scholarly journals SPECIAL TILTING MODULES FOR ALGEBRAS WITH POSITIVE DOMINANT DIMENSION

2020 ◽  
pp. 1-27
Author(s):  
MATTHEW PRESSLAND ◽  
JULIA SAUTER
Keyword(s):  
Global Dimension ◽  
Point Of View ◽  
Tilting Modules ◽  
Endomorphism Algebra ◽  
Dominant Dimension ◽  
Gorenstein Algebras ◽  
Endomorphism Algebras ◽  
Cotilting Modules

Abstract We 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 minimal d-Auslander–Gorenstein algebras and d-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.

scholarly journals Tilting modules and dominant dimension with respect to injective modules

2020 ◽  
Author(s):  
Takahide Adachi ◽  
Mayu Tsukamoto
Keyword(s):  
Projective Dimension ◽  
Sufficient Condition ◽  
Tilting Modules ◽  
Dominant Dimension ◽  
Gorenstein Algebras ◽  
Injective Modules ◽  
Finite Projective Dimension

Abstract In this paper, we study a relationship between tilting modules with finite projective dimension and dominant dimension with respect to injective modules as a generalization of results of Crawley-Boevey–Sauter, Nguyen–Reiten–Todorov–Zhu and Pressland–Sauter. Moreover, we give characterizations of almost n-Auslander–Gorenstein algebras and almost n-Auslander algebras by the existence of tilting modules. As an application, we describe a sufficient condition for almost 1-Auslander algebras to be strongly quasi-hereditary by comparing such tilting modules and characteristic tilting modules.

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

2021 ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Global Dimension ◽  
Group Algebras ◽  
Dominant Dimension ◽  
Schur Algebras ◽  
Homological Dimensions ◽  
Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

scholarly journals PROPERLY STRATIFIED ALGEBRAS AND TILTING

2005 ◽  
Vol 92 (1) ◽  
pp. 29-61 ◽  
Author(s):  
ANDERS FRISK ◽  
VOLODYMYR MAZORCHUK
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Category Of Modules ◽  
Cotilting Modules ◽  
Contravariantly Finite ◽  
Properly Stratified Algebras ◽  
Stratified Algebra ◽  
Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

scholarly journals Arte interativa: quem é o autor e onde está o espectador?

2020 ◽  
Author(s):  
António Cardoso
Keyword(s):  
Linear Structure ◽  
Global Dimension ◽  
Point Of View ◽  
Interactive Art ◽  
Traditional Role ◽  
External Observer ◽  
Theatrical Performance ◽  
Symbolic Meanings ◽  
The Spectator

This article examines how the emergence of interactive art brought about the necessity of re-positioning the traditional role of the spectator and led to the re-formulation of the notion of authorship. The non-linear structure of interactive artworks empowers the spectator with the (apparent) control of its narrative and creates a relational space filled with an input-output dynamic. It is suggested that this elevates the spectator to the realm of co-authorship of the work of art that he/she is interacting with, because the specificity of each interaction generates symbolic meanings that the original author cannot anticipate.It is also argued that the input required from the spectator to interact with the artwork could be compared to a theatrical performance. Thus, an analogy between the spectator-in-action of an interactive artwork and the figure of an actor is established. This brings us to the question concluding the article: as the reception of interactive art implies an action from its spectator, the compatibility between action and contemplation is questioned.Finally, the article concludes that the corpus of an interactive artwork has to include the spectator that acts and creates the input needed for the interaction to be established. Therefore, only from the point of view of an external observer one can gain access to its global dimension, that is, as meta-spectators.

Karl Haushofer as a “Pioneer” of National Socialist Cultural Diplomacy in Fascist Italy

2019 ◽  
Vol 52 (03) ◽  
pp. 424-449
Author(s):  
Nicola Bassoni
Keyword(s):  
Political System ◽  
Cultural Diplomacy ◽  
Global Dimension ◽  
Point Of View ◽  
National Socialist ◽  
Political Activities ◽  
History Of ◽  
Political Model

AbstractThe relevant historiography has largely overlooked the role of Karl Haushofer as a cultural-political actor in National Socialist-Fascist relations. From 1924 to 1944, the German geopolitician dealt extensively with Italy, with an eye to both its geopolitical role in Europe and to the political system of Benito Mussolini's regime. On behalf of Rudolf Hess, he began visiting Italy during the 1930s, aiming to overcome ideological and political misunderstandings between Rome and Berlin. He established a network of contacts with Italian scholars and politicians, passed information back to the so-called deputy Führer, and attempted to influence official German policy toward Italy. He eventually promoted the development of an Italian geopolitics, and, in so doing, achieved one of the most significant cultural-political transfers from National Socialist Germany to fascist Italy. This article analyzes the contacts between Haushofer and Italy, both his political activities and his geopolitical theories. It is a case study of a history of contradictions: a man committed to Pan-Germanist culture and to the defense of German minorities abroad, Haushofer also attempted to improve relations between Berlin and Rome. Moreover, he considered the Axis from a geopolitical point of view—as a realization of the European imperial idea—and from a trilateral perspective, i.e., he viewed Japan not only as an ally, but also as a cultural and political model. The reconstruction of Haushofer's relations with Italy is, therefore, an opportunity to rethink the antinomies, as well as the global dimension, of the National Socialist-Fascist alliance.

scholarly journals TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

2013 ◽  
Vol 94 (1) ◽  
pp. 133-144
Author(s):  
ZHAOYONG HUANG ◽  
XIAOJIN ZHANG
Keyword(s):  
Projective Dimension ◽  
Indecomposable Module ◽  
Global Dimension ◽  
Injective Dimension ◽  
Tilted Algebra ◽  
Gorenstein Algebras

AbstractLet $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

Rigidity dimension of algebras

2019 ◽  
pp. 1-27
Author(s):  
HONGXING CHEN ◽  
MING FANG ◽  
OTTO KERNER ◽  
STEFFEN KOENIG ◽  
KUNIO YAMAGATA
Keyword(s):  
Hochschild Cohomology ◽  
Global Dimension ◽  
Homological Dimension ◽  
Dominant Dimension ◽  
Finite Global Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type

Abstract A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

scholarly journals The dominant dimension of cohomological Mackey functors

2018 ◽  
Vol 17 (12) ◽  
pp. 1850228
Author(s):  
Markus Linckelmann
Keyword(s):  
Finite Group ◽  
Defect Group ◽  
Dominant Dimension ◽  
Text Block ◽  
Symmetric Algebras ◽  
Endomorphism Algebras ◽  
Mackey Functors ◽  
A Finite Group

We show that a separable equivalence between symmetric algebras preserves the dominant dimensions of certain endomorphism algebras of modules. We apply this to show that the dominant dimension of the category [Formula: see text] of cohomological Mackey functors of a [Formula: see text]-block [Formula: see text] of a finite group with a nontrivial defect group is [Formula: see text].

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