Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

The classical localization theorem due to Borel, Atiyah-Segal, and Quillen says that the equivariant cohomology of a space can be recovered, up to inverting some elements, from the equivariant cohomology of the fixed point subspace. A version of this result for topological K-theory was proved by Segal in 1968. In this talk, we discuss the algebraic analog of this result, due to Thomason for schemes, and its extension to algebraic stacks. We also formulate the Atiyah-Bott, Graber-Pandharipande virtual localization formula for the structure sheaf. This is based on a joint work in progress with Adeel Khan and Hyeonjun Park.