On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)

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-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 pp-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 (https://doi.org/10.2422/2036-2145.201912_006)