Constructive commutative algebra: projective modules over polynomial rings and dynamical Gröbner bases
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham ; Heidelberg ; New York ; Dordrecht ; London
Springer
2015
|
| Schriftenreihe: | Lecture notes in mathematics
2138 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VII, 271 Seiten |
| ISBN: | 9783319194936 |
Internformat
MARC
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|---|---|---|---|
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| 100 | 1 | |a Yengui, Ihsen |4 aut | |
| 245 | 1 | 0 | |a Constructive commutative algebra |b projective modules over polynomial rings and dynamical Gröbner bases |c Ihsen Yengui |
| 264 | 1 | |a Cham ; Heidelberg ; New York ; Dordrecht ; London |b Springer |c 2015 | |
| 300 | |a VII, 271 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 2138 | |
| 650 | 0 | 7 | |a Kommutative Algebra |0 (DE-588)4164821-3 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Kommutative Algebra |0 (DE-588)4164821-3 |D s |
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| 830 | 0 | |a Lecture notes in mathematics |v 2138 |w (DE-604)BV000676446 |9 2138 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028336943&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-028336943 | ||
Datensatz im Suchindex
| _version_ | 1804175205661147136 |
|---|---|
| adam_text | Contents
1 Introduction 1
2 Projective Modules Over Polynomial Rings 9
2.1 Quillen’s Proof of Serre’s Problem.............................. 9
2.1.1 Finitely-Generated Projective Modules .................... 9
2.1.2 Finitely-Generated Stably Free Modules............... . 16
2.1.3 Concrete Local-Global Principle ......................... 21
2.1.4 The Patchings of Quillen and Vaserstein................... 27
2.1.5 Horrocks’ Theorem....................................... 28
2.1.6 Quillen Induction Theorem................................. 29
2.2 Suslin’s Proof of Serre’s Problem................................. 31
2.2.1 Making the Use of Maximal Ideals Constructive............. 31
2.2.2 A Reminder About the Resultant .......................... 31
2.2.3 A Lemma of Suslin........................................ 34
2.2.4 A More General Strategy (By ‘‘Backtracking”)......... . 36
2.2.5 Suslin’s Lemma for Rings Containing an Infinite
Field .................................................. 38
2.2.6 Suslin’s Algorithm........................................ 41
2.2.7 Suslin’s Solution to Serre’s Problem................... 47
2.2.8 A Simple Result About Coherent Rings..................... 48
2.3 Constructive Definitions of Krull
Dimension.................................................... 50
2.3.1 Ideals and Filters .................................... 50
2.3.2 Zariski Lattice....................................... 51
2.3.3 Krull Boundary ......................................... 51
2.3.4 Pseudo-Regular Sequences and Krull Dimension.............. 53
2.3.5 Krull Dimension of a Polynomial Ring Over
a Discrete Field........................................ 55
2.3.6 Application to the Stable Range Theorem.............. . 56
2.3.7 SemFs Splitting Theorem and Forster-Swan
Theorem............................................ 57
2.3.8 Support on a Ring and «-Stability ........................ 58
V
YI
CONTENTS
2.4 Projective Modules Over R[Xi,... ,Xrt],
R an Arithmetical Ring......................................... 66
2.4.1 A Constructive Proof of Brewer-Costa-Maroscia
Theorem................................................. 66
2.4.2 The Theorem of Lequain, Simis and Vasconcelos.......... 71
2.5 Suslin’s Stability Theorem..................................... 76
2.5.1 “Obvious” Syzygies ..................................... 76
2.5.2 E2(R) as a Subgroup of SL2(R).......................... 77
2.5.3 Suslin’s Normality Theorem.............................. 83
2.5.4 Unimodular Rows and Elementary Operations .............. 86
2.5.5 Local-Global Principle for Elementary Polynomial
Matrices................................................ 88
2.5.6 A Realization Algorithm for SL3 (R[AT])................ 92
2.5.7 Elementary Unimodular Completion........................ 93
2.6 The Hermite Ring Conjecture.................................... 95
2.6.1 The Hermite Ring Conjecture in Dimension One............ 95
2.6.2 Stably Free Modules Over R[X] of Rank dim R
are Free.............................................. 99
2.6.3 Two New Conjectures.....................................101
3 Dynamical Grobner Bases 105
3.1 Dickson’s Lemma and the Division Algorithm.....................105
3.2 Grobner Rings..................................................112
3.3 Grobner Bases Over Strongly Discrete Coherent Arithmetical
Rings..........................................................117
3.3.1 Grobner Bases Over a Coherent Valuation Ring............117
3.3.2 Grobner Bases Over Z/parL...............................126
3.3.3 When a Valuation Domain Is Grobner?.....................128
3.3.4 When a Coherent Valuation Ring with Zero-Divisors
is Grobner?........................................... 133
3.3.5 Dynamical Grobner Bases Over Grobner Arithmetical
Rings................................................. 138
3.3.6 A Parallelisable Algorithm for Computing Dynamical
Grobner Bases Over Z/mZ via the Chinese Remainder
Theorem.................................................142
3.3.7 A Parallelisable Algorithm for Computing Grobner Bases
Over (Z/paZ) x (Z/paZ) .................................144
3.3.8 Dynamical Grobner Bases Over
F 2[a7b]/{a2-a,b2-b).................................. 147
3.3.9 Dynamical Grobner Bases Over the Integers ........ 149
3.3.10 Grobner Bases Over the Integers via Prime
Factorization......................................... 151
3.3.11 A Branching-Free Algorithm for Computing Grobner Bases
Over the Integers................................... 153
CONTENTS
VII
3.4 Computing Syzygy Modules with Polynomial Rings Over
Gröbner Arithmetical Rings .....................................156
3.4.1 Computing Syzygy Modules with Polynomial Rings
Over Gröbner Valuation Rings.............................156
3.4.2 Computing Dynamically a Generating Set for Syzygies
of Polynomials Over Gröbner Arithmetical Rings...........166
4 Syzygies in Polynomial Rings Over Valuation Domains 173
4.1 Preliminary Tools...............................................174
4.2 Saturation of Finitely-Generated Sub-V-Modules
of V[Xi .................*..........................177
4.3 Saturation of a Finitely-Generated V[X]-Module, with V a Valuation
Domain........................................................ 186
4.4 Computing Syzygies Over R[X], with R a Prüfer Domain............194
4.4.1 The Case of a Valuation Domain...........................194
4.4.2 The Case of a Prüfer Domain..............................196
4.5 The Multivariate Case...........................................196
4.5.1 Hilbert Series......................................... 196
4.5.2 The Saturation Defect Series ............................199
4.5.3 A Saturation Algorithm in the Multivariate Case ...... 202
5 Exercises 207
6 Detailed Solutions to the Exercises 221
Notation List 255
Bibliography 259
Index
269
|
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| id | DE-604.BV042909073 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:12:35Z |
| institution | BVB |
| isbn | 9783319194936 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028336943 |
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| owner_facet | DE-83 DE-824 DE-188 DE-29T DE-355 DE-BY-UBR |
| physical | VII, 271 Seiten |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Yengui, Ihsen aut Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases Ihsen Yengui Cham ; Heidelberg ; New York ; Dordrecht ; London Springer 2015 VII, 271 Seiten txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 2138 Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-19494-3 Lecture notes in mathematics 2138 (DE-604)BV000676446 2138 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028336943&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yengui, Ihsen Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases Lecture notes in mathematics Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4164821-3 |
| title | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases |
| title_auth | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases |
| title_exact_search | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases |
| title_full | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases Ihsen Yengui |
| title_fullStr | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases Ihsen Yengui |
| title_full_unstemmed | Constructive commutative algebra projective modules over polynomial rings and dynamical Gröbner bases Ihsen Yengui |
| title_short | Constructive commutative algebra |
| title_sort | constructive commutative algebra projective modules over polynomial rings and dynamical grobner bases |
| title_sub | projective modules over polynomial rings and dynamical Gröbner bases |
| topic | Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Kommutative Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028336943&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT yenguiihsen constructivecommutativealgebraprojectivemodulesoverpolynomialringsanddynamicalgrobnerbases |