JOURNAL OF ALGEBRA 179, 936᎐963Ž. 1996 ARTICLE NO. 0046
On Modules and Crossed Modules
R. Lavendhomme and Th. Lucas
Uni¨ersite´ Catholique de Lou¨ain, B-1348 Lou¨ain-la-Neu¨e, Belgium
Communicated by Walter Feit
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INTRODUCTION
This is a paper without any result, it is just a meditation on the concept of module. Besides the aesthetic appeal of such questions, we find some justification in that our concept of a crossed module covers classical examples, gives a unified proof that the semi-direct product is an internal category, allows computations of non-abelian cocycles, and gives general meaning to algebras of derivations. Let ⌫ be a familiar equational algebraic theory, such as the theory of groups or the theory of rings. The theory ⌫ is described as a set of equations written in a one-sorted language L. An interpretation of L is given by a non-empty set A and interpretations in A of the operation symbols of A. A model of ⌫ is an interpretation of L which moreover satisfies the axioms of ⌫. On the other hand, the well-known example of modules is often de- scribed as a model of a theory written in a 2-sorted language. We prefer here to consider them as 2-sorted interpretations of the 1-sorted theory of rings. This allows us to present in a unified setting basic notions such as modules, crossed modules, and derivation algebras. One of the intuitions we want to convey to the reader is that the definition of a crossed module is almost forced by the decision to pass from a 1-sorted interpretation to a 2-sorted interpretation of the same 1-sorted language. Section 1 will present our notions of a modulated model of a theory ⌫ and of a crossed ⌫-module. Section 2 is devoted to examples. We show in particular that our notion of a crossed ⌫-module covers the classical examples of Lie crossed mod- ules and crossed modules in groups. Section 3 defines semi-direct products under general conditions, gives examples, and proves that semi-direct products are internal categories.
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0021-8693r96 $12.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. MODULES AND CROSSED MODULES 937
In Section 4, we comment on the notion of two-cocycle and check that with our presentation of crossed modules in groups, the general notion of two-cocycle and the classical notion coincide. In Section 5, we show that Lie R-algebra of derivations, the group of automorphisms of a group, MacLane’s R-algebra of bimultiplications, etc., are all particular instances of the same fundamental concept. The notion of a crossed module originates in algebraic topology and more particularly in homotopy theory: see for example Mac Lane and Whitehead’s basic paperwx 7 . Since then, crossed modules have been extensively used in the cohomology of groups, for example by Brown, Dedecker, and Duskin. A short historical note on the question has been written by Maclane as an appendix of Holt’s paperwx 2 . The notion of a crossed module has also been extended to categories of algebras. See for example Luewx 5 and Lavendhomme and Roisinwx 3 , further developed in wx4 . The most interesting algebraic situations have been highlighted by Orzech inwx 8 under the name ‘‘categories of interest’’.
1. MODULATED ⌫-STRUCTURES AND CROSSED ⌫-MODULES
We start with a usual language L having only one sort of variable and for each n a set Opn of n-ary operation symbols. Terms of L are defined as usual. To introduce the idea of a two-sorted interpretation, we consider two symbols ␣ and  Žto be interpreted later by elements of the algebra and scalars respectively. . An n-ary modulation is a sequence s Ž. ␣  1,..., nn; q1of n q 1 symbols or . We say that nq1is the target Ž. of . If all i 1 F i F n q 1 are ␣, we denote the corresponding modulation by ␣ and call it internal to ␣; similar conventions hold for . An L-modulation M is fixed by giving for each n and each F g Opn a set MF of n-ary modulations such that
 g MF , Ž. if s 1,...,nF; gM then s , Ž. if s 1,...,nFi;␣ gM then s ␣ for some i,1FiFn.
The elements of MF are called modulations admissible for F. From now on, we consider M as being part of the definition of L. To interpret L, we first associate with ␣ a non-empty set <<␣ s A Žthe set underlying the algebra. and with  a non-empty set << s B Žthe set of scalars.Ž. . A modulated interpretation ⌽ of L o¨er A, B , to be denoted by 938 LAVENDHOMME AND LUCAS
Ž. A,B,⌽, is fixed by giving for each n, each F g Opn, and each s Ž. 1,..., nn; q1gMFa mapping << < <иии < < < < F : 1 = = nnª q1, called Ž.- modulated interpretation of F. We extend the notions of admissible modulation and of modulated interpretation to terms of the language L, more exactly to pairs txŽ.of terms t and sequences x of distinct variables containing those of t. Ž. Ž . aIfxis the ith variable of the sequence x, s 1,..., nn; q1is Ž. admissible for xx if and only if nq1 s i and the corresponding modu- lated interpretation is the ith projection
xxŽ.si:<<1=иии = <<niª << . Ž. Ž. Ž. Ž. bIftxsFt1 x иии txkkwith F g Op , and if for each i,1FiFk, Ž. Ž.Ž. i s1,...,ni; is admissible for tx i and s 1,...,kF; gM , Ž. Ž. then s 1,...,n; is admissible for tx and the corresponding modulated interpretation <Ž.< 6 tx 6 <<1=иии = <<n <<. Consider now a theory ⌫ in L, i.e., a set of equations of the form t12s t Ž.we consider only equational theories . We say that the modulated inter- Ž. pretation A, B, ⌽ satisfies or is a model of t12s t iff for every x and every modulation admissible for tx12Ž.and tx Ž., tx12Ž.stx Ž.. ŽComposition with projections shows that this does not really depend on x.Ž.Ž. We say that A, B, ⌽ is a model of ⌫ or a modulated ⌫-model . if Ž.A, B, ⌽ is a model of every element of ⌫. According to our view, the notion of crossed module arises when putting coherence conditions on different modulations admissible for the same operation. Here are the details. Introduce a partial order on the set of n-ary modulations by letting ␣ F  and extending it by X X X Ž.1,..., nn; q11FŽ.,..., nn; q1 iff X X X 11F ,..., nnF and nq1F nq1. MODULES AND CROSSED MODULES 939 For that order, there is a smallest n-ary modulation, namely ␣ , and a greatest n-ary modulation,  . Recall that ␣ is admissible for every F. DEFINITION 1. A crossed ⌫-module Ž.A, B, ⌽, D is determined by a modulated interpretation Ž.A, B, ⌽ of L which is a modulated model of ⌫ and a mapping D : A ª B X such that for every n, every F g OpnF , and every , g M satisfying Ž. XŽXXX . s1,..., nn; q11F s ,..., nn; q1the following diagram commutes: << F 6 << иии << < < 1= = nnq1 D˜˜1=иии =Dn D˜nq1 6 6 << FЈ 6 <<XXиии << <<X 1 = = n nq1 . Ž In the diagram, the convention is that for every i,1FiFnq1, D˜i is the XX. identity if iis and D˜ is D if is ␣ and is  . Note that if ␣ g MF , one of the diagrams of the definition expresses Ž X . that D : A ª B is homomorphic for F take s ␣ and s  .A reason why we do not include ␣ in every MF is that we want to cover some trivial examples as we will show in the next section. The notion of morphism of modulated interpretations makes no difficul- ties; it is an ordered pair X X X Ž.Žf , g : A, B, ⌽ .Žª A , B , ⌽ . such that f is a mapping from A to AX and g is a mapping from B to BX making the following diagram commute for every n, for every F g Opn , for every g MF : << F 6 << иии << < < 1= = nnq1 1=иии =n nq1 6 6 << FЈ 6 <<Ј иии << Ј << Ј 1 = = n nq1 . ŽIn the diagram, the convention is that<< is associated with Ž.A, B, ⌽ , < For crossed ⌫-modules Ž.ŽA, B, ⌽, D and AXXXX, B , ⌽ , D .we add the commutativity of the square D 6 AAЈ f g 6 6 DЈ6 AЈBЈ. Those definitions clearly determine the category of modulated interpreta- tions and the category of crossed ⌫-modules. 2. EXAMPLES 2.1. Some Tri¨ial Examples 2.1.1. Let ⌫ be the theory of monoids with the two operation symbols и and e. If one admits the modulations Ž.Ž., ␣; ␣ and , ;  for и and Ž.;for e, a modulated model of ⌫ reduces to an action of a monoid B on a set A. Indeed, in this case, the modulations of the axioms x и Ž.Ž.y и z s x и y и z and x и e s x s e и x give for Ž., ;  the associativity and neutral element conditions for B and for Ž., ␣; ␣ the conditions ᭙a g A ᭙b12, b g BbŽ.1иb 2)asb 1и Ž.b 2)a and eB ) a s a Ž. where ) denotes the action B = A ª A . Note that  is the only admissible modulation to consider for x s e и x. 2.1.2. Let ⌫ be an equational theory. If one admits only the internal modulations ␣ and  for every operation, a modulated model of ⌫ reduces to an ordered pair of ⌫-algebras and a crossed ⌫-module is simply a homomorphism D : A ª B of ⌫-algebras. 2.1.3. Let ⌫ be the theory of ringsŽ not necessarily commutative or unitary. , in the language q, y,0,иwith the usual axioms. Let us admit for q, y, and 0 the two internal modulations ␣ and  and for и the modulations Ž.Ž., ␣; ␣ and , ;  . A modulated model Ž.A, B, ⌽ of ⌫ will reduce in this case to a ring B acting on an abelian group A with ᭙b g B, ᭙a12, a g Ab)Ž.a1qa 2sb)a 1qb)a 2, ᭙b12,bgB,᭙agAbŽ.1qb 2)asb 1)aqb 2)a MODULES AND CROSSED MODULES 941 Ž.following from the distributivity axioms and ᭙b12, b g B, ᭙a g Ab1)Ž.Ž.b 2)asb1иb 2)a Ž.following from the associativity axiom . This means that A is a left B-module. If moreover we admit the modulation Ž.␣, ; ␣ for и, then A becomes a B-B-bimodule, the key condition Ž.b1212) a ) b s b ) Ž.a) b coming as a modulation of associativity. A crossed ⌫-module incorporates besides that a mapping D : A ª B which, according to the definition of Section 1, must be a homomorphism of abelian groups and must satisfy the only supplementary condition: ᭙b g B ᭙a g A D Ž.b) a s b и D Ž.a . This means that D is a B-linear form on A. 2.2. R-Algebras with Multilinear Axioms Let R be a unitary commutative ring. We are interested in theories ⌫ containing the theory of R-algebrasŽ. associativity not necessarily included and want to determine the modulated models of ⌫ and the crossed ⌫-modules. For what concerns the language of R-modules Ž.q, y, 0 and unary laws , let us admit only the internal modulations ␣ and  . If for multiplication we admit  and Ž., ␣; ␣ , a modulated model of the axioms of R-mod- ules will consist in an R-algebra B,an R-module A, and an R-bilinear Ž. action )1 : B = A ª A. If we also admit the modulation ␣, ; ␣ , we get an R-bilinear action )2 : A = B ª A. In this subsection, we suppose that in addition to the axioms of R-alge- bras, ⌫ contains only multilinear axioms: associativity, commutativity, anticommutativity, and Jacobi’s identity are typical examples. To be spe- cific, we will explicitly consider the case of Lie R-algebras and associative R-algebras. 2.2.1. Lie R-algebras, when R is a unitary commutative ring with a characteristic different from 2, have as specific axioms Jacobi’s identity and anticommutativity.Ž In characteristic 2, one has to add wxx, x s 0, which is not linear and will be dealt with later.. 942 LAVENDHOMME AND LUCAS Forwx , , we admit  and Ž., ␣; ␣ . A modulated model consists in an R-Lie algebra B and a Lie module A. The action ) : B = A ª A satisfies the usual condition wxb12,b)asb 1)Ž.b 2)ayb 2) Ž.b 1)a, coming from a modulation of Jacobi’s identity. If forwx , we also admit Ž.␣, ; ␣ , then we have a right action )2 : A = B ª A, but a modulation of the anticommutativity axiom gives b) a q a)2 b s 0, allowing one to reduce the second action to the first one, so that nothing really new happens. If forwx , we also admit the internal modulation ␣ , then A becomes a Lie R-algebra operating by derivation, the typical property b)wxwa12,asb)a12,a xwqa 1,b)a 2 x coming from a modulation of Jacobi’s identity. Turning now to crossed modules, we recall the classical notion of Lie crossed module. It is a quadruple Ž.A, B, ⌽, D where Ž.1 Aand B are Lie R-algebras; Ž.2 ⌽:B=AªAis bilinear; Ž.3 ⌽is a Lie action in the sense that wxb12,b)asb 1)Ž.b 2)ayb 2) Ž.b 1)a; Ž.4 ⌽operates by derivation; Ž.5 Dis a homomorphism of Lie R-algebras; Ž.6 Dis B-linear, D Ž.b) a s b, D Ž.a ; Ž. Ž. 7wxa12,asDa 1)a 2. PROPOSITION 1Ž. For the Theory ⌫ of Lie R-Algebras . Admitting for wx,the modulations ␣ ,  , and Ž., ␣; ␣ , the notion of a crossed ⌫- module defined in Section 1 coincides with the classical notion of a Lie crossed module. Proof. PointsŽ. 1 to Ž. 4 express what it is to be a modulated model of the theory ⌫ of Lie R-algebras. A crossed R-module incorporates a mapping D : A ª B which is R-linear and such that Ž. i for ␣ F  , the compatibility condition corresponds to condition Ž.5; MODULES AND CROSSED MODULES 943 Ž. Ž . ii for , ␣; ␣ F  , the compatibility condition corresponds to conditionŽ. 6 ; Ž. Ž . iii for ␣ F , ␣; ␣ , the compatibility condition corresponds to conditionŽ. 7 . 2.2.2. Associati¨eR-algebras have associativity as a specific axiom. For multiplication we admit the internal modulations as well as the external modulations Ž.Ž.Ž, ␣; ␣ and ␣, ; ␣ they are the only admissible modu- lations because of the restrictions of the definition. . Modulated models will in this way consist in an R-algebra B and an R-module A together with two R-bilinear actions )1 : B = A ª A )2 : A = B ª A. The modulations of associativity give the internal associativities together with X X b)11Ž.Ž.b ) a s bb )1a Ž.for Ž ,  , ␣ ; ␣ .Ž.1, X X a)22Ž.Žbb s a) b .)2b Ž.for Ž␣ ,  ,  ; ␣ ., X X b)12Ž.Ž.a)bsb) 12a)bŽ.for Ž , ␣ ,  ; ␣ ., X X b)11Ž.Žaa s b) aa .Ž.for Ž␣ ,  , ␣ ; ␣ .Ž.2, X X abŽ.Ž.)12a s a)ba Ž.for Ž␣ ,  , ␣ ; ␣ ., X X aaŽ.Ž.)22b s aa ) b Ž.for Ž␣ , ␣ ,  ; ␣ .. If commutativity is considered, one gets a)21b s b) a and the associativity conditions may be reduced toŽ. 1 and Ž. 2 . This determines the notion of a modulated model of the theory ⌫ of associative R-algebras. To obtain a crossed ⌫-module, we need moreover a homomorphism D : A ª B satisfying D Ž.b)1a s bD Ž.a and D Ž.Ž.a)2b s D ab. 2.3. R-Algebras with Polynomial Axioms The situation is more complex when an axiom is a non-multilinear polynomial. Such is for example the case of Lie R-algebras when axioma- tized with no restriction on characteristic and wxx, x s 0. Slightly abstract- ing, this is essentially the following example. 944 LAVENDHOMME AND LUCAS 2 Consider the theory ⌫ of rings satisfying the identity x s 0. The 2 2 modulations of this axiom reduce to a s 0 and b s 0 for a g A and 2 b g B. However, substituting x q y to x in x s 0, one easily derives in ⌫ that xy q yx s 0, a theorem which admits in general the following modulation: ᭙a g A ᭙b g Ba)21bqb)as01Ž. The problem is that modulated models of ⌫ will not necessarily satisfyŽ. 1 . 2 ŽŽŽ.As an example, take for A the additive group ޚr 2 wxx rŽx .., q and for B the ring ŽŽ.ޚr 2,q,m .where m is the null multiplication; as the left action )11of B on A, take 1) 0 s 1)1x s 0 and 1)11 s x; as the right action )2 of B on A, take the null action; modulations of the axioms of ⌫ are satisfied, but we do not haveŽ. 1 as we compute 1)211 q 1)1 s 0 q x s x / 0.. It seems that the theory to consider is not ⌫ but the theory ⌫˜of rings 2 together with the two identities x s 0 and xy q yx s 0. Of course, we touch here the problem of the sensitivity of our defini- tions with respect to the form of the axioms of ⌫. For which axiomatiza- tions of a theory ⌫ is it true that the modulated models of ⌫ satisfy the modulations of the equational theorems of ⌫? General answers are beyond the scope of this paper, but we suspect that for R-algebras with polynomial axioms, it essentially suffices to add the derived identities: if Ž. Px1,..., xniis0 is an axiom, add all ⌬ P s 0, where ⌬ P is defined by ⌬ Ž.ŽiPx1,..., xni, y .sPx Ž1,..., xiy1, xiiiqy, xq1,..., xn. yPxŽ.1,..., xiy1, xii, xq1,..., xn yPxŽ.1,..., xiy1, yii, xq1,..., xn and iterate the procedure. 2 3 For x s 0, the procedure will lead to xy q yx s 0. For x s 0, the procedure will give: xxy q xyx q xyy q yxx q yxy q yyx s 0 and xzy q zxy q xyz q zyx q yxz q yzx s 0. We may also remark here that the consideration of unary operations makes no special difficulties. As operations, they only admit internal MODULES AND CROSSED MODULES 945 modulations, but in axioms, different occurences may admit different modulations. As a simple example, consider a unary operation u with the Ž.Ž.Ž. axiom ux12,x sux 1иux 2; apart from the internal modulations, ubaABAŽ .subua Ž. Ž. and uabAABŽ .suaub Ž. Ž. are also possible modulations. 2.4. The Case of Groups In the foregoing examples, the trivial modulations are the only ones admissible for the additive structure. What happens in the case of groups? It seems that here also one has to keep an unmodulated law of group, but introduce a second law admitting non-trivial modulations. Here is one way to do it. 1 Besides multiplication и,Ž.y and e introduce ) together with the axiom: Ž.Ža x)yx . sxy. From the point of view of the theory of groups nothing has changed, the axiom simply saying that x operates by inner automorphism. The theory ⌫ consisting in the usual axioms for groups together withŽ. a is equivalent to the theory of groups. AxiomŽ. a admits no other modulations than the trivial ones, because it also is trivial for multiplication. But form ⌫˜ by adding to ⌫ the two consequences: Ž.Žb xy .) z s x) Žy) z .Žaction . Ž.c x) Žyz .s Žx) yx .Ž)z .Žhomomorphy . . 1 We can keep for и,Ž.y , and e the trivial modulations, but for ) we will accept ␣ ,  , and the non-trivial Ž., ␣; ␣ . A modulated model of ⌫˜ appears thus as a pair A, B of groups together with an action )BA : B = A ª A satisfying the following modulations ofŽ. b and Ž. c : X X Ž.bb )BAa s b) BA Žb ) BAa . X X b)BAŽ.Žaa s b)BAa .Žи b)BAa .. For crossed ⌫˜-modules, we have in addition a homomorphism of groups D : A ª B satisfying X X D Ž.a )BAa s a) Aa and D Ž.b)BAa s b)BD Ž.a , 946 LAVENDHOMME AND LUCAS resulting from the two commutative diagrams D=A 6 A = AB=A ) *BA 6 6 A A and B=D 6 B = AB=B *BA *B 6 6 D 6 A B. Those remarks suffice to establish PROPOSITION 2. For the theory ⌫˜ defined abo¨e, the notion of a crossed ⌫-module coincides with the classical notion of a crossed module in groups. We may again observe in this example the dependence of the notion of ⌫-module on the form of the axioms of ⌫ and even on the language L in which the axioms of ⌫ are expressed. 3. SEMI-DIRECT PRODUCTS: INTERNAL CATEGORIES We assume in this paragraph that the algebraic theory ⌫ contains the theory of groups and that the only modulations admissible for the lan- guage of groups are the internal ones. The law of group is denoted additively. Let Ž.A, B, ⌽ be a modulated model of ⌫. We want to indicate cases, and there are plenty of them, where one can define on A = B a structure which turns it into a model of ⌫. The following terminology is useful. Let txŽ.,y be a term of the ŽŽ .Ž. language L of ⌫ with x s x1,..., xm and y s y1,..., yn . We say that tis Ž.Ž.x, ␣ , y,  -modulable if there exists a term txXŽ.,ysuch that ⌫ X proves t s t Ž.in the one-sorted language L and the modulation ␣ ,...,␣, ,...,;␣ x, y s ž/^`_ m ^`_n is admissible for tX. Note that it is possible for a non-x, y-modulable term txŽ.,y to be X equalŽ. modulo ⌫ to a x,y-modulable term txŽ.,y. MODULES AND CROSSED MODULES 947 Ž.Ž. For example, in the theory of rings, the term x11q y и x 22q y y yy 12 does not admit the modulation Ž.␣, ␣, , ; ␣ but is equal modulo ⌫ to Ž xx12qxy 12qyx 12which admits that modulation if multiplication admits all modulations. . If F is a k-ary operation of L, we say that F is crossable if the term FxŽ.Ž.Ž.Ž.qyyFy is x, ␣ , y,  -modulable. For a crossable k-ary F, let TxF Ž.,ybe a term such that in ⌫, Ž.Ž.Ž. FxqyyFysTxF ,yand such that it admits the modulation x, y Ž.henceforth denoted by . One can in this case interpret F in A = B, k FA=B: Ž.A = B ª A = B by letting FaA=BŽ.Ž.Ž.11,b,..., akk,b sŽ.< The term < < DEFINITION 2. Let ⌫ be an equational theory in L such that all operations of L are crossable and let Ž.A, B, ⌽ be a modulated model of ⌫. We say that Ž.A, B, ⌽ has a semi-direct product if A h B is an Ž.ordinary 1-sorted model of ⌫. Here are examples of semi-direct products. EXAMPLE 1. Let ⌫ be a theory of R-algebrasŽ not necessarily associa- tive or unitary. with polynomial axioms. For the structure of R-module, take only internal modulations, and for multiplication, admit all possible modulations. All operations are crossable: Ž.Ž.Ž.Ž.a11q b q a 22q b y b 12q b s a 12q a , raŽ.qbyrb s ra 948 LAVENDHOMME AND LUCAS Ž.modulated by projection , and Ž.Ž.a1q b 1и a 2q b 2y bb 12saa 12qab 12qba 12. The L-structure on A h B is therefore given by Ž.Ž.Ža11, b q a 22, b s a 1q a 21, b q b 2 . raŽ.Ž,bsra, rb . Ž.Ž.Ža1,b 1иa 2,b 2saa 12qab 12qba 12,bb 12.. To examine the axioms, one has to compute in A h B the interpretation of a general term PxŽ.1,..., xn , i.e., of a general polynomial Ž associativity and commutativity of the variables not being assumed. . Let us fix some notation. We recall from Section 2.3 that ⌬ iPxŽ.Ž1,..., xni, y sPx1,..., xiiqy,..., x n . yPxŽ.1,..., xin,..., x yPxŽ.1,..., yin,..., x . We denote by ⌬ PxŽ.,..., x ,..., x , y the expression obtained by aii,b1 ˆˆini Ž. substituting aiito x and b iito y in ⌬ iP. For example, if Px12,x s 23 xxx121Ž ., ⌬PxŽ. a2 x3b a11,b 2121s Ž. 3 3 qŽ.ab11Ž. x 21a q Ž.ab 11 Ž. x 21b 3 3 qŽ.ba11Ž. x 21a q Ž.ba 11 Ž. x 2b 1 2 3 qb121Ž.xa. Another notation.Foraterm QxŽ.1,..., xn,wedenoteby n SQxa, b Ž.1,..., xn the sum of the 2 terms obtained by replacing in X QxŽ.1,..., xniiieach x by a or by b ; and we denote by SQx a,bŽ.1,..., xn the sum of the same terms with the exception of QbŽ.1,...,bn . Ž. Finally, if a s a1,...,an ,wedenotebyaŽi. the sequence Ž.a1,...,aˆin,...,a and similarly exclude more indices. With these notations at hand, we can write the interpretation of PxŽ.in Ž.ŽŽ.Ž.. AhB:PaAhBB,bsQa,b,Pb X QaŽ.,b sSPxa,b Ž. n S ⌬Px qÝaŽi.Ž,bai. ii,bŽ.Ži. is1 S ⌬⌬Px qÝaŽi,j.Ž,bai,j. ii,ba jj,bŽ.Ži,j. i-j qиии ⌬ ⌬ иии ⌬ P. q a11, ba 22,bann,b MODULES AND CROSSED MODULES 949 The proof is by recursion and presents only notational difficulties. What is important and not unexpected is that each term of the sum is a modulation of P or of one of its derived polynomials. Denote by ⌫˜ the theory obtained by adding to ⌫ all formulas R s 0, where R is obtained from an axiom of ⌫ by the process of derivation. PROPOSITION 3Ž For the Theory ⌫ of R-Algebras with Polynomial Axioms. . Admitting all modulations for multiplication and only the internal modulation for the other operations, ifŽ. A, B, ⌽ is a modulated model of ⌫˜, then A h B is a model of ⌫ Ž.and ⌫˜ . Proof. The proof that A h B is an R-algebra is routine. Turning to specific polynomial axioms, let us consider an axiom PxŽ.s0of ⌫. Then Ž.ŽŽ.Ž.. Ž. PaAh BB,bsQa,b,Pb with Qa,b as described above. By the definition of ⌫˜, Ž.A, B, ⌽ satisfies not only the modulations of P s 0 but also the modulations of the derived axioms. Hence QaŽ.,b s0 and Ž.Ž. Ž . PaAh B ,bs0, 0 . Clearly A h B is a usual one-sorted model not only of ⌫ but also of ⌫˜˜since ⌫ and ⌫ have the sameŽ. one-sorted conse- quences. EXAMPLE 2. We mention a few improvements of the preceding result. If the language L has a constant c Žwhich has only the two internal modulations. , that constant is automatically crossable: the term to be Ž.;␣-modulated is simply c y c Žs 0 . . For example, in rings with unity, 1 is crossable. PROPOSITION 4. Proposition 3 holds for the theory ⌫ of unitary R-algebras with polynomial axioms when the unity admits the two internal modulations. Proof. The general construction givesŽ. 0, 1 as unity for A h B. It is also possible to adjoin to L a family of additive unary operations. As an example of a ternary law, consider the case of R-algebras where multiplication as well as addition only admit the internal modulations, but ŽŽ .Ž Ž. where the associator Ax123,x,x obeying the axiom Ax123,x,x s Ž.Ž.. xxx123yxx 123 x admits all modulations. The associator is then cross- able, for AxŽ.112233qy,xqy,xqy yAyŽ 123,y,y. sAxŽ.123,x,x qAxŽ.Ž.Ž.123,x,y qAx 123,y,x qAy 123,x,x qAxŽ.Ž.Ž.123,y,y qAy 123,x,y qAy 12,y,x 3. EXAMPLE 3. An interesting example with non-additive unary opera- tions is given by Grothendieck’s -ringsŽ cf. e.g.wx 1. . 950 LAVENDHOMME AND LUCAS A -ring is a commutative unitary ring A together with a denumerable Žn . family of unary operations : A ª A ng ގ verifying 0 Ž.1 Ž.xs1 1 Ž.2 Ž.xsx Ž.nŽ.ÝnnykŽ.kŽ. 3 xqys ks0 x y. It follows that for all n, ny1 nnnkk Žxqy .y Ž.ysÝy Ž.x Ž.y, ks0 so that each nis crossableŽ provided multiplication admits all possible modulations and each n admits the two internal modulations. . The interpretation of n in FA h B is given by ny1 nnykk n ABBŽ.a,bÝ Ž.Ž.Ž.ab,b. h sž/ ks0 Moreover, PROPOSITION 5. Proposition 3 holds for theories ⌫ containing the axioms of -ringsŽ. and containing otherwise only usual polynomial identities . Proof. Axioms 1 and 2 are trivial. Let us verify Axiom 3, n nnXX ykkXX AhBAŽ.Ž.Ž.aqa,bqbsÝhBAa,bhBa,b, ks0 by observing that both members have as a first component i j X r s X Ý AAŽ.a Ža . B Ž.b B Žb .. iqjqrqssn rqs-n EXAMPLE 4. As a last example, we consider the case ofŽ not necessarily commutative. groups with the structure of Section 2.4. Multiplication is crossable because y1 y1 Ž.Ž.Ž.ab11 ab 22 bb 12sabab 1121 sa112иŽ.b)a MODULES AND CROSSED MODULES 951 which admits the required modulation. So is the law ), because y1 Ž.Ž.Ž.Ž.ab11) ab 22 b 1)b 2 y1 y1 y1 y1 y1 sabab1121bbb 121a 1bb 12b 1 y1 y1 sab11Ž.)abbb 2Ž.Ž. 121)a 1 , and the lawŽ.y1 , because 1 Ž.ab y1 и Žby1 .y 1 1 sbyayb 1 1 sŽ.by )ay. The corresponding structure on A h B is given by Ž.Ž.a11, ba 22,bsŽ.ab 11 Ž)a 2 .,bb 12 y1 1 1 1 Ž.a,bsŽ. Ž.by)ay,by eAhBABsŽ.e,e y1 y1 Ž.Ž.a11,b)a 22,bsŽ.ab 11 Ž)abbb 2 .Ž.Ž. 121)a 1,b 1)b 2. One easily computes that y1 Ž.Ž.Ž.Ž.Ž.a11, b ) a 22, b s ab 1122 a,ba 11,b so that the structure we find on A h B reduces to the well known semi-direct product of groups. Leaving the examples, we suppose now that we have a crossed ⌫-module Ž.Ž.A,B,⌽,Dwhere A, B, ⌽ is a modulated model of ⌫ of the type studied in the examples, i.e., possessing a semi-direct product. We can then give a unified proof of the fact that A h B is an internal category Ž.Ѩ01,Ѩ,,inside the category of ⌫-algebras. Here are some details. PROPOSITION 6. The mapping Ѩ1 A h B ª B D a q bۍ Ž.Ž.a, b is a homomorphism of ⌫-algebras. 952 LAVENDHOMME AND LUCAS Proof. Let F be a k-ary operation of L. Let TF be a term admitting the modulation Ž.␣ ,...,␣, ,...,;␣ and such that in ⌫, s ^`_ k ^`_k TxFŽ.,ysFx Žqy .yFyŽ..2Ž. We then have FaAhBFBŽ.,bsŽ.< Ѩ1Ž.FaAhBFBŽ.,bsDŽ.< By the definition of crossed module, DŽ.< ByŽ. 2 and the fact that B is a ⌫-algebra, we have < Ѩ1Ž.FaAhBBŽ.,bsFŽ.D Ž.aqb sFBŽ.Ѩ11Ž.a,b 1,...,Ѩ 1 Žakk,b . and Ѩ1 is a homomorphism. Besides Ѩ102we can take as Ѩ the projection : A h B ª B which by construction is a homomorphism of ⌫-algebras. For composable maps and composition, we note that the pullback p1 6 PAhB p2 Ѩ1 6 6 Ѩ06 AhBB in the category of ⌫-algebra is given by the pullback in Sets: XX X PsÄ4Ž.Ž.a,b,a,bb X X X Ž.Ž.Ž.a, b , a , b s Ža q a, b .. MODULES AND CROSSED MODULES 953 Finally, for , we consider . Ž.0, bۍ : B ª A h B : b It is easy to check that those definitions satisfy the conditions for an internal category. PROPOSITION 7. The diagram p10Ѩ ªª PªA)B¤B p21Ѩ ªª is an internal category in the category of ⌫-algebras. 4. NON-ABELIAN COCYCLES In this section we want to connect the present approach with that ofwx 4 concerning non-Abelian cohomology without however fully rewriting what is presented there. We concentrate on the notion of a two-cocycle. Let ⌫ be an algebraic theory with semi-direct product. Let X be a ⌫-algebra and Ž.A, B, ⌽, D a crossed ⌫-module. A two-cocycle from X to Ž.A, B, ⌽, D is an ordered pair Ž.Q, ⌬ where Q : X ª B is a map and ⌬ associates to any Ž primitive . n or composite n-ary operation F a map ⌬ F : X ª A h B such that n Ѩ 0 (⌬ FXs Q( F , Ѩ1(⌬ FBs F (Q , and which satisfies compatibility con- ditions with projection and composition: Ž. Ž . Ž . Ž Ž .. aifPx1,..., xnipsx , ⌬ x1,..., xnis 0, Qx ; Ž. Ž .Ž . bifFsG(H1,...,Hmiwith Gm-ary and each Hn-ary , ⌬⌬Ž.H,...,H ,G ⌬ ,...,⌬ . FGsŽ.(1XmXAhBH(Ž.1 Hm An alternative more classical presentation is to define ⌬ F for primitive operations and to postulate aŽ. ‘‘cocycle’’ condition for each equational axiom of ⌫. One of the results ofwx 4 links two-cocycles with extensions. We do not repeat the proof here but want to note that a general notion of crossed product relative to a two-cocycle is implicit inwx 4 . In the frame of the present workŽ assuming on ⌫ the hypotheses of the first paragraph of Section 3.Ž. , the definition would run as follows. If Q, ⌬ is a two-cocycle 954 LAVENDHOMME AND LUCAS with coefficients in Ž.A, B, ⌽, D , we define a structure A hŽQ, ⌬. X on A = X, interpreting F by FaŽ.,xFaŽ.,Qx ⌬ Ž.Ž.x , Fx. AhŽQ,⌬.XsŽ1Ž.AhBFXq The meaning of that identity should be clear: in abelian cohomology, the homomorphism defect of Q is measured by the difference FQxŽ.yQF Ž. x ; it appears here as ⌬ F Ž.x and a general categorical approach already suggested inwx 4 should view it simply as an arrow. From the computations made inwx 4 , we may state the following result: PROPOSITION 8. The structure A hŽQ,⌬. Xisa⌫-algebra. We devote the rest of the present section to verify that our non-standard presentation of groups given in Section 2.4 does not give rise to exotic cocycle conditions. Recall that we have introduced in the theory ⌫ of groups a binary operation ) satisfying the axiom Ž.x) yxsxy. In general, if Ž.Q, ⌬ is a two-cocycle of X with coefficients in the crossed moduleŽ.Ž. in groups A, B, ⌽, D , we may note ⌬ FFXŽ.x s ŽŽ.cx Ž.,QF Ž. x . Of course, the two-cocycle is characterized for groups by Q and the cF for FgÄ4e,y,m,)Ž.efor unit, y for symmetry, and m for multiplication . From the structure of internal category, the pair Ž.a, b g A h B may be viewed as an arrow a b ª D Ž.a и b. In this way, the components of a two-cocycle are c 6 QeŽ. e e, Ž. cx6 QxŽ.y1 y Qx Ž.y1, Ž. cx,y6 QxyŽ. m QxŽ.иQy Ž., c#Ž.x,y6 QxŽ)yQx . Ž.)Qy Ž. and we must give the cocycle conditions in terms of the axioms of the theory. MODULES AND CROSSED MODULES 955 Ž.a The associativity of multiplication gives the condition of commuta- tivity of the diagram QxyzŽ.Ž.Ž.sQxyz Ž. cxymŽ.,z cxmŽ.,yz 6 6 QxyQzŽ.Ž. QxQyz Ž.Ž. cxmŽ.,y 6 Ž. Ž . Qx)cym,z 6 Ž.QxQyŽ. Ž. Qz Ž.sQx Ž. Ž. QyQz Ž. Ž. Ž. Ž . Remark that in the expression Qx)cym ,z,)denotes the action of B on A and that it is indeed an arrow of the internal category, since y1 D Ž.QxŽ.)cymm Ž,z .sQxc Ž. Ž y,zQx . Ž. . In this way, we obtain the classical cocycle condition Ž.QxŽ.)cymm Ž,z .иcx Ž,yz .s cx mm Ž,y .иcxy Ž,z . Ž.3 Ž.b The unity axioms give the commutativity of QexŽ.sQx Ž. cemŽ.,x 6 QeŽ.иQx Ž. e ce 6 eиQxŽ.sQx Ž. which defines cemand gives a condition on c : y1 cemsceŽ.,x .4Ž. Similarly, the commutativity of QxŽ.иesQx Ž. cxmŽ.,e 6 QxŽ.иQe Ž. e Ž. 6 Qx)ce QxŽ.иesQx Ž. gives y1 y1 cems QxŽ.)cx Ž,e . .5Ž. 956 LAVENDHOMME AND LUCAS 1 Ž.c The first action of inverses xxy segives the commutativity of 1 QxŽ.Ž.yxsQe y1 cxmŽ.,x 6 Qxy1иQx Ž.Ž.ce 6 Ž. 6 cxy y1 QxŽ.иQx Ž.se which may be viewed as a definition of cy in terms of cmeand c : 1 и y1y cxyŽ.scemcx Ž,x . .6Ž. 1 Similarly, the second axiom of inverses xxy s e will give y1 QxŽ.)cxy Ž.scxm Ž,x ., but this may be derived directly fromŽ.Ž. 3 ᎐ 6. Ž.d The axiom concerning ) gives the commutativity of QxŽ..Ž.)yиxsQxy Ž. cxm )y,x 6 QxŽ.Ž.yиQx ) cxmŽ.,y Ž. 6 c#x,y 6 Ž.QxŽ.)Qy Ž.иQx Ž.sQx Ž.иQy Ž. which again may be viewed as a definition of c# in terms of cm: y1 c#Ž.x, y s cxmm Ž.Ž,yиcx)y,x .. Summarizing the preceding observations, we obtain: ROPOSITION Ž. P9. The notion of two-cocycle Q, cme, c , cy, c# reduces to the classical notion of two-cocycleŽ. Q, cme, c . Note that in the classical theory, it is customary to normalize cocycles by Ž. taking Qe seand ce s e. MODULES AND CROSSED MODULES 957 5. DERIVATION OBJECTS As has been observed by Porterwx 9 , a crossed moduleŽ in his case, in commutative algebras.Ž. generalizes the notion of ideal: A, B, ⌽, D is viewed as an external version of an ideal of B. Porter also observes that crossed modules simultaneously generalize the notion of B-moduleŽ com- pare this with our Example 2.1.3 of Section 2. . From that point of view, the interesting functor is that which associates to a crossed module Ž.A, B, ⌽, D the algebra B. The category of crossed modules over B is then a general- ization of the set of ideals of B and of the category of B-modules. The other point of view, fixing A, is also worth being considered. Let ⌫ be an algebraic theory, M a fixed modulation and A a ⌫-algebra. We denote by CrŽ.A the category whose objects are crossed ⌫-modules Ž.A,B,⌽,Dand morphisms are morphisms of crossed ⌫-modules of the formŽ. 1A , g . An important concept is embodied in the following definition: DEFINITION 3. The algebra A has a ⌫-algebra of derivations if the category CrŽ.A has a terminal object. That terminal object is denoted by Ž.A,DA, , and is also called the ⌫-algebra of derivations of A. Here are some examples. 1. We begin with Lie R-algebras, from which we borrowed the name of the concept. Let A be a Lie R-algebra. Let Der A be the Lie R-algebra of deriva- tions of A in the usual senseŽ square brackets designating the commuta- tor. . Let ad : A ª Der A : a ¬ wxa, y be the adjoint representation. Let designate the modulated structure defined by the Lie R-algebras A and Der A and the external structure given by evaluation: : Der A = A ª A with Ž.D, a s Da Ž.. It is an easy classical result that ŽA, Der A, ,ad . is a crossed module in Lie R-algebras. PROPOSITION 10. Any Lie R-algebra A has in the sense of Definition 3 a Lie R-algebra of deri¨ations, which is gi¨en byŽ. A, Der A, ,ad. Proof. Let Ž.A, B, ⌽, D be a crossed module in Lie algebras. We have to check that there is a unique morphism Ž.Ž1,A : A,B,⌽,D .Žª A, Der A, ,ad . . 958 LAVENDHOMME AND LUCAS The external homomorphism condition gives the commutative diagram *BA 6 B = AA =1A 1A 6 6 ⑀6 Der A = A A, i.e., Ž.Ž.basb)a, thus fixing . It remains to show that is indeed a morphism of crossed modules: Ž.i Žba .Ž.is linear in a and Ž.baŽ.wxw12,asb)a12,a xwqa 1,b)a 2 x sŽ.Ž.ba12,aqa 1,Ž.Ž.ba 2, so that Ž.b is in Der A. Ž.ii is a homomorphism of Lie R-algebras: linearity is immediate, and from wxb12,b)asb 1) wxb 2)ayb 2) wxb 1)a, Ž .Ž.Ž. wxb12,b swb 1,b 2xfollows. Ž.iii The external homomorphism clause is the definition of itself. Ž.iv The commutativity of 6B D A 6 ad 6 Der A Ž Ž ..Ž . Ž . Ž.Ž. follows from D aa12sDa 1)a 2swxa 12,asad aa 12. 2. The case of groups is analogous. Let G be a group and Aut G be its group of automorphisms. Let : G ª Aut G be the inner representa- Ž. y1 tion of G which to g g G associates g h s ghg . Designate by the modulated structure of which the external component : Aut G = G ª G is given by the evaluation. It is trivial that Ž.G, Aut G, , is a crossed module in groups. PROPOSITION 11. G has in the sense of Definition 3 a group of deri¨ations which is gi¨en byŽ. G, Aut G, , . MODULES AND CROSSED MODULES 959 Proof. It is easy to check that the unique morphism Ž.Ž1,G :G,B,⌽,D .ŽªG, Aut G, , . is given by Ž.Ž.ghsg)h. 3. We study the case ofŽ. not necessarily unitary nor commutative associati¨eR-algebras. Let A be an associative R-algebra. We recall Mac Lane’s construction of the R-algebra BimŽ.A of bimultipliers of A Žcf.wx 6. . An element of BimŽ.A is a pair Ž␥, ␦ .of R-linear mappings from A to A such that X X ␥ Ž.aa s ␥ Ž.a и a Ž.1 X X ␦Ž.aa s a и ␦ Ž.a Ž.2 and X X a и ␥ Ž.a s ␦ Ž.a и a Ž.3 BimŽ.A has an obvious R-module structure and a product X X X X Ž.Ž.Ž␥ , ␦ и ␥ , ␦ s ␥ (␥ , ␦ (␦ ., the value of which is still in BimŽ.A . 2 Suppose that AnnŽ.A s Ž.0 or that A s A. Then BimŽ.A acts on A by 1: BimŽ.A = A ª A : Ž. Ž␥ , ␦ ., a ¬ ␥ Ž.a 2:A=BimŽ.A ª A : Ž.a, Ž␥ , ␦ .¬ ␦ Ž.a Ž. Ž. Ž . Ž. and there is a : A ª Bim A defined by a s ␥a, ␦a with ␥a x s ax Ž. and ␦a x s xa. PROPOSITION 12. Let A be an associati¨eR-algebra such that AnnŽ.A s 2 Ž.0or A s A. ThenŽ. A, Bim A, , is a crossed module in associati¨e R-algebras which is also the associati¨eR-algebra of deri¨ations of A in the sense of Definition 3. Proof. Ž.␣ We first show that ŽA, Bim A, , .is a crossed module. Ž.i Let us check that X Y Y X Y Y Ž.␥ , ␦ и12Ž.Ž.a и Ž␥ , ␦ .s Ž.␥ , ␦ и1a и 2 Ž␥ , ␦ . 960 LAVENDHOMME AND LUCAS Ž.with the obvious notations for the actions 12and . We do it here under the hypothesis that AnnŽ.A s Ž.0 . Let x g A and multiply both members on the left by x: X Y Y x и ŽŽ.␥ , ␦ и12Ž.a и Ž␥ , ␦ . Y X sxиŽ.␥␦Ž.Ž.a YX s␦Ž.xи␦ Ža .Ž.by Ž. 3 Y X s␦Ž.Ž.␦Ž.xиa byŽ. 2 Y X s␦Ž.Ž.xи␥Ž.aby Ž. 3 YX sxи␦Ž.Ž.␥Ž.aby Ž. 2 X YY sxиŽŽ.Ž.␥,␦и12aи Ž␥,␦ .. Since this holds for any x, the hypothesis AnnŽ.A s Ž.0 furnishes the conclusion. The other modulations of associativity are trivial. Ž.ii We verify that ␥Ž.Ž.,␦и1as Ž.Ž.␥,␦иa as follows: ␥Ž.Ž.,␦и1asŽ␥Ž. Ž.a sŽ.␥␥Ža.,␦␥Ža. Ž.Ž.Ž␥,␦иas␥(␥aa,␦(␦ . and ␥␥Ža.Ž.x s ␥ Ž.a и x s␥Ž.ax Ž.by Ž. 1 s␥␥Ž.aŽ.x ␦␥Ža.Ž.xsxи␥ Ž.a s␦Ž.xиa Ž.byŽ. 3 s␦␦aŽ.Ž.x . Similarly Ž.a и2Ž.␥ , ␦ s Ž.Ž.a и ␥ , ␦ and ŽŽ..A, Bim A , , is indeed a crossed module. MODULES AND CROSSED MODULES 961 Ž.Let ŽA, B, ⌽, D .be another crossed module. We have to define Ž.Ž1,A : A,B,⌽,D .ªŽ.A, Bim Ž.A , , . Ž. Ž . Letting b s bb, D one must have the following commutativities: ؒ 1 6 B=AA =1A ؒ 6 16 BimŽ.A = A A and ؒ 2 6 B = AA 1A= ؒ 6 26 A=BimŽ.A A, Ž. Ž. i.e., b a s b и1 a and D b a s a и2 b. This shows how to define and proves its uniqueness. The proof that 6B D A 6 6 BimŽ.A commutes is trivial. The proposition is thus proved under the hypothesis AnnŽ.A s Ž.0 but 2 the proof for A s A is similar. 4. Some reinforcements of the theory of associati¨eR-algebras are also 2 easily dealt with. We still assume AnnŽ.A s Ž.0orAsA. Ž. Ž. Ž . Ž . ␣If A is unitary, then Bim A has a unity ␥11, ␦ s 1AA , 1 and preserves it. Ž.If A is a commutative and Ž␥, ␦ .g Bim Ž.A , then ␥ s ␦. This is because for every x in A: x и ␦ Ž.a s ␦ Ž.a и x s a и ␥ Ž.x s ␥ Ž.x и a s␥Ž.xa s ␥ Ž.ax s ␥ Ž.a и x s x и ␥ Ž.a . 962 LAVENDHOMME AND LUCAS In this case, BimŽ.A may be identified with the R-algebra MA Ž.of multipliers of A. Recall that a multiplier of A is a linear mapping : A ª A such that Ž.Ž.aa12s aa 1 2. In fact, MAŽ.is commutative: for any x g A, X X X x и Ž.Ž.a s Žxa .s Ž.Ž.x a X X sŽ.a Ž.xsxиŽ. Ž.a . X X A mapping : A ª MAŽ.is defined by Ž.Žaa.saa and one has: PROPOSITION 13. If A is a commutati¨e associati¨eR-algebra with 2 AnnŽ.A s Ž.0 or A s A, thenŽŽ.. A, MA,, is a crossed module in commutati¨e associati¨eR-algebras which is also the commutati¨e associati¨e R-algebra of deri¨ations of A in the sense of Definition 3. 5. In some cases, it is also interesting to look at the category M⌫Ž.A of modulated ⌫-models Ž.A, B, ⌽ . We indicate some degenerate cases where M⌫Ž.A has a terminal object. If ⌫ is the theory of monoids and if Ž.Ž., ␣; ␣ and , ;  are the only admitted modulations, then the terminal object of M⌫Ž.A is ŽŽ..Ž.A, End A , where End A is the monoid of mappings of A in A and the external component of is evaluation. 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