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Categories of Modules with Differentials JOURNAL OF ALGEBRA 185, 50]73Ž. 1996 ARTICLE NO. 0312 Categories of Modules with Differentials Paul Popescu Department of Mathematics, Uni¨ersity of Craio¨a, 13, A.I. Cuza st., Craio¨a, 1100, Romania Communicated by Walter Feit CORE Metadata, citation and similar papers at core.ac.uk Received August 1, 1994 Provided by Elsevier - Publisher Connector 1. INTRODUCTION The definitions of the module with arrow Ž.module fleche , the infinitesi- mal module, and the Lie pseudoalgebra, as used here, are considered inwx 7 . Moreover, we define the preinfinitesimal module, called inwx 1 ``un pre- espace d'Elie Cartan regulier.'' Inwx 7 the Lie functor is constructed from the category of differentiable groupoids in the category of Lie algebroids, but it is inwx 2 that the first general and abstract treatment of the algebraic properties of Lie alge- broids is made, giving a clear construction of the morphisms of Lie algebroids. We shall use it fully in this paper to define completely the categories M W A Ž.Ž.modules and arrows , P I M preinfinitesimal modules , IMŽ.Ž.infinitesimal modules , and L P A Lie pseudoalgebras ; we call these categories the categories of modules with differentials. We notice that in wx7 and wx 1 the objects of these categories and the subcategories of modules over a fixed algebra are considered. Inwx 3 the category of L P A of Lie pseudoalgebras is defined and some aspects related to the functional covariance or contravariance correspondence with categories constructed with additional structures on vector bundles are studied. In Section 2 we give a brief description of the category MA , of modules over commutative k-algebrasŽ considered also inwx 3. , and we define a strict category of modules using a suitable functor. In Sections 3 and 4 we define the categories of modules with differen- tials. In Section 5 we study certain wide and strict categories of modules with differentials which are defined in this paper. It is proved that, under suitable conditions on the functor which define them, they admit direct products. 50 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. MODULES WITH DIFFERENTIALS 51 Finally, in Section 6, we study the pullback, which extends the inductor defined inwx 8 , using the ideas from wx 2 . We define the pullback in categories of modules with differentials and in categories of vector bundles with differentialsŽ. where it is in fact a fibred product , extending the definition of the inductor given inwx 2 in the case of Lie algebroids. Using two papers of the authorwx 5, 6 , we consider thoroughly with some examples the study of differential modules, considering categories which we call categories of vector bundles with differentials. They are extensions of the Lie algebroidwx 7, 4, 2 and are very close to it. This paper is the first part of a study about categories with differentials. The second part will contain the study of actions, fibrations and ideals, using the same approach as inwx 2 for Lie algebroids and groupoids. We intend this paper to be useful to those who follow the study of differential geometry using algebraic tools and also to those who use geometric examples in algebra. 2. CATEGORIES OF MODULES The largest category whose objects and morphisms we shall use in this paper is a contravariant category of modules MA. The modules are considered over commutative, associative and unitary algebras. The alge- bras are taken in their turn over a commutative and unitary ring k, fixed from now on. The category MA is considered for example inwx 3 , where it is ¤ denoted as Mod . The commutative, associative, and unitary algebras over k and their morphisms are the objects and the morphisms of a category Ak denoted simply by A. Ž. The objects of the category MA are coupled A, L , where A is an object of A and L is an A-module. The morphisms of two objects Ž AXX, L . X and Ž.A, L are couples Ž.w, c , where w: A ª A is a morphism of A and XX X Ž c:LªAmA Lis a morphism of A -modules. The tensor product is an A-module, considering AX as an A-module through w.. We denote such a Ž XX.Ž. morphism in MA by A , L ªŽw, c . A, L . The composition law of two morphisms in M: ŽA , L . Ž.Ž.ŽA , L , A , L A , L .is 00ªŽw11,c.11 11ªŽw22,c.22 Ž.A,L Ž.A,L, where, if 00ªŽw1212(w,c)c. XL,cŽ.X aXXA L, g01sÝiAm11 ig0mA 1 i cŽ.XXbiXYAL 2 isÝamA22ag1mA 2 a 52 PAUL POPESCU then Ž.Ž.c c X a w bi X Y A L .1Ž. 21) s Ýi1Ž.amA22ag 0mA2 i,a Notice that if A00is an object of A then the category of A -modules is a subcategory M of M . It has as objects all the objects and the morphisms A 0 A of M which have the form Ž.Ž.A , L and id , c , respectively. Taking into A 0 A 0 account the canonical isomorphism A L L, a X aX Ž.2 0mAiAiii00l ÝÝm l ii for every A -module L, there is an isomorphism of categories M and the 0 A0 category of A0-modules. From now on we shall not make any distinction between them. Notice also that M is not a full subcategory of M . A0 A Consider now AX a subcategory of A. In the same manner as above, denote by MAX the subcategory of MA whose objects and morphisms are defined by couples in which the algebra is an object of AX and the morphism is a morphism of AX. Consider now C00an arbitrary category and F: C ª MAa functor; we ŽXŽX.. denote as C0 MA a category which has as objects couples M , FM with MX an object in C X, and as morphisms couples ŽwX, FŽwX .. for wX morphism of C00; we say that C MAis a strict category of modules induced by F.We X notice also that F induces a forgetful functor F0: C ª A that generates a category of algebrasŽ. see Section 5 . An important example of a strict category of modules is given below. It is used frequently in the examples of this paper. ` EXAMPLE 2.1. Take as C0 the category V B of C -vector bundles which have finite dimensional fibers. We denote by j s Ž.E, p, B such a vector bundle which has the total space E, the base space B and the projection p, all smoothŽ i.e., of class C`.. The morphisms of two vector X Ž XX X.Ž.X bundles j s E , p , B and j are couples f00, f , where f : B ª B is a X smooth map and f: E ª E is an f0-morphism of vector bundles. There is induced U X U f00: FŽ.B ª F ŽB ., f Ž.u s u( f0, a morphism of commutative, associative and unitary algebras and U X U Ž X . f : E ª fE0 which is uniquely determined by a F B -module morphism, denoted also by f U : U X U f : GŽ.j ª G Žf0j .. MODULES WITH DIFFERENTIALS 53 Notice that for a smooth manifold B we denote as FŽ.B the real algebra of smooth real functions on B and for the vector bundle j s Ž.E,p,Bwe denote as G Ž.Ž.j the F B -module of sections on j. From the isomorphism of FŽ BX .-modules XXUX FBŽ.mFŽB.G Ž.j ªG Žf0j ., umXªuX, Ž.UUXX Ž . where X g G fE0 is x ª x , Xf Ž x. , it follows that f can be also Ž X .Ž.Ž0 . considered taking values in F B mFŽ B. G j seewx 2 for more details . For k s R a contravariant functor F0: V B ª MA defined by the formula U U F00Ž.j s Ž.F Ž.B , G Ž.j , Ff Ž0,f .s Žf0,f . is obtained. This defines a strict category of modules V B MA , used in the next examples of this paper. 3. CATEGORIES OF MODULES WITH ARROWS In this section and in the following one we define some categories which we call categories of modules with differentials; they contain as subcategories those obtained fixing an algebraŽ seewx 7. The construction is inspired by wx2 ; the examples are fromwx 5 , wx 6 , and wx 2 . Most of the notation fromwx 7 andwx 2 is used, pointing out the extensions given here. Let A be a commutative, associate, and unitary k-algebra. For the p A-modules L and M we denote as Hom AŽ.L, M the A-module of alter- nated A-linear p-forms from L to M and as DerŽ.A the A-module of derivations of A Ž.which is a Lie k-algebra . An A-module with arrow Ž.Žm.w.a. cf.wx 7 module flecheÂÂ.is an A-module M Mtogether with an A-module morphism r s r : M ª DerŽ.A called an anchor. As an example, DerŽ.A and its identity morphism is an m.w.a. If M r s 0 we say that the m.w.a. is tri¨ial.If Mis an A-m.w.a. we make the notation M wxX,aMsrŽ.Ž.Xa, XgM,agA. DEFINITION 3.1wx 7 . A morphism of A-m.w.a.'s L and M is an element 1 Ž.
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