THE FRACTION FIELD OF TOPOLOGICAL K-THEORY

John Rognes

Contents

1. The p-adic integers and the p-adic numbers

2. Tate–Poitou arithmetic duality

3. The connective and the periodic p-adic Adams summands

4. A weak duality

5. The fraction field of the p-adic Adams summand

6. Topological higher arithmetic duality

This poster provides motivation for my talk on “Topological logarithmic structures”, scheduled for Thursday at 16:30.

Typeset by AMS-TEX

1. The p-adic integers and the p-adic numbers

Fix a prime p, work in a p-complete setting for simplicity, and consider Quillen’s localization sequence

K(ℤ/p) ⟶^{i_{∗}} K(ℤ_{p}) ⟶^{j^{∗}} K(ℚ_{p}) .

It is a cofiber sequence of spectra, where i_{∗} is the transfer associated to i: ℤ_{p} → ℤ/p and j^{∗} is the natural map associated to j: ℤ_{p} → ℚ_{p}. We use algebro-geometric conventions for the variance of these maps.

The proven Lichtenbaum–Quillen conjectures concern the étale descent properties of p-completed algebraic K-theory, here denoted K(A)_{p}. The main case is that of Galois descent: If A → B is a G-Galois extension, then the induced map

K(A)_{p} → K(B)_{p}^{hG}

is an equivalence in sufficiently high degrees.

This is more interesting in the case of the fraction field ℚ_{p} than in the case of its valuation ring ℤ_{p}, simply because there are many more Galois extensions of the former than of the latter. For example, the p-th cyclotomic extension

ℚ_{p} → ℚ_{p}(ζ_{p})

is Δ-Galois, where Δ = (ℤ/p)^{×} ≅ ℤ/(p−1), whereas the map of valuation rings

ℤ_{p} → ℤ_{p}[ζ_{p}]

is not Galois, since it is ramified at (p). The canonical map h: ℤ_{p}[ζ_{p}] ⊗_{ℤ_{p}} ℤ_{p}[ζ_{p}] → ∏_{Δ} ℤ_{p}[ζ_{p}] has cokernel a nontrivial finite ℤ_{p}-module.

In the limit, we compare K(ℚ_{p})_{p} with continuous homotopy fixed points of K(\overline{ℚ}_{p})_{p}, and in the algebraically closed case we have Suslin’s theorem saying that

K(\overline{ℚ}_{p})_{p} ≃ ku_{p} .

Note that K(\overline{ℚ}_{p})_{p} is significantly simpler than K(ℚ_{p})_{p} or K(ℚ_{p}^{nr})_{p}, where ℚ_{p}^{nr} is the maximal unramified extension of ℚ_{p}.

2. Tate–Poitou arithmetic duality

The local field ℚ_{p} also has the exceptional feature that its Galois cohomology satisfies a form of Poincaré duality, known as Tate–Poitou arithmetic duality. The mod p Galois cohomology is related to the mod p algebraic K-theory by an Atiyah–Hirzebruch spectral sequence

E_{s,t}^{2} = H_{Gal}^{−s}(ℚ_{p}; 𝔽_{p}(t/2)) ⟹ K_{s+t}(ℚ_{p}; ℤ/p)

(for 2s + t ≥ 0), and the cup product

H_{Gal}^{r}(ℚ_{p}; 𝔽_{p}(i)) ⊗ H_{Gal}^{2−r}(ℚ_{p}; 𝔽_{p}(1 − i))

⟶^{∪} H_{Gal}^{2}(ℚ_{p}; 𝔽_{p}(1)) ≅ [p] Br(ℚ_{p}) ≅ ℤ/p

is a perfect pairing. Here Br(F) is the Brauer group of central simple F - algebras up to Morita equivalence.

Notation

ku_{p} connective p-complete topological K-theory
π_{∗}ku_{p} = ℤ_{p}[u], |u| = 2 ;
KU_{p} periodic p-complete topological K-theory
π_{∗}ku_{p} = ℤ_{p}[u, u^{−1}] ;
ℓ_{p} the connective p-complete Adams summand
π_{∗}ℓ_{p} = ℤ_{p}[v_{1}], |v_{1}| = 2p − 2 ;
L_{p} the periodic p-complete Adams summand
π_{∗}L_{p} = ℤ_{p}[v_{1}, v_{1}^{−1}] .

3. The connective and the periodic p-adic Adams summands

There is a similar localization sequence

K(ℤ_{p}) ⟶^{π_{∗}} K(ℓ_{p}) ⟶^{ρ^{∗}} K(L_{p})

established by Andrew Blumberg and Mike Mandell. Here π_{∗} is the transfer associated to the zero-th Postnikov section π: ℓ_{p} → Hℤ_{p}, and ρ^{∗} is the natural map associated to the localization map ρ: ℓ_{p} → L_{p}.

Also in this case there are more Galois extensions for the periodic commutative S-algebras than for their connective covers. There is a Δ-Galois extension

L_{p} → KU_{p} ,

with Δ as above, but

ℓ_{p} → ku_{p}

is not Galois. The canonical map h: ku_{p} ∧_{ℓ_{p}} ku_{p} → ∏_{Δ} ku_{p} has cofiber a nontrivial ku_{p}-module with finite Postnikov tower. In other words, ℓ_{p} → ku_{p} is ramified at (u) over (v_{1}).

Still, it is hard to find many G-Galois extensions of L_{p}. Andy Baker and Birgit Richter have recently shown that the largest connected one is

\overline{KU}_{p} = KU_{p}^{nr} ,

with

π_{∗}KU_{p}^{nr} = 𝕎\overline{𝔽}_{p}[u, u^{−1}] ,

where 𝕎 denotes the Witt ring functor.

A T C-calculation, like the one mentioned in the next section, shows that the algebraic K-theory K(\overline{KU}_{p})_{p} of this algebraic closure in commutative S-algebras is not any simpler than that of L_{p} and KU_{p}. This is in contrast to Suslin’s theorem, and suggests that the algebraic closure should be formed in a category where also ramified Galois extensions exist.

4. A weak duality

If we assume that there is a motivic cohomology theory for these commutative S-algebras, then we also expect a spectral sequence relating it to algebraic K-theory. This time we prefer to work with V (1)-coefficients, where

V (1) = S/(p, v_{1}) = S ∪_{p} e^{1} ∪_{α_{1}} e^{2p−1} ∪_{p} e^{2p}

is the Smith–Toda complex, and V (1)_{∗}X = π_{∗}(V (1) ∧ X). The Atiyah– Hirzebruch spectral sequence may then take the form

E_{s,t}^{2} = H_{mot}^{−s}(ℓ_{p}; 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K_{s+t}(ℓ_{p})

(for 2s + t ≥ 0), and similarly for L_{p} in place of ℓ_{p}.

By computing with topological cyclic homology, Christian Ausoni and the author determined the abutment of this expected spectral sequence for p ≥ 5, in which case its E^{2}-term has a more-or-less obvious form. There is a v_{2}-self map of degree (2p^{2} − 2) acting on V (1), so v_{2} acts on the spectral sequence by a shift of (p^{2} − 1) in t/2, as well as on the abutment. The remarkable outcome of the computations is that the algebraic K-theory abutment is v_{2}-periodic, i.e., it is v_{2}-torsion free.

In this case, once the motivic cohomology is made periodic in the tgrading (and christened Galois cohomology), there is no difference between the ℓ_{p}- and the L_{p}-case. Furthermore, there is an apparent perfect cup product pairing

H_{Gal}^{r}(L_{p}; 𝔽_{p^{2}}(i)) ⊗ H_{Gal}^{3−r}(L_{p}; 𝔽_{p^{2}}(d − i)) ⟶^{∪} H_{Gal}^{3}(L_{p}; 𝔽_{p^{2}}(d)) ≅ ℤ/p ,

where d = p^{2} +p. However, since in this case the target group of the pairing depends on the prime p (in the twist d), there is little hope of an intrinsic understanding of this group, like the identification of H_{Gal}^{2}(F; 𝔾_{m}) with the Brauer group Br(F) mentioned above.

5. The fraction field of the p-adic Adams summand

It appears that we should not view L_{p} as a (fraction) field in commutative S-algebras, due to the possibility of ramification over (p). So we seek some form of localization of L_{p} away from p, which we will formally denote as

ff L_{p} = p^{−1}L_{p} .

By this we do not mean the usual algebraic Bousfield localization L_{p}[p^{−1}] = Lℚ_{p}, with π_{∗}Lℚ_{p} = ℚ_{p}[v_{1}, v_{1}^{−1}], since this is an Hℚ_{p}-algebra, so K(Lℚ_{p}) is a K(ℚ_{p})-module, and V (1)_{∗}K(ℚ_{p}) is finite, so V (1)_{∗}K(Lℚ_{p}) will only consist of v_{2}-torsion. This completely loses the connection with the v_{2}- periodicity that we think is the essential feature in this story. We are looking for a milder, less drastic, localization.

Instead we can show that ℓ/p = k(1) admits the structure of a noncommutative ℓ_{p}-algebra, so L/p = K(1) is an L_{p}-algebra, and we can form the following commutative 3 × 3-diagram of horizontal and vertical cofiber sequences:

\xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ_{p}) \ar[r]^{j^{∗}} \ar[d]^{π_{∗}} & K(ℚ_{p}) \ar[d]^{π_{∗}} \\ K(ℓ/p) \ar[r]^{i_{∗}} \ar[d]^{ρ^{∗}} & K(ℓ_{p}) \ar[r]^{j^{∗}} \ar[d]^{ρ^{∗}} & K(p^{−1}ℓ_{p}) \ar[d]^{ρ^{∗}} \\ K(L/p) \ar[r]^{i_{∗}} & K(L_{p}) \ar[r]^{j^{∗}} & K(ff L_{p})}

For now, the terms K(p^{−1}ℓ_{p}) and K(ff L_{p}) are just notation for the homotopy cofiber spectra that fit in the diagram. The Thursday talk will start to give some intrinsic sense to these expressions.

We thus view Spec(ff L_{p}) as a kind of complement of Spnc(L/p) in Spec(L_{p}), and similarly for p^{−1}ℓ_{p}.

6. Topological higher arithmetic duality

Further topological cyclic homology calculations by Ausoni and Rognes determine V (1)_{∗}K(ℓ/p), V (1)_{∗}K(p^{−1}ℓ_{p}) and V (1)_{∗}K(ff L_{p}). These remain v_{2}-periodic, and now the expected motivic and Galois cohomology groups of ff L_{p} are better behaved than those of L_{p}, in that there is again an apparent perfect cup product pairing

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

where the twist in the target group is now independent of p.

This is exactly the kind of arithmetic duality as is seen in the Galois cohomology of “higher” local fields, like 𝔽_{p}((s, t)) and ℚ_{p}((t)), for which the local class field theory is expressed in terms of Milnor K-theory. The calculations therefore suggest that the fraction field ff L_{p}, and its analogously defined Δ-Galois extension ff KU_{p}, are brave new 2-dimensional “higher” local fields, invisible to classical algebra.

The valuation ring of ff L_{p} is then p^{−1}ℓ_{p}, with residue field ℚ_{p} = p^{−1}ℤ_{p}. We expect that there are Galois extensions

\xymatrix{\overline{ff L_{p}} \\ ff L\overline{ℤ}_{p} \ar[u]_{I_{v_{1}}} & _{\overline{ℚ}_{p}} \\ ff L_{p}^{nr} \ar[u]_{I_{p}} & ℚ_{p}^{nr} \ar[u]_{I_{p}} & _{\overline{𝔽}_{p}} \\ ff L_{p} \ar[u]_{ℤ̂} \ar[uuu]^{G_{ffLp}} & ℚ_{p} \ar[u]_{ℤ̂} \ar[uu]^{G_{ℚp}} & 𝔽_{p} \ar[u]_{ℤ̂}^{G_{𝔽p}}}

and group extensions

I_{v_{1}} → G_{ffL_{p}} → G_{ℚ_{p}} and I_{p} → G_{ℚ_{p}} → G_{𝔽_{p}} ,

where G_{𝔽_{p}} has p-cohomological dimension 1, G_{ℚ_{p}} has p-cohomological dimension 2, and the hypothetical G_{ffL_{p}} has p-cohomological dimension 3.

Department of Mathematics, University of Oslo, Norway

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