A Comparison Map for Symmetric Homology and Gamma Homology
aa r X i v : . [ m a t h . A T ] M a y A COMPARISON MAP FOR SYMMETRIC HOMOLOGY ANDGAMMA HOMOLOGY
DANIEL GRAVES
Abstract.
We construct a comparison map between the gamma homology theory ofRobinson and Whitehouse and the symmetric homology theory of Fiedorowicz and Aultin the case of an augmented, commutative algebra over a unital commutative groundring.
Introduction
The gamma homology theory of Robinson and Whitehouse [Whi94] and the symmetrichomology theory of Fiedorowicz, [Fie] and [Aul10], are both constructed by building asymmetric group action into Hochschild homology, albeit in very different ways. In thispaper we provide a comparison map for the two theories in the case of an augmented,commutative algebra.Robinson and Whitehouse developed gamma homology , which we will frequently write asΓ-homology, for commutative algebras to encode information about homotopy commu-tativity. Γ-homology is closely related to stable homotopy theory as demonstrated byPirashvili [Pir00] and Pirashvili and Richter [PR00]. When the ground ring contains Q ,Γ-homology coincides with Harrison homology up to a shift in degree. We will define anormalized version of Harrison homology for an augmented, commutative algebra.The symmetric homology theory for associative algebras was first introduced by Fiedorow-icz [Fie] and was extensively developed by Ault, [Aul10] and [Aul14]. It is the homologytheory associated to the symmetric crossed simplicial group [FL91]. Symmetric homologyalso has connections to stable homotopy theory. For instance, the symmetric homology ofa group algebra is isomorphic to the homology of the loop space on the infinite loop spaceassociated to the classifying space of the group [Aul10, Corollary 40].Defining a comparison map between the two homology theories is not straightforward ingeneral. Our results utilize the fact in the case of an augmented, commutative algebrathere exist smaller chain complexes than the standard complexes used to compute eachhomology theory.The paper is arranged as follows. Section 1 collates some prerequisites on symmetricgroups, augmented algebras and chain complexes associated to simplicial k -modules. Werecall the Hochschild complex, the shuffle subcomplex and the Eulerian idempotents asthese will play an important role in defining normalized Harrison homology. Section 2recalls some constructions of functor homology. Symmetric homology is defined in termsof functor homology and Γ-homology can be similarly defined in this setting. In particular,we recall the notion of Tor functors for a small category and a chain complex that computesthem. School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK
Mathematics Subject Classification.
Key words and phrases.
Gamma homology, Symmetric homology, Harrison homology, Functorhomology. e begin Section 3 by recalling the definition of Harrison homology for commutativealgebras. We construct a normalized version of Harrison homology for augmented, com-mutative algebras over a ground ring containing Q . In particular, we prove that underthese conditions there is a smaller chain complex, constructed from the augmentationideal, that computes Harrison homology.In Section 4 we recall the defintion of Γ-homology for a commutative algebra. We recallthe Robinson-Whitehouse complex , whose homology is Γ-homology. For an augmented,commutative algebra we prove that the Robinson-Whitehouse complex splits using a con-struction called the pruning map . One summand of this splitting is constructed anal-ogously to the Robinson-Whitehouse complex using the augmentation ideal. We provethat for a flat algebra over a ground ring containing Q , the homology of this summand isisomorphic to the Γ-homology of the algebra.Section 5 recalls the definition of symmetric homology. We recall Ault’s chain complex forcomputing the symmetric homology of an augmented algebra.In Section 6 we construct a surjective map of chain complexes which gives rise to a longexact sequence connecting the symmetric homology of an augmented commutative algebrawith a direct summand of its gamma homology. We use the material of Sections 3 and4 to prove that when the algebra is flat over a ground ring containing Q we have along exact sequence connecting the symmetric homology of the algebra with its gammahomology.The results in this paper first appeared in the author’s thesis [Gra19]. We note thatthe statements of Theorem 4.18 and Theorem 6.7 in this paper have an extra flatnesscondition that was accidentally omitted in [Gra19, Theorem 32.2.1] and [Gra19, Theorem32.2.2].Throughout the paper, k will denote a unital commutative ring.1. Background
We collect prerequisites for the remainder of the paper. We will require facts about thesymmetric groups for each of Harrison homology, Γ-homology and symmetric homology.We recall some facts about augmented algebras that will be required for our results. Webriefly recall the construction of the Hochschild complex, the shuffle subcomplex and theEulerian idempotents, all of which can be found in [Lod98], as these will be essential indefining a normalized Harrison homology in Section 3.1.1.
Symmetric groups and shuffles.Definition 1.1.
The symmetric group Σ n on n = { , . . . , n } for n > θ i = ( i i + 1) for 1 i n − θ i ◦ θ i +1 ) = id n for 1 i n − Definition 1.2.
For 1 i n −
1, a permutation σ ∈ Σ n is called an i - shuffle if σ (1) < σ (2) < · · · < σ ( i ) and σ ( i + 1) < σ ( i + 2) < · · · < σ ( n ) . Definition 1.3.
We define an element sh i,n − i in the group algebra k [Σ n ] by sh i,n − i = X i − shuffles σ ∈ Σ n sgn ( σ ) σ. That is, we take the signed sum over all the i -shuffles in Σ n . efinition 1.4. We define the total shuffle operator , sh n , in k [Σ n ] to be the sum of theelements sh i,n − i . That is, sh n = n − X i =1 sh i,n − i . Augmented algebras.Definition 1.5. A k -algebra A is said to be augmented if it is equipped with a k -algebrahomomorphism ε : A → k . We define the augmentation ideal of A to be Ker( ε ) and wedenote it by I .Recall from [LV12, Section 1.1.1] that for an augmented k -algebra A there is an isomor-phism of k -modules A ∼ = I ⊕ k . It follows that every element a ∈ A can be written uniquelyin the form y + λ where y ∈ I and λ ∈ k . Definition 1.6.
Let A be an augmented k -algebra with augmentation ideal I . A basictensor in A ⊗ n is an elementary tensor a ⊗ · · · ⊗ a n such that either a i ∈ I or a i = 1 k foreach 1 i n . A tensor factor a i is called trivial if a i = 1 k and is called non-trivial if a i ∈ I .One notes that the k -module A ⊗ n is generated k -linearly by all basic tensors. Definition 1.7. A bimodule over an associative k -algebra A is a k -module M with a k -linear action of A on the left and right satisfying ( am ) a ′ = a ( ma ′ ) where a , a ′ ∈ A and m ∈ M . An A -bimodule M is said to be symmetric if am = ma for all a ∈ A and m ∈ M .One notes that for an augmented k -algebra A , k is an A -bimodule with the structure mapsdetermined by the augmentation.1.3. Hochschild homology.Definition 1.8.
Let X ⋆ be a simplicial k -module. The associated chain complex , denoted C ⋆ ( X ) is defined to have the k -module X n in degree n with the boundary map defined tobe the alternating sum of the face maps of X ⋆ .The degenerate subcomplex , denoted D ⋆ ( X ), is defined to be the subcomplex of C ⋆ ( X )generated by the degeneracy maps of X ⋆ .The normalized chain complex , denoted N ⋆ ( X ), is defined to be the quotient of C ⋆ ( X ) bythe subcomplex D ⋆ ( X ). Remark . The normalized chain complex is isomorphic to the Moore subcomplex of C ⋆ ( X ) [Wei94, Definition 8.3.6]. Definition 1.10.
Let A be an associative k -algebra and let M be an A -bimodule. The Hochschild complex , denoted C ⋆ ( A, M ), is the associated chain complex of the simplicial k -module with C n ( A, M ) = M ⊗ A ⊗ n , the k -module generated k -linearly by all elementarytensors ( m ⊗ a ⊗ · · · ⊗ a n ). The face maps ∂ i : C n ( A, M ) → C n − ( A, M ) are determinedby • ∂ ( m ⊗ a ⊗ · · · ⊗ a n ) = ( ma ⊗ a ⊗ · · · ⊗ a n ), • ∂ i ( m ⊗ a ⊗ · · · ⊗ a n ) = ( m ⊗ a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n ) for 1 i n − • ∂ n ( m ⊗ a ⊗ · · · ⊗ a n ) = ( a n m ⊗ a ⊗ · · · ⊗ a n − ) nd the degeneracy maps insert the multiplicative identity 1 A into the elementary tensor.The boundary map is denoted by b . In the case where M = A we denote the Hochschildcomplex by C ⋆ ( A ). The homology of C ⋆ ( A, M ) is denoted by HH ⋆ ( A, M ) and is calledthe
Hochschild homology of A with coefficients in M . Proposition 1.11.
Let A be an augmented k -algebra with augmentation ideal I . Let M be an A -bimodule which is flat over k . The normalized complex of the Hochschildcomplex is canonically isomorphic to the chain complex C ⋆ ( I, M ) , formed analogously tothe Hochschild complex.Proof. This result is well-known. A full proof can be found in [Gra19, Section 5.2]. (cid:3)
Shuffle complex.
It is shown in [Lod98, Section 4.2.8] that the total shuffle operatorsof Definition 1.4 form a chain map sh n : C n ( A ) → C n ( A ), allowing us to make the followingdefinition. Definition 1.12.
Let a ⊗ a ⊗ · · · ⊗ a n be an elementary tensor of the k -module C n ( A ).Let n − X i =1 sh i,n − i ( a ⊗ a ⊗ · · · ⊗ a n )denote the linear combination of tensors obtained by applying the total shuffle operator.Let Sh n ( A ) denote the submodule of C n ( A ) generated by all such k -linear combinationsobtained from the k -module generators of C n ( A ). The shuffle complex , denoted Sh ⋆ ( A ),is defined to have the module Sh n ( A ) in degree n with boundary map induced from theHochschild differential.1.5. Eulerian idempotents.
In the case where the ground ring k contains Q and A isa commutative k -algebra, we can decompose the Hochschild complex into a direct sum ofsubcomplexes. We recall the Eulerian idempotents from [Lod98, Proposition 4.5.3]. Proposition 1.13.
For n > and i n , there exist pairwise-disjoint idempotentelements e ( i ) n ∈ Q [Σ n ] such that e (1) n + · · · + e ( n ) n is the identity in Q [Σ n ] . These elementsare called the Eulerian idempotents . (cid:3) The Eulerian idempotents act on C n ( A, M ) = M ⊗ A ⊗ n by 1 M ⊗ e ( i ) n . Let ( m ⊗ a ⊗· · ·⊗ a n )be a k -module generator of C n ( A, M ). Let e ( i ) n ( m ⊗ a ⊗ · · · ⊗ a n ) denote the k -linearcombination of tuples obtained by applying the Eulerian idempotent to our generator.This is simply a linear combination of permutations of our generator. The subcomplex e ( i ) ⋆ C ⋆ ( A, M ) is generated in degree n by all linear combinations of the form e ( i ) n ( m ⊗ a ⊗· · · ⊗ a n ), arising from k -module generators of C n ( A, M ). The following is a theorem ofGerstenhaber and Schack [GS87].
Theorem 1.14.
Let k be a commutative ring containing Q . Let A be a k -algebra and let M be a symmetric A -bimodule. The Eulerian idempotents e ( i ) n naturally split the Hochschildcomplex into a direct sum of subcomplexes: C ⋆ ( A, M ) = ∞ M i =1 e ( i ) ⋆ C ⋆ ( A, M ) . Thus, this provides a decomposition of Hochschild homology: H n ( A, M ) = H n (cid:16) e (1) n C ⋆ ( A, M ) (cid:17) ⊕ · · · ⊕ H n (cid:16) e ( n ) n C ⋆ ( A, M ) (cid:17) . (cid:3) . Functor Homology
Symmetric homology [Aul10, Definition 14] is defined in the setting of functor homol-ogy. Gamma homology has a functor homology description due to Pirashvili and Richter[PR00]. In this section we recall the necessary constructions of functor homology, as foundin [PR02] for example. We also recall a chain complex construction of Gabriel and Zisman[GZ67, Appendix 2] used to calculate the homology groups of functors.2.1.
Modules over a category.Definition 2.1.
Let C be a small category. We define the category of left C -modules ,denoted CMod , to be the functor category Fun ( C , kMod ). We define the category ofright C -modules , denoted ModC , to be the functor category Fun ( C op , kMod ). Definition 2.2.
We define the trivial right C -module , k ⋆ , to be the constant functor atthe trivial k -module.It is well-known that the categories CMod and
ModC are abelian, see for example [PR02,Section 1.6].2.2.
Tensor product of functors.Definition 2.3.
Let G be an object of ModC and F be an object of CMod . We definethe tensor product G ⊗ C F to be the k -module L C ∈ Ob( C ) G ( C ) ⊗ k F ( C ) (cid:10) G ( α )( x ) ⊗ y − x ⊗ F ( α )( y ) (cid:11) where (cid:10) G ( α )( x ) ⊗ y − x ⊗ F ( α )( y ) (cid:11) is the k -submodule generated by the set (cid:8) G ( α )( x ) ⊗ y − x ⊗ F ( α )( y ) : α ∈ Hom( C ) , x ∈ src ( G ( α )) , y ∈ src ( F ( α )) (cid:9) . This quotient module is spanned k -linearly by equivalence classes of elementary tensors in L C ∈ Ob( C ) G ( C ) ⊗ k F ( C ) which we will denote by [ x ⊗ y ]. Definition 2.4.
The bifunctor − ⊗ C − : ModC × CMod → kMod is defined on objects by ( G, F ) G ⊗ C F . Given two natural transformationsΘ ∈ Hom
ModC ( G, G ) and Ψ ∈ Hom
CMod ( F, F ) , the morphism Θ ⊗ C Ψ is determined by [ x ⊗ y ] (cid:2) Θ C ( x ) ⊗ Ψ C ( y ) (cid:3) .It is well-known that the bifunctor − ⊗ C − is right exact with respect to both variablesand preserves direct sums, see for example [PR02, Section 1.6]. Definition 2.5.
We denote the left derived functors of the bifunctor −⊗ C − by Tor C ⋆ ( − , − ). .3. Gabriel-Zisman complex.
Recall the nerve of C [Lod98, B.12]. N ⋆ C is the simpli-cial set such that N n C for n > n in C and N C is the set of objects in C . The face maps are defined to either composeadjacent morphisms in the string or truncate the string and the degeneracy maps insertidentity morphisms into the string. We will denote an element of N n C by ( f n , . . . , f )where f i ∈ Hom C ( C i − , C i ).For a small category C and a functor F ∈ CMod there is a simplicial k -module, de-noted C ⋆ ( C , F ) due to Gabriel and Zisman [GZ67, Appendix 2] whose n th homology iscanonically isomorphic to Tor C n ( k ⋆ , F ). Definition 2.6.
Let F ∈ CMod . We define C n ( C , F ) = M ( f n ,...,f ) F ( C )where the sum runs through all elements ( f n , . . . , f ) of N n C and f i ∈ Hom C ( C i − , C i ).We write a generator of C n ( C , F ) in the form ( f n , . . . , f , x ) where ( f n , . . . , f ) ∈ N n C indexes the summand and x ∈ F ( C ). The face maps ∂ i : C n ( C , F ) → C n − ( C , F ) aredetermined by ∂ i ( f n , . . . , f , x ) = ( f n , . . . , f , F ( f )( x )) i = 0 , ( f n , . . . , f i +1 ◦ f i , . . . , f , x ) 1 i n − , ( f n − , . . . , f , x ) i = n. The degeneracy maps insert identity maps into the string. By abuse of notation we willalso denote the associated chain complex by C ⋆ ( C , F ).3. Harrison Homology
Harrison homology.
We recall the definition of the Harrison complex for a commu-tative algebra as a quotient of the Hochschild complex. We recall that when the groundring contains Q there is an alternative description of the Harrison complex as a subcomplexof the Hochschild complex. Definition 3.1.
Let A be a commutative k -algebra. The Harrison complex of A is definedto be the quotient of the Hochschild complex C ⋆ ( A ) by the shuffle complex Sh ⋆ ( A ). Thatis, CHarr ⋆ ( A ) := C ⋆ ( A ) Sh ⋆ ( A )with boundary map induced from the Hochschild boundary map. Definition 3.2.
Let A be a commutative k -algebra. Let M be a symmetric A -bimodule.We define the Harrison complex of A with coefficients in M to be the complex CHarr ⋆ ( A, M ) := M ⊗ A CHarr ⋆ ( A ) . Definition 3.3.
The homology of the chain complex M ⊗ A CHarr ⋆ ( A ) is called the Harrison homology of A with coefficients in M and is denoted by Harr ⋆ ( A, M ).Recall the decomposition of the Hochschild complex from Theorem 1.14. The followingresult is due to Barr [Bar68].
Proposition 3.4.
Let k ⊇ Q . Let A be a commutative k -algebra and let M be an A -bimodule. There is a natural isomorphism of chain complexes e (1) ⋆ C ⋆ ( A, M ) ∼ = CHarr ⋆ ( A, M ) . (cid:3) .2. Normalized Harrison homology.
Suppose k contains Q and let A be an aug-mented commutative k -algebra with augmentation ideal I . Let M be a symmetric A -bimodule which is flat over k . In this section we define a normalized version of Harrisonhomology under these conditions.Forming a normalized Harrison homology is not straightforward in general. Whilst we canform a description of the Harrison complex as a subcomplex of the Hochschild complex,this is because the Hochschild boundary map is compatible with the Eulerian idempotents.In fact, the Eulerian idempotents are not compatible with individual face and degeneracymaps, so the Harrison subcomplex does not arise as a chain complex associated to asimplicial object.Let D ⋆ ( A, M ) denote the degenerate subcomplex of the Hochschild complex. That is, thesubcomplex of C ⋆ ( A, M ) generated by elementary tensors for which at least one tensorfactor is equal to 1 A , the multiplicative identity in A . The Eulerian idempotents split thedegenerate subcomplex in precisely the same way that they split the Hochschild complex.We therefore have a natural isomorphism of chain complexes D ⋆ ( A, M ) ∼ = ∞ M i =1 e ( i ) ⋆ D ⋆ ( A, M ) . Furthermore, since the degenerate subcomplex is acyclic, each summand of the decompo-sition is acyclic.By inclusion, e ( i ) ⋆ D ⋆ ( A, M ) is a subcomplex of e ( i ) ⋆ C ⋆ ( A, M ). In particular, taking i = 1, e (1) ⋆ D ⋆ ( A, M ) is a subcomplex of the Harrison complex. This gives rise to a short exactsequence of chain complexes; for each i > → e ( i ) ⋆ D ⋆ ( A, M ) → e ( i ) ⋆ C ⋆ ( A, M ) Q i −→ e ( i ) ⋆ C ⋆ ( A, M ) e ( i ) ⋆ D ⋆ ( A, M ) → . By a standard construction this gives rise to a long exact sequence in homology. Sincethe complex e ( i ) ⋆ D ⋆ ( A, M ) is acyclic we can deduce that the quotient map Q i is a quasi-isomorphism for each i > Proposition 3.5.
There is an isomorphism of k -modules H n (cid:16) e ( i ) ⋆ C ⋆ ( A, M ) (cid:17) ∼ = H n e ( i ) ⋆ C ⋆ ( A, M ) e ( i ) ⋆ D ⋆ ( A, M ) for each n > and i > . (cid:3) Recall the chain complex C ⋆ ( I, M ) of Proposition 1.11. Once again we can use the Eulerianidempotents to obtain a splitting: C ⋆ ( I, M ) ∼ = ∞ M i =1 e ( i ) ⋆ C ⋆ ( I, M ) . Having formed these two splittings of chain complexes using well-known methods, we provethat the chain complex e ( i ) ⋆ C ⋆ ( I, M ) computes the Harrison homology of an augmentedalgebra.
Lemma 3.6.
For each i > there is an isomorphism of chain complexes f i : e ( i ) ⋆ C ⋆ ( A, M ) e ( i ) ⋆ D ⋆ ( A, M ) → e ( i ) ⋆ C ⋆ ( I, M ) . roof. We can choose representatives such that the quotient complex, in degree n , isgenerated by equivalence classes h e ( i ) ⋆ ( m ⊗ y ⊗ · · · ⊗ y n ) i where each y i is an element of the augmentation ideal I .With this choice of representatives, we have a well-defined map of chain complexes f i : e ( i ) ⋆ C ⋆ ( A, M ) e ( i ) ⋆ D ⋆ ( A, M ) → e ( i ) ⋆ C ⋆ ( I, M )determined by h e ( i ) ⋆ ( m ⊗ y ⊗ · · · ⊗ y n ) i e ( i ) ⋆ ( m ⊗ y ⊗ · · · ⊗ y n ) on generators in degree n .The inverse is given by the map determined by sending a generator e ( i ) ⋆ ( m ⊗ y ⊗ · · · ⊗ y n )of e ( i ) ⋆ C ⋆ ( I, M ) to its equivalence class in the quotient. (cid:3)
Let I i : e ( i ) ⋆ C ⋆ ( I, M ) → e ( i ) ⋆ C ⋆ ( A, M ) denote the inclusion of chain complexes.
Theorem 3.7.
Let k ⊇ Q and let A be a augmented, commutative k -algebra with aug-mentation ideal I . Let M be a symmetric A -bimodule which is flat over k . For each i > there is an isomorphism of chain complexes e ( i ) ⋆ C ⋆ ( A, M ) ∼ = e ( i ) ⋆ C ⋆ ( I, M ) ⊕ e ( i ) ⋆ D ⋆ ( A, M ) . Proof.
One easily checks that the composite f i ◦ Q i ◦ I i is the identity map on the chaincomplex e ( i ) ⋆ C ⋆ ( I, M ). The theorem then follows upon observing that Ker( f i ◦ Q i ) ∼ = e ( i ) ⋆ D ⋆ ( A, M ). (cid:3) Corollary 3.8.
Taking i = 1 in Theorem 3.7, there is an isomorphism of k -modules H n (cid:16) e (1) ⋆ C ⋆ ( I, M ) (cid:17) ∼ = Harr n ( A, M ) for each n > . (cid:3) Definition 3.9.
Let k ⊇ Q . Let A be an augmented, commutative k -algebra with aug-mentation ideal I and let M be a symmetric A -bimodule which is flat over k . We call thechain complex e (1) ⋆ C ⋆ ( I, M ) the normalized Harrison complex .4.
Gamma Homology
We recall the definition of Γ-homology for a commutative k -algebra in the functor homol-ogy setting. We recall the Robinson-Whitehouse complex, whose homology is Γ-homology.In the case of an augmented, commutative k -algebra we define a splitting of the Robinson-Whitehouse complex by defining the pruning map . Furthermore, we utilize the normalizedHarrison homology of Subsection 3.2 to prove that when k ⊇ Q and A is a flat, aug-mented, commutative k -algebra we can compute Γ-homology using only the augmentationideal.4.1. The categories Γ and Ω .Definition 4.1. Let Γ denote the category whose objects are the finite based sets [ n ] = { , , . . . , n } , for n >
0, where 0 denotes that the set is based at 0. The morphisms arebasepoint-preserving maps of sets.
Definition 4.2.
Let Ω denote the category whose objects are the finite sets n = { , . . . , n } for n > .2. Gamma homology as functor homology.Definition 4.3.
Let t denote the right Γ-module Hom Set ⋆ ( − , k ), where the commutativering k is considered to be a based set with basepoint 0. The functor t is sometimes referredto as the based k -cochain functor [Rob18, Section 3.4]. Definition 4.4.
Let F be a left Γ-module. We define the Γ -homology of F by H Γ ⋆ ( F ) := Tor Γ ⋆ ( t, F ) . Definition 4.5.
Let A be a commutative k -algebra and let M be a symmetric A -bimodule.We define the Loday functor L ( A, M )( − ) : Γ → kMod on objects by L ( A, M ) (cid:0) [ n ] (cid:1) = A ⊗ n ⊗ M. For an element f ∈ Hom Γ (cid:0) [ p ] , [ q ] (cid:1) , L ( A, M )( f ) is determined by (cid:0) a ⊗ · · · ⊗ a p ⊗ m (cid:1) Y i ∈ f − (1) a i ⊗ · · · ⊗ Y i ∈ f − ( q ) a i ⊗ Y i ∈ f − (0) a i m where an empty product is understood to be 1 A ∈ A . Definition 4.6.
Let A be a commutative k -algebra and let M be a symmetric A -bimodule.We define the Γ -homology of A with coefficients in M by H Γ ⋆ ( A, M ) := H Γ ⋆ (cid:0) L ( A, M ) (cid:1) . The Robinson-Whitehouse complex.
The Robinson-Whitehouse complex for com-mutative algebras was first defined in the thesis of Sarah Whitehouse [Whi94, DefinitionII.4.1]. Pirashvili and Richter provided a more general construction for all left Γ-modules[PR00, Section 2].
Definition 4.7.
Let N Ω n ( x,
1) denote the set of strings of composable morphisms oflength n in the category Ω whose initial domain is the set x and whose final codomain isthe set 1. An element x f −→ x f −→ · · · f n − −−−→ x n − f n −→ N Ω n ( x,
1) will be denoted (cid:2) f n | · · · | f (cid:3) . Remark . Observe that a morphism f ∈ Hom Ω (cid:0) x, x (cid:1) induces a map f ⋆ : A ⊗ x → A ⊗ x defined by permuting and multiplying the tensor factors. Definition 4.9.
Let (cid:2) f n | · · · | f (cid:3) be an element of N Ω n ( x, i th compo-nent of (cid:2) f n | · · · | f (cid:3) , denoted (cid:2) f in − | · · · | f i (cid:3) to be the string of morphisms correspondingto the preimage of i ∈ x n − , re-indexed such that the domain of the j th morphism is theset c where c is the cardinality of the set f − j . . . f − n − ( i ) and the final codomain is 1. Definition 4.10.
Let A be a commutative k -algebra and let M be a symmetric A -bimodule. We define the simplicial k -module C Γ ⋆ ( A, M ) as follows. Let C Γ ( A, M ) = A ⊗ M and let C Γ n ( A, M ) := M x > k (cid:2) N Ω n ( x, (cid:3) ⊗ A ⊗ x ⊗ M. Let X = (cid:2) f n | · · · | f (cid:3) ⊗ ( a ⊗ · · · ⊗ a n ) ⊗ m be a generator of C Γ n ( A, M ).We define the face maps ∂ i : C Γ n ( A, M ) → C Γ n − ( A, M ) to be determined by ∂ ( X ) = (cid:2) f n | · · · | f (cid:3) ⊗ f ⋆ ( a ⊗ · · · ⊗ a n ) ⊗ m,∂ i ( X ) = (cid:2) f n | · · · | f i +1 ◦ f i | · · · | f (cid:3) ⊗ ( a ⊗ · · · ⊗ a n ) ⊗ m or 1 i n − ∂ n ( X ) = X i ∈ x n − h f in − | · · · | f i i ⊗ (cid:0) a i ⊗ · · · ⊗ a i k (cid:1) ⊗ Y j f − ··· f − n − ( i ) a j m, where { i , . . . , i k } is the ordered preimage of i ∈ x n − under f n − ◦· · · ◦ f . The degeneracymaps insert identity morphisms into the string. By abuse of notation we will also denotethe associated chain complex by C Γ ⋆ ( A, M ). Remark . In the case where A is an augmented, commutative k -algebra, C Γ n ( A, M )is generated by elements of the form (cid:2) f n | · · · | f (cid:3) ⊗ ( a ⊗ · · · ⊗ a n ) ⊗ m where a ⊗ · · · ⊗ a n is a basic tensor in the sense of Definition 1.6. Definition 4.12.
Let A be an augmented k -algebra with augmentation ideal I and let M be a symmetric A -bimodule. We define the chain complex C Γ ⋆ ( I, M ) in degree n by C Γ n ( I, M ) := M x > k (cid:2) N Ω n ( x, (cid:3) ⊗ I ⊗ x ⊗ M with the boundary map induced by that of C Γ ⋆ ( A, M ).4.4.
Pruning map.
Let A be an augmented, commutative k -algebra with augmentationideal I and let M be a symmetric A -bimodule which is flat over k . We will demonstratethat the chain complex C Γ ⋆ ( I, M ) is a direct summand of the chain complex C Γ ⋆ ( A, M ).We do so by providing a splitting map called the pruning map . Definition 4.13.
Let (cid:2) f n | · · · | f (cid:3) ⊗ ( a ⊗ · · · ⊗ a x ) ⊗ m be a generator of C Γ n ( A, M )such that ( a ⊗ · · · ⊗ a x ) ⊗ m is a basic tensor in the sense of Definition 1.6.Let L = { l , . . . , l h } be the set such that a i ∈ I if and only if i ∈ L . Let m i := Im (cid:16) ( f i ◦ · · · ◦ f ) (cid:12)(cid:12) L (cid:17) . Let e f : | L | → | m | denote the map obtained from f by restricting the domain to theset L , restricting the codomain to m and re-indexing both domain and codomain in thecanonical way.For 2 i n let e f i : | m i − | → | m i | denote the map obtained from f i by restrictingthe domain to the set m i − , restricting the codomain to the set m i and re-indexing bothdomain and codomain in the canonical way. Remark . Intuitively, the pruning map removes the trivial tensor factors from the basictensor ( a ⊗ · · · ⊗ a x ) ∈ A ⊗ x and prunes the graph in order to preserve the permutationsand multiplications of the non-trivial tensor factors. Definition 4.15.
Let A be an augmented, commutative k -algebra with augmentationideal I . Let M be a symmetric A -bimodule which is flat over k . We define the pruningmap P ⋆ : C Γ ⋆ ( A, M ) → C Γ ⋆ ( I, M )to be the k -linear map of chain complexes determined in degree n by (cid:2) f n | · · · | f (cid:3) ⊗ ( a ⊗ · · · ⊗ a x ) ⊗ m hf f n | · · · | e f i ⊗ (cid:0) a l ⊗ · · · ⊗ a l h (cid:1) ⊗ m. Further details on the pruning map including examples and a proof that it is well-definedmap of chain complexes can be found in the author’s thesis [Gra19, 18.2, App. A]. heorem 4.16. Let A be an augmented, commutative k -algebra with augmentation ideal I . Let M be a symmetric A -bimodule which is flat over k . Let i : C Γ ⋆ ( I, M ) → C Γ ⋆ ( A, M ) denote the inclusion of the subcomplex. The composite P ⋆ ◦ i : C Γ ⋆ ( I, M ) → C Γ ⋆ ( I, M ) is the identity map.Proof. An element in the image of i is a k -linear combination of generators of the form (cid:2) f n | · · · | f (cid:3) ⊗ ( y ⊗ · · · ⊗ y x ) ⊗ m such that ( y ⊗ · · · ⊗ y x ) ∈ I ⊗ x . In particular, ( y ⊗ · · · ⊗ y x ) contains no trivial factors.By construction, the pruning map P ⋆ is the identity on such elements. (cid:3) Corollary 4.17.
Under the conditions of Theorem 4.16, there is an isomorphism of chaincomplexes C Γ ⋆ ( A, M ) ∼ = C Γ ⋆ ( I, M ) ⊕ Ker ( P ⋆ ) and hence there is an isomorphism of k -modules H Γ n ( A, M ) ∼ = H Γ n ( I, M ) ⊕ H n (cid:0) Ker( P ⋆ ) (cid:1) for each n > . (cid:3) Theorem 4.18.
Let k ⊇ Q . Let A be a flat, augmented, commutative k -algebra withaugmentation ideal I and let M be a symmetric A -bimodule. Then there is an isomorphismof graded k -modules H Γ ⋆ ( I, M ) ∼ = H Γ ⋆ ( A, M ) . Proof.
Whitehouse [Whi94, Theorem III.4.2] proves that under these conditions there isan isomorphism H Γ n − ( A, M ) ∼ = Harr n ( A, M )for n >
1. One may check that the same method yields an isomorphism between the n th homology of the complex e (1) ⋆ C ⋆ ( I, M ) and H Γ n − ( I, M ). The theorem then follows fromCorollary 3.8. (cid:3) Symmetric Homology
We recall the definition of the category ∆ S and the symmetric bar construction. We recallthe definition of reduced symmetric homology and the chain complex that computes it.The material in this section can be found in [Aul10].5.1. The category ∆ S . In order to best define the comparison map we provide a de-scription of the category ∆ S such that it has the same objects as the category Ω. Ourdefinition is isomorphic to the definition found in [FL91] and [Aul10], the isomorphismbeing given by shifting the index. Definition 5.1.
The category ∆ has as objects the sets n = { , . . . , n } for n > Remark . Morphisms in ∆ are generated by the face maps δ i ∈ Hom ∆ (cid:0) n, n + 1 (cid:1) andthe degeneracy maps σ j ∈ Hom ∆ (cid:0) n + 1 , n (cid:1) for n >
1, 1 i n + 1 and 1 j n subjectto the usual relations, see for example [Lod98, Appendix B]. efinition 5.3. The category ∆ S has as objects the sets n = { , . . . , n } for n >
1. Anelement of Hom ∆ S ( n, m ) is a pair ( ϕ, g ) where g ∈ Σ n and ϕ ∈ Hom ∆ ( n, m ).For ( ϕ, g ) ∈ Hom ∆ S ( n, m ) and ( ψ, h ) ∈ Hom ∆ S ( m, l ) the composite is the pair (cid:0) ψ ◦ h ⋆ ( ϕ ) , ϕ ⋆ ( h ) ◦ g (cid:1) where h ⋆ ( ϕ ) and ϕ ⋆ ( h ) are determined by the relations h ⋆ ( δ i ) = δ h ( i ) , h ⋆ ( σ j ) = σ h ( j ) and δ ⋆i ( θ k ) = θ k k < i − id n − k = i − id n − k = iθ k − k > i σ ⋆j ( θ k ) = θ k k < j − θ j θ j − k = j − θ j θ j +1 k = jθ k +1 k > j where the δ i and σ j are the face and degeneracy maps of the category ∆ and the θ k aretranspositions. Remark . The category ∆ S is isomorphic to the category of non-commutative sets asfound in [FT87, A10] and [PR02, Section 1.2]. Remark . A morphism ( ϕ, g ) in ∆ S is an epimorphism if and only if the morphism ϕ in ∆ is an epimorphism. Definition 5.6.
Let Epi∆ S denote the subcategory of ∆ S whose objects are the sets n = { . . . , n } for n > S .5.2. Symmetric homology.Definition 5.7.
Let A be an associative k -algebra. We define the symmetric bar con-struction B symA : ∆ S → kMod to be the functor defined on objects by B symA ( n ) = A ⊗ n . For a morphism ( ϕ, g ) ∈ Hom ∆ S ( n, m ), B symA ( ϕ, g ) is determined by a ⊗ · · · ⊗ a n Y < i ∈ ϕ − a g − ( i ) ⊗ · · · ⊗ Y < i ∈ ϕ − m ) a g − ( i ) where the product is ordered according to the morphism ϕ . An empty product is definedto be the multiplicative unit 1 A . Definition 5.8.
Let A be an augmented, associative k -algebra with augmentation ideal I . We define the functor B symI : Epi∆ S → kMod to be defined on objects by B symI ( n ) = I ⊗ n . For a morphism ( ϕ, g ) ∈ Hom ∆ S ( n, m ) wedefine B symI ( ϕ, g ) = B symA ( ϕ, g ). Definition 5.9.
Let A be an associative k -algebra. For n >
0, we define the n th symmetrichomology of A to be HS n ( A ) := Tor ∆ Sn (cid:0) k ⋆ , B symA (cid:1) , where k ⋆ is the trivial right ∆ S -module of Definition 2.2.By [Aul10, Section 6] we can make the following definition. Definition 5.10.
Let A be an augmented associative k -algebra with augmentation ideal I . We define the reduced symmetric homology of A , denoted g HS ⋆ ( A ) to be the homologyof the chain complex CS ⋆ ( I ) := C ⋆ (cid:0) Epi∆
S, B symI (cid:1) . emark . By [Aul10, Section 6.3], the reduced symmetric homology of the augmented k -algebra A coincides with the symmetric homology of A in non-negative degrees. Fur-thermore, HS ( A ) ∼ = g HS ( A ) ⊕ k. Comparison Map
Let A be an augmented, commutative k -algebra with augmentation ideal I . We define asurjective map of chain complexes N CS ⋆ ( I ) → N C Γ ⋆ ( I, k )between the normalized chain complex of CS ⋆ ( I ) from Definition 5.10 and the normalizedchain complex of C Γ ⋆ ( I, k ) from Definition 4.12. We therefore obtain a long exact sequencein homology connecting the reduced symmetric homology of A with a summand of the Γ-homology of A . By Theorem 4.18 we obtain a long exact sequence in homology connectingthe reduced symmetric homology with the entire Γ-homology when k ⊇ Q and A is flatover k .6.1. A quotient of the symmetric complex.
Recall the chain complex CS ⋆ ( I ) ofDefinition 5.10. This is a chain complex associated to a simplicial k -module so we canform the normalized chain complex N CS ⋆ ( I ). In degree n we have the k -module generatedby equivalence classes of the form (cid:2) ( f n , . . . , f ) , ( y ⊗ · · · ⊗ y x ) (cid:3) where x f −→ · · · f n −→ x n is a string of non-identity morphisms N n (Epi∆ S ) and y ⊗· · ·⊗ y x ∈ B symI ( x ). The bound-ary map is given by the alternating sum of face maps described in Definition 2.6. Definition 6.1.
Denote by
N CS ⋆ ( I ) the subcomplex of N CS ⋆ ( I ) generated by the equiv-alence classes for which x n = 1. Let q : N CS ⋆ ( I ) → N CS ⋆ ( I ) := N CS ⋆ ( I ) N CS ⋆ ( I )denote the quotient map. Remark . Observe that this well-defined. The only face map that affects the finalcodomain in the string is the last face map, which omits the last morphism in the string.Since this last morphism was a non-identity surjection and x n >
1, we see x n − > k -module N CS n ( I ) is generated by equivalence classes (cid:2) ( f n , . . . , f ) , ( y ⊗ · · · ⊗ y x ) (cid:3) where ( f n , . . . , f ) denotes a string of non-identity morphisms x f −→ · · · f n − −−−→ x n − f n −→ N n (Epi∆ S ) and ( y ⊗ · · · ⊗ y x ) ∈ B symI ( x ).6.2. Mapping to the Robinson-Whitehouse complex.
Recall the chain complex C Γ ⋆ ( I, k ) from Definition 4.12. Since C Γ ⋆ ( I, k ) is the chain complex associated to asimplicial k -module we can take the normalized complex, N C Γ ⋆ ( I, k ). A generator of the k -module N C Γ n ( I, k ) is an equivalence class of the form h(cid:2) f n | f n − | · · · | f (cid:3) ⊗ ( y ⊗ · · · ⊗ y x ) ⊗ k i , where (cid:2) f n | f n − | · · · | f (cid:3) denotes a string of non-identity morphisms x f −→ · · · f n − −−−→ x n − f n −→ n N n Ω and ( y ⊗ · · · ⊗ y x ) ∈ I ⊗ x .The category ∆ S is isomorphic to the category of non-commutative sets, F ( as ). We cantherefore consider a morphism in Epi∆ S to be a surjection of finite sets, that is a morphismin Ω, with the extra structure of a total ordering on the preimage of each singleton in thecodomain. Definition 6.3.
Let U : Epi∆ S → Ω denote the forgetful functor which is the identity onobjects and sends a morphism of Epi∆ S to the underlying surjection of sets. Definition 6.4.
Let Φ n : N CS n ( I ) → N C Γ n ( I, k )be the map of k -modules determined by (cid:2) ( f n , . . . , f ) , ( y ⊗ · · · ⊗ y x ) (cid:3) h(cid:2) U ( f n ) | · · · | U ( f ) (cid:3) ⊗ ( y ⊗ · · · ⊗ y x ) ⊗ k i . The proof that the maps Φ n assemble into a map of chain complexes can be found in[Gra19, Appendix D]. Since both chain complexes arise from simplicial k -modules theproof consists of a standard but time-consuming check that the maps Φ n are compatiblewith the face maps.6.3. Main result.Theorem 6.5.
Let A be an augmented, commutative k -algebra with augmentation ideal I . There is a surjective map of chain complexes Φ ◦ q : N CS ⋆ ( I ) → N C Γ ⋆ ( I, k ) . Proof.
Recall from Definition 6.1 that q : N CS ⋆ ( I ) → N CS ⋆ ( I ) is a quotient map and istherefore surjective.A generator h(cid:2) f n | f n − | · · · | f (cid:3) ⊗ ( y ⊗ · · · ⊗ y x ) ⊗ k i of N C Γ n ( I, k ) is the image of (cid:2) ( f n , . . . , f ) , ( y ⊗ · · · ⊗ y x ) (cid:3) in N CS n ( I ) where we take the total orderings on the preimages of each f i to be thecanonical ones. Hence Φ is also a surjective map. (cid:3) Corollary 6.6.
Let A be an augmented, commutative k -algebra with augmentation ideal I . There is a short exact sequence of chain complexes → Ker(Φ ◦ q ) → N CS ⋆ ( I ) Φ ◦ q −−→ N C Γ ⋆ ( I, k ) → , which gives rise to the long exact sequence · · · H n (Ker(Φ ◦ q ) g HS n ( A ) H Γ n ( I, k ) H n − (Ker(Φ ◦ q )) · · · H Γ ( I, k ) H (Ker(Φ ◦ q )) g HS ( A ) H Γ ( I, k ) 0 connecting the reduced symmetric homology of A with a direct summand of the Γ -homologyof A . (cid:3) heorem 6.7. Let k ⊇ Q . Let A be a flat, augmented, commutative k -algebra. There isa long exact sequence · · · H n (Ker(Φ ◦ q ) g HS n ( A ) H Γ n ( A, k ) H n − (Ker(Φ ◦ q )) · · · H Γ ( A, k ) H (Ker(Φ ◦ q )) g HS ( A ) H Γ ( A, k ) 0 connecting the reduced symmetric homology of A with the Γ -homology of A with coefficientsin k .Proof. The theorem follows from Corollary 6.6 and Theorem 4.18. (cid:3)
Acknowledgements
I would like to thank Sarah Whitehouse, my Ph.D. supervisor, for all her guidance andencouragement. I am grateful to James Brotherston for his proof-reading and helpfulsuggestions.
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