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Miller E., Sturmfels B. — Combinatorial Commutative Algebra
Miller E., Sturmfels B. — Combinatorial Commutative Algebra

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Название: Combinatorial Commutative Algebra

Авторы: Miller E., Sturmfels B.

Аннотация:

Combinatorial commutative algebra is an active area of research with thriving connections to other fields of pure and applied mathematics. This book provides a self-contained introduction to the subject, with an emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determinantal rings. The eighteen chapters cover a broad spectrum of topics, ranging from homological invariants of monomial ideals and their polyhedral resolutions, to hands-on tools for studying algebraic varieties with group actions, such as toric varieties, flag varieties, quiver loci, and Hilbert schemes. Over 100 figures, 250 exercises, and pointers to the literature make this book appealing to both graduate students and researchers.


Язык: en

Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 426

Добавлена в каталог: 19.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(n+1)^{n-1}$ theorem      266 363 365 367
$\Bbbk$-algebra, $\mathbb N^n$-graded      4
$\Bbbk$-algebra, $\mathbb Z$-graded      266
$\Bbbk$-algebra, $\mathbb Z^d$-graded      215
$\mathbb N$-grading (arbitrary)      263—265
$\mathbb Z$-grading      See Standard $\mathbb Z$-grading
$\mathbb Z^d$-graded product      219
$\mathbb Z^d$-grading      269
(co)boundary map      9 10 63.
(co)boundary map, of cell complex      91
(co)homological degree      9 66 233 257
(co)homology, Alexander duality on      83—84
(co)homology, classes of subvarieties in      309
(co)homology, commutes with direct limits      252
(co)homology, equivariant      172 208
(co)homology, long exact sequence of      65 235
(co)homology, of contractible space      17 67 74 236
(co)homology, of links      17 101 253 267
(co)homology, reduced      9 10 18 65
(co)homology, relative      92 106 254 258
(co)homology, sheaf      See Sheaf cohomology
3-connected      53
Abelian group      129 149
Abelian group, cyclic      194
Abelian group, divisible      See Divisible group
Abelian group, finite      172 194
Abelian group, finitely generated      129
Abelian group, free      131 133
Abelian group, sequence denes multigrading      149 191
Abelian group, torsion      152 161
Abelian group, torsion-free      151 152 187
Acyclic cover      94
Additive identity      129
Additivity      166 169 172 306 311
Additivity, yields Schubert polynomials      323—324
Adjacent transposition      291 298 303 325
Adjointness      See Functor adjoint
Admissible family      374
Admissible family, universal      374
Admissible ideal      373 374 377
Alexander duality      81 105—106
Alexander duality, as planar map duality      99—100 106
Alexander duality, on $\mathbb N^n$-graded modules      228
Alexander duality, on antidiagonal ideals      318
Alexander duality, on arbitrary ideals      88 89—91 226 269
Alexander duality, on cogeneric ideals      123
Alexander duality, on free and injective resolutions      106
Alexander duality, on free and irreducible resolutions      225
Alexander duality, on free resolutions      See Duality for resolutions
Alexander duality, on generic ideals      122
Alexander duality, on homological invariants      100 102—104
Alexander duality, on irrelevant ideal      199
Alexander duality, on upper bound problems      125
Alexander duality, principle behind      96 126
Alexander duality, simplicial      16 17 81 85 98 105
Alexander duality, squarefree      16 81—82 89 102 226 318
Alexander duality, tight      104
Alexander duality, topological      83 84
Alexander inversion formula      86 106
Algebraic geometry      viii 21 41 106 193 353 355 376
Algebraic shifting      40 45 106
Algebraic torus      21 172 191 197 200 363
Algebraic torus, coordinates on      192
Almost 3-connected      53 54
Almost n-connected      59
Antidiagonal complex      318 319—323 329
Antidiagonal complex, from matrix Schubert variety      323
Antidiagonal complex, is ball or sphere      329
Antidiagonal complex, is shellable      327
Antidiagonal complex, is subword complex      327
Antidiagonal term      280 318
Antidiagonal term, caused by rank condition      321
Associated prime, multigraded      133 152 166
Associated prime, of $\mathbb Z$-graded module      263
Associated prime, of Borel-xed ideal      39
Associated prime, of initial ideal      145
Associated prime, of local cohomology      254 256 270
Associated prime, of principal ideal      147 269
Auslander — Buchsbaum formula      100 264
Ball      145 329 330
Barvinok's algorithm      229 244
Barvinok's Theorem      241
Barycenter      112
Basis weights      158
Bass number      104 106 223 224 228 265
Bass number, of local cohomology      255 270
Bayer, Dave      86 106
Bender — Knuth involution      329
betti diagram      102 103
Betti number      See also Syzygy
Betti number, $\mathbb N^n$-graded      14 15—18
Betti number, characteristic dependence      18 58 80
Betti number, dual to Bass number      104
Betti number, duality for      76 98
Betti number, extremal      See Extremal Betti number
Betti number, from cellular resolution      65—66
Betti number, multigraded      157
Betti number, of Borelxed ideal      30—33 38
Betti number, of generic ideal      53 112
Betti number, of generic lattice ideal      190
Betti number, of lattice ideal      174 175
Betti number, of lex-segment ideal      35
Betti number, of local cohomology      255
Betti number, of monomial ideal      16 85
Betti number, of Stanley — Reisner ideal      17 85
Betti number, of trivariate ideal      53
Betti number, of twisted cubic      174
Betti number, under deformation      119
Betti number, upper bound on      53 119—121
Betti number, upper-semicontinuity      160
Bialynicki — Birula decomposition      363
Bigatti — Hulett Theorem      35 39
Binary complexity model      241
Binet — Cauchy formula      339
Binomial ideal      See Lattice ideal
Bit size      241
Blowup      197 206 372
Borel group      21 299 342 346
Borel — Weil theorem      288
Borel-fixed ideal, advantage of      41
Borel-fixed ideal, generic initial ideal is      24 26 35 38
Borel-fixed ideal, in positive characteristic      40
Borel-fixed ideal, variable-swapping characterizes      23
Boundary (chain)      9 62
Boundary map      See (co)boundary map
Braid relations      305 308
Bridgeland — King — Reid theorem      368
Brion's Formula      229 237—243 246
Bruhat order      295 298
Bruhat order, characterization by ranks      296
Bruhat order, characterization by row switch      297
Bruhat order, on symmetric group      309
Bruhat order, rank function on      308
Bruhat order, respects length      302
Buchberger graph      48 60
Buchberger graph, characterization of genericity by      109
Buchberger graph, embedded in staircase      60
Buchberger graph, is almost n-connected      59
Buchberger graph, of generic ideal      50 111
Buchberger graph, planarity      49—50 58 59 75
Buchberger graph, Scarf edges lie in      110 125
Buchberger map      51 59
Buchberger's Criterion      47 359 370
Buchberger's Second Criterion      48
Canonical module      263 265—266
Canonical module, of normal semigroup ring      235
Canonical module, of polynomial ring      254
Canonical module, of semigroup ring      233 236 267
Caratheodory property      141
Categorical quotient      203 204
Cech complex      250 251 253 260
Cech complex, canonical: $\mathcal \breve C^\dot_\Delta$      259 262 270
Cech complex, from Taylor resolution      260
Cech complex, generalized: $\mathcal\breve C^\dot_{\mathcal F}$      259 260 261
Cech hull      260 261 269 270
Cell complex      62 77.
Cell complex, acyclic      64 66 74 109
Cell complex, colabeled      92
Cell complex, contractible      73 79 235
Cell complex, dual to cocomplex      258
Cell complex, injectively labeled      227
Cell complex, labeled      See Labeled cell complex
Cell complex, locally finite      178 180
Cell complex, pair of      See Cellular pair
Cell complex, pure      96 117
Cell complex, shellable      269
Cell complex, weakly colabeled      92 97
Cell complex, weakly labeled      79
Cellular free resolution      63 79.
Cellular free resolution, acyclicity      64
Cellular free resolution, examples of      67—71
Cellular free resolution, minimal      95—97 99 105
Cellular free resolution, naturally occurring      94
Cellular free resolution, of artinian quotient      94 96 105
Cellular free resolution, of cogeneric ideal      See coScarf complex
Cellular free resolution, of trivariate ideal      99
Cellular free resolution, of unimodular Lawrence ideal      187
Cellular free resolution, simple      69 123
Cellular free resolution, simplicial      69 111 115 123
Cellular free resolution, symmetric      75
Cellular injective resolution      232
Cellular pair      92 258
Cellular pair, colabeled      92
Cellular pair, weakly colabeled      92
Centrally symmetric convex body      243
Chain complex      See also (co)homology
Chain complex, of cell complex      62
Chain complex, reduced      9 62 175 233
Chain complex, relative      235
Chain in poset of Pluecker coordinates      276 278—279 280
Chain in poset, weak order      298 300 305
Character group      192 194
Characteristic p methods      330
Chow class      172
Chow group      309
Chutable rectangle      313
Chute move      313 314 316 319—322 329
Coarse grading      153
Coarsen      8 155 263 265 346
Cocellular monomial matrix      92
Cochain complex      See also (co)homology
Cochain complex, of cell complex      91
Cochain complex, of cellular pair      92
Cochain complex, reduced      10 82 83 253
Cochain complex, relative      124
Cocomplex      See Polyhedral cocomplex
Cocycle (i-cocycle)      11
Cogenerator      255
Cohen — Macaulay condition      262—266
Cohen — Macaulay condition, and associated primes      148
Cohen — Macaulay condition, by Groebner degeneration      160—161 286
Cohen — Macaulay condition, equivalent characterizations      263 270
Cohen — Macaulay condition, for determinantal variety      290
Cohen — Macaulay condition, for generic monomial quotient      114
Cohen — Macaulay condition, for isospectral Hilbert scheme      372
Cohen — Macaulay condition, for matrix Schubert variety      311
Cohen — Macaulay condition, for module      100 269
Cohen — Macaulay condition, for monomial quotient      103
Cohen — Macaulay condition, for normal semigroup ring      267
Cohen — Macaulay condition, for Pluecker algebra      286
Cohen — Macaulay condition, for quiver locus      342 352
Cohen — Macaulay condition, for ring      174 263
Cohen — Macaulay condition, for Schubert determinantal ring      328 330 341
Cohen — Macaulay condition, for semigroup ring      266
Cohen — Macaulay condition, for simplicial complex      101 327
Cohen — Macaulay condition, for Stanley — Reisner ring      101
Cohen — Macaulay condition, for zero-fiber of Hilb      367 378
Cohen — Macaulay condition, over local ring      347
Cohen — Macaulay condition, under localization      342 352
Cohen — Macaulay condition, via depth      104 265
Cohen — Macaulay condition, via Serre's conditions $S_k$      270
Cohen — Macaulay condition, via shellability      266 267 327
Cohull complex      97
coKoszul complex      82 83 250
Colon ideal      90 91 366 398
Compatible fan      199
Complete bipartite graph $K_{r, s$      49 207
Complete fan      199
Complete graph $K_n$      58
Completion (of ring)      154
Complex (of modules)      See also Resolution
Complex (of modules), acyclic      63 64 93 109 347
Complex (of modules), algebraic coScarf      123
Complex (of modules), Cech      See Cech complex
Complex (of modules), cellular free      63 64 73 107 111
Complex (of modules), cellular injective      218 233
Complex (of modules), chain      See Chain complex
Complex (of modules), chain map of      19 162 252
Complex (of modules), cocellular free      93
Complex (of modules), cochain      See Cochain complex
Complex (of modules), coKoszul      See coKoszul complex
Complex (of modules), dualizing      See Dualizing complex
Complex (of modules), Eagon — Northcott      See Eagon — Northcott complex
Complex (of modules), exact      11
Complex (of modules), Ishida      257 258 267 270
Complex (of modules), Koszul      See Koszul complex
Complex (of modules), minimal      12 109
Complex (of modules), of flat modules      251 262
Complex (of modules), of free modules      11
Complex (of modules), of injective modules      216
Complex (of modules), of localizations      250
Complex (of modules), Scarf      See Scarf complex
Complex (of modules), stable Koszul      See Stable Koszul complex
Complex (of modules), Taylor      See Taylor complex
Complex (of modules), total      19 252
Complexity theory      241
Computer software      20 60 75 91 106 132 148 190 246
Cone (over a subcomplex)      18 29 327
Cone (over a subcomplex), is contractible      17
Cone (over a variety)      196
Cone (polyhedral)      134
Cone (polyhedral), dual      200
Cone (polyhedral), in fan      199
Cone (polyhedral), over polytope      230 238
Cone (polyhedral), pointed      134 138—140
Cone (polyhedral), rational      134 137—139
Cone (polyhedral), simplicial      134
Convex polyhedron      72 79 144 177 204 256.
Convex polytope      See Polytope
Coordinate subspace      6 200 323 324
Corner      86. See also Syzygy as
coScarf complex      123 124
Cotangent bundle      375
Cotangent space      362
Cover (of cell complex)      94
Coxeter group      309 329 330
Cross (crossing tile) $+$      312 313
Cross-polytope      See Octahedron
cube      81 82 88 138 198 199 206 230
Cycle (i-cycle)      9 40
Cycle notation      291 308
Deformation      126. See also Specialization
Deformation, generic      115 116
Deformation, of lattice module      188—189
Deformation, of monomial ideal      67 115 117 119
Degeneracy locus      309 353
Degenerativity      166 167 172
Degree map      149
Degree map, fiber of      153
Degree of determinantal variety      306 308
1 2 3 4 5 6
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