2. Tensor Products of Complexes for Nonassociative Algebras with Metagroup Relations
A concept of modules over metagroup algebras is given in
Appendix A.
Definition 1. Assume that is an associative unital ring, G is a metagroup and is a metagroup algebra of G over . Let B also be a unital right A-module (see Definitions A1 and A2 in Appendix A) such that B also has a structure of a two-sided -module. Suppose that B has a decomposition as a two-sided -module, where is a two-sided -module, and satisfies the following conditions: ;
and and ;
;
, ,
for each in G, b and c in , and .
A right A-module B satisfying Conditions – will be called G-graded. Similarly left and two-sided modules are considered, -bimodules. For -bimodules it can also be shortly written A-bimodule or two-sided A-module. Suppose that B is an A-bimodule and
there exists a -bilinear mapping such that
and and and and
for all x, y, z in B, ;
and
for every g, h, s in G, , , .
Then we call B a G-graded algebra over A (or a G-graded A-algebra). The algebra B is called unital if and only if
B has a unit element such that and for each .
Assume that B is the A-algebra and , where is the metagroup algebra. We put
;
;
;
;
;
.
Then , and are called a commutant, a nucleus and a centralizer correspondingly of the algebra B relative to a subset P in B. Instead of , or it will also be written shortly , or correspondingly. We put and for the G-graded A-algebra B, where .
Lemma 1. Let B be a G-graded A-algebra (see Definition 1). Then and is an associative -algebra.
Proof. We put . From Definition 1 it follows that B also has a structure of a -algebra and Y is a -subalgebra in B. From Conditions and we infer that Y is the associative -algebra and .
On the other hand, Conditions – imply that is the associative -subalgebra in B. From Definition 1 we deduce that
.
Assume that , . Then , where , , for each . If , , , then Conditions and imply that . Therefore, , consequently, . Thus . □
Definition 2. Suppose that is a unital -graded -algebra, where is a metagroup algebra for each , is an associative unital ring. Suppose that X is a -graded left -module and Y is a -graded left -module (see also Definition A3).
Suppose also that is a map such that f is
a left -homomorphism and for each , where is a homomorphism of metagroups:
and and
for every g and h in G.
The map satisfying conditions and will be called a -graded left -homomorphism of the left modules X and Y. If the ring is specified, it may be shortly written homomorphism instead of -homomorphism. Symmetrically is defined a -graded right homomorphism of a right -module X and a right -module Y. For a -bimodule X and a -bimodule Y if a map is -graded left and -graded right homomorphism, then f will be called a -graded -homomorphism of bimodules X and Y.
Assume that X is a left -module and Y is a left -module and is a map such that
is a left -homomorphism and , where
is a -homomorphism from into such that
, , is injective, and and
and and
and
for every g and h in , and b in , and s in , where is embedded into as , is a unit element in the unital -algebra , is naturally emebedded into as , is the unit in .
If f satisfies Conditions –, then f will be called a -generic left homomorphism of left modules X and Y. For right modules a -generic right homomorphism is defined analogously. If X is a -bimodule and Y is a -bimodule and f is a -generic left and -generic right homomorphism, then f will be called a -generic homomorphism of bimodules X and Y.
A -generic left homomorphism f of left modules X and Y such that is an epimorphism of onto we will call -epigeneric.
If additionally the homomorphism is surjective in and is the -homomorphism, then is called an isomorphism of with (or automorphism if ).
A -generic left homomorphism f of left modules X and Y such that is the isomorphism of with will be called -exact.
In particular, if , then “-graded” or “-generic” will be shortened to “-graded” or “-generic” correspondingly, etc. If is an automorphism of (or of correspondingly), then the -graded (or the -generic) left -homomorphism from X into Y will be called -exact (or -exact correspondingly). Similarly, -exact or -exact right homomorphisms of right modules and -exact or -exact homomorphisms of bimodules are defined (as shortening of -exact or of -exact homomorphisms of bimodules).
If X and Y are G-graded B-algebras and f is a G-graded (or G-exact or B-generic or B-exact) homomorphism from X into Y considered as B-bimodules and in addition the following condition is satisfied
for each x and v in X,
then f will be called a G-graded (or G-exact or B-generic or B-exact correspondingly) homomorphism of the B-algebras, where for some .
Definition 3. Assume that G is a metagroup, is an associative unital ring, is a metagroup algebra of G over , X is a two-sided B-module, where B is a unital G-graded A-algebra. We denote by the n-fold direct product of G with itself such that is a metagroup, where is a natural number. We consider a two-sided -module for each ,…, in G and a vector indicating an order of pairwise multiplications in the braces (see Definition 1 in [28]). A two-sided B-module X will be called smashly -graded if it satisfies Conditions – given below. Suppose that X has the following decomposition
as the two-sided -module, where , and by induction for each . Assume also that X satisfies the following conditions:
there exists a left -linear and a right -linear isomorphism
such that
for each , where is such that
,
(see also Lemma 1 and Example 2 in [28]); there exist left -linear and right -linear isomorphisms
and
and
and
,
and
and
for every , , elements , ,…,, in the metagroup G, vectors and indicating orders of pairwise multiplications, where .
For shortening the terminology it also will be said “-graded” instead of “smashly -graded”, because the case is specified.
If the module X is -graded and satisfies the following condition: it is a direct sum
of two-sided -submodules , where , then we will say that that X is directly -graded.
Similarly defined are the -graded left and right B-modules.
The -graded left (or right or two-sided) B-module X is called essentially -graded with , if for each ,…, in G there exists in G such that for each , where for each if ; is the symmetric group (i.e., of all permutations σ of the finite set ).
Lemma 2. Let X be a smashly -graded two-sided (or left, or right) B-module, . Then X can be supplied with a smashly -graded two-sided (or left, or right, respectively) B-module structure.
Proof. The smashly -graded two-sided B-module X has the decomposition as the two-sided -module. By virtue of Lemma 1 is the associative -algebra such that and is contained in B. The case is trivial. It remains the case . We put
for
for ,
for . Then we define
.
Notice that if both vectors and in terms correspond to order of multiplication, that is
with the corresponding vectors and , then
,
where . Therefore, from – and – it follows, that Y is a smashly -graded B-bimodule. This construction supplies X with the smashly -graded B-bimodule structure. □
Definition 4. If B is a -graded A-bimodule, , if also B, supplied with the G-graded A-bimodule structure by Lemma 2, is a G-graded A-algebra (see Definition 1), then B will be called a -graded A-algebra.
Lemma 3. Let B be a smashly -graded A-algebra, let also X be a smashly -graded B-bimodule (or left, or right B-module), (see Definitions 3 and 4). Then there exists a smashly -graded B-bimodule (or left, or right, respectively, B-module) Y such that Y can be supplied with a smashly -graded B-bimodule (or left, or right, respectively, B-module) structure relative to which it is isomorphic with X. Moreover, X and Y are isomorphic if considered as -bimodules (or left, or right, respectively, -modules).
Proof. The case of is trivial taking . Let now . We choose Y such that
with
,
for , where ,
for , where ,
for , where .
Notice that the sum in may generally not be direct (see also Definition 3). Formulas – imply that X and Y are isomorphic as -bimodules (or left, or right, respectively, -modules). In view of Lemma 2 this Y can be supplied with a smashly -graded B-bimodule (or left, or right, respectively, B-module) structure. Relative to the latter structure Y is isomorphic with X by –.
By virtue of Lemmas 1 and 2 we infer that , consequently, is an associative -algebra. As the -algebra is the subalgebra in B considered as the -algebra. Then X and Y are the -bimodules (or left, or right, respectively, -modules), since X and Y are the B-bimodules (or left, or right, respectively, B-modules). From Definitions 1 and 3 it follows that for each a and b in and , if X is the left B-module; , if X is the right B-module. Therefore, – imply that X and Y are isomorphic, if considered as the -bimodules (or left, or right, respectively, -modules). □
Definition 5. Assume that is a G-graded B-bimodule (see Definitions 1, 2), where is the metagroup algebra, B is a unital G-graded A-algebra, is a commutative associative unital ring. Assume also that is -graded with a gradation , where is the ring of all integers, . Suppose also that is -graded B-bimodule for each integer n (see Definition 3). Suppose also that is a -graded B-generic homomorphism of degree such that . Then d is called a differential of a G-graded B-complex or . Shortly it may be written a differential complex or , if G and B are specified.
Let be a -graded -complex. If is a homomorphism of degree 0 such that it is -generic (or -exact) with , then ψ is called a homomorphism (or a -exact homomorphism correspondingly) of complexes.
Similarly defined are the complexes and their homomorphisms in other cases: if is a G-graded left B-module or right B-module.
Remark 1. In this work mainly essentially -graded B-modules are considered (see also Proposition 6 below). Lemmas 2 and 3 serve in order to encompass other cases for their simultaneous treatment.
Henceforth, differential G-graded B-complexes and unital G-graded A-algebras B are considered for metagroups G and metagroup algebras , if something other will not be outlined.
Definition 5 means that and , also and for each . Examples of G-graded A-complexes are provided by [28] (see also Proposition 1 and Theorem 1 there), where and are left and right A-homomorphisms, consequently, A-exact, hence A-generic. Then is a module of cycles, is a module of boundaries. They are B-bimodules G-graded and -graded such that , , , for each .
The homomorphism of G-graded B- and -complexes correspondingly means that and for each .
Assume that is a differential G-graded B-complex, is a differential -graded -complex, is a differential -graded -complex.
Then and . This induces homomorphisms and and . Their -homogeneous components are , ,…,. If ψ and ϕ are homomorphisms of into , then is a homomorphism from into such that and and . Moreover, is a homomorphism from into for each with , and .
If is bijective, then it is called a -generic (or -exact) homologism, respectively, or shortly homologism. If , then the differential G-graded B-complex is called null homological. If (), then is called acyclic descending degree n (ascending degree n, respectively).
If is a homomorphisms, then is a homomorphism from into with and and .
Analogously considered are complexes, their homomorphisms and homologisms in other cases: if is a G-graded left B-module or right B-module. Then Lemma 4, Definitions 6, 9, Theorems 1, 2, 10, 11, Propositions 3, 4, 5, Corollary 1 are similarly formulated and proved in these cases.
Suppose thatis an exact sequence, where u and v are homomorphisms of complexes. Lemma 4. For the exact sequence with a -exact homomorphism v and there exists a -generic homomorphism with such that it is a graph of a -generic -graded homomorphism of degree from to .
Proof. From , Definition 5 and Remark 1 it follows that u is a -generic homomorphism and v is -generic homomorphism. From the exactness of , in Definitions 1 and 5 it follows that is injective, hence is an isomorphism. Since is an isomorphism, then also is an isomorphism of -algebras. Therefore, is a homomorphism of -algebras (see ). Thus there exists a homomorphism .
Then , consequently, . On the other hand, , hence . For each let .
Let and . Therefore, there exists such that . Then there is such that , consequently, . Then there exists such that , hence and belongs to . Thus .
Notice that for each element there exists such that and . This means that , hence . The homomorphism is bihomogeneous of degree . Thus is a graph of the -generic homomorphism of degree . This homomorphism is -graded, because complexes , , and homomorphisms u, v are -graded. □
Definition 6. The -generic -graded homomorphism from Lemma 4 of degree from to is called a connecting homomorphism relative to the exact sequence and it is denoted by or , where and are its homogeneous components.
Theorem 1. For the exact sequence of complexes with a -exact homomorphism v there exists an exact sequence
with a -generic homomorphism ,
a -generic homomorphism and
a -generic homomorphism for each n.
Proof. From Lemma 4 it follows that the homomorphism
exists and it is -generic for each n. Then the exact sequence induces the exact sequences
and
with -generic homomorphisms and , -generic homomorphisms and for each n. On the other hand, the sequence
is exact with B-exact homomorphisms , , for each n, similarly for and . Then the exact sequences and induce the exact sequence
with a -generic homomorphism and a -generic homomorphism for each n. Note that the homomorphism is obtained from the -generic homomorphism by restricting on and and then using quotient maps onto and correspondingly by the construction in Lemma 4. Therefore the exact sequences of the types – imply that there exists the exact sequence such that the homomorphism from into coincides with in . □
Definition 7. We consider the cartesian product of G-graded B-bimodlues X and Y (see Definition 2). Let be a G-graded B-bimodule generated from using finite additions of elements and the left and right multiplications on elements such that
and
and and
and (see also ) for each x and in X, and in Y, , .
Suppose that X, Y and Z are G-graded B-bimodules.
Let be a -bilinear map. Let also Λ satisfy the following identities:
and and
for each , , , g and h and s in G. If Λ fulfills Conditions and , then it will be said that the map Λ is G-balanced.
Let C be a G-graded B-bimodule supplied with a -bilinear map denoted by for each and such that
is generated by a set and
if is a G-balanced map of G-graded B-bimodules X, Y and Z, and for each fixed the map and for each fixed the map are G-graded homomorphisms of G-graded B-bimodules, then there exists a G-graded homomorphism of G-graded B-bimodules such that for each and .
If Conditions and are satisfied, then the G-graded B-bimodule C is called a G-smashed tensor product (or shortly tensor product) of X with Y over B and denoted by .
Similarly if X is the G-graded B-bimodule (or right B-module), Y is the G-graded left B-module (or the B-bimodule), then the G-smashed tensor product of X with Y over B is defined and it is the G-graded left B-module (or right B-module correspondingly).
Definition 8. A G-graded B-bimodule X is called flat if for each exact sequence of G-graded right B-modules Y, , and B-epigeneric homomorphisms u, v:
a sequence of -linear homomorphisms
is exact, where and .
Proposition 1. A G-graded B-bimodule X is flat if and only if for each injective B-epigeneric homomorphism of G-graded right B-modules the -linear homomorphism is injective and B-epigeneric.
Proof. If the module X is flat and a homomorphism is B-epigeneric and injective, thence the following sequence
is exact and consequently, the homomorphism is -linear and injective, where .
If the sequence is exact with B-epigeneric homomorphisms, then we put . Let be a canonical embedding and let be such that corresponds to . Then i and p are B-epigeneric, because . Therefore, the following sequence
is exact, where . Hence, the following sequence
also is exact with -linear homomorphisms and , where and . Then , consequently, is -linear with . Since is injective, then , consequently, the sequence is exact. □
Definition 9. Let be a G-graded B-complex and be a -graded -complex and let f and g be two -generic homomorphisms of into . A -generic homomorphism s of -degree 1 from into such that
is called a homotopy relating f with g. It is said that the homomorphisms f and g are homotopic.
Proposition 2. If f and g are homotopic -generic homomorphisms of into (see Definition 9), then .
Proof. If s is a homotopy relating f with g, then by Conditions –, because d is B-generic and is -generic, hence and consequently, . □
Lemma 5. Assume that X is a -graded left -module, Y is a -graded left -module, is a -generic homomorphism, (see Definitions 2 and 3). If is injective, then is isomorphic with . If is bijective onto (i.e., injective and surjective), then and are isomorphic.
Proof. From Conditions and it follows that is isomorphic with , if is injective. On the other hand, f and are the left -homomorphisms. Therefore, is isomorphic with , where is the metagroup algebra (see Definition A2). There exists a (single-valued) left -homomorphism , because is injective. Then Conditions and imply that is isomorphic with .
Therefore, if is bijective from onto , then and are isomorphic as -graded -algebra and -graded -algebra, respectively. □
Proposition 3. Assume that are -graded -complexes and , , , are -generic homomorphisms of complexes with for f and g, for ψ, for η. If s is a homotopy relating f with g, then is a homotopy relating with .
Proof. The composition is a -generic homomorphisms of -degree 1 from into . Then using Definitions 5 and 9 one verifies the assertion of this proposition. □
Corollary 1. Let be -graded -complexes, where , let also and be -generic homomorphisms from into for and . Let be a homotopy of with for and . Then is a homotopy relating with .
Proof. From the conditions of this Corollary and Definition 2 it follows that the homomorphisms , , , , and are -generic. In view of Proposition 3 relates with , while relates with , hence relates with . □
Definition 10. A -generic homomorphism of -graded -complexes, where , is called a homotopism if there exists a -generic homomorphism such that and are homotopic to and , respectively. The complex is homotopic to 0, if is homotopic to .
Proposition 4. If is a homotopism, then it is a homologism. Moreover, if is homotopic to , then also is a homotopism.
Proof. In the notation of Definition 10 by Proposition 2. Similarly . Hence, is bijective and is a homologism (see Remark 1).
Then for and one gets that is homotopic to , consequently, to . Analogously is homotopic to , consequently, to by Proposition 3. Thus, is the homotopism. □
Proposition 5. Suppose that is a G-graded B-complex (see Definition 5). Then the following conditions are equivalent:
there exists a B-generic homotopism of onto ;
there exists a B-generic and -graded of degree 1 endomorphism s of the module for which ;
and are direct multipliers of ;
is a direct sum of subcomplexes, which have length either 0 or 1 and zero homology.
Proof. ⇒. Assume that is a homotopism. Therefore, there exists a morphism of complexes and the B-generic and -graded of degree 1 endomorphism s of the module such that . From and it follows that and consequently, . The latter implies .
⇒. Let s be an endomorphism provided by Condition . Hence, . The maps d and s are B-generic, hence is the -linear B-generic projector from onto . On the other hand, , consequently, is the -linear B-generic projector from onto .
⇒. For each integer n we consider and . Choose -graded B-sub-bimodules and in for which and . By we denote the p-th translate of , where and , . We take and subcomplexes in , because , and . This gives . Note that for each the complex is either nil or of null length, while is either nil or of length one with zero homology. The latter implies .
⇒. Condition is satisfied if the complex is of null length or of length one with zero homology. □
Definition 11. It is said that a G-graded B-complex is split, if it satisfies equivalent conditions of Proposition 5. A B-generic endomorphism s of satisfying is called a splitting of .
Let X be a G-graded B-bimodule, and let be a G-graded B-complex and let be null from the right (or left) and let be a homologism. Then the pair (or respectively) is called a left (or right, respectively) G-graded resolution of X. A length of the G-graded B-complex is called a length of the resolution. If and are two left resolutions (or two right resolutions and ) and is a B-generic or B-exact morphism of complexes such that (or respectively), then f is called a B-generic or B-exact, respectively, morphism of resolutions.
Analogously considered are complexes and splittings in other cases: if is a G-graded left B-module or right B-module and X is a G-graded left B-module or right B-module correspondingly.
Remark 2. Let be a metagroup algebra of a (nonassociative) metagroup G over a commutative associative unital ring , let B be a G-graded A-algebra. We put
for each ,
, and by induction
for each natural number n (see Definition 2.13). Therefore is supplied with a B-bimodule structure.
Proposition 6. The B-bimodule is -graded for each . Moreover, if G, and B are nontrivial, then is essentially -graded for each .
Proof. If and , then it can be supplied with the trivial -gradation such that for each in G. If and , then it is G-graded and by Lemma 3 it can be supplied with the -graded structure.
For each it satisfies the following identities:
, and
and
, and
,
where ;
with ;
with ;
with
where ,
,
, ;
where ,
using shortened notation;
for every in G, , , for each , where is a shortened notation of the left ordered tensor product
,
.
If n is nonnegative, , then from and it follows that
with consisting of all elements which are sums of elements of the form with for each (see Remark 2). From Identities – it follows that the B-bimodule satisfies Conditions –.
If G, and B are nontrivial, then Identities – and Definitions 3, 2.13 imply that is essentially -graded for each . □
Proposition 7. Let the algebra B and the B-bimodules be as in Remark 2. Then an acyclic left B-complex exists.
Proof. We take the B-bimodules for each as in Remark 2 and . In view of Proposition 6 the B-bimodule is -graded for each . The metagroup algebra A is unital and the G-graded A-algebra B is unital such that A has the natural embedding into B as , where is the unit element in B. Therefore (see Definition 1).
Then we describe a boundary -linear operator on for each natural number n. Using the decomposition it is sufficient to give it at first on for every in G, , , for each . Then it has by the -linearity an extension on . Therefore we put:
, where
,
,…,
,
,
;
,…,
for each in G.
Then Formulas and in Definition 1 imply that
for each , where
,…,
for every in G. This means that is a left and right B-homomorphism of B-bimodules, consequently, is B-exact. Particularly,
,
.
Then we define a -linear homomorphism , which has the form:
for every
in
G. From Formula
in Lemma 1 in [
28] and the identities
–
in Proposition 1 in [
28] and
in Definition 1 it follows that
for every in G, for each , .
We put to be a -linear mapping such that
and
for each
and
. Hence Formulas
and
in Proposition 1 in [
28] and
,
imply that
is the identity on
, consequently,
is a monomorphism.
Therefore, from Formulas and , it follows that
,
for every ,…, in G, for each .
Then Formulas – imply the homotopy conditions
for each ,
where I denotes the identity operator on . This leads to identities:
and hence gives the recurrence relation
.
Notice that Formula implies that as the left B-module is generated by . Utilizing the recurrence relation by induction in n we infer:
for each ,
since by Formulas and .
Let be the enveloping algebra of B, where denotes an opposite algebra. The latter as an -linear space is the same, but with the multiplication for each . This permits to consider the -graded B-bimodule as . Therefore, the mapping provides the augmentation .
Thus, according to identities the left complex is acyclic:
. □
Remark 3. For a G-graded left B-module X let be a complex such that for each with for each (see Proposition 7 and Remark 2), while and for each , where denotes a unit operator on X, such that for each . Let a map be defined by the following formula:
for each a and b in B and . Formula above, Conditions – in Definition A3 and the identities , imply that
.
This procedure induces a -graded homomorphism
.
Proposition 8. The map (see Remark 2) is a homotopism of complexes of left B-modules. The complex splits as a G-graded left B-complex and is a left resolution of the G-graded left B-module X.
Proof. For each there exists a -linear map
for each ,…, in B, where and by induction for each . This is a homomorphism from into as right A-modules. Proposition 7, Formula , Lemma 1, Remark 2 and Definitions 1, A1, A2 imply that
for each , because for each and in G, where e is the unit element in G. In particular,
for each and in B. A map such that for each induces a -linear homomorphism . This implies that . From Formulas and it follows that . Then defining , , we infer that and . The homomorphisms , , and are B-generic, since the homomorphisms d, v, and are B-generic. Thus, is a homotopism (see Definition 10). Proposition 7 implies that is a left resolution of the G-graded left B-module X, because is the homotopism. □
Definition 12. The left resolvent for X is called the standard resolvent of the G-graded left B-module X.