ALGEBRAIC K-THEORY OF FINITELY

PRESENTED RING SPECTRA

John Rognes

September 29, 2000

1. Galois extensions

Let S be the sphere spectrum. An S-algebra A is a monoid (A, μ: A ∧ A → A, η: S → A) in a good symmetric monoidal category of spectra, such as the Smodules of Elmendorf, Kriz, Mandell and May [EKMM], the symmetric spectra of Jeff Smith [HSS], or the simplicial functors of Manos Lydakis [Ly]. When A is commutative there is also the notion of an A-algebra (B, μ: B ∧_{A} B → B, η: A → B).

La A → B be a map of commutative S-algebras. (Make the necessary cofibrancy and fibrancy assumptions.) Let G be a grouplike topological monoid acting on B through A-algebra maps.

Definition. A → B is a G-Galois extension if (1) G ≃ π_{0}(G) is finite,

(2) A ≃ B^{hG} = F (EG_{+}, B)^{G}, and

(3) B ∧_{A} B ≃ F (G_{+}, B).

A → B is a G-pro-Galois extension if G is a filtered limit G = lim_{α} G_{α}, B is a filtered colimit B = colim_{α} B_{α} and A → B_{α} is a G_{α}-Galois extension for each α. Then A ≃ B^{hG} and B ∧_{A} B ≃ F (G_{+}, B) where the homotopy fixed points and function spectra are formed in a continuous sense.

Examples.

(1) The trivial G-Galois extension A → B = F (G_{+}, A) takes A to constant maps from G.

(2) When R → T is a G-Galois extension of commutative rings, the map of Eilenberg–Mac Lane ring spectra HR → HT is a G-Galois extension (of commutative S-algebras).

(3) Complexification KO → KU is a C_{2}-Galois extension, and inclusion of the p-local Adams summand L → KU_{(p)} is a (ℤ/p)^{∗}-Galois extension.

(4) More generally EO_{n} → E_{n} is a G-Galois extension when EO_{n} = E_{n}^{hG} for G a maximal finite subgroup of G_{n} = S_{n} ⋊ C_{n}. Here C_{n} = Gal(𝔽_{p^{n}}/𝔽_{p}) is cyclic of order n and S_{n} is the nth Morava stabilizer group of automorphisms of a height n formal group law defined over 𝔽_{p^{n}}. The Lubin–Tate spectrum E_{n} has homotopy E_{n∗} = 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]][u, u^{−1}], where each u_{i} has degree 0 and u has degree 2, and G_{n} acts on E_{n} through S-algebra maps,

Typeset by AMS-TEX

cf. the works of Morava, Hopkins–Miller and Goerss–Hopkins [Mo], [Re], [GH].

(5) The inclusion J_{p} → KU_{p} is a ℤ_{p}^{∗}-pro-Galois extension, with k ∈ ℤ_{p}^{∗} acting as the Adams operation ψ^{k}.

(6) More generally L_{K(n)}S → E_{n} is (most likely) a G_{n}-pro-Galois extension. The assertion L_{K(n)}S ≃ E_{n}^{hG_{n}} is a version of the Morava change of rings theorem, and the equivalence E_{n} ∧_{L_{K(n)}S} E_{n} ≃ F (G_{n+}, E_{n}) is a variation on a result of Devinatz–Hopkins. (Needs some further checking.)

\xymatrix{ &  & _{\overline{E}_{n}} &  & _{\overline{KU}_{p}} \\ MU \ar[rr] &  & E_{n} \ar[u] &  & KU_{p} = E_{1} \ar[u] \\  &  &  & EO_{n} \ar[ul] &  & L_{p} = EO_{1} \ar[ul]_{Δ} \\  &  & L_{K(n)}S \ar[ur] \ar[uu]^{G_{n}} &  & J_{p} = L_{K(1)}S \ar@{-->}[ur]_{Γ} \ar[uu]^{Γ′} & H\overline{ℚ} \\ S \ar[r] \ar[uuu] & … \ar[r] & L_{n}S \ar[r] \ar[u] & … \ar[r] & L_{1}S \ar[r] \ar[u] & Hℚ = L_{0}S \ar[u]^{G_{ℚ}}}

(When the maximal finite subgroup of G_{n} used to form EO_{n} is not normal, EO_{n} will not be Galois over L_{K(n)}S.)

Proposition. Let A → B be G-Galois, M an A-module, N a B-module.

(1) B is strongly dualizable as an A-module. So F_{A}(B, A) ∧_{A} M ≃ F_{A}(B, M). We write D_{A}(B) = F_{A}(B, A) for the A-dual of B.

(2) B is self-dual as an A-module. So B ≃ D_{A}(B).

(3) A is B-complete, i.e., the map A → C(A → B) to the totalization of the cosimplicial spectrum [q] ↦ B ∧_{A} ⋯ ∧_{A} B with (q + 1) copies of B is a homotopy equivalence.

(4) B ∧_{A} N ≃ F (G_{+}, N).

(5) N ∧ G_{+} ≃ F_{A}(B, N). In particular B ∧ G_{+} ≃ F_{A}(B, B).

Question. What can be said when B = A ∧ X for a spectrum X ? Is X suitably self-dual ?

Question. Is B faithfully flat as an A-module ? That is, does B ∧_{A} M ≃ ∗ imply that M ≃ ∗ ?

This holds in many cases, including the trivial Galois extension, Galois extensions of commutative rings, KO → KU, L → KU_{(p)}, EO_{2} → E_{2} and J_{p} → KU_{p}.

For example, if M is a KO-module with KU ∧_{KO} M ≃ ∗ then from the cofiber sequence ΣKO → KO → KU we get that η: ΣM → M is a homotopy equivalence. But η is nilpotent, so M ≃ ∗. A similar argument works for EO_{2} → E_{2} ≃ EO_{2} ∧ DA(1).

Definition. A commutative S-algebra A is connected if we can only factor A as A ≃ A′ × A′′ as commutative S-algebras when A′ or A′′ is contractible.

Definition. A connected commutative S-algebra A is separably closed if it admits no connected G-Galois extension A → B with π_{0}(G) nontrivial. We write \overline{A} for a separable closure of A.

2. Étale maps

Example. Let F → E be a G-Galois extension of number fields. Then the map of number rings O_{F} → O_{E} is G-Galois if and only if F → E is unramified, i.e., if and only if O_{F} → O_{E} is an étale map.

Definition. A map A → B of S-algebras is formally étale if the topological André– Quillen homology T AQ(B/A) ≃ ∗ is contractible.

One definition of T AQ(B/A) is as the B-module spectrum with nth space S^{n}⊗B with the tensor product formed in the category of commutative A-algebras.

The lifts in the diagram

\xymatrix{A \ar[r] \ar[d] & C ∨ M \ar[d] \\ B \ar[r] & C}

where M is a C-module and M → C ∨ M → C a square-zero extension, are the A-linear derivations Der_{A}(B, M) of B with values in M, and Der_{A}(B, M) ≃ F_{B}(T AQ(B/A), M). Dually to the unique lifting property of covering spaces this space is always contractible precisely when A → B is formally étale.

The following criterion is useful.

Proposition. T AQ(B/A) ≃ ∗ if and only if B ≃ HH^{A}(B).

Here HH^{A}(B) is the realization of the simplicial spectrum [q] ↦ B ∧_{A} ⋯ ∧_{A} B with (q + 1) copies of B and Hochschild-type face maps. The special case T HH(B) = HH^{S}(B) is the topological Hochschild homology of B.

Proof. There is a spectral sequence from the symmetric B-algebra of T AQ(B/A) to HH^{A}(B), which when T AQ(B/A) ≃ ∗ collapses to B ≃ HH^{A}(B).

Conversely the identity T AQ(HH^{A}(B)/B) ≃ ΣT AQ(B/A) shows that B ≃ HH^{A}(B) implies T AQ(B/A) ≃ ∗. □

Proposition. A G-Galois extension A → B is formally étale.

Proof. B ∧_{A} B ≃ F (G_{+}, B) is a product of copies of B, so contains B as a retract as a B ∧_{A} B-module. The composite B → HH^{A}(B) ≃ Tor^{B∧_{A}B}(B, B) → Tor^{B∧_{A}B}(B ∧_{A} B, B) ≃ B is an equivalence, and the right hand map is a split injection. Hence all the maps are homotopy equivalences. □

The transitivity sequence for T AQ can be applied to show that A → B is formally étale (if and?) only if B ∧_{A} T HH(A) ≃ T HH(B). Compare Geller and Weibel [GW].

This much indicates that we have the beginnings of a good theory.

3. Galois descent in algebraic K-theory

Let E be an S-algebra. Two important invariants of the category of E-module spectra is the algebraic K-theory K(E) and the topological Hochschild homology T HH(E). When E is commutative, these are also commutative S-algebras.

What are the global structural properties of these invariants ?

Galois descent problem. La A → B be a G-Galois extension of commutative Salgebras. Does K(A) → K(B)^{hG} induce an equivalence (with suitable coefficients, in sufficiently high degrees) ?

This is known to hold for A → B a Galois extension of finite fields by Quillen [Q1], for p-complete algebraic K-theory of p-local number fields (p odd) by Hesselholt and Madsen [HM2], and for 2-local algebraic K-theory of number fields or 2-local number fields by Voevodsky [V] and Rognes–Weibel [RW].

The separably closed case. Is K(\overline{A}) simple to describe when \overline{A} is separably closed ?

Theorem (Quillen, Suslin).

(1) K(\overline{𝔽}_{p})_{p} ≃ Hℤ_{p}.

(2) K(\overline{ℚ})_{p} ≃ ku_{p}.

Note that p^{−1}Hℤ_{p} may deserve the name E_{0}, and v_{1}^{−1}ku_{p} = KU_{p} = E_{1}.

Questions. What is a separable closure \overline{E}_{n} of E_{n}, or equivalently of L_{K(n)}S ? What does the “fundamental theorem of algebra” say in such an S-algebra ?

What is \overline{S} ? If \overline{S} = S this is the S-algebra version of Minkowski’s theorem \overline{ℤ} = ℤ, saying that every number ring other than ℤ is ramified somewhere.

I stated something like the following conjecture at Schloß Ringberg in January 1999.

Optimistic Conjecture. The k-connected covers of K(\overline{E}_{n})_{p} and E_{n+1} are homotopy equivalent for k sufficiently large.

This would allow the recursive definition E_{n+1} = L_{K(n+1)}K(\overline{E}_{n}), in the category of commutative S-algebras.

When Galois descent holds, we get a spectral sequence

E_{st}^{2} = H^{−s}(G; K_{t}(B)) ⟹ K_{s+t}(A)

converging with suitable coefficients and in sufficiently high degrees. Then the complexity of K(A) gets split between the group cohomology of G and the algebraic K-theory of B. When B = \overline{A} is separably closed, and if K(\overline{A}) has a simple form, then the complexity is all in the cohomology of the absolute Galois group G_{A} = Gal(\overline{A}/A).

Conversely, if we can somehow compute K(A) we may estimate H^{∗}(G_{A}; −) and K(\overline{A}). (Differentials in the descent spectral sequence tend to make this harder.) We shall elaborate on this in two examples later.

In the Hopkins–Miller example we are looking at spectral sequences

E_{st}^{2} = H^{−s}(G_{n}; K_{t}(E_{n})) ⟹ K_{s+t}(L_{K(n)}S) .

The relation between L_{n}S and L_{K(n)}S is illuminated by Hopkins’ chromatic splitting conjecture. Letting n grow, we can hope to compare K(S) = A(∗) with lim_{n} K(L_{n}S), but it is not clear how algebraic K-theory interacts with the limit in the chromatic tower.

4. Localization sequences

Here is one strategy for how to compute algebraic K-theory. The maps 𝔽_{p} = ℤ_{p}/p ← ℤ_{p} → p^{−1}ℤ_{p} = ℚ_{p} induce a cofiber sequence of spectra K(𝔽_{p}) → K(ℤ_{p}) → K(ℚ_{p}) due to Quillen [Q2].

Let ku_{p} be the connective p-complete topological K-theory spectrum, and ℓ_{p} its Adams summand. So ku_{p∗} = ℤ_{p}[u] with |u| = 2, and ℓ_{p} = ℤ_{p}[v_{1}] with |v_{1}| = 2p−2.

There are analogous maps Hℤ_{p} = ku_{p}/u ← ku_{p} → u^{−1}ku_{p} = KU_{p} and Hℤ_{p} = ℓ_{p}/v_{1} ← ℓ_{p} → v_{1}^{−1}ℓ_{p} = L_{p} inducing diagrams K(ℤ_{p}) → K(ku_{p}) → K(KU_{p}) and K(ℤ_{p}) → K(ℓ_{p}) → K(L_{p}).

Question. Are these cofiber sequences of spectra ? This requires identifying the algebraic K-theory of u-torsion ku_{p}-modules or v_{1}-torsion ℓ_{p}-modules with K(ℤ_{p}).

Note that K(ℤ_{p}) is known, by calculations of Bökstedt and Madsen for p odd [BM] and by Rognes for p = 2 [R]. Thus if these diagrams are cofiber sequences then it suffices to compute K(ku_{p}) or K(ℓ_{p}), and the transfer map from K(ℤ_{p}), in order to compute K(KU_{p}) or K(L_{p}). This would in turn give an estimate on G_{KU_{p}} or G_{L_{p}}, and thus a hint about the structure of \overline{KU}_{p}.

The spectra ku_{p} and ℓ_{p} are connective. This makes it significantly easier to compute their algebraic K-theory, due to the possibility of comparing with topological cyclic homology.

5. Topological cyclic homology

We briefly recall the topological cyclic homology of an S-algebra E, first constructed in [BHM] by Bökstedt, Hsiang and Madsen.

T HH(E) = HH^{S}(E) is the geometric realization of the simplicial spectrum [q] ↦ E ∧ E ∧ ⋯ ∧ E with (q + 1) copies of E and Hochschild-type face maps. This is a cyclic object in the sense of Connes, and T HH(E) admits an S^{1}-action. Let C_{p^{n}} ⊂ S^{1} be the cyclic group of order p^{n}. Then T C(E; p) is formed as a homotopy limit:

T C(E; p) =

holim ( \xymatrix{… \ar[r]^{R} \ar[r]_{F} & T HH(E)^{C_{p^{n}}} \ar[r]^{R} \ar[r]_{F} & T HH(E)^{C_{p^{n−1}}} \ar[r]^{R} \ar[r]_{F} & … \ar[r]^{R} \ar[r]_{F} & T HH(E)} )

The maps R and F are called restriction and Frobenius maps, respectively, by analogy with similar maps among Witt rings of finite length.

The cyclotomic trace map is a natural transformation trc: K(E) → T C(E; p), and the composite with the canonical map β: T C(E; p) → T HH(E) is the Dennis– Bökstedt trace map tr = β ∘ trc: K(E) → T HH(E).

Theorem (Hesselholt–Madsen, Dundas, McCarthy). Let E be a connective S-algebra with π_{0}(E) a finite module over the Witt vectors of a perfect field of characteristic p, e.g. a finite ℤ_{p}-module, then trc: K(E) → T C(E; p) identifies K(E)_{p} with the connective cover of T C(E; p)_{p}.

In general T C(E; p)_{p} is (−2)-connected, so the homotopy cofiber of trc has the form Σ^{−1}HA for a known group A.

H_{∗}(T HH(E); 𝔽_{p}) is generally quite accessible through the Bökstedt spectral sequence

E_{s∗}^{2} = HH_{s}^{𝔽_{p}}(H_{∗}(E; 𝔽_{p})) ⟹ H_{∗}(T HH(E); 𝔽_{p}) .

Supposing E is commutative, this is a spectral sequence of H_{∗}(E; 𝔽_{p})-algebras and A_{∗}-comodules, where A_{∗} is the dual Steenrod algebra.

We will eventually want to pass over the (inverse) limit defining T C(E; p). One cannot expect to do this in homology, since the correspondence

H_{∗}(T C(E; p); 𝔽_{p}) → Rlim_{n,R,F} (H_{∗}(T HH(E)^{C_{p^{n}}}; 𝔽_{p}))

rarely is an equivalence.

But limits interact well with homotopy, even with finite coefficients, i.e., with coefficients in a finite CW-spectrum V. Let V_{∗}(X) = π_{∗}(V ∧X) be the V -homotopy of X.

Examples.

(1) V = S = V (−1) gives ordinary homotopy.

(2) V = S/p = V (0) (the mod p Moore spectrum) gives mod p homotopy.

(3) For p odd the Smith–Toda complex V (1) is the homotopy cofiber of the Adams map v_{1} : Σ^{2p−2}V (0) → V (0) inducing multiplication by v_{1} in BP - homology and an isomorphism in topological K-theory. Then V (1)-homo-topy may be thought of as mod p and v_{1} homotopy.

So we should choose V to match E so as to make V_{∗}(T HH(E)) computable from H_{∗}(T HH(E); 𝔽_{p}). Presumably we can then also determine V_{∗}(T HH(E)^{C_{p^{n}}}) for all n ≥ 1, and by forming the algebraic limit we obtain V_{∗}(T C(E; p)). This is essentially V_{∗}(K(E)_{p}) by the cited theorem.

In turn, knowing the V -homotopy of T C(E; p) suffices to detect, if not to construct, a completed version of T C(E; p). If X → Y induces V_{∗}(X) ≅ V_{∗}(Y) then X ≃ Y if H_{∗}(V) is infinite, and X_{p} ≃ Y_{p} if H_{∗}(V) contains nontrivial p-torsion.

Example. Bökstedt and Madsen considered the case E = Hℤ_{p}, p odd, using V = S/p = V (0). Using the mod p homotopy of T HH(ℤ_{p}) they computed the mod p homotopy of T C(ℤ_{p}; p), and thus of K(ℤ_{p}) and K(ℚ_{p}). Then they (essentially) produced a map

j_{p} ∨ Σj_{p} ∨ Σku_{p} → K(ℚ_{p})_{p}

inducing an isomorphism between the computed mod p homotopy groups, and could conclude that the map is a homotopy equivalence.

Variants of this argument go through for p = 2, cf. [R].

6. Finitely presented spectra

The extraction of V -homotopy V_{∗}(T HH(E)) from homology H_{∗}(T HH(E); 𝔽_{p}) is most plausible when H_{∗}(V ∧ T HH(E); 𝔽_{p}) has tiny projective dimension as an A_{∗}-comodule, e.g. when it is free, i.e., when V ∧T HH(E) is a wedge of suspensions of H𝔽_{p}. For E commutative and V a ring spectrum, V ∧ T HH(E) is a module spectrum over V ∧ E, so this happens when V ∧ E is a wedge of suspensions of H𝔽_{p}.

A related notion was considered by Mahowald and Rezk [MR]:

Definition. A bounded below, p-complete spectrum E is finitely presented (an fp-spectrum) if H^{∗}(E; 𝔽_{p}) is finitely presented as an A-module. Equivalently there is a nontrivial finite CW spectrum F such that π_{∗}(F ∧ E) = F_{∗}(E) is finite. Then there is a unique integer n, called the fp-type of E, such that F_{∗}(E) is infinite if F has chromatic type ≤ n (K(n)_{∗}(F) ≠ 0), and F_{∗}(E) is finite if F has chromatic type > n (K(n)_{∗}(F) = 0).

We may also define a more refined notion:

Definition. E has pure fp-type n if furthermore F_{∗}(E) is a free finitely generated P (v)-module for some finite CW spectrum F of chromatic type n, with v_{n}-map v: Σ^{d}F → F. (Then the mapping cone V = C_{v} has chromatic type (n + 1) and V_{∗}(E) is finite.)

These definitions are well behaved by thick subcategory considerations.

When E is a finitely presented ring spectrum of fp-type n we choose a finite CW ring spectrum V (of chromatic type n + 1) making V_{∗}(E) as simple as possible. Then V_{∗}(T HH(E)) can be (relatively) easily read off from H_{∗}(V ∧T HH(E); 𝔽_{p}) ≅ H_{∗}(V; 𝔽_{p}) ⊗ H_{∗}(T HH(E); 𝔽_{p}), which is now a H_{∗}(V ∧ E; 𝔽_{p})-module. Then proceed as before to determine V_{∗}(T HH(E)^{C_{p^{n}}}) and pass to the limit to obtain V_{∗}(T C(E; p)).

Examples.

(1) For E = H𝔽_{p} of fp-type −1 use V = S. Hesselholt and Madsen [HM1] computed T C(𝔽_{p}; p) ≃ Hℤ_{p}∨Σ^{−1}Hℤ_{p} recovering Quillen’s result K(𝔽_{p})_{p} ≃ Hℤ_{p}. The answer has pure fp-type 0, i.e., has no p-torsion.

(2) For E = Hℤ_{p} of fp-type 0 use V = S/p = V (0), at least for p odd. Bökstedt and Madsen [BM1], [BM2] computed the mod p homotopy of T C(ℤ_{p}; p) and deduce K(ℤ_{p})_{p} ≃ j_{p} ∨ Σj_{p} ∨ Σ^{3}ku_{p}. Then answer has pure fp-type 1, i.e., its mod p homotopy has no v_{1}-torsion. Similar results hold for p = 2 by [R].

(3) For E = ℓ_{p} = BP ⟨1⟩_{p} of fp-type 1 use V = V (1) for p ≥ 5. Ausoni and Rognes [AR] computed the mod p and v_{1} homotopy of T C(ℓ_{p}; p), and similarly for K(ℓ_{p})_{p}. The result has pure fp-type 2, i.e., its V (1)-homotopy is a free finitely generated P (v_{2})-module on 4p + 4 generators.

(4) Other fp-spectra of fp-type 1 include ku_{p}, ko_{p} and j_{p}.

(5) The connective topological modular forms spectrum eo_{2} with H^{∗}(eo_{2}; 𝔽_{2}) = A//A(2) has fp-type 2.

(6) The spectrum E = BP ⟨n⟩_{p} has fp-type n, but is not known to be a commutative S-algebra for n ≥ 2. The nth Smith–Toda complex V (n) with BP_{∗}(V (n)) = BP_{∗}/(p, …, v_{n−1}) makes V (n) ∧ BP ⟨n⟩_{p} ≃ H𝔽_{p}, but is not known to exist for n ≥ 4. (But other chromatic type (n + 1) ring spectra certainly exist.)

7. Algebraic K-theory of topological K-theory

Theorem (Ausoni–Rognes). For p ≥ 5 let ℓ_{p} = BP ⟨1⟩_{p} be the Adams summand of connective p-complete topological K-theory and let V (1) be the Smith–Toda complex. Let v_{2} = [τ_{2}] ∈ π_{2p^{2}−2}V (1). Then

V (1)_{∗}(T C(ℓ_{p}; p)) ≅ E(λ_{1}, λ_{2}, ∂) ⊗ P (v_{2})
⊕ E(λ_{2}){t^{e}λ_{1} | 0 < e < p} ⊗ P (v_{2})
⊕ E(λ_{1}){t^{ep}λ_{2} | 0 < e < p} ⊗ P (v_{2})

is a free P (v_{2})-module on 4p + 4 generators. Here |∂| = −1, |λ_{1}| = 2p − 1, |λ_{2}| = 2p^{2} − 1 and |t| = −2.

There is an exact sequence

0 → Σ^{2p−3}𝔽_{p} ⟶^{a} V (1)_{∗}K(ℓ_{p}) ⟶^{trc} V (1)_{∗}T C(ℓ_{p}; p) ⟶^{∂} Σ^{−1}𝔽_{p} → 0

determining the V (1)-homotopy of K(ℓ_{p}).

Corollary. T C(ℓ_{p}; p)_{p} is a finitely presented spectrum of pure fp-type 2.

In this sense T C(ℓ_{p}; p) is like eo_{2}, or BP ⟨2⟩_{p} if the latter exists.

Recall that K(ℚ_{p})_{p} has mod p homotopy a free P (v_{1})-module on p+3 generators, where

p + 3 = ∑_{i=1}^{p−1} ∑_{n=0}^{∞} dim_{𝔽_{p}} H^{n}(Gal(ℚ̄_{p}/ℚ_{p}); 𝔽_{p}(i)) ,

and K(ℚ_{p})_{p} is constructed from p + 3 copies of BP ⟨1⟩_{p} = ℓ_{p} up to extensions involving Adams operations. A more precise statement can be obtained by taking the degrees of the P (v_{1})-module generators into account.

Likewise we get that the cofiber of the transfer map K(ℤ_{p}) → K(ℓ_{p}), which most likely is K(L_{p}), has V (1)-homotopy a free P (v_{2})-module on 4p+4 generators, where we estimate

4p + 4 = ∑_{i=1}^{p^{2}−1} ∑_{n=0}^{∞} dim_{𝔽_{p}} H^{n}(Gal(\overline{L}_{p}/L_{p}; 𝔽_{p^{2}}(i)) ,

and K(L_{p}) is constructed from 4p + 4 copies of BP ⟨2⟩_{p}, up to extensions involving BP ⟨2⟩_{p}-operations. Again a more precise statement can be obtained by taking the degrees of the P (v_{2})-module generators into account.

Moral. Algebraic K-theory of topological K-theory is a form of elliptic cohomology.

These calculations generalize to determine V (n)_{∗}K(BP ⟨n⟩_{p}) if BP ⟨n⟩_{p} exists as a commutative S-algebra and V (n) exists as a ring spectrum, in which case the result is of pure fp-type n + 1. Hence we are led to the following:

Chromatic red-shift problem. Let E be an S-algebra of pure fp-type n. Does T C(E; p) have pure fp-type n + 1 ?

So far this is known to be correct for E = Hk with k a finite extension of 𝔽_{p}, for E = HA with A the valuation ring of a finite extension of ℚ_{p}, and for E = ℓ_{p}. One might also consider E = S as a limiting case, of infinite fp-type. Then T C(S; p) contains S ≃ T HH(S) as a retract, so in this case the fp-type of the result is ∞ + 1 = ∞.

References

[AR] C. Ausoni and J. Rognes, Algebraic K-theory of topological K-theory, preprint (2000).

[BHM] M. Bökstedt, W.-C. Hsiang and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 865–940.

[BM1] M. Bökstedt and I. Madsen, Topological cyclic homology of the integers, Asterisque 226 (1994), 57–143.

[BM2] M. Bökstedt and I. Madsen, Algebraic K-theory of local number fields: the unramified case, Prospects in topology (Princeton, NJ, 1994), Ann. of Math. Studies, vol. 138, Princeton University Press, 1995, pp. 28–57.

[EKMM] A.D. Elmendorf, I. Kriz, M.A. Mandell and J.P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997.

[GW] S.C. Geller and C.A. Weibel, Étale descent for Hochschild and cyclic homology, Comment. Math. Helv. 66 (1991), 368–388.

[GH] P.G. Goerss and M.J. Hopkins, Realizing commutative ring spectra as E_{∞} ring spectra, preprint (2000).

[HM1] L. Hesselholt and I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), 29–101.

[HM2] L. Hesselholt and I. Madsen, On the K-theory of local fields, preprint (1999).

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Mathematisches Forschungsinstitut Oberwolfach