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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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Simplicial complex, bijection with squarefree ideals 5 Simplicial complex, Cohen — Macaulay 101 266 327 Simplicial complex, dimension of 4 Simplicial complex, irrelevant See Irrelevant complex Simplicial complex, labeled 13 107. Simplicial complex, of faces missed by polyhedron 205 Simplicial complex, of poset chains See Order complex Simplicial complex, pure 45 267 323 327 329 Simplicial complex, shellable 266 267 270 327 329 330 Simplicial complex, shifted 29 31 Simplicial complex, Stanley — Reisner 142 199 253 318 Simplicial complex, subword See Subword complex Simplicial complex, vertex-decomposable 327 329 330 Simplicial complex, void See Void complex Simplicial complex, without pair of covering faces 121 Simplicial fan 199 204 Slope variety 330 Smith normal form 131 133 148 Smooth fan 199 Smooth point 301 363 Socle 104 265 Socle of local cohomology 255 256 270 Source degree 12 Spec 193 281 Special fiber 158 353 367 Specialization 52. See also Deformation Spector (SpecTor) 202 208 256 281 Spector of Pluecker algebra 287 288 Spectral sequence 20 Spectrum See Spec Sphere 83 84 86 102 145 269 329 330 Spherical variety 288 Squarefree ideal 4 5—8 16—19 Squarefree ideal, advantage of 41 Squarefree ideal, as polarization 44 Squarefree ideal, associated to polytope 70 75 82 259 Squarefree ideal, encodes fan 199 Squarefree ideal, from determinantal ideal 318 Squarefree ideal, from lattice ideal 146 Squarefree ideal, generated in degree d 71 Squarefree ideal, in Pluecker algebra 287 Squarefree module 106 Squarefree vector 5 75 Stable Koszul complex 250 Staircase diagram 42 45 69 74 88 89 98 99 113 126 177 179 185 226 261 356 362 369 Staircase surface 49 50 59 60 68 99 100 105 Standard vii 21 171 230 290 304 361 Standard monomial 158 211 215 285 Standard monomial theory 288 Standard monomial, as basis element 158 357 Stanley — Reisner ideal 3 5 6 18 19 Stanley — Reisner ideal, in Alexander duality 16 82 Stanley — Reisner ideal, of initial complex 142 Stanley — Reisner ideal, of order complex 279 Stanley — Reisner ideal, of real projective plane 69 80 Stanley — Reisner ring 3 5 19 Stanley — Reisner ring, Cohen — Macaulay 101 267 Stanley — Reisner ring, Gorenstein 269 Stanley — Reisner ring, of real projective plane 70 75 Star 105 Steinitz theorem 62 Stick twisted cubic 82 102 Straightening law 288 329 Strand (= lace) 352 Strictly divides 109 117 Strong deformation 52 54 60. Subalgebra basis See Sagbi basis Subdivision See also Triangulation Subdivision, barycentric 69 73 94 110 Subdivision, in finite periodic 180 Subdivision, of polyhedral cell complex 77 Subdivision, of polytope 77 Subdivision, of simplex 77 117 Subdivision, of torus 188 Subdivision, regular See Regular subdivision Subword 326—327 Subword complex 326 327—328 330 Summable See Laurent series summable Support 248 251 252 Support 257 265 Support of a vector 7 Support of a vector, full 77 105 Support-regularity 103 Suspension (of a graph) 53 Sweeping 299 Symmetric function 305 353 355 368 Symmetric function, complete homogeneous 308 Symmetric function, elementary 308 Symmetric function, Stanley 330 Symmetric group 291 325 337 356 364 368 Symmetric product 356 364 372 Symplectic geometry 284 288 System of parameters 262 263—265 269 Syzygy See also Betti number Syzygy as corner 43 49 52 54 56 57 Syzygy as face of cell complex 99 119 Syzygy from Buchberger edge 48 Syzygy from planar map 47 Syzygy in linear free resolution 30 Syzygy module 11 14 48 Syzygy module from Buchberger graph 48 Syzygy of bivariate ideal 43 Syzygy of Borel-fixed ideal 30 Syzygy of lattice ideal 174 Syzygy of lattice module 179 Syzygy of Laurent monomial module 178 187 Syzygy of lex-segment ideal 35 Syzygy of monomial ideal 187 Syzygy of trivariate ideal 53 Syzygy of twisted cubic 174 Syzygy of unimodular Lawrence ideal 187 Tableau 276 280 281 282 288 306 328 Tangent cone 233 234 235 237 238 Tangent space 369 371 Target degree 12 Taylor complex 107 108—110 111 115 Taylor resolution 67 74 80 108 111 Tensor product 15 153 155 182 216 Term order See also Weight order Term order, antidiagonal 280 323 Term order, diagonal 280 282 324 Term order, for free module 27 159 Term order, for polynomial ring 24 279 Term order, lexicographic 26 33 278 Term order, partial 142 282 Term order, position-over-term (POT) 27 31 Term order, refines weight order 148 Term order, reverse lexicographic 26 106 277 282 Term order, term-over-position (TOP) 27 Tessellation 180 Topology, relative cellular 91 Topology, simplicial viii 9 Tor 15 19 20 83 157 175 Toric ideal 148 244 281 282. Toric variety 23 191 198 200—208 246 Toric variety, affine 193—194 196 200 201 Toric variety, as sagbi degeneration 281 Toric variety, as spector 202 Toric variety, determined by equivalent data 202 Toric variety, diagonal embedding 207 Toric variety, from polytope 197—198 202—203 205 207 Toric variety, from quiver 197 208 Toric variety, projective 71 195—198 Toric variety, smooth 196 207 Torus, algebraic See Algebraic torus Torus, as hull complex 186 Totally ordered group 133 152 Translate, 6 15 Translate, 269 Translate, 177 261 Translate, A-graded 153 174 Transposition See Adjacent transposition TREE 69 197 Triangle, lattice point-free 147 Triangulation 50 77 148. Triangulation as Scarf complex 114 Triangulation in polynomial time 243 Triangulation of polytope 377 Triangulation of saturated semigroup 243 Triangulation, regular 113 114 123 144 146 148 232 Twisted cubic 169 170 171 174 Twisted cubic, stick See Stick twisted cubic Underlying cell complex 92 Unit in Laurent series ring 154 Unit in semigroup 133 151 Unit in semigroup ring 133 Universal cover 183 Universal grading 172 308 Upper Bound Theorem 19 119 266 Upper triangular matrices See Borel group Upper-semicontinuity 160 172 Variety, affine 195 Variety, cohomology over quotient 256 Variety, degeneration of 311 Variety, determinantal See Determinantal variety Variety, irreducible 193 295 301 342 Variety, morphism of See Morphism Variety, of 206 Variety, of complexes 351 Variety, of irrelevant ideal 200 205 Variety, of linear maps 331 Variety, of quiver representations 332 Variety, of sequences of linear maps 331 Variety, projective 195 Variety, quasiprojective 358 Variety, smooth See Smooth point Variety, toric See Toric variety Vector bundle 309 353 367 Vector label 217 Veronese subring 194 205 Vertex denominator 237 Vertex figure 105 Vertex K-polynomial 237 Vertex label 62 Vertex semigroup 237 Vertex, axial 99 Vertex, of simplicial complex 4 Void complex 4 10 121 235 Weak order 298—299 300 302 Weak order, induction on 304 306 Weak order, on symmetric group 309 330 Weight order 142 158 159 330 Weight vector 142 148 158 172 Weight vector, generic for I 142 Weight, exponential 307 344 Weight, of a term 142 158 Weight, ordinary 344 345 346 348 Word 326 Young tableau See Tableau Young, Alfred 288 Zariski closed 25 358 Zariski dense 342 Zariski open 25 358 Zariski topology 193 Zelevinsky map 331 334 338 341 352 Zelevinsky map, multigrading on 345 Zelevinsky permutation 337 338—340 348—350 352 Zero set 6 Zero-fiber of Hilb 367 368 378 Zerodivisor 264 269 Zerodivisor, in semigroup ring 131 132 Zorn's lemma 221
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