The derived categorical Atiyah-Segal Completion Theorem

The aim of this project would be to give an extension of the Atiyah-Segal Completion Theory, a classical theorem in topological K-theory, to the more general setting of the derived category of an abelian category.

Given a Lie group G and a G-space X, the Atiyah-Segal theorem relates the G-equivariant K-theory of X to the (regular) K-theory of a certain quotient of X by the action of G, known as the homotopy quotient or Borel construction. When the action of G is free, this is just X/G, and the theorem states that KG(X) = K(X/G). This gives us a way to compute the equivariant K-theory, an object of interest in many areas. However, we may in fact define the K-theory of any exact category, and so want to find the analogue of this theorem in the more general setting.

Namely, if A is an abelian category, we can form the category Ch(A) of chain complexes in A, and the derived category D(A) is the quotient of this by all quasi-isomorphisms. Then our aim is to replace the functor K in the classic formulation of the Atiyah-Segal theorem with the functor D. This consists of three broad steps: firstly, to give the correct definition of the derived category Db(X) of complexes of vector bundles over X; secondly, to define the notion of localisation of this category at a certain ideal, another term appearing in the classical theorem; and finally, to study the corresponding Atiyah-Segal map in this context, and to show (hopefully) it is an equivalence of dg categories.