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Pré-Publication, Document De Travail Année : 2024

Algebraic K-theory of real topological K-theory

Résumé

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.

Dates et versions

hal-04418028 , version 1 (25-01-2024)

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Gabriel Angelini-Knoll, Christian Ausoni, John Rognes. Algebraic K-theory of real topological K-theory. 2024. ⟨hal-04418028⟩
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