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Equivariant sheaves and functors. (English) Zbl 0808.14038
Lecture Notes in Mathematics. 1578. Berlin: Springer-Verlag. 139 p. (1994).
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 \(f^*\), \(f_ *\), \(f^ !\), \(f_ !\), \(\operatorname{Hom}\), \(\otimes\), 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.

MSC:
14L30 Group actions on varieties or schemes (quotients)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
18-02 Research exposition (monographs, survey articles) pertaining to category theory
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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