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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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Degree of monomial 149 Degree of projective variety 149 Degree, 165 166 171 310 Deletion (from simplicial complex) 327 Demazure operator 307 Depth 104 265 Descent 308 Determinant See Determinantal ideal 289 295 309 318 Determinantal ideal, classical 290 308 Determinantal ideal, cogenerated by minor 310 Determinantal ideal, Cohen — Macaulayness of 290 325 328 330 341 Determinantal ideal, combinatorics of 290 312 329 Determinantal ideal, generated by essential minors 294 Determinantal ideal, Grassmannian Schubert 172 Determinantal ideal, ladder 295 309 Determinantal ideal, of diagonal locus 364—365 372 Determinantal ideal, over commutative ring 339 Determinantal ideal, primality of 292 311 323 330 341 Determinantal ideal, Schubert See Schubert determinantal ideal Determinantal ideal, vexillary Schubert 295 309 Determinantal variety See also Matrix Schubert variety Determinantal variety, classical 290 295 306 308 Diagonal locus 356 364 372 Diagonal term 278 280 Diagram (of partial permutation) 294 Diagram of Zelevinsky permutation 337—338 348 Differential 15 19 63 233 Differential operator 365 Differential, horizontal 19 251 Differential, total 19 Differential, vertical 19 251 Dimension vector 332 Direct limit 252 Direct product 219 Directed graph 197 353 Distraction 360 361 369 Divided difference 289 304 305 309 Divided difference, isobaric See Isobaric divided difference Divisible group 192 218 Double quiver polynomial 346 353 Double quiver polynomial, positive formula for 348 Double quiver polynomial, specializes to quiver polynomial 347 Double quiver polynomial, variables in 343 Double Schubert polynomial 304 353 Double Schubert polynomial, as degeneracy locus class 309 Double Schubert polynomial, as multidegree 289 305 309 324 Double Schubert polynomial, double quiver polynomial from 346 Double Schubert polynomial, double Schur polynomial is 330 Double Schubert polynomial, example of 307 Double Schubert polynomial, for inverse permutation 308 Double Schubert polynomial, from quiver polynomial 352 Double Schubert polynomial, indexing of 309 Double Schubert polynomial, is universal multidegree 308 Double Schubert polynomial, is well-defined 305 Double Schubert polynomial, positive formula for 315 324 329 Double Schubert polynomial, quiver polynomial from 344 347 350 Double Schubert polynomial, recursion for 289 304 305 Double Schubert polynomial, variables in 305 Duality for resolutions 91 94—95 106 122 123 126 228 Duality for resolutions, in three variables 99—100 Dualizing complex 233 236 270 Dualizing complex, detects Cohen — Macaulayness 266 Dualizing complex, Dynkin diagram 353 Dualizing complex, for general ring 246 265 270 Dualizing complex, Hilbert series from 239 Dualizing complex, local cohomology from 249 254 Dualizing complex, Matlis dual of 257 Dualizing complex, normalized 233 246 Dualizing complex, of normal semigroup ring 234—236 Eagon — Northcott complex 187 Eagon — Reiner Theorem 101 106 228 Economics 126 Ehrhart polynomial 148 229 Ehrhart polynomial, as Hilbert function 230 Ehrhart polynomial, coefficients of 230 245 Ehrhart polynomial, computing 242 246 Ehrhart polynomial, from lattice point enumerator 240 Ehrhart reciprocity 240 246 Ehrhart's Theorem 229 Eigenvector (of torus action) 192 Elbow joint (tile) 312 313 Embedded prime 136 Embedding dimension 361 Equivalence of categories 183 Equivariant Hilbert polynomial 172 Equivariant multiplicity 172 Essential extension 214 220 222 228 Essential set 294 301 308 309 339 Essential submodule 214 221 Euler characteristic 66 Euler characteristic, 66 74 Euler's formula 53 58 120 Ext 198 252 263 265 268 Exterior algebra 106 Exterior power 274 339 Extremal Betti number 102 103 106 Extremal combinatorics 126 f-vector 8 157 Face label 62 217 Face, as basis vector 63 Face, dimension of 4 Face, empty 4 63 66 Face, flag of 148 Face, injective hull of See Injective hull Face, interior 124 Face, maximal See Facet Face, missed by polyhedron 204 205 Face, of cell complex 62 Face, of cone 134 Face, of semigroup 133 134 Face, of simplicial complex 4 29 Facet of cellular pair 92 Facet of cone 138 Facet of polytope 71 Facet of simplicial complex 5 29 Fan 198—199 Fan, nonsimplicial 207 Farkas' Lemma 134 205 235 Ferrers diagram (shape) 285 288 305 328 356 Fiber product 366 372 374 Field 3 Field, algebraically closed 273 290 Field, characteristic two 70 Field, characteristic zero 21 352 Field, finite 6 Field, of complex numbers C 191 Field, positive characteristic 277 Fine grading 153 Flag (of faces) 73 Flag (of vector spaces) 273 293 Flag variety 80 273 275 288 293 309 330 Flag variety, degenerates to toric variety 281 Flag, homogeneous coordinates for 275 Flat degeneration 286 288 Flat degeneration, Groebner See Groebner degeneration Flat degeneration, sagbi See Sagbi degeneration Flat family 172 360 367 Forest 197 Formal character 368 Fourier transform 246 Free resolution viii 11 Free resolution, 30 Free resolution, cellular See Cellular free resolution Free resolution, compared to injective resolution 211 Free resolution, equivariant 180 187 Free resolution, existence of finite 156 161 Free resolution, from staircase 47 Free resolution, in Cohen — Macaulay criteria 263 Free resolution, linear See Linear free resolution Free resolution, minimal 12 14 19 157 Free resolution, modulo nonzerodivisor 159 160 Free resolution, modulo regular sequence 346 Free resolution, of bivariate ideal 43 Free resolution, of Borel-fixed ideal 27—29 Free resolution, of generic ideal See Scarf complex Free resolution, of generic lattice ideal 188 Free resolution, of generic Laurent monomial module See Scarf complex Free resolution, of lattice ideal 174 181 183 Free resolution, of lattice module 183 Free resolution, of Laurent monomial module 178 Free resolution, of quiver ideal 346 Free resolution, of residue field 14 Free resolution, of semigroup ring 184 Free resolution, of squarefree ideal 116 Free resolution, of trivariate ideal 54 Free resolution, of twisted cubic 174 Free resolution, of zero 19 347 Free resolution, over semigroup ring 209 Free resolution, squarefree 261 Free resolution, support-linear 103 104 Frobenius power 18 78 226 From lattice module 183 Fulton polynomial 344 353 Functor, adjoint 216 Functor, Alexander duality 106 269 Functor, derived 248 Functor, exact 182 218 219 269 Functor, faithful 183 Functor, full 183 Functor, fully faithful 183 Functor, of points 378 Gelfand — Tsetlin pattern 284 285 288 Gelfand — Tsetlin toric variety 330 Generic initial ideal (gin) 26—27 35 40 45 106 Generic matrix 290 332 Generic monomial ideal 107 109 111—119 122 126 187 Generic monomial ideal, characterization of 76 116—117 126 Generic monomial ideal, Cohen — Macaulay quotient by 114 Generic monomial ideal, free resolution of See Scarf complex Generic monomial ideal, resolution by Buchberger map 51 Generic monomial ideal, trivariate 50 Generic quiver representation 333 Geometric invariant theory See GIT Geometric quotient 204 GIT acronym 193 GIT quotient 208 GIT quotient, affine 193 194 195 200 201 203 207 GIT quotient, categorical See Categorical quotient GIT quotient, computing 195 GIT quotient, geometric See Geometric quotient GIT quotient, projective 194 195 196 204—205 Gordan's lemma 137 148 Gorenstein ring 255 266 269 270 365 Gorenstein variety 367 Gotzmann number 376 Graded 0 Graded Betti number 0 Graded component 150 153 Graded degree 0 Graded free resolution 0 Graded Hilbert function 0 Graded Hilbert series 0 Graded homomorphism 0 Graded ideal 0 Graded injective module 0 Graded injective resolution 0 Graded K-polynomial 0 Graded module 0 Graded Nakayama's Lemma 0 Graded polynomial ring 0 Graded translate 0 Graded _ See _ *graded; * “arbitrarily” “finely” “multi” “ “positively “un” “ or and Grassmannian 273 275 288 306 Grassmannian, 281 Grassmannian, 283 Grassmannian, as Hilbert scheme 355 Grassmannian, contained in Hilbert scheme 370 Grassmannian, contains Hilbert scheme 357—358 368 Grassmannian, degenerates to toric variety 281 Grassmannian, Schubert classes on 330 Greatest common divisor 81 92 Green book 19. See [Sta96] Greenlees — May duality 106 270 Groebner basis viii 24 47—48 148 279. Groebner basis, and Hilbert scheme 358 360 363 370 Groebner basis, as straightening law 288 Groebner basis, comprehensive 25 Groebner basis, for determinantal ideal 290 323 Groebner basis, for module 27 Groebner basis, for Pluecker relations 276 277 281 Groebner basis, for quiver ideal 339 353 Groebner basis, for syzygies of bivariate ideal 43 Groebner basis, for syzygies of Borel-fixed ideal 30 Groebner basis, for toric ideal 281—283 Groebner basis, geometric interpretation See Groebner degeneration Groebner basis, minimal 282 Groebner basis, short encoding for toric ideal 244 246 Groebner basis, under weight order 142 Groebner basis, universal, for toric ideal 244 Groebner degeneration 158 286 311 323 Groebner degeneration, partial 353 Groebner degeneration, yields rational curve in Hilb 360 Grothendieck polynomial 309 Grothendieck — Riemann — Roch theorem 172 Group See also Orbit Group action See also Orbit Group action, free 184 357 Group action, left 299 Group action, of 368 Group action, transitive 301 Group algebra 131 161 163 171 181 Group, abelian See Abelian group Group, algebraic viii 287 Group, Borel See Borel group Group, Coxeter See Coxeter group Group, general linear 21 23 208 Group, representation of 287 288 Group, symmetric See Symmetric group Group, torus See Algebraic torus H-polynomial 8 157 266 Hankel matrix 305 Hartshorne's counterexample 255 269 Hasse diagram 276 Hilbert basis 137 138 Hilbert basis theorem 4 24 Hilbert basis, as Laurent polynomial 244 Hilbert basis, associated to sign pattern 180 Hilbert basis, at vertex of polytope 237 Hilbert basis, computing 138—140 141 150 244 Hilbert basis, in two dimensions 138 143 146 Hilbert basis, of antidiagonal semigroup 284 Hilbert basis, of Gelfand — Tsetlin semigroup 285 Hilbert basis, parametrizes GIT quotient 193 Hilbert function See also Hilbert series Hilbert function, 34 40 231 Hilbert function, multigraded 355 373 375 Hilbert function, positively graded 153 173 Hilbert polynomial 165 171 230 231 376 Hilbert scheme 21 355 Hilbert scheme, classical 373 376 Hilbert scheme, connectedness of 40 360—361 370 376 377 Hilbert scheme, irreducibility of 355 359 361 363 370 375 376 Hilbert scheme, isospectral 366 367 Hilbert scheme, local equations for 357—359 369 375 Hilbert scheme, most singular point of 371 Hilbert scheme, multigraded 355 373 374 375—376 378 Hilbert scheme, of 361 Hilbert scheme, of points in 361 368—373 Hilbert scheme, of points in plane 355 356—363 366—367
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