Papers, preprints and other things
· N. Andruskiewitsch, I. Angiono, J. Pevtsova, S. Witherspoon, “Cohomology rings of finite-dimensional pointed Hopf algebras over abelian groups”, Res. Math. Sci. 9 (2022), no. 1, Paper No. 12, 132 pp.
· D. Benson, S. Iyengar, H. Krause, J. Pevtsova, “Stratification and duality for unipotent finite supergroup schemes”, in Equivariant topology and derived algebra, 241-275, London Math. Soc. Lecture Note Ser., 474, Cambridge Univ. Press, Cambridge, (2022)
· C. Negron, J. Pevtsova, “Support
for integrable Hopf algebras via noncommutative
hypersurfaces”, International
Mathematics Research Notices (2021),
rnab264, https://doi.org/10.1093/imrn/rnab264
· D. Benson, S. Iyengar, H. Krause, J. Pevtsova, “Rank varieties and п-points for elementary supergroup schemes”, Trans. Amer. Math. Soc. Ser. B 8 (2021), 971--998
· D. Benson, S. Iyengar, H. Krause, J. Pevtsova,
“Local duality for the singularity category of a finite dimensional Gorenstein algebra”, Nagoya Math. Journal, 244
(2021), 1-24.
· D. Benson, J. Pevtsova, “Representations and cohomology of a family of finite supergroup schemes”, J. Algebra 561 (2020), 84--100, special issue dedicated to the memory of Kai Maagard.
Abstract: We examine the cohomology and representation theory of a certain family of finite supergroup schemes. In particular, we show that a certain relation holds in the cohomology
ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This
special case feeds into the proof of a more general detection theorem for unipotent finite
supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.
· D. Benson, S. Iyengar, H. Krause,
J. Pevtsova, “Detecting nilpotence
and projectivity over finite unipotent supergroup schemes”, Selecta Math, 27 (2021),
no. 2, paper No. 25, 59pp.
Abstract: This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme G over a perfect field k of positive characteristic p>2. It is proved that an element x in the cohomology of G is nilpotent if and only if for every extension field K of k and every elementary sub-supergroup scheme E of GK, the restriction of xK to E is nilpotent. It is also shown that a kG-module M is projective if and only if for every extension field K of k and every elementary sub-supergroup scheme E of GK, the restriction of MK to E is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra.
· D. Benson, S. Iyengar, H. Krause,
J. Pevtsova, “Local duality for
representations of finite group schemes”, Compositio Math. 155 (2019) no. 2, 424-453
Abstract: A duality theorem for the stable module category
of representations of a finite group scheme is proved. One of its consequences
is an analogue of Serre duality, and the existence of Auslander-Reiten
triangles for the ϸ-local and ϸ-torsion subcategories of the
stable category, for each homogeneous prime ideal ϸ in the cohomology ring of the
group scheme.
· J.
Pevtsova, J. Stark, “Varieties of elementary subalgebras of maximal
dimension for modular Lie algebras”, Geometric and topological aspects of the representation theory of
finite groups, 339-375, Springer Proc. Math. Stat., 242, Springer, Cham, (2018)
Abstract. Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties E(r,g) of r-dimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra g. In this paper, we identify the varieties E(r,g) for a reductive restricted Lie algebra g and r the maximal dimension of an abelian p-nilpotent subalgebra of g.
The following code
by J. Stark is referenced in the above paper:
CommutingRootSubgroups.sws
roots.mgm
The CommutingRootSubgroups.sws
file contains code to compute maximal sets of commuting roots and their orbits
in root systems. It is written in
sage and supplied as a worksheet for the sage notebook. You can either download sage here or access it online here. The file roots.mgm has code that checks
whether all abelian lie subalgebras of a reductive lie algebra can be
conjugated to the lie subalgebra generated by its leading terms. It is written in Magma.
· Colocalising subcategories of modules over finite group schemes, (with D. Benson, S. Iyengar, and H. Krause), Annals of K-theory (2), no. 3, (2017), 387-408
Abstract: The Hom closed colocalising subcategories of the stable module category of
a finite group scheme are classified. This complements the classification of
the tensor closed localising subcategories in our
previous work. Both classifications involve 
-points
in the sense of Friedlander and Pevtsova. We identify for each -point
an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding
minimal colocalising subcategory.
· Stratification for module categories of finite group schemes, (with D. Benson, S. Iyengar, and H. Krause), Journal of the AMS (31) no. 1, (2018), 265--302.
Abstract: The tensor ideal localising subcategories of the stable module category of all, including infinite dimensional, representations of a finite group scheme over a field of positive characteristic are classified. Various applications concerning the structure of the stable module category and the behavior of support and cosupport under restriction and induction are presented.
· Stratification and -cosupport:
finite groups, (with D. Benson, S. Iyengar, and H. Krause), Math Zeitschrift 287,
no. 3-4, (2017) 947-965
Abstract. We introduce the notion of -cosupport
as a new tool for the stable module category of a finite group scheme. In the
case of a finite group, we use this to give a new proof of the classification
of tensor ideal localising subcategories. In a sequel
to this paper, we carry out the corresponding classification for finite group
schemes.
· Localizing subcategories for finite group schemes , "Long" abstract for a talk given in Oberwolfach, Oberwolfach Reports 12 (2015).
· J. Pevtsova, Z. Reichstein, “Modular representations of high
essential dimension”, an appendix to ``A numerical invariant for
linear representations of finite groups” by N. Karpenko and Z.
Reichstein, Commentarii Math. Helvetici. 90 (2015), no. 3, pp 667--701
Abstract. We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that in the modular setting ed(ρ) can be arbitrary large (under a mild assumption on G). Here, G is fixed, and ρ is allowed to range over the finite-dimensional representations of G. The appendix gives a constructive version of this result.
· J. Pevtsova, S. Witherspoon, “Tensor Ideals and Varieties for Modules of Quantum Elementary Abelian Groups”, Proceedings of the AMS, 143 (2015), no. 9, pp 3727--3741
Abstract. In a previous paper we constructed rank and support variety theories for "quantum elementary abelian groups," that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.
· Jon F. Carlson, Eric M.
Friedlander, Julia Pevtsova, “Vector
bundles associated to Lie algebras”, J. f\"ur die Reine und Ang. Math. (Crelle), 716 (2016),
pp 147--178
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).
· Elementary subalgebras of Lie algebras (with Jon F. Carlson and Eric M. Friedlander), J. Algebra 442 (2015), pp 155--189
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.
· Elementary subalgebras of Lie algebras (with Jon F. Carlson and Eric M. Friedlander), preprint (2012)*
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".
· Representations of elementary abelian p-groups and bundles on Grassmannians (with Jon F. Carlson and Eric M. Friedlander), Advances in Math. 229 (2012), pp. 2985-3051
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.
· Invariants in modular representation theory , "Long" abstract for a talk given in Oberwolfach, Oberwolfach Reports 7, issue 3 (2010), pp. 1885-1952.
· A realization theorem for modules of constant Jordan type and vector bundles (with D.J. Benson), Trans. Amer. Math. Soc. 364 (2012), pp. 6459--6478.
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.
· Generalized support varieties for finite group schemes, (with E. Friedlander), Documenta Math, Extra Volume Suslin (2010), pp. 197--222
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 $\zeta \in H^\bullet(G,k)$.
· Cohomology of finite dimensional pointed Hopf algebras, (with M. Mastnak, P. Schauenburg and S. Witherspoon), Proceedings of the LMS, 100 (2010), part 2, pp.377--404.
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.
· Constructions for infinitesimal group schemes, (with E. Friedlander), Trans. Amer. Math. Soc. 363 (2011), no. 11, pp. 6007-6061.
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 $P(G)$.
· Modules of constant Jordan type, (with Jon F. Carlson and Eric M. Friedlander), J. f\"ur die Reine und Ang. Math. (Crelle) 614 (2008), 191-234.
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.
· Varieties for Modules of Quantum Elementary Abelian Groups, (with S. Witherspoon), Algebra Represent. Theory (2009) 12, 567-595.
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.
· A note on classification of conjugacy classes of maximal elementary abelian subgroups of GL(4, Fp)
Abstract. This is a >>very<< explicit calculation of conjugacy classes of maximal elementary abelian subgroups inside GL(4, Fp) 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!
· Generic and maximal Jordan types, (with E. Friedlander and A. Suslin), Invent. Math. 168 (2007), pp. 485--522.
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 $p$.
· $\Pi$-supports for modules for finite group schemes over a field , (with E. Friedlander), Duke Math. J., 139 (2007), no. 2, pp. 317--368.
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 $\Proj H*(G,k)$.
· Generic Jordan Type pf modular representations, "Long" abstract for a talk given in Oberwolfach, Oberwolfach Reports 2, issue 3 (2005), pp. 2375–2434.
· Representation-theoretic support spaces for finite group schemes, (with E. Friedlander), American Journal of Math. {127} (2005), pp. 379--420
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.
· Erratum to "Representation-theoretic support spaces for finite group schemes", PDF file
· Support cones for infinitesimal group schemes , Hopf Algebras, 203-213, Lecture Notes in Pure $\&$ Applied Math., {237}, Dekker, New York, 2004
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.
· Infinite Dimensional Modules for Frobenius Kernels, Journal of Pure & Applied Algebra, {173}, no 1, 59-83, 2002
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.
· Thesis "Infinite dimensional modules for
infinitesimal group schemes", 2002.