Smith L., Meyer D.M. — Poincare Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations :: Электронная библиотека попечительского совета мехмата МГУ

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Smith L., Meyer D.M. — Poincare Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations

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Название: Poincare Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations

Авторы: Smith L., Meyer D.M.

Аннотация:

Poincaré duality algebras originated in the work of topologists on the cohomology of closed manifolds, and Macaulay's dual systems in the study of irreducible ideals in polynomial algebras. These two ideas are tied together using basic commutative algebra involving Gorenstein algebras. Steenrod operations also originated in algebraic topology, but may best be viewed as a means of encoding the information often hidden behind the Frobenius map in characteristic p<>0. They provide a noncommutative tool to study commutative algebras over a Galois field. In this Tract the authors skilfully bring together these ideas and apply them to problems in invariant theory. A number of remarkable and unexpected interdisciplinary connections are revealed that will interest researchers in the areas of commutative algebra, invariant theory or algebraic topology.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2005

Количество страниц: 200

Добавлена в каталог: 27.05.2008

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Предметный указатель

$(d_{n,0},...,d_{n,n-1})^{\bot}$       88
$d_{2,0}$       41—42 63—64 94 99 160
$d_{2,1}$       41—42 63—64 94 99 160
$d_{n,0}$       83
$d_{n,i}$       81 82
$D_{n}$       88 89
$L_{n}$       83 85
$P^{k}(d_{n,i})$       82
$Sq(d_{2,0})$       94 96
$Sq(d_{2,1})$       94 96—97
$Sq(d_{n,k})$       109
$Wu_{k}$       60
$\cap$ -product      35 36
$\chi Wu_{k}$       61
$\chi$       56 58
$\Delta_{s,t}$       101
$\gamma$ -operations      33 71
$\mathcal{A}^{*}$       54
$\mathcal{A}^{*}$ -equivalent      75
$\mathcal{A}^{*}$ -indecomposable      6 73 93 139
$\mathcal{A}^{*}$ -invariant      64 97 105 109
$\mathcal{D}$       75—76
$\mathcal{P}$       56
$\mathcal{P}^{**}$       56
$\mathcal{P}^{*}$       54
$\mathcal{P}^{*}$ Double Annihilator Theorem      see “Double Annihilator Theorem
$\mathcal{P}^{*}$ -absorbing      67
$\mathcal{P}^{*}$ -decomposable      65
$\mathcal{P}^{*}$ -equivalence class      68
$\mathcal{P}^{*}$ -equivalent      65
$\mathcal{P}^{*}$ -indecomposable      65 170—171
$\mathcal{P}^{*}$ -invariant      65
$\mathcal{P}^{*}$ -invariant ideal      see “Ideal
$\mathcal{P}^{*}$ -invariant subalgebra      see “Subalgebra
$\mathfrak{D}_{n}$       88 89
$\overline{A}$ -primary      19
$\Phi(-)$       50
$\Xi$       14 15—16 41 95
2-adic expansion      102
2-adic valuation      111—112
Absorbing subspace      67—68
Adem — Wu relations      109
Ancestor ideal      17—18 38—39 133
Ancestor ideal, big      17 18 21 76 137
Ancestor ideal, little      17 137—138
Augmentation ideal      9 42
Big ancestor ideal      see “Ancestor ideal big”
Bilinear pairing      10
Binomial coefficient      98
Block parameters      118 121
Boundedness conjecture      74
Canonical anti-automorphism      56 58
Cartan formula      86
Catalecticant      39 133 136 137—140 155
Catalecticant equations      137
Catalecticant matrix      7 76 78 136
Cauchy — Frobenius lemma      23 24
Change of rings      7 155 157 162
Characteristic classes      60
Cocommutative      65
Cocontraction pairing      155
Codimension one subspace      18—19 158
Coexact      50 51 86—87 151—153 157
Coexact sequence      see “Coexact”
Cofactor matrix      41 84 91
Cofibration      152
Cohen — Macaulay      145
Cohen — Macaulay algebra      12
Cohomology      95
Coinvariants      81 86
Complete intersection      2 11 141
Comultiplication      32
Conjugacy class      27
Conjugate Mitchell — Stong element      167
Conjugate Wu class, total      61 64 117
Conjugate Wu classes      61 62 97 108—109 167
Conjugate Wu classes, k-th      61
Conjugate Wu classes, trivial      61 62
Connected sum      11 49 69 138 141
Contraction pairing      36 6
Coproduct      32
Cramer’s Rule      43
Cup-product      53
Cyclic      48
D(2)      160 168 170 172
D(3)      173
D(n)      81 85 160 171
Degree one      17
Diagonal      32
Dickson algebra      7 81 85 87 160 168 170—173
Dickson algebra, fractal of      87
Dickson coinvariants      6 83 85
Dickson polynomial      6 41 63 81—82 85 94—96 100 105
Dickson polynomial, formulae for      82
Dickson polynomial, power of      93 95—96 105 107—109
Dickson — Euler class      83
Differential operator      53
Divided power algebra      5 32
Divided power operation      32 33 38
Divided powers      see “Divided power operation”
Double annihilator      12
Double Annihilator Theorem      5 37 58 65 134—135
Double Annihilator Theorem, $\mathcal{P}^{*}$ -version      59 67
Double duality      10
Dual principal system      5 37 40 42—44 49 62 89 135
Dual system      5 12 31 35 37 47—49
Elementary symmetric polynomial      42—43 81
Expansion, dyadic      120
Finite extension      146 148 150—151 155—156 158 165 169—170
Finite type      1
fit      149
Fit extension      149 150—151 156 165 169—170
Fitness      154
Flat      44—45
Formal dimension      1 9 17—18 23
Fractal      87
Frobenius automorphism      158
Frobenius functor      44 51
Frobenius homomorphism      5 45 53
Frobenius operator      72
Frobenius periodicity      96 99
Frobenius power      31 41 44 64 77 97 164
Fundamental class      1 6 9—10 15 17—19 44 47 49 66 68 77—78 85—86 90 93—94 100 131 152 170
G-set      25
Galois field      5 23 41—42 53—54 56 90
Gorenstein algebra      5 9 12 13 16 57 146 148 150—151 155
Hall subgroup      29
Hasse — Schmidt differentials      53
Hilbert ideal      81 91
Hit Problem      4—7 54 65 74 93 100 123 160
Homogeneous component      94—95
Hopf algebra      5 12 31—32 35 65 135 162
Ideal quotient      45
Ideal, $\mathcal{A}^{*}$ -invariant      93 95—97 99 105—110 112—113 117 128 131 171
Ideal, $\mathcal{P}^{*}$ -invariant      54 56 62—63 162—163 165 169—170
Ideal, ancestor      see “Ancestor ideal”

Ideal, irreducible      4—5 7 9 12 13 15—19 31—32 35 40 42 46 49—50 53—54 56 62—63 90 133 135—136 139 146 148 150—151 155—156 158 163 169—170
Ideal, over      see “Over ideal”
Ideal, parameter      see “Parameter ideal”
Ideal, principal      13 15
Ideal, regular      7 12 142 171
Ideals, decreasing chain      97
Indecomposable Poincare duality algebra      11
Index sequence      34
Injective hull      135
Instability condition      161
Integral domain      147
Integrally closed      147
Inverse polynomial      31 133—134 137 158
Inverse polynomial algebra      134
Inverse principal system      135
Inverse system      5 31 135 156 158 163 170
Involution      16 27 38 41 95
Irreducible ideal      see “Ideal irreducible”
Isomorphism classes      19
Isotropy subgroup      22
Jordan form      24
Krull dimension      146
Lighthouse      95 100
Linear form      57
Little ancestor ideal      see “Ancestor ideal little”
Lying over      7 145—146 148
m-primary ideal      51
Macaulay dual      6 37 41—42 47—48 85
Macaulay dual of the ideal $(d^{2^{r}}_{2,0}d^{2^{r}}_{2,1})$       101
Macaulay dual of the ideal $(d^{2^{t}}_{2,0},d^{2^{s}-2^{t}}_{2,0})$       101—102
Macaulay inverse      135
Macaulay’s double annihilator correspondence      37
Macaulay’s Double Annihilator Theorem      see “Double Annihilator Theorem”
Macaulay’s dual principal system      see “Dual principal system”
Macaulay’s inverse system      135
Map of degree one      17
Middle associative      10
Middle dimension      100
Mitchell — Stong element      167
Mitchell — Stong element, conjugate      see “Conjugate Mitchell — Stong element”
Modular case      2
Module of $\mathcal{P}^{*}$ -indecomposables      65
Moduli space      19 23
Molien’s theorem      27
Monomial      66 68 70 72 75 78 89 93—94 138
Monomial action      70
Monomial ideal      37 42
Nil radical      32
Node      94
Noether Isomorphism Theorem      44
Noetherian algebra      15
Nondecreasing sequence      100
Nondegenerate      10
Number theoretic function      111
Obstruction      109
Orbit      22 26 39 88
Orbit space      19 23
Over ideal      14 41
P      49 71 74 77
p-adic expansion      34
Palindromic      37 137
Paradigm, $K\subset L$       5 31 40 88—89 113
Parameter ideal      12 13 16 41 47 55 57 145 148—149 151 159
Parameter ideal, $\mathcal{P}^{*}$ -invariant      55
Periodicity map      74
Periodicity operator      49 71 74 77 100
Poincare dual      15—17 35 131 152 154
Poincare duality      17—23 26—29 31 35 37—40 44 49 51 53—54 57—58 60—63 65—66 68—70 73—74 76—79 81 83 85—86 90 93—94 99—100 105 123 131 146—148 151—154 158 160 164 168
Poincare duality algebra      1 2 5 9 11 17—23 26—29 31 35 37—40 44 49—51 53—54 57—58 60—63 65—66 68—70 73—74 76—79 81 83 85—86 90 93—94 99—100 105 123 131 146—148 151—154 158 160 164 168 173
Poincare duality quotient      6 17—18 21 23 39 49 53—54 63 65—66 70 73 77 79 90 93 95 100 170
Poincare duality quotient of $\mathbb{F}[V]$       12
Poincare polynomial      95
Poincare series      20 27—28 99 137 141
Polynomial algebra      155—156
Prime ideal      7 146
Primitive element, Milnor      83
Principal      47
Principal ideal      see “Ideal principal”
Projective space      19
Pseudoreflection group      2
Rank      12
Rank of a Poincare duality quotient of $\mathbb{F}[V]$       12
Rank sequence      138
Regular ideal      see “Ideal regular”
Regular sequence      12 49 66 81 86—87 94 142 145 148
Relative transfer      148
Ring extension, finite      see “Finite extension”
Ring of invariants      81
Row-echelon form      137
Spike      6 93
Split      50—51 147—148 151—152
Sq      64 94
Steenrod algebra      4—7 53—54 58 66 79 100 160 163 169—170
Steenrod operation, total      56 57 71 105 109
Steenrod operations      7 54 66 71
Steenrod square, total      64
Steinberg, R.      2
Stiefel — Whitney class, power of      93
Stiefel — Whitney classes      6 62 93 123
Stong — Tamagawa formulae      82
Stripping operation      35
Subalgebra, $\mathcal{P}^{*}$ -invariant      86—87
Subspace, codimension one      see “Codimension one subspace”
Support      36 70 78
Sylow subgroup      29
symmetric      10
Symmetric coinvariants      6
System of parameters      12 42 146 148
Thom class      56 57 63—64 66 95—96 105 109 165
Thom — Wu formulae      4
Topological space      95
Total Steenrod operation      see “Steenrod operation total”
Total Steenrod square      see “Steenrod square total”
Totalization      13
Transition element      14 15 41 43 142 149 151
Transition invariant      14
Transitive      155
Transversal      27
Trivial conjugate Wu classes      see “Conjugate Wu classes trivial”
Trivial Wu classes      see “Wu classes trivial”
Type, finite      1
Unstable algebra      53—54 56 163
Unstable algebra (over $\mathcal{P}^{*}$ )      54
Unstable Poincare duality algebra      54
Unstable polynomial agebra      162 169—170
Upper triangular      141
Wu class      5—6
Wu class, total      60 64 170
Wu class, total conjugate      61
Wu classes      3 60 64 66 68 71 77 83 86 90 165
Wu classes, conjugate      see “Conjugate Wu classes”
Wu classes, k-th      60
Wu classes, nontrivial      99
Wu classes, trivial      54 60 61 63 65 68—70 73 79 86 93—99 105—109 121 123 125 128 131 169—170 173
Wu Wen Tsuen      54
Wu’s formula      124—125

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