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Let p∈Z be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace K(OF)∧p→TC(OF)∧p after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of Z.
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The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum
Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} .
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- PAR ID:
- 10390343
- Date Published:
- Journal Name:
- Transactions of the American Mathematical Society
- ISSN:
- 0002-9947
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/tran/8822
