Weil Modules and Gauge Bundles

Acta Mathematica Sinica, English Series Jan., 2006, Vol. 22, No. 1, pp. 271–278 Published online: Oct. 16, 2005 DOI: 10.1007/s10114-005-0616-3 Http://www.ActaMath.com

Weil Modules and Gauge Bundles ˇ Miroslav KURES Institute of Mathematics, Brno University of Technology, Technick´ a 2, 61669 Brno, Czech Republic E-mail: [email protected] Abstract Finite dimensional modules over Weil algebras are investigated and corresponding gauge bundle functors, from the category of vector bundles into the category of fibered manifolds, are determined. The equivalence of the two definitions of gauge Weil functors is proved and a number of geometric examples is presented, including a new description of vertical Weil bundles. Keywords Weil algebra, product preserving bundle, gauge bundle functor, jet MR(2000) Subject Classification 58A32, 58A20; 13H99

0

Introduction

In modern categorical approach to differential geometry, if we interpret geometric objects as bundle functors, then natural transformations represent a number of geometric constructions, see [1]. In this context, the result of finding the bijection between natural transformations between two Weil functors T A , T B (generalizing the well-known functors of higher order velocities and, of course, the tangent functor as the first of them), and the corresponding morphisms of Weil algebras A and B, have fundamental importance. Mikulski in [2] presents a continuation of the research in this direction. A gauge bundle functor is a covariant functor F : V B → F M from the category V B of vector bundles into the category F M of fibered manifolds satisfying [G I.] (Base-preservation) BF M ◦ F = BV B . [G II.] (Locality) For every inclusion of an open vector subbundle iE|U : E|U → E, F (E|U ) −1 is the restriction p−1 E (U ) of pE : F E → BV B (E) over U and F iE|U is the inclusion pE (U ) → F E. (BV B : V B → Mf and BF M : F M → Mf being base functors, Mf the category of manifolds). Mikulski obtained the bijection between natural transformations between two so-called gauge Weil functors T A,V , T B,W and the corresponding morphisms of Weil modules (A, V ), (B, W ); for details, see [2]. This new paper has the following resume. Section 1 is an algebraic introduction showing that every A-module is isomorphic to a factor module of the module AL , where L is determined uniquely. The significant module epimorphism is presented and the eventuality of when a Weil algebra stands in the role of a module is investigated, too. The gauge Weil functors are introduced in Section 2 by a new definition. The correctness of it is proved and this concept is exemplified by a number of geometrically important general cases generalizing over and above the modules studied by Shurygin and Smolyakova in [3]. We also summarize a connection of (A, V )-jets with (r, s, q)-jets investigated by Doupovec and Kol´ aˇr in [4]. The relation between vertical Weil bundles and higher order vertical bundles is also introduced. All manifolds and maps are assumed to be of class C ∞ . Received March 4, 2004, Accepted August 16, 2004 The author is supported by GA CR, Grant No. 201/02/0225. The paper represents the extended version of the contribution at the International Conference on Algebras, Modules and Rings, Lisboa 2003. The author’s participation at this conference was also partially supported by the grant of UNESCO–ROSTE

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1

Weil Algebras and Weil Modules

The Weil algebra A is a local commutative R-algebra with identity, the nilpotent ideal n of which has a finite dimension as a vector space and A/n = R. We call the order of A the minimum ord(A) of the integers r ∈ N∪{0} satisfying nr+1 = 0; the integer w(A) = dim(n/n2 ) is called the width of A. As it is well known, one can assume that the Weil algebra is a finite dimensional factor R-algebra of the algebra R[t1 , . . . , tk ] of real polynomials in indeterminates t1 , . . . , tk , where k = w(A), or, equivalently, A is a factor algebra of the jet algebra Drk := J0r (Rk , R) (r-th order jets from Rk to R with the source in 0). Then there is a (non-unique) epimorphism πA : Drk → A. Especially, we have: Proposition 1 (i) ord(A) = r and w(A) = 1 if and only if A = Dr1 ; (ii) ord(A) = 1 and w(A) = k if and only if A = D1k ; (iii) ord(A) = 0 if and only if w(A) = 0 if and only if A = R. Proof (i) For w(A) = 1, every Weil algebra has a form A = R[t]/i, and a monomial of the lowest degree in i can be in the role of the generator of i. (ii) For ord(A) = 1, every Weil algebra has a form A = R[t1 , . . . , tk ]/i, where all monomials of the second or higher degree belong to i. Moreover, k represents the minimal number of indeterminates for expressing of Weil algebra in the form A = R[t1 , . . . , tk ]/i and that is why i does not contain any homogeneous polynomial of the first degree, because its occurrence in i means that one indeterminate is a linear combination of the rest. (iii) The assertion is trivial. However, there are several different Weil algebras from r = k = 2. Let us denote by eα the elements of a basis of A as a real vector space, α = 1, . . . , dimR A < ∞. Then every element a ∈ A is expressible as a = α rα eα , rα ∈ R (in a unique way). It will be necessary to cite some facts from the ring theory, cf. e.g. [5, 6, 7, 8]. Let R be a ring and M an R-module. Let G = {mi ; i ∈ I} be a set of elements of M . Then all elements  of the form m = i ri mi , where ri ∈ R, i ∈ I, represent a submodule of M . This submodule is denoted by span(G ). Elements of G are called generators of M if span(G ) = M and then G is called a generator set of M . An R-module M can have many generator sets, especially, M = span(M ) always holds. For G = {mi ; i ∈ I}, it is clear that mi0 ∈ span(G ) for all i0 ∈ I. / span(G − {mi0 }) is as above, that is, satisfied for all i0 ∈ I, we say that the However, if mi0 ∈ generator set G is minimal. In general, it may not come to that an R-module M has a minimal generator set. Nevertheless, it can also have many minimal generator sets; the cardinalities of  them can be different. Elements {mi ; i ∈ I} are linearly independent if whenever i ri mi = 0 (ri ∈ R) then all ri = 0. A basis of M is a linearly independent minimal set of generators of M . If an R-module M has a basis, we say that M is a free R-module. A ring has the invariant minimality property (IMP) and is called IMP-ring if, for each minimally generated R-module M , the number of elements in each minimal generator set of M is invariant, see [8]. Similarly, a ring is defined to be an invariant basis number ring or IBN-ring if, for every free module, the number of elements in a basis is invariant, see [7]. Let A be a Weil algebra. Proposition 2 (i) Every A-module has a minimal generator set ; (ii) A is an IMP-ring ; (iii) A is an IBN-ring. Proof (i) The Jacobson radical of A identifies with the nilpotent ideal n and A/n = R. The field R is artinian. That is why A is a perfect algebra and therefore every A-module has a minimal generator set; cf. [6, 8]. (ii) The main result of [8] is that a ring R has IMP if and only if R is a local ring. A is a local algebra. (iii) This result follows directly from (ii).

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Let A be a Weil algebra and V an  A-module. Then V is also a real vector space and every element v ∈ V is expressible as v = p rp vp , rp ∈ R, p = 1, . . . , dimR V ≤ ∞ (in a unique way), where vp are  elements of a basis of V as a real vector space. Moreover, we can express v in the form v = i ai gi ai ∈ A, i = 1, . . . , L, where gi ∈ V form a minimal generator set G of V (and hence L is its invariant cardinality; we use L only in this sense from here). Furthermore, we L L A( denotes the direct summand of L copies; we incorporate also write simply AL for L L : (Drk )L → AL for πA . the case of A0 = R) and πA If V is finite dimensional as a real vector space, we name it the Weil A-module. Some important examples of Weil A-modules are mentioned in [9]. Proposition 3 Every Weil A-module V is isomorphic to a factor module of the module AL , where L is the cardinality of minimal generator set of V . For L, dimR V dimR V ≥ L ≥ dimR A is satisfied. Proof The inequality dimR V ≥ L follows from the fact that vp forma generator  The in set. RV follows from the expression of v in the form v = a g = equality L ≥ dim i i i i α riα eα gi , dimR A riα ∈ R. Both inequalities are satisfied for every A-module. A Weil A-module is finite dimensional as a real vector space; it means L is a finite number, too. We recall some useful denotations from [9]. The roman font (Hom, Epi) is used for algebra homomorphisms and epimorphisms, and the italic font (Hom, Epi) for module homomorphisms and epimorphisms from now on. If we write no subscript, we think of the homomorphisms over R. If A, B are two Weil algebras, V a Weil A-module and W a Weil B-module, then we can consider a homomorphism μ from V to W as a homomorphism of R-modules (real vector spaces) and write μ ∈ Hom(V, W ). Apart from that, we can consider an algebra homomorphism α from A to B, i.e. α ∈ Hom(A, B), and a map μ from V to W satisfying μ(v + w) = μ(v) + μ(w), μ(av) = α(a)μ(v) for all a ∈ A, v, w ∈ V . In this case, we write μ ∈ HomA,B (V, W ). (If A = B, we write HomA (V, W ) instead HomA,A (V, W ).) Of course, μ ∈ HomA,B (V, W ) implies μ ∈ Hom(V, W ). It is evident that, for a given μ, the relation μ(av) = α(a)μ(v) is not satisfied for an arbitrary α ∈ Hom(A, B); if it is satisfied, we say that μ is over α. We write a pair of homomorphisms α ∈ Hom(A, B), μ ∈ HomA,B (V, W ), where μ is over α, exclusively in brackets , i.e. α, μ . As every Weil A-module is a factor module of AL , then there is the canonical epimorphism L ρV : AL → V and we can define πA,V : (Drk )L → V , πA,V := ρV ◦ πA . Proposition 4

The module epimorphism πA,V is over the algebra epimorphism πA .

Proof We go from Drk -module (Drk )L ; let a, v1 , . . . , vL ∈ Drk . Then L πA,V (a(v1 , . . . , vL )) = ρV (πA (av1 ), . . . , πA (avL )) (the definitions of πA,V and πA ) = ρV (πA (a)πA (v1 ), . . . , πA (a)πA (vL )) (as πA ∈ Epi(Drk , A)) L L = ρV (πA (a)πA (v1 , . . . , vL )) (the factoring and the definition of πA ) L L = πA (a)ρV (πA (v1 , . . . , vL )) (as ρV ∈ EpiA (A , V ); ρV is over idA ) = πA (a)πA,V (v1 , . . . , vL ) (the definition of πA,V ). Remark 1 We have presented that there exists the module epimorphism πA,V which is over the algebra epimorphism πA . Analogously to the algebra case (where the epimorphism πA : Drk → A is not determined uniquely), the module epimorphisms from (Drk )L to V over πA are not determined uniquely: πA,V is nothing but only an important example of such a epimorphism. Every Weil algebra is a real vector space, i.e. an R-module. Furthermore, for a Weil algebra A  R with a nilpotent ideal n, we say that an element of a ∈ A is maximal, if it vanishes after

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multiplication by an arbitrary b ∈ n. Let MA be a set of all elements of A having a form r1 +r2 a, where r1 , r2 ∈ R and a is a maximal element of A. Evidently, MA is a subalgebra of A. Proposition 5 A is a D1k -module for all k ∈ {1, . . . , dimR MA − 1}. Proof Let 1, eα , α = 1, . . . , k, be elements of a basis of D1k and 1, e¯β , β = 1, . . . , dimR MA − 1, elements of a basis of MA (as real vector spaces). Clearly, for k = dimR MA − 1 the map i: D1k → MA given by i: eβ → e¯β is an isomorphism. If k < dimR MA − 1, we can select k arbitrary elements among e¯β and obtain new basis 1, e¯α , α = 1, . . . , k to represent the basis of a subalgebra, too. However, if MAk denotes such a subalgebra, the map ik : D1k → MAk given by i: eα → e¯α is also an isomorphism. It is well known that if A and B are Weil algebras and if B is a subalgebra of A, then A is a Weil B-module; it was recalled, e.g., in [9]. So, B = D1k and we have proved the assertion. Example 1 Let A = D12 = R[s, t]/m2 and V = R[S, T ]/ S 2 + T 3 + m5 , m being the maximal ideal of R[s, t], R[S, T ], respectively. The elements of A have a form r1 + r2 s + r3 t and elements of V have a form r1 + r2 S + r3 T + r4 S 2 + r5 ST + r6 T 2 + r7 S 3 + r8 S 2 T + r9 ST 2 , with the simultaneous vanishing of all monomials of the fifth or higher order in common with S 4 , S 3 T , S 2 T 2 , S 2 + T 3 , S 3 + ST 3 and S 2 T + S 4 . We observe that the elements S 3 and S 2 T are maximal and that is why the isomorphism i from the foregoing proposition has the form i(s) = S 3 i(t) = S 2 T. V is a Weil A-module; for a ∈ A, v ∈ V the product av is given by i(a)v. The set G = {1, S, T, S 2 , ST, T 2 , ST 2 } forms its generator set as span(G ) = V . The generator set is minimal: this follows from the fact that no generator can be removed. We have proved that L = 7 (we remark that dimR A = 3 and dimR V = 9). The epimorphism ρV : A7 → V is defined by ρV (a1 , a2 , a3 , a4 , a5 , a6 , a7 ):= a1 + a2 S + a3 T + a4 S 2 + a5 ST + a6 T 2 + a7 ST 2 . Now, if we write ai = ri1 + ri2 s + ri3 t for i = 1, . . . , 7, it yields r11 + r12 S 3 + r13 S 2 T + r21 S + r31 T + r41 S 2 + r51 ST + r61 T 2 + r71 ST 2 . G is not a basis of V , we have ker ρV = {(0, r22 s + r23 t, r32 s + r33 t, r42 s + r43 t, r52 s + r53 t, r62 s + r63 t, r72 s + r73 t)} and V ∼ = A7 / ker ρV (the isomorphism I as modules over A; it entails that V and A7 / ker ρV are isomorphic also as vector spaces). Let a = b = (0, 1, 0, 0, 0, 0, 0) ∈ A7 / ker ρV . Then both a, b map to S ∈ V through the isomorphism I. We have immediately that S = I(0, 1, 0, 0, 0, 0, 0) = I(ab) = I(a)I(b) = S 2 , which proves that I is not an algebra isomorphism. 2

Weil Bundles and Gauge Weil Bundles

We denote by W A the category with Weil algebras as objects and algebra homomorphisms of Weil algebras as morphisms. Then the known result of Kainz and Michor, Luciano and Eck reads as follows (see [1]): Product preserving bundle functors from the category Mf of manifolds into the category F M of fibered manifolds are in bijection with objects of W A , and natural transformations between two such functors are in bijection with the morphisms of W A . The correspondence is determined by the following construction of the bundle functor T A from a given Weil algebra A, cf. [1, 10]. Let M be a smooth manifold and let A be a Weil algebra. Two smooth maps g, h: Rk → M are said to determine the same A-jet j A g = j A h, if for every smooth function φ: M → R, πA (j0r (φ ◦ g)) = πA (j0r (φ ◦ h))

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is satisfied. The space T A M of all A-jets on M is fibered over M and is called the Weil bundle. The functor T A from Mf into F M is called the Weil functor. Analogously, we have the category W M , objects of which are Weil modules, and morphisms of which are pairs α, μ consisting of homomorphisms α ∈ Hom(A, B), μ ∈ HomA,B (V, W ), where μ is over α. Mikulski (see [2]) has obtained the following result: Product preserving gauge bundle functors from the category V B of vector bundles into the category F M of fibered manifolds are in bijection with objects of W M , and natural transformations between two such functors are in bijection with the morphisms of W M . The correspondence is determined by the construction of the bundle functor T A,V from a given Weil module. Before such a construction made, we dwell on jets of fibered manifold morphisms. Two morphisms of fibered manifolds determine the same (r, s, q)-jet (r ≤ s, r ≤ q) at a point y if they have the same r-jet in y, their restrictions to the fiber through y have the same s-jet in y, and their base maps have the same q-jet in the base point of y. (See [4] and [1] for details. The determination of the same 0-jet at a point corresponds to possessing identical values in this point.) We present the situation for the case of vector bundles. Let E → M , ¯→M ¯ be two vector bundles. E ¯ are two vector bundle morphisms having the same 0-jet in Proposition 6 If g, h: E → E v ∈ E, then s = 0 or s = ∞. Proof Let xi be base coordinates and y p fiber coordinates of E and analogously, ξ I , let η P be ¯ The morphisms g, h have the coordinate form coordinates of E. g: η P = gˆpP (xi )y p ξI h: For v = (xi0 , y0p ) we suppose

ηP

= g I (xi ), ˆ P (xi )y p = h

ξI

= hI (xi ).

p

ˆ P (xi )y p , gˆpP (xi0 )y0p = h p 0 0

ˆ P (xi ). However, s = 1 reads as but, in general, it does not imply gˆpP (xi0 ) = h p 0  P i p  ˆ P (xi )y p )  ∂(h ∂(ˆ gp (x )y ) p  =    q ∂y ∂y q v

v

ˆ P (xi ). It means restrictions of g, h to the fiber through v are identical and have or gˆpP (xi0 ) = h p 0 the same jets of every order. ¯ are two vector Corollary 1 Let z ∈ M and Ez be the fiber of E over z ∈ M . If g, h: E → E bundle morphisms having the same r-jet in v ∈ E and r ≥ 1, then s = ∞ and, moreover, g, h have the same r-jet at any point w ∈ Ez . ˆ P (xi ) is satisfied here. Proof Of course, gˆpP (xi0 ) = h p 0 Let E → M be a vector bundle and let (A, V ) be a Weil module. The denotation Rm,n is used for the trivial vector bundle Rm × Rn → Rm (including the cases m = 0 of one-point manifold and n = 0 of 0-dimensional fibers). The following construction takes up the abovementioned construction of Weil bundle. Two smooth vector bundle morphisms g, h: Rk,L → E are said to determine the same (A, V )-jet j A,V g = j A,V h, if for every smooth vector bundle morphism φ: E → R1,1 , πA , πA,V (j0r (φ ◦ g)) = πA , πA,V (j0r (φ ◦ h)) is satisfied. The space T A,V E of all (A, V )-jets on E is fibered over M and is called the gauge ¯ we have induced the morphism Weil bundle. For every vector bundle morphism ψ: E → E A,V A,V A,V ¯ A,V A,V A,V E by T ψ: T E →T ψ(j g):= j (ψ ◦ g). The functor T A,V from V B into T F M is called the gauge Weil functor. We have shown that the jets in question are (r, ∞, r)-jets

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in fact. The 0 in j0r reads as (0, . . . , 0, 0, . . . , 0). As to the construction of T A,V used in [2] and       L-times

k-times

[9], we are obligated to prove the following assertion: Proposition 7 Equivalently, T A,V E can be expressed as { α, μ |α ∈ Hom(Cz∞ (M ), A), μ ∈ HomCz∞ (M ),A (Cz∞,f.l. (E), V ), z∈M

where Cz∞ (M ) is the algebra of germs of smooth functions from M into R in z ∈ M , Cz∞,f.l. (E) is the Cz∞ (M )-module of germs of smooth fiber linear functions from E into R in z ∈ M , Hom(Cz∞ (M ), A) is the set of all algebra homomorphisms from Cz∞ (M ) into A and HomCz∞ (M ),A (Cz∞,f.l. (E), V ) is the set of all module homomorphisms from Cz∞,f.l. (E) into V . Proof Let tI be the base coordinates and τ P the fiber coordinates of Rk,L and analogously, let xi , y p be the coordinates of E. The morphism g: Rk,L → E has the coordinate form g:

yp

=

gˆPp (tI )τ P

xi

=

g i (tI ),

and the induced base map g: Rk → M has the coordinate form g: xi = g i (xI ). Let s be a base coordinate and σ a fiber coordinate of R1,1 and let f : E → R be a fiber linear map and F : M → R a map. Their coordinate forms are f : σ = fˆp (xi )y p s and F: Let

A,V E:= Tgerm

z∈M

= f (xi ), s = F (xi ).

{ α, μ |α ∈ Hom(Cz∞ (M ), A), μ ∈ HomCz∞ (M ),A (Cz∞,f.l. (E), V )}.

A,V We have the canonical fibered bijection ιE : T A,V E → Tgerm E over idM given by

ιE : j A,V g → αg , μg , where and

αg ∈ Hom(Cz∞ (M ), A),

αg (germz F ):= πA (j0r (F ◦ g))

μg ∈ HomCz∞ (M ),A (Cz∞,f.l. (E), V )

μg (germz f ):= πA,V (j0r (f ◦ g)),

where z = g(0) ∈ M . The map αg is well defined as αg (germz F1 ) = αg (germz F2 ) implies πA (j0r (F1 ◦ g)) = πA (j0r (F2 ◦ g)); it follows that F1 = F2 . The map μg is also well defined as μg (germz f1 ) = μg (germz f2 ) implies πA,V (j0r (f1 ◦ g)) = πA,V (j0r (f2 ◦ g)); it follows that f1 = f2 . Clearly, both αg and μg are homomorphisms over R. The map αg is an algebra homomorphism, because αg (germz F1 germz F1 ) = αg (germz (F1 F2 )) = πA (j0r (F1 F2 ◦ g)) = πA (j0r ((F1 ◦ g)(F2 ◦ g))) = πA (j0r (F1 ◦ g)j0r (F2 ◦ g)) = πA (j0r (F1 ◦ g))πA (j0r (F2 ◦ g)) = αg (germz F1 )αg (germz F2 ). The map μg is a module homomorphism over αg , because μg (germz F germz f ) = μg (germz F f ) = πA,V (j0r (F f ◦ g)) = πA,V (j0r (F ◦ g) ◦ j0r (f ◦ g)) = (see Proposition 4)= πA (j0r (F ◦ g))πA,V (j0r (f ◦ g)) = αg (germz F )μg (germz f ). In local coordinates, it is not difficult to show that ιE is a bijective correspondence. Let us notice that the composition f ◦ g: Rk,L → R1,1 has the following coordinate form: gPp (tI )τ P f ◦ g: σ = fˆp (gi (tI ))ˆ s

=

f (gi (tI )),

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p p ∂(fˆp (g i (tI ))ˆ gP (tI )τ P ) ∂(fˆp (g i (tI ))ˆ gP (tI )τ P ) ∂(f (g i (tI ))) , , , ∂tJ ∂τ Q ∂tJ

and, after differentiation higher derivatives), we obtain

etc. (second and

   ∂g i  ∂ r gi i   , g |0 , , . . . , g :=  I1 ...Ir ∂tI 0 ∂tI1 . . . ∂tIr 0    ∂ˆ gPp  ∂ r gˆPp p p p  , gˆP |0 , gˆP I := , . . . , gˆP I1 ...Ir := I1 ∂tI 0 ∂t . . . ∂tIr 0 as jet coordinates, which are the same as in the construction of j A,V g as in the construction of αg , μg . Selfsame epimorphisms πA and πA,V are used in identical ways, too. As to local trivialization, we have proved that the map ιE is a bijection. As to the equivalence, T A,V and A,V will not be distinguished from now on. Tgerm Remark 2 Proposition 7 generalizes a similar result for Weil functors, cf. [1, 35.16]. i

g iI :=

Let F → M , G → M be two fibered manifolds over a base M . We denote by F ×M G the fibered product of F , G over M and by ×qM F the fibered product F ×M . . . ×M F . In addition,    q-times

×0B F := M . (The fibered product is also called the Whitney sum.) Example 2 For E = R0,n we have T A,V E ∼ = V n ; for E = Rm,0 we have T A,V E ∼ = Am ; for m,n A,V m n ∼ we have T E = A × V . If M is an arbitrary manifold and E is the trivial E = R vector bundle with 0-dimensional fibers over M , then T A,V E = T A M . After a while we are ready to describe more general examples of gauge Weil bundles. Example 3 For V = AL the epimorphism ρV equals idAL . So, by a direct use of the construction of T A,V , we have obtained L T A,A E ∼ = ×LA T A E, T M

for all L ∈ N ∪ {0}. Remark 3 For L = 0, the formula reads as T A,R E = T A M ×M E. Remark 4 If we take A as Drk in the previous example, then we have obtained, for L = 1, r r the functor T Dk ,Dk sending an object E → M of V B into the (classical) velocities prolongation Tkr E → M as the object in F M . L

Remark 5 If A = R, then AL = RL . The functor T R,R sends E → M to ×L ME → M. However, for A = R every Weil A-module V is isomorphic to RL for some L as real vector space. It will be more analysed at the end of the paper. L

L

Example 4 Let V = B1L1 ⊕ · · · ⊕ Bq q , where Bj j are factor algebras of a Weil algebra A, j = 1, . . . , q. Then V is a Weil A-module. Weil modules studied in [3] represent a special case for q = 2 and B1 = A. We have derived directly that L T A,V E ∼ = ×L1A T B1 E ×T A M · · · ×T A M × qA T Bq E. T M

T M

In closing the paper, we add one noteworthy geometric consequence yet. The bundle T1r M of (one-dimensional) velocities of order r on M is also called the tangent bundle of order r and denoted for short by T r M . It consists of r-jets at 0 ∈ R of differentiable mappings μ: R → M . (For 0 ≤ s < r the projection πsr : T r M → T s M is defined by πsr (j0r μ) = j0s μ.) The tangent bundle of higher order is a cornerstone in the contemporary view to variational calculus and analytical mechanics. For an acquaintance with basic properties of T r M , including a description of the corresponding Weil algebra Dr1 for which it can be used see e.g. author’s paper [11]. Let B be a Weil algebra. In [1] is defined the vertical Weil functor V B from the category F M m of fibered manifolds with m-dimensional bases to the category F M of fibered manifolds by putting V B Y := T B Yz z∈M

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for a fibered manifold Y → M (with fibers Yz over z ∈ M ). If B = Dr1 , we write V r Y for short r instead of V D1 Y . The bundle V r Y is called the vertical bundle of order r. Let E → M be a vector bundle. We have obtained the following identification of arbitrary vertical Weil bundles and higher order vertical bundles: Proposition 8 V B E ∼ = V dimR B−1 E. Proof As to Remark 3, for A = R every Weil A-module V is isomorphic to RL for some L as a real vector space. Then dimR V = L as in Proposition 3 (dimR A = 1, of course); i.e. if dim B V is a Weil algebra B, then dimR B = dimR V = L. The functor T R,R R sends E → M to RB E → M. ×dim M with respect to R consists of 1, t, t2 , . . . , tr−1 . Therefore, the dimension The basis of Dr−1 1 r−1 dimR D1 equals r. Hence the functor V r−1 sends E → M to ×rM E → M . We put r = dimR B and the proof is accomplished. Acknowledgement The author would like to express his thanks to Professor Wlodzimierz M. Mikulski from Jagiellonian University in Cracow for helpful comments and very instructive discussions on the topic of this paper. References [1] Kol´ aˇr, I., Michor, P. W., Slov´ ak, J.: Natural Operations in Differential Geometry, Springer Verlag, Berlin, 1993 [2] Mikulski, W. M.: Product preserving gauge bundle functors on vector bundles. Colloq. Math., 90, 277–285 (2001) [3] Shurygin, V. V., Smolyakova, L. B.: An analog of the Vaisman–Molino cohomology for manifolds modelled on some types of modules over Weil algebras and its application. Lobachevskii Journal of Math., 9, 55–75 (2001) [4] Doupovec, M. Kol´ aˇr, I.: On the jets of fibred manifold morphisms. Cah. Topologie G´ eom. Diff´ er. Cat´ eg., 40(1), 21–30 (1999) [5] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra, Addison Wesley, Mass–London– Don Mills, Ont., 1969 [6] Belshoff, R., Xu, J.: Nonzero injective covers of modules. Missouri J. Math. Sci., 13(3), 163–171 (2001) [7] Faith, C.: Algebra: Rings, Modules and Categories I, Springer Verlag, New York–Heidelberg, 1973 [8] Rant, W. H.: Rings whose modules require an invariant number of minimal generators. Missouri J. Math. Sci., 13(1), 43–46 (2001) [9] Kureˇs, M., Mikulski, W. M.: Liftings of linear vector fields to product preserving gauge bundle functors on vector bundles. Lobachevskii Journal of Math., 12, 51–61 (2003) [10] Kol´ aˇr, I.: Jet-like approach to Weil bundles, Seminar lecture notes, Masaryk University, Brno, 2001 [11] Kureˇs, M.: Torsions of connections on tangent bundles of higher order. Rendiconti del Circ. Matemat. di Palermo, Serie II, 54(Suppl.), 65–73 (1998) [12] Wu. M., Zeng, G. X.: Higher level orderings on modules. Acta Mathematica Sinica, English Series, 21(2), 279–288 (2005)