Approximation of perfectoid rings by Noetherian rings and prisms

The theory of perfectoid towers, introduced by Ishiro-Nakazato-Shimomoto, provides an axiomatic approach to perfectoid theory in commutative algebra through tower-theoretic approximations. In contrast, Bhatt and Scholze introduced prisms as a "deperfection" of perfectoid rings. Our main result shows that a "gradual perfection" of a prism becomes a perfectoid tower. As a consequence, we prove that any p-torsion-free p-adically complete delta-ring that is reduced modulo p admits a perfectoid tower. This approach allows for more systematic construction of perfectoid rings and towers from Noetherian rings than previously possible. We also provide new examples of perfectoid towers arising from certain singularities, which were inaccessible by previous methods. In this talk, we will review the concepts of perfectoid towers, explain our construction from prisms, and demonstrate its applications through these novel examples.