Algebraic K-theory of the fraction field of
topological K-theory
Christian Ausoni and John Rognes
August 4 2009
Overview
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Motivation.
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Algebraic K -theory of `.
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The “fraction field” F(`) of `.
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Algebraic K -theory of F(`).
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Speculations.
Algebraic K -theory of S-algebras
We are interested in computing the algebraic K -theory of
S-algebras, and more particularly of “easy” S-algebras, like
topological K -theory.
Our motivations are
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Waldhausen’s A-theory A(M) ' K (S[ΩM]).
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Interpolating from Z to S and exploring which structural
properties of the algebraic K -theory known for rings also hold
when we climb the chromatic tower.
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Study the arithmetic/algebro-geometric properties of
S-algebras through their algebraic K -theory, serving as “test”
for new concepts.
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Explore the “chromatic red-shift” phenomenon.
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K (ku) represents a form of elliptic cohomology theory derived
from a “geometric” construction : the two-vector bundles of
Baas, Dundas and Rognes.
Trace methods
The main tool we have so far is the theory of trace maps
developed by Waldhausen, Goodwillie, Böktstedt-Hsiang-Madsen,
Dundas, Hesselholt, and many others.
If A is a p-completed connective S-algebra, the Bökstedt trace
map tr : K (A) → THH(A) (topological analogue of the Dennis
trace) factors through the cyclotomic trace map to topological
cyclic homology
trc
K (A) −→ TC (A) → THH(A)
If π0 (A) = Zp or Fp , then there is a cofibre sequence (Dundas,
Hesselholt-Madsen)
trc
K (A)p −→ TC (A) → Σ−1 HZp
allowing one to evaluate K (A)p from TC (A).
Connective K -theory
Let p be an odd prime. We consider the commutative S-algebras
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ku, connective complex K -theory (corresponds to the infinite
loop space BU × Z), and kup , its p-completion.
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`, the (p-completed) Adams summand of kup .
Here ` = kuph∆ (homotopy fixed points), where ∆ ∼
= Z/(p − 1) is
∗
the torsion subgroup of Zp , acting on kup by p-adic Adams
operations.
The coefficient rings are
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ku∗ = Z[u] and kup = Zp [u], with |u| = 2 (Bott class),
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`∗ = Zp [v1 ], with |v1 | = 2p − 2.
The “inclusion” ` → kup induces Zp [v1 ] ⊂ Zp [u], v1 7→ u p−1 .
View ` → kup as a tamely ramified extension with Galois group ∆,
analogous to Zp → Zp [ζp ].
This ramification makes the formulas for TR(ku) more
complicated, so in the sequel we concentrate on `.
Finite coefficients and Bott periodicity
The algebraic K -theory of a number field F or of its integers OF is
much easier to handle if one introduces mod (p) coefficients
K∗ (F ; Z/p) = V (0)∗ K (F ) = π∗ (V (0) ∧ K (F )),
p
where V (0) is the cofibre of S → S (notice that K∗ (F ; Z/p) is
related to K∗ (F ) by a long exact sequence).
The advantages are that
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divisible subgroups of K∗ (F ) vanish,
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V (0)∗ K (F ) is, in high enough degrees, v1 or “Bott” periodic.
The last fact is accounted for by Suslin’s Theorem
K (F̄ ) 'p ku,
together with Galois descent, and is the first example of red-shift
(explaining why there is no good algebraic definition of algebraic
K -theory).
V (1)-homotopy
The p-local stable homotopy category features higher forms of
periodicity, one for each integer n > 0, referred to as vn -periodicity.
It is detected by Morava K -theory K (n), with coefficients
K (0)∗ = Q and K (n)∗ = Fp [vn , vn−1 ] if n > 1.
Here vn -periodicity has period |vn | = 2p n − 2.
Because of red-shift, the algebraic K -theory of ` or other K -theory
spectra is easier to handle by using mod (p, v1 )-coefficients
V (1)∗ K (`) = π∗ (V (1) ∧ K (F )),
v
where V (1) is the cofibre of Σ2p−2 V (0) →1 V (0).
Notice that for p > 5, V (1) is a commutative ring spectrum, and
2
has a periodic map v2 : Σ2p −2 V (1) → V (1). In particular, for any
spectrum X , V (1)∗ X is a module over P(v2 ) = Fp [v2 ].
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From now on, we assume p > 5.
V (1)-homotopy of THH(`)
Theorem (McCLure-Staffeldt)
There is an isomorphism of Fp -algebras
V (1)∗ THH(`) ∼
= E (λ1 , λ2 ) ⊗ P(µ2 ),
where |λ1 | = 2p − 1, |λ2 | = 2p 2 − 1 and |µ2 | = 2p 2 .
Remark
Angeltveit, Hill and Lawson computed THH∗ (`) from
V (1)∗ THH(`) by means of Bockstein spectral sequences. The
answer shows complicated p- and v1 -torsion patterns.
V (1)-homotopy of K (`)
Theorem (A, Rognes)
There is an isomorphism of P(v2 )-modules
V (1)∗ K (`) = P(v2 ) ⊗ E (λ1 , λ2 )
⊕ P(v2 ) ⊗ Fp {∂λ1 , ∂v2 , ∂λ2 , ∂λ1 λ2 }
⊕ P(v2 ) ⊗ E (λ2 ) ⊗ Fp {t d λ1 | 0 < d < p}
⊕ P(v2 ) ⊗ E (λ1 ) ⊗ Fp {t dp λ2 | 0 < d < p}
⊕ Fp {s}
with |∂| = −1, |t| = −2 and |s| = 2p − 3.
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V (1)∗ K (`) is almost a free P(v2 )-module!
Here we wish to interpret λ1 and λ2 as follows: for a suitable
fraction field f : ` → F(`) we ought to have f∗ (λ1 ) = v1 δ1 in
V (0)∗ K (F(`)) and f∗ (λ2 ) = v2 δ2 in V (1)∗ K (F(`)), with
δ1 , δ2 ∈ K1 (F(`)) corresponding to the units p, v1 in F(`).
Lichtenbaum-Quillen Conjecture for `
A second reason to search for F(`) is to try to understand the
above computation of K (`) by means of Galois descent, via a
generalization of the Lichtenbaum-Quillen Conjectures.
This requires to first pass from ` to F(`), because ` itself has not
enough interesting Galois extensions.
Conjectures (Rognes)
1. If Ω is a separable closure of F(`), then there is an equivalence
K (Ω) 'K (2) E2
where E2 is Morava’s E -theory with E2∗ = W (Fp2 )[[u1 ]][u ±1 ].
2. If F(`) → B is a G -Galois extension, then
V (1)∗ K (F(`)) → V (1)∗ K (B)hG
is an isomorphism in high enough degrees.
Periodic K -theory
In view of `∗ = Zp [v1 ] and of our understanding of the stable
homotopy category (chromatic picture), it seems reasonable that it
suffices to “invert” v0 = p and v1 in ` to obtain F(`).
The S-algebras mentioned above all have a corresponding periodic
version :
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KU, with KU∗ = Z[u ±1 ], and KUp , with KUp ∗ = Zp [u ±1 ],
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L, the periodic Adams summand, with L∗ = Zp [v1±1 ].
Thus the canonical algebra map ` → L corresponds to inverting v1
(this will be confirmed later).
What about inverting p ?
The obvious candidate is F(`) = L[ p1 ], having as coefficients the
graded field Qp [v1±1 ]. This would be disappointing since L[ p1 ] is an
HQ-algebra.
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We use algebraic K -theory to test this candidate.
Localization sequences in K -theory
Indeed, we expect K (F(`)) to fit in the following diagram of
localization sequences in algebraic K -theory
K (Fp )
/ K (Zp )
/ K (Qp )
K (`/p)
/ K (`p )
/ K (p −1 `p )
K (L/p)
/ K (Lp )
?
/ K (F(`)) .
The top-row is Quillen’s localization sequence with respect to
p ∈ Zp , and the left and middle rows are the localization sequences
with respect to v1 ∈ ` and v1 ∈ `/p conjectured by Rognes and
established by Blumberg-Mandell.
The other sequences are potentially localization sequences, and
here K (F(`)) is actually defined as the iterated cofibre of the
upper left square. Thus to compute V (1)∗ K (F(`)) it essentially
remains to compute V (1)∗ K (`/p).
Mod p K-theory
In the above diagram, we defined `/p and L/p as the cofibre of the
multiplication-by-p self-maps,
p
p
` → ` → `/p, L → L → L/p,
with `/p ∗ = Z/p[v1 ] and L/p ∗ = Z/p[v1 , v1−1 ].
Notice that `/p and L/p admit an S-algebra structure, but not a
commutative one.
Main Theorem (A, Rognes)
The P(v2 )-module V (1)∗ K (`/p) is a free P(v2 )-module of
rank 2p 2 − 2p + 8, on explicitly given generators.
The V (1)-homotopy of K (F(`))
Corollary
There is an isomorphism of P(v2 )-modules
V (1)∗ K (F(`)) ∼
= N ⊕ T,
where
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N is a free P(v1 )-module of rank (2p 2 + 6),
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T is an Fp -module of rank 4 with v2 T = 0.
Moreover, there are classes δ1 , δ2 ∈ V (1)1 K (F(`)) with v1 δ1 = λ1
and v2 δ2 = λ2 .
Remark
Notice that V (1)∗ K (L[ p1 ]) is a v2 -torsion module, because it is a
module over V (1)∗ K (Qp ), which is v2 -torsion. With the
computation of V (1)∗ K (F(`)) above, this indicates:
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The HQp -algebra L[ p1 ] does not qualify for F(`).
The fraction field
It turns out that the structure of V (1)∗ K (F(`)) fits very nicely
with the Lichtenbaum-Quillen Conjectures for `.
Indeed, we consider a conjectural Galois-descent spectral sequence
−s
2
Es,t
= HGal
(F(`); Fp2 (t/2)) ⇒ v2−1 V (1)s+t K (F(`))
where the coefficient module is
v2−1 V (1)∗ K (Ω) = V (1)∗ E2 = Fp2 [u ±1 ]
with action of GF (`) induced by its action on K (Ω). Working
2 .
backwards, Rognes conjecturally evaluated Es,t
The good surprise is that
I The cohomological dimension is 3,
I H ∗ (F(`); Fp 2 (∗)) has self-duality, analogous to Tate-Poitou
Gal
duality for local fields:
3−n
n
HGal
(F(`); Fp2 (2 − k)) ∼
(F(`); Fp2 (k))∗ .
= HGal
This feature is not present in the case of ` or L. It suggest that
F(`) is a form of a 2-local field in mixed characteristic 0, p and v1 .
Summary of the above computations of K (F(`))
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Use trace methods to compute K (Zp ), K (`/p) and K (`).
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Use iterated cofibre sequences to evaluate K (L/p), K (L) and
K (F(`)).
A direct trace computation of K (F(`)), starting with THH(F(`)),
would be nicer and would shed light on the nature of F(`).
However, notice that the corresponding sequences
THH(Zp ) → THH(`) → THH(L)
and
TC (Zp ) → TC (`) → TC (L)
are NOT cofibre sequences.
Similarly, we do not expect to have a localization sequence
THH(L/p) → THH(L) → THH(F(`)),
and neither for TC .
The case of local fields
Suppose that K is a complete discrete valuation field with perfect
residue field k of odd characteristic p. Let A be the valuation ring.
Hesselholt and Madsen define ad hoc relative versions THH(A|K )
and TC (A|K ) (using a suitable category of modules), which fit in
a diagram of horizontal localization cofibre sequences
K (k)
/ K (A)
/ K (K )
/ TC (A)
/ TC (A|K )
/ THH(A)
/ THH(A|K ).
TC (k)
THH(k)
They compute V (0)∗ TC (A|K ), which turns out to be much more
manageable then V (0)∗ TC (A).
Differentials with logarithmic poles
There is a natural short exact sequence
0 → Ω1A = THH1 (A) → THH1 (A|K ) → THH0 (k) = k → 0.
It can be compared to the natural short exact sequence
1
0 → Ω1A → ω(A,M)
→ k → 0.
1
Here M = A ∩ K ∗ and ω(A,M)
= (Ω1A ⊕ A ⊗Z K ∗ )/ ∼ is the target
of the universal derivation of the log ring (A, M).
Define
I d : A → THH1 (A|K ) as Connes’ operator (cyclic action),
using the identification A = THH0 (A|K ), and
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tr
dlog : M ⊂ K ∗ = K1 (K ) → THH1 (A|K ).
Then (d, dlog ) : (A, M) → THH1 (A|K ) is a derivation, and the
canonical map
∼
=
1
ω(A,M)
−→ THH1 (A|K )
is an isomorphism.
Log differential graded rings
This extends to a natural map of “log differential graded rings”
∗
ω(A,M)
→ THH∗ (A|K )
and to a natural map of “log Witt complexes”
∗
W• ω(A,M)
→ TR∗• (A|K )
over (A, M) (left-hand side is universal).
These maps allow H.-M. to organize the computation of
V (0)∗ THH(A|K ) and V (0)∗ TC (A|K ), and to evaluate
V (0)∗ K (K ).
In particular, they establish an isomorphism of log differential
graded rings
∼
=
∗
ω(A,M)
⊗Z Fp [κ] −→ V (0)∗ THH(A|K ),
where |κ| = 2 and dκ = κ dlog(−p).
In the example of K = Qp , this evaluates as
∼ E (dlog(p)) ⊗ P(κ).
V (0)∗ THH(Zp | Qp ) =
Returning to F(`)
Inspired by the case of local fields, we search for a good
construction of THH(− | −) in the case of ring spectra, leading to
localization cofibre sequences
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THH(Zp ) → THH(`) → THH(` | L)
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THH(Fp ) → THH(`/p) → THH(`/p | L/p)
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THH(`/p | L/p) → THH(` | L) → THH(` | F(`))
and similarly for TC (− | −), such that the localization cofibre
sequences for K , TC and THH are compatible with the traces, as
above.
Then TC (` | F(`)) would be the desired approximation of K (F(`)).
This test-case might lead to a suitable definition of K (F(`)) via
“log algebraic K -theory”.
Modest computational evidence
Some evidence for the existence of “log-THH” is provided by our
computations. First, using iterated cofiber sequences, as for
K (F(`)), we compute
V (1)∗ THH(` | F(`)) ∼
= E (dlog(p), dlog(v1 )) ⊗ P(κ)
with | dlog(p)| = | dlog(v1 )| = 1 and |κ| = 2.
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This fits very well with Hesselholt-Madsen’s result on
V (0)∗ THH(Zp | Qp ) ∼
= E (dlog(p)) ⊗ P(κ).
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Because of the simplicity of V (1)∗ THH(` | F(`)), computing
V (1)∗ TC (` | F(`)) is expected to be easier than V (1)∗ TC (`).
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In our formula for V (1)∗ K (F(`)), it allows us to interpret the
classes δ1 and δ2 as dlog(p) and dlog(v1 ), via the trace map
V (1)∗ K (F(`)) → V (1)∗ THH(` | F(`)).
Log-étale descent
In context of local fields as above, a finite extension K → F with
ramification index e prime to p defines a “log-étale” extension
(A, MA ) → (B, MB ), where B is the integral closure of A in F, as
reflected by the formulas
1
1
∼
ω(B,M
and
= B ⊗A ω(A,M
B )/W (k)
A )/W (k)
THH(B | F ) 'p B ∧A THH(A | K ).
Using the cofibre-definition of log-THH, we established an
equivalence
THH(kup | KUp ) 'p ku ∧` THH(` | L)
This suggests that the “tamely ramified” extension ` → kup
corresponds to a “log-étale” extension (` | L) → (kup | KUp ), and
that THH(− | −) ought also to have log-geometric content in this
context.