TOPOLOGICAL LOGARITHMIC STRUCTURES

John Rognes

March 7th 2008

Contents

1. The fraction field of topological K-theory

2. Topological logarithmic structures

3. Logarithmic TAQ and THH

§1. The fraction field of topological K-theory

1.1. The p-adic integers and the p-adic numbers. Fix a prime p, work in a p-complete setting for simplicity, and consider Quillen’s localization sequence

K(ℤ/p) ⟶^{i_{∗}} K(ℤ_{p}) ⟶^{j^{∗}} K(ℚ_{p}) .

It is a cofiber sequence of spectra, where i_{∗} is the transfer associated to i: ℤ_{p} → ℤ_{p}/p = ℤ/p and j^{∗} is the natural map associated to j: ℤ_{p} → ℤ_{p}[p^{−1}] = ℚ_{p}. We use algebro-geometric conventions for the variance of these maps.

The proven Lichtenbaum–Quillen conjectures concern the étale descent properties of p-adically complete algebraic K-theory, here denoted K(A)_{p}. The main case is that of Galois descent: If A → B is a G-Galois extension, then the induced map K(A)_{p} → K(B)_{p}^{hG} is an equivalence in sufficiently high degrees.

This is more interesting in the case of the fraction field ℚ_{p} than in the case of its valuation ring ℤ_{p}, simply because there are many more Galois extensions of the former than of the latter. For example, the p-th cyclotomic extension ℚ_{p} → ℚ_{p}(ζ_{p}) is Δ-Galois, where Δ = (ℤ/p)^{×} ≅ ℤ/(p−1), whereas the map of valuation rings ℤ_{p} → ℤ_{p}[ζ_{p}] is not Galois, since it is ramified at (p). The canonical map h: ℤ_{p}[ζ_{p}] ⊗_{ℤ_{p}} ℤ_{p}[ζ_{p}] → ∏_{Δ} ℤ_{p}[ζ_{p}] has cokernel a nontrivial finite ℤ_{p}-module.

In the limit, we compare K(ℚ_{p})_{p} with continuous homotopy fixed points of K(\overline{ℚ}_{p})_{p}, and in the algebraically closed case we have Suslin’s theorem saying that K(\overline{ℚ}_{p})_{p} ≃ ku_{p}. Here ku_{p} denotes p-adically completed connective topological Ktheory, with π_{∗}ku_{p} = ℤ_{p}[u].

The local field ℚ_{p} also has the exceptional feature that its Galois cohomology satisfies a form of Poincaré duality, known as Tate–Poitou arithmetic duality. The mod p Galois cohomology is related to the mod p algebraic K-theory by an Atiyah– Hirzebruch spectral sequence

E_{s,t}^{2} = H_{Gal}^{−s}(ℚ_{p}; 𝔽_{p}(t/2)) ⟹ K_{s+t}(ℚ_{p}; ℤ/p)

Typeset by AMS-TEX

(for 2s + t ≥ 0), and the cup product

H_{Gal}^{r}(ℚ_{p}; 𝔽_{p}(i)) ⊗ H_{Gal}^{2−r}(ℚ_{p}; 𝔽_{p}(1 − i)) ⟶^{∪} H_{Gal}^{2}(ℚ_{p}; 𝔽_{p}(1)) ≅ [p] Br(ℚ_{p}) ≅ ℤ/p

(where Br(F) is the Brauer group of central simple F -algebras up to Morita equivalence) is a perfect pairing.

1.2. The connective and the periodic p-adic Adams summands. There is a similar localization sequence

K(ℤ_{p}) ⟶^{π_{∗}} K(ℓ_{p}) ⟶^{ρ^{∗}} K(L_{p})

established by Andrew Blumberg and Mike Mandell. Here ℓ_{p} denotes the Adams summand of ku_{p}, with π_{∗}ℓ_{p} = ℤ_{p}[v_{1}], π_{∗} is the transfer associated to the zero-th Postnikov section π: ℓ_{p} → ℓ_{p}/v_{1} = Hℤ_{p}, and ρ^{∗} is the natural map associated to the localization map ρ: ℓ_{p} → ℓ_{p}[v_{1}^{−1}] = L_{p}.

Also in this case there are more Galois extensions for the periodic commutative S-algebras than for their connective covers. Letting KU_{p} = ku_{p}[u^{−1}] denote padically complete periodic topological K-theory, then L_{p} → KU_{p} is a Δ-Galois extension, with Δ as above, but ℓ_{p} → ku_{p} is not Galois. The canonical map h: ku_{p} ∧_{ℓ_{p}} ku_{p} → ∏_{Δ} ku_{p} has cofiber a nontrivial ku_{p}-module with finite Postnikov tower. In other words, ℓ_{p} → ku_{p} is ramified at (u) over (v_{1}).

Still, it is hard to find many G-Galois extensions of L_{p}. Andy Baker and Birgit Richter have recently shown that the largest (connected) one is KU_{p}^{nr}, with π_{∗}KU_{p}^{nr} = 𝕎\overline{𝔽}_{p}[u, u^{−1}], where 𝕎 denotes the Witt ring functor.

For p odd, it is impossible to adjoin a p-th root of unity to L_{p} to realize L_{p}[ζ_{p}] as a commutative S-algebra, as can be shown by a θ-ring computation. ((Here is the argument I learned from Vigleik Angeltveit: If L_{p}[ζ_{p}] is E_{∞} and K(1)-local, there are operations ψ = ψ^{p} and θ with ψ(x) = x^{p} + pθ(x) and ψ a ring operation. Can write p = q^{p−1} in ℤ_{p}[ζ_{p}], with q a unit times the uniformizer π = 1 − ζ_{p}, but then p = ψ(p) = ψ(q)^{p−1} = (q^{p} + pθ(q))^{p−1} = (p(q + θ(q)))^{p−1} is divisible by p^{2}.))

Attempts to realize other coefficient extensions that ramify at p effectively flounder when one tries to apply the Goerss–Hopkins obstruction theory, since the obstruction groups are André–Quillen cohomology groups related to Kähler differentials, which are far from zero in this case.

So there does not seem to be an interesting algebraic closure \overline{KU}_{p} in commutative S-algebras, whose algebraic K-theory is any simpler than that of KU_{p} itself, in contrast to Suslin’s theorem.

If we assume that there is a motivic cohomology theory for these commutative Salgebras, then we also expect a spectral sequence relating it to algebraic K-theory. This time we prefer to work with V (1)-coefficients, where

V (1) = S/(p, v_{1}) = S ∪_{p} e^{1} ∪_{α_{1}} e^{2p−1} ∪_{p} e^{2p}

is the Smith–Toda complex, and V (1)_{∗}X = π_{∗}(V (1) ∧ X). The Atiyah–Hirzebruch spectral sequence may then take the form

E_{s,t}^{2} = H_{mot}^{−s}(ℓ_{p}; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K_{s+t}(ℓ_{p})

(for 2s + t ≥ 0), and similarly for L_{p} in place of ℓ_{p}.

By computing with topological cyclic homology, Christian Ausoni and the author determined the abutment of this expected spectral sequence for p ≥ 5, in which case its E^{2}-term has a more-or-less obvious form. There is a v_{2}-self map of degree (2p^{2} − 2) acting on V (1), so v_{2} acts on the spectral sequence by a shift of (p^{2} − 1) in t/2, as well as on the abutment. The remarkable outcome of the computations is that the algebraic K-theory abutment is v_{2}-periodic, i.e., it is v_{2}-torsion free. ((Slight caveat here, caused by the very finite coefficients!))

In this case, once the motivic cohomology is made periodic in the t-grading (and christened Galois cohomology), there is no difference between the ℓ_{p}- and the L_{p}-case. Furthermore, there is an apparent perfect cup product pairing

H_{Gal}^{r}(L_{p}; 𝔽_{p^{2}}(i)) ⊗ H_{Gal}^{3−r}(L_{p}; 𝔽_{p^{2}}(d − i)) ⟶^{∪} H_{Gal}^{3}(L_{p}; 𝔽_{p^{2}}(d)) ≅ ℤ/p ,

where d = p^{2}+p. However, since in this case the target group of the pairing depends on the prime p (in the twist d), there is little hope of an intrinsic understanding of this group, like the identification of H_{Gal}^{2}(F; 𝔾_{m}) with the Brauer group Br(F) mentioned above.

1.3. The fraction field of the p-adic Adams summand. It appears that we should not view L_{p} as a (fraction) field in commutative S-algebras, due to the possibility of ramification over (p). So we seek some form of localization of L_{p} away from p, which we will formally denote as

ff L_{p} = p^{−1}L_{p} .

By this we do not mean the usual algebraic Bousfield localization L_{p}[p^{−1}] = Lℚ_{p}, with π_{∗}Lℚ_{p} = ℚ_{p}[v_{1}, v_{1}^{−1}], since this is an Hℚ_{p}-algebra, so K(Lℚ_{p}) is a K(ℚ_{p})- module, and V (1)_{∗}K(ℚ_{p}) is finite, so V (1)_{∗}K(Lℚ_{p}) will only consist of v_{2}-torsion. This completely loses the connection with the v_{2}-periodicity that we think is the essential feature in this story. We are looking for a milder, less drastic, localization.

Instead we can show that ℓ_{p}/p = ℓ/p = k(1) admits the structure of a noncommutative ℓ_{p}-algebra, so L_{p}/p = L/p = K(1) is an L_{p}-algebra, and we can form the following commutative 3 × 3-diagram of horizontal and vertical cofiber sequences:

\xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ_{p}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & K(ℚ_{p}) \ar[d]^{π_{∗}} \\ K(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & K(ℓ_{p}) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & K(p^{−1}ℓ_{p}) \ar[d]^{ρ^{∗}} \\ K(L/p) \ar[r]^{i_{∗}} & K(L_{p}) \ar[r]^{j^{∗}} & K(ff L_{p})}

For now, the terms K(p^{−1}ℓ_{p}) and K(ff L_{p}) are just notation for the homotopy cofiber spectra that fit in the diagram. Hence we wish that Spec(ff L_{p}) is a kind of complement of Spnc(L/p) in Spec(L_{p}), and similarly for p^{−1}ℓ_{p}. It is unpleasant that the terms K(ℓ/p) and K(L/p) should depend on the choice of ℓ_{p}-algebra structure, of which there are uncountably many, but this should not be a devastating problem.

Further topological cyclic homology calculations by Ausoni and Rognes determine V (1)_{∗}K(ℓ/p), and thereby also V (1)_{∗}K(p^{−1}ℓ_{p}) and V (1)_{∗}K(ff L_{p}). These remain v_{2}-periodic, and now the expected motivic and Galois cohomology groups of ff L_{p} are better behaved than those of L_{p}, in that there is again an apparent perfect cup product pairing

H_{Gal}^{r}(ff L_{p}; 𝔽_{p^{2}}(i)) ⊗ H_{Gal}^{3−r}(ff L_{p}; 𝔽_{p^{2}}(2 − i)) ⟶^{∪} H_{Gal}^{3}(ff L_{p}; 𝔽_{p^{2}}(2)) ≅ ℤ/p ,

where the bigrading of the target group is now independent of p. This is exactly the kind of arithmetic duality as is seen in the Galois cohomology of “higher” local fields, like 𝔽_{p}((s, t)) and ℚ_{p}((t)), for which the local class field theory is expressed in terms of Milnor K-theory. The calculations therefore suggest that the fraction field ff L_{p}, and its analogously defined Δ-Galois extension ff KU_{p}, are brave new 2-dimensional “higher” local fields, invisible to classical algebra.

The valuation ring of ff L_{p} is then p^{−1}ℓ_{p}, with residue field ℚ_{p} = p^{−1}ℤ_{p}. We expect that there are Galois extensions

\xymatrix{\overline{ff L_{p}} \\ ff L\overline{ℤ}_{p} \ar[u]_{I_{v_{1}}} & _{\overline{ℚ}_{p}} \\ ff L_{p}^{nr} \ar[u]_{I_{p}} & ℚ_{p}^{nr} \ar[u]_{I_{p}} & _{\overline{𝔽}_{p}} \\ ff L_{p} \ar[u]_{ℤ̂} \ar[uuu]^{G_{ff Lp}} & ℚ_{p} \ar[u]_{ℤ̂} \ar[uu]^{G_{ℚp}} & 𝔽_{p} \ar[u]_{ℤ̂}^{G_{𝔽p}}}

and group extensions

I_{v_{1}} → G_{ff L_{p}} → G_{ℚ_{p}} and I_{p} → G_{ℚ_{p}} → G_{𝔽_{p}} ,

where G_{𝔽_{p}} has p-cohomological dimension 1, G_{ℚ_{p}} has p-cohomological dimension 2, and the hypothetical G_{ff L_{p}} has p-cohomological dimension 3.

§2. Topological logarithmic structures

We propose that a structured ring spectrum version of logarithmic geometry may provide an algebro-geometric context in which the fraction fields ff L_{p}, ff KU_{p}, and some or all of their Galois extensions, will exist.

Let A be a commutative S-algebra. We may view A as an E_{∞} ring spectrum, by which we will really mean an L-spectrum, where L is the linear isometries operad. The underlying space Ω^{∞}A is then an E_{∞} ring space. With respect to the additive structure, Ω^{∞}A_{⊕} is a grouplike E_{∞} space, but this will be of little concern to us. With respect to the multiplicative structure, Ω^{∞}A_{⊗} is a pointed E_{∞} space (= E_{∞} space with zero), by which we mean an L_{0}-space, i.e., an L_{+}-space in pointed spaces. In such spaces, the operadic neutral element 1 is almost always different from the base point 0.

The homotopy invertible components of Ω^{∞}A_{⊗}, denoted GL_{1}(A), constitute a grouplike E_{∞}-space, or L-space, and the inclusion ι: GL_{1}(A) → Ω^{∞}A_{⊗} is a map of E_{∞} spaces. It extends to a map ι_{0} : GL_{1}(A)_{+} → Ω^{∞}A_{⊗} of pointed E_{∞}-spaces.

Definition (prelog and log S-algebras). (i) Let A be a commutative S-algebra. The following definition works best if A is connective: A (pointed) prelog structure (M, α) on A is a pointed E_{∞} space M, and a map

α: M → Ω^{∞}A_{⊗}

of pointed E_{∞} spaces. We usually write (A, M) for A with this prelog structure, and call it a prelog S-algebra.

(ii) Consider the pullback square

[figure: α^{−1}GL_{1}(A); α̃; GL_{1}(A); ι; M; α; Ω^{∞}A_{⊗}]

in (unpointed) E_{∞} spaces. We say that (M, α) is a (pointed) log structure on A, and that (A, M) is a log S-algebra, if the restricted map α̃: α^{−1}GL_{1}(A) → GL_{1}(A) is a weak equivalence.

(iii) If M = N_{+}, where N is an E_{∞} space and zero is the added base point, the restricted E_{∞} map α′: N → Ω^{∞}A_{⊗} defines an unpointed prelog structure (N, α′) on A, making (A, N) an unpointed prelog S-algebra. It is an unpointed log S-algebra if α̃′: α^{′−1}GL_{1}(A) → GL_{1}(A) is a weak equivalence.

Remarks. (i) Note that ι is a fibration, so the pullback square is a homotopy pullback. It is less clear how pullback along ι_{0} makes homotopy theoretic sense.

(ii) The emphasis of logarithmic geometry will be on the part of M that does not map to GL_{1}(B). Hence the log condition is a normalization condition, saying that nothing exceptional happens over the homotopy units: the part of M sitting over GL_{1}(B) is equivalent to GL_{1}(B). It may also be viewed as a fibrancy condition.

(iii) For non-connective A, this definition is too restrictive, since a space may act by an equivalence on a non-connective spectrum, without acting homotopy invertibly on its infinite loop space.

(iv) One might alternatively work with diagram spectra, say by letting A be a connective commutative symmetric ring spectrum. Then n ↦ Ω^{n}A_{n} defines a commutative monoid Ω^{∞}A_{⊗} in pointed I-spaces, where I is the category of finite sets and injective functions, and a prelog structure is a map α: M → Ω^{∞}A_{⊗} in pointed I-spaces. Then GL_{1}(A) should be interpreted as a commutative monoid in pointed I-spaces, following Christian Schlichtkrull.

(v) It is possible to do some work with non-commutative S-algebras and A_{∞} spaces with zero, but we will not do so in these notes.

Examples. (i) In classical algebra, only unpointed log structures are considered, but one might just as well have worked with pointed log structures, since the zero will always be isolated in the discrete topology. If R is a commutative ring, (R, ·) its multiplicative monoid, N some other commutative monoid, and α′: N → (R, ·) a monoid homomorphism, then clearly the induced map N_{+} → Ω^{∞}HR_{⊗} defines a prelog structure on the Eilenberg–Mac Lane ring spectrum HR.

(ii) In topology, we have good reasons to consider pointed log structures. For example, to approximate the localization KU = ku[u^{−1}] by a log structure on ku, we think of KU as obtained from ku by inverting the structure maps Σ^{2}ku → ku. These are induced by the S-module map ū: Σ^{∞}S^{2} → ku representing the generator u ∈ π_{2}ku, and the product on ku. In adjoint terms, we have the basepoint preserving map u: S^{2} → Ω^{∞}ku ≃ BU × ℤ, which extends to a pointed E_{∞} map

β: L_{0}S^{2} → Ω^{∞}ku_{⊗} ,

(β for Bott) where

L_{0}S^{2} = ⋁_{j≥0} L(j) ⋉_{Σ_{j}} S^{2j}

is the free L_{0}-space (= pointed E_{∞} space) on S^{2}.

((In May’s more recent notation, this is the free L_{+}-space L_{+}(S^{2}) on S^{2}, in pointed spaces, which equals the free L-space L(S^{2} ⊔ {1}) on S^{0} ∨ S^{2} = S^{2} ⊔ {1}, in spaces under S^{0} = {0, 1}.))

The associated log S-algebra (ku, (L_{0}S^{2})^{a}) is the topological log-geometric approximation to KU. Later calculations with topological André–Quillen homology show that thinking of u as a free (unpointed) map, and working with the free Lspace L(S^{2} ⊔ {0, 1}), does not yield a satisfactory theory. There is a similar prelog structure

α: L_{0}S^{q} → Ω^{∞}ℓ_{⊗}

(α for Adams) on the p-local Adams summand, where q = 2p − 2.

Definition (logification). (i) To each prelog structure (M, α) on A there is an associated log structure (M^{a}, α^{a}) on A, where M^{a} is the homotopy pushout of the diagram

[figure: (α^{−1}GL_{1}(A))_{+}; α̃_{+}; GL_{1}(A)_{+}; ι_{0}; M; M^{a}]

in pointed E_{∞} spaces, and α^{a}: M^{a} → Ω^{∞}A_{⊗} is the induced map. We write (A, M)^{a} for the log S-algebra associated to (A, M), to emphasize that M^{a} depends on A.

(ii) When M = N_{+} has an isolated zero, M^{a} = (N^{a})_{+}, where N^{a} is the homotopy pushout of the diagram

[figure: α^{−1}GL_{1}(A); α̃′; GL_{1}(A); ι; N; N^{a}]

in E_{∞} spaces.

Remark. In the context of commutative monoids in pointed I-spaces the coproduct is the smash product, so when α^{−1}GL_{1}(A) = 1 we get

M^{a} = GL_{1}(A) ⋉ M

in the pointed case, and

N^{a} = GL_{1}(A) × N

in the unpointed case.

Definition (trivial log structures). The trivial (pointed) prelog structure on A is given by M = 1_{+} = {0, 1}. The associated log structure (A, 1)^{a} equals (A, GL_{1}(A)_{+}), and is called the trivial log structure.

Definition (prelog and log maps). (i) A map (A, M) → (B, N) of prelog Salgebras is a pair (f, f^{♭}), where f: A → B is a map of commutative S-algebras, and f^{♭}: M → N is a map of pointed E_{∞} spaces, such that the diagram

\xymatrix{M \ar[r]^{α} \ar[d]_{f^{♭}} & Ω^{∞}A_{⊗} \ar[d]^{Ω^{∞}f} \\ N \ar[r]^{β} & Ω^{∞}B_{⊗}}

of pointed E_{∞} spaces is commutative. Equivalently, the adjoint diagram

\xymatrix{Σ^{∞}M \ar[r]^{ᾱ} \ar[d]_{Σ^{∞}f^{♭}} & A \ar[d]^{f} \\ Σ^{∞}N \ar[r]^{β̄} & B}

of commutative S-algebras is required to commute.

(ii) A map of log S-algebras is a map of the underlying prelog S-algebras. The log S-algebras generate a reflective full subcategory Log_{S} of the prelog S-algebras, with the logification functor (A, M) ↦ (A, M)^{a} as the reflector, i.e., the left adjoint to the forgetful functor.

Example. There is a map (f, f^{♭}): (ℓ, L_{0}S^{q}) → (ku_{(p)}, L_{0}S^{2}) of prelog S-algebras, where f: ℓ → ku_{(p)} is the usual map and f^{♭}: L_{0}S^{q} → L_{0}S^{2} is the pointed E_{∞}-map that extends the pointed map

S^{q} ⊂ L(p−1) ⋉_{Σ_{p−1}} S^{2(p−1)} ⊂ L_{0}S^{2} .

There is also an induced map of associated log S-algebras.

Remark. Even if ℓ → ku_{(p)} is not étale as a map of commutative S-algebras, we may expect that (ℓ, L_{0}S^{q}) → (ku_{(p)}, L_{0}S^{2}) is étale as a map of (pre)log S-algebras. This would fit with the classical expectations, since ℓ → ku_{(p)} appears to be tamely ramified at (v_{1}) = (u)^{p−1}. An alternative guess is that the prelog structures with M = W ⊔{1} and N = BU_{(p)}⊔{1}, where Ω^{∞}ℓ ≃ W ×ℤ_{(p)}, will have this property.

Definition (localizations). (i) Let (A, M) be a (pre)log S-algebra, and assume M = N_{+} has an isolated zero. Let ΓN ≃ ΩBN be the (multiplicative) group completion of N, and let

A[N^{−1}] = S[ΓN] ∧_{S[N]} A

where S[N] = Σ^{∞}N_{+} = Σ^{∞}M, and similarly for S[ΓN]. Then there are prelog maps

(A, 1_{+}) → (A, N_{+}) → (A[N^{−1}], N_{+})

with logifications

(A, 1_{+})^{a} → (A, N_{+})^{a} → (A[N^{−1}], 1_{+})^{a} ,

since N → Ω^{∞}A_{⊗} → Ω^{∞}A[N^{−1}]_{⊗} maps into GL_{1}(A[N^{−1}]). Hence there are maps of affine topological log schemes

U = Spec(A[N^{−1}]) → Λ = Spec(A, N_{+}) → X = Spec(A)

exhibiting Λ = Spec(A, N_{+}) as an intermediate “compactification” of the Zariski open U in X. The latter topological schemes are viewed as topological log schemes by equipping them with the trivial log structures. For the purposes of differential algebraic geometry, including TAQ, THH and TC, we propose to work with the log scheme Λ in place of the open subscheme U.

(ii) On the other hand, if M = L_{0}T has an isolated unit, for some zero-pointed space T, then the natural map

L(j) ⋉_{Σ_{j}} T^{∧j} ∧ T → L(j + 1) ⋉_{Σ_{j+1}} T^{∧(j+1)}

induces maps M ∧ T → M and σ: Σ^{∞}M ∧ T → Σ^{∞}M. If T = S^{d} for some d ≥ 0, then we can let T^{−1}Σ^{∞}M be the homotopy colimit of the infinite sequence of maps

Σ^{∞}M ⟶^{Σ^{−d}σ} Σ^{−d}Σ^{∞}M ⟶^{Σ^{−2d}σ} Σ^{−2d}Σ^{∞}M → ….

For d > 0, this will usually not be a connective spectrum. ((More generally, it might make sense to let T^{−1}Σ^{∞}M be the homotopy colimit of the adjoint maps

Σ^{∞}M ⟶^{σ̃} F (T, Σ^{∞}M) ⟶^{σ̃} F (T ∧ T, Σ^{∞}M) → …,

but this is less clear.)) By the theory of Bousfield localizations in the category of Σ^{∞}M-modules [EKMM, VIII.2.2], we can take T^{−1}Σ^{∞}M to be a commutative Σ^{∞}M-algebra. Then we let

T^{−1}A = T^{−1}Σ^{∞}M ∧_{Σ^{∞}M} A .

There are prelog maps

(A, 1_{+}) → (A, L_{0}T) → (T^{−1}A, L_{0}T) ,

where T^{−1}A might not be connective. Still, for T = S^{d} we will have that the multiplication map

Σ^{∞}T ∧ T^{−1}A ⟶^{τ̄∧1} A ∧ T^{−1}A ⟶^{μ} T^{−1}A

is an equivalence, so the space T generating M acts by an equivalence on T^{−1}A. Here τ: T → Ω^{∞}A_{∞} is the restriction of α over T ⊂ M, and τ̄ is its left adjoint. We view this as saying that the non-connective prelog S-algebras (T^{−1}A, L_{0}T) and (T^{−1}A, 1_{+}) should be log equivalent. (This example shows that our definition of a log structure is not the right one for non-connective S-algebras.) With this interpretation, we again have maps

U = Spec(T^{−1}A) → Λ = Spec(A, M) → X = Spec(A)

that exhibit Λ as intermediate between U and X.

Example. For A = ku, T = S^{2}, M = L_{0}S^{2} and β the Bott prelog structure, we get T^{−1}A = KU.

Remark. Following Martin Olsson, we can embed topological logarithmic geometry into the topological algebraic geometry of stacks, by taking each topological logarithmic scheme Λ (Zariski or étale locally modeled on Spec(A, M)) to a moduli stack of topological logarithmic schemes Y over Λ, with morphisms the “exact” maps Y → Y′ over Λ. Then notions like smooth, unramified and étale morphisms of topological logarithmic schemes should become special cases of the same notions for topological stacks. Clark Barwick understands this point of view well.

§3. Logarithmic TAQ and THH

We would like to define a cyclotomic spectrum THH(A, M), for (A, M) a log S-algebra, such that there is a cofiber sequence

THH(ℓ/p) ⟶^{i_{∗}} THH(ℓ_{p}) ⟶^{j^{∗}} THH(ℓ_{p}, M)

of cyclotomic spectra, for a suitable log structure α: M → (Ω^{∞}ℓ_{p})_{⊗}. Then we would get a cofiber sequence

TC(ℓ/p; p) ⟶^{i_{∗}} TC(ℓ_{p}; p) ⟶^{j^{∗}} TC(ℓ_{p}, M; p) ,

and trc: K(p^{−1}ℓ_{p}) → TC(ℓ_{p}, M; p) will be a p-adic equivalence in positive degrees. So for the purpose of K-theory calculations with topological cyclic homology, the log S-algebra (ℓ_{p}, M) would be a suitable interpretation for p^{−1}ℓ_{p}.

Thinking of topological Hochschild homology as generalized differential forms, we expect logarithmic THH to be a generalization of differential forms with logarithmic poles. For a discrete (unpointed) log ring (R, N), the log differential forms Ω_{(R,N)}^{∗} are generated by 1-forms dr and d log x, for r ∈ R and x ∈ N, with α(x) · d log x = dα(x).

3.1. Logarithmic topological André–Quillen homology. It is a little clearer how to deal with Kähler differentials, i.e., 1-forms, than general Kähler forms, so we begin by defining logarithmic topological André–Quillen homology, TAQ(A, M), for a log S-algebra (A, M).

Definition (derivations). Let K be an A-module, and consider the square-zero extension pr: A ∨ K → A, with section in: A → A ∨ K. The space of derivations of A with values in K is the mapping space of dashed lifts

Der(A, K) = C_{S} \{ \xymatrix{ & A ∨ K \ar[d]^{pr} \\ A \ar@{-->}[ur]^{d} \ar[r]_{=} & A} \}

in the (model) category of commutative S-algebras. (This equals (C_{S}/A)(A, A∨K), but we find the diagram notation easier to understand.)

Topological André–Quillen homology is constructed so as to corepresent derivations, in the category of A-modules.

Lemma.

M_{A}(TAQ(A), K) ≃ Der(A, K) .

We shall define the logarithmic topological André–Quillen homology TAQ(A, M) as an A-module, corepresenting a space of log derivations.

Remark. Steffen Sagave has thought of how to give an improved definition of TAQ(A, M) as a log A-module. We will not pursue this here.

Definition (inverse image log structures). Let (A, M) be a log S-algebra and K an A-module. We define the inverse image log structure in^{∗}M on A ∨ K, along in: A → A ∨ K, to be the log structure associated to the composite map

M ⟶^{α} Ω^{∞}A_{⊗} ⟶^{Ω^{∞}in} Ω^{∞}(A ∨ K)_{⊗} .

Note that there are homotopy pushout squares

\xymatrix{1_{+} \ar[r] \ar[d] & GL_{1}(A)_{+} \ar[r] \ar[d] & M \ar[d]^{in^{♭}} \\ (1 + Ω^{∞}K)_{⊗+} \ar[r] & GL_{1}(A ∨ K)_{+} \ar[r] & in^{∗}M}

in pointed E_{∞} spaces, so that

in^{∗}M ≃ (1 + Ω^{∞}K)_{⊗} ⋉ M .

Here (1 + Ω^{∞}K)_{⊗} ≃ Ω^{∞}K_{⊕}, since K is a square-zero ideal. There are log maps

(A, M) ⟶^{(in,in^{♭})} (A ∨ K, in^{∗}M) ⟶^{(pr,pr^{♭})} (A, M)

where pr^{♭} collapses (1 + Ω^{∞}K)_{⊗} to a point.

Definition (log derivations). The space of log derivations of (A, M) with values in K is the mapping space of dashed lifts

Der((A, M), K) = Log_{S} \{ \xymatrix{ & (A ∨ K, in^{∗}M) \ar[d]^{(pr,pr^{♭})} \\ (A, M) \ar@{-->}[ur]^{(d,d^{♭})} \ar[r]_{=} & (A, M)} \}

in the (model/infinity) category of log S-algebras.

Lemma. There is a homotopy pullback square

\xymatrix{Der((A, M), K) \ar[r] \ar[d] & Der(A, K) \ar[d]^{ϕ^{∗}} \\ Der^{♭}(M, K) \ar[r]^{ψ^{∗}} & Der(Σ^{∞}M, K)}

where

Der^{♭}(M, K) = L_{0} \{ \xymatrix{ & in^{∗}M \ar[d]^{pr^{♭}} \\ M \ar@{-->}[ur]^{d^{♭}} \ar[r]_{=} & M} \} .

Sketch proof. The condition for (d, d^{♭}) to be a log morphism over (A, M) is that the two composites

M ⟶^{α} Ω^{∞}A_{⊗} ⟶^{Ω^{∞}d} Ω^{∞}(A ∨ K)_{⊗}

and

M ⟶^{d^{♭}} in^{∗}M ⟶^{in^{∗}α} Ω^{∞}(A ∨ K)_{⊗}

agree as maps over Ω^{∞}A_{⊗}, which by adjunction is equivalent to them agreeing in the mapping space of dashed lifts

C_{S} \{ \xymatrix{ & A ∨ K \ar[d]^{pr} \\ Σ^{∞}M \ar@{-->}[ur] \ar[r]_{ᾱ} & A} \} .

This is naturally equivalent to Der(Σ^{∞}M, K). □

Lemma. (i) When M = N_{+} has an isolated zero, in^{∗}N_{+} ≃ (Ω^{∞}K_{⊕} × N)_{+}, so

Der^{♭}(M, K) ≃ L(N, Ω^{∞}K) ≃ M_{A}(A ∧ n, K)

where n = B^{∞}N is the connective spectrum generated by the E_{∞} space N.

(ii) When M = L_{0}T has an isolated unit, and T = S^{d} ((or more generally, T = ΣV is a suspension)), then

Der^{♭}(M, K) ≃ T \{ \xymatrix{ & Ω^{∞}K_{⊕} ⋉ T \ar[d]^{pr} \\ T \ar@{-->}[ur] \ar[r]_{=} & T} \} ≃ Ω^{∞}K_{⊕} ≃ M_{A}(A, K)

where T denotes the category of pointed spaces.

Remark. In part (ii), the homotopy invariance of the mapping space seems troublesome, since pr is not a fibration. The second equivalence assumes that we are talking about strict lifts, not lifts in an equivalent fibration. Since T is a suspension, the space of lifts then deformation retracts to the space of constant lifts.

Definition (log TAQ). (i) When M = N_{+} has an isolated zero, we define TAQ(A, M) as the homotopy pushout

\xymatrix{A ∧_{S[N]} TAQ(S[N]) \ar[r]^{ψ} \ar[d]_{ϕ} & A ∧ n \ar[d] \\ TAQ(A) \ar[r] & TAQ(A, M)}

of A-modules, where n = B^{∞}N, ϕ is induced by ᾱ: Σ^{∞}M → A (corepresenting ϕ^{∗}), and ψ corepresents ψ^{∗}.

(ii) When M = L_{0}T has an isolated unit, and T = S^{d}, we define TAQ(A, M) as the homotopy pushout

\xymatrix{A ∧_{Σ^{∞}M} TAQ(Σ^{∞}M) \ar[r]^{ψ} \ar[d]_{ϕ} & A \ar[d] \\ TAQ(A) \ar[r] & TAQ(A, M)}

of A-modules.

Remarks. (i) By a slight extension of a theorem of Basterra–Mandell, TAQ(S[N]) ≃ S[N]∧n, so A∧_{S[N]}TAQ(S[N]) ≃ A∧n, but ψ is typically only an equivalence when (A, N_{+}) is a trivial log structure, so in general TAQ(A) ≄ TAQ(A, N_{+}). Informally, ψ takes 1 ∧ dα(x) to α(x) ∧ d log x, for x ∈ N.

(ii) Since Σ^{∞}L_{0}T = L_{0}Σ^{∞}T ≅ L(S ∨ Σ^{∞}T) is the free commutative S-algebra on S ∨ Σ^{∞}T, it is easy to see that TAQ(Σ^{∞}L_{0}T) ≃ Σ^{∞}L_{0}T ∧ T, so A ∧_{Σ^{∞}M} TAQ(Σ^{∞}M) ≃ A ∧ T.

Proposition. Let (A, M) be a log S-algebra, with M = N_{+} or M = L_{0}T, and let K be an A-module. Then

M_{A}(TAQ(A, M), K) ≃ Der((A, M), K) .

Hence TAQ(A, M) corepresents log derivations, and deserves the name “log differentials”. We can usually make the maps ψ more explicit:

Proposition. (i) When M = N_{+} has an isolated zero, and n = B^{∞}N is a reduced Thom spectrum T h(ξ)/S with Thom diagonal θ: n → S[N] ∧ n, then

ψ: A ∧ n ≃ A ∧_{S[N]} TAQ(S[N]) → A ∧ n

is the composite

A ∧ n ⟶^{1∧θ} A ∧ S[N] ∧ n ⟶^{1∧ᾱ∧1} A ∧ A ∧ n ⟶^{μ∧1} A ∧ n .

(ii) When M = L_{0}T has an isolated unit, and T = S^{d},

ψ: A ∧ T ≃ A ∧_{Σ^{∞}M} TAQ(Σ^{∞}M) → A

is the composite

A ∧ T ≅ A ∧ Σ^{∞}T ⟶^{1∧τ̄} A ∧ A ⟶^{μ} A .

((The first result assumes that the universal derivation

d^{u}: S[N] → S[N] ∨ TAQ(S[N]) ≃ S[N] ∨ (S[N] ∧ n)

has second component

S[N] ⟶^{π} Σ^{∞}N ⟶^{θ} S[N] ∧ Σ^{∞}N ⟶^{1∧κ̄} S[N] ∧ n

where π collapses 1_{+} in N_{+}, θ is induced by the half-smash diagonal Δ: N → N_{+} ∧ N, and κ̄ is left adjoint to the group completion map κ: N → Ω^{∞}n. Check this!))

Example. If N = LX is the free E_{∞} space on a unit-pointed space X, then n = Σ^{∞}X = S[X]/S, and ψ is the composite

A ∧ X ⟶^{1∧Δ} A ∧ X_{+} ∧ X ≅ A ∧ S[X] ∧ X ⟶^{1∧ᾱ∧1} A ∧ A ∧ X ⟶^{μ∧1} A ∧ X .

Here ᾱ really means the restriction of ᾱ: S[N] → A to S[X] ⊂ S[N].

Example. If A = ℓ_{p} and M = L_{0}S^{q}, with q = 2p − 2, we get a cofiber sequence

TAQ(ℓ_{p}) ⟶ TAQ(ℓ_{p}, M) ⟶^{res} Hℤ_{p}

where Hℤ_{p} is the homotopy cofiber of v_{1} : ℓ_{p} ∧ S^{q} → ℓ_{p}. Similarly, for B = ku_{p} and N = L_{0}S^{2} we get a cofiber sequence

TAQ(ku_{p}) ⟶ TAQ(ku_{p}, N) ⟶^{res} Hℤ_{p}

where Hℤ_{p} is the homotopy cofiber of u: ku_{p} ∧ S^{2} → ku_{p}. There is a transitivity sequence, also for log TAQ, giving a cofiber sequence

Σ^{2}ku_{p}/Σ^{q}ku_{p} ⟶^{∂} TAQ(ku_{p}/ℓ_{p}) → TAQ((ku_{p}, N)/(ℓ_{p}, M))

of relative differentials. A chase of definitions and identifications shows that the left hand term is the homotopy cofiber of the map

Σ^{q}ku_{p} ⟶^{(p−1)u^{p−2}} Σ^{2}ku_{p} ,

i.e., the different of (v_{1}) = (u^{p−1}). So

Σ^{2}ku_{p}/Σ^{q}ku_{p} ≃ Σ^{2}Hℤ_{p} ∨ ⋯ ∨ Σ^{2p−4}Hℤ_{p}

and (f, f^{♭}): (ℓ_{p}, M) → (ku_{p}, N) is (formally) log étale if and only if ∂ is an equivalence. ((Is it?))

3.2. Logarithmic topological Hochschild homology. We do not have as clear a characterization of what we should mean by THH(A, M). For a commutative Salgebra A and an A-module K, thought of as a symmetric A-bimodule, the space of (associative) S-algebra lifts

A_{S} \{ \xymatrix{ & A ∨ K \ar[d]^{pr} \\ A \ar@{-->}[ur] \ar[r]_{=} & A} \}

is corepresented by the A-bimodule I_{A} given by the homotopy fiber of μ: A^{e} = A ∧ A → A, and there is a cofiber sequence

A ∧_{A^{e}} I_{A} → A ⟶^{ζ} THH(A) .

Furthermore, THH(A) is a commutative A-algebra, and there is a Quillen spectral sequence

Symm_{A}^{∗} Σ TAQ(A) ⟹ THH(A)

starting with the extended powers in A-modules of Σ TAQ(A).

Let

A Der^{♭}(M, K) = A_{0} \{ \xymatrix{ & in^{∗}M \ar[d]^{pr} \\ M \ar@{-->}[ur] \ar[r]_{=} & M} \}

be the space of pointed A_{∞} lifts.

Lemma. (i) When M = N_{+} has an isolated zero,

A Der^{♭}(M, K) ≃ T (BN, Ω^{∞}ΣK) ≃ M_{A^{e}}(A^{e} ∧ Σ^{−1}Σ^{∞}BN, K) ,

where BN = Ω^{∞}Σn.

(ii) When M = L_{0}T has an isolated unit, with T = S^{d},

A Der^{♭}(M, K) ≃ Ω^{∞}K_{⊕} ≃ M_{A^{e}}(A^{e}, K) .

This leads to the corepresenting diagrams of A-bimodules

\xymatrix{A^{e} ∧_{Σ^{∞}M^{e}} I_{Σ^{∞}M} \ar[r]^{ψ} \ar[d]_{ϕ} & A^{e} ∧ Σ^{−1}Σ^{∞}BN \\ I_{A}}

for M = N_{+}, and

\xymatrix{A^{e} ∧_{Σ^{∞}M^{e}} I_{Σ^{∞}M} \ar[r]^{ψ} \ar[d]_{ϕ} & A^{e} \\ I_{A}}

for M = L_{0}T, T = S^{d}. The cofiber sequence above and the commutative A-algebra structure on T HH(A) then suggest the following construction.

Definition (log THH). (i) When M = N_{+} has an isolated zero, we define THH(A, M) as the homotopy pushout

\xymatrix{A ∧_{S[N]} THH(S[N]) \ar[r]^{ψ} \ar[d]_{ϕ} & A ∧ BN_{+} \ar[d] \\ THH(A) \ar[r] & THH(A, M)}

in commutative A-algebras, where ϕ is induced by ᾱ: S[N] → A.

(ii) When M = L_{0}T has an isolated unit, with T = S^{d}, we define THH(A, M) as the homotopy pushout

\xymatrix{A ∧_{Σ^{∞}M} THH(Σ^{∞}M) \ar[r]^{ψ} \ar[d]_{ϕ} & A ∧ S_{+}^{1} \ar[d] \\ THH(A) \ar[r] & THH(A, M)}

in commutative A-algebras, where ϕ is induced by ᾱ: Σ^{∞}M → A.

Remarks. (i) THH(S[N]) ≃ S[B^{cy}N] where B^{cy}N is the cyclic bar construction on N, formed with respect to the cartesian product in spaces.

(ii) THH(Σ^{∞}M) ≃ Σ^{∞}B_{∧}^{cy}M, were B_{∧}^{cy}M is the cyclic bar construction on M, formed with respect to the smash product in pointed spaces.

(iii) In order to proceed to a definition of log TC, we would need to endow THH(A, M) with cyclotomic structure. This is presently not apparent from this definition. The algebraic condition d(d log x) = 0 in the logarithmic de Rham complex suggests that the circle action on A ∧ BN_{+} (or A ∧ S_{+}^{1}) is trivial (up to homotopy), which in turn suggests that also the cyclic structure on this part is essentially trivial.

Example. Let A = Hℤ_{p} and M = N_{+} where N = ⟨p⟩ = {p^{j} | j ≥ 0}. Then BN ≃ S^{1}, while

B^{cy}N ≃ {1} ⊔ ∐_{j=1}^{∞} S^{1}

consists of a free 0-N-cell and a free 1-N-cell, and the definition simplifies to a homotopy pushout square

\xymatrix{Hℤ_{p} ∧ S_{+}^{1} \ar[r]^{ψ} \ar[d]_{ϕ} & Hℤ_{p} ∧ S_{+}^{1} \ar[d] \\ THH(ℤ_{p}) \ar[r] & THH(ℤ_{p}, ⟨p⟩_{+})}

of commutative Hℤ_{p}-algebras. Here ψ is induced by the degree p map S^{1} → S^{1}, and ϕ is induced by the inclusion ζ: Hℤ_{p} → THH(ℤ_{p}) and the circle action on the target.

The resulting Künneth spectral sequence in mod p homotopy is interesting. We have (S/p)_{∗}(Hℤ_{p} ∧ S_{+}^{1}) = E(dp) at the upper left, (S/p)_{∗}(Hℤ_{p} ∧ S_{+}^{1}) = E(d log p) at the upper right, and (S/p)_{∗} THH(ℤ_{p}) = E(λ_{1})⊗P (μ_{1}). Here dp and d log p have degree 1, λ_{1} has degree (2p − 1), and μ_{1} has degree 2p. The map ψ takes dp to p · d log p, which is zero mod p. So

E_{∗∗}^{2} = Tor_{∗∗}^{E(dp)}(E(d log p), E(λ_{1}) ⊗ P (μ_{1}))
≅ E(d log p, λ_{1}) ⊗ P (μ_{1}) ⊗ Γ(κ_{0})

where κ_{0} = [dp] has bidegree (1, 1). There is a differential

d^{p}(γ_{p}κ_{0}) = λ_{1}

leaving

E_{∗∗}^{p+1} = E(d log p) ⊗ P (μ_{1}) ⊗ P_{p}(κ_{0}) = E_{∗∗}^{∞} ,

and a multiplicative extension κ_{0}^{p} = μ_{1} in the abutment, so

(S/p)_{∗} THH(ℤ_{p}, ⟨p⟩_{+}) = E(d log p) ⊗ P (κ_{0})

with d log p in degree 1 and κ_{0}. It follows that there is an equivalence

THH(ℤ_{p}, ⟨p⟩_{+}) ≃ THH(ℤ_{p}|ℚ_{p}) ,

where THH(ℤ_{p}|ℚ_{p}) is Hesselholt–Madsen’s relative THH, defined for valuation rings in p-local fields, so as to sit in the desired cofiber sequence

THH(ℤ/p) ⟶^{i_{∗}} THH(ℤ_{p}) ⟶^{j^{∗}} THH(ℤ_{p}|ℚ_{p})

of cyclotomic spectra. Their construction does not obviously generalize to nonclassical rings, unlike ours, which appears to work for a large class of commutative S-algebras. This compatibility computation suggests that both theories are correct. pre-Example. Let A = ℓ_{p} and M = N_{+}, with a pre-log structure α: N_{+} → Ω^{∞}(ℓ_{p})_{⊗}. In order for (ℓ_{p}, M) to model p^{−1}ℓ_{p}, we expect to have π_{0}N ≅ ⟨p⟩, so perhaps

N = ∐_{j≥0} L(j) ×_{Σ_{j}} U^{j}

for a connected, unpointed space U, with α mapping U to the component W_{p} ×{p} of Ω^{∞}ℓ_{p}. We get a homotopy pushout square

\xymatrix{ℓ ∧_{S[N]} S[B^{cy}N] \ar[r]^{ψ} \ar[d]_{ϕ} & ℓ ∧ BN_{+} \ar[d] \\ THH(ℓ_{p}) \ar[r] & THH(ℓ_{p}, N_{+})}

of commutative ℓ_{p}-algebras. We desire an equivalence

THH(ℓ_{p}, N_{+}) ≃ THH(p^{−1}ℓ_{p}) ,

where the right hand side is defined to sit in a cofiber sequence

THH(ℓ/p) ⟶^{i_{∗}} THH(ℓ_{p}) ⟶^{j^{∗}} THH(p^{−1}ℓ_{p}) .

Computations of McClure–Staffeldt and Ausoni–Rognes show that in mod p and v_{1} homotopy, which we write as V (1)-homology,

V (1)_{∗} THH(ℓ_{p}) = E(λ_{1}, λ_{2}) ⊗ P (μ_{2})

and

V (1)_{∗} THH(p^{−1}ℓ_{p}) = E(d log p, λ_{2}) ⊗ P_{p}(κ_{0}) ⊗ P (μ_{2}) .

Also note that

V (1)_{∗}(ℓ ∧_{S[N]} S[B^{cy}N]) = H_{∗}(N\B^{cy}N)

if B^{cy}N is a free N-cell complex, and

V (1)_{∗}(ℓ ∧ BN_{+}) = H_{∗}(BN) ,

where H_{∗} denotes mod p homology. Hence we would like to find an E_{∞} space N, giving a pre-log structure on ℓ_{p}, such that the spectral sequence

E_{∗∗}^{2} = Tor_{∗∗}^{H_{∗}(N\B^{cy}N)}(H_{∗}(BN), E(λ_{1}, λ_{2}) ⊗ P (μ_{2}))

converges to

E(d log p, λ_{2}) ⊗ P_{p}(κ_{0}) ⊗ P (μ_{2}) .

If N = ⟨p⟩ admitted an E_{∞} map to Ω^{∞}(ℓ_{p})_{⊗}, then it would work fine, but nontrivial Dyer–Lashof operations on [p] ∈ H_{0}(ℓ_{p}) may pose an obstruction to this. ((Check this. Lemma II.2.8 in Cohen–Lada–May seems to say that Q^{s}[p] = 0 in spectrum homology, for positive s.)) One guess is that a somewhat larger N may contain extra terms in H_{∗}(N\B^{cy}N) and H_{∗}(BN), that cancel in Tor. pre-Example. Let A = ℓ_{p} and M = L_{0}S^{q}, with the Adams prelog structure α: M → Ω^{∞}(ℓ_{p})_{⊗}. We get a homotopy pushout square

\xymatrix{ℓ_{p} ∧_{Σ^{∞}M} Σ^{∞}B_{∧}^{cy}M \ar[r]^{ψ} \ar[d]_{ϕ} & ℓ_{p} ∧ S_{+}^{1} \ar[d] \\ THH(ℓ_{p}) \ar[r] & THH(ℓ_{p}, M)}

of commutative ℓ_{p}-algebras. In order to have a cofiber sequence

THH(ℤ_{p}) ⟶^{π_{∗}} THH(ℓ_{p}) ⟶^{ρ^{∗}} THH(ℓ_{p}, M)

we should have

V (1)_{∗} THH(ℤ_{p}) = E(λ_{1}, λ_{2}) ⊗ P (μ_{2})

(as before) and

V (1)_{∗} THH(ℓ_{p}, M) = E(d log v_{1}, λ_{1}) ⊗ P (κ_{1})

with d log v_{1} in degree 1 and κ_{1} in degree 2p. Clearly V (1)_{∗}(ℓ_{p} ∧ S^{1}) = E(d log v_{1}), so if

V (1)_{∗}(ℓ_{p} ∧_{Σ^{∞}M} Σ^{∞}B_{∧}^{cy}M) ≅ H_{∗}(M\B_{∧}^{cy}M) ≅ E(dv_{1})

with dv_{1} in degree (2p − 1), then we can indeed have such a cofiber sequence. This requires that κ_{1} = [dv_{1}] has bidegree (1, 2p−1), there is a differential d^{p}(γ_{p}κ_{1}) = λ_{2} and a multiplicative extension κ_{1}^{p} = μ_{2}, quite like in the case of THH(ℤ_{p}, ⟨p⟩_{+}).

We have not yet analyzed the M = L_{0}S^{q}-action on B_{∧}^{cy}M, in this case.

Department of Mathematics, University of Oslo, Norway

   E-mail address: rognes@math.uio.no