ARITHMETIC OF SOME BRAVE NEW RINGS

John Rognes

March 29th 2006

§1. Brave new rings.

Each discrete ring R gives rise to a multiplicative cohomology theory

X ↦ H^{∗}(X; R)

given by singular cohomology with coefficients in R, and the cup product. There are other such multiplicative cohomology theories, for example, topological K-theory

X ↦ KU^{∗}(X)

given by complex vector bundles over X, and their tensor product.

A brave new ring A is a structured ring spectrum, in the sense of algebraic topology, that represents such a multiplicative cohomology theory

X ↦ A^{∗}(X) .

In particular, A consists of a sequence of spaces

A = {n ↦ A_{n}} ,

with additional structure, that represent the graded terms in the multiplicative cohomology theory, so that

A^{n}(X) = [X, A_{n}]

(homotopy classes of maps from X to A_{n}) in each degree n.

A satisfactory description of the additional structure satisfied by this sequence of spaces {n ↦ A_{n}} was only published in 1997. Several solutions exist, and go under the names S-algebras (Elmendorf–Kriz–Mandell–May, 1997), symmetric ring spectra (Hovey–Shipley–Smith, 1999), orthogonal ring spectra (May–Mandell– Shipley–Smith, 2001), functors with smash product (Bökstedt, ca. 1986), etc. They rigidify earlier notions of A_{∞}- and E_{∞} ring spectra.

For example, singular cohomology with R-coefficients is represented by the Eilenberg–Mac Lane spectrum

HR = {n ↦ K(R, n)} ,

with n-th space the Eilenberg–Mac Lane complex K(R, n). Thus discrete rings are special cases of brave new rings.

Typeset by AMS-TEX

Topological K-theory provides a non-algebraic example. As a brave new ring it is represented by the complex K-theory spectrum

KU = \{ \begin{matrix}ℤ × BU & n even \\ U & n odd \end{matrix} ,

with n-th space the group completed union of infinite Grassmannians ℤ × BU for n even, and its loop space U for n odd.

The most fundamental example of a brave new ring is the sphere spectrum

S = {n ↦ S^{n}} ,

with n-th space S^{n}, the n-sphere. It represents stable cohomotopy.

§2. Module categories and K-theory.

The modern description of structured spectra lets us talk about the category

ℳ_{A} = {A-modules}

of modules over a brave new ring A.

For example, the category

ℳ_{HR}

of HR-modules is very similar to the category of chain complexes over a ring R. For another example, the category

ℳ_{S} ≃ {spectra}

of modules over the sphere spectrum S is equivalent to the category of structured spectra, so we may refer to a spectrum as an S-module. This is analogous to viewing abelian groups as ℤ-modules.

There is also a pairing of S-modules, called the smash product

M ∧_{S} N ,

analogous to the tensor product of abelian groups.

A structured ring spectrum A comes with a product map

A ∧_{S} A → A

that makes it an S-algebra, in the same way as a discrete ring is a ℤ-algebra. Thus brave new ring, structured ring spectrum and S-algebra are synonyms.

Algebraic K-theory focuses on a subcategory

h𝒞_{A} ⊂ ℳ_{A}

of finite cell A-modules, and their homotopy equivalences, analogous to the category of finite cell complexes inside of topological spaces. More precisely, we idempotently complete this category by including all retracts, but we shall suppress this point.

For example,

h𝒞_{HR}

is very similar to the category of bounded complexes of finitely generated free Rmodules, and their quasi-isomorphisms.

§3. Algebraic K-theory as arithmetic.

The algebraic K-theory of A is a spectrum K(A) generated by the category h𝒞_{A}. Its 0-th space K(A)_{0} is a group completion of the classifying space Bh𝒞_{A} of this category, as realized by a map

Bh𝒞_{A} → K(A)_{0} .

For a discrete ring R, K(HR) ≃ K(R) is Quillen’s algebraic K-theory. When R = 𝒪_{F} is the ring of integers in a number field,

π_{0}K(𝒪_{F}) = ℤ ⊕ Cl(F) ,
π_{1}K(𝒪_{F}) = 𝒪_{F}^{×}

and in general π_{i}K(𝒪_{F}) carries much arithmetic information about F.

The algebraic K-theory of other brave new rings A is also of significant interest, outside of the field of K-theory. A major motivation for the development of the theory of structured ring spectra was the need to have proper foundations for the definitions above, with applications like the following in mind.

For a manifold X, with loop group ΩX, there is a spherical group ring A = S[ΩX], which is a brave new ring that refines the discrete integral group ring ℤ[π_{1}(X)]. By Waldhausen’s stable parametrized h-cobordism theorem (ca. 1982, writeup in progress at MPI), there is a very close relation between the algebraic K-theory of S[ΩX]

K(S[ΩX]) ∼ B Aut(X)

and the moduli space of manifolds isomorphic to X, or equivalently, the automorphism group of X, in a range of homotopy groups that grows to infinity with the dimension of X. The theory applies for both smooth and topological manifolds and automorphisms. By the geodesic flow methods of Farrell and Jones, it suffices to consider the cases X = ∗ (a point) and X = S^{1} (a circle) to deal with all non-positively curved X.

For a different application, there is a relation (Baas–Dundas–Rognes, 2004) between the algebraic K-theory of topological K-theory

K(KU) ∼ ℰℓℓ ∼ CF T ,

elliptic cohomology and conformal field theory. A 2-vector bundle ℰ over a spacetime X represents an element in K(KU)^{0}(X), and the holonomy of parallel transport in ℰ around a string γ in X defines a “number” in KU, i.e., a virtual vector space, which can be interpreted as the state space of γ.

These theories are thus arithmetic aspects of S[ΩX] and KU, respectively.

§4. The chromatic tower.

We seek a conceptual description of K(S), corresponding to a highly-connected manifold X (like a point, a disc or a sphere) in Waldhausen’s theorem, following an approach outlined by him in 1984.

Fixing a rational prime p, we can for simplicity focus on p-local brave new rings. The short filtration of rings

ℤ_{(p)} ⊂ ℚ

corresponds, at the level of module categories, to the localizations away from the subcategories

{0} ⊂ {p-torsion}

of finitely generated ℤ_{(p)}-modules, or bounded chain complexes of such. This unfolds into an infinite tower of brave new rings

S_{(p)} → ⋯ → L_{n}S → L_{n−1}S → ⋯ → L_{0}S = Hℚ

where

L_{n}S = L_{E(n)}S

is the (implicitly p-local) n-th chromatic localization of the sphere spectrum, which focuses on the n-th kind of periodicity phenomena in stable homotopy theory, i.e., v_{n}-periodicity, which in turn is related to height n formal groups in characteristic p.

At the level of module categories, these correspond to localizations away from the subcategories

{∗} ⊂ ⋯ ⊂ 𝒯_{n} ⊂ 𝒯_{n−1} ⊂ ⋯ ⊂ 𝒯_{0} = {p-torsion}

of h𝒞_{S_{(p)}}, where

𝒯_{n} = {type > n}

is the thick subcategory of p-local finite cell spectra Z of type > n. This means that the first (n + 1) Morava K-theories

K(0)_{∗}(Z), …, K(n)_{∗}(Z)

of Z all vanish. By the Devinatz–Hopkins–Smith nilpotence theorem (1988) these are all the thick subcategories of h𝒞_{S_{(p)}}, so the chromatic tower above is a canonical object to study.

In terms of brave new algebraic geometry, we think of Spec S_{(p)} as being filtered by an increasing sequence of Zariski open subspaces Spec L_{n}S.

In the discrete case, the algebraic K-theory of K(ℤ_{(p)}) is understood in terms of the algebraic K-theory of fields, by the lower cofiber sequence of spectra in the diagram

\xymatrix{K(𝔽_{p}) \ar[dr]_{i_{∗}} \ar[r] & K(ℤ_{p}) \ar[r] & K(ℚ_{p}) \\  & K(ℤ_{(p)}) \ar[r]^{j^{∗}} \ar[u] & K(ℚ) \ar[u]}

Here i_{∗} is the transfer/direct image map, that considers a finitely generated 𝔽_{p}- module as a p-torsion ℤ_{(p)}-module, which goes to zero upon tensoring with ℚ. By Quillen’s dévissage theorem, this map i_{∗} identifies K(𝔽_{p}) with the homotopy fiber of j^{∗}.

In the brave new ring case, there is a similar diagram

\xymatrix{K^{sm}(L̂_{n}S) \ar[dr]_{i_{∗}} \ar[r] & K(L̂_{n}S) \ar[r] & K(L_{n−1}L̂_{n}S) \\  & K(L_{n}S) \ar[r]^{j^{∗}} \ar[u] & K(L_{n−1}S) \ar[u]}

for each n ≥ 1, where

L̂_{n}S = L_{K(n)}S

is the K(n)-localization of S. Here K^{sm}(L̂_{n}S) is the algebraic K-theory of the full subcategory

h𝒞_{L̂_{n}S}^{sm} ⊂ h𝒞_{L̂_{n}S}

of (categorically) small L̂_{n}S-modules. The lower part of the diagram is essentially a cofiber sequence of spectra, except that j^{∗} may not be surjective on π_{0}. We do not claim that the upper part of this diagram is a cofiber sequence.

We think of Spec L̂_{n}S as a completion of Spec L_{n}S away from Spec L_{n−1}S, as Spec ℤ_{p} is the completion of Spec ℤ_{(p)} away away from Spec ℚ. The small L̂_{n}Smodules live over this local neighborhood, and are supported away from the preimage of Spec L_{n−1}S.

§5. Galois extensions.

The algebraic K-theory of fields is quite well understood by the principle of Galois descent, which underlies the now proven Lichtenbaum–Quillen conjectures. To each G-Galois extension F → E there is a map

K(F) → K(E)^{hG}

to the G-homotopy fixed points for the induced G-action on E, and this map is close to a homotopy equivalence, say, with finite coefficients and in sufficiently high degrees. The limiting case, with E = F̄ the separable closure and G = G_{F} the absolute Galois group, is particularly interesting. Taking a common point of view, the presence of this group action separates an arithmetic theory from a geometric theory.

In the case of K(𝔽_{p}), all of this is well understood. In the case of K(ℚ), we understand K(ℚ̄) after completion at a prime p, by a theorem of Suslin, and our fairly detailed knowledge of K(ℚ) feeds back to give information about G_{ℚ}.

There is a theory of Galois extensions of brave new rings (Rognes, MPIM–Bonn preprint 2005–79). A map A → B of commutative brave new rings is a G-Galois extension if the finite group G acts on B over A and two natural maps

A → B^{hG}
B ∧_{A} B → ∏_{G} B

are homotopy equivalences. It is natural to conjecture that the Galois descent principle for algebraic K-theory extends, i.e., that the map

K(A) → K(B)^{hG}

becomes a homotopy equivalence in sufficiently high degrees, after introducing finite coefficients of sufficiently high type.

Conjecture. Let A → B be a K(n)-local G-Galois extension, let F be a type (n + 1) finite cell spectrum, and let T = v_{n+1}^{−1}F be the mapping telescope of any v_{n+1} self-map of F. Then

T ∧ K(A) → T ∧ K(B)^{hG}

is a homotopy equivalence.

§6. The Lubin–Tate spectra.

By the Morava change-of-rings theorem and the Goerss–Hopkins–Miller obstruction theory, there is a K(n)-local pro-𝔾_{n}-Galois extension

L̂_{n}S ⟶^{𝔾_{n}} E_{n}

with 𝔾_{n} the (extended) Morava stabilizer group of automorphisms of the height n Honda formal group law, and E_{n} an even periodic form of E(n), with

π_{0}E_{n} = 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n−1}]]

the complete local ring supporting the universal Lubin–Tate deformation of the Honda formal group law. The commutative S-algebra E_{n} is a Landweber exact complex oriented spectrum.

By Hovey–Strickland (1999), a finite cell L̂_{n}S-module M is small if and only if the smash product

E_{n} ∧_{L̂_{n}S} M ≃ L̂_{n}(E_{n} ∧ M)

is degreewise finite, in the sense that each homotopy group (usually denoted E_{n∗}^{∨}(M)) is finite. We obtain the diagram

\xymatrix{K(K_{n}) \ar[d]_{∼} \ar[dr]^{i_{∗}} \\ K^{df}(E_{n}) \ar[r] & K(E_{n}) \\ K^{sm}(L̂_{n}S) \ar[r] \ar[u]^{𝔾_{n}} & K(L̂_{n}S) \ar[u]^{𝔾_{n}}}

where K^{df}(E_{n}) is the algebraic K-theory of the full subcategory

h𝒞_{E_{n}}^{df} ⊂ h𝒞_{E_{n}}

of degreewise finite E_{n}-modules. There is a map i: E_{n} → K_{n} of brave new rings, where K_{n} is a (non-commutative!) even periodic form of K(n), with π_{0}K_{n} = 𝔽_{p^{n}}. Composing the module action with i defines a functor

h𝒞_{K_{n}} → h𝒞_{E_{n}}^{df}

and an induced map in algebraic K-theory.

Conjecturally then, K^{sm}(L̂_{n}S) → K^{df}(E_{n})^{h𝔾_{n}} is close to a homotopy equivalence, by a form of Galois descent, and K(K_{n}) → K^{df}(E_{n}) is a homotopy equivalence, by a form of dévissage.

We think of Spec E_{n} as a regular covering space, or étale cover, of Spec L̂_{n}S, and the degreewise finite E_{n}-modules live over this covering space, supported away from the preimage of Spec L_{n−1}S. These are then part of an étale topology on Spec S_{(p)}.

More generally, there is a sequence of categories

h𝒞_{E_{n}}^{df} ⊂ ⋯ ⊂ h𝒞_{E_{n}}^{I_{k}-tors} ⊂ ⋯ ⊂ h𝒞_{E_{n}}

for 0 ≤ k ≤ n, where I_{k} = (p, u_{1}, …, u_{k−1}) is an invariant ideal in π_{0}E_{n}, and h𝒞_{E_{n}}^{I_{k}-tors} is the full subcategory of finite cell E_{n}-modules whose homotopy groups are I_{k}-torsion. For k = 0, this equals h𝒞_{E_{n}} and for k = n it equals h𝒞_{E_{n}}^{df}. There is a map E_{n} → E_{n}/I_{k} that makes E_{n}/I_{k} a finite cell E_{n}-module, a functor

h𝒞_{E_{n}/I_{k}} → h𝒞_{E_{n}}^{I_{k}-tors} ,

and an induced map in algebraic K-theory. Optimistically, each of these maps may be homotopy equivalences.

§7. Topological K-theory.

In the first non-algebraic case, n = 1, we have L_{0}S = Hℚ, L_{1}S = J̃_{(p)}, L̂_{1}S = J_{p}, the p-adic image-of-J spectrum, E_{1} = KU_{p}, p-adic topological K-theory, 𝔾_{1} = ℤ_{p}^{×} with k ∈ ℤ_{p}^{×} acting on KU_{p} by the Adams operation ψ^{k}, and K_{1} = KU/p, mod p topological K-theory.

\xymatrix{ & K(KU/p) \ar[d]_{∼} \ar[dr]^{i_{∗}} \\  & K^{df}(KU_{p}) \ar[r] & K(KU_{p}) \ar[r]^{j^{∗}} & K(ff (KU_{p})) \\  & K^{sm}(J_{p}) \ar[r] \ar[dr] \ar[u]^{ℤ_{p}^{×}} & K(J_{p}) \ar[r] \ar[u]^{ℤ_{p}^{×}} & K(Jℚ_{p}) \\ K(S_{(p)}) \ar[r] \ar[d] & K(L_{n}S) \ar[r] & K(J̃_{(p)}) \ar[r] \ar[u] & K(ℚ) \ar[u] \ar[d]^{=} \\ K(ℤ_{(p)}) \ar[rrr] &  &  & K(ℚ)}

To go from K(ℚ) to K(S_{(p)}), the first step is to K(J̃_{(p)}), and the change is K^{sm}(J_{p}), which we expect to obtain by Galois descent from K^{df}(KU_{p}), which we in turn expect is homotopy equivalent to K(KU/p).

We define the algebraic K-theory K(ff (KU_{p})) of the fraction field ff (KU_{p}) of p-adic topological K-theory to be the mapping cone of i_{∗}, so as to have the i_{∗}-j^{∗}-cofiber sequence above. This is a “motivic” definition, for now. No definition of ff (KU_{p}) as a brave new ring is yet known.

We view h𝒞_{KU_{p}}^{df} as a category of coherent sheaves over Spec KU_{p} that are supported over a closed (non-commutative) subobject Spec KU/p. The result of cutting out this subobject is to be Spec ff (KU_{p}).

There is a cofiber sequence

K(ℤ) ⟶^{π_{∗}} K(ku) ⟶^{ρ^{∗}} K(KU)

due to Blumberg–Mandell, where ρ: ku → KU is the connective cover of KU, and π: ku → Hℤ is the 0-th Postnikov section. The same proof will yield similar cofiber sequences for KU_{p} and KU/p. These have the advantage that the algebraic K-theory spectra

K(ku_{p}), K(ku/p), K(ℤ_{p}), K(ℤ/p)

can be calculated by means of the cyclotomic trace map to the topological cyclic homology of Bökstedt–Hsiang–Madsen (1993). This is an invariant derived from topological Hochschild homology, rather like crystalline cohomology in the algebraic case, and for connective p-complete brave new rings, like ku_{p}, ku/p, ℤ_{p} and ℤ/p, the cyclotomic trace map is very close to a homotopy equivalence.

§8. Lichtenbaum–Quillen periodicity.

Such calculations have been made with Christian Ausoni (Uni-Bonn, writeup in progress at MPI) in the last year. Assume p ≥ 5 and let

V (1) = S/(p, v_{1}) = cone(v_{1} : S^{2p−2}/p → S^{0}/p)

be a type 2 finite cell spectrum representing homotopy with mod (p, v_{1}) coefficients, i.e.,

V (1)_{∗}X = π_{∗}(V (1) ∧ X) .

It admits a v_{2}-self map of degree 2(p^{2} − 1), so V (1)_{∗}X is naturally a module over ℤ/p[v_{2}]. There is a long exact sequence

⋯ → π_{i−2p+2}(X; ℤ/p) ⟶^{v_{1}} π_{i}(X; ℤ/p) → V (1)_{i}(X) → …

where π_{i}(X; ℤ/p) denotes mod p homotopy.

Theorem (Ausoni–Rognes, 2005). Each of

V (1)_{∗}K(KU/p), V (1)_{∗}K(KU_{p}), V (1)_{∗}K(ff (KU_{p}))

is a finitely generated free ℤ/p[v_{2}]-module on explicit generators, of rank

2(p − 1)(p^{2} − p + 4), 2(p − 1)(2p + 2), 2(p − 1)(p^{2} + 3)

respectively.

We view this purely calculational result as a verification of a generalization to brave new rings of the Lichtenbaum–Quillen conjecture, to the effect that

π_{∗}(K(F); ℤ/p)

is a finitely generated free ℤ/p[v_{1}]-module, for a local or global number field F, whose v_{1}-localization

v_{1}^{−1}π_{∗}(K(F); ℤ/p) ≅ π_{∗}(K^{ét}(F); ℤ/p)

agrees with the étale K-theory of F, and similarly for the ring 𝒪_{F} of integers in F. For example, π_{∗}(K(ℚ_{p}); ℤ/p) is a finitely generated free ℤ/p[v_{1}]-module of rank (p + 3).

§9. Beilinson–Lichtenbaum motivic truncation.

A more precise form of the Lichtenbaum–Quillen conjecture, again for a field F, is offered by the motivic version of the Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(F; ℤ(t/2)) ⟹ π_{s+t}K(F)

in its mod p form, and the Beilinson–Lichtenbaum conjecture that

H_{mot}^{r}(F; ℤ/p(i)) ≅ \{ \begin{matrix}H_{ét}^{r}(F; ℤ/p(i)) & for r ≤ i \\ 0 & for r > i \end{matrix}

for char(F) ≠ p, where in the field case

H_{ét}^{r}(F; ℤ/p(i)) ≅ H_{Gal}^{r}(F; μ_{p}^{⊗i}) ≅ H_{cont}^{r}(G_{F}; μ_{p}^{⊗i})

is the continuous group cohomology of the absolute Galois group, and

H_{ét}^{r}(F; ℤ/p(∗)) ≅ v_{1}^{−1}H_{mot}^{r}(F; ℤ/p(∗))

where v_{1} ∈ H_{mot}^{0}(F; ℤ/p(p − 1)). This conjecture is now a theorem, by the work of Voevodsky, Rost, Suslin and others.

No definition of motivic cohomology for brave new rings is yet known to me, but if we assume its existence, together with an analogue of the Atiyah–Hirzebruch spectral sequence with mod (p, v_{1}) coefficients, then we can run the spectral sequence backwards to get a prediction for the motivic cohomology of

H_{mot}^{r}(ff (KU_{p}); 𝔽_{p^{2}}(i)) ,

assuming that ff (KU_{p}) is a “brave new field”. Here 𝔽_{p^{2}}(i) = π_{2i}K_{2} = V (1)_{2i}E_{2}.

Pre-theorem.

H_{mot}^{∗}(ff (KU_{p}); 𝔽_{p^{2}}(∗)) ≅ ℤ/p[b] ⊗ Φ_{∗,∗}

Here the “higher Bott element” b is a (p − 1)-st root of v_{2}, in bidegree (r, i) = (0, p + 1), and Φ_{∗,∗} is an explicit finite bigraded ℤ/p-algebra of rank 2(p^{2} + 3) concentrated in the fundamental domain of bidegrees

0 ≤ r ≤ 3, r ≤ i < r + p + 1 .

In particular, ff (KU_{p}) has p-cohomological dimension 3.

Inverting v_{2} = b^{p−1}, we must then have

H_{ét}^{∗}(ff (KU_{p}); 𝔽_{p^{2}}(∗)) ≅ ℤ/p[b, b^{−1}] ⊗ Φ_{∗,∗}

and we can observe that, indeed,

H_{mot}^{r}(ff (KU_{p}); 𝔽_{p^{2}}(i)) ≅ \{ \begin{matrix}H_{ét}^{r}(ff (KU_{p}); 𝔽_{p^{2}}(i)) & for r ≤ i \\ 0 & for r > i. \end{matrix}

We view this calculational conclusion as verifying a generalization of the Beilinson– Lichtenbaum conjecture for the fraction field of p-adic topological K-theory. In particular, it strongly supports the idea that the symbol K(ff (KU_{p})) is really the algebraic K-theory of a field-like object, and that the presumptions made in its construction are justified.

If a similar argument is made to predict the motivic and étale cohomology of KU/p and KU_{p}, then the analogue of the Beilinson–Lichtenbaum conjecture does not hold. This is perhaps to be expected, since KU/p is most like a noncommutative field of characteristic p, and KU_{p} is more like a valuation ring than a field. On the other hand, it also illustrates that something special is going on in the algebraic K-theory and (hypothetical) motivic cohomology of ff (KU_{p}), something that is not shared by most brave new rings.

§10. Tate–Poitou arithmetic duality.

The Galois cohomology of local and global number fields has an additional special property, known as Tate–Poitou arithmetic duality.

In the local number field case, ℚ_{p} ⊂ F, the cup-product pairing

H_{ét}^{r}(F; ℤ/p(i)) ⊗ H_{ét}^{2−r}(F; ℤ/p(1−i)) ⟶^{∪} H_{ét}^{2}(F; ℤ/p(1)) ≅ Br(F)[p] ⟶_{≅}^{inv} ℤ/p

is a perfect pairing, so induces an isomorphism

H_{ét}^{2−r}(F; ℤ/p(1−i)) ≅ H_{ét}^{r}(F; ℤ/p(i))^{∗} .

For higher local fields, say a d-local field F_{d} for d ≥ 0 (with ℚ_{p} ⊂ F_{1}), there is a similar perfect cup-product pairing landing in

H_{ét}^{d+1}(F_{d}; ℤ/p(d)) ≅ ℤ/p

(Deninger–Wingberg, 1986).

Similarly, there is an isomorphism

H_{ét}^{3}(ff (KU_{p}); 𝔽_{p^{2}}(2)) ≅ ℤ/p

and the expected cup-product pairing

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

is visibly perfect.

Pre-theorem. The finite ℤ/p-algebra Φ_{∗,∗} above is a Poincaré algebra.

In this sense, the fraction field of p-adic topological K-theory behaves as a brave new 2-local field. In particular, it has the arithmetic characteristics of such a higher local field. Conceivably, the fraction field ff (E_{n}) of the n-th Lubin–Tate spectrum may behave like a brave new (n + 1)-local field, and the fraction field ff (KU) of integral topological K-theory may have global arithmetic behavior.

Max-Planck-Institut für Mathematik, Bonn

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