THE PRODUCT ON TOPOLOGICAL

HOCHSCHILD HOMOLOGY OF THE INTEGERS

WITH MOD FOUR COEFFICIENTS

John Rognes

Abstract. We compute the natural multiplication on mod four homotopy for the ring spectrum T (ℤ) = T HH(ℤ), and its associated natural module action upon the mod two homotopy.

Introduction

Bökstedt’s topological Hochschild homology T HH(A) = T (A) of a ring A can be thought of as the spectrum defined by forming the Hochschild complex for A, but replacing A with its Eilenberg–Mac Lane spectrum HA everywhere, and replacing all tensor products with smash products of spectra. Intuitively this gives a simplicial spectrum [q] ↦ T (A)_{q} = HA ∧ ⋯ ∧ HA (with q + 1 factors), and although there are technical problems with such a statement, these were overcome by Bökstedt in [B1]. Just like the Hochschild complex is a cyclic complex, T (A) admits a circle action, and can be given the structure of an S^{1}-spectrum. Its 0–simplices T (A)_{0} are HA, and the circle action defines an interesting map

λ: S_{+}^{1} ∧ HA ⟶ T (A)

mapping into the 1-skeleton of T (A). When A is commutative, T (A) is a ring spectrum, so π_{∗}T (A) is an algebra. The inclusion on 0-simplices HA → T (A) is a ring map, so T (A) is a retract of HA ∧ T (A) and thus equivalent to a wedge of Eilenberg–Mac Lane spectra. We will be interested in the case when A equals the integers. Then π_{∗}T (ℤ) is trivial in positive even degrees, so the algebra structure is trivial. Instead we will study the nontrivial algebra structure on its homotopy with finite coefficients.

Ultimately we are interested in studying the trace map tr: K(ℤ) → T (ℤ) defined in [B1] from algebraic K-theory to topological Hochschild homology, and its refinements tr_{p^{n}}: K(ℤ) → T (ℤ)^{C_{p^{n}}} defined in [BHM] for the various finite subgroups C_{p^{n}} ⊂ S^{1}. These refinements combine to define the cyclotomic trace map trc: K(ℤ) → T C(ℤ), which was used in [BM] to compute the (completed) algebraic K-theory of the p-adic integers for odd p. We can extend these calculations to the case p = 2, and an essential ingredient of this program is to algebraically structure the mod two homotopy of the fixed point spectra T (ℤ)^{C_{2^{n}}} for varying n. This note provides the foundations for these calculations, by establishing the properties summarized in Theorem 3.2 below.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 55N45, Secondary: 16E40, 19D55, 55P42.

Key words and phrases. topological Hochschild homology, Moore spectra, products.

1. Homotopy with finite coefficients

Recall the Moore spectrum S^{0}/n defined by the cofiber sequence

S^{0} ⟶^{n} S^{0} ⟶^{i} S^{0}/n ⟶^{j} S^{1}.

The mod n homotopy of a spectrum X is defined as π_{∗}(X; ℤ/n) = π_{∗}(X ∧ S^{0}/n). Given a prime p, we let i_{k} and j_{k} denote the maps i and j above in the case n = p^{k}. If X is a ring spectrum and S^{0}/n has a product map S^{0}/n ∧ S^{0}/n → S^{0}/n, then the mod n homotopy π_{∗}(X; ℤ/n) naturally inherits a product structure from the composite

(X ∧ S^{0}/n) ∧ (X ∧ S^{0}/n) ⟶^{(23)} (X ∧ X) ∧ (S^{0}/n ∧ S^{0}/n) ⟶ X ∧ S^{0}/n.

Here (23) denotes the shuffle map. The mod p Moore spectrum admits such a product map for all odd primes p, but not for p = 2. This is because the cofiber sequence

S^{0}/2 ∧ S^{0} ⟶^{1∧2} S^{0}/2 ∧ S^{0} ⟶^{1∧i_{1}} S^{0}/2 ∧ S^{0}/2 ⟶^{1∧j_{1}} S^{0}/2 ∧ S^{1}

cannot be split, and so there is no (right) unital product map μ_{1} : S^{0}/2 ∧ S^{0}/2 → S^{0}/2 ≅ S^{0}/2 ∧ S^{0}. To see this note that Sq^{1}(a_{0}) = a_{1} in H^{∗}(S^{0}/2; ℤ/2) ≅ ℤ/2{a_{0}, a_{1}}, whence by the Cartan formula Sq^{2}(a_{0}∧a_{0}) = a_{1}∧a_{1} ≠ 0 in H^{∗}(S^{0}/2∧ S^{0}/2; ℤ/2), and so there can be no splitting. It follows easily that twice the identity map on the Moore spectrum S^{0}/2 has the essential factorization

2 = i_{1}ηj_{1} : S^{0}/2 ⟶^{j_{1}} S^{1} ⟶^{η} S^{0} ⟶^{i_{1}} S^{0}/2

up to homotopy. Hence there is no algebra structure on π_{∗}(S^{0}/2), nor a natural algebra structure on π_{∗}(X; ℤ/2) for general ring spectra X. The observations go back to Barratt [Ba].

We are concerned with the two–primary homotopy of T (ℤ) in an equivariant sense. To be precise, we wish to study the mod two homotopy of the C_{2^{n}}-fixed points of T (ℤ) for all n, where C_{2^{n}} ⊂ S^{1} is the cyclic group of order 2^{n}. As noted above T (ℤ) is a wedge of Eilenberg–Mac Lane spectra, so in fact 1 ∧ 2: T (ℤ) ∧ S^{0}/2 → T (ℤ) ∧ S^{0}/2 factors through η: T (ℤ) ∧ S^{1} → T (ℤ), which is inessential. Hence the obstruction to finding a product on T (ℤ) ∧ S^{0}/2 vanishes in this case. But η: T (ℤ)∧S^{1} → T (ℤ) is not equivariantly inessential, in the sense that its restriction even to the C_{2}-fixed sets is essential. Thus it is not clear that T (ℤ) ∧ S^{0}/2 can be given a product structure in any natural equivariant way.

However 2η ≃ ∗, so four times the identity map on S^{0}/2 is inessential. Hence the cofiber sequence

S^{0} ∧ S^{0}/2 ⟶^{4∧1} S^{0} ∧ S^{0}/2 ⟶^{i_{2}∧1} S^{0}/4 ∧ S^{0}/2 ⟶^{j_{2}∧1} S^{1} ∧ S^{0}/2

splits, and there is a pairing of spectra

m: S^{0}/4 ∧ S^{0}/2 ⟶ S^{0}/2

inducing a (left) unital pairing of mod four and mod two homotopy. In a similar way S^{0}/4 admits a multiplication map

μ_{2} : S^{0}/4 ∧ S^{0}/4 ⟶ S^{0}/4.

Oka proved in [O] that μ_{2} can be chosen to be left and right unital and associative up to homotopy, but not homotopy commutative. The commutator satisfies

μ_{2} − μ_{2} ∘ (12) = i_{2} ∘ η^{2} ∘ (j_{2} ∧ j_{2})

up to homotopy, where (12) is the twist map. Hence if X is a ring spectrum such that η^{2} : X ∧ S^{2} → X ∧ S^{0} is equivariantly inessential, then π_{∗}(X; ℤ/4) is equivariantly commutative. This will be the case with T (ℤ) since η^{2} maps to zero in each π_{∗}T (ℤ)^{C_{2^{n}}}, so π_{∗}(T (ℤ)^{C_{2^{n}}}; ℤ/4) is naturally a commutative algebra for all n.

There is also a question of associativity (or transitivity) for the action m, i.e., whether the maps m(μ_{2} ∧ 1) and m(1 ∧ m) are homotopic as maps S^{0}/4 ∧ S^{0}/4 ∧ S^{0}/2 → S^{0}/2. The associator (their difference) factors up to homotopy as

m(μ_{2} ∧ 1) − m(1 ∧ m) = i_{1} ∘ α ∘ (j_{2} ∧ j_{2} ∧ j_{1})

for some map α : S^{3} → S^{0}, and the obstruction to associativity is its mod two residue class in π_{3}(S^{0})/2. For ring spectra X that are algebras over K(ℤ) this obstruction naturally vanishes, since the obstruction (which comes from the stable three-stem) is divisible by two in K_{3}(ℤ) ≅ ℤ/48. Hence in the case of T (ℤ) the action of the mod four homotopy of T (ℤ)^{C_{2^{n}}} upon the mod two homotopy is really associative for all n.

The homotopy Bockstein δ_{k} = i_{k}j_{k}: S^{0}/2^{k} → S^{1} → S^{1}/2^{k} acts on mod 2^{k} homotopy. Oka also proved that μ_{2} can be chosen so that δ_{2} is a derivation, i.e., so that

δ_{2}μ_{2} = μ_{2}(δ_{2} ∧ 1) + μ_{2}(1 ∧ δ_{2})

up to homotopy. There are natural maps ρ: S^{0}/4 → S^{0}/2 (coefficient reduction) and ϵ: S^{0}/2 → S^{0}/4 (coefficient extension) mapping between the following horizontal cofiber sequences

(1.1)  [figure: S^{0}; i_{2}; S^{0}/4; j_{2}; ρ; S^{1}; 2; S^{0}; i_{1}; 2; S^{0}/2; j_{1}; ϵ; S^{1}; S^{0}; i_{2}; S^{0}/4; j_{2}; S^{1}.]

Lemma 1.2. These maps satisfy

ρμ_{2} = m(1 ∧ ρ)  and  ϵm = μ_{2}(1 ∧ ϵ)

up to homotopy.

Proof. Oka constructed m in [O] so as to satisfy the first formula.

The second formula also follows from [O], by the following argument. Let m̂: S^{1}/2 → S^{0}/4 ∧ S^{0}/2 and μ̂_{2} : S^{1}/4 → S^{0}/4 ∧ S^{0}/4 be the homotopy fiber maps of m and μ_{2} respectively. Then μ̂_{2}ϵ = (1 ∧ ϵ)m̂ by Lemma 26 of [O], and there is an induced map of cofibers f: S^{0}/2 → S^{0}/4 satisfying fm = μ_{2}(1 ∧ ϵ). We compute

f = fm(i ∧ 1) = μ_{2}(1 ∧ ϵ)(i ∧ 1) = μ_{2}(i ∧ 1)(1 ∧ ϵ) = 1 ∧ ϵ

as maps S^{0} ∧ S^{0}/2 = S^{0}/2 → S^{0}/4 = S^{0} ∧ S^{0}/4. So f = ϵ and the statement follows. □

Clearly μ_{2}, m, ρ, ϵ and the i_{k}, j_{k} and δ_{k} are compatible with the corresponding pairings and operations in homology (with integral or finite coefficients), under the Hurewicz homomorphisms induced by S^{0} → Hℤ or S^{0}/2^{k} → Hℤ/2^{k} as appropriate. In the diagram

(1.3)  [figure: π_{∗}(T (ℤ); ℤ/4); h_{2}; ρ; H_{∗}(T (ℤ); ℤ/4); ρ; π_{∗}(T (ℤ); ℤ/2); h_{1}; ϵ; H_{∗}(T (ℤ); ℤ/2); ϵ]

the two squares commute, and h_{2} and the right hand ρ are multiplicative. We shall see below that the horizontal maps are injective.

2. Mod two homology of T (ℤ)

We recall Bökstedt’s calculation of H_{∗}(T (ℤ); ℤ/2) (spectrum homology) given in [B2]. The skeleton filtration on T (ℤ) induces a spectral sequence

E_{∗∗}^{2} = HH_{∗}(H_{∗}(Hℤ; ℤ/2)) ⟹ H_{∗}(T (ℤ); ℤ/2)

where H_{∗}(Hℤ; ℤ/2) = \overline{A} = ℤ/2[ξ_{1}^{2}, χξ_{2}, χξ_{3}, …] is contained in the dual of the Steenrod algebra H_{∗}(Hℤ/2; ℤ/2) = A = ℤ/2[ξ_{1}, ξ_{2}, ξ_{3}, …]. Here the ξ_{i} are the Milnor generators in degrees 2^{i} − 1, and χ is the canonical involution on A induced by the twist map (see [Mi]). By the Künneth theorem

E_{∗∗}^{2} ≅ \overline{A} ⊗ E(1 ⊗ ξ_{1}^{2}, 1 ⊗ χξ_{2}, 1 ⊗ χξ_{3}, …)

where E( ) denotes the exterior algebra on the listed generators. Letting [S^{1}] denote the fundamental class in H_{1}(S_{+}^{1}; ℤ/2), the circle action map λ induces a homomorphism

σ: Σ\overline{A} ⟶ H_{∗}(T (ℤ); ℤ/2)

given by σ(x) = λ_{∗}([S^{1}] ⊗ x). Then σ(x) maps to 1 ⊗ x in the spectral sequence above, and so the exterior generators above are σ(ξ_{1}^{2}), σ(χξ_{2}), σ(χξ_{3}) and so on. In fact σ is a derivation because of the form of the second Hochschild boundary:

b_{2}(1 ⊗ x ⊗ y) = xσ(y) − σ(xy) + yσ(x)

As noted T (ℤ) is a Hℤ-algebra, so the spectral sequence consists of (free) \overline{A}- modules. Since E^{2} is multiplicatively generated by classes in filtration 1, clearly d^{r} = 0 for all r ≥ 2 and so the spectral sequence collapses at the E^{2}-term.

Let e_{3} = σ(ξ_{1}^{2}) and e_{2^{i}} = σ(χξ_{i}) for all i ≥ 2 define classes in H_{∗}(T (ℤ); ℤ/2), indexed by their degrees. Our classes differ from those of Bökstedt only in that we apply the involution χ to eliminate the decomposable indeterminacy present in [B2].

Lemma 2.1. In H_{∗}(T (ℤ); ℤ/2) (i) e_{3}^{2} = 0, (ii) e_{2^{i}}^{2} = e_{2^{i+1}} for all i ≥ 2, (iii) β_{1}(e_{4}) = e_{3}, and (iv) Q̂^{4}(e_{3}) = 0.

Proof. T (ℤ) is an E_{∞} ring spectrum, so its homology H_{∗}(T (ℤ); ℤ/2) admits multiplicative Dyer–Lashof operations Q̂^{r} (see [Mad] or [May]). Likewise there are additive Dyer–Lashof operations Q^{r} on H_{∗}(Hℤ; ℤ/2) = \overline{A}, which are recorded in [S, Chapter III, Theorem 2.2]. In particular Q^{2^{i}}(χξ_{i}) = χξ_{i+1}. By Bökstedt’s lemma [B2, Lemma 2.9] these operations are compatible under σ, i.e., Q̂^{r}(σ(x)) = σ(Q^{r}(x)) for all x. Hence we can compute:

e_{3}^{2} = Q̂^{3}(e_{3}) = σ(Q^{3}(ξ_{1}^{2})) = 0

by the Cartan formula. Similarly for i ≥ 2

e_{2^{i}}^{2} = Q̂^{2^{i}}(e_{2^{i}}) = σ(Q^{2^{i}}(χξ_{i})) = σ(χξ_{i+1}) = e_{2^{i+1}}

as claimed. Next χξ_{2} = ξ_{2} + ξ_{1}^{3}, β_{1}(ξ_{1}) = 1 and β_{1}(ξ_{i}) = 0 for i ≥ 2. Hence

β_{1}(e_{4}) = β_{1}(σ(χξ_{2})) = σ(β_{1}ξ_{2} + β_{1}ξ_{1}^{3}) = σ(ξ_{1}^{2}) = e_{3}.

Finally

Q̂^{4}(e_{3}) = σ(Q^{4}(ξ_{1}^{2})) = σ(Q^{2}(ξ_{1})^{2}) = 0,

using the Cartan formula and that σ is a derivation. □

Thus as algebras and free \overline{A}-modules we have

H_{∗}(T (ℤ); ℤ/2) ≅ \overline{A}[e_{3}, e_{4}]/(e_{3}^{2} = 0).

Since T (ℤ) is a wedge of Eilenberg–Mac Lane spectra the integral and mod 2^{k} Hurewicz homomorphisms

h: π_{∗}T (ℤ) → H_{∗}T (ℤ)  and  h_{k}: π_{∗}(T (ℤ); ℤ/2^{k}) → H_{∗}(T (ℤ); ℤ/2^{k})

are all injective. In the mod two case the image consists of \overline{A}-module generators. There is at most one of these in each degree, so under h_{1} we can identify

π_{∗}(T (ℤ); ℤ/2) ≅ ℤ/2[e_{3}, e_{4}]/(e_{3}^{2} = 0).

The algebra structure here is that inherited by π_{∗}(T (ℤ); ℤ/2) as a subgroup of H_{∗}(T (ℤ); ℤ/2) which turns out to be closed under multiplication, and thus becomes a subalgebra. This algebraically defined product needs not a priori be well related to a spectrum level product on T (ℤ) ∧ S^{0}/2. We shall prove instead that it is well related to the natural pairings on homotopy induced by μ_{2} and m.

Remark 2.2. From Lemma 2.1(iv) the higher Bockstein structure on π_{∗}(T (ℤ); ℤ/2) may be deduced. Combining these results with the corresponding odd primary calculations, Bökstedt obtained the integral answer

π_{∗}T (ℤ) = \{ \begin{matrix}ℤ & for ∗ = 0, \\ ℤ/n & for ∗ = 2n − 1, \\ 0 & otherwise. \end{matrix}

3. Homotopy of T (ℤ) with finite coefficients

We recall the additive structure on π_{∗}(T (ℤ); ℤ/4). The mod two homotopy Bockstein δ_{1} is a derivation, because it is detected in homology where it corresponds to the mod two homology Bockstein (β_{1} dual to Sq^{1}), which is a derivation. From β_{1}(e_{4}) = e_{3} we have π_{3}T (ℤ)_{2}^{∧} = ℤ/2. In general π_{4k−1}T (ℤ)_{2}^{∧} is ℤ/2 for k odd, and cyclic of order four or more for k even. So

π_{∗}(T (ℤ); ℤ/4) ≅ \{ \begin{matrix}ℤ/4 & for ∗ ≡ 0, 7 mod 8, \\ ℤ/2 & for ∗ ≡ 3, 4 mod 8, \\ 0 & otherwise. \end{matrix}

We let f_{i} ∈ π_{i}(T (ℤ); ℤ/4) denote an additive generator for each i ≡ 0, 3 mod 4. There is a choice of sign in f_{i} for i ≡ 0, 7 mod 8, which we will specify shortly. The classes f_{i} in odd degrees are integral. We note that 2f_{3} = 2f_{4} = 0.

The homomorphisms ρ and ϵ linking π_{∗}(T (ℤ); ℤ/2) and π_{∗}(T (ℤ); ℤ/4) are easily determined using the maps of cofiber sequences (1.1):

ρ(f_{8k+i}) = \{ \begin{matrix}e_{4}^{2k} & for i = 0, \\ e_{3}e_{4}^{2k} & for i = 3, \\ 0 & for i = 4, \\ e_{3}e_{4}^{2k+1} & for i = 7 \end{matrix}

and

ϵ(e_{3}^{a}e_{4}^{2k+b}) = \{ \begin{matrix}2f_{8k} & for a = 0, b = 0, \\ 0 & for a = 1, b = 0, \\ f_{8k+4} & for a = 0, b = 1, \\ 2f_{8k+7} & for a = 1, b = 1. \end{matrix}

The mod two Bockstein satisfies δ_{2}(f_{4}) = f_{3}, δ_{2}(f_{8}) = ±f_{7} and is a derivation. We agree to choose f_{7} so that δ_{2}(f_{8}) = f_{7}.

We also let g_{i} ∈ π_{i}T (ℤ) denote an integral generator for each i ≡ 3 mod 4, reducing mod four to f_{i} and mod two to e_{i}.

Lemma 3.1. The product on the mod four homotopy of T (ℤ) satisfies the following relations

f_{3} · f_{3} = 0, f_{3} · f_{4} = 2f_{7}, f_{3} · f_{7} = 0, f_{3} · f_{8}^{k} = f_{8k+3}
f_{4} · f_{4} = 0, f_{4} · f_{7} = 0, f_{4} · f_{8}^{k} = f_{8k+4}
f_{7} · f_{7} = 0, f_{7} · f_{8}^{k} = ±f_{8k+7} and f_{8}^{k} = ±f_{8k}

for all k ≥ 0.

Proof. The claims f_{3}^{2} = 0, f_{3}f_{7} = 0 and f_{7}^{2} = 0 are clear since the products lie in trivial groups.

For f_{3}f_{4} note that f_{3} is integral and ϵ is linear with respect to multiplication by integral classes. So f_{3}f_{4} = g_{3} · f_{4} = g_{3} · ϵ(e_{4}) = ϵ(g_{3} · e_{4}) = 2f_{7}. Likewise f_{4}f_{7} = f_{4} · g_{7} = ϵ(e_{4}) · g_{7} = ϵ(e_{4} · g_{7}) = ϵ(e_{3}e_{4}^{2}) = 0.

Because δ_{2} is a derivation δ_{2}(f_{4}^{2}) = 2f_{3}f_{4} = 0, and since δ_{2} is injective in degree eight it follows that f_{4}^{2} = 0.

Next f_{3}f_{8}^{k} is nonzero because its image under ρh_{2} (which is multiplicative) equals e_{3}e_{4}^{2k} which is nonzero. Compare with diagram (1.3). So f_{3}f_{8}^{k} is the only nonzero class in degree 8k + 3, which is the generator f_{8k+3}. Likewise f_{7}f_{8}^{k} is nonzero because its image under ρh_{2} equals e_{3}e_{4}^{2k+1}, and f_{8}^{k} is nonzero because its image e_{4}^{2k} is also nonzero.

Finally we conclude that f_{4}f_{8}^{k} is nonzero by computing δ_{2}(f_{4}f_{8}^{k}) = f_{3}f_{8}^{k} + kf_{4}f_{7}f_{8}^{k−1} = f_{3}f_{8}^{k} which is nonzero. □

Hence we may take f_{i}f_{8}^{k} as the generator in degree 8k + i for all k ≥ 0 and i = 0, 3, 4, 7. Modulo the sign of f_{8}, this precisely pins down a set of generators for π_{∗}(T (ℤ); ℤ/4). The sign of f_{8} can be fixed by insisting that h_{2}(f_{8}) = P (e_{4}) where P is the Pontrjagin squaring operation (see [Mad] or [May]). We omit the details of this argument.

Theorem 3.2. (i) The mod four homotopy algebra of T (ℤ) is

π_{∗}(T (ℤ); ℤ/4) ≅ ℤ/4[f_{3}, f_{4}, f_{7}, f_{8}]/ ∼

with the relations 2f_{3} = 2f_{4} = 0, f_{3}f_{4} = 2f_{7}, and f_{i}f_{j} = 0 for all other i, j < 8. (ii) The pairing

π_{∗}(T (ℤ); ℤ/4) ⊗ π_{∗}(T (ℤ); ℤ/2) ⟶ π_{∗}(T (ℤ); ℤ/2)

induced by m takes x ⊗ y to ρ(x) · y in the subalgebra product on π_{∗}(T (ℤ); ℤ/2) inherited from H_{∗}(T (ℤ); ℤ/2). Thus m(x ⊗ y) = ρ(x) · y.

(iii) The coefficient reduction ρ: π_{∗}(T (ℤ); ℤ/4) → π_{∗}(T (ℤ); ℤ/2) is an algebra homomorphism when mapping to the subalgebra product. The coefficient extension ϵ: π_{∗}(T (ℤ); ℤ/2) → π_{∗}(T (ℤ); ℤ/4) is a π_{∗}(T (ℤ); ℤ/4)-module homomorphism with respect to the actions induced by m and μ_{2}.

Proof. Part (i) summarizes Lemma 3.1 and the conventions following it.

To prove m(x ⊗ y) = ρ(x) · y in π_{∗}(T (ℤ); ℤ/2) it suffices to apply the injection h_{1} and note that h_{1}m(x ⊗ y) = ρh_{2}(x) · h_{1}(y) = h_{1}(ρ(x) · y).

The remaining claims follow from Lemma 1.2. □

References

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Department of Mathematics, University of Oslo, Norway E-mail address: rognes@math.uio.no