Descent and Cohomological Descent
Around 1960, Grothendieck developed the theory of descent. The aim of
this theory is to construct geometric objects on a base space -- in
particular bundles, sheaves and their sections -- in terms a
generalized covering space which is visualized to lie `upstairs' over
the base space. The objects over the base space are obtained by
`descending' similar objects from the covering space.
In late 1960s-early 1970s, Deligne addressed the problem of how to
understand cohomology of the base space via cohomology of a covering,
by this time descending cohomology classes (instead of just descending
global sections of sheaves, which is the case of 0th cohomology). The
theory developed by Deligne, known as `Cohomological Descent' has
found important applications to Hodge theory and to cohomology of
In this expository talk, I will begin with a quick look at
Grothendieck's theory of descent, and then go on to give a brief
introduction to Deligne's theory of Cohomological Descent.