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Bouscaren E. — Model Theory and Algebraic Geometry

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Название: Model Theory and Algebraic Geometry

Автор: Bouscaren E.

Аннотация:

This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski's proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 211

Добавлена в каталог: 18.11.2009

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

Предметный указатель

$acl^{eq}$       see "Imaginary elements"
$dcl^{eq}$       see "Imaginary elements"
$M^{eq}$       see "Imaginary elements"
$T^{eq}$       see "Imaginary elements"
$\delta$ -closed      120
$\delta$ -closure      see "Differential closure"
$\delta$ -connected      135
$\delta$ -definable      131
$\lambda$ -closed      153
$\lambda$ -closed irreducible set      157 161
$\lambda$ -closed set of finite type      158 160 166
$\lambda$ -closed subset of a complete type      160
$\lambda$ -closed subset of a minimal type      163
$\lambda$ -closure      154
Abelian variety      76 85
Abelian variety, Poincare's reducibility theorem      87
Abelian variety, strong rigidity      87 180
Abstract variety      71 72
acl(—)      see "Algebraic closure"
Affine algebraic group      90
Affine quasi-affine variety      70
Affine space      34
Affine variety      70
algebraic      20
Algebraic closure      20
Algebraic group      75
Algebraically closed field      2 61
Algebraically closed fields      2 61
Algebraically closed fields, constructible subset      2
Algebraically closed fields, defined over      67 68
Algebraically closed fields, elimination of imaginaries      63—64
Algebraically closed fields, elimination of quantifiers      10 61
Algebraically closed fields, field of definition      67
Algebraically closed fields, generic type      68
Algebraically closed fields, Morley rank      65 72
Algebraically closed fields, Zariski closed set      65 66
Algebraically closed fields, Zariski closure      69
Almost strongly minimal      36
ample      115 120
Atomic formula      6
Basis of a pregeometry      35
Birationally isomorphic      72
Canonical base      29
Canonical ideal      154
Canonical parameter      21
Categorical      16
Chevalley's Theorem      76 91
Chow coordinates      64
Coheir      33
Compactness Theorem      11 15
Complete theory      9
Complete type      13
Complete variety      73
Complete Zariski geometry      109
Component      see "p-component"
Connected component      47 76
Connected group      47 50
Consistent      9
Constant field      131
Constructible      2 111
Coordinate ring      70
Coordinate ring of affine variety      70
dcl(—)      see "Definable closure"
Definable      8 20
Definable closure      20
Definable group      51
Definable map      19
Definable type      29
Definable, $\wedge$ -definable      14
Definable, $\wedge\wedge$ -definable      14
Definable, 0-definable      19
Degree      see "Morley degree"
Degree of imperfection      144
Derivation      129
Differential closure      134
Differential field      129
Differential ideal      129
Differential polynomial      130
Differential ring      129
Differentially algebraic      130
Differentially closed      131
Differentially closed field      2 131
Differentially closed fields      2 131
Differentially closed fields, elimination of imaginaries      133
Differentially closed fields, elimination of quantifiers      132
Differentially closed fields, Morley rank      134
Differentially finitely generated      130
Differentially transcendental      130
Dimension in algebraic geometry      124
Dimension in Noetherian spaces      108
Dimension of a pregeometry      35
Dimension Theorem      70 109 164
Elementarily equivalent      9
Elementary      9
Elementary embedding      10
Elementary extension      9
Elementary substructure      9
Elimination of imaginaries      22 63
Elimination of imaginaries, in algebraically closed fields      63—64
Elimination of imaginaries, in differentially closed fields      133
Elimination of imaginaries, in separably closed fields      151 157
Elimination of imaginaries, in strongly minimal sets      63
Elimination of quantifiers      10
Elimination of quantifiers, in algebraically closed fields      10 61
Elimination of quantifiers, in differentially closed fields      132
Elimination of quantifiers, in separably closed fields      147
Embedding in model theory      5
Family of plane curves      115
Field definable in a separably closed field      152 153
Field-connected      135
Field-definable      131
Fields definable in algebraically closed fields      81
Fields definable in differentially closed fields      136
Fields definable in separably closed fields      152 153
Finite cover property      202
Finite rank group      92 102 137
Finitely satisfiable      13
Fork      29
Fork, forking      33
Fork, nonforking extension      29 33
Formula      7
Frobenius map      63 143
Function field      95
Function field of affine variety      70
Function field of an affine variety      70
Generic of a group      49
Generic point of a closed set      157
Generic type in algebraically closed fields      68
Generic type in groups      49
Generic type in strongly minimal sets      65
Global type      29
Group definable in a Zariski geometry      114
Group of finite rank      92 102 137
Group, almost strongly minimal subgroups      57
Group, chain conditions      46
Group, connected component      47 76
Group, irreducible component      76
Group, strongly minimal subset      57
Groups definable in $\omega$ -stable theories      51
Groups definable in algebraically closed fields      78
Groups definable in differentially closed fields      135
Groups definable in Zariski geometries      114
Heir      33
Hilbert's Nullstellensatz      67
Homogeneous      12
Homomorphism in model theory      5
Imaginary elements      20 21
Imaginary elements, elimination of imaginaries      22
Indecomposability theorem      53
Indecomposable set      52

Independence in a minimal type      163
Independent      30 35 163
Indiscernible array      118
Infinitely definable      14 51
Infinitely definable group      51
Infinitely definable group in instable theories      51
Irreducible      67 68 70 72 108 157
Irreducible component      68 76 108 157
Irreducible component of a $\lambda$ -closed set      161
Irreducible variety      70 72
Isomorphism in model theory      5
Isomorphism of affine varieties      70
Isomorphism of quasi-affine varieties      71
Isotrivial factor      197
Jacobian of a curve      88—90
Krull dimension      69 125
Lang conjecture      85 91 95 97 101
Lang-type      102
Language      4
Lascar rank      162
Lefschetz principle      4 198
Linear group      76
Local parameters      126
Locally modular      66 115
Locally modular pregeometry      37
Locus      112
Loewenheim — Skolem theorem      12
Manifold (in Zariski geometries)      114
Manin homomorphism      137
Manin kernel      137
Manin — Mumford conjecture      92
Minimal set      153
Minimal type      119 162 174 188
Model      9
Modular pregeometry      37
Monster model      20
Mordell conjecture      85 91 96
Mordell — Lang Conjecture      85 197 see
Morley degree      24 64
Morley rank      23 65
Morley rank, in algebraically closed fields      65 72
Morley rank, in differentially closed fields      134
Morphism in Zariski geometries      114
Morphism of affine varieties      70
Morphism of quasi-affine varieties      71
Morphism of varieties      72
Morphism, p-morphism      77
Multiplicity      30 64
Multiplicity of a type      30
nfcp      202
Noetherian topology      107
Noetherian topology, dimension      108
Nonforking extension      33
Omega-stable field      82
Omega-stable, $\omega$ -stable      25
Omega-stable, $\omega$ -stable field      82
Omega-stable, $\omega$ -stable group      46
Omitting types theorem      15 16
One-based      38 39 54 102 103
One-based group      54 103
One-dimensional, 1-dimensional      38
Ord(f)      see "Order"
Order (of a differential polynomial)      130
Order property      28
Orthogonal      40
p-basis      144
p-component      145
p-free      144
p-independent      144
p-morphism      77
p-rational function      77
p-regular function      77
Partial type      14
Perfect      63 143
Perfect closure      144
Perfect field      63 143
Plane curve      115
Pregeometry      34
Prime differential ideal      130
Principal open set      71
Projective space      73
Projective variety      73
Purely inseparable      144
Quasi-affine variety      70
Radical differential ideal      130
Rank in a minimal type      162
Rational function      63 71 72
Rational, p-rational      77
Realize a Type      13
Reduction mod p      197
Regular at      70
Regular domain      126
Regular map      70
Regular point in Zariski geometries      112
Regular type      34
Regular, p-regular      77
Rigid      184
Rigid, strong rigidity      87 180
Satisfaction of a formula      7
Satisfiable      13
Saturated      12 19
Semi-abelian variety      91
Semi-minimal set      193
Sentence      8
Separable closure      144
Separable closure of an ideal      148
Separable extension      145
Separable ideal      148
Separable over      144
Separable polynomial      144
Separably closed field      2
Separably closed fields      2 144
Separably closed fields, completeness of the theory      146
Separably closed fields, elimination of imaginaries      151 157
Separably closed fields, elimination of quantifiers      147 149
Separably closed fields, generic type      151
Separably closed fields, types      149
Separably closed fields, U-rank      151
Separant      130
Separated variety      72
Signature      4
Simple abelian variety      88 140 182
Smooth      74
Special sort      112
Specialization      117
Stabilizer      93
Stabilizer of a type      48 49 185
Stable      28 102
Stationary      30 31
Stationary type      30
Strong type      23 31
Strongly minimal      24 33 62 65
Strongly minimal subset of a group      57
Strongly minimal, locally modular      37—39 115
Strongly minimal, modular      37
Strongly minimal, trivial      39
Structure      4
Substructure      5
Tarski — Vaught test      10
Term      6
Theory      9
Thin      174
Thin type      174
Torsion points in abelian variety      87
Totally transcendental      25
TYPE      13
Type ideal      149
Type, space of types      15
U-rank      162
Uniformly definable      8

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