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 $P(G)$.
Abstract. We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig's small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, as well as many new pointed Hopf algebras. We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra.
Abstract. 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 subvarieties $\Gamma(G)_M^j$, to a $kG$-module $M$. For $G$ infinitesimal, we construct a finer family of locally closed subvarieties $\Gamma^a(G)_M$ for any partition $a$ of $\dim M$. We give a cohomological interpretation of the varieties $\Gamma^1(G)_M$ for certain modules relating them to generalizations of $Z(\zeta)$, the zero loci of cohomology classes $\zeta \in H^\bullet(G,k)$.
Abstract. Let $E$ be an elementary abelian $p$-group of rank $r$ and let $k$ be a field of characteristic $p$. We introduce functors $\cF_i$ from finitely generated $kE$-modules of constant Jordan type to vector bundles over projective space $\bP^{r-1}$. The fibers of the functors $\cF_i$ encode complete information about the Jordan type of the module. We prove that given any vector bundle $\cF$ of rank s on $P^{r-1}$, there is a kE-module M of stable constant Jordan type $[1]^s$ such that $\cF_1(M)\cong \cF$ if p=2, and such that $\cF_1(M) \cong \cF^*(F)$ if p is odd. Here, $F: P^{r-1}\to P^{r-1}$ is the Frobenius map. We prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type $[1]^s$ then the Chern numbers $c_1, ... ,c_{p-2}$ of $\cF_1(M)$ are divisible by p.
Abstract. We initiate the study of representations of elementary abelian $p$-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by $r$ nilpotent elements, $k[t_1, \ldots, t_r]/(t^p_1, \ldots, t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.
Abstract. We initiate the investigation of the projective variety E(r,g) of elementary subalgebras of dimension r of a (p-restricted) Lie algebra g for some r > 1 and demonstrate that this variety encodes considerable information about the representations of g. For various choices of g and r, we identify the geometric structure of E(r,g). We show that special classes of (restricted) representations of g lead to algebraic vector bundles on E(r,g). For g = Lie(G) the Lie algebra of an algebraic group G, rational representations of G enable us to realize familiar algebraic vector bundles on G-orbits of E(r,g).
* This preprint is now defunct as it has been replaced by two papers: "Elementary subalgebras of Lie algebras" and "Vector bundles associated to Lie algebras".Abstract: This is a survey article covering developments in representation theory of finite group schemes over the last fifteen years. We start with the finite generation of cohomology of a finite group scheme and proceed to discuss various consequences and theories that ultimately grew out of that result. This includes the theory of one-parameter subgroups and rank varieties for infinitesimal group schemes; the π-points and Π-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on varieties closely related to Proj H*(G,k) for an infinitesimal group scheme G. The material is mostly complementary to the article of D. Benson on elementary abelian p-groups in the same volume; we concentrate on the aspects of the theory which either hold generally for any finite group scheme or are specific to finite group schemes which are not finite groups. In the last section we discuss varieties of elementary subalgebras of modular Lie algebras, generalizations of modules of constant Jordan type, and new constructions of bundles on projective varieties associated to a modular Lie algebra.
Abstract: We initiate the investigation of the projective varieties E(r,g) of elementary subalgebras of dimension r of a (p-restricted) Lie algebra g for various r>1. These varieties E(r,g) are the natural ambient varieties for generalized support varieties for restricted representations of g. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional g-module M, the local (r,j)-radical rank and local (r,j)-socle rank, functions which are lower/upper semicontinuous on E(r,g). Examples are given of g-modules for which some of these rank functions are constant.
Abstract: We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra g. These are sheaves on locally closed subvarieties of the projective variety E(r,g) of elementary subalgebras of g of dimension r. We show that representations of constant radical or socle rank studied in [CFP3] which generalize modules of constant Jordan type lead to algebraic vector bundles on E(r,g). For g = Lie(G), the Lie algebra of an algebraic group G, rational representations of G enable us to realize familiar algebraic vector bundles on G-orbits of E(r,g).