The K-theory of perfectoid rings

TL;DR

This paper investigates the properties of p-adic algebraic K-theory for smooth algebras over perfectoid rings, establishing homotopy invariance and relating it to p-adic generic fibres, extending previous results.

Contribution

It proves homotopy invariance of p-adic K-theory for these rings and generalizes known isomorphisms to all degrees in mixed characteristic cases.

Findings

p-adic K-theory is homotopy invariant

K-theory coincides with that of p-adic generic fibre in high degrees

Generalizes Nizio{2}'s result to all degrees in mixed characteristic

Abstract

We establish various properties of the p-adic algebraic K-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the p-adic K-theory of such rings is homotopy invariant, and coincides with the p-adic 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 Nizio{\l}.