Homotopism of Homological Complexes over Nonassociative Algebras with Metagroup Relations

Abstract

The article is devoted to homological complexes. Smashly graded modules and complexes are studied over nonassociative algebras with metagroup relations. Smashed tensor products of homological complexes are investigated. Their homotopisms and homologisms are scrutinized.

1. Introduction

Nonassociative algebras and algebras with group relations are very important in different branches of mathematics and its applications (see, for example, [1,2,3,4,5,6] and references therein). Methods of nonassociative algebras serve for studies in noncommutative geometry, for example, Poisson manifolds, spin manifolds, and octonion manifolds. In turn, it is one of the main tools in quantum field theory and quantum gravity, also in more classical areas such as hydrodynamics or magneto-hydrodynamics. It is worth mentioning that octonions and generalized Cayley–Dickson algebras have played a significant role in mathematics and physics [7,8,9,10,11]. The generalized Cayley–Dickson algebras from a large class are utilized not only in algebraic geometry but also in mathematical analysis, partial differential equations (PDEs), physics of elementary particles, theory of operators and founded applications in natural sciences including physics and quantum field theory (see [7,8,10,12,13,14]).

These algebras possess nonassociative multiplication. Therefore, their canonical generators together with some elements of an underlying ring provide a metagroup instead of a group [15] (see also Appendix A). The metagroups were used for investigations of automorphisms and derivations of nonassociative algebras in [15]. Nonassociative algebras with metagroup relations include as particular cases octonions and the generalized Cayley–Dickson algebras. A remarkable fact was demonstrated in the 20th century about correspondences of nontrivial geometries and unital quasigroups, which may be metagroups (see [16,17,18,19]).

An analysis of PDEs is based on cohomologies and deformed cohomologies [20]. This implies the importance of developing (co)homology theory over metagroup algebras.

This article is devoted to nonassociative algebras with metagroup relations. It is worthwhile to notice that apart from groups, metagroups have specific features: they may be nonassociative or power nonassociative or nonalternative. Then, in the metagroup, left or right inverse elements either may not exist or may not coincide.

For studies of algebras, (co)homology theory is often used. The earlier existing (co)homology theory operated in terms of associative algebras [2,21,22,23]. The aforementioned cohomology theory is not applicable to nonassociative algebras. For some nonassociative algebras (co)homology theory was studied such as Lie algebras, pre-Lie algebras, flexible algebras, alternative algebras (see, for example, [3,24,25,26]). Structures of the latter algebras are quite different from that of the generalized Cayley–Dickson algebras and the nonassociative algebras with metagroup relations.

Earlier in [27], cohomologies of loop spaces on quaternion and octonion manifolds were investigated. They appeared to be different from that of complex manifolds because the quaternion skew field is noncommutative and the octonion algebra is nonassociative. Principles of cohomology theory for nonassociative algebras with metagroup relations were described in [28].

This article is devoted to homological complexes. Smashly graded modules and complexes are studied over nonassociative algebras with metagroup relations. For a metagroup G, a G-smashed tensor product is investigated (see Definition 7). Smashed tensor products of homological complexes are scrutinized in Section 2. Their homotopisms and homologisms are investigated in Section 3. Basics on metagroups and modules over metagroup algebras are given in Appendix A.

All of the main results of this article are new.

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 $\mathcal{T}$ is an associative unital ring, G is a metagroup and $A=\mathcal{T}\left[G\right]$ is a metagroup algebra of G over $\mathcal{T}$. 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 $\mathcal{T}$-module. Suppose that B has a decomposition $B={\sum }_{g\in G}{B}_g$ as a two-sided $\mathcal{T}$-module, where $B_g$ is a two-sided $\mathcal{T}$-module, and satisfies the following conditions:

$\left(1\right)$$B_{g}h=B_{gh}$;

$\left(2\right)$$\left(bh\right)x_g=b\left(h{x}_g\right)$ and $x_g\left(bh\right)=\left(x_{g}h\right)b$ and $b{x}_g=x_{g}b$;

$\left(3\right)$$\left(x_{g}h\right)s={\mathsf{t}}_3(g,h,s)x_g\left(hs\right)$;

$\left(4\right)$$\left(bc\right)x=b\left(cx\right)$, $\left(bx\right)c=b\left(xc\right)$, $\left(xb\right)c=x\left(bc\right)$

for each $h,g,s$ in G, b and c in $\mathcal{T}$, $x_g\in B_g$ and $x\in B$.

A right A-module B satisfying Conditions $\left(1\right)$–$\left(4\right)$ will be called G-graded. Similarly left and two-sided modules are considered, $(A,{}^{1}A)$-bimodules. For $(A,A)$-bimodules it can also be shortly written A-bimodule or two-sided A-module. Suppose that B is an A-bimodule and

$\left(5\right)$ there exists a $\mathcal{T}$-bilinear mapping $B\times B\ni (x,y)\mapsto xy\in B$ such that

$x(y+z)=xy+xz$ and $(y+z)x=yx+zx$ and $\left(bx\right)y=b\left(xy\right)$ and $\left(xb\right)y=x\left(by\right)$ and $\left(xy\right)b=x\left(yb\right)$

for all x, y, z in B, $b\in \mathcal{T}$;

$\left(6\right)$$\left(x_g{y}_h\right)z_s={\mathsf{t}}_3(g,h,s)x_g\left(y_h{z}_s\right)$ and $x_g{y}_h\in B_{gh}$

for every g, h, s in G, $x_g\in B_g$, $y_h\in B_h$, $z_s\in B_s$.

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

$\left(7\right)$ B has a unit element $1=1_B$ such that $1_{B}x=x$ and $x{1}_B=x$ for each $x\in B$.

Assume that B is the A-algebra and $P\subseteq B$, where $A=\mathcal{T}\left[G\right]$ is the metagroup algebra. We put

$\left(8\right)$$Co{m}_B\left(P\right):=\{x\in B:\forall b\in P,xb=bx\}$;

$\left(9\right)$$N_{B,l}\left(P\right):=\{x\in B:\forall b\in P,\forall c\in P,\left(xb\right)c=x\left(bc\right)\}$;

$\left(10\right)$$N_{B,m}\left(P\right):=\{x\in B:\forall b\in P,\forall c\in P,\left(bx\right)c=b\left(xc\right)\}$;

$\left(11\right)$$N_{B,r}\left(P\right):=\{x\in B:\forall b\in P,\forall c\in P,\left(bc\right)x=b\left(cx\right)\}$;

$\left(12\right)$$N_B\left(P\right):=N_{B,l}\left(P\right)\cap N_{B,m}\left(P\right)\cap N_{B,r}\left(P\right)$;

$\left(13\right)$${\mathsf{C}}_B\left(P\right):=Co{m}_B\left(P\right)\cap N_B\left(P\right)$.

Then $Co{m}_B\left(P\right)$, $N_B\left(P\right)$ and ${\mathsf{C}}_B\left(P\right)$ are called a commutant, a nucleus and a centralizer correspondingly of the algebra B relative to a subset P in B. Instead of $Co{m}_B\left(B\right)$, $N_B\left(B\right)$ or ${\mathsf{C}}_B\left(B\right)$ it will also be written shortly $Com\left(B\right)$, $N\left(B\right)$ or $\mathsf{C}\left(B\right)$ correspondingly. We put $P_e=P\cap B_e$ and $P_{\mathsf{C}}:=P\cap B_{\mathsf{C}}$ for the G-graded A-algebra B, where $B_{\mathsf{C}}={\sum }_{g\in \mathsf{C}\left(G\right)}{B}_g$.

Lemma 1.

Let B be a G-graded A-algebra (see Definition 1). Then $N\left(B\right)={\sum }_{g\in N\left(G\right)}{B}_g$ and $N\left(B\right)$ is an associative $\mathcal{T}$-algebra.

Proof. 

We put $Y={\sum }_{g\in N\left(G\right)}{B}_g$. From Definition 1 it follows that B also has a structure of a $\mathcal{T}$-algebra and Y is a $\mathcal{T}$-subalgebra in B. From Conditions $\left(6\right)$ and $\left(12\right)$ we infer that Y is the associative $\mathcal{T}$-algebra and $Y\subset N\left(B\right)$.

On the other hand, Conditions $\left(9\right)$–$\left(12\right)$ imply that $N\left(B\right)$ is the associative $\mathcal{T}$-subalgebra in B. From Definition 1 we deduce that

$N\left(B\right)={\sum }_{g\in G}[B_g\cap N\left(B\right)]$.

Assume that $x\in N\left(B\right)$, $x\ne 0$. Then $x={\sum }_{g\in P\left(x\right)}{x}_g$, where $P\left(x\right)\subset G$, $x_g\in B_g\cap N\left(B\right)$, $x_g\ne 0$ for each $g\in P\left(X\right)$. If $h\in (G\backslash N(G\left)\right)\cap P\left(x\right)$, $y_h\in B_h$, $y_h\ne 0$, then Conditions $\left(1\right)$ and $\left(6\right)$ imply that $y_h\notin N\left(B\right)$. Therefore, $P\left(x\right)\subset N\left(G\right)$, consequently, $N\left(B\right)\subset Y$. Thus $Y=N\left(B\right)$. □

Definition 2.

Suppose that ${}^{j}B$ is a unital ${}^{j}G$-graded ${}^{j}A$-algebra, where ${}^{j}A=\mathcal{T}\left[{}^{j}G\right]$ is a metagroup algebra for each $j\in \{1,2,\dots \}$, $\mathcal{T}$ is an associative unital ring. Suppose that X is a ${}^{1}G$-graded left ${}^{1}B$-module and Y is a ${}^{2}G$-graded left ${}^{2}B$-module (see also Definition A3).

Suppose also that $f:X\to Y$ is a map such that f is

$\left(14\right)$ a left $\mathcal{T}$-homomorphism and $f\left(X_g\right)\subseteq Y_{f_{\iota }\left(g\right)}$ for each $g\in {}^{1}G$, where $f_{\iota }:{}^{1}G\to {}^{2}G$ is a homomorphism of metagroups:

$\left(15\right)$$f_{\iota }\left(gh\right)=f_{\iota }\left(g\right)f_{\iota }\left(h\right)$ and $f_{\iota }(g\backslash h)=f_{\iota }\left(g\right)\backslash f_{\iota }\left(h\right)$ and

$f_{\iota }(g/h)=f_{\iota }\left(g\right)/f_{\iota }\left(h\right)$ for every g and h in G.

The map $f:X\to Y$ satisfying conditions $\left(14\right)$ and $\left(15\right)$ will be called a $({}^{1}G,{}^{2}G)$-graded left $\mathcal{T}$-homomorphism of the left modules X and Y. If the ring is specified, it may be shortly written homomorphism instead of $\mathcal{T}$-homomorphism. Symmetrically is defined a $({}^{3}G,{}^{4}G)$-graded right homomorphism of a right ${}^{3}B$-module X and a right ${}^{4}B$-module Y. For a $({}^{1}B,{}^{3}B)$-bimodule X and a $({}^{2}B,{}^{4}B)$-bimodule Y if a map $f:X\to Y$ is $({}^{1}G,{}^{2}G)$-graded left and $({}^{3}G,{}^{4}G)$-graded right homomorphism, then f will be called a $(({}^{1}G,{}^{2}G),({}^{3}G,{}^{4}G))$-graded $\mathcal{T}$-homomorphism of bimodules X and Y.

Assume that X is a left ${}^{1}B$-module and Y is a left ${}^{2}B$-module and $f:X\to Y$ is a map such that

$\left(16\right)$$f:X\to Y$ is a left $\mathcal{T}$-homomorphism and $f(ax+by)=f_{\iota }\left(a\right)f\left(x\right)+f_{\iota }\left(b\right)f\left(y\right)$, where

$\left(17\right)$$f_{\iota }$ is a $\mathcal{T}$-homomorphism from ${}^{1}B$ into ${}^{2}B$ such that

$f_{\iota }:{}^{1}G\to {}^{2}G$, $f_{\iota }:N\left({}^{1}B\right)\to N\left({}^{2}B\right)$, $f_{\iota }:{}^{1}B\to {}^{2}B$ is injective, $f_{\iota }\left(1_{{}^{1}B}\right)=1_{{}^{2}B}$ and $f_{\iota }:\mathcal{T}\left[N\left({}^{1}G\right)\right]\to \mathcal{T}\left[N\left({}^{2}G\right)\right]$ and

$\left(18\right)$$f_{\iota }\left(ab\right)=f_{\iota }\left(a\right)f_{\iota }\left(b\right)$ and $f_{\iota }(g\backslash h)=f_{\iota }\left(g\right)\backslash f_{\iota }\left(h\right)$ and

$f_{\iota }(g/h)=f_{\iota }\left(g\right)/f_{\iota }\left(h\right)$ and $f_{\iota }(pa+bs)=p{f}_{\iota }\left(a\right)+f_{\iota }\left(b\right)s$

for every g and h in ${}^{1}G$, $a$ and b in ${}^{1}B$, $p$ and s in $\mathcal{T}$, where ${}^{j}A$ is embedded into ${}^{j}B$ as ${}^{j}A{1}_{{}^{j}B}$, $1_{{}^{j}B}$ is a unit element in the unital ${}^{j}A$-algebra ${}^{j}B$, ${}^{j}G$ is naturally emebedded into ${}^{j}A$ as ${}^{j}G{1}_{\mathcal{T}}$, $1_{\mathcal{T}}$ is the unit in $\mathcal{T}$.

If f satisfies Conditions $\left(16\right)$–$\left(18\right)$, then f will be called a $({}^{1}B,{}^{2}B)$-generic left homomorphism of left modules X and Y. For right modules a $({}^{1}B,{}^{2}B)$-generic right homomorphism is defined analogously. If X is a $({}^{1}B,{}^{3}B)$-bimodule and Y is a $({}^{2}B,{}^{4}B)$-bimodule and f is a $({}^{1}B,{}^{2}B)$-generic left and $({}^{3}B,{}^{4}B)$-generic right homomorphism, then f will be called a $(({}^{1}B,{}^{2}B),({}^{3}B,{}^{4}B))$-generic homomorphism of bimodules X and Y.

A $({}^{1}B,{}^{2}B)$-generic left homomorphism f of left modules X and Y such that $f_{\iota }$ is an epimorphism of ${}^{1}B$ onto ${}^{2}B$ we will call $({}^{1}B,{}^{2}B)$-epigeneric.

If additionally the homomorphism $f_{\iota }$ is surjective in $\left(17\right)$ and $f_{\iota }^{-1}:{}^{2}B\to {}^{1}B$ is the $\mathcal{T}$-homomorphism, then $f_{\iota }$ is called an isomorphism of ${}^{1}B$ with ${}^{2}B$ (or automorphism if ${}^{1}B={}^{2}B$).

A $({}^{1}B,{}^{2}B)$-generic left homomorphism f of left modules X and Y such that $f_{\iota }$ is the isomorphism of ${}^{1}B$ with ${}^{2}B$ will be called $({}^{1}B,{}^{2}B)$-exact.

In particular, if ${}^{1}G={}^{2}G$, then “$({}^{1}G,{}^{1}G)$-graded” or “$({}^{1}B,{}^{1}B)$-generic” will be shortened to “${}^{1}G$-graded” or “${}^{1}B$-generic” correspondingly, etc. If $f_{\iota }$ is an automorphism of ${}^{1}G$ (or of ${}^{1}B$ correspondingly), then the ${}^{1}G$-graded (or the ${}^{1}B$-generic) left $\mathcal{T}$-homomorphism from X into Y will be called ${}^{1}G$-exact (or ${}^{1}B$-exact correspondingly). Similarly, ${}^{3}G$-exact or ${}^{3}B$-exact right homomorphisms of right modules and $({}^{1}G,{}^{3}G)$-exact or $({}^{1}B,{}^{3}B)$-exact homomorphisms of bimodules are defined (as shortening of $(({}^{1}G,{}^{1}G),({}^{3}G,{}^{3}G))$-exact or of $(({}^{1}B,{}^{1}B),({}^{3}B,{}^{3}B))$-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

$\left(19\right)$$f\left(vx\right)=f\left(v\right)f\left(x\right)$ 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 $B={}^{j}B$ for some $j\in \{1,2,\dots \}$.

Definition 3.

Assume that G is a metagroup, $\mathcal{T}$ is an associative unital ring, $A=\mathcal{T}\left[G\right]$ is a metagroup algebra of G over $\mathcal{T}$, X is a two-sided B-module, where B is a unital G-graded A-algebra. We denote by $G^n$ the n-fold direct product of G with itself such that $G^n$ is a metagroup, where $n\ge 1$ is a natural number. We consider a two-sided $N\left(B\right)$-module $X_{{\{g_1,\dots ,g_n\}}_{q\left(n\right)}}$ for each $g_1$,…,$g_n$ in G and a vector $q\left(n\right)$ indicating an order of pairwise multiplications in the braces $\{g_1,\dots ,g_n\}$ (see Definition 1 in [28]).

A two-sided B-module X will be called smashly $G^n$-graded if it satisfies Conditions $\left(20\right)$–$\left(24\right)$ given below. Suppose that X has the following decomposition

$\left(20\right)$$X={\sum }_{g_1\in G,\dots ,g_n\in G}{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$

as the two-sided $N\left(B\right)$-module, where ${\left\{g_1\right\}}_{l\left(1\right)}=g_1$, ${\{g_1,g_2\}}_{l\left(2\right)}=g_1{g}_2$ and by induction ${\{g_1,\dots ,g_n,g_{n+1}\}}_{l(n+1)}={\{g_1,\dots ,g_n\}}_{l\left(n\right)}{g}_{n+1}$ for each $n\ge 1$. Assume also that X satisfies the following conditions:

$\left(21\right)$ there exists a left $N\left(B\right)$-linear and a right $N\left(B\right)$-linear isomorphism

$\theta (g_1,\dots ,g_n;q\left(n\right),v\left(n\right)):X_{{\{g_1,\dots ,g_n\}}_{v\left(n\right)}}\to X_{{\{g_1,\dots ,g_n\}}_{q\left(n\right)}}$ such that

$\theta (g_1,\dots ,g_n;q\left(n\right),v\left(n\right))\left(x_{{\{g_1,\dots ,g_n\}}_{v\left(n\right)}}\right)=t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right))x_{{\{g_1,\dots ,g_n\}}_{v\left(n\right)}}$

for each $x_{{\{g_1,\dots ,g_n\}}_{v\left(n\right)}}\in X_{{\{g_1,\dots ,g_n\}}_{v\left(n\right)}}$, where $t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right))\in \mathbf{\Psi }$ is such that

${\{g_1,\dots ,g_n\}}_{q\left(n\right)}=t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right)){\{g_1,\dots ,g_n\}}_{v\left(n\right)}$,

$t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right))=t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right)|id)$

(see also Lemma 1 and Example 2 in [28]);

$\left(22\right)$ there exist left $N\left(B\right)$-linear and right $N\left(B\right)$-linear isomorphisms

${\theta }_l(g_0,g_1,\dots ,g_n;l\left(n\right),l\left(n\right)):g_0{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}\to X_{{\{\left(g_0{g}_1\right),\dots ,g_n\}}_{l\left(n\right)}}$ and

${\theta }_r(g_1,\dots ,g_n,g_{n+1};l\left(n\right),l\left(n\right)):X_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}{g}_{n+1}\to X_{{\{g_1,\dots ,\left(g_n{g}_{n+1}\right)\}}_{l\left(n\right)}}$

$\left(23\right)$$\left(b{g}_0\right)x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}=b(g_0{x}_{{\{g_1,\dots ,g_n\}}_{l(n)}})$ and

$x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}\left(b{g}_{n+1}\right)=(x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}{g}_{n+1})b$ and

$b{x}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}=x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}b$,

$\left(24\right)$$\left(g_0{g}_{n+1}\right)x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}={\mathsf{t}}_3(g_0,g_{n+1},g)g_0(g_{n+1}{x}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}})$ and

$(g_0{x}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}})g_{n+1}={\mathsf{t}}_3(g_0,g,g_{n+1})g_0(x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}{g}_{n+1})$ and

$(x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}{g}_0)g_{n+1}={\mathsf{t}}_3(g,g_0,g_{n+1})x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}\left(g_0{g}_{n+1}\right)$

for every $b\in \mathsf{C}\left(B\right)$, $x_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}\in X_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$, elements $g_0$, $g_1$,…,$g_n$, $g_{n+1}$ in the metagroup G, vectors $q\left(n\right)$ and $v\left(n\right)$ indicating orders of pairwise multiplications, where $g={\{g_1,\dots ,g_n\}}_{l\left(n\right)}$.

For shortening the terminology it also will be said “$G^n$-graded” instead of “smashly $G^n$-graded”, because the case is specified.

If the module X is $G^n$-graded and satisfies the following condition: it is a direct sum

$\left(25\right)$$X={⨁}_{(g_1,\dots ,g_n)\in V}{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$

of two-sided $N\left(B\right)$-submodules $X_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$, where $V\subset G^n$, then we will say that that X is directly $G^n$-graded.

Similarly defined are the $G^n$-graded left and right B-modules.

The $G^n$-graded left (or right or two-sided) B-module X is called essentially $G^n$-graded with $n\ge 1$, if for each $g_1$,…,$g_n$ in G there exists $h_n$ in G such that $X_{\{g_{\sigma \left(1\right)},\dots ,g_{\sigma \left(n\right)}\}}\ne X_{\{h_{\sigma \left(1\right)},\dots ,h_{\sigma \left(n\right)}\}}$ for each $\sigma \in S_n$, where $h_j=g_j$ for each $j\le n-1$ if $n>1$; $S_n$ is the symmetric group (i.e., of all permutations σ of the finite set $\{1,\dots ,n\}$).

Lemma 2.

Let X be a smashly $G^n$-graded two-sided (or left, or right) B-module, $1\le m\le n$. Then X can be supplied with a smashly $G^m$-graded two-sided (or left, or right, respectively) B-module structure.

Proof. 

The smashly $G^n$-graded two-sided B-module X has the decomposition $\left(20\right)$ as the two-sided $N\left(B\right)$-module. By virtue of Lemma 1 $N\left(B\right)$ is the associative $\mathcal{T}$-algebra such that $N\left(B\right)={\sum }_{g\in N\left(G\right)}{B}_g$ and $N\left(B\right)$ is contained in B. The case $m=n$ is trivial. It remains the case $1\le m<n$. We put

$\left(26\right)$$Y_{{\{s,g_{n-m+2},\dots ,g_n\}}_{l\left(m\right)}}={\sum }_{s={\{g_1,\dots ,g_{n-m+1}\}}_{l(n-m+1)};g_1\in G,\dots ,g_{n-m+1}\in G}{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$

for $2<m<n$

$\left(27\right)$$Y_{{\{s,g_n\}}_{l\left(2\right)}}={\sum }_{s={\{g_1,\dots ,g_{n-1}\}}_{l(n-1)};g_1\in G,\dots ,g_{n-1}\in G}{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$

for $2=m<n$,

$\left(28\right)$$Y_{{\left\{s\right\}}_{l\left(1\right)}}={\sum }_{s={\{g_1,\dots ,g_n\}}_{l\left(n\right)};g_1\in G,\dots ,g_n\in G}{X}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$

for $1=m<n$. Then we define

$\left(29\right)$$Y={\sum }_{g_1\in G,\dots ,g_m\in G}{Y}_{{\{g_1,\dots ,g_m\}}_{l\left(m\right)}}$.

Notice that if both vectors $q\left(n\right)$ and $s\left(n\right)$ in $g_1,\dots ,g_{n-m+1}$ terms correspond to $q(n-m+1)$ order of multiplication, that is

$\left(30\right)$${\{g_1,\dots ,g_n\}}_{q\left(n\right)}={\{{\{g_1,\dots ,g_{n-m+1}\}}_{q(n-m+1)},g_{n-m+2},\dots ,g_n\}}_{u\left(m\right)}$

${\{g_1,\dots ,g_n\}}_{v\left(n\right)}={\{{\{g_1,\dots ,g_{n-m+1}\}}_{q(n-m+1)},g_{n-m+2},\dots ,g_n\}}_{w\left(m\right)}$

with the corresponding vectors $u\left(m\right)$ and $w\left(m\right)$, then

$t_n(g_1,\dots ,g_n;q\left(n\right),v\left(n\right))=t_m(p,g_{n-m+2},\dots ,g_n;u\left(m\right),w\left(m\right))$,

where $p={\{g_1,\dots ,g_{n-m+1}\}}_{q(n-m+1)}$. Therefore, from $\left(20\right)$–$\left(24\right)$ and $\left(26\right)$–$\left(30\right)$ it follows, that Y is a smashly $G^m$-graded B-bimodule. This construction supplies X with the smashly $G^m$-graded B-bimodule structure. □

Definition 4.

If B is a $G^n$-graded A-bimodule, $1<n\in \mathbf{N}$, 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 $G^n$-graded A-algebra.

Lemma 3.

Let B be a smashly $G^k$-graded A-algebra, let also X be a smashly $G^m$-graded B-bimodule (or left, or right B-module), $1\le k\le m\le n\in \mathbf{N}$ (see Definitions 3 and 4). Then there exists a smashly $G^n$-graded B-bimodule (or left, or right, respectively, B-module) Y such that Y can be supplied with a smashly $G^m$-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 $N\left(B\right)$-bimodules (or left, or right, respectively, $N\left(B\right)$-modules).

Proof. 

The case of $m=n$ is trivial taking $X=Y$. Let now $1\le m<n$. We choose Y such that

$\left(31\right)$$Y={\sum }_{g_1\in G,\dots ,g_n\in G}{Y}_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}$ with

$\left(32\right)$$Y_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}=X_{{\{h,g_{n-m+2},\dots ,g_n\}}_{l\left(m\right)}}$,

for $2<m$, where $h={\{g_1,\dots ,g_{n-m+1}\}}_{l(n-m+1)}$,

$\left(33\right)$$Y_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}=X_{{\{h,g_n\}}_{l\left(2\right)}}$

for $m=2$, where $h={\{g_1,\dots ,g_{n-1}\}}_{l(n-1)}$,

$\left(34\right)$$Y_{{\{g_1,\dots ,g_n\}}_{l\left(n\right)}}=X_{{\left\{h\right\}}_{l\left(1\right)}}$

for $m=1$, where $h={\{g_1,\dots ,g_n\}}_{l\left(n\right)}$.

Notice that the sum in $\left(31\right)$ may generally not be direct (see also Definition 3). Formulas $\left(31\right)$–$\left(34\right)$ imply that X and Y are isomorphic as $N\left(B\right)$-bimodules (or left, or right, respectively, $N\left(B\right)$-modules). In view of Lemma 2 this Y can be supplied with a smashly $G^m$-graded B-bimodule (or left, or right, respectively, B-module) structure. Relative to the latter structure Y is isomorphic with X by $\left(32\right)$–$\left(34\right)$.

By virtue of Lemmas 1 and 2 we infer that $N\left(B\right)={\sum }_{g\in N\left(G\right)}{B}_g$, consequently, $N\left(B\right)$ is an associative $\mathcal{T}$-algebra. As the $\mathcal{T}$-algebra $N\left(B\right)$ is the subalgebra in B considered as the $\mathcal{T}$-algebra. Then X and Y are the $N\left(B\right)$-bimodules (or left, or right, respectively, $N\left(B\right)$-modules), since X and Y are the B-bimodules (or left, or right, respectively, B-modules). From Definitions 1 and 3 it follows that $a\left(bx\right)=\left(ab\right)x$ for each a and b in $N\left(B\right)$ and $x\in X$, if X is the left B-module; $\left(xb\right)a=x\left(ba\right)$, if X is the right B-module. Therefore, $\left(31\right)$–$\left(34\right)$ imply that X and Y are isomorphic, if considered as the $N\left(B\right)$-bimodules (or left, or right, respectively, $N\left(B\right)$-modules). □

Definition 5.

Assume that $\mathcal{C}$ is a G-graded B-bimodule (see Definitions 1, 2), where $A=\mathcal{T}\left[G\right]$ is the metagroup algebra, B is a unital G-graded A-algebra, $\mathcal{T}$ is a commutative associative unital ring. Assume also that $\mathcal{C}$ is $\mathbf{Z}$-graded with a gradation $({\mathcal{C}}_n:n\in \mathbf{Z})$, where $\mathbf{Z}$ is the ring of all integers, ${\mathcal{C}}^n={\mathcal{C}}_{-n}$. Suppose also that ${\mathcal{C}}^n$ is $G^{\left|n\right|+2}$-graded B-bimodule for each integer n (see Definition 3). Suppose also that $d:\mathcal{C}\to \mathcal{C}$ is a $\mathbf{Z}$-graded B-generic homomorphism of degree $-1$ such that $dd=0$. Then d is called a differential of a G-graded B-complex $(\mathcal{C},d)$ or $\mathcal{C}$. Shortly it may be written a differential complex $(\mathcal{C},d)$ or $\mathcal{C}$, if G and B are specified.

Let $({}^1\mathcal{C},{}^{1}d)$ be a ${}^{1}G$-graded ${}^{1}B$-complex. If $\psi :(\mathcal{C},d)\to ({}^1\mathcal{C},{}^{1}d)$ is a homomorphism of degree 0 such that it is $((B,{}^{1}B),(B,{}^{1}B))$-generic (or $((B,{}^{1}B),(B,{}^{1}B))$-exact) with ${}^{1}d\circ \psi =\psi \circ d$, then ψ is called a homomorphism (or a $((B,{}^{1}B),(B,{}^{1}B))$-exact homomorphism correspondingly) of complexes.

Similarly defined are the complexes and their homomorphisms in other cases: if $\mathcal{C}$ is a G-graded left B-module or right B-module.

Remark 1.

In this work mainly essentially $G^n$-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 $A=\mathcal{T}\left[G\right]$, if something other will not be outlined.

Definition 5 means that $d_n:{\mathcal{C}}_n\to {\mathcal{C}}_{n-1}$ and $d_n{d}_{n+1}=0$, also $d^n:{\mathcal{C}}^n\to {\mathcal{C}}^{n+1}$ and $d^n{d}^{n-1}=0$ for each $n\in \mathbf{Z}$. Examples of G-graded A-complexes are provided by [28] (see also Proposition 1 and Theorem 1 there), where $d_n$ and $d^n$ are left and right A-homomorphisms, consequently, A-exact, hence A-generic.

Then $\mathcal{Z}(\mathcal{C},d)=ker\left(d\right)$ is a module of cycles, $\mathcal{B}(\mathcal{C},d)=Im\left(d\right)$ is a module of boundaries. They are B-bimodules G-graded and $\mathbf{Z}$-graded such that ${\mathcal{Z}}_n(\mathcal{C},d)=ker\left(d_n\right)$, ${\mathcal{B}}_n(\mathcal{C},d)=Im\left(d_{n+1}\right)$, ${\mathcal{Z}}_n(\mathcal{C},d)={\mathcal{Z}}^{-n}(\mathcal{C},d)$, ${\mathcal{B}}_n(\mathcal{C},d)={\mathcal{B}}^{-n}(\mathcal{C},d)$ for each $n\in \mathbf{Z}$.

The homomorphism $\psi :(\mathcal{C},d)\to ({}^1\mathcal{C},{}^{1}d)$ of G-graded B- and ${}^{1}B$-complexes correspondingly means that ${{}^{1}d}_n\circ {\psi }_n={\psi }_{n-1}\circ d_n$ and ${{}^{1}d}^n\circ {\psi }^n={\psi }^{n+1}\circ d_n$ for each $n\in \mathbf{Z}$.

Assume that $(\mathcal{C},d)$ is a differential G-graded B-complex, $({}^1\mathcal{C},{}^{1}d)$ is a differential ${}^{1}G$-graded ${}^{1}B$-complex, $({}^2\mathcal{C},{}^{2}d)$ is a differential ${}^{2}G$-graded ${}^{2}B$-complex.

Then $\psi \left(\mathcal{Z}\left(\mathcal{C}\right)\right)\subset \mathcal{Z}\left({}^1\mathcal{C}\right)$ and $\psi \left(\mathcal{B}\left(\mathcal{C}\right)\right)\subset \mathcal{B}\left({}^1\mathcal{C}\right)$. This induces homomorphisms $\mathcal{Z}\left(\psi \right):\mathcal{Z}\left(\mathcal{C}\right)\to \mathcal{Z}\left({}^1\mathcal{C}\right)$ and $\mathcal{B}\left(\psi \right):\mathcal{B}\left(\mathcal{C}\right)\to \mathcal{B}\left({}^1\mathcal{C}\right)$ and $H\left(\psi \right):H\left(\mathcal{C}\right)\to H\left({}^1\mathcal{C}\right)$. Their $\mathbf{Z}$-homogeneous components are ${\mathcal{Z}}_n\left(\psi \right)$, ${\mathcal{Z}}^n\left(\psi \right)$,…,$H^n\left(\psi \right)$. If ψ and ϕ are homomorphisms of $(\mathcal{C},d)$ into $({}^1\mathcal{C},{}^{1}d)$, then $\psi +\varphi$ is a homomorphism from $(\mathcal{C},d)$ into $({}^1\mathcal{C},{}^{1}d)$ such that $\mathcal{Z}(\psi +\varphi )=\mathcal{Z}\left(\psi \right)+\mathcal{Z}\left(\varphi \right)$ and $\mathcal{B}(\psi +\varphi )=\mathcal{B}\left(\psi \right)+\mathcal{B}\left(\varphi \right)$ and $H(\psi +\varphi )=H\left(\psi \right)+H\left(\varphi \right)$. Moreover, $b\psi$ is a homomorphism from $(\mathcal{C},d)$ into $({}^1\mathcal{C},{}^{1}d)$ for each $b\in \mathcal{T}$ with $\mathcal{Z}\left(b\psi \right)=b\mathcal{Z}\left(\psi \right)$, $\mathcal{B}\left(b\psi \right)=b\mathcal{Z}\left(\psi \right)$ and $H\left(b\psi \right)=bH\left(\psi \right)$.

If $H\left(\psi \right)$ is bijective, then it is called a $((B,{}^{1}B),(B,{}^{1}B))$-generic (or $((B,{}^{1}B),(B,{}^{1}B))$-exact) homologism, respectively, or shortly homologism. If $H\left(\psi \right)=0$, then the differential G-graded B-complex $(\mathcal{C},d)$ is called null homological. If $H_n\left(\mathcal{C}\right)=0$ ($H^n\left(\mathcal{C}\right)=0$), then $(\mathcal{C},d)$ is called acyclic descending degree n (ascending degree n, respectively).

If $\chi :({}^1\mathcal{C},{}^{1}d)\to ({}^2\mathcal{C},{}^{2}d)$ is a homomorphisms, then $\chi \circ \psi$ is a homomorphism from $(\mathcal{C},d)$ into $({}^2\mathcal{C},{}^{2}d)$ with $\mathcal{Z}(\chi \circ \psi )=\mathcal{Z}\left(\chi \right)\circ \mathcal{Z}\left(\psi \right)$ and $\mathcal{B}(\chi \circ \psi )=\mathcal{B}\left(\chi \right)\circ \mathcal{B}\left(\psi \right)$ and $H(\chi \circ \psi )=H\left(\chi \right)\circ H\left(\psi \right)$.

Analogously considered are complexes, their homomorphisms and homologisms in other cases: if $\mathcal{C}$ 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 that

$$\left(35\right)0\to {{}^1\mathcal{C}}_{\overrightarrow{u}}{\mathcal{C}}_{\overrightarrow{v}}{}^2\mathcal{C}\to 0$$

is an exact sequence, where u and v are homomorphisms of complexes.

Lemma 4.

For the exact sequence $\left(35\right)$ with a $((B,{}^{2}B),(B,{}^{2}B))$-exact homomorphism v and $\Gamma =\{x\in \mathcal{C}:dx\in Im(u\left)\right\}$ there exists a $(({}^{3}B,{}^{2}B{\otimes }_{\mathcal{T}}{}^{1}B),({}^{3}B,{}^{2}B{\otimes }_{\mathcal{T}}{}^{1}B))$-generic homomorphism $\eta :\Gamma \to H\left({}^2\mathcal{C}\right)\times H\left({}^1\mathcal{C}\right)$ with ${}^{3}B=u_{\iota }\left({}^{1}B\right)$ such that it is a graph of a $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic $\mathbf{Z}$-graded homomorphism of degree $-1$ from $H\left({}^2\mathcal{C}\right)$ to $H\left({}^1\mathcal{C}\right)$.

Proof. 

From $\left(35\right)$, Definition 5 and Remark 1 it follows that u is a $(({}^{1}B,B),({}^{1}B,B))$-generic homomorphism and v is $((B,{}^{2}B),(B,{}^{2}B))$-generic homomorphism. From the exactness of $\left(35\right)$, in Definitions 1 and 5 it follows that $u_{\iota }:{}^{1}B\to B$ is injective, hence $u_{\iota }:{}^{1}B\to {}^{3}B$ is an isomorphism. Since $v_{\iota }:B\to {}^{2}B$ is an isomorphism, then $v_{\iota }^{-1}:{}^{2}B\to B$ also is an isomorphism of $\mathcal{T}$-algebras. Therefore, $u_{\iota }^{-1}\circ v_{\iota }^{-1}:{}^{2}B\to {}^{1}B$ is a homomorphism of $\mathcal{T}$-algebras (see $\left(17\right)$). Thus there exists a homomorphism ${\eta }_{\iota }:{}^{3}B\to {}^{2}B{\otimes }_{\mathcal{T}}{}^{1}B$.

Then $d\left(u^{-1}\left(dx\right)\right)=u^{-1}\left(dd\left(x\right)\right)=0$, consequently, $u^{-1}\left(dx\right)\in \mathcal{Z}\left({}^1\mathcal{C}\right)$. On the other hand, $dv\left(x\right)=v\left(dx\right)\in Im(v\circ u)=0$, hence $v\left(x\right)\in \mathcal{Z}\left({}^2\mathcal{C}\right)$. For each $x\in \Gamma$ let $\eta \left(x\right)=(v\left(x\right),u^{-1}\left(dx\right))$.

Let $x\in \Gamma$ and $v\left(x\right)\in \mathcal{B}\left({}^2\mathcal{C}\right)$. Therefore, there exists ${}^{2}y\in {}^2\mathcal{C}$ such that $v\left(x\right)=d\left({}^{2}y\right)$. Then there is $y\in \mathcal{C}$ such that ${}^{2}y=v\left(y\right)$, consequently, $v\left(x\right)=v\left(dy\right)$. Then there exists ${}^{1}b\in {}^1\mathcal{C}$ such that $x-dy=u\left({}^{1}b\right)$, hence $dx=u\left(d\left({}^{1}b\right)\right)$ and $u^{-1}\left(dx\right)=d\left({}^{1}b\right)$ belongs to $\mathcal{B}\left({}^1\mathcal{C}\right)$. Thus $u^{-1}\left(dx\right)\in \mathcal{B}\left({}^1\mathcal{C}\right)$.

Notice that for each element ${}^{2}y\in \mathcal{Z}\left({}^2\mathcal{C}\right)$ there exists $x\in \mathcal{C}$ such that ${}^{2}y=v\left(x\right)$ and $v\left(dx\right)=0$. This means that $dx\in Im\left(u\right)$, hence $x\in \Gamma$. The homomorphism $\eta$ is bihomogeneous of degree $(0,-1)$. Thus $\eta (\Gamma )$ is a graph of the $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic homomorphism of degree $-1$. This homomorphism is $\mathbf{Z}$-graded, because complexes $\mathcal{C}$, ${}^1\mathcal{C}$, ${}^2\mathcal{C}$ and homomorphisms u, v are $\mathbf{Z}$-graded. □

Definition 6.

The $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic $\mathbf{Z}$-graded homomorphism from Lemma 4 of degree $-1$ from $H\left({}^2\mathcal{C}\right)$ to $H\left({}^1\mathcal{C}\right)$ is called a connecting homomorphism relative to the exact sequence $(u,v)$ and it is denoted by $\partial (u,v)$ or ${\partial }_{(u,v)}$, where ${\partial }_n(u,v):H_n\left({}^2\mathcal{C}\right)\to H_{n-1}\left({}^1\mathcal{C}\right)$ and ${\partial }^n(u,v):H^n\left({}^2\mathcal{C}\right)\to H^{n-1}\left({}^1\mathcal{C}\right)$ are its homogeneous components.

Theorem 1.

For the exact sequence $\left(35\right)$ of complexes with a $((B,{}^{2}B),(B,{}^{2}B))$-exact homomorphism v there exists an exact sequence

$\left(36\right)$$\dots \to H_{n+1}{\left({}^2\mathcal{C}\right)}_{\overrightarrow{{\partial }_{n+1}(u,v)}}{H}_n{\left({}^1\mathcal{C}\right)}_{\overrightarrow{H_n\left(u\right)}}{H}_n{\left(\mathcal{C}\right)}_{\overrightarrow{H_n\left(v\right)}}$

$H_n{\left({}^2\mathcal{C}\right)}_{\overrightarrow{{\partial }_n(u,v)}}{H}_{n-1}\left({}^1\mathcal{C}\right)\to \dots$

with a $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic homomorphism ${\partial }_n(u,v)$,

a $(({}^{1}B,B),({}^{1}B,B))$-generic homomorphism $H_n\left(u\right)$ and

a $((B,{}^{2}B),(B,{}^{2}B))$-generic homomorphism $H_n\left(v\right)$ for each n.

Proof. 

From Lemma 4 it follows that the homomorphism

$\left(37\right)$${\partial }_n(u,v):H_n\left({}^2\mathcal{C}\right)\to H_{n-1}\left({}^1\mathcal{C}\right)$

exists and it is $(({}^{2}A,{}^{1}A),({}^{2}A,{}^{1}A))$-generic for each n. Then the exact sequence $\left(35\right)$ induces the exact sequences

$\left(38\right)$$0\to {\mathcal{Z}}_n{\left({}^1\mathcal{C}\right)}_{\overrightarrow{{\mathcal{Z}}_n\left(u\right)}}{\mathcal{Z}}_n{\left(\mathcal{C}\right)}_{\overrightarrow{{\mathcal{Z}}_n\left(v\right)}}{\mathcal{Z}}_n\left({}^2\mathcal{C}\right)$ and

$\left(39\right)$${}^1{\mathcal{C}}_n/{\mathcal{B}}_n{\left({}^1\mathcal{C}\right)}_{\overrightarrow{{\overline{u}}_n}}{\mathcal{C}}_n/{\mathcal{B}}_n{\left(\mathcal{C}\right)}_{\overrightarrow{{\overline{v}}_n}}{}^2{\mathcal{C}}_n/{\mathcal{B}}_n\left({}^1\mathcal{C}\right)\to 0$

with $(({}^{1}B,B),({}^{1}B,B))$-generic homomorphisms ${\mathcal{Z}}_n\left(u\right)$ and ${\overline{u}}_n$, $((B,{}^{2}B),(B,{}^{2}B))$-generic homomorphisms ${\mathcal{Z}}_n\left(v\right)$ and ${\overline{v}}_n$ for each n. On the other hand, the sequence

$\left(40\right)$$0\to H_n{\left(\mathcal{C}\right)}_{\overrightarrow{i_n}}{\mathcal{C}}_n/{\mathcal{B}}_n{\left(\mathcal{C}\right)}_{\overrightarrow{d_n}}{\mathcal{Z}}_{n-1}{\left(\mathcal{C}\right)}_{\overrightarrow{p_{n-1}}}{H}_{n-1}\left(\mathcal{C}\right)\to 0$

is exact with B-exact homomorphisms $i_n$, $d_n$, $p_{n-1}$ for each n, similarly for ${}^1\mathcal{C}$ and ${}^2\mathcal{C}$. Then the exact sequences $\left(38\right)$ and $\left(39\right)$ induce the exact sequence

$\left(41\right)$$H_n{\left({}^1\mathcal{C}\right)}_{\overrightarrow{H_n\left(u\right)}}{H}_n{\left(\mathcal{C}\right)}_{\overrightarrow{H_n\left(v\right)}}{H}_n\left({}^2\mathcal{C}\right)$

with a $(({}^{1}B,B),({}^{1}B,B))$-generic homomorphism $H_n\left(u\right)$ and a $((B,{}^{2}B),(B,{}^{2}B))$-generic homomorphism $H_n\left(v\right)$ for each n. Note that the homomorphism ${\partial }_n(u,v)$ is obtained from the $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic homomorphism $u_{n-1}^{-1}\circ d_n\circ v_n^{-1}:{{}^2\mathcal{C}}_n\to {}^1{\mathcal{C}}_{n-1}$ by restricting on ${\mathcal{Z}}_n\left({}^2\mathcal{C}\right)$ and ${\mathcal{Z}}_{n-1}\left({}^1\mathcal{C}\right)$ and then using quotient maps onto $H_n\left({}^2\mathcal{C}\right)$ and $H_{n-1}\left({}^1\mathcal{C}\right)$ correspondingly by the construction in Lemma 4. Therefore the exact sequences of the types $\left(39\right)$–$\left(41\right)$ imply that there exists the exact sequence $\left(36\right)$ such that the homomorphism from $H_n\left({}^2\mathcal{C}\right)$ into $H_{n-1}\left({}^1\mathcal{C}\right)$ coincides with ${\partial }_n(u,v)$ in $\left(37\right)$. □

Definition 7.

We consider the cartesian product $X\times Y$ of G-graded B-bimodlues X and Y (see Definition 2). Let $X{\times }_{B}Y$ be a G-graded B-bimodule generated from $X\times Y$ using finite additions of elements $(x,y)\in X\times Y$ and the left and right multiplications on elements $a\in B$ such that

$\left(42\right)$$(x,y)+(x_1,y_1)=(x+x_1,y+y_1)$ and

$\left(43\right)$$a(x,y)=(ax,ay)$ and $(x,y)a=(xa,ya)$ and

$\left(44\right)$$g(X_e,Y_e)=(X_g,Y_g)$ and $(X_e,Y_e)g=(X_g,Y_g)$ (see also $\left(1\right)$) for each x and $x_1$ in X, $y$ and $y_1$ in Y, $a\in B$, $g\in G$.

Suppose that X, Y and Z are G-graded B-bimodules.

$\left(45\right)$ Let $\Lambda :X\times Y\to Z$ be a $\mathcal{C}\left(B\right)$-bilinear map. Let also Λ satisfy the following identities:

$\left(46\right)$$\Lambda (x_g{b}_h,y_s)={\mathsf{t}}_3(g,h,s)\Lambda (x_g,b_h{y}_s)$ and $\Lambda (cx,y)=c\Lambda (x,y)$ and

$\Lambda (x,yc)=\Lambda (x,y)c$ for each $c\in N\left(B\right)$, $x\in X$, $y\in Y$, g and h and s in G. If Λ fulfills Conditions $\left(45\right)$ and $\left(46\right)$, then it will be said that the map Λ is G-balanced.

Let C be a G-graded B-bimodule supplied with a $\mathcal{C}\left(B\right)$-bilinear map $\xi :X\times Y\to C$ denoted by $\xi (x,y)=x\otimes y$ for each $x\in X$ and $y\in Y$ such that

$\left(47\right)$$X{\otimes }_{A}Y$ is generated by a set $\{x\otimes y:x\in X,y\in Y\}$ and

$\left(48\right)$ if $\Lambda :X\times Y\to Z$ is a G-balanced map of G-graded B-bimodules X, Y and Z, and for each fixed $x\in X$ the map $\Lambda (x,·):Y\to Z$ and for each fixed $y\in Y$ the map $\Lambda (·,y):X\to Z$ are G-graded homomorphisms of G-graded B-bimodules, then there exists a G-graded homomorphism $\psi :C\to Z$ of G-graded B-bimodules such that $\psi (x\otimes y)=\Lambda (x,y)$ for each $x\in X$ and $y\in Y$.

If Conditions $\left(47\right)$ and $\left(48\right)$ 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 $X{\otimes }_{B}Y$.

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 $X{\otimes }_{B}Y$ 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, ${}^{1}Y$, ${}^{2}Y$ and B-epigeneric homomorphisms u, v:

$\left(49\right)$${{}^{1}Y}_{\overrightarrow{u}}{Y}_{\overrightarrow{v}}{}^{2}Y$

a sequence of $\mathbf{Z}$-linear homomorphisms

$\left(50\right)$${{}^{1}Y{⨂}_{B}X}_{\overrightarrow{u\otimes 1}}Y{⨂}_B{X}_{\overrightarrow{v\otimes 1}}{}^{2}Y{⨂}_{B}X$

is exact, where $(u_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$ and $(v_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$.

Proposition 1.

A G-graded B-bimodule X is flat if and only if for each injective B-epigeneric homomorphism $u:{}^{1}Y\to Y$ of G-graded right B-modules the $\mathbf{Z}$-linear homomorphism $u\otimes 1:{}^{1}Y{⨂}_{B}X\to Y{⨂}_{B}X$ is injective and B-epigeneric.

Proof. 

If the module X is flat and a homomorphism $u:{}^{1}Y\to Y$ is B-epigeneric and injective, thence the following sequence

$\left(51\right)$$0\to {{}^{1}Y}_{\overrightarrow{u}}Y$

is exact and consequently, the homomorphism $u\otimes 1$ is $\mathbf{Z}$-linear and injective, where $(u_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$.

If the sequence $\left(49\right)$ is exact with B-epigeneric homomorphisms, then we put $Z_2=v\left(Y\right)$. Let $i:{}^{2}Z\to {}^{2}Y$ be a canonical embedding and let $p:Y\to {}^{2}Z$ be such that $v\left(y\right)$ corresponds to $y\in Y$. Then i and p are B-epigeneric, because $v_{\iota }\left(B\right)=B$. Therefore, the following sequence

$\left(52\right)$${{}^{1}Y}_{\overrightarrow{u}}{Y}_{\overrightarrow{p}}{Z_2}_{\overrightarrow{\xi }}0$

is exact, where $\xi \left(Z_2\right)=\left(0\right)$. Hence, the following sequence

$\left(53\right)$${{}^{1}Y{⨂}_{B}X}_{\overrightarrow{u\otimes 1}}Y{⨂}_B{X}_{\overrightarrow{p\otimes 1}}{Z}_2{⨂}_{B}X$

also is exact with $\mathbf{Z}$-linear homomorphisms $u\otimes 1$ and $p\otimes 1$, where $(u_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$ and $(p_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$. Then $v=i\circ p$, consequently, $v\otimes 1=(i\otimes 1)\circ (p\otimes 1)$ is $\mathbf{Z}$-linear with $(v_{\iota }\otimes 1)\left(B{⨂}_{\mathbf{Z}}B\right)=B{⨂}_{\mathbf{Z}}B$. Since $i\otimes 1$ is injective, then $Ker(v\otimes 1)=Ker(p\otimes 1)=Im(u\otimes 1)$, consequently, the sequence $\left(50\right)$ is exact. □

Definition 9.

Let $(\mathcal{C},d)$ be a G-graded B-complex and $({}^1\mathcal{C},{}^{1}d)$ be a ${}^{1}G$-graded ${}^{1}B$-complex and let f and g be two $((B,{}^{1}B),(B,{}^{1}B))$-generic homomorphisms of $\mathcal{C}$ into ${}^1\mathcal{C}$. A $((B,{}^{1}B),(B,{}^{1}B))$-generic homomorphism s of $\mathbf{Z}$-degree 1 from $\mathcal{C}$ into ${}^1\mathcal{C}$ such that

$\left(54\right)$$g-f={}^{1}d\circ s+s\circ d$

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 $((B,{}^{1}B),(B,{}^{1}B))$-generic homomorphisms of $\mathcal{C}$ into ${}^1\mathcal{C}$ (see Definition 9), then $H\left(f\right)=H\left(g\right)$.

Proof. 

If s is a homotopy relating f with g, then $(g-f)\left(\mathcal{Z}\left(\mathcal{C}\right)\right)=({}^{1}d\circ s+s\circ d)\left(\mathcal{Z}\left(\mathcal{C}\right)\right)=({}^{1}d\circ s)\left(\mathcal{Z}\left(\mathcal{C}\right)\right)\subset \mathcal{B}\left({}^1\mathcal{C}\right)$ by Conditions $\left(16\right)$–$\left(18\right)$, because d is B-generic and ${}^{1}d$ is ${}^{1}B$-generic, hence $H(g-f)=0$ and consequently, $H\left(g\right)=H\left(f\right)$. □

Lemma 5.

Assume that X is a ${}^1{G}^n$-graded left ${}^{1}B$-module, Y is a ${}^2{G}^n$-graded left ${}^{2}B$-module, $f:X\to Y$ is a $({}^{1}B,{}^{2}B)$-generic homomorphism, $n\in \mathbf{N}$ (see Definitions 2 and 3). If $f_{\iota }:{}^{1}B\to {}^{2}B$ is injective, then $f_{\iota }\left({}^{1}B\right)$ is isomorphic with ${}^{1}B$. If $f_{\iota }:{}^{1}B\to {}^{2}B$ is bijective onto (i.e., injective and surjective), then ${}^{1}B$ and ${}^{2}B$ are isomorphic.

Proof. 

From Conditions $\left(16\right)$ and $\left(18\right)$ it follows that $f_{\iota }\left({}^{1}G\right)$ is isomorphic with ${}^{1}G$, if $f_{\iota }:{}^{1}B\to {}^{2}B$ is injective. On the other hand, f and $f_{\iota }$ are the left $\mathcal{T}$-homomorphisms. Therefore, $f_{\iota }\left({}^{1}A\right)$ is isomorphic with ${}^{1}A=\mathcal{T}\left[{}^{1}G\right]$, where ${}^{1}A$ is the metagroup algebra (see Definition A2). There exists a (single-valued) left $\mathcal{T}$-homomorphism $f_{\iota }^{-1}:f_{\iota }\left({}^{1}B\right)\to {}^{1}B$, because $f_{\iota }$ is injective. Then Conditions $\left(17\right)$ and $\left(18\right)$ imply that $f_{\iota }\left({}^{1}B\right)$ is isomorphic with ${}^{1}B$.

Therefore, if $f_{\iota }$ is bijective from ${}^{1}B$ onto ${}^{2}B$, then ${}^{1}B$ and ${}^{2}B$ are isomorphic as ${}^1{G}^n$-graded ${}^{1}A$-algebra and ${}^2{G}^n$-graded ${}^{2}A$-algebra, respectively. □

Proposition 3.

Assume that ${}^k\mathcal{C}$ are ${}^{k}G$-graded ${}^{k}B$-complexes and $f:{}^1\mathcal{C}\to {}^2\mathcal{C}$, $g:{}^1\mathcal{C}\to {}^2\mathcal{C}$, $\psi :{}^3\mathcal{C}\to {}^1\mathcal{C}$, $\eta :{}^2\mathcal{C}\to {}^4\mathcal{C}$ are $(({}^{j}B,{}^{k}B),({}^{j}B,{}^{k}B))$-generic homomorphisms of complexes with $(j,k)=(1,2)$ for f and g, $(j,k)=(3,1)$ for ψ, $(j,k)=(2,4)$ for η. If s is a homotopy relating f with g, then $\eta \circ s\circ \psi$ is a homotopy relating $\eta \circ f\circ \psi$ with $\eta \circ g\circ \psi$.

Proof. 

The composition $\eta \circ s\circ \psi$ is a $(({}^{3}B,{}^{4}B),({}^{3}B,{}^{4}B))$-generic homomorphisms of $\mathbf{Z}$-degree 1 from ${}^3\mathcal{C}$ into ${}^4\mathcal{C}$. Then using Definitions 5 and 9 one verifies the assertion of this proposition. □

Corollary 1.

Let ${}^k\mathcal{C}$ be ${}^{k}G$-graded ${}^{k}B$-complexes, where $k\in \{1,2,3\}$, let also $f^j$ and $g^j$ be $(({}^{j}B,{}^{j+1}B),({}^{j}B,{}^{j+1}B))$-generic homomorphisms from ${}^j\mathcal{C}$ into ${}^{j+1}\mathcal{C}$ for $j=1$ and $j=2$. Let $s^j$ be a homotopy of $f^j$ with $g^j$ for $j=1$ and $j=2$. Then $s^2\circ f^1+g^2\circ s^1$ is a homotopy relating $f^2\circ f^1$ with $g^2\circ g^1$.

Proof. 

From the conditions of this Corollary and Definition 2 it follows that the homomorphisms $s^2\circ f^1$, $f^2\circ f^1$, $g^2\circ f^1$, $g^2\circ s^1$, $g^2\circ f^1$ and $g^2\circ g^1$ are $(({}^{1}B,{}^{3}B),({}^{1}B,{}^{3}B))$-generic. In view of Proposition 3 $s^2\circ f^1$ relates $f^2\circ f^1$ with $g^2\circ f^1$, while $g^2\circ s^1$ relates $g^2\circ f^1$ with $g^2\circ g^1$, hence $s^2\circ f^1+g^2\circ s^1$ relates $f^2\circ f^1$ with $g^2\circ g^1$. □

Definition 10.

A $(({}^{1}B,{}^{2}B),({}^{1}B,{}^{2}B))$-generic homomorphism $f^1:{}^1\mathcal{C}\to {}^2\mathcal{C}$ of ${}^{k}G$-graded ${}^{k}B$-complexes, where $k\in \{1,2\}$, is called a homotopism if there exists a $(({}^{2}B,{}^{1}B),({}^{2}B,{}^{1}B))$-generic homomorphism $f^2:{}^2\mathcal{C}\to {}^1\mathcal{C}$ such that $f^2\circ f^1$ and $f^1\circ f^2$ are homotopic to $1_{{}^1\mathcal{C}}$ and $1_{{}^2\mathcal{C}}$, respectively. The complex ${}^1\mathcal{C}$ is homotopic to 0, if $1_{{}^1\mathcal{C}}$ is homotopic to $0_{{}^1\mathcal{C}}$.

Proposition 4.

If $f^1$ is a homotopism, then it is a homologism. Moreover, if $g^1$ is homotopic to $f^1$, then $g^1$ also is a homotopism.

Proof. 

In the notation of Definition 10 $H\left(f^2\right)\circ H\left(f^1\right)=H(f^2\circ f^1)=H\left(1_{{}^1\mathcal{C}}\right)=1_{H\left({}^1\mathcal{C}\right)}$ by Proposition 2. Similarly $H\left(f^1\right)\circ H\left(f^2\right)=1_{H\left({}^2\mathcal{C}\right)}$. Hence, $H\left(f^1\right)$ is bijective and $f^1$ is a homologism (see Remark 1).

Then for $f^1$ and $g^1$ one gets that $(f^2\circ g^1)$ is homotopic to $(f^2\circ f^1)$, consequently, to $1_{{}^1\mathcal{C}}$. Analogously $(g^1\circ f^2)$ is homotopic to $(f^1\circ f^2)$, consequently, to $1_{{}^2\mathcal{C}}$ by Proposition 3. Thus, $g^1$ is the homotopism. □

Proposition 5.

Suppose that $(\mathcal{C},d)$ is a G-graded B-complex (see Definition 5). Then the following conditions are equivalent:

$\left(55\right)$ there exists a B-generic homotopism of $(\mathcal{C},d)$ onto $\left(H\right(\mathcal{C}),0)$;

$\left(56\right)$ there exists a B-generic and $\mathbf{Z}$-graded of degree 1 endomorphism s of the module $\mathcal{C}$ for which $d=d\circ s\circ d$;

$\left(57\right)$$\mathcal{B}\left(\mathcal{C}\right)$ and $\mathcal{Z}\left(\mathcal{C}\right)$ are direct multipliers of $\mathcal{C}$;

$\left(58\right)$$(\mathcal{C},d)$ is a direct sum of subcomplexes, which have length either 0 or 1 and zero homology.

Proof. 

$\left(55\right)$⇒$\left(56\right)$. Assume that $f:\mathcal{C}\to H\left(\mathcal{C}\right)$ is a homotopism. Therefore, there exists a morphism of complexes $g:H\left(\mathcal{C}\right)\to \mathcal{C}$ and the B-generic and $\mathbf{Z}$-graded of degree 1 endomorphism s of the module $\mathcal{C}$ such that $g\circ f=1_{\mathcal{C}}-s\circ d-d\circ s$. From $d\circ g=0$ and $g\circ 0=0$ it follows that $d\circ g\circ f=0$ and consequently, $d-d\circ s\circ d-d\circ d\circ s=d-d\circ s\circ d=0$. The latter implies $\left(56\right)$.

$\left(56\right)$⇒$\left(57\right)$. Let s be an endomorphism provided by Condition $\left(56\right)$. Hence, $d\circ (1_{\mathcal{C}}-s\circ d)=0$. The maps d and s are B-generic, hence $1_{\mathcal{C}}-s\circ d$ is the $\mathcal{T}$-linear B-generic projector from $\mathcal{C}$ onto $\mathcal{Z}\left(\mathcal{C}\right)$. On the other hand, $d\circ s\circ d=d$, consequently, $d\circ s$ is the $\mathcal{T}$-linear B-generic projector from $\mathcal{C}$ onto $\mathcal{B}\left(\mathcal{C}\right)$.

$\left(57\right)$⇒$\left(58\right)$. For each integer n we consider ${\mathcal{Z}}_n={\mathcal{Z}}_n\left(\mathcal{C}\right)$ and ${\mathcal{B}}_n={\mathcal{B}}_n\left(\mathcal{C}\right)$. Choose $G^{\left|n\right|+2}$-graded B-sub-bimodules ${\mathcal{P}}_n$ and ${\mathcal{Q}}_n$ in ${\mathcal{C}}_n$ for which ${\mathcal{C}}_n={\mathcal{Z}}_n\oplus {\mathcal{Q}}_n$ and ${\mathcal{Z}}_n={\mathcal{B}}_n\oplus {\mathcal{P}}_n$. By $\left(\mathcal{C}\right(p),d(p\left)\right)$ we denote the p-th translate of $(\mathcal{C},d)$, where $\mathcal{C}{\left(p\right)}_n={\mathcal{C}}_{n+p}$ and $\mathcal{C}{\left(p\right)}^n={\mathcal{C}}^{n-p}$, $d\left(p\right)={(-1)}^{p}d$. We take ${\mathcal{S}}_{\left(n\right)}:={\mathcal{P}}_n(-n)$ and ${\mathcal{T}}_{\left(n\right)}={\mathcal{Q}}_n(-n)\oplus {\mathcal{B}}_{n-1}(1-n)$ subcomplexes in $(\mathcal{C},d)$, because $\mathcal{Z}\left(\mathcal{C}\right(p\left)\right)=\mathcal{Z}\left(\mathcal{C}\right)\left(p\right)$, $\mathcal{B}\left(\mathcal{C}\right(p\left)\right)=\mathcal{B}\left(\mathcal{C}\right)\left(p\right)$ and $H\left(\mathcal{C}\right(p\left)\right)=H\left(\mathcal{C}\right)\left(p\right)$. This gives $(\mathcal{C},d)={⨁}_{n\in \mathbf{Z}}({\mathcal{S}}_{\left(n\right)}\oplus {\mathcal{T}}_{\left(n\right)})$. Note that for each $n\in \mathbf{Z}$ the complex ${\mathcal{S}}_{\left(n\right)}$ is either nil or of null length, while ${\mathcal{T}}_{\left(n\right)}$ is either nil or of length one with zero homology. The latter implies $\left(58\right)$.

$\left(58\right)$⇒$\left(55\right)$. Condition $\left(55\right)$ is satisfied if the complex $\mathcal{C}$ is of null length or of length one with zero homology. □

Definition 11.

It is said that a G-graded B-complex $(\mathcal{C},d)$ is split, if it satisfies equivalent conditions of Proposition 5. A B-generic endomorphism s of $\mathcal{C}$ satisfying $\left(56\right)$ is called a splitting of $\mathcal{C}$.

Let X be a G-graded B-bimodule, and let $(\mathcal{P},d)$ be a G-graded B-complex and let $(\mathcal{P},d)$ be null from the right (or left) and let $p:\mathcal{P}\to X$ be a homologism. Then the pair $(\mathcal{P},p)$ (or $(p,\mathcal{P})$ respectively) is called a left (or right, respectively) G-graded resolution of X. A length of the G-graded B-complex $(\mathcal{P},d)$ is called a length of the resolution. If $(\mathcal{P},p)$ and $({}^1\mathcal{P},{}^{1}p)$ are two left resolutions (or two right resolutions $(p,\mathcal{P})$ and $({}^{1}p,{}^1\mathcal{P})$) and $f:\mathcal{P}\to {}^1\mathcal{P}$ is a B-generic or B-exact morphism of complexes such that ${}^{1}p\circ f=p$ (or $f\circ p={}^{1}p$ 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 $\mathcal{C}$ 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 $A=\mathcal{T}\left[G\right]$ be a metagroup algebra of a (nonassociative) metagroup G over a commutative associative unital ring $\mathcal{T}$, let B be a G-graded A-algebra. We put

$\left(59\right)$${\mathcal{K}}_n=0$ for each $n<-1$,

$\left(60\right)$${\mathcal{K}}_{-1}=B$, ${\mathcal{K}}_0=B{\otimes }_{\mathcal{T}}B$ and by induction

$\left(61\right)$${\mathcal{K}}_{n+1}={\mathcal{K}}_n{\otimes }_{\mathcal{T}}B$ for each natural number n (see Definition 2.13). Therefore ${\mathcal{K}}_n$ is supplied with a B-bimodule structure.

Proposition 6.

The B-bimodule ${\mathcal{K}}_n$ is $G^{\left|n\right|+2}$-graded for each $n\in \mathbf{Z}$. Moreover, if G, $\mathcal{T}$ and B are nontrivial, then ${\mathcal{K}}_n$ is essentially $G^{n+2}$-graded for each $n\ge -1$.

Proof. 

If $n<0$ and ${\mathcal{K}}_n=0$, then it can be supplied with the trivial $G^{\left|n\right|+2}$-gradation such that ${\left({\mathcal{K}}_n\right)}_{{\{g_0,\dots ,g_{\left|n\right|+1}\}}_{l(n+2)}}=0$ for each $g_0,\dots ,g_{\left|n\right|+1}$ in G. If $n<0$ and ${\mathcal{K}}_n=B$, then it is G-graded and by Lemma 3 it can be supplied with the $G^{\left|n\right|+2}$-graded structure.

For each $n\in \mathbf{N}$ it satisfies the following identities:

$\left(62\right)$$\forall p\in \mathcal{T}$, $p·(z_{g_0},\dots ,z_{g_{n+1}})=(\left(p{z}_{g_0}\right),\dots ,z_{g_{n+1}})$ and

$(z_{g_0},\dots ,\left(z_{g_{n+1}}p\right))=(z_{g_0},\dots ,z_{g_{n+1}})·p$ and

$\forall j\in \{1,\dots ,n\}$, $p·(z_{g_0},\dots ,z_{g_{n+1}})=(z_{g_0},\dots ,(p{z}_{g_j}),\dots ,z_{g_{n+1}})$ and

$(z_{g_0},\dots ,(z_{g_j}p),\dots ,z_{g_{n+1}})=(z_{g_0},\dots ,z_{g_{n+1}})·p$,

where $0·(z_{g_1},\dots ,z_{g_n})=0$;

$\left(63\right)$$\left(z_g{z}_y\right)·(z_{g_0},\dots ,z_{g_{n+1}})={\mathsf{t}}_3·(z_g·(z_y·(z_{g_0},\dots ,z_{g_{n+1}})))$

with ${\mathsf{t}}_3={\mathsf{t}}_3(g,y,b)$;

$\left(64\right)$${\mathsf{t}}_3·((z_{g_0},\dots ,z_{g_{n+1}})·\left(z_g{z}_y\right))=((z_{g_0},\dots ,z_{g_{n+1}})·z_g)·z_y$ with ${\mathsf{t}}_3={\mathsf{t}}_3(b,g,y)$;

$\left(65\right)$$(z_g·(z_{g_0},\dots ,z_{g_{n+1}}))·z_y={\mathsf{t}}_3·(z_g·((z_{g_0},\dots ,z_{g_{n+1}})·z_y))$ with ${\mathsf{t}}_3={\mathsf{t}}_3(g,b,y)$

$\left(66\right)$$z_g·(z_{g_0},\dots ,z_{g_{n+1}})=$

$t_{n+3}(g,g_0,\dots ,g_{n+1};v_0(n+3);l(n+3))·(\left(z_g{z}_{g_0}\right),z_{g_1},\dots ,z_{g_{n+1}})$

where ${\{g,g_0,\dots ,g_{n+1}\}}_{v_0(n+3)}=g{\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}$,

${\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}={\{g_0,\dots ,g_n\}}_{l(n+1)}{g}_{n+1}$,

${\left\{g_0\right\}}_{l\left(1\right)}=g_0$, ${\left\{g_0{g}_1\right\}}_{l\left(2\right)}=g_0{g}_1$;

where $b={\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}$,

$t_n(g_1,\dots ,g_n;u\left(n\right),w\left(n\right)):=t_n(g_1,\dots ,g_n;u\left(n\right),w\left(n\right)|id)$

using shortened notation;

$\left(67\right)$$(z_{g_0},\dots ,z_{g_{n+1}})·z_g=t_{n+3}(g_0,\dots ,g_{n+1},g;l(n+3),v_{n+2}(n+3))·(z_{g_0},\dots ,z_{g_n},\left(z_{g_{n+1}}{z}_g\right))$

for every $g,y,g_0,\dots ,g_{n+1}$ in G, $z_g\in B_g$, $z_y\in B_y$, $z_{g_j}\in B_{g_j}$ for each $j\in \{0,\dots ,n+1\}$, where $(z_{g_0},\dots ,z_{g_{n+1}})$ is a shortened notation of the left ordered tensor product

$(\dots ((z_{g_0}\otimes z_{g_1})\otimes z_{g_2})\dots \otimes z_{g_n})\otimes z_{g_{n+1}}$,

${\{g_0,\dots ,g_{n+1},g\}}_{v_{n+2}(n+3)}={\{g_0,\dots ,g_n,g_{n+1}g\}}_{l(n+2)}$.

If n is nonnegative, $0\le n\in \mathbf{Z}$, then from $\left(47\right)$ and $\left(48\right)$ it follows that

$\left(68\right)$${\mathcal{K}}_n={\sum }_{g_0\in G,\dots ,g_{n+1}\in G}{\left({\mathcal{K}}_n\right)}_{{\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}}$

with ${\left({\mathcal{K}}_n\right)}_{{\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}}$ consisting of all elements $z_{{\{g_0,\dots ,g_{n+1}\}}_{l(n+2)}}$ which are sums of elements of the form $(z_{g_0},\dots ,z_{g_{n+1}})$ with $z_{g_j}\in B_{g_j}$ for each $j\in \{0,\dots ,n+1\}$ (see Remark 2). From Identities $\left(62\right)$–$\left(67\right)$ it follows that the B-bimodule ${\mathcal{K}}_n$ satisfies Conditions $\left(20\right)$–$\left(24\right)$.

If G, $\mathcal{T}$ and B are nontrivial, then Identities $\left(60\right)$–$\left(68\right)$ and Definitions 3, 2.13 imply that ${\mathcal{K}}_n$ is essentially $G^{n+2}$-graded for each $n\ge -1$. □

Proposition 7.

Let the algebra B and the B-bimodules ${\mathcal{K}}_n$ be as in Remark 2. Then an acyclic left B-complex $\left(\mathcal{K}\right(B),\partial )$ exists.

Proof. 

We take the B-bimodules ${\mathcal{K}}_n$ for each $n\ge 0$ as in Remark 2 and ${\mathcal{K}}_{-1}=B$. In view of Proposition 6 the B-bimodule ${\mathcal{K}}_n$ is $G^{\left|n\right|+2}$-graded for each $n\in \mathbf{Z}$. 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 $A{1}_B$, where $1_B=1$ is the unit element in B. Therefore $\mathcal{T}{1}_B\subset \mathsf{C}\left(B\right)\subset \mathsf{C}\left(A\right)1_B$ (see Definition 1).

Then we describe a boundary $\mathcal{T}$-linear operator ${\partial }_n:{\mathcal{K}}_n\to {\mathcal{K}}_{n-1}$ on ${\mathcal{K}}_n$ for each natural number n. Using the decomposition $\left(68\right)$ it is sufficient to give it at first on $(z_{g_0},\dots ,z_{g_{n+1}})$ for every $g,y,g_0,\dots ,g_{n+1}$ in G, $z_g\in B_g$, $z_y\in B_y$, $z_{g_j}\in B_{g_j}$ for each $j\in \{0,\dots ,n+1\}$. Then it has by the $\mathcal{T}$-linearity an extension on ${\mathcal{K}}_n$. Therefore we put:

$\left(69\right)$${\partial }_n((z_g·(z_{g_0},z_{g_1},\dots ,z_{g_n},z_{g_{n+1}}))·z_y)=$

${\sum }_{j=0}^n{(-1)}^j·t_{n+4}(g,g_0,\dots ,g_{n+1},y;l(n+4),u_{j+1}(n+4))$

$·((z_g·(<z_{g_0},z_{g_1},\dots ,z_{g_{n+1}}{>}_{j+1,n+2}))·z_y)$, where

$\left(70\right)$$<z_{g_0},\dots ,z_{g_{n+1}}{>}_{1,n+2}:=(\left(z_{g_0}{z}_{g_1}\right),z_{g_2},\dots ,z_{g_{n+1}})$,

$\left(71\right)$$<z_{g_0},\dots ,z_{g_{n+1}}{>}_{2,n+2}:=(z_{g_0},\left(z_{g_1}{z}_{g_2}\right),z_{g_3},\dots ,z_{g_{n+1}})$,…,

$\left(72\right)$$<z_{g_0},\dots ,z_{g_{n+1}}{>}_{n+1,n+2}:=(z_{g_0},\dots ,z_{g_{n-1}},\left(z_{g_n}{z}_{g_{n+1}}\right))$,

$\left(73\right)$${\partial }_0(z_g·(z_{g_0},z_{g_1}))·z_y=(z_g·\left(z_{g_0}{z}_{g_1}\right))·z_y$,

$\left(74\right)$${\{g_0,g_1,\dots ,g_{n+1}\}}_{l(n+2)}:=(\dots \left(\left(g_0{g}_1\right)g_2\right)\dots )g_{n+1}$;

$\left(75\right)$${\{g,g_0,\dots ,g_{n+1},y\}}_{u_1(n+4)}:=(g{\{\left(g_0{g}_1\right),g_2,\dots ,g_{n+1}\}}_{l(n+1)})y$,…,

$\left(76\right)$${\{g,g_0,\dots ,g_{n+1},y\}}_{u_{n+1}(n+4)}:=(g{\{g_0,g_1,\dots ,\left(g_n{g}_{n+1}\right)\}}_{l(n+1)})y$

for each $g,g_0,\dots ,g_{n+1},y$ in G.

Then Formulas $\left(1\right)$ and $\left(2\right)$ in Definition 1 imply that

$t_{n+4}(g,g_0,\dots ,g_{n+1},y;l(n+4),u_{j+1}(n+4))=$

$t_{n+2}(g_0,\dots ,g_{n+1};l(n+2),v_{j+1}(n+2))$ for each $j=0,\dots ,n$, where

$\left(77\right)$${\{g_0,\dots ,g_{n+1}\}}_{v_1(n+2)}:={\{\left(g_0{g}_1\right),g_2,\dots ,g_{n+1}\}}_{l(n+1)}$,…,

$\left(78\right)$${\{g_0,\dots ,g_{n+1}\}}_{v_{n+1}(n+2)}:={\{g_0,g_1,\dots ,\left(g_n{g}_{n+1}\right)\}}_{l(n+1)}$

for every $g_0,\dots ,g_{n+1}$ in G. This means that ${\partial }_n$ is a left and right B-homomorphism of B-bimodules, consequently, ${\partial }_n$ is B-exact. Particularly,

$\left(79\right)$${\partial }_1((z_g·(z_{g_0},z_{g_1},z_{g_2}))·z_y)=(z_g·(\left(z_{g_0}{z}_{g_1}\right),z_{g_2}))·z_y-{\mathsf{t}}_3(g_0,g_1,g_2)·(z_g·(z_{g_0},\left(z_{g_1}{z}_{g_2}\right)))·z_y$,

$\left(80\right)$${\partial }_2((z_g·(z_{g_0},z_{g_1},z_{g_2},z_{g_3}))·z_y)=(z_g·(\left(z_{g_0}{z}_{g_1}\right),z_{g_2},z_{g_3}))·z_y-t_4(g_0,\dots ,g_3;l\left(4\right),v_2\left(4\right))·((z_g·(z_{g_0},\left(z_{g_1}{z}_{g_2}\right),z_{g_3}))·z_y)$

$+t_4(g_0,\dots ,g_3;l\left(4\right),v_3\left(4\right))((z_g·(z_{g_0},z_{g_1},\left(z_{g_2}{z}_{g_3}\right)))·z_y)$.

Then we define a $\mathcal{T}$-linear homomorphism ${\mathbf{s}}_n:{\mathcal{K}}_n\to {\mathcal{K}}_{n+1}$, which has the form:

$\left(81\right)$${\mathbf{s}}_n(z_{g_0},\dots ,z_{g_{n+1}})=(1,z_{g_0},\dots .,z_{g_{n+1}})$

for every $g_0,\dots ,g_{n+1}$ in G. From Formula $\left(1\right)$ in Lemma 1 in [28] and the identities $\left(12\right)$–$\left(14\right)$ in Proposition 1 in [28] and $\left(6\right)$ in Definition 1 it follows that

$\left(82\right)$${\mathbf{s}}_n((z_{g_0},\dots ,z_{g_{n+1}})·z_y)=\left({\mathbf{s}}_n(z_{g_0},\dots ,z_{g_{n+1}})\right)·z_y$

for every $g_0,\dots ,g_{n+1},y$ in G, $z_{g_j}\in B_{g_j}$ for each $j\in \{0,\dots ,n+1\}$, $z_y\in B_y$.

We put ${\mathbf{p}}_n:{\mathcal{K}}_{n+1}\to {\mathcal{K}}_n$ to be a $\mathcal{T}$-linear mapping such that

$\left(83\right)$${\mathbf{p}}_n(a\otimes b)=a·b$ and ${\mathbf{p}}_n(b\otimes a)=b·a$ for each $a\in {\mathcal{K}}_n$ and $b\in B$. Hence Formulas $\left(13\right)$ and $\left(14\right)$ in Proposition 1 in [28] and $\left(82\right)$, $\left(83\right)$ imply that ${\mathbf{p}}_n{\mathbf{s}}_n=id$ is the identity on ${\mathcal{K}}_n$, consequently, ${\mathbf{s}}_n$ is a monomorphism.

Therefore, from Formulas $\left(69\right)$ and $\left(79\right)$, $\left(82\right)$ it follows that

$\left(84\right)$$({\partial }_{n+1}{\mathbf{s}}_n+{\mathbf{s}}_{n-1}{\partial }_n)(z_{g_0},\dots ,z_{g_{n+1}})=$

$={\partial }_{n+1}(1,z_{g_0},\dots ,z_{g_{n+1}})+$

${\mathbf{s}}_{n-1}({\sum }_{j=0}^n{(-1)}^j{t}_{n+2}(g_0,\dots ,g_{n+1};l(n+2),v_{j+1}(n+2))·$

$<z_{g_0},\dots ,z_{g_{n+1}}{>}_{j+1,n+2}\left)\right)=$

$={\sum }_{j=0}^{n+1}{(-1)}^j{t}_{n+3}(1,g_0,\dots ,g_{n+1};l(n+3),v_{j+1}(n+3))·$

$<1,z_{g_0},z_{g_1},\dots ,z_{g_{n+1}}{>}_{j+1,n+3}+$

$+{\sum }_{j=0}^n{(-1)}^j{t}_{n+2}(g_0,\dots ,g_{n+1};l(n+2),v_{j+1}(n+2))·$

$<1,z_{g_0},\dots ,z_{g_{n+1}}{>}_{j+2,n+3}=(z_{g_0},\dots ,z_{g_{n+1}})$,

for every $g_0$,…,$g_{n+1}$ in G, $z_{g_j}\in B_{g_j}$ for each $j\in \{0,\dots ,n+1\}$.

Then Formulas $\left(82\right)$–$\left(84\right)$ imply the homotopy conditions

$\left(85\right)$${\partial }_{n+1}{\mathbf{s}}_n+{\mathbf{s}}_{n-1}{\partial }_n=I$ for each $n\ge 0$,

where I denotes the identity operator on ${\mathcal{K}}_n$. This leads to identities:

${\partial }_n{\partial }_{n+1}{\mathbf{s}}_n={\partial }_n(I-{\mathbf{s}}_{n-1}{\partial }_n)={\partial }_n-\left({\partial }_n{\mathbf{s}}_{n-1}\right){\partial }_n={\partial }_n-(I-{\mathbf{s}}_{n-2}{\partial }_{n-1}){\partial }_n$ and hence gives the recurrence relation

$\left(86\right)$${\partial }_n{\partial }_{n+1}{\mathbf{s}}_n={\mathbf{s}}_{n-2}{\partial }_{n-1}{\partial }_n$.

Notice that Formula $\left(81\right)$ implies that ${\mathcal{K}}_{n+1}$ as the left B-module is generated by ${\mathbf{s}}_n{\mathcal{K}}_n$. Utilizing the recurrence relation $\left(86\right)$ by induction in n we infer:

${\partial }_n{\partial }_{n+1}=0$ for each $n\ge 0$,

since ${\partial }_0{\partial }_1=0$ by Formulas $\left(69\right)$ and $\left(73\right)$.

Let $B^e:=B{\otimes }_A{B}^{op}$ be the enveloping algebra of B, where $B^{op}$ denotes an opposite algebra. The latter as an $\mathsf{C}\left(B\right)$-linear space is the same, but with the multiplication $x\circ y=yx$ for each $x,y\in B^{op}$. This permits to consider the $G^2$-graded B-bimodule ${\mathcal{K}}_0$ as $B^e$. Therefore, the mapping ${\partial }_0:{\mathcal{K}}_0\to {\mathcal{K}}_{-1}$ provides the augmentation $ϵ:B^e\to B$.

Thus, according to identities $\left(85\right)$ the left complex $\mathcal{K}\left(B\right)$ is acyclic:

$\left(87\right)$$0\leftarrow B{}_{\overleftarrow{{\partial }_0}}{\mathcal{K}}_0{}_{\overleftarrow{{\partial }_1}}{\mathcal{K}}_1{}_{\overleftarrow{{\partial }_2}}{\mathcal{K}}_2{}_{\overleftarrow{.}}..{}_{\overleftarrow{{\partial }_n}}{\mathcal{K}}_n{}_{\overleftarrow{{\partial }_{n+1}}}{\mathcal{K}}_{n+1}\leftarrow \dots$. □

Remark 3.

For a G-graded left B-module X let $\left(\mathcal{K}\right(B,X),d)$ be a complex such that ${\mathcal{K}}_n(B,X)={\mathcal{K}}_n{⨂}_{B}X$ for each $0\le n\in \mathbf{Z}$ with $d_n={\partial }_n\otimes I_X$ for each $n\ge 1$ (see Proposition 7 and Remark 2), while ${\mathcal{K}}_n(B,X)=0$ and $d_{n+1}=0$ for each $n<0$, where $I_X$ denotes a unit operator on X, such that $I_{X}x=x$ for each $x\in X$. Let a map ${ϵ}_X:{\mathcal{K}}_0(B,X)\to X$ be defined by the following formula:

$\left(88\right)$${ϵ}_X((a\otimes b)\otimes x)=\left(ab\right)x$

for each a and b in B and $x\in X$. Formula $\left(88\right)$ above, Conditions $\left(A3.1\right)$–$\left(A3.3\right)$ in Definition A3 and the identities $\left(73\right)$, $\left(79\right)$ imply that

$\left(89\right)$${ϵ}_X\circ d_1=0$.

This procedure induces a $\mathbf{Z}$-graded homomorphism

$\left(90\right)$${\widehat{ϵ}}_X:\mathcal{K}(B,X)\to X$.

Proposition 8.

The map ${\widehat{ϵ}}_X$ (see Remark 2) is a homotopism of complexes of left B-modules. The complex $\left(\mathcal{K}\right(B,X),d)$ splits as a G-graded left B-complex and $(\mathcal{K}(B,X),{\widehat{ϵ}}_X)$ is a left resolution of the G-graded left B-module X.

Proof. 

For each $n\ge 0$ there exists a $\mathcal{T}$-linear map

$\left(91\right)$$v_n{\left\{(b_0\otimes \dots \otimes b_{n+1})\right\}}_{l(n+2)}={\left\{(1\otimes b_0\otimes \dots \otimes b_{n+1})\right\}}_{l(n+3)}$

for each $b_0$,…,$b_{n+1}$ in B, where ${\left\{(b_0\otimes b_1)\right\}}_{l\left(2\right)}=b_0\otimes b_1$ and by induction ${\left\{(b_0\otimes \dots \otimes b_{n+1})\right\}}_{l(n+2)}={\left\{(b_0\otimes \dots \otimes b_n)\right\}}_{l(n+1)}\otimes b_{n+1}$ for each $n\ge 2$. This $v_n$ is a homomorphism from ${\mathcal{K}}_n$ into ${\mathcal{K}}_{n+1}$ as right A-modules. Proposition 7, Formula $\left(91\right)$, Lemma 1, Remark 2 and Definitions 1, A1, A2 imply that

$\left(92\right)$$d_{n+1}\circ v_n+v_{n-1}\circ d_n=1_{{\mathcal{K}}_n}$

for each $n\ge 1$, because $t_3(e,g_1,g_2)=e$ for each $g_1$ and $g_2$ in G, where e is the unit element in G. In particular,

$\left(93\right)$$d_1\circ v_0(b_0\otimes b_1)=b_0\otimes b_1-1\otimes \left(b_0{b}_1\right)$

for each $b_0$ and $b_1$ in B. A map $\xi :B\to B{⨂}_{\mathcal{T}}B$ such that $\xi \left(b\right)=1\otimes b$ for each $b\in B$ induces a $\mathcal{T}$-linear homomorphism $\widehat{\xi }:B\to \mathcal{K}\left(B\right)$. This implies that ${\widehat{ϵ}}_B\circ \widehat{\xi }=I_B$. From Formulas $\left(92\right)$ and $\left(93\right)$ it follows that $d\circ v+v\circ d=I_{\mathcal{K}\left(B\right)}-\widehat{\xi }\circ {\widehat{ϵ}}_B$. Then defining ${\widehat{\xi }}_X=\widehat{\xi }\otimes I_X$, $d_X=d\otimes I_X$, $v_X=v\otimes I_X$ we infer that ${\widehat{ϵ}}_X\circ {\widehat{\xi }}_X=I_X$ and $d_X\circ v_X+v_X\circ d_X=I_{\mathcal{K}(B,X)}-{\widehat{\xi }}_X\circ {\widehat{\eta }}_X$. The homomorphisms $d_X$, $v_X$, ${\widehat{ϵ}}_X$ and ${\widehat{\xi }}_X$ are B-generic, since the homomorphisms d, v, $\widehat{ϵ}$ and $\widehat{\xi }$ are B-generic. Thus, ${\widehat{ϵ}}_X$ is a homotopism (see Definition 10). Proposition 7 implies that $(\mathcal{K}(B,X),{\widehat{ϵ}}_X)$ is a left resolution of the G-graded left B-module X, because ${\widehat{ϵ}}_X$ is the homotopism. □

Definition 12.

The left resolvent $(\mathcal{K}(B,X),{\widehat{ϵ}}_X)$ for X is called the standard resolvent of the G-graded left B-module X.

3. Smashed Torsion Product

Definition 13.

Let G be a metagroup, $\mathcal{T}$ be a commutative associative unital ring, $A=\mathcal{T}\left[G\right]$ be a metagroup algebra, let B be a G-graded unital A-algebra. Let also $(\mathcal{C},d)$ and $({}^1\mathcal{C},{}^{1}d)$ be G-graded B-complexes, where either $\mathcal{C}$ is the B-bimodule and ${}^1\mathcal{C}$ is the left B-module or $\mathcal{C}$ is the right B-module and ${}^1\mathcal{C}$ is the B-bimodule (see also Definitions 5, 7 and Remark 1). Let a G-smashed tensor product $\mathcal{C}{⨂}_B{}^1\mathcal{C}$ be supplied also with the $\mathbf{Z}$-gradation

$\left(94\right)$${\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)}_n={\sum }_{j+l=n}\left({\mathcal{C}}_j{⨂}_B{}^1{\mathcal{C}}_l\right)$.

We put D to be a $\mathcal{T}$-linear endomorphism of degree $-1$ on $\mathcal{C}{⨂}_B{}^1\mathcal{C}$ such that

$\left(95\right)$$D(x\otimes {}^{1}x)=\left(dx\right)\otimes {}^{1}x+{(-1)}^{j}x\otimes \left({}^{1}dx\right)$

for each $x\in {\mathcal{C}}_j$ and ${}^{1}x\in {}^1{\mathcal{C}}_i$, i and j in $\mathbf{Z}$. The G-graded B-complex $(\mathcal{C}{⨂}_B{}^1\mathcal{C},D)$ is called a G-smashed tensor product (or shortly tensor product) of the complexes $(\mathcal{C},d)$ and $({}^1\mathcal{C},{}^{1}d)$.

Remark 4.

In view of Definitions 5, 13 and A3

$\left(96\right)$$D\circ D=0$,

since $D\circ D(x\otimes {}^{1}x)=(d\circ dx)\otimes {}^{1}x+{(-1)}^{j-1}\left(dx\right)\otimes \left({}^{1}d{}^{1}x\right)$

$+{(-1)}^j\left(dx\right)\otimes \left({}^{1}d{}^{1}x\right)+x\otimes ({}^{1}d\circ {}^{1}d{}^{1}x)$

for each $x\in {\mathcal{C}}_j$ and ${}^{1}x\in {}^1{\mathcal{C}}_i$.

For example, we consider G-graded B-complexes $(\mathcal{C},d)$ and $({}^1\mathcal{C},{}^{1}d)$ like $\left(\mathcal{K}\right(B,X),d)$ and $(\mathcal{K}(B,{}^{1}X),{}^{1}d)$ (see Remark 4). By virtue of Proposition 8 Identities $\left(94\right)$, $\left(95\right)$ and $\left(96\right)$ are satisfied for D naturally induced by d and ${}^{1}d$, since $t_3(g_1,g_2,g_3)\in \mathbf{\Psi }\subset \mathsf{C}\left(G\right)\subset N\left(G\right)$, $t_n(g_1,\dots ,g_n;u\left(n\right),w\left(n\right))\in \mathbf{\Psi }$, $\left(g_1{g}_2\right)y=g_1\left(g_{2}y\right)$, $\left(g_{1}y\right)g_2=g_1\left(y{g}_2\right)$ and $y\left(g_1{g}_2\right)=\left(y{g}_1\right)g_2$ for each $g_1$,…,$g_n$ in G, $y\in N\left(G\right)$, vectors $u\left(n\right)$ and $w\left(n\right)$ indicating an order of multiplications (see Proposition 7).

Thus, this example justifies Definition 13.

In particular, if ${}^1\mathcal{C}$ has ${}^1{\mathcal{C}}_0=X$ and ${}^1{\mathcal{C}}_n=\left(0\right)$ for each $n\ne 0$, then ${\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)}_n={\mathcal{C}}_n{⨂}_{B}X$ for each $n\in \mathbf{Z}$, $D=d\otimes I_X$. Therefore $\mathcal{C}{⨂}_B{B}_s$ is isomorphic with $\mathcal{C}$, where $B_s$ is the algebra B considered as the G-graded left B-module. On the other side, if ${\mathcal{C}}_0=P$ and ${\mathcal{C}}_n=\left(0\right)$ for each $n\ne 0$, then ${\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)}_n=P{⨂}_B{}^1{\mathcal{C}}_n$ for each $n\in \mathbf{Z}$ and $D=I_P\otimes d$.

Proposition 9.

Let $(\mathcal{C},d)$ and $({}^1\mathcal{C},{}^{1}d)$ be G-graded B-complexes and let $\mathcal{C}$ be a B-bimodule and ${}^1\mathcal{C}$ be a left B-module (or $\mathcal{C}$ be a right B-module and ${}^1\mathcal{C}$ be a B-bimodule) (see Definition 13). Then there exists a $\mathcal{T}$-linear $\mathbf{Z}$-graded map $\widehat{h}=\widehat{h}(\mathcal{C},{}^1\mathcal{C})$ of degree 0 from $H\left(\mathcal{C}\right){⨂}_{B}H\left({}^1\mathcal{C}\right)$ into $H\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)$.

Proof. 

Consider $x\in {\mathcal{Z}}_j\left(\mathcal{C}\right)$, $y\in {\mathcal{Z}}_l\left({}^1\mathcal{C}\right)$, where j and l are integers (see Definition 13 and Remark 4). Then the element $x\otimes y$ belongs to ${\mathcal{Z}}_{j+l}\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)$ according to Formula $\left(94\right)$. Therefore, $(x+dw)\otimes (y+dz)=x\otimes y+D(w\otimes y+{(-1)}^j(x+dw)\otimes z)$ for each $w\in {\mathcal{C}}_{j+1}$ and $z\in {}^1{\mathcal{C}}_{l+1}$. This induces a so called canonical $\mathcal{T}$-linear map ${\widehat{h}}_{j,l}:H_j\left(\mathcal{C}\right){⨂}_B{H}_l\left({}^1\mathcal{C}\right)\to H_{j+l}\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)$ such that

$\left(97\right)$${\widehat{h}}_{j,l}(b_g{a}_u,c_v)={\mathsf{t}}_3(g,u,v){\widehat{h}}_{j,l}(b_g,a_u{c}_v)$ and

${\widehat{h}}_{j,l}(a_u{b}_g,c_v)={\mathsf{t}}_3(u,g,v)a_u{\widehat{h}}_{j,l}(b_g,c_v)$

(or ${\widehat{h}}_{j,l}(b_g,c_v{a}_u)={\mathsf{t}}_3(g,v,u){\widehat{h}}_{j,l}(b_g,c_v)a_u$, respectively)

for each $a_g\in {\left(H_j\left(\mathcal{C}\right)\right)}_g$, $c_v\in {\left(H_l\left({}^1\mathcal{C}\right)\right)}_v$, $a_u\in B_u$, g, u and v in G. The left B-bimodule $H\left(\mathcal{C}\right){⨂}_{B}H\left({}^1\mathcal{C}\right)$ (or right, respectively) is supplied with $\mathbf{Z}$-gradation such that

$\left(98\right)$${\left(H\left(\mathcal{C}\right){⨂}_{B}H\left({}^1\mathcal{C}\right)\right)}_n={\sum }_{j+l=n}{H}_j\left(\mathcal{C}\right){⨂}_B{H}_l\left({}^1\mathcal{C}\right)$.

Naturally each $H_j\left(\mathcal{C}\right)$ is the (smashly) $G^{\left|j\right|+2}$-graded B-bimodule and $H_l\left({}^1\mathcal{C}\right)$ is the (smashly) $G^{\left|l\right|+2}$-graded left B-module (or the right B-module and the B-bimodule, respectively), consequently, ${\left(H\left(\mathcal{C}\right){⨂}_{B}H\left({}^1\mathcal{C}\right)\right)}_n$ is the $G^{\left|n\right|+2}$-graded left B-module (or right, respectively) by Lemma 2. Therefore, the family of maps ${\widehat{h}}_{j,l}$ induces a $\mathcal{T}$-linear $\mathbf{Z}$-graded map $\widehat{h}=\widehat{h}(\mathcal{C},{}^1\mathcal{C})$ of degree 0 from $H\left(\mathcal{C}\right){⨂}_{B}H\left({}^1\mathcal{C}\right)$ into $H\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)$. □

Corollary 2.

If the conditions of Proposition 9 are satisfied and the G-graded B-complexes $\mathcal{C}$ and ${}^1\mathcal{C}$ are zero from the right, then the G-graded B-complex $\mathcal{C}{⨂}_B{}^1\mathcal{C}$ is zero from the right and ${\widehat{h}}_{0,0}(\mathcal{C},{}^1\mathcal{C}):H_0\left(\mathcal{C}\right){⨂}_B{H}_0\left({}^1\mathcal{C}\right)\to H_0\left(\mathcal{C}{⨂}_B{}^1\mathcal{C}\right)$ is bijective.

Remark 5.

Assume that $f:(\mathcal{C},d)\to ({}^1\mathcal{C},{}^{1}d)$ and $p:({}^2\mathcal{C},{}^{2}d)\to ({}^3\mathcal{C},{}^{3}d)$ are homomorphisms of G-graded B-complexes, where the pairs $((\mathcal{C},d),({}^2\mathcal{C},{}^{2}d))$ and $(({}^1\mathcal{C},{}^{1}d),({}^3\mathcal{C},{}^{3}d))$ satisfy the conditions of Proposition 9. Then they induce a homomorphism of G-graded B-modules $f\otimes p:\mathcal{C}{⨂}_B{}^2\mathcal{C}\to {}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}$ such that ${(f\otimes p)}_n:{\left(\mathcal{C}{⨂}_B{}^2\mathcal{C}\right)}_n\to {\left({}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}\right)}_n$ for each $n\in \mathbf{Z}$ for $G^{\left|n\right|+2}$-graded B-modules. This means that the homomorphism $f\otimes p$ is $\mathbf{Z}$-graded of zero degree. For derivations D and ${}^{1}D$ of $\mathcal{C}{⨂}_B{}^2\mathcal{C}$ and ${}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}$, respectively, we get that

$(f\otimes p)\left(D(x\otimes y)\right)=f\left(dx\right)\otimes p\left(y\right)+{(-1)}^{j}f\left(x\right)\otimes p\left({}^{2}dy\right)={}^{1}df\left(x\right)\otimes p\left(y\right)+{(-1)}^{j}f\left(x\right)\otimes {}^{3}dp\left(y\right)={}^{1}D(f\left(x\right)\otimes p\left(y\right))$ for each $x\in {\mathcal{C}}_j$, $y\in {}^2{\mathcal{C}}_l$, j and l in $\mathbf{Z}$. In view of Proposition 9 this provides a commutative diagram

$\left(99\right)$${H\left(\mathcal{C}\right){⨂}_{B}H\left({}^2\mathcal{C}\right)}_{\overrightarrow{\widehat{h}(\mathcal{C},{}^2\mathcal{C})}}H\left(\mathcal{C}{⨂}_B{}^2\mathcal{C}\right)$

$H\left(f\right)\otimes H\left(p\right)\downarrow \downarrow H(f\otimes p)$

${H\left({}^1\mathcal{C}\right){⨂}_{B}H\left({}^3\mathcal{C}\right)}_{\overrightarrow{\widehat{h}({}^1\mathcal{C},{}^3\mathcal{C})}}H\left({}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}\right)$.

Proposition 10.

Let $\mathcal{C}$, ${}^1\mathcal{C}$, ${}^2\mathcal{C}$ and ${}^3\mathcal{C}$ be G-graded B-complexes, let the pairs $(\mathcal{C},{}^2\mathcal{C})$ and $({}^1\mathcal{C},{}^3\mathcal{C})$ satisfy the conditions of Proposition 9 and let $f:\mathcal{C}\to {}^1\mathcal{C}$, ${}^{1}f:\mathcal{C}\to {}^1\mathcal{C}$, $p:{}^2\mathcal{C}\to {}^3\mathcal{C}$, ${}^{1}p:{}^2\mathcal{C}\to {}^3\mathcal{C}$ be B-generic homomorphisms of these complexes. Then two homomorphisms $f\otimes p$ and ${}^{1}f\otimes {}^{1}p$ from $\mathcal{C}{⨂}_B{}^2\mathcal{C}$ to ${}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}$ are B-generic.

$\left(100\right)$ If f and p are homotopic to ${}^{1}f$ and ${}^{1}p$, respectively, then two homomorphisms $f\otimes p$ and ${}^{1}f\otimes {}^{1}p$ are homotopic.

$\left(101\right)$ If f and p are homotopisms, then $f\otimes p$ is a homotopism.

$\left(102\right)$ If either $\mathcal{C}$ or ${}^2\mathcal{C}$ is homotopic to zero, then $\mathcal{C}{⨂}_B{}^2\mathcal{C}$ is homotopic to zero.

Proof. 

From $f_{\iota }\left(B\right)=B$, $p_{\iota }\left(B\right)=B$, ${}^{1}f\left(B\right)=B$, ${}^{1}p\left(B\right)=B$ provided by the conditions of this proposition it follows that $(f_{\iota }\otimes p_{\iota })\left(B{\otimes }_{B}B\right)=B{\otimes }_{B}B$ and $({}^1{f}_{\iota }\otimes {}^1{p}_{\iota })\left(B{\otimes }_{B}B\right)=B{\otimes }_{B}B$, consequently, the homomorphisms $f\otimes p$ and ${}^{1}f\otimes {}^{1}p$ are B-generic.

If f and ${}^{1}f$ are homotopic to p and ${}^{1}p$, respectively, then there exist $\mathbf{Z}$-graded B-generic homomorphisms $s:\mathcal{C}\to {}^1\mathcal{C}$ and ${}^{1}s:{}^2\mathcal{C}\to {}^3\mathcal{C}$ of degree 1 such that $f-{}^{1}f=ds+sd$ and $p-{}^{1}p=d{}^{1}s+{}^{1}sd$, where derivations of the G-graded B-complexes $\mathcal{C}$, ${}^1\mathcal{C}$, ${}^2\mathcal{C}$ and ${}^3\mathcal{C}$ are shortly denoted by d. Therefore, there exists a $\mathbf{Z}$-graded homomorphism $S:\mathcal{C}{⨂}_B{}^2\mathcal{C}\to {}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}$ of degree 1 such that $S(x\otimes y)=s\left(x\right)\otimes p\left(y\right)+{(-1)}^j{}^{1}f\left(x\right)\otimes {}^{1}s\left(y\right)$ for each $x\in {\mathcal{C}}_j$, $y\in {}^2{\mathcal{C}}_l$, j and l in $\mathbf{Z}$. Since ${}^{1}f$, p, s, ${}^{1}s$ are B-generic, then S is B-generic. For derivations D of the G-graded B-complexes $\mathcal{C}{⨂}_B{}^2\mathcal{C}$ and ${}^1\mathcal{C}{⨂}_B{}^3\mathcal{C}$ this gives: $(DS+SD)(x\otimes y)=f\left(x\right)\otimes p\left(y\right)-{}^{1}f\left(x\right)\otimes {}^{1}p\left(y\right)$ for each $x\in {\mathcal{C}}_j$, $y\in {}^2{\mathcal{C}}_l$, j and l in $\mathbf{Z}$. Thus $DS+SD=f\otimes p-{}^{1}f\otimes {}^{1}p$, that means that two homomorphisms $f\otimes p$ and ${}^{1}f\otimes {}^{1}p$ are homotopic.

If f and p are homotopisms, then there exist B-generic homomorphisms of complexes $\eta :{}^1\mathcal{C}\to \mathcal{C}$ and ${}^1\eta :{}^3\mathcal{C}\to {}^2\mathcal{C}$ such that $f\circ \eta$, $\eta \circ f$, $p\circ {}^1\eta$, ${}^1\eta \circ p$ are homotopic to $i{d}_{{}^1\mathcal{C}}$, $i{d}_{\mathcal{C}}$, $i{d}_{{}^3\mathcal{C}}$, $i{d}_{{}^2\mathcal{C}}$, respectively. From $\left(100\right)$ it follows that $(f\otimes p)\circ (\eta \otimes {}^1\eta )$ is homotopic to $i{d}_{{}^1\mathcal{C}}\otimes i{d}_{{}^3\mathcal{C}}=i{d}_{{}^1\mathcal{C}\otimes {}^3\mathcal{C}}$, while $(\eta \otimes {}^1\eta )\circ (f\otimes p)$ is homotopic to $i{d}_{\mathcal{C}}\otimes i{d}_{{}^2\mathcal{C}}=i{d}_{\mathcal{C}\otimes {}^2\mathcal{C}}$. This implies that $f\otimes p$ is the homotopism.

The last assertion $\left(102\right)$ of this proposition follows from $\left(101\right)$ in particular for either ${}^2\mathcal{C}=0$ or ${}^3\mathcal{C}=0$, respectively. □

4. Conclusions

The obtained results will be useful for further studies of cohomology theory of nonassociative algebras and noncommutative manifolds with metagroup relations, structure of nonassociative algebras, operator theory, spectral theory over Cayley–Dickson algebras, PDEs, noncommutative analysis, solutions of PDEs with boundary conditions, noncommutative geometry, mathematical physics, and their applications in the sciences. Then there are possible applications in mathematical coding theory, informatics, security of internet resources [29,30,31], because coding frequently uses algebras. This is caused by the fact that conditions imposed on metagroups are weaker than for groups. Utilizing nonassociative algebras with metagroup conditions, it is possible to increase a code complexity comparing it with Lie algebras or group algebras.