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Cohomological descent

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In algebraic geometry, a cohomological descent is, roughly, a "derived" version of a fully faithful descent in the classical descent theory. This point is made precise by the below: the following are equivalent:[1] in an appropriate setting, given a map a from a simplicial space X to a space S,

  • is fully faithful.
  • The natural transformation is an isomorphism.

The map a is then said to be a morphism of cohomological descent.[2]

The treatment in SGA uses a lot of topos theory. Conrad's notes gives a more down-to-earth exposition.

See also

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  • hypercovering, of which a cohomological descent is a generalization

References

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  1. ^ Conrad n.d., Lemma 6.8.
  2. ^ Conrad n.d., Definition 6.5.
  • SGA4 Vbis [1]
  • Conrad, Brian (n.d.). "Cohomological descent" (PDF). Stanford University.
  • P. Deligne, Théorie des Hodge III, Publ. Math. IHÉS 44 (1975), pp. 6–77.
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