Short Exact Sequence Research Articles

Rees short exact sequences and preenvelopes

Abstract In this paper, we consider some properties of commutative diagrams of Rees short exact sequences, and we also investigate the sufficient and necessary condition under which the induced sequences by functors − ⊗ M for the left S-act M. The main conclusions extend some known results. Further, we investigate preenvelopes and precovers in the category 𝓔 S of Rees short exact sequences of right S-acts.

SOME REMARKS ON LOCALLY NOETHERIAN AND LOCALLY ARTINIAN S-ACTS OVER MONOIDS

In the category of S-acts, artinian S-acts are introduced as the acts that satisfy the descending chain condition on its congruences. Noetherian S-acts are also the acts that satisfy the ascending chain condition on its congruences. Rees noetherian and Rees artinian case is defined using Rees congruence. This paper is devoted to study the classes of locally artinian, locally artinian, locally Rees artinian and locally Rees artinian S-acts. An S-act is said to be locally artinian if all its finitely generated subacts artinian. The noetherian, Rees noetherian and Rees artinian case is defined similarly. We give some general properties of these classes of S-acts, specially discuss the behaviour of such acts under Rees short exact sequence and coproduct. Finally, we establish some connections between some classes of S-acts such as projectivity and injectivity with the notions related to locally artinian and locally noetherian.

Locally Coherent Exact Categories

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

A Characterization of Normal Injective and Normal Projective Hypermodules

This study is motivated by the recently published papers on normal injective and normal projective hypermodules. We provide a new characterization of the normal injective and normal projective hypermodules by using the splitting of the short exact sequences of hypermodules. After presenting some of their fundamental properties, we show that if a hypermodule is normal projective, then every exact sequence ending with it is splitting. Moreover, if a hypermodule is normal injective, then every exact sequence starting with it is splitting as well. Finally, we investigate the relationships between semisimple, simple, normal injective, and normal projective hypermodules.

Algebraic computation of short exact sequence and topological invariants using K-theory in general ring of classification spaces

The objective of this study is to calculate the K-Cohomology invariants of certain classification spaces of homotopy groups in short exact sequences, where the computation is done using properties of the K-theoretical features. For these packages we use various applications including K-theory Algebra, which transformed the theory of operator algebra from a smaller branch of functional analysis into a topic with deep and wide application areas in all algebra and topology in mathematics.

A remark on the moduli space of Lie algebroid λ-connections

Let X be a compact Riemann surface of genus g ≥ 3 . Let L = ( L , [ . , . ] , ♯ ) be a holomorphic Lie algebroid over X of rank one and degree ( L ) < 2 − 2 g . We consider the moduli space of holomorphic L λ -connections over X, where λ ∈ C . We compute the Picard group of the moduli space of L λ -connections by constructing an explicit smooth compactification of the moduli space of those L λ -connections whose underlying vector bundle is stable such that the complement is a smooth divisor. We also show that the automorphism group of the moduli space of L λ -connections fits into a short exact sequence that involves the automorphism group of the moduli space of stable vector bundle over X. For λ = 1, we get Lie algebroid de Rham moduli space of L -connections and we determine its Chow group. Communicated by Manuel Reyes

Some new properties on N semigroups

In this study we first show that  satisfies two important homological properties, namely Rees short exact sequence an d short five lemma. In addition, by defining inversive semigroup varieties of  we prove that strictly inverse semigroup  is isomorphic to the spined product of (C)-inversive semigroup and the idempotent semigroup of . Moreover, we give some consequences of the results to make a detailed classification over . It has been recently defined a new semigroup  based on Rees matrix and completely 0-simple semigroups. Further, it has been also proved finiteness conditions and the existence of some fundamental properties over .

2-term averaging L∞-algebras and non-abelian extensions of averaging Lie algebras

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L∞-algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context.

Relation between balanced pairs and TTF triples

Let A be a complete and cocomplete abelian category with the additional assumption that any direct sum and product of short exact sequences are exact. We explore the relation between balanced pairs and TTF triples in A. The main results are: (1) every balanced pair in A induces a TTF triple; (2) if A has projective covers and injective envelopes, then every TTF triple in A gives rise to a balanced pair, and hence there is a bijection between the equivalence classes of balanced pairs and TTF triples in A; (3) a balanced pair in A is quasi admissible if and only if its induced TTF triple is centrally splitting. Our first application of these results provide abundant rings over which every balanced pair is quasi admissible, including local rings, commutative semiperfect rings, and commutative Noetherian rings. Another application is the classification of equivalence classes of cohereditary balanced pairs over arbitrary rings. We also present counterexamples to [15, Open questions 3 and 5]. Finally, we prove that the answers to [15, Open questions 2 and 4] are positive for coherent rings with weak global dimension at most one.

On exactness and splitting

In this paper, we study the concepts of exactness and splitting in exact sequences of modules over a ring. We review some basic definitions and properties of exact sequences, and introduce the notions of short exact sequences, split exact sequences, projective modules, injective modules, and semisimple modules. We prove some important results relating these concepts, such as the splitting lemma. We also give some examples and applications of these results.

Extensions and automorphisms of Rota-Baxter groups

The notion of Rota-Baxter groups was recently introduced by Guo, Lang and Sheng [19] in the geometric study of Rota-Baxter Lie algebras. They are closely related to skew braces as observed by Bardakov and Gubarev. In this paper, we study extensions of Rota-Baxter groups by constructing suitable cohomology theories. Among others, we find relations with the extensions of skew braces. Given an extension of Rota-Baxter groups, we also construct a short exact sequence connecting various automorphism groups, which generalizes the Wells short exact sequence.

Rabinowitz Floer homology of negative line bundles and Floer Gysin sequence

This article is concerned with the Rabinowitz Floer homology of negative line bundles. We construct a refined version of Rabinowitz Floer homology and study its properties. In particular, we build a Gysin-type long exact sequence for this new invariant and discuss an application to the orderability problem for prequantization spaces. We also construct a short exact sequence for the ordinary Rabinowitz Floer homology and provide computational results.

On np-injective rings and Modules

A given R-module is called a right np-injective module if for any non-nilpotent element of , and any right R-homomorphism can be extended to , if is np-injective, then is a right np-injective ring. A given ring is called right weakly np-injective if for each non-nilpotent element of , there exists a positive integer such that any right R-homomorphism can be extended to . A given ring is a right weakly np-injective ring, if . In the matrix ring, we poved that is right np-injective, for some , if is right Weakly np-injective. So, We extended many of the known properties and characterizations of right np-injective rings and modules. Finally, the main result in np-injective module found that the R-module is np-injective module if and only if for any the short exact sequence of R-modules, is also a short exact sequence, where and .

The Atiyah class of generalized holomorphic vector bundles

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.

Rips construction without unique product

Given a finitely presented group $Q,$ we produce a short exact sequence $1\to N \hookrightarrow G \twoheadrightarrow Q \to 1$ such that $G$ is a torsion-free Gromov hyperbolic group without the unique product property and $N$ is without the unique product property and has Kazhdan's Property (T). Varying $Q,$ we show a wide diversity of concrete examples of Gromov hyperbolic groups without the unique product property. As an application, we obtain Tarski monster groups without the unique product property.

Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

A Rota-Baxter Lie algebra gT is a Lie algebra g equipped with a Rota-Baxter operator T:g→g. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra gT by another Rota-Baxter Lie algebra hS. We define the non-abelian cohomology Hnab2(gT,hS) which classifies equivalence classes of such extensions. Given a non-abelian extension [Display omitted] of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in Aut(hS)×Aut(gT) to be induced by an automorphism in Aut(eU) lies in the cohomology group Hnab2(gT,hS). As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras. Finally, we show how these results fit with abelian extensions of Rota-Baxter Lie algebras.

Canonical Witt formal scheme extensions and p-torsion groups

We study the nth arithmetic jet space of the p-torsion subgroup attached to a smooth commutative formal group scheme. We show that the nth jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length n and the p-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.

Chain conditions on (Rees) congruences of S-acts

In this paper, we introduce (Rees) artinian [Formula: see text]-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that (Rees) artinian [Formula: see text]-acts are those acts whose factor acts are all finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian [Formula: see text]-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.

The geometry and DSZ quantization four-dimensional supergravity

We implement the Dirac-Schwinger-Zwanziger integrality condition on four-dimensional classical ungauged supergravity and use it to obtain its duality-covariant, gauge-theoretic, differential-geometric model on an oriented four-manifold $M$ of arbitrary topology. Classical bosonic supergravity is completely determined by a submersion $\pi$ over $M$ equipped with a complete Ehresmann connection, a vertical euclidean metric, and a vertically-polarized flat symplectic vector bundle $\Xi$. Building on these structures, we implement the Dirac-Schwinger-Zwanziger integrality condition through the choice of an element in the degree-two sheaf cohomology group with coefficients in a locally constant sheaf $\mathcal{L}\subset \Xi$ valued in the groupoid of integral symplectic spaces. We show that this data determines a Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}^{n_v}$ whose connections provide the global geometric description of the local electromagnetic gauge potentials of the theory. Furthermore, we prove that the Maxwell gauge equations of the theory reduce to the polarized self-duality condition determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. In addition, we investigate the continuous and discrete U-duality groups of the theory, characterizing them through short exact sequences and realizing the latter through the gauge group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric origin of U-duality, which we explore in several examples, illustrating its dependence on the topology of the fiber bundles $\pi$ and $P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.

Exactness of Proximal Group Homomorphisms

This research introduces groups in proximity spaces which endowed with a proximity relation. Two penultimate choices for such relations are the Efremovic (EF) proximity relation and its extension, namely, the descriptive EF-proximity relation. There is a strong relationship between sets (groups) and set (group) descriptions. Therefore, in this paper we consider this relationship via exactness of descriptive homomorphisms between ordinary descriptive groups and meta-descriptive groups. The definition of a short exact sequence of descriptive homomorphisms is given. Then, results were obtained giving the relationships between the two short exact sequences.