Equivariant cohomology, Koszul duality, and the localization theorem
type: Item
instance of: scholarly article
title: Equivariant cohomology, Koszul duality, and the localization theorem
zbMATH Open document ID: 0897.22009
author: M. Goresky
author: Robert MacPherson
author: Robert Kottwitz
published in: Inventiones Mathematicae
publication date: 1998-10-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22D35
Mathematics Subject Classification ID: 57S10
zbMATH DE Number: 1146049
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: compact Lie group
zbMATH Keywords: Koszul duality
zbMATH Keywords: Lie algebra
reviewed by: Victor P. Snaith
MaRDI profile type: Publication
full work available at URL: s002220050197
OpenAlex ID: W2088536807
DOI: 10.1007/S002220050197
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description: scientific article; zbMATH DE number 1146049