Frobenius Algebras: No. II: Tilted and Hochschild Extension Algebras

631 Pages·2017·3.97 MB·English

by  Andrzej Skowronski

Textbooks in Mathematics
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Frobenius Algebras II
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The text is primarily addressed to graduate students starting research in the a
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ISBN 978-3-03719-159-0
www.ems-ph.org
Skowronski II Cover | Font: Frutiger_Helvetica Neue | Farben: Pantone 116, Pantone 287 | RB 48 mm

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Markus Stroppel, Locally Compact Groups
Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations
Dorothee D. Haroske and Hans Triebel, Distributions, Sobolev Spaces, Elliptic Equations
Thomas Timmermann, An Invitation to Quantum Groups and Duality
Oleg Bogopolski, Introduction to Group Theory
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Tammo tom Dieck, Algebraic Topology
Mauro C. Beltrametti et al., Lectures on Curves, Surfaces and Projective Varieties
Wolfgang Woess, Denumerable Markov Chains
Eduard Zehnder, Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic
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Andrzej Skowron´ski and Kunio Yamagata, Frobenius Algebras I. Basic Representation Theory
Piotr W. Nowak and Guoliang Yu, Large Scale Geometry
Joaquim Bruna and Juliá Cufí, Complex Analysis
Eduardo Casas-Alvero, Analytic Projective Geometry
Fabrice Baudoin, Diffusion Processes and Stochastic Calculus
Olivier Lablée, Spectral Theory in Riemannian Geometry
Dietmar A. Salamon, Measure and Integration
Andrzej Skowron´ski
Kunio Yamagata
Frobenius Algebras II
Tilted and Hochschild Extension Algebras
Authors:
Andrzej Skowron´ski Kunio Yamagata
Faculty of Mathematics Department of Mathematics
and Computer Science Tokyo University of Agriculture
Nicolaus Copernicus University and Technology
Chopina 12/18 Nakacho 2-24-16, Koganei
87-100 Torun´ Tokyo 184-8588
Poland Japan
E-mail: [email protected] E-mail: [email protected]
2010 Mathematics Subject Classification: Primary: 16-01; Secondary: 13E10, 15A63, 15A69, 16Dxx,
16E10, 16E30, 16E40, 16G10, 16G20, 16G60, 16G70, 16S50, 16S70, 18A25, 18E30, 18G15.
Key words: Algebra, module, bimodule, representation, quiver, ideal, radical, simple module,
semisimple module, uniserial module, projective module, injective module, tilting module, hereditary
algebra, tilted algebra, Frobenius algebra, symmetric algebra, selnjective algebra, Hochschild
extension algebra, category, functor, torsion pair, projective dimension, injective dimension, global
dimension, Euler form, Grothendieck group, irreducible homomorphism, almost split sequence,
Auslander–Reiten translation, Auslander–Reiten quiver, stable equivalence, syzygy module, duality
bimodule, Hochschild extension
ISBN 978-3-03719-174-3
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
VII Hereditaryalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Thequiverofanalgebra . . . . . . . . . . . . . . . . . . . . . . 2
2 Thetensoralgebrasofspecies . . . . . . . . . . . . . . . . . . . 19
3 Exactsequences . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 TheEulerforms . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 TheCoxetertransformation . . . . . . . . . . . . . . . . . . . . 75
6 Postprojectiveandpreinjectivecomponents. . . . . . . . . . . . 82
7 HereditaryalgebrasofDynkintype . . . . . . . . . . . . . . . . 96
8 HereditaryalgebrasofEuclideantype. . . . . . . . . . . . . . . 111
9 Hereditaryalgebrasofwildtype. . . . . . . . . . . . . . . . . . 158
10 Representationsofbimodules . . . . . . . . . . . . . . . . . . . 197
11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
VIII Tiltedalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
1 Torsionpairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
2 Tiltingmodules . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3 TheBrenner–Butlertheorem . . . . . . . . . . . . . . . . . . . 242
4 Connectingsequences . . . . . . . . . . . . . . . . . . . . . . . 284
5 Splittingtiltingmodules . . . . . . . . . . . . . . . . . . . . . . 294
6 Tiltedalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
7 ThecriterionofLiuandSkowron´ski . . . . . . . . . . . . . . . 346
8 Reflectionsofhereditaryalgebras . . . . . . . . . . . . . . . . . 362
9 ThetheoremofRingelonregulartiltingmodules. . . . . . . . . 367
10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
IX Auslander–Reitencomponents . . . . . . . . . . . . . . . . . . . . 407
1 Functorsonmodulecategories . . . . . . . . . . . . . . . . . . 407
2 TheIgusa–Todorovtheorem . . . . . . . . . . . . . . . . . . . . 422
3 Degreesofirreduciblehomomorphisms. . . . . . . . . . . . . . 428
4 StableAuslander–Reitencomponents . . . . . . . . . . . . . . . 442
5 GeneralizedstandardAuslander–Reitencomponents . . . . . . . 453
6 Stableequivalence . . . . . . . . . . . . . . . . . . . . . . . . . 456
7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
viii Contents
X SelfinjectiveHochschildextensionalgebras . . . . . . . . . . . . . 477
1 Hochschildcohomologyspaces . . . . . . . . . . . . . . . . . . 477
2 Hochschildextensionalgebras . . . . . . . . . . . . . . . . . . 479
3 Hochschildextensionsbydualitymodules . . . . . . . . . . . . 488
4 Non-FrobeniusselfinjectiveHochschildextensions . . . . . . . . 505
5 Hochschildextensionalgebrasoffinitefieldextensions . . . . . 517
6 Hochschildextensionalgebrasofpathalgebras . . . . . . . . . . 529
7 Hochschildextensionalgebrasofhereditaryalgebras. . . . . . . 540
8 TheAuslander–ReitenquiversofHochschildextensionalgebras. 553
9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
Introduction
Themaintopicofthisbookistherepresentationtheoryoffinitedimensionalas-
sociativealgebraswithanidentityoverafield,whichcurrentlycanberegardedas
thestudyofthecategoriesoftheirfinitedimensionalmodulesandtheassociated
combinatorialandhomologicalinvariants. Aprominentroleintherepresentation
theoryoffinitedimensionalalgebrasoverfieldsisplayedbyFrobeniusalgebras.
TheFrobeniusalgebrasoriginatedinthe 1903papersby Frobeniusandreceived
moderncharacterizationsin a series ofpapersby Brauer,Nesbittand Nakayama
from 1937–1941. In particular, we may say that a finite dimensional algebra A
over a field K is a Frobenius algebra if there exists a nondegenerateK-bilinear
form.(cid:2);(cid:2)/WA(cid:3)A!K whichisassociative,inthesensethat.ab;c/D.a;bc/
for all elements a;b;c of A. Frobenius algebras are selfinjective algebras (pro-
jective and injective modules coincide), and the module category of every finite
dimensionalselfinjectivealgebraoverafieldisequivalenttothemodulecategory
ofaFrobeniusalgebra.
Thebookisdividedintothreevolumesandits mainaimistoprovideacom-
prehensiveintroductiontothemodernrepresentationtheoryoffinitedimensional
algebrasoverfields,withspecialattentiondevotedtotherepresentationtheoryof
Frobeniusalgebras,ormoregenerallyselfinjectivealgebras. Thebookisprimar-
ilyaddressedtograduatestudentsstartingresearchintherepresentationtheoryof
algebras,aswellastomathematiciansworkinginotherrelatedfields. Itishoped
that the bookwill provide a friendly access to the representationtheory of finite
dimensionalalgebras,astheonlyprerequisiteisabasicknowledgeoflinearalge-
bra. Wepresentcompleteproofsofallresultsstatedinthebook. Moreover,arich
supplyofexamplesandexerciseswill helpthe readerunderstandandmasterthe
theorypresentedinthebook.
In the first volume of the book, “Frobenius Algebras I. Basic Representation
Theory” [SY2], divided into six chapters, we provided a generalintroduction to
basicresultsandtechniquesofthemodernrepresentationtheoryoffinitedimen-
sional algebras over fields, including the Morita equivalences and the Morita–
Azumaya dualities for the module categories, and the Auslander–Reiten theory
ofirreduciblehomomorphismsandalmostsplitsequences. The heartofthe first
volume is devoted to presenting fundamental classical as well as recent results
concerningtheselfinjectivealgebrasandtheirmodulecategories. Moreover,two
chaptersofthefirstvolumearedevotedtobasicpropertiesoftwoclassicalclasses
ofFrobeniusalgebrasformedbytheHeckealgebrasoffiniteCoxetergroupsand
thefinitedimensionalHopfalgebras.
In the second volume of the book we continue to present basic results and
techniquesofthemodernrepresentationtheoryoffinitedimensionalalgebrasover