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A Comparison Map for Symmetric Homology and Gamma Homology

We construct a comparison map between the gamma homology theory of Robinson and Whitehouse and the symmetric homology theory of Fiedorowicz and Ault in the case of an augmented, commutative algebra over a unital commutative ground ring.


Introduction
The gamma homology theory of Robinson and Whitehouse [Whi94] and the symmetric homology theory of Fiedorowicz, [Fie] and [Aul10], are both constructed by building a symmetric group action into Hochschild homology, albeit in very different ways. In this paper 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 as Γ-homology, for commutative algebras to encode information about homotopy commutativity. Γ-homology is closely related to stable homotopy theory as demonstrated by Pirashvili [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 a normalized version of Harrison homology for an augmented, commutative algebra.
The symmetric homology theory for associative algebras was first introduced by Fiedorowicz [Fie] and was extensively developed by Ault, [Aul10] and [Aul14]. It is the homology theory associated to the symmetric crossed simplicial group [FL91]. Symmetric homology also has connections to stable homotopy theory. For instance, the symmetric homology of a group algebra is isomorphic to the homology of the loop space on the infinite loop space associated to the classifying space of the group [Aul10, Corollary 40].
Defining a comparison map between the two homology theories is not straightforward in general. Our results utilize the fact in the case of an augmented, commutative algebra there exist smaller chain complexes than the standard complexes used to compute each homology theory.
The paper is arranged as follows. Section 1 collates some prerequisites on symmetric groups, augmented algebras and chain complexes associated to simplicial k-modules. We recall the Hochschild complex, the shuffle subcomplex and the Eulerian idempotents as these will play an important role in defining normalized Harrison homology. Section 2 recalls some constructions of functor homology. Symmetric homology is defined in terms of 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 computes them.
In Section 6 we construct a surjective map of chain complexes which gives rise to a long exact sequence connecting the symmetric homology of an augmented commutative algebra with a direct summand of its gamma homology. We use the material of Sections 3 and 4 to prove that when the algebra is flat over a ground ring containing Q we have a long exact sequence connecting the symmetric homology of the algebra with its gamma homology.
The results in this paper first appeared in the author's thesis [Gra19]. We note that the statements of Theorem 4.18 and Theorem 6.7 in this paper have an extra flatness condition that was accidentally omitted in [ Throughout the paper, k will denote a unital commutative ring.

Background
We collect prerequisites for the remainder of the paper. We will require facts about the symmetric 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. We briefly recall the construction of the Hochschild complex, the shuffle subcomplex and the Eulerian idempotents, all of which can be found in [Lod98], as these will be essential in defining a normalized Harrison homology in Section 3.
Definition 1.1. The symmetric group Σ n on n = {1, . . . , n} for n 1 is generated by the transpositions θ i = (i i + 1) for 1 i n − 1 such that each transposition squares to the identity, disjoint transpositions commute and (θ i • θ i+1 ) 3 = id n for 1 i n − 2.
Definition 1.4. We define the total shuffle operator, sh n , in k[Σ n ] to be the sum of the elements sh i,n−i . That is, 1.2. Augmented algebras.
Definition 1.5. A k-algebra A is said to be augmented if it is equipped with a k-algebra homomorphism ε : A → k. We define the augmentation ideal of A to be Ker(ε) and we denote it by I.
Recall from [LV12, Section 1.1.1] that for an augmented k-algebra A there is an isomorphism of k-modules A ∼ = I ⊕k. It follows that every element a ∈ A can be written uniquely in 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 basic tensor in A ⊗n is an elementary tensor a 1 ⊗ · · · ⊗ a n such that either a i ∈ I or a i = 1 k for each 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. and 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 Hochschild complex by C ⋆ (A). The homology of C ⋆ (A, M ) is denoted by HH ⋆ (A, M ) and is called the 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 Hochschild complex is canonically isomorphic to the chain complex C ⋆ (I, M ), formed analogously to the Hochschild complex.
Proof. This result is well-known. A full proof can be found in [Gra19, Section 5.2].
1.4. Shuffle complex. It is shown in [Lod98, Section 4.2.8] that the total shuffle operators of Definition 1.4 form a chain map sh n : C n (A) → C n (A), allowing us to make the following definition.
Definition 1.12. Let a 0 ⊗ a 1 ⊗ · · · ⊗ a n be an elementary tensor of the k-module C n (A). Let 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 combinations obtained 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 the Hochschild differential.  n (m ⊗ a 1 ⊗ · · · ⊗ a n ) denote the k-linear combination 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  n (m ⊗ a 1 ⊗ · · · ⊗ a n ), arising from k-module generators of C n (A, M ). The following is a theorem of Gerstenhaber 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 Hochschild complex into a direct sum of subcomplexes: Thus, this provides a decomposition of Hochschild homology:

Functor Homology
Symmetric homology [Aul10, Definition 14] is defined in the setting of functor homology. 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 found in [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 of right 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 at the trivial k-module.
It is well-known that the categories CMod and ModC are abelian, see for example [PR02, Section 1.6].

Tensor product of functors.
Definition 2.3. Let G be an object of ModC and F be an object of CMod. We define the tensor product G ⊗ C F to be the k-module is the k-submodule generated by the set

This quotient module is spanned k-linearly by equivalence classes of elementary tensors in
Definition 2.4. The bifunctor It is well-known that the bifunctor − ⊗ C − is right exact with respect to both variables and preserves direct sums, see for example [PR02, Section 1.6].

Gabriel-Zisman complex.
Recall the nerve of C [Lod98,B.12]. N ⋆ C is the simplicial set such that N n C for n 1 consists of all strings of composable morphisms of length n in C and N 0 C is the set of objects in C. The face maps are defined to either compose adjacent morphisms in the string or truncate the string and the degeneracy maps insert identity morphisms into the string. We will denote an element of N n C by (f n , . . For a small category C and a functor F ∈ CMod there is a simplicial k-module, denoted C ⋆ (C, F ) due to Gabriel and Zisman [GZ67, Appendix 2] whose n th homology is canonically isomorphic to Tor C n (k ⋆ , F ). Definition 2.6. Let F ∈ CMod. We define where the sum runs through all elements (f n , . . . , f 1 ) of N n C and The degeneracy maps insert identity maps into the string. By abuse of notation we will also denote the associated chain complex by C ⋆ (C, F ).

Harrison Homology
3.1. Harrison homology. We recall the definition of the Harrison complex for a commutative algebra as a quotient of the Hochschild complex. We recall that when the ground ring contains Q there is an alternative description of the Harrison complex as a subcomplex of the Hochschild complex.
3.2. Normalized Harrison homology. Suppose k contains Q and let A be an augmented commutative k-algebra with augmentation ideal I. Let M be a symmetric Abimodule which is flat over k. In this section we define a normalized version of Harrison homology under these conditions.
Forming a normalized Harrison homology is not straightforward in general. Whilst we can form 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 degeneracy maps, so the Harrison subcomplex does not arise as a chain complex associated to a simplicial object.
Let D ⋆ (A, M ) denote the degenerate subcomplex of the Hochschild complex. That is, the subcomplex of C ⋆ (A, M ) generated by elementary tensors for which at least one tensor factor is equal to 1 A , the multiplicative identity in A. The Eulerian idempotents split the degenerate subcomplex in precisely the same way that they split the Hochschild complex. We therefore have a natural isomorphism of chain complexes Furthermore, since the degenerate subcomplex is acyclic, each summand of the decomposition is acyclic.
By inclusion, e By a standard construction this gives rise to a long exact sequence in homology. Since the complex e Recall the chain complex C ⋆ (I, M ) of Proposition 1.11. Once again we can use the Eulerian idempotents to obtain a splitting: Having formed these two splittings of chain complexes using well-known methods, we prove that the chain complex e Proof. We can choose representatives such that the quotient complex, in degree n, is generated by equivalence classes e (i) ⋆ (m ⊗ y 1 ⊗ · · · ⊗ y n ) 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 ⋆ (m ⊗ y 1 ⊗ · · · ⊗ y n ) on generators in degree n.
The inverse is given by the map determined by sending a generator e

Gamma Homology
We recall the definition of Γ-homology for a commutative k-algebra in the functor homology 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 normalized Harrison homology of Subsection 3.2 to prove that when k ⊇ Q and A is a flat, augmented, commutative k-algebra we can compute Γ-homology using only the augmentation ideal.   Let Ω denote the category whose objects are the finite sets n = {1, . . . , n} for n 1 and whose morphisms are surjections of sets.

Gamma homology as functor homology.
Definition 4.3. Let t denote the right Γ-module Hom Set⋆ (−, k), where the commutative ring k is considered to be a based set with basepoint 0. The functor t is sometimes referred to as the based k-cochain functor [Rob18, Section 3.4].
where an empty product is understood to be 1 A ∈ A. Definition 4.7. Let N Ω n (x, 1) denote the set of strings of composable morphisms of length n in the category Ω whose initial domain is the set x and whose final codomain is the set 1.
Remark 4.8. Observe that a morphism f ∈ Hom Ω x, x 1 induces a map f ⋆ : A ⊗x → A ⊗x 1 defined by permuting and multiplying the tensor factors.
Definition 4.9. Let f n | · · · | f 1 be an element of N Ω n (x, 1). We define the i th component of f n | · · · | f 1 , denoted f i n−1 | · · · | f i 1 to be the string of morphisms corresponding to the preimage of i ∈ x n−1 , re-indexed such that the domain of the j th morphism is the set c where c is the cardinality of the set f −1 j . . . f −1 n−1 (i) and the final codomain is 1. Let X = f n | · · · | f 1 ⊗ (a 1 ⊗ · · · ⊗ a n ) ⊗ m be a generator of CΓ n (A, M ).
Remark 4.11. In the case where A is an augmented, commutative k-algebra, CΓ n (A, M ) is generated by elements of the form f n | · · · | f 1 ⊗ (a 1 ⊗ · · · ⊗ a n ) ⊗ m where a 1 ⊗ · · · ⊗ a n is a basic tensor in the sense of Definition 1.6.

Pruning map. Let
A be an augmented, commutative k-algebra with augmentation ideal I and let M be a symmetric A-bimodule which is flat over k. We will demonstrate that the chain complex CΓ ⋆ (I, M ) is a direct summand of the chain complex CΓ ⋆ (A, M ). We do so by providing a splitting map called the pruning map.
Definition 4.13. Let f n | · · · | f 1 ⊗ (a 1 ⊗ · · · ⊗ a x ) ⊗ m be a generator of CΓ n (A, M ) such that (a 1 ⊗ · · · ⊗ a x ) ⊗ m is a basic tensor in the sense of Definition 1.6.
Let L = {l 1 , . . . , l h } be the set such that a i ∈ I if and only if i ∈ L. Let Let f 1 : |L| → |m 1 | denote the map obtained from f 1 by restricting the domain to the set L, restricting the codomain to m 1 and re-indexing both domain and codomain in the canonical way.
For 2 i n let f i : |m i−1 | → |m i | denote the map obtained from f i by restricting the domain to the set m i−1 , restricting the codomain to the set m i and re-indexing both domain and codomain in the canonical way.
Remark 4.14. Intuitively, the pruning map removes the trivial tensor factors from the basic tensor (a 1 ⊗ · · · ⊗ a x ) ∈ A ⊗x and prunes the graph in order to preserve the permutations and multiplications of the non-trivial tensor factors. to be the k-linear map of chain complexes determined in degree n by f n | · · · | f 1 ⊗ (a 1 ⊗ · · · ⊗ a x ) ⊗ m → f n | · · · | f 1 ⊗ a l 1 ⊗ · · · ⊗ a l h ⊗ m.  is the identity map.
Proof. An element in the image of i is a k-linear combination of generators of the form f n | · · · | f 1 ⊗ (y 1 ⊗ · · · ⊗ y x ) ⊗ m such that (y 1 ⊗ · · · ⊗ y x ) ∈ I ⊗x . In particular, (y 1 ⊗ · · · ⊗ y x ) contains no trivial factors. By construction, the pruning map P ⋆ is the identity on such elements. 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 HΓ n−1 (I, M ). The theorem then follows from Corollary 3.8.

Symmetric Homology
We recall the definition of the category ∆S and the symmetric bar construction. We recall the 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 description of the category ∆S such that it has the same objects as the category Ω. Our definition is isomorphic to the definition found in [FL91] and [Aul10], the isomorphism being given by shifting the index.
Definition 5.1. The category ∆ has as objects the sets n = {1, . . . , n} for n 1 and order-preserving maps as morphisms.
Remark 5.2. Morphisms in ∆ are generated by the face maps δ i ∈ Hom ∆ n, n + 1 and the degeneracy maps σ j ∈ Hom ∆ n + 1, n for n 1, 1 i n + 1 and 1 j n subject to the usual relations, see for example [Lod98,Appendix B].
For (ϕ, g) ∈ Hom ∆S (n, m) and (ψ, h) ∈ Hom ∆S (m, l) the composite is the pair where the δ i and σ j are the face and degeneracy maps of the category ∆ and the θ k are transpositions.
Remark 5.4. The category ∆S is isomorphic to the category of non-commutative sets as found in [FT87,A10] and [PR02, Section 1.2].
Remark 5.5. 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 = {1 . . . , n} for n 1 and whose morphisms are the epimorphisms of ∆S.

Symmetric homology.
Definition 5.7. Let A be an associative k-algebra. We define the symmetric bar construction B sym A : ∆S → kMod to be the functor defined on objects by B sym A (n) = A ⊗n . For a morphism (ϕ, g) ∈ Hom ∆S (n, m), B sym A (ϕ, g) is determined by a 1 ⊗ · · · ⊗ a n →  where the product is ordered according to the morphism ϕ. An empty product is defined to 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  Remark 5.11. 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. Furthermore, HS 0 (A) ∼ = HS 0 (A) ⊕ k.

Comparison Map
Let A be an augmented, commutative k-algebra with augmentation ideal I. We define a surjective map of chain complexes N CS ⋆ (I) → N CΓ ⋆ (I, k) between the normalized chain complex of CS ⋆ (I) from Definition 5.10 and the normalized chain complex of CΓ ⋆ (I, k) from Definition 4.12. We therefore obtain a long exact sequence in homology connecting the reduced symmetric homology of A with a summand of the Γhomology of A. By Theorem 4.18 we obtain a long exact sequence in homology connecting the reduced symmetric homology with the entire Γ-homology when k ⊇ Q and A is flat over k.
6.1. A quotient of the symmetric complex. Recall the chain complex CS ⋆ (I) of Definition 5.10. This is a chain complex associated to a simplicial k-module so we can form the normalized chain complex N CS ⋆ (I). In degree n we have the k-module generated by equivalence classes of the form (f n , . . . , f 1 ), (y 1 ⊗ · · · ⊗ y x ) where x f 1 − → · · · fn − → x n is a string of non-identity morphisms N n (Epi∆S) and y 1 ⊗· · ·⊗y x ∈ B sym I (x). The boundary map is given by the alternating sum of face maps described in Definition 2.6. Remark 6.2. Observe that this well-defined. The only face map that affects the final codomain 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−1 > 1.
The k-module N CS n (I) is generated by equivalence classes (f n , . . . , f 1 ), (y 1 ⊗ · · · ⊗ y x ) where (f n , . . . , f 1 ) denotes a string of non-identity morphisms in N n (Epi∆S) and (y 0 ⊗ · · · ⊗ y x ) ∈ B sym I (x). 6.2. Mapping to the Robinson-Whitehouse complex. Recall the chain complex CΓ ⋆ (I, k) from Definition 4.12. Since CΓ ⋆ (I, k) is the chain complex associated to a simplicial k-module we can take the normalized complex, N CΓ ⋆ (I, k). A generator of the k-module N CΓ n (I, k) is an equivalence class of the form f n | f n−1 | · · · | f 1 ⊗ (y 1 ⊗ · · · ⊗ y x ) ⊗ 1 k , where f n | f n−1 | · · · | f 1 denotes a string of non-identity morphisms