SYMMETRIC HOMOLOGY OF ALGEBRAS: FOUNDATIONS

SYMMETRIC HOMOLOGY OF ALGEBRAS: FOUNDATIONS

arXiv:0902.1274v1 [math.AT] 7 Feb 2009

SHAUN V. AULT Abstract. Symmetric homology of a unital algebra A over a commutative ground ring k is defined using derived functors and the symmetric bar construction of Fiedorowicz. If A = k[G] is a group ring, then HS∗ (k[G]) is related to stable homotopy theory. Two chain complexes that compute HS∗ (A) are constructed, both making use of a symmetric monoidal category ∆S+ containing ∆S. Two spectral sequences are found that aid in computing symmetric homology. In the second spectral sequence, a complex isomorphic ˇ to the suspension of the cycle-free chessboard complex Ωp+1 of Vre´ cica and Zivaljevi´ c plays an important role. Recent results on the connectivity of Ωn imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras.

1. Introduction and Definitions The theory of symmetric homology, in which the symmetric groups Σop k , for k ≥ 0, play the role that the cyclic groups do in cyclic homology, begins with the definition of the category ∆S, containing the simplicial category ∆ as subcategory. Indeed, ∆S is an example of crossed simplicial group [8]. 1.1. The category ∆S. Let ∆S be the category that has as objects, the ordered sets [n] = {0, 1, . . . , n} for n ≥ 0, and as morphisms, pairs (φ, g), where φ : [n] → [m] is a non-decreasing map of sets (i.e., a morphism in ∆), and g ∈ Σop n+1 (the opposite group of the symmetric group acting on [n]). The element g represents an automorphism of [n], and as a set map, takes i ∈ [n] to g −1 (i). Equivalently, a morphism in ∆S is a morphism in ∆ together with a total ordering of the domain [n]. Composition of morphisms is achieved as in [8], namely, (φ, g) ◦ (ψ, h) := (φ · g ∗ (ψ), ψ ∗ (g) · h). Remark 1. Observe that the properties of g ∗ (φ) and φ∗ (g) stated in Prop. 1.6 of [8] are formally rather similar to the properties of exponents. Indeed, the notation g φ := φ∗ (g), φg := g ∗ (φ) will generally be used in lieu of the original notation in what follows. For each n, let τn be the (n + 1)-cycle (0, n, n − 1, . . . , 1) ∈ Σop n+1 . The subgroup generated by τn op is isomorphic to the cyclic group Cn+1 = Cn+1 . Indeed, we may view the cyclic category ∆C as the subcategory of ∆S consisting of all objects [n] for n ≥ 0, together with those morphisms of ∆S of the form (φ, τni ) (cf. [12]). In this way, we get a natural chain of inclusions, ∆ ֒→ ∆C ֒→ ∆S. It is often helpful to represent morphisms of ∆S as diagrams of points and lines, indicating images of set maps. Using these diagrams, we may see clearly how ψ g and g ψ are related to (ψ, g) (see Figure 1). An equivalent characterization of ∆S comes from Pirashvili (cf. [19]), as the category F(as) of ‘noncommutative’ sets. The objects are sets n := {1, 2, . . . , n} for n ≥ 0. By convention, 0 is the empty set. A morphism in MorF(as) (n, m) consists of a map (of sets) f : n → m together with a total ordering Πj on f −1 (j) for all j ∈ m. In such a case, denote by Π the partial order generated by all Πj . If (f, Π) : n → m and (g, Ψ) : m → p, their composition will be (gf, Φ), where Φj is the total ordering on (gf )−1 (j) (for all j ∈ p) induced by Π and Ψ. Explicitly, for each pair i1 , i2 ∈ (gf )−1 (j), we have i1 < i2 under Φ if and only if [f (i1 ) < f (i2 ) under Ψ] or [f (i1 ) = f (i2 ) and i1 < i2 under Π]. There is an obvious inclusion of categories, ∆S ֒→ F(as), taking [n] to n + 1, but there is no object of ∆S that maps to 0. It will be useful to define ∆S+ ⊃ ∆S which is isomorphic to F(as): Definition 2. ∆S+ is the category consisting of all objects and morphisms of ∆S, with the additional object [−1], representing the empty set, and a unique morphism ιn : [−1] → [n] for each n ≥ −1. 1991 Mathematics Subject Classification. Primary 55N35; Secondary 13D03, 18G10. Key words and phrases. symmetric homology, bar construction, spectral sequence, chessboard complex, GAP, cyclic homology. As this work represents my thesis research, I am deeply indebted to my thesis advisor, Zbigniew Fiedorowicz, for overall direction and assistance with finding the resources and writing the proofs. 1

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SHAUN V. AULT

ψ

4 b

3 b

3 b 2 b

2 b 1 b

1 b 0 b

0 b

g

4 b 3 b

3 b

gψ

2 b

2 b 1 b

1 b 0 b

0 b ψg

Figure 1. Morphisms of ∆S: g · ψ = g ∗ (ψ), ψ ∗ (g) = ψ g , g ψ

Remark 3. Pirashvili’s construction is a special case of a more general construction due to May and Thomason b together [17]. This construction associates to any topological operad {C(n)}n≥0 a topological category C b with a functor C → F, where F is the category of finite sets, such that the inverse image of any function Q b is defined using the composition of the operad. f : m → n is the space ni=1 C(#f −1 (i)). Composition in C b May and Thomason refer to C as the category of operators associated to C. They were interested in the case of an E∞ operad, but their construction evidently works for any operad. The category of operators associated to the discrete A∞ operad Ass, which parametrizes monoid structures, is precisely Pirashvili’s construction of F(as), i.e. ∆S+ . One very useful advantage in enlarging our category to ∆S to ∆S+ is the added structure inherent in ∆S+ . Proposition 4. ∆S+ is a permutative category. Proof. Define the monoid product on objects by [n] ⊙ [m] := [n + m + 1], (disjoint union of sets), and on morphisms (φ, g) : [n] → [n′ ], (ψ, h) : [m] → [m′ ], by (φ, g) ⊙ (ψ, h) = (η, k) : [n + m + 1] → [n′ + m′ + 1], where (η, k) is just the morphism (φ, g) acting on the first n + 1 points of [n + m + 1], and (ψ, h) acting on the remaining points. The unit object will be [−1] = ∅. ⊙ is clearly associative, and [−1] acts as two-sided identity. Finally, define the transposition transformation γn,m : [n] ⊙ [m] → [m] ⊙ [n] to be the identity on objects, and on morphisms to be precomposition with the block transposition that switches the first block of size n + 1 with the second block of size m + 1. Remark 5. The fact that ∆S+ is permutative shall be exploited to prove that HS∗ (A) admits homology operations in a forthcoming paper. For the purposes of computation, a morphism α : [n] → [m] of ∆S may be conveniently represented as a tensor product of monomials in the formal non-commuting variables {x0 , x1 , . . . , xn }. Let α = (φ, g), with φ ∈ Mor∆ ([n], [m]) and g ∈ Σop n+1 . The tensor representation of α will have m + 1 tensor factors. Each


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xi will occur exactly once, in the order xg(0) , xg(1) , . . . , xg(n) . The ith tensor factor consists of the product of #φ−1 (i − 1) variables, with the convention that the empty product will be denoted 1. Thus, the ith tensor factor records the total ordering of φ−1 (i). As an example, the tensor representation of the morphism depicted in Fig. 2 is x0 x1 ⊗ x3 x4 ⊗ 1 ⊗ x2 .

4 b 3 b 2 b 1 b 0 b

x1 x0 ⊗ x3 x4 ⊗ 1 ⊗ x2

4 b 3 b

3 b 2 b

2 b 1 b

1 b 0 b

0 b

Figure 2. Morphisms of ∆S in tensor notation With this notation, the composition of two morphisms α = X0 ⊗ X1 ⊗ . . . ⊗ Xm : [n] → [m] and β = Y1 ⊗ Y2 ⊗ . . . Yn : [p] → [n] is given by, αβ = Z0 ⊗ Z1 ⊗ . . . ⊗ Zm , where Zi is determined by replacing each variable in the monomial Xi = xj1 . . . xjs in α by the corresponding monomials Yjk in β. So, Zi = Yj1 . . . Yjs . 1.2. Homological Algebra of Functors. Recall, for a category C , a C -module is covariant functor F : C → k-Mod. Similarly, a C op -module is a contravariant functor G : C → k-Mod. Let M be a C op module and N be a C -module. Following MacLane [13], define the tensor product of functors as a coend, RX M ⊗C N := (M X) ⊗ (N X). That is, M M ⊗C N := M (X) ⊗k N (X)/ ≈, X∈ObjC

where the equivalence ≈ is generated by y ⊗f∗ (x) ≈ f ∗ (y)⊗x for f ∈ MorC (X, Y ), x ∈ N (X) and y ∈ M (Y ). The trivial C -module, resp. C op -module, denoted by k (for either variance), is the functor taking each object to k and each morphism to the identity. Define a product on generators of k[MorC ] by setting f · g := f ◦ g, if the maps are composable, and f · g := 0 otherwise. Extending the product k-linearly, we obtain a ring structure on RC := k[MorC ]. In general RC does not have a unit, only local units; that is, for any finite set of elements of RC , there is an element that acts as identity Melement for these elements. Observe, a C -module M is equivalent to a left RC -module structure on M (X). X∈ObjC

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SHAUN V. AULT

In either characterization, it makes sense to talk about projective C -modules and the torsion products op TorC -module N . (It is also possible to define TorC i (M, N ) for C -module M and C i (M, N ) directly as the derived functors of the categorical tensor product construction; see [8].) Remark 6. Note, the existing literature based on the work of Connes, Loday and Quillen consistently defines the categorical tensor product in the reverse sense: N ⊗C M is the direct sum of copies of N X ⊗k M X modded out by the equivalence x ⊗ f ∗ (y) ≈ f∗ (x) ⊗ y for all C -morphisms f : X → Y . In this context, N is covariant, while M is contravariant. I chose to follow the convention of Pirashvili and Richter [20] in writing tensor products as M ⊗C N so that the equivalence ξ :C -Mod → k[MorC ]-Mod passes to tensor products in a straightforward way: ξ(M ⊗C N ) = ξ(M ) ⊗k[MorC ] ξ(N ). 1.3. The Symmetric Bar Construction. Now that we have defined the category ∆S, the next step should be to define an appropriate bar construction. Recall that the cyclic bar construction is a functor B∗cyc A : ∆C op → k-Mod. We then take the groups Tor∆C (B∗cyc A, k) as our definition of HCi (A) for i ≥ 0. i However the results of [8] show that the cyclic bar construction does not extend to a functor ∆S op → k-Mod. Furthermore, for any functor F : ∆S op → k-Mod, the groups Tor∆S i (F, k) simply compute the homology of the underlying simplicial module of F (given by restricting F to ∆op ). The second author discovered [7] that there is a natural extension of the cyclic bar construction not to a contravariant functor on ∆S, but to a covariant functor. Definition 7. Let A be an associative, unital algebra over a commutative ground ring k. Define a functor B∗sym A : ∆S → k-Mod by: Bnsym A := B∗sym A[n] := A⊗(n+1) B∗sym A(α) : (a0 ⊗ a1 ⊗ . . . ⊗ an ) 7→ α(a0 , . . . , an ), where α : [n] → [m] is represented in tensor notation, and evaluation at (a0 , . . . , an ) simply amounts to substituting each ai for xi and multiplying the resulting monomials in A. If the pre-image α−1 (i) is empty, then the unit of A is inserted. Observe that B∗sym A is natural in A. Remark 8. Fiedorowicz [7] defines the symmetric bar construction functor for morphisms α = (φ, g), where φ ∈ Mor∆([n], [m]) and g ∈ Σop n+1 , via     Y Y ai  ai  ⊗ . . . ⊗  B∗sym A(φ)(a0 ⊗ a1 ⊗ . . . ⊗ an ) =  ai ∈φ−1 (0)

B∗sym A(g)(a0

ai ∈φ−1 (n)

⊗ a1 ⊗ . . . ⊗ an ) = ag−1 (0) ⊗ ag−1 (1) ⊗ . . . ⊗ ag−1 (n)

However, in order that this becomes consistent with earlier notation, we should require B∗sym A(g) to permute the tensor factors in the inverse sense: B∗sym A(g)(a0 ⊗ a1 ⊗ . . . ⊗ an ) = ag(0) ⊗ ag(1) ⊗ . . . ⊗ ag(n) . ∼ =

B∗sym A may be regarded as a simplicial k-module via the chain of functors, ∆op ֒→ ∆C op → ∆C ֒→ ∆S. Here, the isomorphism D : ∆C op → ∆C is the standard duality (see [12]). Fiedorowicz showed in [7] that B∗sym A ◦ D = B∗cyc A. Note that by duality of ∆C, it is equivalent to use the functor B∗sym A, restricted to morphisms of ∆C in order to compute HC∗ (A). Definition 9. The symmetric homology of an associative, unital k-algebra A is denoted HS∗ (A), and is defined as: sym HS∗ (A) := Tor∆S A) ∗ (k, B∗ Remark 10. Since k ⊗∆S M ∼ = colim∆S M , for any ∆S-module M , we can alternatively describe symmetric homology as derived functors of colim. HSi (A) = colim(i) B∗sym A. ∆S

(To see the relation with higher colimits, we need to tensor a projective resolution of B∗sym A with k).


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1.4. The Standard Resolution. Let C be a category. The rank 1 free C -module is k[MorC ], with the left action of composition of morphisms. Now as k-module, k[MorC ] decomposes into the direct sum,   M M  k[MorC ] = k [MorC (X, Y )] . Y ∈ObjC

X∈ObjC

By abuse of notation, denote

M

k [MorC (X, Y )] by k [MorC (X, −)]. So there is a direct sum decompo-

Y ∈ObjC

sition as left C -module),

k[MorC ] =

M

k [MorC (X, −)] .

X∈ObjC

Thus, the submodules k [MorC (X, −)] are projective left C -modules. Similarly, k[MorC ] is the rank 1 free right C -module, with right action of pre-composition of morphisms, and as such, decomposes as: M k[MorC ] = k [MorC (−, Y )] Y ∈ObjC

Again, the notation k [MorC (−, Y )] is shorthand for

M

k [MorC (X, Y )]. The submodules k [MorC (−, Y )]

X∈ObjC

are projective as right C -modules. It will also be important to note that k [MorC (−, Y )] ⊗C N ∼ = N (Y ) as k-module via the evaluation map f ⊗ y 7→ f∗ (y). Similarly, M ⊗C k [MorC (X, −)] ∼ = M (X). Recall (e.g. in Quillen [21], Section 1), for an object Y ∈ ObjC , the category of objects under Y , denoted Y \C , has as objects all pairs (X, φ) such that φ ∈ MorC (Y, X), and as morphisms the commutative triangles. Define a contravariant functor (− \ C ) from C to Cat as follows: An object Y is sent to the under category Y \ C . If ν : Y → Y ′ is a morphism in C , the functor (ν \ C ) : Y ′ \ C → Y \ C is defined on objects by (X, φ) 7→ (X, φν). Thus, (− \ C ) is a C op -category. As noted in [8], the nerve of (− \ C ) is a simplicial C op -set, and the complex L∗ , given by: Ln := k [N (− \ C )n ] op

is a resolution by projective C -modules of the trivial C op -module, k. Here, the boundary map is ∂ := P i i (−1) di , where the di ’s come from the simplicial structure of the nerve of (− \ C ). For completeness, we shall provide a proof of:

Proposition 11. L∗ is a resolution of k by projective C op -modules.

Proof. Fix C ∈ ObjC . Let ǫ : L0 (C) → k be the map defined on generators by ǫ(C → A0 ) := 1k . The complex ǫ ∂ ∂ k ← L0 (C) ← L1 (C) ← . . . is chain homotopic to the 0 complex via the homotopy, id

h−1 : 1 7→ (C → C) id

hn : (C → A0 → . . . → An ) 7→ (C → C → A0 → . . . An ), op Note that each C -module Ln is projective.

for i ≥ 0

Thus, we may compute HS∗ (A) as the homology groups of the following complex: (1)

0 ←− L0 ⊗∆S B∗sym A ←− L1 ⊗∆S B∗sym A ←− L2 ⊗∆S B∗sym A ←− . . .

Denote the complex (1) by Y∗ A. Indeed, Y∗ A is a simplicial k-module, and the assignment A 7→ Y∗ A is a functor k-Alg → k-SimpMod. Corollary 12. For an associative, unital k-algebra A, HS∗ (A) = H∗ (Y∗ A; k). That is, HS∗ (A) = H∗ (k[N (− \ ∆S)] ⊗∆S B∗sym A; k) . Remark 13. By remarks above, it is clear that the related complex k[N (− \ ∆C)] ⊗∆C B∗sym A computes HC∗ (A).

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SHAUN V. AULT

Remark 14. Observe that every element of Ln ⊗∆S B∗sym A is equivalent to one in which the first morphism of the Ln factor is an identity: β1

α

β2

βn

[p] → [q0 ] → [q1 ] → . . . → [qn ] ⊗ (y0 ⊗ . . . ⊗ yp ) id[q

]

β1

β2

βn

0 [q0 ] → [q1 ] → . . . → [qn ] ⊗ α∗ (y0 , . . . , yp ) ≈ [q0 ] −→

Thus, we may consider Ln ⊗∆S B∗sym A to be the k-module generated by the elements n o β1 β2 βn [q0 ] → [q1 ] → . . . → [qn ] ⊗ (y0 ⊗ . . . ⊗ yq0 ) ,

where the tensor product is now over k. The face maps di are defined on generators by: d0 omits β1 and replaces (y0 ⊗ . . . ⊗ yq0 ) by β1 (y0 , . . . , yq0 ); for 0 < i < n, di composes βi+1 with βi ; and dn omits βn . We now have enough tools to compute HS∗ (k). First, we need to show: Lemma 15. N (∆S) is a contractible complex. Proof. Define a functor F : ∆S → ∆S by F : [n] 7→ [0] ⊙ [n], F : f 7→ id[0] ⊙ f, using the monoid multiplication ⊙ defined above. There is a natural transformation id∆S → F given by the following commutative diagram for each f : [m] → [n]: f / [n] [m] δ0 δ0 id⊙f / [n + 1] [m + 1] (k)

Here, δj

: [k − 1] → [k] is the ∆ morphism that misses the point j ∈ [k]. [0]

Consider the constant functor ∆S → ∆S that sends all objects to [0] and all morphisms to id[0] . There is a natural transformation [0] → F given by the following commutative diagram for each f : [m] → [n]. [0] 00 [m + 1]

id

id⊙f

/ [0] 00 / [n + 1]

(k)

Here, 0j : [0] → [k] is the morphism that sends the point 0 to j ∈ [k]. Natural transformations induce homotopy equivalences (see [23] or Prop. 1.2 of [21]), so in particular, the identity map on N (∆S) is homotopic to the map that sends N (∆S) to the nerve of a trivial category. Thus, N (∆S) is contractible. Corollary 16. The symmetric homology of the ground ring k is isomorphic to k, concentrated in degree 0. Proof. HS∗ (k) is the homology of the chain complex generated (freely) over k by the chains n o β1 β2 βn [q0 ] → [q1 ] → . . . → [qn ] ⊗ (1 ⊗ . . . ⊗ 1) ,

where βi ∈ Mor∆S ([qi−1 ], [qi ]). Each chain [q0 ] → [q1 ] → . . . → [qn ] ⊗ (1 ⊗ . . . ⊗ 1) may be identified with the chain [q0 ] → [q1 ] → . . . → [qn ] of N (∆S), and clearly this defines a chain isomorphism to N (∆S). The result now follows from Lemma 15.


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2. Symmetric Homology with Coefficients 2.1. Definitions and Conventions. Following the conventions for Hochschild and cyclic homology in Loday [12], when we need to indicate explicitly the ground ring k over which we compute symmetric homology of A, we shall use the notation: HS∗ (A | k). Furthermore, since the notion “∆S-module” does not explicitly state the ground ring, we shall use the bulkier “∆S-module over k” when the context is ambiguous. Definition 17. If C∗ is a complex that computes symmetric homology of the algebra A over k, and M is a k-module, then the symmetric homology of A over k with coefficients in M is defined by HS∗ (A; M ) := H∗ (C∗ ⊗k M ). Remark 18. Definition 17 is independent of the particular choice of complex C∗ , so we shall generally use the complex Y∗ A = k[N (− \ ∆S)] ⊗∆S B∗sym A of Cor. 12 in this section. Proposition 19. If M is flat over k, then HS∗ (A; M ) ∼ = HS∗ (A) ⊗k M . Proof. Since M is k-flat, the functor − ⊗k M is exact.

Corollary 20. For any Z-agebra A, HSn (A; Q) ∼ = HSn (A) ⊗Z Q. 2.2. Universal Coefficient Theorem for Symmetric Homology. Our goal will be the proof of the following: Theorem 21. If A is a flat k-algebra, and B is a commutative k-algebra, then there is a spectral sequence with E2p,q := Torkp (HSq (A | k), B) ⇒ HS∗ (A; B). To that end, we begin with some preliminary results. Lemma 22. i. If A is a flat k-algebra, then Yn A is flat for each n. ii. If A is a projective k-algebra, then Yn A is projective for each n. Proof. By Remark 14 we may identify: M Yn A ∼ k[N ([m] \ ∆S)n−1 ] ⊗k A⊗(m+1) . = m≥0

Note, k[N ([m] \ ∆S)n−1 ] is free, so if A is flat (resp. projective), then Yn A is also flat (resp. projective).

Proposition 23. If B is a commutative k-algebra, then there is an isomorphism HS∗ (A ⊗k B | B) ∼ = HS∗ (A; B). Proof. Here, we are viewing A ⊗k B as a B-algebra via the inclusion B ∼ = 1A ⊗k B ֒→ A ⊗k B. Observe, there is an isomorphism ∼ =

∼ =

(A ⊗k B) ⊗B (A ⊗k B) −→ A ⊗k A ⊗k (B ⊗B B) −→ (A ⊗k A) ⊗k B. Iterating this for n-fold tensors of A ⊗k B, (A ⊗k B) ⊗B . . . ⊗B (A ⊗k B) ∼ = A ⊗k . . . ⊗k A ⊗k B. {z } | {z } | n

n

B∗sym (A

This shows that the ∆S-module over B, ⊗k B) is isomorphic as k-module to (B∗sym A) ⊗k B over k. The proposition now follows essentially by definition. Let L∗ be the resolution of k by projective ∆S op modules (over k) given by L∗ = k[N (− \ ∆S)]. Taking tensor products (over k) with the algebra B, we obtain a projective resolution of the trivial ∆S op -module over B. Thus, (2) HS∗ (A ⊗k B | B) = H∗ (L∗ ⊗k B) ⊗B[Mor∆S] B∗sym (A ⊗k B); B On the chain level, there are isomorphisms:

(L∗ ⊗k B) ⊗B[Mor∆S] B∗sym (A ⊗k B) ∼ = (L∗ ⊗k B) ⊗B[Mor∆S] (B∗sym A ⊗k B) (3)

∼ = (L∗ ⊗k[Mor∆S] B∗sym A) ⊗k B

The complex (3) computes HS∗ (A; B) by definition.

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SHAUN V. AULT

Remark 24. Since HS∗ (A | k) = HS∗ (A ⊗k k | k), Prop. 23 allows us to identify HS∗ (A | k) with HS∗ (A; k). The construction HS∗ (A; −) is a covariant functor, as is immediately seen on the chain level. In addition, HS∗ (−; M ) is a covariant functor for any k-module, M . Proposition 25. Suppose 0 → X → Y → Z → 0 is a short exact sequence of left k-modules, and suppose A is a flat k-algebra. Then there is an induced long exact sequence in symmetric homology: (4)

. . . → HSn (A; X) → HSn (A; Y ) → HSn (A; Z) → HSn−1 (A; X) → . . .

Moreover, a map of short exact sequences, (α, β, γ), as in the diagram below, induces a map of the corresponding long exact sequences (commutative ladder) 0 (5) 0

/ X

/ Y

/ Z

α

β

γ

/ X′

/ Y′

/ Z′

/ 0

/ 0

Proof. By Lemma 22, the hypothesis A is flat implies that the following is an exact sequence of chain complexes: 0 → Y∗ A ⊗k X → Y∗ A ⊗k Y → Y∗ A ⊗k Z → 0. This induces a long exact sequence in homology . . . → Hn (Y∗ A ⊗k X) → Hn (Y∗ A ⊗k Y ) → Hn (Y∗ A ⊗k Z) → Hn−1 (Y∗ A ⊗k X) → . . . as required. Now let (α, β, γ) be a morphism of short exact sequences, as in diagram (5). Consider the diagram, .. .

.. .

HSn (A; X)

α∗

/ HSn (A; X ′ )

HSn (A; Y )

β∗

/ HSn (A; Y ′ )

HSn (A; Z)

γ∗

/ HSn (A; Z ′ )

(6)

∂′

∂

HSn−1 (A; X)

.. .

α∗

/ HSn−1 (A; X ′ )

.. .

Since HSn (A; −) is functorial, the upper two squares of the diagram commute. Commutativity of the lower square follows from the naturality of the connecting homomorphism in the snake lemma. Remark 26. Any family of additive covariant functors {Tn } between two abelian categories is said to be a long exact sequence of functors if it takes short exact sequences to long exact sequences such as (4) and morphisms of short exact sequences to commutative ladders of long exact sequences such as (6). See [5], Definition 1.1 and also [18], section 12.1. The content of Prop. 25 is that for A flat, {HSn (A; −)}n∈Z is a long exact sequence of functors.


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We may now prove Theorem 21. Proof. Let Tq : k-Mod → k-Mod be the functor HSq (A; −). Observe, since exact sequence of additive covariant functors (Rmk. 26 and Prop. 25); Tq = (indeed, for q < 0); and Tq commutes with arbitrary direct sums. Hence, by Theorem of Dold (2.12 of [5]. See also McCleary [18], Thm. 12.11), there is E2p,q := Torkp (Tq (k), B) ⇒ T∗ (B).

A is flat, {Tq } is a long 0 for sufficiently small q the Universal Coefficient a spectral sequence with

As an immediate consequence, we have the following result. Corollary 27. If f : A → A′ is a k-algebra map between flat algebras which induces an isomorphism in ∼ = symmetric homology, HS∗ (A) → HS∗ (A′ ), then for a commutative k-algebra B, the map f ⊗ idB induces ∼ = an isomorphism HS∗ (A; B) → HS∗ (A′ ; B). Under stronger hypotheses, the universal coefficient spectral sequence reduces to short exact sequences. Recall some notions of ring theory (c.f. the article Homological Algebra: Categories of Modules (200:K), Vol. 1, pp. 755-757 of [10]). A commutative ring k is said to have global dimension ≤ n if for all k-modules X and Y , Extm k (X, Y ) = 0 for m > n. k is said to have weak global dimension ≤ n if for all k-modules X and Y , Torkm (X, Y ) = 0 for m > n. Note, the weak global dimension of a ring is less than or equal to its global dimension, with equality holding for Noetherian rings but not in general. A ring is said to be hereditary if all submodules of projective modules are projective, and this is equivalent to the global dimension of the ring being no greater than 1. Theorem 28. If k has weak global dimension ≤ 1, then the spectral sequence of Thm. 21 reduces to short exact sequences, (7)

0 −→ HSn (A | k) ⊗k B −→ HSn (A; B) −→ Tork1 (HSn−1 (A | k), B) −→ 0.

Moreover, if k is hereditary and and A is projective over k, then these sequences split (unnaturally). Proof. Assume first that k has weak global dimension ≤ 1. So Torkp (Tq (k), B) = 0 for all p > 1. Following Dold’s argument (Corollary 2.13 of [5]), we obtain the required exact sequences, 0 −→ Tn (k) ⊗k B −→ Tn (B) −→ Tork1 (Tn−1 (k), B) −→ 0. Assume further that k is hereditary and A is projective. Then by Lemma 22, Yn A is projective for each n. Theorem 8.22 of Rotman [22] then gives us the desired splitting. Remark 29. The proof given above also proves UCT for cyclic homology. A partial result along these lines exists in Loday ([12], 2.1.16). There, he shows HC∗ (A | k) ⊗k K ∼ = HC∗ (A | K) and HH∗ (A | k) ⊗k K ∼ = HH∗ (A | K) in the case that K is a localization of k, and A is a K-module, flat over k. I am not aware of a statement of UCT for cyclic or Hochschild homology in its full generality in the literature. 2.3. Integral Symmetric Homology and a Bockstein Spectral Sequence. We shall obtain a converse to Cor. 27 in the case k = Z. Theorem 30. Let f : A → A′ be an algebra map between torsion-free Z-algebras. Suppose for B = Q and B = Z/pZ for any prime p, the map f ⊗ idB induces an isomorphism HS∗ (A; B) → HS∗ (A′ ; B). Then f ∼ = also induces an isomorphism HS∗ (A) → HS∗ (A′ ). First, note that if A is flat over k, Prop. 25 allows one to construct the Bockstein homomorphisms βn : HSn (A; Z) → HSn−1 (A; X) associated to a short exact sequence of k-modules, 0 → X → Y → Z → 0. These Bocksteins are natural in the following sense: Lemma 31. Suppose f : A → A′ is a map of flat k-algebras, and 0 → X → Y → Z → 0 is a short exact sequence of k-modules. Then the following diagram is commutative for each n: HSn (A; Z)

β

f∗

HSn (A′ ; Z)

/ HSn−1 (A; X) f∗

β

′

/ HSn−1 (A′ ; X)

10

SHAUN V. AULT

Moreover if the induced map f∗ : HS∗ (A; W ) → HS∗ (A′ ; W ) is an isomorphism for any two of W = X, W = Y , W = Z, then it is an isomorphism for the third. Proof. A and A′ flat imply both sequences of complexes are exact: 0 → Y∗ A ⊗k X → Y∗ A ⊗k Y → Y∗ A ⊗k Z → 0, 0 → Y∗ A′ ⊗k X → Y∗ A′ ⊗k Y → Y∗ A′ ⊗k Z → 0. The map Y∗ A → Y∗ A′ induces a map of short exact sequences, hence induces a commutative ladder of long exact sequences of homology groups. In particular, the squares involving the boundary maps (Bocksteins) must commute. Now, assuming further that f∗ induces isomorphisms HS∗ (A; W ) → HS∗ (A′ ; W ) for any two of W = X, ∼ = W = Y , W = Z, let V be the third module. The 5-lemma implies isomorphisms HSn (A; V ) −→ HSn (A′ ; V ) for each n. We shall now proceed with the proof of Thm. 30. All tensor products will be over Z in what follows. Proof. Let A and A′ be torsion-free Z-modules. Over Z, torsion-free implies flat. Let f : A → A′ be an algebra map inducing isomorphism in symmetric homology with coefficients in Q and also in Z/pZ for any prime p. For m ≥ 2, there is a short exact sequence, p

0 −→ Z/pm−1 Z −→ Z/pm Z −→ Z/pZ −→ 0. Consider first the case m = 2. Since HS∗ (A; Z/pZ) → HS∗ (A′ ; Z/pZ) is an isomorphism, Lemma 31 implies the induced map is an isomorphism for the middle term: ∼ =

f∗ : HS∗ (A; Z/p2 Z) −→ HS∗ (A′ ; Z/p2 Z)

(8)

(Note, all maps induced by f on symmetric homology will be denoted by f∗ .) For the inductive step, fix m > 2 and suppose f induces an isomorphism in symmetric homology, f∗ : ∼ = HS∗ (A; Z/pm−1 Z) −→ HS∗ (A′ ; Z/pm−1 Z). Again, Lemma 31 implies the induced map is an isomorphism on the middle term. ∼ =

f∗ : HS∗ (A; Z/pm Z) −→ HS∗ (A′ ; Z/pm Z)

(9)

Denote Z/p∞ Z := lim Z/pm Z. Note, this is a direct limit in the sense that it is a colimit over a directed −→

system. The direct limit functor is exact (Prop. 5.3 of [24]), so the maps HSn (A; Z/p∞ Z) → HSn (A′ ; Z/p∞ Z) induced by f are isomorphisms, given by the chain of isomorphisms below: f∗ HSn (A; Z/p∞ Z) ∼ = Hn (lim Y∗ A ⊗ Z/pm Z) ∼ = lim H∗ (Y∗ A ⊗ Z/pm Z) −→ −→

−→

lim H∗ (Y∗ A′ ⊗ Z/pm Z) ∼ = H∗ (lim Y∗ A′ ⊗ Z/pm Z) ∼ = HS∗ (A′ ; Z/p∞ Z) −→

−→

(Note, f∗ here stands for lim Hn (Y∗ f ⊗ id).) −→

Finally, consider the short exact sequence of abelian groups, M 0 −→ Z −→ Q −→ Z/p∞ Z −→ 0 p prime

∞

′

The isomorphism f∗ : HS∗ (A; Z/p Z) → HS∗ (A ; Z/p∞ Z) passes to direct sums, giving isomorphisms for each n, ! ! M M ∼ = ′ ∞ ∞ Z/p Z . Z/p Z −→ HSn A ; f∗ : HSn A; p

p

′

Together with the assumption that HS∗ (A; Q) → HS∗ (A ; Q) is an isomorphism, another appeal to Lemma 31 ∼ = gives the required isomorphism in symmetric homology, f∗ : HSn (A) −→ HSn (A′ ). Remark 32. Theorem 30 may be useful for determining integral symmetric homology, since rational computations are generally simpler, and computations mod p may be made easier due to the presence of additional structure.


11

Finally, we state a result along the lines of McCleary [18], Thm. 10.3. (This is a version of the Bockstein spectral sequence for symmetric homology.) Denote the torsion submodule of the graded module H∗ by τ (H∗ ). Theorem 33. Suppose A is free of finite rank over Z. Then there is a singly-graded spectral sequence with E∗1 := HS∗ (A; Z/pZ) ⇒ HS∗ (A)/τ (HS∗ (A)) ⊗ Z/pZ, with differential map d1 = β, the standard Bockstein map associated to 0 → Z/pZ → Z/p2 Z → Z/pZ → 0. Moreover, the convergence is strong. The proof McCleary gives on p. 459 carries over to our case intact. All that is required for this proof is that each Hn (Y∗ A) be a finitely-generated abelian group. The hypothesis that A is finitely-generated, coupled with a result of Section ?, namely Cor. 95, guarantees this. Note, over Z, free of finite rank is equivalent to flat and finitely-generated.

3. Tensor Algebras For a general k-algebra A, the standard resolution is often too difficult to work with. In order to develop workable methods to compute symmetric homology, it will become necessary to find smaller resolutions of k. First, we develop the necessary results for the special case of tensor algebras. Indeed, tensor algebra arguments are also key in the proof of Fiedorowicz’s Theorem (Thm. 1(i) of [7]) about the symmetric homology of group algebras. Let T : k-Alg → k-Alg be the functor sending an algebra A to the tensor algebra generated by A, T A := k ⊕ A ⊕ A⊗2 ⊕ A⊗3 ⊕ . . . There is an algebra homomorphism θ : T A → A, defined by multiplying tensor factors, θ(a0 ⊗ a1 ⊗ . . . ⊗ ak ) := a0 a1 · · · ak . In fact, θ defines a natural transformation T → id. We shall also make use of a k-module homomorphism h, sending the algebra A identically onto the summand A of T A. This map is a natural transformation from the forgetful functor U : k-Alg → k-Mod to the functor U T . The resolution of A by tensor algebras may be regarded as a k-complex, where T n A is viewed in degree n − 1. θ

θ

θ

A T A ←1 T 2 A ←2 . . . 0←A←

(10)

The maps θn for n ≥ 1 are defined in terms of face maps, θn := of θ impies θn is a natural transformation T p+1 → T p .

Pn

i n−i θT i A . i=0 (−1) T

Note that naturality

Remark 34. Note that the complex (10) is nothing more than the complex associated to May’s 2-sided bar construction B∗ (T, T, A) (See chapter 9 of [14]). If we denote by A0 the chain complex consisting of A in degree 0 and 0 in higher degrees, then there is a homotopy hn : Bn (T, T, A) → Bn+1 (T, T, A) that establishes a strong deformation retract B∗ (T, T, A) → A0 . In fact, the homotopy maps are given by hn := hT n+1 A , where h is the natural transformation U → U T discussed above. For each q ≥ 0, if we apply the functor Yq to the complex (10), we obtain the sequence below: (11)

Yq A o

0 o

Yq θA

Yq T A o

Yq θ1

Yq T 2 A o

Yq θ2

Yq T 3 A o

···

L I claim this sequence is exact. Indeed, by Remark 14, Yq A ∼ = n≥0 k [N ([n] \ ∆S)q−1 ] ⊗k A⊗(n+1) . For each n, the k-module map h induces a homotopy h⊗n on each complex, (12)

0 o

A⊗n o

⊗n θA

(T A)⊗n o

θ1⊗n

(T 2 A)⊗n o

Each k [N ([n] \ ∆S)q−1 ] is free as k-module, so tensor products preserve exactness.

···

12

SHAUN V. AULT

Denote by di (A) (or di , when the context is clear) the ith differential map of Y∗ A. Consider the double complex defined by Tp,q := Yq T p+1 A, with vertical differential (−1)p dq .

(13)

.. . Y2 T A o d2 Y1 T A o d1 Y0 T A o

Y2 θ1

Y1 θ1

Y0 θ1

.. . Y2 T 2 A o −d2 Y1 T 2 A o −d1 Y0 T 2 A o

Y2 θ2

Y1 θ2

Y0 θ2

.. . Y2 T 3 A o d2 Y1 T 3 A o d1 Y0 T 3 A o

···

···

···

Consider a second double complex A∗,∗ consisting of the complex Y∗ A concentrated in the 0th column.

(14)

.. .

.. .

.. .

Y2 A o

0 o

0 o

···

0 o

0 o

···

0 o

0 o

···

d2

Y1 A o d1

Y0 A o

Theorem 35. There is a map of double complexes Θ∗,∗ : T∗,∗ → A∗,∗ inducing isomorphism in homology, H∗ (Tot(T )) → H∗ (Tot(A )). Proof. The map Θ∗,∗ is defined by Θ0,q := Yq θA , and Θp,q = 0 for positive p. Functoriality of Y∗ ensures that Y∗ θA is a chain map. The isomorphism follows from the exactness of the sequence (11). Remark 36. Observe that H∗ (Tot(A ); k) = H∗ (Y∗ A; k) = HS∗ (A). This permits the computation of symmetric homology of any given algebra A in terms of tensor algebras: Corollary 37. For an associative, unital k-algebra A, HS∗ (A) ∼ = H∗ (Tot(T ); k), where T∗,∗ is the double complex {Yq T p+1 A}p,q≥0 . The following lemma shows why it is advantageous to work with tensor algebras. Lemma 38. For a unital, associative k-algebra A, there is an isomorphism of k-complexes: M ∼ (15) Y∗ T A = Yn (A), n≥−1

where

Yn (A) =

k [N (∆S)] , n = −1 ⊗(n+1) A , n ≥0 k [N ([n] \ ∆S)] ⊗kΣop n+1


13

Moreover, the differential respects the direct sum decomposition. α

Proof. Any generator of Y∗ T A has the form [p] → [q0 ] → . . . → [qn ] ⊗ u, where   ! ! O O O u= a ⊗ a ⊗ ... ⊗  a , a∈A0

a∈A1

a∈Ap

and A0 , A1 , . . . , Ap are finite ordered lists of elements of A. Indeed, each Aj may be thought of as an element of the set product Amj for some mj . If Aj = ∅, then mj = 0. We use the convention that an empty P tensor product is equal to 1k , and say that the corresponding tensor factor is trivial. Now, let m = ( mj ) − 1. Let π : Am0 × Am1 × . . . × Amp −→ Am+1 be the evident isomorphism. Let Am = π(A0 , A1 , . . . , Ap ). Case 1. If u is non-trivial (i.e., Am 6= ∅), then construct the element O u′ = a a∈Am

Next, construct a ∆-morphism ζu : [m] → [p] as follows: For each j, ζu maps the points ! ! j j−1 j−1 X X X mi − 1 7→ j mi + 1, . . . , mi , i=0

i=0

i=0

αζu

α

′

Observe, (ζu )∗ (u ) = u. Under ∆S-equivalence, [p] → [q0 ] → . . . → [qn ] ⊗ u ≈ [m] → [q0 ] → . . . → [qn ] ⊗ u′ . The choice of u′ and ζu is well-defined with respect to the ∆S-equivalence up to isomorphism of [m] (an element of Σop m+1 ). This shows that any such non-trivial element in Y∗ T A may be written uniquely as an A⊗(m+1) . element of k [N ([m] \ ∆S)] ⊗kΣop m+1 α

⊗(p+1)

id

Case 2. If u is trivial, then u = 1k , and [p] → [q0 ] → . . . → [qn ] ⊗ u ≈ [q0 ] → [q0 ] → . . . → ⊗(q0 +1) [qn ] ⊗ 1k . This element can be identified uniquely with [q0 ] → . . . → [qn ] ∈ k [N (∆S)]. Thus, the isomorphism (15) is proven. Note that for B∗sym T A, the total number of non-trivial tensor factors is preserved under ∆S morphisms. This shows that the differential respects the direct sum decomposition. 4. Symmetric Homology of Monoid Algebras The symmetric homology for the case of a monoid algebra A = k[M ] has been studied by Fiedorowicz in [7]. In the most general formulation (Prop. 1.3 of [7]), we have: Theorem 39. HS∗ (k[M ]) ∼ = H∗ (B(C∞ , C1 , M ); k), where C1 is the little 1-cubes monad, C∞ is the little ∞-cubes monad, and B(C∞ , C1 , M ) is May’s functorial 2-sided bar construction. The proof, found in [7], makes use of a variant of the symmetric bar construction: Definition 40. Let M be a monoid. Define a functor B∗sym M : ∆S → Sets by: Bnsym M := B∗sym M [n] := M n+1 , (set product) B∗sym M (α) : (m0 , . . . , mn ) 7→ α(m0 , . . . , mn ), for α ∈ Mor∆S. where α : [n] → [k] is represented in tensor notation, and evaluation at (m0 , . . . , mn ) is as in definition 7. Definition 41. For a C -set X and C op -set Y , define the C -equivariant set product:   a Y ×C X :=  Y (C) × X(C) / ≈, C∈ObjC

where the equivalence ≈ is generated by the following: For every morphism f ∈ MorC (C, D), and every x ∈ X(C) and y ∈ Y (D), we have y, f∗ (x) ≈ f ∗ (y), x . ∼ =

Note that B∗sym M is a ∆S-set, and also a simplicial set, via the chain of functors ∆op ֒→ ∆C op → ∆C ֒→ ∆S. Let X∗ := N (− \ ∆S) ×∆S B∗sym M .

Proposition 42. X∗ is a simplicial set whose homology computes HS∗ (k[M ]).

14

SHAUN V. AULT

Proof. Since M is a k-basis for k[M ], B∗sym M acts as a k-basis for B∗sym k[M ]. Then, observe that k[N (− \ ∆S) ×∆S B∗sym M ] = k[N (− \ ∆S)] ⊗∆S B∗sym k[M ]. If M = JX+ is a free monoid on a generating set X, then k[M ] = T (X), the (free) tensor algebra over k on the set X. In this case, we have the following: Lemma 43.



HS∗ (T (X)) ∼ = H∗ 

a

n≥−1

where en = X



en ; k  , X

N (∆S), n = −1 n+1 X , n ≥0 N ([n] \ ∆S) ×Σop n+1

Proof. This is essentially a special case of Lemma 38 when the tensor algebra is free, generated by X = {xi | i ∈ I }. This proves Thm. 39 in the special case that M is a free monoid. Indeed, if A = k[JX+ ], we have   a X n+1  HS∗ (A) ∼ N ([n] \ ∆S) ×Σop = H∗ (N (∆S)) ⊕ H∗  n+1 n≥0

Now, the set {N ([n] \ ∆S)}n≥0 has the structure of C∞ -operad, in the sense of May [16], since the augmentated chain complex k ← k [N ([n] \ ∆S)] is acyclic (Prop. 11), and the action of the symmetric group on N ([n] \ ∆S) is free. Let the associated monad be denoted N ∆S . Thus we have by definition, a N ∆S X = X n+1 N ([n] \ ∆S) ×Σop n+1 n≥0

We obtain a chain of equivalences, N

∆S

X ≃ B(N

∆S

, J, JX) ≃ B(C∞ , J, JX) ≃ B(C∞ , C1 , JX).

The first equivalence arises by Prop. 9.9 of May [14], and the last equivalence arises from a weak homotopy equivalence mentioned in Cor. 6.2 of [14] – the James construction J, is equivalent to the little 1-cubes monad, C1 . If M is a group, Γ, then Fiedorowicz [7] found: Theorem 44. HS∗ (k[Γ]) ∼ = H∗ (ΩΩ∞ S ∞ (BΓ); k). This final formulation shows in particular that HS∗ is a non-trivial theory. While it is true that H∗ (Ω∞ S ∞ (X)) = H∗ (QX) is well understood, the same cannot be said of the homology of ΩΩ∞ S ∞ X. Indeed, May states that H∗ (QX) may be regarded as the free allowable Hopf algebra with conjugation over the Dyer-Lashof algebra and dual of the Steenrod algebra (See [3], preface to chapter 1, and also Lemma 4.10). Cohen and Peterson [4] found the homology of ΩΩ∞ S ∞ X, where X = S 0 , the zero-sphere, but there is little hope of extending this result to arbitrary X using the same methods. We shall have more to say about HS1 (k[Γ]) in 11.5. 5. Symmetric Homology Using ∆S+ In this section, we shall show that replacing ∆S by ∆S+ in an appropriate way does not affect the computation of HS∗ . First, we extend the symmetric bar contruction over ∆S+ . Definition 45. For an associative, unital algebra, A, over a commutative ground ring k, define a functor sym B∗ + A : ∆S+ → k-Mod by: sym+ A := B∗sym A[n] := A⊗(n+1) Bn sym+ B−1 A := k, sym+ A(α) : (a0 ⊗ a1 ⊗ . . . ⊗ an ) 7→ α(a0 , a1 , . . . , an ), for α ∈ Mor∆S, B∗ sym B∗ + A(ιn ) : λ 7→ λ(1A ⊗ . . . ⊗ 1A ), (λ ∈ k).


15

Consider the functor Y∗+ : k-Alg→ k-SimpMod given by sym+

Y∗+ A := k[N (− \ ∆S+ )] ⊗∆S+ B∗

(16)

Y∗+ f = id ⊗

A.

sym B∗ + f

Our goal is to prove the following: Theorem 46. For an associative, unital k-algebra A, HS∗ (A) = H∗ (Y∗+ A; k). As the preliminary step, we shall prove the theorem in the special case of tensor algebras. We shall need an analogue of Lemma 38 for ∆S+ . Lemma 47. For a unital, associative k-algebra A, there is an isomorphism of k-complexes: M (17) Y∗+ T A ∼ Yn+ , = n≥−1

where

k [N (∆S+ )] , n = −1 k [N ([n] \ ∆S+ )] ⊗kΣn+1 A⊗(n+1) , n ≥ 0 Moreover, the differential respects the direct sum decomposition. Yn+ =

Proof. The proof follows verbatim as the proof of Lemma 38, only with ∆S replaced with ∆S+ throughout. Lemma 48. There is a chain map JA : Y∗ A → Y∗+ A, which is natural in A. Proof. First observe that the the inclusion ∆S ֒→ ∆S+ induces an inclusion of nerves, N (− \ ∆S) ֒→ N (− \ ∆S+ ), which in turn induces the chain map: (18)

k [N (− \ ∆S)] ⊗∆S B∗sym A −→ k [N (− \ ∆S+ )] ⊗∆S B∗sym A. sym

Now, k [N (− \ ∆S+ )] is both a right ∆S-module and a right ∆S+ -module. Similarly, B∗ + A is both a left sym ∆S-module and a left ∆S+ -module. There is a natural transformation B∗sym A → B∗ + A, again induced by inclusion of categories, hence there is a chain map (19)

sym+

k [N (− \ ∆S+ )] ⊗∆S B∗sym A −→ k [N (− \ ∆S+ )] ⊗∆S B∗

A.

Finally, pass to tensors over ∆S+ : (20)

sym+

k [N (− \ ∆S+ )] ⊗∆S B∗

sym+

A −→ k [N (− \ ∆S+ )] ⊗∆S+ B∗

The composition of maps (18), (19) and (20) defines a chain map JA : Y∗ A → to verify that J is natural in A.

A.

Y∗+ A.

It is straightforward

∼ =

We shall show that JA induces an isomorphism on homology H∗ (Y∗ A; k) −→ H∗ (Y∗+ A; k) by examining the case of tensor algebras. Lemma 49. For a unital, associative k-algebra A, the chain map JT A : Y∗ T A −→ Y∗+ T A is a homotopy equivalence, hence HS∗ (T A) = H∗ (Y∗+ T A; k) . Proof. There is a commutative square of complexes: JT A / Y +T A Y∗ T A ∗ O O ∼ ∼ = = M jn M M Yn n≥−1 / Yn+

n≥−1

n≥−1

The isomorphisms on the left and right follow from Lemmas 38 and 47. The maps jn are defined by inclusion of categories ∆S ֒→ ∆S+ : (21)

j−1 : k [N (∆S)] → k [N (∆S+ )]

16

SHAUN V. AULT

(22)

jn : k [N ([n] \ ∆S)] ⊗kΣn+1 A⊗(n+1) −→ k [N ([n] \ ∆S+ )] ⊗kΣn+1 A⊗(n+1) ,

n ≥ 0.

Observe that N (∆S+ ) is contractible, since [−1] ∈ Obj(∆S+ ) is initial. Thus by Lemma 15, the map j−1 is a homotopy equivalence on the (−1)-component. Now, for n ≥ 0, there is equality N ([n]\∆S+ ) = NL ([n]\∆S), since there are no morphisms [n] → [−1] for n ≥ 0, so jn is a homotopy equivalence. Therefore jn , and hence JT A , is a homotopy equivalence. Remark 50. Observe, this lemma provides our first major departure from the theory of cyclic homology. The proof above would not work over the categories ∆C and ∆C+ , as N (∆C) is not contractible. + Consider a double complex T∗,∗ (A), the analogue of complex (13) for ∆S+ .

.. .

.. .

Y2+ T A (23)

o

Y2+ θ1

.. .

Y2+ T 2 A

Y1+ T A o

Y1+ θ1

Y2+ θ2 Y1+ T 2 A o Y1+ T 3 A o

···

d2

−d1

d1

Y0+ T A o

Y0+ θ1

Y2+ T 3 A

···

−d2

d2

Y2+ θ2

o

o

d1

Y1+ θ2 Y0+ T 2 A o Y0+ T 3 A o

···

+ Consider a second double complex, A∗,∗ (A), the analogue of complex (14) for ∆S+ . It consists of the complex + th Y∗ A as the 0 column, and 0 in all positive columns.

(24)

.. .

.. .

.. .

Y2+ A o

0 o

0 o

···

0 o

0 o

···

0 o

0 o

···

d2

Y1+ A o d1

Y0+ A o

+ + We may think of each double complex construction, A∗,∗ , T∗,∗ , A∗,∗ and T∗,∗ , as a functor from k-Alg to the category of double k-complexes. Each functor takes unital morphisms of algebras to maps of double complexes in the obvious way – for example if f : A → B, then the induced map T∗,∗ (A) → T∗,∗ (B) is defined on the (p, q)-component by the map Yq T p+1 f . The induced map commutes with vertical differentials of A∗,∗ and + + T∗,∗ (resp., A∗,∗ and T∗,∗ ) by naturality of Y∗ (resp. Y∗+ ), and it commutes with the horizontal differentials + + of T∗,∗ and T∗,∗ by naturality of θn . The map J induces a natural transformation J∗,∗ : A∗,∗ → A∗,∗ , defined by JA : Y∗ A → Y∗+ A, p = 0 Jp,∗ (A) = 0, p>0

+ Define a map of bigraded modules, K∗,∗ (A) : T∗,∗ (A) → T∗,∗ (A) by: Kp,∗ (A) = JT p+1 A : Y∗ T p+1 A → Y∗+ T p+1 A. Now, K∗,∗ (A) commutes with the vertical differentials because each JT p+1 A is a chain map.


17

K∗,∗ (A) commutes with the horizontal differentials because of naturality of J. Finally, K∗,∗ defines a natural + transformation T∗,∗ → T∗,∗ , again by naturality of J. A f B

Kp,q (A)

Yq T p+1 A p+1 Yq T f

=JT p+1 A

Kp,q (B)

Yq T p+1 A

=JT p+1 B

/ Yq+ T p+1 A Y + T p+1 f q / Yq+ T p+1 A

Recall by Thm 35, there is a map of double complexes, Θ∗,∗ (A) : T∗,∗ (A) −→ A∗,∗ (A) inducing an isomorphism in homology of the total complexes. We shall need the analogous statement for the double complexes + + T∗,∗ and A∗,∗ . + Theorem 51. For any unital associative algebra, A, there is a map of double complexes, Θ+ ∗,∗ (A) : T∗,∗ (A) → + + + A∗,∗ (A) inducing isomorphism in homology H∗ (Tot(T (A)); k) → H∗ (Tot(A (A)); k). Moreover, Θ+ ∗,∗ + + provides a natural transformation T∗,∗ → A∗,∗ .

Proof. The map Θ+ ∗,∗ (A) is defined as: Θ+ p,q (A)

:=

Yq+ θA , p = 0 0, p>0

This map is a map of double complexes by functoriality of Y∗+ , and the isomorphism follows from the exactness of the sequence (11). Naturality of Θ∗,∗ follows from naturality of θ. Lemma 52. The following diagram of functors and transformations is commutative.

(25)

T∗,∗ K∗,∗ + T∗,∗

Θ∗,∗

Θ+ ∗,∗

/ A∗,∗ J∗,∗ + / A∗,∗

Proof. It suffices to fix an algebra A and examine only the (p, q)-components. Note, if p > 0, then the right hand side of the diagram is trivial, so we may assume p = 0.

(26)

Yq T A (JT A )q + Yq T A

This diagram commutes by naturality of J.

Yq θA

/ Yq A (J ) A q Yq+ θA / Yq+ A

To any double complex B∗,∗ over k, we may associate two spectral sequences: (EI B)∗,∗ , obtained by first taking vertical homology, then horizontal; and (EII B)∗,∗ , obtained by first taking horizontal homology, then vertical. In the case that B∗,∗ lies entirely within the first quadrant, both spectral sequences converge to H∗ (Tot(B); k) (See [18], Section 2.4). Maps of double complexes induce maps of spectral sequences, EI and EII , respectively. Fix the algebra A, and consider the commutative diagram of spectral sequences induced by diagram (25). The induced maps will be indicated by an overline, and explicit mention of the algebra A is suppressed for

18

SHAUN V. AULT

brevity of notation. Θ

EII T (27)

/ EII A

K

J

EII T +

Θ+

/ EII A +

Now, by Thm. 35 and Thm. 51, we know that Θ∗,∗ and Θ+ ∗,∗ induce isomorphisms on total homology, so r r + Θ and Θ also induce isomorphisms on the limit term of the spectral sequences. In fact, both Θ and Θ+ are isomorphisms (EII T )r → (EII A )r for r ≥ 1. This is because taking horizontal homology of T∗,∗ (resp. + T∗,∗ ) kills all components in positive columns, leaving only the 0th column, which is chain-isomorphic to the th + 0 column of A∗,∗ (resp. A∗,∗ ). Consider a second diagram of spectral sequences, with induced maps indicated by a hat. EI T b K EI T +

(28)

b Θ

d + Θ

/ EI A Jb / EI A +

b induces an isomorphism on the limit terms of the sequences EI T and EI T + as a result of Now the map K b r is an isomorphism for r ≥ 1. Lemma 49. As before, K Now, since H∗ (Tot(A ); k) = H∗ (Y∗ A; k) and H∗ (Tot(A + ); k) = H∗ (Y∗+ A; k), we can put together a chain of isomorphisms ∞ b∞ K ∗ Θ ∞ ∞ ∞ / EI T + ∞ H∗ (Y∗ A; k) ∼ = (EII A )∗ o ∼ (EII T )∗ ∼ = (EI T )∗ ∼ ∗ =

=

∼ = EII T +

(29)

∞ ∗

(Θ+ )∞ ∗ ∼ =

/ EII A +

∞ ∗

∼ = H∗ Y∗+ A; k

Commutativity of Diagram (25) ensures that the composition of maps in Diagram (29) is the map induced by JA , hence proving: Theorem 53. For a unital, associative k-algebra A, the chain map JA : Y∗ A → Y∗+ A induces an isomor∼ = phism on homology, H∗ (Y∗ A; k) −→ H∗ (Y∗+ A; k). As a direct consequence, HS∗ (A) ∼ = H∗ (Y∗+ A; k), proving Thm. 46. 6. The Category Epi∆S and a Smaller Resolution The complex (1) is an extremely large and unwieldy for computation. Fortunately, when the algebra A is equipped with an augmentation, ǫ : A → k, complex (1) is homotopic to a much smaller subcomplex. 6.1. Basic and Reduced Tensors. Recall, if A has an augmentation ǫ, then there is an augmentation ǫ ideal I and the sequence 0 → I → A → k → 0 is split exact as k-modules. So A ∼ = I ⊕ k, and every x ∈ A can be decomposed uniquely as x = a + λ for some a ∈ I, λ ∈ k. Definition 54. Define B−1,∅ A = k. For n ≥ 0, if J ⊆ [n], define Bn,J A :=

B0J

⊗

B1J

⊗ ...⊗

BnJ ,

where

BjJ

=

I k

if j ∈ J if j ∈ /J

Lemma 55. For each n ≥ −1, there is a direct sum decomposition of k-modules Bnsym+ A ∼ =

M

Bn,J A.

J⊆[n]

Proof. For n = −1,

sym B−1 + A

= k = B−1,∅ A. For n ≥ 0, Bnsym+ A = (I ⊕ k)⊗(n+1) ∼ =

M

J⊆[n]

Bn,J A.


19

Definition 56. A basic tensor is any tensor w0 ⊗ w1 ⊗ . . . ⊗ wn , where each wj is in I or is equal to the unit of A. Call a tensor factor wj trivial if it is the unit of A. If all factors of a basic tensor are trivial, then the tensor is called trivial, and if no factors are trivial, the tensor is called reduced. It will become convenient to include the object [−1] = ∅ in ∆. Denote the enlarged category by ∆+ . For sym a basic tensor Y ∈ Bn + A, we shall define a map δY ∈ Mor∆+ as follows: If Y is trivial, let δY = ιn . Otherwise, Y has n + 1 non-trivial factors for some n ≥ 0. Define δY : [n] → [n] to be the unique injective map that sends each point 0, 1, . . . , n to a point p ∈ [n] such that Y is non-trivial at factor p. Let Y be the tensor obtained from Y by omitting all trivial factors if such exist, or Y := 1k if Y is trivial. Note, Y is the unique basic tensor such that (δY )∗ (Y ) = Y . sym

Proposition 57. Any chain [q] → [q0 ] → . . . → [qn ] ⊗ Y ∈ k[N (− \ ∆S+ )] ⊗∆S+ B∗ + A, where Y is a basic tensor, is equivalent to a chain [q] → [q0 ] → . . . → [qn ] ⊗ Y , where either Y is reduced or Y = 1k and q = −1. Proof. Let δY and Y be defined as above, and let [q] be the source of δY . Then Y = (δY )∗ (Y ), and φδY

φ

[q] → [q0 ] → . . . → [qn ] ⊗ Y ≈ [q] → [q0 ] → . . . → [qn ] ⊗ Y . 6.2. Reducing to Epimorphisms. We now turn our attention to the morphisms in the chains. Our goal is to reduce to those chains that involve only epimorphisms. Definition 58. Let C be a category. The category EpiC (resp. MonoC ) is the subcategory of C consisting of the same objects as C and only those morphisms f ∈ MorC that are epic (resp. monic). The set of morphisms of EpiC from X to Y may be denoted EpiC (X, Y ). Similarly, the set of morphisms of MonoC from X to Y may be denoted MonoC (X, Y ). Remark 59. A morphism α = (φ, g) ∈ Mor∆S+ is epic (resp. monic) if and only if φ is epic (resp. monic) as morphism in ∆+ . Proposition 60. Any morphism α ∈ Mor∆S+ decomposes uniquely as (η, id) ◦ γ, where γ ∈ Mor(Epi∆S+ ) and η ∈ Mor(Mono∆+ ). Proof. Suppose α has source [−1] and target [n]. Then α = ιn is the only possibility, and this decomposes as ιn ◦ id[−1] . Now suppose the source of α is [p] for some p ≥ 0. Write α = (φ, g), with φ ∈ Mor∆ and g ∈ Σop p+1 . We shall decompose φ as follows: For φ : [p]→[r], suppose φ hits q + 1 distinct points in [r]. Then π : [p] → [q] is induced by φ by maintaining the order of the points hit. η is the obvious order-preserving monomorphism [q] → [r] so that ηπ = φ as morphisms in ∆. To get the required decomposition in ∆S, use: α = (η, id) ◦ (π, g). Now, if (ξ, id) ◦ (ψ, h) is also a decomposition of α, with ξ monic and ψ epic, then (ξ, id) ◦ (ψ, h) = (η, id) ◦ (π, g) implies (ξψ, g −1 h) = (ηπ, id), proving g = h. Uniqueness will then follow from uniqueness of such decompositions entirely within the category ∆. The latter follows from Theorem B.2 of [12], since any monomorphism (resp. epimorphism) of ∆ can be factored uniquely as compositions of δi (resp. σi ). This decompisition will be written: [p] → [r] = [p] ։ im([p] → [r]) ֒→ [r]. Proposition 61. The epimorphism construction is a functor Ep : [p] \ ∆S+ → [p] \ Epi∆S+ . Proof. Fix p ≥ −1. If β is an object of [p] \ ∆S+ (i.e. a morphism [p] → [r1 ]), then E (β) is the epimorphism β

α

α

[p] ։ im([p] → [r]). If [p] → [r1 ] → [r2 ], then there is an induced map im([p] → [r1 ]) → im([p] → [r2 ]) making the diagram commute:

(30)

[r1 ] O dJJJ JJ JJ JJ β

α

/ [r2 ] t: O t t tt tt αβ t t η1 η2 [p] J JJ π t π1 tt 2 J JJ t tt JJ$ ? $ ? ztz t / / im([p] → [r2 ]) im([p] → [r1 ]) α

20

SHAUN V. AULT α

α

α is the epimorphism induced from the map αη1 . Furthermore, for morphisms [p] → [r1 ] →1 [r2 ] →2 [r3 ], we have: α2 α1 = α2 ◦ α1 . Remark 62. Note that if α : [p] → [r] is an epimorphism of ∆S+ , then E (α) = α. Define a variant of the symmetric bar construction: sym+

Definition 63. B∗

I : Epi∆S+ → k-Mod is the functor defined by: sym+ I := I ⊗(n+1) , n ≥ 0, Bn sym+ B−1 I := k,

sym+

B∗

I(α) : (a0 ⊗ a1 ⊗ . . . ⊗ an ) 7→ α(a0 , . . . , an ), for α ∈ Mor(Epi∆S+ )

This definition makes sense since I is an ideal, and α is required to be epimorphic. Note, the simple sym tensors w0 ⊗ . . . ⊗ wn in Bn + I are by definition reduced. Consider the simplicial k-module: sym+

Y∗epi A := k[N (− \ Epi∆S+ )] ⊗Epi∆S+ B∗

(31)

I

There is an obvious inclusion, f : Y∗epi A −→ Y∗+ A. Define a chain map g in the opposite direction as follows. First, by Prop. 57 and observations above, we only need to define g on the chains u = [q] → [q0 ] → . . . → [qn ] ⊗ Y where Y is reduced (or Y = 1k ). In this case, define component maps g(q) := N (Eq ) ⊗ id. A priori, this definition is well-defined only when the tensor product is over k. We would like to assemble the maps g(q) into a chain map g. In order to do this, we must show that the maps are compatible under ∆S+ -equivalence. φψ

φ

Suppose v = [p] −→ [p0 ] → . . . → [pn ] ⊗ Z and v ′ = [q] → [p0 ] → . . . → [pn ] ⊗ ψ∗ (Z), where ψ ∈ Mor∆S+ ([p], [q]). If Z is a basic tensor, then so is ψ∗ (Z). In order to apply g to v or v ′ , each must first be put into a reduced form. Case 1 Suppose Z is trivial. Then v and v ′ both reduce to [−1] → [p0 ] → . . . → [pn ] ⊗ 1, hence g(v) = g(v ′ ). Case 2 Suppose Z is non-trivial. For the sake of clean notation, let W = ψ∗ (Z). Construct δZ , Z, δW and W such that Z = (δZ )∗ (Z) and W = (δW )∗ (W ), as in Prop. 57, and reduce both chains: φψ

(32)

[p] −→ [p0 ] → . . . → [pn ] ⊗ Z reduce ≈ φψδZ

[p] −→ [p0 ] → . . . → [pn ] ⊗ Z

≈

φ

[q] → [p0 ] → . . . → [pn ] ⊗ W reduce ≈ φδW

[q] −→ [p0 ] → . . . → [pn ] ⊗ W

Observe that number of distinct points hit by ψδZ is exactly q + 1; indeed, W = ψ∗ (Z) has q + 1 non-trivial factors. Thus, [q] = im([p] → [q]). Now, Prop. 60 implies that there is precisely one ∆S-epimorphism γ : [p] → [q] making Diagram (33) commute.

(33)

ψ / [q] [p] O ? O W/ ψδ /// Z δW // δZ // // // / [q] [p] OO OOO ???? // OOO ???? // OOO ??? / O OOO ?????? // γ OOO ????// O' ' ? * J im([p] → [q])

That is to say, there exists an epimorphism γ such that γ∗ (Z) = W and ψδZ = δW γ. So we may replace the first morphism of the chain in the lower left of Diagram (32) with φδW γ. Then when we apply g to the


21

chain, the first morphism becomes Ep (φδW γ) = Ep (φδW ) ◦ γ, since γ is epic. Let π := Ep (φδW ). Then the result of applying g to each side Diagram (32) is shown below: (34) φψδZ

[p] −→ [p0 ] → . . . → [pn ] ⊗ Z g(p) πγ [p] → im([p] → [p0 ]) → . . . → im([p] → [pn ]) ⊗ Z

φδW

[q] −→ [p0 ] → . . . → [pn ] ⊗ W g(q) π [q] → im([q] → [p0 ]) → . . . → im([q] → [pn ]) ⊗ W

Observe that there is equality of objects and morphisms, im([p] → [p0 ]) → . . . → im([p] → [pn ]) = im([q] → [p0 ]) → . . . → im([q] → [pn ]). Finally, since γ is epic, the Epi∆S+ -equivalence allows us to identify g(v) ≈ g(v ′ ). This shows that g is well-defined. Now clearly gf = id. Proposition 64. f g ≃ id. sym+

Proof. In what follows, we assume Y is a basic tensor in Bq (n)

hj

(n)

I. Define maps hj

as follows:

([q] → [q0 ] → . . . → [qn ] ⊗ Y ) :=

[q] ։ im([q] → [q0 ]) ։ . . . ։ im([q] → [qj ]) ֒→ [qj ] → . . . → [qn ] ⊗ Y. (n) hj

is well-defined by the functorial properties of the epimorphism construction, and a routine, but tedious, verification shows that h defines a presimplicial homotopy from f g to id. Proposition 65. If A has augmentation ideal I, then sym HS∗ (A) = H∗ Y∗epi A; k = H∗ k[N (− \ Epi∆S+ )] ⊗Epi∆S+ B∗ + I; k .

Proof. The complex (31) has been shown to be chain homotopy equivalent to the complex Y∗+ A, which by Thm. 46, computes HS∗ (A). Remark 66. The condition that A have an augmentation ideal may be lifted (as Richter conjectures), if it can be shown that N (Epi∆S) is contractible. As partial progress along these lines, it can be shown that N (Epi∆S) is simply-connected. 7. A spectral sequence for HS∗ (A)

Fix a unital associative algebra A over commutative ground ring k, and assume A has an augmentation, with augmentation ideal I. Let Y∗ = Y∗epi A be the complex (31). We shall find a useful spectral sequence that computes HS∗ (A) based on a filtration of Y∗ . 7.1. Filtering by Number of Strict Epimorphisms. Appealing to Remark 14, we shall represent a q-chain of Y∗ as [m0 ] → [m1 ] → . . . → [mq ] ⊗ Y , where the tensor product is over k. Consider a filtration of Y∗ by number of strict epimorphisms, or jumps in such chains: Fp Yq is generated by the chains [m0 ] → [m1 ] → . . . → [mq ] ⊗ Y , where mi−1 > mi for no more than p distinct values of i. The face maps of Y∗ only compose or delete morphisms, so Fp is compatible with the differential of Y∗ . The filtration quotients 0 := Fp Yq /Fp−1 Yq is generated by chains [m0 ] → [m1 ] → . . . → [mq ] ⊗ Y , where are easily described: Ep,q 1 0 mi−1 > mi for exactly p distinct values of i. Consider the spectral sequence with Ep,q = Hp+q (Ep,∗ ). Lemma 67. There are chain maps (one for each p): p hY M i ⊗(m0 +1) 0 Epi∆+ [mi−1 ], [mi ] ⊗kΣmp +1 E∗ Σmp +1 , I ⊗k Ep,∗ → m0 >...>mp

i=1

inducing isomorphisms in homology: 1 Ep,q

∼ =

M

m0 >...>mp

p hY i op ⊗(m0 +1) . Epi∆+ [mi−1 ], [mi ] ⊗k Hq Σmp +1 ; I i=1

22

SHAUN V. AULT

We use the convention that I ⊗0 = k, and Σ0 ∼ = 1, the trivial group. Remark 68. Here, we are using the resolution E∗ G of k by free kG-modules, En G = k G-action g.(g0 , g1 , . . . , gn ) = (gg0 , g1 , . . . , gn ), and differential, # "n−1 X (g0 , . . . , gi gi+1 , . . . , gn ) + (g0 , g1 , . . . , gn−1 ). ∂ EG (g0 , g1 , . . . , gn ) =

Qn+1 G , with

i=0

7.2. Proof of Lemma 67. The proof will be broken down into a number of steps. Begin by defining two (m ,...,mp ) (m ,...,mp ) and M∗ 0 . related chain complexes B∗ 0 For m0 > m1 > . . . > mp , define: (m0 ,...,mp )

B∗

:= k

ha i ∼ ∼ ∼ ∼ = = = = [m0 ] → . . . → [m0 ] ։ [m1 ] → . . . → [mp−1 ] ։ [mp ] ⊗ I ⊗(m0 +1) ,

where the coproduct extends over all such chains that begin with 0 or more isomorphisms of [m0 ], followed by a strict epimorphism [m0 ] ։ [m1 ], followed by 0 or more isomorphisms of [m1 ], followed by a strict epimor(m ,...,mp ) phism [m1 ] ։ [m2 ], etc., and the last morphism must be a strict epimorphism [mp−1 ] ։ [mp ]. B∗ 0 0 B is a subcomplex of Ep,∗ with the same induced differential, which we will denote by ∂ . (m0 ,...,mp )

Denote by M∗

, the chain complex consisting of 0 in degrees different from p, and

Mp(m0 ,...,mp ) := I ⊗(m0 +1) ⊗ k

p hY i=1

i Epi∆+ [mi−1 ], [mi ] . (m ,...,m )

p This is the coefficient group that shows up in Lemma 67. Its differential is trivial. Now, Bp 0 is generated by elements of the form [m0 ] ։ [m1 ] ։ . . . ։ [mp ] ⊗ Y , where the chain consists entirely of strict epimorphisms of ∆S+ . Observe that

Bp(m0 ,...,mp ) = k Epi∆S+ ([m0 ], [m1 ]) ⊗ . . . ⊗ k Epi∆S+ ([mp−1 ], [mp ]) ⊗ I ⊗(m0 +1) .

For convenience, the factor I ⊗(m0 +1) should be put on the left: (35)

Bp(m0 ,...,mp ) ∼ = I ⊗(m0 +1) ⊗ k Epi∆S+ ([m0 ], [m1 ]) ⊗ . . . ⊗ k Epi∆S+ ([mp−1 ], [mp ])

Now, each k Epi∆S+ ([m], [n]) is a (kΣm+1 )-(Σn+1 )-bimodule via the action of symmetric group elements as automorphisms of [m] and [n]. Explicitly, for an element (ψ, g) of Epi∆S+ ([m], [n]), and group elements op τ ∈ Σm+1 and σ ∈ Σn+1 , view τ as an automorphism t ∈ Σop m+1 and σ as an automorphism s ∈ Σn+1 . Then (ψ, g).σ := (id, s) ◦ (ψ, g) = (ψ s , gsψ ), and τ.(ψ, g) := (ψ, g) ◦ (id, t) = (ψ, tg). Moreover, view I ⊗(m0 +1) sym as a right (kΣm0 +1 )-module via the identification I ⊗(m0 +1) = Bm0 + I. With this in mind, (35) becomes a op (left) (kΣmp +1 )-module, where the action is the right action of kΣmp +1 on the last tensor factor. I claim that there is a k-module isomorphism, (36)

Mp(m0 ,...,mp ) ∼ = I ⊗(m0 +1) ⊗kG0 k Epi∆S+ ([m0 ], [m1 ]) ⊗kG1 . . . ⊗kGp−1 k Epi∆S+ ([mp−1 ], [mp ]) ,

where Gi is the group Σmi +1 . The isormorphism follows from the following observation. Any element Y ⊗ (ψ1 , g1 ) ⊗ . . . ⊗ (ψp , gp ) in the module on the right in (36) is equivalent to one in which all gi are identities by first writing (ψp , gp ) = gp .(ψp , id) then commuting gp over the tensor to its left and iterating this process to the leftmost tensor factor. Thus, we may write the element uniquely as Z ⊗ φ1 ⊗ . . . ⊗ φp , (m ,...,mp ) where all tensors are now over k, and all morphisms are in Epi∆S+ , that is, Z ⊗ φ1 ⊗ . . . ⊗ φp ∈ Mp 0 . (m0 ,...,mp )

Proposition 69. There is a Σop -equivariant chain map, γ∗ : B∗ ∗ mp +1 (m ,...,mp ) (m ,...,mp ) γ ∼ . isomorphism on homology, H∗ B∗ 0 = H∗ M ∗ 0

(m0 ,...,mp )

→ M∗

, inducing an


23

Proof. γ∗ is defined to be 0 in degrees different from p. In degree p, use the isomorphisms (35) and (36) to define γp as the canonical map, I ⊗(m0 +1) ⊗ k Epi∆S+ ([m0 ], [m1 ]) ⊗ . . . ⊗ k Epi∆S+ ([mp−1 ], [mp ]) I ⊗(m0 +1) ⊗kG0 k Epi∆S+ ([m0 ], [m1 ]) ⊗kG1 . . . ⊗kGp−1 k Epi∆S+ ([mp−1 ], [mp ])

We shall prove that γ∗ induces an isomorphism in homology by induction on p. First, if p = 0, then (m ) (m ) (m ) B∗ 0 = I ⊗(m0 +1) , and concentrated in degree 0. Moreover, γ0 is the identity B0 0 → M0 0 . (m0 ,...,mp−1 ) (m0 ,...,mp−1 ) Next assume γ∗ : B∗ → M∗ induces an isomorphism in homology for any string of p numbers m0 > m1 > . . . > mp−1 . Now assume mp < mp−1 . Let G = Σmp−1 +1 . As graded k-module, there is a degree-preserving isomorphism: (m ,...,mp−1 ,mp ) (m ,...,mp−1 ) (37) θ∗ : B∗ 0 ⊗kG E∗ G ⊗ k G × Epi∆+ ([mp−1 ], [mp ]) −→ B∗ 0

where the degree of an element u⊗(g0 , . . . , gn )⊗(g, φ) is defined recursively: deg (u ⊗ (g0 , . . . , gn ) ⊗ (g, φ)) := (m ) deg(u) + n + 1, and deg(u0 ) = 0 for any u0 ∈ Bn 0 . θ∗ is defined on generators, u ⊗ (g0 , g1 , . . . , gn ) ⊗ (g, φ), by letting g0 act on the right of u, then appending the remaining group elements g1 , . . . , gn onto the chain as automorphisms, and finally appending the morphism (g, φ) to the end. Explicitly, g1

g2

gn

(φ,g)

θ∗ (u ⊗ (g0 , g1 , . . . , gn ) ⊗ (g, φ)) := (u.g0 ) ∗ [mp−1 ] −→ [mp−1 ] −→ . . . −→ [mp−1 ] −→ [mp ], where v ∗ w is the concatenation of chains v and w (The final target of the chain v must agree with the first source of the chain w). If we define a right action of Σmp +1 on k G × Epi∆+ ([mp−1 ], [mp ]) via (g, φ).h := ghφ , φh , then θ∗ becomes a map of right (kΣmp +1 )-modules, since the action defined above simply amounts to postcomposition of the morphism (φ, g) with (id, h). (m ,...,mp−1 ) ⊗kG E∗ G ⊗ k G × Epi∆+ ([mp−1 ], [mp ]) is a chain complex, with differential Observe that B∗ 0 defined on generators by, ∂ u ⊗ (g0 , . . . , gn ) ⊗ (g, φ) = ∂ B (u) ⊗ (g0 , . . . , gn ) ⊗ (g, φ) + (−1)deg(u) u ⊗ ∂ EG (g0 , . . . , gn ) ⊗ (g, φ) . Note, the nth face map of En G ⊗ k G × Epi∆+ ([mp−1 ], [mp ]) is defined by: ∂n (g0 , . . . , gn ) ⊗ (g, φ) = (g0 , . . . , gn−1 ) ⊗ (gn g, φ). It is straightfoward but tedious to verify that θ∗ is a chain map with respect to this differential. θ∗ has a two-sided Σop mp +1 -equivariant inverse, defined by sending g1

g2

gn

(φ,g)

u ∗ [mp−1 ] → [mp−1 ] → . . . → [mp−1 ] → [mp ] 7→ u ⊗ (id, g1 , g2 , . . . , gn ) ⊗ (g, φ). Thus, θ∗ is a chain isomorphism. The next step in this proof is to prove a chain homotopy equivalence, (m ,...,mp−1 ) B∗ 0 ⊗kG E∗ G ⊗ k G × Epi∆+ ([mp−1 ], [mp ]) ≃

(m ,...,mp−1 ) B∗ 0

⊗ k Epi∆+ ([mp−1 ], [mp ])

(m0 ,...,mp−1 )

To that end, we shall define chain maps F∗ and G∗ between the two complexes. Let U∗ := B∗ and S := Epi∆+ ([mp−1 ], [mp ]), and define maps (38)

(39)

F∗ : U∗ ⊗kG E∗ G ⊗ k[G × S] −→ U∗ ⊗ k[S], u.(g0 g) ⊗ φ, if n = 0 F∗ u ⊗ (g0 , . . . , gn ) ⊗ (g, φ) := 0, if n > 0. G∗ : U∗ ⊗ k[S] → U∗ ⊗kG E∗ G ⊗ k[G × S]

,

24

SHAUN V. AULT ∼ =

is the composite, U∗ ⊗ k[S] −→ U∗ ⊗kG G ⊗ k[S] = U∗ ⊗kG E0 G ⊗ k[S] ֒→ U∗ ⊗kG E∗ G ⊗ k[G × S]. The last map is induced by inclusions E0 G ֒→ E∗ G and k[S] ֒→ k[G × S]. Now, F∗ G∗ is the identity, and G∗ F∗ ≃ id via the homotopy, h∗ : u⊗(g0 , . . . , gn )⊗(g, φ) 7→ (−1)deg(u)+n u⊗ (g0 , . . . , gn , g) ⊗ (id, φ). The verification of this claim is rather tedious, but easy. (m ,...,mp ) (m ,...,mp−1 ) , the proof of Prop 69 follows from the fact Since M∗ 0 ⊗ k Epi∆+ ([mp−1 ], [mp ]) ∼ = M∗ 0 that γ∗ decomposes as the following chain of isomorphisms and homotopy equivalences (each of which is also Σop mp +1 -equivariant): i h F∗ ⊗kG E∗ G ⊗ k G × Epi∆+ ([mp−1 ], [mp ]) −→ i γ ⊗id h i ∼ h (m ,...,mp−1 ) (m ,...,mp−1 ) (m ,...,mp ) ∗ = B∗ 0 ⊗ k Epi∆+ ([mp−1 ], [mp ]) −→ M∗ 0 ⊗ k Epi∆+ ([mp−1 ], [mp ]) −→ M∗ 0 −1 (m0 ,...,mp ) θ∗

(m0 ,...,mp−1 )

−→ B∗

B∗

Now, we may prove Lemma 67. Let G = Σmp +1 . Observe, M M 0 ∼ Ep,q Bs(m0 ,...,mp ) ⊗kG Et G, = m0 >...>mp s+t=q

with differential corresponding exactly to the vertical differential defined for E 0 . Note, the outer direct sum respects the differential d0 , so the E 1 term given by: M (m ,...,mp ) 1 0 ⊗kG E∗ G , Hp+q B∗ 0 (40) Ep,q = Hp+q (Ep,∗ )∼ = m0 >...>mp

(m ,...,mp ) B∗ 0

where we view ⊗kG E∗ G as a double complex. In what follows, let (m0 , . . . , mp ) be fixed. In order to take the homology of the double complex, we set up another spectral sequence. From the discussion above, the total differential is given by ∂ total = dv + dh , where dv (u ⊗ (g0 , . . . , gt )) := ∂ B (u) ⊗ (g0 , . . . , gt ), e r , dr } with and dh (u ⊗ (g0 , . . . , gt )) := (−1)deg(u) u ⊗ ∂ EG (g0 , . . . , gt ). Thus, there is a spectral sequence {E ∗,∗ (m ,...,mp ) (m ,...,mp ) e2 ∼ ⊗kG E∗ G . ⊗kG E∗ G, dh , dv ⇒ H∗ B∗ 0 E = H∗,∗ H∗ B∗ 0

e 2 term more closely. Let t be fixed, and take the horizontal differential of the original Let us examine the E (m0 ,...,mp ) e 1 = H∗ B∗(m0 ,...,mp ) ⊗kG Et G, dh ∼ ⊗kG Et G, since Et G B H double complex. We obtain E = ∗ ∗ ∗,t (m ,...,m )

p 1 ∼ e∗,t is flat as left kG-module (in fact, Et G is free). Then, by Prop. 69, E ) ⊗kG Et G. = H∗ (M∗ 0 ( hQ i p I ⊗(m0 +1) ⊗ k i=1 Epi∆+ ([mi−1 ], [mi ]) ⊗kG Et G, in degree p = 0, in degrees different from p

So, the only groups that survive are concentrated in column p. now amounts h QTaking the vertical differential i p op ⊗(m0 +1) to computing the G -equivariant homology of I ⊗k i=1 Epi∆+ [mi−1 ], [mi ] , so #! " p Y op ⊗(m +1) 2 0 e ∼ . Epi ([mi−1 ], [mi ]) ⊗k E = Ht G ; I ∆+

p,t

i=1

2 es,t Since E = 0 for s 6= p, the sequence collapses here. Thus,

Hp+q

(m ,...,mp ) B∗ 0

⊗kG E∗ G ∼ = Hq

op

G ; I

⊗(m0 +1)

⊗k

"

p Y

Epi∆+ ([mi−1 ], [mi ])

i=1

#!

Putting this information back into eq. (40), we obtain the desired isomorphism: #! " p Y M op ⊗(m0 +1) 1 ∼ . Epi ([mi−1 ], [mi ]) ⊗k Hq G ; I E = ∆+

p,q

m0 >...>mp

i=1

A final piece of information needed in order to use Lemma 67 for computation is a description of the 1 horizontal differential d1p,q on Ep,q . This map is induced from the differential d on Y∗ , and reduces the


25

filtration degree by 1. Thus, it is the sum of face maps that combine strict epimorphisms. Let #! " p Y M op ⊗(m0 +1) Epi∆+ ([mi−1 ], [mi ]) ⊗k [u] ∈ Hq Σmp +1 ; I i=1

be represented by a chain, u = Y ⊗ (φ1 , φ2 , . . . , φp ) ⊗ (g0 , . . . , gq ). Then, the face maps di of d1p,q are defined for 0 ≤ i < p by: (φ1 )∗ (Y ) ⊗ (φ2 , . . . , φp ) ⊗ (g0 , . . . , gq ), for i = 0 di (u) := Y ⊗ (φ1 , . . . , φi+1 φi , . . . , φp ) ⊗ (g0 , . . . , gq ), for 0 < i < p.

The last face map dp has the effect of removing the morphism φp by iteratively commuting it past any group φ

g0 g1 ...gi−1

elements to the right of it: dp (u) = Y ⊗ (φ1 , . . . , φp−1 ) ⊗ (g0′ , . . . , gq′ ), where gi′ = gi p involves a change of group from Σmp to Σmp−1 .

. Note that dp

r Proposition 70. The spectral sequence Ep,q above collapses at r = 2.

Proof. This proof relies on the fact that the differential d on Y∗ cannot reduce the filtration degree by more r r than 1. Explicitly, we shall show that dr : Ep,q → Ep−r,q+r−1 is trivial for r ≥ 2. r−1 r r r d is induced by d in the following way. Let Zp = {x ∈ Fp Y∗ | d(x) ∈ Fp−r Y∗ }. Then Ep,∗ = Zpr /(Zp−1 + r−1 r−1 r−1 r−1 r r dZp+r−1 ). Now, d maps Zp → Zp−r and Zp−1 + dZp+r−1 → dZp−1 . Hence, there is an induced map d r making the square below commute. dr is obtained as the composition of d with a projection onto Ep−r,∗ . d Zpr π1 r−1 r−1 r Zp /(Zp−1 + dZp+r−1 )

r / Zp−r π2 d / r−1 r Zp−r /dZp−1 π′ r r−1 r−1 r Ep,∗ Zp−r /(Zp−r−1 + dZp−1 ) OOO OOO OOO dr OOO OOO OOO ' r Ep−r,∗ P In our case, x ∈ Zpr is a sum of the form x = q≥0 ai (Y ⊗ (f1 , f2 , . . . , fq )), where ai 6= 0 for only finitely sym many i, and the sum extends over all symbols Y ⊗(f1 , f2 , . . . , fq ) with Y ∈ B∗ + I, fj ∈ Epi∆S+ composable maps, and at most p of the fj maps are strict epimorphisms. The image of x under π1 looks like π1 (x) = P q≥0 ai [Y ⊗ (f1 , f2 , . . . , fq )], where exactly p of the fj maps are strictly epic. There are, of course, other r−1 relations present as well – those arising from modding out by dZp+r−1 . Consider, dπ1 (x). This should be r the result of lifting π (x) to a representative in Z , then applying π d. One such representative is y = 1 2 p P a (Y ⊗ (f , f , . . . , f )), in which each symbol Y ⊗ (f , f , . . . , f ) has exactly p strict epimorphisms. i 1 2 q 1 2 q q≥0 P Now, d(y) is the sum q≥0 bi (Z ⊗ (g1 , g2 , . . . , gq−1 )), where each symbol Z ⊗(g1 , g2 , . . . , gq−1 ) has either p or r p− 1 strict epimorphisms. Thus, if r ≥ 2, then d(y) ∈ Zp−r implies d(y) = 0. But then, dπ1 (x) = π2 d(y) = 0, r ′ and d = π d is trivial.

7.3. Implications in Characteristic 0. If k is a field of characteristic 0, then for any finite group G and kG-module M , Hq (G, M ) = 0 for all q > 0 (see [1], for example). Thus, by Lemma 67, the E 1 term of the spectral sequence would be concentrated in row 0, and #! " p M Y ⊗(m0 +1) 1 ∼ /Σop , Epi ([mi−1 ], [mi ]) I ⊗k E = ∆+

p,0

m0 >...>mp

mp +1

i=1

that is, the group of co-invariants of the coefficient group, under the right-action of Σmp +1 .

26

SHAUN V. AULT

Since E 1 is concentrated on a row, the spectral sequence collapses at this term. Hence for the k-algebra A, with augmentation ideal I,   #! " p M M Y 1 . /Σop Epi∆+ ([mi−1 ], [mi ]) I ⊗(m0 +1) ⊗ k (41) HS∗ (A) = H∗  mp +1 , d p≥0 m0 >...>mp

i=1

1

This complex is still rather unwieldy as the E term is infinitely generated in each degree. In the next chapter, we shall see another spectral sequence that is more computationally useful. 8. A Second Spectral Sequence

8.1. Reduced Symmetric Homology. Again, we shall assume A is a k-algebra equipped with an augmentation, and whose augmentation ideal is I. Assume further that I is free as k-module, with countable sym basis X. Denote by B∗sym I, the restriction of B∗ + I to ∆S. That is, Bnsym I := I ⊗(n+1) for all n ≥ 0. f∗ := k[N (− \ Epi∆S)] ⊗Epi∆S B∗sym I. Observe that there is a splitting Y∗epi A ∼ f∗ ⊕ k[N (∗)], where Let Y =Y epi Y∗ A is the complex (31) of Section 6, and ∗ is the trivial subcategory of Epi∆S+ consisting of the object [−1] and morphism id[−1] . The fact that I is an ideal ensures that this splitting passes to homology. Hence, f∗ ) ⊕ k0 , where k0 is the graded k-module consisting of k concentrated in degree 0. we have HS∗ (A) ∼ = H∗ (Y f∗ ). e ∗ (A) := H∗ (Y Definition 71. The reduced symmetric homology of A is defined, HS

f∗ is generated, as a k-module, by elements Now, since I = k[X] as k-module and Bnsym I = k[X]⊗(n+1) , Y of the form [m0 ] ։ [m1 ] ։ . . . ։ [mp ] ⊗ (x0 ⊗ x1 ⊗ . . . ⊗ xm0 ), with xi ∈ X. Analogous to Eq. (35), denote e(m0 ,m1 ,...,mq ) := k[X]⊗(m0 +1) ⊗ B

q O

k [Epi∆S ([mi−1 ], [mi ])]

i=1

L e(m0 ,m1 ,...,mq ) . The face maps are given on generators as follows: fq ∼ Then we may observe, Y = m0 ≥...≥mq B  for i = 0  f1 (Y ) ⊗ f2 ⊗ . . . ⊗ fq , Y ⊗ f1 ⊗ . . . ⊗ (fi+1 fi ) ⊗ . . . ⊗ fq , for 1 ≤ i < q di (Y ⊗ f1 ⊗ f2 ⊗ . . . ⊗ fq ) =  Y ⊗ f1 ⊗ . . . ⊗ fq−1 for i = q.

f∗ given by, 8.2. Filtering by Degree. Consider a filtration G∗ of Y M fq = e(m0 ,m1 ,...,mq ) . Gp Y B p≥m0 ≥...≥mq

Observe that G∗ filters the complex by degree of Y ∈ B∗sym I. The only face map that can potentially change the degree of Y is d0 , and since all morphisms are epic, d0 can only reduce the degree. Thus, G∗ is compatible f∗ . The filtration quotients are easily described: with the differential of Y M 0 e(m0 ,m1 ,...,mq ) . Ep,q = B p=m0 ≥...≥mq

0

E splits into a direct sum based on the product of xi ’s in (x0 , . . . , xp ) ∈ X p+1 . For u ∈ X p+1 , let Pu be the set of all distinct permutations of u. Then,  # " q M M M Y 0  Ep,q = Epi∆S ([mi−1 ], [mi ])  . w⊗k u∈X p+1 /Σp+1

p=m0 ≥...≥mq w∈Pu

i=1

8.3. The Categories e Sp , e S′p , Sp and S′p . Before proceeding with the main theorem of this section, we must define four related categories. In the definitions that follow, let {z0 , z1 , z2 , . . . zp } be a set of formal non-commuting indeterminates. Definition 72. e Sp is the category with objects formal tensor products Z0 ⊗ . . . ⊗ Zs , where each Zi is a non-empty product of zi ’s, and every one of z0 , z1 , . . . , zp occurs exactly once in the tensor product. There is a unique morphism Z0 ⊗ . . . ⊗ Zs → Z0′ ⊗ . . . ⊗ Zt′ , if and only if the tensor factors of the latter are products of the factors of the former in some order. In such a case, there is a unique β ∈ Epi∆S so that β∗ (Z0 ⊗ . . . ⊗ Zs ) = Z0′ ⊗ . . . ⊗ Zt′ .


27

e e e′ Sp has initial objects σ∗ (z0 ⊗ z1 ⊗ . . . ⊗ zp ), for σ ∈ Σop p+1 , so N Sp is a contractible complex. Let Sp be the full subcategory of e Sp with all objects σ∗ (z0 ⊗ . . . ⊗ zp ) deleted. Let Sp be a skeletal category equivalent to e Sp . In fact, we may make Sp the quotient category, identifying each object Z0 ⊗ . . . ⊗ Zs with any permutation of its tensor factors, and identifying morphisms φ and ψ if their source and target are equivalent. This category has nerve N Sp homotopy-equivalent to N e Sp . Now, Sp is a poset with unique initial object, z0 ⊗ . . . ⊗ zp . Let S′p be the full subcategory (subposet) of Sp obtained by deleting the object z0 ⊗ . . . ⊗ zp . Clearly, S′p is a skeletal category equivalent to e S′p .

8.4. Main Theorem.

e ∗ (A) with Theorem 73. There is spectral sequence converging (weakly) to HS M 1 ∼ Ep,q Hp+q EGu ⋉Gu |N Sp /N S′p |; k , = u∈X p+1 /Σp+1

where Gu is the isotropy subgroup for the chosen representative of u ∈ X p+1 /Σp+1 . Recall, for a group G, right G-space X, and left G-space Y , X ⋉G Y denotes the equivariant half-smash product. If ∗ is a chosen basepoint for Y having trivial G-action, then X ⋉G Y := (X ×G Y )/(X ×G ∗) = X × Y / ≈, with equivalence relation defined by (x.g, y) ≈ (x, g.y) and (x, ∗) ≈ (x′ , ∗) for all x, x′ ∈ X, y ∈ Y and g ∈ G (cf. [15]). In our case, X is of the form EG, with canonical underlying complex E∗ G, equipped with a right G-action, (g0 , g1 , . . . , gn ).g = (g0 , g1 , . . . , gn g). Observe, both N e Sp and N e S′p carry a left Σp+1 -action (hence also a Gu -action). The action is defined on 0-chains Z0 ⊗ . . . ⊗ Zs by permutation of the individual indeterminates, z0 , z1 , . . . , zp . This action extends to n-chains in the straightforward manner. 0 Define for each u ∈ X p+1 /Σp+1 , the following subcomplex of Ep,q : # " q M M Y Mu := Epi∆S ([mi−1 ], [mi ]) w⊗k p=m0 ≥...≥mq w∈Pu

i=1

∼ = Lemma 74. There is a chain-isomorphism, N e Sp /N e S′p /Gu → Mu .

Proof. Let Cj denote objects of e Sp . As above, we may view each Cj as a morphism of ∆S. By abuse of notation, let Cj also represent a 0-cell of N e Sp /N e S′p . Denote the chosen representative of u again by u (We p+1 /Σp+1 as represented by a (p+1)-tuple whose indices are in non-decreasing view u = (xi0 , xi1 , . . . , xip ) ∈ X order). First define a map N e Sp −→ Mu on n-cells by: φq φ1 (42) α∗ : C0 → . . . → Cq 7→ (C0 (u) ⊗ φ1 ⊗ . . . ⊗ φq ) . (Note, the notation C0 (u) is used in place of the more in order to avoid clutter of notation.) correct (C0 )∗ (u)

I claim α∗ factors through N e Sp /N e S′p . Indeed, if

φ1

φq

C0 → . . . → Cq

∈ Ne S′p , then we cannot have C0 =

σ(z0 ⊗ . . .⊗ zp ) for any symmetric group element σ. That is, C0 , viewed as a morphism, must be strictly epic. Then the degree of of C0 (u) is strictly less than the degree of u, which would make (C0 (u) ⊗ φ1 ⊗ . . . ⊗ φq ) 0 trivial in Ep,q . The map α∗ then factors through N e Sp /N e S′p /Gu , since if γ ∈ Gu , then γ corresponds to an au φq φq φ1 φ1 op tomorphism g ∈ Σp+1 , and by definition, we have: γ. C0 → . . . → Cq = C0 ◦ g → . . . → Cq ◦ g 7→

(C0 (g(u)) ⊗ φ1 ⊗ . . . ⊗ φq ) = (C0 (u) ⊗ φ1 ⊗ . . . ⊗ φq ). (Note, g(u) = u follows from the fact that γ ∈ Gu , the isotropy subgroup for u). For the opposite direction, we begin with a generator of the form, w ⊗ φ1 ⊗ . . . ⊗ φq for some w ∈ Pu . Let τ ∈ Σp+1 so that w = t(u), where t ∈ Σop p+1 corresponds to τ . Define a map sending, φq φ φ2 (43) β∗ : (w ⊗ φ1 ⊗ . . . ⊗ φq ) 7→ t → φ1 t → . . . → φq · · · φ1 t .

28

SHAUN V. AULT

We must check that the definition of β∗ does not depend on choice of τ . Indeed, if w = s(u) also, then u = s−1 t(u), hence s−1 t ∈ Gop u . Thus, φq φq φq φ φ2 φ φ2 φ φ2 s → φ1 s → . . . → φq . . . φ1 s ≈ s−1 t. s → φ1 s → . . . → φq · · · φ1 s = t → φ1 t → . . . → φq · · · φ1 t . The maps α∗ and β∗ are clearly inverse to one another. All that remains is to verify that they are chain maps. We need only check compatibility with the zeroth face maps in either case, since the ith face maps (for i > 0) simply compose the morphisms φi+1 and φi in either chain complex. Consider α∗ first. (The zeroth face maps of either complex will be denoted d0 ). First consider the map alpha∗ . φq φq d0 φ2 φ2 φ1 / C1 → . . . → Cq C0 → C1 → . . . → Cq _ _ α∗ α∗ d0 / (C0 (u) ⊗ φ1 ⊗ φ2 ⊗ . . . ⊗ φq ) (φ1 C0 (u) ⊗ φ2 ⊗ . . . ⊗ φq ) (C1 (u) ⊗ φ2 ⊗ . . . ⊗ φq ) φ1 Sp . The equality in the lower right of the diagram is simply a restatement that C0 → C1 is a morphism of e For the reverse direction, assume w = t(u) as above

(w ⊗ φ1 ⊗ . . . ⊗ φq ) _ β∗

φq φ φ2 t → φ1 t → . . . → φq · · · φ1 t

d0

d0

/ (φ1 (w) ⊗ φ2 . . . ⊗ φq ) _ β∗ φq φ2 / φ1 t → . . . → φq · · · φ1 t

Sp /N e S′p Using this lemma, we identify Mu with the orbit complex N e Sp /N e S′ /G . Now, the complex N e p u Sp /N e S′p /Gu ∼ Sp /N e S′p . (i.e., Gu is a free Gu -complex, so we have an isomorphism: H∗ N e = H∗Gu N e equivariant homology. See [1] for details). Then, by definition, H∗Gu N e Sp /N e S′p = H∗ Gu , N e Sp /N e S′p , which may be computed ithe free resolution, E∗ Gu of k as right Gu -module. The resulting complex i using h h ′ e e k[E∗ Gu ] ⊗kGu k N Sp /k N Sp is a double complex isomorphic to the quotient of two double complexes, namely: i i h h i h ∼ S′p . Sp / E∗ Gu ×Gu N e S′p Sp / k[E∗ Gu ] ⊗kGu k N e k[E∗ Gu ] ⊗kGu k N e = k E∗ Gu ×Gu N e This last complex may be identified with the simplicial complex of the space, S′p | ∼ Sp | / EGu ×Gu |N e EGu ×Gu |N e Sp /N e S′p |. = EGu ⋉Gu |N e

The last piece of the puzzle involves simplifying the spaces |N e Sp /N e S′p |. Since S is a skeletal subcategory of e S, there is an equivalence of categories e S ≃ S, inducing a homotopy equivalence of complexes (hence also of spaces) |N e S| ≃ |N S|. Note that N S inherits a Gu -action from N e S, and the map e S → S is Gu -equivariant. S′p | → Sp | → EGu ×Gu |N Sp | and EGu ×Gu |N e Proposition 75. There are weak equivalences, EGu ×Gu |N e ′ ′ ′ e e EGu ×Gu |N Sp |, inducing a weak equivalence EGu ⋉Gu |N Sp /N Sp | → EGu ⋉Gu |N Sp /N Sp |.

Proof. The case p > 2 will be handled first. Consider the fibration X → EG ×G X → BG associated to a group G and path-connected G-space X. The resulting homotopy sequence breaks up into isomorphisms ∼ = 0 → πi (X) → πi (EG×G X) → 0 for i ≥ 2 and a short exact sequence 0 → π1 (X) → π1 (EG×G X) → G → 0. If there is a G-equivariant homotopy-equivalence f : X → Y for a path-connected G-space Y , then for i ≥ 2,


29

f∗

we have isomorphisms πi (EG ×G X) ← πi (X) → πi (Y ) → πi (EG ×G Y ), and a diagram corresponding to i = 1: / π1 (X) / π1 (EG ×G X) / G / 0 0 f∗

0

(id×f )∗

/ π1 (Y )

/ π1 (EG ×G Y )

/ G

/ 0

Thus, there is a weak equivalence EG ×G X → EG ×G Y . So in our case the desired result will be proved if the spaces |N e S′p | and |N S′p | are path-connected. (Note, |N e Sp | and |N Sp | are path-connected because they are contractible). In fact, we need only check |N S′p |, since this space is homotopy-equivalent to |N e S′p |. let W0 := z0 z1 ⊗z2 ⊗. . .⊗zp . This represents a vertex of N S′p . Suppose W = Z0 ⊗. . .⊗Zi′ z0 z1 Zi′′ ⊗. . .⊗Zs . Then there is a morphism W0 → W , hence an edge between W0 and W . Next, suppose W = Z0 ⊗ . . . ⊗ Zi′ z0 Zi′′ z1 Zi′′′ ⊗ . . . ⊗ Zs . There is a path: Z0 ⊗ . . . ⊗ Zi′ z0 Zi′′ z1 Zi′′′ ⊗ . . . ⊗ Zs Z0 Z1 . . . Zi′ z0 Zi′′ z1 Zi′′′ . . . Zs O Z0 Z1 . . . Zi′ ⊗ z0 ⊗ Zi′′ z1 Zi′′′ . . . Zs z0 ⊗ Z0 Z1 . . . Zi′ Zi′′ z1 Zi′′′ . . . Zs O z0 ⊗ Z0 Z1 . . . Zi′ Zi′′ ⊗ z1 Zi′′′ . . . Zs z0 z1 Zi′′′ . . . Zs ⊗ Z0 Z1 . . . Zi′ Zi′′ O

W0 Similarly, if W = Z0 ⊗ . . . ⊗ Zi′ z1 Zi′′ z0 Zi′′′ ⊗ . . . ⊗ Zs , there is a path to W0 . Finally, if W = Z0 ⊗ . . . ⊗ Zs with z0 occurring in Zi and z1 occurring in Zj for i 6= j, there is an edge to some W ′ in which Zi Zj occurs, and thus a path to W0 . The cases p = 0, 1 and 2 are handled individually: S′0 | = Observe that |N e S′0 | and |N S′0 | are empty spaces, since e S′0 has no objects. Hence, EGu ×Gu |N e ′ EGu ×Gu |N S0 | = ∅. Furthermore, any group Gu must be trivial. Thus there is a chain of homotopy S0 | ≃ |N S0 | ≃ EGu ⋉Gu |N Sp /N S′p |. S0 /N e S′0 | ≃ |N e equivalences, EGu ⋉Gu |N e Next, since |N e S′1 | is homeomorphic to |N S′1 |, each space consisting of the two discrete points z0 z1 and z1 z0 with the same group action, the proposition is true for p = 1 as well. e1 and U e2 that are interchanged by any odd For p = 2, observe that |N e S′2 | has two connected components, U permutation σ ∈ Σ3 . Similarly, |N S′2 | consists of two connected components, U1 and U2 , interchanged by any odd permutation of Σ3 . Now, resticted to the alternating group, A3 , we certianly have weak equivalences ≃ ≃ e2 −→ e1 −→ EHu ×Hu U2 . The EHu ×Hu U1 and EHu ×Hu U for any subgroup Hu ⊆ A3 , EHu ×Hu U

30

SHAUN V. AULT

∼ ∼ = e = e1 −→ action of an odd permutation induces equivariant homeomorphisms U U2 and U1 −→ U2 , and so if we have a subgroup Gu ⊆ Σ3 generated by Hu ⊆ A3 and a transposition, then the two connected components S′2 | ∼ are identified in an AΣ3 -equivariant manner. Thus, if Gu contains a transposition, EGu ×Gu |N e = ′ ∼ e EHu ×Hu U1 ≃ EHu ×Hu U1 = EGu ×Gu |N S2 |. This completes the case p = 2 and the proof of Prop. 75.

Prop. 75 coupled with Lemma 74 produces the required isomorphism in homology, hence proving Thm. 73: M M 1 Ep,q = Hp+q (Mu ) ∼ Sp /N e S′p |; k Hp+q EGu ⋉Gu |N e = u∈X p+1 /Σp+1

∼ =

u∈X p+1 /Σp+1

M

u∈X p+1 /Σp+1

Hp+q EGu ⋉Gu |N Sp /N S′p |; k .

Corollary 76. If the augmentation ideal of A satisfies I 2 = 0, then M M Hn (EGu ⋉Gu N Sp /N S′p ; k). HSn (A) ∼ = p≥0 u∈X p+1 /Σp+1

Proof. This follows from consideration of the original E 0 term of the spectral sequence. E 0 is generated by f∗ when φ0 is an chains Y ⊗ φ0 ⊗ . . . ⊗ φn , with induced differential d0 , agreeing with the differential d of Y isomorphism. When φ0 is a strict epimorphism, however, the zeroth face map of d0 maps the generator to: (φ0 )∗ (Y ) ⊗ φ1 ⊗ . . . ⊗ φn = 0, since (φ0 )∗ (Y ) would have at least one tensor factor that is the product of two or more elements of I. Thus, d0 also agrees with d in the case that φ0 is strictly epic. Hence, the spectral sequence collapses at level 1. (p)

8.5. The complex Sym∗ . Note, for p > 0, there are homotopy equivalences |N Sp /N S′p | ≃ |N Sp | ∨ S|N S′p | ≃ S|N S′p |, since |N Sp | is contractible. |N Sp | is a disjoint union of (p + 1)! p-cubes, identified along certain faces. Geometric analysis of S|N S′p |, however, seems quite difficult. Fortunately, there is an even smaller chain complex homotopic to N Sp /N S′p . Definition 77. Let p ≥ 0 and impose an equivalence relation on k [Epi∆S ([p], [q])] generated by: Z0 ⊗ . . . ⊗ Zi ⊗ Zi+1 ⊗ . . . ⊗ Zq ≈ (−1)ab Z0 ⊗ . . . ⊗ Zi+1 ⊗ Zi ⊗ . . . ⊗ Zq , where Z0 ⊗. . .⊗Zq is a morphism expressed in tensor notation, and a = deg(Zi ) := |Zi |−1, b = deg(Zi+1 ) := |Zi+1 |−1. Here, deg(Z) is one less than the number of factors of the monomial Z. Indeed, if Z = zi0 zi1 . . . zis , then deg(Z) = s. (p) (p) := k [Epi∆S ([p], [p − i])] / ≈. The face maps will be The complex Sym∗ is then defined by Symi defined recursively. On monomials,  i < 0,  0, zj0 . . . zji ⊗ zji+1 . . . zjs , , 0 ≤ i < s, di (zj0 . . . zjs ) =  0, i ≥ s. Then, extend di to tensor products via:

di (W ⊗ V ) = di (W ) ⊗ V + W ⊗ di−deg(W ) (V ),

(44)

where W and V are formal tensors in k [Epi∆S ([p], [q])], and deg(W ) = deg(W0 ⊗. . .⊗Wt ) := Pn Pn−1 (p) (p) The boundary map Symn → Symn−1 is then d = i=0 (−1)i di = i=0 (−1)i di .

Pt

k=0

deg(Wk ).

Remark 78. The result of applying di on any formal tensor product will result in only a single formal tensor product, since in Eq. (44), at most one of the two terms will be non-zero. (p)

(p)

Remark 79. There is an action Σp+1 × Symi → Symi , given by permuting the formal indeterminates zi . Furthermore, this action is compatible with the differential. (p)

Lemma 80. Sym∗

is chain-homotopy equivalent to k[N Sp ]/k[N S′p ].


31

Proof. Let v0 represent the common initial vertex of the p-cubes making up N Sp . Then, as cell-complex, N Sp consists of v0 together with all corners of the various p-cubes and i-cells for each i-face of the cubes. Thus, N Sp consists of (p + 1)! p-cells with attaching maps ∂I p → (N Sp )p−1 defined according to the face maps for N Sp given above. Note that a chain of N Sp is non-trivial in N Sp /N S′p if and only if the initial vertex v0 is included. Thus, any (cubical) k-cell is uniquely determined by the label of the vertex opposite v0 . Label each top-dimensional cell with the permutation induced on the set {0, 1, . . . , p} by the order of indeterminates in final vertex, zi0 zi1 . . . zip . On a given p-cell, for each vertex Z0 ⊗ . . . ⊗ Zs , there is an ordering of the tensor factors so that Z0 ⊗. . .⊗Zs → zi0 zi1 . . . zip preserves the order of formal indeterminates zi . Rewrite each vertex of this p-cell in this order. Now, any p-chain (zi0 ⊗zi1 ⊗. . .⊗zip ) → . . . → zi0 zi1 . . . zip is obtained by choosing the order in which to combine the factors. In fact, the p-chains for this cube are in bijection with the elements of the symmetric group Sp , as in the standard decomposition of a p-cube into p! simplices. A given permutation {1, 2, . . . , p} 7→ {j1 , j2 , . . . , jp } will represent the chain obtained by first combining zj0 ⊗ zj1 into zj0 zj1 , then combining zj1 ⊗ zj2 into zj1 zj2 . In effect, we “erase” the tensor product symbol between zjr−1 and zjr for each jr in order given by the list above. We shall declare that the natural order of combining the factors will be the one that always combines the last two: (zi0 ⊗ . . . ⊗ zip−1 ⊗ zip ) → (zi0 ⊗ . . . ⊗ zip−1 zip ) → (zi0 ⊗ . . . ⊗ zip−2 zip−1 zip ) → . . . → (zi0 . . . zip ). This corresponds to a permutation ρ := {1, . . . , p} 7→ {p, p − 1, . . . , 2, 1}, and this chain will be regarded as positive. A chain Cσ , corresponding to another permutation, σ, will be regarded as positive or negative depending on the sign of the permutation σρ−1 . Finally, the entire p-cell should be identified with the P sum σ∈Sp sign(σρ−1 )Cσ . It is this sign convention that permits the inner faces of the cube to cancel appropriately in the boundary maps. Thus we have a map on the top-dimensional chains: ′ (45) θp : Sym(p) p → k[N Sp ]/k[N Sp ] p . Extend the defintion θ∗ to arbitrary k-cells by sending the k-chain Z0 ⊗ . . . ⊗ Zp−k to the sum of k-length chains with source z0 ⊗ . . . ⊗ zp and target Z0 ⊗ . . . ⊗ Zp−k with signs determined by the natural order of erasing tensor product symbols of z0 ⊗ . . .⊗ zp , excluding those tensor product symbols that never get erased. (4) The following example should clarify the point. Let W = z3 z0 ⊗ z1 ⊗ z2 z4 . This is a 2-cell of Sym∗ . W is obtained from z0 ⊗ z1 ⊗ z2 ⊗ z3 ⊗ z4 = z3 ⊗ z0 ⊗ z1 ⊗ z2 ⊗ z4 by combining factors in some order. There are only 2 erasable tensor product symbols in this example. The natural order (last to first) corresponds to the chain, z3 ⊗ z0 ⊗ z1 ⊗ z2 ⊗ z4 → z3 ⊗ z0 ⊗ z1 ⊗ z2 z4 → z3 z0 ⊗ z1 ⊗ z2 z4 . So, this chain shows up in θ∗ (W ) with positive sign, whereas the chain z3 ⊗ z0 ⊗ z1 ⊗ z2 ⊗ z4 → z3 z0 ⊗ z1 ⊗ z2 ⊗ z4 → z3 z0 ⊗ z1 ⊗ z2 z4 shows up with a negative sign. (p) Now, θ∗ is easily seen to be a chain map Sym∗ → k[N Sp ]/k[N S′p ]. Geometrically, θ∗ has the effect of subdividing a cell-complex (defined with cubical cells) into a simplicial space, so θ∗ is a homotopyequivalence.

Remark 81. As an example, consider |N S2 |. There are 6 2-cells, each represented by a copy of I 2 . The 2-cell labelled by the permutation {0, 1, 2} 7→ {1, 0, 2} consists of the chains z1 ⊗ z0 ⊗ z2 → z1 ⊗ z0 z2 → z1 z0 z2 and −(z1 ⊗z0 ⊗z2 → z1 z0 ⊗z2 → z1 z0 z2 ). Hence, the boundary is the sum of 1-chains: [(z1 ⊗z0 z2 → z1 z0 z2 )−(z1 ⊗ z0 ⊗z2 → z1 z0 z2 )+(z1 ⊗z0 ⊗z2 → z1 ⊗z0 z2 )]−[(z1 z0 ⊗z2 → z1 z0 z2 )−(z1 ⊗z0 ⊗z2 → z1 z0 z2 )+(z1 ⊗z0 ⊗z2 → z1 z0 ⊗z2 )] = (z1 ⊗z0 z2 → z1 z0 z2 )+(z1 ⊗z0 ⊗z2 → z1 ⊗z0 z2 )−(z1 z0 ⊗z2 → z1 z0 z2 )−(z1 ⊗z0 ⊗z2 → z1 z0 ⊗z2 ). This 1-chains correspond to the 4 edges of the square. Thus, in our example this 2-cell of |N Sp | will (2) correspond to z1 z0 z2 ∈ Sym2 , and its boundary in |N Sp /N S′p | will consist of the two edges adjacent to the vertex labeledz0 ⊗ z1 ⊗ z2 , with appropriate signs: (z0 ⊗ z1 ⊗ z2 → z1 ⊗ z0 z2 ) − (z0 ⊗ z1 ⊗ z2 → z1 z0 ⊗ z2 ). (2) The corresponding boundary in Sym1 will be (z1 ⊗ z0 z2 ) − (z1 z0 ⊗ z2 ), matching the boundary map already (p) defined on Sym∗ . See Figs. 3 and 4. Now, with one piece of new notation, we may re-interpret Thm. 73. Definition 82. Let G be a group. Let k0 be the chain complex consisting of k concentrated in degree 0, with trivial G-action. If X∗ is a right G-complex, Y∗ is a left G-complex with k0 ֒→ Y∗ as a G-subcomplex, then define the equivariant half-smash tensor product of the two complexes: ⋉ G Y∗ := (X∗ ⊗kG Y∗ ) / (X∗ ⊗kG k0 ) X∗

32

SHAUN V. AULT

z0 z1 b⊗ z2

z1 z0 b⊗ z2

z2 z0 z1

zb0 z1 z2

b

z2 z1 z0

zb1 z0 z2

b

z0 ⊗ z1 b⊗ z2

z0 ⊗ z1 b⊗ z2

b

b

z1 ⊗ z2 z0

z0 ⊗ z1 z2

b

b

z0 ⊗ z2 z1

z1 ⊗ z0 z2

b

b

z1 z2 z0

z0 z2 z1

Figure 3. |N S2 | consists of six squares, grouped into two hexagons that share a common center vertex b

b

z0 z1 ⊗ z2

b

b

z2 z0 z1

b

z0 z1 z2

b

z1 z0 ⊗ z2

b

b

z2 z1 z0

b

z1 z0 z2

b

b

b b

b

b b

z1 ⊗ z2 z0

b

b

z0 ⊗ z1 z2 b

b

z1 z2 z0

b

z0 ⊗ z2 z1 b

z1 ⊗ z0 z2 b

b

z0 z2 z1

b

b

Figure 4. Sym(2) ≃ N S2 /N S′2 . The center of each hexagon is z0 ⊗ z1 ⊗ z2 . e ∗ (A) with Corollary 83. There is spectral sequence converging (weakly) to HS M 1 ∼ ⋉ Gu Sym(p) Hp+q E∗ Gu Ep,q = ∗ ;k , u∈X p+1 /Σp+1

where Gu is the isotropy subgroup for the chosen representative of u ∈ X p+1 /Σp+1 . (p)

9. Properties of the Complex Sym∗ L (p) as a bigraded differential 9.1. Algebra Structure of Sym∗ . We may consider Sym∗ := p≥0 Sym∗ (p)

(p)

(q)

(p+q+1)

is algebra, where bideg(W ) = (p + 1, i) for W ∈ Symi . The product ⊠ : Symi ⊗ Symj → Symi+j ′ ′ defined by: W ⊠ V := W ⊗ V , where V is obtained from V by replacing each formal indeterminate zr by zr+p+1 for 0 ≤ r ≤ q. Eq. 44 then implies: (46)

d(W ⊠ V ) = d(W ) ⊠ V + (−1)bideg(W )2 W ⊠ d(V ),

where bideg(W )2 is the second component of bideg(W ).


33

Proposition 84. The product ⊠ is defined on the level of homology. Furthermore, this product (on both the chain level and homology level) is skew commutative in a twisted sense: W ⊠ V = (−1)ij τ (V ⊠ W ), where bideg(W ) = (p + 1, i), bideg(V ) = (q + 1, j), and τ is the permutation sending {0, 1, . . . , q, q + 1, q + 2, . . . , p + q, p + q + 1} 7→ {p + 1, p + 2, . . . , p + q + 1, 0, 1, . . . , p − 1, p}. Proof. Eq. (46) implies the product passes to homology classes. Now, suppose W = Y0 ⊗ Y1 ⊗ . . . ⊗ Yp−i ∈ (p) (p) Symi and V = Z0 ⊗ Z1 ⊗ . . . ⊗ Zq−j ∈ Symj . V ⊠ W = V ⊗ W ′ = (−1)α W ′ ⊗ V,

(47)

where W ′ is related to W by replacing each zr by zr+q+1 . The exponent α = deg(V )deg(W ) = ij arises from (p+q+1) . (The fact that deg(V ) = i and deg(W ) = j may be made clear by observing the relations in Symi+j (s)

that the degree of a formal tensor product in Sym∗ is equal to the number of cut points, that is, the number of places where a tensor product symbol may be inserted.) Next, apply the block transformation τ to Eq. (47) to obtain τ (V ⊠ W ) = (−1)α τ (W ′ ⊗ V ) = (−1)α W ⊗ V ′ = (−1)α W ⊠ V , where V ′ is obtained by replacing zr by zr+p+1 in V . (p)

9.2. Computer Calculations. In principle, the homology of Sym∗ In fact, we have the following results up to p = 7:

may be found by using a computer.

(p)

Theorem 85. For 0 ≤ p ≤ 7, the groups H∗ (Sym∗ ) are free abelian and have Poincar´e polynomials (p) Pp (t) := P H∗ (Sym∗ ); t : P0 (t) = 1, P1 (t) = t, P2 (t) = t + 2t2 , P3 (t) = 7t2 + 6t3 , P4 (t) = 43t3 + 24t4 , P5 (t) = t3 + 272t4 + 120t5 , P6 (t) = 36t4 + 1847t5 + 720t6 ,

P7 (t) = 829t5 + 13710t6 + 5040t7. Proof. These computations were performed using scripts written for the computer algebra systems GAP [9] and Octave [6]. (p)

We conjecture that the H∗ (Sym∗ ) is always free abelian. (p)

(p)

9.3. Representation Theory of H∗ (Sym∗ ). By remark 79, the groups Hi (Sym∗ ; k) are kΣp+1 -modules, so it seems natural to investigate the irreducible representations comprising these modules. Proposition 86. Let Cp+1 ֒→ Σp+1 be the cyclic group of order p + 1, embedded into the symmetric group as the subgroup generated by the permutation τp := (0, p, p − 1, . . . , 1). Then there is a Σp+1 -isomorphism: (p) Hp (Sym∗ ) ∼ = ACp+1 ↑ Σp+1 , i.e., the alternating representation of the cyclic group, induced up to the symmetric group. Note, for p even, ACp+1 coincides with the trivial representation ICp+1 . (p) Hp (Sym∗ ) is generated by the elements σ(bp ), for the distinct cosets σCp+1 , where bp := PpMoreover, jp j j=0 (−1) τp (z0 z1 . . . zp ). P (p) Proof. Let w be a general element of Symp , w = σ∈Σp+1 cσ σ(z0 z1 . . . zp ), where cσ are constants in k. (p)

Hp (Sym∗ ) consists of those w such that d(w) = 0. That is,

(48)

0=

p−1 X X (−1)i cσ σ(z0 . . . zi ⊗ zi+1 . . . zp ).

σ∈Σp+1 i=0

Now for each σ, the terms corresponding to σ(z0 . . . zi ⊗ zi+1 . . . zp ) occur in pairs in the above formula. The obvious term of the pair is (−1)i cσ σ(z0 . . . zi ⊗ zi+1 . . . zp ). Not so obviously, the second term of the pair is (−1)(p−i−1)i (−1)p−i−1 cρ ρ(z0 . . . zp−i−1 ⊗ zp−i . . . zp ), where ρ = στpp−i . Thus, if d(w) = 0,

34

SHAUN V. AULT

then we must have (−1)i cσ + (−1)(p−i−1)(i+1) cρ = 0, or cρ = (−1)(p−i)(i+1) cσ . Set j = p − i, so that cρ = (−1)j(p−j+1) cσ = (−1)jp cσ . This shows that the only restrictions on the coefficients cσ are that the absolute values of coefficients corresponding to σ, στp , στp2 , . . . must be the same, and their corresponding signs in w alternate if and only if p is odd; otherwise, they have the same signs. Clearly, the elements σ(bp ) (p) for distinct cosets σCp+1 represents an independent set of generators over k for Hp (Sym∗ ). Observe that bp is invariant under the action of sign(τp )τp , and so bp generates an alternating representation ACp+1 over k. Induced up to Σp+1 , we obtain the representation ACp+1 ↑ Σp+1 of dimension (p + 1)!/(p + 1) = p!, generated by the elements σ(bp ) as in the proposition. Definition 87. For a given proper partition λ = [λ0 , λ1 , λ2 , . . . , λs ] of the p + 1 integers {0, 1, . . . , p}, (p) an element W of Sym∗ will designated as type λ if it equivalent to ±(Y0 ⊗ Y1 ⊗ Y2 ⊗ . . . ⊗ Ys ) with (p) deg(Yi ) = λi − 1. That is, each Yi has λi factors. The notation Symλ or Symλ will denote the k-submodule (p) of Symp−s generated by all elements of type λ. In what follows, |λ| will refer to the number of components of λ. The action of Σp+1 leaves Symλ invariant for any given λ, so the there is a decomposition as kΣp+1 -module: M (p) Symλ . Symp−s = λ⊢(p+1),|λ|=s+1

Proposition 88. For a given proper partition λ ⊢ (p + 1), (a) Symλ contains exactly one alternating representation AΣp+1 iff λ contains no repeated components. (b) Symλ contains exactly one trivial representation IΣp+1 iff λ contains no repeated even components. Proof. Symλ is a quotient of the regular representation, since it is the image of the Σp+1 -map πλ : kΣp+1 → Symλ , σ 7→ ψλ s, where s ∈ Σop p+1 is the ∆S-automorphism of [p] corresponding to σ and ψλ is a ∆-morphism [p] → [ |λ| ] that sends the points 0, . . . , λ0 − 1 to 0, the points λ0 , . . . , λ0 + λ1 − 1 to 1, and so on. Hence, there can be at most 1 copy of AΣp+1 and at most 1 copy of IΣp+1 in Symλ . Let W be the “standard” element of Symλ . That is, the indeterminates zi occur in W in numerical orderP and the degrees of monomials of W are in decreasing order. AΣp+1 exists in Symλ iff the element V = σ∈Σp+1 sign(σ)σ(W ) is non-zero. Suppose that some component of λ is repeated, say λi = λi+1 = ℓ. If W = Y0 ⊗ Y1 ⊗ . . . ⊗ Ys , then deg(Yi ) = deg(Yi+1 ) = ℓ − 1. Now, we know that W = (−1)deg(Yi )deg(Yi+1 ) Y0 ⊗ . . . ⊗ Yi+1 ⊗ Yi ⊗ . . . Ys = −(−1)ℓ α(W ), for the permutation α ∈ Σp+1 that exchanges the indices of indeterminates in Yi with those in Yi+1 in an order-preserving way. In V , the term α(W ) shows up with sign sign(α) = (−1)ℓ , thus cancelling with W . Hence, V = 0, and no alternating representation exists. If, on the other hand, no component of λ is repeated, then no term ±α(W ) can be equivalent to W for α 6= id, so V survives as the generator of AΣp+1 in Symλ . P A similar analysis applies for trivial representations. This time, we examine U = σ∈Σp+1 σ(W ), which would be a generator for IΣp+1 if it were non-zero. As before, if there is a repeated component, λi = λi+1 = ℓ, then W = (−1)ℓ−1 α(W ). However, this time, W cancels with α(W ) only if ℓ − 1 is odd. That is, |λi | = |λi+1 | is even. If ℓ − 1 is even, or if all λi are distinct, then the element U must be non-zero. (p)

Proposition 89. Hi (Sym∗ ) contains an alternating representation for each partition λ ⊢ (p + 1) with |λ| = p − i such that no component of λ is repeated. Proof. This proposition will follow from the fact that d(V ) = 0 for any generator V of an alternating representation in Symλ . Then, by Schur’s Lemma, the alternating representation must survive at the level of homology. P Let V = σ∈Σp+1 sign(σ)σ(W ) be the generator mentioned in Prop. 88. d(V ) consists of individual terms dj (σ(W )) = σ(dj (W )) along with appropriate signs. For a given j, dj (W ) is identical to W except at some monomial Yi , where a tensor product symbol is inserted. We will introduce some notation to make the argument a little cleaner. If Y = zi0 zi1 . . . zir is a monomial, then the notation Y {s, . . . , t} refers to the monomial zis zis+1 . . . zit , assuming 0 ≤ s ≤ t ≤ r. Now, we write (49)

dj (W ) = (−1)a+ℓ Y0 ⊗ . . . ⊗ Yi {0, . . . , ℓ} ⊗ Yi {ℓ + 1, . . . , m} ⊗ . . . ⊗ Ys ,


35

where a = deg(Y0 ) + . . . + deg(Yi−1 ). Use the relations in Sym∗ to rewrite Eq. (49): (50)

(−1)(a+ℓ)+ℓ(m−ℓ−1) Y0 ⊗ . . . ⊗ Yi {ℓ + 1, . . . , m} ⊗ Yi {0, . . . , ℓ} ⊗ . . . ⊗ Ys .

Let α be the permutation that relabels indices in such a way that Yi {0, . . . , m − ℓ − 1} 7→ Yi {ℓ + 1, . . . , m} and Yi {m − ℓ, . . . , m} 7→ Yi {0, . . . , ℓ}, so that the following is equivalent to Eq. (50). (51)

2

(−1)a+mℓ−ℓ α (Y0 ⊗ . . . ⊗ Yi {0, . . . , m − ℓ − 1} ⊗ Yi {m − ℓ, . . . , m} ⊗ . . . ⊗ Ys )

Now, Eq.( 51) also occurs in dj ′ (sign(α)α(W )) for some j ′ . This term looks like: (52) (53)

sign(α)(−1)a+m−ℓ−1 α Y0 ⊗ . . . ⊗ Yi {0, . . . , m − ℓ − 1} ⊗ Yi {m − ℓ, . . . , m} ⊗ . . . ⊗ Ys = (−1)mℓ−ℓ

2

+a−1

α Y0 ⊗ . . . ⊗ Yi {0, . . . , m − ℓ − 1} ⊗ Yi {m − ℓ, . . . , m} ⊗ . . . ⊗ Ys

Comparing the signs of Eq. (53) and Eq. (51), we verify the two terms canel each other out in the sum d(V ). By Proposition 89, it is clear that if p + 1 is a triangular number – i.e., p + 1 is of the form r(r + 1)/2 for some positive integer r, then the lowest dimension in which an alternating representation may occur is p + 1 − r, corresponding to the partition λ = [r, r − 1, . . . , 2, 1]. A little algebra yields the following statement for any p. p (p) Corollary 90. Hi (Sym∗ ) contains an alternating representation in degree p+1−r, where r = ⌊ 2p + 9/4− 1/2⌋. Moreover, there are no alternating representations present for i ≤ p − r. There is not much known about the other irreducible representations occurring in the homology groups of (p) (p) Sym∗ , however computational evidence shows that Hi (Sym∗ ) contains no trivial representation, IΣp+1 , for i ≤ p − r (r as in the conjecture above) up to p = 50. (p) (p) ˇ 9.4. Connectivity of Sym∗ . Quite recently, Vre´cica and Zivaljevi´ c [25] observed that the complex Sym∗ is isomorphic to the suspension of the cycle-free chessboard complex Ωp+1 (in fact, the isomorphism takes (p) + the form k SΩ+ p+1 → Sym∗ , where Ωp+1 is the augmented complex). The m-chains of the complex Ωn are generated by ordered lists, L = {(i0 , j0 ), (i1 , j1 ), . . . , (im , jm )}, where 1 ≤ i0 < i1 < . . . < im ≤ n, all 1 ≤ js ≤ n are distinct integers, and the list L is cycle-free. It may be easier to say what it means for L not to be cycle free: L is not cycle-free if there exists a subset Lc ⊆ L and ordering of Lc so that Lc = {(ℓ0 , ℓ1 ), (ℓ1 , ℓ2 ), . . . , (ℓt−1 , ℓt ), (ℓt , ℓ0 )}. The differential of Ωn is defined on generators by:

d ({(i0 , j0 ), . . . , (im , jm )}) :=

m X

(−1)s {(i0 , j0 ), . . . , (is−1 , js−1 ), (is+1 , js+1 ), . . . , (im , jm )}.

s=0

For completeness, an explicit isomorphism shall be provided: Proposition 91. Let Ω+ n denote the augmented cycle-free (n × n)-chessboard complex, where the unique (−1)-chain is represented by the empty n × n chessboard, and the boundary map on 0-chains takes a vertex (p) to the unique (−1)-chain. For each p ≥ 0, there is a chain isomorphism, ω∗ : k SΩ+ p+1 → Sym∗ .

Proof. Note that we may define the generating m-chains of k [Ωp+1 ] as cycle-free lists L, with no requirement on the order of L, under the equivalence relation: σ.L := {(iσ−1 (0) , jσ−1 (0) ), . . . , (iσ−1 (m) , jσ−1 (m) )} ≈ + sign(σ)L, for σ ∈ Σm+1 . Suppose L is an (m + 1)-chain of SΩ+ p+1 (i.e. an m-chain of Ωp+1 ). Call a subset ′ ′ ′ L ⊆ L a queue if there is a reordering of L such that L = {(ℓ0 , ℓ1 ), (ℓ1 , ℓ2 ), . . . , (ℓt−1 , ℓt )}. L′ is called a maximal queue if it is not properly contained in any other queue. Since L is supposed to be cycle-free, we can partition L into some number of maximal queues, L′1 , L′2 , . . . , L′q . Let σ be a permutation representing the reordering of L into maximal ordered queues. Now, each maximal ordered queue L′i will correspond to a monomial of formal indeterminates zi as follows. (54)

L′s := {(ℓ0 , ℓ1 ), (ℓ1 , ℓ2 ), . . . , (ℓt−1 , ℓt )} 7→ zℓ0 −1 zℓ1 −1 · · · zℓt −1 .

36

SHAUN V. AULT

For each maximal ordered queue, L′s , denote the monomial obtained by formula (54) by Zs . Let k1 , k2 , . . . , ku be the numbers in {0, 1, 2, . . . , p} such that kr + 1 does not appear in any pair (is , js ) ∈ L. Now we may define ω∗ on L = L′1 ∪ L′2 ∪ . . . ∪ L′q . ωm+1 (L) := Z1 ⊗ Z2 ⊗ . . . ⊗ Zq ⊗ zk1 ⊗ zk2 ⊗ . . . ⊗ zku .

(55)

Observe, if L = ∅ is the (−1)-chain of Ω+ p+1 , then there are no maximal queues in L, and so ω0 (∅) = z0 ⊗ z1 ⊗ . . . ⊗ zp . ω∗ is a (well-defined) chain map with inverse given by essentially reversing the process. To each monomial Z = zi0 zi1 · · · zit with t > 0, there is an associated ordered queue L′ = {(i0 +1, i1 +1), (i1 +1, i2 +1), . . . (it−1 + 1, it + 1)}. If the monomial is a singleton, Z = zi0 , the associated ordered queue will be the empty set. Now, (p) given a generator Z1 ⊗ Z2 ⊗ . . . ⊗ Zq ∈ Sym∗ , map it to the list L := L′1 ∪ L′2 ∪ . . . ∪ L′q , preserving the original order of indices. (p)

Theorem 92. Sym∗

is ⌊ 32 (p − 1)⌋-connected.

Proof. See Thm. 10 of [25].

This remarkable fact yields the following useful corollaries: e ∗ (A). Corollary 93. The spectral sequences of Thm. 73 and Cor. 83 converge strongly to HS (p)

Proof. This relies on the fact that the connectivity of the complexes Sym∗ is a non-decreasing func L (p) ⋉ tion of p. Fix n ≥ 0, and consider H Sym

E G , the component of E 1 residing at po∗ ∗ u Gu u n

1 2 3 sition p, q for p + q = n. A priori, the induced differentials whose sources are Ep,q , Ep,q , Ep,q , . . . will L L (p+1) ⋉ Gu Sym(p+2) ⋉ Gu Sym∗ , , u Hn−1 E∗ Gu have as targets certain subquotients of u Hn−1 E∗ Gu ∗ L ⋉ Gu Sym(p+3) , etc. Now, if n − 1 < ⌊(2/3)(p + k − 1)⌋ for some k ≥ 0, then for K > k, we ∗ u Hn−1 E∗ Gu (p+K) ⋉ Gu Sym(p+K) have Hn−1 Sym∗ = 0, hence also, Hn−1 E∗ Gu = 0, using the fibration mentioned ∗

in the proof of Thm. 73 and the Hurewicz Theorem. Thus, the induced differential dk is zero for all k ≥ K. 1 2 3 On the other hand, the induced differentials whose targets are Ep,q , Ep,q , Ep,q , . . . must be zero after stage p, since there are no non-zero components with p < 0.

Corollary 94. For each i ≥ 0, there is a positive integer Ni so that if p ≥ Ni , there is an isomorphism f e i (A). Hi G p Y ∗ ∼ = HS

Corollary 95. If A is finitely-generated over a Noetherian ground ring k, then HS∗ (A) is finitely-generated over k in each degree. Proof. Examination of the E 1 term shows that the nth reduced symmetric A is a sub homology group of L L (p) (p) ⋉ Gu Sym∗ ; k is a finite⋉ Gu Sym∗ ; k . Each Hn E∗ Gu quotient of p≥0 u∈X p+1 /Σp+1 Hn E∗ Gu dimensional k-module. The inner sum is finite as long as X is finite. Thm. 92 shows the outer sum is finite as well. The bounds on connectivity are conjectured to be tight. This is certainly true for p ≡ 1 (mod 3), (3k−1) 6= 0 or based on Thm. 16 of [25]. Corollary 12 of the same paper establishes that either H2k Sym∗ (3k) 6= 0. For k ≤ 2, both statements are true. When the latter condition is true, this gives a H2k Sym∗

tight bound on connectivity for p ≡ 0 (mod 3). When the former is true, there is not enough information (3k−1) for a tight bound, since we are more interested in proving that H2k−1 Sym∗ is non-zero, since for (5) (2) = Z. = Z and H3 Sym∗ k = 1, 2, we have computed the integral homology, H1 Sym∗

SYMMETRIC HOMOLOGY OF ALGEBRAS: FOUNDATIONS (p)

37

(n)

9.5. Filtering Sym∗ by partition types. In 9.3, we saw that Sym∗ decomposes over kΣn+1 as a direct (n) sum of the submodules Symλ for partitions λ ⊢ (n + 1). We may filter Sym∗ by the size of the largest component of the partition, M Symλ , Fp Sym(n) := q λ⊢(n+1),|λ|=n+1−(p+q),λ0 ≤p+1

(n)

where λ = [λ0 , λ1 , . . . , λn−q ], is written in non-increasing order. The differential of Sym∗ respects this filtering, since it can only reduce the size of partition components. With respect to this filtering, we have an E 0 term for a spectral sequence: M 0 ∼ Symλ . Ep,q = λ⊢(n+1),|λ|=n+1−(p+q),λ0 =p+1

0

The vertical differential d is induced from d by keeping only those terms of d(W ) that share largest component size with W . 10. A Partial Resolution As before, k is a commutative ground ring. In this section, we find an explicit partial resolution of the trivial ∆S op -module k by projective modules, allowing the computation of HS0 (A) and HS1 (A) for a unital associative k-algebra A. The resolution will be constructed through a number of technical lemmas. Recall from 1.4, the modules k [Mor∆S (−, [q])] are projective as ∆S op -module. In proving exactness, it suffices to examine the individual sub-k-modules, k [Mor∆S ([n], [q])]. ρ ǫ Lemma 96. For each n ≥ 0, the sequence, 0 ← k ← k [Mor∆S ([n], [0])] ← k Mor∆S ([n], [2])] is exact, where ǫ is defined by ǫ(φ) = 1 for any morphism φ : [n] → [0], and ρ is defined by ρ(ψ) = (x0 x1 x2 )◦ ψ − (x2 x1 x0 )◦ ψ for any morphism ψ : [n] → [2]. Note, x0 x1 x2 and x2 x1 x0 are ∆S morphisms [2] → [0] written in tensor notation. Proof. Clearly, ǫ is surjective. Now, ǫρ = 0, since ρ(ψ) consists of two morphisms with opposite signs. Let φ0 = x0 x1 . . . xn : [n] → [0]. The kernel of ǫ is spanned by elements φ − φ0 for φ ∈ Mor∆S ([n], [0]). So, it suffices to show that the submodule of k [Mor∆S ([n], [0])] generated by (x0 x1 x2 )ψ − (x2 x1 x0 )ψ for ψ : [n] → [2] contains all of the elements φ−φ0 . In other words, it suffices to find a sequence φ =: φk , φk−1 , . . . , φ2 , φ1 , φ0 so that each φi is obtained from φi+1 by reversing the order of 3 (possibly emtpy) blocks, XY Z → ZY X. Let φ = xi0 xi1 . . . xin . If φ = φ0 , we may stop here. Otherwise, we may produce a sequence ending in φ0 by way of a certain family of rearrangements: xk+1 . . . xn xik xi0 xi1 . . . xik−1 , where ik 6= k. That is, a k-rearrangment : xi0 xi1 . . . xik−1 xik xk+1 . . . xn k-rearrangement only applies to those monomials that agree with φ0 in the final n−k indeterminates, but not in the final n − k + 1 indeterminates. If k = n, then this rearrangement reduces to the cyclic rearrangment, xin xi0 xi1 . . . xin−1 . xi0 xi1 . . . xin Beginning with φ, perform n-rearrangements until the final indeterminate is xn . For convenience of notation, let this new monomial be xj0 xj1 . . . xjn . (Of course, jn = n.) If jk = k for all k = 0, 1, . . . , n, then we are done. Otherwise, there will be a number k such that jk 6= k but jk+1 = k + 1, . . . , jn = n. Perform a k-rearrangement followed by enough n-rearrangements so that the final indeterminate is again xn . The net result of these rearrangements is that the ending block xk+1 xk+2 . . . xn remains fixed while the beginning block xj0 xj1 . . . xjk becomes cyclically permuted to xjk xj0 . . . xjk−1 . It is clear that applying this combination of rearrangements repeatedly will finally obtain a monomial xℓ0 xℓ1 . . . xℓk−1 xk xk+1 . . . xn . Now repeat the process, until after a finite number of steps, we finally obtain φ0 . Let Bn := {xi0 xi1 . . . xik−1 ⊗ xik ⊗ xk+1 xk+2 . . . xn : 1 ≤ k ≤ n, ik 6= k}. k[Bn ] is a free submodule of k [Mor∆S ([n], [2])] of size (n + 1)! − 1. Corollary 97. When restricted to k[Bn ], the map ρ of Lemma 96 is surjective onto the kernel of ǫ. Proof. In the proof of Lemma 96, the n-rearrangements correspond to the image of elements xi0 . . . xin−1 ⊗ xin ⊗ 1, with in 6= n. For k < n, k-rearrangements correspond to the image of elements xi0 . . . xik−1 ⊗ xik ⊗ xk+1 . . . xn , with ik 6= k. Lemma 98. k [Mor∆S ([n], [m])] is a free k-module of rank (m + n + 1)!/m!.

38

SHAUN V. AULT

Proof. A morphism φ : [n] → [m] of ∆S is nothing more than an assignment of n + 1 objects into m + 1 compartments, along with a total ordering of the original n + 1 objects, hence #Mor∆S ([n], [m]) = m+n+1 . (n + 1)! = (m+n+1)! m! m

Lemma 99. ρ|k[Bn ] is an isomorphism k[Bn ] ∼ = ker ǫ.

Proof. Since the rank of k [Mor∆S ([n], [0])] is (n + 1)!, the rank of the ker ǫ is (n + 1)! − 1. The isomorphism then follows from Corollary 97. Lemma 100. The relations of the form: XY ⊗ Z ⊗ W + W ⊗ ZX ⊗ Y + Y ZX ⊗ 1 ⊗ W + W ⊗ Y Z ⊗ X ≈ 0

(56) (57)

collapse k Mor∆S ([n], [2]) onto k[Bn ].

and

1⊗X ⊗1≈0

Proof. This proof proceeds in multiple steps. Step 1. [Degeneracy Relations] X ⊗ Y ⊗ 1 ≈ X ⊗ 1 ⊗ Y ≈ 1 ⊗ X ⊗ Y . First, observe that letting X = Y = W = 1 in Eq. (56) yields Z ⊗ 1 ⊗ 1 ≈ 0, since 1 ⊗ Z ⊗ 1 ≈ 0. Then, letting X = Z = W = 1 in Eq. (56) produces 1 ⊗ 1 ⊗ Y ≈ 0. Thus, any formal tensor with two trivial factors is equivalent to 0. Next, let Z = W = 1 in Eq. (56). Then using the above observation, we obtain 1 ⊗ X ⊗ Y + 1 ⊗ Y ⊗ X ≈ 0, that is, 1 ⊗ X ⊗ Y ≈ −(1 ⊗ Y ⊗ X). Then, if we let X = W = 1, we obtain Y ⊗ Z ⊗ 1 + 1 ⊗ Z ⊗ Y ≈ 0, which is equivalent to Y ⊗ Z ⊗ 1 − 1 ⊗ Y ⊗ Z ≈ 0. Finally, let X = Y = 1 in Eq. (56). The expression reduces to Z ⊗ 1 ⊗ W − 1 ⊗ Z ⊗ W ≈ 0. Step 2. [Sign Relation] X ⊗ Y ⊗ Z ≈ −(Z ⊗ Y ⊗ X). Let Y = 1 in Eq. (56), and use the degeneracy relations to rewrite the result as X ⊗ Z ⊗ W + 1 ⊗ W ⊗ ZX + 1 ⊗ ZX ⊗ W + W ⊗ Z ⊗ X ≈ 0. Since 1 ⊗ ZX ⊗ W ≈ −(1 ⊗ W ⊗ ZW ), the desired result follows: X ⊗ Z ⊗ W + W ⊗ Z ⊗ X ≈ 0. Step 3. [Hochschild Relation] XY ⊗ Z ⊗ 1 − X ⊗ Y Z ⊗ 1 + ZX ⊗ Y ⊗ 1 ≈ 0. This relation is named after the similar relation, XY ⊗ Z − X ⊗ Y Z + ZX ⊗ Y , that arises in the Hochschild complex. Let W = 1 in Eq. (56), and use the degeneracy and sign relations to obtain the desired result. n X τnj xi0 xi1 . . . xin−1 ⊗ xin ⊗ 1 ≈ 0, where τn ∈ Σn+1 is the (n + 1)-cycle Step 4. [Cyclic Relation] j=0

(0, n, n − 1, . . . , 2, 1), which acts by permuting the indices. For n = 0, there are no such relations (indeed, no relations at all). For n = 1, the cyclic relation takes the form x0 ⊗ x1 ⊗ 1 + x1 ⊗ x0 ⊗ 1 ≈ 0, which follows from degeneracy and sign relations. Assume now that n ≥ 2. For each k = 1, 2, . . . , n − 1, define:   Ak := xi0 xi1 . . . xik−1 , Bk := xik ,  Ck := xik+1 . . . xin . Pn−1 By the Hochschild relation, 0 ≈ k=1 (Ak Bk ⊗ Ck ⊗ 1 − Ak ⊗ Bk Ck ⊗ 1 + Ck Ak ⊗ Bk ⊗ 1). But for k ≤ n − 2, Ak Bk ⊗ Ck ⊗ 1 = Ak+1 ⊗ Bk+1 Ck+1 ⊗ 1. Thus, after some cancellation, (58)

0 ≈ −A1 ⊗ B1 C1 ⊗ 1 + An−1 Bn−1 ⊗ Cn−1 ⊗ 1 +

n−1 X

Ck Ak ⊗ Bk ⊗ 1.

k=1

Now observe that sign and degeneracy relations imply that −A1 ⊗ B1 C1 ⊗ 1 ≈ xi1 . . . xin ⊗ xi0 ⊗ 1. The term An−1 Bn−1 ⊗ Cn−1 ⊗ 1 is equal to xi0 . . . xin−1 ⊗ xin ⊗ 1, and for 1 ≤ k ≤ n − 1, Ck Ak ⊗ Bk ⊗ 1 = xik+1 . . . xin xi0 . . . xik−1 ⊗ xik ⊗ 1. Thus, Eq. (58) can be rewritten as the cyclic relation, 0 ≈ (xi0 . . . xin−1 ⊗ xin ⊗ 1) +

n−1 X

xik+1 . . . xin xi0 . . . xik−1 ⊗ xik ⊗ 1.

k=0

Step 5. Every element of the form X ⊗ Y ⊗ 1 is equivalent to a linear combination of elements of Bn . To prove this, we shall induct on the size of Y . Suppose Y consists of a single indeterminate. That is, X ⊗ Y ⊗ 1 = xi0 . . . xin−1 ⊗ xin ⊗ 1. Now, if in 6= n, we are done. Otherwise, we use the cyclic relation to Pn write xi0 . . . xin−1 ⊗ xin ⊗ 1 ≈ − j=1 τnj xi0 . . . xin−1 ⊗ xin ⊗ 1 .


39

Now suppose k ≥ 1 and any element Z ⊗ W ⊗ 1 with |W | = k is equivalent to an element of k[Bn ]. Consider X ⊗ Y ⊗ 1 = xi0 . . . xin−k−1 ⊗ xin−k . . . xin ⊗ 1. Let   A := xi0 xi1 . . . xin−k−1 , B := xin−k . . . xin−1 ,  C := xin .

Then, by the Hochschild relation, X ⊗ Y ⊗ 1 = A ⊗ BC ⊗ 1 ≈ AB ⊗ C ⊗ 1 + CA ⊗ B ⊗ 1. But since C has one indeterminate and B has k indeterminates, this last expression is equivalent to an element of k[Bn ] by inductive hypothesis. Step 6. [Modified Hochschild Relation] XY ⊗ Z ⊗ W − X ⊗ Y Z ⊗ W + ZX ⊗ Y ⊗ W ≈ 0, modulo k [Bn ]. First, we show that X ⊗ Y ⊗ W + Y ⊗ X ⊗ W ≈ 0 (mod k [Bn ]). Indeed, if we let Z = 1 in Eq. (56), then sign and degeneracy relations yield: X ⊗ Y ⊗ W + Y ⊗ X ⊗ W ≈ XY ⊗ W ⊗ 1 + Y X ⊗ W ⊗ 1, i.e., by step 5, (59)

X ⊗ Y ⊗ W ≈ −(Y ⊗ X ⊗ W ) (mod k [Bn ]).

Now, using sign and degeneracy relations, Eq. (56) can be re-expressed: XY ⊗ Z ⊗ W − Y ⊗ ZX ⊗ W + Y ZX ⊗ W ⊗ 1 − X ⊗ Y Z ⊗ W ≈ 0. Using Eq. (59), we then arrive at the modified Hochschild relation, (60)

XY ⊗ Z ⊗ W − X ⊗ Y Z ⊗ W + ZX ⊗ Y ⊗ W ≈ 0 (mod k [Bn ]).

Step 7. [Modified Cyclic Relation]

k X j=0

τkj xi0 xi1 . . . xik−1 ⊗ xik ⊗ xik+1 . . . xin ≈ 0, modulo k [Bn ].

Note, the (k + 1)-cycle τk permutes the indices i0 , i1 , . . . , ik , and fixes the rest. Eq. (59) proves the modified cyclic relations for k = 1. The modified cyclic relation for k ≥ 2 follows from the modified Hochschild relation in the same manner as in step 4. Of course, this time all equivalences are taken modulo k [Bn ]. Step 8. Every element of the form X ⊗ Y ⊗ xn is equivalent to an element of k[Bn ]. We shall use the modified cyclic and modified Hochschild relations in a similar way as cyclic and Hochschild relations were used in step 5. Again we induct on the size of Y . If |Y | = 1, then X ⊗ Y ⊗ xn = xi0 . . . xin−2 ⊗ xin−1 ⊗ xn . If in−1 6= n − 1, then we are done. Otherwise, use the modified cyclic relation to re-express X ⊗ Y ⊗ xn as a sum of elements of k [Bn ]. Next, suppose k ≥ 1 and any element Z ⊗ W ⊗ xn with |W | = k is equivalent to an element of k[Bn ]. Consider an element X ⊗ Y ⊗ xn with |Y | = k + 1. Write Y = BC with |B| = k and |C| = 1 and use the modified Hochschild relation to rewrite X ⊗ BC ⊗ xn in terms of two elements whose middle tensor factors are either B or C (modulo k [Bn ]). By inductive hypothesis, the rewritten expression must lie in k[Bn ]. Step 9. Every element of k [Mor∆S ([n], [2])] is equivalent to a linear combination of elements from the following set: (61)

Cn := {X ⊗ xin ⊗ 1 | in 6= n} ∪ {X ⊗ xin−1 ⊗ xn | in−1 6= n − 1} ∪ {X ⊗ Y ⊗ Zxn | |Z| ≥ 1}

Note, the k-module generated by Cn contains k [Bn ]. Let X ⊗ Y ⊗ Z be an arbitrary element of k [Mor∆S ([n], [2])]. If |X| = 0, |Y | = 0, or |Z| = 0, then the degeneracy relations and step 5 imply that X ⊗ Y ⊗ Z is equivalent to an element of k [Bn ]. Suppose now that |X|, |Y |, |Z| ≥ 1. If xn occurs in X, use the relation X ⊗ Y ⊗ W ≈ −(Y ⊗ X ⊗ W ) (mod k [Bn ]) to ensure that xn occurs in the middle factor. If xn occurs in Z, use the sign relation and the above relation to put xn into the middle factor. In any case, it suffices to assume our element has the form: X ⊗U xn V ⊗Z. Using the modified Hochschild relation, X ⊗U xn V ⊗Z ≈ −(Z ⊗V ⊗XU xn )+Z ⊗V X ⊗U xn , (mod k [Bn ]). The first term is certainly in k[Cn ], since |X| ≥ 1. If |U | > 0, the second term also lies in k[Cn ]. If, on the other hand, |U | = 0, then step 8 implies that Z ⊗ V X ⊗ xn is an element of k[Bn ]. Observe that Step 9 proves Lemma 100 for n = 0, 1, 2, since in these cases, any elements that fall within the set {X ⊗ Y ⊗ Zxn | |Z| ≥ 1} must have either |X| = 0 or |Y | = 0, hence are equivalent via the degeneracy relation to elements of k [{X ⊗ xin ⊗ 1 | in 6= n}]. In what follows, assume n ≥ 3.

40

SHAUN V. AULT

Step 10. Every element of k [Mor∆S ([n], [2])] is equivalent, modulo k [Bn ], to a linear combination of elements from the following set: (62)

Dn := {X ⊗ xin−2 ⊗ xn−1 xn | in−2 6= n − 2} ∪ {X ⊗ Y ⊗ Zxn−1 xn | |Z| ≥ 1}.

First, we require a relation that transports xn from the end of a tensor: (63)

W ⊗ Z ⊗ Xxn ≈ W ⊗ xn Z ⊗ X

(mod k [Bn ]).

Letting Y = xn in Eq. (56), and making use of the sign relation, we have: W ⊗ Z ⊗ Xxn ≈ W ⊗ ZX ⊗ xn + xn ZX ⊗ W ⊗ 1 + W ⊗ xn Z ⊗ X. By steps 5 and 8, W ⊗ Z ⊗ Xxn ≈ W ⊗ xn Z ⊗ X, modulo elements of k [Bn ]. Now, let X ⊗ Y ⊗ Z be an arbitrary element of k [Mor∆S ([n], [2])]. Locate xn−1 and use the techniques of Step 9 to re-express X ⊗ Y ⊗ Z as a linear combination of terms of the form: Xj ⊗ Yj ⊗ Zj xn−1 , modulo k [Bn ]. Our goal is to re-express each term as a linear combination of vectors in which xn occurs only in the second tensor factor. If xn occurs in Xj , then observe Xj ⊗ Yj ⊗ Zj xn−1 ≈ −(Yj ⊗ Xj ⊗ Zj xn−1 ), (mod k [Bn ]). If xn occurs in Zj , then first substitute Y = xn−1 into Eq. (56), obtaining the relation: Xxn−1 ⊗ Z ⊗ W + W ⊗ ZX ⊗ xn−1 + xn−1 ZX ⊗ 1 ⊗ W + W ⊗ xn−1 Z ⊗ X ≈ 0 ⇒ W ⊗ Z ⊗ Xxn−1 ≈ W ⊗ ZX ⊗ xn−1 + W ⊗ xn−1 Z ⊗ X

(mod k [Bn ])

By the modified Hochschild relation, W ⊗ xn−1 Z ⊗ X ≈ W xn−1 ⊗ Z ⊗ X + ZW ⊗ xn−1 ⊗ X, (mod k [Bn ]), then using sign relations etc., we obtain: W ⊗ Z ⊗ Xxn−1 ≈ W ⊗ ZX ⊗ xn−1 + Z ⊗ X ⊗ W xn−1 − ZW ⊗ X ⊗ xn−1 ,

(mod k [Bn ]).

Thus, we can express our original element X ⊗ Y ⊗ Z as a linear combination of elements of the form X ′ ⊗ U ′ xn V ′ ⊗ Z ′ xn−1 , (mod k [Bn ]). Then using modified Hochschild etc., rewrite each such term as follows: X ′ ⊗ U ′ xn V ′ ⊗ Z ′ xn−1 ≈ X ′ U ′ ⊗ xn V ′ ⊗ Z ′ xn−1 − U ′ ⊗ xn V ′ X ′ ⊗ Z ′ xn−1 ,

(mod k [Bn ]).

By Eq. (63), we transport xn to the end of each term, so X ′ ⊗ U ′ xn V ′ ⊗ Z ′ xn−1 ≈ X ′ U ′ ⊗ V ′ ⊗ Z ′ xn−1 xn − U ′ ⊗ V ′ X ′ ⊗ Z ′ xn−1 xn , modulo elements of k [Bn ]. If |Z ′ | ≥ 1, then we are done. Otherwise, we have some elements of the form X ′′ ⊗ Y ′′ ⊗ xn−1 xn . Use an induction argument analogous to that in step 8 to re-express this type of element as a linear combination of elements of the form U ⊗ xin−2 ⊗ xn−1 xn such that in−2 6= n − 2, modulo elements of k [Bn ]. Step 11. Every element of k [Mor∆S ([n], [2])] is equivalent to an element of k [Bn ]. We shall use an iterative re-writing procedure. First of all, define sets: Bnj := {A ⊗ xin−j ⊗ xn−j+1 . . . xn | in−j 6= n − j}, Cnj := {A ⊗ B ⊗ Cxn−j+1 . . . xn | |C| ≥ 1}.

Sn−1 Now clearly, Bn = j=0 Bnj . In what follows, ‘reduced’ will always mean reduced modulo elements of k [Bn ]. By steps 9 and 10, we can reduce an arbitrary element X ⊗ Y ⊗ Z to linear combinations of elements in Bn0 ∪ Bn1 ∪ Bn2 ∪ Cn2 . Suppose now that we have reduced elements to linear combinations of elements from the set Bn0 ∪ Bn1 ∪ . . . ∪ Bnj ∪ Cnj , for some j ≥ 2. I claim any element of Cnj can be re-expressed as a linear combination of elements from the set Bn0 ∪ Bn1 ∪ . . . ∪ Bnj+1 ∪ Cnj+1 . Indeed, let X ⊗ Y ⊗ Zxn−j+1 . . . xn , with |Z| ≥ 1. Let w := xn−j+1 . . . xn . We may now think of X ⊗ Y ⊗ Zw as consisting of the ‘indeterminates’ x0 , x1 , . . . , xn−j , w, hence, by step 10, we may reduce this element to a linear combination of elements from the set {X ⊗ xin−j−1 ⊗ xn−j w | in−j−1 6= n − j − 1} ∪ {X ⊗ Y ⊗ Zxn−j w | |Z| ≥ 1}. This implies the element may written as a linear combination of elements from the set Bnj+1 ∪ Cnj+1 , modulo elements of the form A ⊗ B ⊗ 1 and A ⊗ B ⊗ xn−j+1 . . . xn . Since {A ⊗ B ⊗ xn−j+1 xn−j+2 . . . xn } ⊆ Cnj−1 , the inductive hypothesis ensures that the there is set containment {A ⊗ B ⊗ xn−j+1 . . . xn } ⊆ Bn0 ∪ . . . ∪ Bnj . This completes the inductive step. After a finite number of iterations, then, we can re-express any element X ⊗ Y ⊗ Z as a linear combination from the set Bn0 ∪ . . . Bnn−1 ∪ Cnn−1 = Bn ∪ Cnn−1 . But Cnn−1 = {A ⊗ B ⊗ Cx2 . . . xn | |C| ≥ 1}. Any element from this set has either |A| = 0 or |B| = 0, therefore is equivalent to an element of k[Bn ] already.


41

Corollary 101. If 12 ∈ k, then the four-term relation XY ⊗ Z ⊗ W + W ⊗ ZX ⊗ Y + Y ZX ⊗ 1 ⊗ W + W ⊗ Y Z ⊗ X ≈ 0 is sufficient to collapse k [Mor∆S ([n], [2])] onto k [Bn ]. Proof. We only need to modify step 1 of the previous proof. We will establish that X ⊗ 1 ⊗ 1 ≈ 1 ⊗ X ⊗ 1 ≈ 1 ⊗ 1 ⊗ X ≈ 0. Setting three variables at a time equal to 1 in Eq. (56) we obtain, (64)

2(W ⊗ 1 ⊗ 1) + 2(1 ⊗ 1 ⊗ W ) ≈ 0,

(65)

Z ⊗ 1 ⊗ 1 + 3(1 ⊗ Z ⊗ 1) ≈ 0,

(66)

2(Y ⊗ 1 ⊗ 1) + 1 ⊗ Y ⊗ 1 + 1 ⊗ 1 ⊗ Y ≈ 0,

when X = Y = Z = 1. when X = Y = W = 1. when X = Z = W = 1.

Equivalently, we have a system of linear equations,    2 0 2 z1  1 3 0   z2  = 0, 2 1 1 z3

where z1 := X ⊗ 1 ⊗ 1, z2 := 1 ⊗ X ⊗ 1, and z3 := 1 ⊗ 1 ⊗ X. Since the determinant of the coefficient matrix is −4, the matrix is invertible in the ring k as long as 1/2 ∈ k. Lemma 100 together with Lemmas 99 and 96 show the following sequence of k-modules is exact: (67)

(α,β)

ρ

ǫ

0 ← k ← k [Mor∆S ([n], [0])] ← k [Mor∆S ([n], [2])] ←− k [Mor∆S ([n], [3])] ⊕ k [Mor∆S ([n], [0])] ,

where α : k [Mor∆S ([n], [3])] → k [Mor∆S ([n], [2])] is given by composition with the ∆S morphism, x0 x1 ⊗ x2 ⊗ x3 + x3 ⊗ x2 x0 ⊗ x1 + x1 x2 x0 ⊗ 1 ⊗ x3 + x3 ⊗ x1 x2 ⊗ x0 , and β : k [Mor∆S ([n], [0])] → k [Mor∆S ([n], [2])] is induced by 1 ⊗ x0 ⊗ 1. This holds for all n ≥ 0, so we have constructed a partial resolution of k by projective ∆S op -modules: (68)

(α,β)

ρ

ǫ

0 ← k ← k [Mor∆S (−, [0])] ← k [Mor∆S (−, [2])] ←− k [Mor∆S (−, [3])] ⊕ k [Mor∆S (−, [0])] 11. Using the Partial Resolution for Low Degree Computations

Let A be a unital associative algebra over k. Since Eq. (68) is a partial resolution of k, it can be used to find HSi (A) for i = 0, 1. 11.1. Main Theorem. Theorem 102. HSi (A) for i = 0, 1 may be computed as the degree 0 and degree 1 homology groups of the following (partial) chain complex: ∂

∂

2 1 (A ⊗ A ⊗ A ⊗ A) ⊕ A, A ⊗ A ⊗ A ←− 0 ←− A ←−

(69) where

∂1 : a ⊗ b ⊗ c 7→ abc − cba, ∂2 :

a ⊗ b ⊗ c ⊗ d 7→ ab ⊗ c ⊗ d + d ⊗ ca ⊗ b + bca ⊗ 1 ⊗ d + d ⊗ bc ⊗ a, a 7→ 1 ⊗ a ⊗ 1.

Proof. Tensoring the complex (68) with B∗sym A over ∆S, we obtain a complex that computes HS0 (A) and HS1 (A). The statement of the theorem then follows from isomorphisms induced by the evaluation map, ∼ = k [Mor∆S (−, [p])] ⊗∆S B∗sym A −→ Bpsym A.

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11.2. Degree 0 Symmetric Homology. Theorem 103. For a unital associative algebra A over commutative ground ring k, HS0 (A) ∼ = A/([A, A]), where ([A, A]) is the ideal generated by the commutator submodule [A, A]. Proof. By Thm. 102, HS0 (A) ∼ = A/k [{abc − cba}] as k-module. But k {abc − cba} is an ideal of A. Now clearly [A, A] ⊆ k [{abc − bca}]. On the other hand, k [{abc − cba}] ⊆ ([A, A]) since abc − cba = a(bc − cb) + a(cb) − (cb)a. Corollary 104. If A is commutative, then HS0 (A) ∼ = A. Remark 105. Theorem 103 implies that symmetric homology does not preserve Morita equivalence, since for n > 1, HS0 (Mn (A)) = Mn (A)/ ([Mn (A), Mn (A)]) = 0, while in general HS0 (A) = A/([A, A]) 6= 0. 11.3. Degree 1 Symmetric Homology. Using GAP, we have made the following explicit computations of degree 1 integral symmetric homology. See section 13 for a discussion of how computer algebra systems were used in symmetric homology computations. A Z[t]/(t2 ) Z[t]/(t3 ) Z[t]/(t4 ) Z[t]/(t5 ) Z[t]/(t6 ) Z[C2 ] Z[C3 ] Z[C4 ] Z[C5 ] Z[C6 ]

HS1 (A | Z) Z/2Z ⊕ Z/2Z Z/2Z ⊕ Z/2Z (Z/2Z)4 (Z/2Z)4 (Z/2Z)6 Z/2Z ⊕ Z/2Z 0 (Z/2Z)4 0 (Z/2Z)6

Based on these calculations, we conjecture: Conjecture 106. HS1 k[t]/(tn ) =

(k/2k)n , (k/2k)n−1

if n ≥ 0 is even. if n ≥ 1 is odd.

Remark 107. The computations of HS1 Z[Cn ] are consistent with those of Brown and Loday [2]. See 11.5 for a more detailed treatment of HS1 for group rings. Additionally, HS1 has been computed for the following examples. These computations were done using GAP in some cases and in others, Fermat [11] computations on sparse matrices were used in conjunction with the GAP scripts. (e.g. when the algebra has dimension greater than 6 over Z). A Z[t, u]/(t2 , u2 ) Z[t, u]/(t3 , u2 ) Z[t, u]/(t3 , u2 , t2 u) Z[t, u]/(t3 , u3 ) Z[t, u]/(t2 , u4 ) Z[t, u, v]/(t2 , u2 , v 2 ) Z[t, u]/(t4 , u3 ) Z[t, u, v]/(t2 , u2 , v 3 ) Z[i, j, k], i2 = j 2 = k 2 = ijk = −1 Z[C2 × C2 ] Z[C3 × C2 ] Z[C3 × C3 ] Z[S3 ]

HS1 (A | Z) Z ⊕ (Z/2Z)11 Z2 ⊕ (Z/2Z)11 ⊕ Z/6Z Z2 ⊕ (Z/2Z)10 Z4 ⊕ (Z/2Z)7 ⊕ (Z/6Z)5 Z3 ⊕ (Z/2Z)20 ⊕ Z/4Z Z6 ⊕ (Z/2Z)42 Z6 ⊕ (Z/2Z)19 ⊕ Z/6Z ⊕ (Z/12Z)2 Z11 ⊕ (Z/2Z)45 ⊕ (Z/6Z)4 (Z/2Z)8 (Z/2Z)12 (Z/2Z)6 (Z/3Z)9 (Z/2Z)2


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11.4. Splittings of the Partial Complex. Under certain circumstances, the partial complex in Thm.102 splits as a direct sum of smaller complexes. This observation becomes increasingly important as the dimension of the algebra increases. Indeed, some of the computations of the previous section were done using splittings. Definition 108. For a commutative k-algebra A and u ∈ A, define the k-modules: A⊗n u := {a1 ⊗ a2 ⊗ . . . ⊗ an ∈ A⊗n | a1 a2 · . . . · an = u}

Proposition 109. If A = k[M ] for a commutative monoid M , then complex (69) splits as a direct sum of complexes ∂

∂

2 1 (A ⊗ A ⊗ A ⊗ A)u ⊕ (A)u , (A ⊗ A ⊗ A)u ←− 0 ←− (A)u ←− L where u ranges over the elements of M . Thus, for i = 0, 1, we have HSi (A) ∼ = u∈M HSi (A)u .

(70)

Proof. Since M is a commutative monoid, there are direct sum decompositions as k-module: A⊗n = L ⊗n )u . The boundary maps ∂1 and ∂2 preserve the products of tensor factors, so the inclusions u∈M (A ⊗n (A )u ֒→ A⊗n induce maps of complexes, hence the complex itself splits as a direct sum. Definition 110. For each u, the homology groups of complex (70) will be called the u-layered symmetric homology of A, denoted HSi (A)u . We may use layers to investigate the symmetric homology of k[t]. This algebra is monoidal, generated by m m the monoid {1, t, t2 , t3 , . . .}. Now, the t -layerp symmetric homology of k[t] will be the same as the t -layer m+2 symmetric homology of k Mm+1 , where Mq denotes the cyclic monoid generated by an indeterminate s with the property that sp = sq . Using this observation and subsequent computation, we conjecture: Conjecture 111. HS1 (k[t])tm =

0 m = 0, 1 k/2k, m ≥ 2

This conjecture has been verified up to m = 18, in the case k = Z. 11.5. 2-torsion in HS1 . The occurrence of 2-torsion in HS1 (A) for the examples considered in 11.3 and 11.4 comes as no surprise, based on Thm. 44. First consider the following chain of isomorphisms: π2s (BΓ) = h ∼ π1 (ΩΩ∞ S ∞ (BΓ)) ∼ π2 (Ω∞ S ∞ (BΓ)) = = π1 (Ω0 Ω∞ S ∞ (BΓ)) → H1 (Ω0 Ω∞ S ∞ (BΓ)). Here, Ω0 Ω∞ S ∞ (BΓ) denotes the component of the constant loop, and h is the Hurewicz homomorphism, which is an isomorphism since Ω0 Ω∞ S ∞ (BΓ) is path-connected and π1 is abelian (since it is actually π2 of a space). On the other hand, by Thm. 44, HS1 (k[Γ]) ∼ = H1 (ΩΩ∞ S ∞ (BΓ); k) ∼ = H1 (ΩΩ∞ S ∞ (BΓ)) ⊗ k. Note, all ∞ ∞ tensor products will be over Z in this section. Now ΩΩ S (BΓ) consists of disjoint homeomorphic copies of Ω0 Ω∞ S ∞ (BΓ), one for each element of Γ/[Γ, Γ], (where [Γ, Γ] is the commutator subgroup of Γ), so we may write H1 (ΩΩ∞ S ∞ (BΓ)) ⊗ k ∼ = H1 (Ω0 Ω∞ S ∞ (BΓ)) ⊗ k [Γ/[Γ, Γ]] and obtain the following result: Proposition 112. If Γ is a group, then HS1 (k[Γ]) ∼ = π2s (BΓ) ⊗ k [Γ/[Γ, Γ]]. As an immediate corollary, if Γ is abelian, then HS1 (k[Γ]) ∼ = π2s (BΓ) ⊗ k[Γ]. Moreover, by results of s e Γ = (Γ ⊗ Γ) / ≈, where Brown and Loday [2], if Γ is abelian, then π2 (BΓ) is the reduced tensor square, Γ ∧ g ⊗ h ≈ −h ⊗ g for all g, h ∈ Γ. (This construction is notated with multiplicative group action in [2], since they deal with the more general case of non-abelian groups.) Proposition 113. HS1 (k[Cn ]) =

(Z/2Z)n 0

n even. n odd.

Proof. π2s (BCn ) = Z/2Z if n is even, and 0 if n is odd. The result then follows from Prop. 112, as k [Cn /[Cn , Cn ]] ∼ = k[Cn ] ∼ = k n , as k-module.

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12. Relations to Cyclic Homology The relation between the symmetric bar construction and the cyclic bar construction arising from the inclusions ∆C ֒→ ∆S gives rise to a natural map HC∗ (A) → HS∗ (A) Indeed, by remark 13, we may define sym cyclic homology thus: HC∗ (A) = Tor∆C A), where we understand B∗sym A as the restriction of the ∗ (k, B∗ functor to ∆C. Using the partial complex of Thm. 102, and an analogous one for computing cyclic homology (c.f. [12], p. 59), the map HC∗ (A) → HS∗ (A) for degrees 0 and 1 is induced by the following partial chain map: 0 o

A o

0 o

A o

∂1C

∂2C

A⊗A o

A⊗3 ⊕ A

γ1

γ0 =id ∂1S

A⊗3 o

γ2 ∂2S

A⊗4 ⊕ A

In this diagram, the boundary maps in the upper row are defined as follows:

∂2C :

∂1C : a ⊗ b 7→ ab − ba a⊗b⊗c a

7→ ab ⊗ c − a ⊗ bc + ca ⊗ b 7→ 1 ⊗ a − a ⊗ 1

The boundary maps in the lower row are defined as in Thm. 102.

∂2S :

∂1S : a ⊗ b ⊗ c 7→ abc − cba a ⊗ b ⊗ c ⊗ d 7→ ab ⊗ c ⊗ d − d ⊗ ca ⊗ b + bca ⊗ 1 ⊗ d + d ⊗ bc ⊗ a a 7→ 1 ⊗ a ⊗ 1

The partial chain map is given in degree 1 by γ1 (a ⊗ b) := a ⊗ b ⊗ 1. In degree 2, γ2 is defined on the summand A⊗3 via a ⊗ b ⊗ c 7→ (a ⊗ b ⊗ c ⊗ 1 − 1 ⊗ a ⊗ bc ⊗ 1 + 1 ⊗ ca ⊗ b ⊗ 1 + 1 ⊗ 1 ⊗ abc ⊗ 1 − b ⊗ ca ⊗ 1 ⊗ 1) − 2abc − cab, and on the summand A via a 7→ (−1 ⊗ 1 ⊗ a ⊗ 1) + (4a). 12.1. Examples. To provide some examples, consider the maps γ1 : HC1 (Z[t]/(tn )) → HS1 (Z[t]/(tn )). It can be shown (e.g. by direct computation) that HC1 (Z[t]/(t2 )) ∼ = Z/2Z is generated by the 1-chain t ⊗ t. γ1 (t ⊗ t) = t ⊗ t ⊗ 1 ∈ HS1 (Z[t]/(t2 ) is a non-trivial element of Z/2Z ⊕ Z/2Z (which may be verified by direct computation as well). The map HC1 (Z[t]/(t3 )) → HS1 (Z[t]/(t2 )) may be similarly analyzed. Here, the chain t ⊗ t + t ⊗ t2 is a generator of HC1 (Z[t]/(t3 )) ∼ = Z/6Z, which gets sent by γ1 to t ⊗ t ⊗ 1 + t ⊗ t2 ⊗ 1, a non-trivial element of 3 ∼ HS1 (Z[t]/(t )) = Z/2Z ⊕ Z/2Z. The case n = 4 is bit more interesting. Here, HC1 (Z[t]/(t4 )) ∼ = Z/2Z ⊕ Z/12Z, generated by t ⊗ t and t ⊗ t2 + t ⊗ t3 , respectively. The image of the map γ1 in HS1 (Z[t]/(t4 )) is (Z/2Z)2 ⊆ (Z/2Z)4 . 13. Using Computer Algebra Systems for Computing Symmetric Homology The computer algebra systems GAP, Octave and Fermat were used to verify proposed theorems and also to obtain some concrete computations of symmetric homology for some small algebras. A tar-file of the scripts that were created and used for this work is available at http://arxiv.org/e-print/0807.4521v1/. This tar-file contains the following files: • Basic.g - Some elementary functions, necessary for some functions in DeltaS.g • HomAlg.g - Homological Algebra functions, such as computation of homology groups for chain complexes. • Fermat.g - Functions necessary to invoke Fermat for fast sparse matrix computations. • fermattogap, gaptofermat - Auxiliary text files for use when invoking Fermat from GAP. • DeltaS.g - This is the main repository of scripts used to compute various quantities associated with the category ∆S, including HS1 (A) for finite-dimensional algebras A.


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In order to use the functions of DeltaS.g, simply copy the above files into the working directory (such as ~/gap/), invoke GAP, then read in DeltaS.g at the prompt. The dependent modules will automatically be loaded (hence they must be present in the same directory as DeltaS.g). Note, most of the computations involving homology require substantial memory to run. I recommend calling GAP with the command line option “-o mem”, where mem is the amount of memory to be allocated to this instance of GAP. All computations done in this dissertation can be accomplished by allocating 20 gigabytes of memory. The following provides a few examples of using the functions of DeltaS.g [ault@math gap]$ gap -o 20g gap> Read("DeltaS.g"); gap> gap> ## Number of morphisms [6] --> [4] gap> SizeDeltaS( 6, 4 ); 1663200 gap> gap> ## Generate the set of morphisms of Delta S, [2] --> [2] gap> EnumerateDeltaS( 2, 2 ); [ [ [ 0, 1, 2 ], [ ], [ ] ], [ [ 0, 2, 1 ], [ ], [ ] ], [ [ 1, 0, 2 ], [ ], [ ] ], [ [ 1, 2, 0 ], [ ], [ ] ], [ [ 2, 0, 1 ], [ ], [ ] ], [ [ 2, 1, 0 ], [ ], [ ] ], [ [ 0, 1 ], [ 2 ], [ ] ], [ [ 0, 2 ], [ 1 ], [ ] ], [ [ 1, 0 ], [ 2 ], [ ] ], [ [ 1, 2 ], [ 0 ], [ ] ], [ [ 2, 0 ], [ 1 ], [ ] ], [ [ 2, 1 ], [ 0 ], [ ] ], [ [ 0, 1 ], [ ], [ 2 ] ], [ [ 0, 2 ], [ ], [ 1 ] ], [ [ 1, 0 ], [ ], [ 2 ] ], [ [ 1, 2 ], [ ], [ 0 ] ], [ [ 2, 0 ], [ ], [ 1 ] ], [ [ 2, 1 ], [ ], [ 0 ] ], [ [ 0 ], [ 1, 2 ], [ ] ], [ [ 0 ], [ 2, 1 ], [ ] ], [ [ 1 ], [ 0, 2 ], [ ] ], [ [ 1 ], [ 2, 0 ], [ ] ], [ [ 2 ], [ 0, 1 ], [ ] ], [ [ 2 ], [ 1, 0 ], [ ] ], [ [ 0 ], [ 1 ], [ 2 ] ], [ [ 0 ], [ 2 ], [ 1 ] ], [ [ 1 ], [ 0 ], [ 2 ] ], [ [ 1 ], [ 2 ], [ 0 ] ], [ [ 2 ], [ 0 ], [ 1 ] ], [ [ 2 ], [ 1 ], [ 0 ] ], [ [ 0 ], [ ], [ 1, 2 ] ], [ [ 0 ], [ ], [ 2, 1 ] ], [ [ 1 ], [ ], [ 0, 2 ] ], [ [ 1 ], [ ], [ 2, 0 ] ], [ [ 2 ], [ ], [ 0, 1 ] ], [ [ 2 ], [ ], [ 1, 0 ] ], [ [ ], [ 0, 1, 2 ], [ ] ], [ [ ], [ 0, 2, 1 ], [ ] ], [ [ ], [ 1, 0, 2 ], [ ] ], [ [ ], [ 1, 2, 0 ], [ ] ], [ [ ], [ 2, 0, 1 ], [ ] ], [ [ ], [ 2, 1, 0 ], [ ] ], [ [ ], [ 0, 1 ], [ 2 ] ], [ [ ], [ 0, 2 ], [ 1 ] ], [ [ ], [ 1, 0 ], [ 2 ] ], [ [ ], [ 1, 2 ], [ 0 ] ], [ [ ], [ 2, 0 ], [ 1 ] ], [ [ ], [ 2, 1 ], [ 0 ] ], [ [ ], [ 0 ], [ 1, 2 ] ], [ [ ], [ 0 ], [ 2, 1 ] ], [ [ ], [ 1 ], [ 0, 2 ] ], [ [ ], [ 1 ], [ 2, 0 ] ], [ [ ], [ 2 ], [ 0, 1 ] ], [ [ ], [ 2 ], [ 1, 0 ] ], [ [ ], [ ], [ 0, 1, 2 ] ], [ [ ], [ ], [ 0, 2, 1 ] ], [ [ ], [ ], [ 1, 0, 2 ] ], [ [ ], [ ], [ 1, 2, 0 ] ], [ [ ], [ ], [ 2, 0, 1 ] ], [ [ ], [ ], [ 2, 1, 0 ] ] ] gap> gap> ## Generate only the epimorphisms [2] -->> [2] gap> EnumerateDeltaS( 2, 2 : epi ); [ [ [ 0 ], [ 1 ], [ 2 ] ], [ [ 0 ], [ 2 ], [ 1 ] ], [ [ 1 ], [ 0 ], [ 2 ] ], [ [ 1 ], [ 2 ], [ 0 ] ], [ [ 2 ], [ 0 ], [ 1 ] ], [ [ 2 ], [ 1 ], [ 0 ] ] ] gap> gap> ## Compose two morphisms of Delta S.

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SHAUN V. AULT

gap> a := Random(EnumerateDeltaS(4,3)); [ [ 0 ], [ 2, 4, 1 ], [ ], [ 3 ] ] gap> b := Random(EnumerateDeltaS(3,2)); [ [ ], [ 3, 0, 2 ], [ 1 ] ] gap> MultDeltaS(b, a); [ [ ], [ 3, 0 ], [ 2, 4, 1 ] ] gap> MultDeltaS(a, b); Maps incomposeable [ ] gap> gap> ## Examples of using morphisms of Delta S to act on simple tensors gap> A := TruncPolyAlg([3,2]); gap> ## TruncPolyAlg is defined in Basic.g gap> ## TruncPolyAlg([i_1, i_2, ..., i_n]) is generated by gap> ## x_1, x_2, ..., x_n, under the relation (x_j)^(i_j) = 0. gap> g := GeneratorsOfLeftModule(A); [ X^[ 0, 0 ], X^[ 0, 1 ], X^[ 1, 0 ], X^[ 1, 1 ], X^[ 2, 0 ], X^[ 2, 1 ] ] gap> x := g[2]; y := g[3]; X^[ 0, 1 ] X^[ 1, 0 ] gap> v := [ x*y, 1, y^2 ]; gap> ## v represents the simple tensor xy \otimes 1 \otimes y^2. [ X^[ 1, 1 ], 1, X^[ 2, 0 ] ] gap> ActByDeltaS( v, [[2], [], [0], [1]] ); [ X^[ 2, 0 ], 1, X^[ 1, 1 ], 1 ] gap> ActByDeltaS( v, [[2], [0,1]] ); [ X^[ 2, 0 ], X^[ 1, 1 ] ] gap> ActByDeltaS( v, [[2,0], [1]] ); [ 0*X^[ 0, 0 ], 1 ] gap> gap> ## Symmetric monoidal product on DeltaS_+ gap> a := Random(EnumerateDeltaS(4,2)); [ [ ], [ 2, 1, 0 ], [ 3, 4 ] ] gap> b := Random(EnumerateDeltaS(3,3)); [ [ ], [ ], [ ], [ 1, 3, 2, 0 ] ] gap> MonoidProductDeltaS(a, b); [ [ ], [ 2, 1, 0 ], [ 3, 4 ], [ ], [ ], [ ], [ 6, 8, 7, 5 ] ] gap> MonoidProductDeltaS(b, a); [ [ ], [ ], [ ], [ 1, 3, 2, 0 ], [ ], [ 6, 5, 4 ], [ 7, 8 ] ] gap> MonoidProductDeltaS(a, []); [ [ ], [ 2, 1, 0 ], [ 3, 4 ] ] gap> gap> ## Symmetric Homology of the algebra A, in degrees 0 and 1. gap> SymHomUnitalAlg(A); [ [ 0, 0, 0, 0, 0, 0 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 0, 0 ] ] gap> ## ’0’ represents a factor of Z, while a non-zero p represents gap> ## a factor of Z/pZ. gap> gap> ## Using layers to compute symmetric homology gap> C2 := CyclicGroup(2); gap> A := GroupRing(Rationals, DirectProduct(C2, C2));


gap> ## First, a direct computation without layers: gap> SymHomUnitalAlg(A); [ [ 0, 0, 0, 0 ], [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ] ] gap> ## Next, compute HS_0(A)_u and HS_1(A)_u for each generator u. gap> g := GeneratorsOfLeftModule(A); [ (1)* of ..., (1)*f2, (1)*f1, (1)*f1*f2 ] gap> SymHomUnitalAlgLayered(A, g[1]); [ [ 0 ], [ 2, 2, 2 ] ] gap> SymHomUnitalAlgLayered(A, g[2]); [ [ 0 ], [ 2, 2, 2 ] ] gap> SymHomUnitalAlgLayered(A, g[3]); [ [ 0 ], [ 2, 2, 2 ] ] gap> SymHomUnitalAlgLayered(A, g[4]); [ [ 0 ], [ 2, 2, 2 ] ] gap> ## Computing HS_1( Z[t] ) by layers: gap> SymHomFreeMonoid(0,10); HS_1(k[t])_{t^0} : [ ] HS_1(k[t])_{t^1} : [ ] HS_1(k[t])_{t^2} : [ 2 ] HS_1(k[t])_{t^3} : [ 2 ] HS_1(k[t])_{t^4} : [ 2 ] HS_1(k[t])_{t^5} : [ 2 ] HS_1(k[t])_{t^6} : [ 2 ] HS_1(k[t])_{t^7} : [ 2 ] HS_1(k[t])_{t^8} : [ 2 ] HS_1(k[t])_{t^9} : [ 2 ] HS_1(k[t])_{t^10} : [ 2 ] gap> ## Poincare polynomial of Sym_*^{(p)} for small p. gap> ## There is a check for torsion, using a call to Fermat gap> ## to find Smith Normal Form of the differential matrices. gap> PoincarePolynomialSymComplex(2); C_0 Dimension: 1 C_1 Dimension: 6 C_2 Dimension: 6 D_1 SNF(D_1) D_2 SNF(D_2) 2*t^2+t gap> PoincarePolynomialSymComplex(5); C_0 Dimension: 1 C_1 Dimension: 30 C_2 Dimension: 300 C_3 Dimension: 1200 C_4 Dimension: 1800 C_5 Dimension: 720 D_1 SNF(D_1) D_2 SNF(D_2) D_3 SNF(D_3) D_4 SNF(D_4)

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D_5 SNF(D_5) 120*t^5+272*t^4+t^3 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

K. S. Brown. Cohomology of Groups. Springer, New York, 1982. R. Brown and J.-L. Loday. Van Kampen theorems for diagrams of spaces. Topology, 26, No. 3:311–335, 1987. F. R. Cohen, T. J. Lada, and J. P. May. The Homology of Iterated Loop Spaces. Springer-Verlag, Berlin, 1976. F. R. Cohen and F. P. Peterson. On the homology of certain spaces looped beyond their connectivity. Israel Journal of Mathematics, 66:105–131, 1989. A. Dold. Universelle koeffizienten. Math. Zeitschr., 80:63–88, 1962. J. W. Eaton. GNU Octave Manual. Network Theory Limited, 2002. Z. Fiedorowicz. The symmetric bar construction. Preprint. Available at: http://www.math.ohio-state.edu/∼ fiedorow/. Z. Fiedorowicz and J.-L. Loday. Crossed simplicial groups and their associated homology. Trans. A. M. S., 326:57–87, 1991. The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.10, 2007. K. Itˆ o, editor. Encyclopedic Dictionary of Mathematics. The MIT Press, Cambridge, MA, 1993. R. H. Lewis. Fermat computer algebra system. http://home.bway.net/lewis , 2008. J.-L. Loday. Cyclic Homology. Springer-Verlag, Berlin, 1998. S. MacLane. The Milgram bar construction as a tensor product of functors. Lecture Notes in Math., 168:135–152, 1970. J. P. May. The Geometry of Iterated Loop Spaces. Springer-Verlag, Berlin, 1972. J. P. May. H∞ Ring Spectra and their Applications. Springer-Verlag, Berlin, 1986. J. P. May. Definitions: operads, algebras and modules in operads: proceedings of renaissance conferences (hartford/luminy, 1995). Contemp. Math., 202:1–7, 1997. J. P. May and R. W. Thomason. The uniqueness of infinite loop space machines. Topology, 17, No. 3:205–224, 1978. J. McCleary. User’s Guide to Spectral Sequences (2nd Ed.). Cambridge University Press, Cambridge, 2001. T. Pirashvili. On the PROP corresponding to bialgebras. Cah. Topol. G´ eom. Diff´ er. Cat´ eg., 43:221–239, 2002. T. Pirashvili and B. Richter. Hochschild and cyclic homology via functor homology. K-Theory, 25 (1):39–49, 2002. D. Quillen. Higher algebraic K-theory: I. Lecture Notes in Math., 341:77–139, 1973. J. J. Rotman. An Introduction to Homological Algebra. Academic Press, New York, 1979. ´ G. Segal. Classifying spaces and spectral sequences. Publications math´ ematiques de l’I.H.E.S., 34:105–112, 1968. B. Stenstr¨ om. Rings of Quotients: An Introduction to Methods of Ring Theory. Springer-Verlag, Berlin, 1975. ˇ S. T. Vre´ cica and R. T. Zivaljevi´ c. Cycle-free chessboard complexes and symmetric homology of algebras. European Journal of Combinatorics, 30, Issue 2:542–554, 2009.

Department of Mathematics, Fordham University, Bronx, New York, 10461, USA. E-mail address: [email protected]