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| Bruns W., Vetter U. — Determinantal Rings |
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| Предметный указатель |
Absolute invariant 74 86
Absolute semi-invariant 76
Acyclicity criterion 207 208
Algebra generated by the t-minors 132 133 151
Algebra with straightening law (ASL) 38
Algebraic quotient 80
Analytic spread 117
Arithmetical rank 61
Arithmetical rank of a determinantal ideal 62 81
Arithmetical rank of an ideal dening a Schubert cycle 62
Arithmetical rank of an ideal generated by a poset ideal 62
Arithmetical rank of an ideal generated by a poset ideal, in a symmetric ASL 62
ASL 38
ASL, discrete 62
ASL, symmetric 43
ASL-property of a determinantal ring 52
ASL-property of a graded ring derived from a straightening-closed ideal 108 110
ASL-property of a Schubert cycle 53
ASL-property of B[X] on  48
ASL-property of G(X) on  41
Associated graded module 30
Associated graded ring 30 108
Associated graded ring derived from a straightening-closed ideal 109
Associated graded ring derived from an ideal of maximal minors 112 114
Associated graded ring derived from the ideal of t-minors 132
Associated prime ideals 1
Associated prime ideals of a G-stable ideal 148 149
Associated prime ideals of powers of a determinantal ideal 132
Auslander — Bridger dual 214
Auslander — Buchsbaum equation 1
Auslander-module 198
Auslander-module and rigidity 199 200
Bitableau 138
Bitableau, left (right) final 139
Bitableau, left (right) initial 139
Bitableau, standard 138
Canonical (divisor) class 93
Canonical (divisor) class of a determinantal ring 97 98 103
Canonical (divisor) class of a Schubert cycle 97 102
Canonical module 210
Canonical module of a determinantal ring 22 97 98 103 115
Canonical module of a Schubert cycle 97 102 115
Class group see Divisor class group
Cofactors, matrix of 3
Cohen — Macaulay module 209
Cohen — Macaulay property and perfection 210
Cohen — Macaulay property of a determinantal ring 13 25 60
Cohen — Macaulay property of a graded ring derived from an ideal of maximal minors 112
Cohen — Macaulay property of a Schubert cycle 60
Cohen — Macaulay property of an ASL 59
Cohen — Macaulay ring 209
Cohen — Macaulay type of a determinantal ring 115
Complete intersection 198
Content of a tableau (bitableau) 138
D-ideal 145
D-ideal, primary 147
D-ideal, prime 147
D-ideal, principal 146
D-ideal, radical 147
Decomposition of K[X] into irreducible G-modules 143 144
Dehomogenization 211
Dehomogenization and being reduced 212
Dehomogenization of a homogeneous ideal 212 213
Dehomogenization, B[X] as a 48
Dehomogenization, integrity 212
Dehomogenization, normality 212
Dening ideal of an ASL 40
Dening ideal of G(X) 43
Depth see also Grade
Depth of a module 1 206
Derivations, almost perfection 188
Derivations, module of 184
Derivations, perfection 192
Derivations, system of generators 188
Derivations, syzygetic behaviour 197
Derivations, syzygies 189
Determinantal ideal 5 51 155 156 175
Determinantal ideal of a homomorphism 202
Determinantal ring 4 5
Determinantal ring as a 53
Determinantal ring as a ring of absolute invariants 76 86 87 160
Determinantal variety 4 6
Diagram 136
Differentials, depth 180 181 183
Differentials, free locus 178
Differentials, grade 180 183
Differentials, module of 174
Differentials, presentation as a second syzygy 186
Differentials, syzygetic behaviour 182
Differentials, syzygies 181
Dimension of a determinantal ring 57
Dimension of a Schubert cycle 57
Dimension of an ASL 55
Discrete ASL 62
Divisor class group of a determinantal ring 95 96 160
Divisor class group of a Schubert cycle 95
Dual basis 2
Dual of a homomorphism 2
Dual of a module 2
Dualizing module 210
Eagon — Northcott complex 16
Extended Rees algebra 108
Extended Rees algebra derived from a straightening-closed ideal 108 109
Extended Rees algebra derived from an ideal of maximal minors 112 114
Extended symbolic Rees ring 124
Extended symbolic Rees ring derived from a determinantal ideal 124
Extraspecial pair 43
Factoriality of a determinantal ring 95
Factoriality of a Schubert cycle 69 95
Fitting invariants of a module 203
Form ideal, form module 30
Free resolution 18 21
Free resolution of a generic module 172
Free resolution of an ideal, of maximal minors 18
Free resolution of an ideal, submaximal minors in a quadratic matrix 25
Free resolution of symmetric and exterior powers 18 21
Free resolution of the image of a generic map 172
Full semigroup 150
G-decomposition of K[X] 143 144
G-module 80
G-stable ideals of K[X] 146
G-submodule generated by a bitableau 145
Generic map 162
Generic map, free resolution of the image 172
Generic map, perfection of the image 164 165
Generic module 162
Generic module, almost perfection 164
Generic module, free resolution 172
Generic module, perfection 18 167 170
Generic module, projective dimension 171
Generic module, reflexivity 162
Generic module, syzygetic behaviour 171 172
Generic perfection 27
Generic perfection and substitution of indeterminates 31 32
Generic perfection of a determinantal ideal 61
Generic perfection of a Schubert cycle 61
Generic perfection of the canonical module 30
Generic point 73
Generic point of a determinantal ring 74 85 87
Generic point of a Schubert cycle 83 85
Generically perfect ideal 27
Generically perfect module 27
Gorenstein property 210
Gorenstein property of a determinantal ring 22 25 98 104
Gorenstein property of a Schubert cycle 102
Gorenstein ring 210
Grade 206
Grade and projective dimension 207
Grade estimates for modules of invariants 89
Grade estimates for powers of ideals of maximals minors 90 118 119
| Grade estimates for symbolic powers of a determinantal ideal 125
Grade of a determinantal ideal 12 61 159
Grade of a module 206
Grade of an ideal 206
Grade of an ideal dening a Schubert cycle 61
Grade, behaviour along exact sequences 207
Grade, homological description 206
Grade, local description 207
Grassmann variety 6 8
Gulliksen — Negard complex 22
Height of a determinantal ideal 12 61 159 175
Height of a specialization of a perfect ideal 36
Height of an ideal dening a Schubert cycle 61
Height of an ideal generated by minors of a matrix 10 11
Hilbert — Burch theorem 218
Homogenization of an ideal 212 213
Ideal in a partially ordered set 50
Ideal in a partially ordered set, cogenerated by a subset 51
Ideal in a partially ordered set, generated by a subset 51
Ideals of maximal minors 10 105
Ideals of maximal minors, free resolution 18
Indecomposability ob the subspace of (right) initial bitableaux 142
Induction argument for determinantal rings see Localization argument
Integral closure of a G-stable ideal 149
Integrity of a determinantal ring 14 65 158
Integrity of a graded ring derived from an ideal of maximal minors 114
Integrity of a Schubert cycle 65
Invariant 80
Invariant, absolute 74 86
Invariant, theory, main problems 80
Invariant, versus absolutely invariant 75
Isotopic component 142
Kaehler differentials see Differentials
Koszul complex 17
Leading form 30 108
Linear independence of standard monomials in G(X) 43 84
Linearly reductive 80 81
Localization argument for determinantal rings 11 66 122
Localization argument for Schubert cycles 64
Locally upper semi-modular poset 58
Maximal minors 3 105
Maximal minors, ideals of 10 105
Maximal minors, powers of ideals of 106 114
Maximal minors, subalgebra of 106 107 112
Minimal primes of an ideal generated by a poset ideal, in 66
Minimal primes of an ideal generated by a poset ideal, in 66
Minor of a matrix 3
Module of derivations see Derivations
Module of differentials see Differentials
Module, n-th syzygy 214
Module, n-torsionfree 213
Module, n-torsionless 214
Module, reflexive 214
Module, torsionfree 213
Multiplicity of a (maximal minors) determinantal ring 16
Normal subsemigroup 150
Normality criterion 212
Normality of a determinantal ring 15 61 65 150 160
Normality of a graded ring derived from an ideal generated by maximal minors 114
Normality of a Schubert cycle 65
Perfect ideal 209
Perfect module 209
Perfection 209
Perfection and Cohen — Macaulay property 210
Perfection of a determinantal ideal 13 18 25 61 159 175
Perfection of a Schubert cycle 61
Perfection of a specialization 28 29
Pluecker coordinates 6
Pluecker map 6
Pluecker relations 41
POSET 39
Powers of an ideal generated by a regular sequence 15 29
Powers of an ideal of maximal minors 79 114
Powers of determinantal ideals 79 114 130 149
Primary decomposition of a G-stable ideal 148 149
Primary decomposition of products of determinantal ideals 126 130 149
Prime elements in a determinantal ring 97
Principal radical system 155
Principal radical system of determinantal ideals 158
Products of determinantal ideals 126 137
Radical of a D-ideal 147
Radical subsemigroup 150
Rank of a free module 1
Rank of a module 1 204
Rank of a poset element 55
Rank of a subset of a poset 55
Rational representation 80
Rational singularity 81 150
Reduced ASL 54
Reduced determinantal ring 15 55
Reduced Rees algebra 108
Reduced Rees algebra, derived from a straightening-closed ideal 110 111
Reduced Rees algebra, derived from an ideal of maximal minors 112 114
Reduced Rees algebra, extended 108
Reduced ring 1
Reduced Schubert cycle 55
Reflexive module 214
Regular elements in an ASL 55 56
Reynolds operator 81 88
Rigidity 198 200
Ring of invariants, being noetherian 81
Ring of invariants, Cohen — Macaulay property 81
Ring of invariants, existence of a Reynolds operator 81
Ring of invariants, normality 81 115 149
Ring of invariants, rational singularities 81 149
Ring of U-invariants 149
Ring of U-invariants in a determinantal ring 150
Schubert cycle 6 9
Schubert cycle as a ring of absolute invariants 76 86
Schubert variety 6 8
Segre product of ASLs 107
Self-covering poset ideal 110
Serre type criterion for torsionfree modules 213
Serre's condition ( ) for a determinantal ring 71
Serre's condition () for a Schubert cycle 71
Serre's condition ( ) for an ASL 60
Shape of a monomial 135
Singular locus of a determinantal ring 12 70 71 160
Singular locus of a Schubert cycle 68
Special pair 43
Spectrum of a ring 1
Standard basis 38
Standard bitableau 138
Standard monomial 38
Standard monomials as generators of an ASL 39
Standard representation 39
Standard representation of a bitableau 138 139
Standard representation of a monomial 39
Standard tableau 138
Straightening law 38
Straightening relations 38
Straightening-closed ideal 108
Straightening-closed ideal, extended Rees-algebra derived from a 108
Straightening-closed ideal, powers of a 108
Submaximal minor 3
Support of a module 1
Support of a monomial 131
Symbolic graded ring 124
Symbolic graded ring, derived from a determinantal ideal 124
Symbolic powers of determinantal ideals 122 123 124 126 129 130 149
Symbolic powers of determinantal ideals, intersection of 136 137
Symmetric algebra of a straightening-closed ideal 111
Symmetric ASL 43
Syzygies of a generic module 166
Syzygies of an ideal generated by a poset ideal 53
Syzygies of determinantal ideals 164 175
Syzygy module 214
Tableau 137
Tableau, final 139
Tableau, initial 139
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