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On and around some conjectures of Ausoni and Rognes Noel, Justin


Ausoni and Rognes have conjectured that the algebraic \(K\)-theory of commutative \(S\)-algebras should satisfy a form of Galois descent after telescopic localization (at least in special cases). In the case of discrete commutative rings this is a celebrated result of Thomason. Now by a result of Mitchell, the algebraic \(K\)-theory of discrete rings only has chromatic information at heights \(0\) and \(1\) and the proof of Thomason's result at height \(0\) (i.e., rationally) is actually quite easy. In joint work with Dustin Clausen, Akhil Mathew, and Niko Naumann we show that the easy case of Thomason's result combined with May's nilpotence conjecture implies the hard case. Moreover, this approach can be combined with a joint result with Lennart Meier and Niko Naumann to establish important cases of the Ausoni-Rognes conjecture. We also give a new proof of Mitchell's theorem. This proof uses a general technique for bounding the chromatic complexity of commutative \(S\)-algebras using techniques from equivariant homotopy theory. In the case of \(KU\) we are able to show \(K(KU)\) has chromatic information only up to height \(2\) at the primes \(2\), \(3\), and \(5\). Ausoni and Rognes established this result for primes \(5\) and higher by explicit calculation.

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