3437fga3.iifga3.ii.xmlThe existence theorem and the formal theory of modulesFGA3.II1840fga3.ii-original-citationfga3.ii-original-citation.xmlOriginalfga3.iiA. Grothendieck. "Technique de descente et théorèmes d'existence en géométrie algébrique, II: Le théorème d'existence et théorie formelle des modules". Séminaire Bourbaki 12 (1959–60), Talk no. 195. http://www.numdam.org/book-part/SB_1958-1960__5__369_0/1841fga3.ii-afga3.ii-a.xmlRepresentable and pro-representable functorsAfga3.ii1842fga3.ii-a.1fga3.ii-a.1.xmlRepresentable functorsA.1fga3.ii-aLet be a category. For all , let be the contravariant functor from to the category of sets given by If we have a morphism in , then this evidently induces a morphism of functors; is a covariant functor in , i.e. we have defined a canonical covariant functor from to the category of covariant functors from the dual of to the category of sets. We then recall:864fga3.ii-a.1-proposition-1.1fga3.ii-a.1-proposition-1.1.xmlProposition1.1fga3.ii-a.1This functor is fully faithful; in other words, for every pair of objects of , the natural map is bijective.In particular, if a functor is isomorphic to a functor of the form , then is determined up to unique isomorphism. We then say that the functor is representable. The above proposition then implies that the canonical functor defines an equivalence between the category and the full subcategory of consisting of representable functors. This fact is the basis of the idea of a "solution of a universal problem", with such a problem always consisting of examining if a given (contravariant, as here, or covariant, in the dual case) functor from to is representable. Note further that, just by the definition of products in a category [Gro1957], the functor commutes with products whenever they exist (and, more generally, with finite or infinite projective limits, and, in particular, with fibred products, taking "kernels" [], etc., whenever such things exist): we have an isomorphism of functors whenever exists, i.e. we have functorial (in ) bijections In particular, the data of a morphism in (i.e. of a "composition law" in between , , and ) is equivalent to the data of a morphism , i.e. to the data, for all , of a composition law of sets such that, for every morphism in , the system of set maps is a morphism for the two composition laws, with respect to and . In this way, we see that the idea of a "-group" structure, or a "-ring" structure, etc. on an object of can be expressed in the most manageable way (in theory as much as in practice) by saying that, for every , we have a group law (resp. ring law, etc.) in the usual sense on the set , with the maps corresponding to morphisms that should be group homomorphisms (resp. ring homomorphisms, etc.). This is the most intuitive and manageable way of defining, for example, the various classical groups , , , etc. on a prescheme over an arbitrary base, and of writing the classical relations between these groups, or of placing a "vector bundle" structure on the affine scheme over defined by a quasi-coherent sheaf , and of defining and studying the many associated flag varieties (Grassmannians, projective bundles), etc.; the general yoga is purely and simply identifying, using the canonical functor , the objects of with particular contravariant functors (namely, representable functors) from to the category of sets.The usual procedure of reversing the arrows that is necessary, for example, in the case of affine schemes in order to pass from the geometric language to the language of commutative algebra, leads us to dualise the above considerations, and, in particular, to also introduce covariant representable functors , i.e. those of the form .1843fga3.ii-a.2fga3.ii-a.2.xmlPro-representable functors, pro-objectsA.2fga3.ii-aLet be a projective system of objects of ; there is a corresponding covariant functor which can be written more explicitly as which is a functor from to . A functor from to that is isomorphic to a functor of this type with filtered is said to be pro-representable. By the previous section, these are exactly the functors that are isomorphic to filtered inductive limits of representable functors. Let be another filtered projective system in (indexed by another filtered preordered set of indices ). Then we can easily show that we have a canonical bijection (generalising ). This leads to introducing the category of pro-objects of , whose objects are projective systems of objects of (indexed by arbitrary filtered preordered sets of indices), and whose morphisms between objects and are given by with the composition of pro-homomorphisms being evident. By the very construction itself, can be considered as a contravariant functor in , establishing an equivalence between the dual category of the category of pro-objects of and the category of pro-representable covariant functors from to . Of course, an object of canonically defines a pro-object, denoted again by , so that is equivalent to a full subcategory of . Then, if is an arbitrary pro-object of , then (with the above identification) we have that with the projective limit being taken in (since ).We draw attention to the fact that, even if the projective limit of the exists in , it will generally not be isomorphic to the projective limit in , as is already evident in the case where is the category of sets. We note that, by the definition itself, is defined by the condition that the functor in and with values in be representable via , i.e. that it be isomorphic to ; then is already defined in terms of the pro-object , and, in a precise way, depends functorially on the pro-object whenever it is defined; there is therefore no problem with denoting it by . If projective limits in always exist, then is a functor from to , and there is a canonical homomorphism of functors . Since every (covariant, say, for simplicity) functor can be extended in an obvious way to a functor it follows that, if projective limits always exist in , then also canonically defines a composite functor sending to .A pro-object is said to be a strict pro-object if it is isomorphic to a pro-object , where the transition morphisms are epimorphisms; a functor defined by such an object is said to be strictly pro-representable. We can thus further demand that be a filtered ordered set, and that every epimorphism be equivalent to an epimorphism for some suitable (uniquely determined by this condition). Under these conditions, the projective system is determined up to unique isomorphism (in the usual sense of isomorphisms of projective systems). It thus follows that the projective limit of a projective system of strict pro-objects always exists in , and that, with the above notation of and , we have that In particular, if every pro-object of is strict (cf. the previous section), then the extended functor commutes with projective limits.1844fga3.ii-a.3fga3.ii-a.3.xmlCharacterisation of pro-representable functorsA.3fga3.ii-aLet and be categories in which all finite projective limits (i.e. limits over finite, not necessarily filtered, preordered sets) exist, or, equivalently, in which finite products and finite fibred products exist (which implies, in particular, the exists of a "right-unit object" such that consists of only on element for all ). Let be a covariant functor from to . Then the following conditions are equivalent: commutes with finite projective limits; commutes with finite products and with finite fibred products; commutes with finite products, and, for every exact diagram in (cf. FGA 3.I, A, Definition 2.1), the image of the diagram under is exact. We then say that is left exact.In what follows, we assume that finite projective limits always exist in . It is then immediate from the definitions that a representable functor is left exact, and, by taking the limit, that a pro-representable functor is left exact.To obtain a converse, let be a covariant functor, and let and . We say that (or the pair ) is minimal if, for all and all , and for every strict monomorphism (cf. FGA 3.I, §A.2) such that , is an isomorphism. We also say that a pair dominates (where and ) if there exists a morphism such that ; if is minimal, and if is left exact, then this morphism is unique; if is minimal, then is surjective. From this we easily deduce the following proposition:1558fga3.ii-a.3-proposition-3.1fga3.ii-a.3-proposition-3.1.xmlProposition3.1fga3.ii-a.3For to be strictly pro-representable, it is necessary and sufficient that it satisfy the following two conditions: is left exact; and every pair , with , is dominated by some minimal pair. This second condition is trivial if every object of is Artinian (by taking a sub-object of that is minimal amongst those for which there exists some such that is the image of ). Whence:1562fga3.ii-a.3-proposition-3.1-corollaryfga3.ii-a.3-proposition-3.1-corollary.xmlCorollaryfga3.ii-a.3Let be a category whose objects are all Artinian and in which all finite projective limits exist. Then the pro-representable functors from to are exactly the left exact functors, and they are in fact strictly pro-representable.This last fact also implies that every pro-object of is then strict.1845fga3.ii-a.4fga3.ii-a.4.xmlExample: groups of Galois type, pro-algebraic groupsA.4fga3.ii-aIf is the category of ordinary finite groups, then is equivalent to the category of totally disconnected compact topological groups. ([Trans.] Here the word "Hausdorff" is implicit.) It is groups of this type, and their generalisations, obtained by replacing ordinary finite groups with schemes of finite groups over a given base prescheme (for example, finite algebraic groups over a field ), that serve as fundamental groups, homotopy groups, and absolute and relative homology groups for preschemes. In all these examples, the corollary to applies, and it is indeed by the associated functor that the should be defined []. It is the same if we start with the category of algebraic or quasi-algebraic groups over a field (or, more generally, over a Noetherian prescheme): we recover the "pro-algebraic groups" of Serre [Ser1958].1846fga3.ii-a.5fga3.ii-a.5.xmlExample: "formal varieties"A.5fga3.ii-aLet be a Noetherian ring, the category of -algebras that are Artinian modules of finite type over (or, more concisely, Artinian -algebras). The conditions of the corollary to are then satisfied. Here, the category is equivalent to the category of topological algebras over that are isomorphic to topological projective limits of algebras , i.e. those whose topology is linear, separated, and complete, and such that, for every open ideal of , the algebra is an Artinian algebra over . The functor associated to such an algebra is exactly Note also that the category is essentially the product of analogous categories, corresponding to the local rings that are the completions of the for the maximal ideals of ; we can thus, if so desired, restrict to the case where is such a complete local ring. In any case, decomposes canonically as the topological product of its local components, which correspond to the "points" of the formal scheme [] defined by . Such a point is defined by an object of some , where is a field (for example, the residue field of the local component in question), and where two pairs and define the same point if and only if they are both dominated by the same , or if they both dominate the same . (If the are algebraically closed, then it suffices to take the set given by the sum of the ).It is important to give conditions that ensure that the local component of corresponding to some be a Noetherian ring. If is a complete local ring (Noetherian, we recall), then it is equivalent to say that is isomorphic to a quotient ring of a formal series ring . To give such a criterion, we introduce (for every ring ) the -algebra of "dual numbers" of , defined by Let be the augmentation homomorphism, which defines (if ) a map Using the fact that is left exact, we intrinsically define the structure of an -module on the subset consisting of the that are "reducible along "; using the explicit form of in terms of , we find that this -module can be identified with , where is the kernel of the homomorphism , i.e. if is a field, then the maximal ideal of the local component of . From this, we immediately deduce the following proposition:579fga3.ii-a.5-proposition-5.1fga3.ii-a.5-proposition-5.1.xmlProposition5.1fga3.ii-a.5Let , where is a field. For the corresponding local component of to be a Noetherian ring, it is necessary and sufficient that the set of elements of that are reducible along be a vector space of finite dimension over . Under these conditions, we have a canonical isomorphism (where is the maximal ideal of given by the kernel of the homomorphism ), and so, in particular, the dimension of the -vector space is equal to the dimension of the vector space over the field .[Comp.] The formula given above is only correct when is a field; in the general case, we must replace with the quotient of this space by the image of , where is the maximal ideal of .Suppose that is Noetherian, and suppose, for notational simplicity, that is complete and local, and that . ([Comp.] The following definition is correct only when the residue extension is separable; for the general case, see [Gro1960b, III, 1.1].) We say that is simple over if is a finite and étale algebra over the completion algebra of the localisation of at one of its maximal ideals that induces the maximal ideal of ; if the residue extension of over is trivial (for example, if the residue field of is algebraically closed), then this is equivalent to saying that itself is isomorphic to such a formal series algebra. Finally, if we no longer necessarily suppose that is Noetherian, then we again say that is simple over if is isomorphic to a topological projective limit of quotient -algebras that are Noetherian and -simple in the above sense. We can immediately generalise to the case where and are no longer assumed to be local. With this, we have the following proposition:580fga3.ii-a.5-proposition-5.2fga3.ii-a.5-proposition-5.2.xmlProposition5.2fga3.ii-a.5For to be simple over , it is necessary and sufficient that the associated functor send epimorphisms to epimorphisms.If this is the case, then this implies that, for every surjective homomorphism in , the morphism is also surjective. Of course, it suffices to verify this condition in the case where is local, and (proceeding step-by-step) where the ideal of given by the kernel of is annihilated by the maximal ideal of . This leads, in practice, to verifying that a certain obstruction, linked to infinitesimal invariants of the situation that give us a functor , is null; this is a problem of a cohomological nature.To finish, we say some words, in the above context, about rings of definition. Let still be a functor from to , assumed to be pro-representable via a topological -algebra . Then, for every and every , there exists a smallest subring of such that is the image of an element of (which is then uniquely determined): indeed, it suffices to think of as a homomorphism from to , and to take to be the image of under this . We then say that is the ring of definition of the object . If is an algebra homomorphism, and if , then the ring of definition of is the image under of the ring of definition of . If we start with a functor from to , then the existence of rings of definition, along with their properties that we have just discussed, is more or less equivalent to the condition that be pro-representable; that is, they are usually far from being trivial.1847fga3.ii-bfga3.ii-b.xmlThe two existence theoremsBfga3.iiKeeping the notation of , and, given a covariant functor we wish to find manageable criteria for to be pro-representable, i.e. expressible via a -algebra as above. By the corollary of §A, Proposition 3.1, to ensure this, it is necessary and sufficient that be left exact. In the current state of the technique of descent (cf. the questions asked in FGA 3.I, §A.2.c), this criterion is not directly verifiable, in this form, in the most important cases, and we need criteria that seem less demanding.1603fga3.ii-b-theorem-1fga3.ii-b-theorem-1.xmlTheorem1fga3.ii-bFor the functor to be pro-representable, it is necessary and sufficient that it satisfy the two following conditions: commutes with finite products; for every algebra and every homomorphism in such that the diagram is exact (cf. FGA 3.I, §A, Definition 1.2), the diagram is also exact. Furthermore, it suffices to verify condition (ii) in the case where is local, and when, further, we are in one of the two following cases: is a free module over ; the quotient module is an -module of length . 1501#286unstable-286.xmlProoffga3.ii-b-theorem-1 The proof of this theorem is rather delicate, and cannot be sketched here. We content ourselves with pointing out that it relies essentially on a study of equivalence relations (in the sense of categories) in the spectrum of an Artinian algebra (the study of which poses even more problems, whose solutions seems essential for the further development of the theory). In applications, the verification of condition (i) is always trivial. The verification of condition (ii) splits into two cases: case (a), where is a free -module, can be dealt with using the technique of descent by flat morphisms (cf. FGA 1, Theorems 1, 2, and 3), which offers no difficulty; to deal with case (b), we will use the following result:1604fga3.ii-b-theorem-2fga3.ii-b-theorem-2.xmlTheorem2fga3.ii-bLet be a local Artinian ring with maximal ideal , and let be an -algebra containing , and such that , and is exact (which is the case, in particular, if is an -module of length ). Let be the fibred category (cf. FGA 3.I, §A, Definition 1.1) of quasi-coherent sheaves that are flat over varying preschemes. Then the morphism is a strict -descent morphism (cf. FGA 3.I, §A, Definition 1.7). 1488#287unstable-287.xmlProoffga3.ii-b-theorem-2 We prove this by first proving that (cf. FGA 3.I, §A.4.e), with the hypothesis that allowing us to easily reduce to the case where is a field (namely ). We can then apply the equivalences described in FGA 3.I, §A.4.e. In other words, the data of a flat -module is completely equivalent to the data of a flat -module endowed with an -isomorphism from to satisfying the usual transitivity condition for a descent data (loc. cit.).1848fga3.ii-cfga3.ii-c.xmlApplications to some particular casesCfga3.ii1849fga3.ii-c.1fga3.ii-c.1.xmlGeneral remarks on functors represented by preschemesC.1fga3.ii-cLet be a locally Noetherian prescheme. A prescheme over is said to be locally of finite type over if, for all that project to , there exists an affine neighbourhood of of ring , and an affine neighbourhood of (over the aforementioned affine neighbourhood of ) of ring , such that is an -algebra of finite type. There are many important examples of preschemes locally of finite type over , that are not of finite type over , given by solutions of classical universal problems; thus it is important to be able to consider the Picard scheme of a curve as a union of infinitely-many connected components (that we must avoid confusing with the connected component of the identity element, i.e. the "Picard variety"). It is thus sometimes useful to place ourselves in the category of preschemes locally of finite type over , in order to study the question of representability of a contravariant functor . The main goal of these articles is to develop a general technique that allows us to recognise when such a functor is representable, and to study the properties of the corresponding -prescheme by means of the properties of . We note in passing that, in this study, we find non-pathological examples of preschemes over that are not separated over , notably as "Picard preschemes" of excellent -schemes; we must thus refrain from banishing preschemes that are not schemes from algebraic geometry.Let be a prescheme locally of finite type over , and let be the associated contravariant functor. We can consider the restriction of to the subcategory of consisting of preschemes over that are Artinian and finite over : if , then is the category dual to the category of Artinian -algebras considered in . If , where is a local Artinian ring, then consists of a single point living above a closed point of , and an -homomorphism from to (i.e. an element of ) is defined by the data of a point over , along with an -homomorphism from to . If there exists such a homomorphism, then is necessarily a closed point of (since its residue field is algebraic over the residue field of ). This thus shows that the restriction of to "Artinian -algebras" is pro-representable, and is represented by the topological -algebra whose local components are the completions of the local rings of at the points of that are closed and live above closed points of . This shows that only knowing gives precise information about the structure of (that is, the structure of the completions of its local rings at the aforementioned points). We note that, even in the case where is the spectrum of an algebraically closed field, it is only thanks to the systematic consideration of "varieties" such that may admit nilpotent elements (and, in particular, working with the spectra of local Artinian rings) that we can arrive at the "good formulation" of classical universal problems, and understand the "infinitesimal" aspect.If we start with a given functor , and we want to know whether or not it is representable, then studying the functor (using and ) will give quasi-complete hints; either, as is often the case (by simply testing, for example, the nature of the sets and their functorial behaviour, cf. ), is already not pro-representable (which explains the failure of attempts made up until now to define varieties of modules in a reasonably natural way for the classification of vector bundles of rank 1]]>); or we might be able to show that is indeed representable, but that that vector spaces are not of finite dimension, in which case we must be content with the "formal" solution; or it could be the case that is indeed representable by a product of complete Noetherian local rings, which gives very strong assumptions for itself to be representable, and, combined with the analogous properties (but of a more global nature) that we will later develop, will in all likelihood suffice to imply that it is indeed so. Finally, we come across interesting geometric problems (see and below) where we have only the functor (not coming from any "global" functor ), and where we will consider ourselves content if we can associate to it a "formal scheme of modules".To finish these generalities, we note how the theory of schemes explains some apparent anomalies, such as the Igusa surface whose "Picard variety" consists of a single point, and for which, however, ; in this case, is a non-trivial "purely infinitesimal" group, i.e. defined by a local algebra of finite rank over the base field and endowed with a diagonal map corresponding to the multiplicative structure of ; if is the maximal ideal of , then the dual of is canonically isomorphic to (cf. below). It is only when the Picard group is an algebraic group in the classical sense (i.e. simple over the base field ) that the dimension of (which is always equal to that of ) is equal to that of the Picard group.1850fga3.ii-c.2fga3.ii-c.2.xmlThe schemes , , , etc.C.2fga3.ii-cLet and be preschemes over ; for every prescheme over , let and , and consider the set as a contravariant functor in . If it is representable, then we denote by the prescheme over that represents it, and we then have a functorial isomorphism There are variants of this universal problem, the solutions to which can be summarised as follows: a prescheme of -automorphisms of an -prescheme (which will be a prescheme in groups), a prescheme of -homomorphisms from an -prescheme in groups to another (which will be a prescheme in commutative groups if the latter scheme in groups is commutative), etc. We can also generalise the definition of by considering a prescheme over the prescheme over , and the functor (the set of "sections" of over ); if this functor is representable, then the -prescheme that represents it will be denoted by , and we will thus have, by definition, a functorial isomorphism Setting , we recover . From these definitions follows a formula for the new preschemes thus introduced that is as trivial as it is useful, that we will not give here (given that it holds in every category where products and fibred products exist). More serious is the question of existence of schemes of the type . We note first of all that, for fixed , (resp. ) can only exist for all over (resp. for all over ) if is flat over . Furthermore, we can convince ourselves that it is not reasonable to expect the existence of a solution, for general enough , except in the case where is further proper over . It seems, however, that these conditions are sufficient for the existence of and , with the condition that, if necessary, we make some sort of "quasi-projective" hypothesis on (resp. ); this is what we can verify anyway in numerous cases (for example, when is affine over , or, by direct elementary constructions, when is finite over ). Then and give:1752fga3.ii-c.2-proposition-2.1fga3.ii-c.2-proposition-2.1.xmlProposition2.1fga3.ii-c.2Let be a Noetherian ring, and and arbitrary preschemes over . Consider the functor on the category of Artinian -algebras. If is flat over , then this functor is pro-representable.Furthermore, we can show that, for all and all , we have a canonical isomorphism where is the sheaf of Kähler -differentials of with respect to . Taking to be a field, we find, using §A, Proposition 5.1 and the finiteness theorem from , the following corollary:1753fga3.ii-c.2-proposition-2.1-corollaryfga3.ii-c.2-proposition-2.1-corollary.xmlCorollaryfga3.ii-c.2Suppose that is flat and proper over , and that is of finite type over . Then is pro-representable, and the local components of the corresponding topological -algebra are Noetherian rings.1754fga3.ii-c.2-remarksfga3.ii-c.2-remarks.xmlRemarksfga3.ii-c.2The problems considered in this section, and many others, having been generally studied, in the framework of classical algebraic geometry, via the "Chow coordinates" of cycles in projective space, allow us to consider these cycles as points of suitable projective varieties. This procedure, and, more generally, the use of Chow coordinates, seems irredeemably insufficient from the point of view of schemes, since it destroys the nilpotent elements in the parameterised varieties, and, in particular, do not lend themselves to a satisfying study of infinitesimal variations of cycles (without taking its non-intrinsic nature, linked to the projective space, into account). The language of Chow coordinates has sadly been the only one used by many algebraic geometers for the study of families of varieties or families of cycles, which seems to have been a serious obstacle to the understanding of these notions, despite its certain technical interest (probably temporary). If we wish to obtain the analogue of Chow varieties in the theory of schemes, we are led to the following universal problem: let be a prescheme over , and, for every prescheme over , consider the set of closed sub-preschemes of that are flat over ; we want to represent this functor in via some prescheme over . More generally, we can start with a quasi-coherent sheaf on , and take to be the set of quotient sheaves of that are flat over . It seems that there exists a solution to this problem, with a scheme that is locally of finite type over , if is proper over , if is locally Noetherian, and if is furthermore coherent. In any case, supposing only that is locally Noetherian, the restriction of to "Artinian -algebras" is pro-representable, and, if, furthermore, is proper over , and is coherent, then the local components of the corresponding topological ring are Noetherian. Of course, even after having proven the existence of the "Chow scheme" of over , it remains to find a decomposition of it into disjoint open subsets (corresponding to fixing continuous invariants, such as degree and dimension of the cycles that we vary) over , as well as to make precise the relations between this scheme with the classical Chow varieties, and to make precise when a is projective (or at least quasi-projective) over .1755fga3.ii-c.2-remarkfga3.ii-c.2-remark.xmlRemarkfga3.ii-c.2[Comp.] The problems described here are completely resolved in the projective case by "Hilbert schemes" (cf. ). Examples by Nagata and Hironaka show, however, that the functors described are not necessarily representable if we do not make the projective hypothesis, even if we restrict to the classification of subvarieties of dimension of a complete non-singular variety of dimension ; this is linked to the fact that the symmetric square of such a variety does not necessarily exist.1851fga3.ii-c.3fga3.ii-c.3.xmlPicard schemesC.3fga3.ii-c[Comp.] For a more complete study, see .Let be an -prescheme, and consider the multiplicative sheaf of units of the structure sheaf of , along with the group called the relative Picard group of . An element of this group is thus defined by giving an open cover of , along with an invertible sheaf on each , such that is isomorphic to for all , or, at least locally over (i.e. these two sheaves are "equivalent" in the sense of FGA 3.I, §B.4). If admits a section, then is exactly the set of classes of invertible sheaves on up to "equivalence" (loc. cit.). We now set, for all over , which gives a covariant functor in , that we call the Picard functor of ; if this functor is representable, then the prescheme over that represents it is called the Picard prescheme of , and denoted by . In this case, we then have an isomorphism of functors: Taking the Picard prescheme is compatible with extension of the base, and, in particular, the Picard preschemes of the fibres of over (which are preschemes over the residue fields of the ) are the fibres of . Of course, since is a commutative group, the Picard preschemes are preschemes in groups. Note as well that the generalised Jacobians of Rosenlicht are exactly the connected components of the identity in the Picard schemes of complete curves (possibly with singularities), which should make most of their properties clear (once their existence has been proven).1807fga3.ii-c.3-remarkfga3.ii-c.3-remark.xmlRemarkfga3.ii-c.3The definition adopted here is only reasonable when every point of admits an open neighbourhood over which admits a section. In the general case, it is necessary to slightly modify the definition of the Picard functor in order to still obtain an existence theorem.Here, the plausible existence conditions for a Picard prescheme are the following: is proper and flat over ; ; and locally admits a section over . This condition naturally arises in the application of the technique of descent, in eliminating the automorphisms of an invertible sheaf on by endowing them with a marked section over the section (FGA 3.I, §B.4). Notably, we find the following:1808fga3.ii-c.3-proposition-3.1fga3.ii-c.3-proposition-3.1.xmlProposition3.1fga3.ii-c.3Suppose that is flat over , where is Noetherian, and suppose that, for all of finite type over , we have (if is proper and separable and has separable fibres, or if is the spectrum of a field, then it follows from Künneth that the latter condition is equivalent to ). Then the Picard functor of on the category of Artinian -algebras is pro-representable.Furthermore, we then have and, in particular:1809fga3.ii-c.3-proposition-3.1-corollaryfga3.ii-c.3-proposition-3.1-corollary.xmlCorollaryfga3.ii-c.3If is proper over , then the local components of the topological -algebra corresponding to the Picard functor are Noetherian.1810fga3.ii-c.3-remarks-ifga3.ii-c.3-remarks-i.xmlRemarksfga3.ii-c.3We can generalise the definitions and results from this section to the classification of principal bundles on , with structure group being a scheme in groups over that is affine and flat over , and also commutative. In the case where would not be commutative, and thus where the adjoint bundle in groups of a principal bundle (whose sections of the automorphisms of the principal bundle) would no longer be trivial, no longer holds true as it is stated. We can, however, modify the universal problem in such a way that we again obtain a solution (at least, for now, in formal geometry). The golden rule to remember, in the context of the current section and in the following, and every time we are looking for "schemes of modules" for classes of objects that are only defined up to isomorphism, is always the following: eliminate the possible automorphisms of the objects that we want to classify, by introducing, if necessary, auxiliary structures (points or elements of marked sections, fixing differential forms, etc.) that we take to be insignificant enough that we do not substantially modify the initial problem.1811fga3.ii-c.3-remarks-iifga3.ii-c.3-remarks-ii.xmlRemarksfga3.ii-c.3[Comp.] I have recently shown that the formal scheme of modules for an abelian variety over a field is indeed simple over the Witt ring, or, in other words, that every abelian scheme over a local Artinian ring that is the quotient of another such scheme comes, by reduction, from an abelian scheme over . The proof simply uses the variance properties of the obstruction class of the covering, introduced in FGA II, §6. Recall also that the schemes of modules for curves of genus or for polarised abelian schemes have been constructed by Mumford (cf. Séminaire Mumford–Tate, Harvard University (1961–62)).1852fga3.ii-c.4fga3.ii-c.4.xmlFormal modules of a varietyC.4fga3.ii-cLet be a local Noetherian ring of residue field (more often than not, will be equal to , or to a Cohen -ring), and let be a prescheme over . For every local Artinian -algebra , consider the set of isomorphism classes of -preschemes that are flat over , endowed with an isomorphism 1759fga3.ii-c.4-equation-starfga3.ii-c.4-equation-star.xmlEquation*fga3.ii-c.4 where is the residue field of ; of course, the isomorphisms between such flat -preschemes should respect the above isomorphism given in the structure. If is a (not necessary local) Artinian -algebra, with local components , then we take to be the product of the . Then becomes a multiplicative functor in , and we call it the functor of modules for (and ). If this functor is representable, then it has a corresponding local topological -algebra , of residue field , and the formal spectrum of is called the formal scheme of modules for (and ) (cf. for some details on this).Here, if we wish to apply the technique of descent, the "finite" automorphisms of are inoffensive, since they have no influence on the existence of automorphisms (in the precise sense above) of -preschemes ; the necessary and sufficient condition, if is not simply a field, for to not have any non-trivial -automorphisms is that where denotes the sheaf of -derivations (i.e. the tangent sheaf) of . We can easily show (at least, if is simple over ) that We thus conclude, as per usual:1760fga3.ii-c.4-proposition-4.1fga3.ii-c.4-proposition-4.1.xmlProposition4.1fga3.ii-c.4Suppose that . Then the formal scheme of modules for exists. If, furthermore, is proper over , then the formal scheme of modules is Noetherian.1761fga3.ii-c.4-remarksfga3.ii-c.4-remarks.xmlRemarksfga3.ii-c.4If is not assumed to be simple over , then can be identified with a sub--module of where we set , where is considered as a coherent sheaf on via the diagonal morphism , and where denotes the coherent sheaf of ideals on defined by the diagonal morphism. More precisely, an easy globalisation of Hochschild theory shows that the above can be identified with the set of classes, up to isomorphism, of sheaves of -algebras that are flat over , and endowed with an augmentation isomorphism (recall that we set ). The submodule is that which corresponds to the sheaves of commutative algebras. The simplicity hypotheses are thus not essential in the theory of modules, as implies. Recall (loc. cit.) that, in particular, every simple and proper algebraic curve over admits a formal scheme of modules that is simple over , and of relative dimension equal to if the genus is , and to if . These two latter cases no longer figure directly in . We can, however, recover them in the case of elliptic curves () thanks to the remarks that will follow.We can, of course, vary ad libitum by considering systems of schemes over endowed with various structures. Suppose, for example, that is an abelian scheme over , with a marked origin (i.e. is considered as a scheme in groups over ), and let be the set of isomorphism classes of abelian schemes over (i.e. of schemes in groups that are proper and simple over ) endowed with an isomorphism as in of abelian schemes. We can show that imposing a multiplicative structure (or even only a "unit section") eliminates the infinitesimal automorphisms, and that there thus exists a formal scheme of modules that corresponds to a complete local Noetherian ring . We can also show that, if is a proper and simple scheme with "absolutely connected" fibres over a locally Noetherian prescheme , then every multiplicative structure on that admits a unit section is necessarily associative and commutative (provided that it is associative and commutative on one fibre, and provided that is connected), and is furthermore uniquely determined by the knowledge of the unit section. Further, supposing that is the spectrum of a local Artinian ring of residue field , that is proper over and endowed with a section , and finally that is endowed with the structure of an abelian scheme over , admitting the point of corresponding to as the zero element, an easy calculation of obstructions, combined with an argument due to Serre, allows us to prove that there exists on a multiplicative structure admitting the section as the unit section. (From here, using the "existence theorem" of to pass to the case where is complete local Noetherian, and then the technique of descent from for the general case, we can prove the analogous claim for all locally Noetherian connected ). This proves that the functor considered here is isomorphic to the analogous functor defined at the start of this section by abstracting the multiplicative structure on . It then follows that, in particular, if is the maximal ideal of , then is canonically isomorphic to the dual of , and is thus of dimension , where . It would be very interesting to determine if is indeed simple over , i.e. isomorphic to an algebra of formal series in variables over . Now §A, Proposition 5.2 allows us to give an equivalent formulation of this problem as an existence problem of abelian schemes that are reducible along a given abelian scheme. In any case, we see, by a transcendental way, that the answer is affirmative if is of characteristic . In characteristic , it evidently suffices to restrict to the case where is the ring of Witt vectors constructed over an algebraically closed field . This could be the moment for the "Greenberg functor" to prove its worth...1853fga3.ii-c.5fga3.ii-c.5.xmlExtension of coveringsC.5fga3.ii-cLet be a formal Noetherian prescheme [], an open subset of defined locally by the "non-vanishing" of a section of that is not a zero divisor (and thus large enough that every section of over an open subset that is zero on is zero). ([Comp.] It is also necessary to assume that the section defining is not a zero divisor not only on , but also on every .) Let be an "ideal of definition" for , and let , which is thus a ordinary Noetherian prescheme. Then, if and are flat coverings of (i.e. preschemes over defined by sheaves of algebras that are coherent and locally free as sheaves of modules) that are unramified over , the evident map is injective; in particular, an automorphism of that induces the identity on is the identity. This allows us to apply the technique of descent to the situation. We start, in particular, with a flat covering of , unramified over , and let be the set of classes, up to isomorphism (inducing the identity on ), of flat coverings of that induce on (and that are thus necessarily unramified over ). We similarly define for every open subset of , and, more generally, for every formal prescheme over . With this, the results of and imply, first of all, the following results: If varies amongst open subset of , then the form a sheaf on , say . The restriction of this sheaf to is the constant sheaf whose fibres consist of a single element. More generally, describing the fibres of is a question of complete local rings, in a precise way: For all , we have (i.e. isomorphism classes of finite free algebras over endowed with an isomorphism from to , where is the sheaf of algebras on that defines ). We have a canonical isomorphism ; in other words, for every open subset of , we have . Suppose that comes from an ordinary proper scheme over a complete local Noetherian ring that has ideal of definition by taking the -adic completion of , where . Then is canonically isomorphic to the set of classes of flat coverings of the ordinary scheme that are "reducible along ". Figuratively speaking, we can say that (a) and (b) establish the fundamental relations between the local and global aspects of the problem; (c) gives the relations between the "finite" and "infinitesimal" aspects; and finally (d) remembers (under precise conditions) the identity between the "formal" and "algebraic" aspects.Now suppose that is defined by a local complete Noetherian ring , with (and so is a prescheme over ). For every algebra that is finite over , we set This is a covariant functor in , with values in the category of sets, and, by (c), this functor is completely determined by how it acts on Artinian algebras ; it is equivalent to say either that this functor is pro-representable, i.e. of the form where is a topological algebra of the type described in , or that this is true when we restrict to only Artinian algebras . The combination of and then effectively implies:1772fga3.ii-c.5-proposition-5.1fga3.ii-c.5-proposition-5.1.xmlProposition5.1fga3.ii-c.5The above functor is pro-representable.Of course, by (a), if , then consists of a single element for all over , and the functor is then not very interesting (we will have ). It seems that, in practically every other case, the topological local ring is not Noetherian. Its existence, however, shows, in a striking manner, the "continuous" nature of the set of solutions (corresponding intuitively to the fact that there is a "continuous" choice in the way in which the ramification spreads when we take an extension of ). We will compare this result with the point of view of J.-P. Serre [@Ser1958] via local class field theory, drawing attention as well to the continuous character of the topological Galois group of the maximal abelian extension of a "geometric" local field, with the dual group (in the sense of Pontrjagin) appearing as an inductive limit of algebraic (or at least quasi-algebraic) groups; here as well, the classification of extensions is given by infinite-dimensional "varieties". We can also take, in the above, to be the formal spectrum of a complete local ring (of which will be, for example, a Cohen subring), and we might hope that the results of this section can be used in the study of extensions of a local complete ring of dimension 1]]>. Just as much in the local case as in the global case, they might allow us to formulate precise relations between the phenomena of higher ramification and phenomena in characteristic (approachable via a transcendental way). In any case, it is the preliminary analysis of that allows us to extend the methods described in for the study of the fundamental group to the "tamely ramified" case, and to resolve, by a transcendental way, the "problem of three points".To finish, we note that the situation simplifies if is of dimension ; then, by (a) and (b), can be identified with , where the are the points of : we can take arbitrary "local" extensions at the ramification points. Further, if is normal, then we note that the formal scheme of modules guaranteed by is simple over .References3438Ser1958Ser1958.xmlCorps locaux et isogéniesReference1960J.P. SerreSéminaire Bourbaki 11 Talk no. 1853440Gro1960bGro1960b.xmlSéminaire de Géométrie AlgèbriqueReference1960A. GrothendieckParis, Institut des Hautes Études Scientifiques3442Gro1957Gro1957.xmlSur quelques points d'algèbre homologiqueReference1957A. GrothendieckTohoku math. J. 9 pp. 119–221Backlinks3444fga3.vi-3-theorem-3.5fga3.vi-3-theorem-3.5.xmlTheorem3.5fga3.vi-3Under the conditions of , let be such that is simple over (or, equivalently, such that ). Then there exists an open neighbourhood of such that is simple over at the points of , which is thus an open abelian subscheme in . A fortiori, exists. 2260#240unstable-240.xmlProoffga3.vi-3-theorem-3.5 We describe the principle of the proof. The above allows us to reduce to the case where is the spectrum of an Artinian local ring , and we argue by induction on the infinitesimal order of . We can thus suppose that is simple over , and reduce to proving that is simple over . Note that, for this, it suffices to construct an abelian scheme over that extends , along with an invertible module on that extends the invertible module on that arises in the definition of the Picard scheme as the solution to a universal problem. (N.B. We can suppose that is endowed with a section over ...). For this construction, we must use the following key result: every abelian scheme defined over a quotient of an Artinian local ring can be extended (in other words, the absolute "formal scheme of modules" () for an abelian scheme over an algebraically closed field is simple over the ring of Witt vectors over ); this result can be obtained by using the general formal properties of the obstruction to lifting, and the group operations. With this result, we start by extending arbitrarily to ; we then find an obstruction to lifting , found in (where ), and more precisely in the subspace (taking into account the fact that the restriction of to the two factors and is trivial). But this latter space is exactly , where is the tangent sheaf to , and thus also the space that expresses the indeterminacy that there was in the lifting of to (). So we can correct this lifting (in exactly one way, as should be the case) in such a way as to kill the obstruction to lifting . 3445fga3.iii-2fga3.iii-2.xmlQuotient preschemes › Example: finite preschemes over 2fga3.iiiLet be the category of finite preschemes over , which is assumed to be locally Noetherian. Then is equivalent to the opposite category of the category of coherent sheaves of commutative algebras on , or, if is affine of ring , then it is equivalent to the opposite category of the category of finite -algebras over (i.e. those that are modules of finite type over ). We thus immediately conclude that, in , finite projective limits and finite inductive limits exist. This is well known (without any finiteness hypotheses) for the former; the fibre product of preschemes and over corresponds to the tensor product of corresponding algebras, and the kernel of two morphisms , defined by two -algebra homomorphisms , corresponds to the quotient of by the ideal generated by the , etc. For finite inductive limits, it suffices to consider, on one hand, finite sums, which correspond to the ordinary product of -algebras, and, on the other hand, cokernels of pairs of morphisms , which correspond (as we can immediately see) to the sub-ring of given by elements where the homomorphisms agree (with this sub-ring being finite over thanks to the Noetherian hypothesis). We also note that we can show, using the Noetherian hypothesis, that finite inductive limits, and, in particular, quotients, thus constructed in the category of finite preschemes over are, in fact, quotients in the category of all preschemes.As we mentioned in , there are non-effective epimorphisms in (or even non-strict, which is the same, since fibre products exist). I do not know if equivalence relations are still effective if we have no flatness hypothesis. I have only obtained, in this direction, very partial, positive, results, that are vital for the proof of the fundamental theorem of the formal theory of modules (cf. FGA 3.II, §B, Theorem 1). We note that it is easy, in the given problem, to reduce to the case where is the spectrum of a local Artinian ring, with an algebraically closed residue field. But even if is a field, the answer is not known.We can also consider the case of a prescheme over that is no longer assumed to be finite over , but by considering an equivalence relation on such that is a finite morphism. We then say that is a finite equivalence relation. Supposing, for simplicity, that and are affine (which implies that is affine, so that the situation is reduced to one of pure commutative algebra), we do not know, even in this case, if there exists a quotient , and if the canonical morphism is finite. (The most simple case is that where we suppose that is the spectrum of a field , and where is the spectrum of , i.e. the affine line). Of course, if the two problems above turn out to be true, then we can conclude that, in the situation described, is effective. Note that the problem of existence of a quotient and of the finiteness of are stated in exactly the same terms if, instead of an equivalence graph in , we only have an equivalence pregraph in , in the sense of .The question of passing to the quotient by a more or less arbitrary finite equivalence relation arises in the construction of preschemes by "gluing" given preschemes along certain closed sub-preschemes; the gluing law is expressed precisely by a finite equivalence relation on the prescheme given by the sum of the . We also expect that the solutions of the problems stated here, as well as of their many variations, will be a preliminary condition for the clarification of a general technique for non-projective constructions, in the direction introduced in .The only general positive fact known to the author is the following:1765fga3.iii-2-proposition-2.1fga3.iii-2-proposition-2.1.xmlProposition2.1fga3.iii-2Let be a locally Noetherian prescheme, a point of , and an algebraically closed extension of . Consider the corresponding "fibre functor" , that associates, to any -scheme that is finite over , the set of points of with values in . This functor (which is trivially left exact) is right exact, i.e. it commutes with finite inductive limits, and, in particular, with the cokernel of pairs of morphisms.By using this result for all the "geometric points" of , we thus deduce that the "quotient" category of , given by arguing "modulo surjective radicial morphisms" (i.e. by formally adjoining inverses for such morphisms), is a "geometric" category, i.e. it satisfies the same "finite nature" properties as the category of sets. In particular, every equivalence relation is effective. This implies that, if is an equivalence relation on , where is finite over , then the canonical morphism (where ) is radicial and surjective (and, in fact, a surjective closed immersion, since it is a monomorphism).3446fga3.ifga3.i.xmlGeneralities, and descent by faithfully flat morphismsFGA3.I583fga3.i-original-citationfga3.i-original-citation.xmlOriginalfga3.iA. Grothendieck. "Technique de descente et théorèmes d'existence en géométrie algébrique, I: Généralités. Descente par morphismes fidèlement plats". Séminaire Bourbaki 12 (1959–60), Talk no. 190. http://www.numdam.org/book-part/SB_1958-1960__5__299_0/[Comp.] For various details concerning the theory of descent, see also [Gro1960b, VI, VII, and VIII].From a technical point of view, the current article, and those that will follow, can be considered as variations on Hilbert's celebrated "Theorem 90". The introduction of the method of descent in algebraic geometry seems to be due to A. Weil, under the name of "descent of the base field". Weil considered only the case of separable finite field extensions. The case of radicial extensions of height 1 was then studied by P. Cartier. Lacking the language of schemes, and, more particularly, lacking nilpotent elements in the rings that were under consideration, the essential identity between these two cases could not have been formulated by Cartier.Currently, it seems that the general technique of descent that will be explained (combined with, when necessary, the fundamental theorems of "formal geometry", cf. ) is at the base of the majority of existence theorems in algebraic geometry. ([Trans.] [Comp.] It now seems excessive to say that the technique of descent is "at the base of the majority of existence theorems in algebraic geometry". This is true to a large extent for the non-projective techniques that are the object of study of the first two talks of this current series (i.e. "Techniques of descent and existence theorems in algebraic geometry"), but not for the projective techniques (talks IV, V, and VI).) It is worth noting as well that this aforementioned technique of descent can certainly be transported to "analytic geometry", and we can hope that, in the not-too-distant future, specialists will know how to prove the "analytic" analogues of the existence theorems in formal geometry that will be given in talk II. In any case, the recent work of Kodaira–Spencer, whose methods seem unfit for defining and studying "varieties of modules" in the neighbourhood of their singular points, shows reasonably clearly the necessity of methods that are closer to the theory of schemes (which should naturally complement transcendental techniques).In the present talk (namely talk I) we will discuss the most elementary case of descent (the one indicated in the title). The applications of , , and below (in ) are, however, already vast in number. We will restrict ourselves to giving only some of them as examples, without aiming for the maximum generality possible.We will freely use the language of schemes, for which we refer to the already cited article, as well as [GR1958]. We make clear to point out, however, that the preschemes considered in this current article are not necessarily Noetherian, and that the morphisms are not necessarily of finite type. So, if is a local Noetherian ring, with completion , then we will need to consider the non-Noetherian ring , as well as the morphisms of affine schemes that correspond to the inclusions between the rings in question.584fga3.i-afga3.i-a.xmlPreliminaries on categoriesAfga3.i585fga3.i-a.1fga3.i-a.1.xmlFibred categories, descent data, -descent morphismsA.1fga3.i-a586fga3.i-a.1.afga3.i-a.1.a.xmlA.1.afga3.i-a.1587fga3.i-a.1-definition-1.1fga3.i-a.1-definition-1.1.xmlDefinition1.1fga3.i-a.1.aA fibred category with base (or over ) consists of a category for every , a category for every -morphism , a functor , which we also write as for (with being thought of as an "object of over ", i.e. as being endowed with the morphism ) for any two composible morphisms , an isomorphism of functors with the above data being subject to the conditions that is the identity isomorphism if or is an identity isomorphism for any three composible morphisms , the following diagram, given by using the isomorphisms of the form , commutes: 592fga3.i-a.1-example-1fga3.i-a.1-example-1.xmlExample1fga3.i-a.1.aLet be a category where all fibre products exist. We then define a fibred category with base by setting to be the category of objects of over , and the functor corresponding to a morphism being defined by the fibre product .593fga3.i-a.1-example-2fga3.i-a.1-example-2.xmlExample2fga3.i-a.1.aLet be the category of preschemes, and, for , let be the category of quasi-coherent sheaves of modules on . If is a morphism of preschemes, then is the inverse image of sheaves of modules functor. We thus obtain a category fibred over .594fga3.i-a.1.bfga3.i-a.1.b.xmlA.1.bfga3.i-a.1595fga3.i-a.1-definition-1.2fga3.i-a.1-definition-1.2.xmlDefinition1.2fga3.i-a.1.bA diagram of maps of sets is said to be exact if is a bijection from to the subset of consisting of the such that .596fga3.i-a.1-definition-1.3fga3.i-a.1-definition-1.3.xmlDefinition1.3fga3.i-a.1.bLet be a fibred category with base , and consider a diagram of morphisms in such that ; this diagram is said to be -exact if, for every pair of elements of , the diagram of sets 597fga3.i-a.1-definition-1.3-equationfga3.i-a.1-definition-1.3-equation.xmlEquation+fga3.i-a.1-definition-1.3 (where ) is exact.In this diagram above, for simplicity, we have identified with , using .598fga3.i-a.1-definition-1.4fga3.i-a.1-definition-1.4.xmlDefinition1.4fga3.i-a.1.bLet be a fibred category with base , and consider morphisms in . Let . We define a gluing data on (with respect to the pair ) to be an isomorphism from to . If are both endowed with gluing data, then a morphism in is said to be compatible with the gluing data if the following diagram commutes: >> \beta _2^*(\xi ') \\@VVV @VVV \\\beta _1^*(\eta ') @>>> \beta _2^*(\eta '). \end {CD} ]]>With this definition, the objects of that are endowed with gluing data (with respect to and ) then form a category. If is a morphism such that , then, for every , the object of is endowed with a canonical gluing data, since where we again set ; furthermore, if is a morphism in , then is a morphism in that is compatible with the canonical gluing data. We thus obtain a canonical functor from the category to the category of objects of endowed with gluing data with respect to the pair . With this, we can also rephrase by saying that is -exact if the above functor is fully faithful, i.e. if the above functor defines an equivalence between the category and a subcategory of the category of objects of endowed with gluing data with respect to .599fga3.i-a.1-definition-1.5fga3.i-a.1-definition-1.5.xmlDefinition1.5fga3.i-a.1.bWe say that a gluing data on is effective (with respect to ) if , endowed with this data, is isomorphic to for some .In the case where is -exact, the object in is then determined up to unique isomorphism, and the category is equivalent to the category of objects of endowed with effective gluing data.600fga3.i-a.1.cfga3.i-a.1.c.xmlA.1.cfga3.i-a.1The most important case is that where with the being the two projections and from to its two factors (where we now suppose that has all fibre products). We then have two immediate necessary conditions for a gluing data on some to be effective: , where denotes the diagonal morphism, and where we identify with . , where denotes the canonical projection from to the partial product of its th and th factors. 604fga3.i-a.1-definition-1.6fga3.i-a.1-definition-1.6.xmlDefinition1.6fga3.i-a.1.cWe define descent data on , with respect to the morphism , to be a gluing data on with respect to the pair of canonical projections that satisfies conditions (i) and (ii) above.605fga3.i-a.1-definition-1.7fga3.i-a.1-definition-1.7.xmlDefinition1.7fga3.i-a.1.cA morphism is said to be an -descent morphism if the diagram of morphisms is -exact (); we say that is a strict -descent morphism if, further, every descent data () on any object of is effective. This latter condition (of strictness) can also be stated in a more evocative way: "giving an object of is equivalent to giving an object of endowed with a descent data".Note that, if an -descent morphism ([Comp.] It is useless to assume here that is an -descent morphism.) admits a section (i.e. a morphism such that ), then it is a strict -descent morphism: if is endowed with descent data with respect to , then "it comes from" .606fga3.i-a.1.dfga3.i-a.1.d.xmlA.1.dfga3.i-a.1We can present the above notions in a more intuitive manner, by introducing, for an object of over , the set where elements will be denoted by , , etc. Given an object , there then corresponds, to every , an object of , which will also be denoted by . A gluing data on (with respect to ) is then defined by the data, for every over , and every pair of points , of an isomorphism (satisfying the evident conditions of functoriality in ). Conditions (i) and (ii) of can then be written as , for all and all . , for all and all . We can show that (ii bis) implies that , by taking , and thus, since is an isomorphism by hypothesis, implies (i bis), which is thus a consequence of (ii bis) (and so (i) is also a consequence of (ii)). But if we no longer suppose a priori that the are isomorphisms (i.e. that is an isomorphism), then (ii bis) no longer necessarily implies (i bis); the combination of (ii bis) and (i bis), however, does imply that the are isomorphisms (since we then have ).610fga3.i-a.2fga3.i-a.2.xmlExact diagrams and strict epimorphisms, descent morphisms, and examplesA.2fga3.i-a611fga3.i-a.2.afga3.i-a.2.a.xmlA.2.afga3.i-a.2612fga3.i-a.2-definition-2.1fga3.i-a.2-definition-2.1.xmlDefinition2.1fga3.i-a.2.aLet be a category. A diagram of morphisms is said to be exact if, for all , the corresponding diagram of sets is exact (). We then say that (or, by an abuse of language, ) is a kernel of the pair of morphisms.This kernel is evidently determined up to unique isomorphism. If is the category of sets, then the above definition is compatible with . Dually, we define the exactness of a diagram of morphisms in and then say that is a cokernel of the pair of morphisms.613fga3.i-a.2-definition-2.2fga3.i-a.2-definition-2.2.xmlDefinition2.2fga3.i-a.2.aA morphism is said to be a strict epimorphism if it is an epimorphism and, for every morphism , the following necessary condition is also sufficient for to factor as : for every and every pair of morphisms such that , we also have that .If the fibre product exists, then it is equivalent to say that the diagram is exact, i.e. that is a cokernel of the pair . In any case, a cokernel morphism is a strict epimorphism. Note also that, if a strict epimorphism is also a monomorphism, then it is an isomorphism. We leave to the reader the task of developing the dual notion of a strict monomorphism.To make the relation between the ideas of -descent morphisms and strict epimorphisms more precise, we introduce the following definitions:614fga3.i-a.2-definition-2.3fga3.i-a.2-definition-2.3.xmlDefinition2.3fga3.i-a.2.aA morphism is said to be a universal epimorphism (resp. a strict universal epimorphism) if, for every over , the fibre product exists, and the projection is an epimorphism (resp. a strict epimorphism).In very nice categories (such as the category of sets, the category of sets over a topological space, abelian categories, etc.), the four notions of "epijectivity" introduced above all coincide; they are, however, all distinct in a category such as the category of preschemes, or the category of preschemes over a given non-empty prescheme , even if we restrict to -schemes that are finite over .615fga3.i-a.2-definition-2.4fga3.i-a.2-definition-2.4.xmlDefinition2.4fga3.i-a.2.aA morphism is said to be a descent morphism (resp. a strict descent morphism) if it is an -descent morphism (resp. a strict -descent morphism) (cf. ), where denotes the fibred category over of objects of over objects of (cf. ).616fga3.i-a.2-proposition-2.1fga3.i-a.2-proposition-2.1.xmlProposition2.1fga3.i-a.2.aIf has all finite products and (finite) fibre products, then there is an identity between descent morphisms in and strict universal epimorphisms in .617fga3.i-a.2.bfga3.i-a.2.b.xmlA.2.bfga3.i-a.2618#263unstable-263.xmlExamplefga3.i-a.2.bLet be the category of preschemes. Let , and let and be preschemes that are finite over , i.e. that correspond to sheaves of algebras and over that are quasi-coherent (as sheaves of modules) and of finite type (i.e. coherent, if is locally Noetherian). Let be the structure morphism of , and let and be -morphisms from to , defined by algebra homomorphisms , which we also denote by and . Using the fact that a finite morphism is closed (the first Cohen–Seidenberg theorem), we can easily prove that the diagram in 619fga3.i-a.2.b-equationfga3.i-a.2.b-equation.xmlEquation+#263 is exact if and only if the diagram of sheaves on is exact. In particular, if is a finite morphism corresponding to a sheaf of algebras on , then is a strict epimorphism if and only if the diagram of sheaves is exact (it is an epimorphism if and only if is injective). If is affine of ring , then is affine of ring , with finite over , and so is a strict epimorphism if and only if is an isomorphism from to the subring of consisting of the such that (it is an epimorphism if and only if is injective). As we have already mentioned, even if is the scheme of a local Artinian ring, then a finite morphism that is an epimorphism is not necessarily a strict epimorphism. However, we can prove that, if is a Noetherian prescheme, then every finite morphism that is an epimorphism is the composition of a finite sequence of strict epimorphisms (also finite). This also shows that the composition of two strict epimorphisms is not necessarily a strict epimorphism.620fga3.i-a.2.cfga3.i-a.2.c.xmlA.2.cfga3.i-a.2If is an exact diagram of finite morphisms, then, for every flat morphism of prescheme, the diagram induced from by a change of base is again exact. It thus follows that, if and are -preschemes, with flat over , then the following diagram of maps (where and are the inverse images of and over , and and are their inverse images over ) is exact: In particular, if denotes the fibred category (over the category of preschemes) such that, for , is the category of flat -preschemes, then the diagram is -exact. (This result becomes false if we do not impose the flatness hypothesis; in particular, a finite strict epimorphism is not necessarily a descent morphism). We similarly see that is -exact if denotes the fibred category for which is the category of flat quasi-coherent sheaves on the prescheme (here, again, the flatness hypothesis is essential). In either case, the question of effectiveness of a gluing data (and, more specifically, of a descent data, when ) on a flat object over is delicate (and its answer in many particular cases in one of the principal objects of these current articles). The speaker does not know if, for every finite strict epimorphism , every descent data on a flat quasi-coherent sheaf on is effective (even if we suppose that is the spectrum of a local Artinian ring, and we restrict to locally free sheaves of rank ). More generally, let be a ring, and an -algebra (where everything is commutative) such that the diagram is exact, which is equivalent to saying that the corresponding morphism between the spectra of and is an -descent morphism, where is the fibred category of flat quasi-coherent sheaves. Let be a flat -module endowed with a descent data to , i.e. with an isomorphism of -modules that satisfies conditions (i) and (ii) of (which we leave to the reader to write out in terms of modules). Is this data effective (relative to the fibred category of flat quasi-coherent sheaves)? Let be the subset of consisting of the such that which is a sub--module of . The canonical injection defines a homomorphism of -modules . The effectiveness of then implies the following: is a flat -module, and the above homomorphism is an isomorphism.621fga3.i-a.2.c-remarkfga3.i-a.2.c-remark.xmlRemarkfga3.i-a.2.cIn the above, we have imposed no flatness hypotheses on the morphisms of the diagram , and this obliges us, in order to have a technique of descent, to impose flatness hypotheses on the objects over and that we consider. In , we will impose a flatness hypothesis on , which will allow us to have a technique of descent for objects over and that are no longer under any flatness hypotheses. In any case, there is a flatness hypothesis involved somewhere. This is one of the main reasons for the importance of the notion of flatness in algebraic geometry (whose role could not be visible when we restricted to base fields, over which everything, in fact, is flat!).622fga3.i-a.3fga3.i-a.3.xmlApplication to étalementsA.3fga3.i-aLet be a local ring, and a local algebra over whose maximal ideal induces that of . We say that is étalé over (instead of "unramified", as used elsewhere) if it satisfies the following conditions: is flat over ; and is a separable finite extension of (where denotes the maximal ideal of ). If and are Noetherian, and is algebraically closed, then this implies that the homomorphism between the completions that extends is an isomorphism. A morphism of finite type is said to be étale at , or is said to be étalé over at , if is étalé over ; is said to be étale, or an étalement, or is said to be étalé over , if is étale at all . Note that, if is locally Noetherian, then the set of points of where is étale is open, and Zariski's "main theorem" allows us to precisely state the structure of in a neighbourhood around such a point (by an equation of well-known type).If is a scheme of finite type over the field of complex numbers, then there exists a corresponding analytic space (in the sense of Serre [Ser1956]), except for the fact that can have nilpotent elements in its structure sheaf, which changes nothing essential in [Ser1956]. We then easily see that is an étalement if and only if is an étalement, i.e. if every point of admits a neighbourhood on which induces an isomorphism onto an open subset of . In particular, to every étale covering of (i.e. every finite étale morphism ), there is a corresponding étale covering of , which is connected if and only if is connected [Ser1956]. We can also easily see that, if and are étale schemes over , then the natural map is bijective, i.e. the functor from the category of étale schemes over to the category of analytic spaces over is "fully faithful", and thus defines an equivalence between the first category and a subcategory of the second. A theorem of Grauert–Remmert [GR1958] implies that, if is normal, then we thus obtain an equivalence between the category of étale coverings of and the category of (finite) étale coverings of , i.e. every étale covering of is -isomorphic to some , where is an étale covering of . We will show that the Grauert–Remmert theorem remains true without any normality hypotheses on . First let be a finite strict epimorphism, and suppose that the theorem has been proven for ; we will show that it holds for . Let be an étale covering of , and consider its inverse image over , which corresponds to a coherent analytic sheaf of algebras on that is the inverse image of a sheaf of algebras on defining . By hypothesis, on , comes from an étale covering of , i.e. comes from a coherent sheaf of algebras on . Also, is endowed with a canonical descent data with respect to , i.e. with an isomorphism between its two inverse images on (satisfying conditions (i) and (ii)), and this isomorphism comes from, by what has already been said, an isomorphism between the corresponding algebraic sheaves, i.e. from a descent data on with respect to . We can easily show that the latter is effective (since it is effective on , and the effectiveness of a descent data, as described explicitly in the previous section, is something that can be checked locally on the completions of the modules that are involved). From this, we obtain a coherent sheaf of algebras on that defines a covering of , and this is the desired covering. The above result then obviously holds true if is just a composition of a finite number of finite strict epimorphisms, i.e. is just an arbitrary finite epimorphism (by the factorisation result stated in ). It thus follows that the Grauert–Remmert theorem holds true if is a reduced scheme, i.e. such that has no nilpotent elements, as we can see by introducing its normalisation . We can easily pass to the general case.A completely analogous argument, again using the factorisation result for finite strict epimorphisms, and the "formal" nature of the effectiveness of descent data, allows us to prove the following result: let be a locally Noetherian prescheme, and let be a finite, surjective, and radicial morphism (or, equivalently, a morphism of finite type such that, for every over , the morphism is a homeomorphism, which we can also express by saying that is a universal homeomorphism). For every étalé over , consider its inverse image , which is étalé over . Then the functor is an equivalence between the category of preschemes that are étalé over and the category of preschemes that are étalé over . (We use the bijectivity of for preschemes and that are étalé over , which can be proven directly without difficulty. We also use the fact that the stated theorem is true if , where is a nilpotent coherent sheaf of ideals of , cf. [Mur1958, Lemma 6]). Note also that we do not suppose here that the in question are finite over . This result implies, in particular, that the morphism induces an isomorphism between the fundamental group of and the fundamental group of ("topological invariance of the fundamental group of a prescheme").626fga3.i-a.4fga3.i-a.4.xmlRelations to 1-cohomologyA.4fga3.i-a627fga3.i-a.4.afga3.i-a.4.a.xmlA.4.afga3.i-a.4Let be a category such that the product of any two objects always exists, and let . For every finite set , we can consider , and so we obtain a covariant functor from the category of non-empty finite sets to the category , i.e. what we can call a simplicial object of , denoted by . This object depends covariantly on ; also, if and are morphisms , then the corresponding morphisms are homotopic. We say that dominates if , and this gives an (upward) directed preorder on . It follows from the above that, if dominates , then there exists a canonical class (up to homotopy) of homomorphisms of simplicial objects ; in particular, if and are such that and dominate one another, then and are homotopically equivalent. Now let be a (contravariant, to be clear) functor from to an abelian category . Then is a cosimplicial object of , and thus defines, in a well-known way, a (cochain) complex in , of which we can take the cohomology: (we may write a subscript "" on the if there is any possibility for confusion). This is a cohomological functor in , of which the variance for varying follows from what has already been said about the ; more precisely, for fixed and varying in (preordered by the domination relation), the form an inductive system of graded objects of ; in particular, if and are such that each one dominates the other, then and are canonically isomorphic.Suppose that has all fibre products. Then we can, for fixed , apply the above to the category of objects of over ; we then write and instead of and if we wish to make clear that we are working in the category ; then is a cochain complex in that, in degree , is equal to (where there are factors ).Note that, as per usual, we can define without assuming the category to be abelian: it is the kernel (), if it exists, of the pair of morphisms corresponding to the two projections . In particular, we then have the natural morphism (called the augmentation) which is an isomorphism in nice cases (in particular, in the case where is a strict epimorphism and is "left exact"). Similarly, if takes values in the category of groups in a category , then we can also define ; in the case where is the category of sets (i.e. when takes values in the category of non-necessarily-commutative groups), is the quotient of the subgroup of consisting of the such that by the group with operators acting on , and thus, in particular, on the subset , by 628fga3.i-a.4.bfga3.i-a.4.b.xmlA.4.bfga3.i-a.4For example, let be a fibred category with base . Let , and, for all over , let Then is a contravariant functor from to the category of sets. With this setup, saying that the augmentation morphism is an isomorphism for every pair of elements implies that is an -descent morphism ().629fga3.i-a.4.cfga3.i-a.4.c.xmlA.4.cfga3.i-a.4Similarly, for and any object of over we thus define a contravariant functor from to the category of groups. With this setup, we claim that is canonically identified with the set of descent data on with respect to (), and that can be identified with the set of isomorphism classes of objects of endowed with a descent data relative to that are isomorphic, as objects of , to . Then, if is an -descent morphism (cf. ), then contains as a subset the set of isomorphism classes of objects of such that is isomorphic (in ) to ; further, this inclusion is the identity if and only if every descent data on with respect to is effective. (This will be the case, in particular, if is a strict -descent morphism).630fga3.i-a.4.c-remarkfga3.i-a.4.c-remark.xmlRemarkfga3.i-a.4.cThe cochain complexes of the form contain, as particular cases, the majority of standard known complexes (that of Čech cohomology, of group cohomology, etc.), and play an important role in algebraic geometry (notably in the "Weil cohomology" of preschemes).631fga3.i-a.4.dfga3.i-a.4.d.xmlA.4.dfga3.i-a.4632fga3.i-a.4-example-1fga3.i-a.4-example-1.xmlExample1fga3.i-a.4.dLet be an object over , and let be a group of automorphisms of such that is "formally -principal over ", i.e. such that the natural morphism (where denotes the direct sum of copies of ) is an isomorphism. (We suppose that all the necessary direct sums exist in ). Let be a contravariant functor from to the category of abelian groups. Then is canonically isomorphic to the simplicial group of standard homogeneous cochains, and so is canonically isomorphic to .633fga3.i-a.4.efga3.i-a.4.e.xmlA.4.efga3.i-a.4634fga3.i-a.4-example-2fga3.i-a.4-example-2.xmlExample2fga3.i-a.4.eLet be the category of preschemes. We denote by (for "additive group") the contravariant functor from to the category of abelian groups, defined by We similarly define the functor (for "multiplicative group") by (i.e. the group of invertible elements of the ring ), and, more generally, the functor (for "linear group of order ") by which is a functor from to the category of (not-necessary-commutative, if 1]]>) groups; for we recover . We can also think of as an automorphism functor (cf. ) by considering the fibred category with base such that is the category of locally free sheaves on , for , since then . By , it follows that, if is an -descent morphism (cf. ), then contains the set of isomorphism classes of locally free sheaves on whose inverse image on is isomorphic to , and this inclusion is an equality if and only if every descent data on (with respect to ) is effective. If is the spectrum of a local ring, then this implies that , since every locally free sheaf on is then trivial.We note that the following conditions concerning a morphism are equivalent: The augmentation homomorphism is an isomorphism. is an -descent morphism (where is the fibred category over described above). is a strict epimorphism (cf. ). Now suppose that and ; then with the coboundary operator being the alternating sum of the face operators Similarly, can be identified with , with the simplicial operations for being induced by those in . We can write down the simplicial operations for in the same explicit manner. In all the cases known to the speaker, for 0]]>, and, if is local, then , and, more generally, (if is an -descent morphisms, i.e. if the diagram is exact, then compare this with ). We note that Hilbert's "Theorem 90" is exactly the fact that if is a field and is a finite Galois extension of (cf. ), and we can also express it by saying that, in the case in question, is a strict descent morphisms for the fibred category of locally free sheaves of rank . This latter statement is the one that is most suitable to generalise Hilbert's theorem, by varying the hypotheses both on the morphism and on the quasi-coherent sheaves in question.Finally, we note that, for a local Artinian with maximal ideal , and an -algebra (where we denote, for any integer 0]]>, the ring (resp. ) by (resp. )), the following properties are equivalent: for all . for all . for all and all . If is a strict epimorphism, then the above conditions imply that it is a strict descent morphism for free modules (not necessarily of finite type) over .643fga3.i-a.4.e-remarkfga3.i-a.4.e-remark.xmlRemarkfga3.i-a.4.eThe definition of the groups in the case where (resp. ) is a scheme over the field (resp. ) is due to Amitsur. The group is particularly interesting as a "global" variant of the Brauer group, for which we can refer to [GD1960, VII].644fga3.i-bfga3.i-b.xmlDescent by faithfully flat morphismsBfga3.i645fga3.i-b.1fga3.i-b.1.xmlStatement of the descent theoremsB.1fga3.i-b646fga3.i-b.1-definition-1.1fga3.i-b.1-definition-1.1.xmlDefinition1.1fga3.i-b.1A morphism of prescheme is said to be flat if is a flat module over the ring for all (i.e. if is an exact functor in the -module ). A morphism is said to be faithfully flat if it is flat and surjective.For example, if and , then is flat over if and only if is a flat -module, and is faithfully flat over if and only if is a faithfully flat -module (i.e. if and only if is an exact and faithful functor in the -module ); this also implies, in the terminology of Serre [Ser1956], that the pair is flat. If is faithfully flat over , then the inverse image functor of quasi-coherent sheaves on is exact and faithful; in other words, for a sequence of homomorphisms of quasi-coherent sheaves on to be exact, it is necessary and sufficient that its inverse image on be exact (in particular, for a homomorphism of quasi-coherent sheaves on to be a monomorphism (resp. an epimorphism, resp. an isomorphism), it is necessary and sufficient that its inverse image on be a monomorphism (resp. an epimorphism, resp. an isomorphism)). This property holds true if we restrict to an arbitrary open subset of , and then characterise faithfully flat morphisms in this form.647fga3.i-b.1-definition-1.2fga3.i-b.1-definition-1.2.xmlDefinition1.2fga3.i-b.1A morphism is said to be quasi-compact if the inverse image of every quasi-compact open subset of is quasi-compact (i.e. a finite union of affine open subsets).It evidently suffices to verify this property for the affine open subsets of . For example, an affine morphism (i.e. a morphism such that the inverse image of an affine open subset is affine) is quasi-compact.The class of flat (resp. faithfully flat, resp. quasi-compact) morphisms is stable under composition and by "base extension", and of course contains all isomorphisms.648fga3.i-b.1-theorem-1fga3.i-b.1-theorem-1.xmlTheorem1fga3.i-b.1Let be a morphism of preschemes that is faithfully flat and quasi-compact. Then is a strict descent morphism (cf. ) for the fibred category of quasi-coherent sheaves (cf. §A, Example 2).This statement implies two things: If and are quasi-coherent sheaves on , and and their inverse images on , then the natural homomorphism is a bijection from the left-hand side to the subgroup of the right-hand side consisting of homomorphisms that are compatible with the canonical descent data on these sheaves, i.e. whose inverse images under the two projections of to give the same homomorphism . Every quasi-coherent sheaf on endowed with a descent data with respect to the morphism (cf. ) is isomorphic (endowed with this data) to the inverse image of a quasi-coherent sheaf on . Setting in (i), we obtain:652fga3.i-b.1-corollary-1fga3.i-b.1-corollary-1.xmlCorollary1fga3.i-b.1Let be a quasi-coherent sheaf on , with and denoting its inverse images on and (respectively), and let and be the two projections from to . Then the diagram of maps of sets is exact (cf. ).Also, the combination of (i) and (ii) with gives:653fga3.i-b.1-corollary-2fga3.i-b.1-corollary-2.xmlTheorem2fga3.i-b.1Let be as in . Then there is a bijective correspondence between quasi-coherent subsheaves of and quasi-coherent subsheaves of whose inverse images on under the two projections and give the same subsheaf of .Of course, we have an equivalent statement in terms of quotient sheaves. As we have already seen in , should be thought of as a generalisation of Hilbert's "Theorem 90", and implies, as particular cases, various formulations in terms of -cohomology. For the proof, we can easily reduce to the case where and , and, for (i), we can easily restrict to proving , i.e. the exactness of the diagram for every -module , which follows from the more general lemma:654fga3.i-b.1-lemma-1.1fga3.i-b.1-lemma-1.1.xmlLemma1.1fga3.i-b.1Let be a faithfully flat -algebra. Then, for every -module , the -augmented complex (cf. ) is a resolution of . 655#262unstable-262.xmlProoffga3.i-b.1-lemma-1.1 It suffices to prove that the augmented complex induced from the above by extension of the base to satisfies the same conclusions. This leads to proving the statement when we replace by , and by , and so we can restrict to the case where there exists an -algebra homomorphism (or, in geometric terms, the case where over admits a section). In this case, the claim follows from the generalities of . We note, in passing, the following corollary, which generalises a well-known statement in Galois cohomology (compare with ):656fga3.i-b.1-lemma-1.1-corollaryfga3.i-b.1-lemma-1.1-corollary.xmlCorollaryfga3.i-b.1If is faithfully flat over , then , and for .To prove part (ii) of , we proceed, as for (i), by restricting to the case where over admits a section, where the result then follows from (i) (cf. ).We can evidently vary and its corollaries ad libitum by introducing various additional structures on the quasi-coherent sheaves (or systems of sheaves) in question. For example, the data on of a quasi-coherent sheaf of commutative algebras "is equivalent to" the data on of such a sheaf endowed with a descent data relative to . Taking into account the functorial correspondence between such quasi-coherent sheaves on and affine preschemes over , we obtain the second claim of the following theorem:657fga3.i-b.1-theorem-2fga3.i-b.1-theorem-2.xmlTheorem2fga3.i-b.1Let be as in . Then is a (non-strict, in general) descent morphism (cf. §A, Definition 2.4), and it is a strict descent morphism for the fibred category of affine schemes over preschemes (cf. §A, Definition 1.7).The first claim of the theorem implies this: let and be preschemes over , with and their inverse images over , and and their inverse images over ; then the diagram of natural maps is exact, i.e. is a bijection from to the subset of consisting of homomorphisms that are compatible with the canonical descent data on and (i.e. whose inverse images under the two projections from to are equal). This follows easily from and , if we restrict to being affine over ; in the general case, we need to combine with the following result:658fga3.i-b.1-lemma-1.2fga3.i-b.1-lemma-1.2.xmlLemma1.2fga3.i-b.1Let be a faithfully flat and quasi-compact morphism. Then can be identified with a topological quotient space of , i.e. every subset of such that is open, is open.To complete , we must give effectiveness criteria for a descent data on an -prescheme (in the case where is not assumed to be affine over ). Note first of all that such a descent data is not necessarily effective, even if is the spectrum of a field , the spectrum of a quadratic extension of , and a proper algebraic scheme of dimension over (as we can see, due to Serre, by using the non-projective surface of Nagata). For a descent data on with respect to (assumed to be faithfully flat and quasi-compact) to be effective, it is necessary and sufficient that be a union of open subsets that are affine over and "stable" under the descent data on . This is certainly the case (for any and any descent data on ) if the morphism is radicial (i.e. injective, and with radicial residual extensions). We can also show that this is the case if is finite, and every finite subset of that is contained in a fibre of over is also contained in an open subset of that is affine over (this is the Weil criterion). It is, in particular, the case if is quasi-projective, and, in this case, we can show that the "descended" prescheme is also quasi-projective (and projective if is projective). In summary:659fga3.i-b.1-theorem-3fga3.i-b.1-theorem-3.xmlTheorem3fga3.i-b.1Let be faithfully flat and quasi-compact morphism of preschemes. If is radicial, then it is a strict descent morphism. If is finite, then it is a strict descent morphism with respect to the fibred category of quasi-projective (or projective) preschemes over preschemes.660fga3.i-b.1-remarkfga3.i-b.1-remark.xmlRemarkfga3.i-b.1I do not know if, in the second claim above, the hypothesis that be finite is indeed necessary; we can prove that, in any case, we can "formally" replace it by the following, seemingly weaker, hypothesis: for every point of there exists an open neighbourhood , a finite and faithfully flat over , and an -morphism from to . A type of case that is not covered by the above is that where and , with a local Noetherian ring and its completion; or even that where is quasi-finite over (i.e. locally isomorphic to an open subset of a finite -scheme) but not finite. In these two cases, the speaker also does not know the answer to the following question: let be an -scheme such that is projective over ; is it then true that is projective over ?[Comp.] A morphism that is quasi-finite, étale, surjective, or a morphism of the form , is not always a strict descent morphism, even if is the local ring of an algebraic curve over an algebraically closed field and . We can thus find a proper simple morphism that makes into a principal -bundle over , with an elliptic curve, such that is projective, but is not projective. So this is also an example of a homogeneous principal bundle that is non-isotrivial under an abelian scheme.661fga3.i-b.2fga3.i-b.2.xmlApplication to the descent of certain properties of morphismsB.2fga3.i-bLet be a class of morphisms of preschemes. Let be a morphism of preschemes, and let be a morphism of -preschemes, with the inverse image of under . We can then ask if the relation "" implies that "". It appears that the answer is affirmative in many important cases, if we suppose that is faithfully flat and quasi-compact (this latter hypothesis sometimes being overly strong). We can see this directly without difficulty if is the class of surjective (resp. radicial) morphisms (with these two cases following from the surjectivity of ), or flat (resp. faithfully flat, resp. simple) morphisms (with these three cases following from the faithful flatness of ), or morphisms of finite type. Using , , and , we see that it is also true if is one of the following classes: isomorphisms, open immersions, closed immersions, immersions (if is of finite type, and is locally Noetherian), affine morphisms, finite morphisms, quasi-finite morphisms, open morphisms, closed morphisms, homeomorphisms, separated morphisms, or proper morphisms. The only important case not covered here is that of projective or quasi-projective morphisms, which has already been discussed in the remark in .662fga3.i-b.3fga3.i-b.3.xmlDecent by finite faithfully flat morphismsB.3fga3.i-bLet be a finite morphism, corresponding to a sheaf of algebras on that is locally free and of finite type as a sheaf of modules, and everywhere non-zero. Then is a faithfully flat and quasi-compact morphism, to which we can thus apply the above results. The data of a quasi-coherent sheaf on is equivalent to the data of the quasi-coherent sheaf on endowed with its -modules structure (noting that ). For simplicity, we also denote this sheaf on by . The two inverse images of on similarly correspond to the quasi-coherent sheaves of -modules and . The data of an -homomorphism from the former to the latter is equivalent to the data of a homomorphism of -modules, and, taking into account the fact that is locally free, this is equivalent to the data of a homomorphism of -modules: i.e. to the data, for every section of over an open subset , of a homomorphism of -modules that satisfies the conditions 663fga3.i-b.3-equation-3.1fga3.i-b.3-equation-3.1.xmlEquation3.1fga3.i-b.3 where and are (respectively) sections of and over an open subset of that is contained inside . Conditions (i) and (ii) of a descent data (cf. ) can then be written (respectively) as 664fga3.i-b.3-equation-3.2fga3.i-b.3-equation-3.2.xmlEquation3.2fga3.i-b.3 665fga3.i-b.3-equation-3.3fga3.i-b.3-equation-3.3.xmlEquation3.3fga3.i-b.3 In other words, a descent data on is equivalent to a representation of the sheaf of -algebras in the sheaf of -algebras that satisfies the two linearity conditions (). If we have a pairing of quasi-coherent sheaves on : (that we can think of as a pairing of sheaves of -modules on ), and gluing data on the defined by homomorphisms (for ), then these data are equivalent to the given pairing, in the evident meaning of the phrase, if and only if the following condition is satisfied:For every section of over an open subset, and denoting by the section of (where is considered as an -module with its left structure) defined by the formula (where and are sections of over a smaller open subset), we have the formula 666fga3.i-b.3-equation-3.4fga3.i-b.3-equation-3.4.xmlEquation3.4fga3.i-b.3 for every pair of sections of over a smaller subset. (We can express this property by saying that the homomorphisms are compatible with the diagonal map of , with respect to the given pair). In particular, , , , and allow us to understand, in terms of representations of algebras via diagonal maps, the descent data on a quasi-coherent sheaf of algebras on , and thus also (by restricting to commutative algebras) the descent data on an affine -scheme.From here, we obtain an analogous interpretation of descent data on an arbitrary -prescheme : the data of such an is equivalent to the data of a prescheme over endowed with a homomorphism of -algebras and a descent data on is equivalent to the data of a sheaf homomorphism that is compatible with the morphism and that satisfies the conditions analogous to , , , and above.667fga3.i-b.3-example-1fga3.i-b.3-example-1.xmlWeil's exampleExample1fga3.i-b.3Suppose that is a Galois étale covering with Galois group (cf. and ). Then a descent data on a quasi-coherent sheaf on (resp. on an -prescheme ) is equivalent to the data of a representation of by automorphisms of (resp. of ) that is compatible with the action of on . This result is "formal", i.e. it can be proven in terms of categories, but, from the point of view of this section, we also obtain the explicit structure of (endowed with its ring structure, the ring homomorphism , and the diagonal map), which is completely known thanks to the following result: admits, as a left -module, a basis given by the sections of that correspond to elements of .668fga3.i-b.3-example-2fga3.i-b.3-example-2.xmlCartier's exampleExample2fga3.i-b.3Let be a prime number, and suppose that (i.e. that is of characteristic ), that (i.e. that is radicial of height ), and that the sheaf of algebras over locally admits a -basis (i.e. a family of sections such that is generated as an algebra by the under the sole condition that ). We suppose that the set of the is finite, of cardinality . Let be the sheaf of -derivations of , which is a locally free sheaf of rank of -modules, and, furthermore, a sheaf of Lie -algebras over (but not over ) that satisfies the following condition: 669fga3.i-b.3-equation-3.5fga3.i-b.3-equation-3.5.xmlEquation3.5fga3.i-b.3-example-2670fga3.i-b.3-lemmafga3.i-b.3-lemma.xmlLemmafga3.i-b.3 is generated, as an -algebra endowed with an algebra homomorphism , by the sub-left--module , with the following additional relations: 671fga3.i-b.3-equation-3.6fga3.i-b.3-equation-3.6.xmlEquation3.6fga3.i-b.3-lemmaIt follows from the above lemma that a descent data on the quasi-coherent sheaf on is equivalent to the data, for all , of an -endomorphism of that satisfies the following conditions: 672fga3.i-b.3-equation-3.7fga3.i-b.3-equation-3.7.xmlEquation3.7fga3.i-b.3 673fga3.i-b.3-equation-3.8fga3.i-b.3-equation-3.8.xmlEquation3.8fga3.i-b.3 674fga3.i-b.3-equation-3.9fga3.i-b.3-equation-3.9.xmlEquation3.9fga3.i-b.3 675fga3.i-b.3-equation-3.10fga3.i-b.3-equation-3.10.xmlEquation3.10fga3.i-b.3 (This is what we may call a linear connection on , which is further flat and compatible with the -th powers). We can similarly write down the notion of a descent data on an -prescheme , with being replaced by the condition that the are derivations of . Since the morphism is radicial, ensures that every such descent data is effective, and thus defines an -prescheme .Note that we have not needed to impose any flatness, non-singular, or finiteness hypotheses on or .676fga3.i-b.4fga3.i-b.4.xmlApplication to rationality criteriaB.4fga3.i-bLet be an -prescheme such that the direct image of on is ; this property remains true for any flat base extension . If is an invertible sheaf (i.e. locally free of rank ) on , then there is a bijective correspondence between automorphisms of (identified with the invertible sections of ) and invertible sections of . So let be a section of over ; we define a section of over to be a section of the invertible sheaf on . It follows from the above that, if (for ) are invertible sheaves on , each endowed with a section over , and if and are isomorphic, then there exists exactly one isomorphism from to that is compatible with the sections in question (i.e. sending the first to the second). We also, independently of the section , regard two invertible sheaves and on as equivalent if every point of has an open neighbourhood such that the restrictions of and to are isomorphic. Then every invertible sheaf on is equivalent to an invertible sheaf endowed with a marked section over (we take ), and is determined up to isomorphism. In other words, the classification up to equivalence of invertible sheaves on is the same as the classification up to isomorphism of invertible sheaves endowed with a marked section.Since these properties remain true under flat extensions of the base (by replacing the section with its inverse image under ), we thus conclude, taking into account:With the prescheme being as above, and admitting a section , let be a faithfully flat and quasi-compact morphism; let be an invertible sheaf on . For to be equivalent to the inverse image on of an invertible sheaf on , it is necessary and sufficient that its inverse images and on be equivalent. If this is the case, then is determined up to equivalence. (We then say that is rational on ).Considering this principle in the case where is as in and in the previous section, we recover the rationality criteria of Weil and of Cartier. (We note that the authors restrict to the case where and are spectra of fields; a fortiori, is then the spectrum of a local ring, and the equivalence relation introduced above is exactly the relation of being isomorphic). The the first case, is rational on if and only if its images under are all equivalent to . To express the rationality criterion in the second case, we consider, more generally, the diagonal morphism of , with the corresponding sheaf of ideals on , and the sheaf , which can be identified with its inverse image on (the sheaf of -differentials of with respect to ). Since the restrictions of the (for ) to the diagonal are isomorphic (since they are both isomorphic to ), i.e. has a restriction to the diagonal which is trivial, it follows that the restriction of to is given, up to isomorphism, by a well-defined element of Also, being precise, we have , and thus, if is locally free on (as in the Cartier case), then defines a section of on (called the Atiyah–Cartier class of the invertible sheaf on ) whose vanishing is necessary and sufficient for the inverse images of under the two projections of to to be equivalent (where is the sheaf of ideals on defined by the diagonal morphism ). This vanishing is thus trivially necessary for the inverse images of on itself to be equivalent, and thus also for to be equivalent to the inverse image of an invertible sheaf on . The Atiyah–Cartier class can also be understood as the obstruction to the existence, locally over , of a connection of relative to the derivations of , with such a connection further being determined when we know the derivations of corresponding to the natural extensions of derivations of to . From this, and the results of the previous section, we easily conclude that, in the case of the aforementioned , and when admits a section, the vanishing of the Atiyah–Cartier class is also sufficient for to be rational on .677fga3.i-b.5fga3.i-b.5.xmlApplication to the restriction of the base scheme to an abelian schemeB.5fga3.i-bLet be a prescheme. We define an abelian scheme over to be a simple proper scheme over whose fibres at the points are schemes of abelian varieties over the . Suppose that is Noetherian and regular (i.e. that its local rings are regular), then we can show, using the connection theorem of Murre [Mur1958] (at least in the case "of equal characteristics", where the cited theorem is currently proven) that every rational section of over is everywhere defined (i.e. is a section) (which generalises a classical theorem of Weil). It then follows, more generally, that, if is a simple scheme over , then every rational -map from to is everywhere defined. From this, we obtain the following, which generalises a result of Chow–Lang: with Noetherian and regular, and denoting its ring of rational functions (a direct sum of fields), let be an abelian scheme over ; if is isomorphic to a -scheme of the form , where is an abelian scheme over , then is determined up to unique isomorphism.Using the above uniqueness result, we see that the question of restriction of the base to is local on (and thus that it suffices to know how to do the restriction to , where ). In the same way, we see that, if is a simple morphism of finite type, if is the ring of rational functions of , and if is of the form , then is endowed with a canonical descent data with respect to . Taking into account, we thus conclude:678fga3.i-b.5-proposition-5.1fga3.i-b.5-proposition-5.1.xmlProposition5.1fga3.i-b.5Let be an irreducible regular Noetherian prescheme, with field of rational functions , let be a finite extension of that is unramified over , let be the normalisation of in (which is thus an étale cover of ), and let be an abelian scheme over such that is of the form , where is a projective abelian scheme over . Then is of the form , where is a projective abelian scheme over .679fga3.i-b.5-remarkfga3.i-b.5-remark.xmlRemarkfga3.i-b.5The speaker does not know if we can replace the hypothesis that be a surjective étale cover (which allows us to apply ) with the hypothesis that it is instead a simple and surjective morphism of finite type (not even if we suppose that it is an étalement), or if the proposition still holds true without supposing that is projective over (a condition which could be automatically satisfied).680fga3.i-b.6fga3.i-b.6.xmlApplication to local triviality and isotriviality criteriaB.6fga3.i-bLet be a prescheme, a "prescheme of groups" over , a prescheme over on which " acts" (on the right). We say that is formally principal homogeneous for if the well-known morphism (induced from the actions of on ) is an isomorphism. From now on, we assume to be flat over (and thus faithfully flat over ), and we reserve the name of principal homogeneous bundle for for a formally principal homogeneous bundle that is faithfully flat and quasi-compact over . It is immediate that this is equivalent to being able to find a faithfully flat and quasi-compact extension of the base such that the formally principal homogeneous bundle for is trivial, i.e. isomorphic to (i.e. admitting a section); we can take, in particular, . Note also that, if is locally Noetherian, then the faithfully-flat hypothesis on is equivalent to the hypothesis that be faithfully flat over for all (where denotes the completion of the local ring ), as follows from the fact that is faithfully flat over . Also, if is of finite type over , and is locally Noetherian, then the set of points satisfying the above condition is constructible, and so, if is a "Jacobson prescheme" (for example, a scheme of finite type over a field, or, more generally, over a Jacobson ring), then it suffices to verify the condition in question for the closed points of . This leads us to the case where the base is the spectrum of a complete local ring . If (with a complete Noetherian local ring), and if is of finite type over , then the faithful flatness of is also equivalent to the existence of an that is finite and flat over such that is trivial, and, if, further, is simple over , then we can suppose to be étale over . Then, if, further, the residue field of is algebraically closed (the "geometric case"), then is faithfully flat over if and only if it is trivial. Thus, if is an algebraic prescheme over an algebraically closed field, and if is simple and of finite type over , then we see that the faithfully-flat condition on is equivalent to the condition of being analytically trivial (SLF) of Serre [Ser1958a, pp.1–12].We can consider other, stronger, types of conditions on , that have a "local triviality" nature. In particular, we say that is isotrivial (resp. strictly isotrivial) if, for all , there exists an open neighbourhood of , and a finite and faithfully flat morphism (resp. a surjective étale covering) such that is trivial. (We stray from the terminology of Serre [GD1960], which uses "locally isotrivial" for what we call "strictly isotrivial"). Strict isotriviality is mainly useful if is simple over , but is, however, an inadequate notion in other cases.If is affine over , then every principal homogeneous bundle for is affine, by , whence the possibility, thanks to , to "descend" from such bundles by faithfully flat and quasi-compact morphisms. Taking, in particular, , defined by the condition that the functor of -preschemes (with values in the category of groups) can be identified with the functor described in . Using the facts that every principal homogeneous bundle for (resp. every locally free sheaf of rank on ) becomes isomorphic to the "trivial" object (resp. ) under a suitable faithfully flat and quasi-compact extension of ; that we can descend the type of objects in question (principal homogeneous bundles for , resp. locally free sheaves of rank ) by such morphisms; and, finally that the automorphism group of the trivial bundle on any is functorially isomorphic to the automorphism group of the trivial locally free sheaf of rank on , we "formally" conclude that it is "equivalent" to give, on (or on some ) a principal homogeneous bundle for the group , or to give a locally free sheaf of rank . (More precisely, we have an equivalence of fibred categories). We thus conclude, in particular:353fga3.i-b.6-proposition-6.1fga3.i-b.6-proposition-6.1.xmlProposition6.1fga3.i-b.6Every principal homogeneous bundle for the group is locally trivial.By known arguments, we thus conclude the same result for others structure groups such as , , and products of such groups. We thus also conclude that, if is a closed subgroup of that is flat over , and such that the quotient exists, and such that is an isotrivial (resp. strictly isotrivial) principal homogeneous bundle on , of structure group , then every principal homogeneous bundle of structure group is isotrivial (resp. strictly isotrivial). This applies to all the "linear groups" on that have been used up until now, and, in particular, to the case where , with a prescheme over the field , and a linear group (in the classical sense) over (and thus in particular simple). This thus answers, for such groups, a question of Serre (loc. cit.).We also point out that, for most groups (linear or not) that are simple over that we know of, and certainly for all those of the form as above, we can show that every isotrivial principal homogeneous bundle is strictly isotrivial, which answers, in particular, another question of Serre (loc. cit. pp.1–14), taking into account the fact that a homogeneous principal bundle obtained by a descent à la Cartier (cf. ) is obviously isotrivial.354fga3.i-b.6-remarkfga3.i-b.6-remark.xmlRemarkfga3.i-b.6One of the essential difficulties in these questions (setting aside the question of the existence of quotient schemes) is the lack of effectiveness criteria for a descent data along a faithfully flat non-finite morphism.3447indexindex.xmlGrothendieck's "Foundations of Algebraic Geometry" (FGA)Click for PDF version2889translators-notetranslators-note.xmlNote from the translatorindexThis is an English translation of Alexander Grothendieck's "Fondements de la Géometrie Algébrique". The original (French) notes have been scanned and uploaded by the Grothendieck Circle here, though you can also find the individual talks, as well as the errata, on Numdam.The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein. Any notes by the translator are in italics and prefixed with "[Trans.]".You can view the entire source code of this translation (and contribute or submit corrections) in the GitHub repository. Corrections and comments welcome.The translator would like to sincerely thank Steve Hnizdur for catching many typos and mistakes, as well as Jon Sterling for helping with the technical support in getting this translation ported to run on Forester. Foreword Duality theorems for coherent algebraic sheaves Formal geometry and algebraic geometry Technique of descent and existence theorems in algebraic geometry Generalities, and descent by faithfully flat morphisms The existence theorem and the formal theory of modules Quotient preschemes Hilbert schemes Picard schemes: Existence theorems Picard schemes: General properties Bibliography 3448fga3.iii-introductionfga3.iii-introduction.xmlQuotient preschemes › Introductionfga3.iii1600fga3.iii-introduction-remarkfga3.iii-introduction-remark.xmlRemarkfga3.iii-introduction[Comp.] We note that the application (of the theory developed here) in ("Picard schemes: Existence theorems") can equally be replaced by a suitable use of Hilbert schemes (cf. Séminaire Mumford–Tate, Harvard University (1961–62)). As mentioned in , the most important gap in the theory presented here is the lack of an existence criterion for quotients by a non-proper equivalence relation, such as the equivalence relations coming from certain actions of the projective group. An important theorem in this direction has been obtained by Mumford [@Mum1961]. For a refinement of his result, and various applications the the theory, see Séminaire Mumford–Tate, Harvard University (1961–62).The problems discussed in the current talk differ from those discussed in the two previous ones, in that we try to represent certain covariant, no longer contravariant, functors of varying schemes. The procedure of passing to the quotient is, however, essential in many questions of construction in algebraic geometry, including those from and . Indeed, the question of effectiveness of a descent data on a -prescheme , with respect to a faithfully flat and quasi-compact morphism , is equivalent to the question of existence of a quotient of (satisfying reasonable properties that we examine below) by the flat equivalence relation on defined by the descent data; the questions raised in FGA 3.I, §A.2.c can probably be answered at the same time as the questions posed in of this current talk. Similarly, the Picard scheme (for the definition, see FGA 3.II, §C.3) of an -scheme can be defined in many ways, such as as a quotient of certain other schemes (with positive divisors, or immersions into a projective) by flat equivalence relations, with the definition and construction of these auxiliary schemes being also more simple: they are basically schemes of the type , and variants defined in FGA 3.II, §C.2, and their construction will be the subject of the following talk (under suitable hypotheses of projectivity). Thus, combining the results of the current talk with those of the following, we will obtain the construction of Picard schemes, under suitable hypotheses.The problem of passing to the quotient in preschemes again offers unresolved questions. The most important is mentioned in . It currently remains as the only obstacle to the construction of schemes of modules over the integers for curves of arbitrary degree, polarised abelian varieties, etc. That is to say, its solution deserves the efforts of specialists of algebraic groups.3449fga3.iv-introductionfga3.iv-introduction.xmlHilbert schemes › Introductionfga3.ivThe techniques described in and were, for the most part, independent of any projective hypotheses on the schemes in question. Unfortunately, they have not as of yet allowed us to solve the existence problems posed in . In the current article, and the following, we will solve these problems by imposing projective hypotheses. The techniques used are typically projective, and practically make no use of any results from and . Here we will construct "Hilbert schemes", which are meant to replace the use of Chow coordinates, as was mentioned in FGA 3.II, §C.2. In the next article, the theory of passing to the quotient in schemes, developed in , combined with the theory of Hilbert schemes, will allow us, for example, to construct Picard schemes (defined in FGA 3.II, §C.3) under rather general conditions.In summary, we can say that we now have a more or less satisfying technique of projective constructions, apart from the fact that we are still missing a theory of passing to the quotient by groups such as the projective group, acting "without fixed points" (cf. FGA 3.III, §8). The situation even seems slightly better in analytic geometry (if we restrict to the study of "projective" analytic spaces over a given analytic space), since, for analytic spaces, the difficulty of passing to the quotient by a group that acts nicely disappears. Either way, in algebraic geometry, as well as in analytic geometry, it remains to develop a construction technique that works without any projective hypotheses.3450fga3.iv-7fga3.iv-7.xmlHilbert schemes › Supplements and questions7fga3.ivAs remarked by J.-P. Serre, it follows from a well-known example of Nagata that we can find a scheme that is the spectrum of a field , an -scheme that is the spectrum of a quadratic extension of , and finally a simple and proper (but non-projective) -scheme of dimension such that does not exist. This implies a fortiori that the Hilbert scheme does not exist (nor even the -scheme that would represent the étale covers of rank of contained inside , nor a fortiori the symmetric square of , cf. ). This thus imposes serious limitations on the possibilities of non-projective constructions in algebraic geometry. (It is, however, plausible that such limitations do not present themselves in analytic geometry, just as they do not present themselves in formal geometry (cf. )). However, if is a proper scheme over the spectrum of a field , and if is quasi-projective over , then exists, and is a scheme, given by the sum of a sequence of quasi-projective schemes over (as in the projective case ). To see this, we can reduce to the case where is itself projective, by dominating by a projective -scheme ; we will not give here the details of the proof, which also uses the result of factorisation of a finite morphism given in FGA 3.I, §A.2.b. The success of the method is all in the fact that, with the spectrum of a field, the that appears in Chow's lemma will automatically be flat over . I do not know if the result remains true without any hypotheses on , supposing only that is proper and flat over , and that is quasi-projective over . An important case in the applications is that where is a closed subscheme of ; if then exists, it is necessarily a closed subscheme of . We can construct it directly in a relatively simple manner whenever is projective over , without using the theory of Hilbert schemes, and the method used shows more generally that, if is affine over , then exists and is affine over . It equally shows that, if is proper and flat over (but not necessarily projective over ), then, for every vector bundle that is locally trivial on , exists and is a vector bundle on . It would be desirable for these results to be studied again and unified.