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The Continuous K-Groups of Nuclear C_p-Vector Spaces

Author: Wei Yang · Added: August 9, 2026

Article date: August 9, 2026.

Abstract

We compute the homotopy groups of $K(\operatorname{Nuc}(\mathbb{C}_p))$ , together with their Galois-equivariant and condensed refinements. They vanish in negative and positive even degrees, while $K_0\cong\mathbb{Z}$ , $K_1\cong\mathbb{C}_p^\times$ , and, for $k\geq2$ ,

K_{2k-1}^{\mathrm{cont}}\bigl(\operatorname{Nuc}(\mathbb{C}_p)\bigr) \cong \bigoplus_{\ell\ne p}(\mathbb{Q}_\ell/\mathbb{Z}_\ell)(k) \oplus (B_{\mathrm{cris}}^+)^{\varphi=p^k}/\mathbb{Z}_p(k).

This high-odd isomorphism may be chosen for the underlying abstract Galois action. Its prime-to- $p$ splitting is induced by reduction to the special fiber, whereas the crystalline identification is generally noncanonical. In condensed abelian groups we retain the internal continuous Galois action and identify the $p$ -primary term with Fontaine’s quotient topology. The proof uses formal/Tate localization, finite-coefficient rigidity, the rational Beilinson comparison, and the arithmetic gluing map supplied by the cyclotomic trace.