Consider now a theory G in L, i.e., a set of equations of the form t 1 s t 2 Žwe consider only equational theories.. We say that the modulated interpretation Ž A, B, F . satisfies or is a model of t 1 s t 2 iff for every x and every modulation m admissible for t 1Ž x . and t 2 Ž x ., t1Ž x .

m

s t2 Ž x .

m.

ŽComposition with projections shows that this does not really depend on x .. We say that Ž A, B, F . is a model of G Žor a modulated G-model . if Ž A, B, F . is a model of every element of G. 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 a F b and extending it by X Ž j 1 , . . . , j n ; j nq1 . F Ž j 1X , . . . , j nX ; j nq1 .

iff

j 1 F j 1X , . . . , j n F j nX

and

X j nq1 F j nq1 .

MODULES AND CROSSED MODULES

939

For that order, there is a smallest n-ary modulation, namely ma , and a greatest n-ary modulation, mb . Recall that ma is admissible for every F. DEFINITION 1. A crossed G-module Ž A, B, F, D . is determined by a modulated interpretation Ž A, B, F . of L which is a modulated model of G and a mapping

D : AªB such that for every n, every F g Op n , and every m , mX g MF satisfying X . the following diagram m s Ž j 1 , . . . , j n ; j nq1 . F mX s Ž j 1X , . . . , j nX ; j nq1 commutes: < F
D ˜nq1 X < j nq < 1 .

6

< F < m9

6

D˜ 1 = ??? = D˜n

6

< j 1X < = ??? = < j nX <

< j nq1 <

6

< j 1 < = ??? = < j n <

ŽIn the diagram, the convention is that for every i, 1 F i F n q 1, D ˜i is the identity if j i s j iX and D ˜i s D if j i s a and j iX s b .. Note that if ma g MF , one of the diagrams of the definition expresses that D : A ª B is homomorphic for F Žtake m s ma and mX s mb .. A reason why we do not include ma 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 difficulties; it is an ordered pair

Ž f , g . : Ž A, B, F . ª Ž AX , BX , FX . 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 Op n , for every m g MF : < F
wnq1

< j nq 1 <9.

6

< F < m9

6

w 1 = ??? = w n

6

< j 1 <9 = ??? = < j n <9

< j nq1 <

6

< j 1 < = ??? = < j n <

ŽIn the diagram, the convention is that < < is associated with Ž A, B, F ., <
940

LAVENDHOMME AND LUCAS

For crossed G-modules Ž A, B, F, D . and Ž AX , BX , FX , DX . we add the commutativity of the square 6

D

A

A9 g

f

6

6

D9

6

A9 B9. Those definitions clearly determine the category of modulated interpretations and the category of crossed G-modules. 2. EXAMPLES 2.1. Some Tri¨ ial Examples 2.1.1. Let G be the theory of monoids with the two operation symbols ? and e. If one admits the modulations Ž b , a ; a . and Ž b , b ; b . for ? and Ž; b . for e, a modulated model of G reduces to an action of a monoid B on a set A. Indeed, in this case, the modulations of the axioms x ? Ž y ? z. s Ž x ? y. ? z and

x?esxse?x give for Ž b , b ; b . the associativity and neutral element conditions for B and for Ž b , a ; a . the conditions ;a g A ; b1 , b 2 g B and

Ž b1 ? b 2 . ) a s b1 ? Ž b 2 ) a . eB ) a s a

Žwhere ) denotes the action B = A ª A.. Note that mb is the only admissible modulation to consider for x s e ? x. 2.1.2. Let G be an equational theory. If one admits only the internal modulations ma and mb for every operation, a modulated model of G reduces to an ordered pair of G-algebras and a crossed G-module is simply a homomorphism D : A ª B of G-algebras. 2.1.3. Let G be the theory of rings Žnot necessarily commutative or unitary., in the language q, y, 0, ? with the usual axioms. Let us admit for q, y, and 0 the two internal modulations ma and mb and for ? the modulations Ž b , a ; a . and Ž b , b ; b .. A modulated model Ž A, B, F . of G will reduce in this case to a ring B acting on an abelian group A with ; b g B, ;a1 , a2 g A

b) Ž a1 q a2 . s b) a1 q b) a2 ,

; b1 , b 2 g B, ;a g A

Ž b1 q b 2 . ) a s b1 ) a q b 2 ) a

MODULES AND CROSSED MODULES

941

Žfollowing from the distributivity axioms. and ; b1 , b 2 g B, ;a g A

b1 ) Ž b 2 ) a . s Ž b1 ? b 2 . ) a

Žfollowing from the associativity axiom.. This means that A is a left B-module. If moreover we admit the modulation Ž a , b ; a . for ?, then A becomes a B-B-bimodule, the key condition

Ž b1 ) a. ) b 2 s b1 ) Ž a) b 2 . coming as a modulation of associativity. A crossed G-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: ; b g B ;a g A

D Ž b) a . s b ? 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 G containing the theory of R-algebras Žassociativity not necessarily included. and want to determine the modulated models of G and the crossed G-modules. For what concerns the language of R-modules Žq, y, 0 and unary laws., let us admit only the internal modulations ma and mb . If for multiplication we admit mb and Ž b , a ; a ., a modulated model of the axioms of R-modules will consist in an R-algebra B, an R-module A, and an R-bilinear action )1 : B = A ª A. If we also admit the modulation Ž a , b ; a ., we get an R-bilinear action )2 : A = B ª A. In this subsection, we suppose that in addition to the axioms of R-algebras, G contains only multilinear axioms: associativity, commutativity, anticommutativity, and Jacobi’s identity are typical examples. To be specific, 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. ŽIn characteristic 2, one has to add w x, x x s 0, which is not linear and will be dealt with later..

942

LAVENDHOMME AND LUCAS

For w , x, we admit mb and Ž b , a ; a .. 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

w b1 , b 2 x ) a s b1 ) Ž b 2 ) a . y b 2 ) Ž b1 ) a . , coming from a modulation of Jacobi’s identity. If for w , x we also admit Ž a , b ; a ., 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 for w , x we also admit the internal modulation ma , then A becomes a Lie R-algebra operating by derivation, the typical property b) w a1 , a2 x s w b) a1 , a2 x q w a1 , b) a2 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, F, D . where Ž1. A and B are Lie R-algebras; Ž2. F : B = A ª A is bilinear; Ž3. F is a Lie action in the sense that

w b1 , b 2 x ) a s b1 ) Ž b 2 ) a . y b 2 ) Ž b1 ) a . ; Ž4. F operates by derivation; Ž5. D is a homomorphism of Lie R-algebras; Ž6. D is B-linear,

D Ž b) a . s b, D Ž a . ; Ž7. w a1 , a2 x s D Ž a1 .) a2 . PROPOSITION 1 ŽFor the Theory G of Lie R-Algebras.. Admitting for w , x the modulations ma , mb , and Ž b , a ; a ., the notion of a crossed Gmodule defined in Section 1 coincides with the classical notion of a Lie crossed module. Proof. Points Ž1. to Ž4. express what it is to be a modulated model of the theory G of Lie R-algebras. A crossed R-module incorporates a mapping D : A ª B which is R-linear and such that Ž5.;

Ži. for ma F mb , the compatibility condition corresponds to condition

MODULES AND CROSSED MODULES

943

Žii. for Ž b , a ; a . F mb , the compatibility condition corresponds to condition Ž6.; Žiii. for ma F Ž b , a ; a ., the compatibility condition corresponds to condition Ž7.. 2.2.2. Associati¨ e R-algebras have associativity as a specific axiom. For multiplication we admit the internal modulations as well as the external modulations Ž b , a ; a . and Ž a , b ; a . Žthey are the only admissible modulations 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 b)1 Ž bX )1 a . s Ž bbX . )1 a X

Ž for Ž b , b , a ; a . . Ž 1 . , X

a)2 Ž bb . s Ž a)2 b . )2 b

Ž for Ž a , b , b ; a . . ,

b)1 Ž a)2 bX . s Ž b)1 a . )2 bX

Ž for Ž b , a , b ; a . . ,

X

X

Ž for Ž a , b , a ; a . . Ž 2. ,

X

X

a Ž b)1 a . s Ž a)2 b . a

Ž for Ž a , b , a ; a . . ,

a Ž aX )2 b . s Ž aaX . )2 b

Ž for Ž a , a , b ; a . . .

b)1 Ž aa . s Ž b)1 a . a

If commutativity is considered, one gets a)2 b s b)1 a and the associativity conditions may be reduced to Ž1. and Ž2.. This determines the notion of a modulated model of the theory G of associative R-algebras. To obtain a crossed G-module, we need moreover a homomorphism D : A ª B satisfying

D Ž b)1 a . s bD Ž a . and

D Ž a)2 b . s D Ž a . b. 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 axiomatized with no restriction on characteristic and w x, x x s 0. Slightly abstracting, this is essentially the following example.

944

LAVENDHOMME AND LUCAS

Consider the theory G of rings satisfying the identity x 2 s 0. The modulations of this axiom reduce to a2 s 0 and b 2 s 0 for a g A and b g B. However, substituting x q y to x in x 2 s 0, one easily derives in G that xy q yx s 0, a theorem which admits in general the following modulation: ;a g A ; b g B

a)2 b q b)1 a s 0

Ž 1.

The problem is that modulated models of G will not necessarily satisfy Ž1.. ŽAs an example, take for A the additive group ŽZrŽ2.w x xrŽ x 2 ., q. and for B the ring ŽZrŽ2., q, m. where m is the null multiplication; as the left action )1 of B on A, take 1)10 s 1)1 x 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 G are satisfied, but we do not have Ž1. as we compute 1)2 1 q 1)11 s 0 q x s x / 0..

˜ of rings It seems that the theory to consider is not G but the theory G 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 definitions with respect to the form of the axioms of G. For which axiomatizations of a theory G is it true that the modulated models of G satisfy the modulations of the equational theorems of G? 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 P Ž x 1 , . . . , x n . s 0 is an axiom, add all D i P s 0, where D i P is defined by Ž D i P . Ž x 1 , . . . , x n , yi . s P Ž x 1 , . . . , x iy1 , x i q yi , x iq1 , . . . , x n . y P Ž x 1 , . . . , x iy1 , x i , x iq1 , . . . , x n . y P Ž x 1 , . . . , x iy1 , yi , x iq1 , . . . , x n . and iterate the procedure. For x 2 s 0, the procedure will lead to xy q yx s 0. For x 3 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 axiom uŽ x 1 , x 2 . s uŽ x 1 . ? uŽ x 2 .; apart from the internal modulations, u A Ž ba. s u B Ž b . u A Ž a .

and

u A Ž ab . s u A Ž a . u B Ž b .

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. Besides multiplication ?, Ž .y1 and e introduce ) together with the axiom: Ža. Ž x ) y . x s xy. 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 G consisting in the usual axioms for groups together with Ža. is equivalent to the theory of groups. Axiom Ža. admits no other modulations than the ˜ by trivial ones, because it also is trivial for multiplication. But form G adding to G the two consequences: Žb. Ž xy .) z s x )Ž y ) z . Žaction. Žc. x )Ž yz . s Ž x ) y .Ž x ) z . Žhomomorphy.. We can keep for ?, Ž .y1 , and e the trivial modulations, but for ) we will ˜ accept ma , mb , and the non-trivial Ž b , a ; a .. A modulated model of G appears thus as a pair A, B of groups together with an action )B A : B = A ª A satisfying the following modulations of Žb. and Žc.:

Ž bbX . )B A a s b)B A Ž bX )B A a. b)B A Ž aaX . s Ž b)B A a . ? Ž b)B A aX . .

˜ For crossed G-modules, we have in addition a homomorphism of groups D : A ª B satisfying D Ž a . )B A aX s a)A aX and

D Ž b)B A a . s b)B D Ž a . ,

946

LAVENDHOMME AND LUCAS

resulting from the two commutative diagrams

)

B=A

6

D=A

A=A

*BA

6

6 A

A

and B=B

6

B= D

B=A

*B

6

D

6 A

6

*BA

B.

Those remarks suffice to establish

˜ defined abo¨ e, the notion of a crossed PROPOSITION 2. For the theory G G-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 G-module on the form of the axioms of G and even on the language L in which the axioms of G are expressed.

3. SEMI-DIRECT PRODUCTS: INTERNAL CATEGORIES We assume in this paragraph that the algebraic theory G contains the theory of groups and that the only modulations admissible for the language of groups are the internal ones. The law of group is denoted additively. Let Ž A, B, F . be a modulated model of G. 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 G. The following terminology is useful. Let t Ž x, y . be a term of the language L of G Žwith x s Ž x 1 , . . . , x m . and y s Ž y 1 , . . . , yn .. We say that t is Ž x, a ., Ž y, b .-modulable if there exists a term tX Ž x, y . such that G proves t s tX Žin the one-sorted language L. and the modulation

m x , y s ^a ,` ..., a _, b , . . . , b ; a

ž

m

^` _ n

/

is admissible for tX . Note that it is possible for a non-m x, y-modulable term t Ž x, y . to be equal Žmodulo G . to a m x, y-modulable term tX Ž x, y ..

MODULES AND CROSSED MODULES

947

For example, in the theory of rings, the term Ž x 1 q y 1 . ? Ž x 2 q y 2 . y y 1 y 2 does not admit the modulation Ž a , a , b , b ; a . but is equal modulo G to x 1 x 2 q x 1 y 2 q y 1 x 2 which admits that modulation Žif multiplication admits all modulations.. If F is a k-ary operation of L, we say that F is crossable if the term F Ž x q y . y F Ž y . is Ž x, a ., Ž y, b .-modulable. For a crossable k-ary F, let TF Ž x, y . be a term such that in G, F Ž x q y . y F Ž y . s TF Ž x, y . and such that it admits the modulation m x, y Žhenceforth denoted by m .. One can in this case interpret F in A = B, k

FA= B : Ž A = B . ª A = B by letting FA=B Ž Ž a1 , b1 . , . . . , Ž a k , bk . . s Ž < TF < m Ž a, b . , FB Ž b . . . The term < TF < m may be looked upon as a kind of Taylor term corresponding to a ‘‘point’’ a and an ‘‘increment’’ Žor ‘‘action’’. b. It is tempting to write < TF < m Ž a, b . s F Ž a q b . y FB Ž b . but F Ž a q b . does not make sense if A / B. On the other hand, if B s A, the interpretation FA= A of F in A = A turns q: A = A ª A into a homomorphism for F. If all operations of the language L are crossable, one obtains in this way an interpretation of L in A = B, which we denote by A h B. DEFINITION 2. Let G be an equational theory in L such that all operations of L are crossable and let Ž A, B, F . be a modulated model of G. We say that Ž A, B, F . has a semi-direct product if A h B is an Žordinary 1-sorted. model of G. Here are examples of semi-direct products. EXAMPLE 1. Let G be a theory of R-algebras Žnot necessarily associative 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 a1 q b1 . q Ž a2 q b2 . . y Ž b1 q b2 . s a1 q a2 , r Ž a q b . y rb s ra

948

LAVENDHOMME AND LUCAS

Žmodulated by projection., and

Ž a1 q b1 . ? Ž a2 q b 2 . y b1 b 2 s a1 a2 q a1 b 2 q b1 a2 . The L-structure on A h B is therefore given by

Ž a1 , b1 . q Ž a2 , b 2 . s Ž a1 q a2 , b1 q b 2 . r Ž a, b . s Ž ra, rb . Ž a1 , b1 . ? Ž a2 , b 2 . s Ž a1 a2 q a1 b 2 q b1 a2 , b1 b 2 . . To examine the axioms, one has to compute in A h B the interpretation of a general term P Ž x 1 , . . . , x n ., i.e., of a general polynomial Žassociativity and commutativity of the variables not being assumed.. Let us fix some notation. We recall from Section 2.3 that D i P Ž x 1 , . . . , x n , yi . s P Ž x 1 , . . . , x i q yi , . . . , x n . y P Ž x1 , . . . , x i , . . . , x n . y P Ž x 1 , . . . , yi , . . . , x n . . We denote by D a i , b i P Ž x 1 , . . . , ˆ xi, . . . , xn, ˆ yi . the expression obtained by substituting a i to x i and bi to yi in D i P. For example, if P Ž x 1 , x 2 . s x 12 Ž x 23 x 1 ., D a1 , b 1 P Ž x 2 . s a12 Ž x 23 b1 . q Ž a1 b1 . Ž x 23 a1 . q Ž a1 b1 . Ž x 23 b1 . q Ž b1 a1 . Ž x 23 a1 . q Ž b1 a1 . Ž x 23 b1 . q b12 Ž x 23 a1 . . Another notation. For a term Q Ž x 1 , . . . , x n ., we denote by S a, b QŽ x 1 , . . . , x n . the sum of the 2 n terms obtained by replacing in QŽ x 1 , . . . , x n . each x i by a i or by bi ; and we denote by SXa, b QŽ x 1 , . . . , x n . the sum of the same terms with the exception of QŽ b1 , . . . , bn .. Finally, if a s Ž a1 , . . . , a n ., we denote by aŽ i. the sequence Ž a1 , . . . , a ˆi , . . . , an . and similarly exclude more indices. With these notations at hand, we can write the interpretation of P Ž x . in A h B : PA h B Ž a, b . s Ž QŽ a, b ., PB Ž b .. Q Ž a, b . s SXa, b P Ž x . n

q Ý S aŽ i. , b Ž i. D a i , b i P Ž x Ž i. . is1

q Ý S aŽ i , j. , b Ž i , j. D a i , b i D a j , b j P Ž x Ž i , j. . i-j

q ??? q D a1 , b 1 D a 2 , b 2 ??? D a n , b nP.

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 ˜ the theory of P or of one of its derived polynomials. Denote by G obtained by adding to G all formulas R s 0, where R is obtained from an axiom of G by the process of derivation. PROPOSITION 3 ŽFor the Theory G of R-Algebras with Polynomial Axioms.. Admitting all modulations for multiplication and only the internal ˜ modulation for the other operations, if Ž A, B, F . is a modulated model of G, ˜ .. then A h B is a model of G Ž and G Proof. The proof that A h B is an R-algebra is routine. Turning to specific polynomial axioms, let us consider an axiom P Ž x . s 0 of G. Then PA h B Ž a, b . s Ž QŽ a, b ., PB Ž b .. with QŽ a, b . as described above. By the ˜ Ž A, B, F . satisfies not only the modulations of P s 0 but definition of G, also the modulations of the derived axioms. Hence QŽ a, b . s 0 and PA h B Ž a, b . s Ž0, 0.. Clearly A h B is a Žusual one-sorted. model not only ˜ since G and G ˜ have the same Žone-sorted. conseof G but also of G 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; a .-modulated is simply c y cŽs 0.. For example, in rings with unity, 1 is crossable. PROPOSITION 4. Proposition 3 holds for the theory G of unitary R-algebras with polynomial axioms when the unity admits the two internal modulations. Proof. The general construction gives Ž0, 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 where the associator Ž AŽ x 1 , x 2 , x 3 . Žobeying the axiom AŽ x 1 , x 2 , x 3 . s x 1Ž x 2 x 3 . y Ž x 1 x 2 . x 3 . admits all modulations. The associator is then crossable, for AŽ x1 q y1 , x 2 q y 2 , x 3 q y 3 . y AŽ y1 , y 2 , y 3 . s AŽ x1 , x 2 , x 3 . qA Ž x 1 , x 2 , y 3 . q A Ž x 1 , y 2 , x 3 . q A Ž y 1 , x 2 , x 3 . qA Ž x 1 , y 2 , y 3 . q A Ž y 1 , x 2 , y 3 . q A Ž y 1 , y 2 , x 3 . . EXAMPLE 3. An interesting example with non-additive unary operations is given by Grothendieck’s l-rings Žcf. e.g. w1x..

950

LAVENDHOMME AND LUCAS

A l-ring is a commutative unitary ring A together with a denumerable family of unary operations Ž l n : A ª A. ng N verifying Ž1. l0 Ž x . s 1 Ž2. l1 Ž x . s x Ž3. l n Ž x q y . s Ý nks 0 l nyk Ž x . l k Ž y .. It follows that for all n, ny1

ln Ž x q y . y ln Ž y . s

Ý lnyk Ž x . lk Ž y . , ks0

so that each l n is crossable Žprovided multiplication admits all possible modulations and each l n admits the two internal modulations.. The interpretation of l n in FA h B is given by ny1

l nA h B Ž a, b . s

žÝ

/

l nyk Ž a . l kB Ž b . , l n Ž b . .

ks0

Moreover, PROPOSITION 5. Proposition 3 holds for theories G containing the axioms of l-rings Ž and containing otherwise only usual polynomial identities .. Proof. Axioms 1 and 2 are trivial. Let us verify Axiom 3,

l nA h B Ž a q aX , b q bX . s

n

X X k Ý lnyk A h B Ž a, b . l A h B Ž a , b . ,

ks0

by observing that both members have as a first component

Ý

liA Ž a . l Aj Ž aX . l rB Ž b . l Bs Ž bX . .

iqjqrqssn rqs-n

EXAMPLE 4. As a last example, we consider the case of Žnot necessarily commutative. groups with the structure of Section 2.4. Multiplication is crossable because

Ž a1 b1 . Ž a2 b 2 . Ž b1 b 2 .

y1

s a1 b1 a2 by1 1 s a1 ? Ž b1 ) a2 .

951

MODULES AND CROSSED MODULES

which admits the required modulation. So is the law ), because

Ž Ž a1 b1 . ) Ž a2 b2 . . Ž b1 ) b2 . y1 y1 y1 y1 y1 s a1 b1 a2 by1 1 b 1 b 2 b 1 a1 b 1 b 2 b 1 y1 s a1 Ž b1 ) a2 . Ž Ž b1 b 2 by1 1 . ) a1 . ,

and the law Ž .y1 , because

Ž ab .

y1

? Ž by1 .

y1

s by1 ay1 b s Ž by1 . ) ay1 . The corresponding structure on A h B is given by

Ž a1 , b1 . Ž a2 , b 2 . s Ž a1 Ž b1 ) a2 . , b1 b 2 . Ž a, b .

y1

s Ž Ž by1 . ) ay1 , by1 .

eA h B s Ž eA , eB . y1 Ž a1 , b1 . ) Ž a2 , b 2 . s Ž a1 Ž b1 ) a2 . Ž Ž b1 b 2 by1 1 . ) a1 . , b 1 ) b 2 . .

One easily computes that

Ž a1 , b1 . ) Ž a2 , b 2 . s Ž a1 b1 . Ž a2 , b 2 . Ž a1 , b1 .

y1

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 G-module Ž A, B, F, D . where Ž A, B, F . is a modulated model of G 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 ­ 0 , ­ 1 , « , m . inside the category of G-algebras. Here are some details. PROPOSITION 6. The mapping ­1

A h BªB

Ž a, b . { D Ž a. q b is a homomorphism of G-algebras.

952

LAVENDHOMME AND LUCAS

Proof. Let F be a k-ary operation of L. Let TF be a term admitting the modulation m s Ž^a ,` ..., a _, b , . . . , b ; a . and such that in G,

^` _

k

k

TF Ž x, y . s F Ž x q y . y F Ž y . .

Ž 2.

We then have FA h B Ž a, b . s Ž < TF < m Ž a, b . , FB Ž b . . and

­ 1 Ž FA h B Ž a, b . . s D Ž < TF < m Ž a, b . . q FB Ž b . . By the definition of crossed module,

D Ž < TF < m Ž a, b . . s < TF < B Ž D Ž a . , b . . By Ž2. and the fact that B is a G-algebra, we have < TF < B Ž D Ž a . , b . s FB Ž D Ž a . q b . y FB Ž b . , hence

­ 1 Ž FA h B Ž a, b . . s FB Ž D Ž a . q b . s FB Ž ­ 1 Ž a1 , b1 . , . . . , ­ 1 Ž a k , bk . . and ­ 1 is a homomorphism. Besides ­ 1 we can take as ­ 0 the projection p 2 : A h B ª B which by construction is a homomorphism of G-algebras. For composable maps and composition, we note that the pullback AhB ­1

p2

­0

6

6

AhB

6

p1

6

P

B

in the category of G-algebra is given by the pullback in Sets: P s  Ž a, b . , Ž aX , bX . < bX s D Ž a . q b 4 . We then have on P a composition

m Ž Ž a, b . , Ž aX , bX . . s Ž aX q a, b . .

MODULES AND CROSSED MODULES

953

Finally, for « , we consider

« : B ª A h B : b{ Ž 0, b . . It is easy to check that those definitions satisfy the conditions for an internal category. PROPOSITION 7. The diagram p1

ª m

­0

ª «

P ª A) B ¤ B p2

ª

­1

ª

is an internal category in the category of G-algebras.

4. NON-ABELIAN COCYCLES In this section we want to connect the present approach with that of w4x concerning non-Abelian cohomology without however fully rewriting what is presented there. We concentrate on the notion of a two-cocycle. Let G be an algebraic theory with semi-direct product. Let X be a G-algebra and Ž A, B, F, D . a crossed G-module. A two-cocycle from X to Ž A, B, F, D . is an ordered pair Ž Q, D . where Q : X ª B is a map and D associates to any Žprimitive or composite. n-ary operation F a map D F : X n ª A h B such that ­ 0 ( D F s Q( FX , ­ 1 ( D F s FB ( Q n, and which satisfies compatibility conditions with projection and composition: Ža. if P Ž x 1 , . . . , x n . s x i , D p Ž x 1 , . . . , x n . s Ž0, QŽ x i ..; Žb. if F s G(Ž H1 , . . . , Hm . Žwith G m-ary and each Hi n-ary., D F s m Ž D G ( Ž H1 X , . . . , Hm X . , GA h B ( Ž D H 1 , . . . , D H m . . . An alternative more classical presentation is to define D F for primitive operations and to postulate a Ž‘‘cocycle’’. condition for each equational axiom of G. One of the results of w4x 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 in w4x. In the frame of the present work Žassuming on G the hypotheses of the first paragraph of Section 3., the definition would run as follows. If Ž Q, D . is a two-cocycle

954

LAVENDHOMME AND LUCAS

with coefficients in Ž A, B, F, D ., we define a structure A h ŽQ, D . X on A = X, interpreting F by FA h ŽQ ,D . X Ž a, x . s Ž p 1 Ž FA h B Ž a, Qx . q D F Ž x . , FX Ž x . . . The meaning of that identity should be clear: in abelian cohomology, the homomorphism defect of Q is measured by the difference F Ž Qx . y QF Ž x .; it appears here as D F Ž x . and a general categorical approach already suggested in w4x should view it simply as an arrow. From the computations made in w4x, we may state the following result: PROPOSITION 8. The structure A hŽQ,D . X is a G-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 G of groups a binary operation ) satisfying the axiom

Ž x ) y . x s xy. In general, if Ž Q, D . is a two-cocycle of X with coefficients in the crossed module Žin groups. Ž A, B, F, D ., we may note D F Ž x . s Ž c F Ž x . , Q Ž FX Ž x . . . Of course, the two-cocycle is characterized for groups by Q and the c F for F g  e, y, m, )4 Ž e for 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 . ? b. In this way, the components of a two-cocycle are

y1

,

c mŽ x , y .

QŽ x . ? QŽ y . ,

c# Ž x , y .

QŽ x . ) QŽ y .

6

QŽ x ) y .

QŽ x .

6

Q Ž xy .

e,

c yŽ x .

6

Q Ž xy1 .

ce

6

QŽ e.

and we must give the cocycle conditions in terms of the axioms of the theory.

955

MODULES AND CROSSED MODULES

Ža. The associativity of multiplication gives the condition of commutativity of the diagram Q Ž Ž xy . z . s Q Ž x Ž yz . . c mŽ x , yz .

c mŽ xy , z .

6

6

Q Ž xy . Q Ž z .

Q Ž x . Q Ž yz . Q Ž x .) c mŽ y , z .

6

6

c mŽ x , y .

Ž QŽ x . QŽ y . . QŽ z . s QŽ x . Ž QŽ y . QŽ z . . Remark that in the expression QŽ x .) c mŽ y, z ., ) denotes the action of B on A and that it is indeed an arrow of the internal category, since

D Ž Q Ž x . ) c m Ž y, z . . s Q Ž x . c m Ž y, z . Q Ž x .

y1

.

In this way, we obtain the classical cocycle condition

Ž Q Ž x . ) c m Ž y, z . . ? c m Ž x, yz . s c m Ž x, y . ? c m Ž xy, z .

Ž 3.

Žb. The unity axioms give the commutativity of Q Ž ex . s Q Ž x . 6

c mŽ e, x .

QŽ e. ? QŽ x .

e

ce

6

e ? QŽ x . s QŽ x . which defines c e and gives a condition on c m : c e s c m Ž e, x .

y1

.

Ž 4.

Similarly, the commutativity of QŽ x ? e. s QŽ x . 6

c mŽ x , e .

QŽ x . ? QŽ e.

6

Q Ž x .) c e

e

QŽ x . ? e s QŽ x . gives ce s QŽ x .

y1

) c m Ž x, e .

y1

.

Ž 5.

956

LAVENDHOMME AND LUCAS

Žc. The first action of inverses xy1 x s e gives the commutativity of Q Ž xy1 x . s Q Ž e . 6

c mŽ xy1 , x .

6

c yŽ x .

ce

6

Q Ž xy1 . ? Q Ž x .

QŽ x .

y1

? QŽ x . s e

which may be viewed as a definition of cy in terms of c m and c e : cy Ž x . s c e ? c m Ž xy1 , x .

y1

.

Ž 6.

Similarly, the second axiom of inverses xxy1 s e will give Q Ž x . ) cy Ž x . s c m Ž x, xy1 . , but this may be derived directly from Ž3. ] Ž6.. Žd. The axiom concerning ) gives the commutativity of Q Ž x ) y . ? x . s Q Ž xy . 6

c mŽ x ) y , x .

QŽ x ) y . ? QŽ x .

6

c# Ž x , y .

c mŽ x , y .

6

Ž QŽ x . ) QŽ y . . ? QŽ x . s QŽ x . ? QŽ y . which again may be viewed as a definition of c# in terms of c m : c# Ž x, y . s c m Ž x, y . ? c m Ž x ) y, x .

y1

.

Summarizing the preceding observations, we obtain: PROPOSITION 9. The notion of two-cocycle Ž Q, c m , c e , cy, c#. reduces to the classical notion of two-cocycle Ž Q, c m , c e .. Note that in the classical theory, it is customary to normalize cocycles by taking QŽ e . s e and c e s e.

MODULES AND CROSSED MODULES

957

5. DERIVATION OBJECTS As has been observed by Porter w9x, a crossed module Žin his case, in commutative algebras. generalizes the notion of ideal: Ž A, B, F, D . is viewed as an external version of an ideal of B. Porter also observes that crossed modules simultaneously generalize the notion of B-module Žcompare 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, F, D . the algebra B. The category of crossed modules over B is then a generalization 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 G be an algebraic theory, M a fixed modulation and A a G-algebra. We denote by CrŽ A. the category whose objects are crossed G-modules Ž A, B, F, D . and morphisms are morphisms of crossed G-modules of the form Ž1 A , g .. An important concept is embodied in the following definition: DEFINITION 3. The algebra A has a G-algebra of derivations if the category CrŽ A. has a terminal object. That terminal object is denoted by Ž A, DA, « , t . and is also called the G-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 derivations of A in the usual sense Žsquare brackets designating the commutator.. Let ad : A ª Der A : a ¬ w a, yx 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 DŽ a.. It is an easy classical result that Ž A, 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 by Ž A, Der A, « , ad.. Proof. Let Ž A, B, F, D . be a crossed module in Lie algebras. We have to check that there is a unique morphism

Ž 1 A , u . : Ž A, B, F , D . ª Ž A, Der A, « , ad. .

958

LAVENDHOMME AND LUCAS

The external homomorphism condition gives the commutative diagram

u =1 A

6

*BA

B=A

A

1A

6

6

6

e

Der A = A

A,

i.e., u Ž b .Ž a. s b) a, thus fixing u . It remains to show that u is indeed a morphism of crossed modules: Ži. u Ž b .Ž a. is linear in a and

u Ž b . Ž w a1 , a2 x . s w b) a1 , a2 x q w a1 , b) a2 x s u Ž b . Ž a1 . , a2 q a1 , u Ž b . Ž a2 . , so that u Ž b . is in Der A. Žii. u is a homomorphism of Lie R-algebras: linearity is immediate, and from

w b1 , b 2 x ) a s b1 ) w b 2 ) a x y b 2 ) w b1 ) a x , u Žw b1 , b 2 x. s w u Ž b1 ., u Ž b 2 .x follows. Žiii. The external homomorphism clause is the definition of u itself. Živ. The commutativity of 6B D

ad

u

66

A

Der A follows from u Ž D Ž a1 ..Ž a2 . s D Ž a1 .) a2 s w a1 , a2 x s adŽ a1 .Ž a2 .. 2. The case of groups is analogous. Let G be a group and Aut G be its group of automorphisms. Let t : G ª Aut G be the inner representation of G which to g g G associates t g Ž h. s ghgy1 . 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, « , t . 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 by Ž G, Aut G, « , t ..

MODULES AND CROSSED MODULES

959

Proof. It is easy to check that the unique morphism

Ž 1 G , u . : Ž G, B, F , D . ª Ž G, Aut G, « , t . is given by u Ž g .Ž h. s g ) h. 3. We study the case of Žnot necessarily unitary nor commutative. associati¨ e R-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. w6x.. An element of BimŽ A. is a pair Žg , d . of R-linear mappings from A to A such that

g Ž aaX . s g Ž a . ? aX X

X

Ž 1.

d Ž aa . s a ? d Ž a .

Ž 2.

a ? g Ž aX . s d Ž a . ? aX

Ž 3.

and

BimŽ A. has an obvious R-module structure and a product

Ž g , d . ? Ž g X , d X . s Ž g (g X , d ( d X . , the value of which is still in BimŽ A.. Suppose that AnnŽ A. s Ž0. or that A2 s A. Then BimŽ A. acts on A by

« 1 : Bim Ž A . = A ª A : Ž Ž g , d . , a . ¬ g Ž a . « 2 : A = Bim Ž A . ª A : Ž a, Ž g , d . . ¬ d Ž a . and there is a m : A ª BimŽ A. defined by m Ž a. s Žga , da . with gaŽ x . s ax and d aŽ x . s xa. PROPOSITION 12. Let A be an associati¨ e R-algebra such that AnnŽ A. s Ž0. or A2 s A. Then Ž A, Bim A, « , m . is a crossed module in associati¨ e R-algebras which is also the associati¨ e R-algebra of deri¨ ations of A in the sense of Definition 3. Proof. Ž a . We first show that Ž A, Bim A, « , m . is a crossed module. Ži. Let us check that

Ž g , d . ?1 Ž aX ?2 Ž g Y , d Y . . s Ž Ž g , d . ?1 aX . ?2 Ž g Y , d Y .

960

LAVENDHOMME AND LUCAS

Žwith the obvious notations for the actions « 1 and « 2 .. We do it here under the hypothesis that AnnŽ A. s Ž0.. Let x g A and multiply both members on the left by x: x ? Ž Ž g , d . ?1 Ž aX ?2 Ž g Y , d Y . . s x ? Ž g Ž d Y Ž aX . . . s d Ž x . ? d Y Ž aX .

Ž by Ž 3 . .

s d Y Ž d Ž x . ? aX .

Ž by Ž 2 . .

sd

Y

X

Ž x ? g Ža ..

Ž by Ž 3 . .

s x ? d Y Ž g Ž aX . .

Ž by Ž 2 . . X

s x ? Ž Ž Ž g , d . ?1 a . ? 2 Ž g Y , d Y . . Since this holds for any x, the hypothesis AnnŽ A. s Ž0. furnishes the conclusion. The other modulations of associativity are trivial. Žii. We verify that

m Ž Ž g , d . ?1 a . s Ž g , d . ? m Ž a . as follows:

m Ž Ž g , d . ?1 a . s Ž m Ž g Ž a . . s Ž gg Ž a. , dg Ž a. .

Ž g , d . ? m Ž a. s Ž g (ga , d a ( d . and

gg Ž a. Ž x . s g Ž a . ? x s g Ž ax .

Ž by Ž 1 . .

s g Ž ga Ž x . .

dg Ž a. Ž x . s x ? g Ž a . s d Ž x. ? a

Ž by Ž 3 . .

s da Ž d Ž x . . . Similarly

m Ž a ?2 Ž g , d . . s m Ž a . ? Ž g , d . and Ž A, BimŽ A., « , m . is indeed a crossed module.

MODULES AND CROSSED MODULES

961

Ž b . Let Ž A, B, F, D . be another crossed module. We have to define

Ž 1 A , u . : Ž A, B, F , D . ª Ž A, Bim Ž A . , « , m . . Letting u Ž b . s Ž l b , D b . one must have the following commutativities: ?

6

1

B=A

A

u=1 A 1

6

6

?

Bim Ž A . = A

A

and ?

6

2

B=A

A

1 A= u

6

?

6

2

A = Bim Ž A .

A,

i.e., l b Ž a. s b ?1 a and D b Ž a. s a ?2 b. This shows how to define u and proves its uniqueness. The proof that 6B D

A

u

m

66

Bim Ž A . commutes is trivial. The proposition is thus proved under the hypothesis AnnŽ A. s Ž0. but the proof for A2 s A is similar. 4. Some reinforcements of the theory of associati¨ e R-algebras are also easily dealt with. We still assume AnnŽ A. s Ž0. or A2 s A. Ž a . If A is unitary, then BimŽ A. has a unity Žg 1 , d 1 . s Ž1 A , 1 A . and m preserves it. Ž b . If A is a commutative and Žg , d . g BimŽ A., then g s d . This is because for every x in A: x ? d Ž a. s d Ž a. ? x s a ? g Ž x . s g Ž x . ? a s g Ž xa. s g Ž ax . s g Ž a . ? x s x ? g Ž a . .

962

LAVENDHOMME AND LUCAS

In this case, BimŽ A. may be identified with the R-algebra M Ž A. of multipliers of A. Recall that a multiplier of A is a linear mapping l : A ª A such that

lŽ a1 a2 . s lŽ a1 . a2 . In fact, M Ž A. is commutative: for any x g A, x ? lX Ž l Ž a . . s lXlŽ xa. s lX Ž x . lŽ a . s l Ž a . lX Ž x . s x ? l Ž lX Ž a . . . A mapping m : A ª M Ž A. is defined by m Ž a.Ž aX . s aaX and one has: PROPOSITION 13. If A is a commutati¨ e associati¨ e R-algebra with AnnŽ A. s Ž0. or A2 s A, then Ž A, M Ž A., « , m . is a crossed module in commutati¨ e associati¨ e R-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 G Ž A. of modulated G-models Ž A, B, F .. We indicate some degenerate cases where M G Ž A. has a terminal object. If G is the theory of monoids and if Ž b , a ; a . and Ž b , b ; b . are the only admitted modulations, then the terminal object of M G Ž A. is Ž A, EndŽ A., « . where EndŽ A. is the monoid of mappings of A in A and the external component of « is evaluation. If G is the theory of groups and the same modulations are admitted, then the terminal object of M G Ž A. is Ž A, AutŽ A., « . where AutŽ A. is the group of bijections of A on A. If G is the theory of rings, if one admits the internal modulations for the structure of group and Ž b , a ; a . and Ž b , b ; b . for multiplicationn, then the terminal object of M G Ž A. Ž A being here an abelian group. is Ž A, LŽ A., « . where LŽ A. is the ring of additive endomorphisms of A.

REFERENCES 1. M. F. Atiyah and P. O. Tall, Group representations, l-rings, and the J-homomorphism, Topology 8 Ž1969., 253]297. 2. D. F. Holt, An interpretation of the cohomology of groups H n Ž G, M ., J. Algebra 60 Ž1979., 307]320. 3. R. Lavendhomme, and J.-R. Roisin, Note on nonabelian cohomology, in ‘‘Applications of Sheaves,’’ Lecture Notes in Math, Vol. 753, pp. 534]541, Springer-Verlag, New Yorkr Berlin, 1979. 4. R. Lavendhomme and J. R. Roisin, Cohomologie non abelienne de structures algebriques, ´ ´ J. Algebra 67 Ž1980., 385]414.

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5. A. S.-T Lue, Cohomology of groups relative to a variety, Math. Z. 121 Ž1971., 220]232. 6. S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. 2 Ž1958., 316]345. 7. S. Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Natl. Acad. Sci. 36 Ž1950., 41]48. 8. G. Orzech, Obstruction theory in algebraic categories, I and II, J. Pure Appl. Algebra 2 Ž1972., 287]314, 315]340. 9. T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, J. Algebra 109 Ž1987., 415]429.