COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY · COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY ANNA MARIE BOHMANN, TEENA GERHARDT, AMALIE H˜GENHAVEN, BROOKE

COMPUTATIONAL TOOLS FOR TOPOLOGICAL

COHOCHSCHILD HOMOLOGY

ANNA MARIE BOHMANN, TEENA GERHARDT, AMALIE HØGENHAVEN,BROOKE SHIPLEY, AND STEPHANIE ZIEGENHAGEN

Abstract. In recent work Hess and Shipley [14] defined a theory of topo-

logical coHochschild homology (coTHH) for coalgebras. In this paper we de-

velop computational tools to study this new theory. In particular, we prove aHochschild-Kostant-Rosenberg type theorem in the cofree case for differential

graded coalgebras. We also develop a coBokstedt spectral sequence to compute

the homology of coTHH for coalgebra spectra. We use a coalgebra structureon this spectral sequence to produce several computations.

1. Introduction

The theory of Hochschild homology for algebras has a topological analogue,called topological Hochschild homology (THH). Topological Hochschild homologyfor ring spectra is defined by changing the ground ring in the ordinary Hochschildcomplex for rings from the integers to the sphere spectrum. For coalgebras, thereis a theory dual to Hochschild homology called coHochschild homology. Variationsof coHochschild homology for classical coalgebras, or corings, appear in [6, Section30], and [8], for instance, and the coHochschild complex for differential gradedcoalgebras appears in [13]. In recent work [14], Hess and Shipley define a topologicalversion of coHochschild homology (coTHH), which is dual to topological Hochschildhomology.

In this paper we develop computational tools for coTHH. We begin by giving avery general definition for coTHH in a model category endowed with a symmetricmonoidal structure in terms of the homotopy limit of a cosimplicial object. Thislevel of generality makes concrete calculations difficult; nonetheless, we give moreaccessible descriptions of coTHH in an assortment of algebraic contexts.

Classically, one of the starting points in understanding Hochschild homology isthe Hochschild–Kostant–Rosenberg theorem, which, in its most basic form, iden-tifies the Hochschild homology of free commutative algebras. The set up for coal-gebras is not as straightforward as the algebra case and we return to the generalsituation to analyze the ingredients necessary for defining an appropriate notionof cofree coalgebras. In Theorem 3.11 we conclude that the Hochschild–Kostant–Rosenberg theorem for coalgebras in an arbitrary model category boils down toan analysis of the interplay between the notion of “cofree” and homotopy limits.We then prove our first computational result, a Hochschild–Kostant–Rosenberg

Date: October 25, 2016.

2010 Mathematics Subject Classification. Primary:xxxxxxxx; Secondary: xxxx.Key words and phrases. Topological Hochschild homology, coalgebra, Hochschild–Kostant–

Rosenberg.

1

2 BOHMANN, GERHARDT, HØGENHAVEN, SHIPLEY, AND ZIEGENHAGEN

theorem for differential graded coalgebras. A similar result has been obtained byFarinati and Solotar in [10] in the ungraded setting.

Theorem 1.1. Let X be a nonnegatively graded cochain complex over a field k. LetSc(X) be the cofree coaugmented cocommutative coassociative conilpotent coalgebraover k cogenerated by X and V (Sc(X)) the underlying differential graded k-module.Then there is a quasi-isomorphism of differential graded k-modules

coTHH(Sc(X)) ' V (Sc(X))⊗k V (Sc(Σ−1X)).

Definitions of all of the terms here appear in Sections 2 and 3 and this theorem isproved as Theorem 3.15.

We then develop calculational tools to study coTHH(C) for a coalgebra spectrumC. Recall that for topological Hochschild homology, the Bokstedt spectral sequenceis an essential computational tool. For a ring spectrum R, the Bokstedt spectralsequence is of the form

E2∗,∗ = HH∗(H∗(R;Fp))⇒ H∗(THH(R);Fp).

This spectral sequence arises from the skeletal filtration of the simplicial spectrumTHH(R)•. Analogously, for a coalgebra spectrum C, we consider the Bousfield–Kan spectral sequence arising from the cosimplicial spectrum coTHH•(C). We callthis the coBokstedt spectral sequence. We identify the E2-term in this spectralsequence and see that, as in the THH case, the E2-term is given by a classicalalgebraic invariant:

Theorem 1.2. Let C be a coalgebra spectrum that is cofibrant. The Bousfield–Kanspectral sequence for coTHH(C) gives a coBokstedt spectral sequence with E2-page

Es,t2 = coHHs,t(H∗(C;Fp)),that abuts to

Ht−s(coTHH(C);Fp).

As one would expect with a Bousfield–Kan spectral sequence, the coBokstedtspectral sequence does not always converge. However, we identify conditions underwhich it converges strongly. In particular, if the coalgebra spectrum C is a sus-pension spectrum of a simply connected space X, denoted Σ∞+ X, the coBokstedtspectral sequence converges strongly to H∗(coTHH(Σ∞+ X);Fp).

Further, we prove that for C cocommutative this is a spectral sequence of coal-gebras. Using this additional algebraic structure we prove several computationalresults, including the following. The divided power coalgebra Γ[−], the exteriorcoalgebra Λ(−), and the polynomial coalgebra Fp[−] are introduced in detail inSection 5.

Theorem 1.3. Let C be a cocommutative coassociative coalgebra spectrum that iscofibrant, and whose homology coalgebra is

H∗(C;Fp) = ΓFp [x1, x2, . . . ]

where the xi are cogenerators in strictly positive even degrees and there are onlyfinitely many cogenerators in each degree. Then the coBokstedt spectral sequencefor C collapses at E2, and

E2∼= E∞ ∼= ΓFp [x1, x2, . . . ]⊗ ΛFp(z1, z2, . . . ),

with xi in degree (0,deg(xi)) and zi in degree (1,deg(xi)).

COMPUTATIONAL TOOLS FOR TOPOLOGICAL COHOCHSCHILD HOMOLOGY 3

This theorem should thus be thought of as a computational analogue of theHochschild–Kostant–Rosenberg Theorem at the level of homology.

These computational results apply to determine the homology of coTHH of manysuspension spectra, which have a coalgebra structure induced by the diagonal mapon spaces. For simply connected X, topological coHochschild homology of Σ∞+ Xcoincides with topological Hochschild homology of Σ∞+ ΩX, and thus is closely re-lated to algebraic K-theory, free loop spaces, and string topology. Our results givea new way of approaching these invariants, as we briefly recall in Section 5.

The paper is organized as follows. In Section 2, we define coTHH for a coalge-bra in any model category endowed with a symmetric monoidal structure as thehomotopy limit of the cosimplicial object given by the cyclic cobar constructioncoTHH•. We then give conditions for when this homotopy limit can be calculatedefficiently and explain a variety of examples.

In Section 3, we develop cofree coalgebra functors. These are not simply dualto free algebra functors, and we are explicit about the conditions on a symmetricmonoidal category that are necessary to make these functors well behaved. In thiscontext an identification similar to the Hochschild–Kostant–Rosenberg Theoremholds on the cosimplicial level. We also prove a dg-version of the Hochschild–Kostant–Rosenberg theorem.

In Section 4 we define and study the coBokstedt spectral sequence for a coalgebraspectrum. In particular, we analyze the convergence of the spectral sequence, andprove that it is a spectral sequence of coalgebras.

In Section 5 we use the coBokstedt spectral sequence to make computationsof the homology of coTHH(C) for certain coalgebra spectra C. We exploit thecoalgebra structure in the coBokstedt spectral sequence to prove that the spectralsequence collapses at E2 in particular situations and give several specific examples.

1.1. Acknowledgements. The authors express their gratitude to the organizers ofthe Women in Topology II Workshop and the Banff International Research Station,where much of the research presented in this article was carried out. We also thankKathryn Hess and Ben Antieau for several conversations about coTHH and HKRtheorems. The second author was supported by the National Science FoundationCAREER Grant, DMS-1149408. The third author was supported by the DNRFNiels Bohr Professorship of Lars Hesselholt. The fourth author was supported bythe National Science Foundation, DMS-140648.

2. coHochschild homology: Definitions and examples

Let (D,⊗, 1) be a symmetric monoidal category. We will suppress the associa-tivity and unitality isomorphisms that are part of this structure in our notation.The coHochschild homology of a coalgebra in D is defined as the homotopy limitof a cosimplicial object in D. In this section, we define coHochschild homology andthen discuss a variety of algebraic examples.

Definition 2.1. A coalgebra (C,4) in D consists of an object C in D togetherwith a morphism 4 : C → C ⊗ C, called comultiplication, which is coassociative,i.e. satisfies

(C ⊗4)4 = (4⊗ C)4.


Here we use C to denote the identity morphism on the object C ∈ D. A coalgebrais called counital if it admits a counit morphism ε : C → 1 such that

(ε⊗ C)4 = C = (C ⊗ ε)4.Finally, a counital coalgebra C is called coaugmented if there furthermore is acoaugmentation morphism η : 1→ C satisfying the identities

4η = η ⊗ η and εη = 1.

We denote the category of coaugmented coalgebras by CoalgD.The coalgebra C is called cocommutative if

τC,C4 = 4,with τC,D : C ⊗D → D ⊗ C the symmetry isomorphism. The category of cocom-mutative coaugmented coalgebras is denoted by comCoalgD.

We often do not distinguish between a coalgebra and its underlying object.Given a counital coalgebra, we can form the associated cosimplicial coHochschild

object:

Definition 2.2. Let (C,4, ε) be a counital coalgebra in D. We define a cosimplicialobject coTHH•(C) in D by setting

coTHHn(C) = C⊗n+1

with cofaces

δi : C⊗n+1 → C⊗n+2, δi =

C⊗i ⊗4⊗ Cn−i, 0 ≤ i ≤ n,τC,C⊗n+1(4⊗ C⊗n), i = n+ 1,

and codegeneracies

σi : C⊗n+2 → C⊗n+1, σi = C⊗i+1 ⊗ ε⊗ C⊗n−i for 0 ≤ i ≤ n.

Note that if C is cocommutative,4 is a coalgebra morphism and hence coTHH•(C)is a cosimplicial counital coalgebra in D.

Recall that if D is a model category, the category D∆ of cosimplicial objects inD can be equipped with the Reedy model structure and simplicial frames exist inD. Hence we can form the homotopy limit holim∆X

• for X ∈ D∆. More precisely,we use the model given by Hirschhorn [15, Chapters 16 and 19] and in particularapply [15, 19.8.7], so that holim∆(X•) can be computed via totalization as

holim∆X• = TotY•

where Y• is a Reedy fibrant replacement of X•.

Definition 2.3. Let D be a model category and a symmetric monoidal category.Let (C,4, 1) be a counital coalgebra in D. Then the coHochschild homology of Cis defined by

coTHH(C) = holim∆coTHH•(C).

Remark 2.4. Let D be a symmetric monoidal model category. Let f : C → D bea morphism of counital coalgebras and assume that f is a weak equivalence andC and D are cofibrant in D. Then f induces a degreewise weak equivalence ofcosimplicial objects coTHH•(C) → coTHH•(D). Hence the corresponding Reedyfibrant replacements are also weakly equivalent and f induces a weak equivalencecoTHH(C)→ coTHH(D) according to [15, 19.4.2(2)].


A central question raised by the definition of coHochschild homology is that ofunderstanding the homotopy limit over ∆. In the rest of this section, we discussconditions under which one can efficiently compute it, including an assortment ofalgebraic examples which we discuss further in Section 3.

For a cosimpicial object X• in D let Mn(X•) be its nth matching object givenby

Mn(X•) = limα∈∆([n],[a]),α surjective,

a 6=n

Xa.

The matching object Mn(X•) is traditionally also denoted by Mn+1(X•), but westick to the conventions in [15, 15.2]. If D is a model category, X• is called Reedyfibrant if for every n, the morphism

σ : Xn →Mn(X•)

is a fibration, where σ is induced by the collection of maps α∗ : Xn → Xa forα ∈ ∆([n], [a]).

If D is a simplicial model category and X• is Reedy fibrant, holim∆X• is weakly

equivalent to the totalization of X• defined via the simplicial structure; see [15,Theorem 18.7.4]. In this case the totalization is given by the equalizer

Tot(X•) = eq

(∏n≥0

(Xn)∆n

⇒∏

α∈∆([a],[b])

(Xb)∆a

),

where DK denotes the cotensor of an object D of D and a simplicial set K andwhere the two maps are induced by the cosimplicial structures of X• and ∆•. Ifthe category D is not simplicial, one has to work with simplicial frames as in [15,Chapter 19].

In the dual case, if A is a monoid in a symmetric monoidal model category,one considers the cyclic bar construction associated to A, i.e. the simplicial objectwhose definition is entirely dual to the definition of coTHH•(C). This simplicialobject is Reedy cofibrant under mild conditions; see for example [27, Example23.8]. But this doesn’t dualize to coTHH•(C): It is neither reasonable to expectthe monoidal product to commute with pullbacks, nor to expect that pullbacks andthe monoidal product satisfy an analogue of the pushout-product axiom.

Nonetheless, in certain cases coTHH•(C) can be seen to be Reedy fibrant. Inparticular Reedy fibrancy is simple when the symmetric monoidal structure is givenby the cartesian product.

Lemma 2.5. Let D be any category and let C be a coalgebra in (D,×, ∗). Then

σ : coTHHn(C)→Mn(coTHH•(C))

is an isomorphism for n ≥ 2. Hence if D is a model category, then coTHH•(C) isReedy fibrant if C itself is fibrant.

Proof. For surjective α ∈ ∆([n], [n− 1]), let

pα : Mn(coTHH•(C))→ coTHHn−1(C) = C×n


be the canonical projection to the copy of coTHHn−1(C) corresponding to α. Thenan inverse to σ is given by

Mn(coTHH•(C))(pα)α−−−−→ (C×n)×n

((C×n)×n,C×n×(∗)×n−1)−−−−−−−−−−−−−−−−−→ (C×n)n+1

(C×∗n−1)×···×(∗n−1×C)×(∗n−1×C)−−−−−−−−−−−−−−−−−−−−−−−−→ C×n+1.

Intuitively, this map sends (c1, . . . , cn) ∈Mn(coTHH•(C)) with ci = (ci1, . . . , cin) ∈

C×n to (c11, . . . , cnn, c

1n). The Reedy fibration conditions for n = 1, 0 follow from the

fibrancy of C.

In several algebraic examples, degreewise surjections are fibrations and the fol-lowing lemma will imply Reedy fibrancy.

Lemma 2.6. Let k be a field. Let C be a counital coalgebra in the category D ofgraded k-modules. Then the matching maps

σ : coTHHn(C)→Mn(coTHH•(C))

are surjective.

Proof. From Hirschhorn [15, 15.2.6] we know that the matching space Mn(X•) fora cosimplicial graded k-module X• is given by

Mn(X•) = (x0, . . . , xn−1) ∈ (Xn−1)×n | σi(xj) = σj−1(xi) for 0 ≤ i < j ≤ n− 1.This space Mn(X•) can be obtained by iterated pullbacks: For k ≤ n − 1, defineMkn(X•) by

Mkn(X•) = (x0, . . . , xk) ∈ (Xn−1)×k+1 | σi(xj) = σj−1(xi) for 0 ≤ i < j ≤ k

and let σ(k) be the map (σ0, . . . , σk) : Xn →Mkn(X•) . Then Mn−1

n (X•) = Mn(X•)and σ(n−1) = σ.

We show by induction that for X• = coTHH•(C) the maps σ(k) are all surjective.The map σ(0) = σ0 : C ⊗ C → C is clearly surjective. Assume that σ(k−1) issurjective, and consider (x0, . . . , xk) ∈Mk

n(X•). Since (x0, . . . , xk−1) ∈Mk−1n (X•),

we can choose y ∈ C⊗n such that

σ(k−1)(y) = (x0, . . . , xk−1).

We know that σi(xk) = σk−1(xi) = σiσk(y) for i < k.Choose a basis (cl)l∈L of C, such that there is exactly one l0 ∈ L with ε(cl0) = 1,

and such that ε(cl) = 0 for all l 6= l0. Write xk and y in terms of tensors of thebasis elements as

xk =∑

j0,...,jn−1∈Lλ(j0,...,jn−1)cj0 ⊗ · · · ⊗ cjn−1

andy =

∑j0,...,jn∈L

µj0,...,jncj0 ⊗ · · · ⊗ cjn

for some coefficients λj0,...,jn−1and µj0,...,jn in k. If j1, . . . , jk 6= l0, then σi(cj0 ⊗

· · · ⊗ cjn) = 0 for 0 ≤ i ≤ k − 1. Hence the coefficient µj0,...,jn does not contribute

to σ(k−1)(y) and so we can assume that µj0,...,jn = 0 if j1, . . . , jk 6= l0.Define z ∈ C⊗n+1 by

z = y +∑

j1,...,jk 6=l0

∑j0,jk+1,...,jn−1

λj0,...,jn−1cj0 ⊗ · · · ⊗ cjk ⊗ cl0 ⊗ cjk+1

⊗ · · · ⊗ cjn−1.


For i < k and any simple tensor of basis elements cj0⊗· · ·⊗cjn where j1, . . . , jk 6= l0,σi(cj0 ⊗ · · · ⊗ cjn) = 0. Hence for i < k,

σi(z) = σi(y) = xi.

We must show that σk(z) = xk.Fix an n-tuple (j0, . . . , jn−1) ∈ Ln. The terms of z that contribute to the

coefficient of cj0 ⊗ · · · ⊗ cjn−1 in σk(z) are those of the form

cj0 ⊗ · · · ⊗ cjk ⊗ cl0 ⊗ cjk+1⊗ · · · ⊗ cjn−1

.

If ji 6= l0 for all 1 ≤ i ≤ k, then for any l ∈ L, the coefficient µj0,...,jk,l,jk+1,...,jn−1is

zero, and thus the coefficient of cj0 ⊗ · · · ⊗ cjn−1 in σk(z) is λj0,...,jn−1 as required.Suppose js = l0 for some 1 ≤ s ≤ k. The coefficient of cj0 ⊗ · · · ⊗ cjn−1 in σk(z)

is then the coefficient of cj0⊗· · ·⊗cjs−1⊗cjs+1

⊗· · ·⊗cjn−1in σs−1σk(z). Similarly,

the coefficient of cj0 ⊗· · ·⊗ cjn−1in xk is the coefficient of cj0 ⊗· · ·⊗ cjs−1

⊗ cjs+1⊗

· · · ⊗ cjn−1 in σs−1(xk). Since (x0, . . . , xk) ∈Mkn(X•),

σs−1(xk) = σk−1(xs−1)

and by construction, xs−1 = σs−1(z). Thus σs−1(xk) = σk−1σs−1(z) and so by thecosimplicial identities,

σs−1(xk) = σs−1σk(z).

Thus the coefficients of cj1 ⊗ · · · ⊗ cjn in xk and σk(z) are equal.

We now collect a couple of examples of what coTHH is in various categories.

Example 2.7. Let D = (ckmod,⊗, k) be the symmetric monoidal category of cosim-plicial k-modules for a field k. There is a simplicial model structure on D; see [11,II.5] for an exposition. The simplicial cotensor of a cosimplicial k-module M• anda simplicial set K• is given by

(MK)n = Set(Kn,Mn)

with α ∈ ∆([n], [m]) mapping f : Kn →Mn to

Kmα−→ Kn

f−→Mnα∗−−→Mm,

see [12, II.2.8.4]. As in the simplicial case (see [12, III.2.11]), one can check thatthe fibrations are precisely the degreewise surjective maps. Hence by Lemma 2.6coTHH(C) can be computed as the totalization, that is, as the diagonal of thebicosimplicial k-module coTHH•(C). Even though D is not a monoidal modelcategory, the tensor product of weak equivalences is again a weak equivalence, andhence a weak equivalence of counital coalgebras induces a weak equivalence oncoTHH.

Recall that the dual Dold–Kan correspondence is an equivalence of categories

N∗ : ckmod dgmod≥0(k) : Γ•

between cosimplicial k-modules and nonnegatively graded cochain complexes overk. Given M• ∈ ckmod, the normalized cochain complex N∗(M•) is defined by

Na(M•) =

a−1⋂i=0

ker(σi : Ma →Ma−1)


for a > 0 and N0(M•) = M0, with differential d : Na(M•)→ Na+1(M•) given by

a+1∑i=0

(−1)iδi.

Example 2.8. Let k be a field. Let D be the symmetric monoidal category D =(dgmod≥0(k),⊗, k) of nonnegatively graded cochain complexes over k. The dualDold–Kan correspondence yields that D is a simplicial model category, with weakequivalences the quasi-isomorphisms and with fibrations the degreewise surjections.Hence by Lemma 2.6 we have that coTHH•(C) is Reedy fibrant for any counitalcoalgebra C in D. The totalization of a cosimplicial cochain complex M• is givenby

Tot(M•) ∼= N∗(Tot(Γ(M•))) = N∗(diag(Γ(M•))).

Recall that given a double cochain complex B∗,∗ with differentials dh : B∗,∗ →B∗+1,∗ and dv : B∗,∗ → B∗,∗+1, we can form the total cochain complex Totdgmod(B)given by

Totdgmod(B)n =∏

p+q=n

Bp,q.

The projection to the component Bs,n+1−s of the differential applied to (bp ∈Bp,n−p)p is given by dh(bs−1,n−s+1) + (−1)pdv(bs,n−s). A similar construction canbe carried out for double chain complexes.

As in the bisimplicial case, we can turn a bicosimpicial k-module X•,• into adouble complex by applying the normalized cochain functor to both cosimplicialdirections. The homotopy groups of diagX•,• are the homology of the associatedtotal complex, hence

coTHH(C) = TotdgmodN∗(coTHH•(C))

for any counital coalgbra in D. Again weak equivalences of counital coalgebrasinduce weak equivalences on coTHH.

Example 2.9. The category D = (sSet,×, ∗) is a simplicial symmetric monoidalmodel category with its classical model structure. Every object is cofibrant, andby Lemma 2.5, coTHH(C) can be computed as the totalization of the cosimplicialsimplicial set coTHH•(C) when C is fibrant.

Example 2.10. Let D = (skmod,⊗, k) be the category of simplicial k-modules overa field k. Every object is cofibrant in D. Let C be a counital coalgebra. Since everymorphism that is degreewise surjective is a fibration (see [12, III.2.10]), Lemma2.6 yields that coTHH•(C) is Reedy fibrant, and coTHH(C) is given by the usualtotalization of cosimplicial simplicial k-modules.

Example 2.11. The category D = (dgmod≥0(k),⊗, k) of nonnegatively graded chaincomplexes over a ring k with the projective model structure is both a monoidalmodel category as well as a simplicial model category via the Dold–Kan equivalence(N,Γ) with (skmod,⊗, k). If we work over a field, every object is cofibrant, and forany counital coalgebra C in D Lemma 2.6 yields that coTHH•(C) is Reedy fibrant.

However, a word of warning is in order. The definition of coHochschild homol-ogy given in Definition 2.3 does not coincide with the usual notion of coHochschildhomology of a differential graded coalgebra as for example found in [8]: Usually, co-Hochschild homology of a coalgebra C in k-modules or differential graded k-modulesis given by applying the normalized cochain complex functor N∗ to coTHH•(C) and


forming the total complex Totdgmod(N∗(coTHH•(C))) of the double chain complexresulting from interpreting Na(coTHH•(C)) as living in chain complex degree −a.

But for any cosimplicial nonnegatively graded chain complex M•,

Tot(M•) ∼= N∗(Tot(Γ(M•))),

where the second Tot is the totalization of the simplicial cosimplicial k-moduleΓ(M•). As explained in [11, III.1.1.13], we have an identity

N∗(Tot(Γ(M•))) = τ∗(Totdgmod(N∗(M•))),

where τ∗(C) of an unbounded chain complex C is given by

τ∗(C)n =

Cn, n > 0,

ker(C0 → C−1), n = 0,

0, n < 0.

Hence if C is concentrated in degree zero, so is coTHH(C), in contrast to theusual definition as found in [8].

Nonetheless, if we assume that C0 = 0, the above identification of the totalizationyields that the two notions of coHochschild homology coincide.

Example 2.12. The category D = (dgmod,⊗, k) of unbounded chain complexes overa commutative ring k is a symmetric monoidal model category with weak equiva-lences the quasiisomorphisms and fibrations the degreewise surjective morphisms;see [16, 4.2.13].

Let M be a cofibrant chain complex. A simplicial frame of C is given by thesimplicial chain complex fr(M) which is given in simplicial degree m by the internalhom object dgmod(N∗(∆

m),M) (see also [16, 5.6.10]). Again, coTHH•(C) is Reedy

fibrant. This yields that the homotopy limit holim∆coTHH•(C) coincides with thetotal complex of the double complex N∗(coTHH•(C)).

Example 2.13. Both the category of symmetric spectra as well as the category ofS-modules are simplicial symmetric monoidal model categories. Sections 4 and 5discuss coTHH for coalgebra spectra.

3. coHochschild homology of cofree coalgebras in symmetricmonoidal categories

We next turn to calculations of coHochschild homology of cofree coalgebras insymmetric monoidal categories. The first step is to specify what we mean by a“cofree” coalgebra in our symmetric monoidal category D. Intuitively, such a coal-gebra should be given by a right adjoint to the forgetful functor from the category ofcoalgebras in D to D itself. The constructions are the usual ones and can be foundfor example in [22, II.3.7] in the algebraic case, but we specify conditions underwhich we can guarantee the existence of this adjoint. For an operadic backgroundto these conditions we refer the reader to [7].

Let D be complete and cocomplete and admit a zero object 0. Given a coalgebra(C,4) in a symmetric monoidal category D and n ≥ 0, we denote by 4n : C →C⊗n+1 the iterated comultiplication defined inductively by

40 = C, 4n+1 = (4⊗ C⊗n)4n.Note that if (C,4, ε, η) is a coaugmented coalgebra, the cokernel of the coaug-

mentation, coker η, is a coalgebra.


Definition 3.1. We call a (non-coaugmented) coalgebra C conilpotent if the mor-phism

(4n)n≥0 : C →∏n≥0

C⊗n+1,

that is, the morphism given by 4n after projection to C⊗n+1, factors through⊕n≥0 C

⊗n+1 via the canonical morphism⊕n≥0

C⊗n+1 →∏n≥0

C⊗n+1.

We say that a coaugmented coalgebra C is conilpotent if coker η is a conilpotentcoalgebra.

We denote the full subcategory of CoalgD consisting of conilpotent coalgebras

by CoalgconilD . Similarly, the category of cocommutative conilpotent coaugmented

coalgebras is denoted by comCoalgconilD .

Definition 3.2. The symmetric monoidal category (D,⊗, 1) is called cofree-friendlyif it is complete, cocomplete, admits a zero object, and the following additional con-ditions hold:

(1) For all objects D in D, the functor D ⊗− preserves colimits.(2) Finite sums and finite products are naturally isomorphic, that is for every

finite set J and all objects Dj , j ∈ J , the morphism⊕j∈J

Dj →∏j∈J

Dj

induced by the identity morphism on each Dj is an isomorphism.

Proposition 3.3. If D is cofree-friendly, the functor

U : CoalgconilD → D, (C,4, ε, η) 7→ coker η

admits a right adjoint T c, which we call the cofree coaugmented conilpotent coal-gebra functor. The underlying object of T c(X) is given by

⊕n≥0X

⊗n. The comul-tiplication 4 is defined via deconcatenation, i.e. by⊕

n≥0

X⊗n⊕n≥0(X⊗n)a+b=n

−−−−−−−−−−−−→⊕n≥0

∏a,b≥0a+b=n

X⊗n∼=−→⊕n≥0

⊕a,b≥0a+b=n

X⊗n

∼=⊕a,b≥0

X⊗a ⊗X⊗b∼=−→⊕a≥0

X⊗a ⊗⊕b≥0

X⊗b,

where the isomorphism on the first line is provided by condition (2) of Definition 3.2.The counit morphism π : T c(X) → X is the identity on X and the zero morphismon the other summands.

Example 3.4. The categories of chain or cochain complexes discussed in Examples2.8, 2.11 and 2.12 are cofree-friendly, and the cofree coalgebra T c(X) generated byan object in any of these categories is the usual tensor coalgebra.

Similarly, the categories of simplicial or cosimplicial k-modules discussed in Ex-amples 2.7 and 2.10 are cofree-friendly, and the cofree coalgebra generated by X isobtained by applying the tensor coalgebra functor in each simplicial or cosimplicialdegree.


Example 3.5. The categories of simplicial sets, of pointed simplicial sets and ofspectra are not cofree-friendly, since finite coproducts and finite products are notisomorphic in any of these categories. A more abstract way of thinking about thisis given by observing that in these categories coaugmented coalgebras can not bedescribed as coalgebras over a cooperad in the usual way: While the category ofsymmetric sequences in these categories is monoidal with respect to the plethysmthat gives rise to the notion of operads, the plethysm governing cooperad structuresdoes not define a monoidal product. This is discussed in detail by Ching [7]; see inparticular Remark 2.10, Remark 2.20 and Remark 2.21.

Let G be a group with identity element e. Recall that a G-action on an objectX in D consists of morphisms

φg : X → X, g ∈ G,

such that φgφh = φgh and φe = X. The fixed points XG of this actions are givenby the equalizer

XG = eq( X(X)g∈G//

(φg)g∈G

//∏g∈GX ).

Definition 3.6. Recall that for any object X of D the symmetric group Σn actson X⊗n by permuting the factors. We call D permutation-friendly if the morphism

(X⊗a)Σa ⊗ (X⊗b)Σb → (X⊗a+b)Σa×Σb ,

induced by the morphisms (X⊗a)Σa → X⊗a and (X⊗b)Σb → X⊗b, is an isomor-phism for all X in D and all a, b ≥ 0.

Proposition 3.7. If D is cofree-friendly and permutation-friendly, the forgetfulfunctor

U : comCoalgconilD → D, (C,4, ε, η) 7→ coker η

admits a right adjoint Sc, which is called the cofree cocommutative conilpotentcoaugmented coalgebra functor. For a given object X of D, define Sc(X) =⊕

n≥0(X⊗n)Σn . The comultiplication 4 on Sc(X) is defined by⊕n≥0

(X⊗n)Σn

⊕n≥0((X⊗n)Σn )a+b=n

−−−−−−−−−−−−−−−→⊕n≥0

∏a,b≥0,a+b=n

(X⊗n)Σn∼=−→⊕n≥

⊕a,b≥0,a+b=n

(X⊗n)Σn

∼=⊕a,b≥0

(X⊗a ⊗X⊗b)Σa+b

⊕a,b≥0

resa,b

−−−−−−−→⊕a,b≥0

(X⊗a ⊗X⊗b)Σa×Σb

∼=−→⊕a,b≥0

(X⊗a)Σa ⊗ (X⊗b)Σb∼=−→⊕a≥0

(X⊗a)Σa ⊗⊕b≥0

(X⊗b)Σb .

The isomorphism on the first line again exists because of condition (2) of Definition3.2 and the isomorphism at the beginning of the last line is derived from Definition3.6. The map resa,b is the map

resa,b : (X⊗a ⊗X⊗b)Σa+b → (X⊗a ⊗X⊗b)Σa×Σb

induced by the inclusion Σa×Σb → Σa+b. The counit morphism π : Sc(X)→ X isgiven by the identity on X and by the zero morphism on the other summands.


Example 3.8. If k is a field, all the categories discussed in Example 3.4 are permu-tation friendly in addition to being cofree friendly. This means that the categoriesof nonnegatively graded (co)chain complexes and unbounded chain complexes ofk-modules and the categories of simplicial and cosimplicial k-modules all admitwell-defined cofree cocommutative coalgebra functors.

The permutation friendliness of each of these categories can be deduced from thepermutation friendliness of the category of graded k-modules. A proof for gradedk-modules concentrated in even degrees or for k a field of characteristic 2 can befound in Bourbaki [3, p. IV.49]. If the characteristic of k is different from 2 and Mis concentrated in odd degrees, (M⊗n)Σn and (M⊗a⊗M⊗b)Σa×Σb can be describedexplicitly in terms of a basis of M . For an arbitrary graded module M , the claimfollows from writing M as the direct sum of its even and its odd degree part.

More concretely, if M admits a countable basis mi, i ≥ 1, and the mi are evendegree elements, or if the characteristic of k is 2, we identify Sc(M) with the Hopfalgebra Γk[m1,m2, . . . ]. As an algebra, Γk[m1,m2, . . . ] is the divided power algebragenerated by M . The isomorphism to Sc(M) is given by identifying the element

γj1(mi1) · · · γjn(mjn) ∈ Γk[x1, x2, . . . ] with the element∑σ∈sh(j1,...,jn) σ.(m

⊗j1i1⊗

· · · ⊗ mjnin

) ∈ Sc(M), where σ ranges over the set of (j1, . . . , jn)-shuffles. Thecoproduct is given on multiplicative generators by

4(γj(mi)) =∑a+b=j

γa(mi)⊗ γb(mi).

If M is concentrated in odd degrees and the characteristic of k is different from2, we can identify Sc(M) with the Hopf algebra Λk(x1, x2, . . . ). This is the exterioralgebra generated by M , and we identify xi1 · · ·xin with

∑σ∈Σn

sgn(σ)σ · (x1 ⊗· · · ⊗ xn). The coproduct is given by

4(xi) = 1⊗ xi + xi ⊗ 1.

Now that we have established what we mean by cofree coalgebras in D, we turnto an analysis of coHochschild homology for these coalgebras. These computationsshould be thought of as a simple case of a dual Hochschild–Kostant–Rosenbergtheorem for coalgebras.

Let (C,4C , εC , ηC) and (D,4D, εD, ηD) be cocommutative coaugmented coal-gebras. Recall that their product in comCoalgD exists and is given by C ⊗D withcomultiplication (C ⊗ τC,D ⊗D)(4C ⊗4D), counit εC ⊗ εD and coaugmentationηC ⊗ ηD. Here τC,D : C ⊗ D → D ⊗ C is the symmetry isomorphism switching Cand D. The projections C ⊗ D → C and C ⊗ D → D are given by C ⊗ εD andεD ⊗ C.

Since Sc is a right adjoint, we have the following property:

Lemma 3.9. If D is cofree-friendly, permutation-friendly and cocomplete, there isa natural isomorphism

Sc(X × Y )→ Sc(X)⊗ Sc(Y )

in comCoalgD. It is given by the morphisms Sc(X×Y )→ Sc(X) and Sc(X×Y )→Sc(Y ) which are induced by the projections X × Y → X and X × Y → Y .

Every category C that admits finite products gives rise to the symmetric monoidalcategory (C,×, ∗), where ∗ is the terminal object. Every object X in C is then acounital cocommutative coalgebra with respect to this monoidal structure: the


comultiplication A → A × A is the diagonal, the counit A → ∗ is the map to theterminal object. We indicate the monoidal structure we use to form coTHH with asubscript, so that if C is a coalgebra with respect to ⊗, we write coTHH•⊗(C).

Proposition 3.10. Let D be cofree-friendly and permutation-friendly. Then forany object X in D

coTHH•⊗(Sc(X)) ∼= Sc(coTHH•×(X)).

Proof. Applying Lemma 3.9 yields an isomorphism in each cosimplicial degree n:

coTHHn⊗(Sc(X)) = Sc(X)⊗n+1 ∼= Sc(X×n+1) = Sc(coTHHn

×(X)).

To check that the cosimplicial structures agree, it suffices to show that the comul-tiplications and counits on both sides agree. The diagonal X → X ×X induces amorphism of coalgebras

Sc(X)→ Sc(X ×X).

Since Sc(X) is cocommutative, the comultiplication

4Sc(X) : Sc(X)→ Sc(X)⊗ Sc(X) ∼= Sc(X ×X)

is a morphism of coalgebras as well. These maps agree after projection to X ×X.A similar argument yields that X → 0 induces the counit of Sc(X).

The following theorem is a consequence of this proposition. This is the bestresult we obtain about the coHochschild homology of cofree coalgebras withoutmaking further assumptions on our symmetric monoidal category D.

Theorem 3.11. If D is a cofree- and permutation-friendly model category, Propo-sition 3.10 implies that

coTHH⊗(Sc(X)) ∼ holim∆(Sc(coTHH•×(X))).

IfD is a simplicial model category, there is an easy description of Tot(coTHH•×(X))as a free loop object.

Proposition 3.12 (see e.g. [19]). Let D be a simplicial model category and X afibrant object of D. Then

holim∆(coTHH•×(X)) ∼ XS1

.

This result identifies coTHH×(X) as a sort of “free loop space” on X. This isprecisely the case in the category of spaces, as in Example 4.4.

Example 3.13. For the category of cosimplicial k-modules we can choose the diag-onal as a model for the homotopy limit over ∆. Since the cofree coalgebra cogen-erated by a cosimplicial k-module is given by applying Sc degreewise, we actuallyobtain an isomorphism of coalgebras

coTHH⊗(Sc(X)) ∼= Sc(XS1

).

In the case where Sc is a right Quillen functor, we obtain the following corollaryto Theorem 3.11 and Proposition 3.12.

Corollary 3.14. Let D be a simplicial and a monoidal model category. Let X inD be fibrant. If there is a model structure on comCoalgD such that Sc is a rightQuillen functor, then

coTHH⊗(Sc(X)) ∼= RSc(XS1

).


In general, however, Sc is not a right Quillen functor. Up to a shift in degreeswhich is dual to the degree shift in the free commutative algebra functor, themost prominent counterexample is the category of chain complexes over a ring k ofpositive characteristic: The chain complex D2 consists of one copy of k in degrees1 and 2, with differential the identity. This complex is fibrant and acyclic, butSc(D2) is not acyclic. This same counterexample shows Sc is not right Quilleneither for unbounded chain complexes or nonnegatively graded cochain complexes.

Nevertheless, a Hochschild–Kostant–Rosenberg type theorem for cofree coalge-bras still holds for coHochschild homology in the category of nonnegatively gradedcochain complexes dgmod≥0. We denote the underlying differential graded k-module of Sc(X) by V (Sc(X)). The desuspension Σ−1X of a cochain complexX is the cochain complex with (Σ−1X)n = Xn−1.

Theorem 3.15. Let k be a field. Then for X in dgmod≥0 there is a quasi-isomorphism

coTHH⊗(Sc(X))→ V (Sc(X × Σ−1X)) ∼= V (Sc(X))⊗ V (Sc(Σ−1X)).

Note that this is an identification of differential graded k-modules, not of coalge-bras. We’ll determine the corresponding coalgebra structure on coTHH⊗(Sc(X))in certain cases in Proposition 5.1.

This result corresponds to the result of the Hochschild–Kostant–Rosenberg the-orem applied to a free symmetric algebra S(X) generated by a chain complex X, inwhich case the Kahler differentials Ω∗S(X)|k are isomorphic to V (S(X))⊗V (S(ΣX)).

See, for example, [18, Theorems 3.2.2 and 5.4.6].The proof is dual to the proof of the corresponding results for Hochschild ho-

mology. We follow the line of proof given by Loday [18, Theorem 3.2.2]. We beginby proving a couple of lemmas. First we identify coTHH×(X):

Lemma 3.16. For a cochain complex X over a field k,

Totdgmod(N∗(coTHH•×(X))) ∼= X × Σ−1X.

Proof. This follows easily from the fact that

N0(coTHH•×(X)) = X, N1(coTHH•×(X)) = 0×X

and Na(coTHH•×(X)) = 0 for a ≥ 2.

We next prove the special case of Theorem 3.15 where the module of cogeneratorsis one dimensional.

Lemma 3.17. Let k be a field. Let X in dgmod≥0 be concentrated in a singlenonnegative degree and be one dimensional in this degree. Then there is a quasi-isomorphism

coTHH⊗(Sc(X))→ V (Sc(X))⊗ V (Sc(Σ−1X)).

Proof. First let the characteristic of k be 2 or X be concentrated in even degree,so that Sc(X) ∼= Γk[x] for a generator x. Up to the internal degree induced byX, we can use the results of Doi [8, 3.1] to compute coTHH⊗(Sc(X)) using aSc(X)⊗ Sc(X)-cofree resolution of Sc(X). Such a resolution is given by

Sc(X)4 // Sc(X)⊗ Sc(X)

f // Sc(X)⊗X ⊗ Sc(X) // 0,


with f(xi⊗xj) = xi−1⊗x⊗xj−xi⊗x⊗xj−1, where x−1 = 0. Hence the homologyof coTHH⊗(Sc(X)) is concentrated in cosimplicial degrees 0 and 1. An explicitcalculation gives that all elements in N0(coTHH•⊗(Sc(X))) ∼= Sc(X) are cycles,

while generating cycles in N1(coTHH•⊗(Sc(X))) are given by∑ni=1 i · xn−i ⊗ xi

for n ≥ 1. We identify xn in cosimplicial degree zero with xn ⊗ 1 ∈ V (Sc(X)) ⊗V (Sc(Σ−1X)) ∼= Γk[x] ⊗ Λk(σ−1x) and

∑ni=1 i · xn−i ⊗ xi with xn−1 ⊗ σ−1x ∈

V (Sc(X))⊗ V (Sc(Σ−1X)).Now assume that X is concentrated in odd degree and that the characteristic of

k is not 2, so that Sc(X) = Λk(x). Hence a typical element in coTHHa(Sc(X)) isof the form y = xt0⊗xt1⊗· · ·⊗xta with ti ∈ 0, 1. Now y ∈ Na(coTHH•(Sc(X)))if and only if t1 = · · · = ta = 1, and all differentials in N∗(coTHH•(Sc(X)))are trivial. Identifying 1 ⊗ x⊗a with 1 ⊗ (σ−1x)a ∈ V (Sc(X)) ⊗ V (Sc(Σ−1X)) ∼=Λk(x) ⊗ Γk(σ−1x) and x ⊗ x⊗a with x ⊗ (σ−1x)⊗a ∈ V (Sc(X)) ⊗ V (Sc(Σ−1X))yields the result.

The general case follows from the interplay of products and the cofree functorand the following result.

Lemma 3.18 (Cf. [24, p. 719]). Let k be a field and let X be a graded k-module.Then

Sc(X) ∼= colimV⊂X fin dim

Sc(V ),

where the canonical projection πX : Sc(X) ∼= colimV⊂Xfin dim Sc(V ) → X is given

by the colimit of the maps Sc(V )πV−−→ V → X.

Proof of Theorem 3.15. Recall from Example 2.12 that coTHH(C) of a counital

coalgebra in dgmod≥0 can be computed as

coTHH⊗(C) = Totdgmod(N∗coTHH•⊗(C)).

Assume first that X has trivial differential. If X is finite dimensional,

coTHH•⊗(Sc(X)) ∼= coTHH•⊗(Sc(X1))⊗ ...⊗ coTHH•⊗(Sc(Xn))

for one dimensional k-vector spaces Xi. Hence Lemma 3.17 and the dual Eilenberg–Zilber map yield the desired quasi-isomorphism. Lemma 3.18 proves Theorem 3.15for infinite dimensional cochain complexes X with trivial differential.

If X is any nonnegatively graded cochain complex, applying the result for gradedk-modules which we just proved shows that there is a morphism of double complexes

N∗(coTHH•⊗(Sc(X)))→ V (Sc(N∗(coTHH•×(X))))

which is a quasi-isomorphism on each row, that is, if we fix the degree induced bythe grading on X. Hence this induces a quasi-isomorphism on total complexes.

Remark 3.19. If X is concentrated in even degrees or if the characteristic of kis two, the quasi-isomorphism of Theorem 3.15 is indeed a quasi-isomorphism ofcoalgebras. However, this doesn’t hold in general. For certain cases we determinethe coalgebra structure on coTHH⊗(Sc(X)) in Proposition 5.1.

Remark 3.20. Analogous to Proposition [18, 5.4.6], the proof of Theorem 3.15 showsthat the result actually holds for any differential graded cocommutative couaug-mented coalgebra C whenever the underlying graded cocommutative coaugmentedcoalgebra is cofree.


Remark 3.21. A similar result holds for unbounded chain complexes if we defineTotdgmod and hence coTHH as a direct sum instead of a product in each degree.For nonnegatively graded cochain complexes both definitions of Totdgmod agree.However, if we use the homotopically correct definition of coTHH for unboundedchain complexes via the product total complex, Lemma 3.17 doesn’t hold for Xthat are concentrated in degree −1.

4. A coBokstedt Spectral Sequence

While in algebraic cases we are able to understand certain good examples ofcoTHH by an analysis of the definition, in topological examples we require furthertools. In this section, we construct a spectral sequence, which we call the coBokstedtspectral sequence. Recall that for a ring spectrum R, the skeletal filtration on thesimplicial spectrum THH(R)• yields a spectral sequence

E2∗,∗ = HH∗(H∗(R;Fp))⇒ H∗(THH(R);Fp),

called the Bokstedt spectral sequence [2]. Analogously, for a coalgebra spectrumC, we consider the Bousfield-Kan spectral sequence arising from the cosimplicialspectrum coTHH•(C). We show in Theorem 4.6 that this is a spectral sequence ofcoalgebras. Since our spectral sequence is an instance of the Bousfield–Kan spectralsequence, its convergence is not immediate, but in the case where C is a suspensionspectrum Σ∞+ X with X a simply connected space we show in Corollary 4.5 belowthat this spectral sequence converges to H∗(coTHH(Σ∞+ X);Fp).

Let (Spec,∧,S) denote a symmetric monoidal category of spectra, such as thosegiven by [9], [17], or [21]. Let C be a coalgebra in this category with comultiplication4 : C → C ∧S C and counit ε : C → S. Assume that C is cofibrant as a spectrumso that coTHH(C) has the correct homotopy type, as in Remark 2.4. Note thatthe spectral homology of C with coefficients in Fp is a graded Fp-coalgebra withstructure maps

4 : H∗(C;Fp)H∗(4)−−−−→ H∗(C ∧ C;Fp) ∼= H∗(C;Fp)⊗Fp H∗(C;Fp),

ε : H∗(C;Fp)H∗(ε)−−−−→ Fp.

Theorem 4.1. Let C be a coalgebra spectrum that is cofibrant. The Bousfield–Kan spectral sequence for the cosimplicial spectrum coTHH•(C) gives a coBokstedtspectral sequence for calculating Ht−s(coTHH(C);Fp) with E2-page

Es,t2 = coHHFps,t(H∗(C;Fp)).

Proof. The spectral sequence arises as the Bousfield–Kan spectral sequence of acosimplicial spectrum. We briefly recall the general construction. Let X• be aReedy fibrant cosimplicial spectrum and recall that

Tot(X•) = eq( ∏n≥0

(Xn)∆n

⇒∏

α∈∆([a],[b])

(Xb)∆a),

where ∆ is the cosimplicial space whose m-th level is the standard m-simplex ∆m.Let skn∆ ⊂ ∆ denote the cosimplicial subspace whose m-th level is skn∆m, then-skeleton of the m-simplex and set

Totn(X•) := eq( ∏m≥0

(Xm)skn∆m

⇒∏

α∈∆([a],[b])

(Xb)skn∆a).


The inclusions skn∆ → skn+1∆ induce the maps in a tower of fibrations

· · · → Totn(X•)pn−→ Totn−1(X•) −→ · · · → Tot0(X•) ∼= X0.

Let Fnin−→ Totn(X•)

pn−→ Totn−1(X•) denote the inclusion of the fiber and considerthe associated exact couple:

π∗(Tot∗(X•)) π∗(Tot∗(X

•))

π∗(F∗).

p∗

∂i∗

This exact couple gives rise to a cohomologically graded spectral sequence Er, drwith Es,t1 = πt−s(Fs) and differentials dr : Es,tr → Es+r,t+r−1

r . It is a half planespectral sequence with entering differentials.

The fiber Fs can be identified with Ωs(NsX•), where

NsX• = eq(Xs

(σ0,...,σs−1)//∗//∏s−1i=0 X

s−1).

We have isomorphisms

Es,t1 = πt−s(Ωs(NsX•)) ∼= πt(N

sX•) ∼= Nsπt(X•)

and under these isomorphisms the differential d1 : Nsπt(X•) → Ns+1πt(X

•) isidentified with

∑(−1)iπt(δ

i). Since the cohomology of the normalized complexagrees with the cohomology of the cosimplicial object, we conclude that the E2

term is

Es,t2∼= Hs(πt(X

•),∑

(−1)iπt(δi)).

Let C be a coalgebra spectrum and consider the spectral sequence arising fromthe Reedy fibrant replacement R(coTHH•(C) ∧HFp). We have isomorphisms

π∗R(coTHHn(C) ∧HFp) ∼= π∗(coTHHn(C) ∧HFp) ∼= H∗(C;Fp)⊗Fp (n+1)

The map π∗(δi) corresponds to the i’th coHochschild differential under this identi-

fication, therefore

Es,t2 = coHHFps,t(H∗(C;Fp)).

From [5, IX 5.7], we have the following statement about the convergence of thecoBokstedt spectral sequence. In [5, IX 5], the authors restrict to i ≥ 1 so that theyare working with groups. Since we have abelian groups for all i, that restriction isnot necessary here.

Proposition 4.2. If for each s there is an r such that Es,s+ir = Es,s+i∞ , then thecoBokstedt spectral sequence for coTHH(C) converges completely to

π∗TotR(coTHH•(C) ∧HFp).

There is a natural map

P : Tot(RcoTHH•(C)) ∧HFp → TotR(coTHH•(C) ∧HFp).

Since π∗(Tot(RcoTHH•(C)) ∧HFp) ∼= H∗(coTHH(C);Fp), we have the following.


Corollary 4.3. If the conditions on Es,s+ir in Proposition 4.2 hold and the map Pdescribed above induces an isomorphism in homotopy, then the coBokstedt spectralsequence for coTHH(C) converges completely to H∗(coTHH(C);Fp).

Example 4.4. Let X be a simply connected space, let d : X → X ×X denote thediagonal map and let c : X → ∗ be the map collapsing X to a point. We form acosimplicial space X• with Xn = X×(n+1) and with cofaces

δi : X×(n+1) → X×(n+2), δi =

X×i × d×X×(n−i), 0 ≤ i ≤ n,τX,X×(n+1)(d×X×n), i = n+ 1,

and codegeneracies

σi : X×(n+2) → X×(n+1), σi = X×(i+1) × c×X×(n−i) for 0 ≤ i ≤ n.

The cosimplicial space X• is the cosimplicial space coTHH•(X) of Definition 2.2for the coalgebra (X, d, c) in the category of spaces, see Example 2.9. As alreadynoted earlier in Proposition 3.12, the cosimplicial space X• totalizes to the freeloop space LX, see Example 4.2 of [4].

We can consider the spectrum Σ∞+ X as a coalgebra with comultiplication arisingfrom the diagonal map d : X → X × X and counit arising from c : X → ∗. Wehave an identification of cosimplical spectra

coTHH•(Σ∞+ X) = Σ∞+ (X•).

The topological coHochschild homology is the totalization of a Reedy fibrant re-placement of the above cosimplicial spectrum. We have a natural map

Σ∞+ Tot(X•)→ Tot(Σ∞+ X•),

as Malkiewich describes in Section 4 of [20], which after composing with the mapto the totalization of a Reedy fibrant replacement becomes a stable equivalence;see Proposition 4.6 of [20]. Hence we have a stable equivalence

Σ∞+ LX∼=−→ coTHH(Σ∞+ X).

Corollary 4.5. Let X be a simply connected space. The coBokstedt spectral se-quence arising from the coalgebra Σ∞+ X converges strongly to

H∗(coTHH(Σ∞+ X);Fp) ∼= H∗(LX;Fp).

Proof. For X simply connected, the spectral sequence arising from the cosimplicialspace

Y • = hom∗((S1+)•, X) := hom(S1

• , X),

as described by Bousfield in [4, 4.2], converges strongly to H∗(LX;Fp). We con-jecture that this is the same spectral sequence as the coBokstedt spectral sequenceof Theorem 4.1, which would immediately imply strong convergence of the latterspectral sequence.

To prove strong convergence without identifying these spectral sequences, notethat by Bousfield’s results [4, Lemma 2.3, Theorem 3.2], strong convergence forthe spectral sequence for Y • implies that the tower H∗(Totn(Y •);Fp) is pro-constant. In other words, this tower is pro-isomorphic to the constant towerH∗(Tot(Y •);Fp).

Note here that if X is a fibrant simplicial set, then Y • is a Reedy fibrant cosim-plicial space, see Example 2.9. For that reason we have not mentioned fibrantreplacements here; instead, implicitly we assume that X is fibrant.


The tower that determines the convergence of the coBokstedt spectral sequence isπ∗TotnR(coTHH•(Σ∞+ X)∧HFp). Since Totn is a finite homotopy limit functor,and we are working stably, it is also a finite homotopy colimit functor and hencecommutes with applying Σ∞+ (−) ∧ HFp. Using the fact that coTHH•(Σ∞+ X) ∼=Σ∞+ Y

•, this tower is isomorphic to π∗(Σ∞+ Totn(Y •) ∧ HFp) which in turn isisomorphic to H∗(TotnY

•;Fp). Since we have observed in the previous paragraphthat this tower is pro-constant, that is, pro-isomorphic to the constant tower ofH∗(TotY •;Fp) ∼= H∗(Σ

∞+ LX;Fp) ∼= H∗(coTHH(Σ∞+ X);Fp), it follows that the

coBokstedt spectral sequence converges.Note that we could also use Corollary 4.3 to prove strong convergence here. In the

previous paragraphs we show that the tower π∗ TotnR(coTHH•(Σ∞+ X) ∧HFp)is pro-constant. It follows that it is pro-isomorphic to π∗TotR(coTHH•(Σ∞+ X) ∧HFp). Hence π∗TotR(coTHH•(Σ∞+ X)∧HFp) is isomorphic toH∗(coTHH(Σ∞+ X);Fp),and Corollary 4.3 applies.

Our next result is to prove that the coBokstedt spectral sequence of Theorem 4.1is a spectral sequence of coalgebras. We exploit this additional structure in Section5 to make calculations of topological coHochschild homology in nice cases.

We first describe a functor Hom(−, C) : Setop → comCoalg, where C is a co-commutative coalgebra. In particular this will give us a cosimplicial spectrumHom(S1

• , C) that agrees with the cosimplicial spectrum coTHH•(C) defined in Def-inition 2.2.

Let X be a set and C be a cocommutative coalgebra spectrum. Let Hom(X,C)be the indexed smash product ∧

x∈XC.

If f : X → Y is a map of sets, we define a map of spectra Hom(Y,C)→ Hom(X,C)as the product over Y of the component maps

C →∧

x∈f−1(y)

C

given by iterated comultiplication over f−1(y). Here by convention the smashproduct indexed on the empty set is S, and the map to C → S is the counit map.Since C is cocommutative, this map doesn’t depend on a choosen ordering forapplying the comultiplications and for each X, the comultiplication on C extendsto define a cocommutative comultiplication on Hom(X,C). Hence Hom(−, C) is afunctor Setop → comCoalg. If X∗ is a simplicial set, the composite

(∆op)op Xop∗−−−→ Setop Hom(−,C)−−−−−−−→ comCoalg

thus defines a cosimplicial cocommutative coalgebra spectrum.To obtain the cosimplicial coalgebra yielding coTHH(C), we take X∗ to be the

simplicial circle ∆1•/∂∆1

•. Here ∆1• is the simplicial set with ∆1

n = x0, . . . , xn+1where xt is the function [n] → [1] so that the preimage of 1 has order t. The faceand degeneracy maps are given by:

δi(xt) =

xt t ≤ ixt−1 t > i


σi(xt) =

xt t ≤ ixt+1 t > i

The n-simplices of S1∗ are obtained by identifying x0 and xn+1 so that S1

n =x0, . . . , xn. Comparing the coface and codegeneracy maps in Hom(S1

• , C) andcoTHH•(C) shows that the two cosimplicial spectra are the same.

Theorem 4.6. Let C be a cocommutative coalgebra that is cofibrant. Then theBousfield–Kan spectral sequence described in Theorem 4.1 is a spectral sequence ofFp-coalgebras. In particular, for each r > 1 there is a coproduct

ψ : E∗∗r → E∗∗r ⊗Fp E∗∗r ,

and the differentials dr respect the coproduct.

Proof. Below we will construct the coproduct using natural maps of spectral se-quences

E∗∗r∇−−→ D∗∗r

AW←−−−− E∗∗r ⊗Fp E∗∗r .

Recall that the cosimplicial spectrum coTHH•(C) can be identified with the cosim-plicial spectrum Hom(S1

• , C). Let D∗∗r denote the Bousfield–Kan spectral sequencefor the cosimplicial spectrum Hom((S1 q S1)•, C) ∧ HFp. The codiagonal map∇ : S1 q S1 → S1 is a simplicial map and gives a map Hom(S1

• , C) → Hom((S1 qS1)•, C). The map ∇ : E∗∗r → D∗∗r is induced by this codiagonal map. Let X•

denote the cosimplicial HFp-coalgebra spectrum coTHH(C)• ∧ HFp. Note thatthe cosimplicial spectrum Hom((S1 q S1)•, C) ∧HFp is the cosimplicial spectrum[n] → Xn ∧HFp X

n. This is the diagonal cosimplicial spectrum associated to abicosimplicial spectrum, (p, q)→ Xp ∧HFp X

q. We will denote this diagonal cosim-plicial spectrum by diag(X• ∧HFp X

•). The spectral sequence D∗∗r is given by theTot tower for the Reedy fibrant replacement R(diag(X• ∧HFp X

•)).For the cosimplicial spectrum X•, we will use X•X• to denote the “box prod-

uct” of X• with itself. This is a cosimplicial spectrum given by taking the Artin–Mazur codiagonal of the bicosimplicial spectrum (p, q) → Xp ∧HFp X

q. It is dis-cussed in [1, Section 5], [25, Example 9.2.10] and [23, Section 2]. In [23], McClureand Smith show that the box product can either be described as the left Kan ex-tension along the ordinal sum functor ord: ∆ ×∆ → ∆ or described explicitly asthe cosimplicial object whose nth level is the coequalizer

coeq

( ∨p+q=n−1

Xp ∧Xq ⇒∨

p+q=n

Xp ∧Xq

)

where the maps in the coequalizer are δp+1 ∧ id and id ∧ δ0. The cofaces andcodegeneracies are defined in terms of those for X•; the specific formulas are givenin [25] and [23].

There is a canonical map of cosimplicial spectra

X•X•ψ−→ diag(X• ∧HFp X

•).

arising from the universal property of X•X• as a left Kan extension. From thisperspective, ψ is induced by the composite of the natural transformation

id∆×∆ ⇒ diag ord


with the identity on the bisimplicial spectrum X• ∧X•. In terms of the concreteformula for the nth level, the map ψ is induced by the maps Xp∧Xq → Xp+q∧Xp+q

(δp+q+1 · · · δp+1) ∧ (δ0 · · · δ0)

where the right hand smash factor has p composites of δ0. The cosimplicial identitiesimply that this is a map of cosimplicial spectra.

Let T ∗∗r denote the spectral sequence given by the Tot tower for the Reedyfibrant replacement R(X•X•). The map ψ induces a map on the Reedy fibrantreplacements, and hence a map of spectral sequences T ∗∗r → D∗∗r . The spectralsequence T ∗∗r can be identified as E∗∗r ⊗ E∗∗r , so this is a map

E∗∗r ⊗ E∗∗rAW−−→ D∗∗r .

On the r = 1 page the map AW is the Alexander–Whitney map

E∗∗1 ⊗Fp E∗∗1 = N∗(π∗X

•)⊗Fp N∗(π∗X

•)→ N∗(π∗X• ⊗Fp π∗X

•) = D∗∗1 .

By the cosimplicial analog of the Eilenberg–Zilber theorem (see, for example, [11])this induces an isomorphism on cohomology

H∗(E∗,∗1 )⊗Fp H∗(E∗,∗1 ) ∼= H∗(E∗,∗1 ⊗Fp E

∗,∗1 )

∼= // H∗(D∗,∗1 ) ∼= D∗∗2 ,

where the first isomorphism is given by the Kunneth theorem. This composite

is the map AW : E∗,∗2 ⊗Fp E∗,∗2

∼=−→ D∗∗2 . By induction and repeated application

of the Kunneth theorem, we get equivalences AW : E∗,∗r ⊗Fp E∗,∗r

∼=−→ D∗∗r for allr > 1. The composite of ∇ with the inverse of this isomorphism gives us the desiredcoproduct.

5. Computational results

Now that we have developed the coBokstedt spectral sequence and its coalge-bra structure, we use these structures to make computations of the homology ofcoTHH(X) for certain coalgebra spectra. We also prove that this spectral sequencecollapses in certain cases.

We first make some elementary computations of coHochschild homology, whichwe will later use as input to the coHH to coTHH spectral sequence described inTheorem 4.1. The coalgebras we consider are the underlying coalgebras of Hopfalgebras, specifically, of exterior Hopf algebras, polynomial Hopf algebras, and di-vided power Hopf algebras. As mentioned in Example 3.8, the exterior Hopf alge-bra and the divided power Hopf algebras give the free cocommutative coalgebrason graded Fp-modules concentrated in odd degree or even degree, respectively. Re-call from Example 3.8 that ΛFp(y1, y2, . . . ) denotes the exterior Hopf algebra ongenerators yi in odd degrees and that ΓFp [x1, x2, . . . ] denotes the divided powerHopf algebra on generators xi in even degrees. Let Fp[w1, w2, . . . ] denote the poly-nomial Hopf algebra on generators wi in even degree. The coproduct is given by

4(wji ) =∑k

(jk

)wki ⊗ w

j−ki .

In the following computations, we assume that all the Hopf algebras in questiononly have finitely many generators in any given degree. The following computa-tion of coHochschild homology is the main result we use as input for our spectralsequence.


Proposition 5.1. Let the cogenerators xi be of even nonnegative degree and letthe cogenerators yi be of odd nonnegative degree. We evaluate the coHochschildhomology as a coalgebra for the following cases of free cocommutative coalgebras:

coHH∗(ΓFp [x1, x2, . . . ]) = ΓFp [x1, x2, . . . ]⊗ ΛFp(z1, z2, . . . ),

where deg(zi) = (1,deg(xi)).

coHH∗(ΛFp(y1, y2, . . . )) = ΛFp(y1, y2, . . . )⊗ Fp[w1, w2, . . . ]

where deg(wi) = (1,deg(yi)).

Before proving the proposition, we recall some basic homological coalgebra. Con-sider any coalgebra C over a field k. Let M be a right C-comodule and N be aleft C-comodule with corresponding maps ρM : M →M ⊗C and ρN : N → C⊗N .Recall that the cotensor product of M and N over C, MCN , is defined as thekernel of the map

ρM ⊗ 1− 1⊗ ρN : M ⊗N →M ⊗ C ⊗N.

The Cotor functors are the derived functors of the cotensor product. In otherwords, CotornC(M,N) = Hn(MCX) where X is an injective resolution of N as aleft C-comodule.

Let Ce = C ⊗ Cop. Let N be a (C,C)-bicomodule. Then N can be regarded asa right Ce-comodule. As shown in Doi [8], coHHn(N,C) = CotornCe(N,C).

Lemma 5.2. For C = ΓFp [x1, x2, . . .] or C = ΛFp(y1, y2, . . .),

coHH(C) = C ⊗ CotorC(Fp,Fp).

Proof. As above, coHHn(C) = CotornCe(C,C). Observe that C is a Hopf algebra.Let S denote the antipode in C. As in Doi [8, Section 3.3], we consider the coalgebramap

∇ : Ce → C

given by ∇(c⊗dop) = cS(d). Given a (C,C)-comodule M , let M∇ denote M viewedas a right C-comodule via the map ∇.

Let Fp → Y 0 → Y 1 → · · · be an injective resolution of Fp as a left C-comodule.When C is a Hopf algebra, (Ce)∇ is free as a right C-comodule, and (Ce)∇CY n

is injective as a left Ce-comodule [8]. The sequence

(Ce)∇CFp → (Ce)∇CY0 → (Ce)∇CY

1 → · · ·

is therefore an injective resolution of (Ce)∇CFp ∼= C as a left Ce-comodule.Cotensoring over Ce with C we have:

CCe((Ce)∇CY

0)→ CCe((Ce)∇CY

1)→ · · ·

which gives:

C∇CY0 → C∇CY

1 → · · ·Therefore coHHn(C) ∼= CotornCe(C,C) ∼= CotornC(C∇,Fp). In our cases C is cocom-mutative, so C∇ has the trivial C-comodule structure. Therefore CotornC(C∇,Fp) ∼=C ⊗ CotornC(Fp,Fp).

We now want to compute CotorC(Fp,Fp) for the coalgebras C that we are in-terested in.


Proposition 5.3. For C = ΓFp [x1, x2, . . .], CotorC(Fp,Fp) = ΛFp(z1, z2, . . .) wheredeg(zi) = (1,deg(xi)). For C = ΛFp(y1, y2, . . .), CotorC(Fp,Fp) = Fp[w1, w2, . . .]where deg(wi) = (1,deg(yi)).

Proof. Note that in both cases C is a cocommutative coalgebra that in each fixeddegree is a free and finitely generated Fp-module. Let A denote the Hopf algebradual of C. As in [26] we conclude that CotorC(Fp,Fp) is a Hopf algebra, which

is dual to the Hopf algebra TorA(Fp,Fp). Recall that the Hopf algebra dual ofΓFp [x1, x2, . . . ] is Fp[x1, x2, . . . ], and the Hopf algebra ΛFp(y1, y2, . . . ) is self dual.We recall the classical results:

TorFp[x1,x2,...](Fp,Fp) ∼= ΛFp(z1, z2, . . .)

TorΛFp (y1,y2,...)(Fp,Fp) ∼= ΓFp [w1, w2, . . .]

where the degree of zi is (1,deg(xi)) and the degree of wi is (1,deg(yi)). The resultfollows.

We have now proven Proposition 5.1. These results provide the input for thecoBokstedt spectral sequence. They are the starting point for the following theo-rem, which says that this spectral sequence collapses at E2 in the case where thehomology of the input is cofree.

Theorem 5.4. Let C be a cocommutative coassociative coalgebra spectrum that iscofibrant, and whose homology coalgebra is

H∗(C;Fp) = ΓFp [x1, x2, . . . ]

where the xi are cogenerators in strictly positive even degrees and there are onlyfinitely many cogenerators in each degree. Then the coBokstedt spectral sequencefor C collapses at E2, and

E2∼= E∞ ∼= ΓFp [x1, x2, . . . ]⊗ ΛFp(z1, z2, . . . ),

with xi in degree (0,deg(xi)) and zi in degree (1,deg(xi)).

Proof. From Theorem 4.6, the coBokstedt spectral sequence is a spectral sequenceof graded coalgebras, so the differential on the rth page dr : Es,tr → Es+r,t+r−1

r

satisfies the graded co-Leibniz rule

4 dr = (dr ⊗ id + id⊗dr) 4.We have adopted the Koszul sign convention in the above equation, i.e. if f, g : C →D are graded maps and x, y are homogeneous elements then

(f ⊗ g)(x⊗ y) = (−1)deg(g)·deg(x)f(x)⊗ g(y).

By Proposition 5.1 we have an identification of the E2-page:

Es,t2 = coHHFps,t(ΓFp [x1, x2, . . . ]) = ΓFp [x1, x2, . . . ]⊗ ΛFp(z1, z2, . . . )

with xi in degree (0,deg(xi)) and zi in degree (1,deg(xi)).Recall that an element x in a co-augmented coalgebra is called primitive if

4(x) = x ⊗ 1 + 1 ⊗ x. The primitive elements in ΓFp [x1, x2, . . . ] are exactlythe elements in the submodule Fpx1 ⊕ Fpx2 ⊕ · · · and the primitive elements inΛFp(z1, z2, · · · ) are exactly the elements in the submodule Fpz1⊕Fpz2⊕· · · . More-over, if A and B are co-augmented k-coalgebras, then the primitive elements in(A ⊗k B, (id⊗τ ⊗ id) (4A ⊗ 4B)) are the elements of the form a ⊗ 1 or 1 ⊗ b,


where a is a primitive element in A or b is a primitive element in B. Hence thereare no primitive elements in Es,t2 when s ≥ 2.

Fix r ≥ 2. We show that dr is zero because the co-Leibniz rule forces the imageof the lowest degree non-vanishing differential to be primitive. First we see that

4 dr(1) = (dr ⊗ 1 + 1⊗ dr)(1⊗ 1) = dr(1)⊗ 1 + 1⊗ dr(1).

Thus dr(1) is a primitive element in Er,r−12 and therefore dr(1) = 0. Next assume

that all differentials originating from Ei,jr with i + j < t + s vanish. Let y ∈ Es,t.Note that

4(y) = y ⊗ 1 + 1⊗ y +∑

yi1 ⊗ yi2,

where yi1, yi2 ∈ Ei,j with i + j < t + s. It follows from the inductive hypothesisthat

4 dr(y) = (dr ⊗ 1 + 1⊗ dr) 4(y) = dr(y)⊗ 1 + 1⊗ dr(y).

Hence dr(y) is a primitive element in Es+r,t+r−12 . Since there are no primitive

elements in Es,t2 for s ≥ 2, dr = 0 and the spectral sequence collapses at E2.

Recall that for a space X the suspension spectrum Σ∞X is an S-coalgebra.This provides a wealth of examples for which we can use the computational tools ofTheorem 5.4. Before stating particular results, we discuss the motivation for study-ing topological coHochschild homology of suspension spectra. Recall the followingtheorem from Malkiewich [20] and Hess–Shipley [14]:

Theorem 5.5. For X a simply connected space,

THH(Σ∞+ (ΩX)) ' coTHH(Σ∞+ X)

Consequently, for a simply connected space X, to understand THH of Σ∞+ (ΩX),one can instead compute the coTHH of the suspension spectrum of X. In some casesthis topological coHochschild homology computation is more accessible. In partic-ular, to compute the mod p homology H∗(THH(Σ∞+ (ΩX));Fp) using the Bokstedtspectral sequence, the necessary input is the Hochschild homology of the algebraH∗(Σ

∞+ (ΩX);Fp). To compute H∗(coTHH(Σ∞+ X);Fp) the spectral sequence input

is the coHochschild homology of the coalgebra H∗(Σ∞+ X;Fp). In some cases this

homology coalgebra is easier to work with than the homology algebra of the loopspace, making the topological coHochschild homology calculation a more accessiblepath to studying the topological Hochschild homology of based loop spaces. Topo-logical Hochschild homology of based loop spaces is of interest due to connectionsto algebraic K-theory, free loop spaces, and string topology, which we will nowrecall.

Recall Bokstedt and Waldhausen showed that THH(Σ∞+ (ΩX)) ' Σ∞+ LX, whereLX is the free loop space, LX = Map(S1, X). The homology of free loop spaces,H∗(LX), is the main object of study in the field of string topology. Since

coTHH(Σ∞+ X) ' Σ∞+ LX

for simply connected X, topological coHochschild homology gives a new way ofapproaching this homology.

Topological coHochschild homology of suspension spectra also has connectionsto algebraic K-theory. Waldhausen’s A-theory of spaces A(X) is equivalent to


K(Σ∞+ ΩX). Recall that there is a trace map from algebraic K-theory to topologicalHochschild homology

A(X) ' K(Σ∞+ ΩX)→ THH(Σ∞+ ΩX).

When X is simply connected this gives us a trace map to topological coHochschildhomology as well:

A(X) ' K(Σ∞+ ΩX)→ THH(Σ∞+ ΩX) ' coTHH(Σ∞+ X).

In the remainder of this section we will make some explicit computations of thehomology of coTHH of suspension spectra.

Example 5.6. Let X be a product of n copies of CP∞. As a coalgebra

H∗(Σ∞+ X;Fp) ∼= ΓFp(x1, x2, . . . xn)

where deg(xi) = 2. By Theorem 5.4, there is an isomorphism of graded Fp-modules

H∗(coTHH(Σ∞+ X);Fp) ∼= ΓFp(x1, x2, . . . xn)⊗ ΛFp(x1, x2, . . . xn).

Note that any space having cohomology with polynomial generators in even de-grees will give us a suspension spectrum whose homology satisfies the conditions ofTheorem 5.4. Hence this theorem allows us to compute the topological coHochschildhomology of various suspension spectra, as in the example below.

Example 5.7. Theorem 5.4 directly yields the following computations of topologicalcoHochschild homology. The isomorphisms are of graded Fp-modules.

H∗(coTHH(Σ∞+ BU(n));Fp) ∼= ΓFp(y1, . . . yn)⊗ ΛFp(y1, . . . yn)

|yi| = 2i, |yi| = 2i− 1

H∗(coTHH(Σ∞+ BSU(n));Fp) ∼= ΓFp(y2, . . . yn)⊗ ΛFp(y2, . . . yn)

|yi| = 2i, |yi| = 2i− 1

H∗(coTHH(Σ∞+ BSp(n));Fp) ∼= ΓFp(z1, . . . zn)⊗ ΛFp(z1, . . . zn)

|zi| = 4i, |zi| = 4i− 1

With some restrictions on the prime, we also immediately get the mod p ho-mology of the topological coHochschild homology of the suspension spectra of theclassifying spaces BSO(2k), BG2, BF4, BE6, BE7, and BE8. Note that these com-putations also yield the homology of the free loops spaces H∗(LBG;Fp) and the ho-mology H∗(THH(Σ∞+ G);Fp), for G = U(n), SU(n), Sp(n), SO(2k), G2, F4.E6, E7,and E8. The topological coHochschild homology calculations also follow directlyfor products of copies of any of these classifying spaces BG.

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Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville,

TN, 37240, USA

E-mail address: [email protected]

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East

Lansing, MI, 48824, USAE-mail address: [email protected]

Department of Mathematics, Copenhagen University, Universitetsparken 5, Copen-

hagen, DenmarkE-mail address: [email protected]

Department of Mathematics, Statistics, and Computer Science, University of Illinoisat Chicago, 508 SEO m/c 249, 851 S. Morgan Street, Chicago, IL, 60607-7045, USA


KTH Royal Institute of Technology, Department of Mathematics, SE-100 44 Stock-

holm, Sweden


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