Generalized Support Varieties for Finite Group Schemes

  • Julia Pevtsova


We construct two families of refinements of the (projectivized) support variety of a finite dimensional module MM for a finite group scheme GG. For an arbitrary finite group scheme, we associate a family of non-maximal rank varieties Γj(G)M\Gamma^j(G)_M, 1jp11\leq j \leq p-1, to a kGkG-module MM. For GG infinitesimal, we construct a finer family of locally closed subvarieties Va(G)MV^{\underline a}(G)_M of the variety of one parameter subgroups of GG for any partition a\underline a of dimM\dim M. For an arbitrary finite group scheme GG, a kGkG-module MM of constant rank, and a cohomology class ζ\zeta in H1(G,M){H}^1(G,M) we introduce the zero locus Z(ζ)Π(G)Z(\zeta) \subset \Pi(G). We show that Z(ζ)Z(\zeta) is a closed subvariety, and relate it to the non-maximal rank varieties. We also extend the construction of Z(ζ)Z(\zeta) to an arbitrary extension class ζExtGn(M,N)\zeta \in {Ext}^n_G(M,N) whenever MM and NN are kGkG-modules of constant Jordan type.