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

631 Pages·2017·3.97 MB·English
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Textbooks in Mathematics KA un nd ir oz Ye aj mS k ao gw a Andrzej Skowron´ski taro Andrzej Skowron´ski n´ s Kunio Yamagata k Kunio Yamagata i Frobenius Algebras II This is the second of three volumes which will provide a comprehensive introduction T F Frobenius Algebras II to the modern representation theory of Frobenius algebras. The first part of the ilte ro d b book is devoted to fundamental results of the representation theory of finite a e n n dimensional hereditary algebras and their tilted algebras, which allow to describe the d i representation theory of prominent classes of Frobenius algebras. H us o c A h The second part is devoted to basic classical and recent results concerning the s lg c h e Hmoocdhuslceh ciladt eegxoterinessi.o Mnso oref ofivneirt,e t hdeim shenaspieosn oalf acolgnenbercatse bdy c doumapliotyn ebnimtso odfu tlhees satnadb lteh eir ild E bra Tilted and Hochschild Extension Algebras Auslander–Reiten quivers of Frobenius algebras are described. x s te I n I s The only prerequisite in this volume is a basic knowledge of linear algebra and some io n results of the first volume. It includes complete proofs of all results presented and A provides a rich supply of examples and exercises. lg e b r The text is primarily addressed to graduate students starting research in the a s representation theory of algebras as well mathematicians working in other fields. 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 EMS Textbooks in Mathematics EMS Textbooks in Mathematics is a series of books aimed at students or professional mathemati- cians seeking an introduction into a particular field. The individual volumes are intended not only to provide relevant techniques, results, and applications, but also to afford insight into the motivations and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature. Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes 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 Marek Jarnicki and Peter Pflug, First Steps in Several Complex Variables: Reinhardt Domains 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 Capacities 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 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2017 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Toourwifes MiraandTaeko andchildren Magda,Akiko,IkuoandTaketo 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

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This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras
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