Extension Functors Research Articles

$k$-regular sequences to extension functors and local cohomology modules

$k$-regular sequences to extension functors and local cohomology modules

Universal lifting properties of abelian semigroup actions

In this paper, we prove that every action of a commutative cancellative semigroup can be extended to an invertible action in some kinds of categories, and this extension has universal lifting property. And the universal extension functor preserves all limits in the categorical point of view, especially, it preserves inverse limits and products. Moreover, we give some applications in dynamical systems.

Homological Algebra and Its Application: A Descriptive Study

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.

Notes on Relative Extension Functors

The objective of this paper is to study the relative homological properties of contravariantly and covariantly finite subcategories. Some sufficient conditions for are obtained. We also give the conditions under which the stable categories are one‐side triangulated categories.

Extension Functors of Generalized Local Cohomology Modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.

A Functorial Presentation of Units of Burnside Rings

Let \(B^\times \) be the biset functor over \(\mathbb {F}_2\) sending a finite group G to the group \(B^\times (G)\) of units of its Burnside ring B(G), and let \(\widehat{B^\times }\) be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from \(B^\times \) in the dual Burnside functor \(\widehat{\mathbb {F}_2B}\), or equivalently, an explicit set of generators \(\mathcal {G}_\mathcal {S}\) of the kernel L of the natural surjection \(\mathbb {F}_2B\rightarrow \widehat{B^\times }\). This yields a two terms projective resolution of \(\widehat{B^\times }\), leading to some information on the extension functors \(\mathrm{Ext}^1(-,B^\times )\). For a finite group G, this also allows for a description of \(B^\times (G)\) as a limit of groups \(B^\times (T/S)\) over sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor \(B^\times \) is not finitely generated, and that its dual \(\widehat{B^\times }\) is finitely generated, but not finitely presented. The last result of the paper shows in addition that \(\mathcal {G}_\mathcal {S}\) is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.

A categorical approach to operator semigroups

The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. Theory 78(2):473–500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34–70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory.

SPECIAL TILTING MODULES FOR ALGEBRAS WITH POSITIVE DOMINANT DIMENSION

AbstractWe study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimald-Auslander–Gorenstein algebras andd-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.

On relative Auslander algebras

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.

Quiver-graded Richardson orbits

In Lie theory, a dense orbit in the nilpotent radical of a parabolic group under the operation of the parabolic is called a Richardson orbit. We define a quiver-graded version of Richardson orbits generalizing the classical definition in the case of the general linear group. We define a quasi-hereditary algebra called the nilpotent quiver algebra whose isomorphism classes of Δ-filtered modules correspond to orbits in our generalized setting. We translate the existence of a Richardson orbit into the existence of a rigid Δ-filtered module of a given dimension vector. We study an idempotent recollement of this algebra whose associated intermediate extension functor can be used to produce Richardson orbits in some situations. This can be explicitly calculated in examples. We also give examples where no Richardson orbit exists.

Finiteness of extension functors of generalized local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity, an ideal of R, M and X finite R–modules, and t a non-negative integer such that is –cofinite for all i < t. In this article, we show that and are finite, and is finite if and only if is finite. As a consequence, we observe that is a finite set.

Free oriented extensions of subfactor planar algebras

We show that the restriction functor from oriented factor planar algebras to subfactor planar algebras admits a left adjoint, which we call the free oriented extension functor. We show that for any subfactor planar algebra realized as the standard invariant of a hyperfinite [Formula: see text] subfactor, the projection category of the free oriented extension admits a realization as bimodules of the hyperfinite [Formula: see text] factor.

1 × N hybrid radio frequency photonic splitter based on a dual-polarization dual-parallel Mach Zehnder modulator

The notion of (semi)bricks, regarded as a generalization of (semi)simple modules, appeared in a paper of Ringel in 1976. In recent years, there has been several new developments motivated by links to {\tau}-tilting theory studied by Demonet-Iyama-Jasso and Asai. In this paper, we will discuss how to glue semibricks along a recollement with the intermediate extension functor similar to gluing simple modules by Beilinson-Bernstein-Deligne. As an application, we investigate the behavior of {\tau}-tilting finite under recollements of module categories of algebras. Moreover, we give some examples to show the construction of support {\tau}-tilting modules over {\tau}-tilting finite algebras by gluing semibricks via recollements.

Extension Functors of Cousin Cohomology Modules

Let R be a commutative Noetherian ring with non-zero identity, $$\mathcal {F}$$ a filtration of $${{\mathrm{Spec}}}(R)$$ which admits an R-module X, and $${{\mathrm{C}}}_R(\mathcal {F},X)$$ the Cousin complex for X with respect to $$\mathcal {F}$$ . In this paper, we first introduce the Cousin functor and the Cousin spectral sequences. Then for non-negative integers s, t and a finite R-module N, we study the membership of R-modules $${{\mathrm{H}}}^{s-1}({{\mathrm{Ext}}}^t_R(N,{{\mathrm{C}}}_R(\mathcal {F},X)))$$ and $${{\mathrm{Ext}}}^{s}_R(N,{{\mathrm{H}}}^{t-1}({{\mathrm{C}}}_R(\mathcal {F},X)))$$ in Serre subcategories of the category of R-modules and find some conditions for validity of an isomorphism between them. Finally, we use these results to present some facts about the vanishing and finiteness of Cousin cohomology modules.

Bohr Compactifications of Algebras and Structures

This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes—often varieties—are plentiful and varied, and in many cases the associated topological prevariety is standard.

Weyl Groups, Ideal Class Groups and the K-Theory of the Extension Functor

We introduce the notion of the Weyl group of a number ring. After that, we give a matrix approach to the study of the ideal class group of a number field, the Bass exact sequence of K-theory of the extension functor, and the capitulation kernel of an extension of number fields.

A Complex of Modules and Its Applications to Local Cohomology and Extension Functors

Let $(R,m)$ be a commutative Noetherian complete local ring and let $M$ be a non-zero Cohen-Macaulay $R$-module of dimension $n$. It is shown that, if $\operatorname{projdim}_R(M)&lt;\infty$, then $\operatorname{injdim}_R(D(H^n_{\mathfrak{m}}(M)))&lt;\infty$, and if $\operatorname{injdim}_R(M)&lt;\infty$, then $\operatorname{projdim}_R(D(H^n_{\mathfrak{m}}(M)))&lt;\infty$, where $D(-):= \operatorname{Hom}_{R}(-,E)$ denotes the Matlis dual functor and $E := E_R(R/\mathfrak{m})$ is the injective hull of the residue field $R/\mathfrak{m}$. Also, it is shown that if $(R,\mathfrak{m})$ is a Noetherian complete local ring, $M$ is a non-zero finitely generated $R$-module and $x_1,\ldots,x_k$, $(k\geq 1)$, is an $M$-regular sequence, then \[ D(H^k_{(x_1,\ldots,x_k)}(D(H^k_{(x_1,\ldots,x_k)}(M))))\simeq M. \] In particular, $\operatorname{Ann} H^k_{(x_1,\ldots,x_k)}(M)=\operatorname{Ann} M$. Moreover, it is shown that if $R$ is a Noetherian ring, $M$ is a finitely generated $R$-module and $x_1,\ldots,x_k$ is an $M$-regular sequence, then \[ \operatorname{Ext}^{k+1}_R(R/(x_1,\ldots,x_k),M)=0. \]

Derived string topology and the Eilenberg-Moore spectral sequence

Let M be a simply-connected closed manifold of dimension m. Chas and Sullivan have defined (co)products on the homology of the free loop space H*(LM). Felix and Thomas have extended the loop (co)products to those of simply-connected Gorenstein spaces over a field. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. In Algebraic Topology, one of the most important tools for computing the (co)homology of the space of free loops on a space is the (co)homological Eilenberg-Moore spectral sequence. Consider, over any field, the homological Eilenberg-Moore spectral sequence converging to H *(LM). Our description of the loop product enables one to conclude that this spectral sequence is multiplicative with respect to the Chas-Sullivan loop product and that its E 2-term is the Hochschild cohomology of H*(M). This gives a new method to compute the loop products on H *(LS m ) and H *(LℂP r ), the free loop space homology of spheres and complex projective spaces.

On the commutativity of the Clifford and “extension of scalars” functors

We introduce sheaves of A-modules of fractions (or just A-modules of fractions), on a topological space X, with denominator a monoid-subsheaf S of A; as aside worth noting, we remark (Theorem 2.4) that there is an isomorphism between the functors S−1 and (S−1A)⊗_. Moreover, we discuss the classical problem related to the commutativity of the functors: Clifford functor Cl and algebra extension functor of the ground algebra K of a quadratic K-module(M,q). As a particular case, we show (Corollary 3.5) that given a sheaf A of algebras on a topological space X and S as above, the functor ClS−1A commutes with the functor S−1ClA.

Generalized Serre conditions and perverse coherent sheaves

In algebraic geometry, one often encounters the following problem: given a scheme X, find a proper birational morphism Y→X where the geometry of Y is “nicer” than that of X. One version of this problem, first studied by Faltings, requires Y to be Cohen–Macaulay; in this case Y→X is called a Macaulayfication of X. In another variant, one requires Y to satisfy the Serre condition Sr. In this paper, the authors introduce generalized Serre conditions—these are local cohomology conditions which include Sr and the Cohen–Macaulay condition as special cases. To any generalized Serre condition Sρ, there exists an associated perverse t-structure on the derived category of coherent sheaves on a suitable scheme X. Under appropriate hypotheses, the authors characterize those schemes for which a canonical finite Sρ-ification exists in terms of the intermediate extension functor for the associated perversity. Similar results, including a universal property, are obtained for a more general morphism extension problem called Sρ-extension.