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: in an appropriate setting, given a map a from a simplicial space X to a space S,
a ∗ : D + ( S ) → D + ( X ) {\displaystyle a^{*}:D^{+}(S)\to D^{+}(X)} is fully faithful. The natural transformation id D + ( S ) → R a ∗ ∘ a ∗ {\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. 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
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.
External links http://ncatlab.org/nlab/show/cohomological+descent