descent

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descent

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Ideas

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Definitions

For modules

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For categories fibered in groupoids

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In HTT

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In terms of hypercovers

See also hyperdescent.

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Misc

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Descent categories from monads

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Recovering descent for rings/modules

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  • What is descent?
  • What is effective descent
  • What is Unsorted/descent?
  • Types of descent:
    • What is Zariski descent?
    • What is Galois descent?
    • What is Unsorted/etale?
    • What is fppf descent?
    • What is fpqc descent?
    • What is descent for quasicoherent sheaves?
  • What is the connection to Neron models? See Bosch et al
  • How is this related to etale cohomology?

Relation to torsors

See torsor and rational point. For elliptic curves, related to the Selmer group attachments/Pasted%20image%2020220319204120.png

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Applications

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🗓️ Timeline
  • 2021-10-08
    descent: pairs \((M, \phi)\) where $M\in {}_{S}{\mathsf{Mod}} $ and \(\phi: M\otimes_RS { \, \xrightarrow{\sim}\, }S\otimes_R M\) is a twist isomorphism.

    General idea: \(R{\hbox{-}}\)modules \(M\) can be specified by \(S\otimes_R M\) along with Unsorted/descent.

    Refs: stacks vector bundle Unsorted/descent

    faithfully flat descent : there is an equivalence of categories \({}_{R}{\mathsf{Mod}} \to {\mathsf{Desc}}(R\searrow S)\),

  • 2021-10-03

    Cartan-Ehresmann connections, descent attachments/2021-10-03_13-42-38.png

  • 2021-09-19

    The problem is that I don’t really know how to relate the bottom line (whose exactness is the usual condition for sheaves, stacks, etc) to the intermediate steps. This seems like it wants \({\mathcal{F}}({\textstyle\coprod}{-}) = \prod {\mathcal{F}}({-})\), so it commutes with (co?)limits, since probably contravariant functors send coproducts to products. Moreover the bar construction in the 2nd line might form a simplicial object? And the condition of satisfying descent is maybe related to either this being a simplicial object, or its image in the bottom line assembling to a simplicial object, since there are clear degeneracy maps and one would want sections in order to build face maps. Super vague, there are a lot of details missing here!!

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