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









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

: Smith L., Meyer D.M.

:

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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ed2k: ed2k stats

: 2005

: 200

: 27.05.2008

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$(d_{n,0},...,d_{n,n-1})^{\bot}$      88
$d_{2,0}$      4142 6364 94 99 160
$d_{2,1}$      4142 6364 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 9697
$Sq(d_{n,k})$      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
$\mathcal{D}$      7576
$\mathcal{P}$      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 170171
$\mathcal{P}^{*}$-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 1516 41 95
2-adic expansion      102
2-adic valuation      111112
Absorbing subspace      6768
Adem Wu relations      109
Ancestor ideal      1718 3839 133
Ancestor ideal, big      17 18 21 76 137
Ancestor ideal, little      17 137138
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 137140 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      1819 158
Coexact      50 51 8687 151153 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 108109 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
Cramers 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 170173
Dickson algebra, fractal of      87
Dickson coinvariants      6 83 85
Dickson polynomial      6 41 63 8182 85 9496 100 105
Dickson polynomial, formulae for      82
Dickson polynomial, power of      93 9596 105 107109
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 134135
Double Annihilator Theorem, $\mathcal{P}^{*}$-version      59 67
Double duality      10
Dual principal system      5 37 40 4244 49 62 89 135
Dual system      5 12 31 35 37 4749
Elementary symmetric polynomial      4243 81
Expansion, dyadic      120
Finite extension      146 148 150151 155156 158 165 169170
Finite type      1
fit      149
Fit extension      149 150151 156 165 169170
Fitness      154
Flat      4445
Formal dimension      1 9 1718 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 6 910 15 1719 44 47 49 66 68 7778 8586 90 9394 100 131 152 170
G-set      25
Galois field      5 23 4142 5354 56 90
Gorenstein algebra      5 9 12 13 16 57 146 148 150151 155
Hall subgroup      29
Hasse Schmidt differentials      53
Hilbert ideal      81 91
Hit Problem      47 54 65 74 93 100 123 160
Homogeneous component      9495
Hopf algebra      5 12 3132 35 65 135 162
Ideal quotient      45
Ideal, $\mathcal{A}^{*}$-invariant      93 9597 99 105110 112113 117 128 131 171
Ideal, $\mathcal{P}^{*}$-invariant      54 56 6263 162163 165 169170
Ideal, ancestor      see Ancestor ideal
Ideal, irreducible      45 7 9 12 13 1519 3132 35 40 42 46 4950 5354 56 6263 90 133 135136 139 146 148 150151 155156 158 163 169170
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 133134 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 145146 148
m-primary ideal      51
Macaulay dual      6 37 4142 4748 85
Macaulay dual of the ideal $(d^{2^{r}}_{2,0}d^{2^{r}}_{2,1})$      101
Macaulay dual of the ideal $(d^{2^{t}}_{2,0},d^{2^{s}-2^{t}}_{2,0})$      101102
Macaulay inverse      135
Macaulays double annihilator correspondence      37
Macaulays Double Annihilator Theorem      see Double Annihilator Theorem
Macaulays dual principal system      see Dual principal system
Macaulays 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
Moliens theorem      27
Monomial      66 68 70 72 75 78 89 9394 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 8889 113
Parameter ideal      12 13 16 41 47 55 57 145 148149 151 159
Parameter ideal, $\mathcal{P}^{*}$-invariant      55
Periodicity map      74
Periodicity operator      49 71 74 77 100
Poincare dual      1517 35 131 152 154
Poincare duality      1723 2629 31 35 3740 44 49 51 5354 5758 6063 6566 6870 7374 7679 81 83 8586 90 9394 99100 105 123 131 146148 151154 158 160 164 168
Poincare duality algebra      1 2 5 9 11 1723 2629 31 35 3740 44 4951 5354 5758 6063 6566 6870 7374 7679 81 83 8586 90 9394 99100 105 123 131 146148 151154 158 160 164 168 173
Poincare duality quotient      6 1718 21 23 39 49 5354 63 6566 70 73 77 79 90 93 95 100 170
Poincare duality quotient of $\mathbb{F}[V]$      12
Poincare polynomial      95
Poincare series      20 2728 99 137 141
Polynomial algebra      155156
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 8687 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      5051 147148 151152
Sq      64 94
Steenrod algebra      47 5354 58 66 79 100 160 163 169170
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      8687
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 6364 66 9596 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      5354 56 163
Unstable algebra (over $\mathcal{P}^{*}$)      54
Unstable Poincare duality algebra      54
Unstable polynomial agebra      162 169170
Upper triangular      141
Wu class      56
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 6870 73 79 86 9399 105109 121 123 125 128 131 169170 173
Wu Wen Tsuen      54
Wus formula      124125
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