Perfectoid algebras and the almost purity theorem

Abstract

This thesis focuses on the study of perfectoid algebras and their significance in proving the almost purity theorem, an important result in p-adic Hodge theory. We begin with the foundations of non-archimedean analysis and perfectoid fields, providing illustrative examples, and introduce the framework of almost mathematics. We then develop the theory of perfectoid algebras in detail. Using tools such as the cotangent complex, we present a proof of the tilting equivalence between K-perfectoid algebras and K♭-perfectoid algebras, following Scholze’s approach. Finally, leveraging this equivalence, we derive the almost purity theorem for perfectoid fields.