Fantechi B., Kleiman S.L., Illusie L. — Fundamental Algebraic Geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Fantechi B., Kleiman S.L., Illusie L. — Fundamental Algebraic Geometry

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Название: Fundamental Algebraic Geometry

Авторы: Fantechi B., Kleiman S.L., Illusie L.

Аннотация:

In the middle part of the last century, Alexander Grothendieck introduced a number of revolutionary concepts into algebraic geometry. This text for advanced graduate students and beginning researchers offers detailed treatments of the theories and proofs he presented in the collection titled Fondements de la Geometrie Algebrique — commonly referred to as FGA. Four main themes are addressed: descent theory; Hilbert and Quot schemes; the formal existence theorem; and the Picard scheme. The text is based upon course materials developed for a summer school held in 2003 at the International Center for Theoretical Physics in Trieste, Italy.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 339

Добавлена в каталог: 18.11.2009

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Предметный указатель

$Def_{X}$       152
$Def_{\varepsilon}$       152
$div(\mathcal{F})$       161
$D_{F_{0},Q_{0}}$       155
$D_{f}$       156
$h_{f}$       13
$H_{X}$       13 107
$h_{\mathcal{U}}$       30
$\mathcal{F}_{cart}$       55
$\mathfrak{H}ilb$       108 109
$\mathfrak{M}or_{S}$       133
$\mathfrak{Q}uot$       109
$\tau$ -equivalent      291
(Art/k)      143
(CLoc/k)      145
(Loc/k)      145
2-Yoneda Lemma, strong version      60
2-Yoneda Lemma, weak version      59
Abel map      241 263
Abelian function      242
Abelian integral      238
Abelian scheme      222 289
Abelian variety      222 242
Abelian variety, dual      246
Action of a discrete group object      25
Action of a functor into groups on a functor into sets      20
Action of a group object on an object      21
Action of a group object on an object as a group homomorphism      22
Action of a group object on an object, characterization via diagrams      21
Addition theorem      238 241
Adic filtration      181
Adic noetherian ring      181
Adic topology      181
Albanese variety      246
Algebraic space      104 271
Algebraically equivalent      280
Algebraizable coherent sheaf      205
Amplitude, perfect      199
AR zero      189
Arithmetic genus      243 284 318
ARML      189
Arrow, cartesian      41
Arrow, cartesian, properties      42
Arrow, equivariant      21
Arrow, invariant      100
Artin — Rees theorem      189
Artin — Rees zero      189
Artin — Rees — Mittag-Leffler condition      189
Artinian local algebra      143
Base-change formula      196
Base-change homomorphism      91 118 121
Base-change homomorphism, compatibility      92
Base-change isomorphism      118 121 197
Base-change map      91 118 121 195
Base-change property of Hilbert and Quot schemes      112
Base-change theorem      92 121
Base-preserving functor      42
Base-preserving natural transformations      56
Bounded family      291
Brauer group      257
Canonical topology      35
Canonical truncation      200
Cartesian arrow      41
Cartesian arrow, properties      42
Cartier divisor      247 257
Castelnuovo lemma      115
Castelnuovo — Mumford regularity      114 249
Category of artinian local algebras (Art/k)      143
Category of complete local algebras (CLoc/k)      145
Category of local algebras (Loc/k)      145
Category with discrete group object      24
Category with discrete objects      23
Category, comma      11
Category, fibered      42
Category, fibered, $\mathcal{F}_{cart}$       55
Category, fibered, equivalence of, characterization      56
Category, fibered, fiber of      42
Category, fibered, in equivalence relations      58
Category, fibered, in equivalence relations, characterization      58
Category, fibered, in groupoids      52
Category, fibered, in sets      53
Category, fibered, in sets, characterization      53
Category, fibered, in sets, equivalence with functors      54
Category, fibered, morphism of      42
Category, fibered, of quasi-coherent sheaves      50
Category, fibered, of sheaves      49
Category, fibered, of sheaves, as a stack      74
Category, fibered, represent able      59
Category, fibered, represent able, characterization      61
Category, fibered, split      45
Category, opposite      11
Chevalley and Rosenlicht theorem      276
Class of arrows, local      89
Class of arrows, stable      48
Classical topology      26
Classifying stack of a topological group      48
Cleavage      43
Co-equaliser      134
Coherent sheaves, extension of      131
Cohomologically flat sheaf      202 272
Comma category      11
Comma topology      35
Comparison theorem      253
Complete intersection, local      149
Complete intersection, local ring      149
Complete linear system      258
Complete local algebra      145
Completeness of the characteristic system      244 285
Completion, formal      185
Complex, cotangent      226
Complex, Grothendieck      119
Complex, intersection cohomology      173
Complex, perfect      199
Complex, pseudo-coherent      199
Complex, strictly perfect      199
Condition, Artin — Rees — Mittag-Leffler      189
Condition, Mittag-Leffler      189
Connected component of e      275
Contravariant functor      11
Cotangent complex      226
Covering      25 252
Crepant      170
Curve (over a scheme)      219
Curvilinear scheme      172
Deformation functor      143
Deformation functor, $Def_{X}$       152
Deformation functor, $Def_{\varepsilon}$       152
Deformation functor, $D_{F_{0},Q_{0}}$       155
Deformation functor, $D_{f}$       156
Deformation functor, $D_{X,p}$       144
Deformation functor, $h_{R}$       145
Deformation functor, $H_{Z,X}$       144
Deformation functor, examples of      144
Deformation functor, morphism of      143
Deformation functor, pro-represent able      148
Deformation functor, smooth morphism of      150
Deformation of a flat scheme      212
Deformation of a vector bundle      210
Degree      317
Depth      163
Descent data, functoriality of      75
Descent data, object with      68
Descent data, object with, via sieves      71
Descent for affine morphisms      90
Descent for modules      80
Descent for quasicoherent sheaves      82
Descent for sheaves of commutative quasi-coherent algebras      87
Descent morphism      86
Descent, along torsors      103
Descent, via ample invertible sheaves      93

Discrete group object      24
Discrete object      23
Discrete object functor      23
Discrete object functor, preserves finite products      24
Divisor, Cartier      247 257
Divisor, effective      257
Divisor, relative effective      258
Dual, Abelian variety      246 290
Dual, autoduality      246 290
Dual, numbers      281
Effective Cartier divisor $div(\mathcal{F})$       161
Effective Cartier divisor, relative      161
Effective epimorphism      34
Effective quotient      134
Embedding dimension      145
Epimorphism      34
Epimorphism, effective      34
Equalizer      30
Equivalence criteria      246 278
Equivalence of categories      10
Equivalence of fibered categories      56
Equivalence of fibered categories, characterization      56
Equivalence relation, schematic      133
Equivalent topologies      33
Equivalent topologies, has the same stacks      78
Equivalent topologies, have the same sheaves      33
Equivalent, algebraically      280
Equivalent, numerically      291 317
Equivariant, arrow      21
Equivariant, object      63
Essentially surjective functor      10
Essentially zero projective system      189
etale cover      215
Etale topology      26
etale topology, global      27
Example, Grothendieck — Raynaud      289
Example, Hironaka      103 112
Example, Iarrobino      168
Example, Mumford      267
Example, Severi and Zappa      285
Exchange property      203 260
Extension of coherent sheaves      131
Extension, small      144
Faithfully flat morphism of schemes      9
Family, bounded      291
Family, Poincare      244 277
Fiber functor      216
Fiber of a fibered category      42
Fibered category      42
Fibered category in equivalence relations, characterization      58
Fibered category in groupoids      52
Fibered category in sets      53
Fibered category in sets, characterization      53
Fibered category in sets, equivalence with functors      54
Fibered category of quasi-coherent sheaves      50
Fibered category of sheaves      49
Fibered category of sheaves, as a stack      74
Fibered category, $\mathcal{F}cart$       55
Fibered category, equivalence of, characterization      56
Fibered category, representable, characterization      61
Fibered subcategory      55
Filtration, adic      181
Filtration, good      188
Filtration, I-adic      181
Filtration, I-good      188
Flat morphism of schemes      8
Flatness criterion      184
Flatness, generic      122 123
Flattening stratification      123
Fock space      177
Formal completion      185
Formal functions, theorem on      193
Formal scheme (affine noetherian)      182
Formal scheme (locally noetherian)      182
Formal spectrum      182
Formula, base-change      196
fppf topology      27
Fpqc characterization of fpqc sheaves      36
Fpqc characterization of fpqc stacks      83
Fpqc morphism of schemes      28
Fpqc morphism of schemes, properties of      28
Fpqc topology      28
Fpqc topology, local properties in the      29
Fully faithful functor      10
Functor of morphisms $\mathfrak{M}or_{S}$       133
Functor of points      13 107 251
Functor, $Div_{X/S}$       259
Functor, $LinSys_{\mathcal{L}/X/S}$       261
Functor, $Pic_{(X/S)(et)}$       252
Functor, $Pic_{(X/S)(fppf)}$       252
Functor, $Pic_{(X/S)(zar)}$       252
Functor, $Pic_{X/S}$       252
Functor, $Pic_{X}$       252
Functor, base-preserving      42
Functor, contravariant      11
Functor, deformation      143
Functor, deformation, examples of      144
Functor, discrete object      23
Functor, discrete object, preserves finite products      24
Functor, essentially surjective      10
Functor, fiber      216
Functor, fully faithful      10
Functor, Hilbert      108 109
Functor, preserving finite products      11
Functor, Quot      109
Functor, representable      14
Functor, representable is a sheaf in the fpqc topology      34
Functor, representable is a sheaf in the global classical topology      34
Functor, representable, characterization via existence of a universal object      15
Functor, representable, examples of      15
Functor, separated      29
Functor, separated, characterization via sieves      32
Functoriality of descent data      75
Fundamental group      215
Generalized Jacobian      247 289
Generic flatness      122 123
Genus, arithmetic      243 284 318
Genus, geometric      240 243 284
Genus, topological      240
Geometric genus      240 243 284
Gerbe      211
Global classical topology      26
Global etale topology      27
Global Zariski topology      26
Good filtration      188
Grassmannian      111
Grassmannian as a Hilbert scheme      111
Grassmannian as a Quot scheme      110
Grassmannian of a coherent sheaf      111
Grassmannian of a vector bundle      110
Grassmannian, construction of      112
Grothendieck — Murre theorem      270
Grothendieck's specialization theorem      218
Grothendieck, complex      119
Grothendieck, topology      25
Group object      18
Group object, characterization via diagrams      18
Group object, discrete      24
Group object, homomorphism of group objects      20
Group scheme      18
Group, fundamental      215
Groupoid      11
Heisenberg algebra      177
Hilbert functor      108 109
Hilbert polynomial      109
Hilbert scheme      107 108 159 166 171
Hilbert scheme of a smooth curve, smoothness of      167
Hilbert scheme of a smooth surface, partition of      171
Hilbert scheme of a smooth surface, smoothness of      167
Hilbert scheme of hypersurfaces in $\mathbb{P}^{n}$       111

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