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Miller E., Sturmfels B. — Combinatorial Commutative Algebra
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Название: Combinatorial Commutative AlgebraАвторы: 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: 2004Количество страниц: 426Добавлена в каталог: 19.03.2005Операции: Положить на полку |
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Lawrence ideal, unimodular 187 190 Lawrence lifting 187 lcm-lattice 74 79 80 Leading term See Initial term Least common multiple 18 42 61 74 81 92 107 109 110. Length of free resolution 11 Length of lacing diagram 350 Length of module 165 356 Length of partial permutation 289 294 325 Lenstra — Lenstra — Lovasz algorithm 243 Lineality 134 139 233 239 254 257 Linear extension 277 Linear free resolution 30 70 101 105 Linear programming 143 145 148 Link 17 86 101 105 181 253 327 Littlewood-Richardson rule 288 Local cohomology 247 248 249—262 269 Local cohomology, computing 256 262 Local cohomology, finite data structure for 255—256 Local cohomology, grading on 251 Local cohomology, in Cohen — Macaulay criteria 263 265 267 Local cohomology, module structure 269 Local cohomology, not finitely cogenerated 250 255 Local cohomology, not finitely generated 250 Local cohomology, of canonical module 249 254—255 269 Local cohomology, of semigroup ring 255 258 Local cohomology, of Stanley — Reisner ring 253 Local cohomology, via Cech cohomology 251 265 Local cohomology, via Ext 252 Local cohomology, via generalized Cech cohomology 260 Local cohomology, via Ishida complex 258 267 Local cohomology, with maximal support 258 Local cohomology, with Stanley — Reisner support 254 260 Local duality 106 233 265 270 Local duality, with monomial support 270 Localization, along a face 222 Localization, at maximal ideal 346 Localization, exactness of 347 Localization, flatness of 218 Localization, monomial 105 Localization, of 196 Localization, of polynomial ring 342 352 Localization, of semigroup 134 258 Localization, of semigroup ring 216 257 258 Localization, of Stanley — Reisner ring 253 Long word (permutation) 291 304 Lower triangular matrices See Borel group Macaulay's theorem 34 Macdonald polynomial 368 378 Manifold 86 Mapping cocylinder 162 Matching (of a graph) 336 Matlis duality 216 217 257 269 Matlis duality, as Hom into injective hull of 219 Matlis duality, in Alexander duality 226 Matlis duality, in Cohen — Macaulay criterion 263 Matrix Schubert variety 289 290 309 Matrix Schubert variety, associated to Grassmannian 306 Matrix Schubert variety, boundary components of 302—304 306 Matrix Schubert variety, containment among 295 Matrix Schubert variety, dimension of 295 298 301 Matrix Schubert variety, double Schubert polynomial from 305 Matrix Schubert variety, equations defining 291—295 Matrix Schubert variety, fixed by adjacent transposition 302 Matrix Schubert variety, for long word 291 Matrix Schubert variety, for Zelevinsky permutation 342 Matrix Schubert variety, from quiver locus 338 341 Matrix Schubert variety, Groebner degeneration of 323 353 Matrix Schubert variety, is Borel orbit closure 299—302 Matrix Schubert variety, is Cohen — Macaulay 311 Matrix Schubert variety, is irreducible 295 301 Matrix Schubert variety, nonzero function 303 306 Matrix Schubert variety, partial permutations in 296 Matrix Schubert variety, Schubert polynomial from 307 Matrix Schubert variety, Schubert variety from 293 Maximal ideal, in affine semigroup ring 147 257 Maximal ideal, in polynomial ring 87 263 369 Maximal ideal, localization at 346 Maximal ideal, of artinian ring 356 Maximal ideal, of identity in 303 Maximal ideal, of partial permutation 303 Maximal ideal, of partition on Hilb ( 361 Maximal ideal, of smooth point 301 Meet 92 281 Menger's Theorem 57 Minimal generator as facet of cell complex 124 Minimal generator as inner corner 45 57 Minimal generator as vertex of hull complex 73 Minimal generator as vertex of polyhedron 74 Minimal generator of 195 Minimal generator of cogeneric ideal 123 Minimal generator of diagonal locus ideal 364 373 Minimal generator of graded component 150 Minimal generator of irreducible ideal 212 Minimal generator of lattice ideal 175 181 Minimal generator of lattice module 179 Minimal generator of Laurent monomial module 177 178 Minimal generator of maximal ideal 147 361 Minimal generator of module 156 221 Minimal generator of monomial ideal 4 28 42 72 88 358 Minimal generator of pointed semigroup 137 173 Minimal generator of semigroup ring 147 Minimal generator of squarefree ideal 81 Minimal generator, computing for toric ideal 244 Minimal generator, number of 14 38 Minimal generator, upper bound on number of 125 Minkowski sum 245 Minkowski's theorem 243 Minor (miracle) 180 Minor (of a graph) 58 Minor (of a matrix) 274 291 Minor (of a matrix), 187 206 Minor (of a matrix), defined by rank condition 318 Minor (of a matrix), generic 275 Minor (of a matrix), in Groebner basis 323 324 339 353 Minor (of a matrix), in product of generic matrices 333 Minor (of a matrix), in product of matrices 331 Minor (of a matrix), in product of two matrices 335—336 Minor (of a matrix), in sagbi basis 280 324 Minor (of a matrix), maximal 176 274 290 308 Mitosis 314 315 317 329 330 Modest module See Module modest Module, 6 Module, 263 Module, $\mathbb Z^n / L-graded 182 Module, 182 Module, Cohen — Macaulay 100 263 269 Module, equivariant 182—183 Module, filtration by prime quotients 169 Module, finite-length 165 Module, finitely generated 12 14 150 151 153 161 165 Module, flat 218 219 264 373 Module, free 11 27 156 263 Module, graded 153 Module, homologically injective 218 219—221 Module, in finitely generated 152 153 177 Module, injective See Injective module Module, is submodule of injective 220 Module, locally free 373 374 Module, modest 163 164—165 238 Module, of Laurent series 163 Module, positively graded 153 156—158 160—161 373 Module, Q-graded 211 222 224 Module, Q-graded part of 214 Module, ungraded 251 252 258 Module, ungraded free 12 Moebius function 232 243 Moment curve 119 Moment polytope 71 Monomial 3 149 Monomial ideal, artinian 47 50 54 76 77—78 104 Monomial ideal, as initial ideal 144 Monomial ideal, as point on Hilb 356 360 363 367 Monomial ideal, Borel-fixed See Borel-fixed ideal Monomial ideal, cogeneric 107 122 Monomial ideal, Cohen — Macaulay quotient by 103 Monomial ideal, generic See Generic monomial ideal Monomial ideal, graded translate of 177 Monomial ideal, ideal is fixed by torus 22 363 Monomial ideal, in finite periodic 176 Monomial ideal, in polynomial ring 3 Monomial ideal, in semigroup ring 133 135 209 Monomial ideal, irreducible See Irreducible monomial Monomial ideal, maximal 87 147 257 Monomial ideal, most singular on Hilb 371 Monomial ideal, neighborly 121 Monomial ideal, prime 5 81 134 166 Monomial ideal, resolves itself 61 Monomial ideal, rigid 59 Monomial ideal, squarefree See Squarefree ideal Monomial ideal, stable See Ideal stable Monomial ideal, strongly generic 50 60 109 126. Monomial ideal, trivariate 44—47 49—59 67 285 Monomial ideal, with given Hilbert function 377 Monomial label 62 217 Monomial matrix 20 174 Monomial matrix, cellular 62 Monomial matrix, cellular injective 217 233 Monomial matrix, differential without 13 63 64 107 Monomial matrix, for 250 259 Monomial matrix, for free modules 12 215 227 Monomial matrix, for injective modules 215 217 227 Monomial matrix, minimal 12 Monomial order See Term order Monomial, in semigroup ring 133 Monomial, squarefree 4 Monomial, standard See Standard monomial Morphism, fibers of 204 Morphism, G-equivariant 203 Morphism, of schemes 374 Morphism, of varieties 201 351 372 Morphism, projective 195 Mountain topography 52 Multidegree 149 167 172 286 Multidegree, additivity See Additivity Multidegree, degenerativity See Degenerativity Multidegree, existence and uniqueness 166 Multidegree, is polynomial (not integer) 304 310 Multidegree, of codimension r module 169 Multidegree, of graded translate 169 Multidegree, of matrix Schubert variety 289 304—307 Multidegree, of prime monomial quotient 168 Multidegree, of quiver locus 343 Multidegree, of twisted cubic 169—170 171 Multidegree, of variety 167 Multidegree, positivity 171 311 Multidegree, under change of grading 171 Multidegree, universal 172 Multigraded 153 Multigrading 149 172 304 375 Multiple Proj 288 Multiplicity (of M at 165 n! theorem 266 355 363 365 367 368 n-connected 59 Nakayama's lemma 162 Nakayama's lemma, 155 Nakayama's lemma, for semigroup rings 147 Nakayama's lemma, nonstandard version 264 Nakayama's lemma, positively graded 155 Near-cone 29 Nerve (of a cover) 94 95 176 Newton polytope 71 Nilpotent element 356 Nonface 5 17 Nonzerodivisor 262 264 Normal fan 145 146 199 205 Normalization 140 141 147 230 231 octahedron 18 66 68 71 82 146 148 199 230 245 246 Of minimal length ( 338 349 Offspring See Pipe dream offspring One-line notation 291 Open cover 196 208 358 Opposite big cell 345 Opposite Schubert cell 341 345 Optical illusion 81 88 90 Optimal vector 142 Orbit of algebraic group viii 301 Orbit of Borel group 289 300—301 Orbit of general linear group 208 353 Orbit of subgroup of torus 193 194 200 203 204 207 363 Order complex 126 279 Order dimension 60 Order ideal 64 368 Orientation 62 Oriented matriod 72 Outer normal vector 76 205 Parabolic subgroup 39 341 342 Part (of a partition) 285 Partial flag variety 341 345 Partial permutation 289 290 291 Partial permutation, drawing of 312 Partial permutation, extension of 292 293 301 304 312 Partial permutation, family connecting pair of 300 Partial permutation, in lacing diagram 331—332 352 Partial permutation, indexes Borel orbit 300 Partial permutation, length after switching rows 296—297 Partition 285 288 305 328 356 361 364 365 Permutation See also Partial permutation Permutation, conventions for 291 325 Permutation, Grassmannian 308 328 Permutation, vexillary 295 309 Permutation, Zelevinsky See Zelevinsky permutation Permutohedron 68 80 123 Picture space 330 Pipe dream 312 313—324. Pipe dream formula 353 Pipe dream, barren 314 Pipe dream, coordinate subspace from 320 Pipe dream, offspring of 314 Pipe dream, subword from 325—326 Pipe dream, top-justified 320 321 Planar graph 60 Planar map 51 Planar map, axial 99 Planar map, dual of 99 Planar map, labeled 51 52 54 Planar map, radial 99 Plane partition 369 Pluecker algebra 275 Pluecker algebra, is Cohen — Macaulay 286 Pluecker algebra, presentation of 276 Pluecker algebra, spector of 287 Pluecker coordinates 273 275 Pluecker coordinates, as generic minors 275 Pluecker coordinates, form sagbi basis 280 324 Pluecker coordinates, relations among 277 Pluecker coordinates, represent flags 275 293 Pluecker coordinates, represent subspaces 274 357 Pluecker relations 276 277—279 373 Polarization 44—45 59—60 116 Polyhedral cell complex See Cell complex Polyhedral cocomplex 258 Polyhedral subcomplex, in a cover 94 Polyhedral subcomplex, of cone 248 254 Polyhedral subcomplex, of polytope 73 234 235 Polyhedral subdivision See Subdivision Polyhedron See Convex polyedron Polynomial ring 3 Polynomial ring, 4 Polynomial ring, 21 191 230
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