900indexindex.xmlSéminaire de Géométrie Algébrique du Bois-Marie
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SGA 1: Étale covers and the fundamental group
Directed by A. Grothendieck.
Augmented with two exposés by Mme M. Raynaud.
1960–61.
arXiv:math/0206203v2
Introduction ✓
Foreword ✓
Étale morphisms ✓
Smooth morphisms: generalities, differential properties (in progress)
SGA 6: Intersection theory and the Riemann–Roch theorem
Directed by P. Berthelot, A. Grothendieck, and L. Illusie.
With the collaboration of D. Ferrand, J.P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud, and J.P. Serre.
1966–67.
Introduction (in progress)
Outline of a programme for an intersection theory (in progress)
Classes of sheaves and the Riemann–Roch theorem (in progress)
Generalities on finiteness conditions in derived categories (in progress)
890Exposésga6-0rrrsga6-0rrr.xmlSGA 6: Intersection theory and the Riemann–Roch theorem › Classes of sheaves and the Riemann–Roch theorem0RRRsga6
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712sga6-0rrr.isga6-0rrr.i.xml\lambda-rings (formal preliminaries)Isga6-0rrr
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891sga1-forewordsga1-foreword.xmlSGA 1: Étale covers and the fundamental group › Forewordsga1
Each of these written exposés covers the material of multiple consecutive oral exposés.
It did not seem useful to make a note of the dates.
Exposé VII, which is referenced at various points throughout Exposé VIII, has not been written by the speaker, who, in the oral conferences, was limited to outlining the language of descent in general categories, by working from a strictly utilitarian point of view and not entering into the logical difficulties that often arise due to this language.
It seemed that a proper exposé of this language would go beyond the limits of these current notes, even if only due to length.
For a proper exposé of the theory of descent, I refer the reader to an article in preparation by Jean Giraud.
Whilst waiting for its appearance (it is now published: J. Giraud, “Méthode de la Descente”. Bull. Soc. Math. France 2 (1964), viii+150 p.), I think that an attentive reader will have no problems in supplementing, by their own means, the phantom references in Exposé VIII.
Other oral exposés, found after Exposé XI, and to which there are references in certain places of the text, have also not been written down, and were meant to form the substance of an Exposé XII and an Exposé XIII.
The first of these oral exposés covered, in the framework of schemes and analytic spaces with nilpotent elements (as introduced in the Séminaire Cartan 1960/61), the construction of the analytic space associated to a prescheme of locally finite type over a complete valuation field k, GAGA-type theorems in the case where k is the field of complex numbers, and the application to the comparison of the fundamental group defined by transcendental methods and the fundamental group studied in these notes (cf. A. Grothendieck, “Fondements de la Géométrie Algébrique”. Séminaire Bourbaki 190 (December 1959), page 10).
The latter oral exposés outlined the generalisation of methods developed in the text for the study of coverings that admit moderate ramification, and of the structure of the fundamental group of a complete curve minus a finite number of points (cf. loc. cit. 182, page 27, théorème 14).
These exposés do not introduce any essentially new ideas, which is why it did not seem necessary to write them up properly before the appearance of the corresponding chapters of Éléments de Géométrie Algébrique.
(They are included in the present volume in Exposé XII by Mme Raynaud with a different proof from the original given in the oral seminar (cf. ).)
However, the Lefschetz type theorems for the fundamental group and the Picard group, from both a local and a global point of view, were the subject of a separate seminar in 1962, which was completely written down and is available to read.
(Cohomologie étale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), appeared in North Holland Pub. Cie.)
We point out that the results developed, both in the present Séminaire and the seminar from 1962, will be used in an essential manner in the appearance of many key results about the étale cohomology of preschemes, which will be the subject of a Séminaire (led by M. Artin and myself) in 1963/64, currently in preparation.
(Cohomologie étale des schémas (SGA 4), to appear in the same series.)
Exposés I to IV, which are of an essentially local nature, and very elementary, will be absorbed entirely by Chapter IV of Éléments de Géométrie Algébrique, of which the first part is being printed and will probably be published towards the end of 1964.
They can, nevertheless, be useful for a reader who wants to catch up to speed with the essential properties of smooth, étale, or flat morphisms, before diving into the arcana of a systematic treatment.
As for the other exposés, they will be absorbed into Chapter VIII of the Éléments, whose publication can barely even be contemplated for many years.
(In fact, following a change to the initial plan for the Éléments, the study of the fundamental group has been postponed to a later chapter of the Éléments, cf. the Introduction that precedes this Foreword.)Bures, June 1963.892Exposésga6-isga6-i.xmlSGA 6: Intersection theory and the Riemann–Roch theorem › Generalities on finiteness conditions in derived categoriesIsga6756Sectionsga6-i.0sga6-i.0.xmlIntroductionI.0sga6-i
The aim of Exposés I to IV is to develop, with generality suitable for this seminar, the formalism of finiteness conditions in the derived categories of ringed toposes.
As was mentioned in , it is the need to define "Grothendieck groups" that have good variance properties on arbitrary schemes that requires us to generalise and relax the notions of finiteness used up until now.
The classical notion of coherent sheaves on a ringed space (X, \mathcal { O } _X) becomes uninteresting as soon as \mathcal { O } _X is no longer coherent.
We could think of replacing the notion of coherence by the notion of finite presentation, but this presents the inconvenience that the kernel of an epimorphism of modules of finite presentation is not itself, in general, of finite presentation.
We arrive at a satisfying notion by remarking that, if \mathcal { O } _X is coherent, then a coherent sheaf \mathcal { F } is not only of finite presentation but also of finite n-presentation for all n \in \mathbb {N}, where "finite n-presentation" means that there exists, locally, an exact sequence
\mathcal { L } _n \to \mathcal { L } _{n-1} \to \ldots \to \mathcal { L } _0 \to \mathcal { F } \to 0
where the \mathcal { L } _i are free and of finite type.
If we no longer suppose the sheaf \mathcal { O } _X to be coherent, then we say that an \mathcal { O } _X-module \mathcal { F } is pseudo-coherent if it satisfies the above condition, i.e. if it is of finite n-presentation for all n \in \mathbb {N}.
Then the notion of pseudo-coherence possesses, with respect to short exact sequences, the same stability property as the notion of coherence: if two of the terms of a short exact sequence of \mathcal { O } _X-modules are pseudo-coherent, then so too is the third.
We can easily generalise the above notion to complexes of sheaves: a complex \mathcal { F } of \mathcal { O } _X-modules is said to be pseudo-coherent if it is n-pseudo-coherent for all n \in \mathbb {Z}, where "n-pseudo coherent" means that there exists, locally, a quasi-isomorphism \mathcal { L } \xrightarrow { \sim } \mathcal { F }, where \mathcal { L } is a bounded-above complex, whose components in degree \geq n are free sheaves of finite type.
For a complex \mathcal { F } concentrated in degree 0, to say that \mathcal { F } is pseudo-coherent as a complex is equivalent to saying that it is pseudo-coherent as a module.
The notion of a pseudo-coherent complex has excellent stability properties, described in and [sga6-i.2 (?)].
Firstly, it is stable under isomorphism in the derived category D(X) of the category of \mathcal { O } _X-modules;
even better, if two objects of a distinguished triangle in D(X) are pseudo-coherent, then so too is the third: in other words, the full subcategory D(X)_ \mathrm {coh} of D(X) consisting of pseudo-coherent complexes is a triangulated subcategory.
Furthermore, pseudo-coherence is preserved under inverse images and derived tensor products.
In the case where \mathcal { O } _X is coherent, to say that a complex \mathcal { F } is pseudo-coherent simply means that the cohomology of \mathcal { F } is locally bounded above and that all the \mathcal { H } ^i( \mathcal { F } ) are coherent.
If \mathcal { O } _X is a sheaf of regular local rings, then every coherent \mathcal { O } _X-module \mathcal { F } locally admits a finite left resolution by free modules of finite type, i.e. there exists, locally, an exact sequence
0 \to \mathcal { L } _n \to \mathcal { L } _{n-1} \to \ldots \to \mathcal { L } _0 \to \mathcal { F } \to 0
where the \mathcal { L } _i are free and of finite type.
It is thus natural, when \mathcal { O } _X is a sheaf of arbitrary local rings, to consider modules that enjoy the above property;
such modules are called perfect.
More generally, we say that a complex \mathcal { F } of \mathcal { O } _X-modules is perfect if there exists, locally, a quasi-isomorphism \mathcal { L } \xrightarrow { \sim } \mathcal { F }, where \mathcal { L } is a bounded complex of free modules of finite type.
In [sga6-i.3 (?)], we show that the full subcategory D(X)_ \mathrm {perf} of D(X) consisting of perfect complexes is, like D(X)_ \mathrm {coh}, a triangulated subcategory that is "stable" under inverse images and derived tensor products.
We clearly have that D(X)_ \mathrm {perf} \subseteq D(X)_ \mathrm {coh}, but the inclusion is strict in general.
Pseudo-coherence can also be defined "by passing to the limit" from perfectness: a complex \mathcal { F } is pseudo-coherent if and only if, for all n \in \mathbb {Z}, \mathcal { F } can be "locally approximated to order n by a perfect complex", by which we mean that there exists, locally, a distinguished triangle
\mathcal { L } \to \mathcal { F } \to \mathcal { R } \to \mathcal { L } [1]
where \mathcal { L } is perfect, and \mathcal { R } is acyclic in degree \geq n.
Conversely, perfectness can be recovered from pseudo-coherence by an additional regularity condition: a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-dimension (cf. [sga6-i.5 (?)]).
We thus recover the fact that, if the local rings are regular, then every coherent sheaf is perfect, and, more generally, that every pseudo-coherent complex with locally bounded cohomology is perfect.
Pseudo-coherence and perfectness are the two fundamental notions of finiteness with which we will work in this seminar.
Sections I.1 to I.5 of this present exposé are dedicated to their definition, and to establishing their elementary stability properties.
For this, we only use two or three local properties of the category of free modules of finite type with respect to the category of all modules.
It also turns out to be practical — and useful, most notably in [sga6-ii (?)] — to axiomatise the situation, by introducing a notion of pseudo-coherence (resp. perfectness) in a fibred category over a site with respect to a fibred subcategory that respects suitable conditions.
Sections I.6 to I.8 of this present exposé generalise a certain number of well known notions for locally free sheaves of finite type (rank, duality, and trace of an endomorphism) to the setting of perfect complexes.
In [sga6-ii (?)], we examine the problem of "the existence of global resolutions": under what conditions, on a ringed topos (X, \mathcal { O } _X), is a pseudo-coherent (resp. perfect) complex globally isomorphic in D(X) to a complex of locally free modules of finite type?
The importance of this questions rests essentially on the fact that we do not know, as of yet, how to generalise certain usual constructions on locally free sheaves (exterior and symmetric powers, and Chern classes) to perfect complexes (for more details on this, see [sga6-xiv (?)]).
This is why it is convenient to have tractable sufficient conditions for the existence of global resolutions, which allows us to reduce certain questions about pseudo-coherent (resp. perfect) complexes to analogous questions about locally free sheaves of finite type.
Such criteria are given in [sga6-ii (?)].
In particular, we show that there exist global resolutions in the following cases:
X is the Zariski topos of an affine scheme, or, more generally, of a quasi-compact scheme that has an ample invertible sheaf, or an ample family of invertible sheaves in the sense of Kleiman (for example, a regular scheme);
or X is the topos of sheaves of sets on a compact topological space, and \mathcal { O } _X is the sheaf of continuous complex-valued functions.
As an illustration of these methods — and to give evidence for the flexibility of the notions we have introduced — we give, in the appendix, a "purely sheaf-theoretic" definition of the index of a family of elliptic operators.
[sga6-iii (?)] studies the stability of finiteness conditions under the derived direct image.
To obtain reasonable statements, we need to put the notions of pseudo-coherence and perfectness "into perspective".
We place ourselves here in the setting of ordinary schemes, which suffices for the seminar, but there is no doubt that we must sooner or later develop an analogous theory for relative schemes or analytic spaces.
Let p \colon X \to S be an S-scheme locally of finite type, and let \mathcal { F } be a complex of \mathcal { O } _X-modules;
we say that \mathcal { F } is pseudo-coherent (resp. perfect) with respect to p if we can locally embed (by a closed immersion) X into a smooth S-scheme X' in such a way that the extension of \mathcal { F } by zero on X' is pseudo-coherent (resp. perfect).
The notion of pseudo-coherence with respect to p is especially interesting in the case where S is not locally Noetherian, since in the contrary case it agrees with the ordinary notion of pseudo-coherence.
On the other hand, as we might expect, to say that \mathcal { F } is perfect with respect to p is equivalent to saying that \mathcal { F } is pseudo-coherent with respect to p and locally of finite tor-dimension with respect to p (i.e. with respect to the sheaf of rings p^{-1}( \mathcal { O } _S)).
The central theorem of [sga6-iii (?)] is the finiteness theorem, which affirms (in a slightly more precise way) that, if f \colon X \to Y is a proper morphism of S-schemes locally of finite type, then the functor \mathbb {R}f_* sends complexes that are pseudo-coherent with respect to S to complexes that are pseudo-coherent with respect to S;
this theorem is, in reality, a conjecture, but has nevertheless been proven in the two particular following cases:
S is locally Noetherian;
f is projective.
Unfortunately, it seems that the extension to the general case is of the same order of difficulty as the analogous theorem in analytic geometry (Grauert's "theorem").
Combining the finiteness theorem with an essentially trivial formula called the projection formula, we obtain tractable criteria for the stability of relative perfectness under direct images.
We recover, as a corollary, "Grauert's continuity and semi-continuity theorems" (EGA III 7.6).
[sga6-iv (?)] translates the results of Exposés I to III into the language of "Grothendieck groups".
On a ringed space (X, \mathcal { O } _X), we denote by K^ \bullet (X) (resp. K_ \bullet (X)) the "Grothendieck group" of the category of perfect complexes of finite tor-dimension (resp. of pseudo-coherent complexes of bounded cohomology).
The group K^ \bullet (X) is a ring, and K_ \bullet (X) is a module over K^ \bullet (X).
As suggested by the superscript bullet, K^ \bullet is a contravariant functor, whilst K_ \bullet is a covariant functor for proper and pseudo-coherent morphisms (of schemes or of analytic spaces, and under the caveat that we have the finiteness theorem...).
There are also unusual variances (covariance of K^ \bullet and contravariance of K_ \bullet) for morphisms satisfying suitable regularity hypotheses.
Finally, the globalisation criteria of [sga6-ii (?)] allow us to make the link with the "Grothendieck groups" defined naively from coherent sheaves or locally free sheaves of finite type.
758Sectionsga6-i.1sga6-i.1.xmlPreliminary definitionsI.1sga6-i757Sectionsga6-i.1.1sga6-i.1.1.xmlFibred categories with additive (resp. abelian, resp. triangulated) fibresI.1.1sga6-i.1716sga6-i.1.1.1sga6-i.1.1.1.xmlI.1.1.1sga6-i.1.1
Let \mathcal {S} be a category, and let \mathcal {C} be a fibred \mathcal {S}-category ([SGA 1 VI 6.1]) with additive (resp. triangulated) fibres.
We say that \mathcal {C} is an additive (resp. triangulated) \mathcal {S}-category (or that \mathcal {C} is additive (resp. triangulated) over \mathcal {S}) if, for every arrow f \colon X \to Y of \mathcal {S}, the inverse image functor (determined up to unique isomorphism) f^* \colon \mathcal {C}_Y \to \mathcal {C}_X is additive (resp. exact).
We say that \mathcal {C} is an abelian \mathcal {S}-category if \mathcal {C} is an additive \mathcal {S}-category with abelian fibres, and that \mathcal {C} is a flat abelian \mathcal {S}-category if, further, the inverse image functors are exact.
717sga6-i.1.1.2sga6-i.1.1.2.xmlI.1.1.2sga6-i.1.1
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893sga1-introductionsga1-introduction.xmlSGA 1: Étale covers and the fundamental group › Introductionsga1
In the first section of this introduction, we give some details about the contents of the present volume; in the second, about the entirety of the Séminaire de Géométrie Algébrique du Bois-Marie, of which the present volume constitutes the first tome.
670sga1-introduction-1sga1-introduction-1.xml1sga1-introduction
The present volume details the fundamentals of a theory of the fundamental group in algebraic geometry, from the "Kroneckerian" point of view, which allows us to treat the case of an algebraic variety, in the usual sense, and that of a ring of integers of a number field, for example, on an equal footing.
This point of view can only be expressed in a satisfactory manner in the language of schemes, and we will freely use this language, as well as the main results stated in the first three chapters of the Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck (cited as "EGA").
The study of this present volume of the Séminaire de Géométrie Algébrique du Bois-Marie does not require any other knowledge of algebraic geometry, and can thus serve as an introduction to the current techniques of algebraic geometry to a reader who wishes to familiarise themselves with them.
Exposés I to XI of this book are a textual reproduction, practically unchanged, of the mimeographed notes from the oral seminar, which were distributed by the Institut des Hautes Études Scientifiques.
(As well as the notes of the following seminars. Since this method of distribution turned out to be impractical and inadequate in the long term, all the Séminaire de Géométrie Algébrique du Bois-Marie from now on will appear in book form, like the present volume.)
We have restricted ourselves to adding some footnotes to the original text, correcting typos, and making an adjustment to terminology (notably, "simple morphism" being replaced with "smooth morphism", which does not lead to the same confusion).
Exposés I to IV present the local notions of étale and smooth morphisms;
they hardly ever use the language of schemes, as detailed in Chapter I of the Éléments.
(A more complete study is now available in EGA IV, 17 and 18.)
Exposé V presents the axiomatic description of the fundamental group of a scheme, which is useful even in the classical case where the scheme is simply the spectrum of a field, since we then find a strong and convenient reformulation of usual Galois theory.
Exposés VI and VIII present the theory of descent, which has become more and more important in algebraic geometry over the past few years, and which could do the same in analytic geometry and in topology.
We note that Exposé VII was not transcribed, but its contents can be found incorporated into an article by J. Giraud ("Méthode de la Descente". Bull. Soc. Math. France 2 (1964), viii+150 p.).
In Exposé IX, we study more specifically the theory of descent by étale morphisms, obtaining a systematic approach for Van Kampen type theorems for the fundamental group, which appear here as simple translations of theorems of descent.
This essentially involves a calculation of the fundamental group of a connected scheme X endowed with a surjective and proper morphism (say, X' \to X) in terms of the fundamental groups of the connected components of X' and of the fibre products X' \times _X X', X' \times _X X' \times _X X', and the homomorphisms between these groups induced by the canonical simplicial morphisms between the above schemes.
Exposé X gives the theory of specialisation of the fundamental group for a proper and smooth morphism, with the most striking result being the determination (more or less) of the fundamental group of a smooth algebraic curve in characteristic p>0, thanks to the known result obtained by transcendental methods in characteristic zero.
Exposé XI gives some examples and addenda, including a cohomological version of Kummer's theory of coverings, as well as Artin--Schreier's.
For other commentaries on the text, see the Foreword, found after this Introduction.
Since the writing of the seminar in 1961, the language of étale topology, along with a corresponding cohomology theory, has been developed by M. Artin in collaboration with myself;
it is detailed in tome 4 ("Cohomologie étale des schémas") of the Séminaire de Géométrie Algébrique, which will appear in the same series as the present volume.
This language, as well as the results that it has given up until now, give us a particularly supple tool for the study of the fundamental group, allowing us to better understand (and even improve upon) certain results given here.
There will thus be a need to entirely rewrite the theory of the fundamental group from this point of view (in fact, all the key results so far appear in loc. cit.).
This was what was planed for the chapter of Éléments dedicated to the fundamental group, which also had to cover many other ideas which could not find their place here (relying on the technique of resolution of singularities):
calculation of the "local fundamental group" of a complete local ring in terms of a suitable resolution of singularities of the ring, local and global Künneth formulas for the fundamental group without any hypotheses of properness (cf. Exposé XIII), the results of M. Artin on the comparison of the local fundamental groups of an excellent Henselian local ring and of its completion (SGA 4 XIX).
We also note the necessity of developing a theory of the fundamental group of a topos, which would simultaneously cover the ordinary topological theory, the semi-simplicial version, the "profinite" version developed in Exposé V, and the slightly more general pro-discrete version found in [SGA 3 X 7] (adapted to the case of non-normal and non-unibranch schemes).
While we wait for such a rewriting of the whole theory, Exposé XIII, by Mme Raynaud, using the language and results of SGA 4, aims to show the part that we can extract in some typical questions, most notably generalising certain results of Exposé X to non-proper relative schemes.
There we give, in particular, the structure of the "prime at p" fundamental group of a non-complete algebraic curve in arbitrary characteristic (which I announced in 1959, but for which a proof had not been published up until now).
Despite these numerous gaps and imperfections (as others would say: because of these gaps and imperfections), I think that the present volume could be useful for the reader who wishes to familiarise themselves with the theory of the fundamental group, as well as a work of reference, while we await the writing and appearance of a text that avoids the criticisms that I have just listed.
671sga1-introduction-2sga1-introduction-2.xml2sga1-introduction
The present volume constitutes tome 1 of the Séminaire de Géométrie Algébrique du Bois-Marie, whose following volumes are intended to appear in the same series as this one.
The aim of the Séminaire, parallel to the Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck, is to establish the fundamentals of algebraic geometry, following the points of view of the latter.
The standard reference for all the volumes of the Séminaire consists of Chapters I, II, and III of Éléments de Géométrie Algébrique (cited as EGA I, II, and III), and we assume the reader to be have the education in commutative algebra and homological algebra that these chapters imply.
(See the Introduction of EGA I for more precise details.)
Furthermore, in each volume of the Séminaire, we will freely refer as needed to previous volumes of the same Séminaire, or to other chapters of Éléments, either already published or on the brink of being published.
Each volume of the Séminaire is based on a main subject, indicated in the title of the corresponding volume(s);
the oral seminar generally lasts one academic year, sometimes more.
The exposés within each volume of the Séminaire are generally in a logical order of dependence on one another;
however, the different volumes of the Séminaire are largely logically independent of one another.
So the volume "Group schemes" is largely logically independent of the two volumes of the Séminaire that come before it chronologically;
however, it makes frequent reference to results of EGA IV.
Here is the list of the volumes of the Séminaire that should appear (cited as SGA I to SGA 7 in what follows):
SGA 1.
Étale covers and the fundamental group, 1960/61.
SGA 2.
Local cohomology of coherent sheaves and local and global Lefschetz theorems, 1961/62.
SGA 3.
Group schemes, 1963/64 (3 volumes, in collaboration with M. Demazure).
SGA 4.
Topos theory and étale cohomology of schemes, 1963/64 (3 volumes, in collaboration with M. Artin and J.L. Verdier).
SGA 5.
\ell-adic cohomology and L-functions, 1964/65 (2 volumes).
SGA 6.
Intersection theory and the Riemann--Roch theorem, 1966/67 (2 volumes, in collaboration with P. Berthelot and L. Illusie).
SGA 7.
Local monodromy groups in algebraic geometry.
Three of these partial Séminaires have been written in collaboration with other mathematicians, who appear as coauthors on the covers of the corresponding volumes.
As for the other active participants of the Séminaire, whose roles (from as much of an editorial point of view as a mathematical one) have grown over the years, the name of each participant appears at the top of the exposés for which they are responsible, either as speaker or as writer, and the list of those who appear in a given volume can be found on the flyleaf of the volume in question.
It is useful to give some precise remarks concerning the relation between the Séminaire and the Éléments.
The latter was intended, in principle, to give an exposé of a set of ideas and techniques that were judged to be the most fundamental in Algebraic Geometry, as these ideas and techniques emerge all by themselves from the natural requirements of logical and aesthetic coherence.
From this point of view, it was natural to consider the Séminaire as a preliminary version of the Éléments, destined to be included almost entirely, sooner or later, in the latter.
This process had already somewhat started a few years ago, since Exposés I to IV of the current volume (SGA 1) are entirely covered by EGA IV, and Exposés VI to VIII should be, in some years, in EGA VI.
However, given how the work in constructing the Éléments and the Séminaire is developing, and as the proportions of the two become more precise, the initial principle (that the Séminaire should be only a preliminary and provisional version) seems less and less realistic, for the reason (amongst others) of the limits imposed by the long-sighted nature of the length of the human life.
Taking into account the care that is generally given to the writing of the various parts of the Séminaire, there will doubtless be no need to go back over them in the Éléments (or other treaties that would take over from it) until further progress in the writing process would allow us to make very substantial improvements, at the cost of rather big changes.
This is the case as of now for the current seminar SGA 1, as we said above, and also for SGA 2 (thanks to recent results by Mme. Raynaud).
However, nothing currently indicates that this will also be the case in the near future for any of the other parts mentioned above (SGA 3 to SGA 7).
[Trans.] We omit one paragraph from this translation since it simply describes the citation and cross-reference formatting of the book, which we do not precisely follow here.
Finally, for the ease of the reader, every time that it seems necessary, at the end of the volumes of the SGA we attach an index of notation and an index of terminology containing, if necessary, an English translation of the French terms used.
I want to add an extra-mathematical comment to this introduction.
In the month of November, 1969, I discovered that the Institut des Hautes Études Scientifiques, where I have been a professor essentially since its founding, had been receiving subventions from the Ministère des Armées for three years.
Already as a young researcher, I found it extremely regrettable how few qualms the majority of scientists had in agreeing to collaborate in one form or another with military institutions.
My motivations back then were essentially of a moral nature, and thus not very likely to be taken seriously.
Today they acquire a new force and dimension, given the danger of destruction of the human species threatened by the proliferation of military institutions and of the means of mass destruction that they posses.
I have explained my thoughts on these problems, which are much more important than the advancement of any of the sciences (mathematics included), in more detail in other places;
the reader can, for example, consult the article by G. Edwards in the Volume 1 of the journal Survivre (August, 1970), which gives a more detailed summary of these problems than I have done elsewhere.
So I found myself working for three years at an institution even though it was, unbeknownst to me, participating in a financing model that I consider immoral and dangerous.
(It goes without saying that the opinion that I express here is entirely my own, and not that of the publishing house Springer, which is editing this volume.)
Being currently the only person to have this opinion amongst my colleagues at the IHES, which has condemned to fail my efforts to suppress military subventions from the IHES budget, I have taken the necessary decision to leave the IHES on the 30th of September, 1970, and to equally suspend all scientific collaboration with this institute for as long as it continues to accept such subventions.
I have asked M. Motchane, the director of the IHES, for the IHES to abstain, starting from the 1st of October, 1970, from sharing mathematical texts of which I am an author, or which form part of the Séminaire de Géométrie Algébrique du Bois-Marie.
As was mentioned above, the diffusion of this seminar will be undertaken by the publishing house Julius Springer, in the Lecture Notes series.
I am happy to thank here Springer and Mr. K. Peters for the efficient and polite help that they have given me in making this publication possible, in particular for dealing with the typing of the photo-offset of new exposés added to old seminars, and of missing exposés in incomplete seminars.
I equally thank Mr. J.P. Delale, who had the thankless job of compiling the index of notation and the index of terminology.
Massy, August 1970.894sga6-introductionsga6-introduction.xmlSGA 6: Intersection theory and the Riemann–Roch theorem › Introductionsga6
...
895Exposésga6-0sga6-0.xmlSGA 6: Intersection theory and the Riemann–Roch theorem › Outline of a programme for a theory of intersections0sga6
The current exposé is of an introductory nature, and its reading is not logically necessary for the study of the seminar.
It is aimed particularly at readers familiar with the provisional version of the Riemann–Roch theorem, as given in the report by Borel–Serre [@2] or in the report by Grothendieck mentioned previously in the introduction (cited as [RRR]), which is reproduced as an appendix at the end of this current exposé.
687Sectionsga6-0.1sga6-0.1.xml1sga6-0
Recall the Riemann–Roch formula for a proper morphism
f \colon X \to Y
of smooth quasi-projective schemes over a field k and a coherent sheaf \mathcal { F } on X:
679Equationsga6-0.1-equation-1.1sga6-0.1-equation-1.1.xml1.1sga6-0.1 \operatorname {Todd} (T_Y) \operatorname {ch} _Y(f_*( \operatorname {cl} ( \mathcal { F } ))) = f_*( \operatorname {Todd} (T_X) \operatorname {ch} _X( \mathcal { F } )) \tag{1.1}
where \operatorname {cl} ( \mathcal { F } ) denotes the class of \mathcal { F } in the group K(X) of classes of coherent sheaves on X, and \operatorname {ch} _X and \operatorname {ch} _Y denote the Chern characters of on X and Y (resp.), and T_X and T_Y the tangent bundles to X and Y (resp.).
This formula holds in A(Y) \otimes _ \mathbb {Z} \mathbb {Q}, where A(Y) is the Chow ring of Y;
the f_* on the right-hand side is induced by tensoring with \mathbb {Q} the "direct image of cycles" homomorphism
f_* \colon A(X) \to A(Y)
and the f_* on the left-hand side is the Euler–Poincaré characteristic of \mathcal { F } with respect to f:
f_*( \operatorname {cl} ( \mathcal { F } )) = \sum _i (-1)^i \operatorname {cl} ( \operatorname {R} ^i f_*( \mathcal { F } )).
As we know, \operatorname {Todd} (-) and \operatorname {ch} (-) are universal polynomials in the Chern classes of the argument with coefficients in \mathbb {Q}.
Since the constant term of \operatorname {Todd} (-) is 1, it is an invertible element for any value of the argument, so that can be rewritten, after multiplication by \operatorname {Todd} (T_Y)^{-1}, in the form which is more useful for our needs:
680Equationsga6-0.1-equation-1.2sga6-0.1-equation-1.2.xml1.2sga6-0.1 \operatorname {ch} _Y(f_*( \operatorname {cl} ( \mathcal { F } ))) = f_*( \operatorname {Todd} (T_f) \operatorname {ch} _X( \mathcal { F } )) \tag{1.2}
where we set
681Equationsga6-0.1-equation-1.3sga6-0.1-equation-1.3.xml1.3sga6-0.1 T_f = T_X - f^*(T_Y) \in K(X) \tag{1.3}
so that T_f plays the role of a virtual relative tangent bundle of X over Y.
In the case where the morphism f is smooth (i.e. with everywhere-surjective tangent map), we have simply
T_f = T_{X/Y}
(the tangent bundle along the fibres) and so, in the case where f \colon X \to Y is an immersion, we find
T_f = - \check {N}_{X/Y}
where \check {N}_{X/Y} denotes the normal sheaf of X in Y.
One of the main goals of this Seminar is to generalise simultaneously in two directions:
Remove the hypothesis of the existence of a base field k.
Replace the regularity hypotheses on Y and X by a "local regularity" hypothesis on f.
Finally, along the way, we will equally deal with the problem:
Remove the quasi-projectivity hypotheses which, in the absence of a base field, are expressed by the existence of ample invertible modules on X and on Y.
689Sectionsga6-0.2sga6-0.2.xml2sga6-0
We now examine generalisation (a) from sga6-0.2, keeping, however, the hypotheses of regularity and of existence of ample invertible modules on X and on Y.
The definition of K(X) and K(Y), and of the homomorphism f_* \colon K(X) \to K(Y) then gives no new problems, thanks to the fact that X and Y are regular.
The most natural route to giving meaning to thus seems to consist of defining the Chow rings A(X) and A(Y) and a group homomorphism
f_* \colon A(X) \to A(Y)
as well as establishing a theory of Chern classes, providing maps
c_i \colon K(X) \to A(X)
(and similarly for Y), and finally giving a description of an virtual relative tangent bundle element
T_f \in K(X). 688Sectionsga6-0.2.1sga6-0.2.1.xml2.1sga6-0.2
For
TO-DO: finish896sga1sga1.xmlSGA 1: Étale covers and the fundamental group779sga1-introductionsga1-introduction.xmlIntroductionsga1
In the first section of this introduction, we give some details about the contents of the present volume; in the second, about the entirety of the Séminaire de Géométrie Algébrique du Bois-Marie, of which the present volume constitutes the first tome.
670sga1-introduction-1sga1-introduction-1.xml1sga1-introduction
The present volume details the fundamentals of a theory of the fundamental group in algebraic geometry, from the "Kroneckerian" point of view, which allows us to treat the case of an algebraic variety, in the usual sense, and that of a ring of integers of a number field, for example, on an equal footing.
This point of view can only be expressed in a satisfactory manner in the language of schemes, and we will freely use this language, as well as the main results stated in the first three chapters of the Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck (cited as "EGA").
The study of this present volume of the Séminaire de Géométrie Algébrique du Bois-Marie does not require any other knowledge of algebraic geometry, and can thus serve as an introduction to the current techniques of algebraic geometry to a reader who wishes to familiarise themselves with them.
Exposés I to XI of this book are a textual reproduction, practically unchanged, of the mimeographed notes from the oral seminar, which were distributed by the Institut des Hautes Études Scientifiques.
(As well as the notes of the following seminars. Since this method of distribution turned out to be impractical and inadequate in the long term, all the Séminaire de Géométrie Algébrique du Bois-Marie from now on will appear in book form, like the present volume.)
We have restricted ourselves to adding some footnotes to the original text, correcting typos, and making an adjustment to terminology (notably, "simple morphism" being replaced with "smooth morphism", which does not lead to the same confusion).
Exposés I to IV present the local notions of étale and smooth morphisms;
they hardly ever use the language of schemes, as detailed in Chapter I of the Éléments.
(A more complete study is now available in EGA IV, 17 and 18.)
Exposé V presents the axiomatic description of the fundamental group of a scheme, which is useful even in the classical case where the scheme is simply the spectrum of a field, since we then find a strong and convenient reformulation of usual Galois theory.
Exposés VI and VIII present the theory of descent, which has become more and more important in algebraic geometry over the past few years, and which could do the same in analytic geometry and in topology.
We note that Exposé VII was not transcribed, but its contents can be found incorporated into an article by J. Giraud ("Méthode de la Descente". Bull. Soc. Math. France 2 (1964), viii+150 p.).
In Exposé IX, we study more specifically the theory of descent by étale morphisms, obtaining a systematic approach for Van Kampen type theorems for the fundamental group, which appear here as simple translations of theorems of descent.
This essentially involves a calculation of the fundamental group of a connected scheme X endowed with a surjective and proper morphism (say, X' \to X) in terms of the fundamental groups of the connected components of X' and of the fibre products X' \times _X X', X' \times _X X' \times _X X', and the homomorphisms between these groups induced by the canonical simplicial morphisms between the above schemes.
Exposé X gives the theory of specialisation of the fundamental group for a proper and smooth morphism, with the most striking result being the determination (more or less) of the fundamental group of a smooth algebraic curve in characteristic p>0, thanks to the known result obtained by transcendental methods in characteristic zero.
Exposé XI gives some examples and addenda, including a cohomological version of Kummer's theory of coverings, as well as Artin--Schreier's.
For other commentaries on the text, see the Foreword, found after this Introduction.
Since the writing of the seminar in 1961, the language of étale topology, along with a corresponding cohomology theory, has been developed by M. Artin in collaboration with myself;
it is detailed in tome 4 ("Cohomologie étale des schémas") of the Séminaire de Géométrie Algébrique, which will appear in the same series as the present volume.
This language, as well as the results that it has given up until now, give us a particularly supple tool for the study of the fundamental group, allowing us to better understand (and even improve upon) certain results given here.
There will thus be a need to entirely rewrite the theory of the fundamental group from this point of view (in fact, all the key results so far appear in loc. cit.).
This was what was planed for the chapter of Éléments dedicated to the fundamental group, which also had to cover many other ideas which could not find their place here (relying on the technique of resolution of singularities):
calculation of the "local fundamental group" of a complete local ring in terms of a suitable resolution of singularities of the ring, local and global Künneth formulas for the fundamental group without any hypotheses of properness (cf. Exposé XIII), the results of M. Artin on the comparison of the local fundamental groups of an excellent Henselian local ring and of its completion (SGA 4 XIX).
We also note the necessity of developing a theory of the fundamental group of a topos, which would simultaneously cover the ordinary topological theory, the semi-simplicial version, the "profinite" version developed in Exposé V, and the slightly more general pro-discrete version found in [SGA 3 X 7] (adapted to the case of non-normal and non-unibranch schemes).
While we wait for such a rewriting of the whole theory, Exposé XIII, by Mme Raynaud, using the language and results of SGA 4, aims to show the part that we can extract in some typical questions, most notably generalising certain results of Exposé X to non-proper relative schemes.
There we give, in particular, the structure of the "prime at p" fundamental group of a non-complete algebraic curve in arbitrary characteristic (which I announced in 1959, but for which a proof had not been published up until now).
Despite these numerous gaps and imperfections (as others would say: because of these gaps and imperfections), I think that the present volume could be useful for the reader who wishes to familiarise themselves with the theory of the fundamental group, as well as a work of reference, while we await the writing and appearance of a text that avoids the criticisms that I have just listed.
671sga1-introduction-2sga1-introduction-2.xml2sga1-introduction
The present volume constitutes tome 1 of the Séminaire de Géométrie Algébrique du Bois-Marie, whose following volumes are intended to appear in the same series as this one.
The aim of the Séminaire, parallel to the Éléments de Géométrie Algébrique by J. Dieudonné and A. Grothendieck, is to establish the fundamentals of algebraic geometry, following the points of view of the latter.
The standard reference for all the volumes of the Séminaire consists of Chapters I, II, and III of Éléments de Géométrie Algébrique (cited as EGA I, II, and III), and we assume the reader to be have the education in commutative algebra and homological algebra that these chapters imply.
(See the Introduction of EGA I for more precise details.)
Furthermore, in each volume of the Séminaire, we will freely refer as needed to previous volumes of the same Séminaire, or to other chapters of Éléments, either already published or on the brink of being published.
Each volume of the Séminaire is based on a main subject, indicated in the title of the corresponding volume(s);
the oral seminar generally lasts one academic year, sometimes more.
The exposés within each volume of the Séminaire are generally in a logical order of dependence on one another;
however, the different volumes of the Séminaire are largely logically independent of one another.
So the volume "Group schemes" is largely logically independent of the two volumes of the Séminaire that come before it chronologically;
however, it makes frequent reference to results of EGA IV.
Here is the list of the volumes of the Séminaire that should appear (cited as SGA I to SGA 7 in what follows):
SGA 1.
Étale covers and the fundamental group, 1960/61.
SGA 2.
Local cohomology of coherent sheaves and local and global Lefschetz theorems, 1961/62.
SGA 3.
Group schemes, 1963/64 (3 volumes, in collaboration with M. Demazure).
SGA 4.
Topos theory and étale cohomology of schemes, 1963/64 (3 volumes, in collaboration with M. Artin and J.L. Verdier).
SGA 5.
\ell-adic cohomology and L-functions, 1964/65 (2 volumes).
SGA 6.
Intersection theory and the Riemann--Roch theorem, 1966/67 (2 volumes, in collaboration with P. Berthelot and L. Illusie).
SGA 7.
Local monodromy groups in algebraic geometry.
Three of these partial Séminaires have been written in collaboration with other mathematicians, who appear as coauthors on the covers of the corresponding volumes.
As for the other active participants of the Séminaire, whose roles (from as much of an editorial point of view as a mathematical one) have grown over the years, the name of each participant appears at the top of the exposés for which they are responsible, either as speaker or as writer, and the list of those who appear in a given volume can be found on the flyleaf of the volume in question.
It is useful to give some precise remarks concerning the relation between the Séminaire and the Éléments.
The latter was intended, in principle, to give an exposé of a set of ideas and techniques that were judged to be the most fundamental in Algebraic Geometry, as these ideas and techniques emerge all by themselves from the natural requirements of logical and aesthetic coherence.
From this point of view, it was natural to consider the Séminaire as a preliminary version of the Éléments, destined to be included almost entirely, sooner or later, in the latter.
This process had already somewhat started a few years ago, since Exposés I to IV of the current volume (SGA 1) are entirely covered by EGA IV, and Exposés VI to VIII should be, in some years, in EGA VI.
However, given how the work in constructing the Éléments and the Séminaire is developing, and as the proportions of the two become more precise, the initial principle (that the Séminaire should be only a preliminary and provisional version) seems less and less realistic, for the reason (amongst others) of the limits imposed by the long-sighted nature of the length of the human life.
Taking into account the care that is generally given to the writing of the various parts of the Séminaire, there will doubtless be no need to go back over them in the Éléments (or other treaties that would take over from it) until further progress in the writing process would allow us to make very substantial improvements, at the cost of rather big changes.
This is the case as of now for the current seminar SGA 1, as we said above, and also for SGA 2 (thanks to recent results by Mme. Raynaud).
However, nothing currently indicates that this will also be the case in the near future for any of the other parts mentioned above (SGA 3 to SGA 7).
[Trans.] We omit one paragraph from this translation since it simply describes the citation and cross-reference formatting of the book, which we do not precisely follow here.
Finally, for the ease of the reader, every time that it seems necessary, at the end of the volumes of the SGA we attach an index of notation and an index of terminology containing, if necessary, an English translation of the French terms used.
I want to add an extra-mathematical comment to this introduction.
In the month of November, 1969, I discovered that the Institut des Hautes Études Scientifiques, where I have been a professor essentially since its founding, had been receiving subventions from the Ministère des Armées for three years.
Already as a young researcher, I found it extremely regrettable how few qualms the majority of scientists had in agreeing to collaborate in one form or another with military institutions.
My motivations back then were essentially of a moral nature, and thus not very likely to be taken seriously.
Today they acquire a new force and dimension, given the danger of destruction of the human species threatened by the proliferation of military institutions and of the means of mass destruction that they posses.
I have explained my thoughts on these problems, which are much more important than the advancement of any of the sciences (mathematics included), in more detail in other places;
the reader can, for example, consult the article by G. Edwards in the Volume 1 of the journal Survivre (August, 1970), which gives a more detailed summary of these problems than I have done elsewhere.
So I found myself working for three years at an institution even though it was, unbeknownst to me, participating in a financing model that I consider immoral and dangerous.
(It goes without saying that the opinion that I express here is entirely my own, and not that of the publishing house Springer, which is editing this volume.)
Being currently the only person to have this opinion amongst my colleagues at the IHES, which has condemned to fail my efforts to suppress military subventions from the IHES budget, I have taken the necessary decision to leave the IHES on the 30th of September, 1970, and to equally suspend all scientific collaboration with this institute for as long as it continues to accept such subventions.
I have asked M. Motchane, the director of the IHES, for the IHES to abstain, starting from the 1st of October, 1970, from sharing mathematical texts of which I am an author, or which form part of the Séminaire de Géométrie Algébrique du Bois-Marie.
As was mentioned above, the diffusion of this seminar will be undertaken by the publishing house Julius Springer, in the Lecture Notes series.
I am happy to thank here Springer and Mr. K. Peters for the efficient and polite help that they have given me in making this publication possible, in particular for dealing with the typing of the photo-offset of new exposés added to old seminars, and of missing exposés in incomplete seminars.
I equally thank Mr. J.P. Delale, who had the thankless job of compiling the index of notation and the index of terminology.
Massy, August 1970.780sga1-forewordsga1-foreword.xmlForewordsga1
Each of these written exposés covers the material of multiple consecutive oral exposés.
It did not seem useful to make a note of the dates.
Exposé VII, which is referenced at various points throughout Exposé VIII, has not been written by the speaker, who, in the oral conferences, was limited to outlining the language of descent in general categories, by working from a strictly utilitarian point of view and not entering into the logical difficulties that often arise due to this language.
It seemed that a proper exposé of this language would go beyond the limits of these current notes, even if only due to length.
For a proper exposé of the theory of descent, I refer the reader to an article in preparation by Jean Giraud.
Whilst waiting for its appearance (it is now published: J. Giraud, “Méthode de la Descente”. Bull. Soc. Math. France 2 (1964), viii+150 p.), I think that an attentive reader will have no problems in supplementing, by their own means, the phantom references in Exposé VIII.
Other oral exposés, found after Exposé XI, and to which there are references in certain places of the text, have also not been written down, and were meant to form the substance of an Exposé XII and an Exposé XIII.
The first of these oral exposés covered, in the framework of schemes and analytic spaces with nilpotent elements (as introduced in the Séminaire Cartan 1960/61), the construction of the analytic space associated to a prescheme of locally finite type over a complete valuation field k, GAGA-type theorems in the case where k is the field of complex numbers, and the application to the comparison of the fundamental group defined by transcendental methods and the fundamental group studied in these notes (cf. A. Grothendieck, “Fondements de la Géométrie Algébrique”. Séminaire Bourbaki 190 (December 1959), page 10).
The latter oral exposés outlined the generalisation of methods developed in the text for the study of coverings that admit moderate ramification, and of the structure of the fundamental group of a complete curve minus a finite number of points (cf. loc. cit. 182, page 27, théorème 14).
These exposés do not introduce any essentially new ideas, which is why it did not seem necessary to write them up properly before the appearance of the corresponding chapters of Éléments de Géométrie Algébrique.
(They are included in the present volume in Exposé XII by Mme Raynaud with a different proof from the original given in the oral seminar (cf. ).)
However, the Lefschetz type theorems for the fundamental group and the Picard group, from both a local and a global point of view, were the subject of a separate seminar in 1962, which was completely written down and is available to read.
(Cohomologie étale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), appeared in North Holland Pub. Cie.)
We point out that the results developed, both in the present Séminaire and the seminar from 1962, will be used in an essential manner in the appearance of many key results about the étale cohomology of preschemes, which will be the subject of a Séminaire (led by M. Artin and myself) in 1963/64, currently in preparation.
(Cohomologie étale des schémas (SGA 4), to appear in the same series.)
Exposés I to IV, which are of an essentially local nature, and very elementary, will be absorbed entirely by Chapter IV of Éléments de Géométrie Algébrique, of which the first part is being printed and will probably be published towards the end of 1964.
They can, nevertheless, be useful for a reader who wants to catch up to speed with the essential properties of smooth, étale, or flat morphisms, before diving into the arcana of a systematic treatment.
As for the other exposés, they will be absorbed into Chapter VIII of the Éléments, whose publication can barely even be contemplated for many years.
(In fact, following a change to the initial plan for the Éléments, the study of the fundamental group has been postponed to a later chapter of the Éléments, cf. the Introduction that precedes this Foreword.)Bures, June 1963.781Exposésga1-isga1-i.xmlÉtale morphismsIsga1
To simplify the exposition, we assume that all preschemes in the following are locally Noetherian (at least, starting from ).
515Sectionsga1-i.1sga1-i.1.xmlBasics of differential calculusI.1sga1-i
Let X be a prescheme over Y, and \Delta _{X/Y} the diagonal morphism X \to X \times _Y X.
This is an immersion, and thus a closed immersion of X into an open subset V of X \times _Y X.
Let \mathcal { I } _X be the ideal of the closed sub-prescheme corresponding to the diagonal in V (N.B. if one really wishes to do things intrinsically, without assuming that X is separated over Y — a misleading hypothesis — then one should consider the set-theoretic inverse image of \mathcal { O } _{X \times X} in X and denote by \mathcal { I } _X the augmentation ideal in the above ...).
The sheaf \mathcal { I } _X/ \mathcal { I } _X^2 can be thought of as a quasi-coherent sheaf on X, which we denote by \Omega _{X/Y}^1.
This sheaf is of finite type if X \to Y is of finite type, and it behaves well with respect to a base change Y' \to Y.
We also introduce the sheaves \mathcal { O } _{X \times _Y X}/ \mathcal { I } _X^{n+1}= \mathcal { P } ^n_{X/Y}, which are sheaves of rings on X, giving us preschemes denoted by \Delta _{X/Y}^n and called the n-th infinitesimal neighbourhood of X/Y.
The polysyllogism is entirely trivial, even if rather long (cf. EGA IV 16.3);
it seems wise to not discuss it until we use it for something helpful, with smooth morphisms.
516Sectionsga1-i.2sga1-i.2.xmlQuasi-finite morphismsI.2sga1-i316Propositionsga1-i.2.1sga1-i.2.1.xmlI.2.1sga1-i.2
Let A \to B be a local homomorphism (N.B. all rings are now Noetherian), and let \mathfrak {m} be the maximal ideal of A.
Then the following conditions are equivalent:
B/ \mathfrak {m}B is of finite dimension over k=A/ \mathfrak {m}.
\mathfrak {m}B is an ideal of definition, and B/ \mathfrak {r}(B)=k(B) is an extension of k=k(A).
The completion \widehat {B} of B is finite over the completion \widehat {A} of A.
If the conditions of are satisfied, then we say that B is quasi-finite over A.
A morphism f \colon X \to Y is said to be quasi-finite at x (or the Y-prescheme f is said to be quasi-finite at x) if \mathcal { O } _x is quasi-finite over \mathcal { O } _{f(x)}.
This is equivalent to saying that x is isolated in its fibre f^{-1}(x).
A morphism is said to be quasi-finite if it is quasi-finite at each point.
(In [EGA II 6.2.3] we further suppose that f is of finite type.)
317Corollarysga1-i.2.2sga1-i.2.2.xmlI.2.2sga1-i.2
If A is complete, then quasi-finiteness is equivalent to finiteness.
We could also give the usual polysyllogism (i), (ii), (iii), (iv), (v) for quasi-finite morphisms, but that doesn't seem necessary here.
517Sectionsga1-i.3sga1-i.3.xmlUnramified morphismsI.3sga1-i334Propositionsga1-i.3.1sga1-i.3.1.xmlI.3.1sga1-i.3
Let f \colon X \to Y be a morphism of finite type, x \in X, and y=f(x).
Then the following conditions are equivalent:
\mathcal { O } _x/ \mathfrak {m}_y \mathcal { O } _x is a finite separable extension of k(y).
\Omega _{X/Y}^1 is zero at x.
The diagonal morphism \Delta _{X/Y} is an open immersion on a neighbourhood of x.
333Proofsga1-i.3.1
For the implication (i) \implies (ii), we can use Nakayama to reduce to the case where Y= \operatorname {Spec} (k) and X= \operatorname {Spec} (k'), where it is well known, and also trivial by the definition of separable;
(ii) \implies (iii) comes from a nice and easy characterisation of open immersions, using Krull;
(iii) \implies (i) follows as well from reducing to the case where Y= \operatorname {Spec} (k) and the diagonal morphism is everywhere an open immersion.
We must then prove that X is finite with separable ring over k, and this leads us to consider the case where k is algebraically closed.
But then every closed point of X is isolated (since it is identical to the inverse image of the diagonal by the morphism X \to X \times _k X defined by x), whence X is finite.
We can thus suppose that X consists of a single point, with ring A, and so A \otimes _k A \to A is an isomorphism, hence A=k.
335Definitionsga1-i.3.2sga1-i.3.2.xmlI.3.2sga1-i.3
Let f satisfy one of the equivalent conditions of .
Then we say that f is unramified at x, or that X is unramified at x on Y.
Let A \to B be a local homomorphism.
We say that it is unramified, or that B is a local unramified algebra on A, if B/ \mathfrak {r}(A)B is a finite separable extension of A/ \mathfrak {r}(A), i.e. if \mathfrak {r}(A)B= \mathfrak {r}(B) and k(B) is a separable extension of k(A).
(cf. regrets in [III 1.2])
336Remarkssga1-i.3
The fact that B is unramified over A can be seen at the level of the completions of A and B.
Unramified implies quasi-finite.
337Corollarysga1-i.3.3sga1-i.3.3.xmlI.3.3sga1-i.3
The set of points where f is unramified is open.
338Corollarysga1-i.3.4sga1-i.3.4.xmlI.3.4sga1-i.3
Let X' and X be preschemes of finite type over Y, and g \colon X' \to X a Y-morphism.
If X is unramified over Y, then the graph morphism \Gamma _g \colon X' \to X \times _Y X is an open immersion.
Indeed, this is the inverse image of the diagonal morphism X \to X \times _Y X by
g \times _Y \mathrm {id} _{X'} \colon X' \times _Y X \to X \times _Y X.
One can also introduce the annihilator ideal \mathfrak {d}_{X/Y} of \Omega _{X/Y}^1, called the different ideal of X/Y;
it defines a closed sub-prescheme of X which, set-theoretically, is the set of point where X/Y is ramified, i.e. not unramified.
343Propositionsga1-i.3.5sga1-i.3.5.xmlI.3.5sga1-i.3
An immersion is ramified.
The composition of two ramified morphisms is also ramified.
Base extension of a ramified morphisms is also ramified.
We are rather indifferent about (ii) and (iii) (the second seems more interesting to me).
We can, of course, also be more precise, by giving some one-off statements;
this is more general only in appearance (except for in the case of definition (b)), and is boring.
We obtain, as per usual, the corollaries:
348Corollarysga1-i.3.6sga1-i.3.6.xmlI.3.6sga1-i.3
The cartesian product of two unramified morphisms is unramified.
If gf is unramified then so too is f.
If f is unramified then so too is f_ \mathrm {red}.
349Propositionsga1-i.3.7sga1-i.3.7.xmlI.3.7sga1-i.3
Let A \to B be a local homomorphism, and suppose that the residue extension k(B)/k(A) is trivial, with k(A) algebraically closed.
In order for B/A to be unramified, it is necessary and sufficient that \widehat {B} be (as an \widehat {A}-algebra) a quotient of \widehat {A}.
350Remarkssga1-i.3
In the case where we don't suppose that the residue extension is trivial, we can reduce to the case where it is by taking a suitable finite flat extension of A which destroys the aforementioned extension.
Consider the example where A is the local ring of an ordinary double point of a curve, and B a point of its normalisation:
then A \subset B, B is unramified over A with trivial residue extension, and \widehat {A} \to \widehat {B} is surjective but not injective.
We are thus going to strengthen the notion of unramified-ness.
518Sectionsga1-i.4sga1-i.4.xmlÉtale morphisms. Étale coversI.4sga1-i
We are going to suppose that everything concerning flat morphisms that we need to be true is indeed true;
these facts will be proved later, if there is time.
(cf. [sga1-iv (?)].)369Definitionsga1-i.4.1sga1-i.4.1.xmlI.4.1sga1-i.4
Let f \colon X \to Y be a morphism of finite type.
We say that f is étale at x if f is both flat and unramified at x.
We say that f is étale if it is étale at all points.
We say that X is étale at x over Y, or that it is a Y-prescheme which is étale at x etc.
Let f \colon A \to B be a local homomorphism.
We say that f is étale, or that B is étale over A, if B is flat and unramified over A
(cf. regrets in [sga1-iii.1.2 (?)].)
371Propositionsga1-i.4.2sga1-i.4.2.xmlI.4.2sga1-i.4
For B/A to be étale, it is necessary and sufficient that \widehat {B}/ \widehat {A} be étale.
370Proofsga1-i.4.2
This is true individually for both "unramified" and "flat".
372Corollarysga1-i.4.3sga1-i.4.3.xmlI.4.3sga1-i.4
Let f \colon X \to Y be of finite type, and x \in X.
The property of f being étale at x depends only on the local homomorphism \mathcal { O } _{f(x)} \to \mathcal { O } _x, and in fact only on the corresponding homomorphism for the completions.
373Corollarysga1-i.4.4sga1-i.4.4.xmlI.4.4sga1-i.4
Suppose that the residue extension k(A) \to k(B) is trivial, or that k(A) is algebraically closed.
Then B/A is étale if and only if \widehat {A} \to \widehat {B} is an isomorphism.
We can combine flatness with .
375Propositionsga1-i.4.5sga1-i.4.5.xmlI.4.5sga1-i.4
Let f \colon X \to Y be a morphism of finite type.
Then the set of points where f is étale is open.
374Proofsga1-i.4.5
Again, this is true individually for both "unramified" and "flat".
This proposition shows that we can forget about the "one-off" comments in the study of morphisms of finite type that are somewhere étale.
381Propositionsga1-i.4.6sga1-i.4.6.xmlI.4.6sga1-i.4
An open immersion is étale.
The composition of two étale morphisms is étale.
The base extension of an étale morphism is étale.
380Proofsga1-i.4.6
Indeed, (i) is trivial, and for (ii) and (iii) it suffices to note that it is true for "unramified" and "flat".
As a matter of fact, there are also corresponding comments to make about local homomorphisms (without the finiteness condition), which in any case should appear in the multiplodoque (starting with the case of unramified).
([Trans.] Grothendieck's "multiplodoque d'algèbre homologique" was the final version of his Tohoku paper — see (2.1) in "Life and work of Alexander Grothendieck" by Ching-Li Chan and Frans Oort for more information.)382Corollarysga1-i.4.7sga1-i.4.7.xmlI.4.7sga1-i.4
The cartesian product of two étale morphisms is étale.
383Corollarysga1-i.4.8sga1-i.4.8.xmlI.4.8sga1-i.4
Let X and X' be of finite type over Y, and g \colon X \to X' a Y-morphism.
If X' is unramified over Y and X is étale over Y, then g is étale.
358Proofsga1-i.4.8
Indeed, g is the composition of the graph morphism \Gamma _g \colon X \to X \times _Y X' (which is an open immersion by ) and the projection morphism, which is étale since it is just a "change of base" by X' \to Y of the étale morphism X \to Y.
384Definitionsga1-i.4.9sga1-i.4.9.xmlI.4.9sga1-i.4
We say that a cover of Y is étale (resp. unramified) if it is a Y-scheme X that is finite over Y and étale (resp. unramified) over Y.
The first condition means that X is defined by a coherent sheaf of algebras \mathcal { B } over Y.
The second means that \mathcal { B } is locally free over Y (resp. means absolutely nothing) and, further, that, for all y \in Y, the fibre \mathcal { B } (y)= \mathcal { B } _y \otimes _{ \mathcal { O } _y}k(y) is a separable algebra (i.e. a finite composition of finite separable extensions) over k(y).
385Propositionsga1-i.4.10sga1-i.4.10.xmlI.4.10sga1-i.4
Let X be a flat cover of Y of degree n (the definition of this term deserved to figure in ) defined by a locally free coherent sheaf \mathcal { B } of algebras.
We define, as usual, the trace homomorphism \mathcal { B } \to \mathcal { A } (that is, a homomorphism of \mathcal { A }-modules, where \mathcal { A } = \mathcal { O } _Y).
For X to be étale it is necessary and sufficient that the corresponding bilinear form \operatorname {tr} _{ \mathcal { B } / \mathcal { A } }xy define an isomorphism of \mathcal { B } over \mathcal { B }, or, equivalently, that the discriminant section
d_{X/Y} = d_{ \mathcal { B } / \mathcal { A } } \in \Gamma \big (Y, \wedge ^n \check { \mathcal { B } } \otimes _ \mathcal { A } \wedge ^n \check { \mathcal { B } } \big )
is invertible, or that the discriminant ideal defined by this section is the unit ring.
282Proofsga1-i.4.10
We can reduce to the case where Y= \operatorname {Spec} (k), and then it is a well-known criterion of separability, and thus trivial by passing to the algebraic closure of k.
386Remarksga1-i.4
We will have a less trivial statement to make later on (), when we do not suppose a priori that X is flat over Y, but instead require some normality hypothesis.
519Sectionsga1-i.5sga1-i.5.xmlFundamental property of étale morphismsI.5sga1-i441Theoremsga1-i.5.1sga1-i.5.1.xmlI.5.1sga1-i.5
Let f \colon X \to Y be a morphism of finite type.
For f to be an open immersion, it is necessary and sufficient that it be étale and radicial.
321Proofsga1-i.5.1
Recall what "radicial" means: injective, with radicial residual extensions (recall also that it means that the morphism remains injective under any base extension).
The necessity is trivial, and the sufficiency remains to be shown.
We are going to give two different proofs: the first is shorter, the second is more elementary.
A flat morphism is open, and so we can suppose (by replacing Y with f(X)) that f is an onto homeomorphism.
For any base extension, it remains true that f is flat, radicial, and surjective, thus a homeomorphism, and a fortiori closed.
Thus f is proper.
Thus f is finite (reference: Chevalley's theorem), defined by a coherent sheaf \mathcal { B } of algebras.
Now \mathcal { B } is locally free, and further, by hypothesis, of rank 1 everywhere, and so X=Y.
We can suppose that Y and X are affine.
We can further easily reduce to proving the following:
if Y= \operatorname {Spec} (A), with A local, and if f^{-1}(y) is non-empty (where y is the closed point of Y), then X=Y (indeed, this would imply that every y \in f(X) has an open neighbourhood U such that X|U=U).
We will then have that X= \operatorname {Spec} (B), and wish to prove that A=B.
But for this we can reduce to proving the analogous claim where we replace A by \widehat {A}, and B by B \otimes _A \widehat {A}
(taking into account the fact that \widehat {A} is faithfully flat over A).
We can thus suppose that A is complete.
Let x be the point over y.
By , \mathcal { O } _x is finite over A, and is thus (being flat and radicial over A) identical to A.
So X=Y \coprod X' (disjoint sum).
But since X is radicial over Y, X' is empty.
443Corollarysga1-i.5.2sga1-i.5.2.xmlI.5.2sga1-i.5
Let f \colon X \to Y be a morphism that is both a closed immersion and étale.
If X is connected, then f is an isomorphism from X to a connected component of Y.
442Proofsga1-i.5.2
Indeed, f is also an open immersion.
We thus deduce:
444Corollarysga1-i.5.3sga1-i.5.3.xmlI.5.3sga1-i.5
Let X be an unramified Y-scheme, with Y connected.
Then every section of X over Y is an isomorphism from Y to a connected component of X.
There is thus a bijective correspondence between the set of such sections and the set of connected components X_i of X such that the projection X_i \to Y is an isomorphism (or, equivalently, by , surjective and radicial).
In particular, a section is determined by its value at a point.
409Proofsga1-i.5.3
Only the first claim demands a proof;
by , it suffices to note that a section is a closed immersion (since X is separated over Y) and also étale, by .
446Corollarysga1-i.5.4sga1-i.5.4.xmlI.5.4sga1-i.5
Let X and Y be preschemes over S, with X unramified and separated over S, and Y connected.
Let f and g be S-morphisms from Y to X, and suppose that y is a point of Y such that f(y)=g(y)=x, and such that the residue homomorphisms k(x) \to k(y) defined by f and g are identical ("f and g agree geometrically at y").
Then f and g are identical.
445Proofsga1-i.5.4
This follows from by reducing to the case where Y=S, and by replacing X with X \times _S Y.
Here is a particularly important variant of :
448Theoremsga1-i.5.5sga1-i.5.5.xmlI.5.5sga1-i.5
Let S be a prescheme, and let X and Y be S-preschemes.
Let S_0 be a closed sub-prescheme of S that has the same underlying space as S, and let X_0=X \times _S S_0 and Y_0=Y \times _S S_0 be the "restrictions" of X and Y to S_0.
Suppose that X is étale over S.
Then the natural map
\operatorname {Hom} _S(Y,X) \to \operatorname {Hom} _{S_0}(X_0,Y_0)
is bijective.
447Proofsga1-i.5.5
We can again reduce to the case where Y=S, and then this follows from the "topological" description of sections of X/Y given in .
449Scholiumsga1-i.5
Let S be a prescheme, and let X and Y be S-preschemes.
Let S_0 be a closed sub-prescheme of S that has the same underlying space as S, and let X_0=X \times _S S_0 and Y_0=Y \times _S S_0 be the "restrictions" of X and Y to S_0.
Suppose that X is étale over S.
Then the natural map
\operatorname {Hom} _S(Y,X) \to \operatorname {Hom} _{S_0}(X_0,Y_0)
is bijective.
The following form of (which looks more general) is often useful:
451Corollarysga1-i.5.6sga1-i.5.6.xmlExtension of liftingsI.5.6sga1-i.5
Consider a commutative diagram of morphisms
\begin {CD} X @<<< Y_0 \\ @VVV @VVV \\ S @<<< Y \end {CD}
where X \to S is étale, and Y_0 \to Y is a bijective closed immersion.
Then we can find a unique morphism Y \to X such that the two corresponding triangles commute.
450Proofsga1-i.5.6
By replacing S with Y, and X with X \times _S Y, we can reduce to the case where Y=S, and this is then a particular case of for Y=S.
We also note the following immediate consequence of (which we did not give as a corollary, in order to not interrupt the line of ideas developed following ):
452Propositionsga1-i.5.7sga1-i.5.7.xmlI.5.7sga1-i.5
Let X and X' be preschemes that are of finite type and flat over Y, and let g \colon X \to X' be a Y-morphism.
For g to be an open immersion (resp. an isomorphism), it is necessary and sufficient that the induced morphism on the fibres
g \otimes _Y k(y) \colon X \otimes _Y k(y) \to X' \otimes _Y k(y)
be an open immersion (resp. an isomorphism) for all y \in Y.
439Proofsga1-i.5.7
It suffices to prove sufficiency;
since it is true for the property of being a surjection, we can reduce to the case of an open immersion.
By , we have to show that g is radicial (which is trivial) and étale (which follows from below).
454Corollarysga1-i.5.8sga1-i.5.8.xmlI.5.8sga1-i.5
Let X and X' be Y-preschemes, g \colon X \to X' a Y-morphism, x a point of X, and y the projection of x in Y.
For g to be quasi-finite (resp. unramified) at x, it is necessary and sufficient that g \otimes _Y k(y) be so.
453Proofsga1-i.5.8
The two algebras over k(g(x)) that we have to study in order to see whether or not we do indeed have a morphism which is quasi-finite (resp. unramified) at x are the same for g and g \otimes _Y k(y).
456Corollarysga1-i.5.9sga1-i.5.9.xmlI.5.9sga1-i.5
With the notation of , suppose that X and X' are flat and of finite type over Y.
For g to be flat (resp. étale) at x, it is necessary and sufficient that g \otimes _Y k(y) be so.
455Proofsga1-i.5.9
For "flat", the statement only serves as a reminder, since this is one of the fundamental criteria of flatness.
(cf. [sga1-iv.5.9 (?)].)
For "étale", this follows by taking into account.
520Sectionsga1-i.6sga1-i.6.xmlApplication to étale extensions of complete local ringsI.6sga1-i
This section is a particular case of the results on formal preschemes, which should appear in the multiplodoque.
Nevertheless, here we get away without much difficulty, i.e. without the explicit local determination of the étale morphisms in [sga1-i.7 (?)] (using the Main Theorem).
This is perhaps sufficient reason to keep this current section (even in the multiplodoque) where it is.
495Theoremsga1-i.6.1sga1-i.6.1.xmlI.6.1sga1-i.6
Let A be a complete local ring (Noetherian, of course), with residue field k.
For any A-algebra B, let R(B)=B \otimes _Ak be thought of as a k-algebra;
this depends functorially on B.
Then R defines an equivalence between the category of A-algebras that are finite and étale over A and the category of algebras that are finite rank and separable over k.
Firstly, the functor in question is fully faithful, as follows from the more general fact:
496Corollarysga1-i.6.2sga1-i.6.2.xmlI.6.2sga1-i.6
Let B and B' be A-algebras that are finite over A.
If B is étale over A, then the canonical map
\operatorname {Hom} _{ A \mathsf {-alg} }(B,B') \to \operatorname {Hom} _{ k \mathsf {-alg} }(R(B),R(B'))
is bijective.
475Proofsga1-i.6.2
We can reduce to the case where A is Artinian (by replacing A by A/ \frak {m}^n), and then this is a particular case of .
It remains to prove that, for every finite and separable k algebra (or we can simply say "étale", for brevity) L, there exists some B étale over A such that R(B) is isomorphic to L.
We can suppose that L is a separable extension of k, and, as such, it admits a generator x, i.e. it is isomorphic to an algebra k[t]/Fk[t], where F \in k[t] is a monic polynomial.
We can lift F to a monic polynomial F_1 in A[t], and we take B=A[t]/F_1A[t].
521Sectionsga1-i.7sga1-i.7.xmlLocal construction of unramified and étale morphismsI.7sga1-i412Propositionsga1-i.7.1sga1-i.7.1.xmlI.7.1sga1-i.7
Let A be a Noetherian ring, B an algebra which is finite over A, and u a generator of B over A.
Let F \in A[t] be such that F(u)=0 (we do not assume F to be monic), and u'=F'(u) (where F' is the differentiated polynomial).
Let \mathfrak {q} be a prime ideal of B not containing u', and \mathfrak {p} its intersection with A.
Then B_ \mathfrak {q} is unramified over A_ \mathfrak {p}.
In other words, taking Y= \operatorname {Spec} (A), X= \operatorname {Spec} (B), and X_{u'}= \operatorname {Spec} (B_{u'}), we claim that X_{u'} is unramified over Y.
This statement follows from the following, more precise:
414Corollarysga1-i.7.2sga1-i.7.2.xmlI.7.2sga1-i.7
The different ideal of B/A contains u'B, and is equal to u'B if the natural homomorphism A[t]/FA[t] \to B (sending t to u) is an isomorphism.
413Proofsga1-i.7.2
Let J be the kernel of the homomorphism C=A[t] \to B, so that J contains FA[t], and is equal to it in the second case described in .
Since the homomorphism C \to B is surjective, \Omega _{B/A}^1 can be identified with the quotient of \Omega _{C/A}^1 by the sub-module generated by J \Omega _{C/A}^1 and \mathrm {d} (J) (we should have explicitly described the definition of the homomorphism d and the calculation of \Omega ^1 for an algebra of polynomials in ).
Identifying \Omega _{C/A}^1 with C, via the basis \mathrm {d} t, we obtain B/B \cdot J', and so the different ideal is generated by the set J' of images in B of derivatives of G \in J (and it suffices to take G that generate J).
Since F \in J (resp. F is a generator of J), we are done.
416Corollarysga1-i.7.3sga1-i.7.3.xmlI.7.3sga1-i.7
Under the conditions of , and supposing that F is monic and that A[t]/FA[t] \to B is an isomorphism, in order for B_ \mathfrak {q} to be étale over A_ \mathfrak {p} it is necessary and sufficient that \mathfrak {q} not contain u'.
415Proofsga1-i.7.3
Since B is flat over A, being étale is equivalent to being unramified, and we can apply .
418Corollarysga1-i.7.4sga1-i.7.4.xmlI.7.4sga1-i.7
Under the conditions of , in order for B to be étale over A, it is necessary and sufficient that u' be invertible, or for the ideal F' generated by F in A[t] to be the unit ideal.
417Proofsga1-i.7.4
The second claim follows from the first along with Nakayama (in B).
A monic polynomial F \in A[t] that has the property stated in is said to be a separable polynomial (if F is not monic, we must at least require that the coefficient of its leading term be invertible; in the case where A is a field, we recover the usual definition).
420Corollarysga1-i.7.5sga1-i.7.5.xmlI.7.5sga1-i.7
Let B be an algebra which is finite over the local ring A.
Suppose that K(A) is infinite, or that B is local.
Let n be the rank of L=B \otimes _A K(A) over K(A)=k.
For B to be unramified (resp. étale) over A, it is necessary and sufficient that B be isomorphic to a quotient of (resp. isomorphic to) A[t]/FA[t], where F is a separable monic polynomial, which we can assume to be (resp. which is necessarily of) degree n.
419Proofsga1-i.7.5
We only have to prove necessity.
Suppose that B is unramified over A, and thus that L is separable over k.
It then follows from the hypotheses that L/k admits a generator \xi, and so the \xi ^i (for 0 \leq i<n) form a basis for L over k.
Let u \in B be a lift of \xi;
by Nakayama, the u^i (for 0 \leq i<n) generate (resp. form a basis of) the A-module B, and, in particular, we obtain a monic polynomial F \in A[t] such that F(u)=0;
then B is isomorphic to a quotient of (resp. isomorphic to) A[t]/FA[t].
Finally, by applying to L/k, we see that F and F' generate A[t] modulo \mathfrak {m}A[t], and so (by Nakayama in A[t]/FA[t]) F and F' generate A[t], and we are done.
421Theoremsga1-i.7.6sga1-i.7.6.xmlI.7.6sga1-i.7
Let A be a local ring, and A \to \mathcal { O } a local homomorphism such that \mathcal { O } is isomorphic to the localisation of an algebra of finite type over A.
Suppose that \mathcal { O } is unramified over A.
Then we can find an A-algebra B that is integral over A, a maximal ideal \mathfrak {n} of B, a generator u of B over A, and a monic polynomial F \in A[t] such that \mathfrak {n} \not \ni F'(u) and such that \mathcal { O } is isomorphic (as an A-algebra) to B_ \mathfrak {n}.
If \mathcal { O } is étale over A, then we can take B=A[t]/FA[t].
(Of course, these conditions are more than sufficient ...)
Before proving , we first state some nice corollaries:
423Corollarysga1-i.7.7sga1-i.7.7.xmlI.7.7sga1-i.7
For \mathcal { O } to be unramified over A, it is necessary and sufficient that \mathcal { O } be isomorphic to the quotient of an algebra which is unramified and étale over A.
422Proofsga1-i.7.7
We can take \mathcal { O } '=B'_{ \mathfrak {n}'}, where B'=A[t]/FA[t] and where \mathfrak {n}' is the inverse image of \mathfrak {n} in B'.
425Corollarysga1-i.7.8sga1-i.7.8.xmlI.7.8sga1-i.7
Let f \colon X \to Y be a morphism of finite type, and x \in X.
For f to be unramified at x, it is necessary and sufficient that there exist an open neighbourhood U of x such that f|U factors as U \to X' \to Y, where the first arrow is a closed immersion, and the second is an étale morphism.
424Proofsga1-i.7.8
This is a simple translation of .
We will now show how the jargon of follows from the main theorem:
there exists, by , an epimorphism \mathcal { O } ' \to \mathcal { O }, where \mathcal { O } has all the desired properties;
but since \mathcal { O } ' and \mathcal { O } are étale over A, the morphism \mathcal { O } ' \to \mathcal { O } is étale by , and thus an isomorphism.
426Proofsga1-i.7.6-proofsga1-i.7.6-proof.xmlI.7.6sga1-i.7
This mimics a proof from the Séminaire Chevalley.
By the "Main Theorem", we have that \mathcal { O } =B_ \mathfrak {n}, where B is an algebra that is finite over A, and \mathfrak {n} is a maximal ideal.
Then B/ \mathfrak {n}=B( \mathcal { O } ) is a separable, and thus monogenous, extension of k;
if \mathfrak {n}_i (for 1 \leq i \leq r) are maximal ideas of B that are distinct from \mathfrak {n}, then there thus exists an element u of B that belongs to all the \mathfrak {n}_i, and thus whose image in B/ \mathfrak {n} is a generator.
But B/ \mathfrak {n}=B_ \mathfrak {n}/ \mathfrak {n}B_ \mathfrak {n}=B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} (where \mathfrak {m} is the maximal ideal of A).
Suppose, for the moment, that we have both and .
Let n be the rank of the k-algebra L=B \otimes _A k.
By Nakayama, there exists a monic polynomial of degree n in A[t] such that F(u)=0.
Let f be the polynomial induced from F by reduction \mod \mathfrak {m}.
Then L is k-isomorphic to k[t]/fk[t], and so, by , f'( \xi ) is not contained in the maximal ideal of L that corresponds to \mathfrak {n} (where \xi denotes the image of t in L, i.e. the image of u in L).
Since f'( \xi ) is the image of F'(u), we are done.
427Lemmasga1-i.7.9sga1-i.7.9.xmlI.7.9sga1-i.7
Let A be a local ring, B an algebra that is finite over A, \mathfrak {n} a maximal ideal of B, and u an element of B whose image in B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} is a generator as an algebra over k=A/ \mathfrak {m}, and such that u is contained in every maximal ideal of B that is distinct from \mathfrak {n}.
Let B'=B[u] and \mathfrak {n}'= \mathfrak {n}B'.
Then the canonical homomorphism B'_{ \mathfrak {n}'} \to B_ \mathfrak {n} is an isomorphism.
428Lemmasga1-i.7.10sga1-i.7.10.xmlI.7.10sga1-i.7
Let B be a algebra that is finite over A and generated by a single element u, and let \mathfrak {n} be a maximal ideal of B such that B_ \mathfrak {n} is unramified over A.
Then there exists a monic polynomial F \in A[t] such that F(u)=0 and F'(u) \not \in \mathfrak {n}.
N.B. should have appeared as a corollary to , and before (which it implies).
So now follows from the combination of and ;
it remains only to prove .
429Proofsga1-i.7.9-proofsga1-i.7.9-proof.xmlI.7.9sga1-i.7
Let S'=B' \setminus \mathfrak {n}', so that B'S'^{-1}=B'_ \mathfrak {n'}.
Similarly, let S=B \setminus \mathfrak {n}, so that BS^{-1}=B_ \mathfrak {n}.
We then have a natural homomorphism BS'^{-1} \to BS^{-1}=B_ \mathfrak {n};
we will show that this is an isomorphism, i.e. that the elements of S are invertible in BS'^{-1}, i.e. that every maximal ideal \mathfrak {p} of BS'^{-1} does not meet S, i.e. that every maximal ideal of BS'^{-1} induces \mathfrak {n} on B.
\begin {CD} B @>>> BS'^{-1} @>>> BS^{-1} = B_ \mathfrak {n} \\ @AAA @AAA \\ B' @>>> B'S'^{-1} = B'_{ \mathfrak {n}'} \end {CD}
Since BS'^{-1} is finite over B'S'^{-1}=B'_{ \mathfrak {n}'}, \mathfrak {p} induces the unique maximal ideal \mathfrak {n}'B_{ \mathfrak {n}'} of B'_{ \mathfrak {n}'}, and thus induces the maximal ideal \mathfrak {n}' of B';
since B is finite over B', the ideal \mathfrak {q} of B induced by \mathfrak {p}, which lives over \mathfrak {n}', is necessarily maximal, and does not contain u, and is thus identical to \mathfrak {n}.
(We have just used the fact that u belongs to every maximal ideal of B that is distinct from \mathfrak {n}).
We now prove that BS'^{-1} is equal to B'S'^{-1}:
since the former is finite over the latter, we can reduce, by Nakayama, to proving equality modulo \mathfrak {n}'BS'^{-1}, and, a fortiori, it suffices to prove equality modulo \mathfrak {m}BS'^{-1};
but BS'^{-1}/ \mathfrak {m}BS'^{-1}=B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} is generated, over k, by u (here we use the other property of u), and so the image of B' (and, a fortiori, of B'S'^{-1}) inside is everything (as a sub-ring that contains k and the image of u.)
430Remarksga1-i.7
We must be able to state for a ring \mathcal { O } that is only semi-local, so that we also cover :
we make the hypothesis that \mathcal { O } / \mathfrak {m} \mathcal { O } is a monogenous k-algebra;
we can thus find some u \in B whose image in B/ \mathfrak {m}B is a generator, and belongs to every maximal ideal of B that doesn't come from \mathcal { O }.
Both and should be able to be adapted without difficulty.
More generally, ...
529Sectionsga1-i.8sga1-i.8.xmlInfinitesimal lifting of étale schemes. Applications to formal schemesI.8sga1-i522Propositionsga1-i.8.1sga1-i.8.1.xmlI.8.1sga1-i.8
Let Y be a prescheme, Y_0 a sub-prescheme, X_0 an étale Y_0-scheme, and x a point of X_0.
Then there exists an étale Y-scheme X, a neighbourhood U_0 of x in X_0, and a Y_0-isomorphism U_0 \xrightarrow { \sim }X \times _Y Y_0.
503Proofsga1-i.8.1
Let y be the projection of x in Y_0;
applying to the étale local homomorphism A_0 \to B_0 of local rings of y and x in Y_0 and X_0, we obtain an isomorphism
\begin {aligned} B_0 &= (C_0)_{ \mathfrak {n}_0} \\ C_0 &= A_0[t]/F_0A_0[t] \end {aligned}
where F_0 is a monic polynomial, and \mathfrak {n}_0 is a maximal ideal of C_0 not containing the class of F'_0(t) in C_0.
Let A be the local ring of y in Y, let F be a monic polynomial in A[t] that gives F_0 under the surjective homomorphism A \to A_0 (we lift the coefficients of F_0), and let C=A[t]/FA[t], with \mathfrak {n} the maximal ideal of C given by the inverse image of \mathfrak {n}_0 under the natural epimorphism C \to C \otimes _A A_0=C_0.
Let
B = C_ \mathfrak {n}.
It is immediate, by construction and by , that B is étale over A, and that we have an isomorphism B \otimes _A A_0=A_0.
We know that there exists a Y-scheme X of finite type, along with a point z of X over y such that \mathcal { O } _z is A-isomorphic to C;
since the latter is étale over A= \mathcal { O } _y, we can (by taking X to be small enough) assume that X is étale over Y.
Let X'_0=X \times _Y Y_0.
Then the local ring of z in X'_0 can be identified with \mathcal { O } _z \otimes _A A_0=B \otimes _A A_0, and is thus isomorphic to B_0.
This isomorphism is defined by an isomorphism from a neighbourhood U_0 of x in X to a neighbourhood of z in X'_0, and we can assume this to be identical to X'_0 by taking X to be small enough.
524Corollarysga1-i.8.2sga1-i.8.2.xmlI.8.2sga1-i.8
The analogous claim holds for étale covers, if we suppose the residue field k(y) to be infinite.
523Proofsga1-i.8.2
The proof is the same, just replacing by .
525Theoremsga1-i.8.3sga1-i.8.3.xmlI.8.3sga1-i.8
The functor described in is an equivalence of categories.
477Proofsga1-i.8.3
By , it remains only to show that every étale S_0-scheme X_0 is isomorphic to an S_0-scheme X \times _S S_0, where X is an étale S-scheme.
The underlying topological space of X must necessarily be identical to the one of X_0, and with X_0 being identified with a closed sub-prescheme of X.
The problem is thus equivalent to the following:
find, on the underlying topological space |X_0| of X_0, a sheaf of algebras \mathcal { O } _X over f_0^*( \mathcal { O } _S) (where f_0 is the projection X_0 \to S_0, thought of here as a continuous map of the underlying spaces) that makes |X_0| an étale S-prescheme X, as well as an algebra homomorphism \mathcal { O } _X \to \mathcal { O } _{X_0} that is compatible with the homomorphism f_0^*( \mathcal { O } _S) \to f_0^*( \mathcal { O } _{S_0}) on the sheaves of scalars, and that induces an isomorphism \mathcal { O } _X \otimes _{f_0^*( \mathcal { O } _S)*}f_0^*( \mathcal { O } _{S_0}) \xrightarrow { \sim } \mathcal { O } _{X_0}.
(Then X will be an étale S-prescheme that is reduced along X_0, and thus separated over S, since X_0 is separated over S_0, and X satisfies all the desired properties).
If (U_i) is an open cover of X_0, and if we find a solution to the problem on each of the U_i, then it follows from the uniqueness theorem that these solutions glue (i.e. the sheaves of algebras that they define, endowed with their augmentation homomorphisms, glue), and we claim that the ringed space thus constructed over S is an étale S-prescheme X endowed with an isomorphism X \times _S S_0 \xleftarrow { \sim }X_0.
It thus suffices to find a solution locally, which we know is possible by .
527Corollarysga1-i.8.4sga1-i.8.4.xmlI.8.4sga1-i.8
Let S be a locally Noetherian formal prescheme, endowed with an ideal of definition \mathcal { J }, and let S_0=(|S|, \mathcal { O } _S/ \mathcal { J } ) be the corresponding ordinary prescheme.
Then the functor \mathfrak {X} \mapsto \mathfrak {X} \times _S S_0 from the category of étale covers of S to the category of étale covers of S_0 is an equivalence of categories.
526Proofsga1-i.8.4
Of course, we define an étale cover of a formal prescheme S to be a cover of S (i.e a formal prescheme over S defined by a coherent sheaf of algebras \mathcal { B }) such that \mathcal { B } is locally free, and such that the residue fibres \mathcal { B } _s \otimes _{ \mathcal { O } _s}k(s) of \mathcal { B } are separable algebras over k(s).
If we denote by S_n the ordinary prescheme (|S|, \mathcal { O } _S/ \mathcal { J } ^{n+1}), then the data of a coherent sheaf of algebras \mathcal { B } on S is equivalent to the data of a sequence of coherent sheaves of algebras \mathcal { B } _n on the S_n, endowed with a transitive system of homomorphisms \mathcal { B } _m \to \mathcal { B } _n (for m \geq n) defining the isomorphisms \mathcal { B } _m \otimes _{ \mathcal { O } _{S_m}} \mathcal { O } _{S_n} \xrightarrow { \sim } \mathcal { B } _n.
It is immediate that \mathcal { B } is locally free if and only if the \mathcal { B } _n are locally free over the S_n, and that the separability condition is satisfied if and only if it is satisfied for \mathcal { B } _0, or for all the \mathcal { B } _n.
Thus \mathcal { B } is étale over S if and only if the \mathcal { B } _n are étale over the S_n.
Taking this into account, follows immediately from .
528Remarksga1-i.8
It was not necessary to restrict ourselves to the case of covers in , but this is the only case that we will use for the moment.
558Sectionsga1-i.9sga1-i.9.xmlInvariance propertiesI.9sga1-i
Let A \to B be a morphism that is local and étale;
we study here some cases where a certain property of A implies the same property for B, or vice versa.
A certain number of such propositions are already consequences of the simple fact that B is quasi-finite and flat over A, and we content ourselves with "recalling" some of them.
A and B have the same Krull dimension, and the same depth (Serre's "cohomological codimension", in the more modern language).
It also follows, for example, that A is Cohen–Macaulay if and only if B is.
Also, for any prime ideal \mathfrak {q} of B (inducing some \mathfrak {p} of A), B_ \mathfrak {q} is again quasi-finite and flat over A_ \mathfrak {p}, as long as we suppose that B is the localisation of an algebra of finite type over A (this follows from the fact that the set of points where a morphism of finite type is quasi-finite (resp. flat) is open);
furthermore, every prime ideal \mathfrak {p} of A is induced by a prime ideal \mathfrak {q} of B (since B is faithfully flat over A).
It thus follows, for example, that \mathfrak {q} and \mathfrak {p} have the same rank;
also, A has no embedded prime ideals if and only if B has none.
We will thus content ourselves with more specific propositions concerning the case of étale morphisms.
531Propositionsga1-i.9.1sga1-i.9.1.xmlI.9.1sga1-i.9
Let A \to B be an étale local homomorphism.
For A to be regular, it is necessary and sufficient that B be regular.
530Proofsga1-i.9.1
Let k be the residue field of A, and L the residue field of B.
Since B is flat over A, and since L=B \otimes _A k (i.e. \mathfrak {n}= \mathfrak {m}B, where \mathfrak {m} and \mathfrak {n} are the maximal ideals of A and B respectively), the \mathfrak {m}-adic filtration on B is identical to the \mathfrak {n}-adic filtration, and
\operatorname {gr} ^ \bullet (B) = \operatorname {gr} ^ \bullet (A) \otimes _k L.
It follows that \operatorname {gr} ^ \bullet (B) is a polynomial algebra over L if and only if \operatorname {gr} ^ \bullet (A) is a polynomial algebra over K.
(N.B. we have not used the fact that L/k is separable.)
532Corollarysga1-i.9.2sga1-i.9.2.xmlI.9.2sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is regular, then X is regular;
the converse is true if f is surjective.
533PropositionI.9.2sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is reduced, then X is reduced;
the converse is true if f is surjective.
This is equivalent to the following:
535Corollarysga1-i.9.3sga1-i.9.3.xmlI.9.3sga1-i.9
Let f \colon A \to B be an étale local homomorphism, with B isomorphic to the localisation of an A-algebra of finite type over A.
For A to be reduced, it is necessary and sufficient that B be reduced.
534Proofsga1-i.9.3
The necessity is trivial, since A \to B is injective (since B is faithfully flat over A).
For the sufficiency, let \mathfrak {p}_i be the minimal prime ideals of A.
By hypothesis, the natural map A \to \prod _i A/ \mathfrak {p}_i is injective, and so tensoring with the flat A-module B gives that B \to \prod _i B/ \mathfrak {p}_iB is injective, and we can thus reduce to proving that the B/ \mathfrak {p}_iB are reduced.
Since B/ \mathfrak {p}_iB is étale over A/ \mathfrak {p}_i, we can reduce to the case where A is integral.
Let K be the field of fractions of A, so that A \to K is injective, and thus so too is B \to B \otimes _A K (since B is A-flat), and we can thus reduce to proving that B \otimes _A K is reduced.
But B is the localisation of an A-algebra of finite type over A, and thus is the local ring of a point x of a scheme of finite type X= \operatorname {Spec} (C) over Y= \operatorname {Spec} (A) that is also étale over Y, so B \otimes _A K is a localisation (with respect to some suitable multiplicatively stable set) of the ring C \otimes _A K of X \otimes _A K.
Since X \otimes _A K is étale over K, its ring is a finite product of fields (that are separable extensions of K), and thus so too is B \otimes _A K.
537Corollarysga1-i.9.4sga1-i.9.4.xmlI.9.4sga1-i.9
Let f \colon A \to B be an étale local homomorphism, with A analytically reduced (i.e. such that the completion \widehat {A} of A has no nilpotent elements).
Then B is analytically reduced, and a fortiori reduced.
536Proofsga1-i.9.4
Indeed, \widehat {B} is finite and étale over \widehat {A};
we can apply .
541Theoremsga1-i.9.5sga1-i.9.5.xmlI.9.5sga1-i.9
Let f \colon A \to B be a local homomorphism, with B isomorphic to the localisation of an A-algebra of finite type over A.
If f is étale, then A is normal if and only if B is normal.
If A is normal, then f is étale if and only if f is injective and unramified (and then B is normal, by (i)).
We will give two different proofs of (i):
the first using certain properties of quasi-finite flat morphisms (stated at the start of this section) and without using (and thus the Main Theorem);
the second proof does the opposite.
For (ii), it seems like we do indeed need the Main Theorem, no matter what.
546Proofsga1-i.9.5.i-proof-1sga1-i.9.5.i-proof-1.xmlFirst proofI.9.5.isga1-i.9
We use the following necessary and sufficient condition for a local Noetherian ring A of dimension \neq0 to be normal.
545sga1-i.9-serres-criterionsga1-i.9-serres-criterion.xmlSerre's criterionsga1-i.9.5.i-proof-1
For every rank-1 prime ideal \mathfrak {p} of A, A_ \mathfrak {p} is normal (or, equivalently, regular);
For every rank-\geq2 prime ideal \mathfrak {p} of A, the depth of A_ \mathfrak {p} is \geq2.
(cf. EGA IV 5.8.6.)
We assume this criterion here, but it should also appear in the section on flatness.
Its main advantage is that it does not suppose a priori that A is reduced, nor a fortiori that A is integral.
Here, we can already suppose that \dim A= \dim B \neq0.
By the statements at the start of this section, the rank-1 (resp. rank-\geq2) prime ideals \mathfrak {p} of A are exactly the intersections of A with the rank-1 (resp. rank-\geq2) prime ideals \mathfrak {q} of B.
Finally, if \mathfrak {p} and \mathfrak {q} correspond to one another, then B_ \mathfrak {q} is étale over A_ \mathfrak {p}, and thus of the same depth as A_ \mathfrak {p}, and is regular if and only if A_ \mathfrak {p} is (by ).
Applying Serre's criterion, we see that A is normal if and only if B is.
547Proofsga1-i.9.5.i-proof-2sga1-i.9.5.i-proof-2.xmlSecond proofI.9.5.isga1-i.9
Suppose that B is normal, with field of fractions L;
let K be the field of fractions of A (and note that A is integral, since B is integral).
We have already seen, in the proof of , that B \otimes _A K is a finite product of fields;
since it is contained in L, it is a field;
since it contains B, it is equal to L itself.
An element of K that is integral over A is integral over B, and is thus in B, since B is normal, and thus also in A, since B \cap K=A (as follows from the fact that B is faithfully flat over A).
Now suppose that A is normal;
we will prove that B is also normal.
By , we have that B=B'_ \mathfrak {n}, where B'=A[t]/FA[t] (with F and \mathfrak {n} as in ).
Thus L=B \otimes _A K is a localisation of B' \otimes _A K=K[t]/FK[t], and also a product of fields (finite separable extensions of K).
This latter product (B' \otimes _A K) is a direct factor of B'_K (since each time we localise an Artinian ring (here B'_K) with respect to a multiplicatively stable set), and thus corresponds to a decomposition F=F_1F_2 in K[t], with the generator of L corresponding to t being annihilated by F_1.
But, since A is normal, the F_i are in A[t] (supposing that they are monic).
Note that B \to L=B \otimes _A K is injective (since A \to K is, since B is flat over A), and so F_1(u)=0, with u being the class of t in L.
Suppose that F were of minimal degree;
then it would follows that F_2=1.
(N.B. we would have F'(u)=F'_1(u)F_2(u)+F_1(u)F'_2(u)=F'_1(u)F_2(u), since F_1(u)=0, whence F'_1(u) \neq 0 since F'(u) \neq0.)
Thus
L = B \otimes _A K = K[t]/FK[t]
and so F is a separable polynomial in K[T] (but evidently not necessarily in A[t]).
(N.B. for now, we have only shown, essentially, that we can choose F and \mathfrak {n} in such that, with the above notation, B' \to B'_ \mathfrak {n}=B is injective;
for this, we have used the fact that A is normal;
I do not know if this remains true without this normality hypothesis).
Now recall the well-known lemma, taken from Serre's lectures last year:
548Lemmasga1-i.9.6sga1-i.9.6.xmlI.9.6sga1-i.9
Let K be a ring, F \in K[t] a separable monic polynomial, L=K[t]/FK[t], and u the class of t in L (so that F'(u) is an invertible element of L).
Then
\operatorname {tr} _{L/K} u^i/F'(u) = \begin {cases} 0 & \text {if }0 \leq i<n-1 \text {;} \\ 1 & \text {if }i=n-1 \end {cases}
where n= \deg F.
549Corollarysga1-i.9.7sga1-i.9.7.xmlI.9.7sga1-i.9
The determinant of the matrix (u^j \cdot u^i/F'(u))_{0 \leq i,j \leq n-1} is equal to (-1)^{n(n-1)/2}, and thus invertible in every sub-ring A of K.
550Corollarysga1-i.9.8sga1-i.9.8.xmlI.9.8sga1-i.9
Let A be a sub-ring of K, V the A-module generated by the u^i (for 0 \leq i \leq n-1), and V' the sub-A-module of L consisting of the x \in L such that \operatorname {tr} _{L/K}(xy) \in A for all y \in V (i.e. for y of the form u^i, for 0 \leq i \leq n-1).
Then V' is the A-module given by the basis u^i/F'(u) (for 0 \leq i \leq n-1).
551Corollarysga1-i.9.9sga1-i.9.9.xmlI.9.9sga1-i.9
Suppose that K is the field of fractions of an integral normal ring A, with the coefficients of F lying in A.
Then, with the notation of , V' contains the normal closure A' of A in L, which is thus contained in A[u]/F'(u), and a fortiori in A[u][F'(u)^{-1}].
We can apply the above corollary to the situation that we have obtained in the proof: since F'(u) is invertible in B, and since B contains A[u], B contains A'.
By the Main Theorem (or by the fact that B=A[u]_ \mathfrak {n}), B is a localisation of A'.
Since A' is normal, so too is B.
552Proofsga1-i.9.5.ii-proofsga1-i.9.5.ii-proof.xmlI.9.5.iisga1-i.9
We proceed as in the above proof to show that we can choose F in such that we again have
L = B \otimes _A K = K[t]/FK[t]
The only obstacle a priori is that we can no longer prove that B \to L is injective, since B is no longer assumed to be flat over A, and so we can only apply the same argument a priori to the image B_1 of B under the aforementioned homomorphism.
It immediately follows that B_1 is flat over A (since it is the localisation of a free A-algebra).
By , the morphism B \to B_1 is étale, and thus an isomorphism, which finishes the proof.
(From an editorial point of view, we should perform the two proofs above, and place the formal calculations of the lemma and of its corollaries in a separate section).
553Corollarysga1-i.9.10sga1-i.9.10.xmlI.9.10sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is normal, then X is normal;
the converse is true if f is surjective.
555Corollarysga1-i.9.11sga1-i.9.11.xmlI.9.11sga1-i.9
Let f \colon X \to Y be a dominant morphism, with Y normal and X connected.
If f is unramified, then it is also étale, and X is then normal and thus irreducible (since it is connected).
554Proofsga1-i.9.11
Let U be the set of points where f is étale.
Since U is open, it suffices to show that it is also closed and non-empty.
Since U contains the inverse image of the generic point of Y (recall that, for an algebra over a field, unramified = étale), it is non-empty (since X dominates Y).
If x belongs to the closure of U, then it belongs to the closure of an irreducible component U_i of U, and thus to an irreducible component X_i= \bar {U_i} of X which intersects U and which thus dominates Y (since every component of U, being flat over Y, dominates Y).
Then, if y is the projection of x over Y, \mathcal { O } _y \to \mathcal { O } _x is injective (taking into account the fact that \mathcal { O } _y is integral).
Since \mathcal { O } _y is normal and \mathcal { O } _y \to \mathcal { O } _x is unramified, we conclude with the help of (ii) from .
556Corollarysga1-i.9.12sga1-i.9.12.xmlI.9.12sga1-i.9
Let f \colon X \to Y be a dominant morphism of finite type, with Y normal and X irreducible.
Then the set of points where f is étale is identical to the complement of the support of \Omega _{X/Y}^1, i.e. to the complement of the sub-prescheme of X defined by the different ideal \mathfrak {d}_{X/Y}.
( is the "less trivial" statement which was alluded to in the remark in .)
557Remarksga1-i.9
We do not claim that a connected étale cover of an irreducible scheme is itself irreducible if we do not assume the base to be normal;
this question will be studied in .
559Sectionsga1-i.10sga1-i.10.xmlÉtale covers of a normal schemeI.10sga1-i250Propositionsga1-i.10.1sga1-i.10.1.xmlI.10.1sga1-i.10
Let Y be normal and connected of field K, and let X be a separated étale prescheme over Y.
Then the connected components X_i of X are integral, their fields K_i are finite separable extensions of K, and X_i can be identified with a non-empty open subset of the normalisation of X in K_i (and thus X with a dense open subset of the normalisation of Y in R(X)=L= \prod K_i, where R(X) is the ring of rational functions on X).
249Proofsga1-i.10.1
By , X is normal, and a fortiori its local rings are integral, and so the connected components of X are irreducible.
Since X_i is normal, and also finite and dominant over Y, it follows from a particular (almost trivial, actually) case of the Main Theorem that X_i is an open subset of the normalisation of X in the field K_i of X_i.
251Corollarysga1-i.10.2sga1-i.10.2.xmlI.10.2sga1-i.10
Under the conditions of , X is finite over Y (i.e. an étale cover of Y) if and only if X is isomorphic to the normalisation Y' of Y in L=R(X) (the ring of rational functions on X).
246Proofsga1-i.10.2
We know that this normalisation is finite over Y (since Y is normal, and R/K separable);
conversely, if X is finite over Y, then it is also finite over Y', and so its image in Y' is closed (and it is also dense).
An algebra L of finite rank over K is said to be unramified over X (or simply unramified over K if X is evident) if L is a separable algebra over K (i.e. a direct sum of separable extensions K_i) and the normalisation Y' of Y in L (i.e. the disjoint sum of the normalisations of Y in the K_i) is unramified (i.e. étale, by ) over Y.
Thus:
253Corollarysga1-i.10.3sga1-i.10.3.xmlI.10.3sga1-i.10
For every X that is finite over Y and such that every irreducible component of X dominates Y, let R(X) be the ring of rational functions on X (given by the product of the local rings of the generic points of the irreducible components of X), so that X \mapsto R(X) is a functor, with values in algebras of finite rank over K=R(Y).
Then this functor establishes an equivalence between the category of connected étale covers of Y and the category of extensions L of K that are unramified over Y.
252Proofsga1-i.10.3
The inverse functor is the normalisation functor.
Suppose that Y is affine, and thus defined by a normal ring A with field of fractions K.
Let L be a finite extension of K given by a direct sum of fields.
Then, by definition, the normalisation Y' of Y in L is isomorphic to \operatorname {Spec} (A'), where A' is the normalisation of A in L.
To say that L is unramified over Y implies that A' is unramified (or even étale) over A.
If A is local, then it is equivalent to say that the local rings A'_ \mathfrak {n} (where \mathfrak {n} runs over the finite set of maximal ideals of A', i.e. the prime ideals of A' that induce the maximal ideal \mathfrak {m} of A) are unramified (i.e. étale) over the local ring A.
Finally, note that the discriminant criterion of can also be applied to this situation
(more generally, a variant of the aforementioned criterion can be stated thusly, without any preliminary flatness condition when X dominates Y, but with Y still assumed to be locally integral: A \to B and B \to B \otimes _A K are injective — then \operatorname {tr} _{L/K} is defined — and \operatorname {tr} _{L/K}(xy) induces a fundamental bilinear form B \times B \to A, i.e. there exists x_i \in B (for 1 \leq i \leq n= \operatorname {rank} _K L) such that \operatorname {tr} (x_ix_j) \in A for all i,j, and \det ( \operatorname {tr} (x_i x_j))_{1 \leq i,j \leq n} is invertible in A).
The syllogism immediately implies the syllogism of being unramified in the classical case:
258Propositionsga1-i.10.4sga1-i.10.4.xmlI.10.4sga1-i.10
Let Y be a normal integral prescheme, of field K.
Then
K is unramified over Y.
If L is an extension of K that is unramified over Y, and if Y' is a normal prescheme, of field L, that dominates Y (e.g. the normalisation of Y in L), and M an extension of L that is unramified over Y', then M/K is unramified over X (this is the transitivity property).
Let Y' be a normal integral prescheme that dominates Y, of field K'/K;
if L is an extension of K that is unramified over Y, then L \otimes _K K' is an extension of K' that is unramified over Y' (this is the translation property).
Furthermore:
259Corollarysga1-i.10.5sga1-i.10.5.xmlI.10.5sga1-i.10
Under the conditions of (iii) in , if Y= \operatorname {Spec} (A) and Y'= \operatorname {Spec} (A'), then the normalisation \bar {A'} of A' in L'=L \otimes _K K' can be identified with \bar {A} \otimes _A A', where \bar {A} is the normalisation of A in L.
Usually, people (those who are disgusted by the consideration of non-integral rings, even if they are direct sums of fields) state the translation property in the following (weaker) form:
260Corollarysga1-i.10.6sga1-i.10.6.xmlI.10.6sga1-i.10
Under the conditions of (iii) in , let L_1 be a sum extension of L/K (unramified over Y) and of K'/K.
Then L_1/K' is unramified over Y'.
In the case where Y= \operatorname {Spec} (A) and Y'= \operatorname {Spec} (A'), we further have that
\bar {A'} = A[ \bar {A},A']
i.e. the normalisation ring \bar {A'} of A' in L_1 is the A-algebra generated by A' and by the normalisation \bar {A} of A in L.
This latter fact is actually false without the unramified hypothesis, even in the case of extensions given by direct sums of number fields...
To finish this section, we are going to give the intuitive interpretation of the notion of étale covers: there should be the "maximal number" of points over the point y \in Y in question, and, in particular, there should not be "multiple points combined" over y.
To prove results in this sense, in all desirable generality, we will assume here found below (whose proof will be given in the multiplodoque, Chapter IV, Section 15, and uses Chevalley's technique of constructible sets, and a little bit of the theory of descent...).
A morphism of finite type f \colon X \to Y is said to be universally open if, for every base extension Y' \to Y (with Y' locally Noetherian), the morphism f' \colon X'=X \times _Y Y' \to Y' is open, i.e. sends open subsets to open subsets.
We can actually restrict to the case where Y' is of finite type over y (and even to the case where Y' is of the form Y[t_1, \ldots ,t_r], where the t_i are indeterminates).
A universally open morphism is a fortiori open (but the converse is false);
on the other hand, if f is open, and if X and Y are irreducible, then all of the components of all of the fibres of f are of the same dimension (i.e. the dimension of the generic fibre f^{-1}(z), where z is the generic point of Y).
Finally, if Y is normal, then this latter condition already implies that f is universally open (Chevalley's theorem).
It thus follows, for example, that, if f \colon X \to Y is a quasi-finite morphism, with Y normal and irreducible, then f is universally open (or even open) if and only if every irreducible component of X dominates Y.
Recall also that a flat morphism (of finite type) is open, and thus also universally open.
With these preliminaries, "recall" the following:
261Propositionsga1-i.10.7sga1-i.10.7.xmlI.10.7sga1-i.10
Let f \colon X \to Y be a quasi-finite, separated, universally open morphism.
For all y \in Y, let n(y) be the "geometric number of points in the fibre f^{-1}(y)", equal to the sum of the separable degrees of the residue extensions k(x)/k(y) as x runs over the points of f^{-1}(y).
Then the function y \mapsto n(y) on Y is upper semi-continuous.
For it to be constant on a neighbourhood of the point y (i.e. for it to be the case that n(y)=n(z_i), where the z_i are the generic points of the irreducible components of Y that contain y), it is necessary and sufficient for there to exist a neighbourhood U of y such that X|U is finite over U.
(cf. EGA IV 15.5.1.)262Corollarysga1-i.10.8sga1-i.10.8.xmlI.10.8sga1-i.10
If y \mapsto n(y) is constant, and if Y is geometrically unibranch(For the definition, cf. ), then the irreducible components of X are disjoint.
263Propositionsga1-i.10.9sga1-i.10.9.xmlI.10.9sga1-i.10
Let f \colon X \to Y be a separated étale morphism.
With the notation of , the function n \mapsto n(y) is upper semi-continuous.
For it to be constant on a neighbourhood of the point y (i.e. for it to be the case that n(y)=n(z_i), where the z_i are the generic points of the irreducible components of Y that contain y), it is necessary and sufficient that there exist a neighbourhood U of y such that X|U is finite over U, i.e. such that X|U is an étale cover of U.
264Corollarysga1-i.10.10sga1-i.10.10.xmlI.10.10sga1-i.10
For a separated étale morphism f \colon X \to Y (with Y connected) to be finite (i.e. for f to make X an étale cover of Y), it is necessary and sufficient that all the fibres of f have the same geometric number of points.
In and its corollary (), there was no normality hypothesis on Y;
if we make such a hypothesis, then we find the following stronger statement (which is usually taken as the definition of unramified for a cover):
266Theoremsga1-i.10.11sga1-i.10.11.xmlI.10.11sga1-i.10
Let f \colon X \to Y be a separated quasi-finite morphism.
Suppose that Y is irreducible, that every component of X dominates Y, and that X is reduced (i.e. that \mathcal { O } _X has no nilpotent elements).
Let n be the degree of X over Y (i.e. the sum of the degrees, over the field K of Y, of the fields K_i of the irreducible components X_i of X).
Let y be a normal point of Y.
Then the geometric number n(y) of points of X over y is \leq n, with equality if and only if there exists an open neighbourhood U of y such that X|U is an étale cover of U.
265Proofsga1-i.10.11
The "only if" is trivial;
we will prove the "if".
Let z be the generic point of Y.
Then n(z), which is equal to the sum of the separable degrees of the K_i/K, is \leq n, and, by , we have that n(y) \leq n(z);
thus n(y) \leq n, with equality implying that X|U if finite over U, for some suitable neighbourhood U of y.
We can thus suppose that X is finite over Y, and that the function n(y') on Y is constant.
Then, by , X is the disjoint union of its irreducible components, and so, to prove that it is unramified at y, we can restrict to the case where X is irreducible, thus integral.
Finally, we can assume that Y= \operatorname {Spec} ( \mathcal { O } _y).
The theorem thus reduces to the following classical statement:
268Corollarysga1-i.10.12sga1-i.10.12.xmlI.10.12sga1-i.10
Let A be a normal local ring (Noetherian, as always), of field K;
let L be a finite extension of K of degree n, and of separable degree n_s;
let B be a sub-ring of L that is finite over A, with field of fraction L;
let \mathfrak {m} be the maximal ideal of A, and n' the separable degree of B/ \mathfrak {m}B over A/ \mathfrak {m}A=k (which is equal to the sum of the separable deprees of the residue extensions of this ring).
Then n' \leq n_s, and a fortiori n' \leq n.
This latter inequality is an equality if and only if B is unramified (i.e. étale) over A.
267Proofsga1-i.10.12
It remains only to show that, if n'=n, then B is étale over A.
Recall the proof in the case where k is infinite:
we need only show that R=B/ \mathfrak {m}B is separable over k;
if this were not the case, then it would follow (by a known lemma) that there exists an element a of R whose minimal polynomial over k is of degree >n'.
This element would come from an element x of B, whose minimal polynomial over K (as an element of L) is of degree \leq n;
but this minimal polynomial has coefficients in A, since A is normal, and thus gives, by restriction modulo \mathfrak {m}, a monic polynomial F \in k[t] of degree \leq n=n', such that F(a)=0.
But this is a contradiction.
In the general case (where k can be finite), we can again use geometric language:
we consider Y'= \operatorname {Spec} (A[t]), which is faithfully flat over Y, and the generic point y' of the fibre \operatorname {Spec} (k[t]) of Y' over y.
Then X is unramified over Y at y if and only if X'=X \times _Y Y'= \operatorname {Spec} (B[t]) is unramified over Y' at y', as we immediately see.
On the other hand, by the choice of y', its residue field is k(t), and thus infinite.
Since y' is a normal point of Y', we are now in the previous case.
560Sectionsga1-i.11sga1-i.11.xmlVarious addendaI.11sga1-i
We have already said that a connected étale cover of an integral scheme is not necessarily integral.
Here are two examples of this fact.
Let C be an algebraic curve with an ordinary double point x, and let C' be its normalisation, with a and b the two points of C' over x.
Let C'_1 and C'_2 be copies of C', with a_i (resp. b_i) the point of C'_i corresponding to a (resp. b).
In the curve C'_1 \coprod C'_2, we identify a_1 with b_2, and a_2 with b_1 (we leave making this process of identification precise to the reader; it will be explained in Chapter VI of the multiplodoque, but, in the case of curves over an algebraically closed field, is already covered in Serre's book on algebraic curves).
We obtain a curve C'' that is connected and reducible, and is a degree-2 étale cover of C.
The reader can verify that, generally, the "Galois" connected étale covers C'' of C whose inverse images C'' \times _C C' are trivial covers of C' (i.e. isomorphic to the sum of a certain number of copies of C') are "cyclic" of degree n, and, conversely, for every integer n>0, we can construct a cyclic connected étale cover of degree n.
In the language of the fundamental group (which will be developed later), this implies that the quotient of \pi _1(C) by the closed invariant subgroup generated by the image of \pi _1(C') \to \pi _1(C) (the homomorphism induced by the projection) is isomorphic to the compactification of \mathbb {Z}.
More precisely, we should show that the fundamental group of C is isomorphic to the (topological) free product of the fundamental group of C with the compactification of \mathbb {Z}.
We note that is was questions of this sort that gave birth to the "theory of descent" for schemes.
Let A be a complete integral local ring;
we know that its normalisation A' is finite over A (by Nagata), and is thus a complete semi-local ring, and thus local, since it is integral.
Suppose that the residue extension L/k that it defines is non-radicial (in the contrary case, we say that A is geometrically unibranch; cf. below).
This will be the case, for example, for the ring \mathbb {R}[{[s,t]}]/(s^2+t^2) \mathbb {R}[{[s,t]}], where \mathbb {R} is the field of real numbers.
Then let k' be a finite Galois extension of k such that L \otimes _k k' decomposes;
let B be a finite and étale algebra over A corresponding to the residue extension k' (recall that B is essentially unique).
Then the residue algebra of B'=A' \otimes _A B over B is L \otimes _k k', which is not local, and so B' is not a local ring, and thus B has zero divisors (since it is complete).
Now B' is contained in the total ring of fractions of B (since it is free over A', thus torsion free over A', thus torsion free over A, thus contained in B' \otimes _A K=B'_{(K)}=A'_{(K)} \otimes _K B_{(K)}=B_{(K)}, since A'_{(k)}=K), and so B is not integral.
In the case of the ring \mathbb {R}[s,t]/(s^2+t^2) \mathbb {R}[s,t], taking k'/k= \mathbb {C}/ \mathbb {R}, we see that B is the local ring of two secant lines at their point of intersection.
We note also that, if there exists a connected étale cover X of Y that is integral but not irreducible, then every irreducible component of X gives an example of an unramified cover X' of Y that dominates Y but is not étale over Y.
In the case of example (a), we thus see that C' is unramified over C, without being étale at the two points a and b (note that, directly, by inspection of the completions of the local rings at x and a, from the "formal" point of view, C' at the point a can be identified with a closed subscheme of C at the point x, i.e. one of the two "branches" of C passing through x).
In both (a) and (b), we see that the fact that the conclusions of (i) and (ii) in fail to hold is directly linked with the fact that a point of Y "blows up" at distinct points of the normalisation (in (b), the fact that the residue extension is non-radicial should be interpreted geometrically in this way).
More precisely, we say that an integral local ring A is geometrically unibranch if its normalisation has only a single maximal ideal, with the corresponding residue extension being radicial;
a point y of an integral prescheme is said to be geometrically unibranch if its local ring is geometrically unibranch.
Examples: a normal point, an ordinary cusp point of a curve, etc.
It seems that, if Y admits a point which is not unibranch, then there always exists a non-irreducible connected étale cover of Y;
at least, this is what we have shown in case (b), when Y is the spectrum of a complete local ring.
We can show, however, that if all the points of Y are geometrically unibranch, then every unramified connected Y-prescheme that dominates Y is étale and irreducible.
The proof follows that of , using the following generalisation of , which will be proved later by means of the technique of descent:(cf. [sga1-ix.4.10 (?)]. For a more direct demonstration, cf. EGA IV 18.10.3, using a variant of for geometrically unibranch local rings.)
Let Y' \to Y be a finite, radicial, and surjective morphism (i.e. what we could call a "universal homeomorphism").
Consider the functor X \mapsto X \times _Y Y'=X' from Y-preschemes to Y'-preschemes.
This functor induces an equivalence between the category of étale Y-schemes and the category of étale Y'-schemes.
We could apply this, for example, in the case where Y' is the normalisation of Y, with Y assumed to be unibranch (and Y' finite over Y, which is true in all the cases that one encounters in practice), or to to case of some Y'' "sandwiched" between Y and its normalisation (which no longer need be finite over Y).
782Exposésga1-iisga1-ii.xmlSmooth morphisms: generalities, differential propertiesIIsga1729Sectionsga1-ii.1sga1-ii.1.xmlGeneralitiesII.1sga1-ii730Sectionsga1-ii.2sga1-ii.2.xmlSome smoothness criteria of a morphismII.2sga1-ii731Sectionsga1-ii.3sga1-ii.3.xmlPermanence propertiesII.3sga1-ii732Sectionsga1-ii.4sga1-ii.4.xmlDifferential properties of smooth morphismsII.4sga1-ii733Sectionsga1-ii.5sga1-ii.5.xmlCase of a base fieldII.5sga1-ii897sga6sga6.xmlSGA 6: Intersection theory and the Riemann–Roch theorem750sga6-introductionsga6-introduction.xmlIntroductionsga6
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751Exposésga6-0sga6-0.xmlOutline of a programme for a theory of intersections0sga6
The current exposé is of an introductory nature, and its reading is not logically necessary for the study of the seminar.
It is aimed particularly at readers familiar with the provisional version of the Riemann–Roch theorem, as given in the report by Borel–Serre [@2] or in the report by Grothendieck mentioned previously in the introduction (cited as [RRR]), which is reproduced as an appendix at the end of this current exposé.
687Sectionsga6-0.1sga6-0.1.xml1sga6-0
Recall the Riemann–Roch formula for a proper morphism
f \colon X \to Y
of smooth quasi-projective schemes over a field k and a coherent sheaf \mathcal { F } on X:
679Equationsga6-0.1-equation-1.1sga6-0.1-equation-1.1.xml1.1sga6-0.1 \operatorname {Todd} (T_Y) \operatorname {ch} _Y(f_*( \operatorname {cl} ( \mathcal { F } ))) = f_*( \operatorname {Todd} (T_X) \operatorname {ch} _X( \mathcal { F } )) \tag{1.1}
where \operatorname {cl} ( \mathcal { F } ) denotes the class of \mathcal { F } in the group K(X) of classes of coherent sheaves on X, and \operatorname {ch} _X and \operatorname {ch} _Y denote the Chern characters of on X and Y (resp.), and T_X and T_Y the tangent bundles to X and Y (resp.).
This formula holds in A(Y) \otimes _ \mathbb {Z} \mathbb {Q}, where A(Y) is the Chow ring of Y;
the f_* on the right-hand side is induced by tensoring with \mathbb {Q} the "direct image of cycles" homomorphism
f_* \colon A(X) \to A(Y)
and the f_* on the left-hand side is the Euler–Poincaré characteristic of \mathcal { F } with respect to f:
f_*( \operatorname {cl} ( \mathcal { F } )) = \sum _i (-1)^i \operatorname {cl} ( \operatorname {R} ^i f_*( \mathcal { F } )).
As we know, \operatorname {Todd} (-) and \operatorname {ch} (-) are universal polynomials in the Chern classes of the argument with coefficients in \mathbb {Q}.
Since the constant term of \operatorname {Todd} (-) is 1, it is an invertible element for any value of the argument, so that can be rewritten, after multiplication by \operatorname {Todd} (T_Y)^{-1}, in the form which is more useful for our needs:
680Equationsga6-0.1-equation-1.2sga6-0.1-equation-1.2.xml1.2sga6-0.1 \operatorname {ch} _Y(f_*( \operatorname {cl} ( \mathcal { F } ))) = f_*( \operatorname {Todd} (T_f) \operatorname {ch} _X( \mathcal { F } )) \tag{1.2}
where we set
681Equationsga6-0.1-equation-1.3sga6-0.1-equation-1.3.xml1.3sga6-0.1 T_f = T_X - f^*(T_Y) \in K(X) \tag{1.3}
so that T_f plays the role of a virtual relative tangent bundle of X over Y.
In the case where the morphism f is smooth (i.e. with everywhere-surjective tangent map), we have simply
T_f = T_{X/Y}
(the tangent bundle along the fibres) and so, in the case where f \colon X \to Y is an immersion, we find
T_f = - \check {N}_{X/Y}
where \check {N}_{X/Y} denotes the normal sheaf of X in Y.
One of the main goals of this Seminar is to generalise simultaneously in two directions:
Remove the hypothesis of the existence of a base field k.
Replace the regularity hypotheses on Y and X by a "local regularity" hypothesis on f.
Finally, along the way, we will equally deal with the problem:
Remove the quasi-projectivity hypotheses which, in the absence of a base field, are expressed by the existence of ample invertible modules on X and on Y.
689Sectionsga6-0.2sga6-0.2.xml2sga6-0
We now examine generalisation (a) from sga6-0.2, keeping, however, the hypotheses of regularity and of existence of ample invertible modules on X and on Y.
The definition of K(X) and K(Y), and of the homomorphism f_* \colon K(X) \to K(Y) then gives no new problems, thanks to the fact that X and Y are regular.
The most natural route to giving meaning to thus seems to consist of defining the Chow rings A(X) and A(Y) and a group homomorphism
f_* \colon A(X) \to A(Y)
as well as establishing a theory of Chern classes, providing maps
c_i \colon K(X) \to A(X)
(and similarly for Y), and finally giving a description of an virtual relative tangent bundle element
T_f \in K(X). 688Sectionsga6-0.2.1sga6-0.2.1.xml2.1sga6-0.2
For
TO-DO: finish752Exposésga6-0rrrsga6-0rrr.xmlClasses of sheaves and the Riemann–Roch theorem0RRRsga6
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712sga6-0rrr.isga6-0rrr.i.xml\lambda-rings (formal preliminaries)Isga6-0rrr
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759Exposésga6-isga6-i.xmlGeneralities on finiteness conditions in derived categoriesIsga6756Sectionsga6-i.0sga6-i.0.xmlIntroductionI.0sga6-i
The aim of Exposés I to IV is to develop, with generality suitable for this seminar, the formalism of finiteness conditions in the derived categories of ringed toposes.
As was mentioned in , it is the need to define "Grothendieck groups" that have good variance properties on arbitrary schemes that requires us to generalise and relax the notions of finiteness used up until now.
The classical notion of coherent sheaves on a ringed space (X, \mathcal { O } _X) becomes uninteresting as soon as \mathcal { O } _X is no longer coherent.
We could think of replacing the notion of coherence by the notion of finite presentation, but this presents the inconvenience that the kernel of an epimorphism of modules of finite presentation is not itself, in general, of finite presentation.
We arrive at a satisfying notion by remarking that, if \mathcal { O } _X is coherent, then a coherent sheaf \mathcal { F } is not only of finite presentation but also of finite n-presentation for all n \in \mathbb {N}, where "finite n-presentation" means that there exists, locally, an exact sequence
\mathcal { L } _n \to \mathcal { L } _{n-1} \to \ldots \to \mathcal { L } _0 \to \mathcal { F } \to 0
where the \mathcal { L } _i are free and of finite type.
If we no longer suppose the sheaf \mathcal { O } _X to be coherent, then we say that an \mathcal { O } _X-module \mathcal { F } is pseudo-coherent if it satisfies the above condition, i.e. if it is of finite n-presentation for all n \in \mathbb {N}.
Then the notion of pseudo-coherence possesses, with respect to short exact sequences, the same stability property as the notion of coherence: if two of the terms of a short exact sequence of \mathcal { O } _X-modules are pseudo-coherent, then so too is the third.
We can easily generalise the above notion to complexes of sheaves: a complex \mathcal { F } of \mathcal { O } _X-modules is said to be pseudo-coherent if it is n-pseudo-coherent for all n \in \mathbb {Z}, where "n-pseudo coherent" means that there exists, locally, a quasi-isomorphism \mathcal { L } \xrightarrow { \sim } \mathcal { F }, where \mathcal { L } is a bounded-above complex, whose components in degree \geq n are free sheaves of finite type.
For a complex \mathcal { F } concentrated in degree 0, to say that \mathcal { F } is pseudo-coherent as a complex is equivalent to saying that it is pseudo-coherent as a module.
The notion of a pseudo-coherent complex has excellent stability properties, described in and [sga6-i.2 (?)].
Firstly, it is stable under isomorphism in the derived category D(X) of the category of \mathcal { O } _X-modules;
even better, if two objects of a distinguished triangle in D(X) are pseudo-coherent, then so too is the third: in other words, the full subcategory D(X)_ \mathrm {coh} of D(X) consisting of pseudo-coherent complexes is a triangulated subcategory.
Furthermore, pseudo-coherence is preserved under inverse images and derived tensor products.
In the case where \mathcal { O } _X is coherent, to say that a complex \mathcal { F } is pseudo-coherent simply means that the cohomology of \mathcal { F } is locally bounded above and that all the \mathcal { H } ^i( \mathcal { F } ) are coherent.
If \mathcal { O } _X is a sheaf of regular local rings, then every coherent \mathcal { O } _X-module \mathcal { F } locally admits a finite left resolution by free modules of finite type, i.e. there exists, locally, an exact sequence
0 \to \mathcal { L } _n \to \mathcal { L } _{n-1} \to \ldots \to \mathcal { L } _0 \to \mathcal { F } \to 0
where the \mathcal { L } _i are free and of finite type.
It is thus natural, when \mathcal { O } _X is a sheaf of arbitrary local rings, to consider modules that enjoy the above property;
such modules are called perfect.
More generally, we say that a complex \mathcal { F } of \mathcal { O } _X-modules is perfect if there exists, locally, a quasi-isomorphism \mathcal { L } \xrightarrow { \sim } \mathcal { F }, where \mathcal { L } is a bounded complex of free modules of finite type.
In [sga6-i.3 (?)], we show that the full subcategory D(X)_ \mathrm {perf} of D(X) consisting of perfect complexes is, like D(X)_ \mathrm {coh}, a triangulated subcategory that is "stable" under inverse images and derived tensor products.
We clearly have that D(X)_ \mathrm {perf} \subseteq D(X)_ \mathrm {coh}, but the inclusion is strict in general.
Pseudo-coherence can also be defined "by passing to the limit" from perfectness: a complex \mathcal { F } is pseudo-coherent if and only if, for all n \in \mathbb {Z}, \mathcal { F } can be "locally approximated to order n by a perfect complex", by which we mean that there exists, locally, a distinguished triangle
\mathcal { L } \to \mathcal { F } \to \mathcal { R } \to \mathcal { L } [1]
where \mathcal { L } is perfect, and \mathcal { R } is acyclic in degree \geq n.
Conversely, perfectness can be recovered from pseudo-coherence by an additional regularity condition: a complex is perfect if and only if it is pseudo-coherent and locally of finite tor-dimension (cf. [sga6-i.5 (?)]).
We thus recover the fact that, if the local rings are regular, then every coherent sheaf is perfect, and, more generally, that every pseudo-coherent complex with locally bounded cohomology is perfect.
Pseudo-coherence and perfectness are the two fundamental notions of finiteness with which we will work in this seminar.
Sections I.1 to I.5 of this present exposé are dedicated to their definition, and to establishing their elementary stability properties.
For this, we only use two or three local properties of the category of free modules of finite type with respect to the category of all modules.
It also turns out to be practical — and useful, most notably in [sga6-ii (?)] — to axiomatise the situation, by introducing a notion of pseudo-coherence (resp. perfectness) in a fibred category over a site with respect to a fibred subcategory that respects suitable conditions.
Sections I.6 to I.8 of this present exposé generalise a certain number of well known notions for locally free sheaves of finite type (rank, duality, and trace of an endomorphism) to the setting of perfect complexes.
In [sga6-ii (?)], we examine the problem of "the existence of global resolutions": under what conditions, on a ringed topos (X, \mathcal { O } _X), is a pseudo-coherent (resp. perfect) complex globally isomorphic in D(X) to a complex of locally free modules of finite type?
The importance of this questions rests essentially on the fact that we do not know, as of yet, how to generalise certain usual constructions on locally free sheaves (exterior and symmetric powers, and Chern classes) to perfect complexes (for more details on this, see [sga6-xiv (?)]).
This is why it is convenient to have tractable sufficient conditions for the existence of global resolutions, which allows us to reduce certain questions about pseudo-coherent (resp. perfect) complexes to analogous questions about locally free sheaves of finite type.
Such criteria are given in [sga6-ii (?)].
In particular, we show that there exist global resolutions in the following cases:
X is the Zariski topos of an affine scheme, or, more generally, of a quasi-compact scheme that has an ample invertible sheaf, or an ample family of invertible sheaves in the sense of Kleiman (for example, a regular scheme);
or X is the topos of sheaves of sets on a compact topological space, and \mathcal { O } _X is the sheaf of continuous complex-valued functions.
As an illustration of these methods — and to give evidence for the flexibility of the notions we have introduced — we give, in the appendix, a "purely sheaf-theoretic" definition of the index of a family of elliptic operators.
[sga6-iii (?)] studies the stability of finiteness conditions under the derived direct image.
To obtain reasonable statements, we need to put the notions of pseudo-coherence and perfectness "into perspective".
We place ourselves here in the setting of ordinary schemes, which suffices for the seminar, but there is no doubt that we must sooner or later develop an analogous theory for relative schemes or analytic spaces.
Let p \colon X \to S be an S-scheme locally of finite type, and let \mathcal { F } be a complex of \mathcal { O } _X-modules;
we say that \mathcal { F } is pseudo-coherent (resp. perfect) with respect to p if we can locally embed (by a closed immersion) X into a smooth S-scheme X' in such a way that the extension of \mathcal { F } by zero on X' is pseudo-coherent (resp. perfect).
The notion of pseudo-coherence with respect to p is especially interesting in the case where S is not locally Noetherian, since in the contrary case it agrees with the ordinary notion of pseudo-coherence.
On the other hand, as we might expect, to say that \mathcal { F } is perfect with respect to p is equivalent to saying that \mathcal { F } is pseudo-coherent with respect to p and locally of finite tor-dimension with respect to p (i.e. with respect to the sheaf of rings p^{-1}( \mathcal { O } _S)).
The central theorem of [sga6-iii (?)] is the finiteness theorem, which affirms (in a slightly more precise way) that, if f \colon X \to Y is a proper morphism of S-schemes locally of finite type, then the functor \mathbb {R}f_* sends complexes that are pseudo-coherent with respect to S to complexes that are pseudo-coherent with respect to S;
this theorem is, in reality, a conjecture, but has nevertheless been proven in the two particular following cases:
S is locally Noetherian;
f is projective.
Unfortunately, it seems that the extension to the general case is of the same order of difficulty as the analogous theorem in analytic geometry (Grauert's "theorem").
Combining the finiteness theorem with an essentially trivial formula called the projection formula, we obtain tractable criteria for the stability of relative perfectness under direct images.
We recover, as a corollary, "Grauert's continuity and semi-continuity theorems" (EGA III 7.6).
[sga6-iv (?)] translates the results of Exposés I to III into the language of "Grothendieck groups".
On a ringed space (X, \mathcal { O } _X), we denote by K^ \bullet (X) (resp. K_ \bullet (X)) the "Grothendieck group" of the category of perfect complexes of finite tor-dimension (resp. of pseudo-coherent complexes of bounded cohomology).
The group K^ \bullet (X) is a ring, and K_ \bullet (X) is a module over K^ \bullet (X).
As suggested by the superscript bullet, K^ \bullet is a contravariant functor, whilst K_ \bullet is a covariant functor for proper and pseudo-coherent morphisms (of schemes or of analytic spaces, and under the caveat that we have the finiteness theorem...).
There are also unusual variances (covariance of K^ \bullet and contravariance of K_ \bullet) for morphisms satisfying suitable regularity hypotheses.
Finally, the globalisation criteria of [sga6-ii (?)] allow us to make the link with the "Grothendieck groups" defined naively from coherent sheaves or locally free sheaves of finite type.
758Sectionsga6-i.1sga6-i.1.xmlPreliminary definitionsI.1sga6-i757Sectionsga6-i.1.1sga6-i.1.1.xmlFibred categories with additive (resp. abelian, resp. triangulated) fibresI.1.1sga6-i.1716sga6-i.1.1.1sga6-i.1.1.1.xmlI.1.1.1sga6-i.1.1
Let \mathcal {S} be a category, and let \mathcal {C} be a fibred \mathcal {S}-category ([SGA 1 VI 6.1]) with additive (resp. triangulated) fibres.
We say that \mathcal {C} is an additive (resp. triangulated) \mathcal {S}-category (or that \mathcal {C} is additive (resp. triangulated) over \mathcal {S}) if, for every arrow f \colon X \to Y of \mathcal {S}, the inverse image functor (determined up to unique isomorphism) f^* \colon \mathcal {C}_Y \to \mathcal {C}_X is additive (resp. exact).
We say that \mathcal {C} is an abelian \mathcal {S}-category if \mathcal {C} is an additive \mathcal {S}-category with abelian fibres, and that \mathcal {C} is a flat abelian \mathcal {S}-category if, further, the inverse image functors are exact.
717sga6-i.1.1.2sga6-i.1.1.2.xmlI.1.1.2sga6-i.1.1
...
898Exposésga1-iisga1-ii.xmlSGA 1: Étale covers and the fundamental group › Smooth morphisms: generalities, differential propertiesIIsga1729Sectionsga1-ii.1sga1-ii.1.xmlGeneralitiesII.1sga1-ii730Sectionsga1-ii.2sga1-ii.2.xmlSome smoothness criteria of a morphismII.2sga1-ii731Sectionsga1-ii.3sga1-ii.3.xmlPermanence propertiesII.3sga1-ii732Sectionsga1-ii.4sga1-ii.4.xmlDifferential properties of smooth morphismsII.4sga1-ii733Sectionsga1-ii.5sga1-ii.5.xmlCase of a base fieldII.5sga1-ii899Exposésga1-isga1-i.xmlSGA 1: Étale covers and the fundamental group › Étale morphismsIsga1
To simplify the exposition, we assume that all preschemes in the following are locally Noetherian (at least, starting from ).
515Sectionsga1-i.1sga1-i.1.xmlBasics of differential calculusI.1sga1-i
Let X be a prescheme over Y, and \Delta _{X/Y} the diagonal morphism X \to X \times _Y X.
This is an immersion, and thus a closed immersion of X into an open subset V of X \times _Y X.
Let \mathcal { I } _X be the ideal of the closed sub-prescheme corresponding to the diagonal in V (N.B. if one really wishes to do things intrinsically, without assuming that X is separated over Y — a misleading hypothesis — then one should consider the set-theoretic inverse image of \mathcal { O } _{X \times X} in X and denote by \mathcal { I } _X the augmentation ideal in the above ...).
The sheaf \mathcal { I } _X/ \mathcal { I } _X^2 can be thought of as a quasi-coherent sheaf on X, which we denote by \Omega _{X/Y}^1.
This sheaf is of finite type if X \to Y is of finite type, and it behaves well with respect to a base change Y' \to Y.
We also introduce the sheaves \mathcal { O } _{X \times _Y X}/ \mathcal { I } _X^{n+1}= \mathcal { P } ^n_{X/Y}, which are sheaves of rings on X, giving us preschemes denoted by \Delta _{X/Y}^n and called the n-th infinitesimal neighbourhood of X/Y.
The polysyllogism is entirely trivial, even if rather long (cf. EGA IV 16.3);
it seems wise to not discuss it until we use it for something helpful, with smooth morphisms.
516Sectionsga1-i.2sga1-i.2.xmlQuasi-finite morphismsI.2sga1-i316Propositionsga1-i.2.1sga1-i.2.1.xmlI.2.1sga1-i.2
Let A \to B be a local homomorphism (N.B. all rings are now Noetherian), and let \mathfrak {m} be the maximal ideal of A.
Then the following conditions are equivalent:
B/ \mathfrak {m}B is of finite dimension over k=A/ \mathfrak {m}.
\mathfrak {m}B is an ideal of definition, and B/ \mathfrak {r}(B)=k(B) is an extension of k=k(A).
The completion \widehat {B} of B is finite over the completion \widehat {A} of A.
If the conditions of are satisfied, then we say that B is quasi-finite over A.
A morphism f \colon X \to Y is said to be quasi-finite at x (or the Y-prescheme f is said to be quasi-finite at x) if \mathcal { O } _x is quasi-finite over \mathcal { O } _{f(x)}.
This is equivalent to saying that x is isolated in its fibre f^{-1}(x).
A morphism is said to be quasi-finite if it is quasi-finite at each point.
(In [EGA II 6.2.3] we further suppose that f is of finite type.)
317Corollarysga1-i.2.2sga1-i.2.2.xmlI.2.2sga1-i.2
If A is complete, then quasi-finiteness is equivalent to finiteness.
We could also give the usual polysyllogism (i), (ii), (iii), (iv), (v) for quasi-finite morphisms, but that doesn't seem necessary here.
517Sectionsga1-i.3sga1-i.3.xmlUnramified morphismsI.3sga1-i334Propositionsga1-i.3.1sga1-i.3.1.xmlI.3.1sga1-i.3
Let f \colon X \to Y be a morphism of finite type, x \in X, and y=f(x).
Then the following conditions are equivalent:
\mathcal { O } _x/ \mathfrak {m}_y \mathcal { O } _x is a finite separable extension of k(y).
\Omega _{X/Y}^1 is zero at x.
The diagonal morphism \Delta _{X/Y} is an open immersion on a neighbourhood of x.
333Proofsga1-i.3.1
For the implication (i) \implies (ii), we can use Nakayama to reduce to the case where Y= \operatorname {Spec} (k) and X= \operatorname {Spec} (k'), where it is well known, and also trivial by the definition of separable;
(ii) \implies (iii) comes from a nice and easy characterisation of open immersions, using Krull;
(iii) \implies (i) follows as well from reducing to the case where Y= \operatorname {Spec} (k) and the diagonal morphism is everywhere an open immersion.
We must then prove that X is finite with separable ring over k, and this leads us to consider the case where k is algebraically closed.
But then every closed point of X is isolated (since it is identical to the inverse image of the diagonal by the morphism X \to X \times _k X defined by x), whence X is finite.
We can thus suppose that X consists of a single point, with ring A, and so A \otimes _k A \to A is an isomorphism, hence A=k.
335Definitionsga1-i.3.2sga1-i.3.2.xmlI.3.2sga1-i.3
Let f satisfy one of the equivalent conditions of .
Then we say that f is unramified at x, or that X is unramified at x on Y.
Let A \to B be a local homomorphism.
We say that it is unramified, or that B is a local unramified algebra on A, if B/ \mathfrak {r}(A)B is a finite separable extension of A/ \mathfrak {r}(A), i.e. if \mathfrak {r}(A)B= \mathfrak {r}(B) and k(B) is a separable extension of k(A).
(cf. regrets in [III 1.2])
336Remarkssga1-i.3
The fact that B is unramified over A can be seen at the level of the completions of A and B.
Unramified implies quasi-finite.
337Corollarysga1-i.3.3sga1-i.3.3.xmlI.3.3sga1-i.3
The set of points where f is unramified is open.
338Corollarysga1-i.3.4sga1-i.3.4.xmlI.3.4sga1-i.3
Let X' and X be preschemes of finite type over Y, and g \colon X' \to X a Y-morphism.
If X is unramified over Y, then the graph morphism \Gamma _g \colon X' \to X \times _Y X is an open immersion.
Indeed, this is the inverse image of the diagonal morphism X \to X \times _Y X by
g \times _Y \mathrm {id} _{X'} \colon X' \times _Y X \to X \times _Y X.
One can also introduce the annihilator ideal \mathfrak {d}_{X/Y} of \Omega _{X/Y}^1, called the different ideal of X/Y;
it defines a closed sub-prescheme of X which, set-theoretically, is the set of point where X/Y is ramified, i.e. not unramified.
343Propositionsga1-i.3.5sga1-i.3.5.xmlI.3.5sga1-i.3
An immersion is ramified.
The composition of two ramified morphisms is also ramified.
Base extension of a ramified morphisms is also ramified.
We are rather indifferent about (ii) and (iii) (the second seems more interesting to me).
We can, of course, also be more precise, by giving some one-off statements;
this is more general only in appearance (except for in the case of definition (b)), and is boring.
We obtain, as per usual, the corollaries:
348Corollarysga1-i.3.6sga1-i.3.6.xmlI.3.6sga1-i.3
The cartesian product of two unramified morphisms is unramified.
If gf is unramified then so too is f.
If f is unramified then so too is f_ \mathrm {red}.
349Propositionsga1-i.3.7sga1-i.3.7.xmlI.3.7sga1-i.3
Let A \to B be a local homomorphism, and suppose that the residue extension k(B)/k(A) is trivial, with k(A) algebraically closed.
In order for B/A to be unramified, it is necessary and sufficient that \widehat {B} be (as an \widehat {A}-algebra) a quotient of \widehat {A}.
350Remarkssga1-i.3
In the case where we don't suppose that the residue extension is trivial, we can reduce to the case where it is by taking a suitable finite flat extension of A which destroys the aforementioned extension.
Consider the example where A is the local ring of an ordinary double point of a curve, and B a point of its normalisation:
then A \subset B, B is unramified over A with trivial residue extension, and \widehat {A} \to \widehat {B} is surjective but not injective.
We are thus going to strengthen the notion of unramified-ness.
518Sectionsga1-i.4sga1-i.4.xmlÉtale morphisms. Étale coversI.4sga1-i
We are going to suppose that everything concerning flat morphisms that we need to be true is indeed true;
these facts will be proved later, if there is time.
(cf. [sga1-iv (?)].)369Definitionsga1-i.4.1sga1-i.4.1.xmlI.4.1sga1-i.4
Let f \colon X \to Y be a morphism of finite type.
We say that f is étale at x if f is both flat and unramified at x.
We say that f is étale if it is étale at all points.
We say that X is étale at x over Y, or that it is a Y-prescheme which is étale at x etc.
Let f \colon A \to B be a local homomorphism.
We say that f is étale, or that B is étale over A, if B is flat and unramified over A
(cf. regrets in [sga1-iii.1.2 (?)].)
371Propositionsga1-i.4.2sga1-i.4.2.xmlI.4.2sga1-i.4
For B/A to be étale, it is necessary and sufficient that \widehat {B}/ \widehat {A} be étale.
370Proofsga1-i.4.2
This is true individually for both "unramified" and "flat".
372Corollarysga1-i.4.3sga1-i.4.3.xmlI.4.3sga1-i.4
Let f \colon X \to Y be of finite type, and x \in X.
The property of f being étale at x depends only on the local homomorphism \mathcal { O } _{f(x)} \to \mathcal { O } _x, and in fact only on the corresponding homomorphism for the completions.
373Corollarysga1-i.4.4sga1-i.4.4.xmlI.4.4sga1-i.4
Suppose that the residue extension k(A) \to k(B) is trivial, or that k(A) is algebraically closed.
Then B/A is étale if and only if \widehat {A} \to \widehat {B} is an isomorphism.
We can combine flatness with .
375Propositionsga1-i.4.5sga1-i.4.5.xmlI.4.5sga1-i.4
Let f \colon X \to Y be a morphism of finite type.
Then the set of points where f is étale is open.
374Proofsga1-i.4.5
Again, this is true individually for both "unramified" and "flat".
This proposition shows that we can forget about the "one-off" comments in the study of morphisms of finite type that are somewhere étale.
381Propositionsga1-i.4.6sga1-i.4.6.xmlI.4.6sga1-i.4
An open immersion is étale.
The composition of two étale morphisms is étale.
The base extension of an étale morphism is étale.
380Proofsga1-i.4.6
Indeed, (i) is trivial, and for (ii) and (iii) it suffices to note that it is true for "unramified" and "flat".
As a matter of fact, there are also corresponding comments to make about local homomorphisms (without the finiteness condition), which in any case should appear in the multiplodoque (starting with the case of unramified).
([Trans.] Grothendieck's "multiplodoque d'algèbre homologique" was the final version of his Tohoku paper — see (2.1) in "Life and work of Alexander Grothendieck" by Ching-Li Chan and Frans Oort for more information.)382Corollarysga1-i.4.7sga1-i.4.7.xmlI.4.7sga1-i.4
The cartesian product of two étale morphisms is étale.
383Corollarysga1-i.4.8sga1-i.4.8.xmlI.4.8sga1-i.4
Let X and X' be of finite type over Y, and g \colon X \to X' a Y-morphism.
If X' is unramified over Y and X is étale over Y, then g is étale.
358Proofsga1-i.4.8
Indeed, g is the composition of the graph morphism \Gamma _g \colon X \to X \times _Y X' (which is an open immersion by ) and the projection morphism, which is étale since it is just a "change of base" by X' \to Y of the étale morphism X \to Y.
384Definitionsga1-i.4.9sga1-i.4.9.xmlI.4.9sga1-i.4
We say that a cover of Y is étale (resp. unramified) if it is a Y-scheme X that is finite over Y and étale (resp. unramified) over Y.
The first condition means that X is defined by a coherent sheaf of algebras \mathcal { B } over Y.
The second means that \mathcal { B } is locally free over Y (resp. means absolutely nothing) and, further, that, for all y \in Y, the fibre \mathcal { B } (y)= \mathcal { B } _y \otimes _{ \mathcal { O } _y}k(y) is a separable algebra (i.e. a finite composition of finite separable extensions) over k(y).
385Propositionsga1-i.4.10sga1-i.4.10.xmlI.4.10sga1-i.4
Let X be a flat cover of Y of degree n (the definition of this term deserved to figure in ) defined by a locally free coherent sheaf \mathcal { B } of algebras.
We define, as usual, the trace homomorphism \mathcal { B } \to \mathcal { A } (that is, a homomorphism of \mathcal { A }-modules, where \mathcal { A } = \mathcal { O } _Y).
For X to be étale it is necessary and sufficient that the corresponding bilinear form \operatorname {tr} _{ \mathcal { B } / \mathcal { A } }xy define an isomorphism of \mathcal { B } over \mathcal { B }, or, equivalently, that the discriminant section
d_{X/Y} = d_{ \mathcal { B } / \mathcal { A } } \in \Gamma \big (Y, \wedge ^n \check { \mathcal { B } } \otimes _ \mathcal { A } \wedge ^n \check { \mathcal { B } } \big )
is invertible, or that the discriminant ideal defined by this section is the unit ring.
282Proofsga1-i.4.10
We can reduce to the case where Y= \operatorname {Spec} (k), and then it is a well-known criterion of separability, and thus trivial by passing to the algebraic closure of k.
386Remarksga1-i.4
We will have a less trivial statement to make later on (), when we do not suppose a priori that X is flat over Y, but instead require some normality hypothesis.
519Sectionsga1-i.5sga1-i.5.xmlFundamental property of étale morphismsI.5sga1-i441Theoremsga1-i.5.1sga1-i.5.1.xmlI.5.1sga1-i.5
Let f \colon X \to Y be a morphism of finite type.
For f to be an open immersion, it is necessary and sufficient that it be étale and radicial.
321Proofsga1-i.5.1
Recall what "radicial" means: injective, with radicial residual extensions (recall also that it means that the morphism remains injective under any base extension).
The necessity is trivial, and the sufficiency remains to be shown.
We are going to give two different proofs: the first is shorter, the second is more elementary.
A flat morphism is open, and so we can suppose (by replacing Y with f(X)) that f is an onto homeomorphism.
For any base extension, it remains true that f is flat, radicial, and surjective, thus a homeomorphism, and a fortiori closed.
Thus f is proper.
Thus f is finite (reference: Chevalley's theorem), defined by a coherent sheaf \mathcal { B } of algebras.
Now \mathcal { B } is locally free, and further, by hypothesis, of rank 1 everywhere, and so X=Y.
We can suppose that Y and X are affine.
We can further easily reduce to proving the following:
if Y= \operatorname {Spec} (A), with A local, and if f^{-1}(y) is non-empty (where y is the closed point of Y), then X=Y (indeed, this would imply that every y \in f(X) has an open neighbourhood U such that X|U=U).
We will then have that X= \operatorname {Spec} (B), and wish to prove that A=B.
But for this we can reduce to proving the analogous claim where we replace A by \widehat {A}, and B by B \otimes _A \widehat {A}
(taking into account the fact that \widehat {A} is faithfully flat over A).
We can thus suppose that A is complete.
Let x be the point over y.
By , \mathcal { O } _x is finite over A, and is thus (being flat and radicial over A) identical to A.
So X=Y \coprod X' (disjoint sum).
But since X is radicial over Y, X' is empty.
443Corollarysga1-i.5.2sga1-i.5.2.xmlI.5.2sga1-i.5
Let f \colon X \to Y be a morphism that is both a closed immersion and étale.
If X is connected, then f is an isomorphism from X to a connected component of Y.
442Proofsga1-i.5.2
Indeed, f is also an open immersion.
We thus deduce:
444Corollarysga1-i.5.3sga1-i.5.3.xmlI.5.3sga1-i.5
Let X be an unramified Y-scheme, with Y connected.
Then every section of X over Y is an isomorphism from Y to a connected component of X.
There is thus a bijective correspondence between the set of such sections and the set of connected components X_i of X such that the projection X_i \to Y is an isomorphism (or, equivalently, by , surjective and radicial).
In particular, a section is determined by its value at a point.
409Proofsga1-i.5.3
Only the first claim demands a proof;
by , it suffices to note that a section is a closed immersion (since X is separated over Y) and also étale, by .
446Corollarysga1-i.5.4sga1-i.5.4.xmlI.5.4sga1-i.5
Let X and Y be preschemes over S, with X unramified and separated over S, and Y connected.
Let f and g be S-morphisms from Y to X, and suppose that y is a point of Y such that f(y)=g(y)=x, and such that the residue homomorphisms k(x) \to k(y) defined by f and g are identical ("f and g agree geometrically at y").
Then f and g are identical.
445Proofsga1-i.5.4
This follows from by reducing to the case where Y=S, and by replacing X with X \times _S Y.
Here is a particularly important variant of :
448Theoremsga1-i.5.5sga1-i.5.5.xmlI.5.5sga1-i.5
Let S be a prescheme, and let X and Y be S-preschemes.
Let S_0 be a closed sub-prescheme of S that has the same underlying space as S, and let X_0=X \times _S S_0 and Y_0=Y \times _S S_0 be the "restrictions" of X and Y to S_0.
Suppose that X is étale over S.
Then the natural map
\operatorname {Hom} _S(Y,X) \to \operatorname {Hom} _{S_0}(X_0,Y_0)
is bijective.
447Proofsga1-i.5.5
We can again reduce to the case where Y=S, and then this follows from the "topological" description of sections of X/Y given in .
449Scholiumsga1-i.5
Let S be a prescheme, and let X and Y be S-preschemes.
Let S_0 be a closed sub-prescheme of S that has the same underlying space as S, and let X_0=X \times _S S_0 and Y_0=Y \times _S S_0 be the "restrictions" of X and Y to S_0.
Suppose that X is étale over S.
Then the natural map
\operatorname {Hom} _S(Y,X) \to \operatorname {Hom} _{S_0}(X_0,Y_0)
is bijective.
The following form of (which looks more general) is often useful:
451Corollarysga1-i.5.6sga1-i.5.6.xmlExtension of liftingsI.5.6sga1-i.5
Consider a commutative diagram of morphisms
\begin {CD} X @<<< Y_0 \\ @VVV @VVV \\ S @<<< Y \end {CD}
where X \to S is étale, and Y_0 \to Y is a bijective closed immersion.
Then we can find a unique morphism Y \to X such that the two corresponding triangles commute.
450Proofsga1-i.5.6
By replacing S with Y, and X with X \times _S Y, we can reduce to the case where Y=S, and this is then a particular case of for Y=S.
We also note the following immediate consequence of (which we did not give as a corollary, in order to not interrupt the line of ideas developed following ):
452Propositionsga1-i.5.7sga1-i.5.7.xmlI.5.7sga1-i.5
Let X and X' be preschemes that are of finite type and flat over Y, and let g \colon X \to X' be a Y-morphism.
For g to be an open immersion (resp. an isomorphism), it is necessary and sufficient that the induced morphism on the fibres
g \otimes _Y k(y) \colon X \otimes _Y k(y) \to X' \otimes _Y k(y)
be an open immersion (resp. an isomorphism) for all y \in Y.
439Proofsga1-i.5.7
It suffices to prove sufficiency;
since it is true for the property of being a surjection, we can reduce to the case of an open immersion.
By , we have to show that g is radicial (which is trivial) and étale (which follows from below).
454Corollarysga1-i.5.8sga1-i.5.8.xmlI.5.8sga1-i.5
Let X and X' be Y-preschemes, g \colon X \to X' a Y-morphism, x a point of X, and y the projection of x in Y.
For g to be quasi-finite (resp. unramified) at x, it is necessary and sufficient that g \otimes _Y k(y) be so.
453Proofsga1-i.5.8
The two algebras over k(g(x)) that we have to study in order to see whether or not we do indeed have a morphism which is quasi-finite (resp. unramified) at x are the same for g and g \otimes _Y k(y).
456Corollarysga1-i.5.9sga1-i.5.9.xmlI.5.9sga1-i.5
With the notation of , suppose that X and X' are flat and of finite type over Y.
For g to be flat (resp. étale) at x, it is necessary and sufficient that g \otimes _Y k(y) be so.
455Proofsga1-i.5.9
For "flat", the statement only serves as a reminder, since this is one of the fundamental criteria of flatness.
(cf. [sga1-iv.5.9 (?)].)
For "étale", this follows by taking into account.
520Sectionsga1-i.6sga1-i.6.xmlApplication to étale extensions of complete local ringsI.6sga1-i
This section is a particular case of the results on formal preschemes, which should appear in the multiplodoque.
Nevertheless, here we get away without much difficulty, i.e. without the explicit local determination of the étale morphisms in [sga1-i.7 (?)] (using the Main Theorem).
This is perhaps sufficient reason to keep this current section (even in the multiplodoque) where it is.
495Theoremsga1-i.6.1sga1-i.6.1.xmlI.6.1sga1-i.6
Let A be a complete local ring (Noetherian, of course), with residue field k.
For any A-algebra B, let R(B)=B \otimes _Ak be thought of as a k-algebra;
this depends functorially on B.
Then R defines an equivalence between the category of A-algebras that are finite and étale over A and the category of algebras that are finite rank and separable over k.
Firstly, the functor in question is fully faithful, as follows from the more general fact:
496Corollarysga1-i.6.2sga1-i.6.2.xmlI.6.2sga1-i.6
Let B and B' be A-algebras that are finite over A.
If B is étale over A, then the canonical map
\operatorname {Hom} _{ A \mathsf {-alg} }(B,B') \to \operatorname {Hom} _{ k \mathsf {-alg} }(R(B),R(B'))
is bijective.
475Proofsga1-i.6.2
We can reduce to the case where A is Artinian (by replacing A by A/ \frak {m}^n), and then this is a particular case of .
It remains to prove that, for every finite and separable k algebra (or we can simply say "étale", for brevity) L, there exists some B étale over A such that R(B) is isomorphic to L.
We can suppose that L is a separable extension of k, and, as such, it admits a generator x, i.e. it is isomorphic to an algebra k[t]/Fk[t], where F \in k[t] is a monic polynomial.
We can lift F to a monic polynomial F_1 in A[t], and we take B=A[t]/F_1A[t].
521Sectionsga1-i.7sga1-i.7.xmlLocal construction of unramified and étale morphismsI.7sga1-i412Propositionsga1-i.7.1sga1-i.7.1.xmlI.7.1sga1-i.7
Let A be a Noetherian ring, B an algebra which is finite over A, and u a generator of B over A.
Let F \in A[t] be such that F(u)=0 (we do not assume F to be monic), and u'=F'(u) (where F' is the differentiated polynomial).
Let \mathfrak {q} be a prime ideal of B not containing u', and \mathfrak {p} its intersection with A.
Then B_ \mathfrak {q} is unramified over A_ \mathfrak {p}.
In other words, taking Y= \operatorname {Spec} (A), X= \operatorname {Spec} (B), and X_{u'}= \operatorname {Spec} (B_{u'}), we claim that X_{u'} is unramified over Y.
This statement follows from the following, more precise:
414Corollarysga1-i.7.2sga1-i.7.2.xmlI.7.2sga1-i.7
The different ideal of B/A contains u'B, and is equal to u'B if the natural homomorphism A[t]/FA[t] \to B (sending t to u) is an isomorphism.
413Proofsga1-i.7.2
Let J be the kernel of the homomorphism C=A[t] \to B, so that J contains FA[t], and is equal to it in the second case described in .
Since the homomorphism C \to B is surjective, \Omega _{B/A}^1 can be identified with the quotient of \Omega _{C/A}^1 by the sub-module generated by J \Omega _{C/A}^1 and \mathrm {d} (J) (we should have explicitly described the definition of the homomorphism d and the calculation of \Omega ^1 for an algebra of polynomials in ).
Identifying \Omega _{C/A}^1 with C, via the basis \mathrm {d} t, we obtain B/B \cdot J', and so the different ideal is generated by the set J' of images in B of derivatives of G \in J (and it suffices to take G that generate J).
Since F \in J (resp. F is a generator of J), we are done.
416Corollarysga1-i.7.3sga1-i.7.3.xmlI.7.3sga1-i.7
Under the conditions of , and supposing that F is monic and that A[t]/FA[t] \to B is an isomorphism, in order for B_ \mathfrak {q} to be étale over A_ \mathfrak {p} it is necessary and sufficient that \mathfrak {q} not contain u'.
415Proofsga1-i.7.3
Since B is flat over A, being étale is equivalent to being unramified, and we can apply .
418Corollarysga1-i.7.4sga1-i.7.4.xmlI.7.4sga1-i.7
Under the conditions of , in order for B to be étale over A, it is necessary and sufficient that u' be invertible, or for the ideal F' generated by F in A[t] to be the unit ideal.
417Proofsga1-i.7.4
The second claim follows from the first along with Nakayama (in B).
A monic polynomial F \in A[t] that has the property stated in is said to be a separable polynomial (if F is not monic, we must at least require that the coefficient of its leading term be invertible; in the case where A is a field, we recover the usual definition).
420Corollarysga1-i.7.5sga1-i.7.5.xmlI.7.5sga1-i.7
Let B be an algebra which is finite over the local ring A.
Suppose that K(A) is infinite, or that B is local.
Let n be the rank of L=B \otimes _A K(A) over K(A)=k.
For B to be unramified (resp. étale) over A, it is necessary and sufficient that B be isomorphic to a quotient of (resp. isomorphic to) A[t]/FA[t], where F is a separable monic polynomial, which we can assume to be (resp. which is necessarily of) degree n.
419Proofsga1-i.7.5
We only have to prove necessity.
Suppose that B is unramified over A, and thus that L is separable over k.
It then follows from the hypotheses that L/k admits a generator \xi, and so the \xi ^i (for 0 \leq i<n) form a basis for L over k.
Let u \in B be a lift of \xi;
by Nakayama, the u^i (for 0 \leq i<n) generate (resp. form a basis of) the A-module B, and, in particular, we obtain a monic polynomial F \in A[t] such that F(u)=0;
then B is isomorphic to a quotient of (resp. isomorphic to) A[t]/FA[t].
Finally, by applying to L/k, we see that F and F' generate A[t] modulo \mathfrak {m}A[t], and so (by Nakayama in A[t]/FA[t]) F and F' generate A[t], and we are done.
421Theoremsga1-i.7.6sga1-i.7.6.xmlI.7.6sga1-i.7
Let A be a local ring, and A \to \mathcal { O } a local homomorphism such that \mathcal { O } is isomorphic to the localisation of an algebra of finite type over A.
Suppose that \mathcal { O } is unramified over A.
Then we can find an A-algebra B that is integral over A, a maximal ideal \mathfrak {n} of B, a generator u of B over A, and a monic polynomial F \in A[t] such that \mathfrak {n} \not \ni F'(u) and such that \mathcal { O } is isomorphic (as an A-algebra) to B_ \mathfrak {n}.
If \mathcal { O } is étale over A, then we can take B=A[t]/FA[t].
(Of course, these conditions are more than sufficient ...)
Before proving , we first state some nice corollaries:
423Corollarysga1-i.7.7sga1-i.7.7.xmlI.7.7sga1-i.7
For \mathcal { O } to be unramified over A, it is necessary and sufficient that \mathcal { O } be isomorphic to the quotient of an algebra which is unramified and étale over A.
422Proofsga1-i.7.7
We can take \mathcal { O } '=B'_{ \mathfrak {n}'}, where B'=A[t]/FA[t] and where \mathfrak {n}' is the inverse image of \mathfrak {n} in B'.
425Corollarysga1-i.7.8sga1-i.7.8.xmlI.7.8sga1-i.7
Let f \colon X \to Y be a morphism of finite type, and x \in X.
For f to be unramified at x, it is necessary and sufficient that there exist an open neighbourhood U of x such that f|U factors as U \to X' \to Y, where the first arrow is a closed immersion, and the second is an étale morphism.
424Proofsga1-i.7.8
This is a simple translation of .
We will now show how the jargon of follows from the main theorem:
there exists, by , an epimorphism \mathcal { O } ' \to \mathcal { O }, where \mathcal { O } has all the desired properties;
but since \mathcal { O } ' and \mathcal { O } are étale over A, the morphism \mathcal { O } ' \to \mathcal { O } is étale by , and thus an isomorphism.
426Proofsga1-i.7.6-proofsga1-i.7.6-proof.xmlI.7.6sga1-i.7
This mimics a proof from the Séminaire Chevalley.
By the "Main Theorem", we have that \mathcal { O } =B_ \mathfrak {n}, where B is an algebra that is finite over A, and \mathfrak {n} is a maximal ideal.
Then B/ \mathfrak {n}=B( \mathcal { O } ) is a separable, and thus monogenous, extension of k;
if \mathfrak {n}_i (for 1 \leq i \leq r) are maximal ideas of B that are distinct from \mathfrak {n}, then there thus exists an element u of B that belongs to all the \mathfrak {n}_i, and thus whose image in B/ \mathfrak {n} is a generator.
But B/ \mathfrak {n}=B_ \mathfrak {n}/ \mathfrak {n}B_ \mathfrak {n}=B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} (where \mathfrak {m} is the maximal ideal of A).
Suppose, for the moment, that we have both and .
Let n be the rank of the k-algebra L=B \otimes _A k.
By Nakayama, there exists a monic polynomial of degree n in A[t] such that F(u)=0.
Let f be the polynomial induced from F by reduction \mod \mathfrak {m}.
Then L is k-isomorphic to k[t]/fk[t], and so, by , f'( \xi ) is not contained in the maximal ideal of L that corresponds to \mathfrak {n} (where \xi denotes the image of t in L, i.e. the image of u in L).
Since f'( \xi ) is the image of F'(u), we are done.
427Lemmasga1-i.7.9sga1-i.7.9.xmlI.7.9sga1-i.7
Let A be a local ring, B an algebra that is finite over A, \mathfrak {n} a maximal ideal of B, and u an element of B whose image in B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} is a generator as an algebra over k=A/ \mathfrak {m}, and such that u is contained in every maximal ideal of B that is distinct from \mathfrak {n}.
Let B'=B[u] and \mathfrak {n}'= \mathfrak {n}B'.
Then the canonical homomorphism B'_{ \mathfrak {n}'} \to B_ \mathfrak {n} is an isomorphism.
428Lemmasga1-i.7.10sga1-i.7.10.xmlI.7.10sga1-i.7
Let B be a algebra that is finite over A and generated by a single element u, and let \mathfrak {n} be a maximal ideal of B such that B_ \mathfrak {n} is unramified over A.
Then there exists a monic polynomial F \in A[t] such that F(u)=0 and F'(u) \not \in \mathfrak {n}.
N.B. should have appeared as a corollary to , and before (which it implies).
So now follows from the combination of and ;
it remains only to prove .
429Proofsga1-i.7.9-proofsga1-i.7.9-proof.xmlI.7.9sga1-i.7
Let S'=B' \setminus \mathfrak {n}', so that B'S'^{-1}=B'_ \mathfrak {n'}.
Similarly, let S=B \setminus \mathfrak {n}, so that BS^{-1}=B_ \mathfrak {n}.
We then have a natural homomorphism BS'^{-1} \to BS^{-1}=B_ \mathfrak {n};
we will show that this is an isomorphism, i.e. that the elements of S are invertible in BS'^{-1}, i.e. that every maximal ideal \mathfrak {p} of BS'^{-1} does not meet S, i.e. that every maximal ideal of BS'^{-1} induces \mathfrak {n} on B.
\begin {CD} B @>>> BS'^{-1} @>>> BS^{-1} = B_ \mathfrak {n} \\ @AAA @AAA \\ B' @>>> B'S'^{-1} = B'_{ \mathfrak {n}'} \end {CD}
Since BS'^{-1} is finite over B'S'^{-1}=B'_{ \mathfrak {n}'}, \mathfrak {p} induces the unique maximal ideal \mathfrak {n}'B_{ \mathfrak {n}'} of B'_{ \mathfrak {n}'}, and thus induces the maximal ideal \mathfrak {n}' of B';
since B is finite over B', the ideal \mathfrak {q} of B induced by \mathfrak {p}, which lives over \mathfrak {n}', is necessarily maximal, and does not contain u, and is thus identical to \mathfrak {n}.
(We have just used the fact that u belongs to every maximal ideal of B that is distinct from \mathfrak {n}).
We now prove that BS'^{-1} is equal to B'S'^{-1}:
since the former is finite over the latter, we can reduce, by Nakayama, to proving equality modulo \mathfrak {n}'BS'^{-1}, and, a fortiori, it suffices to prove equality modulo \mathfrak {m}BS'^{-1};
but BS'^{-1}/ \mathfrak {m}BS'^{-1}=B_ \mathfrak {n}/ \mathfrak {m}B_ \mathfrak {n} is generated, over k, by u (here we use the other property of u), and so the image of B' (and, a fortiori, of B'S'^{-1}) inside is everything (as a sub-ring that contains k and the image of u.)
430Remarksga1-i.7
We must be able to state for a ring \mathcal { O } that is only semi-local, so that we also cover :
we make the hypothesis that \mathcal { O } / \mathfrak {m} \mathcal { O } is a monogenous k-algebra;
we can thus find some u \in B whose image in B/ \mathfrak {m}B is a generator, and belongs to every maximal ideal of B that doesn't come from \mathcal { O }.
Both and should be able to be adapted without difficulty.
More generally, ...
529Sectionsga1-i.8sga1-i.8.xmlInfinitesimal lifting of étale schemes. Applications to formal schemesI.8sga1-i522Propositionsga1-i.8.1sga1-i.8.1.xmlI.8.1sga1-i.8
Let Y be a prescheme, Y_0 a sub-prescheme, X_0 an étale Y_0-scheme, and x a point of X_0.
Then there exists an étale Y-scheme X, a neighbourhood U_0 of x in X_0, and a Y_0-isomorphism U_0 \xrightarrow { \sim }X \times _Y Y_0.
503Proofsga1-i.8.1
Let y be the projection of x in Y_0;
applying to the étale local homomorphism A_0 \to B_0 of local rings of y and x in Y_0 and X_0, we obtain an isomorphism
\begin {aligned} B_0 &= (C_0)_{ \mathfrak {n}_0} \\ C_0 &= A_0[t]/F_0A_0[t] \end {aligned}
where F_0 is a monic polynomial, and \mathfrak {n}_0 is a maximal ideal of C_0 not containing the class of F'_0(t) in C_0.
Let A be the local ring of y in Y, let F be a monic polynomial in A[t] that gives F_0 under the surjective homomorphism A \to A_0 (we lift the coefficients of F_0), and let C=A[t]/FA[t], with \mathfrak {n} the maximal ideal of C given by the inverse image of \mathfrak {n}_0 under the natural epimorphism C \to C \otimes _A A_0=C_0.
Let
B = C_ \mathfrak {n}.
It is immediate, by construction and by , that B is étale over A, and that we have an isomorphism B \otimes _A A_0=A_0.
We know that there exists a Y-scheme X of finite type, along with a point z of X over y such that \mathcal { O } _z is A-isomorphic to C;
since the latter is étale over A= \mathcal { O } _y, we can (by taking X to be small enough) assume that X is étale over Y.
Let X'_0=X \times _Y Y_0.
Then the local ring of z in X'_0 can be identified with \mathcal { O } _z \otimes _A A_0=B \otimes _A A_0, and is thus isomorphic to B_0.
This isomorphism is defined by an isomorphism from a neighbourhood U_0 of x in X to a neighbourhood of z in X'_0, and we can assume this to be identical to X'_0 by taking X to be small enough.
524Corollarysga1-i.8.2sga1-i.8.2.xmlI.8.2sga1-i.8
The analogous claim holds for étale covers, if we suppose the residue field k(y) to be infinite.
523Proofsga1-i.8.2
The proof is the same, just replacing by .
525Theoremsga1-i.8.3sga1-i.8.3.xmlI.8.3sga1-i.8
The functor described in is an equivalence of categories.
477Proofsga1-i.8.3
By , it remains only to show that every étale S_0-scheme X_0 is isomorphic to an S_0-scheme X \times _S S_0, where X is an étale S-scheme.
The underlying topological space of X must necessarily be identical to the one of X_0, and with X_0 being identified with a closed sub-prescheme of X.
The problem is thus equivalent to the following:
find, on the underlying topological space |X_0| of X_0, a sheaf of algebras \mathcal { O } _X over f_0^*( \mathcal { O } _S) (where f_0 is the projection X_0 \to S_0, thought of here as a continuous map of the underlying spaces) that makes |X_0| an étale S-prescheme X, as well as an algebra homomorphism \mathcal { O } _X \to \mathcal { O } _{X_0} that is compatible with the homomorphism f_0^*( \mathcal { O } _S) \to f_0^*( \mathcal { O } _{S_0}) on the sheaves of scalars, and that induces an isomorphism \mathcal { O } _X \otimes _{f_0^*( \mathcal { O } _S)*}f_0^*( \mathcal { O } _{S_0}) \xrightarrow { \sim } \mathcal { O } _{X_0}.
(Then X will be an étale S-prescheme that is reduced along X_0, and thus separated over S, since X_0 is separated over S_0, and X satisfies all the desired properties).
If (U_i) is an open cover of X_0, and if we find a solution to the problem on each of the U_i, then it follows from the uniqueness theorem that these solutions glue (i.e. the sheaves of algebras that they define, endowed with their augmentation homomorphisms, glue), and we claim that the ringed space thus constructed over S is an étale S-prescheme X endowed with an isomorphism X \times _S S_0 \xleftarrow { \sim }X_0.
It thus suffices to find a solution locally, which we know is possible by .
527Corollarysga1-i.8.4sga1-i.8.4.xmlI.8.4sga1-i.8
Let S be a locally Noetherian formal prescheme, endowed with an ideal of definition \mathcal { J }, and let S_0=(|S|, \mathcal { O } _S/ \mathcal { J } ) be the corresponding ordinary prescheme.
Then the functor \mathfrak {X} \mapsto \mathfrak {X} \times _S S_0 from the category of étale covers of S to the category of étale covers of S_0 is an equivalence of categories.
526Proofsga1-i.8.4
Of course, we define an étale cover of a formal prescheme S to be a cover of S (i.e a formal prescheme over S defined by a coherent sheaf of algebras \mathcal { B }) such that \mathcal { B } is locally free, and such that the residue fibres \mathcal { B } _s \otimes _{ \mathcal { O } _s}k(s) of \mathcal { B } are separable algebras over k(s).
If we denote by S_n the ordinary prescheme (|S|, \mathcal { O } _S/ \mathcal { J } ^{n+1}), then the data of a coherent sheaf of algebras \mathcal { B } on S is equivalent to the data of a sequence of coherent sheaves of algebras \mathcal { B } _n on the S_n, endowed with a transitive system of homomorphisms \mathcal { B } _m \to \mathcal { B } _n (for m \geq n) defining the isomorphisms \mathcal { B } _m \otimes _{ \mathcal { O } _{S_m}} \mathcal { O } _{S_n} \xrightarrow { \sim } \mathcal { B } _n.
It is immediate that \mathcal { B } is locally free if and only if the \mathcal { B } _n are locally free over the S_n, and that the separability condition is satisfied if and only if it is satisfied for \mathcal { B } _0, or for all the \mathcal { B } _n.
Thus \mathcal { B } is étale over S if and only if the \mathcal { B } _n are étale over the S_n.
Taking this into account, follows immediately from .
528Remarksga1-i.8
It was not necessary to restrict ourselves to the case of covers in , but this is the only case that we will use for the moment.
558Sectionsga1-i.9sga1-i.9.xmlInvariance propertiesI.9sga1-i
Let A \to B be a morphism that is local and étale;
we study here some cases where a certain property of A implies the same property for B, or vice versa.
A certain number of such propositions are already consequences of the simple fact that B is quasi-finite and flat over A, and we content ourselves with "recalling" some of them.
A and B have the same Krull dimension, and the same depth (Serre's "cohomological codimension", in the more modern language).
It also follows, for example, that A is Cohen–Macaulay if and only if B is.
Also, for any prime ideal \mathfrak {q} of B (inducing some \mathfrak {p} of A), B_ \mathfrak {q} is again quasi-finite and flat over A_ \mathfrak {p}, as long as we suppose that B is the localisation of an algebra of finite type over A (this follows from the fact that the set of points where a morphism of finite type is quasi-finite (resp. flat) is open);
furthermore, every prime ideal \mathfrak {p} of A is induced by a prime ideal \mathfrak {q} of B (since B is faithfully flat over A).
It thus follows, for example, that \mathfrak {q} and \mathfrak {p} have the same rank;
also, A has no embedded prime ideals if and only if B has none.
We will thus content ourselves with more specific propositions concerning the case of étale morphisms.
531Propositionsga1-i.9.1sga1-i.9.1.xmlI.9.1sga1-i.9
Let A \to B be an étale local homomorphism.
For A to be regular, it is necessary and sufficient that B be regular.
530Proofsga1-i.9.1
Let k be the residue field of A, and L the residue field of B.
Since B is flat over A, and since L=B \otimes _A k (i.e. \mathfrak {n}= \mathfrak {m}B, where \mathfrak {m} and \mathfrak {n} are the maximal ideals of A and B respectively), the \mathfrak {m}-adic filtration on B is identical to the \mathfrak {n}-adic filtration, and
\operatorname {gr} ^ \bullet (B) = \operatorname {gr} ^ \bullet (A) \otimes _k L.
It follows that \operatorname {gr} ^ \bullet (B) is a polynomial algebra over L if and only if \operatorname {gr} ^ \bullet (A) is a polynomial algebra over K.
(N.B. we have not used the fact that L/k is separable.)
532Corollarysga1-i.9.2sga1-i.9.2.xmlI.9.2sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is regular, then X is regular;
the converse is true if f is surjective.
533PropositionI.9.2sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is reduced, then X is reduced;
the converse is true if f is surjective.
This is equivalent to the following:
535Corollarysga1-i.9.3sga1-i.9.3.xmlI.9.3sga1-i.9
Let f \colon A \to B be an étale local homomorphism, with B isomorphic to the localisation of an A-algebra of finite type over A.
For A to be reduced, it is necessary and sufficient that B be reduced.
534Proofsga1-i.9.3
The necessity is trivial, since A \to B is injective (since B is faithfully flat over A).
For the sufficiency, let \mathfrak {p}_i be the minimal prime ideals of A.
By hypothesis, the natural map A \to \prod _i A/ \mathfrak {p}_i is injective, and so tensoring with the flat A-module B gives that B \to \prod _i B/ \mathfrak {p}_iB is injective, and we can thus reduce to proving that the B/ \mathfrak {p}_iB are reduced.
Since B/ \mathfrak {p}_iB is étale over A/ \mathfrak {p}_i, we can reduce to the case where A is integral.
Let K be the field of fractions of A, so that A \to K is injective, and thus so too is B \to B \otimes _A K (since B is A-flat), and we can thus reduce to proving that B \otimes _A K is reduced.
But B is the localisation of an A-algebra of finite type over A, and thus is the local ring of a point x of a scheme of finite type X= \operatorname {Spec} (C) over Y= \operatorname {Spec} (A) that is also étale over Y, so B \otimes _A K is a localisation (with respect to some suitable multiplicatively stable set) of the ring C \otimes _A K of X \otimes _A K.
Since X \otimes _A K is étale over K, its ring is a finite product of fields (that are separable extensions of K), and thus so too is B \otimes _A K.
537Corollarysga1-i.9.4sga1-i.9.4.xmlI.9.4sga1-i.9
Let f \colon A \to B be an étale local homomorphism, with A analytically reduced (i.e. such that the completion \widehat {A} of A has no nilpotent elements).
Then B is analytically reduced, and a fortiori reduced.
536Proofsga1-i.9.4
Indeed, \widehat {B} is finite and étale over \widehat {A};
we can apply .
541Theoremsga1-i.9.5sga1-i.9.5.xmlI.9.5sga1-i.9
Let f \colon A \to B be a local homomorphism, with B isomorphic to the localisation of an A-algebra of finite type over A.
If f is étale, then A is normal if and only if B is normal.
If A is normal, then f is étale if and only if f is injective and unramified (and then B is normal, by (i)).
We will give two different proofs of (i):
the first using certain properties of quasi-finite flat morphisms (stated at the start of this section) and without using (and thus the Main Theorem);
the second proof does the opposite.
For (ii), it seems like we do indeed need the Main Theorem, no matter what.
546Proofsga1-i.9.5.i-proof-1sga1-i.9.5.i-proof-1.xmlFirst proofI.9.5.isga1-i.9
We use the following necessary and sufficient condition for a local Noetherian ring A of dimension \neq0 to be normal.
545sga1-i.9-serres-criterionsga1-i.9-serres-criterion.xmlSerre's criterionsga1-i.9.5.i-proof-1
For every rank-1 prime ideal \mathfrak {p} of A, A_ \mathfrak {p} is normal (or, equivalently, regular);
For every rank-\geq2 prime ideal \mathfrak {p} of A, the depth of A_ \mathfrak {p} is \geq2.
(cf. EGA IV 5.8.6.)
We assume this criterion here, but it should also appear in the section on flatness.
Its main advantage is that it does not suppose a priori that A is reduced, nor a fortiori that A is integral.
Here, we can already suppose that \dim A= \dim B \neq0.
By the statements at the start of this section, the rank-1 (resp. rank-\geq2) prime ideals \mathfrak {p} of A are exactly the intersections of A with the rank-1 (resp. rank-\geq2) prime ideals \mathfrak {q} of B.
Finally, if \mathfrak {p} and \mathfrak {q} correspond to one another, then B_ \mathfrak {q} is étale over A_ \mathfrak {p}, and thus of the same depth as A_ \mathfrak {p}, and is regular if and only if A_ \mathfrak {p} is (by ).
Applying Serre's criterion, we see that A is normal if and only if B is.
547Proofsga1-i.9.5.i-proof-2sga1-i.9.5.i-proof-2.xmlSecond proofI.9.5.isga1-i.9
Suppose that B is normal, with field of fractions L;
let K be the field of fractions of A (and note that A is integral, since B is integral).
We have already seen, in the proof of , that B \otimes _A K is a finite product of fields;
since it is contained in L, it is a field;
since it contains B, it is equal to L itself.
An element of K that is integral over A is integral over B, and is thus in B, since B is normal, and thus also in A, since B \cap K=A (as follows from the fact that B is faithfully flat over A).
Now suppose that A is normal;
we will prove that B is also normal.
By , we have that B=B'_ \mathfrak {n}, where B'=A[t]/FA[t] (with F and \mathfrak {n} as in ).
Thus L=B \otimes _A K is a localisation of B' \otimes _A K=K[t]/FK[t], and also a product of fields (finite separable extensions of K).
This latter product (B' \otimes _A K) is a direct factor of B'_K (since each time we localise an Artinian ring (here B'_K) with respect to a multiplicatively stable set), and thus corresponds to a decomposition F=F_1F_2 in K[t], with the generator of L corresponding to t being annihilated by F_1.
But, since A is normal, the F_i are in A[t] (supposing that they are monic).
Note that B \to L=B \otimes _A K is injective (since A \to K is, since B is flat over A), and so F_1(u)=0, with u being the class of t in L.
Suppose that F were of minimal degree;
then it would follows that F_2=1.
(N.B. we would have F'(u)=F'_1(u)F_2(u)+F_1(u)F'_2(u)=F'_1(u)F_2(u), since F_1(u)=0, whence F'_1(u) \neq 0 since F'(u) \neq0.)
Thus
L = B \otimes _A K = K[t]/FK[t]
and so F is a separable polynomial in K[T] (but evidently not necessarily in A[t]).
(N.B. for now, we have only shown, essentially, that we can choose F and \mathfrak {n} in such that, with the above notation, B' \to B'_ \mathfrak {n}=B is injective;
for this, we have used the fact that A is normal;
I do not know if this remains true without this normality hypothesis).
Now recall the well-known lemma, taken from Serre's lectures last year:
548Lemmasga1-i.9.6sga1-i.9.6.xmlI.9.6sga1-i.9
Let K be a ring, F \in K[t] a separable monic polynomial, L=K[t]/FK[t], and u the class of t in L (so that F'(u) is an invertible element of L).
Then
\operatorname {tr} _{L/K} u^i/F'(u) = \begin {cases} 0 & \text {if }0 \leq i<n-1 \text {;} \\ 1 & \text {if }i=n-1 \end {cases}
where n= \deg F.
549Corollarysga1-i.9.7sga1-i.9.7.xmlI.9.7sga1-i.9
The determinant of the matrix (u^j \cdot u^i/F'(u))_{0 \leq i,j \leq n-1} is equal to (-1)^{n(n-1)/2}, and thus invertible in every sub-ring A of K.
550Corollarysga1-i.9.8sga1-i.9.8.xmlI.9.8sga1-i.9
Let A be a sub-ring of K, V the A-module generated by the u^i (for 0 \leq i \leq n-1), and V' the sub-A-module of L consisting of the x \in L such that \operatorname {tr} _{L/K}(xy) \in A for all y \in V (i.e. for y of the form u^i, for 0 \leq i \leq n-1).
Then V' is the A-module given by the basis u^i/F'(u) (for 0 \leq i \leq n-1).
551Corollarysga1-i.9.9sga1-i.9.9.xmlI.9.9sga1-i.9
Suppose that K is the field of fractions of an integral normal ring A, with the coefficients of F lying in A.
Then, with the notation of , V' contains the normal closure A' of A in L, which is thus contained in A[u]/F'(u), and a fortiori in A[u][F'(u)^{-1}].
We can apply the above corollary to the situation that we have obtained in the proof: since F'(u) is invertible in B, and since B contains A[u], B contains A'.
By the Main Theorem (or by the fact that B=A[u]_ \mathfrak {n}), B is a localisation of A'.
Since A' is normal, so too is B.
552Proofsga1-i.9.5.ii-proofsga1-i.9.5.ii-proof.xmlI.9.5.iisga1-i.9
We proceed as in the above proof to show that we can choose F in such that we again have
L = B \otimes _A K = K[t]/FK[t]
The only obstacle a priori is that we can no longer prove that B \to L is injective, since B is no longer assumed to be flat over A, and so we can only apply the same argument a priori to the image B_1 of B under the aforementioned homomorphism.
It immediately follows that B_1 is flat over A (since it is the localisation of a free A-algebra).
By , the morphism B \to B_1 is étale, and thus an isomorphism, which finishes the proof.
(From an editorial point of view, we should perform the two proofs above, and place the formal calculations of the lemma and of its corollaries in a separate section).
553Corollarysga1-i.9.10sga1-i.9.10.xmlI.9.10sga1-i.9
Let f \colon X \to Y be an étale morphism.
If Y is normal, then X is normal;
the converse is true if f is surjective.
555Corollarysga1-i.9.11sga1-i.9.11.xmlI.9.11sga1-i.9
Let f \colon X \to Y be a dominant morphism, with Y normal and X connected.
If f is unramified, then it is also étale, and X is then normal and thus irreducible (since it is connected).
554Proofsga1-i.9.11
Let U be the set of points where f is étale.
Since U is open, it suffices to show that it is also closed and non-empty.
Since U contains the inverse image of the generic point of Y (recall that, for an algebra over a field, unramified = étale), it is non-empty (since X dominates Y).
If x belongs to the closure of U, then it belongs to the closure of an irreducible component U_i of U, and thus to an irreducible component X_i= \bar {U_i} of X which intersects U and which thus dominates Y (since every component of U, being flat over Y, dominates Y).
Then, if y is the projection of x over Y, \mathcal { O } _y \to \mathcal { O } _x is injective (taking into account the fact that \mathcal { O } _y is integral).
Since \mathcal { O } _y is normal and \mathcal { O } _y \to \mathcal { O } _x is unramified, we conclude with the help of (ii) from .
556Corollarysga1-i.9.12sga1-i.9.12.xmlI.9.12sga1-i.9
Let f \colon X \to Y be a dominant morphism of finite type, with Y normal and X irreducible.
Then the set of points where f is étale is identical to the complement of the support of \Omega _{X/Y}^1, i.e. to the complement of the sub-prescheme of X defined by the different ideal \mathfrak {d}_{X/Y}.
( is the "less trivial" statement which was alluded to in the remark in .)
557Remarksga1-i.9
We do not claim that a connected étale cover of an irreducible scheme is itself irreducible if we do not assume the base to be normal;
this question will be studied in .
559Sectionsga1-i.10sga1-i.10.xmlÉtale covers of a normal schemeI.10sga1-i250Propositionsga1-i.10.1sga1-i.10.1.xmlI.10.1sga1-i.10
Let Y be normal and connected of field K, and let X be a separated étale prescheme over Y.
Then the connected components X_i of X are integral, their fields K_i are finite separable extensions of K, and X_i can be identified with a non-empty open subset of the normalisation of X in K_i (and thus X with a dense open subset of the normalisation of Y in R(X)=L= \prod K_i, where R(X) is the ring of rational functions on X).
249Proofsga1-i.10.1
By , X is normal, and a fortiori its local rings are integral, and so the connected components of X are irreducible.
Since X_i is normal, and also finite and dominant over Y, it follows from a particular (almost trivial, actually) case of the Main Theorem that X_i is an open subset of the normalisation of X in the field K_i of X_i.
251Corollarysga1-i.10.2sga1-i.10.2.xmlI.10.2sga1-i.10
Under the conditions of , X is finite over Y (i.e. an étale cover of Y) if and only if X is isomorphic to the normalisation Y' of Y in L=R(X) (the ring of rational functions on X).
246Proofsga1-i.10.2
We know that this normalisation is finite over Y (since Y is normal, and R/K separable);
conversely, if X is finite over Y, then it is also finite over Y', and so its image in Y' is closed (and it is also dense).
An algebra L of finite rank over K is said to be unramified over X (or simply unramified over K if X is evident) if L is a separable algebra over K (i.e. a direct sum of separable extensions K_i) and the normalisation Y' of Y in L (i.e. the disjoint sum of the normalisations of Y in the K_i) is unramified (i.e. étale, by ) over Y.
Thus:
253Corollarysga1-i.10.3sga1-i.10.3.xmlI.10.3sga1-i.10
For every X that is finite over Y and such that every irreducible component of X dominates Y, let R(X) be the ring of rational functions on X (given by the product of the local rings of the generic points of the irreducible components of X), so that X \mapsto R(X) is a functor, with values in algebras of finite rank over K=R(Y).
Then this functor establishes an equivalence between the category of connected étale covers of Y and the category of extensions L of K that are unramified over Y.
252Proofsga1-i.10.3
The inverse functor is the normalisation functor.
Suppose that Y is affine, and thus defined by a normal ring A with field of fractions K.
Let L be a finite extension of K given by a direct sum of fields.
Then, by definition, the normalisation Y' of Y in L is isomorphic to \operatorname {Spec} (A'), where A' is the normalisation of A in L.
To say that L is unramified over Y implies that A' is unramified (or even étale) over A.
If A is local, then it is equivalent to say that the local rings A'_ \mathfrak {n} (where \mathfrak {n} runs over the finite set of maximal ideals of A', i.e. the prime ideals of A' that induce the maximal ideal \mathfrak {m} of A) are unramified (i.e. étale) over the local ring A.
Finally, note that the discriminant criterion of can also be applied to this situation
(more generally, a variant of the aforementioned criterion can be stated thusly, without any preliminary flatness condition when X dominates Y, but with Y still assumed to be locally integral: A \to B and B \to B \otimes _A K are injective — then \operatorname {tr} _{L/K} is defined — and \operatorname {tr} _{L/K}(xy) induces a fundamental bilinear form B \times B \to A, i.e. there exists x_i \in B (for 1 \leq i \leq n= \operatorname {rank} _K L) such that \operatorname {tr} (x_ix_j) \in A for all i,j, and \det ( \operatorname {tr} (x_i x_j))_{1 \leq i,j \leq n} is invertible in A).
The syllogism immediately implies the syllogism of being unramified in the classical case:
258Propositionsga1-i.10.4sga1-i.10.4.xmlI.10.4sga1-i.10
Let Y be a normal integral prescheme, of field K.
Then
K is unramified over Y.
If L is an extension of K that is unramified over Y, and if Y' is a normal prescheme, of field L, that dominates Y (e.g. the normalisation of Y in L), and M an extension of L that is unramified over Y', then M/K is unramified over X (this is the transitivity property).
Let Y' be a normal integral prescheme that dominates Y, of field K'/K;
if L is an extension of K that is unramified over Y, then L \otimes _K K' is an extension of K' that is unramified over Y' (this is the translation property).
Furthermore:
259Corollarysga1-i.10.5sga1-i.10.5.xmlI.10.5sga1-i.10
Under the conditions of (iii) in , if Y= \operatorname {Spec} (A) and Y'= \operatorname {Spec} (A'), then the normalisation \bar {A'} of A' in L'=L \otimes _K K' can be identified with \bar {A} \otimes _A A', where \bar {A} is the normalisation of A in L.
Usually, people (those who are disgusted by the consideration of non-integral rings, even if they are direct sums of fields) state the translation property in the following (weaker) form:
260Corollarysga1-i.10.6sga1-i.10.6.xmlI.10.6sga1-i.10
Under the conditions of (iii) in , let L_1 be a sum extension of L/K (unramified over Y) and of K'/K.
Then L_1/K' is unramified over Y'.
In the case where Y= \operatorname {Spec} (A) and Y'= \operatorname {Spec} (A'), we further have that
\bar {A'} = A[ \bar {A},A']
i.e. the normalisation ring \bar {A'} of A' in L_1 is the A-algebra generated by A' and by the normalisation \bar {A} of A in L.
This latter fact is actually false without the unramified hypothesis, even in the case of extensions given by direct sums of number fields...
To finish this section, we are going to give the intuitive interpretation of the notion of étale covers: there should be the "maximal number" of points over the point y \in Y in question, and, in particular, there should not be "multiple points combined" over y.
To prove results in this sense, in all desirable generality, we will assume here found below (whose proof will be given in the multiplodoque, Chapter IV, Section 15, and uses Chevalley's technique of constructible sets, and a little bit of the theory of descent...).
A morphism of finite type f \colon X \to Y is said to be universally open if, for every base extension Y' \to Y (with Y' locally Noetherian), the morphism f' \colon X'=X \times _Y Y' \to Y' is open, i.e. sends open subsets to open subsets.
We can actually restrict to the case where Y' is of finite type over y (and even to the case where Y' is of the form Y[t_1, \ldots ,t_r], where the t_i are indeterminates).
A universally open morphism is a fortiori open (but the converse is false);
on the other hand, if f is open, and if X and Y are irreducible, then all of the components of all of the fibres of f are of the same dimension (i.e. the dimension of the generic fibre f^{-1}(z), where z is the generic point of Y).
Finally, if Y is normal, then this latter condition already implies that f is universally open (Chevalley's theorem).
It thus follows, for example, that, if f \colon X \to Y is a quasi-finite morphism, with Y normal and irreducible, then f is universally open (or even open) if and only if every irreducible component of X dominates Y.
Recall also that a flat morphism (of finite type) is open, and thus also universally open.
With these preliminaries, "recall" the following:
261Propositionsga1-i.10.7sga1-i.10.7.xmlI.10.7sga1-i.10
Let f \colon X \to Y be a quasi-finite, separated, universally open morphism.
For all y \in Y, let n(y) be the "geometric number of points in the fibre f^{-1}(y)", equal to the sum of the separable degrees of the residue extensions k(x)/k(y) as x runs over the points of f^{-1}(y).
Then the function y \mapsto n(y) on Y is upper semi-continuous.
For it to be constant on a neighbourhood of the point y (i.e. for it to be the case that n(y)=n(z_i), where the z_i are the generic points of the irreducible components of Y that contain y), it is necessary and sufficient for there to exist a neighbourhood U of y such that X|U is finite over U.
(cf. EGA IV 15.5.1.)262Corollarysga1-i.10.8sga1-i.10.8.xmlI.10.8sga1-i.10
If y \mapsto n(y) is constant, and if Y is geometrically unibranch(For the definition, cf. ), then the irreducible components of X are disjoint.
263Propositionsga1-i.10.9sga1-i.10.9.xmlI.10.9sga1-i.10
Let f \colon X \to Y be a separated étale morphism.
With the notation of , the function n \mapsto n(y) is upper semi-continuous.
For it to be constant on a neighbourhood of the point y (i.e. for it to be the case that n(y)=n(z_i), where the z_i are the generic points of the irreducible components of Y that contain y), it is necessary and sufficient that there exist a neighbourhood U of y such that X|U is finite over U, i.e. such that X|U is an étale cover of U.
264Corollarysga1-i.10.10sga1-i.10.10.xmlI.10.10sga1-i.10
For a separated étale morphism f \colon X \to Y (with Y connected) to be finite (i.e. for f to make X an étale cover of Y), it is necessary and sufficient that all the fibres of f have the same geometric number of points.
In and its corollary (), there was no normality hypothesis on Y;
if we make such a hypothesis, then we find the following stronger statement (which is usually taken as the definition of unramified for a cover):
266Theoremsga1-i.10.11sga1-i.10.11.xmlI.10.11sga1-i.10
Let f \colon X \to Y be a separated quasi-finite morphism.
Suppose that Y is irreducible, that every component of X dominates Y, and that X is reduced (i.e. that \mathcal { O } _X has no nilpotent elements).
Let n be the degree of X over Y (i.e. the sum of the degrees, over the field K of Y, of the fields K_i of the irreducible components X_i of X).
Let y be a normal point of Y.
Then the geometric number n(y) of points of X over y is \leq n, with equality if and only if there exists an open neighbourhood U of y such that X|U is an étale cover of U.
265Proofsga1-i.10.11
The "only if" is trivial;
we will prove the "if".
Let z be the generic point of Y.
Then n(z), which is equal to the sum of the separable degrees of the K_i/K, is \leq n, and, by , we have that n(y) \leq n(z);
thus n(y) \leq n, with equality implying that X|U if finite over U, for some suitable neighbourhood U of y.
We can thus suppose that X is finite over Y, and that the function n(y') on Y is constant.
Then, by , X is the disjoint union of its irreducible components, and so, to prove that it is unramified at y, we can restrict to the case where X is irreducible, thus integral.
Finally, we can assume that Y= \operatorname {Spec} ( \mathcal { O } _y).
The theorem thus reduces to the following classical statement:
268Corollarysga1-i.10.12sga1-i.10.12.xmlI.10.12sga1-i.10
Let A be a normal local ring (Noetherian, as always), of field K;
let L be a finite extension of K of degree n, and of separable degree n_s;
let B be a sub-ring of L that is finite over A, with field of fraction L;
let \mathfrak {m} be the maximal ideal of A, and n' the separable degree of B/ \mathfrak {m}B over A/ \mathfrak {m}A=k (which is equal to the sum of the separable deprees of the residue extensions of this ring).
Then n' \leq n_s, and a fortiori n' \leq n.
This latter inequality is an equality if and only if B is unramified (i.e. étale) over A.
267Proofsga1-i.10.12
It remains only to show that, if n'=n, then B is étale over A.
Recall the proof in the case where k is infinite:
we need only show that R=B/ \mathfrak {m}B is separable over k;
if this were not the case, then it would follow (by a known lemma) that there exists an element a of R whose minimal polynomial over k is of degree >n'.
This element would come from an element x of B, whose minimal polynomial over K (as an element of L) is of degree \leq n;
but this minimal polynomial has coefficients in A, since A is normal, and thus gives, by restriction modulo \mathfrak {m}, a monic polynomial F \in k[t] of degree \leq n=n', such that F(a)=0.
But this is a contradiction.
In the general case (where k can be finite), we can again use geometric language:
we consider Y'= \operatorname {Spec} (A[t]), which is faithfully flat over Y, and the generic point y' of the fibre \operatorname {Spec} (k[t]) of Y' over y.
Then X is unramified over Y at y if and only if X'=X \times _Y Y'= \operatorname {Spec} (B[t]) is unramified over Y' at y', as we immediately see.
On the other hand, by the choice of y', its residue field is k(t), and thus infinite.
Since y' is a normal point of Y', we are now in the previous case.
560Sectionsga1-i.11sga1-i.11.xmlVarious addendaI.11sga1-i
We have already said that a connected étale cover of an integral scheme is not necessarily integral.
Here are two examples of this fact.
Let C be an algebraic curve with an ordinary double point x, and let C' be its normalisation, with a and b the two points of C' over x.
Let C'_1 and C'_2 be copies of C', with a_i (resp. b_i) the point of C'_i corresponding to a (resp. b).
In the curve C'_1 \coprod C'_2, we identify a_1 with b_2, and a_2 with b_1 (we leave making this process of identification precise to the reader; it will be explained in Chapter VI of the multiplodoque, but, in the case of curves over an algebraically closed field, is already covered in Serre's book on algebraic curves).
We obtain a curve C'' that is connected and reducible, and is a degree-2 étale cover of C.
The reader can verify that, generally, the "Galois" connected étale covers C'' of C whose inverse images C'' \times _C C' are trivial covers of C' (i.e. isomorphic to the sum of a certain number of copies of C') are "cyclic" of degree n, and, conversely, for every integer n>0, we can construct a cyclic connected étale cover of degree n.
In the language of the fundamental group (which will be developed later), this implies that the quotient of \pi _1(C) by the closed invariant subgroup generated by the image of \pi _1(C') \to \pi _1(C) (the homomorphism induced by the projection) is isomorphic to the compactification of \mathbb {Z}.
More precisely, we should show that the fundamental group of C is isomorphic to the (topological) free product of the fundamental group of C with the compactification of \mathbb {Z}.
We note that is was questions of this sort that gave birth to the "theory of descent" for schemes.
Let A be a complete integral local ring;
we know that its normalisation A' is finite over A (by Nagata), and is thus a complete semi-local ring, and thus local, since it is integral.
Suppose that the residue extension L/k that it defines is non-radicial (in the contrary case, we say that A is geometrically unibranch; cf. below).
This will be the case, for example, for the ring \mathbb {R}[{[s,t]}]/(s^2+t^2) \mathbb {R}[{[s,t]}], where \mathbb {R} is the field of real numbers.
Then let k' be a finite Galois extension of k such that L \otimes _k k' decomposes;
let B be a finite and étale algebra over A corresponding to the residue extension k' (recall that B is essentially unique).
Then the residue algebra of B'=A' \otimes _A B over B is L \otimes _k k', which is not local, and so B' is not a local ring, and thus B has zero divisors (since it is complete).
Now B' is contained in the total ring of fractions of B (since it is free over A', thus torsion free over A', thus torsion free over A, thus contained in B' \otimes _A K=B'_{(K)}=A'_{(K)} \otimes _K B_{(K)}=B_{(K)}, since A'_{(k)}=K), and so B is not integral.
In the case of the ring \mathbb {R}[s,t]/(s^2+t^2) \mathbb {R}[s,t], taking k'/k= \mathbb {C}/ \mathbb {R}, we see that B is the local ring of two secant lines at their point of intersection.
We note also that, if there exists a connected étale cover X of Y that is integral but not irreducible, then every irreducible component of X gives an example of an unramified cover X' of Y that dominates Y but is not étale over Y.
In the case of example (a), we thus see that C' is unramified over C, without being étale at the two points a and b (note that, directly, by inspection of the completions of the local rings at x and a, from the "formal" point of view, C' at the point a can be identified with a closed subscheme of C at the point x, i.e. one of the two "branches" of C passing through x).
In both (a) and (b), we see that the fact that the conclusions of (i) and (ii) in fail to hold is directly linked with the fact that a point of Y "blows up" at distinct points of the normalisation (in (b), the fact that the residue extension is non-radicial should be interpreted geometrically in this way).
More precisely, we say that an integral local ring A is geometrically unibranch if its normalisation has only a single maximal ideal, with the corresponding residue extension being radicial;
a point y of an integral prescheme is said to be geometrically unibranch if its local ring is geometrically unibranch.
Examples: a normal point, an ordinary cusp point of a curve, etc.
It seems that, if Y admits a point which is not unibranch, then there always exists a non-irreducible connected étale cover of Y;
at least, this is what we have shown in case (b), when Y is the spectrum of a complete local ring.
We can show, however, that if all the points of Y are geometrically unibranch, then every unramified connected Y-prescheme that dominates Y is étale and irreducible.
The proof follows that of , using the following generalisation of , which will be proved later by means of the technique of descent:(cf. [sga1-ix.4.10 (?)]. For a more direct demonstration, cf. EGA IV 18.10.3, using a variant of for geometrically unibranch local rings.)
Let Y' \to Y be a finite, radicial, and surjective morphism (i.e. what we could call a "universal homeomorphism").
Consider the functor X \mapsto X \times _Y Y'=X' from Y-preschemes to Y'-preschemes.
This functor induces an equivalence between the category of étale Y-schemes and the category of étale Y'-schemes.
We could apply this, for example, in the case where Y' is the normalisation of Y, with Y assumed to be unibranch (and Y' finite over Y, which is true in all the cases that one encounters in practice), or to to case of some Y'' "sandwiched" between Y and its normalisation (which no longer need be finite over Y).