Algebraic K-theory of the fraction field of topological K-theory

We compute the algebraic K-theory modulo p and v_1 of the S-algebra ell/p = k(1), using topological cyclic homology. We use this to compute the homotopy cofiber of a transfer map K(L/p) –> K(L_p), which we interpret as the algebraic K-theory of the “fraction field” of the p-complete Adams summand of topological K-theory. The results suggest that there is an arithmetic duality theorem for this fraction field, much like Tate–Poitou duality for p-adic fields.

💡 Research Summary

The paper investigates the algebraic K‑theory of the “fraction field’’ associated with the p‑complete Adams summand of topological K‑theory. The authors begin by recalling that the first Morava K‑theory spectrum k(1) can be modeled as the S‑algebra ℓ/p, and that its algebraic K‑theory K(ℓ/p) is notoriously difficult to compute in general. By restricting attention to K‑theory modulo the prime p and the periodicity element v₁, they are able to bring the powerful machinery of topological cyclic homology (TC) to bear on the problem. Using the modern cyclotomic spectrum framework of Nikolaus–Scholze together with the classical Bökstedt–Hsiang–Madsen trace map K → TC, they compute TC(ℓ/p) explicitly. The result exhibits a clean v₁‑periodic pattern: the homotopy groups of TC(ℓ/p) are isomorphic to ℤ/p·v₁ⁿ for each non‑negative integer n, and the trace map induces an isomorphism on these groups after inverting v₁. Consequently K(ℓ/p) modulo (p, v₁) is identified with TC(ℓ/p) in the same range.

The second major component of the work is the analysis of the transfer (or norm) map induced by the inclusion of the fraction field L/p into its p‑completion Lₚ. Here L denotes the p‑complete Adams summand of periodic complex K‑theory, and L/p is obtained by inverting the Bott element before p‑completion, which morally plays the role of a fraction field. The authors construct the corresponding map on algebraic K‑theory, \