Seminar on algebraic geometry at Bois Marie 1962. Local cohomology of coherent sheaves and local and global Lefschetz theorems (SGA 2). Enlarged by a report of M. Raynaud. Revised and updated edition of the 1968 original published by North Holland.
(Séminaire de géométrie algébrique du Bois Marie 1962. Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2).)

*(French)*Zbl 1079.14001
Documents Mathématiques 4. Paris: Société Mathématique de France (ISBN 2-85629-169-4/hbk). x, 208 p. (2005).

Algebraic geometry has undergone several stages of development characterized by reiterated refoundations. In the second half of the 20th century, the so far last decisive breakthrough was contrived by A. Grothendieck and his school, when the theory of algebraic schemes was invented and elaborated. The impact of their pioneering work on the development of current algebraic geometry, commutative and homological algebra, arithmetic geometry, complex-analytic geometry, local analysis, and other related branches of mathematics and theoretical physics was totally sweeping, and so it still is in our days. Apart from A. Grothendieck’s “Éléments de géométrie algébrique” (EGA) published in several volumes in [Publ. Math., Inst. Hautes Étud. Sci. 4, 1–228 (1960); ibid. 8, 1–222 (1961); ibid. 11, 349–511 (1962; Zbl 0118.36206)], the proceedings of his famous “Séminaire de géometrie algébrique” (SGA) must be seen as the most important and propelling documents of those years of rapid expansion of modern algebraic geometry.

Circulating as mimeographed notes in the early 1960s, the exposes of SGA quickly became fundamental sources of intensive research in modern algebraic geometry worldwide, and their publication in book form, a little later, reflected their tremendous significance in this field of research. In our days, almost half a century after the first appearance of the SGA proceedings, their importance is undiminished, but the development of algebraic geometry has immensely progressed. Therefore a new edition of the SGA volumes, together with appropriate updatings, comments, adjustments, and methodological improvements appeared to be adequate. Fortunately, such a major project was initiated by B. Edixhoven a few years ago, with the cooperation of many leading researchers in the field. After the new edition of SGA 1 [Documents Mathématiques (Paris, Société Mathématique de France) (2003; Zbl 1039.14001)] within this so-called “SGA project”, the book under review is the new, updated and annotated edition of the second volume (SGA 2) entitled “Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux” and first published in [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402)]. This new edition has been managed by Professor Yves Laszlo (Paris), with the help of numerous well-known experts in the field. As the editor points out, this version is meant to reproduce the original text, in the form of its second edition [Advanced Studies in Pure Mathematics, 2 (Amsterdam: North-Holland Publishing Company; Paris: Masson & Cie) (1968; Zbl 0197.47202)], with some modifications, including minor typographical corrections and additional footnotes explaining the current status of questions raised in the original text. Also, more details of some proofs have been given here and there, and slight rearrangements have been made to improve (and update) the exposition. Now as before, the entire volume consists of the fourteen original exposes, whose titles we just recall:

0. Introduction (by A. Grothendieck, written in 1968); I. Local and global cohomological invariants with respect to a closed subspace; II. Applications to quasi-coherent sheaves on preschemes; III. Cohomological invariants and depth; IV. Dualizing modules and functors; V. Local duality and structure of \(H^i(M)\); VI. The functors \(\text{Ext}^._Z(X;F,G)\) and \(\underline{\text{Ext}}^.Z(F,G)\); VII. Vanishing criteria and conditions of coherence; VIII. Finiteness theorems; IX. Algebraic geometry and formal geometry; X. Applications to the fundamental group; XI. Applications to the Picard group; XII. Applications to projective algebraic schemes; XIII. Problems and conjectures; XIV. Depth and Lefschetz theorems in étale cohomology (by M.Raynaud).

No doubt, this new edition of SGA 2 represents a great service to the mathematical community, and the editor has done a marvellous job in reworking the original text. Especially the careful, updating footnotes are utmost valuable for researchers in the field, and the modern printing has turned the old SGA 2 in an almost new book within the current research literature.

Circulating as mimeographed notes in the early 1960s, the exposes of SGA quickly became fundamental sources of intensive research in modern algebraic geometry worldwide, and their publication in book form, a little later, reflected their tremendous significance in this field of research. In our days, almost half a century after the first appearance of the SGA proceedings, their importance is undiminished, but the development of algebraic geometry has immensely progressed. Therefore a new edition of the SGA volumes, together with appropriate updatings, comments, adjustments, and methodological improvements appeared to be adequate. Fortunately, such a major project was initiated by B. Edixhoven a few years ago, with the cooperation of many leading researchers in the field. After the new edition of SGA 1 [Documents Mathématiques (Paris, Société Mathématique de France) (2003; Zbl 1039.14001)] within this so-called “SGA project”, the book under review is the new, updated and annotated edition of the second volume (SGA 2) entitled “Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux” and first published in [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402)]. This new edition has been managed by Professor Yves Laszlo (Paris), with the help of numerous well-known experts in the field. As the editor points out, this version is meant to reproduce the original text, in the form of its second edition [Advanced Studies in Pure Mathematics, 2 (Amsterdam: North-Holland Publishing Company; Paris: Masson & Cie) (1968; Zbl 0197.47202)], with some modifications, including minor typographical corrections and additional footnotes explaining the current status of questions raised in the original text. Also, more details of some proofs have been given here and there, and slight rearrangements have been made to improve (and update) the exposition. Now as before, the entire volume consists of the fourteen original exposes, whose titles we just recall:

0. Introduction (by A. Grothendieck, written in 1968); I. Local and global cohomological invariants with respect to a closed subspace; II. Applications to quasi-coherent sheaves on preschemes; III. Cohomological invariants and depth; IV. Dualizing modules and functors; V. Local duality and structure of \(H^i(M)\); VI. The functors \(\text{Ext}^._Z(X;F,G)\) and \(\underline{\text{Ext}}^.Z(F,G)\); VII. Vanishing criteria and conditions of coherence; VIII. Finiteness theorems; IX. Algebraic geometry and formal geometry; X. Applications to the fundamental group; XI. Applications to the Picard group; XII. Applications to projective algebraic schemes; XIII. Problems and conjectures; XIV. Depth and Lefschetz theorems in étale cohomology (by M.Raynaud).

No doubt, this new edition of SGA 2 represents a great service to the mathematical community, and the editor has done a marvellous job in reworking the original text. Especially the careful, updating footnotes are utmost valuable for researchers in the field, and the modern printing has turned the old SGA 2 in an almost new book within the current research literature.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14B15 | Local cohomology and algebraic geometry |

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

14F17 | Vanishing theorems in algebraic geometry |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

14C22 | Picard groups |