Fundamental Algebraic Geometry: Grothendieck's FGA Explained

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American Mathematical Soc., 2005 - Mathematics - 339 pages
2 Reviews
Alexander Grothendieck introduced many concepts into algebraic geometry; they turned out to be astoundingly powerful and productive and truly revolutionized the subject. Grothendieck sketched his new theories in a series of talks at the Seminaire Bourbaki between 1957 and 1962 and collected his write-ups in a volume entitled ``Fondements de la Geometrie Algebrique,'' known as FGA. Much of FGA is now common knowledge; however, some of FGA is less well known, and its full scope is familiar to few. The present book resulted from the 2003 ``Advanced School in Basic Algebraic Geometry'' at the ICTP in Trieste, Italy. The book aims to fill in Grothendieck's brief sketches. There are four themes: descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. Most results are proved in full detail; furthermore, newer ideas are introduced to promote understanding, and many connections are drawn to newer developments. The main prerequisite is a thorough acquaintance with basic scheme theory. Thus this book is a valuable resource for anyone doing algebraic geometry.
  

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I haven't read this book, but in spite of what the pop-up description says, I am reasonably confident it is not "an enquiry into the use and status of English in medieval England..." Somebody needs to fix that!

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Page 173 has some interesting ideas about varieties expressible as product of fibers relative to a particular morphism.

Contents

Grothendieck topologies fibered categories and descent theory
1
Introduction
3
Preliminary notions
7
12 Category theory
10
Contravariant functors
13
22 Group objects
18
23 Sheaves in Grothendieck topologies
25
Fibered categories
41
63 Nonprorepresentable functors
150
64 Examples of tangentobstruction theories
152
65 More tangentobstruction theories
157
Hilbert Schemes of Points
159
71 The symmetric power and the HilbertChow morphism
160
72 Irreducibility and nonsingularity
166
73 Examples of Hilbert schemes
169
74 A stratification of the Hilbert schemes
170

32 Examples of fibered categories
48
33 Categories fibered in groupoids
52
34 Functors and categories fibered in sets
53
35 Equivalences of fibered categories
56
36 Objects as fibered categories and the 2Yoneda Lemma
59
37 The functors of arrows of a fibered category
61
38 Equivariaht object to fibired categories
63
Stacks
67
42 Descent theory for quasicoherent sheaves
79
43 Descent for morphisms of schemes
88
44 Descent along torsors
99
Construction of Hilbert and Quot schemes
105
Construction of Hilbert and Quot schemes
107
51 The Hilbert and Quot functors
108
52 CastelnuovoMumford regularity
114
53 Semicontinuity and basechange
118
54 Generic flatness and flattening stratification
122
55 Construction of Quot schemes
126
56 Some variants and applications
130
Local properties and Hilbert schemes of points
139
Introduction
141
Elementary Deformation Theory
143
62 Prorepresentable functors
148
75 The Betti numbers of the Hilbert schemes of points
173
76 The Heisenberg algebra
175
Grothendiecks existence theorem in formal geometry with a letter of JeanPierre Serre
179
Grothendiecks existence theorem in formal geometry
181
82 The comparison theorem
187
83 Cohomological flatness
196
84 The existence theorem
204
85 Applications to lifting problems
208
86 Serres examples
228
87 A letter of Serre
231
The Picard scheme
235
The Picard scheme
237
92 The severed Picard functors
252
93 Relative effective divisors
257
94 The Picard scheme
262
95 The connected component of the identity
275
96 The torsion component of the identity
291
Answers to all the exercises
301
Basic intersection theory
313
Bibliography
323
Index
333
Copyright

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