Descent and vanishing in chromatic algebraic \(K\)-theory via group actions. (Descente et annulation en \(K\)-théorie chromatique via les actions des groupes.) (English. French summary) Zbl 07904675
Summary: We prove some \(K\)-theoretic descent results for finite group actions on stable \(\infty\)-categories, including the \(p\)-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes’s redshift philosophy: in particular, we show that if \(R\) is an \(\mathbb{E}_\infty\)-ring spectrum with \(L_{T(n)}R=0\), then \(L_{T(n+1)}K(R)=0\). Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.
MSC:
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 55P42 | Stable homotopy theory, spectra |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
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