Cotorsion modules over tame finite-dimensional hereditary algebras

1981 ◽  
pp. 263-269 ◽  
Author(s):  
Frank Okoh
Keyword(s):  
Finite Dimensional ◽  
Hereditary Algebras

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

1996 ◽  
Vol 39 (1) ◽  
pp. 111-114
Author(s):  
F. Okoh
Keyword(s):  
Rational Functions ◽  
Indecomposable Module ◽  
Torsion Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
The Status ◽  
Hereditary Algebras ◽  
Divisible Modules ◽  
Torsion Free

AbstractIf R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

scholarly journals A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

2020 ◽  
pp. 1-36
Author(s):  
O. MENDOZA ◽  
M. ORTÍZ ◽  
C. SÁENZ ◽  
V. SANTIAGO
Keyword(s):  
Classical Theory ◽  
General Class ◽  
General Setting ◽  
Finite Dimensional ◽  
Classical Notion ◽  
Hereditary Algebras

Abstract We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

scholarly journals Tilting modules over tame hereditary algebras

2012 ◽  
Vol 2013 (682) ◽  
pp. 1-48
Author(s):  
Lidia Angeleri Hügel ◽  
Javier Sánchez
Keyword(s):  
Complete Classification ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Infinite Dimensional ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Tame Hereditary Algebra

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.

scholarly journals Parametrizing torsion pairs in derived categories

2021 ◽  
Vol 25 (23) ◽  
pp. 679-731
Author(s):  
Lidia Angeleri Hügel ◽  
Michal Hrbek
Keyword(s):  
Natural Extension ◽  
Commutative Rings ◽  
Derived Categories ◽  
Derived Category ◽  
Content Type ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Zariski Spectrum ◽  
Compactly Generated

We investigate parametrizations of compactly generated t-structures, or more generally, t-structures with a definable coaisle, in the unbounded derived category D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) of a ring A A . To this end, we provide a construction of t-structures from chains in the lattice of ring epimorphisms starting in A A , which is a natural extension of the construction of compactly generated t-structures from chains of subsets of the Zariski spectrum known for the commutative noetherian case. We also provide constructions of silting and cosilting objects in D ( M o d - A ) \mathrm {D}({\mathrm {Mod}}\text {-}A) . This leads us to classification results over some classes of commutative rings and over finite dimensional hereditary algebras.

scholarly journals PURITY RELATIVE TO CLASSES OF FINITELY PRESENTED MODULES

2013 ◽  
Vol 12 (08) ◽  
pp. 1350050 ◽  
Author(s):  
AKEEL RAMADAN MEHDI
Keyword(s):  
General Description ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Left And Right ◽  
Finitely Presented ◽  
Tame Hereditary Algebras ◽  
Finite Dimensional Algebras

We investigate purities determined by classes of finitely presented modules including the correspondence between purities for left and right modules. We show some cases where purities determined by matrices of given sizes are different. Then we consider purities over finite-dimensional algebras, giving a general description of the relative pure-injectives which we make completely explicit in the case of tame hereditary algebras.

scholarly journals Torsion Pairs and Ringel Duality for Schur Algebras

2021 ◽  
Author(s):  
Karin Erdmann ◽  
Stacey Law
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Schur Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Functorial Approach

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

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.

THE AUSLANDER–REITEN SEQUENCES ENDING AT GABRIEL–ROITER FACTOR MODULES OVER TAME HEREDITARY ALGEBRAS

2007 ◽  
Vol 06 (06) ◽  
pp. 951-963 ◽  
Author(s):  
BO CHEN
Keyword(s):  
Closed Field ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Type Formula ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Almost All ◽  
Euclidean Type ◽  
Tame Hereditary Algebras

Let Λ = kQ be a finite dimensional hereditary algebra over an algebraically closed field k with Q a quiver of Euclidean type [Formula: see text], [Formula: see text], or [Formula: see text]. We study the Auslander–Reiten sequences terminating at Gabriel–Roiter factor modules and show that for almost all but finitely many Gabriel–Roiter factor modules, the Auslander–Reiten sequences have indecomposable middle terms.

scholarly journals FINITISTIC DIMENSIONS AND PIECEWISE HEREDITARY PROPERTY OF SKEW GROUP ALGEBRAS

2014 ◽  
Vol 57 (3) ◽  
pp. 509-517 ◽  
Author(s):  
LIPING LI
Keyword(s):  
Global Dimension ◽  
Hereditary Property ◽  
Finite Dimensional ◽  
Homological Characterization ◽  
Complete Set ◽  
Hereditary Algebras ◽  
A Finite Group ◽  
Skew Group Algebras ◽  
Finitistic Dimensions

AbstractLet Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.

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