The $\text{K}$-theory of perfectoid rings

Abstract

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.