Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the canonical, fpqc, and finite topologies on affine schemes. The
  state of the art   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.

16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org