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

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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.

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  • 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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