GENERIC JORDAN TYPE OF MODULAR REPRESENTATIONS

JULIA PEVTSOVA

Elementary abelian p-subgroups of a finite group G capture significant aspects of the cohomology and representation theory of G. For example, if k is a field of characteristic p > 0, then a theorem of D. Quillen [9] asserts that the Krull dimension of the cohomology algebra H^{•}(G, k) is equal to the maximum of the ranks of elementary abelian p-subgroups of G and a theorem of L. Chouinard [3] asserts that a kG-module is projective if and only if its restrictions to all elementary abelian p-subgroups of G are projective. Quillen’s geometric description of Spec H^{•}(G, k)

[9] provides the basis for interesting invariants of kG-modules, most notably the cohomological support variety |G|_{M} of a kG-module M.

The geometric methods developed by Quillen were further applied to study representations of other algebraic structures, such as restricted Lie algebras. E. Friedlander and B. Parshall developed a theory of support varieties for finite dimensional p-restricted Lie algebras g over a field k of characteristic p > 0 (e.g., [4]). For restricted Lie algebras, the role of the group algebra kG of the finite group G is played by the restricted enveloping algebra u(g). Indeed, restricted Lie algebras lead one to more interesting geometrical structure than do finite groups, and seemingly lead to stronger results. For example, the theorem of G. Avrunin and L. Scott [1] identifying the cohomological support variety |E|_{M} of a finite dimensional kE module M for an elementary abelian p-group E with the rank variety of J. Carlson [2] admits a formulation in the case of a restricted Lie algebras g in terms of closed subvarieties of the p-nilpotent cone of g (cf. [4], [8], [11]).

A uniform approach to the study of the cohomology and related representation theory of all finite group schemes was presented in [5], [6]. This approach involves the use of π-points of G, which are finite flat maps of K-algebras K[t]/t^{p} → KG for field extensions K/k; these play the role of “cyclic shifted subgroups” in the case that G is an elementary abelian p-group and the role of 1-parameter subgroups in the case that G is an infinitesimal group scheme. In [6], the space Π(G) of equivalence classes [α_{K}] of π-points α_{K} : K[t]/t^{p} → KG of G is given a scheme structure without reference to cohomology such that Π(G) is isomorphic as a scheme to Proj H^{•}(G, k). In particular, there is a natural bijection between such equivalence classes of π-points and homogeneous prime ideals of H^{•}(G, k).

The purpose of the talk, though, is to demonstrate that one can go further in the search of information about modules encoded geometrically.

For a finite group scheme G over a field k of characteristic p > 0, we associate new invariants to a finite dimensional kG-module M. Namely, for each generic point of the projectivized cohomological variety Proj H^{•}(G, k) we exhibit a “generic Jordan type” of M. We prove the following theorem which both describes the invariant and demonstrates that it is well-defined.

Theorem 1. Let G be a finite group scheme, let M be a finite dimensional Gmodule and let α_{K} : Kℤ/p → KG be a π-point of G which represents a generic point [α_{K}] ∈ Π(G). Then the Jordan type of α_{K}(t) viewed as a nilpotent operator on M_{K} depends only upon [α_{K}] and not the choice of α_{K} representing [α_{K}].

The Jordan type of such α_{K}(t) is called the generic Jordan type of M. In a special case when G = E is an elementary abelian p-group, the theorem specializes to the non-trivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E.

We verify that sending a module M to its generic Jordan type [α_{K}]^{∗}(M_{K}) for generic [α_{K}] ∈ Proj H^{•}(G, k) determines a well-defined tensor triangulated functor on stable module categories

[α_{K}]^{∗} : stmod(kG) → stmod(K[t]/t^{p}).

The second invariant we present is the non-maximal support variety of a finitedimensional kG-module M, Γ(G)_{M} ⊂ Proj H^{•}(G, k). The non-maximal support variety coincides with the “classical” support when the module is generically projective but gives a new non-tautological invariant in the case when the “classical” support variety of M is the entire cohomological spectrum: for example, when the dimension of M is not divisible by p.

This is a joint work with E. Friedlander and A. Suslin.

References

[1] G. Avrunin, L. Scott, Quillen stratification for modules, Inventiones Math. 66 (1982) 277-286.

[2] J. Carlson, The varieties and the cohomology ring of a module, J. Algebra 85 (1983) 104-143.

[3] L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 287-303.

[4] E. Friedlander, B. Parshall. Support varieties for restricted Lie algebras, Inventiones Math. 86 (1986), 553–562.

[5] E. Friedlander, J. Pevtsova, Representation-theoretic support spaces for finite group schemes, American Journal of Math 127 (2005), 379-420.

[6] E. Friedlander, J. Pevtsova, Π-supports for modules for finite group schemes over a field. Preprint

[7] E. Friedlander, A. Suslin, Cohomology of finite group scheme over a field, Invent. Math 127 (1997), 235-253.

[8] J.C. Jantzen, Kohomologie von p-Lie-Algebren und nilpotente Elemente, Abh. Math.Sem. Univ. Hamburg 56 (1986), 191-219.

[9] D. Quillen The spectrum of an equivariant cohomology ring: I, II, Ann. Math. 94 (1971), 549-572, 573-602.

[10] A. Suslin, E. Friedlander, C. Bendel, Infinitesimal 1-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997) 693-728.

[11] A. Suslin, E. Friedlander, C. Bendel Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), 729-759.