---
date: 2021-04-29
modification date: Saturday 14th May 2022 22:10:37
title: "2021-04-29_2 Yves Andre On the canonical, fpqc and finite topologies"
aliases: ["Yves Andre On the canonical, fpqc and finite topologies"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #projects/notes/seminars #web/blog 
- Refs:
	- Yves André (CNRS), "On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)". Princeton/IAS NT seminar.
		- Slides: <https://www.ias.edu/sites/default/files/Princeton%20Yves.pdf#page=1> #resources/slides 
		- Video of talk: <https://youtu.be/rGNP9IsgRVE?t=27> #resources/videos 
- Links:
	- [[talk note index]]

---


## Yves André (CNRS), "On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)"


> Reference: Yves André (CNRS), "On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)". Princeton/IAS NT seminar.


![image_2021-04-29-16-36-39](attachments/image_2021-04-29-16-36-39.png)

- What does it mean for an algebra to be [faithfully flat](faithfully%20flat.md) over another algebra?

- [p-adic Hilbert functor](Unsorted/Bhatt-Lurie.md): see [Bhatt-Lurie](Bhatt-Lurie).

  - Use this to get "*almost results*", then use [[prismatic cohomology]] techniques (where one has Frobenius) to remove the "almost".

- Extracting $p$th roots? 
  Passing from $k[g_1, \cdots, g_n]$ to $k[g_1^{1/p}, \cdots, g_n^{1/p}]$, I think...

- F-pure and strongly F-regular singularities are characteristic $p$ analogs of [log canonical](Unsorted/log%20discrepancy.md) and [log terminal](Unsorted/log%20discrepancy.md) singularities in the [minimal model program](minimal%20model%20program.md).

- [[Tilting]] : pass from mixed characteristic to characteristic $p$.
  Try to use simpler proofs/theorems from characteristic $p$ situation.

  - Going forward: some limiting process after inverting $p$..?
    Going backward: take [[Witt Vectors]].

- What properties of schemes descend along [faithfully flat](faithfully%20flat.md) morphism? See EGA.
  However, what properties descend for the [fpqc](fpqc) topology?

  - What is a [faithfully flat](Unsorted/faithfully%20flat.md) morphism?

- See Faltings' [[almost purity]] theorem.

- Commutative algebra: see excellent regular domains, integral vs algebraic closures.
	- See [Cohen-Macaulay](Unsorted/Cohen-Macaulay.md) rings and modules

- Can have $\fpqc$ coverings that are not [fppf](fppf) coverings.

- What is a [regular scheme](Unsorted/regular%20scheme.md)?

  - **Theorem**: any finite covering of a regular scheme is an $\fpqc$ covering.

  - Very nontrivial in characteristic zero.

  - [Noether normalization](Unsorted/Noether%20normalization.md) can show some finite coverings of $\AA^3_{/k}$ are not $\fppf$ coverings.

- Sometimes local or coherent cohomology classes

- See Grothendieck's [descent](Unsorted/descent.md)?

	[faithfully flat](Unsorted/faithfully%20flat.md) implies something is an equivalence.