Grothendieck《基础代数几何学(FGA)》解读（影印版）

Alexander Grothendieck以极其深刻、极富创造性的思想，使得代数几何学发生了里程碑式的变革。他在1957年到1962年的布尔巴基讨论班上给出了他的新理论的一个概述，然后将这些讲义整理成一系列的文章，编成了著名的《基础代数几何学》(Fondements dela géométrie algébrique)，即我们熟知的FGA。 FGA中的许多内容目前已广为人知，然而仍有一些知识是大家所不了解的，只有少数几何学家熟悉它的全部内容。本书源自2003年在意大利的里雅斯特(Trieste)开设的基础代数几何高级学校，目的就是完善Grothendieck对于其理论过于简要的概述。本书讨论的四个重要主题为： 下降理论、Hilbert和Quot概形、形式存在定理和Picard概形。作者们给出了主要结果的完整证明，在必要时使用较新的概念以使读者更好理解，并且阐述了FGA的理论与新近发展的联系。 本书适合于对代数几何学感兴趣的研究生和专业研究人员阅读。学习本书需要全面扎实的基础概形理论知识。

Preface Part 1． Grothendieck topologies~ fibered categories and descent theory Introduction Chapter 1．Preliminary notions 1．1．Algebraic geometry 1．2．Category theory Chapter 2．Contravariant functors 2．1．Representable functors and the Yoneda Lemma 2．2．Group objects 2．3．Sheaves in Grothendieck topologies Chapter 3．Fibered categories 3．1．Fibered categories 3．2．Examples of fibered categories 3．3．Categories fibered in groupoids 3．4．Functors and categories fibered in sets 3．5．Equivalences of fibered categories 3．6．Objects as fibered categories and the 2-Yoneda Lemma 3．7．The functors of arrows of a fibered category 3．8．Equivariant objects in fibered categories Chapter 4．Stacks 4．1．Descent of objects of fibered categories 4．2．Descent theory for quasi-coherent sheaves 4．3．Descent for morphisms of schemes 4．4．Descent along torsors Part 2． Construction of Hilbert and Quot schemes Chapter 5．Construction of Hilbert and Quot schemes Introduction 5．1．The Hilbert and Quot functors 5．2．Castelnuovo-Mumford regularity 5．3．Semi-continuity and base-change 5．4，Generic flatness and flattening stratification 5．5．Construction of Quot schemes 5．6．Some variants and applications Part 3． Local properties and Hilbert schemes of points Introduction Chapter 6．Elementary Deformation Theory 6．1．Infinitesimal study of schemes 6．2．Pro-representable functors 6．3．Non-pro-representable functors 6．4，Examples of tangent-obstruction theories 6．5．More tangent-obstruction theories Chapter 7．Hilbert Schemes of Points Introduction 7．1．The symmetric power and the Hilbert-Chow morphism 7．2．Irreducibility and nonsingularity 7．3．Examples of Hilbert schemes 7．4．A stratification of the Hilbert schemes 7．5．The Betti numbers of the Hilbert schemes of points 7．6．The Heisenberg algebra Part 4． Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre Chapter 8．Grothendieck's existence theorem in formal geometryIntroduction 8．1．Locally noetherian formal schemes 8．2．The comparison theorem 8．3．Cohomological flatness 8．4．The existence theorem 8．5．Applications to lifting problems 8．6．Serre's examples 8．7．A letter of Serre Part 5． The Picard scheme Chapter 9．The Picard scheme 9．1．Introduction 9．2．The several Picard functors 9．3．Relative effective divisors 9．4．The Picard scheme 9．5．The connected component of the identity 9．6．The torsion component of the identity Appendix A．Answers to all the exercises Appendix B．Basic intersection theory Bibliography Index

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