arXiv subjects

Eric M. Friedlander

Publications and source records attributed to Eric M. Friedlander.

At least 19 recordsLinked to original sources

arXiv2026

The Stable Adams Conjecture

We provide proofs of several variants for spectra of the original Adams Conjecture. Our approach uses F-spaces (also known as Segal Gamma-spaces), resulting in establishing both unstable and stable homotopy equivalences between maps of spectra associated to the J-map and the Adams operations. We employ a rigid version of Artin-Mazur etale homotopy theory applied to commutative diagrams of simplicial schemes defined over the Witt vectors of fields of positive characteristic. Much of this text is devoting to developing a theory of X-fibrations and l-complete X-fibrations over the F-spaces we consider.

math.AT↗

arXiv2025

Invariants for $\mathbb G_{(r)}$-modules

We revisit the constructions given by J. Pevtsova and the author of refined invariants for finite dimensional representations of infinitesimal group schemes $\mathbb G_{(r)}$ over a field $k$ of characteristic $p>0$. Our focus is on the universal $p$-nilpotent operator seen as an element in the group algebra of the group scheme $\mathbb G_{(r),X}$ over $X$, where $X$ is either the moduli space $V_r(\mathbb G)$ of height $r$ $1$-parameter subgroups of $\mathbb G$ or the moduli space $\mathcal C_r(\mathcal N_p(\mathfrak g))$ of $r$-tuples of $p$-nilpotent, pair-wise commuting elements of the Lie algebra of $\mathbb G$. We formalize Jordan type function using several variants of the continuous function $JT_{\mathbb G,r,M}(-): \mathbb P V_r(\mathbb G) \to \mathcal Y$ where $\mathcal Y$ is the poset of Young diagrams with $p$-columns. One of these variants is designed to be more conducive to computation. The vector bundle construction given by J. Pevtsova and the author is extended to all finite dimensional $\mathbb G_{(r)}$-modules, producing coherent sheaves on $X$ which are locally free on the strata of $X$ associated to $JT_{\mathbb G,r,M}(-)$.

math.RT↗

arXiv2024

Infinite dimensional modules for linear algebraic groups

We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider the abelian subcategory $CoMod(C) \subset Mod(\mathbb G)$ and the left exact functor $(-)_C: Mod(\mathbb G) \to CoMod(C)$ that is right adjoint to the inclusion functor. The class of cofinite $\mathbb G$-modules is formulated using finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ and the new invariant of "cofinite type" is introduced. We are particularly interested in mock injective $\mathbb G$-modules, $\mathbb G$-modules which are not seen by earlier support theories. Various properties of these ghostly $\mathbb G$-modules are established. The stable category $StMock(\mathbb G)$ is introduced, enabling mock injective $\mathbb G$-modules to fit into the framework of tensor triangulated categories.

math.RT↗

arXiv2024

Filtrations and Growth of $\mathbb G$-modules

We investigate infinite dimensional modules for an affine group scheme $\mathbb G$ of finite type over a field of positive characteristic $p$. For any subspace $X \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider the abelian subcategory $Mod(\mathbb G,X) \subset Mod(\mathbb G)$ of ``$X$-comodules" and the left exact functor $(-)_X: Mod(\mathbb G) \to Mod(\mathbb G,X)$ which is right adjoint to the inclusion functor. We employ ``ascending converging sequences" $\{ X_i \}$ of subspaces of $\mathcal O(\mathbb G)$ to provide functorial filtrations $\{ M_{X_i }\}$ of each $\mathbb G$-module $M$. A $\mathbb G$-module $M$ is injective if and only if each $M_{X_i}$ is an injective $X_i$-comodule for some (or, equivalently, for all) such $\{ X_i \}$. We consider the explicit ascending converging sequence $ \{ \mathcal O(\mathbb G)_{\leq d,ϕ} \}$ of finite dimensional subcoalgebras of $\mathcal O(\mathbb G)$ depending upon a closed embedding $ϕ: \mathbb G \ \hookrightarrow \ GL_N$. Of particular interest to us are mock injective $\mathbb G$-modules, modules whose support varieties are empty. Restrictions of a $\mathbb G$-module to each $\mathcal O(\mathbb G)_{\leq d,ϕ}$ provide new invariants for $\mathbb G$-modules. For cofinite $\mathbb G$-modules $M$, we explore the the growth of $d \mapsto M_{\cal O(\mathbb G)_{\leq d,ϕ}}$.

math.RT↗

arXiv2023

Reformulation of the stable Adams conjecture

We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur \'etale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "\'etale functor" which commutes up to homotopy with products of simplicial schemes. In order to prove the Stable Adams Conjecture, we construct the universal $\mathbb Z/\ell$-completed $X$-fibrations for various pointed simplicial sets $X$. Thus, two maps from a given $\mathcal F$-space $\underline{\mathcal B}$ to the base $\mathcal F$-space of the universal $\mathbb Z/\ell$-completed $X$-fibration $\pi_{X,\ell}: \underline {\mathcal B} (G_\ell(X),X_\ell) \to \underline {\mathcal B} G_\ell(X)$ determine homotopy equivalent maps of spectra if and only they correspond via pull-back of $\pi_{X,\ell}$ to fiber homotopy equivalent $\mathbb Z/\ell$-completed $X$-fibrations over $\underline {\mathcal B}$. For the proof of the Stable Adams Conjecture, we consider maps of $\mathcal F$-spaces $\underline {\mathcal B }\to \underline {\mathcal B} G_\ell(S^2)$ where $\underline {\mathcal B}$ is an $\mathcal F$-space model of connective $\ell$-completed connective $K$-theory.

math.AT↗

arXiv2022

Support Varieties and stable categories for algebraic groups

We consider rational representations of a connected linear algebraic group $\mathbb G$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto Π(\mathbb G)_M$ to $\mathbb G$-modules of the $π$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more "intrinsic" and "explicit" theory $M \mapsto \mathbb P\mathfrak C(\mathbb G)_M$ of supports for an algebraic group of exponential type, a theory which uses 1-parameter subgroups $\mathbb G_a \to \mathbb G$. We extend our support theory to bounded complexes of $\mathbb G$-modules, $C^\bullet \mapsto Π(\mathbb G)_{C^\bullet}$. We introduce the tensor triangulated category $StMod(\mathbb G)$, the Verdier quotient of the bounded derived category $D^b(Mod(\mathbb G))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the standard properties" for a theory of supports for $StMod(\mathbb G)$. As an application, we employ $C^\bullet \mapsto Π(\mathbb G)_{C^\bullet}$ to establish the classification of $(r)$-complete, thick tensor ideals of $stmod(\mathbb G)$ in terms of $stmod(\mathbb G)$-realizable subsets of $Π(\mathbb G)$ and the classification of $(r)$-complete, localizing subcategories of $StMod(\mathbb G)$ in terms of $StMod(\mathbb G)$-realizable subsets of $Π(\mathbb G)$.

math.RT↗

arXiv2021

Support theory for Drinfeld doubles of some infinitesimal group schemes

Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $π$-point style rank variety for the Drinfeld double, identify $π$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.

math.RT↗

arXiv2021

Support Theory for Extended Drinfeld Doubles

Following earlier work with Cris Negron on the cohomology of Drinfeld doubles $D(\mathbb G_{(r)})$, we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" $\tilde D(\mathbb G_{(r)})$ of Frobenius kernels $\mathbb G_{(r)}$ of smooth linear algebraic groups $\mathbb G$ over a field $k$ of characteristic $p > 0$. To a $\tilde D(\mathbb G_{(r)})$-module $M$ we associate the space $Π(\tilde D(\mathbb G_{(r)}))_M$ of equivalence classes of "pairs of $π$-points" and prove most of the desired properties of $M \mapsto Π(\tilde D(\mathbb G_{(r)}))_M$. Namely, this association satisfies the "tensor product property" and admits a natural continuous map $Ψ_{\tilde D}$ to cohomological support theory. Moreover, for $M$ finite dimensional and with suitable conditions on $\mathbb G_{(r)}$, this association provides a "projectivity test", $Ψ_{\tilde D}$ is a homeomorphism, and identifies $Π(\tilde D(\mathbb G_{(r)}))_M$ with the cohomological support variety of $M$ for various classes of $\tilde D(\mathbb G_{(r)})$-modules $M$.

math.RT↗

arXiv2019

Cohomology of unipotent group schemes

We verify that universal classes in the cohomology of $GL_N$ determine explicit cohomology classes of Frobenius kernels $G_{(r)}$ of various linear algebraic groups $G$ . We consider the relationship of $\varprojlim_r H^*(U_{(r)},k)$ to the rational cohomology $H^*(U,k)$ of many unipotent algebraic groups $U$. The second half of this paper investigates in detail the cohomology of Frobenius kernels $(U_3)_{(r)}$ of the Heisenberg group $U_3 \subset GL_3$.

math.RT↗

arXiv2019

Geometric Invariants of Representations of Finite Groups

J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator" for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p > 0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$ associated to a $G$-module $M$. Of special interest is the fact that these coherent sheaves are vector bundles if $M$ is of constant Jordan type. In this paper, we provide similar invariants for a finite group $τ$ which recover the invariants earlier obtained for elementary abelian $p$-groups. To do this, we replace the analogue of 1-parameter subgroups by a refined version of equivalence classes of $π$-points for $kτ$. More generally, we provide a construction of vector bundles for the semi-direct product $G\rtimes τ$ of an infinitesimal group scheme $G$ and a finite group $τ$. A major motivation for this study is to further our understanding of the relationship between representations of $\mathbb G(\mathbb F_p)$ and $\mathbb G_{(r)}$ associated to a finite dimensional rational $\mathbb G$-module $M$, where $\mathbb G$ is a reductive group with $r$-th Fobenius kernel $\mathbb G_{(r)}$. Using vector bundles, we extend and sharpen earlier results comparing support varieties.

math.RT↗

arXiv2017

Rational Cohomology and Supports for Linear Algebraic Groups

This paper is an extended version of four lectures at PIMS in Vancouver given June 27 - 30, 2016. The primary goal of these lectures was to publicize the author's recent efforts to extend to representations of linear algebraic groups the "theory of support varieties" which has proved successful in the study of representations of finite group schemes. The lectures offer readers an introduction to the subject together with "homework problems, simplify and clarify some points in the literature, and mention some directions for future research.

math.RT↗

arXiv2016

An approach to intersection theory on singular varieties using motivic complexes

We introduce techniques of Suslin, Voevodsky, and others into the study of singular varieties. Our approach is modeled after Goresky-MacPherson intersection homology. We provide a formulation of perversity cycle spaces leading to perversity homology theory and a companion perversity cohomology theory based upon generalized cocycle spaces. These theories lead to conditions on pairs of cycles which can be intersected and a suitable equivalence relation on cocycles/cycles enabling pairings on equivalence classes. We establish suspension and splitting theorems, as well as a localization property. Some examples of intersections on singular varieties are computed.

math.KT↗

arXiv2015

Filtrations, 1-parameter Subgroups, and Rational Injectivity

We investigate rational $G$-modules $M$ for a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic $p > 0$ using filtrations by sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the special case of the additive group $\mathbb G_a$, interesting structures and examples are revealed. The "degree" filtration we consider for unipotent algebraic groups leads to a "filtration by exponential degree" applicable to rational $G$ modules for any linear algebraic group $G$ of exponential type; this filtration is defined in terms of 1-parameter subgroups and is related to support varieties introduced recently by the author for such rational $G$-modules. We formulate in terms of this filtration a necessary and sufficient condition for rational injectivity for rational $G$-modules. Our investigation leads to the consideration of two new classes of rational $G$-modules: those that are "mock injective" and those that are "mock trivial".

math.RT↗

arXiv2014

Elementary subalgebras of Lie algebras

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 $r > 0$ 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 $E(r, g)$.

math.RA↗

arXiv2014

Elementary Subalgebrs of Lie Algebras

We initiate the investigation of the projective varieties $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These varieties $\mathbb E(r,\mathfrak g)$ are the natural ambient varieties for generalized support varieties for restricted representations of $\mathfrak g$. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional $\mathfrak u(\mathfrak g)$-module $M$, the local $(r,j)$-radical rank and local $(r,j)$-socle rank, functions which are lower/upper semicontinuous on $\mathbb E(r,\mathfrak g)$. Examples are given of $\mathfrak u(\mathfrak g)$-modules for which some of these rank functions are constant.

math.RT↗

arXiv2014

Vector Bundles Associated to Lie Algebras

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak g$. These are sheaves on locally closed subvarieties of the projective variety $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of $\mathfrak g$ of dimension $r$. We show that representations of constant radical or socle rank studied in \cite{CFP3} which generalize modules of constant Jordan type lead to algebraic vector bundles on $\mathbb E(r,\mathfrak g)$. For $\mathfrak g = Lie(G)$, the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $\mathbb E(r, \mathfrak g)$.

math.AG↗

arXiv2014

Support varieties for rational representations

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p > 0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M \mapsto V(G)_M$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_M$ and the family of support varieties $V_r(G)_M$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)} \subset G$. These support varieties seem particularly appropriate for the investigation of infinite dimensional rational $G$-modules.

math.RT↗

arXiv2011

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} $Γ^j(G)_M$, $1\leq j \leq p-1$, 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 $ζ$ in $\HHH^1(G,M)$ we introduce the {\it zero locus} $Z(ζ) \subset Π(G)$. 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 $ζ\in \Ext^n_G(M,N)$ whenever $M$ and $N$ are $kG$-modules of constant Jordan type.

math.RT↗