ALGEBRAIC K-THEORY OF THE FRACTION

FIELD OF TOPOLOGICAL K-THEORY

John Rognes

Ringberg proposal: October 28th 2005

On joint work with Christian Ausoni.

(1) Let ku be the connective complex K-theory spectrum. There is a pushout square of associative ku-algebras

\xymatrix{ku \ar[r]^{i} \ar[d]^{π} & ku/p \ar[d]^{π} \\ Hℤ \ar[r]^{i} & Hℤ/p}

where ku/p is not commutative. We imagine localizations p^{−1}ku and p^{−1}KU such that there is a 3 × 3 square of cofiber sequences

\xymatrix{K(ℤ/p) \ar[r]^{i_{∗}} \ar[d]^{π_{∗}} & K(ℤ) \ar[r] \ar[d]^{π_{∗}} & K(ℤ[1/p]) \ar[d]^{π_{∗}} \\ K(ku/p) \ar[r]^{i_{∗}} \ar[d] & K(ku) \ar[r] \ar[d] & K(p^{−1}ku) \ar[d] \\ K(KU/p) \ar[r]^{i_{∗}} & K(KU) \ar[r] & K(p^{−1}KU)}

where i_{∗} and π_{∗} are the algebraic K-theory transfer maps. We propose that inverting all primes p in KU this way yields the S-algebraic fraction field ff (KU) of KU.

(2) To make computations, it is convenient to p-complete, and to restrict attention to the Adams summands ℓ_{p} and L_{p} of ku_{p} and KU_{p}, respectively. Here π_{∗}ℓ_{p} = ℤ_{p}[v_{1}] and π_{∗}L_{p} = ℤ_{p}[v_{1}^{±1}]. There are similar diagrams as above, replacing ku/p and KU/p by ℓ/p = k(1) and L/p = K(1), respectively. Then p^{−1}L_{p} = ff (L_{p}) is the fraction field of the Adams summand of p-adic topological K-theory. We view this as a “brave new” 2-local field, with valuation ring p^{−1}ℓ_{p} and residue field ℚ_{p}. Let V (1) be the Smith–Toda complex. It is a ring spectrum for p ≥ 5, and

Typeset by AMS-TEX

represents homotopy with mod (p, v_{1}) coefficients. Using topological Hochschild homology, topological cyclic homology and the cyclotomic trace map we compute

V (1)_{∗}K(ff (L_{p})) ≡ P (v_{2}) ⊗ E(∂v_{2}, δ_{1}, δ_{2})
⊕ P (v_{2}) ⊗ E(δ_{2}) ⊗ 𝔽_{p}{t^{i}λ_{1} | 0 < i < p}
⊕ P (v_{2}) ⊗ E(δ_{2}) ⊗ 𝔽_{p}{t^{i}v_{2}δ_{1} | 0 < i < p^{2} − p, (i, p) = 1}
⊕ P (v_{2}) ⊗ E(δ_{1}) ⊗ 𝔽_{p}{t^{ip}λ_{2} | 0 < i < p}

up to a tiny error term. This is a free module over P (v_{2}) of rank (2p^{2} + 6) and zero Euler characteristic. The symbols δ_{1} = d log p and δ_{2} = d log v_{1} play the role of logarithmic poles.

(3) We can arrange these groups as the E^{2} = E^{∞} term of a hypothetical motivic spectral sequence

E_{s,t}^{2} = H_{mot}^{−s}(ff (L_{p}); 𝔽_{p^{2}}(t/2)) ⟹ V (1)_{s+t}K(ff (L_{p})) .

Then the analogue of the Bloch–Kato conjecture holds: Letting the Galois cohomology groups

H_{Gal}^{n}(ff (L_{p}); 𝔽_{p^{2}}(k)) = v_{2}^{−1}H_{mot}^{n}(ff (L_{p}); 𝔽_{p^{2}}(k))

be the v_{2}-inverted motivic groups, one recovers the motivic groups by the truncation formula

H_{mot}^{n}(ff (L_{p}); 𝔽_{p^{2}}(k)) = \{ \begin{matrix}H_{Gal}^{n}(ff (L_{p}); 𝔽_{p^{2}}(k)) & for n ≤ k, \\ 0 & for n > k. \end{matrix}

(4) Also, the analogue for 2-local fields of Tate–Poitou local arithmetic duality holds: There is an isomorphism

H_{Gal}^{3}(ff (L_{p}); 𝔽_{p^{2}}(2)) ⟶^{inv} ℤ/p

taking ∂δ_{1}δ_{2} to the generator, and the Galois cohomology cup product

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

is visibly a perfect pairing.

(5) For each local number field F containing ℚ_{p}, with ring of integers O_{F}, there exist non-unique associative ℓ_{p}-algebras ℓO_{F} with π_{∗}ℓO_{F} = ℓO_{F} [v_{1}], by the A_{∞} obstruction theory of Robinson. When F is unramified, ℓO_{F} is uniquely a commutative ℓ_{p}-algebra. In the imagined localizations, there are nontrivial Kähler differentials and derived obstruction groups at p, but we hope that p^{−1}ℓO_{F} is uniquely a commutative p^{−1}ℓ_{p}-algebra, also for ramified F. Let ℤ_{p}^{nr} and ℤ̄_{p} be the ring of integers in the maximal unramified extension of ℚ and the separable closure of ℚ_{p}, respectively. Let Ω_{1} be the K(1)-local separable closure of ff (L_{p}). We would then have a tower of extensions

ff (L_{p}) ⟶^{G_{𝔽p}} ff (Lℤ_{p}^{nr}) ⟶^{I_{p}} ff (Lℤ̄_{p}) ⟶^{I_{v_{1}}} Ω_{1}

where I_{p} → G_{ℚ_{p}} → G_{𝔽_{p}} = ℤ̂ and I_{v_{1}} → G_{ff (L_{p})} → G_{ℚ_{p}}. We then expect the Galois cohomology of ff (L_{p}) introduced above to be the continuous cohomology of the absolute Galois group G_{ff (L_{p})} = Gal(Ω_{1}/ff (L_{p})). The discrepancy between the Galois cohomology of ℚ_{p} and that computed for ff (L_{p}) suggests where to find the remaining extensions from ff (Lℤ̄_{p}) to Ω_{1}.

Max Planck institute for Mathematics, Bonn