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

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 raisonné" 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 present in geometric terms the state of the art about them. 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 which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma. ContentsPart II. Finite coverings with regard to the canonical, fpqc and fppf topologies 4. Finite coverings which are not coverings for the canonical topology 5. Finite coverings which are coverings for the canonical topology but not for the fpqc topology 6. Finite coverings which are coverings for the fpqc topology but not for the fppf topology 7. On "weak functoriality" of coverings for the fpqc topology Part III. The finite topology on affine schemes. Splinters and their fpqc analogs 8. The finite and qfh topologies 9. When every finite covering is canonical: splinters 10. Fpqc analogs of splinters? References

Vanishing of Tors of absolute integral closures in equicharacteristic zero

“…Note the parallels to propositions 4.2, 4.3 from Section 4. In fact, the above proposition is true even without the module finite assumption and follows from a theorem of Bhatt, Iyengar, and Ma and has been observed by Ma, see [AF21,Theorem 10.4]. As remarked in an earlier version of this document we prove similarly that, under this assumption, i.e.…”

“…In particular under respective assumptions it strengthens a recent result of Ma-Schwede [MS20] who use (deJong's) alterations instead of finite covers or R + . The analogy with Kunz's theorem is bolstered by application of our techniques to a question of André and Fiorot [AF21]. In their study of finite covers of (affine) schemes using Grothendieck topologies, they show using Kunz's theorem that in positive characteristic the only (F-finite) 'fpqc analogues of splinters' are regular rings.…”

“…This question has appeared in literature before. André and Fiorot ask whether there is a countably generated S-algebra which is R-free given R → S is a finite extension of complete local domains [AF21,Remark 6.3]. If the answer to Question 3.19 is zero then it answers this question positively.…”

“…Since S is normal we have R is normal. Proposition 6.3 (see [AF21,Theorem 10.4 (2) and (1)]). Let R be an excellent noetherian domain which is a 'fpqc analogue of splinter' (Definition 2.17).…”

“…In Section 5 we state natural questions which arise when we try to apply our techniques in higher dimensions. In Section 6 we use our techniques to make remarks on a question of André and Fiorot [AF21] and in particular answer it under several strong assumptions.…”