Cohomological descent

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,

• $a^{*}:D^{+}(S)\to D^{+}(X)$ is fully faithful.
• The natural transformation $\operatorname {id}_{{D^{+}(S)}}\to Ra_{*}\circ a^{*}$ 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

• hypercovering, of which a cohomological descent is a generalization

References

1. Conrad, Lemma 6.8.
2. Conrad, Definition 6.5.

• SGA4 V^bis
• Brian Conrad, Cohomological descent
• P. Deligne, Théorie des Hodge III, Publ. Math. IHES 44 (1975), pp. 6–77.

External links

• http://ncatlab.org/nlab/show/cohomological+descent