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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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266 363 365 367 4 266 215 263—265 See Standard 219 269 (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 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 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, 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: 259 262 270 Cech complex, from Taylor resolution 260 Cech complex, generalized: 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 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 49 207 Complete fan 199 Complete graph 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
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