#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 197-222

** Eric M. Friedlander and Julia Pevtsova **
Generalized Support Varieties
for Finite Group Schemes

We construct two families of refinements of the (projectivized) support
variety of a finite dimensional module $M$ for a finite group scheme $G$.
For an arbitrary finite group scheme, we associate a family of {\it non-maximal
rank varieties} $\Gamma^j(G)_M$, $1<= j leq p-1$, to a $kG$-module $M$.
For $G$ infinitesimal, we construct a finer family of locally closed subvarieties
$V^{\ul a}(G)_M$ of the variety of one parameter subgroups of $G$ for any
partition $\ul a$ of $\dim M$. For an arbitrary finite group scheme $G$,
a $kG$-module $M$ of constant rank, and a cohomology class $\zeta$ in $\HHH^1(G,M)$
we introduce the {\it zero locus} $Z(\zeta) \subset \Pi(G)$. We show that
$Z(\zeta)$ is a closed subvariety, and relate it to the non-maximal rank
varieties. We also extend the construction of $Z(\zeta)$ to an arbitrary
extension class $\zeta \in \Ext^n_G(M,N)$ whenever $M$ and $N$ are $kG$-modules
of constant Jordan type.

2010 Mathematics Subject Classification: 16G10, 20C20, 20G10

Keywords and Phrases:

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