Equivariant cohomology, Koszul duality, and the localization theorem
This paper studies the real cohomology and equivariant cohomology of a (possibly) singular space acted upon by a compact Lie group \(K\). The equivariant cohomology of \(X\) is defined to be the (real) cohomology of the Borel construction, \(EK \times_KX\). Call \(X\) equivariantly formal of the spectral sequence for \(H^*(EK \times_KX; \mathbb{R})\) collapses at \(E_2\). The authors give a number of sufficient conditions for equivariant formality and show how to calculate the equivariant cohomology when \(K\) is a torus and \(X\) has only a finite number of orbits of dimensions 0 and 1. A Koszul duality equivalence is established between the derived categories of differential graded modules over the Koszul complex of a torus, \(K\), and over the symmetric algebra of the dual Lie algebra. Equivariant formality is characterized in terms of the Koszul duality, when \(K\) is a torus, and the authors explain how to use these derived categories to translate from equivariant to ordinary cohomology and back.
scientific article; zbMATH DE number 1146049