Equivariant sheaves and functors
type: Item
instance of: scholarly article
title: Equivariant sheaves and functors
zbMATH Open document ID: 0808.14038
author: Joseph Bernstein
author: Valery A. Lunts
published in: Lecture Notes in Mathematics
publication date: 1994-07-13T00:00:00Z
review text: Let \(f:X \to Y\) be a continuous map between two locally compact topological spaces. Assume that a topological group \(G\) acts on \(X\) and on \(Y\) in such a way that \(f\) is a \(G\)-map. Then one can define the categories \(Sh_ G (X)\) and \(Sh_ G (Y)\) of \(G\)-equivariant sheaves. In the case when the group \(G\) is trivial one can define the derived categories \(D(X)\) and \(D(Y)\) of the abelian categories \(Sh(X)\) and \(Sh(Y)\) of sheaves on \(X\) and \(Y\) respectively, together with natural functors \(f^*\), \(f_ *\), \(f^ !\), \(f_ !\), \(\Hom\), \(\otimes\), etc. The main aim of the book under review is to define and study the triangulated categories \(D_ G (X)\) and \(D_ G (Y)\) together with similar functors and with a forgetful functor For: \(D_ G \to D\), satisfying certain natural compatibility conditions. Simple examples show that, unless the group \(G\) is discrete, the derived category of \(Sh_ G\) is not the right candidate for the triangulated category \(D_ G\). Therefore the theory developed here is very subtle. After supplying the necessary details for the construction of \(D_ G\), the authors give an algebraic description of the triangulated category \(D_ G(pt)\) when \(G\) is a connected Lie group, in terms of \(DG\)-modules over a natural \(DG\)- algebra \(A_ G\). This description is particularly important for the main applications of the theory. In the last part of the book the authors illustrate their theory with a computation of the equivariant intersection cohomology (with compact supports) of toric varieties.
reviewed by: Lucian Bădescu
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 605663
zbMATH Keywords: equivariant sheaves
zbMATH Keywords: derived categories
zbMATH Keywords: triangulated categories
zbMATH Keywords: equivariant intersection cohomology
zbMATH Keywords: toric varieties
MaRDI profile type: Publication
DOI: 10.1007/BFB0073549
description: scientific article; zbMATH DE number 605663