The mod 2 homology of the general linear goup of a 2-adic local field

Stephen A. Mitchell^{∗}

October 2000

1 Introduction

Let F be a finite extension of ℚ_{2}, of degree d. Our first main theorem gives an explicit computation of the mod two homology Hopf algebra of the infinite general linear group GLF. The answer is formulated in terms of the well-known homology algebras of the infinite unitary group U, its classifying space BU, and the classifying space BO of the infinite orthogonal group. Let P denote the subalgebra of H_{∗}BO generated by the primitive elements; thus P is a polynomial algebra on generators of odd degree. Here and throughout and the paper, H_{∗}(−) means H_{∗}(−; 𝔽_{2}). Let FP denote the subalgebra generated by the squares of the primitive elements; thus FP is a polynomial algebra on generators in degrees congruent to 2 mod 4. (F denotes the Frobenius operator on a bicommutative Hopf algebra.) Then there is a canonical embedding FP⟶H_{∗}BU with image the subalgebra P′ generated by the primitives of H_{∗}BU. Hence we can form the tensor products

B = P ⊗FP H_{∗}BU C = H_{∗}BO ⊗FP B

Theorem 1.1 Let F be a finite extension of ℚ_{2}, of degree d. Then there are the following isomorphisms of Hopf algebras over the Steenrod algebra A:

If d is even and \sqrt{−1} ∈ F:

H_{∗}GLF ≅ (H_{∗}BU)^{2} ⊗ (H_{∗}U)^{d+2}

If d is even and \sqrt{−1} ∉ F:

H_{∗}GLF ≅ H_{∗}BO/FP ⊗ B ⊗ (H_{∗}U)^{d}

If d is odd:

H_{∗}GLF ≅ C ⊗ (H_{∗}U)^{d}

∗Supported by a grant from the National Science Foundation

The group homology H_{∗}GLF can be identified with the singular homology H_{∗}BGLF^{+}, where BGLF^{+} is Quillen’s plus construction on the classifying space of GLF; our calculation is based on an analysis of the homotopy-type of the plus construction. The division into three cases ultimately rests on the behaviour of the Hilbert symbol {−, −} : (F^{×}/(F^{×})^{2}) × (F^{×}/(F^{×})^{2})⟶μ_{2}; this symbol is symplectic if and only if \sqrt{−1} ∈ F, while \{−1, −1\} = −1 if and only if d is odd. In all cases, H_{∗}BGLF^{+} is (H_{∗}BU)^{2} ⊗ (H_{∗}U)^{d+2} up to Hopf algebra extensions. If \sqrt{−1} ∈ F, the two copies of H_{∗}BU are linked to two corresponding copies of H_{∗}U by higher Bockstein operations. These two H_{∗}BU-H_{∗}U pairs arise as copies of the homology of BGL𝔽_{q}^{+} and its delooping, for a suitable odd prime power q.

The cohomology ring structure can be read off from this theorem by dualizing the individual factors. Of course H^{∗}U is an exterior algebra, while H^{∗}BU and H^{∗}BO are polynomial on the universal Chern classes c_{k} and universal Stiefel-Whitney classes w_{k}, respectively. The result is (see Corollary 6.3):

Corollary 1.2 There are isomorphisms of algebras:

If d is even and \sqrt{−1} ∈ F:

H^{∗}GLF ≅ (H^{∗}BU)^{2} ⊗ (H^{∗}U)^{d+2}

If d is even and \sqrt{−1} ∉ F:

H^{∗}GLF ≅ 𝔽_{2}[q_{1}, q_{2}, ...] ⊗ 𝔽_{2}[c_{2}, c_{4}, ...] ⊗ H^{∗}BU ⊗ (H^{∗}U)^{d+1}

where | q_{i} |= 2i − 1.

If d is odd:

H^{∗}GLF ≅ H^{∗}BO ⊗ 𝔽_{2}[c_{1}^{2}, c_{2}^{2}, ...] ⊗ (H^{∗}U)^{d+1}

Our second main theorem and its corollaries address two related questions: (i) to what extent the stable cohomology of GL_{n}F is detected on the diagonal matrices; and (ii) the role played by the two most basic invariants of F, namely its unit group and Brauer group.

Theorem 1.3 Let D ⊂ H_{∗}BGLF^{+} denote the subalgebra generated by BGL_{1}F. Then there is a short exact sequence of A-Hopf algebras

𝔽_{2}⟶D⟶H_{∗}BGLF^{+}⟶H_{∗}BU/P′⟶𝔽_{2}

This result yields a complete, explicit answer to question (i). The ideal of cohomology classes that vanish on diagonal matrices is nothing more than the kernel of the dual map H^{∗}BGLF^{+}⟶D^{∗}. Therefore it is completely determined by the corresponding Hopf kernel, namely (H^{∗}BU/P′)^{∗} ≅ ℤ/2[c_{1}^{2}, c_{2}^{2}, ...] (see Corollary 6.5).

To explain the second question, recall that BGLF^{+} × ℤ has a ring space structure arising from tensor product of vector spaces; hence its homology algebra has a secondary product making it into a so-called Hopf ring. Now each unit defines an element of π_{1}BGLF^{+} and hence an element of H_{1}BGLF^{+}. We also have a canonical map i : Bμ_{F} ⟶BGLF^{+}, where μ_{F} is the group of 2-power roots of unity in F. Then it is not hard to show (see Corollary 6.8) that the subalgebra generated by the image of H_{∗}i, the units, and their Hopf ring products is the algebra D defined above.

To account for the missing piece H_{∗}BGLF^{+}/D, we take a slightly different point of view. The ring space structure makes BGLF^{+} a module over the homotopy ring π_{∗}KF, where KF is the K-theory spectrum of F. We can then reformulate the result just mentioned as the assertion that D is generated by the map i together with the action of K_{m}F, m ≤ 1. But after completion at 2, BGLF^{+} is also the zero-th space of the étale K-theory spectrum K^{ét}F. Moreover, π_{0}K^{ét}F fits into a canonically split short exact sequence

0⟶T Br F ⟶π_{0}K_{ét}F ⟶π_{0}KF^{∧}⟶0

where T Br F = Hom (ℤ/2^{∞}, Br F) ≅ ℤ_{2} is the Tate module associated to the Brauer group. We show (see Theorem 6.9 for the precise statement) that H_{∗}BGLF^{+} is generated by i together with the action of K_{m}^{ét}F, m ≤ 1. In particular, the missing piece described above arises from the action of T Br F on i. In cohomological terms, this means that H^{∗}BGLF^{+} is detected by “generalized tori”; see Corollary 6.10. It would be very interesting to have a more natural description of this “Brauer group action”.

The homology of GLF is accessible because of the landmark work of Voevodsky [23]. Voevodsky’s theorem and the methods of [20] imply the strong form of the 2-adic Lichtenbaum-Quillen conjectures for F, which asserts that the algebraic and étale K-theory spectra of F have equivalent 0-connected covers. (The case F = ℚ_{2} was proved independently by Rognes [19], using entirely different methods.) The author then determined the homotopy type of the 2-adic completions KF^{∧} and (BGLF^{+})^{∧} in [13]. This makes it possible to compute the homology, using the methods of [5] and [12]. When \sqrt{−1} ∈ F, most of the necessary calculations are implicit in [5]. The general case is substantially more complicated, however, because the homotopy-type is more complicated and there are many nontrivial Hopf algebra extensions to deal with.

Organization of the paper: The preliminary sections §2-5 isolate the purely homotopytheoretic part of the analysis.

In §2 we recall some well-known formulae [4] [20] for Hopf rings and Hopf modules, in a form convenient for our applications. For our homology calculations we work in the abelian category H of bicommutative Hopf algebras of finite type over a field [22].

The infinite loop spaces we consider are all module spaces over the real “Image of J” space, defined here as Ω^{∞}L̂S^{0} (L̂(−) denotes Bousfield localization with respect to mod 2 complex K-theory). In particular, the homology algebras that arise are all Hopf modules over the Hopf ring H_{∗}Ω^{∞}L̂S^{0} and hence also Hopf modules over H_{∗}BO. This module structure, discussed in §3, allows us to reduce most of our calculations to dimensions 1 and 2.

In §4 we introduce the building blocks out of which H_{∗}BGLF^{+} is constructed: H_{∗}BO, H_{∗}BU, H_{∗}U, P, P′, B, C. Hopf algebra extensions of the form (H_{∗}BU)^{a}⟶M⟶(H_{∗}U)^{b} are studied.

In §5 we compute the homology of certain excellent spaces Ω_{0}^{∞}X. These are the zero-th spaces of spectra arising as cofibres of suitable maps between wedges of copies of K̂, the periodic complex K-theory spectrum completed at 2. The main result, Theorem 5.3, shows how to read off the A-Hopf algebra H_{∗}Ω_{0}^{∞}X directly from K̂^{∗}X as a module over the Adams operations. As an amusing application, we give an instant calculation of the homology of the delooping of a complex J-space.

The main theorems can be found in §6.

We will freely use the notation and results of [13].

2 Hopf rings and Hopf modules

In this section we set up our notation for Hopf modules and review some classical formulae (see [4], [20] for more background).

Fix a base field k. In the applications k will invariably be the field ℤ/2, but for the time being we may as well proceed more generally. Recall [20] that a Hopf ring A over k is a ring object in the category of coalgebras over k. In particular, A is a bicommutative Hopf algebra with diagonal Δ : A⟶A ⊗ A and multiplication m : A ⊗ A⟶A. The product m will be indicated on elements by juxtaposition: m(x ⊗ y) = xy, while the diagonal will be written Δx = ∑ x′_{i} ⊗ x′′_{i}. The product ρ : A ⊗ A⟶A giving the Hopf ring structure will be written as ρ(x ⊗ y) = x ∘ y. (The reader who is new to Hopf rings is advised that in this context, the term “Hopf algebra” is an anachronism; a better term would be Hopf group, meaning a group object in the category of coalgebras.)

There is an analogous notion of Hopf module. Given a Hopf ring A, a left Hopf module over A is a bicommutative Hopf algebra M equipped with a coalgebra map θ : A ⊗ M⟶M, such that the following diagrams commute:

Unit law:

[figure: A ⊗ M; k ⊗ M; M; θ; η⊗1_{M}; =]

where η is the Hopf ring unit.

Associative law:

[figure: A ⊗ A ⊗ M; A ⊗ M; A ⊗ M; M; 1⊗θ; ρ⊗1; θ; θ]

Left distributive law:

[figure: A ⊗ M ⊗ M; M ⊗ M; A ⊗ M; M; ψ; 1⊗m_{M}; m_{M}; θ]

where ψ is the composite

A ⊗ M ⊗ M ⟶^{Δ_{A}⊗1} A ⊗ A ⊗ M ⊗ M ⟶^{1⊗τ⊗1} A ⊗ M ⊗ A ⊗ M ⟶^{θ⊗θ} M ⊗ M

Here τ switches the two middle factors, with the usual signs. Right distributive law:

[figure: A ⊗ A ⊗ M; M ⊗ M; A ⊗ M; M; ξ; m_{A}⊗1; m_{M}; θ]

where ξ is the composite

A ⊗ A ⊗ M ⟶^{1⊗1⊗Δ_{M}} A ⊗ A ⊗ M ⊗ M ⟶^{1⊗τ⊗1} A ⊗ M ⊗ A ⊗ M ⟶^{θ⊗θ} M ⊗ M

We also record explicitly the assumption that θ is a coalgebra map, in the Diagonal law:

[figure: A ⊗ M; A ⊗ A ⊗ M ⊗ M; M; M; Δ_{A}⊗Δ_{M}; θ; ζ; Δ_{M}]

where ζ is θ ⊗ θ composed with 1 ⊗ τ ⊗ 1. Remark: The Hopf ring unit η should not be confused with the unit i : k⟶A of the underlying Hopf algebra. The latter is really the “zero”, so that the composite

k ⊗ M ⟶^{i⊗1} A ⊗ M ⟶^{θ} M

is the zero map in the abelian category of bicommutative Hopf algebras; that is, the composite M⟶k⟶M of the counit and unit of M.

On elements we write θ(a ⊗ x) = a□x. The five laws then take the form η(1)□x = x

(a ∘ b)□x = a□(b□x)
a□(xy) = ∑(−1)^{|x||a′′_{i}|}(a′_{i}□x)(a′′_{i}□y)
(ab)□x = ∑(−1)^{|b||x′_{i}|}(a□x′_{i})(b□x′′_{i})
Δ(a□x) = ∑(−1)^{|a′′_{i}||x′_{j}|}(a′_{i}□x′_{j}) ⊗ (a′′_{i}□x′′_{j})

We will usually take M to be a connected Hopf algebra. A Hopf ring, on the other hand, can never be connected unless the additive and multiplicative units coincide; i.e., “0=1”. But there is an obvious notion of “Hopf ring without identity”, and in that context it is sensible to consider connected A.

The main example we have in mind is the following: Let E be a ring spectrum and X an E-module spectrum. Then Ω^{∞}E is a ring space and Ω^{∞}X is a module space over Ω^{∞}E. Of course the definition of module space is exactly analogous to the definition of Hopf module, substituting smash product for tensor product throughout. Applying H_{∗}(−; k) to the appropriate space diagrams then yields a Hopf ring structure on H_{∗}(Ω^{∞}E; k) and an H_{∗}(Ω^{∞}E; k) Hopf module structure on H_{∗}(Ω^{∞}X; k). Note that Ω_{0}^{∞}X is a submodule space and hence H_{∗}(Ω_{0}^{∞}X; k) is a connected sub-Hopf module of H_{∗}(Ω^{∞}X; k). Finally, H_{∗}(Ω_{0}^{∞}E; k) is a Hopf ring without identity, and H_{∗}(Ω_{0}^{∞}X; k) is a Hopf module over it. For the most part, we will use only this last structure.

Unless otherwise mentioned, the term decomposable always refers to the underlying Hopf algebra (or “Pontrjagin”) product.

Proposition 2.1 Let M be a connected Hopf module over the connected Hopf ring (without identity) A. Then

a) If either a or x is primitive, so is a□x.

b) If one of a, x is primitive and the other is decomposable, a□x = 0.

Part (a) follows easily from the Diagonal law, while (b) follows easily from the two distributive laws. This check is left to the reader, with the following caution: The diagonal on a primitive in a Hopf algebra is often written as Δa = 1 ⊗ a + a ⊗ 1. But of course the “1” in this notation is i(1), not η(1); cf. the remark above. A safer notation is i(1) = [0], η(1) = [1].

Note that one case of (a) shows that the subalgebra generated by the primitives in M is a sub-Hopf module.

From now on the base field is the field with two elements. Let H denote the abelian category of connected bicommutative Hopf algebras of finite type over ℤ/2 [22]. For each A ∈ H we let F denote the Frobenius and V the Verschiebung on A.

Proposition 2.2 Let M be a Hopf module over the Hopf ring A. Then for all a ∈ A and x ∈ M we have

a□x^{2} = ((Va)□x)^{2}

a^{2}□x = (a□Vx)^{2}

Proof: By the Distributive Law,

a□x^{2} = ∑(a′_{i}□x)(a′′_{i})

Now the coproduct on a can be written as a sum of terms of the form b ⊗ c + c ⊗ b if | a | is odd, while if | a |= 2n, it is a sum of such terms together with terms of the form b_{i} ⊗ b_{i}, | b_{i} |= n. So

a□x^{2} = ∑(b_{i}□x)^{2} = (∑ b_{i}□x)^{2} = ((∑ b_{i})□x)^{2}

But ∑ b_{i} is precisely Va. The other case is similar.

In particular, the image of each k-th iterate F^{k} of the Frobenius is automatically a sub-Hopf module.

All of the examples we are interested in will also be equipped with an action of the Steenrod algebra A. There are evident categories HA, H_{rng}A, HA_{mod}A of Hopf algebras, Hopf rings, and A-Hopf modules over A, where in the last case A ∈ H_{rng}A. In all cases the A-action is understood to satisfy the Cartan formula with respect to the appropriate product operations. In the applications the Cartan formula will be obvious, since the products arise from pairings of spaces X × Y ⟶Z or X ∧ Y ⟶Z.

It is clear that the operations direct sum, cokernel, etc. in H extend automatically to HA and HA_{mod}A. For example, given a diagram M_{1} ⟵ M⟶M_{2} in HA_{mod}A, the pushout in H is an object in HA_{mod}A, and is also the pushout in the latter category.

We conclude by recalling that if E is a ring spectrum, then the [1]-component Ω_{1}^{∞}E inherits an H-space structure from the ring multiplication. Translation by ±[1] then yields another H-space structure on Ω_{0}^{∞}E, often called the multiplicative H-space structure. Thus there is yet another homology product on Ω_{0}^{∞}E, which we will write as a ∗ b. It is related to the ∘-product by the formula (ignoring signs, since we are now in characteristic 2)

a ∗ b = ∑(a′_{i} ∘ b′_{j})a′′_{i}b′′_{j}

This formula also goes back to [4]; for further review in the present context, see [12]. We mainly need the special case in which both a and b are primitive; in that case we have (bearing in mind that [0] ∘ [0] = [0])

a ∗ b = a ∘ b + ab

and in particular a ∗ a = a ∘ a + a^{2}.

Here is the example that will be used frequently in the sequel: Let E = KR be the K-theory spectrum of a commutative ring R. The natural map j_{0} : BGL_{1}R⟶BGLR^{+} = Ω_{0}^{∞}KR has a left inverse given by the determinant, so H_{∗}BGL_{1}R ⊂ H_{∗}BGLR^{+}.

Proposition 2.3 For all a ∈ H̃_{∗}BGL_{1}R, a ∗ a = 0. Hence if | a |= 1, a ∘ a = a^{2}.

Proof: The ∗-product corresponds under translation to the ∘-product on BGLR^{+} × {1}, by definition, and the translated map j_{1} : BGL_{1}R⟶BGLR^{+} × {1} is an H-map with respect to the multiplicative H-space structure on the target. So it is enough to show that if A is any abelian group, and ∗ denotes the homology product arising from the H-space structure on BA, then a ∗ a = 0 for all a ∈ H̃_{∗}BA. This is a well-known and easy calculation (first reducing to the case A finitely-generated, and then to the case A cyclic).

3 Module structures over real J-spaces

Let L̂S^{0} denote the K̂-localization of the sphere spectrum, and let JO(q) denote the fibre of ψ^{q} − 1 : KO⟶KO. Spectra of the form JO(q) were exhaustively analyzed by Fiedorowicz and Priddy in [6], while L̂S^{0} was studied by Ravenel in [17]. We briefly review the relevant results here, introduce an H_{∗}BO-Hopf module structure on certain K-local spaces, and discuss the applications to algebraic K-theory.

Proposition 3.1 If q = ±3 mod 8, then L̂S^{0} ≅ JO(q)^{∧}.

The proof is easy, but note that there are two choices of equivalence, with no reason to prefer one over the other. The point is that there is a short exact sequence

0⟶π_{1}KO^{∧}⟶π_{0}JO(q)^{∧}⟶π_{0}KO^{∧}⟶0

and hence there are exactly two distinct lifts of the unit map S^{0}⟶KO^{∧} to JO(q)^{∧}. Either one of these maps induces an isomorphism on K̂^{∗}, provided q = ±3 mod 8. This ambiguity is only a minor annoyance, however, since L̂S^{0} is the real object of interest. The spectrum JO(q)^{∧} plays an auxiliary role; in particular, we can immediately read off the homotopy groups of L̂S^{0} from the fibre sequence defining JO(q).

Some remarks are in order concerning π_{0} and π_{1}. First of all, L̂S^{0} is a ring spectrum, and the short exact sequence

0⟶ℤ/2⟶π_{0}L̂S^{0}⟶ℤ_{2}⟶0

arising from the canonical map to KO^{∧} is canonically split by the ring spectrum unit. Furthermore, if α is the element of order two, then α^{2} = 0. This can be seen in several ways; the cheapest is to observe that α cannot be idempotent, so that α^{2} = 0 by default. Note that multiplication by the unit 1 + α defines an additive involution of L̂S^{0}, interchanging the two equivalences L̂S^{0} ≅ JO(3)^{∧}.

Second, there is a short exact sequence

0⟶ℤ/2⟶π_{1}L̂S^{0}⟶ℤ/2⟶0

again arising from S^{0}⟶KO. And again, it is canonically split by the unit map. Thus π_{1} has a canonical basis ϵ, η, where ϵ generates the kernel of the above sequence and η is the nontrivial map S^{1}⟶S^{0}⟶L̂S^{0}. One can show that αϵ = ϵ, and hence multiplication by 1 + α interchanges η and η + ϵ.

We next discuss maps of ℝP^{∞} into L̂S^{0}, and more generally from BV to L̂S^{0} where V is an elementary abelian 2-group. There are several interesting ways to compute [BV, L̂S^{0}]. One is to note that the transfer t : ℝP^{∞}⟶S^{0} induces an equivalence L̂ℝP^{∞} ≅ L̂S^{0} ([17], 9.1); then one can exploit the π_{0} calculation already done above. Another is to recall that by the Atiyah-Segal theorem ([1]; see the penultimate paragraph) there are isomorphisms KO^{0}BV_{+} ≅ R_{ℝ}V and

KO^{−1}BV ≅ Hom (π_{1}BV, π_{1}SO) = Hom (V, ℤ/2)

In any case, it is easy to prove:

Proposition 3.2 Let V be an elementary abelian 2-group of order e. Then there is a canonically split short exact sequence

0⟶Hom (π_{1}BV, π_{1}L̂S^{0}/(η))⟶[BV, L̂S^{0}]⟶[BV, KO]⟶0

where the first term is isomorphic to (ℤ/2)^{e} and the last to ℤ_{2}^{e−1}.

The application we need is the following. Regard the suspension spectrum of ℝP_{+}^{∞} as a ring spectrum in the usual way, with multiplication arising from the H-space structure on ℝP^{∞}.

Proposition 3.3 Let f : ℝP_{+}^{∞}⟶L̂S^{0} denote the composite

ℝP_{+}^{∞} ⟶^{t−1} S^{0} ⟶^{i} L̂S^{0}

Then f is a map of ring spectra.

Proof: It is clear that f preserves the unit, since t has degree 2 on the bottom cell. The commutivity of the diagram

[figure: ℝP_{+}^{∞} ∧ ℝP_{+}^{∞}; L̂S^{0} ∧ L̂S^{0}; ℝP_{+}^{∞}; L̂S^{0}; f∧f; f]

is easily checked using Proposition 3.2.

Remark: The map t − 1 itself is not a map of ring spectra. In fact, the Segal conjecture [3] and a simple calculation in the Burnside ring of ℤ/2 × ℤ/2 show that the only ring map ℝP_{+}^{∞}⟶S^{0} is the augmentation.

Now observe that there is an extension of A-Hopf algebras

H_{∗}SO⟶H_{∗}Ω_{0}^{∞}L̂S^{0}⟶H_{∗}BO

in which the second map is the map of Hopf rings arising from the unit map S^{0}⟶KO. (To see this, replace L̂S^{0} by JO(3)^{∧} and observe that the generating complex ℝP^{∞}⟶BO for H_{∗}BO lifts to Ω_{0}^{∞}JO(3).)

Corollary 3.4 H_{∗}Ω_{0}^{∞}L̂S^{0}⟶H_{∗}BO is canonically split as Hopf rings over the Steenrod algebra.

Proof: Consider the commutative diagram

[figure: S(H̃_{∗}ℝP^{∞}); H_{∗}QℝP^{∞}; H_{∗}BO; H_{∗}Ω_{0}^{∞}L̂S^{0}; ≅; f]

in which the unlabelled maps are the obvious ones. Then all maps in the diagram are maps of Hopf rings over the Steenrod algebra. Chasing around the diagram yields the desired splitting.

Now if X is any K̂-local spectrum, then X is a L̂S^{0}-module spectrum, and any map of such spectra is a map of module spectra. Thus the functor X ↦ H_{∗}Ω_{0}^{∞}X takes values in Hopf modules over H_{∗}Ω_{0}^{∞}L̂S^{0}. In view of Corollary 3.4, this Hopf module structure restricts to a Hopf module structure over H_{∗}BO. This is convenient because we do not want to worry about H_{∗}SO ⊂ H_{∗}Ω_{0}^{∞}L̂S^{0}. More generally, we have the following:

Proposition 3.5 Consider the full subcategory of spectra X such that X⟶L̂X is an equivalence on 0-connected covers. Then on this subcategory, the functor X ↦ H_{∗}Ω_{0}^{∞}X takes values in H − H_{∗}BO_{mod}A.

The H_{∗}BO-Hopf modules which can occur are subject to a further condition on products b_{1}□x, x ∈ H_{1}Ω_{0}^{∞}X. Let ℏ denote the mod 2 Hurewicz map, and let x̃ ∈ π_{1}Ω_{0}^{∞}X be any class with ℏx̃ = x.

Proposition 3.6 For all x ∈ H_{1}Ω_{0}^{∞}X,

b_{1}□x = ℏ(ηx̃) = x^{2}

Proof: The first equality is clear; we need only prove the second. This is a general fact about integral homology of one-fold loop spaces. Let W be a connected loop space; say W = ΩY, and let x̃ : S^{1}⟶W. Then we can form the canonical extension g : ΩS^{2}⟶W. Now H_{∗}(ΩS^{2}; ℤ) = ℤ[b], and b^{2} is spherical; in fact if η′ : S^{2}⟶ΩS^{2} is adjoint to the Hopf map then b^{2} = h(η′), where h is the integral Hurewicz map. It follows that h(x̃)^{2} = h(gη′).

Now take W = Ω_{0}^{∞}X. The element gη′ ∈ π_{2}W corresponds the composite S^{2} ⟶^{η} S^{1} ⟶^{x̃} X, or equivalently S^{1} ∧ S^{1} ⟶^{η∧x̃} S^{0} ∧ X⟶X. Hence ℏ(ηx̃) = x^{2}.

An H_{∗}BO-Hopf module M will be called geometric if b_{1}□x = x^{2} for all x ∈ M_{1}.

We conclude with some remarks on these module structures as they apply to algebraic Ktheory spaces. By work of Rognes-Weibel [21] and Bökstedt [2], the strong form of the 2-adic Lichtenbaum-Quillen conjectures holds for ℤ[1/2]. (Recent work of Østvær [15] establishes the strong form of the 2-adic Lichtenbaum-Quillen conjectures for all number rings, again relying on [21] but not on [2].) It follows that the natural map Kℤ[1/2]^{∧}⟶L̂Kℤ[1/2] is an equivalence on (-1)-connected covers P_{0}(−). This in turn yields a unique factorization f in the diagram

[figure: S^{0}; Kℤ[1/2]^{∧}; P_{0}L̂S^{0}; f]

Furthermore, f is a map of ring spectra (see [9] for a general discussion). It follows that any module spectrum Y over Kℤ[1/2]^{∧}—in particular, any suspension of the completed Ktheory spectrum of a ring or scheme—is a module spectrum over P_{0}L̂S^{0}. For such spectra Y we therefore have that H_{∗}Ω^{∞}Σ^{k}Y, k ∈ ℤ, is naturally a Hopf module over the Hopf ring H_{∗}Ω^{∞}L̂S^{0}. Usually we will only consider H_{∗}Ω_{0}^{∞}Σ^{k}Y as a Hopf module over H_{∗}Ω_{0}^{∞}L̂S^{0}, bearing in mind that the latter is a Hopf ring without identity.

It is worth noting that H_{∗}SO plays no role at all in algebraic K-theory; Corollary 3.4 can be replaced by:

Proposition 3.7 Let Y be any module spectrum over Kℤ[1/2]^{∧}. Then the natural action of the Hopf ring H_{∗}Ω_{0}^{∞}L̂S^{0} on H_{∗}Ω_{0}^{∞}Y factors uniquely through an action of the Hopf ring H_{∗}BO.

Proof: It is enough to show that the composite map

H_{∗}SO⟶H_{∗}Ω_{0}^{∞}L̂S^{0}⟶H_{∗}BGLℤ[1/2]^{+}

is the trivial map. Since H_{∗}SO is generated by a map g : ℝP^{∞}⟶SO, it is enough to show that the composite

ℝP^{∞} ⟶^{g} SO⟶Ω_{0}^{∞}L̂S^{0}⟶(BGLℤ[1/2]^{+})^{∧}

is null. Now g represents the nonzero element of KO^{−1}ℝP^{∞} = ℤ/2. On the other hand, by the Lichtenbaum-Quillen conjectures together with Theorem 8.1 of [10] we have an isomorphism

[BG_{+}, BGLℤ[1/2]^{+} × ℤ] ≅ R̂_{ℚ}G

for any finite 2-group G, where R_{ℚ}G is the rational representation ring and R̂ denotes completion with respect to the augmentation ideal. In particular, [BG, BGLℤ[1/2]^{+}] is torsion-free. Taking G = ℤ/2, the proposition follows.

4 Some basic examples

In this section we introduce and study the basic Hopf algebras that serve as building blocks for H_{∗}BGLF^{+}. When writing down short exact sequences in H we omit the end terms 𝔽_{2}; thus by a short exact sequence or extension A⟶B⟶C we mean that

𝔽_{2}⟶A⟶B⟶C⟶𝔽_{2}

is exact in H.

We begin with the most basic examples of all:

The homology of BO and BU. As an A-Hopf algebra, H_{∗}BO is completely described by the canonical isomorphism

S(H̃_{∗}ℝP^{∞}) = 𝔽_{2}[b_{1}, b_{2}, ...] ⟶^{≅} H_{∗}BO

As Hopf algebra H_{∗}BO is self-dual, with H^{∗}BO = ℤ/2[w_{1}, w_{2}, ...]. The primitives P rim H_{∗}BO have a basis consisting of the Newton polynomials s_{k} in the b_{i}’s, k ≥ 1. Since BO = Ω_{0}^{∞}KO and KO is a ring spectrum, H_{∗}BO is a Hopf ring without identity. It is easy to write down an explicit formula for the ∘-product, but all we need is the useful special case

b_{n} ∘ b_{1} = s_{n+1}

In particular, regarding H_{∗}BO as Hopf module over itself (or over H_{∗}(BO × ℤ), if we wish to include the case [1] ∘ b_{1} = b_{1} = s_{1}), the primitives are generated by b_{1}. Furthermore, there is a natural map ΣℝP_{+}^{∞}⟶BO inducing an isomorphism onto the primitives in homology. Let P denote the subalgebra generated by the primitives; thus P is a polynomial algebra on generators of odd degree. We then have a complete description of P as an object of HH_{∗}BO_{mod}A: It is the symmetric algebra on H̃_{∗}ΣℝP_{+}^{∞} modulo the relations s_{2k} = s_{k}^{2}.

The Hopf algebra H_{∗}BU is just the “double” of H_{∗}BO. Thus

H_{∗}BU = S(H̃_{∗}ℂP^{∞}) = 𝔽_{2}[b′_{1}, b′_{2}, ...]

where | b′_{i} |= 2i. Furthermore, the primitives are the Newton polynomials s′_{k} in the b′_{i}, b′_{n} ∘ b′_{1} = s′_{n+1}, etc.

The complexification map H_{∗}BO⟶H_{∗}BU is a map of Hopf rings taking b_{2k+1} to zero and b_{2k} to b′_{k}; hence there is an extension of Hopf algebras

P⟶H_{∗}BO⟶H_{∗}BU

The realification map H_{∗}BU⟶H_{∗}BO maps b′_{k} to b_{k}^{2}; in other words, there is an isomorphism H_{∗}BU ≅ FH_{∗}BO. It is not a map of Hopf rings, but rather a map of Hopf modules over H_{∗}BO.

Proposition 4.1

EndHH_{∗}BO_{mod}AH_{∗}BO = EndHAH_{∗}BO = ℤ_{2}

and hence

AutHH_{∗}BO_{mod}A(H_{∗}BO)^{n} = AutHA(H_{∗}BO)^{n} = GL_{n}ℤ_{2}

and similarly for BU.

Proof: It is enough to show that the natural map i : ℤ_{2}⟶EndHAH_{∗}BO is an isomorphism. This is well-known, but for lack of a reference we sketch a proof: By a theorem of Husemoller [7], H_{∗}BO splits in H as an infinite tensor product A_{1} ⊗ A_{2} ⊗ ..., where A_{1}, for example, is a polynomial algebra on generators of degree 2^{n}, n ≥ 1. Furthermore

HomH(A_{i}, A_{j}) = \{ \begin{matrix}ℤ_{2} & if i = j \\ 0 & if i ≠ j \end{matrix}

Now A_{1} is not an A-submodule, but nevertheless we have the natural map

r : EndHAH_{∗}BO⟶EndHA_{1}

with ri the identity. So it is enough to show that r is injective. But if S = {x_{1}, x_{2}, ..., x_{n}, ...} is any set of indecomposable elements in H_{∗}BO with | x_{n} |= 2^{n}, it is easy to see that S (or indeed any infinite subset of S) generates H_{∗}BO as A-algebra. Hence an A-algebra endomorphism of H_{∗}BO is determined by its restriction to A_{1}, completing the proof.

The homology of the infinite unitary group. We have

H_{∗}U = \overline{S}(H̃_{∗}ΣℂP_{+}^{∞}) = 𝔽_{2}⟨x_{1}, x_{2}, ...⟩

where | x_{i} |= 2i − 1. \overline{S}(V) denotes the strict symmetric algebra on a graded vector space V, in which the ideal generated by the squares of odd-dimensional elements has been factored out. Since ΣK is a module spectrum over K, H_{∗}U is a Hopf module over H_{∗}BU, and hence also over H_{∗}BO by pullback along complexification. We then have

b_{2n}□x_{1} = b′_{n}□x_{1} = x_{2n+1}

(Compare [12], example 1.3.)

Note also that there is an extension of A-Hopf algebras FP⟶P⟶H_{∗}U.

Proposition 4.2

EndHH_{∗}BO_{mod}AH_{∗}U = End_{HH_{∗}BO_{mod}}H_{∗}U = 𝔽_{2}

and hence

AutHH_{∗}BO_{mod}A(H_{∗}U)^{n} = Aut_{HH_{∗}BO_{mod}}(H_{∗}U)^{n} = GL_{n}𝔽_{2}

The proof is trivial, since the primitive generators are all linked to x_{1} by the H_{∗}BOmodule action.

The homology of complex J-spaces. Consider the quotient H_{∗}BO/FP. Note that there is a commutative diagram of A-Hopf algebras, with rows and columns short exact in H:

[figure: FP; FP; 𝔽_{2}; P; H_{∗}BO; H_{∗}BU; H_{∗}U; H_{∗}BO/FP; H_{∗}BU; =; =]

The lower row splits as algebras, but not as coalgebras. In fact the dual (H_{∗}BO/FP)^{∗} is a polynomial algebra on Chern classes c_{2k} of degree 4k coming from BU, and generators q_{k} of degree 2k − 1. The Quillen classes q_{k} are related to the natural Chern classes by the formula

q_{n}^{2} = c_{2n−1} + c_{1}c_{2n−2} + ... + c_{n−1}c_{n}

Let J(q) denote the fibre of ψ^{q} − 1 : K⟶K, where q is odd and q ≠ ±1. Let μ_{2^{a}} denote the group of 2^{a} − th roots of unity in ℂ, where a = ν_{2}(q − 1). Note there is a canonical map (Bμ_{2^{a}})_{+}⟶J(q). Recall that \overline{S}(V) denotes the strict symmetric algebra on V.

Theorem 4.3 [16]

H_{∗}Ω_{0}^{∞}J(q) = \overline{S}(H̃_{∗}Bμ_{2^{a}})

as A-Hopf algebras. Hence if q = 3 mod 4, H_{∗}Ω_{0}^{∞}J(q) ≅ H_{∗}BO/FP, and H^{∗}Ω_{0}^{∞}J(q) = 𝔽_{2}[q_{1}, q_{2}, ...] ⊗ 𝔽_{2}[c_{2}, c_{4}, ...]. If q = 1 mod 4, H_{∗}Ω_{0}^{∞}J(q) is self-dual as Hopf algebra.

Note that if we complete at 2, an analogous J(α) is defined for any 2-adic unit α. Moreover, J(α) is an excellent spectrum in the sense of [5], 4.10, with

K̂^{p}J(α) ≅ \{ \begin{matrix}Λ′/(ψ^{α} − 1) & if p = 0 \\ 0 & if p = −1 \end{matrix}

From this it is easy to see that the homotopy type of J(α) depends only on the topological equivalence class of α; that is, the closed subgroup generated by α. In particular, the theorem above computes H_{∗}Ω_{0}^{∞}J(α) for all such α.

The Hopf algebra B. Define a bicommutative Hopf algebra B by the pushout diagram

[figure: FP; H_{∗}BU; P; B]

where the top map is the canonical isomorphism onto P′, the subalgebra generated by the primitives. Then B is in fact an H_{∗}BO-Hopf module over the Steenrod algebra, and the diagram is a pushout in HH_{∗}BO_{mod}A. (Of course the pushout is just the tensor product, but it is convenient to have the maps displayed.) Note that the diagram embeds in a larger diagram with short exact rows and columns:

[figure: FP; H_{∗}BU; H_{∗}BU/P′; P; B; H_{∗}BU/P′; H_{∗}U; H_{∗}U; 𝔽_{2}; =; =]

In particular, the middle column exhibits B as an extension that splits as coalgebras—since H^{∗}BU is polynomial—but not as algebras, as A-modules, or as H_{∗}BO-Hopf modules. Note that in effect, B is obtained by adjoining primitive square roots to the indecomposable primitives of H_{∗}BU. The middle column extension is in fact the only nontrivial such extension:

Lemma 4.4 Let H_{∗}BU⟶M ⟶^{p} H_{∗}U be an extension in HH_{∗}BO_{mod}A, and suppose M is geometric (Proposition 3.6). Then either the extension splits, or M ≅ B. The first alternative occurs if and only if the generator x ∈ M_{1} satisfies x^{2} = 0.

Proof: Let V denote the vector space spanned by the primitive elements b_{2n}□x. Then p induces an isomorphism of V onto P rim H_{∗}U.

Now suppose that x^{2} = 0. Then b_{1}□x = 0 and hence (b_{2n}□x)^{2} = b_{4n}□x^{2} = 0. Here we have used Proposition 3.6, Proposition 2.2, and the easy calculation Vb_{2n} = b_{n} in H_{∗}BO. Thus V generates a sub-Hopf algebra mapping isomorphically to H_{∗}U, yielding a splitting as Hopf algebras. Moreover, b_{2n+1}□x = b_{2n}□(b_{1}□x) = 0, so we have a splitting as H_{∗}BO-Hopf modules. It also follows that V = H_{∗}ΣℂP_{+}^{∞} as A-modules, so we have a splitting of H_{∗}BO-Hopf modules over the Steenrod algebra.

Conversely, suppose that x^{2} ≠ 0. Then x^{2} = b′_{1} ∈ H_{2}BU by default, and hence

(b_{2n}□x)^{2} = b_{4n}□x^{2} = b′_{2n} ∘ b′_{1} = s′_{2n+1}

So there is a natural map in B⟶M in HH_{∗}BO_{mod}A. It is surjective by construction, and hence an isomorphism by dimension counting.

More generally, consider an object M of HH_{∗}BO_{mod}A fitting into an extension of the form

(H_{∗}BU)^{a}⟶M⟶(H_{∗}U)^{b}

Define the splitting defect δ = δ_{M} to be the rank of the squaring map M_{1}⟶M_{2}.

Proposition 4.5 M ≅ B^{δ} ⊗(H_{∗}U)^{b−δ} ⊗(H_{∗}BU)^{a−δ} as H_{∗}BO-Hopf modules over the Steenrod algebra; i.e., in HH_{∗}BO_{mod}A.

Proof: Using Proposition 4.1 and Proposition 4.2, it is clear that up to automorphism of the two end terms we can assume that the matrix of the squaring map is

( \begin{matrix}I_{δ} & 0 \\ 0 & 0 \end{matrix} )

The proposition now follows from the lemma.

The Hopf algebra C: Define a bicommutative Hopf algebra C by the pushout diagram

[figure: FP; B; H_{∗}BO; C]

where the top map is the canonical map arising from the definition of B. As usual, the diagram is actually a pushout in HH_{∗}BO_{mod}A. Note that this diagram embeds in a larger diagram

[figure: FP; B; E; H_{∗}BO; C; E; H_{∗}BO/FP; H_{∗}BO/FP; 𝔽_{2}; =; =]

with short exact rows and columns, where E = (H_{∗}BU/P′) ⊗ H_{∗}U. As an algebra, C is generated by the b_{i}, the b′_{2i}, and primitives d_{i} of dimension i, i odd, subject to the relations d_{i}^{2} = s_{i}^{2}. Here the s_{i} are, as always, the Newton polynomials in the b_{i}. Thus the middle column above splits as algebras (since (d_{i} + s_{i})^{2} = 0), but not as coalgebras.

To describe the dual C^{∗} as algebra, we consider the middle row of the diagram. This row dualizes to an extension

H^{∗}U ⊗ (H_{∗}BU/P′)^{∗}⟶C^{∗}⟶H^{∗}BO

where (H_{∗}BU/P′)^{∗} = FH^{∗}BU is polynomial on generators of dimension 4k, and the extension splits as algebras. Thus

C^{∗} ≅ H^{∗}BO ⊗ 𝔽_{2}[c_{1}^{2}, c_{2}^{2}, ...] ⊗ H^{∗}U

5 Homology of certain excellent spaces

Let X be an excellent spectrum of odd type [5]. This means that X is K̂-local, K̂^{0}X = 0, and K̂^{−1}X is a finitely-generated Λ′-module of projective dimension one. Choose a minimal resolution

0⟶Λ′^{a} ⟶^{C^{t}} Λ′^{b}⟶K̂^{−1}X

Thus C is an a × b matrix with entries in the maximal ideal M′ = (2, T, 1 − σ) of Λ′, with transpose C^{t}, and there is a fibre sequence

⋁^{b} Σ^{−2}K̂ ⟶^{Σ^{−2}C} ⋁^{a} Σ^{−2}K̂⟶X⟶ ⋁^{b} Σ^{−1}K̂ ⟶^{Σ^{−1}C} ⋁^{a} Σ^{−1}K̂

This fibre sequence can be rewritten in the form

⋁^{b} K̂ ⟶^{C(1)} ⋁^{a} K̂⟶X⟶ ⋁^{b} ΣK̂⟶ ⋁^{a} ΣK̂

where C(1) is the entrywise Tate twist of C (in terms of Adams operations, this amounts to replacing ψ^{k} by kψ^{k}). If we had started with K̂^{1}X instead of K̂^{−1}X, no twist would be needed; but as K̂^{−1}X occurs naturally in practice, it is best to proceed as indicated. Note that since twisting preserves the maximal ideal, C(1) is again a matrix with coefficients in M′.

For the moment, we are only interested in such spectra X with π_{0}X torsion-free.

Proposition 5.1 The following conditions are equivalent:

a) π_{0}X is torsion-free

b) π_{0}C(1) is zero

c) Each matrix entry C(1)_{ij} = f_{ij} + g_{ij}σ has f_{ij}(0) + g_{ij}(0) = 0

d) H̃_{∗}Ω_{0}^{∞}C(1) = 0

Proof: Since π_{0}K̂ is ℤ_{2}(0), the trivial Γ′-module, the equivalence of (b) and (c) is immediate. Clearly (b) implies (a). Conversely, suppose some entry f_{ij}(0) + g_{ij}(0) is nonzero. Then this element must be divisible by two, since otherwise C(1)_{ij} is a unit, contrary to our assumption that it lies in the maximal ideal. Hence π_{0}X has nontrivial torsion.

Finally, note that Γ′ also acts trivially on H_{∗}BU, and therefore the action of Λ′ on H_{∗}BU factors as Λ′⟶ℤ_{2} ⊂ End H_{∗}BU. This proves the equivalence of (c) and (d).

Assume now that π_{0}X is torsion-free. Then we have a fibre sequence

(BU^{∧} × ℤ_{2})^{b}⟶(BU^{∧})^{a}⟶Ω_{0}^{∞}X⟶(U^{∧})^{b}

The Rothenberg-Steenrod spectral sequence of the first three terms collapses (cf. [5], proof of Lemma 10.9), yielding

E^{∞}H_{∗}Ω_{0}^{∞}X ≅ (H_{∗}BU)^{a} ⊗ (H_{∗}U)^{b}

This in turn forces the collapse of the Serre spectral sequence of the second three terms, and we have:

Lemma 5.2 There is an extension in HH_{∗}BO_{mod}A

(H_{∗}BU)^{a}⟶H_{∗}Ω_{0}^{∞}X⟶(H_{∗}U)^{b}

To determine the extension, we have only to compute the splitting defect δ (Proposition 4.5). In some cases δ is easy to compute from Propositon 3.6, but it will be convenient to have a formula in terms of K̂^{−1}X. Let M = (2, T) denote the maximal ideal of Λ. Note that for any finitely-generated Λ′-module N, N/MN is a finitely-generated ℤ/2[σ]-module, and therefore is the direct sum of a free ℤ/2[σ]-module and a trivial ℤ/2[σ]-module.

Theorem 5.3 Let X be an excellent spectrum of odd type, with π_{0}X torsion-free. Let r denote the dimension of a maximal trivial summand in the ℤ/2[σ]-module K̂^{−1}X/MK̂^{−1}X. Then r = δ and

H_{∗}Ω_{0}^{∞}X ≅ B^{r} ⊗ (H_{∗}U)^{b−r} ⊗ (H_{∗}BU)^{a−r}

as H_{∗}BO-Hopf modules over the Steenrod algebra.

Proof: We will show that δ = r. Reducing the matrix C^{t} : Λ′^{a}⟶Λ′^{b} mod M, we obtain a b × a-matrix \overline{C}^{t} over ℤ/2[σ]. Since C has entries in M′ by assumption, every entry of \overline{C}^{t} is either zero or 1 + σ. It follows easily that after a change of basis in (Λ′/MΛ′)^{a} and (Λ′/MΛ′)^{b}, \overline{C}^{t} has the form

( \begin{matrix}(1 + σ)I_{r} & 0 \\ 0 & 0 \end{matrix} )

where I_{r} is the r×r-identity matrix. Now since Λ′ is local, GL_{n}Λ′⟶GL_{n}(Λ′/J) is surjective for any ideal J and all n. Hence the change of basis lifts to Λ′^{a}, Λ′^{b}, and we can assume that the original matrix C^{t} has mod M reduction as indicated above.

Now consider the corresponding resolution

⋁^{b} K̂⟶ ⋁^{a} K̂⟶X ⟶^{j} ⋁^{b} ΣK̂ ⟶^{C} ⋁^{a} ΣK̂

and let α_{1}, ..., α_{b} denote the corresponding basis for π_{1}X. Write π_{1}X = V′ ⊕ V′′, where V′ = ⟨α_{1}, ..., α_{r}⟩ and V′′ = ⟨α_{r+1}, ..., α_{b}⟩. Then we will show that for α ∈ π_{1}X, ηα = 0 mod 2 if and only if α ∈ V′′. This will complete the proof.

Let α = ∑ c_{i}α_{i}, and suppose ηα = 0 mod 2. This means that the mod 2 reduction ᾱ : S^{1}⟶X ∧ M, where M is the mod 2 Moore spectrum, extends over the mapping cone C_{η}:

[figure: S^{2}; S^{1}; X ∧ M; ΣC_{η}; η; ᾱ; β]

Now (j ∧ M)β = (λ_{1}, ..., λ_{b}) with λ_{i} ∈ (K̂ ∧ M)^{0}C_{η} = Λ′/M = ℤ/2[σ]. Moreover C ∧ M acts on ((K̂ ∧ M)^{0}C_{η})^{b} as \overline{C}. Hence (1 + σ)λ_{i} = 0 for i ≤ r. But if c_{i} is nonzero mod 2, λ_{i} is nontrivial on the bottom cell of C_{η}; equivalently, λ_{i} is a generator of (K̂ ∧ M)^{0}C_{η} as ℤ/2[σ]-module. This forces α_{i} = 0 for i ≤ r, or α ∈ V′′.

Conversely, suppose α ∈ V′′. Let f : ΣC_{η}⟶ ⋁^{b} ΣK ∧ M have components (c_{1}λ, ..., c_{b}λ) where λ is a fixed generator of (K̂ ∧ M)^{0}C_{η} as ℤ/2[σ]-module. Then since c_{i} = 0 mod 2 for i ≤ r, there is a lift

[figure: X ∧ M; ΣC_{η}; ⋁^{b} ΣK ∧ M; f]

that provides the desired nullhomotopy of ηα mod 2.

To illustrate, and because we need this computation later, we show how Proposition 5.3 applies to the delooping of a complex J-space. Fix q as in Proposition 4.3 (where more generally q could be replaced by any 2-adic unit not equal to ±1), and consider Ω_{0}^{∞}ΣJ(q). Thus Ω_{0}Ω_{0}^{∞}ΣJ(q) = Ω_{0}^{∞}J(q). If q is an odd prime power, then Ω_{0}^{∞}ΣJ(q) can also be identified with Quillen’s Q-construction for the field 𝔽_{q}. Let j : Σ(Bμ_{2^{a}})_{+}⟶Ω_{0}^{∞}ΣJ(q) denote the adjoint of the canonical map i : B(μ_{2^{a}})_{+}⟶Ω^{∞}J(q).

Theorem 5.4 a) If q = 1 mod 4, there is a canonical isomorphism

H_{∗}Ω_{0}^{∞}ΣJ(q) ≅ H_{∗}BU ⊗ H_{∗}U

in HH_{∗}BO_{mod}A, and H^{∗} ≅ H_{∗} as Hopf algebras.

b) If q = 3 mod 4, H_{∗}Ω_{0}^{∞}ΣJ(q) ≅ B in HH_{∗}BO_{mod}A.

c) In all cases, H_{∗}j is an isomorphism H_{∗}ΣBC_{+} ≅ P rim H_{∗}Ω_{0}^{∞}ΣJ(q).

Proof: ΣJ(q)^{∧} is an excellent spectrum of odd type, with K̂^{−1}ΣJ(q) ≅ Λ′/(ψ^{q} − 1) and π_{0}ΣJ(q)^{∧} ≅ ℤ_{2}; thus the theory above applies. Now

Λ′/(ψ^{q} − 1) = \{ \begin{matrix}Λ′/((1 + T)^{2^{k}} − 1) & if q = 1 mod 4 \\ Λ′/(σ(1 + T)^{2^{k}} − 1) & if q = 3 mod 4 \end{matrix}

Reducing mod M = (2, T) we have

K̂^{−1}ΣJ(q)/M = \{ \begin{matrix}ℤ/2[σ] & if q = 1 mod 4 \\ ℤ/2 & if q = 3 mod 4 \end{matrix}

Hence (a) and (b) follow from Theorem 5.3. Part (c) follows from Theorem 4.3 by a Serre spectral sequence argument; see [8].

6 The homology of BGLF as A-Hopf algebra

In this section we prove the main theorems. We begin with a brief review of the Hilbert symbol, here denoted {a, b}, which plays a key role. In particular, the parity distinction in Theorem 6.2 stems from the fact that {−1, −1} = −1 if and only if d is odd.

6.1 The Hilbert symbol

Recall that there is a nonsingular bilinear map, or symbol,

{−, −} : (F^{×}/(F^{×})^{2}) × (F^{×}/(F^{×})^{2})⟶μ_{2}

that can be defined in at least five equivalent ways (all of which have some relevance for this paper). Where necessary, identify μ_{2} with ℤ/2 below.

The Hilbert symbol: Let L be the maximal abelian extension of F with exponent 2. Local class field theory provides a canonical isomorphism ψ : F^{×}/(F^{×})^{2}⟶G(L/F). Kummer theory, on the other hand, tells us that L is obtained by adjoining square roots to all the elements of (F^{×}/(F^{×})^{2}). We then set \{a, b\} = \frac{ψ(a)(\sqrt{b})}{\sqrt{b}}. See [14] for an introduction to the Hilbert symbol.

The power norm residue symbol: Let H(a, b) denote the quaternion algebra of type a, b. Identify μ_{2} with the subgroup of order two in the Brauer group Br F = ℚ/ℤ. Then set {a, b} = H(a, b).

Cup product: We have H^{2}(F; ℤ/2) = ℤ/2. Then take {−, −} to be the cup product pairing

(F^{×}/(F^{×})^{2}) × (F^{×}/(F^{×})^{2}) = H^{1}(F; ℤ/2) × H^{1}(F; ℤ/2)⟶H^{2}(F; ℤ/2)

K-theory product: We have K_{2}F/2 = ℤ/2. Take {−, −} to be the ring spectrum pairing on homotopy groups

(F^{×}/(F^{×})^{2}) × (F^{×}/(F^{×})^{2}) = (K_{1}F/2) × (K_{1}F/2)⟶K_{2}F/2

Elementary form: Define {a, b} = 1 if ax^{2} + by^{2} = z^{2} has a nontrivial solution in F, and {a, b} = −1 otherwise.

Note that a nonsingular bilinear form ⟨−, −⟩ on a 𝔽_{2}-vector space V is uniquely determined by the annihilators of v ∈ V. In the case of the symbol, the annihilator of a ∈ F^{×} is the group of all b which are norms in the extension F \sqrt{a}/F.

Note also that ⟨v, v⟩ is linear in v (here we are thinking of V as an additive 𝔽_{2}-vector space), and hence there is a unique element ϵ ∈ V such that ⟨ϵ, v⟩ = ⟨v, v⟩ for all v. If ϵ = 0, then the form is symplectic and m = dim V is even. If ϵ ≠ 0, then the form is diagonalizable, and so is isomorphic to the standard Euclidean form. If e_{1}, ..., e_{n} is a diagonalizing basis, then ϵ = e_{1} + ... + e_{n}. Hence ⟨ϵ, ϵ⟩ is nonzero if and only if m is odd. Taking V = F^{×}/(F^{×})^{2} and reverting to multiplicative notation, we have:

Proposition 6.1 a) ϵ = −1, and hence the Hilbert symbol is symplectic if and only if \sqrt{−1} ∈ F;

b) {−1, −1} = −1 if and only if d is odd.

Proof: a) Since −a is a norm in F \sqrt{a}/F, \{a, −a\} = 1. Hence \{−1, a\} = \{a, a\} for all a ∈ F^{×}. b) F^{×}/(F^{×})^{2} has dimension d + 2.

It is enlightening to examine part (b) in the context of the various interpretations of the symbol given above. For example, -1 is not a sum of two squares in ℚ_{2} (although it is the sum of four squares!); this verifies that {−1, −1} = −1 in ℚ_{2}. Using the power norm residue symbol we can then check (b) by recalling that Br ℚ_{2}⟶Br F is multiplication by d on ℚ/ℤ.

6.2 Homology

Recall that B = P ⊗FP H_{∗}BU, and C = H_{∗}BO ⊗FP B. Here P is the subalgebra generated by the primitives in H_{∗}BO, and F is the Frobenius map.

Theorem 6.2 Let F be a finite extension of ℚ_{2}, of degree d. Then there are the following isomorphisms of Hopf algebras over the Steenrod algebra:

If d is even and \sqrt{−1} ∈ F:

H_{∗}BGLF^{+} ≅ (H_{∗}BU)^{2} ⊗ (H_{∗}U)^{d+2}

If d is even and \sqrt{−1} ∉ F:

H_{∗}BGLF^{+} ≅ H_{∗}BO/FP ⊗ B ⊗ (H_{∗}U)^{d}

If d is odd:

H_{∗}BGLF^{+} ≅ C ⊗ (H_{∗}U)^{d}

Proof: We will use the results of [13] describing the homotopy type of KF^{∧}. In all cases there is a commutative diagram of fibre sequences

[figure: K^{red}F; KF^{∧}; K𝔽_{q}^{∧}; L̂K^{red}F; L̂KF; L̂K𝔽_{q}; r]

in which the vertical maps are equivalences on 0-connected covers, and L̂K^{red}F is an excellent spectrum of odd type. Here we have made a choice of admissible odd prime power q and reduction map r, but the homotopy type of K^{red}F does not depend on this choice. We recall also [13] that the reduction map is defined as a composite KF^{∧}⟶KE^{∧} ≅ K𝔽_{q}^{∧}, where E is a certain non-canonical algebraic extension of F.

If F is non-exceptional (i.e., G(F_{∞}/F) ≡ Γ′_{F} is topologically cyclic), then both fibre sequences split and

K^{red}F ≅ ΣK𝔽_{q}^{∧} ⋁(Σku^{∧})^{d}

where ku is the connective cover of K. Hence

H_{∗}BGLF^{+} ≅ H_{∗}Ω_{0}^{∞}K𝔽_{q} ⊗ H_{∗}Ω_{0}^{∞}ΣK𝔽_{q} ⊗ (H_{∗}U)^{d}

Since q = 1 mod 4 if \sqrt{−1} ∈ F, and q = 3 mod 4 otherwise, the non-exceptional case of the theorem now follows from Theorem 4.3 and Theorem 5.4.

Now suppose that F is exceptional. Then

K̂^{−1}K^{red}F ≅ L[k, k] ⊕ (Λ′)^{d−1}

where the Λ′-module L[k, k] has a presentation with generators e_{1}, e_{2} and relation (1+σ)e_{2} = f_{k}e_{1}. Here f_{k} = (1 + T)^{2^{k}} − c_{0}^{2^{k}}, with k = a_{F} − 2, a_{F} = ν_{2}μ_{F}. Hence

K̂^{−1}K^{red}F/M ≅ (ℤ/2[σ])^{d} ⊕ ℤ/2

and therefore Theorem 5.3 yields

H_{∗}Ω_{0}^{∞}K^{red}F ≅ B ⊗ (H_{∗}U)^{d}

Now consider the fibre sequence

Ω_{0}^{∞}K^{red}F ⟶(BGLF^{+})^{∧}⟶Ω_{0}^{∞}K𝔽_{q}^{∧}

Then the associated Serre spectral sequence collapses, since q = 3 mod 4 and the composite ℝP^{∞}⟶BGLF^{+} ⟶^{r} BGLF_{q}^{+} is a generating complex. (This last fact follows easily from the construction of r in [13]). Hence there is an extension in HH_{∗}BO_{mod}A

B ⊗ (H_{∗}U)^{d} ⟶^{j} H_{∗}BGLF^{+} ⟶^{ρ} H_{∗}BO/FP

and a map of Hopf rings ϕ : H_{∗}BO⟶H_{∗}BGLF^{+} such that ρϕ is the quotient map. Now suppose d is even. Then

ϕ(b_{1})^{2} = ϕ(b_{1}) ∘ ϕ(b_{1}) = ℏ{−1, −1} = 0

and

ϕ(s_{2k+1}^{2}) = ϕ(b_{2k} ∘ b_{1})^{2} = ϕ(b_{4k} ∘ b_{1}^{2}) = ϕ(b_{4k}) ∘ ϕ(b_{1})^{2} = 0

Here we have used Proposition 2.2. Thus ϕ factors through a map H_{∗}BO/FP⟶H_{∗}BGLF^{+} that splits r in HH_{∗}BO_{mod}A. This completes the proof of the theorem for d even.

Finally, suppose d is odd. Then as before we have ϕ(b_{1})^{2} = ℏ{−1, −1}, but now {−1, −1} is nonzero in K_{2}F/2. Furthermore ℏ : K_{2}F/2⟶H_{2}BGLF^{+} is injective, so ϕ(b_{1})^{2} is a nonzero primitive in B ⊗ (H_{∗}U)^{d}. This forces ϕ(b_{1})^{2} = b′_{1} ∈ H_{2}BU ⊂ B. Then

ϕ(s_{2k+1}^{2}) = ϕ(b_{4k}□b′_{1}) = s′_{2k+1}

and hence we have a map

C = H_{∗}BO ⊗FP B⟶H_{∗}BGLF^{+}

By construction the resulting map C ⊗ (H_{∗}U)^{d}⟶H_{∗}BGLF^{+} is onto, and therefore is an isomorphism by dimension counting.

The cohomology ring structure is easily read off from the homology. We will be content to describe the ring structure only, but the A-Hopf algebra structure is also determined by Theorem 6.2.

Corollary 6.3 There are isomorphisms of algebras:

If d is even and \sqrt{−1} ∈ F:

H^{∗}BGLF^{+} ≅ (H^{∗}BU)^{2} ⊗ (H^{∗}U)^{d+2}

If d is even and \sqrt{−1} ∉ F:

H^{∗}BGLF^{+} ≅ 𝔽_{2}[q_{1}, q_{2}, ...] ⊗ 𝔽_{2}[c_{2}, c_{4}, ...] ⊗ H^{∗}BU ⊗ (H^{∗}U)^{d+1}

where | q_{i} |= 2i − 1.

If d is odd:

H^{∗}BGLF^{+} ≅ H^{∗}BO ⊗ 𝔽_{2}[c_{1}^{2}, c_{2}^{2}, ...] ⊗ (H^{∗}U)^{d+1}

6.3 Units and cohomology

Let j_{0} : BGL_{1}F ⟶BGLF^{+} denote the natural map. Recall that the determinant provides a left inverse to j_{0} and therefore H_{∗}BGL_{1}F ⊂ H_{∗}BGLF^{+}.

Theorem 6.4 Let D ⊂ H_{∗}BGLF^{+} denote the subalgebra generated by BGL_{1}F. Then there is a short exact sequence of A-Hopf algebras

D⟶H_{∗}BGLF^{+}⟶H_{∗}BU/P′

Now let I_{Δ} denote the ideal in H^{∗}GLF consisting of all classes that vanish on the diagonal subgroup D_{n} ⊂ GL_{n}F for all n. Then H^{∗}GLF/I_{Δ} is a Hopf algebra and we can form the Hopf kernel J_{Δ} of the quotient map H^{∗}GLF ⟶H^{∗}GLF/I_{Δ}. Thus I_{Δ} is generated by the positive-dimensional elements of J_{Δ}.

Corollary 6.5

J_{Δ} = (H_{∗}BU/P′)^{∗} ≅ ℤ/2[c_{1}^{2}, c_{2}^{2}, ...]

Proof: Clearly D is generated by the images of the maps H_{∗}D_{n}⟶H_{∗}GL_{n}F ⟶H_{∗}GLF. Thus the Hopf annihilator of D is J_{Δ}. But by the theorem, this Hopf annihilator is (H_{∗}BU/P′)^{∗}.

We now turn to the proof of Theorem 6.4. The argument will read more smoothly if we assume that d is odd, and later indicate the minor changes needed to handle the case d even.

Let \overline{U} = F^{×}/(F^{×})^{2}, regarded as a ℤ/2-vector space of dimension d + 2. Our first step is to choose a convenient basis for \overline{U}. Note that \overline{U} contains a distinguished subspace \overline{V} of dimension 2, namely \overline{V} = \{ū : \sqrt{u} ∈ F_{∞}\}. There is also a canonical homomorphism δ : \overline{U}⟶ℤ/2 given by (see Proposition 6.1)

δ(ū) ≡ \{\overline{−1}, ū\} = \{ū, ū\}

Note that since we are assuming d odd, \sqrt{−1} ∉ F and \overline{−1} is nonzero in \overline{U}; hence δ is nontrivial. Finally, there is a noncanonical homomorphism λ = λ_{E} : \overline{U}⟶ℤ/2 induced by our fixed choice of reduction map r: π_{1}r : π_{1}KF^{∧}⟶π_{1}K𝔽_{q}^{∧} = ℤ/2. This homomorphism depends on the choice of r. Note, however, that the restriction of λ to V is independent of this choice. For if v̄ ∈ \overline{V} is the unique nontrivial element such that σγ_{F} (\sqrt{v}) = \sqrt{v}, then λ(v̄) = 1; while λ(\overline{−1}) = −1. Apart from this restriction, λ can be arbitrary.

Lemma 6.6 λ ≠ δ. Hence W ≡ Ker λ ∩ Ker δ has dimension d.

Proof: Since d is odd we have {−1, −1} = −1. Interpreting this in terms of the Hilbert symbol, we have that Ψ_{F_{∞}/F} (−1) ∈ G(F_{∞}/F) ≡ Γ′_{F} is nontrivial since it acts nontrivially on \sqrt{−1}. Here Ψ_{F_{∞}/F} : (F^{×})^{∧}⟶Γ′_{F} is the canonical map of local class field theory. Since σ is the only element of order two in Γ′_{F} , it follows that Ψ_{F_{∞}/F} (−1) = σ. Now if {−1, v̄} = 1, then σ(\sqrt{v}) = \sqrt{v}. But then \sqrt{v} ∈ F, a contradiction. Thus δ(v̄) = −1; since λ(v̄) = 1, this proves the lemma.

From now on we fix a basis \overline{−1}, ū_{0}, ū_{1}, ..., ū_{d} for \overline{U}with ū_{0} = v̄ and ū_{1}, ..., ū_{d} a basis for W.

Now let S = S(H̃_{∗}BGL_{1}F), regarded as a sub-Hopf ring of H_{∗}Q_{0}BGL_{1}F. Then the natural map of ring spaces Q_{0}BGL_{1}F ⟶BGLF^{+} induces a map of Hopf rings τ : S⟶H_{∗}BGLF^{+}, with τ(S) = D.

As in Lemma 3.4 of [12], we have:

Lemma 6.7 S is a polynomial algebra on the generator b_{n} (n ≥ 1) and b_{n} ∘ ū_{i_{1}} ∘ ... ∘ ū_{i_{k}} (n ≥ 0, interpreting b_{0} as the Hopf ring unit [1]), and k ≤ d + 1.

Of course the polynomial generators in the lemma do not remain independent in H_{∗}BGLF^{+}. First of all, since the descent spectral sequence for π_{∗}KF^{∧} vanishes above filtration 2, any threefold product of homotopy classes vanishes. Hence for any units a, b, c we have ā∘b̄∘c̄ = 0. Second, let ζ denote the unique nonzero spherical class in H_{2}BGLF^{+}. Then for any a, b we have that ā ∘ b̄ is either zero or ζ. This yields:

Corollary 6.8 D is generated as an H_{∗}BO-Hopf module by b_{n} (n ≥ 1), ū_{i} (n ≥ 0, 0 ≤ i ≤ d), and b_{2n} ∘ ζ (n ≥ 0).

Now by the proof of Theorem 6.2, the sub-Hopf module generated by ū_{1}, ..., ū_{d} is the canonical (even unique) Hopf summand isomorphic to (H_{∗}U)^{d}, while the sub-Hopf module on ū_{0} generates P ⊂ B ⊂ C. Hence H_{∗}GLF/D = H_{∗}BU/P′ as claimed. This completes the proof of Theorem 6.4 for d odd.

If d is even and \sqrt{−1} ∉ F, Lemma 6.6 is clear since δ(−1) = 1, λ(−1) = −1. As ū_{0} we can take any element of Ker λ not in W. Lemma 6.7 and its corollary go through unchanged. The final step is virtually identical to the case d odd, with C replaced by H_{∗}BO/FP ⊗ P ⊗FP H_{∗}BU.

If \sqrt{−1} ∈ F, let ξ = ξ_{2^{a}} be a primitive 2^{a}-th root of unity, where a = a_{F}. Let λ : \overline{U}⟶ℤ/2 be induced by π_{1}KF^{∧}⟶π_{1}K𝔽_{q}/2 = ℤ/2. Then as basis for \overline{U} we take ξ̄, ū_{0}, ..., ū_{d}, where the ū_{i} form a basis for Ker λ. Now ū^{2} = 0 for all units u, and we neither have nor need Lemma 6.6. In Corollary 6.8 we replace b_{n} by b_{2n} and and include ξ̄ as a generator. The sub-Hopf ring generated by b_{2n} and ξ̄ is just H_{∗}BGL𝔽_{q} ≅ H_{∗}BU ⊗ H_{∗}U ⊂ H_{∗}BGLF^{+}. The sub-H_{∗}BO-Hopf module generated by ū_{0}, ..., ū_{d} is (H_{∗}U)^{d+1} ⊂ H_{∗}Ω_{0}^{∞}K^{red}F, and the sub-Hopf module generated by ζ is P ⊂ H_{∗}BU as before.

6.4 The role of the Brauer group

We conclude by showing how the quotient H_{∗}BGLF^{+}/D is related to the Brauer group of F. First observe that Corollary 6.8 can be interpreted in another way: If E is any ring spectrum, the graded group of self-maps [E, E]_{∗} is a module over π_{∗}E. Passing to zero-th spaces, ⊕_{n≥0}[Σ^{n}Ω^{∞}E, Ω^{∞}E] becomes a module over π_{∗}Ω^{∞}E; note that Ω_{0}^{∞}E is a submodule for this action. Then Corollary 6.8 says that D is generated by the canonical map ℝP_{+}^{∞}⟶(BGLF^{+})^{∧} × ℤ_{2} together with the action of π_{i}KF^{∧}, i ≤ 1. We write α ⋆ f for the action of α ∈ π_{0} on a map f.

Now (BGLF^{+})^{∧} is also Ω_{0}^{∞}K^{ét}F ≅ Ω_{0}^{∞}L̂KF, and there is a canonically split short exact sequence

0⟶T Br F ⟶π_{0}Ω_{0}^{∞}L̂KF ⟶ℤ_{2}⟶0

Here T Br F is the Tate module associated to the Brauer group: T Br F ≡ Hom (ℤ/2^{∞}, Br F) = ℤ_{2}. Thus π_{0}Ω_{0}^{∞}L̂KF also acts on (BGLF^{+})^{∧}; in particular, letting α_{0} denote a generator of T Br F, multiplication by α_{0} defines a self-map α_{0}⋆ : (BGLF^{+})^{∧}⟶(BGLF^{+})^{∧}. It is clear that α_{0}⋆ is trivial on homotopy, because α_{0} has filtration two in the descent spectral sequence. But α_{0}⋆ is highly nontrivial on homology:

Theorem 6.9 There is a commutative diagram of A-Hopf algebras

[figure: S(H̃_{∗}Bμ_{F}); H_{∗}BGLF^{+}; H_{∗}BU; H_{∗}BU/P′]

in which the top arrow is induced by α_{0} ⋆i, the diagonal arrow is injective, and the remaining arrows are the obvious quotient maps.

Proof: We assume F is exceptional, so that Bμ_{F} = ℝP^{∞}; the non-exceptional case is similar. Since L̂ℝP^{∞} ≅ L̂S^{0}, and ℝP^{∞} is 0-connected, there is a short exact sequence

0⟶T Br F ⟶[ℝP^{∞}, BGLF^{+}]⟶ℤ_{2}⟶0

Clearly α_{0} ⋆ i is a generator of the subgroup T Br F, and moreover the short exact sequence can be identified with

0⟶[ℝP^{∞}, Ω_{0}^{∞}K^{red}F]⟶[ℝP^{∞}, BGLF^{+}]⟶[ℝP^{∞}, BGL𝔽_{q}^{+}]⟶0

for any choice of reduction map KF^{∧}⟶K𝔽_{q}^{∧}. The presentation of K̂^{−1}K^{red}F described above (see the proof of Theorem 6.2) determines a fibre sequence

⋁^{d} K̂ ⟶^{f} L̂K^{red}F ⟶ ⋁^{d+1} K̂⟶ ⋁^{d} ΣK̂

such that the first component of f is essential and remaining components vanish. We then obtain a lift g:

[figure: ℝP^{∞}; ∏^{d} BU^{∧}; Ω_{0}^{∞}L̂K^{red}F; ∏^{d+1} U^{∧}; g; α_{0}⋆; Ω_{0}^{∞}f]

Here we can assume that the first component of g is a generator of K^{0}ℝP^{∞} and that the remaining components vanish. In particular we have the upper commutative triangle of the theorem. On the other hand the proof of Theorem 6.2 shows that the first component of H_{∗}Ω_{0}^{∞}f is injective, and yields the lower commutative triangle. (Incidentally, this shows that we have constructed a canonical H_{∗}BU ⊂ H_{∗}BGLF^{+}, independent of the choice of reduction map r, resolution of K̂^{−1}K^{red}F, etc.)

Translating this result into cohomology sheds some light on Corollary 6.5. Call a map B(GL_{1}F)^{n}⟶(BGLF^{+})^{∧} a generalized torus if it is the n-fold sum of copies of j_{0} and α_{0} ⋆j_{0}.

Corollary 6.10 H^{∗}BGLF^{+} is detected by generalized tori.

Finally, we speculate that there should be a more natural, algebraic construction of the “Brauer group action”. Let F(X, Y) denote the function spectrum associated to spectra X,Y. It is easy to see that there are natural weak equivalences

F(X, Y)^{∧} ⟶^{≅} F(X, Y^{∧}) ⟵^{≅} F(X^{∧}, Y^{∧})

Furthermore, for any spectrum W there is a universal coefficient sequence

0⟶Ext (ℤ/2^{∞}, π_{n}W)⟶π_{n}W^{∧}⟶Hom (ℤ/2^{∞}, π_{n−1}W)⟶0

Taking W = F(KF, KF) and n = 0 suggests that the homomorphism T Br F ⟶[KF^{∧}, KF^{∧}] arises from a homomorphism Br F ⟶[KF, ΣKF].

References

[1] Atiyah, M.F., and Segal, G.B., Equivariant K-theory and completion, J. Differential Geometry (1969), 1-18.

[2] Bökstedt,M., The rational homotopy type of ΩW h^{Diff}(∗), Springer Lecture Notes in Math. 1051 (1984), 25-37.

[3] Carlsson, G., Segal’s Burnside ring conjecture for (ℤ/2ℤ)^{k}, Topology 22 (1983), 83-103.

[4] Cohen, F.R., Lada, T.J., and May, J.P., The homology of iterated loop spaces, Springer Lecture Notes in Math. 533 (1976).

[5] Dwyer, W., and Mitchell, S.A., On the K-theory spectrum of a ring of algebraic integers, K-theory 14 (1998), 201-263 .

[6] Fiedorowicz,Z., and Priddy, S.B., Homology of Classical Groups over Finite Fields and their associated Infinite Loop Spaces, Springer Lecture Notes in Math. 674 (1978).

[7] Husemoller,D., The structure of the Hopf algebra H*BU over a Z(p)-algebra, Amer. J. Math. 43 (1971), 329-349.

[8] Mitchell, S.A., The rank filtration of Quillen’s Q-construction, in Advances in Homotopy Theory, London Math. Soc. Lecture Notes 139 (1989), 127-142.

[9] Mitchell, S.A., On the Lichtenbaum-Quillen conjectures from a stable homotopytheoretic viewpoint, in Algebraic Topology and Its Applications, MSRI Publications 27, pp. 163-240, Springer-Verlag, New York, 1994.

[10] Mitchell, S.A., On the plus construction for BGLZ[1/2] at the prime 2, Math. Zeit. 209 (1992), 205-222.

[11] Mitchell,S.A., Class groups and general linear group cohomology for rings of algebraic integers, Math. Z. 223 (1996), 101-122.

[12] Mitchell,S.A., Units and general linear group cohomology for rings of algebraic integers, Math. Z. 228, 207-220 (1998).

[13] Mitchell,S.A., The algebraic K-theory spectrum of a 2-adic local field, preprint 2000.

[14] Neukirch, J., Algebraic Number Theory, Springer, New York, 1999.

[15] Østvær, P.A., Étale descent for real number fields, University of Oslo thesis, 2000.

[16] Quillen, D., On the cohomology and K-theory of the general linear group over a finite field, Annals of Math. 96 (1972), 552-586.

[17] Ravenel,D.C., Localization with respect to certain periodic homology theories, Amer. J. Math. 106(1984), 351-414.

[18] Ravenel,D.C., and Wilson,W.S., The Hopf ring for complex cobordism, J. Pure and Applied Algebra 9 (1977), 241-280.

[19] Rognes, J., Algebraic K-theory of the two-adic integers, J. Pure Appl. Algebra 134 (1999), 287-326.

[20] Rognes,J., and Weibel,C., Etale descent for two-primary algebraic K-theory of totally imaginary number rings, K-theory 16 (1999), 101-104.

[21] Rognes,J., and Weibel,C., Two-primary algebraic K-theory of rings of integers in number fields, J. Amer. Math. Soc. 13 (2000), 1-54.

[22] Schoeller,C., Etude de la categorie des algebres de Hopf commutatives connexes sur un corps, Manusc. Math. 3 (1970), 133-155.

[23] Voevodsky, V., The Milnor conjecture, preprint 1996.