GALOIS EXTENSIONS OF

“BRAVE NEW RINGS”

John Rognes

Contents

1. Integral group rings

2. “Brave new rings”

3. Chromatic localization

4. Galois descent

Typeset by AMS-TEX

Integral group rings

First consider:

• X a based space;

• X̃ → X its universal covering space;

• π = π_{1}(X) the fundamental group;

• ℤ[π] the integral group ring. Many classical invariants of topological spaces

are algebraic invariants of the integral group ring ℤ[π].

The cell structure on a finite CW complex X lifts to one on X̃, with π permuting the cells freely. The cellular chain complex

0 → C_{n}(X̃) ⟶^{d_{n}} … ⟶^{d_{1}} C_{0}(X̃) → 0

is a chain complex of free ℤ[π]-modules.

The Whitehead group

When C_{∗}(X̃) is exact, its Whitehead torsion is an invariant in the Whitehead group

W h_{1}(π) = K_{1}(ℤ[π])/(±π) .

For any ring R the first algebraic K-group is the maximal abelian quotient

K_{1}(R) = GL(R)/[GL(R), GL(R)]

of the infinite general linear group of R. For R = ℤ[π], ±π sits inside GL_{1}(R) ⊂ GL(R), and maps to K_{1}(R).

The h-cobordism theorem

A cobordism on a closed n-manifold M is a compact (n + 1)-manifold W containing M as a component of the boundary ∂W.

It is an h-cobordism if the inclusion M ⊂ W is a homotopy equivalence.

Theorem (Smale; Barden; Mazur; Stallings). Let M be a closed n-manifold, n ≥ 5, with π = π_{1}(M). There is a one-to-one correspondence between

(1) isomorphism classes of h-cobordisms on M, and

(2) elements of the Whitehead group W h_{1}(π), which takes [W] to the Whitehead torsion of the exact complex C_{∗}(W̃, M̃).

Implies Poincaré conjecture in dimensions ≥ 6. Fundamental to classification of manifolds by

surgery theory.

“Brave new rings”

Many modern invariant of topological spaces are invariants of “brave new rings”, alias 𝕊-algebras, which are a homotopy theoretic generalization of the classical notion of a ring.

• 𝕊 the sphere spectrum;

• ℳ_{𝕊} the category of 𝕊-modules

(= spectra). ℳ_{𝕊} has a symmetric monoidal pairing ∧, called

the smash product. An 𝕊-algebra A is an 𝕊module with an associative and unital product

μ: A ∧ A → A

and unit η: 𝕊 → A. These represent generalized cohomology theories with cup product.

Analogous to: the integers ℤ, the category of ℤ-modules (= abelian groups), the tensor product ⊗, and ℤ-algebras (= rings).

𝕊-algebras generalize rings; 𝕊 rather than ℤ is the initial object.

The Whitehead space

Now consider:

• X a based space;

• P X → X its path space fibration;

• ΩX the loop group;

• 𝕊[ΩX] the group 𝕊-algebra. The algebraic K-theory space K(A) can be

defined for any 𝕊-algebra A, extending algebraic K-theory of rings.

The (smooth) Whitehead space W h(X) sits in a (split) fiber sequence

Q(X_{+}) ⟶ K(𝕊[ΩX]) ⟶ W h(X) .

So π_{1}(W h(X)) ≅ W h_{1}(π_{1}(X)).

The stable parametrized h-cobordism theorem

Theorem (Waldhausen). Let M be a compact manifold. There is a weak equivalence between

(1) the space of h-cobordisms on M × I^{k}, for k ≫ 0, and

(2) the looped Whitehead space ΩW h(M).

Enriches h-cobordism theorem from a statement about isomorphism classes to one about parameter spaces.

Leads to a description of the spaces of homeomorphisms, resp. diffeomorphisms, of manifolds.

Commutative 𝕊-algebras

Desire to understand the algebraic K-theory of group 𝕊-algebras. Like group rings, these are generally non-commutative.

Theorem (Farrell, Jones). Let M be a closed Riemannian manifold, with sectional curvature K ≤ 0 everywhere. (Other geometric hypotheses also suffice.) Then W h(M) can be assembled up to weak equivalence from

(1) W h({p}) for every point p ∈ M, and

(2) W h(γ) for every closed geodesic γ ⊂ M.

Reduces to the cases 𝕊[Ω{p}] ≅ 𝕊 and 𝕊[Ωγ] ≅ 𝕊[ΩS^{1}] ≃ 𝕊[z, z^{−1}], which are commutative 𝕊algebras.

Remark. Commutative 𝕊-algebras are only commutative up to homotopy, but in a coherent way, hence are somewhat like non-commutative deformations of commutative rings.

Zariski localization

Algebraic K-theory K(R) of a ring R is formed from

• finitely generated projective R-modules, or when R is commutative,

• finite rank locally free sheaves over Spec R. The excision property of topological K-theory

has an analogue in the Zariski localization property of algebraic K-theory, leading to a spectral sequence

E_{s,t}^{2} = H_{Zar}^{−s}(Spec R; K_{t}) ⟹ K_{s+t}(R)

expressing K(R) in terms of the K-theory sheaf K_{∗}. The sheaf is determined by its stalks at Zariski points, i.e., at local rings. (Can restrict further to Hensel local rings, and often to fields.)

Spectra as sheaves

Algebraic K-theory K(A) of an 𝕊-algebra A is formed from

• finite cell A-modules and their retracts. View such modules as sheaves over some (yet undefined) geometric object Spec A when A is commutative, e.g. for A = 𝕊.

The Hurewicz homomorphism h: 𝕊 → ℤ induces a map

ah: Spec ℤ ⟶ Spec 𝕊

which inflates each closed point (p) to a highly complex, infinite dimensional variety in Spec 𝕊 !

Like Spec ℤ, Spec 𝕊 is a cofinite union of local pieces Spec 𝕊_{(p)} for all rational primes p, glued along the generic point Spec ℚ.

What does Spec 𝕊_{(p)} look like ?

The chromatic filtration

A localization functor is thought of as restricting a sheaf to an open subset.

Theorem (Devinatz, Hopkins, Smith). The localization functors on the category of finite cell 𝕊_{(p)}-modules form an infinite chain of functors L_{n}, for 0 ≤ n < ∞, with natural transformations between them:

id → ⋯ → L_{n} → L_{n−1} → ⋯ → L_{1} → L_{0} .

Interpret L_{n} as restricting a sheaf over Spec 𝕊_{(p)} to an open subset U_{n} = {v_{0}, v_{1}, …, v_{n}} of chromatic points:

Spec 𝕊_{(p)} ⊃ ⋯ ⊃ U_{n} ⊃ U_{n−1} ⊃ ⋯ ⊃ U_{1} ⊃ U_{0} .

The infinite nested chain shows that Spec 𝕊_{(p)} is not Noetherian.

Chromatic localization

Let L_{K(n)} be Bousfield localization with respect to the n-th Morava K-theory K(n), for 0 ≤ n < ∞.

Theorem (Hovey, Strickland). The category of L_{n}-local 𝕊_{(p)}-modules with trivial L_{n−1}-localization is equivalent to the category of K(n)-local 𝕊_{(p)}- modules.

Restricted to finite cell modules, this gives a fibration

K(L_{K(n)}𝕊_{(p)}) → K(L_{n}𝕊_{(p)}) → K(L_{n−1}𝕊_{(p)}) .

Together with the Hopkins–Ravenel chromatic convergence theorem, this leads to a spectral sequence

E_{s,t}^{2} = H_{chr}^{−s}(Spec 𝕊; K_{t}) ⟹ K_{s+t}(𝕊)

expressing K(𝕊) in terms of the K-theory sheaf K_{∗}. The sheaf is determined by its stalks at chromatic points, i.e., at the commutative 𝕊-algebras L_{K(n)}𝕊_{(p)}.

Here L_{K(0)}𝕊_{(p)} = ℚ, while L_{K(1)}𝕊_{(p)} = J_{p}^{∧} is the well-known p-complete non-connective imageof-J spectrum.

Galois extensions of commutative rings

Consider:

• R → T a (pro-)finite Galois extension of commutative rings;

• G = Gal(T/R) its (pro-)finite Galois group;

• M a finitely generated projective R-module. The tensor product T ⊗_{R} M is a finitely gener-

ated projective T -module. Get a map

η: K(R) ⟶ K(T)^{hG} = F (EG_{+}, K(T))^{G}

to the G-homotopy fixed points of K(T). Here EG is a free, contractible G-space.

Galois descent for commutative rings

The Lichtenbaum–Quillen conjectures suppose that Galois descent holds for algebraic K-theory, i.e., that η induces an isomorphism on homotopy groups with finite coefficients in sufficiently high degrees. These conjectures have been verified in several cases.

Leads to spectral sequence

E_{s,t}^{2} = H_{Gal}^{−s}(Spec R; K̂_{t}) ⟹ K̂_{s+t}(R)

converging in high degrees. Here Galois-cohomology is the continuous cohomology of the absolute Galois group G_{R} = Gal(\overline{R}/R). The “hat” denotes ℓ-adic completion, for some prime ℓ with 1/ℓ ∈ R. The ℓ-adically completed K-theory sheaf K̂_{∗} is determined by its stalks at Galois closed points, i.e., at separably closed fields, and is known:

Theorem (Suslin). For a separably closed field \overline{F} of characteristic ≠ ℓ, K̂_{∗}(\overline{F}) ≅ π_{∗}(ku)_{ℓ}^{∧} = ℤ̂_{ℓ}[v] with deg(v) = 2. Thus K̂_{t} = ℤ̂_{ℓ}(t/2) for t ≥ 0 even, and = 0 otherwise.

Galois extensions of commutative 𝕊-algebras

Definition. Let A → B be a map of commutative 𝕊-algebras, and let G be a finite group acting on B by A-algebra homomorphisms. Suppose that the natural maps A → B^{hG} and B ∧_{A} B → F (G_{+}, B) are weak equivalences. Then A → B is a G-Galois extension of commutative 𝕊-algebras.

Galois extensions are faithful: if M is an Amodule with B ∧_{A} M ≃ ∗, then M ≃ ∗ is contractible.

Galois extensions are formally étale: the topological André–Quillen homology T AQ(B/A) ≃ ∗ vanishes.

The fundamental theorem holds: if H ⊂ G is a subgroup, then B^{hH} → B is an H-Galois extension. If H is normal, then A → B^{hH} is G/H-Galois.

Minkowski’s theorem on discriminants extends: there are no non-trivial G-Galois extensions 𝕊 → B with B connected (not split as a product of nontrivial 𝕊-algebras).

Examples

(1) A G-Galois extension R → T of commutative rings is a G-Galois extension of commutative 𝕊-algebras.

(2) The complexification map KO → KU from real to complex topological K-theory is a C_{2}-Galois extension.

(3) The map J_{p}^{∧} → KU_{p}^{∧} is a ℤ_{p}^{∗}-pro-Galois extensions. This is the case n = 1 of the following more general example:

(4) E_{n} is the Hopkins–Miller commutative 𝕊algebra with coefficient ring

π_{∗}(E_{n}) = 𝕎_{𝔽_{p^{n}}} [[u_{1}, …, u_{n−1}]][u, u^{−1}]

representing the Lubin–Tate universal deformation of the Honda formal group law F_{n} of height n, over 𝔽_{p^{n}}. The pro-finite group C_{n} ⋉ 𝕊_{n} acts on E_{n}, where C_{n} = Gal(𝔽_{p^{n}}/𝔽_{p}) and 𝕊_{n} is the Morava stabilizer group of automorphisms of F_{n}. There is a map

L_{K(n)}𝕊_{(p)} ⟶ E_{n}

which is a (C_{n} ⋉ 𝕊_{n})-pro-Galois extension of commutative 𝕊-algebras.

Galois descent for commutative 𝕊-algebras

Can extend Lichtenbaum–Quillen conjectures to suppose that Galois descent holds for algebraic Ktheory of commutative 𝕊-algebras.

Conjecture. Let A → B be a G-Galois extension of K(n)-local commutative 𝕊-algebras. Let V be a finite cell 𝕊-module of chromatic filtration (n + 1), so L_{n}V ≃ ∗. Then the natural map

η: K(A) ⟶ K(B)^{hG}

induces an isomorphism on homotopy groups in sufficiently high degrees, after smashing with V.

This leads to a spectral sequence

E_{s,t}^{2} = H_{Gal}^{−s}(Spec A; K_{t}(−; V)) ⟹ K_{s+t}(A; V)

converging in high degrees to the algebraic K-theory of A with finite coefficients in V, i.e., to K_{∗}(A; V) = π_{∗}(K(A) ∧ V).

Conclusion

This would explain K(A), with suitable finite coefficients, in terms of

(1) the group cohomology of the absolute Galois group G_{A} = Gal(\overline{A}/A) of A, where the separable closure \overline{A} is a maximal connected pro-Galois extension of A, and

(2) the values of the K-theory sheaf at Galois closed points, i.e., at separably closed K(n)-local commutative 𝕊-algebras.

In the case A = L_{K(n)}𝕊_{(p)} it is plausible that the Hopkins–Miller spectrum E_{n} is close to \overline{A} (after extending 𝕎_{𝔽_{p^{n}}} to 𝕎_{\overline{𝔽}_{p}}).

It is also plausible that the fairly formal techniques proving Suslin’s theorem for separably closed fields can be applied to separably closed K(n)-local commutative 𝕊-algebras.

Such results for K(L_{K(n)}𝕊_{(p)}) would then assemble by the chromatic filtration to explain the global object K(𝕊), and thus the Whitehead space W h(∗).