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 these two expository lectures, 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.

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Last updated: 29 Feb 2024