σ=uϕ∧σ+-1ϕϕ∧uσ. With the interior product (62), one defines a derivation (64)Luϕ=duϕ+udϕ,ϕ∈O∗dA,Luϕ∧σ=Luϕ∧σ+ϕ∧Luσ, of an N-graded ring O∗[dA] for any u∈dA. Then one can think of elements of O∗[dA] as being differential forms over A.The minimal Chevalley–Eilenberg differential calculusO∗A over a ring A consists of the monomials a0da1∧⋯∧dak, ai∈A. Its de Rham complex(65)0⟶K⟶A⟶dO1A⟶d⋯OkA⟶d⋯is called the de Rham complex of a K-ring A.


### 3.4. Differential Calculus overC∞(X)

LetX be a smooth manifold (Remark 35) and C∞(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∞(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 sheafCX0 of continuous functions (Example A.3), a sheaf CX∞ of smooth real functions on X is defined. Its stalk Cx∞ at x∈X has a unique maximal ideal of germs of functions vanishing at x. Therefore, CX∞ is a local-ringed space (Definition A.2). Though a sheaf CX∞ exists on a topological space X, it fixes a unique smooth manifold structure on X as follows.Theorem 36.
LetX be a paracompact topological space and (X,R) a local-ringed space. Let X admit an open cover {Ui} such that a sheaf   R restricted to each Ui is isomorphic to a local-ringed space (Rn,CRn∞). Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X,R) and (X,CX∞).One can think of this result as being an equivalent definition of smooth real manifolds in terms of local-ringed spaces. A smooth manifoldX also is algebraically reproduced as a certain subspace of the spectrum of a real ring C∞(X) of smooth real functions on X [7, 14].Moreover, the well-known Serre–Swan theorem (Theorem37) states the categorial equivalence between the vector bundles over a smooth manifold X and projective modules of finite rank over the ring C∞(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.
LetX be a smooth manifold. A C∞(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→X over X.In particular, the derivation module of a real ringC∞(X) coincides with a C∞(X)-module T1(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∞(X)-module of finite rank. Its C∞(X)-dual O1(X)=T1(X)∗ is the structure module O1(X) of the cotangent bundle T∗X of X which is a module of one-form on X and, conversely, T1(X)=O1(X)∗ (Theorem 10). It follows that the Chevalley–Eilenberg differential calculus over a real ring C∞(X) is exactly the DGR (O∗(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 a real ring C∞(X) is the de Rham complex of a DGR O∗(X) of exterior forms on X. The cohomology of O∗(X) is called the de Rham cohomology HDR∗(X) of a manifold X.LetY→X be a vector bundle and Y(X) its structure module. An r-order jet manifold JrY of Y→X (Appendix B) also is a smooth vector bundle, and its structure module JrY(X) is exactly the r-order jet module Jr(Y(X)) of a C∞(X)-module Y(X) (Definition 32) [3, 4].In view of this fact and by virtue of Theorem33, a liner k-order differential operator (Definition 30) on a projective C∞(X)-module P of finite rank with values in a projective C∞(X)-module Q of finite rank is represented by a linear bundle morphism JkY→E over X of a jet bundle JkY→X to a vector bundle E→X, where Y→X and E→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.
LetY→X and E→X be smooth fibre bundles. A bundle morphism Δ:JkY→E over X is called the E-valued k-order differential operator on Y. This differential operator sends each section s of Y→X to the section Δ∘Jks of E→X.Jet manifoldsJkY of a fibre bundle Y→X constitute the inverse sequence (B.7) whose inverse limit is an infinite order jet manifold J∞Y (Definition B.1). Then any k-order E-valued differential operator Δ on a fibre bundle Y (Definition 38) is defined by a continuous bundle map (66)Δ∘πr∞:J∞Y⟶XE. 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 Or∗=O∗(JrY) of exterior forms on finite order jet manifolds(67)O∗X⟶π∗O∗Y⟶π01∗O1∗⟶⋯Or-1∗⟶πr-1r∗Or∗⟶⋯,where πr-1r∗ are the pull-back monomorphisms. Its direct limit(68)O∞∗Y=lim→Or∗ (Definition 11) consists of all exterior forms on finite order jet manifolds modulo the pull-back identification. In accordance with Theorem 13, O∞∗Y is a DGR which inherits operations of the exterior differential d and the exterior product ⋀ of DGRs Or∗.Theorem 39.
The cohomologyH∗(O∞∗Y) of the de Rham complex(69)0⟶R⟶O∞0Y⟶dO∞1Y⟶d⋯of a DGR O∞∗Y equals the de Rham cohomology HDR∗(Y) of a fibre bundle Y.Proof.
The result follows from the fact thatY is a strong deformation retract of any jet manifold JrY and, consequently, the de Rham cohomology of JrY equals that of Y [23].One can think of elements ofO∞∗Y as being differential forms on an infinite order jet manifold J∞Y. A DGR O∞∗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].


## 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 HomK(P,Q) of K-module homomorphisms Φ:P→Q can be endowed with two different A-module structures(33)aΦp=aΦp,Φ•ap=Φap,a∈A,p∈P.We refer to the second one as an A•-module structure. Let us put δaΦ=aΦ-Φ•a, a∈A.Definition 30.
An elementΔ∈HomK(P,Q) is called the linear s-order Q-valued differential operator on P if (δa0∘⋯∘δas)Δ=0 for any tuple of s+1 elements a0,…,as of A. A set Diffs(P,Q) of these operators inherits the A- and A•-module structures (33).In particular, linear zero-order differential operators obey a condition(34)δaΔp=aΔp-Δap=0,a∈A,p∈P, and, consequently, they coincide with A-module morphisms P→Q. A linear first-order differential operator Δ satisfies a relation (35)δb∘δaΔp=baΔp-bΔap-aΔbp+Δabp=0,a,b∈A. Of course, an s-order differential operator is of (s+1)-order. Therefore, there is a direct sequence(36)Diff0P,Q⟶Diff1P,Q⋯⟶DiffrP,Q⟶⋯of linear Q-valued differential operators on an A-module P. Its direct limit is an A-A•-module Diff∞(P,Q) of all linear Q-valued differential operators on P.In particular, letP=A. Any linear zero-order Q-valued differential operator Δ on A is defined by its value Δ(1). Then there is an A-module isomorphism Diff0(A,Q)=Q via the association q→Δq, where Δq is given by an equality Δq(1)=q. A linear first-order differential operator Δ on A fulfils a condition (37)Δab=bΔa+aΔb-baΔ1,a,b∈A.Definition 31.
It is called aQ-valued derivation of A if Δ(1)=0; that is, it obeys the Leibniz rule(38)Δab=Δab+aΔb,a,b∈A.If∂ is a derivation of A, then a∂ is well for any a∈A. Hence, derivations of A constitute an A-module d(A,Q), called the derivation module of A.IfQ=A, the derivation module dA=d(A,A) of A also is a Lie algebra over a ring K with respect to a Lie bracket(39)u,u′=u∘u′-u′∘u,u,u′∈dA.


## 3.2. Jets of Modules

A linears-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 anA-module P, let A⊗KP be a tensor product of K-modules A and P. We put (40)δba⊗p=ba⊗p-a⊗bp,p∈P,a,b∈A. Let us denote by μk+1 a submodule of A⊗KP generated by elements (41)δb0∘⋯∘δbka⊗p=aδb0∘⋯∘δbk1⊗p.Definition 32.
Ak-order jet module Jk(P) of a module P is defined as the quotient of a K-module A⊗KP by μk+1. We denote its elements c⊗kp.In particular, a first-order jet moduleJ1(P) consists of elements c⊗1p modulo the relations (42)δa∘δb1⊗1p=ab⊗1p-b⊗1ap-a⊗1bp+1⊗1abp=0.AK-module Jk(P) is endowed with the A- and A•-module structures (43)ba⊗kp=ba⊗kp,b•a⊗kp=a⊗kbp. There exists a module morphism(44)Jk:P∋p⟶1⊗kp∈JkPof an A-module P to an A•-module Jk(P) such that Jk(P), seen as an A-module, is generated by elements Jkp, p∈P. One can show the following [3, 4].Theorem 33.
Any lineark-order Q-valued differential operator Δ on an A-module P uniquely factorizes as (45)Δ:P⟶JkJkP⟶fΔQ through the morphism Jk (44) and some A-module homomorphism fΔ:Jk(P)→Q. The correspondence Δ→fΔ defines an A-module isomorphism(46)DiffkP,Q=HomAJkP,Q.Due to monomorphismsμr→μs, r>s, there exist A-module epimorphisms of jet modules (47)πii+1:Ji+1P⟶JiP,π01:J1P∋a⊗1p⟶ap∈P. Thus, there is an inverse sequence(48)P⟵π01J1P⋯⟵πr-1rJrP⟵⋯of jet modules. Its inverse limit J∞(P) is an A-module together with A-module morphisms (49)πr∞:J∞P→JrP,πr<s∞=πrs∘πs∞.In particular, let us consider a moduleP together with the morphisms Jr (44) which obey compatibility conditions Jr(p)=πrr+k∘Jr+k(p), p∈P. Then it follows from Example 15 that there exists an A-module morphism(50)J∞:P∋p⟶p,J1p,…,Jrp,…∈J∞Pso that Jr(p)=πr∞∘J∞(p). The inverse sequence (48) yields a direct sequence(51)HomAP,Q⟶π01∗HomAJ1P,Q⋯⟶πr-1r∗HomAJrP,Q⋯,where (52)πr-1r∗:HomAJr-1P,Q∋Δ⟶Δ∘πr-1r∈HomAJrP,Q is the pull-back A-module morphism (11). Its direct limit is an A-module HomA(J∞(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(53)Diff∞P,Q=HomAJ∞P,Qof their direct limits in accordance with Theorem 14.Proof.
Any elementΔ∞=Δ∈Diff∞(P,Q) factorizes as (54)Δ∞:P⟶J∞J∞P⟶f∞ΔQ through the morphism J∞ (50) and an A-module homomorphism f∞Δ=fΔ∘πk∞ (Example 16) in accordance with the commutative diagram(55)


## 3.3. Chevalley–Eilenberg Differential Calculus over Commutative Rings

Since the derivation moduledA 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 LieK-algebra dA, let us consider the Chevalley–Eilenberg complex C∗[dA;A] (31) of dA with coefficients in a ring A regarded as a dA-module [5, 7]. This complex contains a subcomplex O∗dA of A-multilinear skew-symmetric maps(56)OkdA=HomA⨉kdA,A∋ϕ:⨉kdA⟶Awith respect to the Chevalley–Eilenberg coboundary operator (32):(57)dϕu0,…,uk=∑i=0k-1iuiϕu0,…,ui^,…,uk+∑i<j-1i+jϕui,uj,u0,…,u^i,…,u^j,…,uk.Indeed, it is readily justified that if ϕ (56) is an A-multilinear map, dϕ (57) is well. In particular, (58)dau=ua,a∈O0dA=A. It follows that d(1)=0; that is, d is an O1[dA]-valued derivation of A.Let us define anN-graded A-module(59)O∗dA=⨁i∈NOidA.It is provided with the structure of an N-graded A-ring with respect to a product(60)ϕ∧ϕ′u1,…,ur+s=∑i1<⋯<ir;j1<⋯<jssgn1⋯r+si1⋯irj1⋯jsϕui1,…,uirϕ′uj1,…,ujs,ϕ∈OrdA,ϕ′∈OsdA,uk∈dA, where sgn…… denotes the sign of a permutation. This product obeys relations(61)ϕ∧ϕ′=-1ϕϕ′ϕ′∧ϕ,dϕ∧ϕ′=dϕ∧ϕ′+-1ϕϕ∧dϕ′,ϕ,ϕ′∈O∗dA.By the first one, O∗[dA] is an N-graded commutative ring. Relation (61) shows that O∗[dA] is a DGR (Definition 29), called the Chevalley–Eilenberg differential calculus over a K-ring A.SinceO1[dA]=HomA(dA,A)=dA∗ and, consequently, dA⊂dA∗∗=O1dA∗, we have the interior product u⌋ϕ=ϕ(u), u∈dA, ϕ∈O1[dA]. It is extended as(62)uϕu1,…,uk-1=kϕu,u1,…,uk-1,u∈dA,ϕ∈O∗dA,to a DGR (O∗[dA],d), and obeys a relation (63)uϕ∧σ=uϕ∧σ+-1ϕϕ∧uσ. With the interior product (62), one defines a derivation (64)Luϕ=duϕ+udϕ,ϕ∈O∗dA,Luϕ∧σ=Luϕ∧σ+ϕ∧Luσ, of an N-graded ring O∗[dA] for any u∈dA. Then one can think of elements of O∗[dA] as being differential forms over A.The minimal Chevalley–Eilenberg differential calculusO∗A over a ring A consists of the monomials a0da1∧⋯∧dak, ai∈A. Its de Rham complex(65)0⟶K⟶A⟶dO1A⟶d⋯OkA⟶d⋯is called the de Rham complex of a K-ring A.


## 3.4. Differential Calculus overC∞(X)

LetX be a smooth manifold (Remark 35) and C∞(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∞(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 sheafCX0 of continuous functions (Example A.3), a sheaf CX∞ of smooth real functions on X is defined. Its stalk Cx∞ at x∈X has a unique maximal ideal of germs of functions vanishing at x. Therefore, CX∞ is a local-ringed space (Definition A.2). Though a sheaf CX∞ exists on a topological space X, it fixes a unique smooth manifold structure on X as follows.Theorem 36.
LetX be a paracompact topological space and (X,R) a local-ringed space. Let X admit an open cover {Ui} such that a sheaf   R restricted to each Ui is isomorphic to a local-ringed space (Rn,CRn∞). Then X is an n-dimensional smooth manifold together with a natural isomorphism of local-ringed spaces (X,R) and (X,CX∞).One can think of this result as being an equivalent definition of smooth real manifolds in terms of local-ringed spaces. A smooth manifoldX also is algebraically reproduced as a certain subspace of the spectrum of a real ring C∞(X) of smooth real functions on X [7, 14].Moreover, the well-known Serre–Swan theorem (Theorem37) states the categorial equivalence between the vector bundles over a smooth manifold X and projective modules of finite rank over the ring C∞(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.
LetX be a smooth manifold. A C∞(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→X over X.In particular, the derivation module of a real ringC∞(X) coincides with a C∞(X)-module T1(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∞(X)-module of finite rank. Its C∞(X)-dual O1(X)=T1(X)∗ is the structure module O1(X) of the cotangent bundle T∗X of X which is a module of one-form on X and, conversely, T1(X)=O1(X)∗ (Theorem 10). It follows that the Chevalley–Eilenberg differential calculus over a real ring C∞(X) is exactly the DGR (O∗(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 a real ring C∞(X) is the de Rham complex of a DGR O∗(X) of exterior forms on X. The cohomology of O∗(X) is called the de Rham cohomology HDR∗(X) of a manifold X.LetY→X be a vector bundle and Y(X) its structure module. An r-order jet manifold JrY of Y→X (Appendix B) also is a smooth vector bundle, and its structure module JrY(X) is exactly the r-order jet module Jr(Y(X)) of a C∞(X)-module Y(X) (Definition 32) [3, 4].In view of this fact and by virtue of Theorem33, a liner k-order differential operator (Definition 30) on a projective C∞(X)-module P of finite rank with values in a projective C∞(X)-module Q of finite rank is represented by a linear bundle morphism JkY→E over X of a jet bundle JkY→X to a vector bundle E→X, where Y→X and E→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.
LetY→X and E→X be smooth fibre bundles. A bundle morphism Δ:JkY→E over X is called the E-valued k-order differential operator on Y. This differential operator sends each section s of Y→X to the section Δ∘Jks of E→X.Jet manifoldsJkY of a fibre bundle Y→X constitute the inverse sequence (B.7) whose inverse limit is an infinite order jet manifold J∞Y (Definition B.1). Then any k-order E-valued differential operator Δ on a fibre bundle Y (Definition 38) is defined by a continuous bundle map (66)Δ∘πr∞:J∞Y⟶XE. 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 Or∗=O∗(JrY) of exterior forms on finite order jet manifolds(67)O∗X⟶π∗O∗Y⟶π01∗O1∗⟶⋯Or-1∗⟶πr-1r∗Or∗⟶⋯,where πr-1r∗ are the pull-back monomorphisms. Its direct limit(68)O∞∗Y=lim→Or∗ (Definition 11) consists of all exterior forms on finite order jet manifolds modulo the pull-back identification. In accordance with Theorem 13, O∞∗Y is a DGR which inherits operations of the exterior differential d and the exterior product ⋀ of DGRs Or∗.Theorem 39.
The cohomologyH∗(O∞∗Y) of the de Rham complex(69)0⟶R⟶O∞0Y⟶dO∞1Y⟶d⋯of a DGR O∞∗Y equals the de Rham cohomology HDR∗(Y) of a fibre bundle Y.Proof.
The result follows from the fact thatY is a strong deformation retract of any jet manifold JrY and, consequently, the de Rham cohomology of JrY equals that of Y [23].One can think of elements ofO∞∗Y as being differential forms on an infinite order jet manifold J∞Y. A DGR O∞∗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 overN-Graded Commutative Rings

The differential calculus overN-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 Z2-graded commutative ring endowed with a fixed N-graded structure (Remark 44).LetK be a commutative ring without a divisor of zero and A an N-graded commutative K-ring. Let P and Q be N-graded A-modules. An N-graded K-module HomK(P,Q) of N-graded K-module homomorphisms Φ:P→Q can admit the two N-graded A-module structures(70)aΦp=aΦp,Φ•ap=Φap,a∈A,p∈P,called A- and A•-module structures, respectively. Let us put(71)δaΦ=aΦ--1aΦΦ•a,a∈A.Definition 40.
An elementΔ∈HomK(P,Q) is said to be the linear Q-valued N-graded differential operator of order s on P if(72)δa0∘⋯∘δasΔ=0for any tuple of s+1 elements a0,…,as of A. A set Diffs(P,Q) of these operators inherits the N-graded A-module structures (70).For instance, zero-orderN-graded differential operators obey a condition (73)δaΔp=aΔp--1aΔΔap=0,a∈A,p∈P; that is, they coincide with graded A-module morphisms P→Q. A first-order N-graded differential operator Δ satisfies a condition (74)δa∘δbΔp=abΔp--1b+ΔabΔap--1bΔaΔbp+-1bΔ+Δ+baΔbap=0,a,b∈A,p∈P.Graded differential operators on anN-graded A-module P form a direct system of N-graded (A-A•)-modules.(75)Diff0P,Q⟶inDiff1P,Q⋯⟶inDiffrP,Q⟶⋯.Its limit Diff∞(P,Q) is a Z2-graded module of all Q-valued graded differential operators on P.In particular, letP=A. Any zero-order Q-valued N-graded differential operator Δ on A is defined by its value Δ(1). Then there is a graded A-module isomorphism Diff0(A,Q)=Q via the association q→Δq, where Δq is given by the equality Δq(1)=q. A first-order Q-valued N-graded differential operator Δ on A fulfils a condition (76)Δab=Δab+-1aΔaΔb--1b+aΔabΔ1,a,b∈A.Definition 41.
It is called theQ-valued N-graded derivation of A if Δ(1)=0, that is, if it obeys the graded Leibniz rule(77)Δab=Δab+-1aΔaΔb,a,b∈A.Any first-orderN-graded differential operator on A falls into a sum (78)Δa=Δ1a+Δa-Δ1a of a zero-order graded differential operator Δ(1)a and an N-graded derivation Δ(a)-Δ(1)a. If ∂ is an N-graded derivation of A, then a∂ is so for any a∈A. Hence, N-graded derivations of A constitute an N-graded A-module d(A,Q), called the graded derivation module.IfQ=A, the N-graded derivation module dA=d(A,A) also is a Lie K-superalgebra (Remark 24) with respect to the superbracket(79)u,u′=u∘u′--1uu′u′∘u,u,u′∈A.Example 42.
LetΛ be a Grassmann-graded ring provided with an odd generating basis {ci}. Its N-graded derivations are defined in full by their action on the generating elements ci. Let us consider odd derivations (80)∂icj=δij,∂i∘∂j=-∂j∘∂i. Then any N-graded derivation of Λ takes a form(81)u=ui∂i,ui∈A.Graded derivations (81) constitute the free N-graded Λ-module dΛ 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 anN-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 (82)∂ab=∂ab+a∂b (cf. (38)) in the noncommutative differential calculus.Since the graded derivation moduledA of an N-graded commutative ring A is a Lie K-superalgebra, one can consider the Chevalley–Eilenberg complex(83)0⟶K⟶A⟶dC1dA;A⟶d⋯CkdA;A⟶d⋯,where a K-ring A is seen as a dA-module [5, 21]. Its cochains are A-modules (84)CkdA;A=HomK⋀kdA,A of K-linear N-graded morphisms of graded exterior products ⋀kdA of an N-graded K-module dA to A. Let us bring homogeneous elements of ⋀kdA into a form (85)ε1∧⋯εr∧ϵr+1∧⋯∧ϵk,εi∈dA0,ϵj∈dA1. Then a Chevalley–Eilenberg coboundary operator d of the complex (83) reads(86)dcε1∧⋯∧εr∧ϵ1∧⋯∧ϵs=∑i=1r-1i-1εicε1∧⋯ε^i⋯∧εr∧ϵ1∧⋯ϵs+∑j=1s-1rϵjcε1∧⋯∧εr∧ϵ1∧⋯ϵ^j⋯∧ϵs+∑1≤i<j≤r-1i+jcεi,εj∧ε1∧⋯ε^i⋯ε^j⋯∧εr∧ϵ1∧⋯∧ϵs+∑1≤i<j≤scϵi,ϵj∧ε1∧⋯∧εr∧ϵ1∧⋯ϵ^i⋯ϵ^j⋯∧ϵs+∑1≤i<r,1≤j≤s-1i+r+1cεi,ϵj∧ε1∧⋯ε^i⋯∧εr∧ϵ1∧⋯ϵ^j⋯∧ϵs, where the caret  ^  denotes omission.It is easily justified that complex (83) contains a subcomplex O∗[dA] of A-linear N-graded morphisms. The N-graded module O∗[dA] is provided with the graded exterior product(87)ϕ∧ϕ′u1,…,ur+s=∑i1<⋯<ir;j1<⋯<jssgn1⋯r+si1⋯irj1⋯js⁡ϕui1,…,uirϕ′uj1,…,ujs,ϕ∈OrdA,ϕ′∈OsdA,uk∈dA, where u1,…,ur+s are graded-homogeneous elements of dA and (88)u1∧⋯∧ur+s=sgn1⋯r+si1⋯irj1⋯js⁡ui1∧⋯∧uir∧uj1∧⋯∧ujs. The graded Chevalley–Eilenberg differential d (86) and the graded exterior product ⋀ (87) bring O∗[dA] into a differential bigraded ring (henceforth, DBGR) whose elements obey relations(89)ϕ∧ϕ′=-1ϕϕ′+ϕϕ′ϕ′∧ϕ,dϕ∧ϕ′=dϕ∧ϕ′+-1ϕϕ∧dϕ′.It is called the graded Chevalley–Eilenberg differential calculus over an N-graded commutative K-ring A. In particular, we have(90)O1dA=HomAdA,A=dA∗.One can extend this duality relation to the graded interior product of u∈dA with any element ϕ∈O∗[dA] by the rules (91)ubda=-1ubbua,a,b∈A,uϕ∧ϕ′=uϕ∧ϕ′+-1ϕ+ϕuϕ∧uϕ′. As a consequence, any graded derivation u∈dA of A yields a graded derivation(92)Luϕ=udϕ+duϕ,ϕ∈O∗dA,u∈dA,Luϕ∧ϕ′=Luϕ∧ϕ′+-1uϕϕ∧Luϕ′, termed the graded Lie derivative of a differential bigraded ring O∗[dA].The minimal graded Chevalley–Eilenberg differential calculusO∗A⊂O∗[dA] over a N-graded commutative ring A consists of monomials a0da1∧⋯∧dak, ai∈A. The corresponding complex(93)0⟶K⟶A⟶dO1A⟶d⋯OkA⟶d⋯is called the bigraded de Rham complex of an N-graded commutative K-ring A.Remark 44.
Let us note that ifA=A0 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 Z2-graded modules, and their N-graded homomorphisms are Z2-graded homomorphisms of this Z2-graded structure. Moreover, N-graded commutativity conditions in fact are the Z2-graded ones. Therefore, the differential calculus over N-graded rings is the Z2-graded differential calculus when the associated N-structure hold fixed. In particular, any N-graded differential operator also is a Z2-graded differential operator, but the converse need not be true. For instance, the differential calculus over a Grassmann-graded K-ring A∗ (Example 42) is the differential calculus over an associated Grassmann algebra A∗. Namely, the derivations (81) of A∗ also are derivations of a Grassmann algebra A∗, and any Z2-graded differential operator on A∗ is a composition of these derivations.


## 5.N-Graded Manifolds

As was mentioned above, we define anN-graded manifold by analogy with smooth and Z2-graded manifolds as a local-ringed 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 aZ2-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 A=A0⊕A1 is a sheaf of real Grassmann algebras such that(i)
there is the exact sequence of sheaves(94)0⟶R⟶A⟶σCZ∞⟶0,R=A1+A12,whereCZ∞ is the sheaf of smooth real functions on Z;(ii)
R/R2 is a locally free sheaf of CZ∞-modules of finite rank (with respect to pointwise operations), and the sheaf A is locally isomorphic to the exterior product ⋀CZ∞(R/R2).A sheafA is called the structure sheaf of a Z2-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 Z2-graded manifold (Z,A). They make up a Z2-graded commutative C∞(Z)-ring A(Z) called the structure ring of (Z,A).Given aZ2-graded manifold (Z,A), by the sheaf dA of graded derivations of A is meant a subsheaf of endomorphisms of the structure sheaf A such that any section u∈dA(U) of dA over an open subset U⊂Z is a graded derivation of the real Z2-graded commutative ring A(U), that is, u∈d(A(U)). Conversely, one can show that, given open sets U′⊂U, there is a surjection of the graded derivation derivation modules d(A(U))→d(A(U′)) [8]. It follows that any graded derivation of a local Z2-graded commutative ring A(U) also is a local section over U of a sheaf dA. Global sections of dA are called graded vector fields on a Z2-graded manifold (Z,A). They make up a graded derivation module dA(Z) of a real Z2-graded commutative ring A(Z). This module is a real Lie superalgebra (Remark 24) with respect to the superbracket (79).Turn now toN-graded manifolds. By virtue of the well-known Batchelor theorem [5, 8], Z2-graded manifolds possess the following structure.Theorem 45.
Let(Z,A) be a Z2-graded manifold. There exists a vector bundle E→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 AE=⋀EZ∗ of germs of sections of the exterior bundle(95)⋀E∗=Z×R⊕Z⁡E∗⊕Z⁡⋀E∗⊕Z⁡⋀2E∗⋯,whose typical fibre is the Grassmann algebra Λ=⋀V∗ in Theorem 28.It should be emphasized that Batchelor’s isomorphism in Theorem45 fails to be canonical.Combining Batchelor Theorem45 and classical Serre–Swan Theorem 37, we come to the following Serre–Swan theorem for Z2-graded manifolds [7].Theorem 46.
LetZ be a smooth manifold. A Z2-graded commutative C∞(Z)-ring A is isomorphic to the structure ring of a Z2-graded manifold with a body Z iff it is the exterior algebra of some projective C∞(Z)-module of finite rank.In fact, the structure sheafAE of a Z2-graded manifold (Z,A) in Theorem 45 is a sheaf in Grassmann-graded rings Λ∗, whose N-graded structure is fixed. Therefore, we come to the following notion of N-graded manifolds.Definition 47.
AnN-graded manifold is defined to be a Z2-graded manifold whose Batchelor isomorphism (Z,AE) is fixed.Thus, Theorem45 states that a Z2-graded manifold is isomorphic to the N-graded one.AnN-graded manifold (Z,AE) is said to be modelled over a vector bundle E→Z, and E is called its characteristic vector bundle. Its structure ring AE is the structure module(96)AE=AEZ=⋀E∗Zof sections of the exterior bundle ⋀E∗ (95). Automorphisms of an N-graded manifold (Z,AE) are restricted to those induced by automorphisms of its characteristic vector bundles E→Z.Accordingly, Serre–Swan Theorem46 can be formulated for the N-graded ones.Theorem 48.
LetZ be a smooth manifold. An N-graded commutative C∞(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∞(Z)-module of finite rank.Remark 49.
One can treat a local-ringed space(Z,A0=CZ∞) as a trivial N-graded manifold whose characteristic vector bundle is E=Z×{0}. Its structure module is a ring C∞(Z) of smooth real functions on Z.Given anN-graded manifold (Z,AE), every trivialization chart (U;zA,ya) of its characteristic vector bundle E→Z yields a splitting domain (U;zA,ca) of (Z,AE). Graded functions on such a chart are Λ-valued functions(97)f=∑k=0m1k!fa1⋯akzca1⋯cak,where fa1⋯ak(z) are smooth functions on U and {ca} is the fibre basis for E∗. One calls {zA,ca} the local basis for an N-graded manifold (Z,AE). Transition functions y′a=ρba(zA)yb of bundle coordinates on E→Z induce the corresponding transformation c′a=ρba(zA)cb of the associated local basis for an N-graded manifold (Z,AE) and the according coordinate transformation law of graded functions (97).The following is an essential peculiarity of anN-graded manifold (Z,AE) in comparison with the Z2-graded ones.Theorem 50.
Derivations of the structure moduleAE of an N-graded manifold (Z,AE) are represented by sections of the vector bundle (102).Proof.
Due to the canonical splittingVE=E×E, the vertical tangent bundle VE of E→Z can be provided with fibre bases {∂/∂ca}, which are the duals of bases {ca}. Then graded derivations of AE on a splitting domain (U;zA,ca) of (Z,AE) read(98)u=uA∂A+ua∂∂ca,where uλ,ua are local graded functions on U. In particular, (99)∂∂ca∘∂∂cb=-∂∂cb∘∂∂ca,∂A∘∂∂ca=∂∂ca∘∂A. The graded derivations (98) act on graded functions f∈AE(U) (97) by a rule(100)ufa⋯bca⋯cb=uA∂Afa⋯bca⋯cb+ukfa⋯b∂∂ckca⋯cb.This rule implies the corresponding transformation law (101)u′A=uA,u′a=ρjauj+uA∂Aρjacj of graded derivations (98). It follows that they can be represented by sections of a vector bundle(102)VE=⋀E∗⊗E⁡TE⟶Z.Thus, the graded derivation module dAE is isomorphic to the structure module VE(Z) of global sections of the vector bundle VE→Z (102).Given the structure ringAE of graded functions on an N-graded manifold (Z,AE) and the real Lie superalgebra dAE of its graded derivations, let us consider the graded Chevalley–Eilenberg differential calculus(103)S∗E;Z=O∗dAE0⟶R⟶AE⟶dS1E;Z⟶d⋯SkE;Z⟶d⋯, over S0[E;Z]=AE. Since a module dAE is isomorphic to the structure module of sections of a vector bundle VE→Z, elements of S∗[E;Z] are represented by sections of the exterior bundle ⋀V¯E of the AE-dual (104)V¯E=⋀E∗⊗E⁡T∗E⟶Zof VE. With respect to the dual fibre bases {dzA} for T∗Z and {dcb} for E∗, sections of V¯E (104) take a coordinate form (105)ϕ=ϕAdzA+ϕadca,ϕa′=ρ-1ϕabb,ϕA′=ϕA+ρ-1∂abAρjaϕbcj. The duality relation S1[E;Z]=dAE∗ (90) is given by a graded interior product (106)uϕ=uAϕA+-1ϕauaϕa.Elements ofS∗[E;Z] are called graded differential forms on an N-graded manifold (Z,AE). Seen as an AE-ring, the DBGR S∗[E;Z] (103) on a splitting domain (zA,ca) is locally generated by the graded one-form dzA, dci such that (107)dzA∧dci=-dci∧dzA,dci∧dcj=dcj∧dci. Accordingly, the graded Chevalley–Eilenberg coboundary operator d (86), termed the graded exterior differential, reads (108)dϕ=dzA∧∂Aϕ+dca∧∂∂caϕ, where derivations ∂λ, ∂/∂ca act on coefficients of graded exterior forms by formula (100), and they are graded commutative with graded forms dzA and dca. Formulas (89), (90), and (92) hold.Theorem 51.
The DBGRS∗[E;Z] (103) is a minimal differential calculus over AE; that is, it is generated by elements df, f∈AE.Proof.
SincedAE=VE(Z), it is a projective C∞(Z)- and AE-module of finite rank, and so is its AE-dual S1[E;Z] (Theorem 10). Hence, dAE is the AE-dual of S1[E;Z] and, consequently, S1[E;Z] is generated by elements df, f∈AE.Cohomology of the DBGRS∗[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∗(Z)→S∗[E;Z] and a body epimorphism S∗[E;Z]→O∗(Z).


## 6.N-Graded Bundles and Jet Manifolds

A morphism of (bothZ2- and N-) graded manifolds (Z,A)→(Z′,A′) is defined as that of local-ringed spaces ϕ:Z→Z′, Φ^:A′→ϕ∗A, where ϕ is a manifold morphism and Φ^ is a sheaf morphism of A′ to the direct image ϕ∗A of A onto Z′ (Appendix A). This morphism of graded manifolds is said to be (i) a monomorphism if ϕ is an injection and Φ^ is an epimorphism and (ii) an epimorphism if ϕ is a surjection and Φ^ is a monomorphism.An epimorphism of graded manifolds(Z,A)→(Z′,A′), where Z→Z′ is a fibre bundle is called the graded bundle [26, 27]. In this case, a sheaf monomorphism Φ^ induces a monomorphism of canonical presheaves A¯′→A¯, which associates to each open subset U⊂Z the ring of sections of A′ over ϕ(U). Accordingly, there is a pull-back monomorphism of the structure rings A′(Z′)→A(Z) of graded functions on graded manifolds (Z′,A′) and (Z,A).In particular, let(Y,A) be a graded manifold whose body is a fibre bundle Y→X. Let us consider a trivial graded manifold (X,A0=CX∞) (Remark 49). Then we have a graded bundle(109)Y,A⟶X,CX∞.Let us denote it by (X,Y,A). Given a graded bundle (X,Y,A), the local basis for a graded manifold (Y,A) can take a form (xλ,yi,ca), where (xλ,yi) are bundle coordinates of Y→X.Definition 52.
One agrees to call the graded bundle (109) over a trivial graded manifold (X,CX∞) the graded bundle over a smooth manifold.IfY→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.
LetY→X be a fibre bundle. Then a trivial graded manifold (Y,CY∞) together with a real ring monomorphism C∞(X)→C∞(Y) is a graded bundle (X,Y,CY∞) of trivial graded manifolds (Y,CY∞)→(X,CX∞).Remark 54.
A graded manifold(X,A) itself can be treated as the graded bundle (X,X,A) (109) associated with the identity smooth bundle X→X.LetE→Z and E′→Z′ be vector bundles and Φ:E→E′ their bundle morphism over a morphism ϕ:Z→Z′. Then a section s∗ of the dual bundle E′∗→Z′ yields the pull-back section Φ∗s∗ of the dual bundle E∗→Z by a law (110)vzΦ∗s∗z=Φvzs∗φz,vz∈Ez. A bundle morphism (Φ,ϕ) induces a morphism of N-graded manifolds(111)Z,AE⟶Z′,AE′.This is a pair (ϕ,Φ^=ϕ∗∘Φ∗) of a morphism ϕ of body manifolds and the composition ϕ∗∘Φ∗ of the pull-back AE′∋f→Φ∗f∈AE of graded functions and the direct image ϕ∗ of a sheaf AE onto Z′. Relative to local bases (zA,ca) and (z′A,c′a) for (Z,AE) and (Z′,AE′), the morphism (111) of N-graded manifolds reads z′=ϕ(z), Φ^(c′a)=Φba(z)cb.The graded manifold morphism (111) is a monomorphism (resp., epimorphism) if Φ is a bundle injection (resp., surjection). In particular, the graded manifold morphism (111) is a graded bundle if Φ is a fibre bundle. Let AE′→AE be the corresponding pull-back monomorphism of the structure rings. By virtue of Theorem 51 it yields a monomorphism of the DBGRs(112)S∗E′;Z′⟶S∗E;Z.Let(Y,AF) be an N-graded manifold modelled over a vector bundle F→Y. This is an N-graded bundle (X,Y,AF):(113)Y,AF⟶X,CX∞,modelled over a composite bundle(114)F⟶Y⟶X.The structure ring of graded functions on an N-graded manifold (Y,AF) is the graded commutative C∞(X)-ring AF=⋀F∗(Y) (96). Let the composite bundle (114) be provided with adapted bundle coordinates (xλ,yi,qa) possessing transition functions (115)x′λxμ,y′ixμ,yj,q′a=ρbaxμ,yjqb. Then the corresponding basis for an N-graded manifold (Y,AF) is (xλ,yi,ca) together with transition functions c′a=ρba(xμ,jj)cb. We call it the local basis for an N-graded bundle (X,Y,AF).As was shown above, the differential calculus on a fibre bundleY→X is formulated in terms of jet manifolds J∗Y of Y. Being fibre bundles over X, they can be regarded as trivial graded bundles (X,JkY,CJkY∞). We describe their partners in a case of N-graded bundles as follows.Let us note that, given anN-graded manifold (X,AE) and its structure ring AE, one can define the jet module J1AE of a C∞(X)-ring AE [5, 7]. It is a module of global sections of the jet bundle J1(⋀E∗). A problem is that J1AE 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,AE) be an N-graded manifold modelled over a vector bundle E→X. Let us consider a k-order jet manifold JkE of E. It is a vector bundle over X. Then let (X,AJkE) be an N-graded manifold modelled over JkE→X.Definition 55.
One calls(X,AJkE) the graded jet manifold of an N-graded manifold (X,AE).Given a splitting domain(U;xλ,ca) of an N-graded manifold (Z,AE), the adapted splitting domain of a graded jet manifold (X,AJkE) reads (116)U;xλ,ca,cλa,cλ1λ2a,…,cλ1⋯λka,cλλ1⋯λr′a=ρbaxcλλ1⋯λra+∂λρbaxcλ1⋯λra.As was mentioned above, a graded manifold is a particular graded bundle over its body (Remark54). Then Definition 55 of graded jet manifolds is generalized to N-graded bundles over smooth manifolds as follows. Let (X,Y,AF) be the N-graded bundle (113) modelled over the composite bundle (114). It is readily observed that the jet manifold JrF of F→X is a vector bundle JrF→JrY coordinated by (xλ,yΛi,qΛa), 0≤Λ≤r. Let (JrY,AJrF) be an N-graded manifold modelled over this vector bundle. Its local generating basis is (xλ,yΛi,cΛa), 0≤Λ≤r.Definition 56.
One calls(JrY,AJrF) the graded jet manifold of an N-graded bundle (X,Y,AF).In particular, letY→X be a smooth bundle seen as a trivial graded bundle (X,Y,CY∞) modelled over a composite bundle Y×{0}→Y→X. Then its graded jet manifold is a trivial graded bundle (X,JrY,CJrY∞), that is, the jet manifold JrY 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 bundlesJr+1Y→JrY (B.7) and the corresponding fibre bundles Jr+1F→JrF also yield the N-graded bundles (117)Jr+1Y,AJr+1F⟶JrY,AJrF, including the sheaf monomorphisms(118)πrr+1∗AJrF⟶AJr+1F,where πrr+1∗Ar is the pull-back onto Jr+1Y of the continuous fibre bundle AJrF→JrY. The sheaf monomorphism (118) induces a monomorphism of canonical presheaves A¯JrF→A¯Jr+1F, which associates to each open subset U⊂Jr+1Y the ring of sections of AJr+1F over πrr+1(U). Accordingly, there is a pull-back monomorphism of the structure rings(119)Sr0F;Y⟶Sr+10F;Y,Sk0F;Y=S0JkF;JkY,of graded functions on graded manifolds JrY,AJrF and Jr+1Y,AJr+1F. As a consequence, we have the inverse sequence of graded manifolds (120)Y,AF⟵J1Y,AJ1F⟵⋯Jr-1Y,AJr-1F⟵JrY,AJrF⟵⋯. One can think of its inverse limit (J∞Y,AJ∞F) as being the graded infinite order jet manifold whose body is an infinite order jet manifold J∞Y and whose structure sheaf AJ∞F is a sheaf of germs of graded functions on N-graded manifolds (J∗Y,AJ∗F). However (J∞Y,AJ∞F) fails to be an N-graded manifold in a strict sense because the inverse limit J∞Y of the sequence (B.7) is a Fréche manifold, but not the smooth one.By virtue of Theorem51, the differential calculus Sr∗[F;Y] of graded differential forms on N-graded manifolds (JrY,AJrF) is minimal. Therefore, the monomorphisms of structure rings (119) yield the pull-back monomorphisms (112) of DBGRs(121)πrr+1∗:Sr∗F;Y⟶Sr+1∗F;Y,Sk∗F;Y=S∗JkF;JkY.As a consequence, we have a direct system of DBGRs(122)S∗F;Y⟶π∗S1∗F;Y⟶⋯Sr-1∗F;Y⟶πr-1r∗Sr∗F;Y⟶⋯.The DBGR S∞∗[F;Y] associated with an N-graded bundle (X,Y,AF) is defined as the direct limit(123)S∞∗F;Y=lim⟶⁡Sr∗F;Yof the direct system (122). It includes all graded differential forms ϕ∈Sr∗[F;Y] on graded manifolds (JrY,AJrF) modulo the monomorphisms (121). Its elements obey relations (89).MonomorphismsO∗(JrY)→Sr∗[F;Y] yield a monomorphism of direct system (67) to direct system (122) and, consequently, a monomorphism(124)O∞∗Y⟶S∞∗F;Yof their direct limits. Accordingly, body epimorphisms Sr∗[F;Y]→Or∗Y yield an epimorphism(125)S∞∗F;Y⟶O∞∗Y.It is readily observed that the morphisms (124)–(125) are cochain morphisms between the de Rham complex (69) of O∞∗Y and the de Rham complex(126)0⟶R⟶S∞0F;Y⟶dS∞1F;Y⋯⟶dS∞kF;Y→⋯of a DBGR S∞∗[F;Y]. Moreover, the corresponding homomorphisms of cohomology groups of these complexes are isomorphisms as follows [9].Theorem 57.
There is an isomorphismH∗(S∞∗[F;Y])=HDR∗(Y) of the cohomology H∗(S∞∗[F;Y]) of the de Rham complex (126) to the de Rham cohomology HDR∗(Y) of Y.As was mentioned above, theS∞∗[F;Y] (123) is split into a graded variational bicomplex which provides Lagrangian theory in Grassmann-graded (even and odd) variables [1, 2, 9].



---

*Source: 8271562-2017-01-17.xml*