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,

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