**Mathematical Surveys and Monographs**

Volume: 123;
2005;
339 pp;
Softcover

MSC: Primary 14; 13; 18;

**Print ISBN: 978-0-8218-4245-4
Product Code: SURV/123.S**

List Price: $91.00

AMS Member Price: $72.80

MAA Member Price: $81.90

**Electronic ISBN: 978-1-4704-1350-7
Product Code: SURV/123.S.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

#### Supplemental Materials

# Fundamental Algebraic Geometry: Grothendieck’s FGA Explained

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*Barbara Fantechi; Lothar Göttsche; Luc Illusie; Steven L. Kleiman; Nitin Nitsure; Angelo Vistoli*

Alexander Grothendieck's concepts turned out to be
astoundingly powerful and productive, truly revolutionizing algebraic
geometry. He sketched his new theories in talks given at the
Séminaire Bourbaki between 1957 and 1962. He then collected
these lectures in a series of articles in Fondements de la
géométrie algébrique (commonly known as
FGA).

Much of FGA is now common knowledge. However, some of it is less
well known, and only a few geometers are familiar with its full
scope. The goal of the current book, which resulted from the 2003
Advanced School in Basic Algebraic Geometry (Trieste, Italy), is to
fill in the gaps in Grothendieck's very condensed outline of his
theories. The four main themes discussed in the book are descent
theory, Hilbert and Quot schemes, the formal existence theorem, and
the Picard scheme. The authors present complete proofs of the main
results, using newer ideas to promote understanding whenever
necessary, and drawing connections to later developments.

With the main prerequisite being a thorough acquaintance with basic
scheme theory, this book is a valuable resource for anyone working in
algebraic geometry.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry.

#### Reviews & Endorsements

All together, this book must be seen as a highly valuable addition to Grotherndieck's fundamental classic FGA, and as a superb contribution to the propagation of his pioneering work just as well. It is fair to say that, for the first time, the wealth of Grotherndieck's FGA has been made accessible to the entire community of algebraic geometers, including non-specialist, young researchers, and seasoned graduate students. The authors have endeavoured to elaborate Grothendieck's ingenious, epoch-making outlines in the greatest possible clarity and detailedness, with complete proofs given throughout...ought to be in the library of anyone using modern algebraic geometry in his (or her) research.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Fundamental Algebraic Geometry: Grothendieck's FGA Explained

- Contents iii4 free
- Preface vii8 free
- Part 1. Grothendieck topologies, fibered categories and descent theory 112 free
- Introduction 314
- Chapter 1. Preliminary notions 718
- Chapter 2. Contravariant functors 1324
- Chapter 3. Fibered categories 4152
- 3.1. Fibered categories 4152
- 3.2. Examples of fibered categories 4859
- 3.3. Categories fibered in groupoids 5263
- 3.4. Functors and categories fibered in sets 5364
- 3.5. Equivalences of fibered categories 5667
- 3.6. Objects as fibered categories and the 2-Yoneda Lemma 5970
- 3.7. The functors of arrows of a fibered category 6172
- 3.8. Equivariant objects in fibered categories 6374

- Chapter 4. Stacks 6778

- Part 2. Construction of Hilbert and Quot schemes 105116
- Part 3. Local properties and Hilbert schemes of points 139150
- Introduction 141152
- Chapter 6. Elementary Deformation Theory 143154
- Chapter 7. Hilbert Schemes of Points 159170
- Introduction 159170
- 7.1. The symmetric power and the Hilbert–Chow morphism 160171
- 7.2. Irreducibility and nonsingularity 166177
- 7.3. Examples of Hilbert schemes 169180
- 7.4. A stratification of the Hilbert schemes 170181
- 7.5. The Betti numbers of the Hilbert schemes of points 173184
- 7.6. The Heisenberg algebra 175186

- Part 4. Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre 179190
- Part 5. The Picard scheme 235246
- Appendix A. Answers to all the exercises 301312
- Appendix B. Basic intersection theory 313324
- Bibliography 323334
- Index 333344