ALGEBRAIC K-THEORY AND THE CHROMATIC FILTRATION

John Rognes

March 20th 2006

1. Brave new schemes and stacks.

By a brave new ring, we mean a (usually commutative) structured ring spectrum. This can variously interpreted as an S-algebra, a symmetric ring spectrum, an orthogonal ring spectrum, or perhaps, an A_{∞}/E_{∞} ring spectrum. We will generically talk about 𝕊-algebras, where 𝕊 denotes the sphere spectrum.

A commutative 𝕊-algebra A can be viewed as an affine brave new (or derived, homotopical or topological) scheme (Spec A, 𝒪), with underlying space the prime ideal spectrum Spec π_{0}(A), but locally brave new ringed by the commutative 𝕊-algebra A[1/f] with π_{∗}(A[1/f]) = π_{∗}(A)[1/f] over the open subset U_{f} = Spec π_{0}(A)[1/f], for non-zero divisors f ∈ π_{0}(A).

We say that A is connective if π_{i}(A) = 0 for i < 0. For connective A the Hurewicz map A → Hπ_{0}(A) maps Spec π_{0}(A) into Spec A, and we can view the brave new scheme as a thickening, or deformation, of the underlying algebraic scheme. More generally, the Postnikov tower

A → ⋯ → P^{n}A → P^{n−1}A → ⋯ → P^{0}A = Hπ_{0}(A)

define an increasing sequence of infinitesimal deformations Spec P^{n}A of Spec π_{0}(A), and to the extent that we can identify A = holim_{n} P^{n}A with this tower, Spec A is the formal deformation of Spec π_{0}(A) given by this increasing union. These brave new schemes are locally brave new ringed by the Postnikov sections P^{n}A[1/f] over U_{f}. For example, this applies to the connective commutative 𝕊-algebras 𝕊, ku (connective K-theory) and Hℤ (the Eilenberg–Mac Lane spectrum of the integers).

We say that A is even periodic if there exists a unit u ∈ π_{2}(A) and π_{1}(A) = 0, so π_{∗}(A) ≅ π_{0}(A)[u^{±1}]. Then Spec π_{0}(A) is locally ringed by the even periodic spectra A[1/f] with π_{∗}(A[1/f]) = π_{0}(A)[1/f][u^{±1}]. The connective cover P_{0}A → A defines a canonical morphism Spec A → Spec P_{0}A of affine brave new schemes. ((Such a structure on the formal deformation Spec P_{0}A of Spec π_{0}(A) might deserve a name.)) For example, this applies to the even periodic commutative 𝕊-algebras KU (complex K-theory) and E_{n} (the Lubin–Tate spectra), where π_{0}E_{n} = 𝕎𝔽_{p^{n}}[[u_{1}, …, u_{n}]] is the complete local ring that supports the universal deformation of the height n Honda formal group Γ_{n} over 𝔽_{p^{n}}. In the case n = 1, E_{1} = KU_{p} is p-adic K-theory.

Globalizing, a brave new scheme X with a Zariski topology is glued together from affine brave new schemes Spec A along open subobjects like Spec A[1/f]. A presheaf over X is glued together from presheaves over the affine pieces. There is a

Typeset by AMS-TEX

(closed) model structure on these, with cofibrations given locally over open affine pieces, and weak equivalences given stalkwise. The sheaves over X are the fibrant presheaves.

The prime ideal spectrum of π_{0}(A) is not always the most interesting brave new geometry underlying A. We say that a map A → B of commutative 𝕊-algebras is a G-Galois extension if G is a finite group acting on B through commutative A-algebra maps, such that the canonical maps i: A → B^{hG} and h: B ∧_{A} B → ∏_{G} B are homotopy equivalences. In symbols, h(b_{1} ∧ b_{2}): g ↦ b_{1}g(b_{2}). Then we wish to view Spec B → Spec A as a regular covering space, with group G of deck transformations. However, it is not necessarily the case that π_{0}(A) → π_{0}(B) is a G-Galois extension of rings, i.e., that the orbit scheme for the G-action on Spec B is realized by Spec A as defined above. In this case, we will instead interpret Spec A = [Spec B/G] as the orbit stack for this G-action, i.e., as the translation category, so that Spec A is a brave new stack, rather than a brave new scheme. For example, this applies to the ℤ/2-Galois extension KO → KU (of the real K-theory spectrum) given by complexification map and the action by complex conjugation, where π_{0}(KO) = π_{0}(KU) = ℤ, so the geometry underlying Spec KO is the orbit category for the ℤ/2-action on Spec KU.

Given a homology theory E_{∗}, represented by a spectrum E, there is also the notion of an E-local G-Galois extension A → B, where the two maps i and h above are only assumed to be E_{∗}-isomorphisms, or equivalently, to become homotopy equivalences after applying the E-localization functor L_{E}. We are mostly interested in this when E = K(n) is the n-th Morava K-theory spectrum, with π_{∗}K(n) = 𝔽_{p}[v_{n}^{±1}]. There is also a notion of a pro-G-Galois extension A → B, given as a filtered direct limit of G_{α}-Galois extensions A → B_{α}, with G = lim_{α} G_{α}. For example, the n-th extended Morava stabilizer group 𝔾_{n} = Aut(Γ_{n}, 𝔽_{p^{n}}) ⋊ Gal(𝔽_{p^{n}}/𝔽_{p}) acts on the even periodic spectrum E_{n}, and there is a K(n)-local pro-𝔾_{n}-Galois extension L_{K(n)}𝕊 → E_{n}. We might then think of Spec L_{K(n)}𝕊 as the orbit stack [Spec E_{n}/𝔾_{n}], keeping in mind that 𝔾_{n} is a profinite group. In the case n = 1, L_{K(1)}𝕊 = J_{p} is the p-adic image of J spectrum, 𝔾_{1} = ℤ_{p}^{×} and k ∈ ℤ_{p}^{×} acts on E_{1} = KU_{p} by the p-adic Adams operation ψ^{k}.

Piecing together affine brave new schemes along Galois covers of open subschemes yields a brave new scheme with an étale topology. We are particularly interested in the chromatic tower from stable homotopy theory. As usual, we fix a rational prime p, implicit in the notation, and write L_{n}X = L_{E(n)}X and L̂_{n}X = L_{K(n)}X, where E(n) is the n-th Johnson–Wilson spectrum with π_{∗}E(n) = ℤ_{(p)}[v_{1}, …, v_{n}^{±1}]. Then there is a tower

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

that can be viewed as an infinite unfolding, in brave new geometric terms, of the short algebraic filtration

ℤ_{(p)} → ℚ .

By chromatic convergence, 𝕊_{(p)} ≃ holim_{n} L_{n}𝕊, so to the extent that we can identify this tower with its homotopy limit, then Spec 𝕊_{(p)} is the increasing union of the open subobjects Spec L_{n}𝕊. However, for n ≥ 1 the spectra L_{n}𝕊 are neither connective nor even periodic, and the geometric structure of this tower appears to be very much richer than that of Spec 𝕊_{(p)} for 𝕊_{(p)} viewed as a connective brave new ring, i.e., Spec ℤ_{(p)} = {(p), (0)} with the local brave new rings 𝕊_{(p)} and 𝕊_{(p)}[1/p] ≃ Hℚ.

For there is a homotopy cartesian square of commutative 𝕊-algebras

\xymatrix{L_{n}𝕊 \ar[r] \ar[d] & L_{n−1}𝕊 \ar[d] \\ L̂_{n}𝕊 \ar[r] & L_{n−1}L̂_{n}𝕊}

for each chromatic height 1 ≤ n < ∞, that is analogous to the arithmetic square

\xymatrix{ℤ_{(p)} \ar[r] \ar[d] & ℚ \ar[d] \\ ℤ_{p} \ar[r] & ℚ_{p}}

of ordinary rings. Furthermore, L_{n−1}𝕊 ∧_{L_{n}𝕊} L̂_{n}𝕊 ≃ L_{n−1}L̂_{n}𝕊, like ℚ ⊗_{ℤ_{(p)}} ℤ_{p} ≅ ℚ_{p}.

We view L_{n}𝕊 → L̂_{n}𝕊 as a completion map, so that Spec L̂_{n}𝕊 is a formal neighborhood in Spec L_{n}𝕊, and the brave new arithmetic square expresses Spec L_{n}𝕊 as being glued together from the open/formal neighborhoods Spec L_{n−1}𝕊 and Spec L̂_{n}𝕊 along their meet Spec L_{n−1}L̂_{n}𝕊. And as remarked above, Spec L̂_{n}𝕊 is geometrically the orbit stack [Spec E_{n}/𝔾_{n}]. Presumably, Spec L_{n−1}L̂_{n}𝕊 is a reasonable open subobject of this orit stack. So from this point of view, Spec 𝕊_{(p)} is to be interpreted as the ind-object given by an increasing union of brave new stacks Spec L_{n}𝕊, where the n-th object is obtained from the (n − 1)-st by attaching an orbit stack along an open subobject.

2. Algebraic K-theory.

To understand these geometric objects, we might study the brave new algebraic analogues of the classical homotopy invariants from algebraic topology, such as the fundamental group, cohomology and topological K-theory. The algebraic analogue of the fundamental group is given by the Galois groups, and a Galois theory for commutative 𝕊-algebras, based on the definitions above, has been developed. It suffices for the study of unramified extensions, but does not directly handle ramified extensions. For this, an appropriate machinery for localization away from the ramification “locus” will be needed.

The theory of étale cohomology or motivic cohomology for commutative 𝕊algebras has not yet been properly developed. In a sufficiently local situation, étale cohomology might specialize to Galois cohomology, i.e., the continuous cohomology of the absolute Galois group, which is then handled by the Galois theory just mentioned, but in general the correct passage to a local situation needs to be found.

However, the algebraic K-theory of brave new rings (commutative or not) was already defined quite long ago. To each 𝕊-algebra A there is a category ℳ_{A} of A-modules, and a full subcategory 𝒞_{A} of semi-finite cell A-modules. The latter are those A-modules that can be obtained from A in finitely many steps by the formation of mapping cones, retracts and desuspensions, and passage to homotopy equivalent A-modules. In other words, 𝒞_{A} is the thick subcategory of ℳ_{A} generated by A. For example, ℳ_{𝕊} is the category of spectra, and 𝒞_{𝕊} is the subcategory of finite spectra. ((For connective A we can work with cell spectra or CW spectra interchangably, but for non-connective A this is not so clear.))

Let h𝒞_{A} ⊂ 𝒞_{A} be the subcategory of homotopy equivalences. Then the algebraic K-theory of A, a space K(A), is defined as a suitable group completion of the nerve of this subcategory, with a map |h𝒞_{A}| → K(A). For a discrete ring R, with Eilenberg–Mac Lane ring spectrum HR, this recovers Quillen’s algebraic K-theory as K(R) ≃ K(HR). The ability to talk about such module categories for more general 𝕊-algebras A, not just the associated homotopy categories, was one of the original motivations for developing the theory of structured ring spectra.

As in algebraic geometry, one might expect an analogue of the Atiyah–Hirzebruch spectral sequence from singular cohomology to topological K-theory, in the form of a spectral sequence from a motivic cohomology of a commutative 𝕊-algebra to its algebraic K-theory. With suitably finite coefficients, one might also expect an étale descent spectral sequence from an étale cohomology to étale K-theory. In a sufficiently local situation, this might reduce to a Galois descent spectral sequence for étale K-theory, which might agree with algebraic K-theory in sufficiently high degrees, or after the introduction of suitable periodicity by localization.

The definition of K(A) in terms of semi-finite cell A-modules seems appropriate for connective or even periodic 𝕊-algebras A. For more general A, like A = L̂_{n}𝕊, it may also be appropriate to study the full subcategory 𝒟_{A} ⊂ ℳ_{A} of dualizable A-modules, i.e., those A-modules X with the property that ν: F_{A}(X, A) ∧_{A} X → F_{A}(X, X) is a homotopy equivalence, perhaps after some E-localization. For connective A, A = L_{n}𝕊 or A = E_{n}, these are the same as the semi-finite A-modules. But for A = L̂_{n}𝕊, the inclusion 𝒞_{A} ⊂ 𝒟_{A} is proper. In fact, for the map K(L̂_{n}𝕊) → K(E_{n})^{h𝔾_{n}} to be close to a homotopy equivalence, the source should probably be interpreted as a group completion of |h𝒟_{L̂_{n}𝕊}|.

One class of important examples is given by the 𝕊-algebra A = 𝕊[ΩM] = Σ^{∞}ΩM_{+} given as a spherical group ring for the loop group ΩM on a manifold M. By Waldhausen’s stable parametrized h-cobordism theorem, the algebraic Ktheory K(𝕊[ΩM]) of these 𝕊-algebras is tightly connected to the automorphism groups of M and the moduli space of manifolds in the isomorphism class of M. ((Elaborate?)) In the smooth category this is interesting already for M = D^{m}, an m-dimensional disc, when 𝕊[ΩD^{m}] ≃ 𝕊, so the algebraic K-theory in question is K(𝕊), that of the sphere spectrum.

As a first approximation, the linearization map 𝕊 → Hℤ induces a rational equivalence

K(𝕊) → K(ℤ)

so Borel’s rational calculation of K(ℤ) yields a rational calculation of K(𝕊) and thus of the diffeomorphism groups of discs, Diff(D^{m}), in a stable range. It is known that K(𝕊) has finite type, like K(ℤ), so to address the finer question of intergral homotopy types, it suffices to treat one rational prime p at a time, either by p-completion or p-localization.

On one hand, there is a homotopy cartesian square

\xymatrix{K(𝕊)_{p} \ar[r] \ar[d] & K(ℤ)_{p} \ar[d] \\ T C(𝕊)_{p} \ar[r] & T C(ℤ)_{p}}

due to Dundas, where T C(A)_{p} is the p-complete topological cyclic homology of an 𝕊-algebra A. Using this, a calculation of K(𝕊)_{p} can be given for p = 2 or p an odd regular prime. However, the answer is about as complicated as the description of the stable homotopy groups of ℂP^{∞}. It might be desirable to find a more conceptual description that separates these stable homotopy groups into periodic families, by way of the chromatic filtration.

For this, one may work p-locally. There is a homotopy equivalence

hofib(K(𝕊) → K(ℤ))_{(p)} ≃ hofib(K(𝕊_{(p)}) → K(ℤ_{(p)}))

due to Waldhausen. This is almost the same as the homotopy fiber of the composite map K(𝕊_{(p)}) → K(ℚ) from the chromatic tower, due to the localization sequence

K(𝔽_{p}) → K(ℤ_{(p)}) → K(ℚ)

where K(𝔽_{p}) is well understood, and K(𝔽_{p})_{(p)} ≃ Hℤ_{(p)}. In more detail, the homotopy fiber of K(ℤ_{(p)}) → K(ℚ) can be decribed as the algebraic K-theory K^{tors}(ℤ_{(p)}) of the category of finite torsion ℤ_{(p)}-modules, and by the dévissage theorem, this is equivalent to the algebraic K-theory of 𝔽_{p}, when each 𝔽_{p}-module is viewed as a torsion ℤ_{(p)}-module.

We are therefore interested in obtaining an understanding of the algebraic Ktheory chromatic tower

K(𝕊_{(p)}) → ⋯ → K(L_{n}𝕊) → K(L_{n−1}𝕊) → ⋯ → K(L_{0}𝕊) ≃ K(ℚ) . To be fair, we should point out that we do not yet know if this tower converges, i.e., if K(𝕊_{(p)}) ≃ holim_{n} K(L_{n}𝕊), but this seems plausible in view of the chromatic convergence theorem.

3. Monochromatic layers.

We therefore study the diagram in algebraic K-theory

\xymatrix{ & K(L_{n}𝕊) \ar[r] \ar[d] & K(L_{n−1}𝕊) \ar[d] \\ K^{sm}(L̂_{n}𝕊) \ar[r] \ar[d]^{𝔾_{n}} \ar[ur] & K(L̂_{n}𝕊) \ar[r] \ar[d]^{𝔾_{n}} & K(L_{n−1}L̂_{n}𝕊) \\ K^{fin}(E_{n}) \ar[r] & K(E_{n})}

where the upper right hand square is derived from the following square of categories

\xymatrix{h𝒞_{L_{n}𝕊} \ar[r] \ar[d] & h𝒞_{L_{n−1}𝕊} \ar[d] \\ h𝒞_{L̂_{n}𝕊} \ar[r] & h𝒞_{L_{n−1}L̂_{n}𝕊} .}

and receives a map from the upper right-hand square in the following diagram

\xymatrix{h𝒞_{L_{n}𝕊}^{e_{n−1}} \ar[r] \ar[d]^{≃} & h𝒞_{L_{n}𝕊} \ar[r] \ar[d] & e_{n−1}𝒞_{L_{n}𝕊} \ar[d] \\ h𝒞_{L̂_{n}𝕊}^{e_{n−1}} \ar[r] \ar[d] & h𝒞_{L̂_{n}𝕊} \ar[r] \ar[d] & e_{n−1}𝒞_{L̂_{n}𝕊} \\ h𝒞_{E_{n}}^{fin} \ar[r] & h𝒞_{E_{n}}}

The prefix e_{n−1} indicates the subcategory of E(n − 1)-equivalences, and the superscript e_{n−1} indicates the full subcategory of E(n − 1)-acyclic objects. So h𝒞_{L_{n}𝕊}^{e_{n−1}} is the category of semi-finite cell E(n)-local spectra of type ≥ n, which by Hovey– Strickland is equivalent to the category of (categorically) small K(n)-local spectra, h𝒞_{L̂_{n}𝕊}^{e_{n−1}}.

The map e_{n−1}𝒞_{L_{n}𝕊} → h𝒞_{L_{n−1}𝕊} is an idempotent completion, i.e., it only adjoins retracts of objects, so the induced map in algebraic K-theory is 0-coconnected, i.e., injective on π_{0} and an isomorphism in higher degrees. By Waldhausen’s generic fibration theorem, there is therefore a homotopy fiber sequence

K^{sm}(L̂_{n}𝕊) → K(L_{n}𝕊) → K(L_{n−1}𝕊)

where the right hand map is perhaps not surjective on π_{0}, and K^{sm}(L̂_{n}𝕊) denotes the algebraic K-theory of the category of small K(n)-local spectra.

The lower vertical maps are induced by base change in the K(n)-local pro-𝔾_{n}- Galois extension L̂_{n}𝕊 → E_{n}. The small K(n)-local spectra X are characterized by their base changes, E_{n} ∧_{L̂_{n}𝕊} X ≃ L̂_{n}(E_{n} ∧ X), whose homotopy is denoted E_{n∗}^{∨}(X). Namely, X is small if and only if E_{n∗}^{∨}(X) is finite in each degree. Hence there is a natural map

K^{sm}(L̂_{n}𝕊) → K^{fin}(E_{n})^{h𝔾_{n}}

where K^{fin}(E_{n}) is the algebraic K-theory of the semi-finite cell E_{n}-modules that have degreewise finite homotopy groups, h𝒞_{E_{n}}^{fin}. By the brave new analogue of the Lichtenbum–Quillen conjecture, it is then to be expected that this map and the Galois descent map

K(L̂_{n}𝕊) → K(E_{n})^{h𝔾_{n}}

(where the source might really be K^{d}(L̂_{n}𝕊) formed from h𝒟_{L̂_{n}𝕊}, but this is not critical for this argument) are both close to equivalences. More precisely, these maps should be homotopy equivalences in v_{n+1}-periodic homotopy, obtained by smashing with a finite type n + 1 spectrum F (n + 1) and forming the mapping telescope for the v_{n+1}-self map, or equivalently, smashing with T (n + 1) = v_{n+1}^{−1}F (n + 1).

Let K_{n} = E_{n}/I be the even periodic Morava K-theory spectrum, with π_{∗}(K_{n}) = 𝔽_{p^{n}}[u^{±1}]. The 𝕊-algebra homomorphism π: E_{n} → K_{n} makes K_{n} a finite cell E_{n}-module spectrum, and more generally, composition with π makes each semifinite cell K_{n}-module a semi-finite cell E_{n}-module with degreewise finite homotopy groups. Optimistically, this inclusion h𝒞_{K_{n}} → h𝒞_{E_{n}}^{fin} induces a dévissage equivalence

K(K_{n}) → K^{fin}(E_{n})

in algebraic K-theory.

These arguments support the idea that the difference between K(L_{n}𝕊) and K(L_{n−1}𝕊) in the algebraic K-theory chromatic tower consists of a π_{0}-contribution, arising from the presence of properly semi-finite (not finite) cell spectra, and a relative term K^{sm}(L̂_{n}𝕊) given by the algebraic K-theory of the small K(n)-local spectra. The belief in Galois descent for algebraic K-theory, along the K(n)-local pro-Galois extension L̂_{n}𝕊 → E_{n}, translates this into a contribution from K^{fin}(E_{n}) given by the algebraic K-theory of the semi-finite cell E_{n}-module spectra that have degreewise finite homotopy. This may, by dévissage, amount to the algebraic K-theory of the (non-commutative) 𝕊-algebra K_{n}, the even periodic variant of Morava’s K(n).

For n = 1, the diagram can be rewritten as

\xymatrix{ & K(J̃_{(p)}) \ar[r] \ar[d] & K(ℚ) \ar[d] \\ K^{sm}(J_{p}) \ar[r] \ar[d]^{ℤ_{p}^{×}} \ar[ur] & K(J_{p}) \ar[r] \ar[d]^{ℤ_{p}^{×}} & K(Jℚ_{p}) \\ K^{fin}(KU_{p}) \ar[r] & K(KU_{p})}

where J̃_{(p)} = L_{1}𝕊, J_{p} = L̂_{1}𝕊, Jℚ_{p} = J_{p}[1/p] ≃ Hℚ_{p} ∨ Σ^{−1}Hℚ_{p}, and KU/p = K_{1} maps K(KU/p) → K^{fin}(KU_{p}).

4. Algebraic K-theory of topoplogical K-theory.

We are thus led to study K(KU_{p}) and K(KU/p). There is a fiber sequence

K(ℤ) → K(ku) → K(KU)

due to Blumberg–Mandell, and the same proof should yield fiber sequences

K(ℤ_{p}) → K(ku_{p}) → K(KU_{p})

and

K(ℤ/p) → K(ku/p) → K(KU/p) .

These are analogous to the dévissage fiber sequence for K(ℤ_{p}) → K(ℚ_{p}), except that p-torsion is replaced by u-torsion. The advantage of this, is that the terms on the left and in the middle are connective 𝕊-algebras A with π_{0} = ℤ/p or ℤ_{p}, for which the cyclotomic trace map identifies K(A)_{p} with the connective cover of T C(A)_{p}.