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Jannsen U. — Mixed Motives And Algebraic K-Theory
Jannsen U. — Mixed Motives And Algebraic K-Theory



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Название: Mixed Motives And Algebraic K-Theory

Автор: Jannsen U.

Аннотация:

The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varieties.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2008

Количество страниц: 264

Добавлена в каталог: 04.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A^{j}(X)$, $A_{i}(X)$      115 117
$A_{0}(X)$, Alb(X)      157
$A_{f}$      85
$CH^{j}(X)_{0}$, $CH_{i}(Z)_{0}$      155
$CH^{j}(X)_{num}$      178
$CH^{r}(X)$      57
$Ch_{b}(X)$      105 107
$c\ell^{'}$, $c\ell^{'}_{j}$      140 156
$c\ell^{r}_{DR}$, $c\ell^{r}_{\sigma}$, $c\ell^{r}_{\mathscr{AH}}$, $\Gamma_{\mathscr{AH}}$      58 59 62
$c\ell^{r}_{\ell}$, $\Gamma_{\ell}$, $\Gamma_{DR}$, $\Gamma_{\mathscr{H}}$      57 62
$dim_{a}k$, L(V, s)      115
$f_{*}$, $\alpha^{*}$, $\cap$, $\eta_{X}$      81
$G(\sigma)$, $MG(\sigma)$, $\underline{Rep} G$      49
$Gr^{W}_{m}$, $\underline{R}_{k}$      12
$G_{S}$, $WRep_{c}(G_{k}$, $\mathbb{Q}_{\ell})$      88
$H \times_{k}k^{'}$, $R_{k^{'}/k}$      16 75
$H\otimes H^{'}$, $\underline{Hom}(H, H^{'})$      12 13
$H^{i}(X, j)$      82
$H^{i}_{AH}$, $H^{AH}_{a}$      94
$H^{i}_{cont}$      70
$H^{i}_{Y}(X, j)$, $H_{a}(X, b)$      80
$H^{i}_{\'{e}t}(X/R;R_{0}, F)$      185
$H^{i}_{\mathscr{D}}$      68
$H^{i}_{\mathscr{I}}$      174
$H^{i}_{\mathscr{MM}}$(=$H^{i}_{\mathscr{I}}$ for $\mathscr{I} = \mathscr{MM}$)      180
$H^{n}(U)$      35
$H^{n}_{DR}(X)$, $H^{n}_{DR}(U)$      1 25
$H^{n}_{\ell}(X)$, $H^{n}_{\ell}(U)$      1 32
$H^{n}_{\sigma}(X)$, $H^{n}_{\sigma}(U)$      1 32
$H^{v}$      15
$H^{\'{e}t}_{a}(\overline{X}, \mathbb{Z}_{\ell}(b))$      86
$H^{\mathscr{M}}_{a}(X, \mathbb{Q}(b))$      104
$H_{DR}$, $H_{\ell}$, $H_{\sigma}$      10
$Ind^{G_{k}}_{G_{k^{'}}}$      17
$I_{\ell}$, $\bar{\sigma}$      1 10 34
$I_{\infty,\sigma}$      1 10 33
$K^{'}_{0}(X)$, $K^{'}_{a}(X)^{(b)}$      104
$K^{AH}_{\cdot}$(Z.), $K^{\cdot}_{AH, c}$(Z.)      101
$K^{Z}_{m}(X)$, $ch^{Z}$      122
$K^{\cdot}_{AH}$(X., Y.)      98
$K_{m}(U)^{(j)}$, $H^{i}_{\mathscr{M}}(U, \mathbf{Q}(j))$      67
$K_{m}(X)$, $ch_{i, j}$      65
$K_{X}$      133
$N^{i}H^{n}$      76 162
$N^{Z}H^{i}$, $N^{Z}CH^{j}$, $N^{i}CH^{j}$      161 162
$Rep_{c}(G_{k}, -)$      86
$r^{'}_{a, b}$, $\tau_{a, b}$      127 128
$R^{p}\Gamma$      79
$T_{\ell}B$      167
$U(\sigma)$, $sp_{\ell}$, $Msp_{\ell}$      50
$WRep_{c}(G_{k}$, $\mathbb{Z}_{\ell})$      89
$W_{m}$      10 83 87
$X_{(p)}$, div, tame      105
$Y^{(q)}$ (two meanings)      27 76
$Z_{i}(X)$, $Z^{j}(X)$      107 108
$Z_{i}(X)_{0}$, $Z^{j}(X)_{0}$      140
$\Delta/\ell^{n}(j)$, $H^{i}_{fine}$      208 209
$\gamma$      14 81 125
$\hat{A} = \underset{n}{\underleftarrow{lim}} A/\ell^{n}$      70 71
$\kappa(x)$      76
$\mathbb{Z}_{\mathscr{M}}(j)$, $CH^{i}(X, j)$      209
$\mathscr{IMR}_{k}$      94
$\mathscr{K}_{m}$      106
$\mathscr{M}\mathscr{M}$, $H^{i}_{MM}$, $H^{MM}_{a}$      180
$\mathscr{V}$, $\mathscr{I}$      79
$\Omega^{p}_{X}$, $\Omega^{p}_{X}<Y>$      25 26
$\overline{U}$, $\overline{X}$      36 57
$\sigma U^{an}$      32
$\sigma X$, $\sigma U$      2 32
$\underline{1}$ (identity object)      13 80
$\underline{1}(n)$, H(n)      17
$\underline{MM}_{k}$, $\underline{M}_{k}$      43 46
$\underline{MR}_{k}$      9
$\underline{Vec}_{F}$      15 125
$\underline{\overset{{\circ}}{V}}_{k}$      25
$\widetilde{H}^{i}_{\'{e}t}(X, —)$, $\widetilde{H}_{a}(X, —)$      184 206
$\widetilde{H}^{\nu}(k, V)$, $\widetilde{H}^{\nu}(k_{\infty}, V)$      199
$\widetilde{r}^{'}_{a, b}$      204
$\widetilde{r}_{i, j}$      200 205
$^{'}N^{\nu}$      172
${\ell}$-adic (co)homology      1 4 32 36—40 69 86 89—90 97 101 115—117 126 149—151 172 184—186 191 215—219
${\ell}$-adic (co)homology and motivic (co)homology      127 184 189—221
${\ell}$-adic Chern character      69 74 190 200 205 210
A-$\mathscr{MH}$      92
Abel — Jacobi map      140—143 151 153—178 183 204
Absolute (co)homology theory      153 174 182 183
Absolute Hodge cycle      3 14 59—65 72 73 127
Adams operators      67 104
Albanese variety      157—160 177
Algebraic cycle      57 107 108 115—121 139—141 155 165 168 170 173—180
Arithmetic $\mathbb{Q}_{\ell}$-representation, -sheaf      199 204
Arithmetical dimension      115—117
Artin motives      49 53
Base change      41 89 116 117 186 200 218
Base extension      16 75
Beilinson complexes      209
Beilinson conjecture on Chow groups      178—182
Beilinson/Bloch conjecture      158 168
Betti (co)homology      92
Birch and Swinnerton-Dyer      168 169
Bloch — Ogus theory      see "Poincare duality theory"
Bloch's conjecture on zero cycles      177
Bloch's theorem on zero cycles      158
Bloch's theorem on zero cycles generalizations      170 182
Canonical filtration      28
Chern class, character      65 67—75 122—126 154 190 200 205 206 209 210 216
Chow group      57—59 105—107 109 118 121 122 154—182 189 204 212 220
Comparison isomorphism      1—5 11 14 33 34 40 41 58 59 65 97 126
Continuous etale cohomology      70 149—151 153
CYCLE      see "Algebraic cycle"
Cycle map      57—59 107—109 113 115 117 122 126 140 143 151—154 162 174 175 189 219
de Rham (co)homology      1 8 25 30 58 93 96 125
de Rham complex      25—31 96 97 101
Deligne (co)homology      68 69 131 132 152 153 182 183 186
Effective (Hodge structures or ${\ell}$-adic representations)      172
Extension class      139—143 150 176 180—183 204
Fibre functor      14 49 125
Filtration by coniveau      76 162—164 168—173 182 211
Filtration on Chow groups      178—182
Fundamental class      81 107 110 123 126
Galois descent      74 75 216
Geometric cohomology      129 153 177 183
Good proper cover      113 114 118 119 163
Grothendieck motive      180
Grothendieck — Riemann — Roch      122
Grothendieck/Serre conjecture      5 61 191
H      10 43 46
Higher Chow groups      209 212
Hodge conjecture classical      58 77
Hodge conjecture for arbitrary varieties      63 108 114
Hodge conjecture generalized (Grothendieck)      172
Hodge cycle      60 61 121
Hodge filtration      10 18 30 32 33 97
Hodge structure      1 58 172
Homologous to zero/homological equivalence      139 140 168 176 179
Identity object      13 80 83
Intermediate Jacobian      141 157
Internal hom      13
Intersection of cycles      174—178
K-cohomology      106 121
K-theory, $K^{'}$-theory      65 67—79 104—107 121—128 131—138 181 189 190 210 220
L-function      115—121 168 169 220
Level filtration      182
Lichtenbaum complexes      212
Linear varieties      217—221
Mixed ${\ell}$-adic sheaf      89—91 117 120
Mixed absolute Hodge complex      98—102
Mixed Hodge structure      10—13 32 34 64 92 93 100 141—143 152 186—188
Mixed motive      43—56 181—184 188
Mixed realization (for absolute Hodge cycles) of a smooth variety      35 55
Mixed realization (for absolute Hodge cycles) of an arbitrary variety      94—104 115 125 175 177 217
Mixed realization (for absolute Hodge cycles), abstract      9—24 43 94
Motive as defined by Grothendieck      180
Motive for absolute Hodge cycles      1—4 46—49 56
Motive, attached to a modular form      5—9
Motivic (co)homology      67 104—107 181 182 189 209—213 215 216
Motivic (co)homology and ${\ell}$-adic (co)homology      127 184 189—221
Motivic (co)homology and other (co)homology theories      126—130 154—156 182—184
Mumford — Tate group      61 62
Mumford's counterexample      157
Numerical equivalence      178—180
One-semi-simple      193 199
Parshin's conjecture      189
Poincare duality      3 8 45 82 108 118 128 134 140 190 215
Poincare duality theory (twisted)      79—107 121—126 173—176 182 186 194 208 210 214—216
Poincare duality theory (twisted) with weights      85 89 92 94 109—113 129 130 139 154—156 161 180
Potential ${\ell}$-adic sheaf      185 199
Projection formula      19 81 111 130 171
Pull back morphism (in homology)      215
Pure (of weight m) ${\ell}$-adic representation/sheaf      87 116
Pure (of weight m) Hodge structure      10
Pure (of weight m) object      83—85
Pure (of weight m) realization      12 15 46 63 64
purity      38 77 208 210
r, $r'$      126 154 182
Realizations for absolute Hodge cycles      12 15 16 46 55 see
Realizations, attached to a modular form      5—9
Realizations, Betti (Hodge), de Rham, ${\ell}$-adic and others      1 12 73 95 96 103 115 182—184 186
Regulator map      184
Representable $A_{0}(X)$      177 182
Resolution of singularities      25 56 93 95 112 114 191 192 194 195
Restriction      16 75
Riemann — Roch theorem      122 123
Riemann — Roch transformations      123—128 154 190 204 206
Roitman's theorems on zero cycles      157 159 160
Semi-simplicity      50 51 63 113 180 191—195 199 204 218 see
Simplicial variety      93 95—97 100—102 192
Smooth ${\ell}$-adic sheaf      87 116 199 201 218
Specialization map      201—203 219
Spectral sequence, associated to a filtration      29
Spectral sequence, Bloch — Ogus      197
Spectral sequence, Brown — Gersten      106
Spectral sequence, Hochschild — Serre      70 143 150 151 153 203 206
Spectral sequence, hypercohomology/ext      144 151 152 180—183 211
Spectral sequence, Leray      32 41 185—187
Spectral sequence, Quillen      71 76 105 131 133 220
Standard conjectures      179 181
T(X), $p_{g}(X)$      157
Tannakian category      14 15 43—56 61
Tate conjecture for arbitrary varieties      63 109 114 116—121
Tate conjecture, classical      57 77 115
Tate conjecture, generalized (Grothendieck)      172
Tate realization, object      17 181
Tate twist      2 17 18 58 92
Td(X), ch, $\tau$      123
Tensor category      14 80 82 83 86 see
Tensor category with weights      83—85 88 92 94 180—183
Todd class      123—126
Twisted Poincare duality theory      see "Poincare duality theory"
Variety      93
Weight      see "Pure of weight m"
Weight filtration on (co)homology      30—32 89 92 97
Weight filtration on a mixed realization      10 12 94
Weight filtration on a tensor category      83 84
Weight filtration on an adic representation      87—91
Weight spectral sequence      30—43
Weights occurring in (co)homology      66 85 89 92 116 117
Weights occurring in an object      66 83
Weil cohomology      181
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