The algebraic K-theory of the chromatic filtration and the telescope conjecture
We develop tools for understanding the algebraic K-theory of categories such as those coming from the chromatic filtration of the stable homotopy category, and apply these tools to improve our understanding of the large scale structure of stable homotopy theory and understand Ravenel's telescope conjecture. More specifically, in joint work with Burklund, we prove a general devissage result which in particular identifies the algebraic K-theory of certain coconnective ring spectra satisfying suitable regularity and flatness hypotheses with the K-theory of their pi₀. Using this and an extension of the Dundas--Goodwillie--McCarthy theorem to —1-connective ring spectra, we obtain a formula for the algebraic K-theory of the K(1)-local sphere in terms of topological cyclic homology of a ring spectrum j_zeta, and in particular find that its algebraic K-groups are not all finitely generated. In joint work with Lee, we extend these computations to understand the algebraic K-theory of the K(1)-local sphere in the stable range using THH, where we observe phenomena such as the failure of Zₚ Galois descent for THH for an extension of j_zeta. In joint work with Burklund, Hahn, and Schlank, we show that the failure of Zₚ-descent also happens for the T(2)-local TC of this extension. Combining this with the cyclotomic redshift result of Ben-Moshe--Carmeli--Schlank--Yanovski, this implies that the T(2)-local algebraic K-theory of the K(1)-local sphere is not K(2)-local, and hence a counterexample to the height 2 telescope conjecture. We also give similar counterexamples to the height n telescope conjecture for all n≥2 and all primes, and show that Zₚ Galois hyperdescent for chromatically localized algebraic K-theory generically fails.
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