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This work concerns the stable module category of a finite group over a field of characteristic dividing the group order. The minimal localising tensor ideals correspond to the non-maximal homogeneous prime ideals in the cohomology ring of the group. Given such a prime ideal, a number of characterisations of the dualisable objects in the corresponding tensor ideal are given. One characterisation of interest is that they are exactly the modules whose restriction along a corresponding $\pi$-point are finite dimensional plus projective. A key insight is the identification of a special property of the stable module category that controls the cohomological behaviour of local dualisable objects. This property, introduced in this work for general triangulated categories and called local regularity, is related to strong generation. A major part of the paper is devoted to developing this notion and investigating its ramifications for various special classes of objects in tensor triangulated categories., Comment: 34 pages

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full subcategory of all $G$-modules projective over $R$ generated by the lattices. The stable category of such $G$-modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of $G$. Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices., Comment: 35 pages. The Introductions, and sections 2 and 5 have been rewritten significantly

We discuss dualisable objects in minimal subcategories of compactly generated tensor triangulated categories, paying special attention to the derived category of a commutative noetherian ring. A cohomological criterion for detecting these local dualisable objects is established. Generalisations to other related contexts are discussed., Comment: 15 pages

Consider an almost-simple algebraic group G and a choice of complex root of unity q. We study the category of quasi-coherent sheaves $\mathscr{X}_q$ on the half-quantum flag variety, which itself forms a sheaf of tensor categories over the classical flag variety G/B. We prove that the category of small quantum group representations for G at q embeds fully faithfully into the global sections of $\mathscr{X}_q$, and that the fibers of $\mathscr{X}_q$ over G/B recover the tensor categories of representations for the small quantum Borels. These relationships hold both at an abelian and derived level. Subsequently, reduction arguments, from the small quantum group to its Borels, appear algebrogeometrically as "fiber checking" arguments over $\mathscr{X}_q$. We conjecture that $\mathscr{X}_q$ also contains the category of dg sheaves over the Springer resolution as a full monoidal subcategory, at the derived level, and hence provides a monoidal correspondence between the Springer resolution and the small quantum group. We relate this conjecture to a known equivalence between dg sheaves on the Springer resolution and the principal block in the derived category of quantum group representations [ABG04, BL07]., Comment: 55 pages, replaces Part I of arXiv:2203.10764

Given a finite cocommutative Hopf algebra $A$ over a commutative regular ring $R$, the lattice of localising tensor ideals of the stable category of Gorenstein projective $A$-modules is described in terms of the corresponding lattices for the fibres of $A$ over the spectrum of $R$. Under certain natural conditions on the cohomology of $A$ over $R$, this yields a stratification of the stable category. These results apply when $A$ is the group algebra over $R$ of a finite group, and also when $A$ is the exterior algebra on a finite free $R$-module., Comment: 26 pages. Clarified connections to the work of Lau (arXiv:2101.01446). The introduction has been rewritten a little

We consider the small quantum group u_q(G), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the small quantum group in type A admits a continuous surjection P(\~N) \to Spec(stab u_q(G)) from the (projectivized) Springer resolution. This surjection is shown to be a homeomorphism over a dense open subset in the spectrum. In type A_1 we calculate the Balmer spectrum precisely, where it is shown to be the projectivized nilpotent cone. Our results extend to arbitrary Dynkin type provided certain conjectures hold for the small quantum Borel. At the conclusion of the paper we touch on relations with geometric representation theory and logarithmic TQFTs, as represented in works of Arkhipov-Bezrukavnikov-Ginzburg and Schweigert-Woike respectively., Comment: 83 pages, comments welcome

We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type $A$, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme $G$, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components $\pi_0(G)$ in $G$., Comment: 43 pages, minor changes to the presentation

We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Q\to R$ by a Noetherian algebra $Q$ which is of finite global dimension. We show that hypersurface support defines a support theory for the big singularity category $Sing(R)$, and that the support of an object in $Sing(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz' support theory for (commutative) local complete intersections. In a companion piece, we employ hypersurface support, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras., Comment: 20 pages

We survey some methods developed in a series of papers, for classifying localising subcategories of tensor triangulated categories. We illustrate these methods by proving a new theorem, providing such a classification in the case of the stable module category of a unipotent finite supergroup scheme.

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic $p\ge 3$, starting with a definition of a $\pi$-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and $\pi$-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra $k[t,\tau]/(t^p-\tau^2)$, where $t$ has even degree and $\tau$ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme., Comment: 25 pages

We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$., Comment: 52 pages, minor changes to text

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one., Comment: 120 pages; the last version has only exposition changes

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. 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. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes., Comment: 19 pages

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology., Comment: 19 pages

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. 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 $E\subseteq G_K$, the restriction of $x_K$ 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 $E\subseteq G_K$, the restriction of $M_K$ 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., Comment: 46 pages; Sections 12 on Z-graded group schemes and the Steenrod algebra is revised compared to the previous version

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 $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme., Comment: 24 pages. This version corrects a mistake in the statement of Theorem 3.1; see also Theorem 1.4, and Examples 3.6 and 3.7. References have been updated

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 pi-points in the sense of Friedlander and Pevtsova. We identify for each pi-point an endofinite module which both generates the corresponding minimal localising subcategory and cogenerates the corresponding minimal colocalising subcategory., Comment: 17 pages, final version to appear in Annals of K-Theory. The duality statement in Theorem 3.1 of v1 has been removed since it is incorrect, and some subsequent arguments were modified

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., Comment: 38 pages. Minor changes from the previous version. This paper will appear in the Journal of the American Mathematical Society

We introduce the notion of $\pi$-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., Comment: 17 pages; minor changes. This will appear in the Math. Zeitschrift

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., Comment: 31 pages

The article covers developments in the 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 $\pi$-points and $\Pi$-support spaces for finite group schemes, modules of constant rank and constant Jordan type, and construction of bundles on projective varieties associated with cohomology ring of an infinitesimal group scheme $G$. 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., Comment: 31 page

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., Comment: Replaces 1st half of arXiv:1207.5898 (with same title). To appear in the Journal of Algebra

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 $\mathbb E(r, \mathfrak g)$., Comment: Replaces second half of arXiv:1207.5898 . To appear in Crelle

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., Comment: 15 pages, minor corrections to Section 4, to appear in Proceedings of the American Mathematical Society

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 $E(r, g)$., Comment: The paper has been split into two at the request of the referee, the first part has the same title as the older combined version

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, $k[t_1,..., t_r]/(t^p_1,..., t_r^p)$. 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.

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} $\Gamma^j(G)_M$, $1\leq j \leq p-1$, 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 $\zeta$ in $\HHH^1(G,M)$ we introduce the {\it zero locus} $Z(\zeta) \subset \Pi(G)$. We show that $Z(\zeta)$ is a closed subvariety, and relate it to the non-maximal rank varieties. We also extend the construction of $Z(\zeta)$ to an arbitrary extension class $\zeta \in \Ext^n_G(M,N)$ whenever $M$ and $N$ are $kG$-modules of constant Jordan type.

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors F_i from finitely generated kE-modules of constant Jordan type to vector bundles over projective space of dimension r-1. The fibers of these functors encode complete information about the Jordan type of the module. We prove that given any vector bundle of rank s on P^{r-1}, there is a kE-module M of stable constant Jordan type [1]^s such that the functor F_1 applied to M yields the original vector bundle for p=2 and the Frobenius twist of the original vector bundle for p>2. We also 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 F_1(M) are divisible by p., Comment: 14 pages

Let G be an infinitesimal group scheme over a field k of positive characteristic p. We introduce the global p-nilpotent operator $\Theta_G: k[G] \to k[V(G)]$, where V(G) is the scheme which represents 1-parameter subgroups of G. This operator applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain G-modules (including those of constant Jordan type), we employ the global p-nilpotent operator to associate various algebraic vector bundles on the projective scheme $\bP(G)$, the projectivization of the scheme of one-parameter subgroups of 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 $\bP(G)$., Comment: 55 pages. This is a final version to appear in Transactions of the AMS. Significantly revised from the original preprint. In particular, last section is completely rewritten with some K_0 calculations added

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., Comment: 47 pages

We define a rank variety for a module of a noncocommutative Hopf algebra $A = \Lambda \rtimes G$ where $\Lambda = k[X_1, ..., X_m]/(X_1^{\ell}, ..., X_m^{\ell})$, $G = ({\mathbb Z}/\ell{\mathbb Z})^m$, 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., Comment: 30 pages, submitted