Local cohomology and support

This website provides a collection of material on the notion of local cohomology and support for triangulated categories. In May 2010, there will be an Oberwolfach Seminar "Representations of Finite Groups: Local Cohomology and Support". As preparation for this seminar, the organizers are collecting material on this webpage; at this stage it is far from being complete. Please contribute and send comments to Henning Krause.

The topic and some basic concepts

Local cohomology: The concept has its origin in algebraic geometry and was introduced by Grothendieck. In algebraic terms, the idea is fairly simple: given an ideal in a commutative ring, one considers for each module the submodule of elements annihilated by some power of this ideal. This operation yields a functor that usually is not exact. Local cohomology measures the failure of exactness in terms of its right derived functors.

Support: In algebraic geometry the support of a sheaf is the collection of points of the underlying space where the stalk does not vanish. This notion carries over to modules (and complexes of modules) over commutative rings, where the underlying space is the spectrum of prime ideals. Following ideas of Quillen, Carlson, and others, there is an analogue for representations of finite groups which is based on the fact that the cohomology ring of a finite group is a graded commutative noetherian ring. Taking as underlying space the set of its graded prime ideals, the support of each representation is defined in terms of its cohomology.

Literature on local cohomology and support

Basic material: representation theory, commutative algebra, triangulated categories

1. D. J. Benson, Representations and cohomology of finite groups I,II, Cambridge University Press.
2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press.
3. R. Hartshorne, Local cohomology: A seminar given by A. Grothendieck (Harvard, 1961), Lecture Notes in Math. 41, Springer Verlag.
4. S. B. Iyengar, Modules and cohomology over group algebras. One commutative algebraist's perspective.
    
in: Trends in commutative algebra (Berkeley, September 2002), 
MSRI Publ. Vol. 51, Cambridge Univ. Press, Cambridge, 2004; 51–86.
5. S. B. Iyengar et al., Twenty-four hours of local cohomology, Graduate Studies in Mathematics, American Mathematical Society.
6. H. Krause, Support of noetherian modules over commutative rings, unpublished notes.
7. H. Krause, Derived categories, resolutions, and Brown representability, Contemp. Math. 436, Amer. Math. Soc. (2007), arXiv:math/0511047.
8. H. Krause, Localization theory for triangulated categories, to appear in:  LMS Lecture Notes Series, Cambridge University Press, arXiv:0806.1324.
9. A. Neeman, Triangulated categories, Princeteon University Press.

1. L. Alonso Tarrío, A. Jeremías López, M. J. Souto Salorio, Bousfield Localization on formal schemes, J. Algebra 278 (2004) 585–610.
2. P. Balmer, The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math., 588 (2005), 149–168.
3. D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.
4. D. J. Benson, S. B. Iyengar, and H. Krause, Local cohomology and support for triangulated categories, Ann. Scient. Ec. Norm. Sup. (4) 41 (2008), arXiv:math/0702610.
5. D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying modular representations of finite groups, arXiv:0810.1339.
6. D. J. Benson, S. B. Iyengar, and H. Krause, Stratifying triangulated categories, arXiv:0910.0642.
7. E. Friedlander and J. Pevtsova, Π-supports for modules for finite group schemes over a field, Duke Math. J. 139 (2007), 317–368.
8. M. Hopkins: Global methods in homotopy theory, in: Homotopy theory (Durham, 1985), London Math. Soc. Lecture Note Ser. 117, Cambridge Univ. Press, Cambridge, 1987, 73–96.
9. M. Hovey, J. Palmieri, and N. Strickland, Axiomatic stable homotopy theory. Mem. Amer. Math. Soc. 610, Amer. Math. Soc., (1997).
10. A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532.
11. R. Takahashi, Classifying thick subcategories of the stable category of Cohen-Macaulay modules, arXiv:0908.0107.
12. R. W. Thomason, The classification of triangulated subcategories, Compositio Math. 105 (1) (1997) 1–27.

1. Mini-Workshop: Thick Subcategories - Classifications and Application, Oberwolfach Report (2008).
2. Mini-Workshop: Support Varieties, Oberwolfach Report (2010).
3. S. B. Iyengar, Stratifying derived categories associated to finite groups and to commutative rings, Workshop "Algebraic triangulated categories and related topics", Research Institute for Mathematical Sciences, Kyoto University, Japan, July 2009.
4. H. Krause, Cosupport and colocalizing subcategories of modules and complexes, Presentation, ICTP Trieste (2010).

Oberwolfach Seminar

Topic: Representations of finite groups: local cohomology and support

Organisers: Dave Benson, Srikanth Iyengar, Henning Krause

Dates: May 23rd - May 29th, 2010

Programme: The seminar discusses some unifying themes in commutative algebra, representation theory, and homotopy theory. The underlying concept is the notion of support which provides a geometric approach for studying various algebraic structures. In terms of applications, the focus will be on modular representations of finite groups. In particular, classifications of thick and localizing subcategories of the stable module categories associated to a finite group will be presented.

There will be three series of lectures, starting with background material from group representation theory, commutative algebra, and the theory of derived categories. The goal is to explain local cohomology functors for specific triangulated categories and to discuss their applications.

Schedule: Here one finds a tentative schedule.

Prerequisites: A solid background in algebra, including the basic theory of rings and modules, basic commutative algebra (first chapters of Atiyah/MacDonald), and basic homological algebra (derived functors, Ext, Tor).

Applications: See the Oberwolfach Seminar website. Deadline is April 1st, 2010.