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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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Polynomial ring, 182 Polynomial ring, bivariate 42 Polynomial ring, is semigroup ring 129 Polynomial ring, multigraded 149 191 Polynomial ring, positively graded 151 152 153 173 195 262 263 Polynomial ring, trivariate 41 Polynomial ring, with real exponents 52 115 Polynomial time 241—245 Polytope 61 62 77 Polytope, 3-dimensional 62 Polytope, boundary of 145 Polytope, cyclic 114 119 120—121 Polytope, interior lattice points in 236—238 240 Polytope, lattice 148 197 199 229—232 237 240—242 246 Polytope, lattice points in 153 229 236—243 Polytope, neighborly 120 121 Polytope, normal 232 245 Polytope, polar 145 258 Polytope, rational 245 246 Polytope, section of cone 233 254 Polytope, simple 70 71 75 82 122 145 Polytope, simplex 231 Polytope, simplicial 82 102 122 Polytope, transportation 207 Polytope, triangulating 231 Polytope, volume of 230 246 Poset, 276—277 278—279 281 286 Poset, Bruhat See Bruhat order Poset, face 72 178 232 243 258 Poset, of injective submodules 221 Poset, of least common multiples See lcm-lattice Poset, pointed semigroup as 137 Poset, product of intervals 90 Poset, weak order See Weak order Positive multigrading 151 194 198 375 Positively graded See Graded Power series, modest 164 Power series, supported on semigroup 154 Power sum 364 365 368 Precedes ( 64 Presentation (of group) 192 Primary decomposition 24 146 Primary decomposition, characteristic dependent 131 Primary decomposition, in semigroup ring 133 135—137 Prime avoidance 263 Primitive integer vector 205 Primitive lattice vector 180 181 Proj 195 256 281 Projective dimension 100 160—161 Projective dimension, dual to regularity 102—104 Projective dimension, of generic ideal 114 Projective dimension, of lattice ideal 175—176 Projective space 6 198 200 206 275 287 Projective space 273 274 Projective spectrum See Proj Proper extension 214 Q-set 147 Quasi-polynomial 245 quiver 353 Quiver ideal 331 333 352 Quiver ideal, determined by lace array 335 Quiver ideal, multigrading for 343 Quiver ideal, primality 336 341—343 352 Quiver locus 331 333 352 Quiver locus, Cohen — Macaulayness 341—343 352 Quiver locus, for other types of quiver 353 Quiver locus, matrix Schubert variety from 338 341 Quiver polynomial 288 331 343 353 Quiver polynomial from 343 Quiver polynomial from, to Schubert determinantal ideal 339 Quiver polynomial, double See Double quiver polynomial Quiver polynomial, K-theoretic 353 Quiver polynomial, positive formula for 347 349 350 353 Quiver representation 208 332—333 334—335 349 Quiver representation, finite type 352 Quiver representation, indecomposable 333 334 352 Quiver variety 208. See also Quiver locus and Toric variety from Quotient semigroup 134 Radical component 371 372 Radical ideal 171 Radical ideal, corresponds to algebraic set 6 Radical ideal, failure to be 369 Radical ideal, from squarefree initial ideal 307 323 Radical ideal, of n points 356 361 369 Radical ideal, support on 248 Radical locus 357 360 362 369 370 371 Rank array (for partial permutation) 290 296—298 Rank array (for partial permutation), of Zelevinsky permutation 340 Rank array (for quiver) 333 334—335 Rank array (for quiver), from permutation 344 352 Rank array (for quiver), irreducible 335 Rank array (for quiver), minimal lacing diagram for 350 352 Rank array (for quiver), prime quiver ideal from 342 Rank array (for quiver), quiver polynomial from 343 Rank array (for quiver), Zelevinsky permutation from 337—338 Rank-nullity theorem 66 156 Ratio formula 353 Rational curve 359 360 Rational function 163 Rational function, as generating function 173 229 Rational function, equated with Laurent polynomial 237 Rational function, equated with Laurent series 154 Rational function, in power series ring 165 167 Rational function, short 229 243 244 246 rc-graph 329 Rectangle array 333 334 337 352 Reduced expression 305 311 325 329 Reduced expression, in Coxeter group 330 Reduced expression, reverse triangular 326 Reduced Groebner basis 25 Reduced Groebner basis, computing for toric ideal 244 Reduced Groebner basis, for module 27 156 Reduced Groebner basis, for Pluecker relations 282 283 Reduced Groebner basis, for syzygies of Borel-fixed ideal 31 Reduced Groebner basis, for toric ideal $J_n=\mathrm{in}_\le(I_n) 282 Reduced Groebner basis, is homogeneous 172 Reduced Groebner basis, under weight order 142 Reduced Groebner basis, uniqueness of 39 148 Reduced homology or cohomology See (co)homology reduced Reduced pipe dream 312 326 329 Reduced pipe dream, as coordinate subspace 318 Reduced pipe dream, as facet complement 318 322 323 Reduced pipe dream, as Gelfand — Tsetlin face 288 330 Reduced pipe dream, as monomial ideal generator 318 Reduced pipe dream, as prime component 311 318 Reduced pipe dream, as Young tableau 328 Reduced pipe dream, bottom 328 Reduced pipe dream, crossing tiles 328 Reduced pipe dream, double Schubert monomial from 324 Reduced pipe dream, for long word 312 317 326 Reduced pipe dream, for Zelevinsky permutation 348—350 Reduced pipe dream, generating all 314—315 Reduced pipe dream, involution on 315 317 Reduced pipe dream, offspring of 317 Reduced pipe dream, quiver monomial from 348 349 Reduced pipe dream, reduced subword from 326 Reduced pipe dream, Schubert monomial from 315 317 Reduced pipe dream, top 321 328 Rees algebra 372—373 Regular function 341 342 367 Regular local 301 304 Regular sequence 262 Regular sequence, criterion for being a 347 Regular sequence, depolarization by 44 60 Regular sequence, in Cohen — Macaulay criteria 263—265 342 Regular sequence, quotient by preserves acyclicity 346 Regular subdivision 78. See also Triangulation regular Regularity 45 101 102—104 Regularity, multigraded 378 Reisner's criterion 101 102 106 266 267 Reisner, Gerald 106 Representation of abelian group 192 Representation of quiver See Quiver representation Representation of symmetric group 368 Representation theory 191 284 287 298 353 Resolution See also Complex (of modules) Resolution of singularities 196 Resolution, Alexander duality for See Duality for resolutions Resolution, by planar map 51 52 54 67 99 Resolution, cellular _ See Cellular _ resolution Resolution, cocellular 93 94—98 Resolution, cohull 97 98 105 123 Resolution, coScarf 123—125 126 Resolution, Eliahou — Kervaire 33 40 Resolution, free See Free resolution Resolution, hull See Hull resolution Resolution, injective See Injective resolution Resolution, irreducible See Irreducible resolution Resolution, Scarf See Scarf complex Resolution, Taylor See Taylor resolution Resolution, weakly cellular 79 98 105 Resolution, weakly cocellular 93 94 96 97 Restriction (of simplicial complex) 85 86 Reverse square word 326 327 Ridge 233 329 Rigid embedding 60 Ring, arbitrary (commutative) 84 129 208 215 216 227 250 251 252 304 309 339 352 373 Ring, Cohen — Macaulay 263 Ring, cohomology 208 288 Ring, determinantal See Ring Schubert Ring, face 19 248 269 Ring, group See Group algebra Ring, K- 208 309 Ring, Laurent polynomial See Laurent polynomial ring Ring, local 302 303 347 356 362 Ring, of global sections 367 Ring, of invariants 193 364 Ring, polynomial See Polynomial ring Ring, power series 6 154 Ring, regular 209 Ring, Schubert determinantal 286 325 328 Ring, semigroup See Semigroup ring Ring, Stanley — Reisner See Stanley — Reisner ring Robinson — Schensted — Knuth correspondence 329 Rook placement 291 Root of unity 194 s-pair 47 48 121 Sagbi acronym 279 288 Sagbi basis 273 279 281 288 Sagbi basis, for Pluecker algebra 280 Sagbi degeneration 281 286 sans serif font 125 Saturation of a semigroup See Semigroup saturation Saturation of an ideal 132 207 366 398 Scalar entries 12 Scarf complex 107 110 111—114 Scarf complex, algebraic 111 Scarf complex, can be disconnected 110 Scarf complex, characteristic independence of 112 Scarf complex, is contained in hull complex 111 Scarf complex, of deformation 115—117 Scarf complex, of generic artinian ideal 113 228 Scarf complex, of lattice 190 Scarf complex, of lattice ideal 188 Scarf complex, of Laurent monomial module 188 Scarf complex, of monomial ideal 187 Scarf complex, purity of 113 Scarf triangulation 227 Scheme 202 207 352 356 366 367 369 374 376 Scheme, projective 375 Schlegel diagram 73 77 114 Schreyer's algorithm 32 156 Schubert determinantal ideal 290 292 293—295. Schubert determinantal ideal, for Zelevinsky permutation 339 Schubert determinantal ideal, from quiver ideal 336 339 Schubert determinantal ideal, primality of 323 Schubert determinantal ideal, universal multigrading for 308 Schubert determinantal ring See Ring Schubert Schubert polynomial 304 309. Schubert polynomial, and Gelfand — Tsetlin patterns 288 Schubert polynomial, as multidegree 307 323 Schubert polynomial, coefficients of 311 Schubert polynomial, combinatorics of 286 312 Schubert polynomial, positive formula for 315 323 329 Schubert polynomial, quantum 353 Schubert polynomial, recursion for 304 305 311 Schubert polynomial, Schur polynomial is 305 328 Schubert polynomial, stable 330 Schubert polynomial, universal 353 Schubert variety 288 289 293 309 330 Schubert variety, in partial flag variety 341 345 Schur function 172 368 Schur polynomial 290 305 306 328 330 353 Sector 249 255 256 Sector partition 255 256 270 Segre variety 206 Semi-invariant 353 Semigroup 129. See also Semigroup ring Semigroup ring 129. See also Lattice ideal Semigroup ring, 201 203—205 Semigroup ring, (anti)diagonal 284 Semigroup ring, affine 133—137 140—141 236 248—250 256—259 266 Semigroup ring, Cohen — Macaulay 266—267 Semigroup ring, completion of 154 Semigroup ring, dimension of 131 Semigroup ring, from lattice module 181 Semigroup ring, Gelfand — Tsetlin 286 Semigroup ring, integral domain 131 Semigroup ring, intersecting ideals in 135—136 Semigroup ring, normal 140 150 193 232 254—256 267 270 286 Semigroup ring, normalization of See Normalization Semigroup ring, over the integers 129 154 Semigroup ring, presentation of 130—131 Semigroup ring, vertex 237 Semigroup, 200 204 205 Semigroup, affine 129 131 133—135 137—141 148 173 176 187 209 233 247 Semigroup, antidiagonal 284 286 Semigroup, can be a group 133 Semigroup, cancellative 129 Semigroup, characteristic function of 376 377 Semigroup, cone over polytope 230 232 Semigroup, embedding in 139 Semigroup, embedding in $\mathbb N^{\mathrm{rank}} 140 152 Semigroup, finitely generated 129 Semigroup, Gelfand — Tsetlin 284—286 Semigroup, holes in 148 Semigroup, nonsimplicial 270 Semigroup, pointed 133 134 140 141 148 152 154 171 173 178 254 257 Semigroup, saturated 137 140 141 147 150 233—236 254 269 286 Semigroup, saturation of 140 141 147 230 268 Semigroup, unsaturated 270 Semigroup, vertex See Vertex semigroup Semistandard monomial 279 283 Semistandard tableau See Tableau Serre's condition 148 270 Shadow 200 SHAPE See Ferrers diagram Sheaf 71 172 208 367 374 Sheaf cohomology 71 256 269 330 Shear 144 350 Shelling 267 Shuffle 277 Sign convention 9 Simple reflection 325 Simplex in simplicial complex See Face Simplex polytope See Polytope simplex Simplicial complex 4 9—11 Simplicial complex, antidiagonal See Antidiagonal complex Simplicial complex, as polyhedral cell complex 62 Simplicial complex, associated to affine semigroup 175
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