RTAA
Representation Theory of Algebras and Applications
Fridays at 16:00 (
Zoom)
Organizers: Kostiantyn Iusenko, John MacQuarrie and Eduardo Marcos.
Seminar Mailing List: RTAA — Representações das Álgebras e Aplicações
FORTHCOMING WEBINARS
PAST WEBINARS
24 June, 2022 |
Marcelo MoreiraUniversidade Federal de AlfenasO estudo da estrutura das (co)homologias de Hochschild da classe de álgebras de dimensão finita $A$ com: $Hom_{A-A} (DA,A) =0$O interesse sobre essa classe de álgebras nasceu na tentativa da generalização das sequências exatas curtas de grupos de cohomologia de Hochschild dos trabalhos [1] e [2]: \begin{equation*} 0 \longrightarrow H^0(TA,DA) \longrightarrow HH^0(TA) \longrightarrow HH^0(A) \longrightarrow 0 \end{equation*} \begin{equation*} 0 \longrightarrow H^1(TA,DA)\oplus\mathcal{E}(DA) \longrightarrow HH^1(TA) \longrightarrow HH^1(A) \longrightarrow 0 \;, \end{equation*} em que $\mathcal{E}(DA)$ é o $k$-subespaço de $Hom_{A\text{-}A}(DA, A)$ formado por todos os morfismos de $A$-bimódulos $f \colon DA \to A$ tais que, para quaisquer $x, y \in DA$, temos que $f(x)y+xf(y) =0$. |
17 June, 2022 |
Marlon StefanoUFMGO método de Butler aplicado a $\mathbb Z_p[C_p\times C_p]$-módulos de permutação.Seja $p$ um número primo e $G$ um $p$-grupo finito com subgrupo normal $N$ de ordem $p$. Vamos denotar por $\mathbb Z_p$ o anel dos inteiros $p$-ádicos. Em um trabalho recente, Zalesskii e MacQuarrie forneceram uma caracterização dos $\mathbb Z_pG$-módulos de permutação em termos de módulos para $\mathbb Z_p[G/N]$. A caracterização depende de duas condições, mas a necessidade dessas condições não era conhecida. Em um trabalho junto com MacQuarrie, aplicamos uma correspondência devida a Butler, que permite associar reticulados a $n$-uplas de $\mathbb Z_pG$-módulos, para mostrar a necessidade dessas condições por exibir um contraexemplo não trivial para a afirmação: se ambos os $N$-invariantes e $N$-coinvariantes de um certo reticulado $U$ são de permutação, então $U$ também o é. |
10 June, 2022 |
Flávio CoelhoIME-USPUma trisecção das componentes do quiver de Auslander-Reiten. O conceito de trisecções na categoria de módulos sobre uma álgebra tem sido bastante utilizado na teoria de representações. Uma dessas trisecções envolve as subcategorias ${\mathcal L}_{\Lambda}$ e ${\mathcal R}_{\Lambda}$ definidas como segue. Para uma álgebra $\Lambda$,
$$ {\mathcal L}_{\Lambda} \ = \ \{ X \in \mbox{ ind}\Lambda \colon \mbox{ pd}_\Lambda Y \leq 1 \mbox{ quando existir um caminho } Y \leadsto X \} $$
$$ {\mathcal R}_{\Lambda} \ = \ \{ X \in \mbox{ ind}\Lambda \colon \mbox{ id}_\Lambda Y \leq 1 \mbox{ quando existir um caminho} X \leadsto Y \} $$
onde $M \leadsto N$ significa que existe um caminho de morfismos n\~ao nulos de $M$ a $N$. Relembre também que pd$_\Lambda N$ e id$_\Lambda N$ indicam, respectivamente, as dimensões projetiva e injetiva de um módulo $N$. Desta forma, $({\mathcal L}_{\Lambda}\setminus {\mathcal R}_{\Lambda}, {\mathcal L}_{\Lambda} \cap {\mathcal R}_{\Lambda}, {\mathcal R}_{\Lambda} \setminus {\mathcal L}_{\Lambda})$ induz uma trisecção na categoria ind$\Lambda$ para classes de álgebras como quasitilted e shod. Vamos discutir quando é que $({\mathcal L}_{\Lambda}\setminus {\mathcal R}_{\Lambda}, {\mathcal L}_{\Lambda} \cap {\mathcal R}_{\Lambda}, {\mathcal R}_{\Lambda} \setminus {\mathcal L}_{\Lambda})$ induz uma trisecção no quiver de Auslander-Reiten de uma álgebra.
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03 June, 2022 |
Manuel SaorínUniversity of Murcia$\mathcal{P}^{<\infty}$-contravariant finiteness and strong tilting iteration via corner algebrasA sufficient condition for the verification of the finitistic dimension conjecture for an Artin algebra $\Lambda$ is that the subcategory $\mathcal{P}^{<\infty}(mod-\Lambda)$ of (finitely generated) modules of finite projective dimension be contravariantly finite in the category $mod-\Lambda$. This last condition is in turn equivalent to the existence of a strong tilting $\Lambda$-module. Two natural problems arise: 1) Give methods to construct Artin algebras $\Lambda$ with the mentioned contravariant finiteness property; 2) Assuming that property, and hence also the existence of a strong tilting $\Lambda$-module $T$, when is it true that the (strongly tilted) endomorphism algebra $\tilde{\Lambda}=End(T_\Lambda)$ has the mentioned contravariant property? and, if so, can this process of strong tilting be iterated indefinitely?. We will show that, by an adequate choice of an idempotent $e\in\Lambda$, the answers to these questions can already be checked by looking at the corner algebra $e\Lambda e$. Show abstract |
27 May, 2022 |
Thomas BrüstleUniversité de SherbrookeHomological approximations in persistence theory Multiparameter persistence modules are defined over a wild algebra and therefore they do not admit a complete discrete invariant. One thus tries in persistence theory to “approximate” such a module by a more manageable class of modules. Using that approach we define a class of invariants for persistence modules based on ideas from homological algebra. |
20 May, 2022 |
Lorna GregoryUniversità degli Studi della Campania Luigi VanvitelliMaranda's theorem for pure-injective modules Let $R$ be a discrete valuation domain with maximal ideal generated by $\pi$ and let $\Lambda$ be an $R$-order inside a separable algebra. For example, $R=\mathbb{Z}_{(p)}$ and $\Lambda=\mathbb{Z}_{(p)}G$ where $G$ is a finite group. |
13 May, 2022 |
Sibylle SchrollUniversity of CologneFull exceptional sequences in the derived category of gentle algebrasIn general it is difficult to determine whether the bounded derived category of a finite dimensional algebras has full exceptional sequences or not. In this talk I will focus on the class of gentle algebras and determine for which gentle algebras the bounded derived category admits full exceptional sequences in terms of an underlying geometric surface model. Gentle algebras are finite dimensional algebras that are connected to many different areas of mathematics such as cluster theory, N=2 gauge theory and homological mirror symmetry of surfaces. Show abstract |
29 April, 2022 |
José Armando ViveroUniversidad de La RepúblicaTriangular LIT algebrasIn this talk I am going to present some results concerning Lat-Igusa-Todorov algebras (LIT algebras for short). The notion of LIT algebra, given by D. Bravo, M. Lanzilotta, O. Mendoza and J. Vivero in [1], is a way of generalizing the concept of Igusa-Todorov algebra given by J. Wei in [2]. The results I am going to present can be found in [2103.12120] Triangular Lat-Igusa-Todorov algebras (arxiv.org). |
08 April, 2022 |
Eduardo MarcosIME-USP$S$-homogenous triples and aplications to algebras with 2-relationsThis talk is on a joint work with Yuri Volkov. We define the notion os S-homogeneous triples and $S$-homogeneous algebras and give an equivalence of categories which we apply to study koszulity, etc... The results are used for the partial classifications of homogeneous algebras whose defining ideal is generated by two relations. Show abstract |
01 April, 2022 |
Calin ChindrisUniversity of MissouriQuiver Radial Isotropy and Applications to the Paulsen Problem in Frame TheorySigma-critical representations are quiver representations that satisfy certain matrix equations. They arise naturally in the context of Kempf-Ness theorem on closed orbits in Invariant Theory. After introducing all the relevant concepts, I will first present the quiver radial isotropy theorem which gives necessary and sufficient conditions for the orbit of a quiver representation to contain a sigma-critical representation. I will then explain how this result can be used to solve the Paulsen Problem for matrix frames. This is based on joint work with Jasim Ismaeel. Show abstract |
03 December, 2021 |
Mark KleinerSyracuse UniversityGraph monoids and preprojective roots of Coxeter groupsA graph monoid $\mathfrak{M}$ of a finite undirected graph $\Gamma$ without loops and multiple edges was introduced in 1969 by Cartier and Foata in order to give a new proof and a noncommutative version of the celebrated MacMahon Master Theorem, which is a result in enumerative combinatorics and linear algebra. If $\Gamma$ is given an acyclic orientation $\Lambda$ and thus becomes a quiver $(\Gamma, \Lambda)$, a certain subset (not a submonoid) of $\mathfrak{M}$ is closely related to the $(+)$-admissible sequences of vertices introduced in 1973 by Bernstein, Gelfand, and Ponomarev in their seminal paper on quiver representations. If $\Gamma$ is the Coxeter graph of a Coxeter group $\mathcal{W}$, we use a canonical surjective monoid homomorphism $\mathfrak{M} \rightarrow \mathcal{W}$ to study preprojective roots of $\mathcal{W}$. Show abstract |
26 November, 2021 |
Rosanna LakingUniversità degli Studi di VeronaWide intervals and mutationTorsion pairs in the category $\textrm{mod} A$ of finite-dimensional modules over a finite-dimensional algebra have played an important role in many aspects of modern representation theory, including the study of cluster theory, $t$-structures and stability conditions. If we consider the collection of all torsion pairs ordered by inclusion of the torsion class, then we obtain a complete lattice. In recent work by Asai and Pfeiffer, the so-called `wide' intervals in this lattice are studied from the perspective of lattice theory. These intervals arise naturally both in relation to stability conditions and also in $\tau$-tilting theory. In this talk we will discuss how the torsion pairs in $\textrm{mod} A$ are parametrised by 2-term cosilting complexes in the unbounded derived category $D(\textrm{Mod} A)$ and how wide intervals correspond to their mutations. This talk is based on ongoing joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitória. Show abstract |
19 November, 2021 |
Alex Sierra CárdenasUniversidade Federal do ParáOn Brauer configurations induced by finite groupsAspects of the representation theory of a Brauer configuration algebra, such as the Cartan matrix and the module length of the respective indecomposable modules, have shown the existence of combinatorial relations that are satisfied by any finite group. The aim of this talk is to present both the Cartan matrix of a Brauer configuration algebra and the module length of any indecomposable projective module associated to a Brauer configuration algebra. Then using these aspects and the concept of subgroup-occurrence of an element in a group, we demonstrate a couple of combinatorial relations satisfied by any finite group, when considering the Brauer configuration induced by a finite group of an order different from a prime number. Show abstract |
12 November, 2021 |
Marco A. PérezInstituto de Matemática y Estadística "Prof. Ing. Rafael Laguardia"Relative strongly Gorenstein objects in abelian categoriesThe purpose of this talk is to present a relativization of the concept of strongly Gorenstein projective module, introduced by Bennis and Mahdou in 2007. Given a pair $(\mathcal{A,B})$ of full subcategories $\mathcal{A}$ and $\mathcal{B}$ of an abelian category $\mathcal{C}$, we say that an object $C \in \mathcal{C}$ is periodic (or strongly) $(\mathcal{A,B})$-Gorenstein projective if there exists a short exact sequence $C \rightarrowtail A \twoheadrightarrow C$, with $A \in \mathcal{A}$, which remains exact after applying the contravariant functor $\text{Hom}(-,B)$ for every $B \in \mathcal{B}$. We shall show several properties, characterizations and examples of these objects in the case where $(\mathcal{A,B})$ is a hereditary pair (meaning that $\textrm{Ext}^i(A,B) = 0$ for every $A \in \mathcal{A}$, $B \in \mathcal{B}$ and $i \geq 1$). For instance, we shall see a relation with relative Gorenstein objects (2021 - Becerril, Mendoza and Santiago) and periodic objects (2020 - Bazzoni, Cortés-Izurdiaga and Estrada). Concerning specific examples, we shall exhibit a characterization of noetherian rings in terms of periodic Gorenstein injective modules relative to the pair (absolutely pure modules, injective modules). If time allows, we shall comment on a more general version of periodic $(\mathcal{A,B})$-Gorenstein projective objects, namely, $(\mathcal{A,B})$-Gorenstein projective objects with period $m \geq 1$. For these families of objects, an object is $(\mathcal{A,B})$-Gorenstein projective with periods $m$ and $n$ if,
and only if, it is $(\mathcal{A,B})$-Gorenstein projective with period $\mathrm{gcd}(m,n)$. |
05 November, 2021 |
Hipolito TreffingerUniversität Bonn$\tau$-tilting theory and stratifying systemsOn the one hand, tau-tilting theory was introduced by Adachi, Iyama and Reiten in 2012 and quickly became central in representation theory of finite dimensional algebras. In this talk, after giving a brief overview of some central results in this theory, we will explain how every tau-rigid object in a module category induces at least one stratifying system, as defined by Erdmann and Sáenz. Time permitting, we will use these results to give a new proof of the fact that every complete stratifying system in the module category of a hereditary algebra is an exceptional sequence, first proven by Cadavid and Marcos. Show abstract |
29 October, 2021 |
Luis Gabriel Rodríguez ValdésUniversidad Nacional Autónoma de MéxicoHomological theory of k-idempotent ideals in dualizing varietiesIn this talk, we develop the theory of $k$-idempotent ideals in the setting of dualizing varieties. Several results given previously by M. Auslander, M. I. Platzeck, and G. Todorov are extended to this context. Given an ideal $I$ (which is the trace of a projective module), we construct a canonical recollement which is the analogous to a well-known recollement in categories of modules over artin algebras. Moreover, we study the homological properties of the categories involved in such a recollement. Consequently, we find conditions on the ideal $\mathcal{I}$ to obtain quasi-hereditary algebras in such a recollement. Applications to bounded derived categories are also given. (Joint work with V. Santiago Vargas and M. L. S. Sandoval). Show abstract |
22 October, 2021 |
Valente SantiagoUniversidad Nacional Autónoma de MéxicoTriangular matriz categories and recollementsWe define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M \in$ $\operatorname{Mod}\left(\mathcal{U} \otimes \mathcal{T}^{o p}\right)$ we will construct the triangular matrix category $\boldsymbol{\Lambda}:=\left[\begin{array}{ll}\mathcal{T} & 0 \\ M & \mathcal{U} \end{array}\right]$ and we prove that there is an equivalence $(\operatorname{Mod}(\mathcal{T}), \mathbb{G M o d}(\mathcal{U})) \simeq \operatorname{Mod}(\boldsymbol{\Lambda})$. We will show that if $\mathcal{U}$ and $\mathcal{T}$ are dualizing $K$-varieties and $M \in \operatorname{Mod}\left(\mathcal{U} \otimes \mathcal{T}^{o p}\right)$ satisfies certain conditions then $\boldsymbol{\Lambda}:=\left[\begin{array}{ll}\mathcal{T} & 0 \\ M& \mathcal{U}\end{array}\right]$ is a dualizing variety. Finally, we will show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in [Q. Chen, M. Zheng. Recollements of abelian categories and special types of comma categories. J. Algebra. 321 (9), 2474-2485 (2009), Theorem 4.4]. Show abstract |
15 October, 2021 |
Oliver LorscheidUniversity of GroningenRepresentation type via quiver GrassmanniansThe representation type of a quiver $Q$ can be characterized by the geometric properties of the associated quiver Grassmannians:
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08 October, 2021 |
Jeremy RickardUniversity of BristolGeneration of the unbounded derived category of a ringIf $R$ is a ring, recall that the derived category $D(R)$ is a category whose objects are chain complexes of $R$-modules. In recent decades, $D(R)$ has been recognized to be of great importance in studying the homological algebra of R.
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01 October, 2021 |
Robinson-Julian SernaUPTC, ColombiaA link between representations of peak posets and cluster algebrasIn this talk, we show a link between the theory of cluster algebras and the theory of representations of posets. Particularly, we give a geometric interpretation of the category of finitely generated socle-projective representations over the incidence algebra of a poset of type $A$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the realization of the cluster category of type $A$ as the category of all diagonals by Caldero-Chapoton-Schiffler. Moreover, given a poset $P$ of type $A$, we define a subalgebra $B(P)$ of a cluster algebra $B$ and we establish a sufficient condition to conclude $B=B(P)$. This is a joint work with R. Schiffler. Show abstract |
24 September, 2021 |
Fernando dos Reis NavesUFMGUm relacionamento funtorial entre álgebras e seus quocientes pelas $n$-ésimas potências do radical de JacobsonA famosa construção de Gabriel provou que uma álgebra de dimensão finita pode ser vista como um quociente de uma álgebra de caminhos por um ideal admissível. A construção permitiu olhar para teoria de representação de álgebras de dimensão finita de um modo combinatório. Iusenko, MacQuarrie e Quirino exibiram uma adjunção que explica de maneira muito precisa a filosofia da construção de Gabriel. A adjunção deles é obtida utilizando uma relação de equivalência nos morfismos das álgebras. Nesta palestra, mostraremos que a construção de Gabriel pode ser obtida utilizando uma relação mais fina. Além disso, através de uma adjunção, provaremos que há uma classe de álgebras que podem ser aproximadas por álgebras quadráticas, isto é, um quociente de uma álgebra de caminhos por um ideal homogêneo de grau 2. Show abstract |
17 September, 2021 |
Andrea SolotarUniversidad de Buenos AiresInvariants of gentle algebrasGentle and locally gentle algebras are quadratic Koszul algebras. The isomorphism classes of gentle
algebras are in bijection with dissections of certain types of surfaces. |
10 September, 2021 |
Claudiano Henrique da Cunha MeloUFMGÁlgebras seriais sem ciclos truncadas são derivadamente equivalentes a álgebras de incidência de posetsDada uma álgebra serial sem ciclos truncada $\Lambda$, denotamos sua categoria derivada por $\mathcal{D}^b(\Lambda).$ Nesta palestra apresentaremos a construção de um complexo inclinante $T_{\Lambda}$ e definiremos um conjunto parcialmente ordenado $S_{\Lambda}$ tais que a álgebra de endomorfismos $End_{\mathcal{D}^b(\Lambda)} T_{\Lambda}$ é isomorfa à álgebra de incidência de poset $kS_{\Lambda}^{op}$. Com isso, usando o Teorema de Rickard podemos concluir que $\Lambda$ é derivadamente equivalente a $kS_{\Lambda}^{op}$. Show abstract |
03 September, 2021 |
Edson Ribeiro AlvaresUniversidade Federal do ParanáAção do grupo de tranças em sequências excepcionais e aplicações para álgebras hereditárias por partesNesta palestra, vamos introduzir o conceito de mutação de sequências excepcionais em categorias de feixes coerentes. Estas mutações permitem definir a ação do grupo de tranças sobre sequências excepcionais. Com o uso eficiente do Teorema de Riemann-Roch e a Teoria de Auslander-Reiten, podemos tirar bons resultados sobre as órbitas desta ação. Estes resultados nos trazem um melhor entendimento sobre as álgebras que são derivadamente equivalente a categoria de feixes coerentes sobre retas projetivas com peso. Show abstract |
30 July, 2021 |
Gustavo MataUniversidad de la República, UruguayTopics on the Igusa-Todorov functionsOne of the most important conjecture in the representation theory of Artin algebras is the finitistic dimension conjecture. It states that $sup\{pd(M) : M\ \text{is a f.g module of finite projective dimension}\}$ is finite. In an attempt to prove the conjecture this conjecture Igusa and Todorov defined in “On finitistic global dimension conjecture for artin algebras”, two functions from the objects of modA to the natural numbers, which generalize the notion of projective dimension. Nowadays they are known as the Igusa-Todorov functions. In this talk I will introduce the Igusa-Todorov functions and their main properties. We will also see how to use these functions to prove the finitistic dimension conjecture in several contexts. Show abstract |
23 July, 2021 |
Danilo Dias da SilvaUniversidade Federal de SergipeFeixes instantons e representações de aljavasO seminário terá como tema apresentar uma nova compactificação para o espaço de módulos (moduli spaces) de feixes intantons de carga $1$ sobre $\mathbb P^3$. Isto será feito identificando estes feixes com representações de uma aljava com três vértices e oito flechas que atendem certas propriedades e olhando essas representações no espaço de módulos de King de representações teta-estáveis sobre esta aljava. Show abstract |
16 July, 2021 |
Mariano Suárez-ÁlvarezUniversidad de Buenos AiresOn the modular automorphism of twisted Calabi-Yau algebrasA twisted Calabi-Yau algebra $A$ is, by definition, endowed with an automorphism $\sigma:A\to A$, well-defined only up to inner automorphisms, that controls the duality theory for representations of $A$. The algebra is Calabi-Yau, and not only twistedly so, precisely when $\sigma$ can be taken to be inner and even the identity map of $A$, but in general it cannot. The objective of the talk will be to present some results that give information about this automorphism in the general situation, and show how it can be used to construct interesting and non-trivial invariants of the algebra, of automorphisms and derivations of the algebra, and of related objects. Some applications of these invariants to the problem of computing Hochschild cohomology and the automorphism groups of our algebras will be discussed. Show abstract |
25 June, 2021 |
Eduardo MarcosIME-USPVariety defined by the Groebner basis of a moduleThis is a joint project with Ed. Green and Schroll Sibylle. |
18 June, 2021 |
Ricardo FranquizUFMGÁlgebra de Árvore de Brauer e Blocos de Grupos ProfinitosSejam $G$ um grupo profinito , $k$ um corpo de característica $p> 0$. A álgebra de grupo completa de G, $k[[G]]$, possui uma decomposição como um produto direto de álgebras pseudocompactas indecomponíveis, que chamaremos de blocos. Cada bloco de $k[[G]]$ tem associado um pro-$p$ subgrupo de $G$ chamado grupo de defeito. O grupo de defeito proporciona uma medida que tão próximo estará um bloco de ser uma álgebra semisimples. Nesta palestra explicaremos como associar a cada bloco um grupo de defeito e descreveremos os blocos cujo grupo de defeito é um grupo cíclico usando a estrutura de álgebra de árvore de Brauer.
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11 June, 2021 |
Joseph GrantUniversity of East AngliaPreprojective algebras: classical and higherI will give an introduction to representations of quivers, aimed at non-experts, and then introduce preprojective algebras of quivers. Iyama showed how many nice aspects of the representations of quivers generalise to certain algebras of higher global dimension, and the preprojective algebra can be defined in this setting. I will explain some results on higher preprojective algebras obtained in joint work with Osamu Iyama. Show abstract |
04 June, 2021 |
Valeriano LanzaUFFModuli of flags of sheaves: a quiver descriptionIn [1], the moduli spaces of framed flags of sheaves on $\mathbb P^2$ were described by means of representations of the so-called enhanced ADHM quiver. We first review those results, with a recent refinement concerning the chamber structure in the space of stability parameters. We shall then discuss the obstruction theory for these moduli spaces, showing in general that they have a perfect obstruction theory, and providing for specific choices of invariants a class of unobstructed points. Finally, open problems and possible further developments will be presented. This is a joint work with Rodrigo von Flach and Marcos Jardim.
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14 May, 2021 |
Vitor GuliszUFPRHigher Auslander-Reiten theory: what is it and how to understand itHigher Auslander-Reiten theory was introduced by Osamu Iyama in the 2000s as a generalization of the theory developed by Maurice Auslander and Idun Reiten in the 1970s. In this talk, we will discuss what is this theory and how to understand it. In doing so, we will examine both the definitions of $n$-almost split sequence and of $n$-cluster tilting subcategory. In the end, we will also look at some open problems of the theory. Show abstract |
07 May, 2021 |
John MacQuarrieUFMGPropriedades homológicas de álgebras preservadas por certas extensõesUma extensão de álgebras associativas é simplesmente uma álgebra associativa $A$ com subálgebra $B$. Em uma sequência de artigos recentes, Cibils, Lanzilotta, Marcos e Solotar colocaram condições sobre a extensão (extensões "limitadas") e mostraram que, sobre essas condições, duas propriedades homológicas valem para $B$ se, e somente se, valem para $A$: nomeadamente "dimensão global finita" e "suporte da homologia de Hochschild finita". Em particular, segue disso que a "Conjectura de Han" vale para $B$ se, e somente se, vale para $A$.
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