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

Hilbert scheme of points, $S^{[n]}$ , stratification of, $S^{[n]}_{\nu}$       171
Hilbert scheme of points, $X^{[n]}$       159
Hilbert scheme of points, $X^{[n]}$ , incidence correspondence      166
Hilbert scheme, examples of      169
Hilbert — Chow morphism of a smooth curve is an isomorphism      169
Hilbert — Chow morphism, $\rho$       159
Hilbert — Chow morphism, existence of      166
Hilbert's fifteenth problem      248
Hilbert's theorem      90 256
Hodge index theorem      319
Holomorphic symplectic variety      170
Homomorphism of deformations      210
Homomorphism of group objects      20
Homomorphism, specialization      217
Hull, pro-represent able      150
Hull, pro-represent able, characterization of      150
Hull, pro-represent able, existence of      151
Hull, pro-represent able, unicity of      151
I-adic filtration      181
I-adic topology      181
I-good filtration      188
Ideal of definition of a locally noetherian forma scheme      183
Ideal of definition of an adic ring      181
Intersection, cohomology      173
Intersection, number      314
Intersection, symbol      314
Invariant arrow      100
Irregularity      243 284
Jacobi inversion problem      239
Jacobian      241 245
Jacobian, generalized      247
Jointy surjective      26
K3 surface      225
Kodaira vanishing theorem      279
Lemma on generic flatness      122 123
Lemma, 2-Yoneda, strong version      60
Lemma, 2-Yoneda, weak version      59
Lemma, Castelnuovo      115
Lemma, Yoneda, dual form      18
Lemma, Yoneda, strong version      14
Lemma, Yoneda, weak version      13
Lie algebra      273 283
Lie — Kolchin theorem      277
Lifting of a flat scheme      212
Local algebra      145
Local algebra, Artinian      143
Local algebra, Artinian, category of, (Art/k)      143
Local algebra, category of, (Loc/k)      145
Local algebra, complete      145
Local algebra, complete, category of, (CLoc/k)      145
Local class of arrows      89
Local complete intersection      149
Local complete intersection in a scheme      157
Local Picard scheme      272
Locally of finite presentation morphism of schemes      7
m-regular sheaf      114
Macdonald's formula      174
Mittag-Leffler condition      189
ML      189
Morphism of deformation functors      143
Morphism of fibered categories      42
Morphism of formal schemes      182
Morphism of formal schemes, smooth      213
Morphism of schemes, faithfully flat      9
Morphism of schemes, flat      8
Morphism of schemes, fpqc      28
Morphism of schemes, fpqc, properties of      28
Morphism of schemes, locally of finite presentation      7
Morphism of schemes, projective      126
Morphism of schemes, quasi-compact      7
Morphism of schemes, quasi-projective      126
Morphism of schemes, strongly projective      126
Morphism of schemes, strongly quasi-projective      126
Morphism of sheaves      29
Morphism, Hilbert — Chow morphism      159
Morphism, Hilbert — Chow morphism, existence of      166
Natural transformation, base-preserving      56
Neron — Severi      244
Numerically equivalent      291
Object with descent data, effective      73
Object with descent data, via sieves      71
Object, discrete      23
Object, universal      15
Obstruction      210 213
Obstruction space, $T_{2}$       147
Oort devissage      270 271 273
Opposite category      11
Path between fiber functors      216
Perfect amplitude      199
Perfect complex      199
Picard functor, absolute      252
Picard functor, relative      252
Picard functor, the      257
Picard group      237
Picard integral      243
Picard number      244
Picard scheme      237 262
Picard scheme, local      272
Pluecker coordinates      114
Pluecker relations      114
Poincare, family      244 277
Poincare, sheaf      262
Polynomial, Hilbert      109
Prestack      72
Prestack, characterization of      73
Pretopology      26
Pro-representable functor      148
Pro-representable hull      150
Pro-representable hull, characterization of      150
Pro-representable hull, existence of      151
Pro-representable hull, unicity of      151
Projection isomorphism      197
Projective morphism      126
Projective system, essentially zero      189
Projective system, strict      189
Property of exchange      203 260
Pseudo-coherent complex      199
Pseudo-functor      44
Pullback      41
Punctual Hilbert scheme, $H_{n}$       171
Punctual Hilbert scheme, $H_{n}$ of curvilinear subschemes, punctual Hilbert scheme, $H_{n}^{c}$       172
Quasi-compact morphism of schemes      7
Quasi-functor      58
Quasi-functor, characterization      58
Quasi-projective morphism      126
Quot functor      109
Quot scheme      107
Quotient by a group action      160
Quotient by a group action, existence of      160
Quotient by a schematic equivalence relation      134
Quotient, effective      134
Quotient, universal      114
Ramanujam — Samuel theorem      274 280
refinement      33
Refinement, characterization via sieves      33
Regular sheaf      114
Regularity, Castelnuovo — Mumford      114 249
Representable, fibered category      59
Representable, fibered category, characterization      61
Representable, functor      14
Representable, functor, characterization via existence of a universal object      15
Representable, functor, examples of      15
Representable, functor, is a sheaf in the fpqc topology      34
Representable, functor, is a sheaf in the global classical topology      34
Riemann Theorem      318
Riemann — Roch theorem      242 318
Rigidification      255
Ring, adic noetherian      181
Samuel's conjecture      272
Saturated topology      34

Saturation of a topology      34
Saturation of a topology, properties of      34
Schematic equivalence relation      133
Scheme, $G^{0}$       275
Scheme, $G^{\tau}$       295
Scheme, $Pic_{X/S}$       262
Scheme, Abelian      222
Scheme, Chevalley      248
Scheme, formal      182
Schlessinger theorem      151
Semiregular      284
Separated functor      29
Separated functor, characterization via sieves      32
Set, $Pic^{\varphi}_{X/S}$       299
Sheaf      29
Sheaf, characterization of fpqc      36
Sheaf, characterization via sieves      32
Sheaf, cohomologically flat      202 272
Sheaf, m-regular      114
Sheaf, Poincare      262
Sheaf, regular      114
Sheaf, universal      262
Sheafification      39
Sheafification, existence of      39
Sieve      30
Sieve, associated with a set of arrows      30
Sieve, belonging to a topology      31
Site      25
Site, small etale      26
Site, subcanonical      35
Small etale site      26
Small extension      144
Smooth morphism of deformation functors      150
Smooth morphism of formal schemes      213
Smooth topology      33
Specialization homomorphism      217
Spectrum, formal      182
Split fibered category      45
Splitting      45
Stable class of arrows      48
STACK      72
Stack, characterization of      73
Stack, characterization of fpqc      83
Stack, characterization via sieves      77
Stein factorization      193
Stratification, flattening      123
Strict, projective system      189
Strictly perfect complex      199
Strictly semismall      173
Strongly projective morphism      126
Strongly quasi-projective morphism      126
Subcanonical site      35
Subcanonical topology      35
Subcategory, fibered      55
Subordinate topology      33
Subordinate topology, characterization via sieves      33
Substack      79
Surface over a scheme      224
Surface, K3      225
Symmetric power, $X^{(n)}$       159
Tangent space at 0      281
Tangent space of a deformation functor, $T_{1}$       147
Tangent space of a ring      145
Tangent-obstruction theory      147
Tangent-obstruction theory of a coherent sheaf      157
Tangent-obstruction theory of a locally complete intersection subscheme      157
Tangent-obstruction theory of a section of the Hilbert scheme      156
Tangent-obstruction theory of a section of the Quot scheme      155
Tangent-obstruction theory, generalized      147
Tangent-obstruction theory, unicity of      147 148
The module $\mathcal{Q}$       260
Theorem of generic representability      270
Theorem on Betti numbers of the Hilbert schemes of a smooth surface      174
Theorem on formal functions      193
Theorem, a representable functor is a sheaf in the fpqc topology      34
Theorem, addition      238 241
Theorem, Artin      271
Theorem, Artin — Rees      189
Theorem, base-change      92 121
Theorem, Briancon      172
Theorem, Cartier      278 279 283 285
Theorem, characterization of $\tau$ -equivalence      291
Theorem, characterization of numerical equivalence      291
Theorem, Chevalley and Rosenlicht      276
Theorem, comparison      253
Theorem, computation of the Lie algebra      281
Theorem, Deligne      291
Theorem, descent along torsors      103
Theorem, descent for affine morphisms      90
Theorem, descent for modules      80
Theorem, descent for quasicoherent sheaves      82
Theorem, descent for sheaves of commutative quasicoherent algebras      87
Theorem, descent, via ample invertible sheaves      93
Theorem, existence of $Pic_{X/k}^{0}$       269 276
Theorem, existence of $Pic_{X/S}$       263 272 290
Theorem, existence of $Pic_{X/S}^{0}$       290
Theorem, existence of $Pic_{X/S}^{\tau}$       297
Theorem, existence of a split fibered category equivalent to a given fibered category      61
Theorem, existence of flattening stratifications      123
Theorem, existence of Hilbert — Chow morphisms      166
Theorem, existence of pro-represent able hulls      151
Theorem, existence of quotients by a finite group action      160
Theorem, existence of quotients by flat projective equivalence relations      135
Theorem, existence of sheafifications      39
Theorem, finiteness of $Pic^{\tau}_{X/S}$       297
Theorem, finiteness of $Pic^{\varphi}_{X/S}$       299
Theorem, flattener      271
Theorem, generic representability      270
Theorem, Grothendieck — Murre      270
Theorem, Grothendieck's specialization      218
Theorem, Hodge index      319
Theorem, Kodaira vanishing      279
Theorem, Lie — Kolchin      277
Theorem, Mumford      116 269
Theorem, properties of intersection numbers      315
Theorem, Ramanujam — Samuel      274 280
Theorem, Raynaud      291
Theorem, relative representability      271
Theorem, representability of $Div_{X/S}$       259
Theorem, representability of $LinSys_{\mathcal{L}/X/S}$       261
Theorem, representability of functors of morphisms      133
Theorem, representability of Quot functors      127
Theorem, Riemann      318
Theorem, Riemann — Roch      242 318
Theorem, Schlessinger      151
Theorem, smoothness of the Hilbert scheme of a smooth curve      167
Theorem, smoothness of the Hilbert scheme of a smooth surface      167
Theorem, Snapper      314
Theorem, the Hibert — Chow morphism of a smooth surface is strictly semismall      173
Theorem, the Hilbert — Chow morphism of a curve is an isomorphism      169
Theorem, Zariski's connectedness      193
Theorem, Zariski's Main      194
Thickening      183
Topological genus      240
Topological group      18
Topology, adic      181
Topology, canonical      35
Topology, classical      26
Topology, classical, global      26
Topology, comma      35
Topology, equivalent      33
Topology, equivalent, has the same sheaves      33
Topology, equivalent, has the same stacks      78
Topology, Etale      26
Topology, fppf      27
Topology, fpqc      28
Topology, fpqc, local properties in the      29
Topology, global etale      27
Topology, global Zariski      26
Topology, Grothendieck      25

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