3248fga3.iiifga3.iii.xmlQuotient preschemesFGA3.III2005fga3.iii-original-citationfga3.iii-original-citation.xmlOriginalfga3.iiiA. Grothendieck. "Technique de descente et théorèmes d'existence en géométrie algébrique, III: Préschemas quotients". Séminaire Bourbaki 13 (1960–61), Talk no. 212. http://www.numdam.org/book-part/SB_1960-1961__6__99_0/2006fga3.iii-introductionfga3.iii-introduction.xmlIntroductionfga3.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.2007fga3.iii-1fga3.iii-1.xmlEquivalence relations, effective equivalence relations1fga3.iiiLet be a category, and an object of . A pair of morphisms is said to be an "equivalence pair" in , with target and source , if, for every object of , the corresponding maps (where we set for any object of ) define a map that induces a bijection from to the graph of an equivalence relation on the set . We introduce an evident equivalence relation on equivalence pairs with target , and call an equivalence class an equivalence -relation on , or simply an equivalence relation if no confusion may arise.If exists, then the data of an equivalence relation on is equivalent to the data of a sub-object of such that, for every object of , the subset of that corresponds to is the graph of an equivalence relation on . Denoting the morphisms from to induced by the projections and by and (respectively), the above condition says that is an equivalence pair. We can also express the axioms of a set-theoretical equivalence relation for the in the diagrammatically in (under the assumption that both and the fibre product exist), following the general principle of FGA 3.II, §A.1. We will not need this.Every time that we have a pair of morphisms with the same source and the same target , we can define the cokernel of the pair as an object of that represents the contravariant (in ) functor which denotes the set of morphisms from to such that . If exists, then it is determined up to unique isomorphism. We will denote it by , or, by an abuse of notation, , with the latter mostly being used when is an equivalence pair: it is then common to identify, in notation, the equivalence relation defined by the pair with the one defined by . Note that, if we consider as a quotient of , then it depends only on the equivalence relation defined by the pair We now start with a morphism which allows us to consider as an "object over ", and we suppose that the fibre product exists. Let and be its projections. Then is an equivalence pair, and is said to be associated with the morphism . It thus defines an equivalence relation, which is said to be associated with the morphism .We say that a pair of morphisms with target , and source , is an effective equivalence pair if the cokernel exists ; the fibre product exists ; and the morphism with components and is an isomorphism. Then the pair is indeed an equivalence pair. We also say that the equivalence relation that it defines is an effective equivalence relation.We say that a morphism is an effective epimorphism if the fibre product exists ; the quotient exists, where and are the projections from to ; and the morphism induced by is an isomorphism. Then is indeed an epimorphism, and even a strict epimorphism (cf.FGA 3.I, §A.2.c), with the converse being true if the fibre product exists. We also say that the quotient object of defined by the epimorphism is an effective quotient of .The above definitions imply the following "Galois correspondence":1598fga3.iii-1-proposition-1.1fga3.iii-1-proposition-1.1.xmlProposition1.1fga3.iii-1There is a bijective correspondence, respecting the natural orders, between the set of effective equivalence relations on and the set of effective quotients of , with such an corresponding to the effective quotient , and such a corresponding to the effective equivalence relation defined by the canonical projection (which is defined by the fibre product endowed with its two projections).In very nice categories (sets, sheaves of sets, etc.), every quotient is effective, and every equivalence relation is effective. This is no longer true in categories such as the category of preschemes over a given prescheme , not even if is the spectrum of field, nor even if we restrict to finite schemes over . The question of effectiveness, and even (in the case of non-finite preschemes over ) the question of existence of quotients, more often than not turn out to be delicate.2008fga3.iii-2fga3.iii-2.xmlExample: 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).2009fga3.iii-3fga3.iii-3.xmlThe case of a group with operators3fga3.iiiWe now suppose that is an arbitrary category. Let and be objects of , and suppose that is a -group with operators on the object . This implies (cf. FGA 3.II, §A.1) that, for every object of , we have a group structure on , and the structure of an operator domain on acting on , such that, for variable , the structures in question "vary functorially" in . If the products and exist in , then such a structure can also be defined as a pair of morphisms subject to the condition that, for every object of , the corresponding composition laws for the sets and make into a group acting on . Translating this axiom into the commutativity of certain diagrams in is easy, but tedious, and, in fact, perfectly useless in all cases known to me.Suppose that exists, and consider the two morphisms with We immediately note that the pair is an equivalence pair if, and only if, for every object of , the map defined by this pair is injective, i.e. if the group acts freely on the set , i.e. if , , and , then is the identity element of the group . We then say that acts freely on (or that is a principal -space under ). The equivalence relation associated to the pair is then called the equivalence relation defined by the group acting freely on . If also exists, and we consider the morphism defined by the pair , then the condition that acts freely implies that is a monomorphism.Of course, even if dose not act freely on , we still wish to have existence criteria for a quotient of by , i.e. for the cokernel of the above pair .The cokernel in question will often be denoted by , or by if acts on the left (with the previous notation being reserved for when acts on the right). We note that, even if the "image" of under exists (this image being defined, for example, as the smallest sub-object of through which we can factor ), say, , then this is usually not an equivalence relation on . If we then try to pass directly to the quotient under (or, more precisely, under the pair of morphisms from to induced by the two projections ), then we lose the particular characteristics of the original pair . It is thus important to find a generalisation of the notion of equivalence relations, appealing directly to the pair defined by a -group with operators.2010fga3.iii-4fga3.iii-4.xmlEquivalence pre-relations4fga3.iiiRecall that a groupoid is defined to be a category where all the morphisms are isomorphisms. A category should be defined as consisting of two base sets, and , with the former being the set of objects and the latter the set of arrows, endowed with the following structures: a pair of maps called the source map and the target map ; a map called the composition map. These data should satisfy well-known axioms, which we will not repeat here, and which can be expressed in terms of the commutativity of certain diagrams along with the existence of a (necessarily unique) map that makes two other diagrams commute, where corresponds to passing from an object to the corresponding identity map, and satisfies To say that a category is a groupoid then, implies the existence of a (necessarily unique) map called the symmetry of , that sends every arrow to an inverse arrow, which can be expressed in terms of the commutativity of four other diagrams, built from , , and the above data, and of which the first two can be written as Having recalled these notions, the general definitions in FGA 3.II, §A.1 show, in particular, what we should mean by "the structure of a -category" (resp. -groupoid) on a pair of objects of an arbitrary category : it is, by definition, the data, for every object in , of the structure of a category (resp. groupoid) in the set-theoretic sense, whose set of objects is , and set of arrows is , with these structures "varying functorially" in . This thus implies the definition of two morphisms called the source morphism and the target morphism, and, if the fibre product in question exists, a morphism called the composition morphism; these three morphisms then suffice to determine the structure of a category (resp. groupoid) on , with the condition to place on them being the following: for every , the three corresponding morphisms for and define the structure of a category (resp. groupoid) on the pair of sets . If necessary, this can be expressed in terms of the commutativity of certain diagrams, implying a well-determined morphism and, in the case of groupoids, a well-determined morphism where the diagrams are as in the "set-theoretic" case. This tedious interpretation of the axioms is thankfully useless in practice, with the only theoretical interest in the possibility of being able to express the data and the axioms using morphisms and equalities of morphisms between certain fibre products being the following: if we have a left-exact functor (i.e. a functor that commutes with finite products and fibre products), then it sends -categories (resp. -groupoids) to a -categories (resp. -groupoids) (under the condition that finite products and fibre products exist in ).It is important, in practice, to know how to understand the morphisms , , , , and as simplicial operations in a suitable semi-simplicial or simplicial objects of (or, at least when fibre products exist in ). To fix terminology, we introduce the category of simplex types as the category whose objects are finite sets of the form for , where denotes the interval of integers from to (inclusive), and whose morphisms are arbitrary maps between these finite sets. We note that the category is equivalent to the category of non-empty finite sets, where we take the morphisms to be maps between finite sets. In , the sum of a non-empty finite family of objects clearly exists, as does the amalgamated sum of two objects over a third (the dual operation to the fibre product). We denote by the subcategory of that has the same objects, but where the morphisms are increasing maps between the . This category is equivalent to the category of non-empty finite totally ordered sets. In this category, the sum of two objects never exists, and the amalgamated sum of two objects and over a third does not exist in general (take, for example, , and , with the two structure maps and being the equal). However, in certain cases, the amalgamated sum does exist; consider which is such that A simplicial object (resp. semi-simplicial object) in a category is defined to be a contravariant functor from (resp. ) to . A simplicial object thus defines a semi-simplicial object by restriction, but the former differs from the latter essentially by the presence of symmetry operations in the , which correspond to the images under the functor of the elements of the symmetric group on elements (considered as the automorphism group of in ).With the above, for all , let (resp. ) be the finite category whose set of objects is , and whose set of arrows is defined by the "chaotic order" relation (resp. the natural total order relation) on (i.e. the set of arrows is the graph of the order relation). It is clear that (resp. ) depends functorially on the object of (resp. ). So if is a category, then (resp. ) is, for varying , a functor from the category (resp. ) to the category of sets, i.e. a simplicial set (resp. semi-simplicial set), which is said to be associated to the category , and denoted by (resp. ). We also have an obvious natural homomorphism from the semi-simplicial set associated to to , and this is an isomorphism if and only if is a groupoid. Then:1863fga3.iii-4-proposition-4.1fga3.iii-4-proposition-4.1.xmlProposition4.1fga3.iii-4The functor from the category of categories to the category of semi-simplicial sets is fully faithful, and defines an equivalence between the category of categories and the category of semi-simplicial sets, i.e. contravariant functors from to that send amalgamated sums (of the type described above) to fibre products of sets.Similarly, the functor from the category of groupoids to the category of simplicial sets is fully faithful, and defines an equivalence between the category of groupoids and the category of simplicial sets, i.e. contravariant functors from to that send amalgamated sums to fibre products.We can thus consider categories as specific examples of semi-simplicial sets, and groupoids as specific examples of simplicial sets, with, of course, the condition that we argue "up to isomorphism", as is rigorous when we interpret certain structures in terms of others. The usual procedure of reduction to the set-theoretic case then implies:1864fga3.iii-4-corollary-4.2fga3.iii-4-corollary-4.2.xmlCorollary4.2fga3.iii-4The above claim remains true when we replace categories, groupoids, and simplicial sets with -categories, -groupoids, and -simplicial objects (respectively), provided that fibre products exist in .The semi-simplicial object in associated to a category in can be made explicit by considering the component of as being the -th fibre product of over , or, even better, by the inductive formula where the (for ) are the natural projections from to (which can also be defined inductively). In this way, , , , , and can be understood as simplicial operations that correspond to morphisms in , namely: the face of , the face of , the face of , the degeneracy , and the symmetry of (respectively). Every other semi-simplicial (resp. simplicial) operation can be formally obtained from the four (resp. five) aforementioned operations by composition and fibre products.We now define an equivalence pre-relation on an object of a category to be the data of a groupoid whose object of objects is . Such a data gives, amongst other things, an object along with two morphisms But we note that only these data alone do not determine the structure in question, contrary to what happens for equivalence pairs. In this talk, we are interested in this notion with the aim of obtaining criteria for the possibility of passing to the quotient, i.e. for being able to form the cokernel of the pair . The statement of this problem thus makes no reference to the additional data inherent to a groupoid. In the proof of the results that will follow, we will, however, make use of this additional data, and, in particular, of the simplicial operations (including the symmetry operations) up to dimension (the fourth fibre power of over will appear).An equivalence relation on an object of defines an equivalence pre-relation: it suffices to show this in the set-theoretic case, and we then associate, to an equivalence relation on a set , the groupoid whose set of objects is , and whose set of arrows is the graph set of the equivalence relation.A -monoid acting on an object of defines a -category whose basic objects are and (under the condition that exists), and that is a -groupoid if and only if is a group. It again suffices to prove this in the set-theoretic case. We then define the composition of arrows and as being i.e. if then is, by definition, the transporter of to , and morphisms compose thanks to the composition of elements of .1865fga3.iii-4-remarkfga3.iii-4-remark.xmlRemarkfga3.iii-4We can avoid the logical difficulties that arise in a statement such as by implicitly assuming that all the objects in question can be found in a fixed "universe" (that is itself a set).2011fga3.iii-5fga3.iii-5.xmlQuotient by a finite and flat equivalence relation5fga3.iii1874fga3.iii-5-theorem-5.1fga3.iii-5-theorem-5.1.xmlTheorem5.1fga3.iii-5Let be an affine scheme, an equivalence pre-relation on , whose component is affine: say, . We suppose that the first projection is a finite and locally free morphism, i.e. that the corresponding homomorphism of rings makes a projective -module of finite type. Let be the subring of given by the kernel of the pair of homomorphisms (i.e. the set of elements such that ). Let , and the morphism defined by the embedding of into . Under these conditions: is integral over , i.e. is an integral morphism. The morphism is surjective, and its fibres are the set-theoretic equivalence classes in modulo , and the topology of is the quotient of that of . is the quotient of by in the category of preschemes. If comes from an equivalence relation, then the morphism is finite and locally free (i.e. is a projective -module of finite type), and the equivalence relation is effective, i.e. is an isomorphism. This theorem generalises the well-known theorem concerning the case of a finite group acting by automorphisms on the ring , and with ring of invariants, and the proof is analogous to the known proof. We can make (iii) more precise as follows:1880fga3.iii-5-corollary-5.2fga3.iii-5-corollary-5.2.xmlCorollary5.2fga3.iii-5The canonical morphism is surjective.Let continue to be a "finite and locally free" equivalence pre-relation on , but with now being an arbitrary prescheme. Suppose that we can find a prescheme and a morphism such that , and further such that the sequence of homomorphisms of sheaves of rings on is exact (where ). It then follows from the theorem that we have conclusions (i) to (iv) analogous to those of the theorem, and, in particular, by (iii), is the quotient of by , and thus determined up to unique isomorphism. Under these conditions, we say that the equivalence pre-relation on is admissible. With this definition:1881fga3.iii-5-theorem-5.3fga3.iii-5-theorem-5.3.xmlTheorem5.3fga3.iii-5Let be a prescheme, and an equivalence pre-relation on such that is a finite and locally free morphism. For to be admissible, it is necessary and sufficient that every set-theoretic equivalence class in modulo be contained in an affine open subset (a condition that is always satisfied if every finite subset of is contained in an affine open subset, for example if is quasi-projective over an affine scheme).We can in fact easily show that every equivalence class modulo in is then contained in an affine open subset that is stable under , and we construct the quotient by gluing the pieces obtained by applying .1882fga3.iii-5-corollary-5.4fga3.iii-5-corollary-5.4.xmlCorollary5.4fga3.iii-5Suppose that this condition is satisfied, and, further, that comes from an equivalence relation. Then the equivalence relation is question is effective, i.e. is an isomorphism, and is a finite and locally free morphism.We then immediately conclude, by descent, the following:1883fga3.iii-5-corollary-5.5fga3.iii-5-corollary-5.5.xmlCorollary5.5fga3.iii-5Under the conditions of , for to be everywhere of rank over , it is necessary and sufficient that be everywhere of rank over . If and are -preschemes, and and are -morphisms (and thus a -prescheme), then is flat over if and only if is flat over .In summary:1884fga3.iii-5-scholiumfga3.iii-5-scholium.xmlScholiumfga3.iii-5The data of a finite, locally free, and surjective morphism of preschemes is equivalent to the data of a prescheme endowed with an equivalence relation such that is finite and locally free, and such that every class is contained in an affine open subset.1885fga3.iii-5-remarks-5.6fga3.iii-5-remarks-5.6.xmlRemarks5.6fga3.iii-5We have not needed to make any Noetherian hypothesis. This idea of passing to the quotient contains, as a particular case, the "inseparable descent" of Cartier, which corresponds to the determination of finite and locally free morphisms such that admits a -basis with respect to (where is a given prescheme whose sheaf is annihilated by the prime number 0]]>). We note that this result can also be easily expressed without any regularity hypothesis on the local rings, and without supposing that is an algebraic scheme over a field. The theory of Jacobson–Bourbaki is obtained by taking to be the spectrum of a field of characteristic . Gabriel had already obtained a particular case of in the theory of passing to the quotient for finite commutative groups over a field . (Compare with ).2012fga3.iii-6fga3.iii-6.xmlQuotient by a proper and flat equivalence relation6fga3.iii1915fga3.iii-6-theorem-6.1fga3.iii-6-theorem-6.1.xmlTheorem6.1fga3.iii-6Let be a locally Noetherian prescheme, a quasi-projective -scheme, and an equivalence pre-relation on , such that: is proper and flat; and is a finite morphism (or, equivalently, by (a), a morphism with finite fibres, which is a condition that is automatically satisfied if comes from an equivalence relation). Under these conditions: exists, and (if is Noetherian) is quasi-projective over . ([Trans.] [Comp.] The fact that is quasi-projective over has only been proven, for now, in the case where comes from an equivalence relation.) The canonical morphism is surjective, proper, and open, and its fibres are the equivalence classes in modulo , and so can be identified with the topological quotient space of by the set-theoretic equivalence relation defined by . Finally, is surjective. If comes from an equivalence relation, then the equivalence relation in question is effective, i.e. is an isomorphism, and, further, is flat (and thus faithfully flat). 1908#261unstable-261.xmlProoffga3.iii-6-theorem-6.1 For the proof, we can reduce to by considering suitable quasi-sections of for , with the proof being analogous to the construction of algebraic quotient groups in the Séminaire Chevalley. In summary:1916fga3.iii-6-scholiumfga3.iii-6-scholium.xmlScholiumfga3.iii-6Let be quasi-projective over , with locally Noetherian. Then the data of a proper, faithfully flat, and surjective morphism from to an -prescheme is equivalent to the data of an equivalence relation on such that is proper and flat.The same method gives the following result:1917fga3.iii-6-theorem-6.2fga3.iii-6-theorem-6.2.xmlTheorem6.2fga3.iii-6Let be a Noetherian prescheme, a prescheme of finite type over , and an equivalence pre-relation on the -prescheme . Suppose that is flat and of finite type; and the morphism is quasi-finite (i.e. has finite fibres). Then there exists a dense open subset of that is saturated for , such that: If is the equivalence pre-relation induced by on , then exists, and is of finite type over . The canonical morphism is surjective and open, and its fibres are the set-theoretic equivalence classes for (and thus is a topological quotient space of by the set-theoretic equivalence relation defined by ). Finally, the morphism is surjective. If comes from an equivalence relation, then we can suppose that is faithfully flat and that is effective. This is a result of an essentially "birational" nature.1925fga3.iii-6-remarks-6.3fga3.iii-6-remarks-6.3.xmlRemarks6.3fga3.iii-6I do not know if, in and , hypothesis (b) is useless. In practice, it obliges us, in the passage to the quotient by groups, to restrict to he case where the stabilisers are all finite groups. We can ask if there are results analogous to and without any flatness hypothesis. I have no counter example in this direction. However, even keeping the flatness hypothesis, and restricting to equivalence relations such that is flat and quasi-finite (but not finite), and with affine, it can still be the case that is not effective: take the equivalence relations induced on the affine open subsets covering the Nagata variety (or a group with two elements acting in a "non-admissible" way).2013fga3.iii-7fga3.iii-7.xmlApplications7fga3.iiiAs we said in the introduction, the most important application of is the construction of Picard schemes, as well as solutions to various other problems of "modules", to which we will later return.We obtain a simple proof of the following result of Shimura:1936fga3.iii-7-proposition-7.1fga3.iii-7-proposition-7.1.xmlProposition7.1fga3.iii-7Let be an abelian scheme defined over a discrete valuation ring with field of fractions . Then every abelian scheme over that is isogenous to a quotient of "simplifies well for ", i.e. is isomorphic to some , where is an abelian scheme over (essentially unique, we recall). 1934#260unstable-260.xmlProoffga3.iii-7-proposition-7.1 We can suppose that is the quotient of by a subscheme in groups . (N.B. will not, in general, be "reduced", i.e. its local rings will have nilpotent elements). Consider the closed subscheme of given by "the closure" of , i.e. the smallest closed subscheme of such that contains . Then , and, since is a discrete valuation ring, we easily deduce that is a subscheme in groups of over . Since is proper over , so too is . Further, is projective over . We can thus apply in order to construct , which is the desired . Finally, essentially known arguments allow us to extract from the following result:1937fga3.iii-7-theorem-7.2fga3.iii-7-theorem-7.2.xmlTheorem7.2fga3.iii-7Let be the spectrum of an Artinian ring, and group schemes of finite type over , and a homomorphism of group schemes over . Suppose that is flat over ; and the kernel of is finite. Under these conditions, the quotient scheme exists, and the canonical morphism is surjective and open, and its fibres are the set-theoretic equivalence classes defined by the right action of on . Finally, if is a monomorphism, then the morphism is flat, and the morphism is an isomorphism, or, in other words, is a principal homogeneous space over , with structure group (acting on the right), or rather considered as a group scheme over (cf. FGA 3.I, §B.6).1941fga3.iii-7-corollary-7.3fga3.iii-7-corollary-7.3.xmlCorollary7.3fga3.iii-7Under these conditions, for to be flat over , it is necessary and sufficient that be flat over . If this condition is satisfied, then the passage to the quotient by commutes with every extension of the base , and if is an invariant subgroup of , then can be endowed with the structure of a quotient group of by .The situation is particularly simple if is the spectrum of a field, since then every -prescheme is automatically flat over . We find:1942fga3.iii-7-corollary-7.4fga3.iii-7-corollary-7.4.xmlCorollary7.4fga3.iii-7Let and be group schemes of finite type over a field , and let be a homomorphism of -groups. Then factors as , where is a homomorphism given by passing to the quotient by the closed subgroup of , and where is a group homomorphism that is a closed immersion. The quotient exists. The usual formalism (as in the Noether theorems) holds amongst algebraic groups over .This result allows us to treat the passage to the quotient in a uniform way for algebraic (in the classical sense, i.e. irreducible over and simple over ) groups, and the passage to the quotient by "infinitesimal" subgroups considered by Cartier. It is advantageous to consider the "hyperalgebras" introduced by Cartier, following from the work of Dieudonné on formal groups, as groups in the category of formal schemes over , and, if necessary (if they correspond to hyperalgebras of finite rank over ), as algebraic groups that are finite over .2014fga3.iii-8fga3.iii-8.xmlA conjecture8fga3.iii1962fga3.iii-8-remark-ifga3.iii-8-remark-i.xmlRemarkfga3.iii-8[Comp.] It now appears that the conjectures in this section are false, even for non-singular varieties over a field of characteristic , both for the existence and the quasi-projectivity of the quotient, and even when acts with a closed graph.The conjecture in question concerns the need of knowing how to pass to the quotient by the projective group acting on certain subschemes of "Hilbert schemes" (with these "Hilbert schemes" replacing, in the theory of schemes, Chow varieties).Let be a prescheme, and an integer. To every prescheme over , we associate the group of invertible matrices with values in the ring of sections of . We thus obtain a contravariant functor in , which can can easily show to be representable, and so the functor corresponds to a group scheme over (which is further affine over ) which we denote by . Its construction is compatible with change of base, so that, in reality, everything comes from a group scheme over , denoted by . The group , called the multiplicative group, and often denoted by , corresponds to the functor , with the latter being the group of "units" over . We have an evident homomorphism , and we can easily construct the quotient group, denoted by , and called the projective group of degree over . It represents the functor that sends to the group of sections of the sheaf , where denotes the sheaf of germs of sections of over . (Note that sections of over do not, in general, come from sections of over !) Note that we can prove that equally represents the functor (where is the projective-type scheme of relative dimension over ), at least when is Noetherian. It is in this way that it appears in the theory of modules.Let be a Noetherian scheme, which we can, if we want, suppose to be affine, and let be a quasi-projective -prescheme endowed with an invertible sheaf that is very ample with respect to . Suppose that the group acts on and simultaneously (in a way that is compatible with its action on ), and that it acts freely on .1963fga3.iii-8-conjecture-8.1fga3.iii-8-conjecture-8.1.xmlConjecture8.1fga3.iii-8Under the above conditions: The equivalence relation defined by is effective, the quotient is of finite type over , and the canonical morphism is flat and surjective (and thus becomes a homogeneous principal bundle on , with group ). Let be the invertible sheaf on induced from by "faithfully flat descent" under (cf. FGA 3.I, §B, Theorem 1). Then is "pre-ample" on with respect to , i.e. there exists an integer and a quasi-finite morphism from to some suitable projective-type scheme such that is isomorphic to the inverse image of .We note that, even if is separated over , then it can be the case that is not separated over (a situation that arises in not-at-all-pathological "module problems"). If (1) is satisfied, then is separated if and only if the equivalence relation defined by has a closed graph, i.e. if has a closed image (and is thus a closed immersion). If is separated, then is pre-ample on with respect to if and only if it is ample, i.e. if a suitable tensor power defines a projective immersion. In the module problems mentioned in the introduction, we can show that the equivalence relation to which we arrive does indeed have a closed graph.1964fga3.iii-8-remarks-8.2fga3.iii-8-remarks-8.2.xmlRemarks8.2fga3.iii-8We have assumed that to give a concrete example, and because it is currently the most important case in practice. The reasonable hypothesis to make on seems rather to be that be one of the "forms" on of a Tohokû group (whose construction over the integers has also been made by Chevalley). The only positive fact that is known to me concerning the above conjecture is the following: Let be an affine scheme over a field of characteristic , on which the group or acts freely. Then the equivalence relation defined by is effective, the quotient is affine, and the morphism is flat and surjective. The proof uses the following fact (that, for now, has only been proven in characteristic ): if we let act on the affine ring of , considered as a vector space over , then the trivial representation of only appears once (in a composition series of a vector subspace of finite dimension over that is stable under ). It seems possible that a systematic use of the theory of linear representations of would give a proof of the conjecture, or at least when we work over a base field. When we are no longer working over a base field, the author knows nothing.1965fga3.iii-8-remark-iifga3.iii-8-remark-ii.xmlRemarkfga3.iii-8[Comp.] As we note at the end of the next talk, the above conjecture is decidedly false. The "positive fact" mentioned in the above remark seems to have been proven simultaneously by various authors (Nagata, Rosenlicht, Grothendieck, ...).Backlinks3249fga3.v-5-remarks-5.1fga3.v-5-remarks-5.1.xmlRemarks5.1fga3.v-5The method that we followed is essentially that of Matsusaka for the projective construction of Picard varieties. The result that we invoke from that allows us to pass to the effective quotient can also easily be deduced from the existence theorem for Hilbert schemes (cf. for example [Mumford–Tate seminar, 1962]). (Classically, these quotients are constructed by using Chow coordinates). Note that the formation of the open of and its decomposition into opens that are quasi-projcetive over following the Hilbert polynomials for the divisors that define the invertible modules in question, is compatible with base change (which allows us to apply the technique of descent).3250indexindex.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 3251fga3.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.