Lecture notes on local rings:
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| Format: | Book |
| Language: | English |
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Singapore
World Scientific
2014
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 213 S. graph. Darst. |
| ISBN: | 9789814603652 |
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| adam_text | Titel: Lecture notes on local rings
Autor: Iversen, Birger
Jahr: 2014
Contents Preface v 1. Dimension of a Local Ring 1 1.1 Nakayama’s lemma...................... 1 1.2 Prime ideals.......................... 2 1.3 Noetherian modules ..................... 4 1.4 Modules of finite length................... 6 1.5 Hilbert’s basis theorem.................... 8 1. G Graded rings......................... 9 1.7 Filtered rings......................... 11 1.8 Local rings .......................... 18 1.9 Regular local rings...................... 17 2. Modules over a Local Ring 19 2.1 Support of a module..................... 19 2.2 Associated prime ideals ................... 20 2.3 Dimension of a module.................... 22 2.4 Depth of a module...................... 24 2.5 Cohen Macaulay modules.................. 25 2. (i Modules of finite projective dimension........... 27 2.7 The Koszul complex..................... 29 2.8 Regular local rings...................... 31 2.9 Projective dimension and depth............... 32 2.10 3-depth............................ 84 2.11 The acyclicity theorem.................... 36 2.12 An example.......................... 39 VÜ
Contents viii 3. Divisor Theory 3.1 Discrete valuation rings................... 43 3.2 Normal domains ....................... 44 3.3 Divisors............................ 46 3.4 Unique factorization..................... 47 3.5 Torsion modules ....................... 4 3.6 The first Chern class..................... 49 3.7 Regular local rings...................... 51 3.8 Picard groups......................... 51 3.9 Dodekind domains...................... 54 4. Completion 57 4.1 Exactness of the completion functor ............ 57 4.2 Separation of the 3-adic topology.............. 59 4.3 Complete filtered rings.................... 60 4.4 Completion of local rings .................. 61 4.5 Structure of complete local l ings.............. 63 5. Injective Modules 65 5.1 Injective module s....................... 65 5.2 Injective envelopes...................... 67 5.3 Decomposition of injective modules............. 68 5.4 Matlis duality......................... 70 5.5 Minimal injective resolutions ................ 73 5.6 Modules of finite injective dimension............ 74 5.7 Gore iistein rings ....................... 77 6. Local Cohomology 81 (i.l Basic prope rties........................ 81 0.2 Local cohomology and elime iision.............. 84 0.3 Local cohomology and di pth ................ 84 0.4 Support in the maximal ideal................ 85 0.5 Local duality for Gore iiste in rings.............. 87 7. Dualizing Comple xe s 89 7.1 Comi)l( X( s of injective modules............... 89 7.2 Complexes with finitely generated cohomology ...... 93 7.3 The e valuatiem map ..................... 96
Contents ix 7.4 Existence of dualizing complexes.............. 98 7.5 The codimension function.................. 100 7.6 Complexes of flat modules.................. 102 7.7 Generalized evaluation maps ................ 105 7.8 Uniqueness of dualizing complexes............. 107 8. Local Duality 109 8.1 Poincare series ........................ 109 8.2 Grotliendieck’s local duality theorem............ 113 8.3 Duality for Cohen-Macaulay modules ........... 117 8.4 Dualizing modules...................... 119 8.5 Locally factorial domains .................. 121 8.6 Conductors.......................... 122 8.7 Formal fibers......................... 125 9. Amplitude and Dimension 129 9.1 Depth of a complex...................... 130 9.2 The dual of a module .................... 136 9.3 The amplitude formula.................... 137 9.4 Dimension of a complex................... 139 9.5 The tensor product formula................. 142 9.6 Depth inequalities ...................... 144 9.7 Condition S r of Scrre .................... 148 9.8 Factorial rings and condition S, ................ 152 9.9 Condition S .......................... 155 9.10 Specialization of Poincare series............... 158 10. Intersection Multiplicities 161 10.1 Introduction to Serre s conjectures............. 161 10.2 Filtration of the Koszul complex.............. 163 10.3 Euler characteristic of the Koszul complex......... 167 10.4 A projection formula..................... 170 10.5 Power series over a field................... 171 10.6 Power series over a discrete valuation ring......... 175 10.7 Application of Cohen s structure theorem......... 178 10.8 The amplitude inequality .................. 181 10.9 Translation invariant operators............... 182 10.10 Todd operators........................ 184 10.11 Serre’s conjecture in the graded case............ 187
Contents 11. Complexes of Free Modules 189 11.1 McCoy s theorem....................... 189 11.2 The rank of a linear map .................. 191 11.3 The Eisenbud -Buchsbaum criterion ............ 194 11.4 Fitting’s ideals........................ 196 11.5 The Euler characteristic................... 199 11.6 McRae s invariant ...................... 203 11.7 The integral character of McRae’s invariant........ 205 Bibliography 207 Index 211
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| spelling | Iversen, Birger Verfasser aut Lecture notes on local rings by Birger Iversen ; ed. by Holger Andreas Nielsen Singapore World Scientific 2014 X, 213 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Stellenring (DE-588)4183087-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Stellenring (DE-588)4183087-8 s DE-604 Nielsen, Holger edt HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027475781&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Lecture notes on local rings |
| title_auth | Lecture notes on local rings |
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| title_full | Lecture notes on local rings by Birger Iversen ; ed. by Holger Andreas Nielsen |
| title_fullStr | Lecture notes on local rings by Birger Iversen ; ed. by Holger Andreas Nielsen |
| title_full_unstemmed | Lecture notes on local rings by Birger Iversen ; ed. by Holger Andreas Nielsen |
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