Grothendieck《基礎(chǔ)代數(shù)幾何學(xué)（FGA）》解讀（影印版）
定 價：135 元
- 作者：（意）凡泰基（Barbara Fantechi）等著
- 出版時間：2019/1/1
- ISBN：9787040510126
- 出 版 社：高等教育出版社
- 中圖法分類：O187
- 頁碼：339
- 紙張：膠版紙
- 版次：1
- 開本：16K
Alexander Grothendieck以極其深刻、極富創(chuàng)造性的思想，使得代數(shù)幾何學(xué)發(fā)生了里程碑式的變革。他在1957年到1962年的布爾巴基討論班上給出了他的新理論的一個概述，然后將這些講義整理成一系列的文章，編成了著名的《基礎(chǔ)代數(shù)幾何學(xué)》(Fondements dela géométrie algébrique)，即我們熟知的FGA。 FGA中的許多內(nèi)容目前已廣為人知，然而仍有一些知識是大家所不了解的，只有少數(shù)幾何學(xué)家熟悉它的全部內(nèi)容。本書源自2003年在意大利的里雅斯特(Trieste)開設(shè)的基礎(chǔ)代數(shù)幾何高級學(xué)校，目的就是完善Grothendieck對于其理論過于簡要的概述。本書討論的四個重要主題為： 下降理論、Hilbert和Quot概形、形式存在定理和Picard概形。作者們給出了主要結(jié)果的完整證明，在必要時使用較新的概念以使讀者更好理解，并且闡述了FGA的理論與新近發(fā)展的聯(lián)系。 本書適合于對代數(shù)幾何學(xué)感興趣的研究生和專業(yè)研究人員閱讀。學(xué)習(xí)本書需要全面扎實的基礎(chǔ)概形理論知識。
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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