##
**Fundamental algebraic geometry: Grothendieck’s FGA explained.**
*(English)*
Zbl 1085.14001

Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3541-6/hbk). x, 339 p. (2005).

In the course of its history, algebraic geometry has undergone several fundamental changes with respect to its conceptual foundations, methods, and techniques. A brilliant analysis of the historical developments in algebraic geometry can be found in the first volume of J. Dieudonné’s “History of algebraic geometry”
[“Cours de Géometrie Algébrique”, Presses Universitaires de France, Paris 1974, Engl. translation by Judith D. Sally, Wadsworth (1985; Zbl 0629.14001)], according to which the history of algebraic geometry is characterized by about seven main epoches. The so far most recent epoch of algebraic geometry began in the 1950s, when A. Grothendieck launched his revolutionary program of refounding the entire theory by means of a completely new, much more general conceptual framework, including arbitrary algebraic schemes, sheaves and their cohomology, methods of categorical and homological algebra, relative geometry, classifying spaces, and other powerful tools. Grothendieck first sketched his new theories, which turned over a new leaf in the development of algebraic geometry, in a series of talks at the Seminaire Bourbaki between 1957 and 1962. The notes of these talks were then published in the famous volume [“Fondements de la Géométrie Algébrique” (Paris: Secrétariat mathématique) (1962; Zbl 0239.14004)], commonly abbreviated FGA. These mimeographed notes became of central importance in the sequel, as they contained the only available outlines of Grothendieck’s fundamental new constructions such as descent theory, Hilbert schemes and Quot schemes, formal algebraic geometry, Picard schemes, and other pioneering approaches toward a better understanding of classical problems in algebraic geometry.

Grothendieck’s various theories sketched in FGA were of crucial significance for the tremendous progress in algebraic geometry thereafter, and most of them even became indispensable ingredients in allied fields of contemporary mathematics such as arithmetic, complex-analytic geometry, mathematical physics, and others. Certainly, much of Grothendieck’s FGA is now common knowledge, or at least utilized folklore, but a good deal of it is still less well-known, and perhaps only a few experts are familiar with its full scope.

In view of the undiminished significance of Grothendieck’s ideas outlined in FGA for the current and future research in algebraic geometry, the book under review aims at explaining its rich contents in full detail, thereby taking into account recent developments and improvements. Written by six experts in the field, this book contains the elaborated lectures delivered by the authors at the “Advanced School in Basic Algebraic Gecmetry”, which was held at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, from July 7 to July 18, 2003, and which addressed advanced graduate students as well as beginning researchers in algebraic geometry.

As the authors point out, this book is not intended to replace Grothendieck’s celebrated FGA, after more than forty years. Rather, it is to fill in Grothendieck’s ingenious sketches with detailed proofs, on the one hand, and to present both newer ideas and more recent developments wherever appropriate, on the other hand. In accordance with Grothendieck’s original FGA, the present book consists of five parts written by different authors, in which the following topics are treated: (1) theory of descent; (2) Hilbert schemes and Quot schemes; (3) local properties of Hilbert schemes of points; (4) Grothendieck’s existence theorem in formal geometry; and (5) Picard schemes.

Part 1 was written by A. Vistoli (Bologna). This part comes with the headline “Grothendieck topologies, fibered categories, and descent theory” and is subdivided into four chapters. Differing from Grothendieck’s original approach in FGA, descent theory is here presented in the language of Grothendieck topologies, which Grothendieck actually introduced somewhat later. Chapter 1 reviews some basic notions of scheme-theoretic algebraic geometry and of category theory, whereas chapter 2 continues the warm-up with a brief introduction to representable functors, Grothendieck topologies and their sheaves, and to group objects in categories. Chapter 3 discusses fibered categories, which provide the appropriate abstract set-up for explaining descent theory in its full generality. The main example, in this context is the category of quasi-coherent sheaves over the category of schemes. Chapter 4 develops descent theory for algebraic stacks, that is for those fibered categories in which it properly works. The author discusses, in full detail, the various ways of defining descent data, including those for quasi-coherent sheaves, morphisms of schemes, and quasi-coherent sheaves along torsors. Overall, the author has tried to develop descent theory in its greatest generality, beyond the scope of FGA, which is certainly necessary for deeper understanding and more advanced applications.

Part 2, comprising Chapter 5, explains Grothendieck’s construction of Hilbert schemes and Quot schemes. Written by N. Nitsure, this chapter provides a modern treatment of these basic constructions using more recent basic tools such as faithtully flat descent, flattening stratifications, semi-continuity techniques, and Castelnuovo-Mumford regularity of sheaves. This chapter concludes with some concrete applications due to D. Mumford, A. Altman and S. Kleiman, A. Grothendieck himself, and others.

Part 3 is formed by chapters 6 and 7. Containing the lectures by B. Pantechi and L. Göttsche (Trieste), this chapter discusses local properties and Hilbert schemes of points. Chapter 6 introduces elementary deformation theory in algebraic geometry, involving the infinitesimal study of schemes, the infinitesimal deformation functor, a tangent-obstruction theory for such a functor, and local moduli spaces. The theory is thoroughly worked out in some special cases, and sketched in a few others. This is used in chapter 7, where the Hilbert scheme of points on a smooth quasi-projective variety is studied. The reader meets here the Hilbert-Chow morphism, stratifications of Hilbert schemes of points, Betti numbers of Hilbert schemes, and further more recent cohomological results in this direction.

Part 4, written by L. Illusie (Paris), is devoted to Grothendieck’s FGA exposé, where he established a fundamental comparison theorem of GAGA-type between algebraic geometry and his newly created formal geometry, on the one hand, and outlined some possible applications to the theory of the algebraic fundamental group and to infinitesimal deformation theory, on the other hand. Shortly afterwards, A. Grothendieck gave a detailed account of this particular topic in EGA III [Publ. Math., Inst. Hautes Étud. Sci. 17, 137–223 (1963; Zbl 0122.16102)] and in SGA1 [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Here, in part 4 of the book under review, the author revisits this topic in great detail, and in a more introductory manner for non-experts, ending with a discussion of J.-P. Serre’s celebrated examples of varieties in positive characteristic that do not lift to characteristic zero.

Being one of the leading experts in this realm, the author has taken the opportunity to give various improvements and updatings of Grothendieck’s original approach, mainly by using the toolkit of derived categories and, especially, perfect complexes, and he has enhanced the entire discussion by several more recent applications to lifting problems in algebraic or formal geometry, respectively. Part 4 comprises chapter 8 of the book, whereas the final part 5 is identical with chapter 9. This concluding part, written by S. Kleiman (Boston), is perhaps the show-piece of the entire collection. In a masterly manner, beginning with a highly enlightening introduction of 15 pages, the author develops in great detail most of Grothendieck’s theory of the Picard scheme, together with its further developments later on.

After the extensive historical introduction, which is also of independent cultural interest, the author describes the four common relative Picard functors in their comparison, and proves then Grothendieck’s existence theorem for the Picard scheme. This is followed by the study of both the connected component of the identity and the torsion component of the identity of the Picard scheme, including the related deep finiteness theorems, and the entire treatise closes with two appendices. Appendix A provides detailed answers to all the exercises scattered in this chapter, and appendix B contains an elementary treatment of basic intersection theory, as far as it is used freely in some proofs.

All together, this book must be seen as a highly valuable addition to Grothendieck’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 Grothendieck’s FGA has been made accessible to the entire community of algebraic geometers, including non-specialists, 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, and with various improvements, simplifications, updatings, and topical hints wherever appropriate. Due to this rewarding undertaking, FGA has come down to earth that is much closer to the community of all algebraic geometers, and therefore the book under review ought to be in the library of anyone using modern algebraic geometry in his research.

Grothendieck’s various theories sketched in FGA were of crucial significance for the tremendous progress in algebraic geometry thereafter, and most of them even became indispensable ingredients in allied fields of contemporary mathematics such as arithmetic, complex-analytic geometry, mathematical physics, and others. Certainly, much of Grothendieck’s FGA is now common knowledge, or at least utilized folklore, but a good deal of it is still less well-known, and perhaps only a few experts are familiar with its full scope.

In view of the undiminished significance of Grothendieck’s ideas outlined in FGA for the current and future research in algebraic geometry, the book under review aims at explaining its rich contents in full detail, thereby taking into account recent developments and improvements. Written by six experts in the field, this book contains the elaborated lectures delivered by the authors at the “Advanced School in Basic Algebraic Gecmetry”, which was held at the International Centre for Theoretical Physics (ICTP) in Trieste, Italy, from July 7 to July 18, 2003, and which addressed advanced graduate students as well as beginning researchers in algebraic geometry.

As the authors point out, this book is not intended to replace Grothendieck’s celebrated FGA, after more than forty years. Rather, it is to fill in Grothendieck’s ingenious sketches with detailed proofs, on the one hand, and to present both newer ideas and more recent developments wherever appropriate, on the other hand. In accordance with Grothendieck’s original FGA, the present book consists of five parts written by different authors, in which the following topics are treated: (1) theory of descent; (2) Hilbert schemes and Quot schemes; (3) local properties of Hilbert schemes of points; (4) Grothendieck’s existence theorem in formal geometry; and (5) Picard schemes.

Part 1 was written by A. Vistoli (Bologna). This part comes with the headline “Grothendieck topologies, fibered categories, and descent theory” and is subdivided into four chapters. Differing from Grothendieck’s original approach in FGA, descent theory is here presented in the language of Grothendieck topologies, which Grothendieck actually introduced somewhat later. Chapter 1 reviews some basic notions of scheme-theoretic algebraic geometry and of category theory, whereas chapter 2 continues the warm-up with a brief introduction to representable functors, Grothendieck topologies and their sheaves, and to group objects in categories. Chapter 3 discusses fibered categories, which provide the appropriate abstract set-up for explaining descent theory in its full generality. The main example, in this context is the category of quasi-coherent sheaves over the category of schemes. Chapter 4 develops descent theory for algebraic stacks, that is for those fibered categories in which it properly works. The author discusses, in full detail, the various ways of defining descent data, including those for quasi-coherent sheaves, morphisms of schemes, and quasi-coherent sheaves along torsors. Overall, the author has tried to develop descent theory in its greatest generality, beyond the scope of FGA, which is certainly necessary for deeper understanding and more advanced applications.

Part 2, comprising Chapter 5, explains Grothendieck’s construction of Hilbert schemes and Quot schemes. Written by N. Nitsure, this chapter provides a modern treatment of these basic constructions using more recent basic tools such as faithtully flat descent, flattening stratifications, semi-continuity techniques, and Castelnuovo-Mumford regularity of sheaves. This chapter concludes with some concrete applications due to D. Mumford, A. Altman and S. Kleiman, A. Grothendieck himself, and others.

Part 3 is formed by chapters 6 and 7. Containing the lectures by B. Pantechi and L. Göttsche (Trieste), this chapter discusses local properties and Hilbert schemes of points. Chapter 6 introduces elementary deformation theory in algebraic geometry, involving the infinitesimal study of schemes, the infinitesimal deformation functor, a tangent-obstruction theory for such a functor, and local moduli spaces. The theory is thoroughly worked out in some special cases, and sketched in a few others. This is used in chapter 7, where the Hilbert scheme of points on a smooth quasi-projective variety is studied. The reader meets here the Hilbert-Chow morphism, stratifications of Hilbert schemes of points, Betti numbers of Hilbert schemes, and further more recent cohomological results in this direction.

Part 4, written by L. Illusie (Paris), is devoted to Grothendieck’s FGA exposé, where he established a fundamental comparison theorem of GAGA-type between algebraic geometry and his newly created formal geometry, on the one hand, and outlined some possible applications to the theory of the algebraic fundamental group and to infinitesimal deformation theory, on the other hand. Shortly afterwards, A. Grothendieck gave a detailed account of this particular topic in EGA III [Publ. Math., Inst. Hautes Étud. Sci. 17, 137–223 (1963; Zbl 0122.16102)] and in SGA1 [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Here, in part 4 of the book under review, the author revisits this topic in great detail, and in a more introductory manner for non-experts, ending with a discussion of J.-P. Serre’s celebrated examples of varieties in positive characteristic that do not lift to characteristic zero.

Being one of the leading experts in this realm, the author has taken the opportunity to give various improvements and updatings of Grothendieck’s original approach, mainly by using the toolkit of derived categories and, especially, perfect complexes, and he has enhanced the entire discussion by several more recent applications to lifting problems in algebraic or formal geometry, respectively. Part 4 comprises chapter 8 of the book, whereas the final part 5 is identical with chapter 9. This concluding part, written by S. Kleiman (Boston), is perhaps the show-piece of the entire collection. In a masterly manner, beginning with a highly enlightening introduction of 15 pages, the author develops in great detail most of Grothendieck’s theory of the Picard scheme, together with its further developments later on.

After the extensive historical introduction, which is also of independent cultural interest, the author describes the four common relative Picard functors in their comparison, and proves then Grothendieck’s existence theorem for the Picard scheme. This is followed by the study of both the connected component of the identity and the torsion component of the identity of the Picard scheme, including the related deep finiteness theorems, and the entire treatise closes with two appendices. Appendix A provides detailed answers to all the exercises scattered in this chapter, and appendix B contains an elementary treatment of basic intersection theory, as far as it is used freely in some proofs.

All together, this book must be seen as a highly valuable addition to Grothendieck’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 Grothendieck’s FGA has been made accessible to the entire community of algebraic geometers, including non-specialists, 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, and with various improvements, simplifications, updatings, and topical hints wherever appropriate. Due to this rewarding undertaking, FGA has come down to earth that is much closer to the community of all algebraic geometers, and therefore the book under review ought to be in the library of anyone using modern algebraic geometry in his research.

Reviewer: Werner Kleinert (Berlin)

### MathOverflow Questions:

Does the Jacobian functor respect deformations?Exposition of Grothendieck’s mathematics

Reference for torsion-freeness of the group of correspondences on a smooth projective variety

Representability of Hom of two finite flat group schemes

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14C20 | Divisors, linear systems, invertible sheaves |

14D15 | Formal methods and deformations in algebraic geometry |

14K30 | Picard schemes, higher Jacobians |

18D30 | Fibered categories |