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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:

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

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Предметный указатель

Hilbert scheme, of points on surface      378
Hilbert scheme, of points on threefold      378
Hilbert scheme, of subschemes of toric variety      378
Hilbert scheme, radical ideal locus      See Radical locus
Hilbert scheme, smoothness of      355 359 361—363 369 375 377
Hilbert scheme, tangent space to      369
Hilbert scheme, toric      376 377 378
Hilbert scheme, universal property of      369 374—375
Hilbert series      See also K-polynomial
Hilbert series, $\mathbb N^n$ -graded      6 7 8
Hilbert series, $\mathbb Z$ -graded      6
Hilbert series, $\mathbb Z$ -graded, of semigroup ring      230
Hilbert series, $\mathbb Z$ -graded, of Stanley — Reisner ring      8
Hilbert series, additivity on exact sequence      264
Hilbert series, characteristic independence      18
Hilbert series, coarse      6
Hilbert series, counts torus weight spaces      288
Hilbert series, finely graded      6 154
Hilbert series, in exact sequence      156
Hilbert series, in nonpositive grading      149
Hilbert series, modulo (non)zerodivisor      264
Hilbert series, multigraded      153 154—157
Hilbert series, of admissible module      373
Hilbert series, of affine semigroup with units      239
Hilbert series, of bivariate ideal      42
Hilbert series, of canonical module      239
Hilbert series, of Cohen — Macaulay module      263
Hilbert series, of determinantal ideal      330
Hilbert series, of graded translate      6 155 157
Hilbert series, of ideal in semigroup ring      228
Hilbert series, of indecomposable injective      239
Hilbert series, of irreducible quotient      228
Hilbert series, of lattice ideal      181 244
Hilbert series, of local cohomology      247 253—254 269
Hilbert series, of modest module      163—165 238
Hilbert series, of monomial quotient      74
Hilbert series, of pointed semigroup      173
Hilbert series, of polynomial ring      6 154
Hilbert series, of quiver locus      347
Hilbert series, of saturated semigroup      243
Hilbert series, of semigroup ring      181 186 230
Hilbert series, of Stanley — Reisner ideal      86
Hilbert series, of Stanley — Reisner ring      7 8
Hilbert series, of subword complex      330
Hilbert series, of tangent cone      237
Hilbert series, of twisted cubic      174
Hilbert syzygy theorem      11 116 156 175 178
Hilbert — Burch theorem      174
Hilbert — Frobenius series      368
Hochster's formula      17 19 85 86 98 102
Hochster, Melvin      106
Hom      215 216
Homogeneous coordinate ring (of toric variety)      71 163 172 202 208
Homogeneous polynomial      192
Homogenization      158—159 162
Homological algebra      viii ix 269
Homological degree      See (co)homological degree
Homology      See (co)homology
Homomorphism, $\mathbb N^n$ -graded      11 215
Homomorphism, $\mathbb Z$ -graded      215
Homomorphism, $\mathbb Z^d$ -graded      215
Homomorphism, A-graded      153
Homomorphism, homogeneous      215
Homomorphism, minimal      12
Homomorphism, ungraded      215
Hosten — Morris number      121
Hull complex      79 177
Hull complex, characteristic independence      75
Hull complex, computing      181
Hull complex, contains Scarf complex      111—112
Hull complex, of artinian ideal      76—78
Hull complex, of generic ideal      111 117
Hull complex, of lattice module      180—181 185
Hull complex, of Laurent monomial module      178
Hull complex, of monomial ideal      73
Hull resolution      71—78 109
Hull resolution, not every cellular resolution is      98
Hull resolution, of lattice module      184
Hull resolution, of Laurent monomial module      178 188
Hull resolution, of monomial ideal      73
Hull resolution, of semigroup ring      184
Hyperplane arrangement      79 80
Hypersimplex      180
Ideal      See also Monomial ideal
Ideal, -fixed      23 33
Ideal, $\mathbb N^n$ -graded      4
Ideal, admissible      See Admissible ideal
Ideal, antidiagonal      318 319 321—323 330
Ideal, binomial      See Lattice ideal
Ideal, Borel-fixed      See Borel-fixed ideal
Ideal, determinantal      See Determinantal ideal
Ideal, face      19 248 255 269
Ideal, finitely generated      4
Ideal, G-stable      193
Ideal, in semigroup      133
Ideal, initial      See Initial ideal
Ideal, irrelevant      See Irrelevant ideal
Ideal, Jacobian      363
Ideal, lattice      See Lattice ideal
Ideal, lex-segment      34—39
Ideal, maximal      See Maximal ideal
Ideal, monomial      See Monomial ideal
Ideal, multigraded      193
Ideal, permutohedron      68 69 75 97 99 123
Ideal, prime      135 195 288
Ideal, principal      136 147 148 174 209 228 269
Ideal, pure      171
Ideal, radical      See Radical ideal
Ideal, squarefree      See Squarefree ideal
Ideal, stable      28 40
Ideal, Stanley — Reisner      See Stanley — Reisner ideal
Ideal, torus-fixed      See Monomial ideal
Ideal, tree      68 69 80 97 99 109 123
Inclusion-exclusion      42—43 67 74 210
Indecomposable injective module      103 212 213—214 247 256
Indecomposable injective module, arbitrarily graded      228
Indecomposable injective module, homomorphism of      216
Indecomposable injective module, is homologically injective      219
Indecomposable injective module, supported on $I_\Delta$       248
Independent paths      57
Initial algebra      279 281
Initial algebra, of Pluecker algebra      280 281 286
Initial complex      142 144—146
Initial form      142
Initial ideal      24
Initial ideal, for Pluecker relations      277—279 282
Initial ideal, of determinantal ideal      311 323
Initial ideal, of distraction      360
Initial ideal, of lattice ideal      142—146 148
Initial ideal, of toric ideal      148
Initial ideal, squarefree      187 307 323
Initial ideal, under weight order      142
Initial module      27 30 166
Initial module as special fiber      158
Initial module under weight order      158 159—161
Initial term      24 279
Initial term, of Pluecker product      278—279 280
Injective hull, arbitrarily graded      228
Injective hull, as essential extension      214
Injective hull, of face      212 213 216 221 233 239
Injective hull, of module      221 222
Injective hull, of residue field $\Bbbk$       104 265
Injective module      211 213 214—223
Injective module, arbitrarily graded      227
Injective module, characterizations of      220
Injective module, decomposition as direct sum      223
Injective module, finely graded      228
Injective module, homomorphism of      216 217
Injective module, indecomposable      See Indecomposable injective module
Injective module, product of      219

Injective module, ungraded      251 258
Injective resolution      103—104 222
Injective resolution, $\mathbb Z^d$ -graded      247
Injective resolution, $\mathbb Z^n$ -graded      126
Injective resolution, arbitrarily graded      227
Injective resolution, compared to free resolution      211
Injective resolution, computing      270
Injective resolution, existence and uniqueness      222
Injective resolution, finite      265
Injective resolution, from cell complex      227
Injective resolution, homological invariants from      222
Injective resolution, in Alexander duality      95
Injective resolution, in Cohen — Macaulay criterion      263
Injective resolution, in finite      265
Injective resolution, local cohomology from      248
Injective resolution, minimal      222 223 224 265
Injective resolution, of canonical module      233 235 266
Injective resolution, of generic monomial ideal      228
Injective resolution, ungraded      252
Inner normal vector      77 78 120 139 197
Integer programming      143 148 190
Integral domain      131
Interior lattice point enumerator      236
Intron      316 317
Intron mutation      316 317 329
Invariant theory      191 378
Irreducible      87 91 211 225
Irreducible component      87 88
Irreducible component as Bass number      223 224
Irreducible component as facet      58 96 117 226
Irreducible component as outer corner      47 100 226
Irreducible component as upper bound for syzygy degree      117
Irreducible component as vertex of cell complex      124
Irreducible component of artinian ideal      104
Irreducible component of generic ideal      114
Irreducible component, under specialization      125
Irreducible component, upper bound on number of      120
Irreducible decomposition      87 106
Irreducible decomposition, computing      87 91 106 212 228
Irreducible decomposition, from cellular resolution      96—97 125
Irreducible decomposition, from staircase      47
Irreducible decomposition, in semigroup ring      211 212
Irreducible decomposition, irredundant      87 105
Irreducible decomposition, of bivariate ideal      43
Irreducible decomposition, of Borel-fixed ideal      24
Irreducible decomposition, of generic ideal      114 117
Irreducible decomposition, of non-monomial ideal      91 228
Irreducible decomposition, uniqueness      90 211
Irreducible hull      222
Irreducible monomial ideal, from indecomposable injective      214
Irreducible monomial ideal, heuristic illustration of      215
Irreducible monomial ideal, in polynomial ring      43 87
Irreducible monomial ideal, in semigroup ring      137 210 225
Irreducible monomial ideal, testing for      212
Irreducible quotient      210 222
Irreducible representation      288
Irreducible resolution      210 228 270
Irreducible resolution, computing      228
Irreducible resolution, existence and uniqueness      211 223 224
Irreducible resolution, over polynomial ring      225
Irrelevant complex      4 121
Irrelevant complex, (co)homology of      9 10 14 83
Irrelevant complex, as a link      17
Irrelevant complex, Euler characteristic of      66
Irrelevant ideal, of $\mathbb N$ -graded ring      195 206
Irrelevant ideal, of compatible fan      199 202 205 208
Irrelevant ideal, of Pluecker algebra      287
Irrelevant ideal, of quotient ring      202
Irrelevant ideal, of toric variety      71 260
Isobaric divided difference      307
Jacobi — Trudi formula      305
Join $\vee$       92 282
K-polynomial      172. See also Hilbert series
K-polynomial, $\mathbb N^n$ -graded      7
K-polynomial, additivity on exact sequence      169 306
K-polynomial, as Euler characteristic      67 74 113
K-polynomial, at vertex of polytope      237
K-polynomial, calculation of      106 158
K-polynomial, from staircase      47
K-polynomial, multigraded      161
K-polynomial, of bivariate ideal      43
K-polynomial, of Borel-fixed ideal      28
K-polynomial, of free module      161
K-polynomial, of Frobenius power I[t]      18 78
K-polynomial, of generic ideal      113
K-polynomial, of matrix Schubert variety      307 309
K-polynomial, of modest module      164—165
K-polynomial, of prime monomial quotient      168
K-polynomial, of quiver locus      347
K-polynomial, of residue field $\Bbbk$       158 161
K-polynomial, of semigroup ring      173 186
K-polynomial, of Stanley — Reisner ring      7 86
K-polynomial, of subword complex      106
K-polynomial, of trivariate ideal      58
K-polynomial, positively graded      157
K-polynomial, records Betti numbers      157
K-polynomial, under change of grading      171
K-polynomial, under Groebner degeneration      162
K-polynomial, universal      172
K-theory      148 157
K-theory, equivariant      106 172 246
Koszul complex      13 30 264.
Koszul complex, as hull complex      74 76
Koszul complex, as linear free complex      31
Koszul complex, as minimal free resolution      14 158 168 175
Koszul complex, compared to coKoszul complex      82 83
Koszul complex, Matlis dual of      218
Koszul simplicial complex, lower: $K_{\mathbf b}(I)$       84
Koszul simplicial complex, upper: $K^{\mathbf b}(I)$       16 18 31 52 84 117
Krull dimension      142 231 254 262 301
l'Hopital's rule      240 241
Labeled cell complex      62 92. labeled
Lace      332 352
Lace array      333 334—335 337 338
Lace array, from permutation      344
Lacing diagram      331 332 333—336
Lacing diagram, as component of degeneration      353
Lacing diagram, double Schubert product from      350
Lacing diagram, from pipe dream      349—350 353
Lacing diagram, minimal      350 352
Lagrange interpolation      242
Laplace expansion      293
Lattice (combinatorial), Boolean      81
Lattice (combinatorial), distributive      92 281—282 288
Lattice (in $\mathbb Z^n$ )      130
Lattice (in $\mathbb Z^n$ ), generic      188 189 190
Lattice (in $\mathbb Z^n$ ), unimodular      187
Lattice ideal      129 130 139 176
Lattice ideal, as point on toric Hilbert scheme      377
Lattice ideal, computation of      132 148 244
Lattice ideal, from meet-join lattice      288
Lattice ideal, generic      188 189
Lattice ideal, in Laurent polynomial ring      192
Lattice ideal, initial ideal of      See Initial ideal of
Lattice ideal, is $\langle x_1\dots x_n \rangle$ -saturated      132
Lattice ideal, prime      131
Lattice ideal, toric ideal is      See Toric ideal
Lattice module      179 180—187
Lattice module, generic      188
Lattice module, presentation of      179
Lattice point enumerator      236 237—243
Laurent monomial module      176 177—180 190
Laurent monomial module, generic      187
Laurent polynomial      157 163
Laurent polynomial ring      131 176 179
Laurent polynomial ring, as coordinate ring of torus      192
Laurent series      153 154—155 163
Laurent series, summable      163 238 239
Laurent series, supported on translates      155 156
Lawrence ideal      139 147 148

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