Structural properties and tilting correspondences of perfectoid towers

We prove that every perfectoid tower can be realized as the fiber product of a diagram involving perfectoid towers that are either $p$-torsion free or perfect of characteristic $p$.

As an application, we conclude that separated perfectoid towers are reduced.

We also establish the tilting invariance of étale cohomology and Koszul homology for perfectoid towers.

As further applications, we prove that tilting preserves fundamental properties of Noetherian local rings such as being Cohen--Macaulay, Gorenstein, complete intersection, or regular.