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

TL;DR

This paper systematically studies finite coverings of affine schemes under canonical and fpqc topologies, revealing their properties, examples, and the conditions under which schemes are splinters, connecting to recent advances in algebraic geometry.

Contribution

It provides a comprehensive analysis of finite coverings in various Grothendieck topologies, including new results on splinters and the behavior of regular schemes under these coverings.

Findings

Finite coverings of regular schemes are coverings for the canonical and fpqc topologies.

Splinters are characterized as schemes where all finite coverings are canonical.

In prime characteristic, splinters are all regular schemes.

Abstract

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