Relative Functor Research Articles

Crossed modules with action

UDC 512.5 Following the idea of a group with action, in a higher dimension, we consider crossed modules with a given crossed-module action upon itself. A category of these objects X M o d * is given by replacing the set of morphisms in the category G r * of groups with action by the set of a modified form of φ -crossed homomorphisms. Moreover, the connections of X M o d * with some other known categories, such as G r * and X M o d , are given by describing some related functors.

On a 2-Relative Entropy.

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a ‘2-relative entropy’ satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.

The function ring functors of pointfree topology revisited

This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${mathfrak R}^*$ and ${mathfrak Z}^*$ corresponding to the bounded functions. Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.

Relative cohomology of complexes based on cotorsion pairs

Let [Formula: see text] be an associative ring with identity. The purpose of this paper is to establish relative cohomology theories based on cotorsion pairs in the setting of unbounded complexes of modules over [Formula: see text]. Let [Formula: see text] be a complete hereditary cotorsion pair in [Formula: see text]-Mod. Then [Formula: see text] and [Formula: see text] are complete hereditary cotorsion pairs in the category of [Formula: see text]-complexes. For any complexes [Formula: see text] and [Formula: see text] and any [Formula: see text], we define the [Formula: see text]th relative cohomology groups [Formula: see text] and [Formula: see text] by special [Formula: see text]-precovers of [Formula: see text] and by special [Formula: see text]-preenvelopes of [Formula: see text], respectively. They are common generalizations of absolute cohomology groups and Gorenstein cohomology groups of complexes. Some induced exact sequences concerning relative cohomology groups are considered. It is also shown that the relative cohomology functor of complexes we considered is balanced.

Symmetric powers, Steenrod operations and representation stability

Working over the prime field Fp, the structure of the indecomposables Q⁎ for the action of the algebra A(p) of Steenrod reduced powers on the symmetric power functors S⁎ is studied by exploiting the theory of strict polynomial functors.In particular, working at the prime 2, representation stability is exhibited for certain related functors, leading to a conjectural representation stability theoretic description of quotients of Q⁎ arising from the polynomial filtration of symmetric powers.

Balance for relative cohomology of complexes

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

Tate homology with respect to cotorsion pairs

ABSTRACTGiven two complete hereditary cotorsion pairs (𝒬,ℛ) and in the category of modules which satisfy the conditions 𝒬′⊆𝒬, , and ℛ′ is enveloping, a kind of Tate homology of modules is introduced and investigated in this article based on complete -resolutions. It is shown that a module admits complete -resolutions precisely when it has finite 𝒬-projective dimension. In particular, an Avramov–Martsinkovsky type exact sequence, which is natural in both variables, is constructed to connect such Tate homology functors and relative homology functors. Applications given for the subcategory 𝒢ℱ of Gorenstein flat modules go in three directions. The first is to improve the Avramov–Martsinkovsky type exact sequence appeared in Liang’s work [Tate homology of modules of finite Gorenstein flat dimension. Algebr. Represent. Theory 16:1541–1560 (2013)] by showing that it is indeed natural in both variables. The second is to characterize the finiteness of Gorenstein flat (resp., flat) dimension by using vanishing of Tate homology functors . Finally, by virtue of a conclusion provided by Enochs et al., some balance results are established for .

Localization of cofibration categories and groupoid C∗–algebras

We prove that relative functors out of a cofibration category are essentially the same as relative functors which are only defined on the subcategory of cofibrations. As an application we give a new construction of the functor that assigns to a groupoid its groupoid $C^*$-algebra and thereby its topological $K$-theory spectrum.

Relative left derived functors of tensor product functors

We introduce and study the relative left derived functor Torn(F,F′)(−,−) in the module category, which unifies several related left derived functors. Then we give some criteria for computing the F-resolution dimensions of modules in terms of the properties of TorTorn(F,F′)(−,−). We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.

On Balance for Relative Homology

Let R be a ring. It is reasonable to use a strict 𝒲𝒳-resolution of a right R-module M of finite 𝒳-projective dimension, where 𝒳 denotes a subcategory of right R-modules closed under extensions and admits an injective cogenerator 𝒲, to define the relative homology functor . A general balance result is established for such relative homology functor that encompasses a theorem of Emmanouil on Gorenstein projective R-modules and extends a theorem of Holm on Gorenstein flat R-modules. We also consider the above relative homology functor with respect to subcategories arising from an arbitrary but fixed semidualizing R-module. Inspired by the idea of Sather-Wagstaff et al. [25], we obtain the corresponding balance results when R is Cohen–Macaulay with a dualizing R-module as applications of our balance result.

Axioms of separation in semitopological groups and related functors

We prove that for every semitopological group G and every i∈{0,1,2,3,3.5}, there exists a continuous homomorphism φG,i:G→H onto a Ti (resp., Ti&T1 for i⩾3) semitopological group H such that for every continuous mapping f:G→X to a Ti- (resp., Ti&T1- for i⩾3) space X, one can find a continuous mapping h:H→X satisfying f=h∘φG,i. In other words, the semitopological group H=Ti(G) is a Ti-reflection of G. It turns out that all Ti-reflections of G are topologically isomorphic. These facts establish the existence of the covariant functors Ti for i=0,1,2,3,3.5, as well as the functors Reg and Tych in the category of semitopological groups and their continuous homomorphisms.We also show that the canonical homomorphisms φG,i of G onto Ti(G) are open for i=0,1,2 and provide an internal description of the groups T0(G) and T1(G) by finding the exact form of the kernels of φG,0 and φG,1. It is also established that the functors Reg and Ti∘T3, for i=0,1,2 are naturally equivalent.

A characterization of simplicial localization functors and a discussion of DK equivalences

In a previous paper, we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of “relative categories”. Here, we characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author. We employ this characterization to obtain a more explicit description of the weak equivalences in the model category of relative categories mentioned above by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.

Composition factors of functors

Let X be a family of finite groups satisfying certain conditions and K be a field. We study composition factors, radicals, and socles of biset and related functors defined on X over K . For such a functor M and for a group H in X , we construct bijections between some classes of maximal (respectively, simple) subfunctors of M and some classes of maximal (respectively, simple) K Out ( H ) -submodules of M ( H ) . We use these bijections to relate the multiplicity of a simple functor S H , V in M to the multiplicity of V in a certain K Out ( H ) -module related to M ( H ) . We then use these general results to study the structure of one of the important biset and related functors, namely the Burnside functor B K which assigns to each group G in X its Burnside algebra B K ( G ) = K ⊗ Z B ( G ) where B ( G ) is the Burnside ring of G. We find the radical and the socle of B K in most cases of X and K . For example, if K is of characteristic p > 0 and X is a family of finite abelian p-groups, we find the radical and the socle series of B K considered as a biset functor on X over K . We finally study restrictions of functors to nonfull subcategories. For example, we find some conditions forcing a simple deflation functor to remain simple as a Mackey functor. For an inflation functor M defined on abelian groups over a field of characteristic zero, we also obtain a criterion for M to be semisimple, in terms of the images of inflation and induction maps on M.

Galois functors and entwining structures

Galois comodules over a coring can be characterised by properties of the relative injective comodules. They motivated the definition of Galois functors over some comonad (or monad) on any category and in the first section of the present paper we investigate the role of the relative injectives (projectives) in this context. Then we generalise the notion of corings (derived from an entwining of an algebra and a coalgebra) to the entwining of a monad and a comonad. Hereby a key role is played by the notion of a grouplike natural transformation g : I → G generalising the grouplike elements in corings. We apply the evolving theory to Hopf monads on arbitrary categories, and to opmonoidal monads with antipode on autonomous monoidal categories (named Hopf monads by Bruguières and Virelizier) which can be understood as an entwining of two related functors. As well known, for any set G the product G × − defines an endofunctor on the category of sets and this is a Hopf monad if and only if G allows for a group structure. In the final section the elements of this case are generalised to arbitrary categories with finite products leading to Galois objects in the sense of Chase and Sweedler.

Strong Gorenstein Test Modules

ABSTRACT Let (R, 𝔪, k) be a commutative Noetherian local ring with maximal ideal 𝔪. Recently, Asadollahi and Salarian used relative cohomology functors to define Gorenstein test modules. In this article it is shown that every nonzero finitely generated R-module is a Gorenstein test module.

On higher order analogues of de Rham cohomology

In [A.M. Vinogradov, Some homological systems associated with the differential calculus in commutative algebras, Russian Math. Surveys 34 (6) (1979) 250–255] for any commutative K-algebra A, K being a commutative ring, any sequence σ of positive integers and any differentially closed (see Section 3) subcategory D of A− Mod, higher analogues dR σ D of the standard de Rham complex dR D ≡ dR (1,…,1,…) D and Spencer complexes were defined. In this paper a detailed exposition of all related functors of differential calculus over general commutative algebras is given for the first time together with some useful working techniques. In the second part of the paper, these techniques are then applied to prove that all complexes dR σ D are quasi-isomorphic under a smoothness assumption on the differentially closed subcategory D . This extends to arbitrary smooth categories of modules the quasi-isomorphism theorem for smooth manifolds and “regular” dR σ complexes proved in [G. Vezzosi, A.M. Vinogradov, Infinitesimal Stokes' formula for higher-order de Rham complexes, Acta Appl. Math. 49 (3) (1997) 311–329].

On Translation Functors for General Linear and Symmetric Groups

In this paper we study the rational representation theory of the general linear group G = GLn(F) over an algebraically closed field F of characteristic p. Given α ∈ Z/pZ, we define functors Trα and Trα, which, roughly speaking, are given by tensoring with the natural G-module V and its dual V* respectively, and then projecting onto certain blocks determined by the residue α. In fact, these functors can be viewed as special cases of Jantzen's translation functors. We prove a number of fundamental properties about these functors and also certain closely related functors that arise in the modular representation theory of the symmetric group. 1991 Mathematics Subject Classification: 20G05, 20C05.

On Relative Projective Functors

On Relative Projective Functors

Relative Lie algebra cohomology revisited

In this paper it is shown that relative Lie algebra cohomology is related to relative cohomology for restricted Lie algebras by a spectral sequence. Also an interpretation of the relative cohomology groups for Lie algebras in terms of the relative right derived functors of a certain relative derivative functor is given.

Cointegrations, relative cohomology for comodules, and coseparable corings

In this paper we introduce cointegrations from one bicomodule into another, a notion extending that of coderivation used by Nakajima [ 1 1 ] and Doi [3]. We show that in the category of comodules over a coring, the relative functor Ext is the derieved functor of cointegrations and, as a corollary, that the Hochschild cohomology of the coring is the derived functor of coderivations. These statements are analogues of the result of Kleiner [7], who proved that the relative functor Ext is the derived functor of integrations in the category of modules over an associative ring, and of the Barr-Rinehart [l] theorem stating that the Hochschild cohomology of an associative algebra is the derived functor of derivations. The proof is based on the existence of couniversal cointegrations and coderivations into an arbitrary comodule. As an application of the results on cohomology, we give a characterization of coseparable corings (Theorem 3.10). Coseparable, coalgebras have been studied in papers by Larson [IS], Nakajima [ 10, 111, and Doi [3], where coseparability is characterized in terms of coderivations. Our theorem extends these characterizations to corings, and gives some new ones, of which (vii) is particularly nice because it is an analogue of the characterization of finite-dimensional separable algebras over a field (cf, Cohn [2, Theorem 10.10.71). The cotensor product was first introduced by Milnor and Moore [9] for comodules over a coalgebra with a field as the ground ring. However, its behavior for corings is not well known, and coseparability seems to play an important role here (see Proposition 1.2, or the fact that cotensor product is not associative in general, but becomes so under some conseparability assumptions). The same remark applies to the functor Corn. (A detailed 211 OO21-8693/89 $3.OO