Category Of Left R-modules Research Articles

Kerler–Lyubashenko Functors on 4-Dimensional 2-Handlebodies

Abstract We construct a braided monoidal functor $J_4$ from Bobtcheva and Piergallini’s category $4\textrm {HB}$ of connected 4-dimensional 2-handlebodies (up to 2-deformations) to an arbitrary unimodular ribbon category $\mathscr {C}$, which is not required to be semisimple. The main example of target category is provided by ${H}\textrm{-mod}$, the category of left modules over a unimodular ribbon Hopf algebra $H$. The source category $4\textrm {HB}$ is freely generated, as a braided monoidal category, by a Bobtcheva--Piergallini Hopf (BPH) algebra object, and this is sent by the Kerler–Lyubashenko functor $J_4$ to the end $\int _{X \in \mathscr {C}} X \otimes X^*$ in $\mathscr {C}$, which is given by the adjoint representation in the case of ${H}\textrm{-mod}$. When $\mathscr {C}$ is factorizable, we show that the construction only depends on the boundary and signature of handlebodies and thus projects to a functor $J_3^{\sigma }$ defined on Kerler’s category $3\textrm {Cob}^{\sigma }$ of connected framed 3-dimensional cobordisms. When $H^*$ is not semisimple and $H$ is not factorizable, our functor $J_4$ has the potential of detecting diffeomorphisms that are not 2-deformations.

Superized Troesch complexes and cohomology for strict polynomial superfunctors

We adapt a construction due to Troesch to the category of strict polynomial superfunctors in order to construct complexes of injective objects whose cohomology is isomorphic to Frobenius twists of the (super)symmetric power functors . We apply these complexes to construct injective resolutions of the even and odd Frobenius twist functors, to investigate the structure of the Yoneda algebra of the Frobenius twist functor, and to compute other extension groups between strict polynomial superfunctors. By an equivalence of categories, this also provides cohomology calculations in the category of left modules over Schur superalgebras.

Homology of twisted quiver bundles with relations

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space (X,OX), allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of OX-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups.

FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

Triangle equivalences involving Gorenstein FP-injective modules

ABSTRACTRecently, Hu et al. [16] introduced the notion of Gorenstein FP-injective modules and investigated a notion of Gorenstein FP-injective dimension for complexes. Let R be a left coherent ring and Db(R-Mod) the bounded derived category of left R-modules. It is proved that the quotient triangulated category of the subcategory of Db(R-Mod) consisting of complexes with both finite Gorenstein FP-injective dimension and FP-projective dimension by the bounded homotopy category of FP-projective-injective left R-modules is triangle-equivalent to the stable category of the Frobenius category of all Gorenstein FP-injective and FP-projective left R-modules. Note that a similar argument is also valid for the case of Gorenstein injective left R-modules. We extend a triangle equivalence established by Beligiannis involving Gorenstein injective left R-modules from rings with finite left Gorenstein global dimension to arbitrary rings.

English

Let (A, B) be a complete and hereditary cotorsion pair in the category of left R-modules. In this paper, the so-called Gorenstein projective complexes with respect to the cotorsion pair (A, B) are introduced. We show that these complexes are just the complexes of Gorenstein projective modules with respect to the cotorsion pair (A, B). As an application, we prove that both the Gorenstein projective modules with respect to cotorsion pairs and the Gorenstein projective complexes with respect to cotorsion pairs possess stability.

Relative singularity categories with respect to gorenstein flat modules

Let R be a right coherent ring and D b (R-Mod) the bounded derived category of left R-modules. Denote by \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\) the subcategory of D b (R-Mod) consisting of all complexes with both finite Gorenstein flat dimension and cotorsion dimension and K b (F ∩ C) the bounded homotopy category of flat cotorsion left R-modules. We prove that the quotient triangulated category \({D^b}{\left( {R - Mod} \right)_{\widehat {\left[ {GF,C} \right]}}}\)/K b (F ∩ C) is triangle-equivalent to the stable category GF ∩ C of the Frobenius category of all Gorenstein flat and cotorsion left R-modules.

A Natural Extension of the Universal Enveloping Algebra Functor to Crossed Modules of Leibniz Algebras

The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category $\mathcal {LM}$ in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.

Universal enveloping crossed module of Leibniz crossed modules and representations

The universal enveloping algebra functor UL: Lb → Alg, defined by Loday and Pirashvili [1], is extended to crossed modules. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Note that the procedure followed in the proof for the Lie case cannot be adapted, since the actor in the category of Leibniz crossed modules does not always exist.

A conception of zero-divisor graph for categories of modules

We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R- MOD . The vertices of Γ(R- MOD ) consist of all nonzero morphisms in R- MOD which are not isomorphisms. Two vertices f and g are adjacent if f ◦ g = 0 or g ◦ f = 0. We observe that these graphs are connected and their diameter is equal or less than four. We prove that Γ(R- MOD ) = 3 if and only if R is a right and left perfect ring and R/J(R) is simple artinian. We also characterize all vertices with complements and that when a kernel or a co-kernel can be a complement for a morphism. Some discussions will be made on radius of these graphs, their clique and chromatic numbers.

Enveloping Algebras and Cohomology of Leibniz Pairs

We introduce the enveloping algebra for a Leibniz pair, and show that the category of modules over a Leibniz pair is isomorphic to the category of left modules over its enveloping algebra. Consequently, we show that the cohomology theory for a Leibniz pair introduced by Flato, Gerstenhaber, and Voronov can be interpreted by Ext-groups of modules over the enveloping algebra.

C-RINGS, COPRODUCTS, AND REFLECTION FUNCTORS

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the category of left modules over the coproduct of R and S op in an appropriate category. Letting K denote a field, we apply this to show that the category of K-representations of a quiver that is reflection equivalent to a path algebra KΓ is equivalent to a full subcategory of left and right K-representations of KΓ.

Quotients of Preradicals on a Module Category

If σ and τ are preradical functors on the category of left modules over a ring R, it is possible to define a “quotient” endofunctor by for all modules M. The collection of all such endofunctors (denoted is shown to constitute a semigroup under the operation of functor composition (modulo the congruence relation which identifies two functors ε, δ if ε(M) ≅ δ(M) for all modules M). Multiplication tables for this semigroup are constructed in some simple cases. An important subsemigroup of is identified, and several important classes of rings are characterized in terms of .

Universal extensions

In the category of left modules over a unital ring we show that a left exact reflector determines, for each n ⩾ 1 , a torsion theoretic setting in which universal extensions of length n exist. Combined with recent work of Rodelo and van der Linden (2011) [9] this establishes the existence of universal central extensions of groups and Lie algebras. Interpreted in the homotopy category of topological spaces, it provides a new perspective on existing results about Quillenʼs plus construction and its effect on homotopy groups.

When is the Diagonal Functor Frobenius?

Given a complete, cocomplete category 𝒞, we investigate the problem of describing those small categories I such that the diagonal functor Δ: 𝒞 → Functors(I, 𝒞) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → 𝒞 are naturally isomorphic. We find necessary conditions on I for a certain class of categories 𝒞, and, as an application, we give both necessary and sufficient conditions in the two special cases 𝒞 =Set or R ℳ, the category of left modules over a ring R.

Right and left modules over the Frobenius skew polynomial ring in theF-finite case

AbstractThe main purposes of this paper are to establish and exploit the result that, over a complete (Noetherian) local ringRof prime characteristic for which the Frobenius homomorphismfis finite, the appropriate restrictions of the Matlis-duality functor provide an equivalence between the category of left modules over the Frobenius skew polynomial ringR[x,f] that are Artinian asR-modules and the category of rightR[x,f]-modules that are Noetherian asR-modules.

Asymmetry in the Converse of Schur's Lemma

We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.

Quasi-localizations of ℤ

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has \(R \subseteq M \subseteq \mathbb{Q} \otimes _\mathbb{Z} R\). We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R+, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family \(\mathcal{F}\) of submodules and thus \((M,\mathcal{F})\). We will restrict our discussion to the case M = R such that \((R,\mathcal{F}) = R\), and in this case we call the family \(\mathcal{F}\) of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.

Decisive dimension and other related torsion theoretic dimensions

The paper is concerned with the study of the decisive dimension defined on the category of left modules over a ring R . We compare the decisive dimension with the Gabriel dimension and other dimensions recently introduced. We give module theoretic as well as lattice theoretic characterizations of rings with decisive dimension. As an application we obtain characterizations of some classes of rings.

On the Atomic Dimension in Module Categories

ABSTRACT This article is concerned with the study of atomic dimension defined on the category of left modules over a ring R. A module theoretic as well as torsion theoretic characterization of rings with atomic dimension is provided. The atomic and Gabriel dimensions are compared and necessary and sufficient conditions for these two dimensions to coincide are established.