DOI: 10.1017/s0010437x26103418 ISSN: 0010-437X
Algebraic
K
-theory of real topological
K
-theory
Gabriel Angelini-Knoll, Christian Ausoni, John Rognes 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
double struck upper F 2 left bracket v 2 Superscript 4 Baseline right bracket
F
2
[
v
2
4
]
$\mathbb{F}_2[v_2^4]$
-module of rank 52, on explicit generators in stems
negative 1 less than or slanted equals asterisk less than or slanted equals 30
−
1
⩽
∗
⩽
30
$-1 \leqslant \ast \leqslant 30$
. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn, Raksit and Wilson, extending work of Bhatt, Morrow and Scholze from the case of classical commutative rings to
double struck upper E Subscript normal infinity
E
∞
$\mathbb{E}_\infty$
-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.