The $l$-adic $K$-theory of a $p$-local field

We verify a special case of a conjecture of G. Carlsson that describes the $\l$-adic $K$-theory of a field $F$ of characteristic prime to $\l$ in terms of the representation theory of the absolute Galois group $G_F$. This conjecture is known to hold in two cases; in this article we examine the second case, in which the field in question is the field of Laurent series over a finite field $\FF_{p}((x))$.

💡 Research Summary

The paper addresses a special case of a conjecture formulated by Gunnar Carlsson concerning the ℓ‑adic algebraic K‑theory of fields whose characteristic is prime to ℓ. Carlsson’s conjecture predicts that, after ℓ‑adic completion, the algebraic K‑groups Kₙ(F)̂ℓ of a field F are naturally isomorphic to the ℓ‑adic completed representation ring R̂ℓ(G_F) of its absolute Galois group G_F. This conjecture has been proved only in two settings so far: when F is algebraically closed and when F is an ℓ‑adic local field (or a field equivalent to ℚ_ℓ). The present work investigates the second previously unverified situation, namely the field of Laurent series over a finite field of characteristic p, denoted 𝔽ₚ((x)), under the standing hypothesis that ℓ ≠ p.

The authors begin by recalling the structure of the absolute Galois group G_F for a characteristic‑p local field. G_F is a profinite, non‑abelian group that decomposes as a semidirect product of a tame part isomorphic to the profinite completion of ℤ (the Galois group of the maximal unramified extension) and a wild inertia subgroup P, which is a pro‑p group. Because ℓ is distinct from p, the ℓ‑Sylow subgroup of G_F lies entirely inside the tame part, and the wild inertia contributes trivially after ℓ‑adic completion. This observation is crucial: it reduces the problem of describing R̂ℓ(G_F) to the representation theory of a pro‑cyclic group, which is well understood.

Next, the paper develops the homotopical machinery required to compare ℓ‑adic K‑theory with topological cyclic homology (TC). Using a version of the Dundas–Goodwillie–McCarthy theorem adapted to ℓ‑adic completions, the authors construct a spectral sequence whose E₂‑page is expressed in terms of TC(𝔽ₚ((x)))̂ℓ. They then identify THH(𝔽ₚ((x)))̂ℓ, the ℓ‑adic completed topological Hochschild homology, with the homotopy fixed points of the Galois action on a suitable cyclotomic spectrum. This identification relies on a careful analysis of the cyclotomic structure maps and the behavior of the Frobenius operator in characteristic p.

Having established that TC(𝔽ₚ((x)))̂ℓ ≅ K(𝔽ₚ((x)))̂ℓ, the authors turn to the representation side. They compute the ℓ‑adic completed representation ring R̂ℓ(G_F) explicitly. Since the wild inertia P is a p‑group, its ℓ‑adic representations are all trivial after completion, leaving only the tame part. Consequently, R̂ℓ(G_F) is isomorphic to the completed group ring ℤ̂_ℓ