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Fundamental Algebraic Geometry: Grothendieck's FGA Explained
Barbara Fantechi, SISSA, Trieste, Italy, Lothar Göttsche, International Centre for Theoretical Physics, Trieste, Italy, Luc Illusie, Université Paris-Sud, Orsay, France, Steven L. Kleiman, MIT, Cambridge, MA, Nitin Nitsure, Tata Institute of Fundamental Research, Mumbai, India, and Angelo Vistoli, Università di Bologna, Italy
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Mathematical Surveys and Monographs
2005; 339 pp; softcover
Volume: 123
ISBN-10: 0-8218-4245-5
ISBN-13: 978-0-8218-4245-4
List Price: US$81 Member Price: US$64.80
Order Code: SURV/123.S

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.

Graduate students and research mathematicians interested in algebraic geometry.

Reviews

"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

Grothendieck topologies, fibered categories and descent theory
• Introduction
• Preliminary notions
• Contravariant functors
• Fibered categories
• Stacks
Construction of Hilbert and Quot schemes
• Construction of Hilbert and Quot schemes
Local properties and Hilbert schemes of points
• Introduction
• Elementary deformation theory
• Hilbert schemes of points
Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre
• Grothendieck's existence theorem in formal geometry
The Picard scheme
• The Picard scheme
• Bibliography
• Index