The \(\text{K}\)-theory of perfectoid rings (Q2099422)

Summary: We establish various properties of the \(p\)-adic algebraic \(\text{K}\)-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the \(p\)-adic \(\text{K}\)-theory of such rings is homotopy invariant, and coincides with the \(p\)-adic \(\text{K}\)-theory of the \(p\)-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł.