Rolf Farnsteiner

Module Varieties

The term "module variety" is somewhat ambiguous as it also refers to the variety of modules of a given dimension. In our context this notion is based on conical varieties CM and RM one associates to a module M of an algebraic group (defined over an algebraically closed field k of positive characteristic p).

A special case of the cohomological support varieties CM first occurred in Quillen's 1971 papers on the spectrum of the even cohomology ring of a finite group G. Quillen studied the variety Ck of the trivial G-module k and showed that it admits a stratification by varieties coming from p-elementary abelian subgroups. About ten years later, Carlson defined the varieties CM for arbitrary G-modules and introduced the more tractable rank variety RM in case G is a p-elementary abelian group. He showed that dim CM = dim RM equals the complexity of the G-module M. His conjecture, that CM and RM are in fact isomorphic, was settled by Avrunin and Scott shortly thereafter.

The definition of CM and RM was subsequently transferred to modules over restricted Lie algebras by Friedlander-Parshall and to infinitesimal group schemes by Suslin-Friedlander-Bendel. In each case, the definition of CM was based on an analog (due to Friedlander-Parshall and Friedlander-Suslin, respectively) of the Evens-Venkov theorem concerning the finite generation of the even cohomology ring. 

Since the dimensions of CM and RM coincide with the complexity of the module M, these homeomorphic varieties are useful tools in the representation theory of algebraic groups. By basic properties of rank varieties, all modules belonging to a connected component of the stable Auslander-Reiten quiver give rise to the same variety.  

In recent work, Friedlander and Pevtsova have generalized the above to the setting of finite group schemes via their theory of p-points. Aside from unifying the previous approaches, this new point of view also affords the definition of finer invariants, leading to the consideration of new classes of modules, such as those of constant Jordan type.



1. Periodicity and representation type of modular Lie algebras.
      J. reine angew. Math. 464 (1995), 47-65

  2. On the distribution of AR-components of restricted Lie algebras.
      Contemp. Math. 229 (1998), 139-157

  3. On support varieties of Auslander-Reiten components.
       Indag. Math. 10 (1999), 221-234

  4. On cocommutative Hopf algebras of finite representation type.
      Adv. in. Math. 155 (2000), 1-22  (with Detlef Voigt)

  5. Auslander-Reiten components for Lie algebras of reductive groups.
      Adv. in. Math. 155 (2000), 49-83 

  6. Schemes of tori and the structure of tame restricted Lie algebras.
       J. London Math. Soc. 63 (2000), 553-570  (with Detlef Voigt)

  7. On the Auslander-Reiten quiver of an infinitesimal group.
      Nagoya. Math. J. 160 (2000), 103-121 

  8. Representations of infinitesimal groups of characteristic 2.
      Arch. Math. 78 (2002), 184-188

  9. Almost split sequences of Verma modules.
      Math. Ann. 322 (2002), 701-743  (with Gerhard Röhrle)

10. On infinitesimal groups of tame representation type.
      Math. Z. 244 (2003), 479-513  (with Detlef Voigt)

11. Auslander-Reiten Components for G1T-Modules.
      J. Algebra Appl. 4 (2005), 739-759

12. Tameness and complexity of finite group schemes.
      Bull. London Math. Soc 39 (2007), 63-70

13. Support spaces and Auslander-Reiten components.
      Contemp. Math. 442 (2007), 61-87