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Refs:
 Yves André (CNRS), “On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)”. Princeton/IAS NT seminar.
 Links:
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.

What does it mean for an algebra to be faithfully flat over another algebra?

padic Hilbert functor: see BhattLurie.
 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…

Fpure and strongly Fregular singularities are characteristic \(p\) analogs of log canonical and log terminal singularities in the minimal model program.

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 morphism? See EGA. However, what properties descend for the fpqc topology?
 What is a faithfully flat morphism?

See Faltings’ almost purity theorem.

Commutative algebra: see excellent regular domains, integral vs algebraic closures.
 See CohenMacaulay rings and modules

Can have \({\operatorname{fpqc}}\) coverings that are not fppf coverings.

What is a regular scheme?

Theorem: any finite covering of a regular scheme is an \({\operatorname{fpqc}}\) covering.

Very nontrivial in characteristic zero.

Noether normalization can show some finite coverings of \({\mathbf{A}}^3_{/k}\) are not \({\operatorname{fppf}}\) coverings.


Sometimes local or coherent cohomology classes

See Grothendieck’s descent?
faithfully flat implies something is an equivalence.