# 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 C_{M} and R_{M} 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
* C_{M} first occurred in Quillen's 1971 papers on the spectrum of the even cohomology ring of a finite
group G. Quillen studied the variety C_{k}
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 C_{M }for arbitrary G-modules and introduced the
more tractable *rank variety *R_{M}
in case G is a p-elementary abelian group. He showed that dim C_{M}
= dim R_{M} equals the * complexity* of the G-module M. His conjecture,
that C_{M} and R_{M} are in fact isomorphic, was settled by
Avrunin and Scott shortly thereafter.

The definition of C_{M} and R_{M} 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 C_{M} 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 C_{M} and R_{M}
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*.

# Publications

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 G _{1}T-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