de Gruyter Expositions in Mathematics 41
Editors V. P. Maslov, Academy of Sciences, Moscow W. D. Neumann, Columbia University, New York R. O. Wells, Jr., International University, Bremen
Approximations and Endomorphism Algebras of Modules by
Rüdiger Göbel and Jan Trlifaj
≥
Walter de Gruyter · Berlin · New York
Authors Rüdiger Göbel Fachbereich 6, Mathematik Universität Duisburg-Essen 45117 Essen Germany E-Mail:
[email protected] Jan Trlifaj Katedra algebry MFF Univerzita Karlova v Praze Sokolovska´ 83 186 75 Prague 8 Czech Republic E-Mail:
[email protected] Mathematics Subject Classification 2000: First: 1602, Second: 03C60, 03Exx, 13-XX, 16-XX, 20Kxx Key words: approximations of modules, infinite dimensional tilting theory, prediction principles, realizations of algebras as endomorphism algebras, modules with distinguished submodules, E-rings.
앝 Printed on acid-free paper which falls within the guidelines 앪 of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data Goebel, Ruediger. Approximations and endomorphism algebras of modules / by Ruediger Goebel and Jan Trlifaj. p. cm ⫺ (De Gruyter expositions in mathematics ; 41) Includes bibliographical references and index. ISBN-13: 978-3-11-011079-1 (alk. paper) ISBN-10: 3-11-011079-2 (alk. paper) 1. Modules (Algebra) 2. Moduli theory. 3. Approximation theory. I. Trlifaj, Jan. II. Title. QA247.G63 2006 5121.42⫺dc22 2006018289
ISSN 0938-6572 ISBN-13: 978-3-11-011079-1 ISBN-10: 3-11-011079-2 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at ⬍http://dnb.ddb.de⬎. 쑔 Copyright 2006 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Cover design: Thomas Bonnie, Hamburg.
For our wives Heidi and Kateˇrina and children Ines, and Lucie, Justina, Magdalena, Šimon and Daniel
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1 Some useful classes of modules . . . . . . . . . . . . . . . . . . . . 1.1 S–completions . . . . . . . . . . . . . . . . . . . . . . . . . . — a first step . . . . . . . . . . . . Support of elements in B Uncountable S in completions . . . . . . . . . . . . . . . . . Modules of cardinality ≤ 2ℵ0 . . . . . . . . . . . . . . . . . 1.2 Pure–injective modules . . . . . . . . . . . . . . . . . . . . . Direct limits, finitely presented modules and pure submodules Characterizations of pure–injective modules . . . . . . . . . 1.3 Locally projective modules . . . . . . . . . . . . . . . . . . . 1.4 Factors of products and slender modules . . . . . . . . . . . . 1.5 Slender modules over Dedekind domains . . . . . . . . . . . .
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2 Approximations of modules . . . 2.1 Preenvelopes and precovers . 2.2 Cotorsion pairs and Tor–pairs 2.3 Minimal approximations . . .
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3 Complete cotorsion pairs . . . . . . . . . . 3.1 Ext and direct limits . . . . . . . . . . 3.2 The variety of complete cotorsion pairs 3.3 Ext and inverse limits . . . . . . . . .
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4 Deconstruction of cotorsion pairs . . . . . . . . . . . . . . . . . 4.1 Approximations by modules of finite homological dimensions 4.2 Hill Lemma and Kaplansky Theorem for cotorsion pairs . . . 4.3 Closure properties providing for completeness . . . . . . . . The tilting case . . . . . . . . . . . . . . . . . . . . . . . . The cotilting case . . . . . . . . . . . . . . . . . . . . . . 4.4 Matlis cotorsion and strongly flat modules . . . . . . . . . .
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134 134 142 149 150 157 163
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4.5
Strongly flat modules over valuation domains . . . . . . . . . . 173 The closure of a cotorsion pair . . . . . . . . . . . . . . . . . . . 178 Direct limits of modules of projective dimension ≤ 1 . . . . . 184
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Tilting approximations . . . . . . . . . . . . . . . 5.1 Tilting modules . . . . . . . . . . . . . . . . 5.2 Classes of finite type . . . . . . . . . . . . . . Deconstruction to countable type . . . . . . Definability and the Mittag–Leffler condition Finite type and resolving subcategories . . . 5.3 Injectivity properties of tilting modules . . . .
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188 188 201 202 208 213 219
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1–tilting modules and their applications . . . . . . . . . . . . . . 6.1 Tilting torsion classes . . . . . . . . . . . . . . . . . . . . . . 6.2 The structure of tilting modules and classes over particular rings 1–tilting classes over artin algebras . . . . . . . . . . . . . . Tilting modules and classes over Prüfer domains . . . . . . . The case of valuation and Dedekind domains . . . . . . . . . 6.3 Matlis localizations . . . . . . . . . . . . . . . . . . . . . . .
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224 224 228 228 231 240 242
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Tilting approximations and the finitistic dimension conjectures . . . 7.1 Finitistic dimension conjectures and the tilting module Tf . . . . 7.2 A formula for the little finitistic dimension of right artinian rings 7.3 Artinian rings with P
