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Levine M., Morel F. — Algebraic Cobordism
Levine M., Morel F. — Algebraic Cobordism

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Название: Algebraic Cobordism

Авторы: Levine M., Morel F.

Аннотация:

Following Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. Surprisingly, this theory satisfies the analogues of Quillen's theorems: the cobordism of the base field is the Lazard ring and the cobordism of a smooth variety is generated over the Lazard ring by the elements of positive degrees. This implies in particular the generalized degree formula conjectured by Rost. The book also contains some examples of computations and applications.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2007

Количество страниц: 244

Добавлена в каталог: 22.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$G_0$ as oriented Borel — Moore functor      27
$G_0$ as oriented Borel — Moore homology theory      145
$G_0$, formal group law      28
$K^0$ and algebraic cobordism      6 115 110
$K^0$ as oriented cohomology      5
$K^0$ formal group law      5
$K^0$ universal property      6 114
Additive fund or      2
Admissible subcategory      1
Admissible subcategory, l.c.i. closed      142
Algebraic cobordism      VII 7
Algebraic cobordism and $K^0$      6 140 216
Algebraic cobordism and complex cobordism      8 13
Algebraic cobordism and MGL      14
Algebraic cobordism and the Chow ring      12 136 139 140 216
Algebraic cobordism and the Lazard ring      7 119
Algebraic cobordism as oriented Borel — Moore homology theory      215 221
Algebraic cobordism as oriented cohomology theory      215 221
Algebraic cobordism in degree 0      47
Algebraic cobordism, birational invariance      121
Algebraic cobordism, computations in low degree      139 140
Algebraic cobordism, definition      38
Algebraic cobordism, degree formula      11 129 230 231
Algebraic cobordism, extended      81
Algebraic cobordism, generalized degree formula      10 216 217 230
Algebraic cobordism, higher degree formula      131 231
Algebraic cobordism, homotopy property      76
Algebraic cobordism, localization property      7
Algebraic cobordism, made additive      103
Algebraic cobordism, made multiplicative      115 216
Algebraic cobordism, projective bundle formula      79
Algebraic cobordism, refined      160 165
Algebraic cobordism, topological Miration      139
Algebraic cobordism, universality      7 92 215 222
Algebraic pre-cobordism      37
Axiom, (BM1)      144
Axiom, (BM2)      144
Axiom, (BM3)      144
Axiom, (CD')      145
Axiom, (CD)      145 156
Axiom, (Dim)      26 33 36 91
Axiom, (EH)      2 91 145
Axiom, (FGL)      27 91
Axiom, (Loc')      85
Axiom, (Loc)      84 91 94
Axiom, (Nilp)      27 91
Axiom, (PB)      2 91 144
Axiom, (Sect)      26 29 37
Axiom, (Triv)      84
Blow-up formula      43 109
Cellular decomposition property      145
Chern character      6
Chern character operator      110
Chern class operator      19 22 35
Chern class operator for vector bundles      93
Chern class operator, refined      164
Chern class operator, Whitney product formula      95
Chern classes      3
Chern classes and Chern class operators      94
Chern classes in an oriented cohomology theory      94
Chow groups and algebraic cobordism      216
Chow groups as oriented Borel-Moore functor      27
Chow groups as oriented Borel-Moore homology theory      145
Chow groups, formal group law      27
Chow groups, universal property      216
Chow ring and algebraic cobordism      12 136 139 216
Chow ring as oriented cohomology      5
Chow ring, universal property      6
Cobordism cycle      20
Cobordism cycle, Chern class operator      22
Cobordism cycle, external product      23
Cobordism cycle, pull-back      22
Cobordism cycle, push-forward      21
Complex cobordism      VII 7—9 12 117
Conner — Floyd theorem      6 14 115 216
Conner — Floyd, Chern class endomorphism      99 225—227
Deformation diagram      56 193 205 219 221 222
Deformation diagram, extended      202
Deformation to the normal bundle      41 168
Deformation to the normal cone      56
Degree formula      11 122
Degree formula for algebraic cobordism      129 134 230 231
Degree formula, generalized      125 127—129 216 217
Degree homomorphism      9
Distinguished lifting      184 IV.
Distinguished lifting of divisor classes      188
Distinguished lifting, uniqueness      187
Divisor class      53 58 218
Divisor class, refined      164 167
Elementary cobordism      38
Excess intersection formula      209 211
Extended homotopy property      3 85
Extended homotopy property as axiom      85 91 145
Extended homotopy property in algebraic cobordism      81
External product      144
Face      53
Formal group law      4 26 44
Formal group law of $CH_*$      27
Formal group law of $G_0$      28
Formal group law of $K^0$      5
Formal group law of an oriented Borel — Moore functor of geometric type      26
Formal group law of an oriented theory      4
Formal group law of the Chow ring      5
Formal group law, additive      5
Formal group law, geometric construction      87
Formal group law, logarithm      103
Formal group law, multiplicative      5
Formal group law, periodic      92
Formal group law, universal      VII
Fulton, William      VIII X 5 28 41 73 111 113 145 159 160 201 210
Fundamental class      146
General position      162 165
General position with respect to D      162
General position with respect to D|D'      165
Generalized degree formula      10 125 216
Generalized degree formula for algebraic cobordism      230
Geometric cobordism      29
Good position      162 165
Good position with respect to D      162
Good position with respect to D|D'      165
Grothendieck — Niemann — Roch theorem      6
Grothendieck. Alexander      VIII 3 93
Gysin morphism      194
Gysin morphism, functoriality      197
Gysin morphism, properties      194—197
Gysin morphism, refined      202 203
Gysin morphism, refined, functoriality      205
Higher degree formula      134 231
Hironaka, Heisuki      X 119 184 233
Homotopy property as axiom      145
Homotopy property in algebraic cobordism      76 81
Iintersection homomorphism      170
Iintersection homomorphism, commutativity      182
Iintersection homomorphism, moving lemma      184
Iintersection homomorphism, refined      170
Inverse Todd class operator      97
Inverse Todd class operator, universal      99 227
l.c.i. closed admissible subcategory      142
l.c.i. morphism      141
l.c.i. morphism, pull-back for      198—200
l.c.i. scheme      143
Landweber — Novikov operations      102 130 229
Lazard ring      VII 4
Lazard ring and algebraic cobordism      7 118 119
Lazard ring, grading      5
Localization property      125
Localization theorem      63
MacPherson, Robert      41
Milnor hypersurface      44
Milnor hypersurface, relation with a formal group law      44 45
Moving Lemma      190
Naive cobordism      38
Normal cone      201
Operational structure of an oriented theory      148
Oriented Borel — Moore functor      18
Oriented Borel — Moore functor and oriented cohomological functors      26
Oriented Borel — Moore functor of geometric type      26 91
Oriented Borel — Moore functor of geometric type, birational invariance      110
Oriented Borel — Moore functor of geometric type, examples      27
Oriented Borel — Moore functor with product      23
Oriented Borel — Moore functor with product, universal      24
Oriented Borel — Moore functor, detected by smooth schemes      19
Oriented Borel — Moore functor, generated by smooth schemes      19
Oriented Borel — Moore functor, quotient      25
Oriented Borel — Moore homology theory      143
Oriented Borel — Moore homology theory and oriented Borel — Moore weak homology theories      153
Oriented Borel — Moore homology theory and oriented cohomology theories      149
Oriented Borel — Moore homology theory, examples      145
Oriented Borel — Moore homology theory, operational structure      148
Oriented Borel — Moore homology theory, twisting      227—228
Oriented Borel — Moore R-functor      24
Oriented Borel — Moore weak homology theory      IX 91
Oriented Borel — Moore weak homology theory, associated to an oriented Borel — Moore homology theory      153
Oriented Borel — Moore weak homology theory, associated to an oriented cohomology theory      155
Oriented Borel — Moore weak homology theory, genetically constant      123
Oriented Borel — Moore weak homology theory, twisting      99
Oriented cohomological functor      26
Oriented cohomological functor and oriented Borel — Moore functors      26
Oriented cohomological functor with product      26
Oriented cohomology theory      VII 2 92
Oriented cohomology theory and oriented Borel — Moore homology theories      149
Oriented cohomology theory and oriented weak cohomology theories      152
Oriented cohomology theory, multiplicative      6
Oriented cohomology theory, ordinary      6
Oriented cohomology theory, periodic      6
Oriented weak cohomology theory      IX 92
Oriented weak cohomology theory, associated to an oriented cohomology theory      152
Projective bundle      X
Projective bundle formula      3
Projective bundle formula as axiom      84 85 91 144
Pseudo-divisor      160
Pseudo-divisor, defining equation      160
Pseudo-divisor, pull-back      192
Pseudo-divisor, support      160
Pull-back for pseudo-divisors      192
Pull-back, l.c.i.      198—200
Pull-back, refined      201 207
Pull-back, refined, commutativity      211
Pull-back, refined, excess intersection formula      209 211
Pull-back, refined, properties      208
Quillen's theorem      VII
Quillen, Daniel      VII VIII 2 14 20 83 99 102 117
Refined algebraic cobordism      160
Refined intersection      201 212
Refined pullback      201 207
Refined pullback, commutativity      211
Refined pullback, excess intersection formula      209 211
Refined pullback, properties      208
Regular embedding      141
Relative dimension      1
Resolution of singularities      IX X 12 40 51 60 63 76 79 81 83 116 119—121 167 169 201 215 217 229 233—234
Rost's degree formula      134
Rost, Markus      VIII 83 122 129 131
Splitting principle      85
Standard cycle      24
Strict normal crossing divisor      53
Strict normal crossing divisor, class in $A^*$      218
Strict normal crossing divisor, cobordism class      53 58
Strict normal crossing divisor, cobordism class, refined      164 167
Strict normal crossing divisor, face      53
Tangent bundle of a smooth morphism      100
Tangent bundle, virtual      101 228
Todd class operator      100
Todd class operator, inverse      97
Todd class operator, inverse, universal      99
Tor-independent morphism      1
Transversality theorem      66
Transverse morphism      1
Twisting construction      99
Twisting construction, examples      101—103
Twisting construction, first form      99
Twisting construction, Grothendieck — Riemann — Roch      115
Twisting construction, Landweber — Novikov operations      102
Twisting construction, second form      101
Unoriented cobordism      13
Very good position      162 165
Very good position with respect to D      162
Very good position with respect to D|D'      165
Virtual normal bundle      100 101
Weak factorization      12 40 74 83 116 119—121 234—235
Whitney product formula      3 95
Zigzag diagram      194
Zigzag diagram, extended      203
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