Homological Detection by Perfectoid Algebras

We first establish estimates for the projective and injective dimensions for quotients of perfectoid algebras by radical ideals.

As applications, we obtain homological characterizations of modules of finite injective dimension, (big) Cohen--Macaulay modules, Gorenstein local rings, and regular local rings in mixed characteristic in terms of perfectoid algebras.

These are mixed characteristic analogues of the corresponding characterizations via Frobenius morphisms in positive characteristic.