aa r X i v : . [ m a t h . QA ] S e p MODULE CATEGORIES OVER AFFINE SUPERGROUPSCHEMES
SHLOMO GELAKI
Abstract.
Let k be an algebraically closed field of characteristic 0or p ą
2. Let G be an affine supergroup scheme over k . We classifythe indecomposable exact module categories over the tensor cat-egory sCoh f p G q of (coherent sheaves of) finite dimensional O p G q -supermodules in terms of p H , Ψ q -equivariant coherent sheaves on G . We deduce from it the classification of indecomposable geomet-rical module categories over sRep p G q . When G is finite, this yieldsthe classification of all indecomposable exact module categoriesover the finite tensor category sRep p G q . In particular, we obtaina classification of twists for the supergroup algebra k G of a finitesupergroup scheme G , and then combine it with [EG2, Corollary4.1] to classify finite dimensional triangular Hopf algebras with theChevalley property over k . introduction Let k be an algebraically closed field of characteristic 0 or p ą
2. Let G be a finite group scheme over k . Consider the finite tensor categoryCoh p G q of finite dimensional O p G q -modules over k , and the finite ten-sor category Rep p G q of finite dimensional rational representations of G over k . In [G2] we classified the indecomposable exact module cate-gories over Rep p G q , generalizing the classification of Etingof and Ostrik[EO] for constant groups G . In particular, we obtained the classifica-tion of twists for the group algebra kG , reproducing the classificationgiven by Movshev for constant groups G in zero characteristic [Mo].The goal of this paper is to extend [G2] to the super case, and thencombine it with [EG2, Corollary 4.1] to classify finite dimensional tri-angular Hopf algebras with the Chevalley property over k (as promisedin [EG2, Remark 1.5(3)]).Let G be a finite supergroup scheme over k . Following [G2], we firstclassify the indecomposable exact module categories over sRep p O p G qq ,where O p G q is the coordinate Hopf superalgebra of G , and then use Date : September 25, 2019.
Key words and phrases. affine supergroup scheme; tensor category; module cat-egory; twist; triangular Hopf algebra. the fact that they are in bijection with the indecomposable exact mod-ule categories over sRep p G q [EO] to get the classification of the latterones. The reason we approach it in this way is that sRep p O p G qq istensor equivalent to the tensor category sCoh f p G q “ sCoh p G q (of coher-ent sheaves) of finite dimensional O p G q -supermodules with the tensorproduct of convolution of sheaves, which allows us to use geometrictools and arguments.In fact, in Theorem 5.1 we classify the indecomposable exact mod-ule categories over sCoh f p G q , where G is any affine supergroup schemeover k (i.e., G is not necessarily finite). The classification is given interms of certain p H , Ψ q -equivariant coherent sheaves on G (see Defini-tion 4.1). However when G is not finite, not all indecomposable exactmodule categories over sRep p G q are obtained from those over sCoh f p G q (see Theorem 6.6 and Remark 6.7); we refer to those which are as ge-ometrical . So the classification of exact module categories (even fiberfunctors) over sRep p G q for infinite affine supergroup schemes G remainsunknown (even when G is a linear algebraic group over C , see [G2]).As a consequence of our results, combined with [AEGN, EO], weobtain in Corollary 7.1 that gauge equivalence classes of twists forthe supergroup algebra k G of a finite supergroup scheme G over k are parameterized by conjugacy classes of pairs p H , J q , where H Ď G is a closed supergroup subscheme and J is a non-degenerate twistfor k H (just as in the case of abstract finite groups). Furthermore,using Proposition 7.5 we show in Proposition 7.6 that a twist for G is non-degenerate if and only if it is minimal (again, as for abstractfinite groups). Finally, in Theorem 7.8 we classify finite dimensionaltriangular Hopf algebras with the Chevalley property over k . Acknowledgments.
The author is grateful to Pavel Etingof forstimulating and helpful discussions.2.
Preliminaries
Throughout the paper we fix an algebraically closed field k of char-acteristic 0 or p ą
2. We refer the reader to the book [EGNO] for thegeneral theory of tensor categories.2.1.
Affine supergroup schemes.
We refer the reader to, e.g. [W],for preliminaries on affine group schemes over k , and to [Ma] for pre-liminaries on affine supergroup schemes over k .Let G be an affine supergroup scheme over k , with unit morphisme : Spec p k q Ñ G , inversion morphism i : G Ñ G , and multiplicationmorphism m : G ˆ G Ñ G , satisfying the usual group axioms. Recall ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 3 that the coordinate algebra O p G q of G is a supercommutative Hopfsuperalgebra over k , and G is the functor from the category of su-percommutative k -superalgebras to the category of groups defined by R ÞÑ G p R q : “ Hom
SAlg p O p G q , R q (so-called functor of points). Notethat any affine supergroup scheme is the inverse limit of affine super-group schemes of finite type .A closed supergroup subscheme H of G is the spectrum of the Hopfquotient O p H q : “ O p G q{ I p H q by a Hopf ideal I p H q Ď O p G q . The ideal I p H q is referred to as the defining ideal of H in O p G q . For example, the even part of G is the closed group subscheme G Ď G with the definingideal I p G q “ x O p G q y , i.e., G is an ordinary affine group scheme withcoordinate algebra O p G q “ O p G q{x O p G q y . In particular, we have asurjective Hopf algebra map π : O p G q ։ O p G q .Let g “ g ‘ g be the Lie superalgebra of G , i.e., g is the space ofleft-invariant derivations of O p G q , g is the space of even derivationsof O p G q , and g is the space of odd derivations of O p G q . We have g “ p m { m q ˚ , where m Ă O p G q is the kernel of the augmentation map,and g “ Lie p G q is the Lie algebra of G .Recall that G acts on g via the adjoint action. Let a : G ˆ g ˚ Ñ g ˚ be the coadjoint action of G on g ˚ . Then ^ g ˚ is an O p G q -comodulealgebra with structure map a ˚ : ^ g ˚ Ñ O p G q b ^ g ˚ .Since O p G q is a quotient Hopf algebra of O p G q , it follows that O p G q has a canonical structure of a left O p G q -comodule algebra with struc-ture map p π b id q ∆. It is known [Ma, Theorem 4.5] that the subalgebraof O p G q -coinvariants in O p G q is isomorphic to ^ g ˚ , and that we havea tensor decomposition(2.1) O p G q – ^ g ˚ b O p G q of O p G q -supercomodule counital superalgebras. In particular, we have abelian equivalencessRep p O p G qq – sRep p^ g ˚ q b sVect sRep p O p G qq (2.2) – sRep p^ g ˚ q b Rep p O p G qq (2.3)such that Rep p O p G qq can be identified with a full tensor subcategoryof sRep p O p G qq in the obvious way.Recall that we haveRep p O p G qq “ Coh f p G q “ à g P G p k q Coh f p G q g , where Coh f p G q g is the abelian subcategory of sheaves supported at g , with unique simple object δ g and indecomposable projective object Some authors use k r G s instead. SHLOMO GELAKI P g : “ { O p G q g in the pro-completion category, where O p G q g is thecompletion of O p G q at g [G2, Section 3.1]. Thus by (2.2), we have(2.4) sRep p O p G qq – à g P G p k q sRep p^ g ˚ q b Coh f p G q g as abelian categories.Recall that closed supergroup subschemes H Ď G are in bijectionwith pairs p H , h q , where H Ď G is a closed group subscheme, h Ď g is an H -invariant subspace, and r h , h s Ď h : “ Lie p H q (see, e.g.,[MS, Section 6.2]).Let Ψ : G ˆ G Ñ G m be a normalized even 2-cocycle. Equivalently,Ψ P O p G q b O p G q is a twist for O p G q , i.e., Ψ is an invertible evenelement satisfying the equations p ∆ b id qp Ψ qp Ψ b q “ p id b ∆ qp Ψ qp b Ψ q , p ε b id qp Ψ q “ p id b ε qp Ψ q “ . Finally, recall that a finite supergroup scheme G is an affine super-group scheme whose function algebra O p G q is finite dimensional. Inthis case, k G : “ O p G q ˚ is a supercocommutative Hopf superalgebra(called the group algebra of G ).2.2. Module categories over tensor categories.
Let C be a tensorcategory over k . Let Ind p C q and Pro p C q be the categories of Ind-objectsand Pro-objects of C , respectively. It is well known that the tensorstructure on C extends to a tensor structure on Ind p C q and Pro p C q .However Ind p C q and Pro p C q are not rigid, but the rigid structure on C induces two duality functors Pro p C q Ñ Ind p C q (“continuous dual”)and Ind p C q Ñ Pro p C q (“linear dual”), which we shall both denote by X ÞÑ X ˚ ; they are antiequivalence inverses of each other. It is alsoknown that Ind p C q has enough injectives.Recall that a (left) module category M over C is a locally finiteabelian category equipped with a (left) action b M : C b M Ñ M ,such that the bifunctor b M is bilinear on morphisms and biexact. Re-call also that M is exact if any additive module functor M Ñ M from M to any other C -module category M is exact, and that M is indecomposable if M is not equivalent to a direct sum of two nontrivialmodule subcategories. It is also known that the C -module structure on M extends to a module structure on Ind p M q over Ind p C q . Moreover, M is exact if and only if for any M P M and any injective object I P Ind p C q (resp., projective object P P Pro p C q ), I b M is injective inInd p M q (resp., P b M is projective in Pro p M q ) (see [EO, Propositions3.11, 3.16], [G2, Proposition 2.4]). ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 5
Following [EO], we say that two simple objects M , M P M are related if there exists an object X P C such that M appears as asubquotient in X b M M . This defines an equivalence relation, and M decomposes into a direct sum M “ ‘ M i of indecomposable ex-act module subcategories indexed by the equivalence classes (see [EO,Lemma 3.8 & Proposition 3.9] and [G2, Proposition 2.5]).Assume M is exact. Recall that an object δ P M generates M if forany M P M there exists X P C such that Hom M p X b M δ, M q ‰
0. Itis known that δ generates M if and only if for any M P M there exists X P C such that M is a subquotient of X b M δ (cf. [EO]). Thus if M is indecomposable and δ is simple, then δ P M generates M .Finally recall that for every two objects M , M P M , we have anobject Hom p M , M q P Pro p C q satisfyingHom M p M , X b M M q – Hom
Pro p C q p Hom p M , M q , X q , X P C (the dual internal Hom). For every M P M , the pro-object Hom p M, M q has a canonical structure of a coalgebra. In terms of internal Hom’s[EO], the algebra Hom p M, M q in Ind p C q is isomorphic to the dual al-gebra p Hom p M, M qq ˚ under the duality functor ˚ : Pro p C q Ñ Ind p C q .Now if M is indecomposable and exact, we have a C -module equiva-lence M – Comod
Pro p C q p Hom p M, M qq .3. The tensor category sCoh f p G q Let G be an affine supergroup scheme over k , and let O p G q “ O p G q ˆ k x u y be the Radford’s biproduct ordinary Hopf algebra, where u is a grou-plike element of order 2 acting on O p G q by parity, and∆ p x q “ ÿ p x b u | x | q b p x b q for every homogeneous element x P O p G q , where ∆ p x q “ ř x b x .Recall that we have an equivalence of tensor categoriesRep p O p G qq – sRep p O p G qq . In particular, Rep p O p G qq is a tensor subcategory of Rep p O p G qq . Definition 3.1.
Let sCoh f p G q (resp., sQCoh p G q ) be the tensor category(resp., monoidal category) of finite dimensional (resp., all) representa-tions of the Hopf algebra O p G q . The purely even case is treated in [G2, Section 3.1].
SHLOMO GELAKI
By definition, we have equivalences of tensor and monoidal categoriessCoh f p G q – sRep p O p G qq and sQCoh p G q – SRep p O p G qq , respectively, where sRep p O p G qq and SRep p O p G qq are the categories of finite dimensional and all representations of the Hopf superalgebra O p G q on k -supervector spaces, respectively.We have that Ind p sCoh f p G qq is the category of locally finite quasi-coherent sheaves of O p G q -supermodules (i.e., representations in whichevery vector generates a finite dimensional subrepresentation). Remark 3.2.
By a quasi-coherent sheaf on G we will mean a quasi-coherent sheaf of O p G q - supermodules , and by a finite quasi-coherentsheaf on G we will mean a quasi-coherent sheaf of finite dimensional O p G q - supermodules . Note that finite quasi-coherent sheaves on G areautomatically supported on finite sets in G . Thus, one can thinkof sCoh f p G q and sQCoh p G q as the k -linear abelian categories of fi-nite quasi-coherent sheaves and quasi-coherent sheaves on G , respec-tively (which explains our notation). In particular, the tensor productsin sCoh f p G q and sQCoh p G q correspond to the convolution product ofsheaves(3.1) X b Y : “ m ˚ p X b Y q (where m ˚ is the direct image functor of m). Notice that the tensorcategory Coh f p G q is identified with the tensor subcategory of sCoh f p G q consisting of sheaves on which odd elements act trivially.We will also consider the following categories. Definition 3.3.
Let Coh f p G q (resp., QCoh p G q ) be the abelian categoryof finite dimensional (resp., all) representations of the algebra O p G q .Note that Coh f p G q is not a tensor category when G is not even,and that we have a tensor equivalence sCoh f p G q – Coh f p G q b sVect.However, we do have the following. Lemma 3.4.
The abelian category
Coh f p G q has a natural structure ofa left module category over sCoh f p G q , given by sCoh f p G q b Coh f p G q Ñ Coh f p G q , X b Y ÞÑ m ˚ p X b Y q . Proof.
The claim follows from the fact that O p G q Ď O p G q is a leftcoideal subalgebra. (cid:3) For every g P G p k q , let sCoh f p G q g : “ sRep p^ g ˚ q b Coh f p G q g . By(2.2), we have an abelian equivalence(3.2) sCoh f p G q – à g P G p k q sCoh f p G q g . ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 7
We will need the following result.
Lemma 3.5.
Every tensor subcategory of sCoh f p G q is either of the form sCoh f p H q for some closed supergroup subscheme H Ď G , or Coh f p H q for some closed subgroup scheme H Ď G .Proof. It is known that every tensor subcategory of Rep p O p G qq corre-sponds to a Hopf quotient of O p G q . Now if u is mapped to 1 in thequotient, then we get the second case (as all odd elements must act byzero). Otherwise, we get the first case. (cid:3) Remark 3.6.
The class of tensor categories sCoh f p G q can be extendedto a larger class of tensor categories sCoh f p G , Ω q in exactly the sameway as in the even case [G2, Section 5]. Namely, let G be an affinesupergroup scheme over k , and let Ω P Z p G , G m q be a normalizedeven 3-cocycle. Equivalently, Ω is a Drinfeld associator for O p G q , i.e.,Ω P O p G q b is an invertible even element satisfying the equations p id b id b ∆ qp Ω qp ∆ b id b id qp Ω q “ p b Ω qp id b ∆ b id qp Ω qp Ω b q and p ε b id b id qp Ω q “ p id b ε b id qp Ω q “ p id b id b ε qp Ω q “ . Then sCoh f p G , Ω q is the abelian category sCoh f p G q equipped with thetensor product given by convolution and associativity constraint givenby the action of Ω (viewed as an invertible element in O p G q b ).4. Equivariant quasi-coherent sheaves
Let G be an affine supergroup scheme over k , let H Ď G be a closedsupergroup subscheme (see Section 2.1), and let ι “ ι H : H ã Ñ G be theinclusion morphism. Let µ : G ˆ H Ñ G be the free action of H on G by right translations (in other words, the free actions of H p R q on G p R q by right translations that are functorial in R , R a supercommutative k -superalgebra). Set η : “ µ p id ˆ m q “ µ p µ ˆ id q : G ˆ H ˆ H Ñ G , and letp : G ˆ H Ñ G , p : G ˆ H ˆ H Ñ G , p : G ˆ H ˆ H Ñ G ˆ H be the obvious projections. We clearly have p ˝ p “ p as morphisms G ˆ H ˆ H Ñ G .Now let Ψ : H ˆ H Ñ G m be a normalized even 2-cocycle, i.e.,Ψ P O p H q b is a twist for O p H q (see Section 2.1), and let O p H q Ψ be the (“twisted”) supercoalgebra with underlying supervector space The purely even case is treated in [G2, Section 3.2].
SHLOMO GELAKI O p H q and comultiplication ∆ Ψ given by ∆ Ψ p f q : “ ∆ p f q Ψ, where ∆is the standard comultiplication of O p H q . Note that Ψ defines anautomorphism of any quasi-coherent sheaf on H ˆ H by multiplication. Definition 4.1.
Let Ψ : H ˆ H Ñ G m be a normalized even 2-cocycleon a closed supergroup subscheme H Ď G .(1) An p H , Ψ q -equivariant quasi-coherent sheaf on G is a pair p S, λ q ,where S P sQCoh p G q and λ : p ˚ p S q – ÝÑ µ ˚ p S q is an isomorphismof sheaves on G ˆ H , such that the diagram of morphisms ofsheaves on G ˆ H ˆ H p ˚ p S q p id ˆ m q ˚ p λ q (cid:15) (cid:15) p ˚ p λ q / / p µ ˝ p q ˚ p S q p µ ˆ id q ˚ p λ q (cid:15) (cid:15) η ˚ p S q id b Ψ / / η ˚ p S q is commutative.(2) Let p S, λ S q and p T, λ T q be two p H , Ψ q -equivariant quasi-coherentsheaves on G . A morphism φ : S Ñ T in sQCoh p G q is said tobe p H , Ψ q -equivariant if the diagram of morphisms of sheaveson G ˆ H p ˚ p S q λ S (cid:15) (cid:15) p ˚ p Ψ q / / p ˚ p T q λ T (cid:15) (cid:15) µ ˚ p S q µ ˚ p φ q / / µ ˚ p T q is commutative.(3) Let sCoh p H , Ψ q f p G q be the k -abelian category of p H , Ψ q -equivariant coherent sheaves on G with finite support in G { H (i.e., sheavessupported on finitely many H -cosets), with p H , Ψ q -equivariantmorphisms.Replacing sQCoh p G q with QCoh p G q everywhere in Definition 4.1,we define the notion of an p H , Ψ q -equivariant O p G q -module and the k -abelian category Coh p H , Ψ q f p G q of finitely generated p H , Ψ q -equivariant O p G q -modules. Example 4.2.
We havesCoh pt u , q f p G q “ sCoh f p G q and Coh pt u , q f p G q “ Coh f p G q . Remark 4.3.
Let p H , Ψ q be another pair consisting of a closed super-group subscheme H Ď G and an even normalized 2-cocycle Ψ on H . ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 9
By considering the free right action of H ˆ H on G given by g p a, b q : “ a ´ gb , we can similarly define pp H , Ψ q , p H , Ψ qq -biequivariant quasi-coherent sheaves on G , pp H , Ψ q , p H , Ψ qq -equivariant O p G q -modules,and the k -abelian categories sCoh pp H , Ψ q , p H , Ψ qq f p G q , Coh pp H , Ψ q , p H , Ψ qq f p G q . Remark 4.4.
Retain the notation from Remark 3.6. Let H Ď G be aclosed supergroup subscheme, and let Ψ P C p H , G m q be a normalizedeven 2-cochain such that d Ψ “ Ω | H . Then similarly to sCoh p H , Ψ q f p G q (the case Ω “ p H , Ψ q f p G , Ω q of p H , Ψ q -equivariant coherent sheaves on p G , Ω q with finite support in G { H , and the category Coh p H , Ψ q f p G , Ω q of finitelygenerated p H , Ψ q -equivariant O p G , Ω q -modules, where O p G , Ω q is theobviously defined quasi-Hopf algebra.Consider now the supercoalgebra O p H q Ψ in sCoh p G q , and let { O p H q Ψ be its profinite completion with respect to the superalgebra structureof O p H q (see [G2, Example 2.4]). Then { O p H q Ψ is a supercoalgebraobject in both Pro p sCoh p G qq and Pro p sCoh f p G qq . Lemma 4.5.
We have abelian equivalences sCoh p H , Ψ q f p G q – Comod
Pro p sCoh f p G qq p { O p H q Ψ q and Coh p H , Ψ q f p G q – Comod
Pro p Coh f p G qq p { O p H q Ψ q . Proof.
We prove the first equivalence, the proof of the second one beingsimilar.For every S P Pro p sCoh f p G qq , we have a natural isomorphismHom G ˆ H p µ ˚ p S q , p ˚ p S qq – Hom G p S, µ ˚ p ˚ p S qq (“adjunction”). Since µ ˚ p ˚ p S q – S b { O p H q , we can assign to anyisomorphism λ : µ ˚ p S q Ñ p ˚ p S q a morphism ρ λ : S Ñ S b { O p H q . It isnow straightforward to verify that ρ λ : S Ñ S b { O p H q Ψ is a comodulemap if and only if p S, λ ´ q is an p H , Ψ q -equivariant coherent sheaf on G with finite support in G { H . (cid:3) The next proposition will be very useful in the sequel.
Proposition 4.6.
Let H Ď G be a closed supergroup subscheme, andlet Ψ be an even normalized -cocycle on H . Then the following hold: (1) The structure sheaf O p H q of H admits a canonical structure ofan p H , Ψ q -equivariant coherent sheaf on H , making it the simpleobject of sCoh p H , Ψ q f p H q – sVect , and the regular O p H q -module admits a canonical structure of an p H , Ψ q -equivariant O p H q -module, making it the simple object of Coh p H , Ψ q f p H q – Vect . (2) The sheaf ι ˚ O p H q is a simple object in sCoh p H , Ψ q f p G q , andthe O p G q -module ι ˚ O p H q P Coh p G q is a simple object in Coh p H , Ψ q f p G q . (3) For every X P sCoh f p G q , we have m ˚ p X b M q P sCoh p H , Ψ q f p G q , M P sCoh p H , Ψ q f p G q , and m ˚ p X b M q P Coh p H , Ψ q f p G q , M P Coh p H , Ψ q f p G q . Proof.
We will prove the proposition for sheaves, the proof for modulesbeing similar.(1) Consider the isomorphism ϕ : “ p m , p q : H ˆ H – ÝÑ H ˆ H .Since p ˝ ϕ “ m, it follows that p p ˝ ϕ q ˚ O p H q “ m ˚ O p H q . Now,multiplication by Ψ defines an isomorphismm ˚ O p H q “ p p ˝ ϕ q ˚ O p H q “ p ϕ ˚ ˝ p ˚ q O p H q Ψ ÝÑ p ϕ ˚ ˝ p ˚ q O p H q , and since we have p ˚ O p H q “ O p H q b O p H q , we get an isomorphism λ : p ˚ O p H q “ O p H q b O p H q ϕ ˚ ÝÑ ϕ ˚ p O p H q b O p H qq Ψ ´ ÝÝÑ m ˚ O p H q . The fact that p O p H q , λ q is an p H , Ψ q -equivariant coherent sheaf on H can be checked now in a straightforward manner using the tensor de-composition (2.1). Clearly, p O p H q , λ q is a simple object in sCoh p H , Ψ q f p H q .Let δ : “ p O p H q , λ q , and consider the simple object δ ´ : “ k | b δ (via id b ∆). It is clear that δ fl δ ´ in sCoh p H , Ψ q f p H q .Now let M be any object in sCoh p H , Ψ q f p H q , and let X : “ M co O p H q .We claim that M – M co O p H q b k δ : “ X b k δ ‘ X b k δ ´ in sCoh p H , Ψ q f p H q , where X denotes the underlying vector space of X .Indeed, let α : M co O p H q b k O p H q Ñ M be the action map, and let β : M Ñ M co O p H q b k O p H q , m ÞÑ ÿ S ´ p m q ¨ m b m . Then it is straightforward to check that α and β are inverse to eachother. Hence, sCoh p H , Ψ q f p H q is semisimple of rank 2, as claimed. The superrepresentation of O p G q on O p H q coming from ι . The representation of O p G q on O p H q coming from ι . ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 11 (2) Since ι is affine, the commutative diagrams H ˆ H ι ˆ id (cid:15) (cid:15) p / / H ι (cid:15) (cid:15) G ˆ H p / / G H ˆ H ι ˆ id (cid:15) (cid:15) m / / H ι (cid:15) (cid:15) G ˆ H µ / / G yield isomorphisms(4.1) p ˚ ι ˚ O p H q – ÝÑ p ι ˆ id q ˚ p ˚ O p H q and(4.2) p ι ˆ id q ˚ m ˚ O p H q – ÝÑ µ ˚ ι ˚ O p H q (“base change”).Let λ : p ˚ O p H q – ÝÑ m ˚ O p H q be the isomorphism constructed in Part(1). Since ι is H -equivariant, we get an isomorphism(4.3) p ι ˆ id q ˚ p ˚ O p H q p ι ˆ id q ˚ p λ q ÝÝÝÝÝÝÑ p ι ˆ id q ˚ m ˚ O p H q . It is now straightforward to check, using the tensor decomposition (2.1),that the composition of isomorphisms (4.1), (4.3) and (4.2)p ˚ ι ˚ O p H q – ÝÑ µ ˚ ι ˚ O p H q endows ι ˚ O p H q with a structure of an p H , Ψ q -equivariant coherentsheaf on G . Clearly, ι ˚ O p H q is simple.(3) Consider the right action id ˆ µ : G ˆ G ˆ H Ñ G ˆ G of H on G ˆ G . If M P sCoh p H , Ψ q f p G q , it is clear that X b M P sCoh f p G ˆ G q is an p H , Ψ q -equivariant coherent sheaf on G ˆ G (here we identify H with thesupergroup subscheme t u ˆ H Ď G ˆ G ). But since m : G ˆ G Ñ G is H -equivariant, m ˚ carries p H , Ψ q -equivariant coherent sheaves on G ˆ G to p H , Ψ q -equivariant coherent sheaves on G . (cid:3) Exact module categories over sCoh f p G q In this section we extend [G2, Section 3.3] to the super case.Let G , H , ι and Ψ be as in Section 4. Set M p H , Ψ q : “ sCoh p H , Ψ q f p G q , M ˝ p H , Ψ q : “ Coh p H , Ψ q f p G q , and let V p H , Ψ q : “ Comod
Pro p sCoh f p G qq p { O p H q Ψ q , V ˝ p H , Ψ q : “ Comod
Pro p Coh f p G qq p { O p H q Ψ q be the abelian categories of right comodules over { O p H q Ψ in Pro p sCoh f p G qq and Pro p Coh f p G qq , respectively. Proposition 5.1.
Fix a pair p H , Ψ q as above, and let δ : “ ι ˚ O p H q P sCoh p H , Ψ q f p G q , δ ˝ : “ ι ˚ O p H q P Coh p H , Ψ q f p G q . The following hold: (1)
Set M : “ M p H , Ψ q and M ˝ : “ M ˝ p H , Ψ q . The bifunctors b M : sCoh f p G q b M Ñ M , X b M ÞÑ m ˚ p X b M q and b M ˝ : sCoh f p G q b M ˝ Ñ M ˝ , X b M ÞÑ m ˚ p X b M q define on M and M ˝ structures of indecomposable sCoh f p G q -module categories. (2) Set V : “ V p H , Ψ q and V ˝ : “ V ˝ p H , Ψ q . The bifunctors b V : sCoh f p G q b V Ñ V , X b V ÞÑ m ˚ p X b V q and b V ˝ : sCoh f p G q b V ˝ Ñ V ˝ , X b V ÞÑ m ˚ p X b V q define on V and V ˝ structures of sCoh f p G q -module categories. (3) We have equivalences M – V and M ˝ – V ˝ of module cat-egories over sCoh f p G q . In particular, Hom p δ, δ q – { O p H q Ψ assupercoalgebras in Pro p sCoh f p G qq , and Hom p δ ˝ , δ ˝ q – { O p H q Ψ as coalgebras in Pro p Coh f p G qq .Proof. We prove it for M and V , the proof for M ˝ and V ˝ being similar.(1) Since m p m ˆ id q “ m p id ˆ m q and Ψ is an even 2-cocycle, it fol-lows from Lemma 4.6 that b M defines on M a structure of a sCoh f p G q -module category. Clearly, sCoh f p H q Ď sCoh f p G q consists of those ob-jects X for which X b M δ is a sum of multiples of δ and k | b δ , andany object M P M is of the form X b M δ for some X P sCoh f p G q . Inparticular, the simple object δ (see Proposition 4.6) generates M , so M is indecomposable.(2) By definition, an object in V is a pair p V, ρ V q consisting of anobject V P Pro p sCoh f p G qq and a morphism ρ V : V Ñ V b { O p H q Ψ in Pro p sCoh f p G qq satisfying the comodule axioms. It is clear that forevery X P sCoh f p G q , we have m ˚ p X b V q P Pro p sCoh f p G qq and that ρ m ˚ p X b V q : “ id X b ρ V is a morphism in Pro p sCoh f p G qq defining onm ˚ p X b V q a structure of a right comodule over { O p H q Ψ .(3) Follows from Lemma 4.5. (cid:3) Example 5.2.
Let G be an affine supergroup scheme over k .(1) M pt u , q “ sCoh f p G q is the regular module.(2) M p G , q “ sVect is the usual superfiber functor on sCoh f p G q . ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 13 (3) M ˝ p G , q “ Vect is the usual fiber functor on sCoh f p G q . Proposition 5.3.
The indecomposable module categories M p H , Ψ q and M ˝ p H , Ψ q over sCoh f p G q are exact.Proof. We prove it for M p H , Ψ q , the proof for M ˝ p H , Ψ q being similar.Set M : “ M p H , Ψ q . It suffices to show that for every projective P P Pro p sCoh f p G qq and X P M , P b M X is projective (see Section2.2). Clearly, it suffices to show it for X : “ δ “ δ p H , Ψ q . Moreover, sinceany projective in Pro p sCoh f p G qq is a completed direct sum of P g, ˘ (seeSection 3.1), it suffices to check that P g b M δ is projective. Furthermore,since P g “ δ g b P , and δ g b M ? is an autoequivalence of M as anabelian category (since δ g is invertible), it suffices to do so for g “ P b M δ “ { O p H q b k P p δ q , where P p δ q isthe projective cover of δ (i.e., the unique indecomposable projective inthe block of Pro p M q containing δ ; as a sheaf on G , it is the functionalgebra on the formal neighborhood of H ), and hence projective asdesired. (cid:3) We are now ready to state and prove the main result of this section.
Theorem 5.4.
Let G be an affine supergroup scheme over k . Thereis a correspondence between conjugacy classes of pairs p H , Ψ q and equivalence classes of indecomposable exact module categories over sCoh f p G q , assigning p H , Ψ q to M p H , Ψ q and M ˝ p H , Ψ q .Proof. By Proposition 5.3, it remains to show that any indecomposableexact module category M over sCoh f p G q has the form M p H , Ψ q or M ˝ p H , Ψ q . To this end, let δ be a simple object generating M , let δ ´ : “ k | b M δ , and consider the full subcategories C : “ t X P sCoh f p G q | X b M δ “ dim k p X q δ ‘ dim k p X q δ ´ u and C ˝ : “ t X P sCoh f p G q | X b M δ “ dim k p X q δ u of sCoh f p G q . It is clear that C “ C ˝ if and only if δ – δ ´ in M , and that C ˝ Ď sCoh f p G q is a tensor subcategory. We claim that C Ď sCoh f p G q is also a tensor subcategory. Indeed, we have p X b Y q b M δ – X b M p Y b M δ q– X b M p dim k p Y q δ ‘ dim k p Y q δ ´ q– dim k p Y q X b M δ ‘ dim k p Y q Π p X q b M δ – dim k p Y qp dim k p X q δ ‘ dim k p X q δ ´ q ‘ dim k p Y qp dim k p X q δ ‘ dim k p X q δ ´ q“ dim k p Y b X ‘ Y b X q δ ‘ dim k p Y b X ‘ Y b X q δ ´ , for every X, Y P C , as required.Assume C ‰ C ˝ . By Lemma 3.5, we can assume that C “ sCoh f p H q for some closed supergroup subscheme H Ď G , or C “ Coh f p H q forsome closed subgroup scheme H Ď G . Moreover, in both cases thefunctor F : C Ñ Vect , F p X q “ Hom M p δ ‘ δ ´ , X b M δ q , together with the tensor structure F p¨q b F p¨q – ÝÑ F p¨ b ¨q coming fromthe associativity constraint, is a fiber functor on C . But, letting X denote the underlying vector space of X (where we view X as an O p H q -supermodule or O p H q -module), we see that F p X q “ X . We thereforeget a functorial isomorphism X b Y – ÝÑ X b Y , which is nothing butan invertible even element Ψ of O p H q b or O p H q b , taking values in G m p k q . Clearly, Ψ is a twist for O p H q or O p H q .To summarize, assuming that δ fl δ ´ in M , we have obtained that if C “ sCoh f p H q then the C -submodule category x δ, δ ´ y Ď M consistingof all direct sums of multiples of δ and δ ´ is equivalent to sCoh p H , Ψ q f p H q ,and if C “ Coh f p H q then the C -submodule category x δ y Ď M consist-ing of all multiples of δ is equivalent to Coh p H , Ψ q f p H q .Now assume C fl C ˝ (the proof being similar when C – C ˝ ), andsuppose C “ sCoh f p H q . Let X P sCoh f p G q and X H P sCoh f p H q bethe maximal subsheaf of X which is scheme-theoretically supportedon H (i.e., X H consists of all vectors in X which are annihilated bythe defining ideal of H in O p G q ). Now, on the one hand, since forany g P G p k q , δ g b M δ and δ g b M δ ´ are simple, and one of them isisomorphic to δ and the other one to δ ´ if and only if g P H p k q , it isclear that Hom Pro p sCoh f p G qq p Hom p δ ‘ δ ´ , δ ‘ δ ´ q , X q“ Hom M p δ ‘ δ ´ , X b M p δ ‘ δ ´ qq “ X H (since it holds for any simple X ). On the other hand, it is clear thatHom Pro p sCoh f p G qq p { O p H q Ψ , X q “ X H . ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 15
Thus by Yoneda’s lemma, the two supercoalgebras Hom p δ ‘ δ ´ , δ ‘ δ ´ q and { O p H q Ψ in Pro p sCoh f p G qq are isomorphic. This implies that M is equivalent to Comod Pro p sCoh f p G qq p { O p H q Ψ q as a module category oversCoh f p G q (as M is indecomposable, exact, and generated by δ ‘ δ ´ ),hence to M p H , Ψ q by Proposition 5.1, as claimed.Finally, one shows similarly that if C “ Coh f p H q , M – M p H , Ψ q ,as desired. (cid:3) Example 5.5.
Observe that Theorem 5.4 reduces to [G2, Theorem3.9] in the even case since when G is an affine group scheme over k , wehave sCoh f p G q “ Coh f p G q b sVect. Example 5.6.
Let V be a n -dimensional odd k -vector space, n ě f p V q are in 2 : 1 correspondence with equiva-lence classes of pairs p W, B q , where W Ď V is a super subspace and B P S W ˚ . For example, if n “ f p V q “ sVect: Vectand sVect. Also, if n “ p W, B q : p , q , p V, q and p V, B q , where B p v, v q “ v a fixed basis for V ). Thus, there are six non-equivalent indecom-posable exact module categories over sCoh f p V q (in agreement with[EO, Theorem 4.5]). More precisely, we have M p , q – sCoh f p V q and M ˝ p , q – Coh f p V q , M p V, q , M p V, B q , which are semisimple ofrank 2, and M ˝ p V, q , M ˝ p V, B q , which are semisimple of rank 1. Remark 5.7.
Retain the notation from Remark 4.4. Similarly, thecategories sCoh p H , Ψ q f p G , Ω q and Coh p H , Ψ q f p G , Ω q admit a structure of anindecomposable exact module category over sCoh f p G , Ω q given by con-volution of sheaves, and furthermore, there is a 1 : 2 correspondencebetween conjugacy classes of pairs p H , Ψ q and equivalence classes ofindecomposable exact module categories over sCoh f p G , Ω q , assigning p H , Ψ q to sCoh p H , Ψ q f p G , Ω q and Coh p H , Ψ q f p G , Ω q .6. Exact module categories over sRep p G q In this section we extend [G2, Section 4] to the super case.Let C be a tensor category. Given two exact module categories M , N over C , let Fun C p M , N q denote the abelian category of C -functors from M to N . The dual category of C with respect to M is the category C ˚ M : “ End C p M q of C -endofunctors of M . If M is indecomposable, C ˚ M is a tensor category, and M is an indecomposable exact modulecategory over C ˚ M . Also, Fun C p M , N q is an exact module category over C ˚ M via the composition of functors. Module categories.
Retain the notation from Sections 4 and 5.Set M pp G , q , p H , Ψ qq : “ sCoh pp G , q , p H , Ψ qq f p G q and M ˝ pp G , q , p H , Ψ qq : “ Coh pp G , q , p H , Ψ qq f p G q . Recall that the 2-cocycle Ψ determines a central extension H Ψ of H by G m . By an p H , Ψ q -superrepresentation of H we will mean arational representation of the affine supergroup scheme H Ψ on a k -supervector space on which G m acts with weight 1 (i.e., via the identitycharacter). Let us denote the category of finite dimensional p H , Ψ q -superrepresentations of H Ψ by N p H , Ψ q . Clearly, we have an equiva-lence of abelian categories N p H , Ψ q – sComod p O p H q Ψ q . Similarly, let N ˝ p H , Ψ q be the category of finite dimensional p H , Ψ q - representations of H Ψ . We have an equivalence of abelian categories N ˝ p H , Ψ q – Comod p O p H q Ψ q . Lemma 6.1.
The following hold: (1)
We have abelian equivalences
Fun sCoh f p G q p M ˝ p G , q , M p H , Ψ qq – M ˝ pp G , q , p H , Ψ qq and Fun sCoh f p G q p M ˝ p G , q , M ˝ p H , Ψ qq – M pp G , q , p H , Ψ qq . In particular, we have a tensor equivalence sCoh f p G q ˚ M ˝ p G , q – sRep p G q . (2) We have sRep p G q -module equivalences Fun sCoh f p G q p M ˝ p G , q , M p H , Ψ qq – N ˝ p H , Ψ q and Fun sCoh f p G q p M ˝ p G , q , M ˝ p H , Ψ qq – N p H , Ψ q . Proof.
We prove the theorem for functors to M p H , Ψ q , the proof forfunctors to M ˝ p H , Ψ q being similar.(1) Since M ˝ p G , q “ Vect, a functor M ˝ p G , q Ñ M p H , Ψ q is justan p H , Ψ q -equivariant sheaf X on G . The fact that the functor is asCoh f p G q -module functor means that we have functorial isomorphisms µ S : S b X – ÝÑ S b X in M p H , Ψ q , S P sCoh f p G q . Thus, µ gives X a commuting G -equivariant structure for the left action of G on itself,i.e., X is a pp G , q , p H , Ψ qq -biequivariant sheaf on G . In particular, ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 17 for S “ k | , we have an isomorphism µ k | : X – ÝÑ X , hence X corresponds to pp G , q , p H , Ψ qq -biequivariant O p G q -module, as desired.Conversely, it is clear that any pp G , q , p H , Ψ qq -biequivariant O p G q -module X defines a sCoh f p G q -module functor M ˝ p G , q Ñ M p H , Ψ q determined by k ÞÑ X with X “ X .Finally, the category of p G , G q -biequivariant sheaves on G is equiva-lent to the category sRep p G q as a tensor category, and the second claimfollows.(2) If X is a pp G , q , p H , Ψ qq -biequivariant O p G q -module, then theinverse image sheaf e ˚ p X q on Spec p k q (“the stalk at 1”) acquires astructure of an p H , Ψ q -representation via the action of the element p h, h ´ q in G ˆ H , i.e., it is an object in N ˝ p H , Ψ q . We have thusdefined a functor M ˝ pp G , q , p H , Ψ qq Ñ N ˝ p H , Ψ q , X ÞÑ e ˚ p X q . Conversely, an p H , Ψ q -representation V can be spread out over G andmade into a p G , p H , Ψ qq -biequivariant O p G q -module. In other words, wehave the functor N ˝ p H , Ψ q Ñ M ˝ pp G , q , p H , Ψ qq , V ÞÑ O p G q b k V. Finally, it is straightforward to verify that the two functors con-structed above are inverse to each other. (cid:3)
Similarly, we have the following result.
Lemma 6.2.
The following hold: (1)
We have abelian equivalences
Fun sCoh f p G q p M p G , q , M p H , Ψ qq – M pp G , q , p H , Ψ qq and Fun sCoh f p G q p M p G , q , M ˝ p H , Ψ qq – M ˝ pp G , q , p H , Ψ qq . In particular, we have a tensor equivalence sCoh f p G q ˚ M p G , q – sRep p G q . (2) We have sRep p G q -module equivalences Fun sCoh f p G q p M p G , q , M p H , Ψ qq – N p H , Ψ q and Fun sCoh f p G q p M p G , q , M ˝ p H , Ψ qq – N ˝ p H , Ψ q . (cid:3) Example 6.3.
We have the following:(1) N pt u , q “ sVect is the usual superfiber functor on sRep p G q . (2) N ˝ pt u , q “ Vect is the usual fiber functor on sRep p G q . Lemma 6.4.
The following hold: (1)
We have a tensor equivalence sRep p G q ˚ N ˝ pt u , q – sCoh f p G q . (2) We have sCoh f p G q -module equivalences Fun sRep p G q p N ˝ pt u , q , N p H , Ψ qq – M ˝ p H , Ψ q and Fun sRep p G q p N ˝ pt u , q , N ˝ p H , Ψ qq – M p H , Ψ q . Proof.
The proof is similar to the proof of Lemma 6.1. (cid:3)
Lemma 6.4 prompts the following definition.
Definition 6.5.
An indecomposable exact module category N oversRep p G q is called geometrical if Fun sRep p G q p N ˝ pt u , q , N q ‰ p G q form afull 2-subcategory Mod geom p sRep p G qq of the 2-category Mod p sRep p G qq .We can now deduce from Lemmas 6.1, 6.4 the main result of thissection, which says that geometrical module categories over sRep p G q are precisely those exact module categories which come from exactmodule categories over sCoh f p G q . More precisely, we have the followinggeneralization of [G2, Theorem 4.5]. Theorem 6.6.
Let G be an affine supergroup scheme over k . Then the -functors Mod p sCoh f p G qq Ñ Mod geom p sRep p G qq , M ÞÑ Fun sCoh f p G q p M ˝ p G , q , M q , and Mod geom p sRep p G qq Ñ Mod p sCoh f p G qq , N ÞÑ Fun sRep p G q p N ˝ pt u , q , N q , are inverse to each other. In particular, there is a correspondencebetween conjugacy classes of pairs p H , Ψ q and equivalence classes ofindecomposable geometrical module categories over sRep p G q , assigning p H , Ψ q to N p H , Ψ q and N ˝ p H , Ψ q . (cid:3) Remark 6.7. If G is not finite, sRep p G q may very well have non-geometrical module categories (see [G2, Remark 4.6]). Remark 6.8.
Retain the notation from Remark 5.7. Similarly to theeven case [G2], we can define supergroup scheme-theoretical categories C p G , H , Ω , Ψ q and C ˝ p G , H , Ω , Ψ q as the dual categories of sCoh f p G , Ω q with respect to sCoh p H , Ψ q f p G , Ω q and Coh p H , Ψ q f p G , Ω q ), respectively. We ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 19 then have that C p G , H , Ω , Ψ q is equivalent to the tensor category of pp H , Ψ q , p H , Ψ qq -biequivariant coherent sheaves on p G , Ω q , supportedon finitely many left H -cosets (equivalently, right H -cosets), withtensor product given by convolution of sheaves. For example, the center Z p sCoh f p G qq of sCoh f p G q is supergroup scheme-theoretical since Z p sCoh f p G qq – p sCoh f p G q b sCoh f p G q op q ˚ sCoh f p G q , so Z p sCoh f p G qq – C p G ˆ G , G , , q as tensor categories, where G is viewed as a closed supergroup sub-scheme of G ˆ G via the diagonal morphism ∆ : G Ñ G ˆ G .Moreover, we can define indecomposable geometrical module cate-gories over C : “ C p G , H , Ω , Ψ q , and obtain that the 2-functorsMod p sCoh f p G , Ω qq Ñ Mod geom p C q , M ÞÑ Fun sCoh f p G , Ω q p M p H , Ψ q , M q , and Mod geom p C q Ñ Mod p sCoh f p G , Ω qq , N ÞÑ Fun C p M p H , Ψ q , N q , are 2-equivalences which are inverse to each other. In particular, theequivalence classes of geometrical module categories over C are in 2 : 1correspondence with the conjugacy classes of pairs p H , Ψ q such that H Ď G is a closed supergroup subscheme and Ψ P C p H , G m q satisfies d Ψ “ Ω | H . (The analogs for C ˝ p G , H , Ω , Ψ q are obvious.)6.2. Semisimple module categories of rank . Recall that the setof equivalence classes of semisimple module categories over sRep p G q ofrank 1 is in bijection with the set of equivalence classes of tensor struc-tures on the forgetful functor sRep p G q Ñ Vect. Therefore, Theorem6.6 implies that the conjugacy class of any pair p H , Ψ q for which thecategory sComod p O p H q Ψ q or Comod p O p H q Ψ q is semisimple of rank 1gives rise to an equivalence class of a tensor structure on the forgetfulfunctor sRep p G q Ñ Vect. Clearly, for such pair p H , Ψ q , H must be a finite supergroup subscheme of G (as a simple coalgebra must be finitedimensional). This observation suggests the following definition. Definition 6.9.
Let H be a finite supergroup scheme over k . Wecall an even 2-cocycle Ψ : H ˆ H Ñ G m (equivalently, a twist Ψ for O p H q “ p k H q ˚ ) non-degenerate if the category sComod p O p H q Ψ q orComod p O p H q Ψ q is equivalent to Vect.We thus have the following corollary. Corollary 6.10.
The conjugacy class of a pair p H , Ψ q , where H Ď G is a finite closed supergroup subscheme and Ψ : H ˆ H Ñ G m is a non-degenerate even -cocycle, gives rise to an equivalence class of aneven Hopf -cocycle for O p G q . (cid:3) Remark 6.11.
Finite supergroup schemes having a non-degenerateeven 2-cocycle may be called supergroup schemes of central type inanalogy with the even case [G2, Remark 4.9].6.3.
Exact module categories over finite supergroup schemes.
Thanks to [EO, Theorem 3.31], Theorem 6.6 can be strengthened inthe finite case to give a canonical bijection between exact module cate-gories over sCoh f p G q “ sCoh p G q and sRep p G q (i.e., for finite supergroupschemes, every exact module category over sRep p G q is geometrical).Namely, we have the following result. Theorem 6.12.
Let G be a finite supergroup scheme over k . The -functors Mod p sCoh p G qq Ñ Mod p sRep p G qq , M ÞÑ Fun sCoh p G q p M ˝ p G , q , M q , and Mod p sRep p G qq Ñ Mod p sCoh p G qq , N ÞÑ Fun sRep p G q p N ˝ pt u , q , N q , are inverse to each other. In particular, the equivalence classes of inde-composable exact module categories over sRep p G q “ sRep p k G q are parameterized by the conjugacy classes of pairs p H , Ψ q , where H Ď G isa closed supergroup subscheme and Ψ : H ˆ H Ñ G m is a normalizedeven -cocycle. (cid:3) Example 6.13.
Let V be a one-dimensional odd vector space, andconsider the purely odd finite supergroup scheme G : “ V . By Example5.6 and Theorem 6.12, the tensor category sRep p V q has exactly six non-equivalent indecomposable exact left module categories correspondingto the pairs p , q , p V, q and p V, B q , where B p v, v q “
1. Namely, thecategories N p , q “ sVect, N ˝ p , q “ Vect, N p V, q “ sMod p^ V q , N ˝ p V, q “ Mod p^ V q , N p V, B q “ sMod p k Z q “ Vect (here k p Z { Z q is viewed as a superalgebra, where the generator of Z { Z is odd), and N ˝ p V, B q “
Mod p Z { Z q “ sVect.7. The classification of triangular Hopf algebras withthe Chevalley property
In Sections 7.1, 7.2 we assume that G is a finite supergroup schemeover k . (The even case is treated in [G2, Sections 6.1-6.3].) ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 21
Twists for k G . By [AEGN, Theorem 5.7], there is a bijectionbetween non-degenerate twists for k G and non-degenerate twists for O p G q . Hence, as a consequence of Theorem 6.12, we deduce the fol-lowing strengthening of Corollary 6.10. Corollary 7.1.
Let G be a finite supergroup scheme over k . The fol-lowing four sets are in canonical bijection with each other: (1) Equivalence classes of tensor structures on the forgetful functor sRep p G q Ñ Vect . (2) Gauge equivalence classes of twists for k G . (3) Conjugacy classes of pairs p H , Ψ q , where H Ď G is a closedsupergroup subscheme and Ψ : H ˆ H Ñ G m is a non-degenerateeven -cocycle. (4) Conjugacy classes of pairs p H , J q , where H Ď G is a closedsupergroup subscheme and J is a non-degenerate twist for k H . (cid:3) Remark 7.2.
If moreover, r g , g s “ k G “ k G ˙^ g ), then each one of the above four sets is in bijection withthe set of conjugacy classes of quadruples p H , ψ, h , B q , where H Ď G is a closed subgroup scheme, ψ : H ˆ H Ñ k is a non-degenerate2-cocycle, Y Ď g is an H -invariant subspace, and B P S h ˚ is non-degenerate (see Section 2.1). Remark 7.3.
Corollary 7.1 was proved for ´etale group schemes in[Mo, EG1, AEGN], for finite supergroups G such that k G “ k G ˙ ^ g in [EO], and for finite group schemes in [G2]. Example 7.4.
Retain the notation from Example 6.13. Then the ten-sor category sRep p V q has exactly two non-equivalent fiber functors toVect corresponding to the left module categories N ˝ p , q and N p V, B q .7.2. Minimal twists for k G . Recall that a twist J for k G is called minimal if the triangular Hopf superalgebra pp k G q J , J ´ J q is minimal,i.e., if the left (right) tensorands of J ´ J span k G [R].By [G2, Proposition 6.7], a twist for a finite group scheme is minimalif and only if it is non-degenerate. In this section we extend this resultto the super case, using the following result (see [EG2, Lemma A.8]and [B, Proposition 1]). Proposition 7.5.
Let D and E be symmetric tensor categories over k , and suppose there exists a surjective symmetric tensor functor F : D Ñ E . If D is finitely tensor-generated and (super-)Tannakian,then so is E . (cid:3) I.e., any object X P E is isomorphic to a subquotient of F p V q for some V P D . We can now state and prove the first main result of this section.
Proposition 7.6.
Let G be a finite supergroup scheme over k , andlet J be a twist for k G . Then J is minimal if and only if it is non-degenerate.Proof. Suppose J is minimal. By Corollary 7.1, there exist a closedsupergroup subscheme H Ď G and a non-degenerate twist J for k H ,such that the image of J under the embedding p k H q J ã Ñ p k G q J is J .Since J is minimal and H Ď G , it follows that H “ G .Conversely, suppose J is non-degenerate. Let p A , J ´ J q be theminimal triangular Hopf sub-superalgebra of pp k G q J , J ´ J q . The re-striction functor sRep p G q ։ sRep p A q is a surjective symmetric tensorfunctor. Thus by Proposition 7.5, sRep p A q is equivalent to sRep p H , u q ,as a symmetric tensor category, for some closed supergroup subscheme H Ď G . Now, it is a standard fact (see, e.g., [G1]) that such an equiv-alence functor gives rise to a twist I P p k H q b and an isomorphism oftriangular Hopf superalgebras pp k H q I , I ´ I q – ÝÑ p A , J ´ J q .We therefore get an injective homomorphism of triangular Hopf su-peralgebras pp k H q I , I ´ I q ã Ñ pp k G q J , J ´ J q , which implies that J I ´ is a symmetric twist for k G . But by [DM, Theorem 3.2], this impliesthat J I ´ is gauge equivalent to 1 b
1. Therefore, the triangular Hopfsuperalgebras pp k G q J I ´ , I J ´ J I ´ q and p k G , b q are isomorphic.In other words, pp k G q I , I ´ I q and pp k G q J , J ´ J q are isomorphic astriangular Hopf superalgebras, i.e., the pairs p G , J q and p H , I q areconjugate. We thus conclude from Corollary 7.1 that H “ G , andhence that J is a minimal twist, as required. (cid:3) Remark 7.7.
Corollary 7.1 and Proposition 7.6 extend [G2, Corollary6.3 & Proposition 6.7] to the super case.7.3.
Triangular Hopf algebras.
Let p H, R q be a finite dimensionaltriangular Hopf algebra with the Chevalley property over k . Recallthat by [EG2, Corollary 4.1], p H, R q is twist equivalent to a finite di-mensional triangular Hopf algebra with R -matrix of rank ď p H, R q corresponds to a unique pair p G , ǫ q , where G isa finite supergroup scheme over k (see Section 2.1). Thus Corollary 7.1implies the following classification result, which extends [EG2, Theo-rem 5.1] to arbitrary finite dimensional triangular Hopf algebras withthe Chevalley property over k . Theorem 7.8.
The following three sets are in canonical bijection witheach other:
ODULE CATEGORIES OVER AFFINE SUPERGROUP SCHEMES 23 (1)
Isomorphism classes of finite dimensional triangular Hopf alge-bras p H, R q with the Chevalley property over k . (2) Conjugacy classes of quadruples p G , H , J , ǫ q , where G is a finitesupergroup scheme over k , H Ď G is a closed supergroup sub-scheme, J is a minimal twist for k H , and ǫ P G p k q is a centralelement of order ď acting by ´ on g . (3) Conjugacy classes of quadruples p G , H , Ψ , ǫ q , where G is a fi-nite supergroup scheme over k , H Ď G is a closed supergroupsubscheme, Ψ is a non-degenerate even -cocycle on H with co-efficients in G m , and ǫ P G p k q is a central element of order ď acting by ´ on g . Remark 7.9.
The correspondence between (1) and (2) in Theorem7.8 is given by p H, R q “ pp k G q J , ǫ q (see [AEG, Theorem 3.3.1]; see alsoSection 3 above). A 2-cocycle Ψ on H as in Theorem 7.8(3) determinesa module category over sRep p G q of rank 1, i.e., a tensor structure onthe forgetful functor sRep p G q Ñ Vect, thus a twist J for k G supportedon H . References [AEG] N. Andruskiewitch, P. Etingof and S. Gelaki. Triangular Hopf algebraswith the Chevalley property.
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