The homotopy invariance of dihedral homology of involutive $A_\infty$-algebras over rings
S.V. Lapin
TL;DR
This paper constructs a dihedral homology functor for involutive $A_ abla$-algebras over rings and proves it is invariant under homotopy equivalences, extending the understanding of algebraic invariants in homotopical algebra.
Contribution
It introduces a dihedral homology functor for involutive $A_ abla$-algebras and proves its homotopy invariance over rings, a novel extension in the field.
Findings
Constructed the dihedral homology functor for involutive $A_ abla$-algebras.
Proved the functor preserves homotopy equivalences as isomorphisms.
Extended the invariance properties of dihedral homology to a broader algebraic context.
Abstract
The dihedral homology functor from the category of involutive -algebras over any commutative unital ring to the category of graded -modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Topics in Algebra
The homotopy invariance of dihedral homology of involutive -algebras over rings.
S.V. Lapin
Abstract
The dihedral homology functor from the category of involutive -algebras over any commutative unital ring to the category of graded -modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules.
In [1], on the basis of the combinatorial and homotopy technique of differential modules with -simplicial faces [2]-[8] and -differential modules [9]-[17] the dihedral triple complex of an involutive -algebra over any commutative unital ring was constructed. This triple complex generalizes the dihedral triple complex [18]-[20] of an involutive associative algebra given over an arbitrary commutative unital ring. Further, in [1], dihedral homology of any involutive -algebra over an arbitrary commutative unital ring was defined as the homology of the chain complex associated with the dihedral triple complex of this involutive -algebra. The dihedral homology of involutive -algebras over commutative unital rings introduced in [1] generalizes the dihedral homology of involutive associative algebras over commutative unital rings defined in [18]. It is well known [18]-[20] that over fields of characteristic zero the dihedral homology introduced in [18] is isomorphic to the dihedral homology defined in [21] by using the complex of coinvariants for the action of dihedral groups. Similar to this, in [1], it was shown that over fields of characteristic zero the dihedral homology of involutive -algebras introduced in [1] is isomorphic to the dihedral homology of involutive -algebras defined in [22] by using the complex of coinvariants for the action of dihedral groups. Moreover, in [1], for involutive homotopy unital -algebras over any commutative unital rings, the analogue of the Krasauskas-Lapin-Solov’ev exact sequence was constructed.
On the other hand, in [23], it was established that the homology of the -construction of an -algebra is homotopy invariant, i.e., this homology is invariant under homotopy equivalences of -algebras. Now note that the dihedral triple complex of an involutive -algebra constructed in [1] is the dihedral analogue of the -construction of an -algebra, and the dihedral homology of an involutive -algebra is defined in [1] as the homology of this dihedral analogue of the -construction. This gives rise to an interesting natural question: do the dihedral homology of involutive -algebras is homotopy invariant under the homotopy equivalences of involutive -algebras? In present paper a positive answer to this question is given.
The paper consists of three paragraphs. In the first paragraph, we first recall necessary definitions related to the notion of a dihedral module with -simplicial faces or, more briefly, an -module [1], which homotopy generalizes the notion of a dihedral module with simplicial faces [18]-[20]. After that, the category of -modules is defined, namely, the notion of a morphism of -modules is introduced, and it is shown that the composition of morphisms of -modules is a morphism of -modules. Next, the concept of a homotopy between morphisms of -modules and the notion of a homotopy equivalence of -modules are introduced.
In the second paragraph, we first recall necessary definitions related to the notion of a dihedral homology of -modules [1]. Next, it is shown that the dihedral homology of -modules defines the functor from the category of -modules to the category of graded modules. In addition, it is shown that this functor sends homotopy equivalences of -modules into isomorphisms of graded modules.
In the third paragraph, we first recall necessary definitions related to the notion of an -algebra [23] and an involutive -algebra [22]. After that we give the definitions of a morphism of involutive -algebras and a homotopy between morphisms of involutive -algebras and, moreover, we introduce the notion of a homotopy equivalence of involutive -algebras. Next, we recall the concept of a dihedral homology of involutive -algebras over an arbitrary commutative unital rings [1]. Then, by using results of the second paragraph, it is shown that the dihedral homology of involutive -algebras defines the functor from the category of involutive -algebras to the category of graded modules. Moreover, it is shown that this functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules. As a corollary, we obtain that the dihedral homology of an involutive -algebra over any field is isomorphic to the dihedral homology of the involutive -algebra of homologies for the source involutive -algebra. In particular, it is obtained that the dihedral homology of an involutive associative differential algebra over any field is isomorphic to the dihedral homology of the involutive -algebra of homologies for the source involutive associative differential algebra.
We proceed to precise definitions and statements. All modules and maps of modules considered in this paper are, respectively, -modules and -linear maps of modules, where is any unital (i.e., with unit) commutative ring.
§ 1. Dihedral modules with -simplicial faces and
their morphisms and homotopies
In what follows, by a bigraded module we mean any bigraded module , , , and by a differential bigraded module, or, briefly, a differential module , we mean any bigraded module endowed with a differential of bidegree .
Recall that a differential module with simplicial faces is defined as a differential module together with a family of module maps , , which are maps of differential modules and satisfy the simplicial commutation relations , . The maps are called the simplicial face operators or, more briefly, the simplicial faces of the differential module .
Now, we recall the notion of a differential module with -simplicial faces [2] (see also [3]-[8]), which is a homotopy invariant analogue of the notion of a differential module with simplicial faces.
Let be the symmetric group of permutations on a -element set. Given an arbitrary permutation and any -tuple of nonnegative integers , where , we consider the -tuple , where acts on the -tuple in the standard way, i.e., permutes its components. For the -tuple , we define a -tuple by the following formulae
[TABLE]
where each is the number of those elements of on the right of that are smaller than .
A differential module with -simplicial faces or, more briefly, an -module is defined as a differential module together with a family of module maps
[TABLE]
[TABLE]
which satisfy the relations
[TABLE]
where is the set of all partitions of the -tuple into two tuples and , , such that the conditions and holds.
The family of maps is called the -differential of the -module . The maps that form the -differential of an -module are called the -simplicial faces of this -module.
It is easy to show that, for , relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
It is easy to check that, for any permutation and any -tuple , where , the conditions and are equivalent to the conditions and . This readily implies that the -tuple , which specified in , coincides with the -tuple .
Simplest examples of differential modules with -simplicial faces are differential modules with simplicial faces. Indeed, given any differential module with simplicial faces , we can define the -differential by setting , , and , , thus obtaining the differential module with -simplicial faces .
It is worth mentioning that the notion of an differential module with -simplicial faces specified above is a part of the general notion of a differential -simplicial module introduced in [4] by using the homotopy technique of differential Lie modules over curved colored coalgebras.
Now we recall [18]-[20] that a dihedral differential module with simplicial faces is defined as a differential module with simplicial faces equipped with two families of module maps
[TABLE]
which satisfy the following relations:
[TABLE]
[TABLE]
[TABLE]
Note that if in the definition of a dihedral module with simplicial faces we remove the family of automorphisms , , then we obtain the definition of a cyclic module with simplicial faces [24].
Now, let us recall [1] that a dihedral differential module with -simplicial faces or, more briefly, an -module is defined as any -module together with two families of module maps , , and , , which satisfy the following relations:
[TABLE]
[TABLE]
[TABLE]
Note that if in the definition of a -module we remove the family of automorphisms , , then we obtain the definition of a -module, i.e., cyclic module with -simplicial faces [25] (see also [26]). Moreover, it is worth mentioning that the notion of a -module specified above is a part of the general notion of a dihedral -simplicial module introduced in [27] by using the homotopy technique of differential modules over curved colored coalgebras.
The family of maps is called the -differential of the -module . The maps are called the -simplicial faces of this -module.
Simplest examples of -modules are dihedral differential modules with simplicial faces. Indeed, given any dihedral differential module with simplicial faces, we can define the -differential by setting , , and , , thus obtaining the -module .
Now, we recall that a map of differential modules with simplicial faces is defined as a map of differential modules that satisfies the relations , .
Let us consider the notion of a morphism of differential modules with -simplicial faces [2] (see also [5]), which homotopically generalizes the notion of a map differential modules with simplicial faces.
A morphism of -modules is defined as a family of module maps
[TABLE]
[TABLE]
(at we will use the denotation ), which satisfy the relations
[TABLE]
[TABLE]
[TABLE]
where is the same as in . The maps are called the components of the morphism .
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now, we recall [2] that a composition of an arbitrary given morphisms of -modules and is defined as a morphism of -modules whose components are defined by
[TABLE]
where is the set of all partitions of the -tuple into two tuples and , , such that the conditions and holds.
For example, at the formulae take, respectively, the following form
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Definition 1.1. We define a morphism of -modules
[TABLE]
as any morphism of -modules whose components satisfy the following conditions:
[TABLE]
[TABLE]
Note that if in the definition of a morphism of -modules we remove the family of automorphisms , , then we obtain the definition of a morphism of -modules, i.e., a morphism of cyclic modules with -simplicial faces [28]. It means that any morphism of -modules always is the morphism of -modules .
By using the fact that any morphism of -modules is a morphism of -modules we define the composition of morphisms of -modules as a composition of morphisms of -modules.
Theorem 1.1. The composition of morphisms of -modules is a morphism of -modules.
Proof. Given any morphisms of -modules and , we need to check that components of the morphism of -modules satisfy the relations and . In [28] it was shown that the composition of morphisms of -modules is a morphism of -modules. It implies that components of the morphism of -modules satisfy the relations . Now, we check that the module maps satisfy the relations . It is clearly that at we have . At by the definition of a composition of morphisms of -modules we have
[TABLE]
[TABLE]
[TABLE]
Let us show that each summand on the right-hand side of is equal to some summand on the right-hand side of . Given any fixed permutation , consider the summand
[TABLE]
on the right-hand side of . By using the relations we obtain
[TABLE]
[TABLE]
where
[TABLE]
Now, given the permutation and the partition
[TABLE]
we define the permutation of the collection by the following formulae:
[TABLE]
[TABLE]
For more clarity, it is worth saying that the permutation is the product of the following permutations:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Comparing the tuples and , we see that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Since , the right-hand side of contains the summand
[TABLE]
It is clear that the following equality holds:
[TABLE]
[TABLE]
Thus, we have shown that each summand on the right-hand side of is equal to a summand on the right-hand side of . It follows that the right-hand sides of and are equal, because the number of summands on the right-hand side of equals that on the right-hand side of and, moreover, the specified above permutations and uniquely determine one another.
It is clear that the associativity of the composition of morphisms of -modules implies the associativity of the composition of morphisms of -modules. Moreover, for each -module , there is the identity morphism
[TABLE]
where and for all . Thus, the class of all -modules over any commutative unital ring and their morphisms is a category, which we denote by .
Now, we recall that a differential homotopy or, more briefly, a homotopy between morphisms of differential modules with simplicial faces is defines as a differential homotopy between morphisms of differential modules , which satisfies the relations , .
Let us consider the notion of a homotopy between morphisms of -modules [2] (see also [5]), which homotopically generalizes the notion of a homotopy between morphisms of differential modules with simplicial faces.
A homotopy between morphisms of -modules is defined as a family of module maps
[TABLE]
[TABLE]
(at we will use the denotation ), which satisfy the relations
[TABLE]
[TABLE]
[TABLE]
where is the same as in . The maps are called the components of the homotopy .
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Definition 1.2. We define a homotopy between an arbitrary morphisms of -modules as any homotopy between morphisms of -modules whose components satisfy the following conditions:
[TABLE]
[TABLE]
Proposition 1.1. For any -modules and , the relation between morphisms of -modules of the form defined by the presence of a homotopy between them is an equivalence relation.
Proof. Suppose given an arbitrary morphism of -modules. Then we have the homotopy between morphisms of -modules and . Suppose given a homotopy between morphisms of -modules and . Then the family of maps is a homotopy between morphisms of -modules and . Suppose given a homotopy between morphisms of -modules and and, moreover, given a homotopy between morphisms of -modules and . Then the family of maps is a homotopy between morphisms of -modules and .
By using specified in Proposition 1.1 the equivalence relation between morphisms of -modules the notion of a homotopy equivalence of -modules is introduced in the usual way. Namely, a morphism of -modules is called a homotopy equivalence of -modules, when this morphism have a homotopy inverse morphism of -modules.
§ 2. The homotopy invariance of dihedral homology of -modules.
First, recall that a -differential module [9] (sees also [10]-[17]) or, more briefly, a -module is defined as a module together with a family of module maps satisfying the relations
[TABLE]
It is worth noting that a -module can be equipped with any -grading, i.e., , where and , and the module maps can have any -degree for each , i.e., .
For , the relations have the form , and hence is a differential module. In [9] the homotopy invariance of the -module structure over any unital commutative ring under homotopy equivalences of differential modules was established. Later, it was shown in [33] that the homotopy invariance of the -module structure over fields of characteristic zero can be established by using the Koszul duality theory.
It is also worth saying that in [9] by using specified above homotopy invariance of the -differential module structure the relationship between -differential modules and spectral sequences was established. More precisely, in [9] was shown that over an arbitrary field the category of -differential modules is equivalent to the category of spectral sequences.
Now, we recall [9] that a -module is said to be stable if, for any , there exists a number such that for each . Any stable -module determines the differential defined by . The map is indeed a differential because relations imply the equality . It is easy to see that if the stable -module is equipped with a -grading , where , and maps , , have -degree satisfying the condition , then there is the chain complex defined by the following formulae:
[TABLE]
It was shown in [2] that any -module determines the sequence of stable -modules , , equipped with the bigrading , , , and defined by the following formulae:
[TABLE]
Let us recall [25] the construction of the chain bicomplex that is defined by the -module . Given any -module , consider the two -modules and defined by for , and the two families of maps
[TABLE]
[TABLE]
Moreover, we consider the chain complexes and that corresponded to the specified above -modules and , where
[TABLE]
Also consider the two families of maps
[TABLE]
In [25] it was shown that the following relations holds:
[TABLE]
[TABLE]
It follows from these relations that any -module determines the chain bicomplex
\vdots$$\vdots$$\vdots$$\vdots$$\overline{X}_{n+1}$$\overline{X}_{n+1}$$\overline{X}_{n+1}$$\overline{X}_{n+1}$$\dotsbbb**b{}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}bbb**b{}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}$${}^{-b^{{}^{\prime}}}$$\overline{X}_{n}$$\overline{X}_{n}$$\overline{X}_{n}$$\overline{X}_{n}$$\dots$$\overline{X}_{n-1}$$\overline{X}_{n-1}$$\overline{X}_{n-1}$$\overline{X}_{n-1}$$\dots$${}^{1-\overline{T}_{n-1}}$${}^{1-\overline{T}_{n}}$${}^{1-\overline{T}_{n+1}}$${}^{\overline{N}_{n-1}}$${}^{\overline{N}_{n}}$${}^{\overline{N}_{n+1}}$${}^{1-\overline{T}_{n-1}}$${}^{1-\overline{T}_{n}}$${}^{1-\overline{T}_{n+1}}$${}^{\overline{N}_{n-1}}$${}^{\overline{N}_{n}}$${}^{\overline{N}_{n+1}}$$\vdots$$\vdots$$\vdots$$\vdots
We denote this chain bicomplex by , where , , , , ,
[TABLE]
The chain complex associated with the chain bicomplex we denote by , where .
Recall [25] that the cyclic homology of a -module is defined as the homology of the chain complex associated with the chain bicomplex .
Now, suppose given any -module . Since can be considered as the -module , we have the specified above chain complexes and and also the families of module maps , , and , . Given the -module , consider the family of module maps
[TABLE]
where
[TABLE]
In [1] it was shown that the following relations holds:
[TABLE]
Since the -module can be considered as the -module , the -module always determines the chain bicomplex . The relations say us that there is a left action of the group on the chain bicomplex of any -module . This left action is defined by means of the automorphism of the order two, which, at any element , is given by the following rule:
[TABLE]
Recall [1] that the dihedral homology of any -module is defined as the hyperhomology of the group with coefficients in the relative to the specified above action of the group on the chain bicomplex .
The hyperhomology is defined as the homology of the chain complex that is associated with the triple chain complex , where is a groups -algebra of the group , the chain complex is any projective resolvente of the trivial -module , and the differential is defined by
[TABLE]
If we take as the projective resolvente the standard free resolvente
[TABLE]
then we obtain that the dihedral homology of any -module is the homology of the chain complex that is associated with the triple chain complex , where , , , , the differentials
[TABLE]
[TABLE]
were defined above, and the differential is given by
[TABLE]
We denote by , where , the chain complex associated to the triple chain complex .
Now, we investigate functorial and homotopy properties of the dihedral homology of -modules.
Theorem 2.1. The dihedral homology of -modules over any commutative unital ring determines the functor from the category of -modules to the category of graded -modules . This functor sends homotopy equivalences of -modules into isomorphisms of graded modules.
Proof. First, show that every morphism of -modules induces a module map of dihedral homology of these -modules. Given an arbitrary morphism of -modules , consider the family of module maps
[TABLE]
For the family of maps , by using we obtain the relations
[TABLE]
where and are sequences of -modules respectively defined by for -modules and . Direct calculations with using show that the families maps and satisfy the relations
[TABLE]
For , the equalities imply that maps of graded modules
[TABLE]
are chain maps , . The equalities follows that the chain maps and satisfy the relations
[TABLE]
For chain bicomplexes and , consider the map of bigraded modules , , , defined by the rule
[TABLE]
Since the chain maps and satisfy the relations , the map of bigraded modules is the map of chain bicomlexes . Now we consider the chain complexes and together with the above-specified action of the group . Let us show that the map is an -equivariant map. Indeed, given any collection , the conditions imply the equality
[TABLE]
Moreover, given any collection , the relations and imply the equality
[TABLE]
Last two equalities follows that the specified above families of maps and satisfy the relations
[TABLE]
By using these relations we obtain the equalities
[TABLE]
These equalities imply the relation . Thus the map of chain bicomplexes is a -equivariant map of chain bicomplexes and, consequently, the map induces the map of the hyperhomology
[TABLE]
Thus, each morphism of -modules defines the map of the dihedral homology graded modules
[TABLE]
Now, we consider the composition of morphisms of -modules and . Let us show that there is the equality of maps . Consider the family of maps , , , defined by . By using the relations we obtain that the following relations holds:
[TABLE]
where is the same as in . It is clear that
[TABLE]
Moreover, given any collections
[TABLE]
there are a collection and a permutation such that , . Therefore, the last above-specified relations imply the relations
[TABLE]
For , by using these relations we obtain the equalities and . These equalities imply the equality of maps of chain bicomlexes. It is clear that the map is a -equivariant map. Since the hyperhomology is a functor by the second argument, the obtained above equality implies the required equality of maps of graded dihedral homology modules.
Given any homotopy between morphisms of -modules, in the same way as above, we obtain the -equivariant homotopy between -equivariant maps and of chain bicomplexes. Since the functor of the hyperhomology sends homotopic -equivariant maps of chain bicomplexes into equal maps of graded modules, we obtain the equality of maps of graded dihedral homology modules. This equality implies that if the map is a homotopy equivalence of -modules, then the induces map of graded dihedral homology modules is a isomorphism of graded modules.
§ 3. The homotopy invariance of dihedral homology
of involutive -algebras.
First, following [23] and [29] (see also [30]), we recall necessary definitions related to the notion of an -algebra.
An -algebra is any differential module with , , , , equipped with a family of maps , , satisfying the following relations for any integer :
[TABLE]
where . For example, at the relations take the forms
[TABLE]
[TABLE]
A morphism of -algebras is defined as a family of module maps , which, for all integers , satisfy the relations
[TABLE]
[TABLE]
where and
[TABLE]
[TABLE]
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
Under a composition of morphisms of -algebras and we mean the morphism of -algebras
[TABLE]
defined by
[TABLE]
where and are the same as in . For example, at the formulae take, respectively, the following view
[TABLE]
[TABLE]
A homotopy between morphisms of -algebras is defined as a family of module maps , which, for all integers , satisfy the relations
[TABLE]
[TABLE]
where and is the same as in ;
[TABLE]
For example, at the relations take, respectively, the following view
[TABLE]
[TABLE]
[TABLE]
The origin of the signs in formulae – is described in detail in [31].
Now, we recall [22] (see also [1]) that an involutive -algebra is defined as any -algebra together with the automorphism of graded modules (the notation for ), which, at any elements and , satisfies the conditions
[TABLE]
[TABLE]
where
[TABLE]
and means that .
Definition 3.1. We define a morphism of involutive -algebras
[TABLE]
as any morphism of -algebras , which, at any elements , satisfies the conditions
[TABLE]
where
[TABLE]
By using the fact that any morphism of involutive -algebras is a morphism of -algebras we define the composition of morphisms of involutive -algebras as a composition of morphisms of -algebras. By using the formulae and it is easy to check that the composition of morphisms of involutive -algebras is a morphism of involutive -algebras. Clearly, the composition of morphisms of involutive -algebras is an associative operation. Moreover, for each involutive -algebra , there is the identity morphism of involutive -algebras
[TABLE]
where and , . Thus, the class of all involutive -algebras over any commutative unital ring and their morphisms is a category, which we denote by .
Definition 3.2. We define a homotopy between morphisms of involutive -algebras as any homotopy between morphisms of -algebras , which, at any elements , satisfies the conditions
[TABLE]
where is the same as in .
For any involutive -algebras and , the relation between morphisms of involutive -algebras of the form defined by the presence of a homotopy between them is an equivalence relation. By using this equivalence relation between morphisms of involutive -algebras the notion of a homotopy equivalence of involutive -algebras is introduced in the usual way. Namely, a morphism of involutive -algebras is called a homotopy equivalence of involutive -algebras, when this morphism have a homotopy inverse morphism of involutive -algebras.
Now, let us proceed to dihedral homology of involutive -algebras. In [1] it was shown that any involutive -algebra defines the tensor -module , which is given by
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and the family of module maps
[TABLE]
[TABLE]
is defined by
[TABLE]
[TABLE]
Now we recall [1] that the dihedral homology of an involutive -algebra is defined as the dihedral homology of the corresponding -module .
In the following assertion we describe functorial and homotopy properties of the dihedral homology of involutive -algebras.
Theorem 3.1. The dihedral homology of involutive -algebras over an arbitrary commutative unital ring determines the functor from the category of involutive -algebras to the category of graded -modules . This functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules.
Proof. First, show that every morphism of involutive -algebras induces a morphism of corresponding tensor -modules. Given any morphism of involutive -algebras , we define the family of module maps
[TABLE]
[TABLE]
by the following rules:
1). If , then
[TABLE]
2). If and ,
[TABLE]
[TABLE]
then
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
3). If and
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
then
[TABLE]
For example, if we consider the map , then by we obtain
[TABLE]
If we consider the map , , then by and we obtain
[TABLE]
[TABLE]
It is worth noting that any collection of integers , , always can be written in the form specified in the rule 2) or in the rule 3).
Now we show that the above-specified family of maps is the morphism of -modules
[TABLE]
In [32] it was shown that defined by the formulae – the family of maps is the morphism of -modules
[TABLE]
Therefore, we need to check that the maps satisfy the relations . Clearly, at , the the equality is true. Consider the maps , where and , that are given by at . Suppose that
[TABLE]
In this case, on the one hand, at any element , we have the equalities
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
On the other hand, we have the equalities
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Since , we obtain the required relation
[TABLE]
In the similar way it is checked that relations holds for all maps , where and , which are defined by at . Now, consider the maps , where and , that are given by the formulae at . Suppose that
[TABLE]
where . In this case by , at , we have
[TABLE]
By using the relations we obtain
[TABLE]
[TABLE]
[TABLE]
In [32] it was shown that the maps satisfy the relations . Therefore, we obtain
[TABLE]
[TABLE]
[TABLE]
By using the formulae we have
[TABLE]
Since , we obtain the required relation
[TABLE]
In the similar way it is checked that the relations holds for all maps , where and , which are given by at . Thus, all module maps satisfy the relations . It follows that the specified above morphism of -modules is the morphism of -modules .
Now, we consider the composition of an arbitrary morphisms and of involutive -algebras. In the same way as in [32] it is checked that the equality of morphisms of -modules is true.
The above follows that there is the functor . The required functor we define as a composition of the functor and the functor specified in Theorem 2.1.
Now we show that the functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules. Taking into account Theorem 2.1, it suffices to show that the functor sends homotopy equivalences into homotopy equivalences of -modules. With this purpose we show that each homotopy between morphisms of involutive -algebras induces a homotopy between corresponding morphisms of tensor -modules.
Given any homotopy between morphisms of involutive -algebras , we define a family of module maps
[TABLE]
[TABLE]
by the following rules:
1*′*). If , then
[TABLE]
2*′*). If and the collection
[TABLE]
is the same as in the above rule 2) defining the formula , then
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where and are the same as in ;
3*′*). If and the collection
[TABLE]
[TABLE]
is the same as in the above rule 3) defining the formula , then
[TABLE]
In similar way as it was done above in the case of morphisms of -modules , it is proved that defined by any homotopy between morphisms of involutive -algebras the family of module maps is a homotopy between corresponding morphisms of -modules . It follows that if is a homotopy equivalence of involutive -algebras, then the corresponding morphism is a homotopy equivalence of -modules. Thus, the functor sends homotopy equivalences of involutive -algebras into homotopy equivalences of -modules and, consequently, the functor sends homotopy equivalences of involutive -algebras into isomorphisms of graded modules.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] S. V. Lapin, Dihedral and reflexive modules with ∞ \infty -simplicial faces and dihedral and reflexive homology of involutive \A \A \A -algebras over unital commutative rings, ar Xiv:1809.07510 v 1 [math.AT] 20 Sep 2018, p. 1-25.
- 2[2] S. V. Lapin, Homotopy simplicial faces and the homology of realizations of simplicial topological spaces (in Russian), Mat. Zametki 94 (5), 661–681 (2013); translation in Math. Notes 94 (5–6), 619–635 (2013).
- 3[3] S. V. Lapin, Homotopy properties of differential Lie modules over curved coalgebras and Koszul duality (in Russian), Mat. Zametki 94 (3), 354–372 (2013); translation in Math. Notes 94 (3–4), 335–350 (2013).
- 4[4] S. V. Lapin, Differential Lie modules over curved colored coalgebras and ∞ \infty -simplicial modules (in Russian), Mat. Zametki 96 (5), 709–731 (2014); translation in Math. Notes 96 (5–6), 698–715 (2014).
- 5[5] S. V. Lapin, Chain realization of differential modules with ∞ \infty -simplicial faces and the B 𝐵 B -construction over \A \A \A -algebras (in Russian), Mat. Zametki 98 (1), 101–124 (2015); translation in Math. Notes 98 (1–2), 111–129 (2015).
- 6[6] S. V. Lapin, Homotopy properties of differential modules with simplicial F ∞ subscript 𝐹 F_{\infty} -faces and D ∞ subscript 𝐷 D_{\infty} -differential modules, Georgian Math. J. 2015; 22(4): 543-562.
- 7[7] S. V. Lapin, Homotopy properties of ∞ \infty -simplicial coalgebras and homotopy unital supplemented \A \A \A -algebras (in Russian), Mat. Zametki 99 (1), 55–77 (2016); translation in Math. Notes 99 (1–2), 63–81 (2016).
- 8[8] S. V. Lapin, Differential modules with ∞ \infty -simplicial faces and A ∞ subscript 𝐴 A_{\infty} -algebras, ar Xiv:1809.01853 v 1 [math.AT] 6 Sep 2018, p. 1-26.