Cohomological descent

{{Short description|Derived descent}}

{{citation style|date=February 2015}}

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:{{harvnb|Conrad|n.d.|loc=Lemma 6.8.}} in an appropriate setting, given a map a from a simplicial space X to a space S,

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

The map a is then said to be a morphism of cohomological descent.{{harvnb|Conrad|n.d.|loc=Definition 6.5.}}

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