Reading seminar on six functors for equivariant cohomology

Time/Venue

Time/venue: Thursday 10:15-12:00, Max Planck Institute for Mathematics, seminar room

Seminar description

Sheaves and cohomology are ubiquitous in geometry and topology. The derived category of sheaves on a space, together with the so-called "six functors" (and the various relations between them), form an "enhancement" of the cohomology groups of spaces that provides more insight into the structure behind these cohomology groups (see e.g. [A1]).

Equivariant cohomology is a cohomology theory for G-spaces (spaces equipped with a group action), that remembers much more about the G-space than just the singular cohomology of its orbit space (rather, it should be thought of as cohomology of the "quotient stack"). In the book Equivariant sheaves and functors [1], Bernstein and Lunts construct a generalization of the aforementioned derived category and six functors for G-spaces, that forms an "enhancement" of equivariant cohomology. 

The aim of this reading seminar will be to get to understand parts I and II of this book, with as our end-goal Theorem 12.7.2.  Before turning to this book, we will spend some time to first learn about the more classical story for sheaves on spaces (without a group action). 

Prerequisites

Some knowledge about abelian categories will probably be assumed. A basic knowledge of Lie groups is also useful. 

Seminar organization

The seminar will have weekly talks of 1:00-1:30 hours. 

Seminar plan

1. Introduction to sheaf theory (23 March 2023)
   Sheaves as pre-sheaves and as étalé spaces, local systems, Serre-Swan, push-forward, pull-back, lower shriek.  
2. Introduction to triangulated categories (30 March 2023)
   Definition of triangulated categories, the category K(A). (Weibel 10.1 and 10.2)
3. The derived category of an abelian category (6 April 2023)
   Localization, construction of the derived category,  derived functors (Weibel 10.3-10.5)
4. Verdier duality for sheaves on nice topological spaces I: preparation (13 April 2023)
   Composition formulas (including base change), soft sheaves, cohomological dimension.
5. Verdier duality for sheaves on nice topological spaces II: proof   (27 April 2023)
   Proof of Verdier duality, examples.
6. Further properties of six functors for topological spaces (4 May 2023)
   Properties of upper-shriek, smooth base change, acyclic maps and remainder. (Bernstein-Lunts Chapter 1)
7. Definition of the bounded equivariant derived category  (11 May 2023)
   (Bernstein-Lunts 2.1-2.3, parallel part of 2.9)
8. Further properties of the bounded equivariant derived category  (18 May 2023)
   Description in terms of fibered categories, structure of triangulated category (Bernstein-Lunts 2.4, 2.5, 2.7, parallel part of 2.9)
9. Introduction to t-structures on triangulated categories  (25 May 2023)
   To be decided...
10. T-structure on the bounded equivariant derived category  (1 June 2023)
    Simplicial description of the equivariant derived category, the t-structure and its heart (Berstein-Lunts Appendix B to Chapter 2)
11. To be decided...  (... June 2023)

The dates of the session are subject to changes and not strict since some topics may take more time than an entire meeting and some less. 

Literature

Textbooks

[1] J. Bernstein and V. Lunts, Equivariant sheaves and functors, Springer-Verlag 1994.

[2] A. Borel et al. , Intersection cohomology, Birkhauser 1984. 

[3] B. Iversen, Cohomology of sheaves, Springer-Verlag 1986.

[4] C.A. Weibel, An introduction to homological algebra, Cambridge University Press 1994. 

Additional literature

[A1] Martin Gallauer, https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf [3]

More to be added along the way.