Yves André (CNRS), “On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)”
Tags: #seminar_notes #blog
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?
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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”.
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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…
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F-pure and strongly F-regular singularities are characteristic \(p\) analogs of log-canonical and log-terminal singularities in the minimal model program.
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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]].
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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?
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See Faltings almost purity theorem.
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Commutative algebra: see excellent regular domains, integral vs algebraic closures. - Cohen-Macaulay rings and modules
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Can have \(\mathrm{\operatorname{fpqc}}\) coverings that are not fppf coverings.
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What is a regular scheme?
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Theorem: any finite covering of a regular scheme is an \(\mathrm{\operatorname{fpqc}}\) covering.
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Very nontrivial in characteristic zero.
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Noether normalization can show some finite coverings of \({\mathbb{A}}^3_{/k}\) are not \(\mathrm{\operatorname{fppf}}\) coverings.
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Sometimes local or coherent cohomology classes
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Grothendieck’s descent?
faithfully flat implies something is an equivalence.