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Cohomologie Locale Des Faisceaux Cohérents et Théorèmes de Lefschetz Locaux et Globaux (SGA 2)

A. Grothendieck, editor
Société Mathématique de France
Publication Date: 
Number of Pages: 
Documents Mathématiques
[Reviewed by
Fernando Q. Gouvêa
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This is the second volume in the republication, in the Documents Mathématiques series of the French mathematical society, of Grothendieck's famous Séminaire de Géométrie Algébrique. See my review of SGA 1 for a discussion of the history and importance of the series.

SGA 2 was the only of the original SGA volumes to be republished, after its first incarnation as "private" seminar notes, by North-Holland in its series of "Advanced Studies in Mathematics" instead of Springer's "Lecture Notes in Mathematics." As such, it has probably been the hardest one to find. Thus, it is especially nice to have it back in print. The new edition has been reset in TeX and augmented by many notes updating the material. The original page numbers are indicated in the margin, which is very helpful for those tracing references to the older printing.

This volume of SGA contains material first presented in 1962, dealing with the local cohomology of coherent sheaves and "Lefschetz theorems," i.e., generalizations of the "hard Lefschetz theorem" relating the cohomology (or homotopy) groups of an algebraic variety with those of certain closed subvarieties. Particularly interesting is section XIII, which is a collection of problems and conjectures, most of which have now been resolved. (The editor's notes are very useful at this point!)

After the original mimeographed publication of the notes, étale cohomology was developed (it was the topic of SGA 4) and it became clear that it led to even more useful Lefschetz-style theorems. There is a note to this effect in section XIII, added by Grothendieck when the notes were first published in 1963. The last section of this book, which was written by Michèlle Raynaud in 1967 for the North-Holland reprint, develops that theory.

The Documents Mathématiques series is a truly valuable contribution, and republishing the SGA volumes is a great project. Both algebraic geometers and historians of modern mathematics will be interested in this series of books.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.


  • Introduction
  • Les invariants cohomologiques globaux et locaux relatifs à un sousespace fermé
  • Applications aux faisceaux quasi-cohérents sur les préschémas
  • Invariants cohomologiques et profondeur
  • Modules et foncteurs dualisants
  • Dualité locale et structure des H$^{\textrm{i}}$(M)
  • Les foncteurs $Ext^\bullet_{\textrm{ Z}}$(X; F, G) et $\underline{\textrm{Ext}}^\bullet_{\textrm{Z}}$(F, G)
  • Critères de nullité, condituions de cohérence des faisceaux $\underline{\textrm{Ext}}^{\textrm{i}}_Y$(F, G)
  • Le théorème de finitude
  • Géométrie algébrique et géométrie formelle
  • Application au groupe fondamental
  • Application au groupe de Picard
  • Applications aux schémas algébriques projectifs
  • Problèmes et conjectures
  • Profondeur et théorèmes de Lefschetz en cohomologie étale
  • Index des notations
  • Index terminologique