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: in an appropriate setting, given a map a from a simplicial space X to a space S,

• [math]\displaystyle{ a^*: D^+(S) \to D^+(X) }[/math] is fully faithful.
• The natural transformation [math]\displaystyle{ \operatorname{id}_{D^+(S)} \to Ra_* \circ a^* }[/math] 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 V^bis [1]
• Conrad, Brian (n.d.). "Cohomological descent". http://math.stanford.edu/~conrad/papers/hypercover.pdf. 
• P. Deligne, Théorie des Hodge III, Publ. Math. IHÉS 44 (1975), pp. 6–77.

External links

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