Abstract. We prove that the projectivity of an arbitrary (possibly infinite dimensional) module for a Frobenius kernel can be detected by restrictions to one-parameter subgroups. Building upon this result, we introduce the support cone of such a module, extending the construction of support variety for a finite dimensional module, and show that such support cones satisfy most of the familiar properties of support varieties. We also verify that our representation-theoretic definition of support cones admits an interpretation in terms of Rickard idempotent modules associated to thick subcategories of the stable category of finite dimensional modules.
Abstract. We verify that the construction of "support cone" for infinite dimensional modules extends to modules over any infinitesimal group scheme and satisfies all good properties of support varieties for finite dimensional modules, thereby extending the results of the author for infinite dimensional modules of Frobenius kernels \cite{P}. We show, using an alternative description of support cones in terms of Rickard idempotents, that for an algebraic group $G$ over an algebraically closed field $k$ of positive characteristic $p$ and a point $s$ in the cohomological support variety of a Frobenius kernel $\Gr$, the orbit $G\cdot s$ can be realized as a support cone of a rational $G$-module.
Abstract. We introduce the space $P(G)$ of abelian $p$-points of a finite group scheme over an algebraically closed field of characteristic $p > 0$. We construct a homeomorphism $\Psi_G: P(G) \to \Proj |G|$ from $P(G)$ to the projectivization of the cohomology variety for any finite group $G$. For an elementary abelian $p$-group (respectively, an infinitesimal group scheme), $P(G)$ can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional $G$-module $M$, $\Psi_G$ restricts to a homeomorphism $P(G)_M \to \Proj |G|_M$, thereby giving a representation-theoretic interpretation of the cohomological support variety.
Abstract. We introduce the space $\Pi(G)$ of equivalence classes of $\pi$-points of a finite group scheme $G$. The study of $\pi$-points can be viewed as the study of the representation theory of $G$ in terms of ``elementary subalgebras" of a very specific and simple form, or as an investigation of flat maps to the group algebra of $G$ utilizing the representation theory of $G$. Our results extend to arbitrary finite group schemes $G$ over arbitrary fields $k$ of positive characteristic and to arbitrarily large $G$-modules the basic results about ``cohomological support varieties" and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite dimensional) $G$-module can be detected by its restriction along $\pi$-points of $G$. We establish that $\Pi(G)$ is homeomorphic to $\Proj H*(G,k)$, and using this homeomorphism we determine up to inseparable isogeny the best possible field of definition of an equivalence class of $\pi$-points. Unlike the cohomological invariant $M \mapsto \Proj H*(G,k)$, the invariant $M \mapsto \Pi(G)_M$ satisfies good properties for all $G$-modules,thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite dimensional $kG$-modules. Finally, using the stable module category of $G$, we provide $\Pi(G)$ with the structure of a ringed space which we show to be isomorphic to the scheme $\Proj H*(G,k)$.
Abstract. 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 $M$. In the very special case in which $G = E$ is an elementary abelian $p$-group, our construction 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 $E$. Furthermore, we construct the non-maximal support variety $\Gamma(G)_M$, a closed subset of $Proj H*(G,k)$ which is non-tautological even when the dimension of $M$ is not divisible by $p$.
Abstract. This is a >>very<< explicit calculation of conjugacy classes of maximal elementary abelian subgroups inside $GL(4,F_p)$ worked out together with Eric Friedlander and Steve Smith. We had different motivations coming to this question from different directions but what we certainly agree upon is that we did not expect the answer to be that complicated!
Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell})$, $G = (\Zl)^m$, and $\text{char } k$ does not divide $\ell$, in terms of certain subalgebras of $A$ playing the role of ``cyclic shifted subgroups". We show that the rank variety of a finitely generated module $M$ is homeomorphic to the support variety of $M$ defined in terms of the action of the cohomology algebra of $A$. As an application we derive a theory of rank varieties for the algebra $\Lambda$. When $\ell=2$, rank varieties for $\Lambda$-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for $\Lambda$-modules coincide with those of Erdmann and Holloway.
Abstract. We introduce the class of modules of constant Jordan type for a finite group scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.
Abstract. Let $G$ be an infinitesimal group scheme over a field $k$ of characteristic $p >0$. We introduce the universal $p$-nilpotent operator $\Theta_G \in \Hom_k(k[G],k[V(G)])$, where $V(G)$ is the scheme which represents 1-parameter subgroups of $G$. This operator $\Theta_G$ applied to $M$ encodes the local Jordan type of $M$, and leads to computational insights into the representation theory of $G$. For certain $kG$-modules $M$ (including those of constant Jordan type), we employ $\Theta_G$ to associate various algebraic vector bundles on $P(G)$, the projectivization of $V(G)$. These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on $P(G)$.