Ideals and reality: projective modules and number of generators of ideals
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIV, 336 S. graph. Darst. |
| ISBN: | 3540230327 9783540230328 |
Staff View
MARC
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| 100 | 1 | |a Ischebeck, Friedrich |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Ideals and reality |b projective modules and number of generators of ideals |c Friedrich Ischebeck ; Ravi A. Rao |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2005 | |
| 300 | |a XIV, 336 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a Springer monographs in mathematics | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Algebraische K-Theorie - Kommutative Algebra - Projektiver Modul - Serre-Vermutung - Vollständiger Durchschnitt | |
| 650 | 4 | |a Commutative rings | |
| 650 | 4 | |a Ideals (Algebra) |x Generators | |
| 650 | 4 | |a Projective modules (Algebra) | |
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Record in the Search Index
| _version_ | 1804137168237494272 |
|---|---|
| adam_text | Contents
1 Basic
Commutative
Algebra 1............................. 1
1.1 The Spectrum .......................................... 1
1.2 Modules................................................ 6
1.3
Localization
............................................ 15
1.3.1
Multiplicatively Closed Subsets .....................
15
1.3.2 Rings and Modules
of Fractions.....................
17
1.3.3
Localization Technique
............................. 20
1.3.4
Prime Ideals of a Localized Ring
.................... 21
1.4
Integral Ring Extensions
................................. 23
1.4.1
Integral Elements
................................. 23
1.4.2
Integrality and Primes
............................. 27
1.5
Direct Sums and Products
................................ 30
1.6
The Tensor Product
..................................... 38
1.6.1
Definition
........................................ 38
1.6.2
Functoriality
...................................... 42
1.6.3
Exactness
........................................ 43
1.6.4
Flat Algebras
..................................... 45
1.6.5
Exterior Powers
................................... 46
xii Contents
2
Introduction
to
Projective
Modules
........................ 49
2.1
Generalities on
Projective
Modules
........................ 49
2.2
Rank
.................................................. 52
2.3
Special Residue Class Rings
............................... 56
2.4
Projective
Modules of Rank
1............................. 59
3
Stably Free Modules
....................................... 69
3.1
Generalities
............................................. 69
3.2
Localized Polynomial Rings
............................... 74
3.3
Action of
GLn(iï) on Umn(iî)
............................ 75
3.4
Elementary Action on Unimodular Rows
................... 76
3.5
Examples of
Completatile
Rows
........................... 80
3.6
Direct sums of a stably free module
........................ 82
3.7
Stable Freeness over Polynomial Rings
..................... 83
3.7.1
Schanueľs
Lemma
................................. 83
3.7.2
Proof of Stable Freeness
............................ 84
4
Serre s Conjecture
......................................... 87
■4.1
Elementary Divisors
..................................... 87
4.2
Horrocks
Theorem
...................................... 89
4.3
Quillen s Local Global Principle
........................... 91
4.4
Suslin s Proof
........................................... 98
4.5
Vaserstein s Proof
.......................................101
5
Continuous Vector Bundles
................................105
5.1
Categories and Functors
..................................105
5.2
Vector Bundles
..........................................108
5.3
Vector Bundles and
Projective
Modules
....................114
5.4
Examples
...............................................121
5.5
Vector Bundles and Grassmannians
........................126
5.5.1
The Direct Limit and Infinite Matrices
...............126
Contents xiii
5.5.2 Metrization
of the Set of Continuous Maps
...........128
5.5.3
Correspondence of Vector Bundles and Classes of Maps
129
5.6
Projective
Spaces
........................................131
5.7
Algebraization of Vector Bundles
..........................135
5.7.1
Projective
Modules over Topological Rings
...........136
5.7.2
Projective
Modules as Pull-Backs
....................140
5.7.3
Construction of a Noetherian
Subalgebra
.............143
6
Basic Commutative Algebra II
.............................149
6.1
Noetherian Rings and Modules
............................149
6.2
Irreducible Sets
.........................................152
6.3
Dimension of Rings
......................................154
6.4
Artinian Rings
..........................................156
6.5
Small Dimension Theorem
................................158
6.6
Noether Normalization
...................................162
6.7
Affine
Algebras
..........................................164
6.8
Hubert s
Nullstellensatz..................................168
6.9
Dimension of a Polynomial ring
...........................171
7
Splitting Theorem and Lindel s Proof
......................175
7.1
Serre s Splitting Theorem
.................................175
7.2
Lindel s Proof
...........................................180
8
Regular Rings
.............................................185
8.1
Definition
..............................................185
8.2
Regular Residue Class Rings
..............................187
8.3
Homological Dimension
..................................189
8.4
Associated Prime Ideals
..................................194
8.5
Homological Characterization
.............................196
8.6
Dedekind Rings
.........................................200
8.7
Examples
...............................................203
8.8
Modules over Dedekind Rings
.............................205
8.9
Finiteness of Class Numbers
..............................209
xiv Contents
9
Number of Generators
.....................................213
9.1
The Problems
...........................................213
9.2
Regular sequences
.......................................216
9.3
Forster-Swan Theorem
...................................221
9.3.1
Basic elements
....................................226
9.3.2
Basic elements and the Forster-Swan theorem
.........227
9.3.3
Forster-Swan theorem via K-theory
.................228
9.4
Eisenbud-Evans theorem
.................................231
9.5
Varieties as Intersections of
η
Hypersurfaces
................236
9.6
The Eisenbud-Evans conjectures
...........................238
10
Curves as Complete Intersection
...........................241
10.1
A Motivation of Serre s Conjecture
........................241
10.2
The Conormal Module
...................................243
10.3
Local Complete Intersection Curves
........................245
10.4
Cowsik-Nori Theorem
....................................248
10.4.1
A Projection Lemma
...............................248
10.4.2
Proof of Cowsik-Nori
..............................251
A Normality of
E„
in
GL„...................................253
В
Some Homological Algebra
................................257
B.I Extensions and Ext1
.....................................257
B.2 Derived functors
.........................................264
С
Complete intersections and Connectedness
................269
D
Odds and Ends
............................................271
E
Exercises
..................................................289
References
.....................................................325
Index
..........................................................333
|
| adam_txt |
Contents
1 Basic
Commutative
Algebra 1. 1
1.1 The Spectrum . 1
1.2 Modules. 6
1.3
Localization
. 15
1.3.1
Multiplicatively Closed Subsets .
15
1.3.2 Rings and Modules
of Fractions.
17
1.3.3
Localization Technique
. 20
1.3.4
Prime Ideals of a Localized Ring
. 21
1.4
Integral Ring Extensions
. 23
1.4.1
Integral Elements
. 23
1.4.2
Integrality and Primes
. 27
1.5
Direct Sums and Products
. 30
1.6
The Tensor Product
. 38
1.6.1
Definition
. 38
1.6.2
Functoriality
. 42
1.6.3
Exactness
. 43
1.6.4
Flat Algebras
. 45
1.6.5
Exterior Powers
. 46
xii Contents
2
Introduction
to
Projective
Modules
. 49
2.1
Generalities on
Projective
Modules
. 49
2.2
Rank
. 52
2.3
Special Residue Class Rings
. 56
2.4
Projective
Modules of Rank
1. 59
3
Stably Free Modules
. 69
3.1
Generalities
. 69
3.2
Localized Polynomial Rings
. 74
3.3
Action of
GLn(iï) on Umn(iî)
. 75
3.4
Elementary Action on Unimodular Rows
. 76
3.5
Examples of
Completatile
Rows
. 80
3.6
Direct sums of a stably free module
. 82
3.7
Stable Freeness over Polynomial Rings
. 83
3.7.1
Schanueľs
Lemma
. 83
3.7.2
Proof of Stable Freeness
. 84
4
Serre's Conjecture
. 87
■4.1
Elementary Divisors
. 87
4.2
Horrocks'
Theorem
. 89
4.3
Quillen's Local Global Principle
. 91
4.4
Suslin's Proof
. 98
4.5
Vaserstein's Proof
.101
5
Continuous Vector Bundles
.105
5.1
Categories and Functors
.105
5.2
Vector Bundles
.108
5.3
Vector Bundles and
Projective
Modules
.114
5.4
Examples
.121
5.5
Vector Bundles and Grassmannians
.126
5.5.1
The Direct Limit and Infinite Matrices
.126
Contents xiii
5.5.2 Metrization
of the Set of Continuous Maps
.128
5.5.3
Correspondence of Vector Bundles and Classes of Maps
129
5.6
Projective
Spaces
.131
5.7
Algebraization of Vector Bundles
.135
5.7.1
Projective
Modules over Topological Rings
.136
5.7.2
Projective
Modules as Pull-Backs
.140
5.7.3
Construction of a Noetherian
Subalgebra
.143
6
Basic Commutative Algebra II
.149
6.1
Noetherian Rings and Modules
.149
6.2
Irreducible Sets
.152
6.3
Dimension of Rings
.154
6.4
Artinian Rings
.156
6.5
Small Dimension Theorem
.158
6.6
Noether Normalization
.162
6.7
Affine
Algebras
.164
6.8
Hubert's
Nullstellensatz.168
6.9
Dimension of a Polynomial ring
.171
7
Splitting Theorem and Lindel's Proof
.175
7.1
Serre's Splitting Theorem
.175
7.2
Lindel's Proof
.180
8
Regular Rings
.185
8.1
Definition
.185
8.2
Regular Residue Class Rings
.187
8.3
Homological Dimension
.189
8.4
Associated Prime Ideals
.194
8.5
Homological Characterization
.196
8.6
Dedekind Rings
.200
8.7
Examples
.203
8.8
Modules over Dedekind Rings
.205
8.9
Finiteness of Class Numbers
.209
xiv Contents
9
Number of Generators
.213
9.1
The Problems
.213
9.2
Regular sequences
.216
9.3
Forster-Swan Theorem
.221
9.3.1
Basic elements
.226
9.3.2
Basic elements and the Forster-Swan theorem
.227
9.3.3
Forster-Swan theorem via K-theory
.228
9.4
Eisenbud-Evans theorem
.231
9.5
Varieties as Intersections of
η
Hypersurfaces
.236
9.6
The Eisenbud-Evans conjectures
.238
10
Curves as Complete Intersection
.241
10.1
A Motivation of Serre's Conjecture
.241
10.2
The Conormal Module
.243
10.3
Local Complete Intersection Curves
.245
10.4
Cowsik-Nori Theorem
.248
10.4.1
A Projection Lemma
.248
10.4.2
Proof of Cowsik-Nori
.251
A Normality of
E„
in
GL„.253
В
Some Homological Algebra
.257
B.I Extensions and Ext1
.257
B.2 Derived functors
.264
С
Complete intersections and Connectedness
.269
D
Odds and Ends
.271
E
Exercises
.289
References
.325
Index
.333 |
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| id | DE-604.BV022934718 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:55:55Z |
| indexdate | 2024-07-09T21:08:00Z |
| institution | BVB |
| isbn | 3540230327 9783540230328 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016139485 |
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| physical | XIV, 336 S. graph. Darst. |
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| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Ischebeck, Friedrich Verfasser aut Ideals and reality projective modules and number of generators of ideals Friedrich Ischebeck ; Ravi A. Rao Berlin [u.a.] Springer 2005 XIV, 336 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Includes bibliographical references and index Algebraische K-Theorie - Kommutative Algebra - Projektiver Modul - Serre-Vermutung - Vollständiger Durchschnitt Commutative rings Ideals (Algebra) Generators Projective modules (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd rswk-swf Projektiver Modul (DE-588)4175892-4 gnd rswk-swf Vollständiger Durchschnitt (DE-588)4188587-9 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Serre-Vermutung (DE-588)4181056-9 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 s Projektiver Modul (DE-588)4175892-4 s Serre-Vermutung (DE-588)4181056-9 s Algebraische K-Theorie (DE-588)4141839-6 s Vollständiger Durchschnitt (DE-588)4188587-9 s DE-604 Rao, Ravi A. Verfasser aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016139485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ischebeck, Friedrich Rao, Ravi A. Ideals and reality projective modules and number of generators of ideals Algebraische K-Theorie - Kommutative Algebra - Projektiver Modul - Serre-Vermutung - Vollständiger Durchschnitt Commutative rings Ideals (Algebra) Generators Projective modules (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd Projektiver Modul (DE-588)4175892-4 gnd Vollständiger Durchschnitt (DE-588)4188587-9 gnd Kommutative Algebra (DE-588)4164821-3 gnd Serre-Vermutung (DE-588)4181056-9 gnd |
| subject_GND | (DE-588)4141839-6 (DE-588)4175892-4 (DE-588)4188587-9 (DE-588)4164821-3 (DE-588)4181056-9 |
| title | Ideals and reality projective modules and number of generators of ideals |
| title_auth | Ideals and reality projective modules and number of generators of ideals |
| title_exact_search | Ideals and reality projective modules and number of generators of ideals |
| title_exact_search_txtP | Ideals and reality projective modules and number of generators of ideals |
| title_full | Ideals and reality projective modules and number of generators of ideals Friedrich Ischebeck ; Ravi A. Rao |
| title_fullStr | Ideals and reality projective modules and number of generators of ideals Friedrich Ischebeck ; Ravi A. Rao |
| title_full_unstemmed | Ideals and reality projective modules and number of generators of ideals Friedrich Ischebeck ; Ravi A. Rao |
| title_short | Ideals and reality |
| title_sort | ideals and reality projective modules and number of generators of ideals |
| title_sub | projective modules and number of generators of ideals |
| topic | Algebraische K-Theorie - Kommutative Algebra - Projektiver Modul - Serre-Vermutung - Vollständiger Durchschnitt Commutative rings Ideals (Algebra) Generators Projective modules (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd Projektiver Modul (DE-588)4175892-4 gnd Vollständiger Durchschnitt (DE-588)4188587-9 gnd Kommutative Algebra (DE-588)4164821-3 gnd Serre-Vermutung (DE-588)4181056-9 gnd |
| topic_facet | Algebraische K-Theorie - Kommutative Algebra - Projektiver Modul - Serre-Vermutung - Vollständiger Durchschnitt Commutative rings Ideals (Algebra) Generators Projective modules (Algebra) Algebraische K-Theorie Projektiver Modul Vollständiger Durchschnitt Kommutative Algebra Serre-Vermutung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016139485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ischebeckfriedrich idealsandrealityprojectivemodulesandnumberofgeneratorsofideals AT raoravia idealsandrealityprojectivemodulesandnumberofgeneratorsofideals |