LEADER 07444nam 2200781Ia 450 001 9910825810303321 005 20230802005652.0 010 $a1-283-85645-X 010 $a3-11-021811-9 024 7 $a10.1515/9783110218114 035 $a(CKB)2550000000711066 035 $a(EBL)893905 035 $a(OCoLC)823387924 035 $a(SSID)ssj0000784696 035 $a(PQKBManifestationID)12344651 035 $a(PQKBTitleCode)TC0000784696 035 $a(PQKBWorkID)10763937 035 $a(PQKB)10247035 035 $a(MiAaPQ)EBC893905 035 $a(DE-B1597)36530 035 $a(OCoLC)948656006 035 $a(OCoLC)953308212 035 $a(DE-B1597)9783110218114 035 $a(Au-PeEL)EBL893905 035 $a(CaPaEBR)ebr10634561 035 $a(CaONFJC)MIL416895 035 $a(EXLCZ)992550000000711066 100 $a20120730h20122012 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aApproximations and endomorphism algebras of modules /$fRudiger Gobel, Jan Trlifaj 205 $a2nd rev. and extended ed. 210 $aBerlin ;$aBoston $cDe Gruyter$dc2012 215 $a1 online resource (1002 p.) 225 0 $aDe Gruyter expositions in mathematics,$x0938-6572 ;$v41 300 $aDescription based upon print version of record. 311 $a3-11-021810-0 320 $aIncludes bibliographical references (p. [419]-449, p. [933]-963) and indexes. 327 $t Frontmatter -- $tContents -- $tIntroduction -- $tList of Symbols -- $tPart I. Some useful classes of modules -- $tChapter 1. S-completions -- $tChapter 2. Pure-injective modules -- $tChapter 3. Mittag-Leffler modules -- $tChapter 4. Slender modules -- $tPart II. Approximations and cotorsion pairs -- $tChapter 5. Approximations of modules -- $tChapter 6. Complete cotorsion pairs -- $tChapter 7. Hill lemma and its applications -- $tChapter 8. Deconstruction of the roots of Ext -- $tChapter 9. Modules of projective dimension one -- $tChapter 10. Kaplansky classes and abstract elementary classes -- $tChapter 11. Independence results for cotorsion pairs -- $tChapter 12. The lattice of cotorsion pairs -- $tPart III. Tilting and cotilting approximations -- $tChapter 13. Tilting approximations -- $tChapter 14. 1-tilting modules and their applications -- $tChapter 15. Cotilting classes -- $tChapter 16. Tilting and cotilting classes over commutative noetherian rings -- $tChapter 17. Tilting approximations and the finitistic dimension conjectures -- $tBibliography -- $tIndex -- $tPart IV Prediction principles -- $tChapter 18. Survey of prediction principles using ZFC and more -- $tChapter 19. Prediction principles in ZFC: the Black Boxes and others -- $tPart V. Endomorphism algebras and automorphism groups -- $tChapter 20. Realising algebras - by algebraically independent elements and by prediction principles -- $tChapter 21. Automorphism groups of torsion-free abelian groups -- $tChapter 22. Modules with distinguished submodules -- $tChapter 23. R-modules and fields from modules with distinguished submodules -- $tChapter 24 Endomorphism algebras of ?n-free modules -- $tPart VI. Modules and rings related to algebraic topology -- $tChapter 25. Localisations and cellular covers, the general theory for R-modules -- $tChapter 26. Tame and wild localisations of size ? 2 ?0 -- $tChapter 27. Tame cellular covers -- $tChapter 28. Wild cellular covers -- $tChapter 29. Absolute E-rings -- $tPart VII. Cellular covers, localisations and E(R)-algebras -- $tChapter 30. Large kernels of cellular covers and large localisations -- $tChapter 31. Mixed E(R)-modules over Dedekind domains -- $tChapter 32. E(R)-modules with cotorsion -- $tChapter 33. Generalised E(R)-algebras -- $tChapter 34. Some more useful classes of algebras -- $tBibliography -- $tIndex 330 $aThis second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2). It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C. The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory. 410 3$aDe Gruyter Expositions in Mathematics 606 $aModules (Algebra) 606 $aModuli theory 606 $aApproximation theory 610 $aApproximations of Modules. 610 $aCotorsion Pair. 610 $aE-Ring. 610 $aEndomorphism Algebra. 610 $aFiltration. 610 $aInfinite Dimensional Tilting Theory. 610 $aPrediction Principle. 615 0$aModules (Algebra) 615 0$aModuli theory. 615 0$aApproximation theory. 676 $a512.42 676 $a512/.42 700 $aGo?bel$b R$g(Ru?diger),$f1940-$060070 701 $aTrlifaj$b Jan$01631517 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910825810303321 996 $aApproximations and endomorphism algebras of modules$93970339 997 $aUNINA