A Berkovich Approach to Perfectoid Spaces

Abstract

Since their inception perfectoid spaces have catalyzed a revolution in p-adic geometry. We redevelop the foundations of perfectoid spaces from the point of view of Berkovich Spaces, where the underlying topological space of an affinoid perfectoid space is a compact Hausdorff space – closely resembling the situation in complex geometry. The key technical ingredient in our construction is arc pi descent for perfectoid Banach algebras. Along the way, we establish various foundational results for arc pi sheaves, notably a form of the Gerritzen-Grauert theorem.