Cohomological descent explained

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)} \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.

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 http://library.msri.org/books/sga/sga/pdf/sga4-2.pdf
• Web site: Brian . Conrad . Cohomological descent . Stanford University . n.d..
• P. Deligne, Théorie des Hodge III, Publ. Math. IHÉS 44 (1975), pp. 6–77.

External links

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