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Algebraic K-theory of real topological K-theory
- Part of
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- Journal:
- Compositio Mathematica / Volume 162 / Issue 8 / August 2026
- Published online by Cambridge University Press:
- 30 September 2026, pp. 1760-1817
- Print publication:
- August 2026
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- Article
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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
$\mathbb{F}_2[v_2^4]$-module of rank 52, on explicit generators in stems 
$-1 \leqslant \ast \leqslant 30$. 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 
$\mathbb{E}_\infty$-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.
