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The p-adic K-theory of smooth algebras over perfectoid rings

Author: Wei Yang · Added: September 26, 2026

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Abstract

Let $A$ be a perfectoid ring and let $R$ be a smooth $A$-algebra. We prove that the $p$-completed algebraic $K$-theory of $R$ agrees with its $p$-completed homotopy $K$-theory. If $R/pR$ has relative dimension at most $d$ over $A/pA$, we also prove that the fiber of $K(R;\mathbf{F}_p)\to K(R[1/p];\mathbf{F}_p)$ has no homotopy groups above degree $d$. This answers Question 5.14 of Antieau—Mathew—Morrow. The proof constructs a perfectoid coefficient extension which induces injections on the homotopy groups of $K$-theory modulo $p$ with support on $p$ and receives a map from a mixed-characteristic rank-one perfectoid valuation ring.