On the étale descent problem for topological cyclic homology and algebraic \(K\)-theory
The author studies the descent properties of topological Hochschild homology, topological cyclic homology, and algebraic K-theory. He shows that THH after completion at a prime \(p\) has étale descent for any ring \(R\) which does not have \(p\)-torsion. If \(R\) has no torsion at all, THH has étale descent for \(R\). For \(R\) a Frobenius \(p\)-ring possessing a system of étale coverings by Frobenius \(p\)-rings which is cofinal among \(R\)'s étale coverings, he shows that topological cyclic homology completed at \(p\) has étale descent. Finally, for Frobenius \(p\)-rings with a cover as described above, with the additional condition that the cyclotomic trace with mod-\(p^\nu\) coefficients \[ K_i(R;{\mathbb Z}/p^\nu)\to \text{TC}_i(R;{\mathbb Z}/p^\nu) \] is an injection for some \(i\) and \(\nu\), he shows that K-theory with coefficients in \({\mathbb Z}/p^\nu\) has étale descent for that \(\nu\). If \(R\) is the ring of integers in a \(p\)-adic field which is unramified or tamely ramified, these conditions hold for \(R\) with any \(\nu\geq 1\), and the author deduces that algebraic K-theory has étale descent for \(R\).
scientific article; zbMATH DE number 1563256