## Fundamental Algebraic Geometry: Grothendieck's FGA ExplainedAlexander 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 |

323 | |

333 | |

### Common terms and phrases

5-scheme affine Assume base change bijective canonical category fibered closed subscheme coherent sheaf cohomology commutative complete composite condition construction Corollary corresponding curve defined definition deformation functor denote descent data diagram dimension effective divisor EGAIII1 embedding equivalent etale cover exact sequence example Exercise exists fibered category finite type formal scheme fppf fpqc topology function functor F geometric fibers given Grothendieck Hence Hilbert polynomial Hilbert schemes homomorphism ideal induced injective integral inverse image invertible sheaf irreducible isomorphism Lemma Let F lifting locally free locally noetherian scheme locally of finite module morphism of schemes natural transformation noetherian scheme nonsingular open subscheme open subset Picard functor Picard scheme Picx/fc Picx/s projective proof proper Proposition prove pullback quasi-coherent sheaves quasi-compact quasi-projective quotient relative effective divisor representable residue field restriction ring smooth Spec stack Suppose surjective Theorem theory torsor unique vector Zariski topology