Descent and vanishing in chromatic algebraic \(K\)-theory via group actions (Q6596072)

Let \(p\) be a prime and let \(T(n)\) denote a telescope of height \(n \geq 0\) at the prime \(p\) in the sense of chromatic homotopy theory. Denote the \(T(n)\)-localization functor by \(L_{T(n)}\). The authors prove a number of results relating to the following questions:\N\begin{itemize}\N\item[1.] Given a ring spectrum \(R\), when is \(L_{T(n)}K(R) = 0\)?\N\item[2.] Let \(G\) be a group and let \(\mathcal{C}\) be a stable \(\infty\)-category with a \(G\)-action. When is the coassembly map \(L_{T(n)}K(\mathcal{C}^{hG}) \to (L_{T(n)}K(\mathcal{C}))^{hG}\) an equivalence?\N\end{itemize}\NThe first set of results concerns the case of \(\mathbb{E}_\infty\)-ring spectra \(R\). Here, the authors prove that the questions are connected in the sense that \(T(n)\)-local vanishing of \(R\) combined with the \(T(n)\)-local vanishing of the K-theory of the Tate construction \(R^{tC_p}\) implies a positive answer to the second quesiton for all \(p\)-groups and all idempotent-complete \(R\)-linear stable \(\infty\)-categories \(\mathcal{C}\), and that this in turn implies \(T(n+1)\)-local vanishing of the K-theory of \(R\). By a theorem of Hahn, the \(T(n+1)\)-local vanishing of \(K(R)\) implies also the vanishing of \(L_{T(m)}K(R)\) for \(m \geq n+1\).\N\NThis is used to show for example that\N\begin{itemize}\N\item[1.] \(L_{T(n+1)}K(R) = 0\) if \(L_{T(n)}R = 0\);\N\item[2.] if \(L_{T(n)}(R^{tC_p}) = 0\), then the coassembly map for \(L_{T(n+1)}K\) is an equivalence for idempotent-complete \(R\)-linear stable \(\infty\)-categories \(\mathcal{C}\) with an action of a \(p\)-group;\N\end{itemize}\NThrough different arguments, they also provide conditions when the coassembly map \(K(\mathcal{C}^{hG}) \to K(\mathcal{C})^{hG}\) is a \(T(n)\)-local equivalence for a stably monoidal \(\infty\)-category with \(G\)-action.\N\NIn addition, they study the analogous question for the coassembly map with respect to non-trivial families of subgroups. Here, they show that for an \(\mathbb{E}_\infty\)-ring spectrum \(R\), the respective coassembly map is a \(T(n)\)-local equivalence for every \(R\)-linear stable \(\infty\)-category and every finite group if the corresponding coassembly map for the rationalized representation ring over \(R\) is an equivalence. Moreover, such an induction statement for \(C_p^{\times n}\) is shown to imply the \(T(n)\)-local vanishing of \(K(R)\), which in particular provides a new proof of Mitchell's theorem [\textit{S. A. Mitchell}, \(K\)-Theory 3, No. 6, 607--626 (1990; Zbl 0709.55011)].