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Fundamental Algebraic Geometry: Grothendieck's FGA Explained

Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli
American Mathematical Society
Publication Date: 
Number of Pages: 
Mathematical Surveys and Monographs 123
[Reviewed by
Fernando Q. Gouvêa
, on

The early days of the "Grothendieck revolution" in algebraic geometry must have been heady times. Over a short span, less than a decade, the face of a whole subject was changed. Powerful new ideas were introduced that remain of fundamental importance. And while there are many today who have tried to pull the discipline back to concrete examples and make it a little more down-to-earth, no one really proposes that we can live without schemes, representable functors, or descent theory.

Many of Grothendieck's ideas first appeared in a series of talks, mostly at the Séminaire Bourbaki. Write-ups of these talks were later collected in a legendary volume usually known as "FGA" (not to be confused with "EGA" or "SGA" — this period seems to have generated many such abbreviations). The book's title was Fondements de la Géométrie Algébrique. Despite the sketchy nature of these articles, they have been used and referred to over and over.

They have not always been read and understood, however. The original publication was not easy to find, and it was very hard going even if found. The authors of the articles in this volume have tried to come to our rescue, producing expository accounts of the material in FGA. These accounts are much easier to read (and, of course, much easier to get a copy of) than the original. That doesn't mean they make for easy reading; not at all! But at least it's not hopeless.

There should be more of this. I am sure there are many examples of famously unreadable papers out there, and I am also sure that for each of these there is someone (probably many someones) who has done the heavy lifting and is now in a position to explain. Publishers should encourage them to do that.

For now, I'm glad to have in my shelves something a little less imposing than FGA to refer to when I want, say, to find out how one constructs the Picard scheme. My thanks to the authors!

Fernando Q. Gouvêa remembers being told that the place to look for a proof of the existence of the Picard scheme was FGA… but also being told that he might be better off just taking it for granted. He is now professor of mathematics at Colby College in Waterville, ME.

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 scheme of points

  • Introduction
  • Elementary deformation theory
  • Hilbert schemes of points

Grothendieck's existence theorem in formal geometry

  • Grothendieck's existence theorem in formal geometry with a letter of Jean-Pierre Serre

The Picard scheme

  • The Picard scheme
  • Bibliography
  • Index