On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

This is a systematic study of the behaviour finite coverings (affine) schemes with regard to two Grothendieck topologies: canonical topology and fpqc topology. The history problem takes roots in foundations topologies, passes through main strides Commutative Algebra leads new Mathematics up perfectoids prisms. We first review affine show, keeping Olivier's lost work, that it coincides effective descent topology; covering maps are given by universally injective ring maps, which we discuss detail. then give catalogue raisonne examples separate canonical, fppf topologies. key result regular for topology, even (but not necessarily topology). We weakly functorial aspect this result. Splinters those Noetherian every also investigate their mysterious analogs, prove prime characteristic, they all regular. us regularity (non-necessarily flat) morphisms $f$ settled thanks recent theorem Bhatt-Iyengar-Ma.