Recollements induced from tilting modules over tame hereditary algebras

Abstract. We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form where is a union of tubes, and denotes the universal localization of R at in the sense of Schofield and Crawley-Boevey. Here is a direct sum of the Prüfer modules corresponding to the tubes in . Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class consists of all modules without indecomposable preprojective summands. Over an arbitrary tame hereditary algebra, T can have finite dimensional summands, but the infinite dimensional part of T is still built up from universal localizations, Prüfer modules and (localizations of) the Lukas tilting module. We also recover the classification of the infinite dimensional cotilting R-modules due to Buan and Krause.

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Separating Splitting Tilting Modules and Hereditary Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1987-025-3 ◽

1987 ◽

Vol 30 (2) ◽

pp. 177-181 ◽

Cited By ~ 3

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.

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Radically filtered quasi-hereditary algebras and rigidity of tilting modules

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116001006 ◽

2017 ◽

Vol 163 (2) ◽

pp. 265-288

Author(s):

AMIT HAZI

Keyword(s):

Closed Field ◽

Tilting Module ◽

Tilting Modules ◽

Hereditary Algebra ◽

Algebraically Closed Field ◽

Hereditary Algebras ◽

Composition Factors

AbstractLetAbe a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to show that the restricted tilting modules forSL4(K) are rigid, whereKis an algebraically closed field of characteristicp≥ 5.

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CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498808002874 ◽

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].

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PROPERLY STRATIFIED ALGEBRAS AND TILTING

Proceedings of the London Mathematical Society ◽

10.1017/s0024611505015431 ◽

2005 ◽

Vol 92 (1) ◽

pp. 29-61 ◽

Cited By ~ 8

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.

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Wakamatsu tilting modules over ring extension

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501986 ◽

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.

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