Twenty-four hours of local cohomology:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Providence, R.I
American Math. Soc.
2007
|
| Series: | Graduate studies in mathematics
87 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 282 S. graph. Darst. |
| ISBN: | 9780821841266 |
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MARC
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| 245 | 1 | 0 | |a Twenty-four hours of local cohomology |c Srikanth B. Iyengar ... |
| 264 | 1 | |a Providence, R.I |b American Math. Soc. |c 2007 | |
| 300 | |a XVIII, 282 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 87 | |
| 650 | 4 | |a Sheaf theory | |
| 650 | 4 | |a Algebra, Homological | |
| 650 | 4 | |a Group theory | |
| 650 | 4 | |a Cohomology operations | |
| 650 | 0 | 7 | |a Lokale Kohomologie |0 (DE-588)4168108-3 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)4173536-5 |a Patentschrift |2 gnd-content | |
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| 700 | 1 | |a Iyengar, Srikanth |d 1970- |e Sonstige |0 (DE-588)137542011 |4 oth | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-2117-5 |
| 830 | 0 | |a Graduate studies in mathematics |v 87 |w (DE-604)BV009739289 |9 87 | |
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Record in the Search Index
| _version_ | 1804138017807400961 |
|---|---|
| adam_text | Contents
Preface
xiii
Introduction
xv
Lecture
1.
Basic Notions
1
§1.
Algebraic sets
1
§2.
Krull dimension
of
a ring
3
§3.
Dimension
of an algebraic set
6
§4.
An
extended example
9
§5.
Tangent
spaces and regular
rings
10
§6.
Dimension
of
a module
12
Lecture
2.
Cohomology
15
§1.
Sheaves
16
§2.
Čech
cohomology
18
§3.
Calculus
versus
topology
23
§4.
Čech
cohomology
and derived functors
26
Lecture
3.
Resolutions and Derived Functors
29
§1.
Eree,
projective,
and flat modules
29
§2.
Complexes
32
§3.
Resolutions
34
§4.
Derived functors
36
Lecture
4.
Limits
41
§1.
An example from topology
41
vii
viii Contents
§2.
Direct
limits
42
§3.
The category of diagrams
44
§4.
Exactness
45
§5.
Diagrams over diagrams
48
§6.
Filtered posets
49
§7.
Diagrams over the pushout
poset
52
§8.
Inverse limits
53
Lecture
5.
Gradings, Filtrations, and
Gröbner
Bases
55
§1.
Filtrations and associated graded rings
55
§2.
Hilbert polynomials
57
§3.
Monomial orders and initial forms
59
§4.
Weight vectors and flat families
61
§5.
Buchberger s algorithm
62
§6. Gröbner
bases and syzygies
65
Lecture
6.
Complexes from a Sequence of Ring Elements
67
§1.
The
Koszul
complex
67
§2.
Regular sequences and depth: a first look
69
§3.
Back to the
Koszul
complex
70
§4.
The
Čech
complex
73
Lecture
7.
Local Cohomology
77
§1.
The torsion functor
77
§2.
Direct limit of Ext modules
80
§3.
Direct limit of
Koszul
cohomology
81
§4.
Return of the
Čech
complex
84
Lecture
8.
Auslander-Buchsbaum Formula and Global Dimension
87
§1.
Regular sequences and depth redux
87
§2.
Global dimension
89
§3.
Auslander-Buchsbaum formula
91
§4.
Regular local rings
92
§5.
Complete local rings
96
Lecture
9.
Depth and Cohomological Dimension
97
§1.
Depth
97
§2.
Cohomological dimension
100
Contents ix
§3.
Arithmetic rank
101
Lecture
10.
Cohen-Macaulay Rings
105
§1.
Noether normalization
106
§2.
Intersection multiplicities
108
§3.
Invariant theory
110
§4.
Local cohomology
115
Lecture
11.
Gorenstein Rings
117
§1.
Bass numbers
118
§2.
Recognizing Gorenstein rings
120
§3.
Injective resolutions of Gorenstein rings
123
§4.
Local duality
123
§5.
Canonical modules
126
Lecture
12.
Connections with Sheaf Cohomology
131
§1.
Sheaf theory
131
§2.
Flasque
sheaves
137
§3.
Local cohomology and sheaf cohomology
139
Lecture
13.
Projective
Varieties
141
§1.
Graded local cohomology
141
§2.
Sheaves on
projective
varieties
142
§3.
Global sections and cohomology
144
Lecture
14.
The Hartshorne-Lichtenbaum Vanishing Theorem
147
Lecture
15.
Connectedness
153
§1.
Mayer-Vietoris sequence
153
§2.
Punctured spectra
154
Lecture
16.
Polyhedral Applications
159
§1.
Polytopes and faces
159
§2.
Upper bound theorem
161
§3.
The h-vector of a simplicial complex
163
§4.
Stanley-Reisner rings
164
§5.
Local cohomology of Stanley-Reisner rings
166
§6.
Proof of the upper bound theorem
168
Lecture
17.
D-modules
171
x
Contents
§1.
Rings of differential operators
171
§2.
The Weyl algebra
173
§3.
Holonomic modules
176
§4. Gröbner
bases
177
Lecture
18.
Local Duality Revisited
179
§1.
Poincaré
duality
179
§2.
Grothendieck duality
180
§3.
Local duality
181
§4.
Global canonical modules
183
Lecture
19. De Rham
Cohomology
191
§1.
The real case:
de Rham s
theorem
192
§2.
Complex manifolds
195
§3.
The algebraic case
198
§4.
Local and
de Rham
cohomology
200
Lecture
20.
Local Cohomology over Semigroup Rings
203
§1.
Semigroup rings
203
§2.
Cones from semigroups
205
§3.
Maximal support: the Ishida complex
207
§4.
Monomial support: Zd-graded injectives
211
§5.
Hartshorne s example
213
Lecture
21.
The Frobenius Endomorphism
217
§1.
Homological properties
217
§2.
Probenius action on local cohomology modules
221
§3.
A vanishing theorem
225
Lecture
22.
Curious Examples
229
§1.
Dependence on characteristic
229
§2.
Associated primes of local cohomology modules
233
Lecture
23.
Algorithmic Aspects of Local Cohomology
239
§1.
Holonomicity of localization
239
§2.
Local cohomology as a D-module
241
§3.
Bernstein-Sato polynomials
242
§4.
Computing with the Probenius morphism
246
Lecture
24.
Holonomic Rank and Hypergeometric Systems
247
Contents xi
§1. GKZ .A-hypergeometric
systems
247
§2.
Rank vs. volume
250
§3. Euler-Koszul homology 251
§4. Holonomic
families
254
Appendix. Injective Modules
and Matlis Duality
257
§1.
Essential extensions
257
§2. Noetherian rings 260
§3. Artinian rings 263
§4. Matlis
duality
265
Bibliography
269
Index 277
|
| adam_txt |
Contents
Preface
xiii
Introduction
xv
Lecture
1.
Basic Notions
1
§1.
Algebraic sets
1
§2.
Krull dimension
of
a ring
3
§3.
Dimension
of an algebraic set
6
§4.
An
extended example
9
§5.
Tangent
spaces and regular
rings
10
§6.
Dimension
of
a module
12
Lecture
2.
Cohomology
15
§1.
Sheaves
16
§2.
Čech
cohomology
18
§3.
Calculus
versus
topology
23
§4.
Čech
cohomology
and derived functors
26
Lecture
3.
Resolutions and Derived Functors
29
§1.
Eree,
projective,
and flat modules
29
§2.
Complexes
32
§3.
Resolutions
34
§4.
Derived functors
36
Lecture
4.
Limits
41
§1.
An example from topology
41
vii
viii Contents
§2.
Direct
limits
42
§3.
The category of diagrams
44
§4.
Exactness
45
§5.
Diagrams over diagrams
48
§6.
Filtered posets
49
§7.
Diagrams over the pushout
poset
52
§8.
Inverse limits
53
Lecture
5.
Gradings, Filtrations, and
Gröbner
Bases
55
§1.
Filtrations and associated graded rings
55
§2.
Hilbert polynomials
57
§3.
Monomial orders and initial forms
59
§4.
Weight vectors and flat families
61
§5.
Buchberger's algorithm
62
§6. Gröbner
bases and syzygies
65
Lecture
6.
Complexes from a Sequence of Ring Elements
67
§1.
The
Koszul
complex
67
§2.
Regular sequences and depth: a first look
69
§3.
Back to the
Koszul
complex
70
§4.
The
Čech
complex
73
Lecture
7.
Local Cohomology
77
§1.
The torsion functor
77
§2.
Direct limit of Ext modules
80
§3.
Direct limit of
Koszul
cohomology
81
§4.
Return of the
Čech
complex
84
Lecture
8.
Auslander-Buchsbaum Formula and Global Dimension
87
§1.
Regular sequences and depth redux
87
§2.
Global dimension
89
§3.
Auslander-Buchsbaum formula
91
§4.
Regular local rings
92
§5.
Complete local rings
96
Lecture
9.
Depth and Cohomological Dimension
97
§1.
Depth
97
§2.
Cohomological dimension
100
Contents ix
§3.
Arithmetic rank
101
Lecture
10.
Cohen-Macaulay Rings
105
§1.
Noether normalization
106
§2.
Intersection multiplicities
108
§3.
Invariant theory
110
§4.
Local cohomology
115
Lecture
11.
Gorenstein Rings
117
§1.
Bass numbers
118
§2.
Recognizing Gorenstein rings
120
§3.
Injective resolutions of Gorenstein rings
123
§4.
Local duality
123
§5.
Canonical modules
126
Lecture
12.
Connections with Sheaf Cohomology
131
§1.
Sheaf theory
131
§2.
Flasque
sheaves
137
§3.
Local cohomology and sheaf cohomology
139
Lecture
13.
Projective
Varieties
141
§1.
Graded local cohomology
141
§2.
Sheaves on
projective
varieties
142
§3.
Global sections and cohomology
144
Lecture
14.
The Hartshorne-Lichtenbaum Vanishing Theorem
147
Lecture
15.
Connectedness
153
§1.
Mayer-Vietoris sequence
153
§2.
Punctured spectra
154
Lecture
16.
Polyhedral Applications
159
§1.
Polytopes and faces
159
§2.
Upper bound theorem
161
§3.
The h-vector of a simplicial complex
163
§4.
Stanley-Reisner rings
164
§5.
Local cohomology of Stanley-Reisner rings
166
§6.
Proof of the upper bound theorem
168
Lecture
17.
D-modules
171
x
Contents
§1.
Rings of differential operators
171
§2.
The Weyl algebra
173
§3.
Holonomic modules
176
§4. Gröbner
bases
177
Lecture
18.
Local Duality Revisited
179
§1.
Poincaré
duality
179
§2.
Grothendieck duality
180
§3.
Local duality
181
§4.
Global canonical modules
183
Lecture
19. De Rham
Cohomology
191
§1.
The real case:
de Rham's
theorem
192
§2.
Complex manifolds
195
§3.
The algebraic case
198
§4.
Local and
de Rham
cohomology
200
Lecture
20.
Local Cohomology over Semigroup Rings
203
§1.
Semigroup rings
203
§2.
Cones from semigroups
205
§3.
Maximal support: the Ishida complex
207
§4.
Monomial support: Zd-graded injectives
211
§5.
Hartshorne's example
213
Lecture
21.
The Frobenius Endomorphism
217
§1.
Homological properties
217
§2.
Probenius action on local cohomology modules
221
§3.
A vanishing theorem
225
Lecture
22.
Curious Examples
229
§1.
Dependence on characteristic
229
§2.
Associated primes of local cohomology modules
233
Lecture
23.
Algorithmic Aspects of Local Cohomology
239
§1.
Holonomicity of localization
239
§2.
Local cohomology as a D-module
241
§3.
Bernstein-Sato polynomials
242
§4.
Computing with the Probenius morphism
246
Lecture
24.
Holonomic Rank and Hypergeometric Systems
247
Contents xi
§1. GKZ .A-hypergeometric
systems
247
§2.
Rank vs. volume
250
§3. Euler-Koszul homology 251
§4. Holonomic
families
254
Appendix. Injective Modules
and Matlis Duality
257
§1.
Essential extensions
257
§2. Noetherian rings 260
§3. Artinian rings 263
§4. Matlis
duality
265
Bibliography
269
Index 277 |
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| id | DE-604.BV035069985 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:03:52Z |
| indexdate | 2024-07-09T21:21:30Z |
| institution | BVB |
| isbn | 9780821841266 |
| language | English |
| lccn | 007060786 |
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| series2 | Graduate studies in mathematics |
| spelling | Twenty-four hours of local cohomology Srikanth B. Iyengar ... Providence, R.I American Math. Soc. 2007 XVIII, 282 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 87 Sheaf theory Algebra, Homological Group theory Cohomology operations Lokale Kohomologie (DE-588)4168108-3 gnd rswk-swf (DE-588)4173536-5 Patentschrift gnd-content Lokale Kohomologie (DE-588)4168108-3 s DE-604 Iyengar, Srikanth 1970- Sonstige (DE-588)137542011 oth Erscheint auch als Online-Ausgabe 978-1-4704-2117-5 Graduate studies in mathematics 87 (DE-604)BV009739289 87 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Twenty-four hours of local cohomology Graduate studies in mathematics Sheaf theory Algebra, Homological Group theory Cohomology operations Lokale Kohomologie (DE-588)4168108-3 gnd |
| subject_GND | (DE-588)4168108-3 (DE-588)4173536-5 |
| title | Twenty-four hours of local cohomology |
| title_auth | Twenty-four hours of local cohomology |
| title_exact_search | Twenty-four hours of local cohomology |
| title_exact_search_txtP | Twenty-four hours of local cohomology |
| title_full | Twenty-four hours of local cohomology Srikanth B. Iyengar ... |
| title_fullStr | Twenty-four hours of local cohomology Srikanth B. Iyengar ... |
| title_full_unstemmed | Twenty-four hours of local cohomology Srikanth B. Iyengar ... |
| title_short | Twenty-four hours of local cohomology |
| title_sort | twenty four hours of local cohomology |
| topic | Sheaf theory Algebra, Homological Group theory Cohomology operations Lokale Kohomologie (DE-588)4168108-3 gnd |
| topic_facet | Sheaf theory Algebra, Homological Group theory Cohomology operations Lokale Kohomologie Patentschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738380&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT iyengarsrikanth twentyfourhoursoflocalcohomology |