Algebraic K-theory of real topological K-theory

TL;DR

This paper computes the A(1)-homotopy of the topological cyclic homology of connective real K-theory (ko), revealing a detailed algebraic structure and extending prismatic and syntomic cohomology techniques to E-infinity rings.

Contribution

It extends prismatic and syntomic cohomology methods to E-infinity rings and computes the A(1)-homotopy of TC(ko), revealing new differentials in the motivic spectral sequence.

Findings

The associated graded is a free F_2[v_2^4]-module of rank 52.

Explicit generators identified in stems -1 to 30.

Nonzero differentials in the motivic spectral sequence from syntomic cohomology to TC.

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.