Chennai Mathematical Institute


3.30 pm
Descent and Cohomological Descent

Nitin Nitsure
TIFR, Mumbai.


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 algebraic stacks.
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.