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Matsumura H. — Commutative ring theory
Matsumura H. — Commutative ring theory

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Название: Commutative ring theory

Автор: Matsumura H.

Аннотация:

In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.


Язык: en

Рубрика: Математика/Алгебра/Учебники/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1986

Количество страниц: 320

Добавлена в каталог: 12.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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 $HS_k(A)$      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
1 2
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