Auslander-Reiten-quivers of functorially finite subcategories
|
|
- Preston Lynch
- 5 years ago
- Views:
Transcription
1 Auslander-Reiten-quivers of functorially finite subcategories Matthias Krebs University of East Anglia August 13, 2012
2 Preliminaries
3 Preliminaries Let K be an algebraically closed field,
4 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity,
5 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity, A-mod the category of all finitely generated left A-modules,
6 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity, A-mod the category of all finitely generated left A-modules, Ω a full subcategory of A-mod closed under direct sums, direct summands and isomorphisms.
7 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y
8 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0
9 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0 Then X τ(y ) = τ Ω (Y ) I such that I is an Ext-injective module and τ Ω (Y ) is indecomposable, but not Ext-injective.
10 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0 Then X τ(y ) = τ Ω (Y ) I such that I is an Ext-injective module and τ Ω (Y ) is indecomposable, but not Ext-injective. Definition τ Ω (Y ) is the relative Auslander-Reiten translate of Y in Ω.
11 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation,
12 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation,
13 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation, functorially finite, if it is both contravariantly and covariantly finite.
14 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation, functorially finite, if it is both contravariantly and covariantly finite. If Ω is contravariantly finite and closed under extensions, then Ω has an Auslander-Reiten-quiver.
15 Resolving subcategories Definition Ω is called resolving if
16 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions,
17 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms,
18 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω.
19 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω. If T is a generalized cotilting module, then Ext i A (, T ) = 0 is a contravariantly finite resolving subcategory.
20 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω. If T is a generalized cotilting module, then Ext i A (, T ) = 0 is a contravariantly finite resolving subcategory. Let Ω be a functorially finite resolving subcategory.
21 Brauer-Thrall conjectures
22 Brauer-Thrall conjectures Brauer-Thrall 1 Ω is representation finite if and only if the Jordan-Hölder length of its indecomposable modules is bounded.
23 Brauer-Thrall conjectures Brauer-Thrall 1 Ω is representation finite if and only if the Jordan-Hölder length of its indecomposable modules is bounded. Brauer-Thrall 1.5 Let Ω be a functorially finite resolving subcategory such that there exist a positive integer n such that there are N N 0 non-isomorphic indecomposable modules of Jordan-Hölder length n in Ω, where N 0 is the cardinality of an infinite, countable set. Then there are infinitely many positive integers n 1, n 2,... with N non-isomorphic indecomposable modules of length n i in Ω for all i N.
24 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2.
25 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]
26 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]
27 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]
28 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω.
29 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω. Moreover, the cosets of all sectional paths from X 0 to X n in A-mod such that all modules along these paths other than X 0 and X n are not in Ω are linearly independent in rad Ω (X 0, X n )/rad 2 Ω (X 0, X n ).
30 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω. Moreover, the cosets of all sectional paths from X 0 to X n in A-mod such that all modules along these paths other than X 0 and X n are not in Ω are linearly independent in rad Ω (X 0, X n )/rad 2 Ω (X 0, X n ). Does the converse hold as well, i.e is a morphism in Ω given by a non-sectional path in A-mod reducible?
31 Sectional paths X n X 0
32 Sectional paths X n X 0
33 Sectional paths Y 1 Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 X n Y 2 Y 3 X 0 Y 4
34 Sectional paths Y 1 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Y 2 Y 3 X 0 Y 4
35 Sectional paths Y 1 Y 2 Y 3 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Every morphism g : X 0 X n factors through f X 0 X 0 g f Y X n Y 4
36 Sectional paths Y 1 Y 2 Y 3 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Every morphism g : X 0 X n factors through f X 0 X 0 g f Y X n There are no irreducible morphisms from X 0 to X n Y 4
37 Sectional paths X n X 0
38 Sectional paths Y 1 Let Y = Y 1 Y 2 Y 3 X n X 0 Y 2 Z Y 3
39 Sectional paths X 0 Y 2 Y 1 X n Let Y = Y 1 Y 2 Y 3 There is a commutative diagram 0 Z Y X n 0 f Z 0 X Z f Y X Y X n 0 Z Y 3
40 Sectional paths X 0 Z Y 2 Y 3 Y 1 X n Let Y = Y 1 Y 2 Y 3 There is a commutative diagram 0 Z Y X n 0 f Z f Y 0 X Z X Y X n 0 If X Z contains only one direct summand that is not Ext-injective, then there are no Ω-irreducible morphisms from X 0 to X n.
41 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ]
42 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.
43 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.
44 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.
45 A counterexample [P 1 ] [P 2 I 2 ] [P 3 S 2 I 1 ] [P 4 [P 5
46 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω.
47 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω. Definition A sectional subgraph Σ is a connected subgraph of Γ Ω such that all subpaths in Σ are sectional. Σ is called full, if any connected subgraph Σ of Γ Ω such that Σ Σ is not a sectional subgraph. The undirected graph Σ associated to Σ is called the type of Σ.
48 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω. Definition A sectional subgraph Σ is a connected subgraph of Γ Ω such that all subpaths in Σ are sectional. Σ is called full, if any connected subgraph Σ of Γ Ω such that Σ Σ is not a sectional subgraph. The undirected graph Σ associated to Σ is called the type of Σ. For certain left or right stable components the type of a full sectional subgraph is independent from the choice of a path. In these components we define the left and right subgraph type of a component as the type of an arbitrary full sectional subgraph respectively.
49 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent:
50 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite.
51 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite. The left subgraph types of all connected components of Γ l are given by Dynkin diagrams.
52 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite. The left subgraph types of all connected components of Γ l are given by Dynkin diagrams. The right subgraph types of all connected components of Γ r are given by Dynkin diagrams.
Auslander-Reiten theory in functorially finite resolving subcategories
Auslander-Reiten theory in functorially finite resolving subcategories arxiv:1501.01328v1 [math.rt] 6 Jan 2015 A thesis submitted to the University of East Anglia in partial fulfilment of the requirements
More informationExamples of Semi-Invariants of Quivers
Examples of Semi-Invariants of Quivers June, 00 K is an algebraically closed field. Types of Quivers Quivers with finitely many isomorphism classes of indecomposable representations are of finite representation
More informationAUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS. Piotr Malicki
AUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS Piotr Malicki CIMPA, Mar del Plata, March 2016 3. Irreducible morphisms and almost split sequences A algebra, L, M, N modules in mod A A homomorphism
More informationRepresentation type and Auslander-Reiten theory of Frobenius-Lusztig kernels
Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels Julian Külshammer University of Kiel, Germany 08.2012 Notation Denote by: k an algebraically closed field (of characteristic
More informationMODULE CATEGORIES WITH INFINITE RADICAL SQUARE ZERO ARE OF FINITE TYPE
MODULE CATEGORIES WITH INFINITE RADICAL SQUARE ZERO ARE OF FINITE TYPE Flávio U. Coelho, Eduardo N. Marcos, Héctor A. Merklen Institute of Mathematics and Statistics, University of São Paulo C. P. 20570
More informationRELATIVE HOMOLOGY. M. Auslander Ø. Solberg
RELATIVE HOMOLOGY M. Auslander Ø. Solberg Department of Mathematics Institutt for matematikk og statistikk Brandeis University Universitetet i Trondheim, AVH Waltham, Mass. 02254 9110 N 7055 Dragvoll USA
More informationGorenstein algebras and algebras with dominant dimension at least 2.
Gorenstein algebras and algebras with dominant dimension at least 2. M. Auslander Ø. Solberg Department of Mathematics Brandeis University Waltham, Mass. 02254 9110 USA Institutt for matematikk og statistikk
More informationDedicated to Helmut Lenzing for his 60th birthday
C O L L O Q U I U M M A T H E M A T I C U M VOL. 8 999 NO. FULL EMBEDDINGS OF ALMOST SPLIT SEQUENCES OVER SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM A S S E M (SHERBROOKE, QUE.) AND DAN Z A C H A R I A (SYRACUSE,
More informationOn the Genus of the Graph of Tilting Modules
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 45 (004), No., 45-47. On the Genus of the Graph of Tilting Modules Dedicated to Idun Reiten on the occasion of her 60th birthday
More informationExtensions of covariantly finite subcategories
Arch. Math. 93 (2009), 29 35 c 2009 Birkhäuser Verlag Basel/Switzerland 0003-889X/09/010029-7 published online June 26, 2009 DOI 10.1007/s00013-009-0013-8 Archiv der Mathematik Extensions of covariantly
More informationHigher dimensional homological algebra
Higher dimensional homological algebra Peter Jørgensen Contents 1 Preface 3 2 Notation and Terminology 5 3 d-cluster tilting subcategories 6 4 Higher Auslander Reiten translations 10 5 d-abelian categories
More informationPowers of translation quivers. Coralie Ducrest Master in Mathematics
Powers of translation quivers Coralie Ducrest Master in Mathematics June 3, 008 Contents 1 Quiver representations 9 1.1 General principles........................ 9 1. Path algebras of finite representation
More informationOn Auslander Reiten components for quasitilted algebras
F U N D A M E N T A MATHEMATICAE 149 (1996) On Auslander Reiten components for quasitilted algebras by Flávio U. C o e l h o (São Paulo) and Andrzej S k o w r o ń s k i (Toruń) Abstract. An artin algebra
More informationA visual introduction to Tilting
A visual introduction to Tilting Jorge Vitória University of Verona http://profs.sci.univr.it/ jvitoria/ Padova, May 21, 2014 Jorge Vitória (University of Verona) A visual introduction to Tilting Padova,
More informationCONTRAVARIANTLY FINITE SUBCATEGORIES CLOSED UNDER PREDECESSORS
CONTRAVARIANTLY FINITE SUBCATEGORIES CLOSED UNDER PREDECESSORS IBRAHIM ASSEM, FLÁVIO U. COELHO, AND SONIA TREPODE Abstract. Let A be an artin algebra, moda the category of finitely generated right A-modules,
More informationGeneralized Matrix Artin Algebras. International Conference, Woods Hole, Ma. April 2011
Generalized Matrix Artin Algebras Edward L. Green Department of Mathematics Virginia Tech Blacksburg, VA, USA Chrysostomos Psaroudakis Department of Mathematics University of Ioannina Ioannina, Greece
More informationThe Diamond Category of a Locally Discrete Ordered Set.
The Diamond Category of a Locally Discrete Ordered Set Claus Michael Ringel Let k be a field Let I be a ordered set (what we call an ordered set is sometimes also said to be a totally ordered set or a
More informationCombinatorial Restrictions on the AR Quiver of a Triangulated Category
AR Quiver of a University of Minnesota November 22, 2014 Outline Definitions and This is work done with Marju Purin and Kos Diveris. C is a k = k-linear triangulated category which is Hom-finite, connected
More informationTRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS
J. Aust. Math. Soc. 94 (2013), 133 144 doi:10.1017/s1446788712000420 TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS ZHAOYONG HUANG and XIAOJIN ZHANG (Received 25 February
More informationRELATIVE THEORY IN SUBCATEGORIES. Introduction
RELATIVE THEORY IN SUBCATEGORIES SOUD KHALIFA MOHAMMED Abstract. We generalize the relative (co)tilting theory of Auslander- Solberg [9, 1] in the category mod Λ of finitely generated left modules over
More informationInfinite dimensional tilting theory
Infinite dimensional tilting theory Lidia Angeleri Hügel Abstract. Infinite dimensional tilting modules are abundant in representation theory. They occur when studying torsion pairs in module categories,
More information6. Dynkin quivers, Euclidean quivers, wild quivers.
6 Dynkin quivers, Euclidean quivers, wild quivers This last section is more sketchy, its aim is, on the one hand, to provide a short survey concerning the difference between the Dynkin quivers, the Euclidean
More informationCLUSTER-TILTED ALGEBRAS ARE GORENSTEIN AND STABLY CALABI-YAU
CLUSTER-TILTED ALGEBRAS ARE GORENSTEIN AND STABLY CALABI-YAU BERNHARD KELLER AND IDUN REITEN Abstract. We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein
More informationOn the Homology of the Ginzburg Algebra
On the Homology of the Ginzburg Algebra Stephen Hermes Brandeis University, Waltham, MA Maurice Auslander Distinguished Lectures and International Conference Woodshole, MA April 23, 2013 Stephen Hermes
More informationarxiv:math/ v2 [math.rt] 9 Feb 2004
CLUSTER-TILTED ALGEBRAS ASLAK BAKKE BUAN, ROBERT J. MARSH, AND IDUN REITEN arxiv:math/040075v [math.rt] 9 Feb 004 Abstract. We introduce a new class of algebras, which we call cluster-tilted. They are
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationGorenstein Homological Algebra of Artin Algebras. Xiao-Wu Chen
Gorenstein Homological Algebra of Artin Algebras Xiao-Wu Chen Department of Mathematics University of Science and Technology of China Hefei, 230026, People s Republic of China March 2010 Acknowledgements
More informationON SPLIT-BY-NILPOTENT EXTENSIONS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 98 2003 NO. 2 ON SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM ASSEM (Sherbrooke) and DAN ZACHARIA (Syracuse, NY) Dedicated to Raymundo Bautista and Roberto
More informationPartial orders related to the Hom-order and degenerations.
São Paulo Journal of Mathematical Sciences 4, 3 (2010), 473 478 Partial orders related to the Hom-order and degenerations. Nils Nornes Norwegian University of Science and Technology, Department of Mathematical
More informationLADDER FUNCTORS WITH AN APPLICATION TO REPRESENTATION-FINITE ARTINIAN RINGS
An. Şt. Univ. Ovidius Constanţa Vol. 9(1), 2001, 107 124 LADDER FUNCTORS WITH AN APPLICATION TO REPRESENTATION-FINITE ARTINIAN RINGS Wolfgang Rump Introduction Ladders were introduced by Igusa and Todorov
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationGENERALIZED BRAUER TREE ORDERS
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 2 GENERALIZED BRAUER TREE ORDERS BY K. W. R O G G E N K A M P (STUTTGART) Introduction. p-adic blocks of integral group rings with cyclic defect
More information1. Quivers and their representations: Basic definitions and examples.
1 Quivers and their representations: Basic definitions and examples 11 Quivers A quiver Q (sometimes also called a directed graph) consists of vertices and oriented edges (arrows): loops and multiple arrows
More informationA BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Shanghai , P. R. China
A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES PU ZHANG Department of Mathematics, Shanghai 200240, P. R. China Shanghai Jiao Tong University Since Eilenberg and Moore [EM], the relative homological
More informationHigher dimensional homological algebra
Higher dimensional homological algebra Peter Jørgensen Contents 1 Preface 3 2 Notation and Terminology 6 3 d-cluster tilting subcategories 7 4 Higher Auslander Reiten translations 12 5 d-abelian categories
More informationCoils for Vectorspace Categories
Coils for Vectorspace Categories Bin Zhu Department of mathematical science, Tsinghua University, Beijing 100084, P. R. China e-mail: bzhu@math.tsinghua.edu.cn Abstract. Coils as components of Auslander-Reiten
More informationThe real root modules for some quivers.
SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will
More informationSTANDARD COMPONENTS OF A KRULL-SCHMIDT CATEGORY
STANDARD COMPONENTS OF A KRULL-SCHMIDT CATEGORY SHIPING LIU AND CHARLES PAQUETTE Abstract. First, for a general Krull-Schmidt category, we provide criteria for an Auslander-Reiten component with sections
More informationMODULE CATEGORIES WITHOUT SHORT CYCLES ARE OF FINITE TYPE
proceedings of the american mathematical society Volume 120, Number 2, February 1994 MODULE CATEGORIES WITHOUT SHORT CYCLES ARE OF FINITE TYPE DIETER HAPPEL AND SHIPING LIU (Communicated by Maurice Auslander)
More informationWeakly shod algebras
Journal of Algebra 265 (2003) 379 403 www.elsevier.com/locate/jalgebra Weakly shod algebras Flávio U. Coelho a,,1 and Marcelo A. Lanzilotta b,1 a Departamento de Matemática-IME, Universidade de São Paulo,
More informationDESINGULARIZATION OF QUIVER GRASSMANNIANS FOR DYNKIN QUIVERS. Keywords: Quiver Grassmannians, desingularizations.
DESINGULARIZATION OF QUIVER GRASSMANNIANS FOR DYNKIN QUIVERS G. CERULLI IRELLI, E. FEIGIN, M. REINEKE Abstract. A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers
More informationThe preprojective algebra revisited
The preprojective algebra revisited Helmut Lenzing Universität Paderborn Auslander Conference Woodshole 2015 H. Lenzing Preprojective algebra 1 / 1 Aim of the talk Aim of the talk My talk is going to review
More informationTHE GLOBAL DIMENSION OF THE ENDOMORPHISM RING OF A GENERATOR-COGENERATOR FOR A HEREDITARY ARTIN ALGEBRA
C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (3) 2008, pp. 89 96 THE GLOBAL DIMENSION OF THE ENDOMORPHISM RING OF A GENERATOR-COGENERATOR FOR A HEREDITARY ARTIN ALGEBRA VLASTIMIL DLAB AND CLAUS MICHAEL RINGEL
More informationA course on cluster tilted algebras
Ibrahim Assem Département de mathématiques Université de Sherbrooke Sherbrooke, Québec Canada JK R A course on cluster tilted algebras march 06, mar del plata Contents Introduction 5 Tilting in the cluster
More informationA BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES. Department of Mathematics, Shanghai Jiao Tong University Shanghai , P. R.
A BRIEF INTRODUCTION TO GORENSTEIN PROJECTIVE MODULES PU ZHANG Department of Mathematics, Shanghai Jiao Tong University Shanghai 200240, P. R. China Since Eilenberg and Moore [EM], the relative homological
More informationClassification of discrete derived categories
CEJM 1 (2004) 1 31 Central European Journal of Mathematics Classification of discrete derived categories Grzegorz Bobiński 1, Christof Geiß 2, Andrzej Skowroński 1 1 Faculty of Mathematics and Computer
More informationEXACT CATEGORY OF MODULES OF CONSTANT JORDAN TYPE
EXACT CATEGORY OF MODULES OF CONSTANT JORDAN TYPE JON F. CARLSON AND ERIC M. FRIEDLANDER To Yuri Manin with admiration Abstract. For a finite group scheme G, we continue our investigation of those finite
More informationarxiv: v1 [math.rt] 11 Dec 2015
Some applications of τ-tilting theory arxiv:1512.03613v1 [math.rt] 11 Dec 2015 Shen Li, Shunhua Zhang School of Mathematics, Shandong University, Jinan, 250100,P.R.China Abstract Let A be a finite dimensional
More informationTilting categories with applications to stratifying systems
Journal of Algebra 302 (2006) 419 449 www.elsevier.com/locate/jalgebra Tilting categories with applications to stratifying systems Octavio Mendoza a,, Corina Sáenz b a Instituto de Matemáticas, UNAM, Circuito
More informationA functorial approach to modules of G-dimension zero
A functorial approach to modules of G-dimension zero Yuji Yoshino Math. Department, Faculty of Science Okayama University, Okayama 700-8530, Japan yoshino@math.okayama-u.ac.jp Abstract Let R be a commutative
More informationCluster Tilting for Representation-Directed Algebras
U.U.D.M. Report 8: Cluster Tilting for Representation-Directed Algebras Laertis Vaso Filosofie licentiatavhandling i matematik som framläggs för offentlig granskning den 9 november 8, kl., sal, Ångströmlaboratoriet,
More informationarxiv:math/ v1 [math.rt] 27 Jul 2005
arxiv:math/0507559v1 [mathrt] 27 Jul 2005 Auslander-Reiten Quivers which are Independent of the Base Ring By Markus Schmidmeier Fix a poset P and a natural number n For various commutative local rings
More informationarxiv: v1 [math.rt] 13 May 2015
arxiv:1505.03547v1 [math.rt] 13 May 2015 Artin algebras of finite type and finite categories of-good modules Danilo D. da Silva Universidade Federal de Sergipe - UFS Departamento de Matematica - DMA Sao
More informationarxiv: v1 [math.rt] 15 Aug 2016
GENDO-SYMMETRIC ALGEBRAS, DOMINANT DIMENSIONS AND GORENSTEIN HOMOLOGICAL ALGEBRA RENÉ MARCZINZIK arxiv:1608.04212v1 [math.rt] 15 Aug 2016 Abstract. This article relates dominant and codominant dimensions
More informationCONTRAVARIANTLY FINITE RESOLVING SUBCATEGORIES OVER A GORENSTEIN LOCAL RING
CONTRAVARIANTLY FINITE RESOLVING SUBCATEGORIES OVER A GORENSTEIN LOCAL RING RYO TAKAHASHI Introduction The notion of a contravariantly finite subcategory (of the category of finitely generated modules)
More informationCLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE
CLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE ASLAK BAKKE BUAN, ROBERT J. MARSH, AND IDUN REITEN Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically
More informationCALDERO-CHAPOTON ALGEBRAS
CALDERO-CHAPOTON ALGEBRAS GIOVANNI CERULLI IRELLI, DANIEL LABARDINI-FRAGOSO, AND JAN SCHRÖER Abstract. Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and
More informationENUMERATING m-clusters USING EXCEPTIONAL SEQUENCES
ENUMERATING m-clusters USING EXCEPTIONAL SEQUENCES KIYOSHI IGUSA Abstract. The number of clusters of Dynkin type has been computed case-by-case and is given by a simple product formula which has been extended
More informationCluster-Concealed Algebras
Cluster-Concealed Algebras Claus Michael Ringel (Bielefeld/Germany) Nanjing, November 2010 Cluster-tilted algebras Let k be an algebraically closed field Let B be a tilted algebra (the endomorphism ring
More informationREPRESENTATION DIMENSION AS A RELATIVE HOMOLOGICAL INVARIANT OF STABLE EQUIVALENCE
REPRESENTATION DIMENSION AS A RELATIVE HOMOLOGICAL INVARIANT OF STABLE EQUIVALENCE ALEX S. DUGAS Abstract. Over an Artin algebra Λ many standard concepts from homological algebra can be relativized with
More informationSTABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1. Kunio Yamagata 2
STABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1 Kunio Yamagata 2 Abstract. A finite dimensional self-injective algebra will be determined when it is stably equivalent to a positive self-injective
More informationIntroduction to the representation theory of quivers Second Part
Introduction to the representation theory of quivers Second Part Lidia Angeleri Università di Verona Master Program Mathematics 2014/15 (updated on January 22, 2015) Warning: In this notes (that will be
More information4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;
4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length
More informationOn the number of terms in the middle of almost split sequences over cycle-finite artin algebras
Cent. Eur. J. Math. 12(1) 2014 39-45 DOI: 10.2478/s11533-013-0328-3 Central European Journal of Mathematics On the number of terms in the middle of almost split sequences over cycle-finite artin algebras
More informationQuiver Representations
Quiver Representations Molly Logue August 28, 2012 Abstract After giving a general introduction and overview to the subject of Quivers and Quiver Representations, we will explore the counting and classification
More informationDimension of the mesh algebra of a finite Auslander-Reiten quiver. Ragnar-Olaf Buchweitz and Shiping Liu
Dimension of the mesh algebra of a finite Auslander-Reiten quiver Ragnar-Olaf Buchweitz and Shiping Liu Abstract. We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over
More informationRelations for the Grothendieck groups of triangulated categories
Journal of Algebra 257 (2002) 37 50 www.academicpress.com Relations for the Grothendieck groups of triangulated categories Jie Xiao and Bin Zhu Department of Mathematical Sciences, Tsinghua University,
More informationIndecomposable Quiver Representations
Indecomposable Quiver Representations Summer Project 2015 Laura Vetter September 2, 2016 Introduction The aim of my summer project was to gain some familiarity with the representation theory of finite-dimensional
More informationTILTING THEORY AND CLUSTER COMBINATORICS
TILTING THEORY AND CLUSTER COMBINATORICS ASLAK BAKKE BUAN, ROBERT MARSH, MARKUS REINEKE, IDUN REITEN, AND GORDANA TODOROV Abstract We introduce a new category C, which we call the cluster category, obtained
More informationCategories of noncrossing partitions
Categories of noncrossing partitions Kiyoshi Igusa, Brandeis University KIAS, Dec 15, 214 Topology of categories The classifying space of a small category C is a union of simplices k : BC = X X 1 X k k
More informationA COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998)
A COURSE IN HOMOLOGICAL ALGEBRA CHAPTER 11: Auslander s Proof of Roiter s Theorem E. L. Lady (April 29, 1998) A category C is skeletally small if there exists a set of objects in C such that every object
More informationAbelian categories. triangulated categories: Some examples.
Abelian categories versus triangulated categories: Some examples Claus Michael Ringel Trondheim, 000 For Idun Reiten on the occasion of her 70th birthday Comparison Abelian categories Rigidity Factorizations
More informationREALISING HIGHER CLUSTER CATEGORIES OF DYNKIN TYPE AS STABLE MODULE CATEGORIES. 1. Introduction
REALISING HIGHER CLUSTER CATEGORIES OF DYNKIN TYPE AS STABLE MODULE CATEGORIES THORSTEN HOLM AND PETER JØRGENSEN Abstract. We show that the stable module categories of certain selfinjective algebras of
More informationModules of the Highest Homological Dimension over a Gorenstein Ring
Modules of the Highest Homological Dimension over a Gorenstein Ring Yasuo Iwanaga and Jun-ichi Miyachi Dedicated to Professor Kent R. Fuller on his 60th birthday We will study modules of the highest injective,
More informationREPRESENTATION DIMENSION OF ARTIN ALGEBRAS
REPRESENTATION DIMENSION OF ARTIN ALGEBRAS STEFFEN OPPERMANN In 1971, Auslander [1] has introduced the notion of representation dimension of an artin algebra. His definition is as follows (see Section
More informationAuslander-Reiten Quiver of the Category of Unstable Modules over a Sub-Hopf Algebra of the Steenrod Algebra
Auslander-Reiten Quiver of the Category of Unstable Modules over a Sub-Hopf Algebra of the Steenrod Algebra Gabriel Angelini-Knoll Abstract For small sub-hopf algebras of the mod-2 Steenrod algebra A 2,
More informationCLUSTER ALGEBRA STRUCTURES AND SEMICANONICAL BASES FOR UNIPOTENT GROUPS
CLUSTER ALGEBRA STRUCTURES AND SEMICANONICAL BASES FOR UNIPOTENT GROUPS CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstract. Let Q be a finite quiver without oriented cycles, and let Λ be the associated
More informationJournal of Algebra 330 (2011) Contents lists available at ScienceDirect. Journal of Algebra.
Journal of Algebra 330 2011) 375 387 Contents lists available at ciencedirect Journal of Algebra www.elsevier.com/locate/jalgebra Higher Auslander algebras admitting trivial maximal orthogonal subcategories
More informationFROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS II
FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS II PHILIPPE CALDERO AND BERNHARD KELLER Abstract. In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster
More informationOne-point extensions and derived equivalence
Journal of Algebra 264 (2003) 1 5 www.elsevier.com/locate/jalgebra One-point extensions and derived equivalence Michael Barot a, and Helmut Lenzing b a Instituto de Matemáticas, UNAM, Mexico 04510 D.F.,
More informationDedicated to Professor Klaus Roggenkamp on the occasion of his 60th birthday
C O L L O Q U I U M M A T H E M A T I C U M VOL. 86 2000 NO. 2 REPRESENTATION THEORY OF TWO-DIMENSIONAL BRAUER GRAPH RINGS BY WOLFGANG R U M P (STUTTGART) Dedicated to Professor Klaus Roggenkamp on the
More informationDerived Canonical Algebras as One-Point Extensions
Contemporary Mathematics Derived Canonical Algebras as One-Point Extensions Michael Barot and Helmut Lenzing Abstract. Canonical algebras have been intensively studied, see for example [12], [3] and [11]
More informationA NOTE ON THE RADICAL OF A MODULE CATEGORY. 1. Introduction
A NOTE ON THE RADICAL OF A MODULE CATEGORY CLAUDIA CHAIO AND SHIPING LIU Abstract We characterize the finiteness of the representation type of an artin algebra in terms of the behavior of the projective
More informationCLASSIFICATION OF ABELIAN HEREDITARY DIRECTED CATEGORIES SATISFYING SERRE DUALITY
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 5, May 2008, Pages 2467 2503 S 0002-9947(07)04426-1 Article electronically published on October 30, 2007 CLASSIFICATION OF ABELIAN HEREDITARY
More informationRepresentation type, boxes, and Schur algebras
10.03.2015 Notation k algebraically closed field char k = p 0 A finite dimensional k-algebra mod A category of finite dimensional (left) A-modules M mod A [M], the isomorphism class of M ind A = {[M] M
More informationRepresentations of Quivers
MINGLE 2012 Simon Peacock 4th October, 2012 Outline 1 Quivers Representations 2 Path Algebra Modules 3 Modules Representations Quiver A quiver, Q, is a directed graph. Quiver A quiver, Q, is a directed
More informationON THE NON-PERIODIC STABLE AUSLANDER REITEN HELLER COMPONENT FOR THE KRONECKER ALGEBRA OVER A COMPLETE DISCRETE VALUATION RING
ON THE NON-PERIODIC STABLE AUSLANDER REITEN HELLER COMPONENT FOR THE KRONECKER ALGEBRA OVER A COMPLETE DISCRETE VALUATION RING KENGO MIYAMOTO Abstract. We consider the Kronecker algebra A = O[X, Y ]/(X
More informationSUPPORT VARIETIES FOR SELFINJECTIVE ALGEBRAS
SUPPORT VARIETIES FOR SELFINJECTIVE ALGEBRAS KARIN ERDMANN, MILES HOLLOWAY, NICOLE SNASHALL, ØYVIND SOLBERG, AND RACHEL TAILLEFER Abstract. Support varieties for any finite dimensional algebra over a field
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationThe Auslander bijections: how morphisms are determined by modules
Bull. Math. Sci. (2013) 3:409 484 DOI 10.1007/s13373-013-0042-2 The Auslander bijections: how morphisms are determined by modules Claus Michael Ringel Received: 7 January 2013 / Revised: 3 June 2013 /
More informationHochschild Cohomology and Representation-finite Algebras. Ragnar-Olaf Buchweitz and Shiping Liu. Introduction
Hochschild Cohomology and Representation-finite Algebras Ragnar-Olaf Buchweitz and Shiping Liu Dedicated to Idun Reiten to mark her sixtieth birthday Introduction Hochschild cohomology is a subtle invariant
More informationarxiv: v2 [math.rt] 17 Mar 2013
REDUCTION OF τ-tilting MODULES AND TORSION PAIRS GUSTAVO JASSO arxiv:0.709v [math.rt] 7 Mar 0 Abstract. The class of support τ-tilting modules was introduced recently by Adachi, Iyama and Reiten. These
More informationSIMPLE-MINDED SYSTEMS, CONFIGURATIONS AND MUTATIONS FOR REPRESENTATION-FINITE SELF-INJECTIVE ALGEBRAS
SIMPLE-MINDED SYSTEMS, CONFIGURATIONS AND MUTATIONS FOR REPRESENTATION-FINITE SELF-INJECTIVE ALGEBRAS AARON CHAN, STEFFEN KOENIG, AND YUMING LIU Abstract. Simple-minded systems of objects in a stable module
More informationOn Representation Dimension of Tame Cluster Tilted Algebras
Bol. Soc. Paran. Mat. (3s.) v. 00 0 (0000):????. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.37626 On Representation Dimension of Tame Cluster Tilted Algebras
More informationarxiv:math/ v1 [math.rt] 10 Nov 2004
arxiv:math/0411238v1 [math.rt] 10 Nov 2004 Quivers with relations and cluster tilted algebras Philippe Caldero, Frédéric Chapoton, Ralf Schiffler Abstract Cluster algebras were introduced by S. Fomin and
More informationON THE RELATION BETWEEN CLUSTER AND CLASSICAL TILTING. 0. Introduction
ON THE RELATION BETWEEN CLUSTER AND CLASSICAL TILTING THORSTEN HOLM AND PETER JØRGENSEN Abstract. Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian;
More informationOriented Exchange Graphs & Torsion Classes
1 / 25 Oriented Exchange Graphs & Torsion Classes Al Garver (joint with Thomas McConville) University of Minnesota Representation Theory and Related Topics Seminar - Northeastern University October 30,
More informationAN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE
AN AXIOMATIC CHARACTERIZATION OF THE GABRIEL-ROITER MEASURE HENNING KRAUSE Abstract. Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
More information