Verschiebung maps among K-groups of truncated polynomial algebras

Abstract

Let p be a prime number, and let A be a ring in which p is nilpotent . In this paper, we consider the maps K q + 1 ( A [ x ] / ( x m ) , ( x ) ) → K q + 1 ( A [ x ] / ( x m n ) , ( x ) ) , induced by the ring homomorphism A [ x ] / ( x m ) → A [ x ] / ( x m n ) , x ↦ x n . We evaluate these maps, up to extension, for general A in terms of topological Hochschild homology, and for regular F p -algebras A , in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative K -group of O K / p O K for certain perfectoid fields K .