SGA — English Translation

SGA 4-I — Front Matter

Translated from the cleaned OCR transcription of the image-only SGA 4-I scan. Page markers refer to PDF pages. Mathematical notation and the damaged synoptic diagram should still be checked against the scan.

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Seminaire de Geometrie Algebrique du Bois-Marie, 1963-1964

Etale Cohomology of Schemes

A seminar directed by M. Artin, A. Grothendieck, and J.-L. Verdier, with the collaboration of N. Bourbaki, P. Deligne, and B. Saint-Donat.

Tome 1. Theory of Topoi
Exposes I to IV.

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Preface to the Second Edition

1. As the reader will see in the foreword to the first incomplete edition of the present Seminaire, the initial aim of the Seminaire was to develop the theory of etale cohomology of schemes. On the other hand, since the year in which the oral seminar was developed, the importance of the language of topologies and topoi in algebraic geometry has continued to grow. This language provides a convenient and intuitive framework for the techniques of descent and passage to quotients, indispensable in practically every question concerning the construction of schemes. It also serves to develop other cohomology theories for schemes, better adapted to certain questions than etale cohomology, such as fppf cohomology or crystalline cohomology.

It now seems probable that this language will also be useful to algebraists and topologists. For this reason the need for a fairly systematic exposition of this language, with notation and terminology as close as possible to those already used elsewhere, directly inspired by topology, has likewise continued to grow. The present edition therefore aims, in addition to its initial purpose, to give such an exposition, one that can serve both as a reference text and for the mathematician wishing to learn the language of topoi, while awaiting the day when a more complete treatise is available. It is this change of perspective that is expressed by the change in the title of the Seminaire relative to the original title, “Etale Cohomology of Schemes”.

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For the preceding reasons, we have entirely recast Exposes I to VI of the original Seminaire, devoted to the language of topologies and topoi. Our guiding principle has been to develop a language and notation that are those already effectively used in the various applications, so as not to lose contact with the “geometric” or “topological” content of the various functors one is led to consider between sites. For this purpose, the notions of topos and of morphism of topoi seem to be the indispensable guiding thread, and they should be given the central place, the notion of site becoming an auxiliary technical notion. This led us in particular to enlarge considerably Expose IV, devoted to these notions, and to rewrite completely Expose VI, devoted to fibered sites and topoi, in this spirit. We have also expanded Expose I, devoted to general notions on categories and presheaves, in order to provide the internal references needed for the revised form of the Seminaire.

2. In addition to these substantial modifications and additions relative to the Seminaire in its initial form, let us point out that the present edition also differs from it by the presence of Exposes XVII and XVIII on cohomology with proper supports and the global duality theorem. These exposes, due to P. Deligne, differ rather substantially from A. Grothendieck’s oral exposes in various respects. Thus P. Deligne uses J.-L. Verdier’s method in topology for the direct construction of the functor $f_{!}$ without a preliminary smoothification of the morphism $f$, and he gives a direct construction method

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for the functor $R{f}_{!}$ of cohomology with proper supports, without a preliminary compactification of the separated morphism $f:X\to Y$ of finite type, thanks to the use of the very useful “theory of cohomological descent”. This latter theory, which belongs to general topology, is the subject of a new Expose VI, written by B. Saint-Donat, and its application to the direct construction of $R{f}_{!}$ is explained by Saint-Donat in an appendix to Expose XVII.

On the other hand, we are indebted to P. Deligne for a careful verification of very many compatibilities, which had been admitted without further ado by his less scrupulous predecessor. This moreover gave him the opportunity, in Expose XVII, to develop rather systematically a certain number of complements to homological algebra and to the formalism of derived categories, especially by making the formalism of derived functors more flexible through the systematic use of the technique of ind- and pro-objects. In this respect, paragraphs 1, 2, and 4 of XVII should be regarded as standard references in homological algebra.

3. In addition to these contributions of presentation and clarification by P. Deligne, this re-edition of the Seminaire also contains a certain number of results due to him, of which the following are the principal ones.

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a) The theory of cohomological descent, already mentioned, which is the subject of Saint-Donat’s Expose VI. This theory has already proved to be an effective instrument for passing from local to global, and for reducing certain cohomological properties of schemes with singularities to the case of regular schemes, when resolution of singularities is available. The same principle should be usable for analytic spaces of all kinds. In this way it was able to be used in the recent development, by P. Deligne, of a Hodge theory for complex algebraic varieties with arbitrary singularities.

b) A certain number of results in “general topology”, that is, on topoi, notably: the existence of enough points for a locally coherent topos (VI 9), the stability of the notion of flatness of a module on a topos under inverse image functors (V 8), which gives him the occasion to develop a useful technique of “local inductive limits”, and the existence of 2-fiber products of topoi (IV 15).

c) A “symmetric Kunneth formula” (XVII 5.5.2.1), giving the cohomology of a symmetric power in etale or coherent cohomology by means of the derived functors of the functor $TS$, the symmetric tensor product, inspired by the well-known formula in topology due to Dold. Contrary to what one might suppose, the formula in etale cohomology does not result from “general nonsense”. It proceeds by successive reductions to the case of constant coefficients on a proper and smooth curve over an algebraically closed field, in which case

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a specialization argument and the Lefschetz principle allow one to reduce to the transcendental case, where the formula follows from the triangulability of the space under consideration. The symmetric Kunneth formula was established by P. Deligne with a view toward various applications to formulas for $L$-functions in cases not covered by the original exposes of SGA 5, and which will no doubt be included in the re-edition of SGA 5, notably for the case where the base ring for the coefficients is not a field, such as ${\mathbf{Z}}_{\ell }$ or $\mathbf{Z}/{\ell }^n\mathbf{Z}$, where $\ell$ is the prime field $\mathbf{Z}/p\mathbf{Z}$, $p$ being the characteristic of the base field of the variety.

d) A theory of “symmetric powers of torsors” (XVII 6.3), developed with a view toward defining the trace of a torsor by a finite locally free morphism $f:X\to Y$, under conditions more general than the usual conditions, where the group $G$ over $Y$ is smooth or $f$ is etale.

e) A theory of “integration of torsors” (XVIII 1.3), and a “universal coefficients” formula, which are presented as elements of a duality theory for continuous coefficients, relative to a morphism $f:X\to S$ that is smooth and of relative dimension 1. Let us point out that the notions introduced by P. Deligne suggest various refinements of Riemann-Roch type formulas in etale cohomology, obtained by writing canonical isomorphisms in suitable categories of “stable objects” formed with constructible etale sheaves, and possible generalizations in higher dimensions. They thus open an interesting field of research. Let us also point out that, as a technical tool, P. Deligne proves an interesting structure theorem for certain torsors on certain smooth relative schemes $f:X\to S$, including in particular projective bundles (XVIII 1.2.2), a theorem that also raises

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interesting questions, connected in particular with the theory of deformations of torsors under a flat group scheme.

Bures, November 1969.

[Synoptic diagram of the re-edition: the OCR of this page is too degraded for a reliable transcription. Consult the scan.]

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Foreword

1. The principal aim of the present Seminaire is to develop the formalism of the “Weil cohomology” of schemes. Starting essentially from the results proved here, well-known arguments, due moreover to Weil himself, make it possible to deduce part of the Weil conjectures on the $L$-functions of nonsingular projective varieties over a finite field (*) (**). This deduction will moreover be presented in a seminar that will follow this one (SGA 5 XIII).

In the present Seminaire, we restrict ourselves to the study of the cohomology of schemes relative to the etale topology. By its description, this topology is very close to the topologies of the usual topological varieties, and one will see that most of the classical results concerning the cohomology of ordinary topological spaces, various spectral sequences, finiteness theorems, Kunneth, duality, Lefschetz theorems, can be formulated and proved in the new context, provided one restricts oneself, where necessary, to torsion sheaves prime to the residual characteristics of the schemes under consideration.

(*) At the moment of writing these lines, it has not been proved that the eigenvalues of the Frobenius homomorphism acting on the $H^i(X,{\mathbf{Z}}_{\ell })$ are algebraic integers, nor a fortiori that their absolute values are equal to $q^{i/2}$.

(**) Added October 1968. For an exposition taking stock of the present state of the Weil conjectures, cf. Kleiman’s expose, “Algebraic cycles and the Weil conjectures”, in J. Giraud et al., Dix exposes sur la cohomologie des schemas, North-Holland. Added in August 1969: the fact that the eigenvalues are algebraic integers was recently proved by P. Deligne (cf. SGA 7 XIX 5).

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One obtains a cohomological theory “with coefficients of characteristic 0”, as requested by Weil, by an essentially trivial passage to the projective limit (cf. SGA 5 VI), making it possible to define cohomology with coefficients in the ring ${\mathbf{Z}}_{\ell }$ of $\ell$-adic integers from the coefficients $\mathbf{Z}/{\ell }^n\mathbf{Z}$, $n\ge 1$. When $\ell$ is prime to the residual characteristics, this cohomology has all the usual good properties of classical cohomology with coefficients $\mathbf{Z}$, and is therefore suited to the formulation of the Weil conjectures.

Moreover, when the schemes under consideration are of finite type over the field of complex numbers $\mathbf{C}$, we shall see that cohomology with torsion coefficients is essentially identical to the classical cohomology of the corresponding analytic spaces. This would make it possible, on the one hand, to apply the results obtained by purely algebraic means to the usual cohomology of algebraic varieties defined over $\mathbf{C}$, and to generalize in this way various classical results, generally proved by transcendental means under nonsingularity conditions, and sometimes to simplify certain proofs, notably in Lefschetz theory. On the other hand, this would make it possible, where appropriate, to apply transcendental results to the cohomology of algebraic varieties in characteristic $p$ that lift to characteristic 0.

By contrast, the essentially new phenomena, original to characteristic $p>0$ and concerning $p$-torsion coefficients, which should play an essential role for example in the theories of local or global class field theory, almost totally escape

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the point of view of etale cohomology adopted in these notes. Their study will no doubt require the use of a finer topology, probably the fppf topology, whose study has been undertaken only in a very particular case by Schatz, “Cohomology of artinian group schemes over local fields”, Annals of Mathematics, vol. 79, no. 3, May 1964, pp. 411-449 (). This seems at present to be the most urgent extension to make to etale cohomological theory (). Let us also point out the problem of extending to schemes the study of homotopy invariants such as the ${\pi }_i$, both from the point of view of the etale topology and from that of the fppf topology; one may hope that Hurewicz-type theorems, still to be formulated, together with our rather good knowledge of cohomology and of the fundamental group, will make it possible to obtain results in this direction ().

2. In addition to the evident influence of Weil’s ideas, let us also point out the links of this Seminaire with various earlier investigations that influenced one or another of us.

a) J. Tate’s theory of the cohomological dimension of fields [C6], itself motivated by the point of view of etale cohomology, was in turn a tool and a model for the theory of the cohomological dimension of schemes.

(*) Cf. also A. Grothendieck, “Le groupe de Brauer III”, § 11, in Dix exposes sur la cohomologie des schemas (North-Holland).

(**) Since these lines were written, it has appeared that the fppf topology does not satisfy all the hoped-for properties, and the “most urgent extension” would rather lie in the development of crystalline cohomology, initiated in A. Grothendieck, Crystals and the De Rham cohomology of schemes, inspired by the work of Monsky-Washnitzer and Manin.

(**) Cf. note () on the following page.

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b) Igusa’s work on “vanishing cycles”, which will only be discussed in a later Seminaire, and Igusa’s proof of the Picard inequality $\rho \le b_2$ in every characteristic, which we obtain here in a rather formal way, together with the finiteness theorem for Neron-Severi, once an explicit cohomological theory is in hand.

c) The results of Ogg and Shafarevich on the cohomology of function fields in one variable with coefficients in abelian varieties. These authors obtain in particular an analogue of the duality theorem for algebraic curves, together with an Euler-Poincare characteristic formula. For an exposition in terms of etale cohomology, cf. M. Raynaud, Seminaire Bourbaki, December 1964 [C].

(*) The problem of the study of the “etale homotopy type” mentioned above has been solved in principle, since the writing of this introduction, by recent work of Michael Artin and Barry Mazur; see their 1966 seminar on this subject, published in M. Artin and B. Mazur, Etale homotopy, Lecture Notes no. 100, Springer, 1969.

(**) Cf. SGA 7 III, IX…

(***) Cf. SGA 6 XIII 5.2, 5.3.

(****) Reproduced in the collection cited in the note on page IX.

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3. Exposes I to VI take up the theory of sheaves from the point of view of sites, or better, of topoi, completing the 1962 Artin seminar (Harvard) on many points. The presentation, and the results not contained in Artin, are due essentially to J. Giraud and J.-L. Verdier.

Exposes VII and VIII develop the first definitions and properties relative to the etale topology and etale cohomology.

Exposes IX, X give the first elementary quantitative results relative to torsion coefficients, concerning in particular the cohomology of algebraic curves, cohomological dimension, and the notion of constructible sheaf.

The results of Exposes XII to XVI (*) are centered around the two “base change theorems”, which technically constitute the central result of this Seminaire. The key to both of these theorems lies in certain properties of the fundamental group, which in our context are most naturally stated in terms of cohomology with coefficients in sheaves of not necessarily commutative groups. Thus, in the statement of the principal cohomological results, we have systematically also taken noncommutative cohomology into consideration, which would make it possible to recover, in a form sometimes more general and more convenient from the technical point of view, the principal results on the fundamental group of SGA 1961 (**).

(*) Obtained between September 1962 and March 1963 by M. Artin and A. Grothendieck.

(**) Cf. Mme M. Raynaud’s expose in SGA 1 XIII.

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Exposes XVII and XVIII (*) present the theory of cohomology “with compact supports” and the global duality theorem, which imply Poincare duality for nonsingular algebraic varieties.

Finally, Expose XIX presents certain local results (**), using resolution of singularities, and until further notice therefore applicable only in characteristic zero.

4. We shall make unrestrained use of the results of EGA I to IV, and of SGA 1 and SGA 2.

(*) Obtained in February-March 1963 by A. Grothendieck, at the same time as the “local duality theorem” that will be presented in SGA 5.

(**) Obtained by Artin in 1964.

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Table of Contents

Page numbers below are the printed page numbers of the volume, not PDF page numbers.

Expose I — Presheaves

By A. Grothendieck and J.-L. Verdier.

Expose II — Topologies and Sheaves

By J.-L. Verdier.

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Expose III — Functoriality of Categories of Sheaves

By J.-L. Verdier.

Expose IV — Topoi

By A. Grothendieck and J.-L. Verdier.

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Contents of this volume

• Exposé I. Presheaves
• Exposé II. Topologies and Sheaves
• Exposé III. Functoriality of Categories of Sheaves
• Exposé IV. Topoi
• Index of terminology
• Index of notations
• Translation ledger