# zbMATH — the first resource for mathematics

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
Full Text: