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Matsumura H. — Commutative ring theory |
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Предметный указатель |
(0-) etale 193 204
(0-) smooth 193 204 233
(0-) unramified 193
(A-) algebra 269
(I-) adic completion 57
(I-) adic topology 57
(I-) adically ideal separated 174
(I-) etale 214
(I-) smooth 213 217 233
(I-) smooth w.r.t. k 217
(I-) unramified 214
(M-) regular element 38 123
(M-) sequence 123
(p-) basis 202
(p-) independent 202
Absolute p-basis 203
Acceptable ring 260
Additive valuation 75
Akizuki’s theorem 16
Algebra, skew-commutative graded 285
Almost integral element 69
Analytically independent elements 107
Annihilator ann(M) 6
Archimedean ordered group 76
Artin — Rees lemma 59 98
Artinian (ring, module) 12 14 16 22 30 63 94
Ascending chain condition (A.C.C.) ix 14 39
Associated graded ring 93 120
Associated prime ideal, Ass(M) 38 156 179
Associated separated module 55
Bass conjecture 152
Cartier equality 205
Catenary ring 31 117 118 250
Catenary ring, formally 252
Catenary ring, universally 118 137 251
Centre of a valuation ring 73
Characteristic p xiii 190 196 241
Characteristic, equal 215
Characteristic, unequal 215 223 228
Closed fibre 116
CM module 134
CM ring 136 188
Cocycle 221
Codepth 139
Coefficient field 215
Coefficient ring 225
Cofinite subfield 241
Coheight of a prime ideal 30
Cohen — Macaulay (CM) x 133 134 137 180 182 188
Cohen’s theorem 17
Combinatorial dimension 30
Complete intersection (c.i.) ring 171
Complete local ring 45 215 224 225
Complete module 55
Completely integrally closed 69
Completion ix 55 93 157 169
COMPLEX 127 274
Complex, double 275
Complex, Koszul 111 127 151 156 169 286
Complexes, tensor product of 281
Composite of valuation rings 72
Composition series 12 15
Contracted ideal 21
Coprime ideals 3
Dedekind ring (domain) ix 82 89
Degree of a homogeneous element 92
Depth x 130 154
Derivation 190 230
Derivation, higher 207
Derived functor 280
Descending chain condition (D.C.C) 14
Determinantal ideals 103
Differential 192
Differential basis 201
Differential ideal 206
Differential module of 190 192
Differential of a complex 274
Dimension (Krull dimension) 20 30 98
Dimension (Krull dimension) of intersection 261
Dimension (Krull dimension), embedding 104 156 169
Dimension (Krull dimension), formula 119
Dimension (Krull dimension), global 155
Dimension (Krull dimension), inequality 118
Dimension (Krull dimension), injective 139 280
Dimension (Krull dimension), projective 155 280
Dimension (Krull dimension), theory ix 32 92 98
Direct limit 270
Direct system 269
Directed family of subfields 241
Directed set 55 269
Discrete module 221
Discrete valuation ring (DVR) 71 78 106 223
Discriminant 198
Divisor class group 165
Divisorial fractional ideal 91
Domain see “Integral domain”
Dominates 72
Double complex 275
DVR see “Discrete valuation ring”
Eakin — Nagata theorem 18
Eisenstein extension 229
Embedded prime 39 42 136
Embedding dimension 104 156 169
Equal characteristic 215 261
Equicharacteristic 215 261
Equidimensional 250
Equidimensional, formally 251
Essential extension 149 281
Essentially of finite type 232
Euler number 111 159
Exact functor 26 268
Exact sequence 12 45 51 53 268
Excellent ring xi 260
Ext 53 129 139 154 185 189 279
Extended ideal 21
Extension of a ring by a module 191
Extension of scalars 268 284
Extension, essential 281
Extension, trivial or split 191
Exterior algebra 169 286
Exterior product 283
Factorial ring see “Unique factorisation domain”
Faithful A-module 6
Faithfully flat 45 47
Fibre over a prime 47 116 178
Fibre, closed 116
Fibre, generic 116
Field 2 106
Filtration 93
Finite free resolution (FFR) 159
Finite length 12 15 94
Finite module 7 15 26
Finite presentation, module of 12 14 19 52
Finite type, essentially of 232
Finitely generated ideal 5 15 50 94 271
Finitely generated module see “Finite module”
Flat (module, algebra) x 45 116 173 186 268
Flat, normally 188
Formal fibre xi 255
Formal power series ring A[X] 4 16 53 61 117 157 165 168 177 181
Formally catenary ring 252
Formally equidimensional 251
Forster — Swan theorem 35
Fractional ideal 19 80 166
Free module 9 13 51
Free module, stably 161 163
G-ring (Grothendieck ring) 256
Generic fibre 116
Generic point 36
Geometrically regular 219 255
| Global dimension 155 182
Going-down Theorem 45 67 116
Going-up Theorem 67 116
Gorenstein ring xi 139 142 145 181 189
Grade 131
Graded module 92 177
Graded ring 92 101
Greatest common divisor (g.c.d.) 163
Height of a prime ideal 20 30 100
Hensel ring x
Hensel’s Lemma 58
Higher derivation 207
Hilbert function 92 95 101
Hilbert nullstellensatz 33
Hilbert polynomial 95 138
Hilbert ring see “Jacobson ring”
Hilbert series 94
Hochschild formula 190 197
Homogeneous element 92
Homogeneous ideal, submodule 92 101
Homology 127 170 274
Horn 6 52 129 140 154 189
I-adic completion 57 272
I-adic topology 57
I-adically ideal-separated 174
Ideal ix 1
Ideal of definition 62 97
Ideal, contracted 21
Ideal, extended 21
Ideal, fractional 19 80
Ideal, maximal 2
Ideal, parameter 138
Ideal, perfect 132
Ideal, primary 21
Ideal, prime 2
Ideal, proper 1
Imperfection module 205
Indecomposable module 145
Independent in the sense of Lech 160
Injective dimension 139 155 280
Injective hull 145 280
Injective module 144 277
Injective resolution 144 185 278
integral 64
Integral, closure ix 64 73 261
Integral, domain 2
Integral, element, almost 69
Integral, extension 45 64
Integrally closed domain 64 73
Inverse limit 55 271
Inverse system 55 271
Invertible fractional ideal 19 80 167
Irreducible closed set 29 39
Irreducible element 5 162
Irreducible submodule, ideal 40 142
Isolated associated prime 39
Iterative higher derivation 209
Jacobian condition, weak (WJ) 239 259
Jacobian criterion 233 237 239 244
Jacobson radical see “Radical”
Jacobson ring 34 115
Koszul complex 111 127 151 156 169 286
Krull dimension 30
Krull intersection theorem 60
Krull ring x 86 165
Krull — Akizuki theorem 84
Least common multiple (l.c.m.) 163
Lech’s lemma 110
Length 12 84 94
Lies over 25 66
Lifting 191
Linear topology 55 93
Linearly disjoint subfields 200
Local homomorphism 48
Local property 26
Local ring 3
Localisation x 20 48 63 65 157
Matlis theory 144
Maximal condition see “Ascending chain condition”
Maximal ideal 2 24
Maximal spectrum m-Spec A 24
Minimal associated prime 40
Minimal basis 8
Minimal condition see “Descending chain condition”
Minimal free resolution 153
Minimal injective resolution 281
Module 7 268
Module of differentials 190 192
Module of finite presentation 12
Module, finite 7
Module, imperfection 205
Module, injective 144 277
Module, projective 277
Multiplicative set 2 20
Multiplicative set, saturation of 23
Multiplicative valuations 71
Multiplicity x 92 104 108 138
Nagata criterion for openness (NC) 186
Nagata ring ix 264
NAK (Nakayama’s lemma) 8
Neat see “Unramified”
Nilpotent element 1 3 6 21
Nilradical nil(A) 3
Noether normalisation 262
Noetherian graded ring 94
Noetherian module 14
Noetherian ring ix 14 17 22 37 84
Noetherian topological space 29
Normal ring 64 82 88 157 183
Normally flat 188
Null-form 107 112
Null-form, ideal of 112
Nullstellensatz 33
Number of generators of a module 35 170
Open loci 172 186 237 245 260
Ordered group 75
Ordered group of rank 1 77
Ordered group, Archimedean 76
p-ring 223
Parameter ideal 138
Parameters, system of 104
Perfect field 200
Perfect ideal 132
Picard group of a ring Pic(A) 166
Polynomial ring A[X] 4 16 31 58 89 117 157 161 168 177 181 192 269
Primary components (of a submodule) 41
Primary decomposition ix 37 41
Primary ideal 21
Primary submodule 39 40
Prime divisor of an ideal 38 100
Prime element 5
Prime ideal 2
Primitive polynomial 168
Principal ideal 5 161 164
Principal ideal, theorem 92 100 162
Projective dimension 155 182 280
Projective module 9 52 80 161 166 277
Projective resolution 51 158 277
Proper ideal 1
Prufer domain 86
Pure submodule 53
Quasi-coefficient field 215
Quasi-coefficient ring 225
Quasi-excellent ring 260
Quasi-regular sequence 124
Quasi-unmixed see “Formally equidimensional”
Quotient of regular local ring 169 171
Radical of an ideal 3
Radical rad(A), Jacobson 3 8 57 98
Ramified local ring 228
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