Introduction

We present here, in a revised and completed form, a photo-offset reissue of the second Séminaire de Géométrie Algébrique of the Institut des Hautes Études Scientifiques, held in 1962 (mimeographed).

The reader is referred to the Introduction to the first of these Séminaires (cited below as SGA 1) for the aims that these seminars pursue and their relations with the Éléments de Géométrie Algébrique.

The text of Exposés I through XI was written up at the time, from my oral lectures and handwritten notes, by a group of auditors comprising I. Giorgiutti, J. Giraud, Mlle M. Jaffe (now Mme M. Hakim), and A. Laudal. These notes were originally regarded as provisional and intended for very limited circulation, pending their absorption into the EGA (an absorption that has by now become problematic, to say the least, just as for the other parts of the SGA). As stated in the avertissement of the original edition, this "confidential" character of the notes was supposed to excuse certain "weaknesses of style", which are doubtless more manifest in the present SGA 2 than in the other Séminaires. I have tried as far as possible to remedy this in the present reissue, by a relatively close revision of the original text. In particular, I have harmonized the numbering systems for statements used across the various Exposés by introducing everywhere the same decimal system, which had already been used in most of the original Exposés of SGA 2, as well as in all the other parts of the SGA. This led me, in particular, to rework entirely the numbering¹ of statements in Exposés III through VIII (and, consequently, of references to those Exposés).² I have also tried to extirpate from the original text the principal errors of typing or syntax (which were numerous and distracting). In addition, Mme M. Hakim kindly agreed to rewrite Exposé IV in a less telegraphic style than the original. As in the other reissues of the SGA, I have likewise added a certain number of footnotes, either to give additional references or to indicate the state of a question on which progress has been made since the original text was written. Finally, this Séminaire has been augmented by a new Exposé, namely Exposé XIV, written by Mme Michèle Raynaud in 1967, which takes up and completes suggestions contained in the "Comments on Exposé XIII" (XIII 6) (written in March 1963). That Exposé takes up Lefschetz-type theorems from the viewpoint of étale cohomology, using the results on étale cohomology expounded in SGA 4 and SGA 5 (to appear in the same Series in Pure Mathematics);³ it is, on that account, less "elementary" in nature than the other Exposés of the present volume, which scarcely use more than the substance of Chapters I through III of EGA.

Here is a sketch of the contents of the present volume.

Exposé I contains the sorites of the "cohomology with supports in $Y$", $H_Y^{\bullet }(X,F)$, where $Y$ is a closed subset of a space $X$ — a cohomology that can be interpreted as a cohomology of $X$ modulo the open set $X-Y$, and that is the abutment of a most useful "local-to-global spectral sequence" I 2.6, involving sheaves of cohomology "with supports in $Y$", ${\mathcal{H}}_Y^{\bullet }(F)$.⁴ This formalism can play, in many questions, a role of "localization" analogous to the one played in differential geometry by the consideration of "tubular" neighborhoods of $Y$. Exposé II studies the preceding notions in the case of quasi-coherent sheaves on preschemes; Exposé III gives their relation with the classical notion of depth (III 3.3).

Exposés IV and V give notions of local duality, which one may compare with Serre's projective duality theorem (XII 1.1); let us note that these two types of duality theorems are substantially generalized in Hartshorne's seminar (cited in a footnote at the end of Exposé IV).⁵

Exposés VI and VII give some easy technical notions, used in Exposé VIII to prove the finiteness theorem (VIII 2.3), which gives necessary and sufficient conditions, for a coherent sheaf $F$ on a noetherian scheme $X$, in order that the local cohomology sheaves ${\mathcal{H}}_Y^i(F)$ be coherent for $i⩽n$ (or equivalently, that the sheaves $R^i{f}_{\ast }(F|X-Y)$ be coherent for $i⩽n-1$, where $f:X-Y\to X$ is the inclusion). This theorem is one of the central technical results of the Séminaire, and we show in Exposé IX how a theorem of this nature can be used to establish a "comparison theorem" and an "existence theorem" in formal geometry, by tracing and generalizing the use made in (EGA III §§ 4 and 5) of the finiteness theorem for a proper morphism.

These last results are applied in X and XI, devoted respectively to Lefschetz-type theorems for the fundamental group and for the Picard group.

These theorems consist in comparing, under certain conditions, the invariants (${\pi }_1$ or Pic) attached respectively to a scheme $X$ and to a subscheme $Y$ (playing the role of a hyperplane section), and in giving in particular conditions under which they are isomorphic. Roughly speaking, the hypotheses made serve to pass from $Y$ to the formal completion of $X$ along $Y$, and to be able to apply afterwards the results of IX to pass from there to an open neighborhood $U$ of $Y$ in $X$. To pass from $U$ to $X$, one needs additional information ("purity" or "parafactoriality" type) for the local rings of $X$ at the points of $Z=X-U$, (which is a finite discrete set in the cases envisaged). This explains the interaction in the proofs of Exposés X, XI, XII between local and global results, in particular in certain inductions. The principal results obtained in X and XI are the theorems of local nature X 3.4 (purity theorem) and XI 3.14 (parafactoriality theorem). One should note that these theorems are proved by cohomological techniques, of essentially global nature. In XII one obtains, using the preceding local results, the global variants of these results for projective schemes over a field, or more generally over a more or less arbitrary base scheme; among the typical statements, let us point out XII 3.5 and XII 3.7.

In XIII, we review some of the many problems and conjectures suggested by the results and methods of the Séminaire. The most interesting are perhaps those concerning the cohomological and homotopical Lefschetz-type theorems for complex analytic spaces, cf. XIII pages 26 and following.⁶ In the context of the étale cohomology of schemes, the corresponding conjectures are proved in XIV by a duality technique that should apply equally in the complex analytic case (cf. the comments XIII p. 25 and XIV 6.4). But the corresponding homotopical statements in the case of analytic spaces (and more particularly the statements involving the fundamental group) seem to require entirely new techniques (cf. XIV 6.4).

I am happy to thank all those who, in various capacities, have helped in the appearance of the present volume, among them the collaborators already cited in this Introduction. In particular, I wish to thank Mlle Chardon for the good grace with which she has discharged the thankless task of preparing the final manuscript materially for photo-offset.

Bures-sur-Yvette, April 1968.

A. Grothendieck.