nLab cohomological descent

Idea

The theory of cohomological descent deals with the question if the derived analogue of the (co)monadic comparison functor is fully faithful (or more rarely an equivalence of categories) when formulated at the level of total derived functors and derived categories, and usually taken with respect to hypercovers.

Related concepts

• descent

  • cover

  • cohomological descent

  • descent morphism

  • monadic descent,

    • Sweedler coring

    • higher monadic descent

    • descent in noncommutative algebraic geometry

• sheaf, (2,1)-sheaf, 2-sheaf (∞,1)-sheaf

References

The notion has been introduced in

• Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.

A summary is also in

• Donu Arapura, Building mixed Hodge structures, in: The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 13–32, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000.

A readable introduction:

• Brian Conrad: Cohomological descent [pdf, pdf]

Closely related is the monadic descent for Karoubian triangulated categories in the sense of pages 36–37 in

• A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf

MathOverflow: Looking for reference talking about relationship between descent theory and cohomological descent