# 2021-04-29_2 Yves Andre On the canonical, fpqc and finite topologies

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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.

## 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?

• p-adic Hilbert functor: see 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 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?

• See Faltings’ almost purity theorem.

• Commutative algebra: see excellent regular domains, integral vs algebraic closures.

• 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 $${\mathbb{A}}^3_{/k}$$ are not $${\operatorname{fppf}}$$ coverings.

• Sometimes local or coherent cohomology classes

• See Grothendieck’s descent?

faithfully flat implies something is an equivalence.

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