Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety looks like a flag of vector spaces (Noether); understanding this « up to » is a primary motivation for a fine study of finite coverings.

The aim of this talk is to give a bird-eye view of some fundamendal questions about them, which took root in Algebraic Geometry (descent problems etc.), then motivated major trends in Commutative Algebra (F-singularities etc.), and recently found complete solutions by methods relying on P. Scholze’s perfectoid geometry.

This is based on joint work with Luisa Fiorot (https://doi.org/10.2422/2036-2145.201912_006)