Éléments de géométrie algébrique

I. The Language of Schemes

by A. GROTHENDIECK written in collaboration with J. DIEUDONNÉ

Institut des Hautes Études Scientifiques 1960

Publications mathématiques de l'I.H.É.S., tome 4 (1960), pp. 5–228.

Introduction

To Oscar Zariski and André Weil.

This memoir, and the many others intended to follow it, are meant to form a treatise on the foundations of algebraic geometry. They do not, in principle, presuppose any particular knowledge of the subject; indeed, it has turned out that such knowledge — despite its obvious advantages — can sometimes (through the overly exclusive habit of the birational viewpoint it implies) hinder a reader who wants to become familiar with the viewpoint and the techniques presented here. On the other hand, we assume that the reader has a good knowledge of the following subjects:

a) Commutative algebra, as set out for instance in the forthcoming volumes of the Éléments of N. Bourbaki (and, until those volumes appear, in Samuel–Zariski [13] and Samuel [11], [12]).

b) Homological algebra, for which we refer to Cartan–Eilenberg [2] (cited as (M)) and Godement [4] (cited as (G)), as well as A. Grothendieck's recent article [6] (cited as (T)).

c) Sheaf theory, where our principal references will be (G) and (T); this theory supplies the indispensable language for interpreting the essential notions of commutative algebra in "geometric" terms and for "globalizing" them.

d) Finally, the reader will find it useful to have some familiarity with the functorial language, which will be constantly employed throughout this treatise, and for which he or she may consult (M), (G), and especially (T); the principles of this language and the main results of the general theory of functors will be expounded in more detail in a work now in preparation by the authors of this treatise.

It is not the place, in this Introduction, to offer a more or less summary description of the "scheme" viewpoint in algebraic geometry, nor the long list of reasons that have made its adoption necessary — in particular, the systematic acceptance of nilpotent elements in the local rings of the "varieties" we shall consider (a step that necessarily pushes the notion of rational map into the background, in favor of the notion of regular map, or "morphism"). The present treatise aims precisely to develop the language of "schemes" systematically and, we hope, to demonstrate its necessity. While it would be easy to do so,

we shall not attempt here to give an "intuitive" introduction to the notions developed in Chapter I. The reader who wants a preliminary glimpse of the material of this treatise may consult the lecture given by A. Grothendieck at the International Congress of Mathematicians in Edinburgh in 1958 [7], and the exposé [8] by the same author. The work [14] (cited as (FAC)) of J.-P. Serre may also be regarded as an intermediate exposition between the classical viewpoint and the scheme-theoretic viewpoint in algebraic geometry; for that reason, reading it can be excellent preparation for reading our Éléments.

For the reader's information, we give below the general plan envisaged for this treatise, subject to later modification, particularly as regards the last chapters:

• Chapter I. — The language of schemes.
• — II. — Elementary global study of certain classes of morphisms.
• — III. — Cohomology of coherent algebraic sheaves. Applications.
• — IV. — Local study of morphisms.
• — V. — Elementary procedures for constructing schemes.
• — VI. — Descent techniques. General method of constructing schemes.
• — VII. — Group schemes; principal fiber bundles.
• — VIII. — Differential study of fiber bundles.
• — IX. — The fundamental group.
• — X. — Residues and duality.
• — XI. — Intersection theories, Chern classes, the Riemann–Roch theorem.
• — XII. — Abelian schemes and Picard schemes.
• — XIII. — Weil cohomology.

In principle, every chapter is regarded as open, and additional paragraphs may always be added later; such paragraphs will appear as separate fascicles, to reduce the inconveniences of the mode of publication adopted. When such a paragraph is foreseen or in preparation at the time a chapter is published, it will be mentioned in that chapter's table of contents, even if, owing to priorities, its actual publication is to be appreciably later. For the reader's convenience, we have collected in a "Chapter 0" various complements from commutative algebra, homological algebra, and sheaf theory used in the course of this treatise — results that are more or less well known, but for which we have not been able to give convenient references. We recommend that the reader refer to Chapter 0 only in the course of reading the treatise proper, and only insofar as the results we cite are not sufficiently familiar to him. In this way, we think, reading this treatise may serve a beginner as a good route to becoming familiar with commutative algebra and homological algebra, whose study, when not accompanied by tangible applications, is judged tedious, even depressing, by a fair number of people.

It is beyond our competence to give in this Introduction even a summary historical sketch of the notions and results expounded. The text will contain only those references judged particularly useful for understanding it, and we shall indicate the origin of only the most important results. Formally at least, the subjects treated in this new work are rather new, which explains the sparseness of references to the Fathers of algebraic geometry of the nineteenth and early twentieth centuries, whose work we know only by hearsay. It is nevertheless fitting to say a few words here about the works that have most directly influenced the authors and contributed to the development of the scheme-theoretic viewpoint. The fundamental work (FAC) of J.-P. Serre must be cited first of all: it has served as an introduction to algebraic geometry for more than one young convert (one of the authors of the present treatise included), put off by the dryness of A. Weil's classical Foundations [18]. There, for the first time, it is shown that the "Zariski topology" of an "abstract" algebraic variety is perfectly suited to applying certain techniques of algebraic topology and, in particular, gives rise to a cohomology theory. Moreover, the definition of an algebraic variety given there is the one that lends itself most naturally to the extension of that notion which we develop here.¹ Serre had already noticed himself that the cohomology theory of affine algebraic varieties carries over without difficulty if one replaces affine algebras over a field by arbitrary commutative rings. Chapters I and II of this treatise, and the first two paragraphs of Chapter III, may therefore be regarded, in essentials, as easy transpositions of the principal results of (FAC) and of a subsequent article by the same author [15] to this broader framework. We have also drawn considerable profit from the Séminaire de Géométrie algébrique of C. Chevalley [1]; in particular, the systematic use of "constructible sets", introduced by him, has proved very useful in scheme theory (cf. Chapter IV). From him we have also borrowed the study of morphisms from the standpoint of dimension (Chapter IV), which carries over to the scheme-theoretic framework with no significant change. It is worth noting, moreover, that the notion of "scheme of local rings", introduced by Chevalley, naturally lends itself to an extension of algebraic geometry (although without the full flexibility and generality we intend to give it here); on the relation between this notion and our theory, see Chapter I, §8. Such an extension has been developed by M. Nagata in a series of memoirs [9] containing many special results on algebraic geometry over Dedekind rings.²

Finally, it goes without saying that a book on algebraic geometry, and especially one on its foundations, is necessarily influenced — if only by way of intermediaries — by mathematicians such as O. Zariski and A. Weil. In particular, Zariski's Theory of holomorphic functions [20], suitably made more flexible by cohomological methods and completed by an existence theorem (Chapter III, §§4 and 5), is — together with the descent technique exposed in Chapter VI — one of the principal tools used in this treatise, and seems to us one of the most powerful at our disposal in algebraic geometry.

The general technique in which it fits may be sketched as follows (a typical example will be given in Chapter IX, in the study of the fundamental group). One has a proper morphism (Chapter II) $f:X\to Y$ from one algebraic variety to another (more generally, from one scheme to another) that one wishes to study in a neighborhood of a point $y\in Y$, with a view to solving a problem $P$ relating to a neighborhood of $y$. One proceeds in successive steps:

1° One may assume that $Y$ is affine, so that $X$ becomes a scheme defined over the affine ring $A$ of $Y$, and one may even replace $A$ by the local ring of $y$. This reduction is always easy in practice (Chapter V) and reduces us to the case where $A$ is a local ring.

2° One studies the problem under consideration when $A$ is a local Artinian ring. For the problem to retain a meaning when $A$ is not assumed to be an integral domain, it is sometimes necessary to reformulate $P$; this reformulation often yields a better understanding of the problem, which is "infinitesimal" in nature at this stage.

3° The theory of formal schemes (Chapter III, §§3, 4, and 5) allows one to pass from the case of an Artinian ring to that of a complete local ring.

4° Finally, if $A$ is an arbitrary local ring, consideration of "multiform sections" over suitable schemes over $X$, approximating a given "formal" section (Chapter IV), will often allow one to pass from a result known for the scheme obtained from $X$ by extension of scalars to the completion of $A$, to an analogous result for a suitably simple finite extension (for example, an unramified one) of $A$.

This sketch shows the importance of the systematic study of schemes defined over an Artinian ring $A$. Serre's viewpoint in his formulation of local class field theory, and the recent work of Greenberg, suggest that such a study might be undertaken by attaching functorially to such a scheme $X$ a scheme $X^{\prime }$ over the residue field $k$ of $A$ (assumed perfect) of dimension equal (in favorable cases) to $n\cdot \dim X$, where $n$ is the length of $A$.

As for the influence of A. Weil, let it suffice to say that one of the principal motivations behind the writing of this treatise has been the need to develop the apparatus necessary to formulate, with all the desired generality, the definition of "Weil cohomology", and to undertake the proof³ of all the formal properties needed to establish his famous conjectures in Diophantine geometry [19]; this motivation stands alongside the desire to find the natural framework for the customary notions and methods of algebraic geometry, and to give the authors the occasion to understand those notions and techniques.

To conclude, we believe it useful to warn the reader that, just like the authors themselves, he will doubtless have some difficulty before becoming accustomed to the language of schemes and convincing himself that the usual constructions suggested by geometric intuition can be transcribed, essentially in only one reasonable way, into this language. As in many parts of modern mathematics, the initial intuition appears to drift further and further from the language appropriate to express it with the precision and generality required. Here, the psychological difficulty lies in the need to carry over, to the objects of a category already rather different from the category of sets (namely, the category of preschemes, or the category of preschemes over a given prescheme), notions familiar for sets: Cartesian products, group laws, ring laws, module laws, fiber bundles, principal homogeneous fiber bundles, and so on. It will doubtless be hard for the mathematician of the future to escape this new effort of abstraction — perhaps, all told, a fairly small one compared with that made by our fathers as they were becoming familiar with set theory.

References will be given following the decimal system; for instance, in III, 4.9.3, the numeral III indicates the chapter, the 4 the paragraph, and the 9.3 the item within the paragraph. Within the same chapter, the chapter prefix is omitted.