Material for Masterclass on Stratifications and duality in modular representation theory

Notes and Exercises

Please click on the relevant talk or exercise on the schedule page.

References

Here we collect a number of references for the Masterclass. The bold items are the ones most relevant for the courses. 

Background material

We assume that the participants are familiar with some basic concepts from commutative algebra, homological algebra, and representation theory. A collection of useful sources covering this background and much more can be found here.

Commutative algebra

• Atiyah, MacDonald, Introduction to commutative algebra, Addison-Wesley series in mathematics, 1994
• Eisenbud, Commutative algebra, GTM 150
• Bruns and Herzog, Cohen-Macaulay rings, Cambridge University Press

Homological algebra and category theory

• Weibel, An introduction to homological algebra, Cambridge University Press
• Neeman, Triangulated categories, Princeton University Press

Representation theory

• Benson, Representations and cohomology of finite groups I, II, Cambridge University Press

Lecture Course I: Stratifications

A comprehensive survey of the stratification of the stable module category of a finite group can be found in the Oberwolfach notes by Benson, Iyengar, Krause. 

The stable module category

• Auslander, Bridger, Stable module theory, AMS Memoirs 94
• Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings, notes
• Hovey, Palmieri, Strickland, Axiomatic stable homotopy theory, AMS memoirs 610
• Benson, Krause, Complexes of injective kG-modules
• Krause, The stable derived category of a Noetherian scheme

Support, cosupport, and stratification theory:

• Benson, Iyengar, Krause, Local cohomology and support for triangulated categories
• Krause, Localization theory for triangulated categories
• Benson, Iyengar, Krause, Stratifying triangulated categories
• Benson, Iyengar, Krause, Colocalizing subcategories and cosupport
• Benson, Iyengar, Krause, Pevtsova, Stratification and \pi-cosupport: finite groups
  

Stratifications of finite groups and finite group schemes

• Benson, Iyengar, Krause, Stratifying modular representations of finite groups
• Benson, Iyengar, Krause, Localising subcategories for cochains on the classifying space of a finite group
• Benson, Iyengar, Krause, Pevtsova, Stratification for module categories of finite group schemes
• Benson, Greenlees, Stratifying the derived category of cochains on BG for G a compact Lie group, JPAA, 2014
• Benson, Iyengar, Krause, Pevtsova, Colocalising subcategories of modules over finite group schemes

Lecture Course II: Local duality

There are many excellent sources available on local duality results in commutative algebra and algebraic geometry. The following list contains a representative selection. 

General treatments of (local) duality

• Hartshorne, Residues and duality, LNM 20
• Iyengar et. al., 24 hours of local duality, Graduate studies in mathematics 87 
• Dwyer, Greenlees, Iyengar, Duality in algebra and topology
• Balmer, Dell'Ambrogio, Sanders, Grothendieck-Neeman duality and the Wirthmüller isomorphism
• Barthel, Heard, Valenzuela, Local duality in algebra and topology

Local duality in modular represention theory

• Benson, Modules with injective cohomology, and local duality for a finite group, New York Journal of Mathematics, 2001
• Benson, Krause, Pure injectives and the spectrum of the cohomology ring of a finite group
• Benson, Greenlees, Localizations and duality in topology and modular representation theory
• Benson, Idempotent kG-modules with injective cohomology, JPAA, 2008
• Barthel, Heard, Valenzuela, Local duality for structured ring spectra
• Benson, Iyengar, Krause, Pevtsova, Local duality for representations of finite group schemes