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 non maximal rank varieties Γj (G)M, 1 ≤ j ≤ p −1, to a kG-module M. For G infinitesimal, we construct a finer family of locally closed subvarieties V a (G)M of the variety of one parameter subgroups of G for any partition a of dimM. For an arbitrary finite group scheme G, a kG-module M of constant rank, and a cohomology class ζ in H1 (G, M) we introduce the zero locus Z(ζ) ⊂ Π(G). We show that Z(ζ) is a closed subvariety, and relate it to the non-maximal rank varieties. We also extend the construction of Z(ζ) to an arbitrary extension class ζ ∈ Extn G (M,N) whenever M and N are kG-modules of constant Jordan type