Algebraic & Geometric Topology [Logo here]

Volume X (20XX) 1–XXX Published: XX Xxxember 20XX

Hopf algebra structure on topological Hochschild homology

Vigleik Angeltveit John Rognes

Abstract The topological Hochschild homology T HH(R) of a commutative S -algebra (E_{∞} ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Bökstedt spectral sequence converging to the mod p homology of T HH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S -algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of T HH(ku) and T HH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K -theory of S -algebras, by means of the cyclotomic trace map to topological cyclic homology.

AMS Classification 55P43, 55S10, 55S12, 57T05 ; 13D03, 55T15, 55T99.

Keywords Topological Hochschild homology, commutative S -algebra, coproduct, Hopf algebra, topological K -theory, image-of-J spectrum, Bökstedt spectral sequence, Steenrod operations, Dyer–Lashof operations.

1 Introduction

The topological Hochschild homology T HH(R) of an S -algebra R (or an A_{∞} ring spectrum, or a functor with smash product) was constructed in the mid-1980’s by Bökstedt [Bo1], as the natural promotion of the classical Hochschild homology of an algebra in the category of vector spaces (equipped with tensor product) to one in the category of spectra (equipped with smash product). It is the initial ingredient in the construction by Bökstedt, Hsiang and Madsen [BHM93] of the topological cyclic homology T C(R; p), which in many cases closely approximates the algebraic K -theory K(R) [Mc97], [Du97], [HM97].

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When R = HO_{E} is the Eilenberg–Mac Lane spectrum of the valuation ring in a local number field, systematic computations of the topological cyclic homology of R were made in [HM03], thereby verifying the Lichtenbaum–Quillen conjectures for the algebraic K -theory of these fields. Particular computations for other commutative S -algebras, like connective complex K -theory ku and its p-local Adams summand ℓ, have revealed a more general pattern of how algebraic K -theory creates a “red-shift” in chromatic filtration and satisfies a Galois descent property [AR02], [Au].

These results indicate that the algebraic K -theory of a commutative S -algebra is governed by an associated “S -algebro-geometric” structure space, for which the smashing localizations of the chromatic filtration are related to a Zariski topology, and étale covers and Galois extensions are related to an étale topology. There are also variant geometries associated to the less strictly commutative S - algebras known as E_{n} ring spectra, for 1 ≤ n ≤ ∞. The extreme cases n = 1 and n = ∞ correspond to the associative and the commutative S -algebras, respectively. We shall encounter examples of the intermediate E_{n} ring spectrum structures in Section 5, when we consider the Brown–Peterson spectrum BP (which is an E_{4} ring spectrum according to Basterra and Mandell) and the Johnson–Wilson spectra BP ⟨m−1⟩. By convention, we let BP = BP ⟨∞⟩.

Further systematic computations of the topological Hochschild homology, topological cyclic homology and algebraic K -theory of S -algebras can be expected to shed new light on these geometries and on the algebraic K -theory functor. The present paper advances the algebraic topological foundations for such systematic computations, especially by taking into account the Hopf algebra structure present in the topological Hochschild homology of commutative S - algebras. This program is continued in [BR05], which analyzes the differentials in the homological homotopy fixed point spectral sequence that approximates the cyclic fixed points of topological Hochschild homology, and in [L-N05], which identifies the action by Steenrod operations on the (continuous co-)homology of these fixed- and homotopy fixed point spectra.

When R is a commutative S -algebra (or an E_{∞} ring spectrum, or perhaps a commutative FSP), there is an equivalence T HH(R) ≃ R ⊗ S^{1} of commutative R-algebras, due to McClure, Schwänzl and Vogt [MSV97], where S^{1} is the topological circle and (−) ⊗ S^{1} refers to the topologically tensored structure in the category of commutative S -algebras. The pinch map S^{1} → S^{1} ∨ S^{1} and reflection S^{1} → S^{1} then induce maps ψ: T HH(R) → T HH(R) ∧_{R} T HH(R) and χ: T HH(R) → T HH(R), which make T HH(R) a Hopf algebra over R in the homotopy category [EKMM97, IX.3.4]. See also Theorem 3.9.

We wish to apply this added structure for computations. Such computations are usually made by starting with the simplicial model [q] ↦ R^{∧(q+1)} = R ⊗ S_{q}^{1} for T HH(R), where now [q] ↦ S_{q}^{1} is the simplicial circle. The resulting skeleton filtration on T HH(R) gives rise to the Bökstedt spectral sequence in homology

E_{∗∗}^{2}(R) = HH_{∗}(H_{∗}(R; 𝔽_{p})) ⟹ H_{∗}(T HH(R); 𝔽_{p}) ,

as we explain in Section 4. Compare [BM94, §2] and [MS93, 3.1]. But the coproduct and conjugation maps are not induced by simplicial maps in this model, so some adjustment is needed in order for these structures to carry over to the spectral sequence. This we arrange in Section 3, by using a doubly subdivided simplicial circle to provide an alternative simplicial model for T HH(R), for which the Hopf algebra structure maps can be simplicially defined. The verification of the Hopf algebra relations then also involves a triply subdivided simplicial circle.

In Section 4 we transport the Hopf algebra structure on T HH(R) to the Bökstedt spectral sequence, showing in Theorem 4.5 that if its initial term E_{∗∗}^{2}(R) is flat as a module over H_{∗}(R; 𝔽_{p}), then this term is an A_{∗}-comodule H_{∗}(R; 𝔽_{p})-Hopf algebra and the d^{2}-differentials respect this structure. Furthermore, if every E^{r} -term is flat over H_{∗}(R; 𝔽_{p}), then the Bökstedt spectral sequence is one of A_{∗}-comodule H_{∗}(R; 𝔽_{p})-Hopf algebras. We also discuss, in Proposition 4.3, the weaker algebraic structure available in the Bökstedt spectral sequence when R is just an E_{2}- or E_{3} ring spectrum.

Thereafter we turn to the computational applications. The mod p homology of the topological Hochschild homology of the Eilenberg–Mac Lane S -algebras H𝔽_{p} and Hℤ was already computed by Bökstedt [Bo2]. The first non-algebraic example, namely the topological Hochschild homology of the Adams summand ℓ = BP ⟨1⟩ of p-local connective topological K -theory, was computed for p odd by McClure and Staffeldt in [MS93].

We show in Section 5 how to extend these computations to include the case of ku_{(2)} = BP ⟨1⟩ at p = 2, and the more general Brown–Peterson and Johnson– Wilson S -algebras BP ⟨m−1⟩, for primes p and symbols 0 ≤ m ≤ ∞ such that these are E_{3} ring spectra. As a sample result we have part of Theorem 5.12(a):

H_{∗}(T HH(BP); 𝔽_{2}) ≅ H_{∗}(BP; 𝔽_{2}) ⊗ E(σξ̄_{k}^{2} | k ≥ 1) .

Here and below, E(−) and P (−) denote the exterior and polynomial algebras on the indicated generators, respectively, and the classes ξ̄_{k} are the conjugates of Milnor’s generators for the dual Steenrod algebra A_{∗}.

To resolve the multiplicative extensions in the Bökstedt spectral sequence we provide a proof of Bökstedt’s formula saying that the suspension map σ: ΣR → T HH(R) takes the Dyer–Lashof operations Q^{k} on the homology of the commutative S -algebra R compatibly to the corresponding operations on the homology of the commutative S -algebra T HH(R). See Proposition 5.9, which also makes precise what happens when R is just an E_{n+1} ring spectrum.

In Section 6 we do the same for the higher real commutative S -algebras ko and tmf at p = 2. As sample results we have Corollary 5.14(a) and Theorem 6.2(a):

H_{∗}(T HH(ku); 𝔽_{2}) ≅ H_{∗}(ku; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})

and

H_{∗}(T HH(ko); 𝔽_{2}) ≅ H_{∗}(ko; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) .

In the more demanding Section 7 we proceed to the p-local real and complex image-of-J spectra j and ju, which are connective, commutative S -algebras. At odd primes the two are homotopy equivalent. We identify the mod p homology algebra of ju at p = 2 and of j = ju at odd primes in Proposition 7.12(a) and (b), and make essential use of our results about Hopf algebra structures to show that the corresponding Bökstedt spectral sequences for T HH collapse at the E^{2}- and E^{p}-terms, respectively, in Proposition 7.13(a) and (b). In Theorem 7.15 we resolve the algebra extension questions to obtain H_{∗}(T HH(ju); 𝔽_{p}) as an algebra, both for p = 2 and for p odd. This proof involves a delicate comparison with the case of T HH(ku) for p = 2, and with T HH(ℓ) for p odd. Again as a sample result, we have Theorem 7.15(a):

H_{∗}(T HH(ju); 𝔽_{2}) ≅ H_{∗}(ju; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) ⊗ Γ(σb) .

Here Γ(σb) = E(γ_{2^{k}}(σb) | k ≥ 0) is the divided power algebra on a class σb in degree 4.

The algebra structure of H_{∗}(j; 𝔽_{2}) is described as a split square-zero extension of (A//A_{2})_{∗} in Proposition 7.12(c):

0 → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → H_{∗}(j; 𝔽_{2}) → (A//A_{2})_{∗} → 0 .

Here A_{2} = ⟨Sq^{1}, Sq^{2}, Sq^{4}⟩ ⊂ A, and K_{∗} ⊂ A_{2∗} is dual to a cyclic A_{2}-module K of rank 17 over 𝔽_{2}. The symbol □ denotes the cotensor product of comodules [MiMo65, 2.2]. The A-module structure of H^{∗}(j; 𝔽_{2}) was given in [Da75], but this identification of the algebra structure seems to be new. The E^{2}-term of the Bökstedt spectral sequence for j is described in Proposition 7.13(c), but it is not flat over H_{∗}(j; 𝔽_{2}), so the coproduct on H_{∗}(T HH(j); 𝔽_{2}) is not conveniently described by this spectral sequence. We have therefore not managed to evaluate the homology of T HH(j) at p = 2 by these methods.

Next we consider the passage from the homology of T HH(R) to its homotopy, with suitably chosen finite coefficients. This has been a necessary technical switch in past computations of topological cyclic homology T C(R; p), since T C is defined as the homotopy inverse limit of a diagram of fixed-point spectra derived from T HH, and the interaction between inverse limits and homology was thought to be difficult to control. The homotopy groups of an inverse limit are much better behaved. Nonetheless, it may be that future computations of the topological cyclic homology of S -algebras will follow a purely homological approach, see [BR05] and [L-N05].

In Section 8 we follow the strategy of [MS93] to compute the homotopy groups of T HH(ku) ∧ M, where M = C_{2} is the mod 2 Moore spectrum, and of T HH(ko)∧Y , where Y = C_{2}∧C_{η} is the 4-cell spectrum employed by Mahowald [Mah82]. The results appear in Theorems 8.13 and 8.14, respectively. In each case the method is to use the Adams spectral sequence to pass from homology to homotopy, and to use a comparison with a Morava K(1)-based Bökstedt spectral sequence to obtain enough information about the v_{1}-periodic towers in the abutment to completely determine the differential structure of the Adams spectral sequence.

The present paper started out as the first author’s Master’s thesis [An02] at the University of Oslo, supervised by the second author. Both authors are grateful to the referee for his careful reading of the paper.

2 Hochschild and topological Hochschild homology

Let k be a graded field, i.e., a graded commutative ring such that every graded k-module is free, and Λ a graded k-algebra (always unital and associative). We recall the definition of the Hochschild homology of Λ, e.g. from [Mac75, X.4]. The Hochschild complex C_{∗}(Λ) = C_{∗}^{k}(Λ) is the chain complex of graded k-modules with C_{q}(Λ) = Λ^{⊗(q+1)} in degree q (all tensor products are over k) and boundary homomorphisms ∂: C_{q}(Λ) → C_{q−1}(Λ) given by

∂(λ_{0} ⊗ ⋯ ⊗ λ_{q}) =

∑_{i=0}^{q−1}(−1)^{i}λ_{0} ⊗ ⋯ ⊗ λ_{i}λ_{i+1} ⊗ ⋯ ⊗ λ_{q} + (−1)^{q+ϵ}λ_{q}λ_{0} ⊗ ⋯ ⊗ λ_{q−1}

where ϵ = |λ_{q}|(|λ_{0}| + ⋯ + |λ_{q−1}|). The Hochschild homology HH_{∗}(Λ) = HH_{∗}^{k}(Λ) is defined to be the homology of this chain complex. It is bigraded, first by the Hochschild degree q and second by the internal grading from Λ. When Λ is commutative (always in the graded sense) the shuffle product of chains defines a product

ϕ: HH_{∗}(Λ) ⊗_{Λ} HH_{∗}(Λ) → HH_{∗}(Λ)

that makes HH_{∗}(Λ) a commutative Λ-algebra, with unit corresponding to the inclusion of 0-chains Λ → HH_{∗}(Λ).

When Λ is commutative there is also a chain level coproduct ψ: C_{∗}(Λ) → C_{∗}(Λ) ⊗_{Λ} C_{∗}(Λ) given in degree q by

ψ(λ_{0} ⊗ λ_{1} ⊗ ⋯ ⊗ λ_{q}) =

∑_{i=0}^{q}(λ_{0} ⊗ λ_{1} ⊗ ⋯ ⊗ λ_{i}) ⊗_{Λ} (1 ⊗ λ_{i+1} ⊗ ⋯ ⊗ λ_{q}) . (2.1)

It is essential to tensor over Λ in the target of this chain map. When HH_{∗}(Λ) is flat as a Λ-module, the chain level coproduct ψ induces a coproduct

ψ: HH_{∗}(Λ) → HH_{∗}(Λ) ⊗_{Λ} HH_{∗}(Λ)

on Hochschild homology. Here the right hand side is identified with the homology of C_{∗}(Λ) ⊗_{Λ} C_{∗}(Λ) by the Künneth theorem. We shall now compare this chain level definition of the coproduct on HH_{∗}(Λ) with an equivalent definition given in more simplicial terms.

Let B_{∗}(Λ) = B_{∗}(Λ, Λ, Λ) be the two-sided bar construction [Mac75, X.2] for the k-algebra Λ. It has B_{q}(Λ) = Λ⊗Λ^{⊗q} ⊗Λ in degree q, and is a free resolution of Λ in the category of Λ-bimodules. We use the bar notation λ_{0}[λ_{1}| … |λ_{q}]λ_{q+1} for a typical generator of B_{q}(Λ). The Hochschild complex is obtained from the two-sided bar construction by tensoring it with Λ viewed as a Λ-bimodule: C_{∗}(Λ) = Λ ⊗_{Λ−Λ} B_{∗}(Λ).

When Λ is commutative, B_{∗}(Λ) is the chain complex Ch(Λ⊗Δ^{1}) associated to the simplicial Λ-bimodule [q] ↦ Λ ⊗ Δ_{q}^{1}. Here Δ^{1} is the simplicial 1-simplex, and Λ ⊗ Δ_{q}^{1} denotes the tensor product of one copy of Λ for each element of Δ_{q}^{1}. See Section 3 below for more on this notation. The Λ-bimodule structure on B_{∗}(Λ) is derived from the inclusion of the two boundary points ∂Δ^{1} → Δ^{1}, and C_{∗}(Λ) equals the chain complex Ch(Λ ⊗ S^{1}) associated to the simplicial Λ-module [q] ↦ Λ ⊗ S_{q}^{1}, where S^{1} = Δ^{1}/∂Δ^{1} is the simplicial circle.

We now discuss three maps of Λ-bimodule chain complexes. First, there is a canonical chain level coproduct ψ: B_{∗}(Λ) → B_{∗}(Λ) ⊗_{Λ} B_{∗}(Λ) of Λ-bimodules, given in degree q by

ψ(λ_{0}[λ_{1}| … |λ_{q}]λ_{q+1}) = ∑_{i=0}^{q} λ_{0}[λ_{1}| … |λ_{i}]1 ⊗_{Λ} 1[λ_{i+1}| … |λ_{q}]λ_{q+1} .

When Λ is commutative the chain level coproduct ψ on C_{∗}(Λ) is derived from this, as the obvious composite map

\xymatrix{C_{∗}(Λ) = Λ ⊗_{Λ−Λ} B_{∗}(Λ) \ar[r]^{1⊗ψ} & Λ ⊗_{Λ−Λ} (B_{∗}(Λ) ⊗_{Λ} B_{∗}(Λ)) \ar[d]^{ψ′} \\  & C_{∗}(Λ) ⊗_{Λ} C_{∗}(Λ) .}

Second, there is a shuffle equivalence sh: B_{∗}(Λ) ⊗_{Λ} B_{∗}(Λ) → dB_{∗}(Λ) of Λbimodules, by the Eilenberg–Zilber theorem [Mac75, VIII.8.8] applied to two copies of the simplicial Λ-bimodule Λ ⊗ Δ^{1}. Here

dB_{∗}(Λ) = Ch(Λ ⊗ dΔ^{1}) = Ch((Λ ⊗ Δ^{1}) ⊗_{Λ} (Λ ⊗ Δ^{1}))

is the chain complex associated to the simplicial tensor product of two copies of Λ ⊗ Δ^{1}, considered as simplicial Λ-modules by way of the right and left actions, respectively. This simplicial tensor product equals Λ ⊗ dΔ^{1}, where the “double 1-simplex” dΔ^{1} = Δ^{1} ∪_{Δ^{0}} Δ^{1} is the union of two 1-simplices that are compatibly oriented. (So dΔ^{1} is the 2-fold edgewise subdivision of Δ^{1} [BHM93, §1].) More explicitly,

sh(x ⊗_{Λ} y) = ∑_{(μ,ν)} sgn(μ, ν)(s_{ν}(x) ⊗_{Λ} s_{μ}(y)) ,

where x ∈ B_{i}(Λ), y ∈ B_{q−i}(Λ), the sum is taken over all (i, q − i)-shuffles (μ, ν), sgn(μ, ν) is the sign of the associated permutation and s_{ν}(x) and s_{μ}(y) are the appropriate iterated degeneracy operations on x and y, respectively.

Third, there is a chain equivalence π: dB_{∗}(Λ) → B_{∗}(Λ) of Λ-bimodules, induced by the simplicial map π: dΔ^{1} = Δ^{1} ∪_{Δ^{0}} Δ^{1} → Δ^{1} that collapses the second Δ^{1} in dΔ^{1} to a point. It is given by

π(x ⊗_{Λ} y) = x · ϵ(y)

for x, y ∈ B_{q}(Λ), where ϵ(λ_{0}[λ_{1}| … |λ_{q}]λ_{q+1}) = λ_{0}λ_{1} … λ_{q}λ_{q+1} is the augmentation.

Lemma 2.2 Let Λ be a commutative k-algebra. The maps sh ∘ ψ: B_{∗}(Λ) → dB_{∗}(Λ) and π: dB_{∗}(Λ) → B_{∗}(Λ) of Λ-bimodule chain complexes are mutual chain inverses. Hence the induced composite

\xymatrix{dC_{∗}(Λ) = Λ ⊗_{Λ−Λ} dB_{∗}(Λ) \ar[r]^{1⊗π} & C_{∗}(Λ) \ar[r]^{1⊗(sh∘ψ)} & dC_{∗}(Λ)}

is chain homotopic to the identity.

Proof All three chain complexes B_{∗}(Λ), B_{∗}(Λ) ⊗_{Λ} B_{∗}(Λ) and dB_{∗}(Λ) are free Λ-bimodule resolutions of Λ, and the maps ψ, sh and π are Λ-bimodule chain maps, so it suffices to verify that the composite π ∘ sh ∘ ψ: B_{∗}(Λ) → B_{∗}(Λ) covers the identity on Λ. In degree zero, ψ(λ_{0}[]λ_{1}) = λ_{0}[]1 ⊗_{Λ} 1[]λ_{1}, sh(λ_{0}[]1 ⊗_{Λ} 1[]λ_{1}) = λ_{0}[]1 ⊗_{Λ} 1[]λ_{1} and π(λ_{0}[]1 ⊗_{Λ} 1[]λ_{1}) = λ_{0}[]λ_{1}, as required. If desired, the explicit formulas can be composed also in higher degrees, to show that (π∘sh∘ψ)(x) ≡ x modulo simplicially degenerate terms, for all x ∈ B_{q}(Λ). In either case, we see that π ∘ sh ∘ ψ is chain homotopic to the identity. The remaining conclusions follow by uniqueness of inverses.

Note that dC_{∗}(Λ) defined above is the chain complex associated to the simplicial Λ-module Λ⊗d′S^{1}, where the “double circle” d′S^{1} = dΔ^{1}/∂dΔ^{1} is the quotient of the double 1-simplex dΔ^{1} = Δ^{1} ∪_{Δ^{0}} Δ^{1} by its two end-points ∂dΔ^{1}. The chain equivalence 1 ⊗ π: dC_{∗}(Λ) = Ch(Λ ⊗ d′S^{1}) → Ch(Λ ⊗ S^{1}) = C_{∗}(Λ) obtained by tensoring down the Λ-bimodule chain equivalence induced from the collapse map π: dΔ^{1} → Δ^{1}, is then more directly obtained from the collapse map π: d′S^{1} → S^{1} that collapses the second of the two non-degenerate 1simplices in d′S^{1} to a point.

There is also a simplicial pinch map ψ: d′S^{1} → S^{1} ∨ S^{1} to the one-point union (wedge sum) of two circles, that collapses the 0-skeleton of d′S^{1} to a point. It induces a map

ψ′: dC_{∗}(Λ) = Ch(Λ ⊗ d′S^{1}) → Ch(Λ ⊗ (S^{1} ∨ S^{1})) .

The target is the chain complex associated to the simplicial tensor product of two copies of Λ⊗S^{1}, considered as a simplicial Λ-module. It is therefore also the target of a shuffle equivalence, namely sh: C_{∗}(Λ)⊗_{Λ}C_{∗}(Λ) → Ch(Λ⊗(S^{1}∨S^{1})).

Proposition 2.3 Let Λ be a commutative k-algebra. The composite map

\xymatrix{dC_{∗}(Λ) \ar[r]^{1⊗π}_{≃} & C_{∗}(Λ) \ar[r]^{ψ} & C_{∗}(Λ) ⊗_{Λ} C_{∗}(Λ) \ar[r]^{sh}_{≃} & Ch(Λ ⊗ (S^{1} ∨ S^{1}))}

of the chain level coproduct ψ in C_{∗}(Λ) (see formula (2.1)), with the chain equivalence 1⊗π induced by the simplicial collapse map π: d′S^{1} → S^{1} and the shuffle equivalence sh, is chain homotopic to the map

\xymatrix{dC_{∗}(Λ) = Ch(Λ ⊗ d′S^{1}) \ar[r]^{ψ′} & Ch(Λ ⊗ (S^{1} ∨ S^{1}))}

induced by the simplicial pinch map ψ: d′S^{1} → S^{1} ∨ S^{1}.

Hence, when HH_{∗}(Λ) is flat over Λ, the coproduct ψ on Hochschild homology agrees, via the identifications induced by 1⊗π and sh, with the map ψ′ induced by the simplicial pinch map.

Proof Consider the following diagram.

\xymatrix{C_{∗}(Λ) \ar[r]^{1⊗ψ} & Λ ⊗_{Λ−Λ} (B_{∗}(Λ) ⊗_{Λ} B_{∗}(Λ)) \ar[r]^{ψ′} \ar[d]^{1⊗sh} & C_{∗}(Λ) ⊗_{Λ} C_{∗}(Λ) \ar[d]^{sh} \\  & dC_{∗}(Λ) \ar[r]^{ψ′} \ar[ul]^{1⊗π} & Ch(Λ ⊗ (S^{1} ∨ S^{1}))}

The composite along the upper row is the coproduct ψ, the composite around the triangle is chain homotopic to the identity by Lemma 2.2, and the square commutes by naturality of the shuffle map with respect to the pinch map ψ′. A diagram chase then provides the claimed chain homotopy.

We shall make use of the following standard calculations of Hochschild homology. The formulas for the coproduct ψ follow directly from the chain level formula (2.1) above. Let P (x) = k[x] and E(x) = k[x]/(x^{2}) be the polynomial and exterior algebras over k in one variable x, and let Γ(x) = k{γ_{i}(x) | i ≥ 0} be the divided power algebra with multiplication

γ_{i}(x) · γ_{j}(x) = (i, j) γ_{i+j}(x) ,

where (i, j) = (i + j)!/i!j! is the binomial coefficient.

Proposition 2.4 For x ∈ Λ let σx ∈ HH_{1}(Λ) be the homology class of the cycle 1 ⊗ x ∈ C_{1}(Λ) in the Hochschild complex. For Λ = P (x) there is a P (x)-algebra isomorphism

HH_{∗}(P (x)) = P (x) ⊗ E(σx) .

The class σx is coalgebra primitive, i.e., ψ(σx) = σx⊗1+1⊗σx. For Λ = E(x) there is an E(x)-algebra isomorphism

HH_{∗}(E(x)) = E(x) ⊗ Γ(σx) .

The ith divided power γ_{i}(σx) is the homology class of the cycle 1⊗x⊗⋯⊗x ∈ C_{i}(Λ). The coproduct is given by

ψ(γ_{k}(σx)) = ∑_{i+j=k} γ_{i}(σx) ⊗ γ_{j}(σx) .

There is a Künneth formula

HH_{∗}(Λ_{1} ⊗ Λ_{2}) ≅ HH_{∗}(Λ_{1}) ⊗ HH_{∗}(Λ_{2}) .

Let P_{h}(x) = k[x]/(x^{h}) be the truncated polynomial algebra of height h. We write P (x_{i} | i ≥ 0) = P (x_{0}, x_{1}, …) = P (x_{0}) ⊗ P (x_{1}) ⊗ …, and so on. When k is of prime characteristic p it is a standard calculation with binomial coefficients that

Γ(x) = P_{p}(γ_{p^{i}}(x) | i ≥ 0)  (2.5)

as a k-algebra.

We have already noted that the Hochschild complex C_{∗}(Λ) is the chain complex associated to a simplicial graded k-module [q] ↦ C_{q}(Λ) = Λ^{⊗(q+1)}, with face maps d_{i} corresponding to the individual terms in the alternating sum defining the Hochschild boundary ∂. In fact, this is a cyclic graded k-module in the sense of Connes, with cyclic structure maps t_{q} that cyclically permute the (q+1) tensor factors in C_{q}(Λ), up to sign. It follows that the geometric realization HH(Λ) = |[q] ↦ C_{q}(Λ)| admits a natural S^{1}-action α: HH(Λ)∧S_{+}^{1} → HH(Λ), and that the Hochschild homology groups are the homotopy groups of this space: HH_{∗}(Λ) = π_{∗}HH(Λ).

The basic idea in the definition of topological Hochschild homology is to replace the ground ring k by the sphere spectrum S, and the symmetric monoidal category of graded k-modules under the tensor product ⊗ = ⊗_{k} by the symmetric monoidal category of spectra, interpreted as S -modules, under the smash product ∧ = ∧_{S}. A monoid in the first category is a graded k-algebra Λ, which then gets replaced by a monoid in the second category, i.e., an S -algebra R. To make sense of this we will work in the framework of [EKMM97], but we could also use [HSS00] or any other reasonable setting that gives a symmetric monoidal category of spectra.

The original definition of topological Hochschild homology was given by Bökstedt in the mid 1980’s [Bo1], inspired by work and conjectures of Goodwillie and Waldhausen. The following definition is not the one originally used by Bökstedt, since he did not have the symmetric monoidal smash product from [EKMM97] or [HSS00] available, but it agrees with the heuristic definition that his more complicated definition managed to make sense of, with the more elementary technology that he had at hand.

Definition 2.6 Let R be an S -algebra, with multiplication μ: R∧R → R and unit η: S → R. The topological Hochschild homology of R is the geometric realization T HH(R) of the simplicial S -module T HH_{•}(R) with

T HH_{q}(R) = R^{∧(q+1)}

in simplicial degree q. The simplicial structure is like that on the simplicial k-module underlying the Hochschild complex. More precisely, the i-th face map d_{i}: R^{∧(q+1)} → R^{∧q} equals id_{R}^{i} ∧ μ ∧ id_{R}^{q−i−1} for 0 ≤ i < q, while d_{q} = (μ∧id_{R}^{q−1})t_{q}, where t_{q} cyclically permutes the (q+1) smash factors R by moving the last factor to the front. The j -th degeneracy map s_{j}: R^{∧(q+1)} → R^{∧(q+2)} equals id_{R}^{j+1} ∧ η ∧ id_{R}^{q−j}, for 0 ≤ j ≤ q.

Furthermore, the cyclic operators t_{q}: T HH_{q}(R) → T HH_{q}(R) make T HH_{•}(R) a cyclic S -module, so that its geometric realization has a natural S^{1}-action

α: T HH(R) ∧ S_{+}^{1} → T HH(R) .

The topological Hochschild homology groups of R are defined to be the homotopy groups π_{∗}T HH(R).

When R is a commutative S -algebra there is a product on T HH(R) that makes it an augmented commutative R-algebra, with unit corresponding to the inclusion of 0-simplices R → T HH(R). See (3.3) below. There is a discussion of alternative definitions of T HH(R) in [EKMM97, IX].

3 Hopf algebra structure on T HH

Already Bökstedt noted that the simplicial structure on T HH(R), as defined above, is derived from the simplicial structure on the standard simplicial circle S^{1} = Δ^{1}/∂Δ^{1}. This can be made most precise in the case when R is a commutative S -algebra, in which case there is a formula T HH(R) ≅ R ⊗ S^{1} in terms of the simplicial tensor structure on the category of commutative S -algebras. A corresponding formula T HH(R) ≃ R ⊗ |S^{1}| in terms of the topological tensor structure was discussed by McClure, Schwänzl and Vogt in [MSV97]. We shall stick to the simplicial context, since we will make use of the resulting skeletal filtrations to form spectral sequences.

We now make this “tensored structure” explicit. Let R be a commutative S -algebra and X a finite set, and let

R ⊗ X = ⋀_{x∈X} R

be the smash product of one copy of R for each element of X. It is again a commutative S -algebra. Now let f: X → Y be a function between finite sets, and let R ⊗ f: R ⊗ X → R ⊗ Y be the smash product over all y ∈ Y of the maps

R ⊗ f^{−1}(y) = ⋀_{x∈f^{−1}(y)} R → R = R ⊗ {y}

that are given by the iterated multiplication from the #f^{−1}(y) copies of R on the left to the single copy of R on the right. If f^{−1}(y) is empty, this is by definition the unit map η: S → R. Since R is commutative, there is no ambiguity in how these iterated multiplications are to be formed.

Note that the construction R ⊗ X is functorial in both R and X. Given an injection X → Y and any function X → Z, there is a natural isomorphism

R ⊗ (Y ∪_{X} Z) ≅ (R ⊗ Y) ∧_{(R⊗X)} (R ⊗ Z) .

There is also a natural map

R ∧ X_{+} = ⋁_{x∈X} R → ⋀_{x∈X} R = R ⊗ X

whose restriction to the wedge summand indexed by x ∈ X is R ⊗ i_{x}, where i_{x}: {x} → X is the inclusion.

By naturality, these constructions all extend degreewise to simplicial finite sets X: [q] ↦ X_{q}, simplicial maps f: X → Y , etc. In particular, we can define the simplicial commutative S -algebra

R ⊗ X = ([q] ↦ R ⊗ X_{q})

with structure maps R ⊗ d_{i}, R ⊗ s_{j}, etc. There is then a useful natural map

ω: R ∧ X_{+} → R ⊗ X .  (3.1)

Consider the special case X = S^{1} = Δ^{1}/∂Δ^{1}. Here Δ_{q}^{1} has (q + 2) elements {x_{0}, …, x_{q+1}} where x_{t}: [q] → [1] has #x_{t}^{−1}(0) = t. The quotient S_{q}^{1} has (q +1) elements, obtained by identifying x_{0} ∼ x_{q+1}. Then d_{i}(x_{t}) = x_{t} for t ≤ i and d_{i}(x_{t}) = x_{t−1} for t > i, while s_{j}(x_{t}) = x_{t} for t ≤ j and s_{j}(x_{t}) = x_{t+1} for t > j. A direct check shows that there is a natural isomorphism

T HH(R) ≅ R ⊗ S^{1}  (3.2)

of (simplicial) commutative S -algebras. In degree q it is the obvious identification R^{∧(q+1)} ≅ R ⊗ S_{q}^{1}.

There are natural maps η: ∗ → S^{1}, ϵ: S^{1} → ∗ and ϕ: S^{1} ∨ S^{1} → S^{1} that map to the base point of S^{1}, retract to ∗ and fold two copies of S^{1} to one, respectively. By naturality, these induce the following maps of commutative S -algebras:

\begin{aligned}η & : R → T HH(R) \\ ϵ & : T HH(R) → R \\ ϕ & : T HH(R) ∧_{R} T HH(R) → T HH(R) . \end{aligned}  (3.3)

In the last case, the product map involves the identification T HH(R) ∧_{R} T HH(R) ≅ R ⊗ (S^{1} ∪_{∗} S^{1}), where S^{1} ∪_{∗} S^{1} = S^{1} ∨ S^{1}. Taken together, these maps naturally make T HH(R) an augmented commutative R-algebra.

There is also a natural map

ω: R ∧ S_{+}^{1} → T HH(R) ,  (3.4)

derived from (3.1), which captures part of the circle action upon T HH(R). More precisely, the map ω admits the following factorization:

ω = α ∘ (η ∧ id): R ∧ S_{+}^{1} → T HH(R) ∧ S_{+}^{1} → T HH(R) .  (3.5)

This is clear by inspection of the definition of the circle action α on the 0simplices of T HH(R).

We would like to have a coproduct on T HH(R), coming from a pinch map S^{1} → S^{1} ∨ S^{1}, but there is no such simplicial map with our basic model for S^{1}. To fix this we again consider a “double model” for S^{1}, denoted dS^{1}, with

dS^{1} = (Δ^{1} ⊔ Δ^{1}) ∪_{(∂Δ^{1}⊔∂Δ^{1})} ∂Δ^{1} .

Here ∂Δ^{1} ⊔ ∂Δ^{1} → ∂Δ^{1} is the identity map on each summand, so the two nondegenerate 1-simplices of dS^{1} have opposing orientations in the geometrically realized circle. It is the quotient of the barycentric subdivision of Δ^{1} by its boundary.

Then we have a simplicial pinch map ψ: dS^{1} → S^{1} ∨ S^{1} that collapses ∂Δ^{1} ⊂ dS^{1} to ∗, as well as a simplicial flip map χ: dS^{1} → dS^{1} that interchanges the two copies of Δ^{1}.

Remark 3.6 The simplicial set dS^{1} introduced here differs from the double circle d′S^{1} considered in Section 2, in that the orientation of the second 1simplex has been reversed. The switch is necessary here to make the flip map χ simplicial. In principle, we could have used the same dS^{1} in Section 2 as here, but this would have entailed the cost of discussing the anti-simplicial involution λ_{0}[λ_{1}| … |λ_{q}]λ_{q+1} ↦ ±λ_{q+1}[λ_{q}| … |λ_{1}]λ_{0} of B_{∗}(Λ), and complicating the formula (2.1) for the chain level coproduct ψ. We choose instead to suppress this point.

We define a corresponding “double model” for T HH(R), denoted dT HH(R), by

dT HH(R) = R ⊗ dS^{1} .

The pinch and flip maps now induce the following natural maps of commutative S -algebras:

\begin{aligned}ψ′ & : dT HH(R) → T HH(R) ∧_{R} T HH(R) \\ χ′ & : dT HH(R) → dT HH(R) . \end{aligned}  (3.7)

Lemma 3.8 Let R be cofibrant as an S -module. Then the collapse map π: dS^{1} → S^{1} that takes the second Δ^{1} to ∗ induces a weak equivalence

\xymatrix{π: dT HH(R) \ar[r]^{≃} & T HH(R) .}

Proof Consider the commutative diagram

[figure: B(R); R ∧ R; B(R); B(R); R ∧ R; R]

of commutative S -algebras. Here B(R) = B(R, R, R) = R ⊗ Δ^{1} is the twosided bar construction, its augmentation B(R) → R is a weak equivalence, and the inclusion R ∧ R → B(R) is a cofibration of S -modules. From [EKMM97, III.3.8] we know that the categorical pushout (balanced smash product) in this case preserves weak equivalences. Pushout along the upper row gives dT HH(R) and pushout along the lower row gives T HH(R), so the induced map π: dT HH(R) → T HH(R) is indeed a weak equivalence.

Theorem 3.9 Let R be a commutative S -algebra. Its topological Hochschild homology T HH(R) is naturally an augmented commutative R-algebra, with unit, counit and product maps η, ϵ and ϕ defined as in (3.3) above. In the stable homotopy category, these maps, the coproduct map

ψ = ψ′ ∘ π^{−1} : T HH(R) → T HH(R) ∧_{R} T HH(R)

and the conjugation map

χ = π ∘ χ′ ∘ π^{−1} : T HH(R) → T HH(R)

naturally make T HH(R) an R-Hopf algebra.

Proof To check that T HH(R) is indeed a Hopf algebra over R in the stable homotopy category we must verify that a number of diagrams commute. We will do one case that illustrates the technique, and leave the rest to the reader.

Let T = T HH(R). In order to show that the diagram

\xymatrix{T \ar[r]^{ψ} \ar[dr]^{ϵ} \ar[d]_{ψ} & T ∧_{R} T \ar[dr]^{id∧χ} \\ T ∧_{R} T \ar[dr]_{χ∧id} & R \ar[dr]^{η} & T ∧_{R} T \ar[d]^{ϕ} \\  & T ∧_{R} T \ar[r]_{ϕ} & T}  (3.10)

commutes in the stable homotopy category it suffices to check that the diagram of simplicial sets

\xymatrix{tS^{1} \ar[r]^{ψ_{1}} \ar[dr]^{ϵ} \ar[d]_{ψ_{2}} & S^{1} ∨ dS^{1} \ar[dr]^{id∨χ′} \\ dS^{1} ∨ S^{1} \ar[dr]_{χ′∨id} & ∗ \ar[dr]^{η} & S^{1} ∨ dS^{1} \ar[d]^{ϕ(id∨π)} \\  & dS^{1} ∨ S^{1} \ar[r]_{ϕ(π∨id)} & S^{1}}

homotopy commutes (simplicially).

Here tS^{1} = ∂Δ^{2} is a “triple model” for S^{1}, with three non-degenerate 1simplices. The pinch map ψ_{1} identifies the vertices 0 and 1 in ∂Δ^{2} and takes the face δ_{0} to the first Δ^{1} in dS^{1}. Then the composite ϕ(id ∨ π)(id ∨ χ′)ψ_{1} factors as

\xymatrix{∂Δ^{2} ⊂ Δ^{2} \ar[r]^{s_{1}} & Δ^{1} → S^{1} ,}

and Δ^{1} is simplicially contractible. Similarly, ψ_{2} identifies the vertices 1 and 2 and takes δ_{2} to the first Δ^{1} in dS^{1}. Then ϕ(π ∨ id)(χ′ ∨ id)ψ_{2} factors as

\xymatrix{∂Δ^{2} ⊂ Δ^{2} \ar[r]^{s_{0}} & Δ^{1} → S^{1} ,}

and again this map is simplicially contractible.

Finally we use a weak equivalence tT HH(R) = R ⊗ tS^{1} → T HH(R), as in Lemma 3.8, to deduce that the diagram (3.10) indeed homotopy commutes.

Remark 3.11 As noted above, T HH(R) is commutative as an R-algebra. The pinch map ψ: dS^{1} → S^{1} ∨ S^{1} is not homotopy cocommutative, so we do not expect that the coproduct ψ: T HH(R) → T HH(R) ∧_{R} T HH(R) will be cocommutative in any great generality. The Hopf algebras arising in this fashion will therefore be commutative, but not cocommutative.

The inclusion of the base point η: ∗ → S^{1} induces a cofiber sequence of Rmodules

\xymatrix{R = R ∧ ∗_{+} \ar[r]^{1∧η_{+}} & R ∧ S_{+}^{1} \ar[r]^{j} & R ∧ S^{1} = ΣR}

which is canonically split by the retraction map

\xymatrix{R ∧ S_{+}^{1} \ar[r]^{1∧ϵ_{+}} & R ∧ ∗_{+} = R .}

Hence in the stable homotopy category there is a canonical section κ: ΣR → R ∧ S_{+}^{1} to the map labeled j above. We let

σ = ω ∘ κ: ΣR → R ∧ S_{+}^{1} → T HH(R)  (3.12)

be the composite stable map. It induces an operator

σ: H_{∗}(ΣR; 𝔽_{p}) → H_{∗}(T HH(R); 𝔽_{p}) ,

which we in Proposition 4.9 shall see is compatible with that of Proposition 2.4.

4 The Bökstedt spectral sequence

Let R be an S -algebra. To calculate the mod p homology H_{∗}(T HH(R); 𝔽_{p}) of its topological Hochschild homology, Bökstedt constructed a strongly convergent spectral sequence

E_{s,∗}^{2}(R) = HH_{s}(H_{∗}(R; 𝔽_{p})) ⟹ H_{s+∗}(T HH(R); 𝔽_{p}) ,  (4.1)

using the skeleton filtration on T HH(R). In fact,

E_{s,∗}^{1}(R) = H_{∗}(R^{∧(s+1)}; 𝔽_{p}) ≅ H_{∗}(R; 𝔽_{p})^{⊗(s+1)} = C_{s}(H_{∗}(R; 𝔽_{p}))

equals the Hochschild s-chains of the algebra Λ = H_{∗}(R; 𝔽_{p}) over k = 𝔽_{p}, and the d^{1}-differential can as usual be identified with the Hochschild boundary operator ∂. To be quite precise, the E^{1}-term is really the associated normalized complex Λ ⊗ Λ̄^{⊗s}, with Λ̄ = Λ/k, but this change does not affect the E^{2}-term.

This is naturally a spectral sequence of A_{∗}-comodules, where A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) is the dual of the mod p Steenrod algebra, since it is obtained by applying mod p homology to a filtered spectrum. If R is a commutative S -algebra, the spectral sequence admits more structure.

Proposition 4.2 Let R be a commutative S -algebra. Then H_{∗}(T HH(R); 𝔽_{p}) is an augmented commutative A_{∗}-comodule H_{∗}(R; 𝔽_{p})-algebra, and the Bökstedt spectral sequence E_{∗∗}^{r}(R) is an augmented commutative A_{∗}-comodule H_{∗}(R; 𝔽_{p})-algebra spectral sequence, converging to H_{∗}(T HH(R); 𝔽_{p}).

Proof We know that T HH(R) is an augmented commutative R-algebra by Theorem 3.9, and the relevant structure maps (3.3) are all maps of simplicial spectra. Hence they respect the skeleton filtration on T HH(R), and we have in particular a composite map of spectral sequences

\xymatrix{E_{∗∗}^{r}(R) ⊗_{Λ} E_{∗∗}^{r}(R) → ′E_{∗∗}^{r} \ar[r]^{ϕ} & E_{∗∗}^{r}(R)}

with ′E_{∗∗}^{r} the spectral sequence associated to the skeleton filtration on the smash product T HH(R) ∧_{R} T HH(R). The left hand map is induced by the usual homology cross product from the E^{1}-term and onwards. This defines the algebra structure on E_{∗∗}^{r}(R), and the remaining claims are straightforward.

In fact, we can settle for less than strict commutativity to have an algebra structure in the Bökstedt spectral sequence. We shall make use of this for some examples related to the Brown–Peterson spectrum in Section 5. An E_{n} ring spectrum [BMMS86, I.4] is an S -module with an action by the little ncubes operad C_{n}, or one that is related to it by a chain of Σ-equivariant weak equivalences. Fiedorowicz and Vogt [FV, 3.4] show that any E_{n+1} ring spectrum is weakly equivalent to an associative S -algebra with an action by the little ncubes operads, i.e., to a C_{n}-algebra in the category of associative S -algebras. Basterra and Mandell (unpublished) have obtained the same result by a different method.

Thus any continuous functor from associative S -algebras takes an E_{n+1} ring spectrum to a C_{n}-algebra in its target category. This applies, for example, to the functor T HH_{•}(−) from associative S -algebras to cyclic S -modules, and to its geometric realization T HH(−) from associative S -algebras to S^{1}-equivariant S -modules. Therefore T HH of an E_{n+1} ring spectrum R is an S^{1}-equivariant E_{n} ring spectrum, and the unit η: R → T HH(R) is a map of E_{n} ring spectra. We do not discuss whether an E_{n+1} ring spectrum structure on R, for some finite n, suffices to define a more-or-less strictly coassociative coproduct or Hopf algebra structure on T HH(R).

Proposition 4.3 Let R be an E_{2} ring spectrum. Then the Bökstedt spectral sequence E_{∗∗}^{r}(R) converging to H_{∗}(T HH(R); 𝔽_{p}) is an A_{∗}-comodule 𝔽_{p}- algebra spectral sequence. If R is an E_{3} ring spectrum, then E_{∗∗}^{r}(R) is a commutative H_{∗}(R; 𝔽_{p})-algebra spectral sequence in A_{∗}-comodules.

Proof To make sense of T HH(R), we have implicitly replaced R with a weakly equivalent associative S -algebra. By the discussion above, for R an E_{2} ring spectrum there is a C_{1}-algebra action on the cyclic S -module T HH_{•}(R), on the A_{∗}-comodule H_{∗}(T HH(R); 𝔽_{p}) and on the Bökstedt spectral sequence E_{∗∗}^{r}(R). This C_{1}-action induces the desired A_{∗}-comodule 𝔽_{p}-algebra structures. For example, the spectral sequence pairing is the composite

\xymatrix{E_{∗∗}^{r}(R) ⊗_{𝔽_{p}} E_{∗∗}^{r}(R) → ′E_{∗∗}^{r} \ar[r]^{μ} & E_{∗∗}^{r}(R) ,}

with ′E_{∗∗}^{r} the spectral sequence associated to T HH(R) ∧ T HH(R) and μ induced by any point in C_{1}(2). When R is an E_{3} ring spectrum, T HH(R) is an E_{2} ring spectrum and homotopy commutative, so the algebra structures are commutative. The unit S -algebra map η: R → T HH(R) is then homotopy central, and supplies the H_{∗}(R; 𝔽_{p})-algebra structure.

Here is a typical application of this algebra structure.

Corollary 4.4 If R is an E_{2} ring spectrum and H_{∗}(R; 𝔽_{p}) is a polynomial algebra over 𝔽_{p}, then E_{∗∗}^{r}(R) collapses at the E^{2}-term, so E_{∗∗}^{2}(R) = E_{∗∗}^{∞}(R). Furthermore, there are no nontrivial (left or right) H_{∗}(R; 𝔽_{p})-module extensions.

Proof If H_{∗}(R; 𝔽_{p}) = P (x_{i}) is a polynomial algebra, where i ranges through some indexing set I, then E_{∗∗}^{2}(R) = HH_{∗}(P (x_{i})) = P (x_{i})⊗E(σx_{i}) by Proposition 2.4 (and passage to colimits). All of the 𝔽_{p}-algebra generators are in filtration ≤ 1, so all differentials on these classes are zero, since the Bökstedt spectral sequence is a right half plane homological spectral sequence. Thus there are no further differentials in this spectral sequence, and E_{∗∗}^{2}(R) = E_{∗∗}^{∞}(R). Furthermore, the E^{∞}-term is a free H_{∗}(R; 𝔽_{p})-module, so also H_{∗}(T HH(R); 𝔽_{p}) ≅ H_{∗}(R; 𝔽_{p}) ⊗ E(σx_{i}) is a free H_{∗}(R; 𝔽_{p})-module.

There may, as we shall see in Section 5, be multiplicative extensions between E_{∗∗}^{∞}(R) and H_{∗}(T HH(R); 𝔽_{p}), as well as A_{∗}-comodule extensions.

A flatness hypothesis is required for the spectral sequence to carry the coproduct and full Hopf algebra structure. Our Sections 5, 6 and 7 will show many examples of Bökstedt spectral sequences with this structure.

Theorem 4.5 Let R be a commutative S -algebra and write Λ = H_{∗}(R; 𝔽_{p}).

(a) If H_{∗}(T HH(R); 𝔽_{p}) is flat over Λ, then there is a coproduct

ψ: H_{∗}(T HH(R); 𝔽_{p}) → H_{∗}(T HH(R); 𝔽_{p}) ⊗_{Λ} H_{∗}(T HH(R); 𝔽_{p}) and H_{∗}(T HH(R); 𝔽_{p}) is an A_{∗}-comodule Λ-Hopf algebra.

(b) If each term E_{∗∗}^{r}(R) for r ≥ 2 is flat over Λ, then there is a coproduct

ψ: E_{∗∗}^{r}(R) → E_{∗∗}^{r}(R) ⊗_{Λ} E_{∗∗}^{r}(R)

and E_{∗∗}^{r}(R) is an A_{∗}-comodule Λ-Hopf algebra spectral sequence. In particular, the differentials d^{r} respect the coproduct ψ.

Proof Write T = T HH(R). There is a Künneth spectral sequence with

E_{∗∗}^{2} = Tor_{∗∗}^{Λ}(H_{∗}(T; 𝔽_{p}), H_{∗}(T; 𝔽_{p})) ⟹ H_{∗}(T ∧_{R} T; 𝔽_{p}) .

When H_{∗}(T; 𝔽_{p}) is flat over Λ the higher Tor-groups vanish, the spectral sequence collapses, and the map

ψ: H_{∗}(T; 𝔽_{p}) → H_{∗}(T ∧_{R} T; 𝔽_{p}) ≅ H_{∗}(T; 𝔽_{p}) ⊗_{Λ} H_{∗}(T; 𝔽_{p})

induces the coproduct in part (a).

For part (b) let dE_{∗∗}^{r} be the spectral sequence associated to the skeleton filtration on dT HH(R) = R ⊗ dS^{1}, and let ′E_{∗∗}^{r} be the spectral sequence associated to T ∧_{R} T = R ⊗ (S^{1} ∨ S^{1}), as in the proof of Proposition 4.2. Then there are natural maps of spectral sequences

\xymatrix{E_{∗∗}^{r}(R) & dE_{∗∗}^{r} \ar[l]_{π} \ar[r]^{ψ′} & ′E_{∗∗}^{r} & E_{∗∗}^{r}(R) ⊗_{Λ} E_{∗∗}^{r}(R) . \ar[l]_{sh}}  (4.6)

Here π: dE_{∗∗}^{r} → E_{∗∗}^{r}(R) is an isomorphism for r ≥ 2, by the algebraic analogue of Lemma 3.8. The map ψ′: dE_{∗∗}^{r} → ′E_{∗∗}^{r} is induced by the simplicial pinch map ψ′ from (3.7). As regards the final (shuffle) map sh, we have

′E_{s,∗}^{1} ≅ Λ^{⊗(s+1)} ⊗_{Λ} Λ^{⊗(s+1)} ≅ E_{s,∗}^{1}(R) ⊗_{Λ} E_{s,∗}^{1}(R)

by the collapsing Künneth spectral sequence for

H_{∗}(T HH_{s}(R) ∧_{R} T HH_{s}(R); 𝔽_{p}) ,

and the map sh for r = 1 is the shuffle equivalence from the bigraded tensor product [E_{∗∗}^{1}(R) ⊗_{Λ} E_{∗∗}^{1}(R)]_{s,∗}. By assumption E_{∗∗}^{2}(R) = HH_{∗}(Λ) is flat over Λ, so by the algebraic Künneth spectral sequence and the Eilenberg–Zilber theorem

\xymatrix{sh: E_{∗∗}^{2}(R) ⊗_{Λ} E_{∗∗}^{2}(R) ≅ H_{∗}(E_{∗∗}^{1}(R) ⊗_{Λ} E_{∗∗}^{1}(R)) \ar[r]^{≅} & H_{∗}(′E_{∗∗}^{1}) = ′E_{∗∗}^{2}}

is an isomorphism. Inductively, suppose that sh is an isomorphism for a fixed r ≥ 2, and that E_{∗∗}^{r+1}(R) is flat over Λ. Then by the algebraic Künneth spectral sequence again

\xymatrix{sh: E_{∗∗}^{r+1}(R) ⊗_{Λ} E_{∗∗}^{r+1}(R) ≅ H_{∗}(E_{∗∗}^{r}(R) ⊗_{Λ} E_{∗∗}^{r}(R)) \ar[r]^{≅} & H_{∗}(′E_{∗∗}^{r}) = ′E_{∗∗}^{r+1}}

is also an isomorphism, as desired. The coproduct ψ on E_{∗∗}^{r}(R) is then the composite map (sh)^{−1} ∘ ψ′ ∘ π^{−1}, for r ≥ 2. The conjugation χ on E_{∗∗}^{r}(R) is more simply defined, as the composite map π ∘ χ′ ∘ π^{−1}.

By Proposition 2.3, the coproduct (and Hopf algebra structure) on the E^{2}-term E_{∗∗}^{2}(R) = HH_{∗}(H_{∗}(R; 𝔽_{p})) that is derived from the R-Hopf algebra structure on T HH(R) agrees with the algebraically defined structure on the Hochschild homology HH_{∗}(Λ) of the commutative algebra Λ = H_{∗}(R; 𝔽_{p}).

Remark 4.7 The same proof shows that if only the initial terms

E_{∗∗}^{2}(R), …, E_{∗∗}^{r_{0}}(R)

of the Bökstedt spectral sequence are flat over Λ, for some integer r_{0}, then these are all A_{∗}-comodule Λ-Hopf algebras and the corresponding differentials d^{2}, …, d^{r_{0}} all respect that structure.

There are natural examples that show that the flatness hypothesis is not always realistic. For example, Ausoni [Au] has studied the case of R = ku at an odd prime p, where H_{∗}(ku; 𝔽_{p}) = H_{∗}(ℓ; 𝔽_{p}) ⊗ P_{p−1}(x) for a class x in degree 2, and already the E^{2}-term E_{∗∗}^{2}(ku) = HH_{∗}(H_{∗}(ku; 𝔽_{p})) is not flat over H_{∗}(ku; 𝔽_{p}). We shall also see in Proposition 7.13(c) that for R = j, the connective, real image-of-J spectrum at p = 2, the E^{2}-term E_{∗∗}^{2}(j) = HH_{∗}(H_{∗}(j; 𝔽_{2})) is not flat over H_{∗}(j; 𝔽_{2}).

We say that the graded k-algebra Λ is connected when it is trivial in negative degrees and the unit map η: k → Λ is an isomorphism in degree 0. We write ν: H_{∗}(X; 𝔽_{p}) → A_{∗} ⊗ H_{∗}(X; 𝔽_{p}) for the A_{∗}-comodule coaction map, and say that a class x ∈ H_{∗}(X; 𝔽_{p}) is A_{∗}-comodule primitive if ν(x) = 1 ⊗ x.

Proposition 4.8 Let R be a commutative S -algebra with Λ = H_{∗}(R; 𝔽_{p}) connected and such that HH_{∗}(Λ) is flat over Λ. Then the E^{2}-term of the first quadrant Bökstedt spectral sequence

E_{∗∗}^{2}(R) = HH_{∗}(Λ)

is an A_{∗}-comodule Λ-Hopf algebra, and a shortest non-zero differential d_{s,t}^{r} in lowest total degree s+t, if one exists, must map from an algebra indecomposable to a coalgebra primitive and A_{∗}-comodule primitive, in HH_{∗}(Λ).

Proof If d^{2}, …, d^{r−1} are all zero, then E_{∗∗}^{2}(R) = E_{∗∗}^{r}(R) is still an A_{∗}- comodule Λ-Hopf algebra. If d^{r}(xy) ≠ 0, where xy is decomposable (a product of classes of positive degree), then the Leibniz formula

d^{r}(xy) = d^{r}(x)y ± xd^{r}(y)

implies that d^{r}(x) ≠ 0 or d^{r}(y) ≠ 0, so xy cannot be in the lowest possible total degree for the source of a differential. Dually, if d^{r}(z) is not coalgebra primitive, with ψ(z) = z ⊗ 1 + 1 ⊗ z + ∑_{i} z′_{i} ⊗ z′′_{i}, then the co-Leibniz formula

ψ ∘ d^{r} = (d^{r} ⊗ 1 ± 1 ⊗ d^{r})ψ

(tensor products over Λ) implies that some term d^{r}(z′_{i}) ≠ 0 or d^{r}(z′′_{i}) ≠ 0, so z cannot be in the lowest possible total degree. Finally, if d^{r}(z) is not A_{∗}-comodule primitive, with ν(z) = 1 ⊗ z + ∑_{i} a_{i} ⊗ z_{i}, then the co-linearity condition

ν ∘ d^{r} = (1 ⊗ d^{r})ν

implies that some term d^{r}(z_{i}) ≠ 0, so z cannot be in the lowest possible total degree.

The last two arguments are perhaps easier to visualize in the 𝔽_{p}-vector space dual spectral sequence. The 𝔽_{p}-dual of the coproduct ψ maps from the cotensor product

HH_{∗}(Λ)^{∗} □_{Λ^{∗}} HH_{∗}(Λ)^{∗} ⊂ HH_{∗}(Λ)^{∗} ⊗_{𝔽_{p}} HH_{∗}(Λ)^{∗}

to HH_{∗}(Λ)^{∗}.

Proposition 4.9 Let R be a commutative S -algebra. For each element x ∈ H_{t}(R; 𝔽_{p}) the image σx ∈ H_{t+1}(T HH(R); 𝔽_{p}) is the coalgebra primitive

ψ(σx) = σx ⊗ 1 + 1 ⊗ σx

that is represented in E_{∗∗}^{∞}(R) by the class σx = [1 ⊗ x] ∈ E_{1,t}^{2}(R) = HH_{1}(Λ)_{t}.

Proof Note that the coproduct on T HH(R) is compatible under ω: R∧S_{+}^{1} → T HH(R) with the pinch map R ∧ dS_{+}^{1} → R ∧ (S^{1} ∨ S^{1})_{+}, and thus under σ: ΣR → T HH(R) with the pinch map ΣR → ΣR ∨ ΣR. The claims then follow by inspection of the definitions in Section 3.

See also [MS93, 3.2] for the last claim above.

5 Differentials and algebra extensions

We now apply the Bökstedt spectral sequence (4.1) to compute the mod p homology of T HH(R) for the Brown–Peterson and Johnson–Wilson S -algebras R = BP ⟨m−1⟩ for 0 ≤ m ≤ ∞. In each case we can replace R by its plocalization R_{(p)} or p-completion R_{p} without changing the mod p homology of T HH(R), so we will sometimes do so without further comment. The cases R = H𝔽_{p} and R = Hℤ were first treated by Bökstedt [Bo2], and the case R = ℓ_{p} ⊂ ku_{p} (the Adams summand of p-complete connective K -theory) is due to McClure and Staffeldt [MS93, 4.2].

5.1 The dual Steenrod algebra

Let A = H^{∗}(H𝔽_{p}; 𝔽_{p}) be the Steenrod algebra, with generators Sq^{i} for p = 2 and β and P^{i} for p odd. We recall the structure of its dual A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) from [Mi60, Thm. 2]. When p = 2 we have

A_{∗} = P (ξ_{k} | k ≥ 1) = P (ξ̄_{k} | k ≥ 1)

where ξ_{k} has degree 2^{k} − 1 and ξ̄_{k} = χ(ξ_{k}) is the conjugate class. Most of the time it will be more convenient for us to use the conjugate classes. The coproduct is given by

ψ(ξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{2^{i}} ,

where as usual we read ξ̄_{0} to mean 1. When p is odd we have

A_{∗} = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 0)

with ξ̄_{k} = χ(ξ_{k}) in degree 2(p^{k} − 1) and τ̄_{k} = χ(τ_{k}) in degree 2p^{k} − 1. The coproduct is given by

ψ(ξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ ξ̄_{j}^{p^{i}}  and  ψ(τ̄_{k}) = 1 ⊗ τ̄_{k} + ∑_{i+j=k} τ̄_{i} ⊗ ξ̄_{j}^{p^{i}} .

The mod p homology Bockstein satisfies β(τ̄_{k}) = ξ̄_{k}.

Any commutative S -algebra R has a canonical structure as an E_{∞} ring spectrum [EKMM97, II.3.4]. In particular, its mod p homology H_{∗}(R; 𝔽_{p}) admits natural Dyer–Lashof operations

Q^{k}: H_{∗}(R; 𝔽_{2}) → H_{∗+k}(R; 𝔽_{2})

for p = 2 and

Q^{k}: H_{∗}(R; 𝔽_{p}) → H_{∗+2k(p−1)}(R; 𝔽_{p})

for p odd. Their formal properties are summarized in [BMMS86, III.1.1], and include Cartan formulas, Adem relations and Nishida relations. For p = 2, Q^{k}(x) = 0 when k < |x| and Q^{k}(x) = x^{2} when k = |x|. For p odd, Q^{k}(x) = 0 when k < 2|x| and Q^{k}(x) = x^{p} when k = 2|x|. In the special case of R = H𝔽_{p}, the Dyer–Lashof operations in A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}) satisfy

Q^{p^{k}}(ξ̄_{k}) = ξ̄_{k+1}

for all primes p, and

Q^{p^{k}}(τ̄_{k}) = τ̄_{k+1}

for p odd. These formulas were first obtained by Leif Kristensen (unpublished), and appeared in print in [BMMS86, III.2.2 and III.2.3].

When R is just an E_{n} ring spectrum, for 1 ≤ n < ∞, only some of these operations are naturally defined. For p = 2 the operations Q^{k}(x) are defined for k ≤ |x| + n − 1, and for p odd they are defined for 2k ≤ |x| + n − 1. In the case of equality, the “top” operation Q^{k}(x) (denoted ξ_{n−1}(x) in op. cit.) is not additive, and special care is called for. We refer to [BMMS86, III.3.1–3] for detailed information about the formal properties that are satisfied in the homology of an E_{n} ring spectrum.

5.2 The Johnson–Wilson spectra BP ⟨m−1⟩

For any prime p let BP = BP ⟨∞⟩ be the Brown–Peterson spectrum and let BP ⟨m−1⟩ for 0 ≤ m < ∞ be the spectrum introduced by Johnson and Wilson in [JW73], with mod p cohomology

H^{∗}(BP; 𝔽_{p}) ≅ A//E = A ⊗_{E} 𝔽_{p}

and

H^{∗}(BP ⟨m−1⟩; 𝔽_{p}) ≅ A//E_{m−1} = A ⊗_{E_{m−1}} 𝔽_{p} .

Here E ⊂ A is the exterior subalgebra generated by the Milnor primitives Q_{k} for k ≥ 0, while E_{m−1} is the exterior subalgebra generated by Q_{0}, …, Q_{m−1}. These elements are inductively defined by Q_{0} = Sq^{1} and Q_{k} = [Sq^{2^{k}}, Q_{k−1}] for p = 2, and by Q_{0} = β and Q_{k+1} = [P^{p^{k}}, Q_{k}] for p odd, see [Mi60, §1]. We note the special cases BP ⟨−1⟩ = H𝔽_{p}, BP ⟨0⟩ = Hℤ_{(p)} and BP ⟨1⟩ = ℓ ⊂ ku_{(p)}. For p = 2 we have ℓ = ku_{(2)}.

The Brown–Peterson and Johnson–Wilson spectra have homotopy groups

π_{∗}BP = ℤ_{(p)}[v_{k} | k ≥ 1]

for m = ∞ and

π_{∗}BP ⟨m−1⟩ = ℤ_{(p)}[v_{1}, …, v_{m−1}] = π_{∗}BP/(v_{k} | k ≥ m)

for 0 ≤ m < ∞, with v_{0} = p. The class v_{k} is detected in the Adams spectral sequence E_{2}^{∗∗} = Ext_{A_{∗}}^{∗∗}(𝔽_{p}, H_{∗}(BP; 𝔽_{p})) for π_{∗}BP by the normalized cobar cocycle

∑_{i+j=k+1}[ξ̄_{i}]ξ̄_{j}^{2^{i}}

for p = 2 (the term for i = 0 is zero) and

− ∑_{i+j=k}[τ̄_{i}]ξ̄_{j}^{p^{i}}

for p odd [Ra04, p. 63]. Under the change-of-rings isomorphism to E_{2}^{∗∗} ≅ Ext_{E_{∗}}^{∗∗}(𝔽_{p}, 𝔽_{p}), where E_{∗} = E(ξ̄_{k} | k ≥ 1) for p = 2 and E_{∗} = E(τ̄_{k} | k ≥ 0) for p odd, these cobar cocycles correspond to [ξ̄_{k+1}] and −[τ̄_{k}], respectively. Modulo decomposables, we have ξ̄_{k+1} ≡ ξ_{k+1} for p = 2 and −τ̄_{k} ≡ τ_{k} for p odd.

In each case 0 ≤ m ≤ ∞, the spectrum BP ⟨m−1⟩ admits the structure of an associative S -algebra. More precisely, there is a tower of associative MU - algebras

BP → ⋯ → BP ⟨m⟩ → BP ⟨m−1⟩ → ⋯ → H𝔽_{p} ,

which by omission of structure is also a tower of S -algebras. See [BJ02, 3.5]. It is well known that H𝔽_{p} and Hℤ_{(p)} admit unique structures as commutative S -algebras, and that the p-complete Adams summand ℓ_{p} ⊂ ku_{p} admits at least one such structure [MS93, §9]. In fact each of ℓ, ku, ℓ_{p} and ku_{p} admits a unique commutative S -algebra structure [BaRi].

It remains a well-known open problem whether BP admits a commutative S - algebra structure. Basterra and Mandell (unpublished) have shown that BP is at least an E_{4} ring spectrum. We shall make use of the existence of an E_{3} ring spectrum structure on BP for the calculations in this section.

For p = 2 and 2 < m < ∞, Strickland [Str99, 6.5] has shown that BP ⟨m−1⟩ is not a homotopy commutative MU -ring spectrum with the most common (e.g. Araki or Hazewinkel) choices of maps MU → BP ⟨m−1⟩, but he also shows that there is a less familiar replacement BP ⟨m−1⟩′ that may serve as a substitute [Str99, 2.10]. We do not know for which p and m the spectrum BP ⟨m−1⟩ admits an E_{2}- or E_{3} ring spectrum structure.

Proposition 5.3 For each p and 0 ≤ m ≤ ∞ the unique S -algebra map BP ⟨m−1⟩ → H𝔽_{p} induces the identifications

H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) = P (ξ̄_{1}^{2}, …, ξ̄_{m}^{2}, ξ̄_{k} | k ≥ m + 1)

when p = 2, and

H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ m)

when p is odd, as 𝔽_{p}-subalgebras of A_{∗}.

Proof See [Wi75, 1.7] and dualize. In the case m = ∞, the notation is meant to imply that there are no classes ξ̄_{k} with k ≥ m + 1, and no classes τ̄_{k} with k ≥ m.

In particular,

H_{∗}(Hℤ; 𝔽_{2}) = P (ξ̄_{1}^{2}, ξ̄_{k} | k ≥ 2)
H_{∗}(ku; 𝔽_{2}) = P (ξ̄_{1}^{2}, ξ̄_{2}^{2}, ξ̄_{k} | k ≥ 3)
H_{∗}(BP; 𝔽_{2}) = P (ξ̄_{k}^{2} | k ≥ 1)

for p = 2, and

H_{∗}(Hℤ; 𝔽_{p}) = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 1)
H_{∗}(ℓ; 𝔽_{p}) = P (ξ̄_{k} | k ≥ 1) ⊗ E(τ̄_{k} | k ≥ 2)
H_{∗}(BP; 𝔽_{p}) = P (ξ̄_{k} | k ≥ 1)

for p odd.

The E^{2}-term of the Bökstedt spectral sequence for BP ⟨m−1⟩ can now be computed from Proposition 2.4. It is

E_{∗∗}^{2}(BP ⟨m−1⟩) = H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}, σξ̄_{k} | k ≥ m + 1)

when p = 2, and

E_{∗∗}^{2}(BP ⟨m−1⟩) = H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ m)

when p is odd. Note that in all cases for p = 2, as well as the case m = ∞ for p odd, the E^{2}-term is generated as an 𝔽_{p}-algebra by classes in filtration ≤ 1.

5.4 Odd-primary differentials

In the remaining cases, with p odd and 0 ≤ m < ∞, there are non-trivial d^{p−1}- differentials in the Bökstedt spectral sequence. These can all be determined by naturality with respect to the map of S -algebras BP ⟨m−1⟩ → H𝔽_{p}, since the map of E^{2}-terms

E_{∗∗}^{2}(BP ⟨m−1⟩) → E_{∗∗}^{2}(H𝔽_{p}) = A_{∗} ⊗ E(σξ̄_{k} | k ≥ 1) ⊗ Γ(στ̄_{k} | k ≥ 0)

is injective. Now H𝔽_{p} is a connective, commutative S -algebra, and the E^{2}- term is free, hence flat, over A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}), so we are in the situation of Proposition 4.8. The A_{∗}-algebra generators (indecomposables) of the E^{2}-term are in filtrations p^{i} for i ≥ 0, by formula (2.5), and the A_{∗}-coalgebra primitives are all in filtration 1, by Proposition 2.4, so the shortest non-zero differential in lowest total degree, if any, must go from some filtration p^{i} for i ≥ 1 to filtration 1.

Indeed, Bökstedt [Bo2, 1.3] found that in the case R = H𝔽_{p} there are nontrivial d^{p−1}-differentials in his spectral sequence computing H_{∗}(T HH(R); 𝔽_{p}). In our notation, they are given by the formula

d^{p−1}(γ_{j}(στ̄_{k})) = σξ̄_{k+1} · γ_{j−p}(στ̄_{k})  (5.5)

for j ≥ p, up to a unit in 𝔽_{p}. This way of writing Bökstedt’s formula first appears in [MS93, p. 21]. A proof of a more general result that implies (5.5) was published by Hunter [Hu96, Thm. 1], as recalled below.

Proposition 5.6 Let R be a commutative S -algebra, and let x ∈ H_{2i−1}(R; 𝔽_{p}) for some i ≥ 1. Then in the Bökstedt spectral sequence E_{∗∗}^{r}(R) the differentials d^{r} all vanish for 2 ≤ r ≤ p − 2, and there is a differential

d^{p−1}(γ_{p}(σx)) = σ(βQ^{i}(x))

up to a unit in 𝔽_{p}.

A short proof of Bökstedt’s formula in particular cases was later found by Ausoni [Au, 4.3]. More recently, the first author has given a construction of T HH of any A_{∞} ring spectrum as a kind of geometric realization using cyclohedra [Sta97], similar to Stasheff’s bar construction of an A_{∞} H - space in [Sta63]. This construction exhibits the d^{p−1}-differential on γ_{p}(σx) as being represented by σ(⟨x, …, x⟩) in the Bökstedt spectral sequence, where ⟨x, …, x⟩ ∈ H_{2pi−2}(R; 𝔽_{p}) denotes the p-fold Massey product. In this formulation only the strict associativity of R is needed.

The proposition above applies with R = H𝔽_{p}, x = τ̄_{k}, i = p^{k} and βQ^{i}(x) = βτ̄_{k+1} = ξ̄_{k+1}, to establish Bökstedt’s formula (5.5) in the special case of j = p. The general case follows from this by induction on j ≥ p and the coalgebra structure on the Bökstedt spectral sequence. In more detail, a comparison of ψd^{p−1}(γ_{j}(στ̄_{k})) = (d^{p−1} ⊗ 1 + 1 ⊗ d^{p−1})(ψγ_{j}(στ̄_{k})) and ψ(σξ̄_{k+1} · γ_{j−p}(στ̄_{k})) shows that the difference d^{p−1}(γ_{j}(στ̄_{k}))−σξ̄_{k+1} ·γ_{j−p}(στ̄_{k}) must be a coalgebra primitive, and there are none such other than zero in its bidegree when j > p.

By naturality, the formula (5.5) for d^{p−1} holds also in the Bökstedt spectral sequence for BP ⟨m−1⟩ at odd primes p, whether this S -algebra is commutative or not. We view its E^{2} = E^{p−1}-term as the tensor product of the complexes E(σξ̄_{k+1}) ⊗ Γ(στ̄_{k}) for k ≥ m and the remaining terms H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}). Applying the Künneth formula, we compute its homology to be

E_{∗∗}^{p}(BP ⟨m−1⟩) = H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P_{p}(στ̄_{k} | k ≥ m) .

At this point we note that also for 0 ≤ m < ∞ and p odd, the E^{p}-term of the Bökstedt spectral sequence is generated as an 𝔽_{p}-algebra by classes in filtration ≤ 1.

To proceed, we shall need to assume that the Bökstedt spectral sequence is an algebra spectral sequence. We know that the hypothesis of the following proposition is satisfied in the commutative cases 0 ≤ m ≤ 2 and in the E_{4} ring spectrum case m = ∞.

Proposition 5.7 Let p and 0 ≤ m ≤ ∞ be such that BP ⟨m−1⟩ admits the structure of an E_{2} ring spectrum. Then its Bökstedt spectral sequence collapses at the E^{p} = E^{∞}-term, which equals

E_{∗∗}^{∞}(BP ⟨m−1⟩) = H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}, σξ̄_{k} | k ≥ m + 1)

when p = 2, and

E_{∗∗}^{∞}(BP ⟨m−1⟩) = H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P_{p}(στ̄_{k} | k ≥ m)

when p is odd. In each case the E^{∞}-term exhibits H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) as a free H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-module.

Proof By Proposition 4.3 we know that the Bökstedt spectral sequence is an 𝔽_{p}-algebra spectral sequence, and by the calculations above we know that its E^{p}-term is generated as an 𝔽_{p}-algebra by classes in filtrations ≤ 1, which must be infinite cycles since the spectral sequence is concentrated in the first quadrant. Thus there is no room for any further differentials and E^{p} = E^{∞}. In each case the E^{∞}-term is free as an H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-module, so there is no room for additive (module-)extensions, either.

5.8 Algebra extensions

To determine H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) as an H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-algebra, we need to resolve the possible multiplicative extensions. For this we will use the first nontrivial Dyer–Lashof operation after the Pontryagin power, the existence of which presupposes that T HH(BP ⟨m−1⟩) is an E_{2} ring spectrum. We shall therefore assume that p and m are such that BP ⟨m−1⟩ is an E_{3} ring spectrum.

For a commutative S -algebra R, the map σ: ΣR → T HH(R) relates the Dyer–Lashof operations on R to those of T HH(R) by the following formula of Bökstedt [Bo2, 2.9]:

Q^{k}(σx) = σQ^{k}(x) .

See also [MS93, p. 22]. There appears to be no published proof of this key relation, so we offer the following slightly more general result.

Recall from Definition 2.6 the S^{1}-action map α: T HH(R) ∧ S_{+}^{1} → T HH(R), inducing the homomorphism

α: H_{∗}(T HH(R); 𝔽_{p}) ⊗ H_{∗}(S_{+}^{1}; 𝔽_{p}) ≅

H_{∗}(T HH(R) ∧ S_{+}^{1}; 𝔽_{p}) → H_{∗}(T HH(R); 𝔽_{p})

in homology. Let s_{1} ∈ H_{1}(S_{+}^{1}; 𝔽_{p}) be the canonical generator. Also recall that any commutative S -algebra R is an E_{∞} ring spectrum, so that the following proposition applies to such an R with n = ∞, and thus for all integers k.

Proposition 5.9 Let R be an E_{n+1} ring spectrum, so that T HH(R) is an E_{n} ring spectrum. Then we have

Q^{k}(α(x ⊗ s_{1})) = α(Q^{k}(x) ⊗ s_{1})

for all classes x ∈ H_{∗}(T HH(R); 𝔽_{p}) and integers k such that the operation Q^{k}(x) is naturally defined, i.e., for k ≤ |x| + n − 1 when p = 2 and for 2k ≤ |x| + n − 1 when p is odd. In particular, we have

Q^{k}(σx) = σQ^{k}(x)

for all classes x ∈ H_{∗}(R; 𝔽_{p}) and the same integers k.

Proof By [FV, 3.4], or the definition T HH(R) ≅ R ⊗ S^{1} in the strictly commutative case, the circle acts on T HH(R) by C_{n}-algebra maps. Thus the right adjoint map

α̃: T HH(R) → F (S_{+}^{1}, T HH(R))

is a map of C_{n}-algebras, where the product structure on the right is given by pointwise multiplication, using the strictly cocommutative diagonal map S_{+}^{1} → S_{+}^{1} ∧ S_{+}^{1}.

Let DS_{+}^{1} = F (S_{+}^{1}, S) be the functional dual of S_{+}^{1}, also with the pointwise multiplication. It has mod p homology H_{∗}(DS_{+}^{1}; 𝔽_{p}) ≅ H^{−∗}(S_{+}^{1}; 𝔽_{p}) = E(ι_{1}) for a canonical class ι_{1} ∈ H^{1}(S_{+}^{1}; 𝔽_{p}) dual to the class s_{1} ∈ H_{1}(S_{+}^{1}; 𝔽_{p}). The Dyer–Lashof operations Q^{k} on H_{∗}(DS_{+}^{1}; 𝔽_{p}) correspond [BMMS86, III.1.2 and VIII.3] to the Steenrod operations P^{−k} on H^{−∗}(S_{+}^{1}; 𝔽_{p}), hence are trivial for k ≠ 0. The same conclusions hold for p = 2, with somewhat different notation.

There is a canonical map of C_{n}-algebras

ν: T HH(R) ∧ DS_{+}^{1} → F (S_{+}^{1}, T HH(R)) ,

given by the composition of functions

F (S, T HH(R)) ∧ F (S_{+}^{1}, S) → F (S_{+}^{1}, T HH(R)) .

Compare [LMS86, III.1]. The map ν is an equivalence since S_{+}^{1} is a finite CW complex (i.e., by Spanier–Whitehead duality). Hence there are homomorphisms

\xymatrix{H_{∗}(T HH(R); 𝔽_{p}) \ar[r]^{α̃} & H_{∗}(F (S_{+}^{1}, T HH(R)); 𝔽_{p}) \\  & H_{∗}(T HH(R); 𝔽_{p}) ⊗ H_{∗}(DS_{+}^{1}; 𝔽_{p}) \ar[u]_{ν}^{≅}}

that take x ∈ H_{∗}(T HH(R); 𝔽_{p}) to

ν^{−1}α̃(x) = x ⊗ 1 + α(x ⊗ s_{1}) ⊗ ι_{1} .

Since α̃ and ν are maps of C_{n}-algebras, we have

Q^{k}(ν^{−1}α̃(x)) = ν^{−1}α̃(Q^{k}(x))

when k is such that Q^{k} is naturally defined on x ∈ H_{∗}(T HH(R); 𝔽_{p}). The external Cartan formula for Dyer–Lashof operations [BMMS86, III.1.1(6)] then gives us

Q^{k}(x ⊗ 1 + α(x ⊗ s_{1}) ⊗ ι_{1}) = Q^{k}(x) ⊗ 1 + Q^{k}(α(x ⊗ s_{1})) ⊗ ι_{1}

since Q^{i}(1) = 0 and Q^{i}(ι_{1}) = 0 for i ≠ 0. Stringing these formulas together yields

Q^{k}(x) ⊗ 1 + Q^{k}(α(x ⊗ s_{1})) ⊗ ι_{1} = Q^{k}(x) ⊗ 1 + α(Q^{k}(x) ⊗ s_{1}) ⊗ ι_{1} ,

and we can read off Q^{k}(α(x ⊗ s_{1})) = α(Q^{k}(x) ⊗ s_{1}), as desired.

Specializing to classes η(x) ∈ H_{∗}(T HH(R); 𝔽_{p}) that are images under the E_{n} ring spectrum map η: R → T HH(R) of elements x ∈ H_{∗}(R; 𝔽_{p}), we have σx = α(η(x) ⊗ s_{1}) and σQ^{k}(x) = α(η(Q^{k}(x)) ⊗ s_{1}) = α(Q^{k}(η(x)) ⊗ s_{1}). Thus we obtain Bökstedt’s formula Q^{k}(σx) = σQ^{k}(x), valid for the same integers k as above.

The same ideas can be used to prove that σ: H_{∗}(R; 𝔽_{p}) → H_{∗+1}(T HH(R); 𝔽_{p}) is a graded derivation, when R is sufficiently commutative for this to make sense.

Proposition 5.10 Let R be an E_{2} ring spectrum, so that T HH(R) is an S -algebra. Then we have

α(xy ⊗ s_{1}) = x · α(y ⊗ s_{1}) + (−1)^{|y|}α(x ⊗ s_{1}) · y

for x, y ∈ H_{∗}(T HH(R); 𝔽_{p}). In particular, we have the Leibniz rule

σ(x · y) = x · σ(y) + (−1)^{|y|}σ(x) · y

for x, y ∈ H_{∗}(R; 𝔽_{p}).

Proof We keep the notation of the proof of Proposition 5.9. Since α̃ and ν are maps of C_{1}-algebras,

ν^{−1}α̃: H_{∗}(T HH(R); 𝔽_{p}) → H_{∗}(T HH(R); 𝔽_{p}) ⊗ E(ι_{1})

is an 𝔽_{p}-algebra homomorphism. Thus

xy ⊗ 1 + α(xy ⊗ s_{1}) ⊗ ι_{1} = (x ⊗ 1 + α(x ⊗ s_{1}) ⊗ ι_{1}) · (y ⊗ 1 + α(y ⊗ s_{1}) ⊗ ι_{1})

for x, y ∈ H_{∗}(T HH(R); 𝔽_{p}). Multiplying out and comparing ι_{1}-coefficients gives the claimed formulas.

We also wish to describe the A_{∗}-comodule structure on H_{∗}(T HH(R); 𝔽_{p}). In the cases R = BP ⟨m−1⟩ the following observations will suffice. The A_{∗}- comodule coaction map

ν: H_{∗}(R; 𝔽_{p}) → A_{∗} ⊗ H_{∗}(R; 𝔽_{p})

is in each case given by restricting the coproduct

ψ: A_{∗} → A_{∗} ⊗ A_{∗}

given in Subsection 5.1, to the subalgebra H_{∗}(R; 𝔽_{p}) ⊂ A_{∗}, since the latter inclusion is induced by a spectrum map R → H𝔽_{p}. The operator

σ: H_{∗}(ΣR; 𝔽_{p}) → H_{∗}(T HH(R); 𝔽_{p})

is also induced by a spectrum map, hence is an A_{∗}-comodule homomorphism. Hence the coaction map

ν: H_{∗}(T HH(R); 𝔽_{p}) → A_{∗} ⊗ H_{∗}(T HH(R); 𝔽_{p})

satisfies

ν ∘ σ = (1 ⊗ σ)ν .  (5.11)

As before, we know that the hypotheses of the following proposition are all satisfied in the commutative cases 0 ≤ m ≤ 2, and for parts (a) and (c) in the E_{4} ring spectrum case m = ∞.

Theorem 5.12 (a) Let p and 0 ≤ m ≤ ∞ be such that BP ⟨m−1⟩ admits the structure of an E_{3} ring spectrum. Then for m < ∞ we have

H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{2}) =

H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}) ⊗ P (σξ̄_{m+1})

when p = 2, and

H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) = H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P (στ̄_{m})

when p is odd, as H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-algebras. For m = ∞ we have

H_{∗}(T HH(BP); 𝔽_{2}) = H_{∗}(BP; 𝔽_{2}) ⊗ E(σξ̄_{k}^{2} | k ≥ 1)

when p = 2, and

H_{∗}(T HH(BP); 𝔽_{p}) = H_{∗}(BP; 𝔽_{p}) ⊗ E(σξ̄_{k} | k ≥ 1)

when p is odd, as H_{∗}(BP; 𝔽_{p})-algebras.

(b) When p and m are such that BP ⟨m−1⟩ is a commutative S -algebra, then these are isomorphisms of primitively generated H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-Hopf algebras.

(c) The A_{∗}-comodule coaction on H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) is given on the tensor factor H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) by restricting the coproduct on A_{∗}. For p = 2 the algebra generators σξ̄_{k}^{2} for 1 ≤ k ≤ m are A_{∗}-comodule primitives, while

ν(σξ̄_{m+1}) = 1 ⊗ σξ̄_{m+1} + ξ̄_{1} ⊗ σξ̄_{m}^{2} .

For p odd the algebra generators σξ̄_{k} for 1 ≤ k ≤ m are A_{∗}-comodule primitives, while

ν(στ̄_{m}) = 1 ⊗ στ̄_{m} + τ̄_{0} ⊗ σξ̄_{m} .

As usual, ξ̄_{0} is read as 1 in such formulas. Thus for m = 0, σξ̄_{1} and στ̄_{0} are also primitive.

Proof (a) We first resolve the algebra extensions in the E^{∞}-terms from Proposition 5.7. In each case the Bökstedt spectral sequence is one of commutative H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-algebras, by the E_{3} ring spectrum hypothesis and Proposition 4.3. By the functoriality of the zeroth Postnikov section, the S -algebra maps BP ⟨m−1⟩ → Hℤ_{(p)} → H𝔽_{p} and their composite are E_{3} ring spectrum maps. (We only need this for m ≠ 0.) Since the induced map in homology is injective, we can read off the E_{3} ring spectrum Dyer–Lashof operations in H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) from the formulas in A_{∗} = H_{∗}(H𝔽_{p}; 𝔽_{p}). This tells us that Q^{2^{k+1}−1}(ξ̄_{k}^{2}) = 0 (by the Cartan formula) and Q^{2^{k}}(ξ̄_{k}) = ξ̄_{k+1} for p = 2, and that Q^{p^{k}}(τ̄_{k}) = τ̄_{k+1} for p odd.

We will also use the formulas x^{2} = Q^{k}(x) for |x| = k and p = 2, and x^{p} = Q^{k}(x) for |x| = 2k and p odd, from [BMMS86, III.1.1(4)]. Likewise we will use the formulas Q^{k}(σx) = σQ^{k}(x) for |x| = k − 1 and p = 2, and Q^{k}(σx) = σQ^{k}(x) for |x| = 2k − 1 and p odd, from Proposition 5.9. These are all are valid in the homology of the E_{2} ring spectrum T HH(BP ⟨m−1⟩).

With these preliminaries, we are ready to compute. For p = 2 the squares in H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{2}) of the algebra generators in E_{∗∗}^{∞}(BP ⟨m−1⟩) are

(σξ̄_{k}^{2})^{2} = Q^{2^{k+1}−1}(σξ̄_{k}^{2}) = σQ^{2^{k+1}−1}(ξ̄_{k}^{2}) = 0

for k = 1, …, m and

(σξ̄_{k})^{2} = Q^{2^{k}}(σξ̄_{k}) = σQ^{2^{k}}(ξ̄_{k}) = σξ̄_{k+1}

for k ≥ m + 1.

For p odd the classes σξ̄_{k} remain exterior, since they are of odd degree in the graded commutative algebra H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}). The p-th powers of the truncated polynomial generators in E_{∗∗}^{∞}(BP ⟨m−1⟩) are

(στ̄_{k})^{p} = Q^{p^{k}}(στ̄_{k}) = σQ^{p^{k}}(τ̄_{k}) = στ̄_{k+1}

for k ≥ m. Hence these truncated polynomial algebras assemble to a polynomial algebra on the single generator στ̄_{m}.

(b) In the strictly commutative case, Proposition 4.9 tells us that all the algebra generators are coalgebra primitive, since they are of the form form σx in Hochschild filtration 1.

(c) To compute the A_{∗}-comodule structure we use that σ is a graded derivation, in view of Proposition 5.10, so σ(y^{p}) = 0 and σ(1) = 0. Then for p = 2 and 1 ≤ k ≤ m we have

ν(σξ̄_{k}^{2}) = ∑_{i+j=k} ξ̄_{i}^{2} ⊗ σξ̄_{j}^{2^{i+1}} = 1 ⊗ σξ̄_{k}^{2}

while

ν(σξ̄_{m+1}) = ∑_{i+j=m+1} ξ̄_{i} ⊗ σξ̄_{j}^{2^{i}} = 1 ⊗ σξ̄_{m+1} + ξ̄_{1} ⊗ σξ̄_{m}^{2} .

For p odd and 1 ≤ k ≤ m we get

ν(σξ̄_{k}) = ∑_{i+j=k} ξ̄_{i} ⊗ σξ̄_{j}^{p^{i}} = 1 ⊗ σξ̄_{k}

while

ν(στ̄_{m}) = 1 ⊗ στ̄_{m} + ∑_{i+j=m} τ̄_{i} ⊗ σξ̄_{j}^{p^{i}} = 1 ⊗ στ̄_{m} + τ̄_{0} ⊗ σξ̄_{m} .

We can get similar conclusions under different hypotheses, by use of naturality with respect to the maps BP → BP ⟨m−1⟩ → H𝔽_{p}. Strickland’s result [Str99, 2.10] suggests that it may be doable, but nontrivial, to relate a given E_{2} ring spectrum structure on BP ⟨m−1⟩ to the one on BP by such a structured map.

Corollary 5.13 Let p and 2 < m < ∞ be such that there is an E_{2} ring spectrum map BP → BP ⟨m−1⟩. Then there is a split extension of associative 𝔽_{p}-algebras

E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}) → H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{2}) → H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) ⊗ P (σξ̄_{m+1})

when p = 2, and

E(σξ̄_{1}, …, σξ̄_{m}) → H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) → H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ P (στ̄_{m})

when p is odd.

If furthermore H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) is commutative, for which it suffices that T HH(BP ⟨m−1⟩) is homotopy commutative, then these extensions are trivial and

H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{2}) =

H_{∗}(BP ⟨m−1⟩; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, …, σξ̄_{m}^{2}) ⊗ P (σξ̄_{m+1})

for p = 2, and

H_{∗}(T HH(BP ⟨m−1⟩); 𝔽_{p}) = H_{∗}(BP ⟨m−1⟩; 𝔽_{p}) ⊗ E(σξ̄_{1}, …, σξ̄_{m}) ⊗ P (στ̄_{m})

for p odd, as commutative H_{∗}(BP ⟨m−1⟩; 𝔽_{p})-algebras.

Proof For brevity we write R = BP ⟨m−1⟩. The zeroth Postnikov section provides an E_{2} ring spectrum map R → Hℤ_{(p)}, which we can continue to H𝔽_{p}. The E_{2} ring spectrum maps BP → R → H𝔽_{p} then induce S -algebra maps

T HH(BP) → T HH(R) → T HH(H𝔽_{p}) .

The Bökstedt spectral sequences at the two ends were determined in Theorem 5.12, as the cases m = ∞ and m = 0, respectively: H_{∗}(T HH(BP); 𝔽_{p}) = H_{∗}(BP; 𝔽_{p}) ⊗ E(σξ̄_{k} | k ≥ 1) and H_{∗}(T HH(H𝔽_{p}); 𝔽_{p}) = A_{∗} ⊗ P (στ̄_{0}), for p odd. The case p = 2 is similar, and will be omitted.

The left hand map embeds the exterior algebra E(σξ̄_{1}, …, σξ̄_{m}) as a graded commutative subalgebra of H_{∗}(T HH(R); 𝔽_{p}). It maps trivially (by the augmentation) under the second map.

The right hand map embeds H_{∗}(R; 𝔽_{p}) in A_{∗}. It also embeds the remaining factor P_{p}(στ̄_{k} | k ≥ m) in the Bökstedt E^{∞}-term as the polynomial subalgebra of P (στ̄_{0}) that is generated by the single class στ̄_{m} = (στ̄_{0})^{p^{m}}. This uses the multiplicative relations (στ̄_{k})^{p} = στ̄_{k+1} for k ≥ 0 in H_{∗}(T HH(H𝔽_{p}); 𝔽_{p}).

The product of the unit map R → T HH(R) and the suspension operator σ provide a splitting H_{∗}(R; 𝔽_{p}) ⊗ P (στ̄_{m}) → H_{∗}(T HH(R); 𝔽_{p}). In the commutative case, this splitting amounts to a trivialization.

From here on we only consider the topological Hochschild homology of strictly commutative S -algebras, not of E_{n} ring spectra for any finite n.

For later reference we extract from Theorem 5.12 the following special cases, which correspond to m = 2. Recall that H_{∗}(ku; 𝔽_{2}) = (A//E_{1})_{∗} ⊂ A_{∗} and H_{∗}(ℓ; 𝔽_{p}) = (A//E_{1})_{∗} ⊂ A_{∗}.

Corollary 5.14 (a) There is an isomorphism

H_{∗}(T HH(ku); 𝔽_{2}) ≅ H_{∗}(ku; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})

of primitively generated H_{∗}(ku; 𝔽_{2})-Hopf algebras.

The A_{∗}-comodule coaction ν: H_{∗}(T HH(ku); 𝔽_{2}) → A_{∗} ⊗ H_{∗}(T HH(ku); 𝔽_{2}) is given on H_{∗}(ku; 𝔽_{2}) by restricting the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗}, and on the algebra generators by ν(σξ̄_{1}^{2}) = 1 ⊗ σξ̄_{1}^{2}, ν(σξ̄_{2}^{2}) = 1 ⊗ σξ̄_{2}^{2} and

ν(σξ̄_{3}) = 1 ⊗ σξ̄_{3} + ξ̄_{1} ⊗ σξ̄_{2}^{2} .

(b) There is an isomorphism

H_{∗}(T HH(ℓ); 𝔽_{p}) ≅ H_{∗}(ℓ; 𝔽_{p}) ⊗ E(σξ̄_{1}, σξ̄_{2}) ⊗ P (στ̄_{2})

of primitively generated H_{∗}(ℓ; 𝔽_{p})-Hopf algebras.

The A_{∗}-comodule coaction ν: H_{∗}(T HH(ℓ); 𝔽_{p}) → A_{∗} ⊗ H_{∗}(T HH(ℓ); 𝔽_{p}) is given on H_{∗}(ℓ; 𝔽_{p}) by restricting the coproduct ψ: A_{∗} → A_{∗} ⊗ A_{∗}, and on the algebra generators by ν(σξ̄_{1}) = 1 ⊗ σξ̄_{1}, ν(σξ̄_{2}) = 1 ⊗ σξ̄_{2} and

ν(στ̄_{2}) = 1 ⊗ στ̄_{2} + τ̄_{0} ⊗ σξ̄_{2} .

6 The higher real cases

For p = 2 there are a few more known examples of commutative S -algebras such that H^{∗}(R; 𝔽_{p}) is a cyclic A-module. Let ko be the connective real K -theory spectrum, with H^{∗}(ko; 𝔽_{2}) ≅ A//A_{1} = A ⊗_{A_{1}} 𝔽_{2}, and let tmf be the Hopkins– Mahowald topological modular forms spectrum, with H^{∗}(tmf; 𝔽_{2}) ≅ A//A_{2} = A ⊗_{A_{2}} 𝔽_{2}. See e.g. [Re01, 21.5]. Here A_{n} ⊂ A is the subalgebra generated by Sq^{1}, …, Sq^{2^{n}}, so A_{1} has rank 8 and A_{2} has rank 64. It is well known that ko is a commutative S -algebra, and in the case of tmf this is a consequence of the Hopkins–Miller theory, as being presented in the commutative case by Goerss and Hopkins.

Proposition 6.1 There are maps of commutative S -algebras ko → H𝔽_{2} and tmf → H𝔽_{2} that induce the following identifications:

H_{∗}(ko; 𝔽_{2}) = P (ξ̄_{1}^{4}, ξ̄_{2}^{2}, ξ̄_{k} | k ≥ 3)

and

H_{∗}(tmf; 𝔽_{2}) = P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ̄_{k} | k ≥ 4) .

Proof The maps can be obtained from the zeroth Postnikov sections ko → Hℤ and tmf → Hℤ. The homology computations are then immediate by dualization from H^{∗}(ko; 𝔽_{2}) ≅ A//A_{1}, cf. [Sto63], and H^{∗}(tmf; 𝔽_{2}) ≅ A//A_{2}.

We now follow the outline of Section 5. By Proposition 2.4 the E^{2}-terms of the respective Bökstedt spectral sequences are

E_{∗∗}^{2}(ko) = H_{∗}(ko; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3)

and

E_{∗∗}^{2}(tmf) = H_{∗}(tmf; 𝔽_{2}) ⊗ E(σξ̄_{1}^{8}, σξ̄_{2}^{4}, σξ̄_{3}^{2}, σξ̄_{k} | k ≥ 4) .

By Corollary 4.4 both spectral sequences collapse at the E^{2}-term, so E_{∗∗}^{2}(R) = E_{∗∗}^{∞}(R). To resolve the algebra extensions we use the Dyer–Lashof operations and Proposition 5.9. The squares in H_{∗}(T HH(ko); 𝔽_{2}) of the algebra generators in E_{∗∗}^{∞}(ko) are

(σξ̄_{1}^{4})^{2} = Q^{5}(σξ̄_{1}^{4}) = σQ^{5}(ξ̄_{1}^{4}) = 0
(σξ̄_{2}^{2})^{2} = Q^{7}(σξ̄_{2}^{2}) = σQ^{7}(ξ̄_{2}^{2}) = 0

by the formula Q^{k}(y^{2}) = 0 for p = 2 and k odd, and

(σξ̄_{k})^{2} = Q^{2^{k}}(σξ̄_{k}) = σQ^{2^{k}}(ξ̄_{k}) = σξ̄_{k+1}

for all k ≥ 3. Similar calculations show that (σξ̄_{1}^{8})^{2} = 0, (σξ̄_{2}^{4})^{2} = 0 and (σξ̄_{3}^{2})^{2} = 0 in H_{∗}(T HH(tmf); 𝔽_{2}), while (σξ̄_{k})^{2} = σξ̄_{k+1} for all k ≥ 4. The A_{∗}-comodule coaction map ν on the resulting algebra generators is obtained from the coproduct on A_{∗} and formula (5.11), as in the proof of Theorem 5.12. The result is as follows:

Theorem 6.2 (a) There is an isomorphism

H_{∗}(T HH(ko); 𝔽_{2}) ≅ H_{∗}(ko; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})

of primitively generated H_{∗}(ko; 𝔽_{2})-Hopf algebras. The A_{∗}-comodule structure is given on H_{∗}(ko; 𝔽_{2}) by restricting the coproduct on A_{∗}, and on the algebra generators by ν(σξ̄_{1}^{4}) = 1 ⊗ σξ̄_{1}^{4} and

ν(σξ̄_{2}^{2}) = 1 ⊗ σξ̄_{2}^{2} + ξ̄_{1}^{2} ⊗ σξ̄_{1}^{4}
ν(σξ̄_{3}) = 1 ⊗ σξ̄_{3} + ξ̄_{1} ⊗ σξ̄_{2}^{2} + ξ̄_{2} ⊗ σξ̄_{1}^{4} .

(b) There is an isomorphism

H_{∗}(T HH(tmf); 𝔽_{2}) ≅ H_{∗}(tmf; 𝔽_{2}) ⊗ E(σξ̄_{1}^{8}, σξ̄_{2}^{4}, σξ̄_{3}^{2}) ⊗ P (σξ̄_{4})

of primitively generated H_{∗}(tmf; 𝔽_{2})-Hopf algebras. The A_{∗}-comodule structure is given on H_{∗}(tmf; 𝔽_{2}) by restricting the coproduct on A_{∗}, and on the algebra generators by ν(σξ̄_{1}^{8}) = 1 ⊗ σξ̄_{1}^{8} and

ν(σξ̄_{2}^{4}) = 1 ⊗ σξ̄_{2}^{4} + ξ̄_{1}^{4} ⊗ σξ̄_{1}^{8}
ν(σξ̄_{3}^{2}) = 1 ⊗ σξ̄_{3}^{2} + ξ̄_{1}^{2} ⊗ σξ̄_{2}^{4} + ξ̄_{2}^{2} ⊗ σξ̄_{1}^{8}
ν(σξ̄_{4}) = 1 ⊗ σξ̄_{4} + ξ̄_{1} ⊗ σξ̄_{3}^{2} + ξ̄_{2} ⊗ σξ̄_{2}^{4} + ξ̄_{3} ⊗ σξ̄_{1}^{8} .

Proof Assemble the computations above.

7 The real and complex image-of-J

We now turn to the various image-of-J spectra that are commutative S -alge-bras. Their mod p cohomology is no longer cyclic as a module over the Steenrod algebra, but of rank 2, so extra work is needed to describe their homology as an A_{∗}-comodule algebra.

7.1 The image-of-J spectra

For any prime p let the p-local, connective complex image-of-J spectrum be ju = K(𝔽_{r})_{(p)}, where r = 3 for p = 2 and r is a prime power that topologically generates the p-adic units for p odd. Being the localized algebraic K -theory of a field, ju is a commutative S -algebra [May77, VIII.3.1]. For p = 2 there is a cofiber sequence of spectra

\xymatrix{ju \ar[r]^{κ} & ku_{(2)} \ar[r]^{ψ^{3}−1} & bu_{(2)} ,}  (7.2)

where bu ≃ Σ^{2}ku is the 1-connected cover of ku and ψ^{3} is the Adams operation. For odd p the cofiber sequence appears as

\xymatrix{ju \ar[r]^{κ} & ℓ \ar[r]^{ψ^{r}−1} & Σ^{q}ℓ}  (7.3)

with q = 2p − 2, where ℓ ⊂ ku_{(p)} is the p-local, connective Adams summand and ψ^{r} is the r-th Adams operation [May77, V.5.16].

Let the 2-local, connective real image-of-J spectrum be j = KN (𝔽_{3})_{(2)}, as defined in [May77, VIII.3.1]. Being the localized algebraic K -theory of a symmetric bimonoidal category, j is a commutative S -algebra. There is a cofiber sequence of spectra

\xymatrix{j \ar[r]^{κ} & ko_{(2)} \ar[r]^{ψ^{3}−1} & bspin_{(2)} ,}  (7.4)

where bspin ≃ Σ^{4}ksp is the 3-connected cover of ko [May77, V.5.16].

The fiber map κ: ju → ℓ is a map of commutative S -algebras, at least after p-adic completion, because there is a discrete model K(k′)_{p} for ℓ_{p} with k′ a suitable subfield of the algebraic closure of 𝔽_{r}, and applying the functor K(−)_{p} to the field inclusions 𝔽_{r} ⊂ k′ ⊂ \overline{𝔽}_{r} produces the commutative S -algebra maps

\xymatrix{ju_{p} \ar[r]^{κ} & ℓ_{p} ⊂ ku_{p} .}

See [May77, VIII.3.2] and [MS93, §9]. Similarly, κ: j → ko becomes a map of commutative S -algebras after 2-adic completion, since there is a discrete model KO(\overline{𝔽}_{3})_{2} for ko_{2}, and κ can be identified with the natural map

KN (𝔽_{3})_{2} → KO(\overline{𝔽}_{3})_{2} .

See [May77, VIII.2.6 and 3.2].

7.5 Cohomology modules

Recall that

H^{∗}(ℓ; 𝔽_{p}) = A//E_{1} ≅ 𝔽_{p}{1, P^{1}, …, P^{p}, … } ,

where E_{1} = E(Q_{0}, Q_{1}) ⊂ A is the exterior algebra generated by Q_{0} = β and Q_{1} = [P^{1}, β], and that

H^{∗}(ko; 𝔽_{2}) = A//A_{1} ≅ 𝔽_{2}{1, Sq^{4}, Sq^{2}Sq^{4} ≡ Sq^{6}, Sq^{1}Sq^{2}Sq^{4} ≡ Sq^{7}, … } ,

where A_{1} = ⟨Sq^{1}, Sq^{2}⟩ ⊂ A is the subalgebra generated by Sq^{1} and Sq^{2}. There are also A-module isomorphisms

H^{∗}(bo; 𝔽_{2}) ≅ ΣA/ASq^{2}
H^{∗}(bso; 𝔽_{2}) ≅ Σ^{2}A/ASq^{3}
H^{∗}(bspin; 𝔽_{2}) ≅ Σ^{4}A/A{Sq^{1}, Sq^{2}Sq^{3}} .

See [AP76, 2.5, 2.4 and p. 501]. Here Sq^{2}Sq^{3} = Sq^{5} + Sq^{4}Sq^{1} in admissible form, but the shorter expression is perhaps more memorable. For p odd we let A_{n} ⊂ A be the subalgebra generated by β, P^{1}, …, P^{p^{n−1}} . In particular, A_{1} = ⟨β, P^{1}⟩ contains E_{1}, and

A//A_{1} ≅ 𝔽_{p}{1, P^{p}, P^{1}P^{p} ≡ −P^{p+1}, βP^{1}P^{p} ≡ Q_{2}, … } .

Lemma 7.6 (a) For p = 2 the map ψ^{3} − 1: ku_{(2)} → Σ^{2}ku_{(2)} induces right multiplication by Sq^{2} on mod 2 cohomology:

(ψ^{3} − 1)^{∗} = Sq^{2} : Σ^{2}A//E_{1} → A//E_{1} .

(b) For p odd the map ψ^{r} − 1: ℓ → Σ^{q}ℓ induces right multiplication by P^{1} on mod p cohomology:

(ψ^{r} − 1)^{∗} = P^{1} : Σ^{q}A//E_{1} → A//E_{1} .

(c) For p = 2 the map ψ^{3} − 1: ko_{(2)} → bspin_{(2)} induces right multiplication by Sq^{4} on mod 2 cohomology:

(ψ^{3} − 1)^{∗} = Sq^{4} : Σ^{4}A/A{Sq^{1}, Sq^{2}Sq^{3}} → A//A_{1} .

Case (c) is due to Mahowald and Milgram [MaMi76, 3.4].

Proof The S -algebra unit map e: S_{(p)} → ju is well-known to be 2-connected for p = 2 and (pq−2)-connected for p odd, since this is the degree of the first element β_{1} in the p-primary cokernel of J [Ra04, 1.1.14]. Hence e^{∗}: H^{∗}(ju; 𝔽_{p}) → H^{∗}(S; 𝔽_{p}) = 𝔽_{p} is cohomologically 2-connected for p = 2 and cohomologically (pq − 2)-connected for p odd, meaning that the homomorphism is injective in the stated degree and an isomorphism in lower degrees. In particular, H^{2}(ju; 𝔽_{2}) = 0 for p = 2 and H^{q}(ju; 𝔽_{p}) = 0 for p odd.

So in the long exact cohomology sequence

\xymatrix{Σ^{2}A//E_{1} \ar[r]^{(ψ^{3}−1)^{∗}} & A//E_{1} \ar[r]^{κ^{∗}} & H^{∗}(ju; 𝔽_{2})}  (7.7)

associated to the cofiber sequence (7.2), the non-zero class of Sq^{2} in A//E_{1} maps to zero under κ^{∗}, hence is in the image of (ψ^{3} − 1)^{∗}. The latter is a left A-module homomorphism, and can only take Σ^{2}(1) to Sq^{2}, hence is given by right multiplication by Sq^{2}. This proves (a). For part (b) we use the same argument for the exact sequence

\xymatrix{Σ^{q}A//E_{1} \ar[r]^{(ψ^{r}−1)^{∗}} & A//E_{1} \ar[r]^{κ^{∗}} & H^{∗}(ju; 𝔽_{p})}  (7.8)

associated to (7.3), in cohomological degree q. The non-zero class of P^{1} in A//E_{1} maps to zero under κ^{∗}, hence must equal the image of Σ^{q}(1) under (ψ^{r} − 1)^{∗}. This proves (b).

The unit map e: S_{(2)} → j is likewise well-known to be 6-connected, since this is the degree of the first element ν^{2} in the 2-primary cokernel of J, so e^{∗}: H^{∗}(j; 𝔽_{2}) → H^{∗}(S; 𝔽_{2}) = 𝔽_{2} is cohomologically 6-connected. In particular, H^{4}(j; 𝔽_{2}) = 0. So in the long exact cohomology sequence associated to the cofiber sequence (7.4)

\xymatrix{Σ^{4}A/A\{Sq^{1}, Sq^{2}Sq^{3}\} \ar[r]^{(ψ^{3}−1)^{∗}} & A//A_{1} \ar[r]^{κ^{∗}} & H^{∗}(j; 𝔽_{2})}  (7.9)

the non-zero class of Sq^{4} in A//A_{1} maps to zero under κ^{∗}, hence is in the image of (ψ^{3} − 1)^{∗}. The only class that can hit it is Σ^{4}(1), which proves (c).

Lemma 7.10 (a) For p = 2 there is a uniquely split extension of A-modules

0 → A//A_{1} → H^{∗}(ju; 𝔽_{2}) → Σ^{3}A//A_{1} → 0 .

Hence there is a canonical A-module isomorphism H^{∗}(ju; 𝔽_{2}) ≅ A//A_{1}{1, x}, with x a class in degree 3.

(b) For p odd there is a non-split extension of A-modules

0 → A//A_{1} → H^{∗}(ju; 𝔽_{p}) → Σ^{pq−1}A//A_{1} → 0 .

As an A-module, H^{∗}(ju; 𝔽_{p}) is generated by two classes 1 and x in degrees 0 and (pq − 1), respectively, with β(x) = P^{p}(1).

(c) There is a unique non-split extension of A-modules

0 → A//A_{2} → H^{∗}(j; 𝔽_{2}) → A ⊗_{A_{2}} Σ^{7}K → 0 .

The cyclic A_{2}-module K = A_{2}/A_{2}{Sq^{1}, Sq^{7}, Sq^{4}Sq^{6} + Sq^{6}Sq^{4}} has rank 17 over 𝔽_{2}. As an A-module, H^{∗}(j; 𝔽_{2}) is generated by two classes 1 and x in degrees 0 and 7, respectively, with Sq^{1}(x) = Sq^{8}(1).

For case (b), see also [Rog03, 5.1(b)]. Case (c) is due to Davis [Da75, Thm. 1], who also shows that H^{∗}(j; 𝔽_{2}) is a free A//A_{3}-module.

Proof (a) In the long exact sequence extending (7.7) the A-module homomorphism (ψ^{3} − 1)^{∗} is induced up from the A_{1}-module homomorphism

Sq^{2} : Σ^{2}A_{1}//E_{1} → A_{1}//E_{1} = 𝔽_{2}{1, Sq^{2}}

with kernel Σ^{2}𝔽_{2}{Sq^{2}} and cokernel 𝔽_{2}{1}. Since A is flat (in fact free) over A_{1}, it follows that ker(ψ^{3} − 1)^{∗} ≅ Σ^{4}A//A_{1} and cok(ψ^{3} − 1)^{∗} ≅ A//A_{1}. Hence there is an extension of A-modules

0 → A//A_{1} → H^{∗}(ju; 𝔽_{2}) → Σ^{3}A//A_{1} → 0 ,

as asserted. The group of such extensions is trivial, by the change-of-rings isomorphism

Ext_{A}^{1}(Σ^{3}A//A_{1}, A//A_{1}) ≅ Ext_{A_{1}}^{1}(Σ^{3}𝔽_{2}, A//A_{1}) .

For in any A_{1}-module extension

0 → A//A_{1} → E → Σ^{3}𝔽_{2} → 0

let x ∈ E be the unique class in degree 3 that maps to Σ^{3}(1). Then Sq^{2}x = 0 since A//A_{1} = 𝔽_{2}{1, Sq^{4}, Sq^{6}, Sq^{7}, … } is trivial in degree 5. Furthermore Sq^{1}x = Sq^{4}(1) would contradict the Adem relation Sq^{2}Sq^{2} = Sq^{3}Sq^{1}, since Sq^{3}Sq^{4} ≡ Sq^{7} in A//A_{1}. So Sq^{1}x = 0 and the extension E is trivial.

Two choices of splitting maps for the trivial extension describing H^{∗}(ju; 𝔽_{2}) differ by an A-module homomorphism Σ^{3}A//A_{1} → A//A_{1}, which must be zero since A//A_{1} is trivial in degree 3. Therefore the splitting is unique, as claimed.

(b) Similarly, in (7.8) the A-module homomorphism (ψ^{r} − 1)^{∗} is induced up from the A_{1}-module homomorphism

P^{1} : Σ^{q}A_{1}//E_{1} → A_{1}//E_{1} = 𝔽_{p}{1, P^{1}, …, P^{p−1}}

with kernel Σ^{q}𝔽_{p}{P^{p−1}} and cokernel 𝔽_{p}{1}. As above it follows that ker(ψ^{r} − 1)^{∗} ≅ Σ^{pq}A//A_{1} and cok(ψ^{r} − 1)^{∗} ≅ A//A_{1}. Hence there is an extension of A-modules

0 → A//A_{1} → H^{∗}(ju; 𝔽_{p}) → Σ^{pq−1}A//A_{1} → 0 ,

as asserted. This time the group of extensions is non-trivial; in fact it is isomorphic to ℤ/p and generated by the extension above. To see this, we again use the change-of-rings isomorphism

Ext_{A}^{1}(Σ^{pq−1}A//A_{1}, A//A_{1}) ≅ Ext_{A_{1}}^{1}(Σ^{pq−1}𝔽_{p}, A//A_{1})

and consider A_{1}-module extensions

0 → A//A_{1} → E → Σ^{pq−1}𝔽_{p} → 0 .

Let x ∈ E be the unique class that maps to Σ^{pq−1}(1). Then P^{1}x = 0 since A//A_{1} = 𝔽_{p}{1, P^{p}, P^{1}P^{p}, … } is trivial in degree (p + 1)q − 1. But βx is a multiple of P^{p}, and this multiple in ℤ/p classifies the extension.

To see that βx is non-zero in the case of H^{∗}(ju; 𝔽_{p}), recall again that the first class β_{1} in the cokernel of J is in degree (pq − 2) and has order p. Let c → S_{(p)} → ju be the usual cofiber sequence. Then by the Hurewicz and universal coefficient theorems, the lowest class x in H^{∗}(Σc; 𝔽_{p}) sits in degree (pq − 1) and supports a non-trivial mod p Bockstein βx ≠ 0. Furthermore, H^{∗}(Σc; 𝔽_{p}) ≅ H^{∗}(ju; 𝔽_{p}) in positive degrees ∗ > 0, so also in H^{∗}(ju; 𝔽_{p}) we have βx ≠ 0.

(c) In (7.9), the A-module homomorphism (ψ^{3} − 1)^{∗} is induced up from the A_{2}-module homomorphism

Sq^{4} : Σ^{4}A_{2}/A_{2}{Sq^{1}, Sq^{2}Sq^{3}} → A_{2}//A_{1} .

A direct calculation shows that A_{2}/A_{2}{Sq^{1}, Sq^{2}Sq^{3}} has rank 24 and A_{2}//A_{1} has rank 8, as 𝔽_{2}-vector spaces. The cokernel of the homomorphism Sq^{4} is A_{2}//A_{2} = 𝔽_{2}{1}, of rank 1, so its kernel Σ^{8}K has rank 17. Here

Σ^{4}K = 𝔽_{2}{Sq^{4}, Sq^{6}, Sq^{7}, Sq^{6}Sq^{2}, Sq^{9}, Sq^{10} + Sq^{8}Sq^{2}, Sq^{7}Sq^{3},

Sq^{11} + Sq^{9}Sq^{2}, Sq^{10}Sq^{2}, Sq^{13} + Sq^{10}Sq^{3}, Sq^{11}Sq^{2}, Sq^{11}Sq^{3},

Sq^{13}Sq^{2} +Sq^{12}Sq^{3}, Sq^{13}Sq^{3}, Sq^{17} +Sq^{15}Sq^{2}, Sq^{17}Sq^{2} +Sq^{16}Sq^{3}, Sq^{17}Sq^{3}}

as a submodule of A_{2}/A_{2}{Sq^{1}, Sq^{2}Sq^{3}}. By another direct calculation, Σ^{4}K is in fact the cyclic A_{2}-submodule generated by Sq^{4}. The annihilator ideal turns out to be generated by Sq^{1}, Sq^{7} and Sq^{4}Sq^{6}+Sq^{6}Sq^{4} = Sq^{10}+Sq^{8}Sq^{2}+Sq^{7}Sq^{3} (in admissible form), so

K ≅ A_{2}/A_{2}{Sq^{1}, Sq^{7}, Sq^{4}Sq^{6} + Sq^{6}Sq^{4}} .

Hence there is an extension of A-modules

0 → A//A_{2} → H^{∗}(j; 𝔽_{2}) → A ⊗_{A_{2}} Σ^{7}K → 0

with A ⊗_{A_{2}} Σ^{7}K ≅ Σ^{7}A/A{Sq^{1}, Sq^{7}, Sq^{4}Sq^{6} + Sq^{6}Sq^{4}}. The group of such A-module extensions is

Ext_{A}^{1}(A ⊗_{A_{2}} Σ^{7}K, A//A_{2}) ≅ ℤ/2 ,

and the extension is determined by the action of Sq^{1} on the generator x in degree 7 that maps to Σ^{7}(1). (Sq^{7}x = 0 by the Adem relation Sq^{1}Sq^{7} = 0 and the fact that Sq^{1} acts injectively from degree 14 of A//A_{2}. (Sq^{4}Sq^{6} + Sq^{6}Sq^{4})x = 0 since A//A_{2} is trivial in degree 17.)

To see that Sq^{1}(x) = Sq^{8}(1) ≠ 0 in H^{∗}(j; ℤ/2), we once again use the cofiber sequence c → S_{(2)} → j and the fact that c is 5-connected with π_{6}(c) = ℤ/2{ν^{2}}. Hence the lowest class x in H^{∗}(Σc; 𝔽_{2}) sits in degree 7 and supports a non-trivial Sq^{1}x ≠ 0. Again, H^{∗}(Σc; 𝔽_{2}) ≅ H^{∗}(j; 𝔽_{2}) in positive degrees, so also in H^{∗}(j; 𝔽_{2}) we have Sq^{1}x ≠ 0. The only possible nonzero value is the Sq^{8} from A//A_{2}.

We display the A_{2}-module Σ^{4}K ⊂ A_{2}/A_{2}{Sq^{1}, Sq^{2}Sq^{3}} below. Here (i) or (i, j) denotes an admissible class with lexicographically leading term Sq^{i} or Sq^{i}Sq^{j}, respectively. The arrows indicate the Sq^{1}- and Sq^{2}-operations. The Sq^{4}-operations can be deduced from the relations Sq^{4}(6) = (10) and Sq^{4}(13) = (17), but are omitted to avoid cluttering the diagram.

\xymatrix{(4) \ar[rr] &  & (6) \ar[r] & (7) \ar[rr] & (6, 2) \ar[r] \ar[rr] & (9) & (7, 3) \\  & (11) \ar[rr] &  & (13) \ar[dr] \\ (10) \ar[ur] \ar[rr] &  & (10, 2) \ar[dr] \ar[rr] &  & (11, 3) \ar[rr] &  & (13, 3) \\  &  &  & (11, 2) \ar[rr] &  & (13, 2) \ar[ur] \\ (17) \ar[rr] &  & (17, 2) \ar[r] & (17, 3)}

7.11 Homology algebras

Let us write (A//A_{1})_{∗} ⊂ A_{∗} for the A_{∗}-comodule subalgebra dual to the quotient A-module coalgebra A//A_{1} of A. For p = 2 we recall from Proposition 6.1 that

(A//A_{1})_{∗} = P (ξ̄_{1}^{4}, ξ̄_{2}^{2}, ξ̄_{k} | k ≥ 3) ≅ H_{∗}(ko; 𝔽_{2}) .

For p odd

(A//A_{1})_{∗} = P (ξ̄_{1}^{p}, ξ̄_{k} | k ≥ 2) ⊗ E(τ̄_{k} | k ≥ 2) ,

but there is no spectrum with mod p homology realizing (A//A_{1})_{∗}. For p = 2 and for p odd there are extensions of A_{∗}-comodules

\xymatrix{0 → Σ^{3}(A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{2}) \ar[r]^{κ} & (A//A_{1})_{∗} → 0}

and

\xymatrix{0 → Σ^{pq−1}(A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{p}) \ar[r]^{κ} & (A//A_{1})_{∗} → 0 ,}

dual to the A-module extensions of Lemma 7.10(a) and (b), respectively, where in both cases κ is an A_{∗}-comodule algebra homomorphism.

Likewise, we write (A//A_{2})_{∗} ⊂ A_{∗} for the A_{∗}-comodule subalgebra dual to the quotient A-module coalgebra A//A_{2} of A. For p = 2 we recall from Proposition 6.1 that

(A//A_{2})_{∗} = P (ξ̄_{1}^{8}, ξ̄_{2}^{4}, ξ̄_{3}^{2}, ξ̄_{k} | k ≥ 4) ≅ H_{∗}(tmf; 𝔽_{2}) .

There is an extension of A_{∗}-comodules

\xymatrix{0 → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → H_{∗}(j; 𝔽_{2}) \ar[r]^{κ} & (A//A_{2})_{∗} → 0 ,}

dual to the A-module extension of Lemma 7.10(c), where κ is an A_{∗}-comodule algebra homomorphism. Here K_{∗} ⊂ A_{2∗} is the A_{2∗}-comodule dual to the cyclic A_{2}-module K = A_{2}/A_{2}{Sq^{1}, Sq^{7}, Sq^{4}Sq^{6} + Sq^{6}Sq^{4}}.

Proposition 7.12 (a) For p = 2, let b ∈ H_{3}(ju; 𝔽_{2}) be the image of Σ^{3}(1) in Σ^{3}(A//A_{1})_{∗}. Then there is an A_{∗}-comodule algebra isomorphism

H_{∗}(ju; 𝔽_{2}) ≅ (A//A_{1})_{∗} ⊗ E(b) ,

where (A//A_{1})_{∗} has the subalgebra structure from A_{∗} and b is A_{∗}-comodule primitive.

The Dyer–Lashof operations in H_{∗}(ju; 𝔽_{2}) satisfy Q^{4}(b) = 0, Q^{5}(ξ̄_{1}^{4}) = 0, Q^{7}(ξ̄_{2}^{2}) = 0 and Q^{2^{k}}(ξ̄_{k}) = ξ̄_{k+1} for all k ≥ 3, so H_{∗}(ju; 𝔽_{2}) is generated by ξ̄_{1}^{4}, ξ̄_{2}^{2}, ξ̄_{3} and b as an algebra over the Dyer–Lashof algebra.

(b) For p odd, let b ∈ H_{pq−1}(ju; 𝔽_{p}) be the image of Σ^{pq−1}(1) in Σ^{pq−1}(A//A_{1})_{∗}. There is an algebra isomorphism

(A//A_{1})_{∗} ⊗ E(b) ≅ H_{∗}(ju; 𝔽_{p})

that takes the algebra generators ξ̄_{1}^{p}, ξ̄_{k}, τ̄_{k} (for k ≥ 2) and b to classes ξ̃_{1}^{p}, ξ̃_{k}, τ̃_{k} and b in H_{∗}(ju; 𝔽_{p}), respectively. These map under κ to ξ̄_{1}^{p}, ξ̄_{k}, τ̄_{k} and 0, respectively.

The Dyer–Lashof operations on H_{∗}(ju; 𝔽_{p}) satisfy Q^{pq/2}(b) = 0 and Q^{p^{k}}(τ̃_{k}) = τ̃_{k+1} for all k ≥ 2, and β(τ̃_{k}) = ξ̃_{k} for all k ≥ 2. Thus H_{∗}(ju; 𝔽_{p}) is generated as an algebra over the Dyer–Lashof algebra by ξ̃_{1}^{p}, ξ̃_{2}, τ̃_{2} and b.

The A_{∗}-comodule structure is determined by

ν(b) = 1 ⊗ b
ν(ξ̃_{1}^{p}) = 1 ⊗ ξ̃_{1}^{p} − τ_{0} ⊗ b + ξ̄_{1}^{p} ⊗ 1
ν(ξ̃_{2}) = 1 ⊗ ξ̃_{2} + ξ̄_{1} ⊗ ξ̃_{1}^{p} + τ_{1} ⊗ b + ξ̄_{2} ⊗ 1
ν(τ̃_{2}) = 1 ⊗ τ̃_{2} + τ̄_{0} ⊗ ξ̃_{2} + τ̄_{1} ⊗ ξ̃_{1}^{p} − τ_{0}τ_{1} ⊗ b + τ̄_{2} ⊗ 1 .

(The class b ∈ H_{pq−1}(ju; 𝔽_{p}) maps under the connecting map for the cofiber sequence c → S_{(p)} → ju to the mod p Hurewicz image of β_{1} ∈ π_{pq−2}(c), so the letter b is chosen to correspond to β.)

(c) There is a square-zero extension of A_{∗}-comodule algebras

\xymatrix{0 → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → H_{∗}(j; 𝔽_{2}) \ar[r]^{κ} & (A//A_{2})_{∗} → 0 ,}

where κ is split as an algebra homomorphism. As an (A//A_{2})_{∗}-module,

ker(κ) = A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} ≅ (A//A_{2})_{∗} ⊗ Σ^{7}K_{∗}

is free of rank 17. There is an algebra isomorphism

H_{∗}(j; 𝔽_{2}) ≅ (A//A_{2})_{∗} ⊗ (𝔽_{2} ⊕ Σ^{7}K_{∗})

where 𝔽_{2} ⊕ Σ^{7}K_{∗} is the split square-zero extension of 𝔽_{2} with kernel Σ^{7}K_{∗} of rank 17.

Proof (a) Let x ∈ H^{3}(ju; 𝔽_{2}) be the class that maps to Σ^{3}(1) in the uniquely split A-module extension of Lemma 7.10(a). Then ψ(x) = x ⊗ 1 + 1 ⊗ x since H^{∗}(ju; 𝔽_{2}) = 0 for 0 < ∗ < 3, so E(x) = 𝔽_{2}{1, x} (no algebra structure is implied) is a sub-coalgebra of H^{∗}(ju; 𝔽_{2}) and there a surjective composite A-module coalgebra homomorphism

A ⊗ E(x) → A ⊗ H^{∗}(ju; 𝔽_{2}) → H^{∗}(ju; 𝔽_{2}) .

Since the A-module extension is split, the generators Sq^{1} and Sq^{2} of A_{1} act trivially on 1 and x in H^{∗}(ju; 𝔽_{2}), so the surjection factors through a surjection

A//A_{1} ⊗ E(x) → H^{∗}(ju; 𝔽_{2}) ,

which by a dimension count must be an A-module coalgebra isomorphism.

Dually, let b ∈ H_{3}(ju; 𝔽_{2}) be the image of Σ^{3}(1) in the split A_{∗}-comodule extension

\xymatrix{0 → Σ^{3}(A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{2}) \ar[r]^{κ} & (A//A_{1})_{∗} → 0 .}

Then b is dual to x, and the dual of the above isomorphism is an A_{∗}-comodule algebra isomorphism

H_{∗}(ju; 𝔽_{2}) ≅ (A//A_{1})_{∗} ⊗ E(b) .

In particular, the unique A_{∗}-comodule splitting (A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{2}) is an algebra map, b is an A_{∗}-comodule primitive, and b^{2} = 0.

To determine the Dyer–Lashof operations that we will need in H_{∗}(ju; 𝔽_{2}) we make use of the Nishida relations and the known A_{∗}-comodule structure. Some of the Nishida relations that we shall use are

Sq_{∗}^{1}Q^{s} = \begin{cases}Q^{s−1} & for s even \\ 0 & for s odd, \end{cases}

and

Sq_{∗}^{2}Q^{s} = \begin{cases}Q^{s−2} + Q^{s−1}Sq_{∗}^{1} & for s ≡ 0, 1 mod 4 \\ Q^{s−1}Sq_{∗}^{1} & for s ≡ 2, 3 mod 4. \end{cases}

See [BMMS86, III.1.1(8)].

The Dyer–Lashof operation Q^{4}(b) lands in H_{7}(ju; 𝔽_{2}) ≅ 𝔽_{2}{ξ̄_{3}, ξ̄_{1}^{4}b}. From the A_{∗}-comodule structure we can read off the dual Steenrod operations Sq_{∗}^{1}(ξ̄_{3}) = ξ̄_{2}^{2} and Sq_{∗}^{4}(ξ̄_{1}^{4}b) = b, since Sq^{i} is dual to ξ_{1}^{i}. These are linearly independent, so Q^{4}(b) is determined by its images under Sq_{∗}^{1} and Sq_{∗}^{4}. By a Nishida relation we get that Sq_{∗}^{1}Q^{4}(b) = Q^{3}(b), and Q^{3}(b) = b^{2} = 0 since |b| = 3 and b is an exterior class. By another Nishida relation Sq_{∗}^{4}Q^{4}(b) = Q^{2}Sq_{∗}^{2}(b), and Sq_{∗}^{2}(b) = 0 since b is A_{∗}-comodule primitive. Thus Sq_{∗}^{1}Q^{4}(b) = 0 and Sq_{∗}^{4}Q^{4}(b) = 0, and the only possibility is that Q^{4}(b) = 0.

The operation Q^{5}(ξ̄_{1}^{4}) lands in H_{9}(ju; 𝔽_{2}) ≅ 𝔽_{2}{ξ̄_{2}^{2}b}. By a Nishida relation Sq_{∗}^{2}Q^{5}(ξ̄_{1}^{4}) = (Q^{3} +Q^{4}Sq_{∗}^{1})(ξ̄_{1}^{4}) = 0, while Sq_{∗}^{2}(ξ̄_{2}^{2}b) = ξ̄_{1}^{4}b ≠ 0. So Q^{5}(ξ̄_{1}^{4}) must be zero.

The operation Q^{7}(ξ̄_{2}^{2}) lands in H_{13}(ju; 𝔽_{2}) ≅ 𝔽_{2}{ξ̄_{2}^{2}ξ̄_{3}, ξ̄_{1}^{4}ξ̄_{2}^{2}b}. Its image under κ in H_{∗}(ku; 𝔽_{2}) ⊂ A_{∗} must vanish, by the Cartan formula, so in fact Q^{7}(ξ̄_{2}^{2}) ∈ 𝔽_{2}{ξ̄_{1}^{4}ξ̄_{2}^{2}b}. By a Nishida relation Sq_{∗}^{2}Q^{7}(ξ̄_{2}^{2}) = Q^{6}Sq_{∗}^{1}(ξ̄_{2}^{2}) = 0, while Sq_{∗}^{2}(ξ̄_{1}^{4}ξ̄_{2}^{2}b) = (ξ̄_{1}^{4})^{2}b ≠ 0, so Q^{7}(ξ̄_{2}^{2}) must be zero.

The claim that Q^{2^{k}}(ξ̄_{k}) = ξ̄_{k+1} follows in the same way as in A_{∗} = H_{∗}(H𝔽_{2}; 𝔽_{2}), see [BMMS86, §III.6]. It suffices to show that Sq_{∗}^{2^{m}}Q^{2^{k}}(ξ̄_{k}) = Sq_{∗}^{2^{m}}(ξ̄_{k+1}) for 0 ≤ m ≤ k, because the only A_{∗}-comodule primitives in H_{∗}(ju; 𝔽_{2}) are 1 and b. The right hand side equals ξ̄_{k}^{2} for m = 0, and is zero otherwise, by the formula for ψ(ξ̄_{k+1}). The Nishida relations imply that Sq_{∗}^{1}Q^{2^{k}}(ξ̄_{k}) = Q^{2^{k}−1}(ξ̄_{k}) = ξ̄_{k}^{2}, while Sq_{∗}^{2}Q^{2^{k}}(ξ̄_{k}) = Q^{2^{k}−1}Sq_{∗}^{1}(ξ̄_{k}) = Q^{2^{k}−1}(ξ̄_{k−1}^{2}) = 0 by the Cartan formula. For 2 ≤ m ≤ k we have Sq_{∗}^{2^{m}}Q^{2^{k}}(ξ̄_{k}) = Q^{2^{k}−2^{m−1}}Sq_{∗}^{2^{m−1}}(ξ̄_{k}) = Q^{2^{k}−2^{m−1}}(0) = 0 by the formula for ψ(ξ̄_{k}) in A_{∗}. Hence Q^{2^{k}}(ξ̄_{k}) = ξ̄_{k+1} in H_{∗}(ju; 𝔽_{2}) is the only possibility.

(b) Let x ∈ H^{pq−1}(ju; 𝔽_{p}) be the class that maps to Σ^{pq−1}(1) in the A-module extension of Lemma 7.10(b). Then ψ(x) = x ⊗ 1 + 1 ⊗ x since H^{∗}(ju; 𝔽_{p}) = 0 for 0 < ∗ < pq − 1, so E(x) = 𝔽_{p}{1, x} is a sub-coalgebra of H^{∗}(ju; 𝔽_{p}) and there a surjective composite A-module coalgebra homomorphism

A ⊗ E(x) → A ⊗ H^{∗}(ju; 𝔽_{p}) → H^{∗}(ju; 𝔽_{p}) .

Dually, let b ∈ H_{pq−1}(ju; 𝔽_{p}) be the image of Σ^{pq−1}(1) in the A_{∗}-comodule extension

\xymatrix{0 → Σ^{pq−1}(A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{p}) \ar[r]^{κ} & (A//A_{1})_{∗} → 0 .}

Then E(b) is a quotient algebra of H_{∗}(ju; 𝔽_{p}) and the dual of the surjection above is an injective A_{∗}-comodule algebra homomorphism

H_{∗}(ju; 𝔽_{p}) → A_{∗} ⊗ E(b) .

We shall describe H_{∗}(ju; 𝔽_{p}) in terms of its image under this injection.

Since (A//A_{1})_{∗} is a free graded commutative algebra, we can choose an algebra section s: (A//A_{1})_{∗} → H_{∗}(ju; 𝔽_{p}) to the surjection κ: H_{∗}(ju; 𝔽_{p}) → (A//A_{1})_{∗}. We write ξ̃_{1}^{p} = s(ξ̄_{1}^{p}), ξ̃_{k} = s(ξ̄_{k}) and τ̃_{k} = s(τ̄_{k}) for the lifted classes in H_{∗}(ju; 𝔽_{p}). Since Σ^{pq−1}(A//A_{1}) vanishes in the degrees of ξ̄_{1}^{p} and τ̄_{2}, the respective lifts ξ̃_{1}^{p} and τ̃_{2} are unique, and we can use the commutative S -algebra structure on ju and the resulting Dyer–Lashof operations on H_{∗}(ju; 𝔽_{p}) to fix the other lifts by the formulas ξ̃_{k} = β(τ̃_{k}) and τ̃_{k+1} = Q^{p^{k}}(τ̃_{k}) for k ≥ 2. This specifies the algebra section s uniquely. (But beware, s is not an A_{∗}-comodule homomorphism!)

Since |b| = pq − 1 is odd we have b^{2} = 0 and the exterior algebra E(b) is a subalgebra of H_{∗}(ju; 𝔽_{p}). Writing i: E(b) → H_{∗}(ju; 𝔽_{p}) for the inclusion, we obtain an algebra map

s ⊗ i: (A//A_{1})_{∗} ⊗ E(b) → H_{∗}(ju; 𝔽_{p}) ,

which we claim is an isomorphism. By a dimension count it suffices to show that its composite with the injection H_{∗}(ju; 𝔽_{p}) → A_{∗} ⊗ E(b) is injective. We have a diagram of algebra maps

\xymatrix{(A//A_{1})_{∗} ⊗ E(b) \ar[r]^{s⊗i} \ar[d] & H_{∗}(ju; 𝔽_{p}) \ar[r] \ar[d]^{κ} & A_{∗} ⊗ E(b) \ar[d] \\ (A//A_{1})_{∗} \ar[r]^{=} & (A//A_{1})_{∗} \ar[r] & A_{∗}}

where the vertical maps take b to zero. The lower map takes ξ ∈ (A//A_{1})_{∗} to ξ ∈ A_{∗}, so the upper composite takes ξ ⊗ 1 to ξ ⊗ 1 (mod b). Hence the latter takes ξ ⊗ b to ξ ⊗ b (mod b^{2} = 0), and it follows that the upper composite indeed is injective.

In low degrees, H^{∗}(ju; 𝔽_{p}) ≅ 𝔽_{p}{1, x, P^{p}(1), P^{1}P^{p}(1), βP^{1}P^{p}(1), … } is dual to H_{∗}(ju; 𝔽_{p}) ≅ 𝔽_{p}{1, b, −ξ̃_{1}^{p}, ξ̃_{2}, τ̃_{2}, … }. By Lemma 7.10(b) we have β(x) = P^{p}(1), so dually the A_{∗}-comodule coactions are given by

ν(b) = 1 ⊗ b
ν(ξ̃_{1}^{p}) = 1 ⊗ ξ̃_{1}^{p} − τ_{0} ⊗ b + ξ̄_{1}^{p} ⊗ 1
ν(ξ̃_{2}) = 1 ⊗ ξ̃_{2} + ξ̄_{1} ⊗ ξ̃_{1}^{p} + τ_{1} ⊗ b + ξ̄_{2} ⊗ 1
ν(τ̃_{2}) = 1 ⊗ τ̃_{2} + τ̄_{0} ⊗ ξ̃_{2} + τ̄_{1} ⊗ ξ̃_{1}^{p} − τ_{0}τ_{1} ⊗ b + τ̄_{2} ⊗ 1 .

In particular, the images in A_{∗} ⊗E(b) of these classes are 1⊗b, −τ_{0} ⊗b+ξ̄_{1}^{p} ⊗1, τ_{1} ⊗ b + ξ̄_{2} ⊗ 1 and −τ_{0}τ_{1} ⊗ b + τ̄_{2} ⊗ 1, respectively.

The Dyer–Lashof operation Q^{pq/2}(b) lands in H_{p^{2}q−1}(ju; 𝔽_{p}) = 𝔽_{p}{ξ̃_{1}^{pq/2}b}. Here P_{∗}^{pq/2}(ξ̃_{1}^{pq/2}b) = b is non-zero, in view of the formula above for ν(ξ̃_{1}^{p}). By a Nishida relation P_{∗}^{pq/2}Q^{pq/2}(b) = Q^{q/2}P_{∗}^{q/2}(b) = 0, so Q^{pq/2}(b) = 0.

(c) Let x ∈ H^{7}(j; 𝔽_{2}) be the class that maps to Σ^{7}(1) in the A-module extension of Lemma 7.10(c). Then ψ(x) = x ⊗ 1 + 1 ⊗ x since H^{∗}(j; 𝔽_{2}) = 0 for 0 < ∗ < 7, so E(x) = 𝔽_{2}{1, x} is a sub-coalgebra of H^{∗}(j; 𝔽_{2}) and there is a surjective composite A-module coalgebra homomorphism

A ⊗ E(x) → A ⊗ H^{∗}(j; 𝔽_{2}) → H^{∗}(j; 𝔽_{2}) .

Dually, let b ∈ H_{7}(j; 𝔽_{2}) be the image of Σ^{7}(1) in the A_{∗}-comodule extension

\xymatrix{0 → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → H_{∗}(j; 𝔽_{2}) \ar[r]^{κ} & (A//A_{2})_{∗} → 0 .}

Here K_{∗} ⊂ A_{2∗} has rank 17, and contains 1 ∈ K_{∗} ⊂ A_{2∗} in degree 0. The dual of the surjection above is an injective A_{∗}-comodule algebra homomorphism

H_{∗}(j; 𝔽_{2}) → A_{∗} ⊗ E(b)

that may otherwise be described as the composite of the A_{∗}-comodule coaction H_{∗}(j; 𝔽_{2}) → A_{∗} ⊗ H_{∗}(j; 𝔽_{2}) and the algebra surjection H_{∗}(j; 𝔽_{2}) → E(b).

We obtain a vertical map of A_{∗}-comodule extensions

\xymatrix{A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} \ar[r] \ar[d] & H_{∗}(j; 𝔽_{2}) \ar[r]^{κ} \ar[d] & (A//A_{2})_{∗} \ar[d] \\ A_{∗}\{b\} \ar[r] & A_{∗} ⊗ E(b) \ar[r] & A_{∗}}

where the right hand square consists of A_{∗}-comodule algebra homomorphisms, and the vertical maps are injective. At the left hand side we find the composite map

A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → A_{∗} □_{A_{2∗}} Σ^{7}A_{2∗} ≅ Σ^{7}A_{∗} ≅ A_{∗}{b}

that is dual to the surjection A{x} ≅ Σ^{7}A → A ⊗_{A_{2}} Σ^{7}K.

In the lower row the ideal A_{∗}{b} has square zero, since b^{2} = 0, so also the ideal ker(κ) = A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} is a square-zero ideal [CE56, XIV.2] in H_{∗}(j; 𝔽_{2}). Its module action by H_{∗}(j; 𝔽_{2}) therefore descends to one by (A//A_{2})_{∗}. It can be described in terms of the algebra product ϕ on A_{∗} by the following commutative diagram with injective vertical maps:

\xymatrix{(A//A_{2})_{∗} ⊗ (A_{∗} □_{A_{2∗}} Σ^{7}K_{∗}) \ar[r] \ar[d] & (A_{∗} □_{A_{2∗}} Σ^{7}K_{∗}) \ar[d] \\ A_{∗} ⊗ Σ^{7}A_{∗} \ar[r]^{Σ^{7}ϕ} & Σ^{7}A_{∗}}

The proof is easy given the algebra embedding of H_{∗}(j; 𝔽_{2}) into A_{∗} ⊗ E(b).

In fact, the square-zero ideal A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} is a free (A//A_{2})_{∗}-module of rank 17. For A_{2} ⊂ A is a direct summand as an A_{2}-module, so dually A_{∗} → A_{2∗} admits an A_{2∗}-comodule section s: A_{2∗} → A_{∗}. For example, the image of s may be 𝔽_{2}{ξ̄_{1}^{i}ξ̄_{2}^{j}ξ̄_{3}^{k} | i < 8, j < 4, k < 2} ⊂ A_{∗}. We then have a map

s □ id: Σ^{7}K_{∗} ≅ A_{2∗} □_{A_{2∗}} Σ^{7}K_{∗} → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} .

Its composite with the inclusion A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} → A_{∗} □_{A_{2∗}} Σ^{7}A_{2∗} ≅ Σ^{7}A_{∗} factors as the two inclusions Σ^{7}K_{∗} → Σ^{7}A_{2∗} → Σ^{7}A_{∗}.

Combining s□id with the (A//A_{2})_{∗}-module action on A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} we obtain the left hand map f in a commuting diagram

\xymatrix{(A//A_{2})_{∗} ⊗ Σ^{7}K_{∗} \ar[r] \ar[d]^{f} & (A//A_{2})_{∗} ⊗ Σ^{7}A_{2∗} \ar[d]^{≅} \\ A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} \ar[r] & A_{∗} □_{A_{2∗}} Σ^{7}A_{2∗} .}

Here A_{∗}□_{A_{2∗}}Σ^{7}A_{2∗} ≅ Σ^{7}A_{∗} and the right hand isomorphism exhibits Σ^{7}A_{∗} as a free (A//A_{2})_{∗}-module on the generators given by the section s: Σ^{7}A_{2∗} → Σ^{7}A_{∗}. (It is a case of the Milnor–Moore comodule algebra theorem [MiMo65, 4.7].) The upper map is injective, hence so is

f: (A//A_{2})_{∗} ⊗ Σ^{7}K_{∗} → A_{∗} □_{A_{2∗}} Σ^{7}K_{∗} .

But both sides have the same, finite dimension over 𝔽_{2} in each degree, so in fact f is an isomorphism of (A//A_{2})_{∗}-modules.

The fact that κ splits as an algebra homomorphism is clear since (A//A_{2})_{∗} is a free graded commutative algebra over 𝔽_{2}. However, the splitting is not an A_{∗}- comodule homomorphism, and the (A//A_{2})_{∗}-module isomorphism f is not an A_{∗}-comodule isomorphism. The A_{∗}-comodule algebra structure on H_{∗}(j; 𝔽_{2}) may, if desired, be obtained by describing the image of the algebra generators under the A_{∗}-comodule algebra embedding H_{∗}(j; 𝔽_{2}) → A_{∗} ⊗ E(b).

Proposition 7.13 (a) For p = 2 the Bökstedt spectral sequence E_{∗∗}^{r}(ju) collapses at the E^{2}-term, with

E_{∗∗}^{∞}(ju) ≅ H_{∗}(ju; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3) ⊗ Γ(σb) .

(b) For p odd the Bökstedt spectral sequence E_{∗∗}^{r}(ju) collapses at the E^{p}-term, with

E_{∗∗}^{∞}(ju) ≅ H_{∗}(ju; 𝔽_{p}) ⊗ E(σξ̃_{1}^{p}, σξ̃_{2}) ⊗ P_{p}(στ̃_{k} | k ≥ 2) ⊗ Γ(σb) .

(c) The Bökstedt spectral sequence for j at p = 2 has E^{2}-term

E_{∗∗}^{2}(j) ≅ HH_{∗}((A//A_{2})_{∗}) ⊗ HH_{∗}(𝔽_{2} ⊕ Σ^{7}K_{∗})

where

HH_{∗}((A//A_{2})_{∗}) ≅ (A//A_{2})_{∗} ⊗ E(σξ̄_{1}^{8}, σξ̄_{2}^{4}, σξ̄_{3}^{2}, σξ̄_{k} | k ≥ 4)

and

HH_{q}(𝔽_{2} ⊕ Σ^{7}K_{∗}) ≅ [(Σ^{7}K_{∗})^{⊗q}]^{C_{q}} ⊕ [(Σ^{7}K_{∗})^{⊗q+1}]_{C_{q+1}} .

In particular, E_{∗∗}^{2}(j) is not flat as a module over H_{∗}(j; 𝔽_{2}).

Proof (a) The Bökstedt spectral sequence for ju at p = 2 begins

E_{∗∗}^{2}(ju) = H_{∗}(ju; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3) ⊗ Γ(σb) .

Proposition 4.8 applies, so a shortest non-zero differential must map from an algebra indecomposable to a coalgebra primitive and A_{∗}-comodule primitive.

Here we are referring to the A_{∗}-comodule H_{∗}(ju; 𝔽_{2})-Hopf algebra structure on E_{∗∗}^{2}(ju). The only possible algebra indecomposables are the γ_{2^{k}}(σb) in degrees 2^{k+2}, for k ≥ 2. The coalgebra primitives are H_{∗}(ju; 𝔽_{2}){σb, σξ̄_{1}^{4}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3}, all in filtration s = 1. A calculation with ν(σξ̄_{2}^{2}) and ν(σξ̄_{3}) shows that among these, the A_{∗}-comodule primitives are equal to

E(b) ⊗ 𝔽_{2}{σb, σξ̄_{1}^{4}, σξ̄_{k} | k ≥ 4} .

These live in degrees 4, 5, 7, 8, 2^{k} and 2^{k} + 3, for k ≥ 4. The image of a differential on γ_{2^{k}}(σb) must be in total degree 2^{k+2} − 1, for k ≥ 2, but these degrees do not contain any simultaneous coalgebra- and comodule primitives. Therefore there are no non-zero differentials, and the spectral sequence collapses at the E^{2}-term.

(b) For p odd the spectral sequence begins

E_{∗∗}^{2}(ju) = H_{∗}(ju; 𝔽_{p}) ⊗ E(σξ̃_{1}^{p}, σξ̃_{k} | k ≥ 2) ⊗ Γ(σb, στ̃_{k} | k ≥ 2) .

By Proposition 5.6 we have E^{2} = E^{p−1} and there are differentials

d^{p−1}(γ_{p}(στ̃_{k})) = σξ̃_{k+1}

for k ≥ 2. This uses the relation βQ^{p^{k}}(τ̃_{k}) = β(τ̃_{k+1}) = ξ̃_{k+1}. There is also a potential differential

d^{p−1}(γ_{p}(σb)) = σ(βQ^{pq/2}(b)) ,

but βQ^{pq/2}(b) is in degree p^{2}q − 2 of H_{∗}(ju; 𝔽_{p}), which is a trivial group, so this differential is zero. Hence

E_{∗∗}^{p}(ju) = H_{∗}(ju; 𝔽_{p}) ⊗ E(σξ̃_{1}^{p}, σξ̃_{2}) ⊗ P_{p}(στ̃_{k} | k ≥ 2) ⊗ Γ(σb) .

Proposition 4.8 applies again, so a shortest differential must map from one of the algebra indecomposables γ_{p^{k}}(σb) in degrees p^{k+1}q, for k ≥ 2. Its target must be among the coalgebra primitives, which are H_{∗}(ju; 𝔽_{p}){σb, σξ̃_{1}^{p}, σξ̃_{2}, στ̃_{k} | k ≥ 2}, all in filtration s = 1. The target must also be A_{∗}-comodule primitive. The formulas for the A_{∗}-comodule structure on H_{∗}(ju; 𝔽_{p}) imply the following formulas:

ν(σb) = 1 ⊗ σb
ν(σξ̃_{1}^{p}) = 1 ⊗ σξ̃_{1}^{p} − τ_{0} ⊗ σb
ν(σξ̃_{2}) = 1 ⊗ σξ̃_{2} + ξ̄_{1} ⊗ σξ̃_{1}^{p} + τ_{1} ⊗ σb
ν(στ̃_{2}) = 1 ⊗ στ̃_{2} + τ̄_{0} ⊗ σξ̃_{2} + τ̄_{1} ⊗ σξ̃_{1}^{p} − τ_{0}τ_{1} ⊗ σb .

The στ̃_{k} are A_{∗}-comodule primitives for k ≥ 3, in view of the relations (στ̃_{k})^{p} = στ̃_{k+1} for k ≥ 2, the formula for ν(στ̃_{2}) and the fact that τ̄_{0}, τ̄_{1} and τ_{0}τ_{1} all square to zero. A calculation then shows that the simultaneous coalgebra- and comodule primitives are equal to

E(b) ⊗ 𝔽_{p}{σb, στ̃_{k} | k ≥ 3} .

These live in degrees pq, 2pq − 1, 2p^{k} and 2p^{k} + pq − 1 for k ≥ 3. The image of a differential d^{r}(γ_{p^{k}}(σb)) is in degree p^{k+1}q − 1 for k ≥ 2, which contains none of the possible target classes. Hence there are no further differentials, and the spectral sequence collapses at the E^{p}-term.

(c) By the Künneth formula

E_{∗∗}^{2}(j) = HH_{∗}(H_{∗}(j; 𝔽_{2})) ≅ HH_{∗}((A//A_{2})_{∗}) ⊗ HH_{∗}(𝔽_{2} ⊕ Σ^{7}K_{∗}) .

Here the first tensor factor was identified in the discussion of tmf in Section 6. By the following Lemma 7.14,

HH_{q}(𝔽_{2} ⊕ Σ^{7}K_{∗}) ≅ [Σ^{7}K_{∗}^{⊗q}]^{C_{q}} ⊕ [Σ^{7}K_{∗}^{⊗(q+1)}]_{C_{q+1}} ,

and e.g. HH_{1}(𝔽_{2} ⊕ Σ^{7}K_{∗}) is not flat as an 𝔽_{2} ⊕ Σ^{7}K_{∗}-module.

Lemma 7.14 Let k be a field and V a graded k-vector space. The Hochschild homology of the split square-zero extension k ⊕ V , with unit (1, 0) and multiplication (k_{1}, v_{1}) · (k_{2}, v_{2}) = (k_{1}k_{2}, k_{1}v_{2} + k_{2}v_{1}), is

HH_{q}(k ⊕ V) ≅ [V^{⊗q}]^{C_{q}} ⊕ [V^{⊗(q+1)}]_{C_{q+1}}

where [V^{⊗q}]^{C_{q}} ⊂ V^{⊗q} denotes the invariants of the cyclic group C_{q} of order q acting by cyclic permutations on V^{⊗q}, and [V^{⊗q}]_{C_{q}} denotes the coinvariants of this action. When dim_{k} V ≥ 2 the Hochschild homology HH_{∗}(k ⊕ V) is not flat as a module over k ⊕ V .

Proof We compute HH_{∗}(k⊕V) as the homology of the normalized Hochschild complex NC_{∗}(k ⊕ V) with

NC_{q}(k ⊕ V) = (k ⊕ V) ⊗ V^{⊗q} ≅ V^{⊗q} ⊕ V^{⊗(q+1)} .

Here we remind the reader that the normalized Hochschild complex of a kalgebra Λ is given by NC_{q}(Λ) = Λ ⊗ (Λ/k)^{⊗q}. We are considering the case Λ = k ⊕ V , with Λ/k = V .

Since V is a square-zero ideal, the Hochschild boundary ∂ is the direct sum over q ≥ 1 of the operators

1 + (−1)^{q}t_{q}: V^{⊗q} → V^{⊗q} ,

i.e., 1+(−1)^{q}t_{q} on the V^{⊗q} -summand and zero on the V^{⊗(q+1)}-summand, where t_{q}(v_{1} ⊗ ⋯ ⊗ v_{q}) = (−1)^{ϵ}v_{q} ⊗ v_{1} ⊗ ⋯ ⊗ v_{q−1} with ϵ = |v_{q}|(|v_{1}| + ⋯ + |v_{q−1}|).

Let the generator T ∈ C_{q} act on V^{⊗q} as (−1)^{q+1}t_{q}, so ∂ is the sum of the operators 1 − T (and T^{q} acts as the identity). Clearly, then, the Hochschild homology is the direct sum over q ≥ 1 of the kernels

ker(1 − T) = [V^{⊗q}]^{C_{q}}

in degree q, and the cokernels

cok(1 − T) = [V^{⊗q}]_{C_{q}}

in degree (q − 1), plus the term k = V^{⊗0} in degree 0.

For an example of the failure of flatness, let V = k{x, y} with x, y in odd degrees and q = 1. Then HH_{1}(k ⊕ V) ≅ V ⊕ V_{C_{2}}^{⊗2} ≅ k{σx, σy, xσx, xσy ≡ yσx, yσy} is not flat over k ⊕ V .

Theorem 7.15 (a) For p = 2 there is an isomorphism

H_{∗}(T HH(ju); 𝔽_{2}) ≅ H_{∗}(ju; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) ⊗ Γ(σb)

of H_{∗}(ju; 𝔽_{2})-Hopf algebras.

(b) For p odd there is an isomorphism

H_{∗}(T HH(ju); 𝔽_{p}) ≅ H_{∗}(ju; 𝔽_{p}) ⊗ E(σξ̃_{1}^{p}, σξ̃_{2}) ⊗ P (στ̃_{2}) ⊗ Γ(σb)

of H_{∗}(ju; 𝔽_{p})-Hopf algebras.

Proof (a) In view of Proposition 7.13(a) we must identify the possible algebra extensions between the E^{∞}-term

E_{∗∗}^{∞}(ju) ≅ H_{∗}(ju; 𝔽_{2}) ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3) ⊗ Γ(σb)

and the abutment H_{∗}(T HH(ju); 𝔽_{2}). Here H_{∗}(ju; 𝔽_{2}) ≅ (A//A_{1})_{∗} ⊗ E(b) by Proposition 7.12.

In H_{∗}(T HH(ju); 𝔽_{2}) we have (σb)^{2} = Q^{4}(σb) = σQ^{4}(b) = 0, (σξ̄_{1}^{4})^{2} = 0, (σξ̄_{2}^{2})^{2} = 0 and (σξ̄_{k})^{2} = σξ̄_{k+1} for all k ≥ 3, by Proposition 5.9 and the statement about Dyer–Lashof operations in Proposition 7.12(a). It remains to prove that we can find classes

γ_{2^{k}} ∈ H_{4·2^{k}}(T HH(ju); 𝔽_{2})

that are represented by γ_{2^{k}}(σb) in E_{∗∗}^{∞}(ju) and satisfy γ_{2^{k}}^{2} = 0, for all k ≥ 0. We have just seen that we can take γ_{1} = σb. So fix a number k ≥ 1, and assume inductively that we have chosen classes γ_{2^{m}} for 0 ≤ m < k that are represented by γ_{2^{m}}(σb) and satisfy γ_{2^{m}}^{2} = 0.

We shall prove below that a class γ_{2^{k}} representing γ_{2^{k}}(σb) can be chosen so that its square γ_{2^{k}}^{2} is both an H_{∗}(ju; 𝔽_{2})-coalgebra primitive and an A_{∗}-comodule primitive. We saw in the proof of Proposition 7.13(a) that the simultaneous coalgebra- and comodule primitives are

E(b) ⊗ 𝔽_{2}{σb, σξ̄_{1}^{4}, σξ̄_{m} | m ≥ 4} .

When k ≥ 1, the only such class in the degree of γ_{2^{k}}^{2} is σξ̄_{k+3} = (σξ̄_{k+2})^{2}. So either γ_{2^{k}}^{2} = 0 or γ_{2^{k}}^{2} = (σξ̄_{k+2})^{2}. In the latter case we change γ_{2^{k}} by subtracting σξ̄_{k+2}, which does not alter the representative at the E^{∞}-term. Thereby we have achieved γ_{2^{k}}^{2} = 0, which will complete the inductive step.

To show that γ_{2^{k}}^{2} can be arranged to be a coalgebra- and comodule primitive, we make use of the maps of E^{∞}-terms and abutments induced by the commutative S -algebra homomorphism κ: ju → ku_{(2)}. The target E^{∞}-term is

E_{∗∗}^{∞}(ku) ≅ H_{∗}(ku; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}, σξ̄_{k} | k ≥ 3)

with abutment

H_{∗}(T HH(ku); 𝔽_{2}) ≅ H_{∗}(ku; 𝔽_{2}) ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) .

Here H_{∗}(ku; 𝔽_{2}) ≅ (A//E_{1})_{∗}. Note that in degrees less than that of γ_{2^{k}}, κ maps H_{∗}(T HH(ju); 𝔽_{2}) modulo classes that square to zero, which by the inductive hypothesis is (A//A_{1})_{∗} ⊗ P (σξ̄_{3}), injectively into H_{∗}(T HH(ku); 𝔽_{2}) modulo classes that square to zero, which is (A//E_{1})_{∗} ⊗ P (σξ̄_{3}). We shall refer to this property as the “near-injectivity of κ”.

So choose any class γ_{2^{k}} in H_{∗}(T HH(ju); 𝔽_{2}) that is represented by γ_{2^{k}}(σb) in E_{∗∗}^{∞}(ju). We shall arrange that its image κγ_{2^{k}} in H_{∗}(T HH(ku); 𝔽_{2}) squares to zero. If not, we can write

κγ_{2^{k}} ≡ c · (σξ̄_{3})^{ℓ}

with c ∈ (A//E_{1})_{∗} not equal to zero, modulo classes in H_{∗}(T HH(ku); 𝔽_{2}) that square to zero, and modulo similar terms with c of lower degree (or equivalently, with higher exponent ℓ). So c · (σξ̄_{3})^{ℓ} is the “leading term” of κγ_{2^{k}} in H_{∗}(T HH(ku); 𝔽_{2}) modulo classes that square to zero.

We divide into three cases. First, if ℓ = 0 and κγ_{2^{k}} ≡ c with c ∈ (A//E_{1})_{∗}, we can apply the Hopf algebra counit ϵ: T HH(R) → R to see that c = ϵ(κγ_{2^{k}}) = κ(ϵγ_{2^{k}}) is in the image of κ: H_{∗}(ju; 𝔽_{2}) → H_{∗}(ku; 𝔽_{2}), so that in fact c ∈ (A//A_{1})_{∗}. We can then replace γ_{2^{k}} by γ_{2^{k}} −c without altering its representative in E^{∞}, and thus eliminate the case ℓ = 0.

Second, if ℓ = 1 and κγ_{2^{k}} ≡ c · σξ̄_{3} with c ∈ (A//E_{1})_{∗}, we need to argue that c ∈ (A//A_{1})_{∗}. If not, we can write c = ξ̄_{1}^{2} · d with d ∈ (A//A_{1})_{∗}. Here ν(c) = ψ(c) ≡ c ⊗ 1 + d ⊗ ξ̄_{1}^{2} modulo terms with first tensor factor in degree less than |d|. Note from the structure of H_{∗}(T HH(ku); 𝔽_{2}) that κγ_{2^{k}} ≡ c · σξ̄_{3} modulo classes that square to zero, and similar terms e · (σξ̄_{3})^{ℓ} with ℓ ≥ 2, so |e| < |d|. So ν(κγ_{2^{k}}) ≡ c ⊗ σξ̄_{3} + d ⊗ ξ̄_{1}^{2}σξ̄_{3} modulo classes that square to zero, and terms with first factor in lower degree. This must be the image of ν(γ_{2^{k}}) under 1 ⊗ κ, so

ν(γ_{2^{k}}) ≡ c ⊗ σξ̄_{3} + d ⊗ x

for some class x ∈ H_{∗}(T HH(ju); 𝔽_{2}) with κ(x) ≡ ξ̄_{1}^{2}σξ̄_{3}. But there exists no such class x, since H_{∗}(T HH(ju); 𝔽_{2}) modulo classes that square to zero is trivial in degree 10. (This is clear by inspection of the Bökstedt spectral sequence, where the group that could cause a problem, E_{2,8}^{2}(ju), is in fact zero.) This contradiction shows that c ∈ (A//A_{1})_{∗}, and we can alter γ_{2^{k}} by c · σξ̄_{3} without altering the representative at E^{∞}, and thus eliminate the case ℓ = 1.

Third, for the remaining cases ℓ ≥ 2 we consider the coalgebra coproduct. We find

ψ(κγ_{2^{k}}) ≡ ∑_{i+j=ℓ} c · (σξ̄_{3})^{i} ⊗ (σξ̄_{3})^{j}

in H_{∗}(T HH(ku); 𝔽_{2}) ⊗_{H_{∗}(ku;𝔽_{2})} H_{∗}(T HH(ku); 𝔽_{2}), modulo classes that square to zero and similar terms with c of lower degree. Since ℓ ≥ 2 this sum includes some terms c · (σξ̄_{3})^{i} ⊗ (σξ̄_{3})^{j} with both i and j positive, so that c · (σξ̄_{3})^{i} and (σξ̄_{3})^{j} are both in degree less than that of γ_{2^{k}}. This is the image under κ⊗κ of ψ(γ_{2^{k}}) in H_{∗}(T HH(ju); 𝔽_{2}) ⊗_{H_{∗}(ju;𝔽_{2})} H_{∗}(T HH(ju); 𝔽_{2}), so it follows by the near-injectivity of κ that c in fact lies in (A//A_{1})_{∗} ⊂ (A//E_{1})_{∗}.

Then we can change the chosen γ_{2^{k}} in H_{∗}(T HH(ju); 𝔽_{2}) by subtracting c · (σξ̄_{3})^{ℓ} from it, and thus remove the “leading” term c · (σξ̄_{3})^{ℓ} from κγ_{2^{k}}. By repeating this process we can arrange that γ_{2^{k}} has be chosen so that κγ_{2^{k}} is zero modulo classes that square to zero, i.e., that κγ_{2^{k}}^{2} = 0.

Then ψ(γ_{2^{k}}) ≡ γ_{2^{k}} ⊗ 1 + 1 ⊗ γ_{2^{k}} modulo classes that square to zero. For ψ(κγ_{2^{k}}^{2}) = 0 so any other terms in ψ(γ_{2^{k}}) must map under κ to classes that square to zero, hence square to zero themselves by the near-injectivity of κ. Hence γ_{2^{k}}^{2} is a coalgebra primitive.

Next consider the A_{∗}-comodule coaction. If ν(γ_{2^{k}}) ≡ 1 ⊗ γ_{2^{k}} modulo classes that square to zero, then ν(γ_{2^{k}}^{2}) = 1⊗γ_{2^{k}}^{2} and γ_{2^{k}}^{2} is an A_{∗}-comodule primitive, as desired. Otherwise, we can write

ν(γ_{2^{k}}) ≡ a ⊗ (σξ̄_{3})^{ℓ}

with a ∈ A_{∗} in positive degree, modulo classes in A_{∗} ⊗ H_{∗}(T HH(ju); 𝔽_{2}) that square to zero, and modulo similar terms with a of lower degree. Then ν(γ_{2^{k}}^{2}) ≡ a^{2} ⊗ (σξ̄_{3})^{2ℓ} modulo terms with a of lower degree. Applying κ yields a^{2} ⊗ (σξ̄_{3})^{2ℓ} = 0, since κγ_{2^{k}}^{2} = 0, so a^{2} = 0. This is impossible for a ≠ 0, so we conclude that γ_{2^{k}}^{2} is indeed an A_{∗}-comodule primitive.

This completes the proof.

(b) Also in the odd primary case we must identify the algebra extensions between the E^{∞}-term from Proposition 7.13(b)

E_{∗∗}^{∞}(ju) = H_{∗}(ju; 𝔽_{p}) ⊗ E(σξ̃_{1}^{p}, σξ̃_{2}) ⊗ P_{p}(στ̃_{k} | k ≥ 2) ⊗ Γ(σb)

and the abutment H_{∗}(T HH(ju); 𝔽_{p}). In the latter we have (σb)^{p} = Q^{pq/2}(σb) = σQ^{pq/2}(b) = 0 and (στ̃_{k})^{p} = Q^{p^{k}}(στ̃_{k}) = σQ^{p^{k}}(τ̃_{k}) = στ̃_{k+1}, by Propositions 5.9 and 7.12(b). The classes σξ̃_{1}^{p} and σξ̃_{2} are in odd degree, and therefore have square zero, since p is odd.

It remains to prove that we can find classes γ_{p^{k}} ∈ H_{p^{k+1}q}(T HH(ju); 𝔽_{p}) that are represented by γ_{p^{k}}(σb) in E_{∗∗}^{∞}(ju) and satisfy γ_{p^{k}}^{p} = 0, for all k ≥ 0. We have just verified this for k = 0.

The remaining inductive proof follows exactly the same strategy as in the p = 2 case. Instead of working modulo classes that square to zero we work modulo classes with p-th power equal to zero. The S -algebra map κ: ju → ℓ induces an algebra homomorphism in homology that has the required “near– injectivity” property, etc. In the tricky case when κγ_{p^{k}} ≡ c · στ̃_{2}, modulo classes in H_{∗}(T HH(ℓ); 𝔽_{p}) that have trivial p-th power, and modulo similar terms with c of lower degree, we have c ∈ (A//E_{1})_{∗} and must argue that in fact c ∈ (A//A_{1})_{∗}. Writing c = ξ̄_{1}^{f} · d, with d ∈ (A//A_{1})_{∗} and 0 ≤ f < p we assume 0 < f < p and reach a contradiction. The coaction ν(κγ_{p^{k}}) contains a term fξ̄_{1}^{f−1}d⊗ξ̄_{1}στ̄_{2}, and we must check that there is no class x ∈ H_{∗}(T HH(ju); 𝔽_{p}) with κx = ξ̄_{1}στ̄_{2}. This is again clear by the Bökstedt spectral sequence. The rest of the proof can safely be omitted.

8 Topological K-theory revisited

We now wish to pass from the homology of the spectra T HH(ku) and T HH(ko) to their homotopy. The first case is the analogue for p = 2 of the discussion in [MS93, §§5–7], where McClure and Staffeldt compute the mod p homotopy groups π_{∗}(T HH(ℓ); ℤ/p) of the Adams summand ℓ ⊂ ku_{(p)} for odd primes p.

The idea is to compute homotopy groups using the Adams spectral sequence

E_{2}^{s,t} = Ext_{A_{∗}}^{s,t}(𝔽_{p}, H_{∗}(X; 𝔽_{p})) ⟹ π_{t−s}(X)_{p}^{∧} ,

which is strongly convergent for bounded below spectra X of ℤ_{p}-finite type. This is difficult for X = T HH(ku) or T HH(ko) at p = 2, just as for T HH(ℓ) at p odd, but becomes manageable after introducing suitable finite coefficients, e.g. after smashing T HH(ku) with the mod 2 Moore spectrum M = C_{2}, or smashing T HH(ko) with the Mahowald spectrum Y = C_{2} ∧ C_{η} [Mah82]. Here C_{f} denotes the mapping cone of a map f. Neither of these finite CW spectra are ring spectra, so an extra argument is needed to have products on T HH(ku)∧M and T HH(ko) ∧ Y . We shall manage with the following very weak version of a ring spectrum.

Definition 8.1 A μ-spectrum (or an A_{2} ring spectrum) is a spectrum R with a unit η: S → R and multiplication μ: R ∧ R → R that is left and right unital, but not necessarily associative or commutative.

The following result is similar to [Ok79, 1.5], but slightly easier.

Lemma 8.2 Let R be a μ-spectrum and let

\xymatrix{S^{k} \ar[r]^{f} & S^{0} \ar[r]^{i} & C_{f} \ar[r]^{π} & S^{k+1}}

be a cofiber sequence such that id_{R} ∧ id_{C_{f}} ∧ f: R ∧ Σ^{k}C_{f} → R ∧ C_{f} is nullhomotopic. Then there exists a multiplication μ: (R∧C_{f})∧(R∧C_{f}) → (R∧C_{f}) that makes R ∧ C_{f} a μ-spectrum and id_{R} ∧ i: R → R ∧ C_{f} a map of μ-spectra.

Proof A choice of null-homotopy provides a splitting m: R∧C_{f} ∧C_{f} → R∧C_{f} in the cofiber sequence

\xymatrix{R ∧ Σ^{k}C_{f} \ar[r]^{id∧id∧f} & R ∧ C_{f} \ar[r]^{id∧id∧i} & R ∧ C_{f} ∧ C_{f} \ar[r]^{id∧id∧π} & R ∧ Σ^{k+1}C_{f}}

which satisfies m(id ∧ id ∧ i) ≃ id (right unitality). The difference id − m(id ∧ i ∧ id): R ∧ C_{f} → R ∧ C_{f} restricts trivially over id ∧ i: R → R ∧ C_{f}, so extends over id ∧ π: R ∧ C_{f} → R ∧ S^{k+1} to a map ϕ: R ∧ S^{k+1} → R ∧ C_{f}. There is a short exact sequence

\xymatrix{[R ∧ Σ^{k+1}C_{f}, R ∧ C_{f}] \ar[r]^{i^{∗}} & [R ∧ S^{k+1}, R ∧ C_{f}] \ar[r]^{f^{∗}} & [R ∧ S^{2k+1}, R ∧ C_{f}]}

where the homomorphism f^{∗} = (id ∧ f ∧ id)^{∗} is trivial, because it can be identified with (id ∧ id ∧ f)^{∗}: [R ∧ C_{f}, R] → [R ∧ Σ^{k}C_{f}, R] under Spanier– Whitehead duality, and we have assumed that this map id ∧ id ∧ f is nullhomotopic. (The Spanier–Whitehead dual of C_{f} is Σ^{−(k+1)}C_{f}.) Thus i^{∗} = (id ∧ Σ^{k+1}i)^{∗} is surjective and ϕ extends further over id ∧ Σ^{k+1}i to a map ψ: R ∧ Σ^{k+1}C_{f} → R ∧ C_{f}. Then we can subtract ψ ∘ (id ∧ id ∧ π) from m so as to make m(id ∧ i ∧ id) ≃ id (left unitality), without destroying the right unitality. Once this is achieved, the required multiplication is defined as the composite

\xymatrix{(R ∧ C_{f}) ∧ (R ∧ C_{f}) \ar[r]^{id∧τ∧id} & R ∧ R ∧ C_{f} ∧ C_{f} \ar[d]^{μ∧id∧id} \\  & R ∧ C_{f} ∧ C_{f} \ar[r]^{m} & (R ∧ C_{f}) ,}

where τ denotes the twist map.

In the case f = 2: S^{0} → S^{0}, the Moore spectrum M = C_{2} has cohomology H^{∗}(M; 𝔽_{2}) = E(Sq^{1}), which equals E_{1}//E(Q_{1}) as an E_{1}-module. By the Cartan formula Sq^{2} acts nontrivially on H^{∗}(M ∧ M; 𝔽_{2}), and therefore M does not split off from M ∧ M. So M is not a μ-spectrum, and the map 2: M → M is not null-homotopic. It must therefore factor as the composite

\xymatrix{M \ar[r]^{π} & S^{1} \ar[r]^{η} & S^{0} \ar[r]^{i} & M .}  (8.3)

See e.g. [AT65, 1.1].

Similarly, in the case f = η: S^{1} → S^{0} the mapping cone C_{η} has cohomology H^{∗}(C_{η}; 𝔽_{2}) = E(Sq^{2}). By the Cartan formula Sq^{4} acts nontrivially on H^{∗}(C_{η}∧ C_{η}; 𝔽_{2}), and therefore C_{η} does not split off from C_{η} ∧ C_{η}. In particular, the map η: ΣC_{η} → C_{η} is not null-homotopic, and must factor as the composite

\xymatrix{ΣC_{η} \ar[r]^{π} & S^{3} \ar[r]^{ν} & S^{0} \ar[r]^{i} & C_{η} .}  (8.4)

Here ν ∈ π_{3}(S) is the Hopf invariant one class. The Mahowald spectrum Y = C_{2} ∧ C_{η} has cohomology H^{∗}(Y; 𝔽_{2}) = E(Sq^{1}, Sq^{2}), which equals A_{1}//E(Q_{1}) as an A_{1}-module.

Lemma 8.5 (a) T HH(ku) ∧ M is a μ-spectrum.

(b) T HH(ko) ∧ C_{η} and T HH(ko) ∧ Y are μ-spectra.

Proof (a) Let T = T HH(ku). By (8.3) the map 1 ∧ 2: T ∧ M → T ∧ M factors through 1 ∧ η: ΣT → T, which in turn factors as

\xymatrix{ΣT \ar[r]^{1∧η} & T ∧ ku → T ,}

since T = T HH(ku) is a ku-module spectrum. But η ∈ π_{1}(S) maps to zero in π_{1}(ku), so this map is null-homotopic, and Lemma 8.2 applies.

(b) Let T = T HH(ko) and R = T HH(ko)∧C_{η}. By (8.4) the map 1∧η: ΣR → R factors through 1 ∧ ν: Σ^{3}T → T, which in turn factors as

\xymatrix{Σ^{3}T \ar[r]^{1∧ν} & T ∧ ko → T ,}

since T = T HH(ko) is a ko-module spectrum. But ν ∈ π_{3}(S) maps to zero in π_{3}(ko), so 1 ∧ η: ΣR → R is null-homotopic, and Lemma 8.2 applies again to prove that R is a μ-spectrum under T.

Using once more that 1 ∧ η: ΣR → R is null-homotopic we get that 1 ∧ 2: R ∧ M → R ∧ M is null-homotopic by (8.3), and so R ∧ M = T HH(ko) ∧ Y is a μ-spectrum under R.

Lemma 8.6 Let B ⊂ A be a sub Hopf algebra and let N be an A_{∗}-comodule algebra. Then there is an isomorphism of A_{∗}-comodule algebras

(A//B)_{∗} ⊗ N ≅ A_{∗} □_{B_{∗}} N .

Here B_{∗} is the quotient Hopf algebra of A_{∗} dual to B, and (A//B)_{∗} = A_{∗}□_{B_{∗}}𝔽_{p} is dual to A ⊗_{B} 𝔽_{p}.

Proof This is analogous to the usual G-homeomorphism G/H ×X ≅ G×_{H} X for a G-space X and subgroup H ⊂ G. Let i: (A//B)_{∗} → A_{∗} be the inclusion and ν: N → A_{∗} ⊗ N the coaction. The composite homomorphism

\xymatrix{(A//B)_{∗} ⊗ N \ar[r]^{i⊗ν} & A_{∗} ⊗ A_{∗} ⊗ N \ar[r]^{ϕ⊗1} & A_{∗} ⊗ N}

equalizes the two maps to A_{∗} ⊗ B_{∗} ⊗ N, and hence factors uniquely through A_{∗} □_{B_{∗}} N. An explicit inverse can be constructed using the Hopf algebra conjugation χ on A_{∗}.

Recall the n-th connective Morava K -theory spectrum k(n), with homotopy π_{∗}k(n) = 𝔽_{p}[v_{n}] where |v_{n}| = 2(p^{n} − 1), and cohomology H^{∗}(k(n); 𝔽_{p}) ≅ A//E(Q_{n}) [BM72]. Dually, H_{∗}(k(n); 𝔽_{p}) ≅ (A//E(Q_{n}))_{∗} ⊂ A_{∗}. In particular, for n = 1 and p = 2 we have k(1) ≃ ku∧M with H_{∗}(k(1); 𝔽_{2}) ≅ (A//E(Q_{1}))_{∗} ≅ P (ξ̄_{1}, ξ̄_{2}^{2}, ξ̄_{k} | k ≥ 3). The n-th periodic Morava K -theory spectrum K(n) is the telescope v_{n}^{−1}k(n) of the iterated maps v_{n}: k(n) → Σ^{2(1−p^{n})}k(n), with homotopy π_{∗}K(n) = 𝔽_{p}[v_{n}, v_{n}^{−1}]. These are (non-commutative) S -algebras for all p and n ([Rob89], [La03]).

Proposition 8.7 (a) There are A_{∗}-comodule algebra isomorphisms

H_{∗}(T HH(ku) ∧ M; 𝔽_{2}) ≅ (A//E(Q_{1}))_{∗} ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})
≅ H_{∗}(k(1); 𝔽_{2}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ) .

Here ν(σξ̄_{1}^{2}) = 1 ⊗ σξ̄_{1}^{2}, ν(σξ̄_{2}^{2}) = 1 ⊗ σξ̄_{2}^{2}, and ν(σξ̄_{3}) = 1 ⊗ σξ̄_{3} + ξ̄_{1} ⊗ σξ̄_{2}^{2}.

The classes λ_{1} = σξ̄_{1}^{2}, λ_{2} = σξ̄_{2}^{2} and μ = σξ̄_{3} + ξ̄_{1} · σξ̄_{2}^{2} (in degrees 3, 7 and 8, respectively) are A_{∗}-comodule primitives.

(b) There are A_{∗}-comodule algebra isomorphisms

H_{∗}(T HH(ko) ∧ Y; 𝔽_{2}) ≅ (A//E(Q_{1}))_{∗} ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})
≅ H_{∗}(k(1); 𝔽_{2}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ) .

Here ν(σξ̄_{1}^{4}) = 1 ⊗ σξ̄_{1}^{4}, ν(σξ̄_{2}^{2}) = 1 ⊗ σξ̄_{2}^{2} + ξ̄_{1}^{2} ⊗ σξ̄_{1}^{4} and ν(σξ̄_{3}) = 1 ⊗ σξ̄_{3} + ξ̄_{1} ⊗ σξ̄_{2}^{2} + ξ̄_{2} ⊗ σξ̄_{1}^{4}.

The exterior classes λ_{1} = σξ̄_{1}^{4} and λ_{2} = σξ̄_{2}^{2} + ξ̄_{1}^{2} · σξ̄_{1}^{4} are A_{∗}-comodule primitives, while μ = σξ̄_{3} + ξ̄_{1} · σξ̄_{2}^{2} has

ν(μ) = 1 ⊗ μ + ξ̄_{1}^{2} ⊗ ξ̄_{1} · λ_{1} + (ξ̄_{2} + ξ̄_{1}^{3}) ⊗ λ_{1} .

The squared class μ^{2} = (σξ̄_{3})^{2} is A_{∗}-comodule primitive.

In each case, the homology algebra on the left hand side has the unit and product induced by the μ-spectrum structure from Lemma 8.5. This product is in fact associative and graded commutative, in view of the exhibited additive and multiplicative isomorphism with the associative and graded commutative algebra on the right hand side.

Proof (a) By Corollary 5.14(a) there is an A_{∗}-comodule algebra isomorphism

H_{∗}(T HH(ku) ∧ M; 𝔽_{2}) ≅ (A//E_{1})_{∗} ⊗ (E_{1}//E(Q_{1}))_{∗} ⊗ E(σξ̄_{1}^{2}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3})

with the diagonal A_{∗}-comodule structure on the first two tensor factors, and the claimed coaction on the remaining generators.

Since H_{∗}(M; 𝔽_{2}) ≅ (E_{1}//E(Q_{1}))_{∗} is in fact an A_{∗}-comodule algebra, there is an A_{∗}-comodule algebra isomorphism

(A//E_{1})_{∗} ⊗ (E_{1}//E(Q_{1}))_{∗} ≅ A_{∗} □_{E_{1∗}} (E_{1}//E(Q_{1}))_{∗} ≅ (A//E(Q_{1}))_{∗}

by Lemma 8.6. We are free to replace the polynomial generator σξ̄_{3} by the primitive class μ, since their difference ξ̄_{1} · σξ̄_{2}^{2} has square zero.

(b) By Theorem 6.2(a) there is an A_{∗}-comodule algebra isomorphism

H_{∗}(T HH(ko) ∧ Y; 𝔽_{2}) ≅ (A//A_{1})_{∗} ⊗ (A_{1}//E(Q_{1}))_{∗} ⊗ E(σξ̄_{1}^{4}, σξ̄_{2}^{2}) ⊗ P (σξ̄_{3}) , with the claimed coaction on the exterior and polynomial generators. Again, by Lemma 8.6 there is an isomorphism of A_{∗}-comodule algebras

(A//A_{1})_{∗} ⊗ (A_{1}//E(Q_{1}))_{∗} ≅ A_{∗} □_{A_{1∗}} (A_{1}//E(Q_{1}))_{∗} ≅ (A//E(Q_{1}))_{∗} .

The classes λ_{1}, λ_{2} and μ are defined as in the statement of the proposition, and their coactions are obtained by direct calculation.

Lemma 8.8 (a) The E_{2}-term of the Adams spectral sequence converging to π_{∗}(T HH(ku) ∧ M) is

E_{2}^{∗∗} ≅ P (v_{1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

with λ_{1}, λ_{2}, μ and v_{1} = [ξ_{2}] in bidegrees (0, 5), (0, 7), (0, 8) and (1, 3), respectively.

(b) The E_{2}-term of the Adams spectral sequence for π_{∗}(T HH(ko) ∧ Y) is

E_{2}^{∗∗} ≅ (P (v_{1}) ⊗ E(λ_{2}, λ_{3}) ⊕ E(λ_{2}){λ_{1}}) ⊗ P (μ^{2})

with λ_{1}, λ_{2}, λ_{3} = λ_{1}μ, μ^{2} and v_{1} = [ξ_{2}] in bidegrees (0, 5), (0, 7), (0, 13), (0, 16) and (1, 3), respectively. It is the homology of the algebra

P (v_{1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

with respect to the differential d(μ) = v_{1}λ_{1}, with cycles λ_{1}, λ_{2} and v_{1}.

Proof (a) By change-of-rings, the E_{2}-term of the Adams spectral sequence for X = T HH(ku) ∧ M is

E_{2}^{∗∗} = Ext_{A_{∗}}^{∗∗}(𝔽_{2}, (A//E(Q_{1}))_{∗} ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ Ext_{E(Q_{1})_{∗}}^{∗∗}(𝔽_{2}, E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ P (v_{1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

as a graded algebra. Here E(Q_{1})_{∗} = E(ξ_{2}) and Ext_{E(ξ_{2})}^{∗∗}(𝔽_{2}, 𝔽_{2}) = P (v_{1}) with v_{1} = [ξ_{2}] in the cobar complex.

(b) The E_{2}-term of the Adams spectral sequence for X = T HH(ko) ∧ Y is

E_{2}^{∗∗} = Ext_{A_{∗}}^{∗∗}(𝔽_{2}, (A//E(Q_{1}))_{∗} ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ Ext_{E(Q_{1})_{∗}}^{∗∗}(𝔽_{2}, E(λ_{1}, λ_{2}) ⊗ P (μ))
≅ Ext_{E(Q_{1})_{∗}}^{∗∗}(𝔽_{2}, 𝔽_{2}{1, λ_{1}, λ_{2}, μ, λ_{1}λ_{2}, λ_{1}μ, λ_{2}μ, λ_{1}λ_{2}μ} ⊗ P (μ^{2})) .

Here

𝔽_{2}{1, λ_{1}, λ_{2}, μ, λ_{1}λ_{2}, λ_{1}μ, λ_{2}μ, λ_{1}λ_{2}μ} ≅ 𝔽_{2}{1, λ_{2}, λ_{1}μ, λ_{1}λ_{2}μ} ⊕ E(Q_{1})_{∗}{λ_{1}, λ_{1}λ_{2}} as E(Q_{1})_{∗}-comodules. For λ_{1}, λ_{2} and μ^{2} are A_{∗}-comodule primitives, while ν(μ) maps to 1 ⊗ μ + ξ_{2} ⊗ λ_{1} in E(Q_{1})_{∗} ⊗ H_{∗}(T HH(ko) ∧ Y; 𝔽_{2}). So

E_{2}^{∗∗} = (P (v_{1}) ⊗ E(λ_{2}, λ_{1}μ) ⊕ E(λ_{2}){λ_{1}}) ⊗ P (μ^{2}) .

We let λ_{3} = λ_{1}μ to obtain the claimed formula.

To determine the differentials in the Adams spectral sequence for π_{∗}(T HH(ku)∧ M) or π_{∗}(T HH(ko)∧Y) we first compute the v_{1}-periodic homotopy. This is in turn easy to derive from the K(1)-homology. Recall that BP_{∗}BP ≅ BP_{∗}[t_{k} | k ≥ 1] with |t_{k}| = 2(2^{k} − 1) and K(1)_{∗} = 𝔽_{2}[v_{1}, v_{1}^{−1}] with |v_{1}| = 2.

Lemma 8.9 For p = 2 there are isomorphisms of K(1)_{∗}-algebras

K(1)_{∗}(ku) ≅ K(1)_{∗}[t_{k} | k ≥ 1]/(v_{1}t_{k}^{2} = v_{1}^{2^{k}}t_{k}) ≅ K(1)_{∗}[u_{k} | k ≥ 1]/(u_{k}^{2} = u_{k})

and

K(1)_{∗}(ko) ≅ K(1)_{∗}[t_{k} | k ≥ 2]/(v_{1}t_{k}^{2} = v_{1}^{2^{k}}t_{k}) ≅ K(1)_{∗}[u_{k} | k ≥ 2]/(u_{k}^{2} = u_{k}) .

Proof We first follow the proof of [MS93, 5.3]. We have K(1)_{∗}(BP) ≅ K(1)_{∗} ⊗_{BP_{∗}} BP_{∗}(BP) ≅ K(1)_{∗}[t_{k} | k ≥ 1], where |t_{k}| = 2(2^{k} − 1). The spectrum ku_{(2)} = BP ⟨1⟩ can be constructed from BP by Baas–Sullivan cofiber sequences killing the classes v_{k+1} for k ≥ 1, which map to

η_{R}(v_{k+1}) ≡ v_{1}t_{k}^{2} + v_{1}^{2^{k}}t_{k} mod (η_{R}(v_{2}), …, η_{R}(v_{k}))

by [Ra04, 6.1.13]. So each η_{R}(v_{k+1}) ∈ K(1)_{∗}(BP) is not a zero divisor mod (η_{R}(v_{2}), …, η_{R}(v_{k})), and

K(1)_{∗}(ku) ≅ K(1)_{∗}[t_{k} | k ≥ 1]/(v_{1}t_{k}^{2} = v_{1}^{2^{k}}t_{k}) .

Substituting u_{k} = v_{1}^{1−2^{k}}t_{k}, the relations become u_{k}^{2} = u_{k} for each k ≥ 1.

The cofiber sequence

\xymatrix{Σko \ar[r]^{η} & ko \ar[r]^{c} & ku \ar[r]^{∂} & Σ^{2}ko}

induces a short exact sequence

\xymatrix{0 → K(1)_{∗}(ko) \ar[r]^{c_{∗}} & K(1)_{∗}(ku) \ar[r]^{∂_{∗}} & Σ^{2}K(1)_{∗}(ko) → 0}

since multiplication by η is zero in K(1)-homology. The connecting map ∂ right multiplies by Sq^{2} in mod 2 cohomology, so right “comultiplies” with the dual class ξ̄_{1}^{2} in mod 2 homology, which corresponds to t_{1} in BP_{∗}BP [Za72, p. 488]. From the coproduct formula for Δ(t_{j}) [Ra04, A2.1.27(e)] it follows that ∂_{∗}(t_{1}) = Σ^{2}(1) while ∂_{∗}(t_{k}) = 0 for k ≥ 2. Hence we can identify K(1)_{∗}(ko) with the claimed subalgebra of K(1)_{∗}(ku), via the K(1)_{∗}-algebra homomorphism c_{∗}.

Lemma 8.10 The unit maps R → T HH(R) induce isomorphisms

\xymatrix{K(1)_{∗}(ku) \ar[r]^{≅} & K(1)_{∗}T HH(ku)}

and

\xymatrix{K(1)_{∗}(ko) \ar[r]^{≅} & K(1)_{∗}T HH(ko) .}

Proof The proof of [MS93, 5.3] continues as follows. There is a K(n)-based Bökstedt spectral sequence

E_{∗∗}^{2} = HH_{∗}^{K(n)_{∗}}(K(n)_{∗}(R)) ⟹ K(n)_{∗}T HH(R)

for every S -algebra R, derived like the one in (4.1), but by applying K(n)- homology to the skeleton filtration of T HH(R). The identification of the E^{2}- term uses the Künneth formula for Morava K -theory. When K(n)_{∗}(R) is concentrated in degrees ∗ ≡ 0 mod |v_{n}|, then K(n)_{∗}(R) ≅ K(n)_{∗}⊗_{𝔽_{p}}K(n)_{0}(R) and we can rewrite the E^{2}-term as

E_{∗∗}^{2} ≅ K(n)_{∗} ⊗_{𝔽_{p}} HH_{∗}^{𝔽_{p}}(K(n)_{0}(R)) .

This is the case for n = 1, p = 2 and R = ku, when K(1)_{0}(ku) is the colimit over m of the algebras 𝔽_{2}[u_{k} | 1 ≤ k ≤ m]/(u_{k}^{2} = u_{k}) ≅ ∏_{i=1}^{2^{m}} 𝔽_{2}. Then for each m the unit map ∏_{i=1}^{2^{m}} 𝔽_{2} → HH_{∗}^{𝔽_{2}}(∏_{i=1}^{2^{m}} 𝔽_{2}) is an isomorphism, so by passage to the colimit the unit map

K(1)_{0}(ku) → HH_{∗}^{𝔽_{2}}(K(1)_{0}(ku))

is an isomorphism. Hence the K(1)-based Bökstedt spectral sequence collapses at the edge s = 0, and the unit map ku → T HH(ku) induces the asserted isomorphism.

Likewise, K(1)_{0}(ko) is the colimit of the algebras 𝔽_{2}[u_{k} | 2 ≤ k ≤ m]/(u_{k}^{2} = u_{k}) ≅ ∏_{i=1}^{2^{m−1}} 𝔽_{2}, and the same argument shows that the unit map ko → T HH(ko) induces an isomorphism in K(1)-homology.

The mod 2 Moore spectrum M admits a degree 8 self-map v_{1}^{4} : M → Σ^{−8}M that induces multiplication by v_{1}^{4} in ku_{∗}(M) = k(1)_{∗} = P (v_{1}). Smashing with C_{η} yields a self-map v_{1}^{4} : Y → Σ^{−8}Y , which admits a fourth root v_{1} : Y → Σ^{−2}Y (up to nilpotent maps). It induces multiplication by v_{1} in ko_{∗}(Y) = k(1)_{∗} = P (v_{1}). See [DM82, 1.2].

Lemma 8.11 Let X be a spectrum such that K(1)_{∗}(X) = 0. Then v_{1}^{−1}π_{∗}(X∧ Y) = 0 and v_{1}^{−4}π_{∗}(X ∧ M) = 0.

Proof Recall that j is the homotopy fiber of the map ψ^{3}−1: ko_{(2)} → bspin_{(2)}. The unit map e: S → j induces an equivalence of mapping telescopes

\xymatrix{v_{1}^{−1}(S ∧ Y) \ar[r]^{≃} & v_{1}^{−1}(j ∧ Y) .}

See e.g. the case n = 0 of [Mah82, 1.4]. Here v_{1}^{−1}(ko_{(2)} ∧ Y) ≃ v_{1}^{−1}k(1) = K(1) and likewise v_{1}^{−1}(bspin_{(2)} ∧ Y) ≃ K(1), so there is a cofiber sequence of spectra

\xymatrix{v_{1}^{−1}Y → K(1) \ar[r]^{ψ} & K(1) .}

Furthermore, v_{1}^{−1}Y ≃ v_{1}^{−4}Y sits in a cofiber sequence

\xymatrix{v_{1}^{−4}ΣM \ar[r]^{η} & v_{1}^{−4}M → v_{1}^{−4}Y}

where η is nilpotent (η^{4} = 0).

From the first cofiber sequence it follows that if K(1)_{∗}(X) = 0, then π_{∗}(X ∧ v_{1}^{−1}Y) = v_{1}^{−1}π_{∗}(X ∧ Y) = 0. From the second cofiber sequence it then follows that multiplication by η is an isomorphism on π_{∗}(X ∧v_{1}^{−4}M) = v_{1}^{−4}π_{∗}(X ∧M). Since η is nilpotent this implies that v_{1}^{−4}π_{∗}(X ∧ M) = 0.

Corollary 8.12 The unit maps induce isomorphisms

\xymatrix{K(1)_{∗} = v_{1}^{−4}π_{∗}(ku ∧ M) \ar[r]^{≅} & v_{1}^{−4}π_{∗}(T HH(ku) ∧ M)}

and

\xymatrix{K(1)_{∗} = v_{1}^{−1}π_{∗}(ko ∧ Y) \ar[r]^{≅} & v_{1}^{−1}π_{∗}(T HH(ko) ∧ Y) .}

Proof Apply Lemmas 8.10 and 8.11 to the cofiber of the unit map R → T HH(R), for R = ku and R = ko, respectively.

Theorem 8.13 Consider the Adams spectral sequence

E_{2}^{∗∗} = P (v_{1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

for π_{∗}(T HH(ku)∧M), with λ_{1} = σξ̄_{1}^{2}, λ_{2} = σξ̄_{2}^{2}, μ = σξ̄_{3}+ξ̄_{1}·σξ̄_{2}^{2} and v_{1} = [ξ_{2}] in bidegrees (s, t) = (0, 3), (0, 7), (0, 8) and (1, 3), respectively. Recursively define

λ_{n} = λ_{n−2}μ^{2^{n−3}}

for n ≥ 3. Likewise define r(1) = 2, r(2) = 4, r(n) = 2^{n} + r(n − 2), s(1) = 3, s(2) = 7 and s(n) = 2^{n} + s(n − 2), for n ≥ 3. So λ_{n} has bidegree (0, s(n)) and 2r(n) + s(n) = 2^{n+2} − 1.

Then the Adams spectral sequence has differentials generated by

d^{r(n)}(μ^{2^{n−1}}) = v_{1}^{r(n)}λ_{n}

for all n ≥ 1. This leaves the E_{∞}-term

E_{∞}^{∗∗} = P (v_{1}){1} ⊕ ⨁_{n=1}^{∞} P_{r(n)}(v_{1}){λ_{n}} ⊗ E(λ_{n+1}) ⊗ P (μ^{2^{n}}) .

Hence π_{∗}(T HH(ku)∧M) = π_{∗}(T HH(ku); ℤ/2) is generated as a P (v_{1})-module by elements 1, x_{n,m} = λ_{n}μ^{2^{n}m} and x′_{n,m} = λ_{n}λ_{n+1}μ^{2^{n}m} for n ≥ 1 and m ≥ 0. Here |x_{n,m}| = s(n)+2^{n+3}m and |x′_{n,m}| = s(n)+s(n+1)+2^{n+3}m. The module structure is generated by the relations v_{1}^{r(n)}x_{n,m} = 0 and v_{1}^{r(n)}x′_{n,m} = 0 for n ≥ 1, m ≥ 0.

Proof The classes λ_{1}, λ_{2} and μ were introduced in Proposition 8.7, and the Adams spectral sequence E_{2}-term was found in Lemma 8.8. By Corollary 8.12 the abutment π_{∗}(T HH(ku) ∧ M) is all v_{1}-torsion, except the direct summand π_{∗}(ku ∧ M) = P (v_{1}), which is included by the unit map. Hence every class λ_{n} is v_{1}-torsion, so there is some integer r(n) such that v_{1}^{r(n)}λ_{n} is hit by a differential.

Suppose by induction that the d^{r(k)}-differentials for 1 ≤ k < n have been found, leaving the term

E_{r(n−1)+1}^{∗∗} = P (v_{1}) ⊗ E(λ_{n}, λ_{n+1}) ⊗ P (μ^{2^{n−1}})

⊕ ⨁_{k=1}^{n−1} P_{r(k)}(v_{1}){λ_{k}} ⊗ E(λ_{k+1}) ⊗ P (μ^{2^{k}}) .

Consider the P (v_{1})-module generated by λ_{n}. Let r be minimal such that v_{1}^{r}λ_{n} is a boundary. The source x of such a differential cannot be divisible by v_{1}, since r is minimal, so d^{r}(x) = v_{1}^{r}λ_{n} where x has Adams filtration s = 0 and even total degree. Furthermore, v_{1}^{r}λ_{n} is not v_{1}-torsion at this term, so x cannot be v_{1}-torsion. Likewise, λ_{n+1} does not annihilate v_{1}^{r}λ_{n}, so λ_{n+1} cannot annihilate x either. This forces x ∈ P (μ^{2^{n−1}}). By Lemma 8.5, T HH(ku)∧M is a μ-spectrum, so d^{r} is a derivation and the Leibniz rule shows that x = μ^{2^{n−1}} , since d^{r} on any higher power of μ^{2^{n−1}} must be divisible by μ^{2^{n−1}} . Hence d^{r}(μ^{2^{n−1}}) = v_{1}^{r}λ_{n}, and by a degree count we must have r = r(n).

To complete the induction step, we must compute the E_{r(n)+1}-term of the Adams spectral sequence. The d^{r(n)}-differential does not affect the summands P_{r(k)}(v_{1}){λ_{k}} ⊗ E(λ_{k+1}) ⊗ P (μ^{2^{k}}) for 1 ≤ k < n, and is zero on E(λ_{n+1}) ⊗ P (μ^{2^{n}}). It acts on P (v_{1}){1, λ_{n}, μ^{2^{n−1}}, λ_{n}μ^{2^{n−1}}}, leaving P_{r(n)}(v_{1}){λ_{n}}⊕P (v_{1})⊗ E(λ_{n+2}), where by definition λ_{n+2} = λ_{n}μ^{2^{n−1}} . This shows that the term P (v_{1}) ⊗ E(λ_{n}, λ_{n+1}) ⊗ P (μ^{2^{n−1}}) at the E_{r(n−1)+1}-term gets replaced by the direct sum of P (v_{1}) ⊗ E(λ_{n+1}, λ_{n+2}) ⊗ P (μ^{2^{n}}) and P_{r(n)}(v_{1}){λ_{n}} ⊗ E(λ_{n+1}) ⊗ P (μ^{2^{n}}).

Theorem 8.14 Consider the Adams spectral sequence

E_{2}^{∗∗} = (P (v_{1}) ⊗ E(λ_{2}, λ_{3}) ⊕ E(λ_{2}){λ_{1}}) ⊗ P (μ^{2})

for π_{∗}(T HH(ko) ∧ Y), with λ_{1} = σξ̄_{1}^{4}, λ_{2} = σξ̄_{2}^{2} + ξ̄_{1}^{2} · σξ̄_{1}^{4}, λ_{3} = σξ̄_{1}^{4}(σξ̄_{3} + ξ̄_{1} · σξ̄_{2}^{2}), μ^{2} = (σξ̄_{3})^{2} and v_{1} = [ξ_{2}] in bidegrees (s, t) = (0, 5), (0, 7), (0, 13), (0, 16) and (1, 3), respectively. Recursively define

λ_{n} = λ_{n−2}μ^{2^{n−3}}

for n ≥ 4. Likewise define r(1) = 1, r(2) = 4, r(n) = 2^{n} + r(n − 2), s(1) = 5, s(2) = 7 and s(n) = 2^{n} + s(n − 2), for n ≥ 3. So λ_{n} has bidegree (0, s(n)) and 2r(n) + s(n) = 2^{n+2} − 1.

Then the Adams spectral sequence has differentials generated by

d^{r(n)}(μ^{2^{n−1}}) = v_{1}^{r(n)}λ_{n}

for all n ≥ 2. This leaves the E_{∞}-term

E_{∞}^{∗∗} = P (v_{1}){1} ⊕ ⨁_{n=1}^{∞} P_{r(n)}(v_{1}){λ_{n}} ⊗ E(λ_{n+1}) ⊗ P (μ^{2^{n}}) .

Hence π_{∗}(T HH(ko) ∧ Y) = π_{∗}(T HH(ko); Y) is generated as a P (v_{1})-module by elements 1, x_{n,m} = λ_{n}μ^{2^{n}m} and x′_{n,m} = λ_{n}λ_{n+1}μ^{2^{n}m} for n ≥ 1 and m ≥ 0. Here |x_{n,m}| = s(n)+2^{n+3}m and |x′_{n,m}| = s(n)+s(n+1)+2^{n+3}m. The module structure is generated by the relations v_{1}^{r(n)}x_{n,m} = 0 and v_{1}^{r(n)}x′_{n,m} = 0 for n ≥ 1, m ≥ 0.

Proof Starting with an imagined E_{1}-term

E_{1}^{∗∗} = P (v_{1}) ⊗ E(λ_{1}, λ_{2}) ⊗ P (μ)

and differential d_{1}(μ) = v_{1}λ_{1}, the proof is the same as for Theorem 8.13.

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Department of Mathematics, Massachusetts Institute of Technology, USA and Department of Mathematics, University of Oslo, Norway

Email: vigleik@math.mit.edu and rognes@math.uio.no