Combinatorial commutative algebra:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2005
|
| Series: | Graduate texts in mathematics
227 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 417 S. graph. Darst. |
| ISBN: | 0387237070 9780387223568 9780387237077 |
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MARC
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| 100 | 1 | |a Miller, Ezra |d 1974- |e Verfasser |0 (DE-588)1043618740 |4 aut | |
| 245 | 1 | 0 | |a Combinatorial commutative algebra |c Ezra Miller ; Bernd Sturmfels |
| 264 | 1 | |a New York, NY |b Springer |c 2005 | |
| 300 | |a XIV, 417 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 227 | |
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Record in the Search Index
| _version_ | 1804133201866653696 |
|---|---|
| adam_text | Contents
Preface
vii
I Monomial Ideals
1
1
Squarefree monomial ideals
3
1.1
Equivalent descriptions
..................... 3
1.2
Hubert series
.......................... 6
1.3
Simplicial complexes and homology
.............. 9
1.4
Monomial matrices
....................... 11
1.5
Betti
numbers
.......................... 14
Exercises
............................ 18
Notes
.............................. 19
2
Borei-
fixed monomial ideals
21
2.1
Group actions
.......................... 21
2.2
Generic initial ideals
...................... 24
2.3
The Eliahou-Korvaire resolution
............... 27
2.4
Lex-segment ideals
....................... 33
Exercises
............................ 39
Notes
.............................. 40
3
Three-dimensional staircases
41
3.1
Monomial ideals in two variables
............... 42
3.2
An example with six monomials
................ 44
3.3
The Buchberger graph
..................... 47
3.4
Gencricity and deformations
.................. 49
3.5
The planar resolution algorithm
................ 53
Exercises
............................ 58
Notes
.............................. 60
4
Cellular resolutions
61
4.1
Construction and exactness
.................. 62
4.2
Betti
numbers and
/■í-polynoinials
.............. 65
4.3
Examples of cellular resolutions
................ 67
4.4
The hull resolution
....................... 71
4.5
Subdividing the simplex
.................... 76
CONTENTS
Exercises
............................ 78
Notes .............................. 79
5 Alexander
duality
81
5.1 Simplicial Alexander
duality
.................. 81
5.2 Generators versus
irreducible components...........
87
5.3
Duality for resolutions
..................... 91
5.4
Cohull resolutions and other applications
.......... 95
5.5
Projective
dimension and regularity
............. 100
Exercises
............................ 104
Notes
.............................. 105
6
Generic monomial ideals
107
6.1
Taylor complexes and genericity
................ 107
6.2
The Scarf complex
....................... 110
6.3
Genericity by deformation
................... 115
6.4
Bounds on
Betti
numbers
................... 119
6.5
Cogeneric monomial ideals
................... 122
Exercises
............................ 125
Notes
.............................. 126
II Toric Algebra
127
7
Semigroup rings
129
7.1
Semigroups and lattice ideals
................. 129
7.2 Affine
semigroups and polyhedral cones
........... 133
7.3
Hubert bases
.......................... 137
7.4
Initial ideals of lattice ideals
.................. 142
Exercises
............................ 146
Notes
.............................. 148
8
Multigraded polynomial rings
149
8.1
Multigradings
.......................... 149
8.2
Hubert series and J-i-polyiiomials
............... 153
8.3
Multigraded
Betti
numbers
.................. 157
8.4
ii-polynomials in
nonpositive gradings............ 161
8.5
Multidegrees
.......................... 165
Exercises
............................ 170
Notes
.............................. 172
9
Syzygies of lattice ideals
173
9.1
Betti
numbers
.......................... 173
9.2
Laurent monomial modules
.................. 176
9.3
Free resolutions of lattice ideals
................ 181
9.4
Genericity and the Scarf complex
............... 187
Exercises
............................ 189
Notes
.............................. 190
CONTENTS
. xiii
10
Toric varieties
191
10.1
Abelian group actions
..................... 191
10.2
Projective
quotients
...................... 194
10.3
Constructing toric varieties
................... 198
10.4
Toric varieties as quotients
................... 203
Exercises
............................ 207
Notes
.............................. 208
11
Irreducible and injective resolutions
209
11.1
Irreducible resolutions
..................... 209
11.2
Injective modules
........................ 212
11.3
Monomial matrices revisited
.................. 215
11.4
Essential properties of injectives
............... 218
11.5
Injective hulls and resolutions
................. 221
Exercises
............................ 225
Notes
.............................. 227
12 Ehrhart
polynomials
229
12.1 Ehrhart
from Hubert
...................... 229
12.2
Dualizing complexes
...................... 232
12.3
Brion s Formula
......................... 236
12.4
Short rational generating functions
.............. 241
Exercises
............................ 245
Notes
.............................. 246
13
Local cohomology
247
13.1
Equivalent definitions
..................... 247
13.2
Hubert series calculations
................... 253
13.3
Toric local cohomology
..................... 256
13.4
Cohen-Macaulay conditions
.................. 262
13.5
Examples of Cohen-Macaulay rings
.............. 266
Exercises
............................ 268
Notes
.............................. 269
III Determinants
. 271
14
Pliicker coordinates
273
14.1
The complete flag variety
................... 273
14.2
Quadratic Pliicker relations
.................. 275
14.3
Minors form sagbi bases
.................... 279
14.4
Gelfand Tsetlin semigroups
.................. 284
Exercises
............................ 286
Notes
.............................. 287
xiv CONTENTS
15 Matrix Schubert
varieties
289
15.1 Schubert
detcrminantal ideals
................. 290
15.2
Essential sets
.......................... 294
15.3 Bruhat
and weak orders
.................... 295
15.4
Borei
group orbits
....................... 299
15.5
Schubert polynomials
...................... 304
Exercises
............................ 308
Notes
.............................. 309
16 Antidiagonal
initial ideals
311
16.1
Pipe dreams
........................... 312
16.2
A combinatorial formula
.................... 315
16.3 Antidiagonal
simplicial complexes
............... 318
16.4
Minors form
Gröbncr
bases
.................. 323
16.5
Subword
complexes
....................... 325
Exercises
............................ 328
Notes
.............................. 329
17
Minors in matrix products
331
17.1
Quiver ideals and quiver loci
................. 331
17.2
Zelevinsky map
......................... 336
17.3
Primality and Colien-Macaulayness
............. 341
17.4
Quiver polynomials
....................... 343
17.5
Pipes to laces
.......................... 348
Exercises
............................ 351
Notes
.............................. 352
18
Hilbert schemes of points
355
18.1
Ideals of points in the plane
.................. 355
18.2
Connectedness and smoothness
................ 359
18.3
Haiman s theory
........................ 363
18.4
Ideals of points in d-space
................... 368
18.5
Multigraded Hilbert schemes
................. 373
Exercises
............................ 377
Notes
.............................. 377
References
379
Glossary of notation
397
Index
401
|
| any_adam_object | 1 |
| author | Miller, Ezra 1974- Sturmfels, Bernd 1962- |
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| classification_rvk | SK 230 |
| classification_tum | MAT 050f MAT 130f |
| ctrlnum | (OCoLC)633168387 (DE-599)BVBBV019737198 |
| discipline | Mathematik |
| format | Book |
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| institution | BVB |
| isbn | 0387237070 9780387223568 9780387237077 |
| language | English |
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| physical | XIV, 417 S. graph. Darst. |
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| series2 | Graduate texts in mathematics |
| spelling | Miller, Ezra 1974- Verfasser (DE-588)1043618740 aut Combinatorial commutative algebra Ezra Miller ; Bernd Sturmfels New York, NY Springer 2005 XIV, 417 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 227 Kombinatorische Algebra (DE-588)4735525-6 gnd rswk-swf Kommutative Algebra (DE-588)4164821-3 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Kommutative Algebra (DE-588)4164821-3 s Kombinatorische Algebra (DE-588)4735525-6 s DE-604 Sturmfels, Bernd 1962- Verfasser (DE-588)121330435 aut Erscheint auch als Online-Ausgabe 978-0-387-27103-3 Graduate texts in mathematics 227 (DE-604)BV000000067 227 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013063978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Miller, Ezra 1974- Sturmfels, Bernd 1962- Combinatorial commutative algebra Graduate texts in mathematics Kombinatorische Algebra (DE-588)4735525-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| subject_GND | (DE-588)4735525-6 (DE-588)4164821-3 (DE-588)4123623-3 |
| title | Combinatorial commutative algebra |
| title_auth | Combinatorial commutative algebra |
| title_exact_search | Combinatorial commutative algebra |
| title_full | Combinatorial commutative algebra Ezra Miller ; Bernd Sturmfels |
| title_fullStr | Combinatorial commutative algebra Ezra Miller ; Bernd Sturmfels |
| title_full_unstemmed | Combinatorial commutative algebra Ezra Miller ; Bernd Sturmfels |
| title_short | Combinatorial commutative algebra |
| title_sort | combinatorial commutative algebra |
| topic | Kombinatorische Algebra (DE-588)4735525-6 gnd Kommutative Algebra (DE-588)4164821-3 gnd |
| topic_facet | Kombinatorische Algebra Kommutative Algebra Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013063978&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT millerezra combinatorialcommutativealgebra AT sturmfelsbernd combinatorialcommutativealgebra |