Reid M. — Undergraduate commutative algebra :: Электронная библиотека попечительского совета мехмата МГУ

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Reid M. — Undergraduate commutative algebra

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Название: Undergraduate commutative algebra

Автор: Reid M.

Аннотация:

These are notes from a commutative algebra course taught at the University of Warwick several times since 1978. In addition to standard material, the book contrasts the methods and ideology of abstract algebra as practiced in the 20th century with its concrete applications in algebraic geometry and algebraic number theory.

Язык:

Рубрика: Математика/Алгебра/Учебники/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1995

Количество страниц: 164

Добавлена в каталог: 18.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$S^{-1}$ is exact      90—92
$\mathbb Z$ versus       5—10 22
$\mathrm{Spec}\mathbb{Z}[Y]$       25
1-dimensional local integral domain      115
a.c.c. (ascending chain condition)      11 13 49 52 56 109
Abstract versus concrete      11 12 23 49 51 95 115 136
Algebraic curve      7 11
Algebraic number theory      8—11 16—18 25 58 112
Algebraic variety      see variety
Algebraically closed field      3 70
Algebraically independent elements      63
Algebraically nonclosed field      72 73 82
Anal germs      33 35 52 112 127
Analytic arc      127
Annihilator Ann^ M      96
Artinian local ring      56
Artinian module      56
Artinian ring      51 55 56
Assassin Ass M      98—109
Associated prime Ass M      98
Axiom of Choice      25 26 36 50 56
Basis of free module      40
Bourbaki, Nicholas      95 98
Cayley — Hamilton theorem      41—43 47
Chain conditions      11 18 26 49—53 76 103 108
Characteristic char k      7
Codimension      1
Codimension, prime ideal      see minimal nonzero prime ideal
Codimension, subvariety      5 112 121
Cokernel coker $\varphi$       45
Completion of DVR      126
Composite valuation      118 127
Coordinate ring k[V]      3 12 70 80 92 121
Correspondences V and I      72
Cusp       7—8 11 62
Cyclic module      41
d.c.c. (descending chain condition)      51 56
Dedekind domain      134
Determinant trick      41—44 60 116
Devissage      103
DIMENSION      115
Dimension theory      2 131
Direct sum of modules      40 45 53—54
Direct sum of rings $A_1\oplus A_2$       2 4 28—29
Disassembling a module      103 111
Discrete valuation      113
Division with remainder      6 14 16
DVR (discrete valuation ring)      12 112—117 119—123
Easy Nullstellensatz      31 73 78 81
Elementary symmetric functions      68 124 125
Empty set theory      20 36
Endomorphism of module      38 41—43
Equal characteristic      7
Euclidean ring      6 14 16—18
Evaluation map $f\mapsto f(P)$       3 21 35 36 70—72 78 81
Exact sequence      44—48 54
Existence of assassin      99 102 115
Existence of maximal ideals      26
Existence of primary decomposition      106 108—109
Existence of prime ideals      25—27
f.g. (finitely generated) algebra      49 55 63 70 75 80
Factorisation      2—3 10 18 95
Faithful A-module      38
Field extension      15 25 58 62 66—67 82 122
Field of fractions K = Frac A      2 84
Finite A-algebra      53 58—61 63 69 93 122—124
Finite and integral      58—61 68
Finite branched cover      132
Finite module      40 43—44 47 53—54 123
Finitely generated ideal      49—51
Finiteness conditions      40 49 58—60
Finiteness of normalisation      13 122—124 136
Fitting's lemma      56 109 111
Formal power series ring       16 32 56 127 130
Free module      40—41
Function field k(V)      92 93
Galois theory      10 14 72 82 123
Gauss' lemma      15 22
Generators of module      40 43
Geometric ring      80
Geometric view      1 3—5 12 21 23—25 29 70 78 106
Geometry      70 75 95
Geometry of Spec A      77—82 97
Germ      33 52
Golden ratio $\tau$       18 59 67
Hilbert basis theorem      49 54—56
Hilbert, David      11 49
history      11 49 142—144
Homological algebra      48
Hypersurface $V(F) \subset k^n$       3
Ideal      19
Ideal, in localisation      87 109
Ideal, maximal      see maximal ideal
Ideal, of denominators      119
Ideal, prime      see prime ideal
Idempotent      2 4 28
Image $\mathrm{im}\varphi$       38 44
Indecomposable ideal      108—109
Inseparable field extension      128 137
Integral algebra      59
Integral and finite      58—61 68
Integral closure $\tilde A$       61
Integral dependence      58
Integral domain      2 99
Integral element over A      58—61
Integral extension      59
Integrally closed      61
Intersection of DVRs      12 121—122
Intersection of minimal prime ideals      29 77 79
Intersection of prime ideals      28—30 95 105
Invertible element      2 26 86
Irreducible components      28—29 76—77 82 95 101 106
Irreducible element      2 8
Irreducible variety      74—75 82 83 92
Isomorphism theorem      38—39 48
Kernel $\mathrm{ker}\varphi$       19 38—39 45
Koszul complex      46—47
Linear algebra      37 41

Local parameter of DVR      113
Local property      118
Local ring      2 31—33 35 47 82 84 88 112 118
Localisation       31—32 82 84—92
Maximal condition      56
Maximal element      25 49 50
Maximal ideal      3 20—21 35 70 72
Maximal spectrum m-Spec A      21 80 119
Minimal element of Supp M      101
Minimal nonzero prime ideal      115 119—121
Minimal polynomial      15 124 125
Minimal prime ideal      28 79 95 98 105
Mixed characteristic      7 16
Module      37
Module, of fractions $S^{-1}M$       89—90 96
Monic polynomial      14 58 66
Multiplicative set      20 27—28 31 34 84—85
Nakayama's lemma      43—44 47 69
Newton's rule      125
Nilpotent      4
Nilpotent element      2 4 27—29 86 104
Nilpotent ideal      56
Nilradical nilradA      2 27 29 35 81
Noether normalisation      58 63—67 122 132
Noether, Emmy      11 49 63 95
Noetherian Induction      49—57 76—77 79 108
Noetherian module      52—54
Noetherian ring      49—51 54—57 95 99 135—141
Noetherian topology      76
Non-Noetherian ring      51
Noncommutative ring      37 38
Nonsingular      62 133
Normal integral domain      61 114
Normal Noetherian ring      118—122
Normalisation $\tilde A$       61—63 68 122—124 133
Nullstellensatz      3 5 31 58 70—75 80 82
Number field      62
Number field and algebraic curve      7—10 62—63
Number field and normal variety      112
Ordered group      116—118
Partially ordered set      13 25 36 49
Picture      13 29 121 141 146
PID (principal ideal domain)      13 14 22
Plane curve      7
Poles      see zeros and poles
Polynomial map      80 83
Polynomial ring $A[x_1, \ldots, x_n]$       3 15 35 49 55 63 106
Presentation of module      41 55
Primary decomposition      12 88 95—111 115
Primary ideal      95 103—111
Prime element      2
Prime element of DVR      113
Prime ideal      20—29 74 78 87 95 100
Prime spectrum      see spectrum Spec A
Primitive extension $K \subset K(\alpha)$       14 15 124
Primitive polynomial      15 22
Principal ideal      6 14 22 112 114
Product of ideals IJ      34
Quadratic number field $\mathbb Q(\sqrt{n})$       62 68
Quadratic reciprocity      16
Quadric cone      5 107
Quotient module M/N      38
Quotient ring A/I      20 91
Radical ideal I = rad I      30 77—78 95
Radical rad I      29 104
Radius of convergence $\rho$       33
Reduced ideal      80
Reduced polynomial      4 82
Reduced ring      27 35
Reducible variety $X=X_1\cup X_2$       4 74
Reduction to local ring      31 84 88 102 110 120
Regular local ring      131
Remainder theorem      14 36
Residue field at P      21 91
Resolution of singularities      13 62—63 133
Ring of analytic functions      35
Ring of fractions $S^{-1}A$       12 84—94
Ring of functions      1 12 52 80 81 89
Ring of germs      33
Ring of integers $\mathcal O_K$       8—10 13 16—18 58 62
Scheme theory      141
Separable field extension      122—126 128
Set theory      see Zorn's lemma
Short exact sequence (s.e.s.)      45—47
Shortest primary decomposition      105
Singular plane curve      7 62
Snake lemma      48
Spec $\mathbb Z[X]$       22
Spectrum Spec A      12 21—25 75 77—82 115 130
Split exact sequence      45—46 48
Strongly coprime ideals      34
Subring $A' \subset \mathrm{End} M$       38 42—43
Subvariety      4
Support Supp M      96—98 100—103
Tangent plane      5
Taylor series      29 130
Total order      116—117
Trace $Tr_{L/K}$       123—126
Transfinite induction      25 36
UFD (unique factorisation domain)      3 13—15 18
UFD is normal      15 62
UMP (universal mapping property)      14 86 93—94
Unequal characteristic      7
Unique factorisation      2—3 14
Uniqueness of primary decomposition      109—110
UNIT      2 26 86
Vacuity      20 36
Valuation ring      113 116—118
Value group of valuation      117
Variety $V \subset k^n$       3 11 70—77 80 89 92 121
Variety $\mathcal V(I)\subset \mathrm{Spec}\, A$       78—79 96—97
Weak Nullstellensatz      3—5 67 70—72
Weight of monomial w(Ym)      65
Zariski topology      75—81 83 96—97 100 130
Zerodivisor      1—2 4 28—29 79 84 85 92
Zerodivisor, for M      95 96 100
Zeros and poles      12 112 121
Zeros theorem      73
Zorn's lemma      13 25—27 36 56

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