Regular rings and perfect(oid) algebras

We prove a p-adic analog of Kunz’s theorem: a p-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a more precise statement on detecting finiteness of projective dimension of finitely generated modules over noetherian rings via maps to perfectoid rings. We also establish a version of the p-adic Kunz’s theorem where the flatness hypothesis is relaxed to almost flatness.