Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety just 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’s-eye view of some fundamental 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 using p-adic methods (perfectoids). Rather than going into detail of the latter, the emphasis will be on synthesizing, from the geometric viewpoint, a rather scattered theme.
This is based on joint work with Luisa Fiorot. See this arXiv paper for more details.
This video is part of the Institute for Advanced Study‘s Number theory seminar.
