Restriction Functors Research Articles

ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS

AbstractIn this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

Lefschetz theorems for tamely ramified coverings

As is well known, the Lefschetz theorems for the \'etale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let $D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $Y$ is an ample regular hyperplane intersecting $D$ transversally, the restriction functor from tame \'etale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre (The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Springer LNM 208). The authors showed that one can lift tame coverings from $Y\setminus D\cap Y$ to the complement of $D\cap Y$ in the formal completion of $X$ along $Y$. One has then to further lift to $X\setminus D$.

Twisted Frobenius Extensions of Graded Superrings

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that $A$ is a twisted Frobenius extension of $B$ if and only if induction of $B$-modules to $A$-modules is twisted shifted right adjoint to restriction of $A$-modules to $B$-modules. A large (non-exhaustive) class of examples is given by the fact that any time $A$ is a Frobenius graded superalgebra, $B$ is a graded subalgebra that is also a Frobenius graded superalgebra, and $A$ is projective as a left $B$-module, then $A$ is a twisted Frobenius extension of $B$.

Towers of graded superalgebras categorify the twisted Heisenberg double

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double, introduced in [9], and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilCoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.

Weil restriction of noncommutative motives

The Weil restriction functor, introduced in the late fifties, was recently extended by Karpenko to the category of Chow motives with integer coefficients. In this article we introduce the noncommutative (=NC) analogue of the Weil restriction functor, where schemes are replaced by dg algebras, and extend it to Kontsevich's categories of NC Chow motives and NC numerical motives. Instead of integer coefficients, we work more generally with coefficients in a binomial ring. Along the way, we extend Karpenko's functor to the classical category of numerical motives, and compare this extension with its NC analogue. As an application, we compute the (NC) Chow motive of the Weil restriction of every smooth projective scheme whose category of perfect complexes admits a full exceptional collection. Finally, in the case of central simple algebras, we describe explicitly the NC analogue of the Weil restriction functor using solely the degree of the field extension. This leads to a “categorification” of the classical corestriction homomorphism between Brauer groups.

On the modular representation theory of the partition algebra

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine exactly for which values of the parameter the partition algebras are semisimple over an arbitrary field. Furthermore, we show that the blocks of the partition algebra over an arbitrary field categorify weight spaces of an action of the quantum groups $$\mathcal {U}_q(\widehat{\mathfrak {sl}_p})$$Uq(slp^) and $$\mathcal {U}_q(\mathfrak {sl}_\infty )$$Uq(sl?) on an analogue of the Fock space. In particular, we recover the block structure which was recently determined by Bowman et al. In order to do so, we use induction and restriction functors as well as analogues of Jucys---Murphy elements. The description of decomposition numbers will be in terms of combinatorics of partitions but can also be given a Lie theoretic interpretation in terms of a Weyl group of type $$A$$A: a simple module $$L(\mu )$$L(μ) is a composition factor of a cell module $$\Delta (\lambda )$$Δ(?) if and only if $$\lambda $$? and $$\mu $$μ differ by the action of a transposition.

Decomposition numbers of Brauer algebras in non-dividing characteristic

We determine decomposition matrices of Brauer algebras over a field of characteristic 0 and for fields of prime characteristic p with p larger than the degree of the Brauer algebra. We reprove and extend work of Martin as well as by Cox and De Visscher over the complex field by using new methods which uniformly treat the characteristic 0 and prime characteristic cases. These methods are inspired by the theory of Jucys–Murphy elements and the use of refined induction and restriction functors which are generalizations of similar functors for symmetric groups.

Hamiltonian reduction and nearby cycles for mirabolic [formula omitted]-modules

We study holonomic D-modules on SLn(C)×Cn, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus.We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

INDUCTION AND RESTRICTION FUNCTORS FOR CYCLOTOMIC q-SCHUR ALGEBRAS

We define the induction and restriction functors for cyclotomic $q$-Schur algebras, and study some properties of them. As an application, we categorify a higher level Fock space by using the module categories of cyclotomic $q$-Schur algebras.

Hecke algebras, finite general linear groups, and Heisenberg categorification

We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q -deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting. Erratum: Due to a technical error, many lines in the diagrams were printed without indication of the corresponding direction (i.e. as simple lines instead of arrows). A corrected electronic version of the article can be downloaded .

A Note on the First Rigid Cohomology Group for Geometrically Unibranch Varieties

We prove a vanishing of the first rigid cohomology group for geometrically unibranch varieties with supports in a proper closed subset and apply it to the full faithfulness problem of the restriction functors of overconvergent isocrystals. As an application, we prove that the first rigid cohomology group is pure of weight 1 for proper and geometrically unibranch varieties over a finite field. We also establish a comparison result of rigid cohomology groups between a geometrically unibranch variety and its normalization.

Categories of unitary representations of Banach–Lie supergroups and restriction functors

We prove two results which show that the categories of smooth and analytic unitary representations of a Banach‐Lie supergroup are well-behaved. The first result states that the restriction functor corresponding to any homomorphism of Banach‐Lie supergroups is well-defined. The second result shows that the category of analytic representations is isomorphic to a subcategory of the category of smooth representations. These facts are needed as a crucial first step to a rigorous treatment of the analytic theory of unitary representations of Banach‐Lie supergroups. They extend the known results for finite-dimensional Lie supergroups. In the infinite-dimensional case the proofs require several new ideas. As an application, we give an analytic realization of the oscillator representation of the restricted orthosymplectic Banach‐Lie supergroup.

Normal coideal subalgebras of semisimple Hopf algebras

The restriction functor from the category of representations of a semisimple Hopf algebra to the category of representations of a normal coideal subalgebra is studied. It is shown that this functor has a similar behavior to the restriction functor to the category of representations of a normal Hopf subalgebra. Commutator subalgebras as normal left (right) coideal subalgebras are also studied.

On the wrapped Fukaya category and based loops

Given an exact relatively Pin Lagrangian embedding $Q \subset M$, we construct an $A^∞$ restriction functor from the wrapped Fukaya category of $M$ to the category of modules on the differential graded algebra of chains over the based loop space of $Q$. If $M$ is the cotangent bundle of $Q$, this functor induces an $A^∞$ equivalence between the wrapped Floer cohomology of a cotangent fibre and the chains over the based loop space of $Q$, extending a result proved by Abbondandolo and Schwarz at the level of homology.

Purity for overconvergence

Let \({X \hookrightarrow \overline{X}}\) be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and \({\overline{Z}\subseteq Z \subseteq \overline{X}}\) closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-isocrystals \({F-{\rm Isoc}^\dagger(X,\overline{X}) \longrightarrow F-{\rm Isoc}^\dagger(X{\setminus}Z, \overline{X}{\setminus}\overline{Z})}\) is an equivalence of categories. We also give an application of our result to the equivalence of certain categories.

On Normal Hopf Subalgebras of Semisimple Hopf Algebras

A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka’s criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor.

DERIVED H-MODULE ENDOMORPHISM RINGS

AbstractLet H be a Hopf algebra, A/B be an H-Galois extension. Let D(A) and D(B) be the derived categories of right A-modules and of right B-modules, respectively. An object M⋅ ∈ D(A) may be regarded as an object in D(B) via the restriction functor. We discuss the relations of the derived endomorphism rings EA(M⋅) = ⊕i∈ℤHomD(A)(M⋅, M⋅[i]) and EB(M⋅) = ⊕i∈ℤHomD(B)(M⋅, M⋅[i]). If H is a finite-dimensional semi-simple Hopf algebra, then EA(M⋅) is a graded sub-algebra of EB(M⋅). In particular, if M is a usual A-module, a necessary and sufficient condition for EB(M) to be an H*-Galois graded extension of EA(M) is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.

Generalized lax epimorphisms in the additive case

In this paper we call generalized lax epimorphism a functor defined on a ring with several objects, with values in an abelian AB5 category, for which the associated restriction functor is fully faithful. We characterize such a functor with the help of a conditioned right cancellation of another functor, constructed in a canonical way from the initial one. As consequences we deduce a characterization of functors inducing an abelian localization and also a necessary and sufficient condition for a morphism of rings with several objects to induce an equivalence at the level of two localizations of the respective module categories.

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.