Infinitely generated complements to partial tilting modules

2002 ◽  
Vol 132 (1) ◽  
pp. 89-96 ◽  
Author(s):  
LIDIA ANGELERI-HÜGEL ◽  
FLÁVIO ULHOA COELHO
Keyword(s):  
Finitely Generated ◽  
Tilting Module ◽  
Tilting Modules ◽  
Artin Algebra ◽  
Arbitrary Ring

We study the existence of complements to a partial tilting module over an arbitrary ring. As a consequence, we show that a finitely generated partial tilting module over an artin algebra has a (possibly infinitely generated) complement.

Separating Splitting Tilting Modules and Hereditary Algebras

1987 ◽  
Vol 30 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Exact Sequence ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Hereditary Algebras

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

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.

Wakamatsu tilting modules over ring extension

2019 ◽  
Vol 18 (10) ◽  
pp. 1950198
Author(s):  
Zhen Zhang ◽  
Jiaqun Wei
Keyword(s):  
Ring Extension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Wakamatsu Tilting Module ◽  
Extension Ring

For a ring [Formula: see text], an extension ring [Formula: see text], and a fixed right [Formula: see text]-module [Formula: see text], we prove the induced left [Formula: see text]-module [Formula: see text] is a Wakamatsu tilting module when [Formula: see text] is a Wakamatsu tilting module.

TILTING MODULES OF FINITE PROJECTIVE DIMENSION: SEQUENTIALLY STATIC AND COSTATIC MODULES

2002 ◽  
Vol 01 (03) ◽  
pp. 295-305 ◽  
Author(s):  
ALBERTO TONOLO
Keyword(s):  
Projective Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Finite Projective Dimension

In [5], Miyashita introduced tilting modules of finite projective dimension. A tilting module AV of projective dimension less or equal than r furnishes r + 1 equivalences between subcategories of A-Mod and End V-Mod: we call static and costatic the modules in A-Mod and End V-Mod, respectively, involved in these equivalences. In this paper we characterize the modules in A-Mod and End V-Mod which have a filtration with static and costatic factors, respectively.

scholarly journals From Jantzen to Andersen filtration via tilting equivalence

2012 ◽  
Vol 110 (2) ◽  
pp. 161 ◽  
Author(s):  
Johannes Kübel
Keyword(s):  
Verma Module ◽  
The Other ◽  
Tilting Module ◽  
Tilting Modules ◽  
Projective Object ◽  
Other Hand

The space of homomorphisms between a projective object and a Verma module in category $\mathcal O$ inherits an induced filtration from the Jantzen filtration on the Verma module. On the other hand there is the Andersen filtration on the space of homomorphisms between a Verma module and a tilting module. Arkhipov's tilting functor, a contravariant self-equivalence of a certain subcategory of $\mathcal O$, which maps projective to tilting modules induces an isomorphism of these kinds of Hom-spaces. We show that this equivalence identifies both filtrations.

On Algebras Stably Equivalent to an Hereditary Artin Algebra

1978 ◽  
Vol 30 (4) ◽  
pp. 817-829 ◽  
Author(s):  
María Inés Platzeck
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finitely Generated Module ◽  
Artin Ring ◽  
Finitely Generated Modules ◽  
Hereditary Artin Algebra

Let Λ be an artin algebra, that is, an artin ring that is a finitely generated module over its center C which is also an artin ring. We denote by mod Λ the category of finitely generated left Λ-modules. We recall that the category of finitely generated modules modulo projectives is the category given by the following data: the objects are the finitely generated Λ-modules.

Simple reflexive modules over Artin algebras

2019 ◽  
Vol 18 (10) ◽  
pp. 1950193
Author(s):  
René Marczinzik
Keyword(s):  
Simple Module ◽  
Positive Answer ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Large Classes ◽  
Gorenstein Algebras ◽  
Artin Algebras ◽  
Reflexive Modules

Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.

scholarly journals Finite Generation of the Extension Algebra Ext(M, M)

1995 ◽  
Vol 59 (3) ◽  
pp. 366-374
Author(s):  
Rainer Schulz
Keyword(s):  
Finitely Generated ◽  
Artin Algebra ◽  
Finite Generation ◽  
Extension Algebra

AbstractFor a module M Over an Artin algebra R, we discuss the question of whether the Yoneda extension algebra Ext(M, M) is finitely generated as an algebra. We give an answer for bounded modules M. (These are modules whose syzygies have direct summands of bounded lengths.)

Wide subcategories of finitely generated Λ-modules

2018 ◽  
Vol 17 (05) ◽  
pp. 1850082 ◽  
Author(s):  
Eduardo N. Marcos ◽  
Octavio Mendoza ◽  
Corina Sáenz ◽  
Valente Santiago
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Artin Algebra ◽  
Necessary And Sufficient

We explore some properties of wide subcategories of the category [Formula: see text] of finitely generated left [Formula: see text]-modules, for some artin algebra [Formula: see text] In particular we look at wide finitely generated subcategories and give a connection with the class of standard modules and standardly stratified algebras. Furthermore, for a wide class [Formula: see text] in [Formula: see text] we give necessary and sufficient conditions to see that [Formula: see text] for some projective [Formula: see text]-module [Formula: see text] and finally, a connection with ring epimorphisms is given.

Sign in / Sign up

Export Citation Format

Share Document