Algebraic K-theory of real topological K-theory
Abstract
We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.
Keywords
Cite
@article{arxiv.2309.11463,
title = {Algebraic K-theory of real topological K-theory},
author = {Gabriel Angelini-Knoll and Christian Ausoni and John Rognes},
journal= {arXiv preprint arXiv:2309.11463},
year = {2026}
}
Comments
59 pages, 13 figures, 1 table. Conjecture B from v1 is proved in v2, along with other improvements. A new Section 3 is added in v3 to provide further exposition, along with other improvements based on useful feedback from referees. Cosmetic edits have been made, v4 was accepted for publication in Compositio Mathematica