# 4 The existence theorem and the formal theory of modules

Grothendieck, A.
“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.

(*[Trans.]*
Sections 4.1 to 4.5 were numbered A.1 to A.5 in the original;
sections 4.6 to 4.10 were numbered C.1 to C.5.)

**A.** Representable and pro-representable functors

## 4.1 Representable functors

Let *canonical covariant functor*

This functor *faithfully flat*;
in other words, for every pair *bijective*.

In particular, if a functor * X is determined up to unique isomorphism*.
We then say that the functor

*representable*. The above proposition then implies that the canonical functor

*equivalence*between the category

*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

*composition law*” in

*sets*

*the general yoga is purely and simply identifying, using the canonical functor*h , the objects of {\mathcal{C}} with particular contravariant functors (namely, representable functors) from {\mathcal{C}} 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 {\mathcal{C}}\to\mathtt{Set}*, i.e. those of the form

## 4.2 Pro-representable functors, pro-objects

Let *with I 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

*category*\operatorname{Pro}({\mathcal{C}}) of pro-objects of {\mathcal{C}} , whose objects are projective systems of objects of

*equivalence between the dual category of the category*\operatorname{Pro}({\mathcal{C}}) of pro-objects of {\mathcal{C}} and the category of pro-representable covariant functors from {\mathcal{C}} to \mathtt{Set} . Of course, an object

*. Then, if*{\mathcal{C}} is equivalent to a full subcategory of \operatorname{Pro}({\mathcal{C}})

*taken in*\operatorname{Pro}({\mathcal{C}}) (since

We draw attention to the fact that, even if the projective limit of the *exists in {\mathcal{C}}*, it will generally

*not*be isomorphic to the projective limit

*, and, in a precise way, depends functorially on the pro-object*\lim{}_{{\mathcal{C}}}X_i is already defined in terms of the {\mathcal{X}}

*pro-object*A pro-object *strict pro-object* if it is isomorphic to a pro-object *epimorphisms*;
a functor defined by such an object is said to be *strictly pro-representable*.
We can thus further demand that *ordered* set, and that every epimorphism *up to unique isomorphism* (in the usual sense of isomorphisms of projective systems).
It thus follows that *the projective limit of a projective system {\mathcal{X}}^{(\alpha)} of strict pro-objects always exists in \operatorname{Pro}({\mathcal{C}})*, and that, with the above notation of

## 4.3 Characterisation of pro-representable functors

Let

F commutes with finite projective limits;F commutes with finite products and with finite fibred products;F commutes with finite products, and, for every exact diagramX\to X'\rightrightarrows X'' in{\mathcal{C}} (cf. [12]), the image of the diagram underF F(X)\to F(X')\rightrightarrows F(X'') is exact.

We then say that *left exact*.

In what follows, we assume that finite projective limits always exist in *a pro-representable functor is left exact*.

To obtain a converse, let
*minimal* if, for all *dominates* *if \xi is minimal, and if F is left exact, then this morphism v is unique*;

*if*\xi'' is minimal, then v is surjective. From this we easily deduce the following proposition:

For

F is left exact; and- every pair
(X,\xi) , with\xi\in F(X) , is dominated by some*minimal*pair.

This second condition is trivial if every object of

Let

This last fact also implies that *every pro-object of {\mathcal{C}} is then strict*.

## 4.4 Example: groups of Galois type, pro-algebraic groups

If ^{9} groups.
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 *pro-algebraic groups*” of Serre [26].

## 4.5 Example: “formal varieties”

Let *Artinian \Lambda-algebras*).
The conditions of the corollary to Proposition 3.1 are then satisfied.
Here, the category

*topological algebras*

*linear*,

*separated*, and

*complete*, and such that, for every open ideal

*Artinian*algebra over

*local components*, which correspond to the “points” of the

*formal scheme*[11] defined by

*field*(for example, the residue field of the local component in question), and where two pairs

It is important to give conditions that ensure that the local component *Noetherian* ring.
If *quotient ring of a formal series ring \Lambda[[t_1,\ldots,t_n]]*.
To give such a criteria, we introduce (for every ring

Let *Noetherian* ring, it is necessary and sufficient for the set *finite dimension* over

*[Comp.]* The formula given above is only correct when

Suppose that *[Comp.]* The following definition is correct only when the residue extension *separable*; for the general case, see SGA III, 1.1.)
We say that * O is simple over \Lambda* if

*if*O is simple over \Lambda

For

If this is the case, then this implies that, for every *surjective* homomorphism *surjective*.
Of course, it suffices to verify this condition in the case where *local*, and (proceeding step-by-step) where the ideal of *infinitesimal* invariants of the situation that give us a functor *cohomological* nature.

To finish, we say some words, in the above context, about *rings of definition*.
Let *smallest* subring *ring of definition of the object \xi\in F(A)*.
If

*equivalent*to the condition that

**B.** The two existence theorems

Keeping the notation of §A.5, and, given a covariant functor
*left exact*.
In the current state of the technique of descent (cf. the questions asked in [12]), this criterion is not directly verifiable, in this form, in the most important cases, and we need criteria that seem less demanding.

For the functor

F commutes with finite products;- for every algebra
A\in{\mathcal{C}} and every homomorphismA\to A' in{\mathcal{C}} such that the diagramA \to A' \rightrightarrows A'\otimes_A A' is exact (cf. [12]), the diagramF(A) \to F(A') \rightrightarrows F(A'\otimes_A A') is also exact.

Furthermore, it suffices to verify condition (ii) in the case where

A is a*free*module overA ;- the quotient module
A'/A is anA -module*of length*.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 *technique of descent by flat morphisms* (cf. [12]), which offers no difficulty;
to deal with case (b), we will use the following result:

Let *exact* (which is the case, in particular, if *strict {\mathcal{F}}-descent morphism* (cf. [12]).

In other words, the data of a flat

We prove Theorem 2 by first proving that

**C.** Applications to some particular cases

## 4.6 General remarks on functors represented by preschemes

Let *locally of finite type* over *The main goal of these exposés is to develop a general technique that allows us to recognise when such a functor F is representable, and to study the properties of the corresponding S-prescheme X by means of the properties of F.*
We note in passing that, in this study, we find non-pathological examples of preschemes over

Let *local* Artinian ring, then *the restriction F_0 of F to “Artinian Y-algebras” is pro-representable, and is represented by the topological Y-algebra whose local components are the completions \widehat{{\mathscr{O}}_x} of the local rings of X at the points x of X that are closed and live above closed points of Y*.
This shows that only knowing

If we start with a given functor

To finish these generalities, we note how the theory of schemes explains some apparent anomalies, such as the Igusa surface

## 4.7 The schemes \operatorname{Hom}_S(X,Y) , \prod_{X/S}Z , \underline{\operatorname{Aut}}(X) , etc.

Let * S-automorphisms of* an

*groups*), a prescheme of

*from an*S -homomorphisms

*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

*existence*of schemes of the type

*flat*over

*proper*over

*affine*over

*finite*over

Let

Furthermore, we can show that, for all

Suppose that *Noetherian* rings.

The 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 *flat* over *projective* (or at least quasi-projective) over

*[Comp.]*
The problems described here are completely resolved in the projective case by “Hilbert schemes” (cf. FGA 3.IV).
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

## 4.8 Picard schemes

*[Comp.]* For a more complete study, see FGA 3.V.

Let *relative Picard group* of *Picard functor* of *Picard prescheme* of *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).

The definition adopted here is only reasonable when every point of

Here, the plausible existence conditions for a Picard prescheme are the following: *proper* and *flat* over *in eliminating the automorphisms of an invertible sheaf {\mathscr{L}} on X by endowing them with a marked section over the section s* ([12]).
Notably, we find the following:

Suppose that

Furthermore, we then have

If

We can generalise the definitions and results from this section to the classification of principal bundles on *affine* and *flat over S*, and also

*commutative*. In the case where

*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.

*[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 *Séminaire Mumford–Tate*, Harvard University (1961–62)).

## 4.9 Formal modules of a variety

Let *local* Noetherian ring of residue field *flat* over

where *functor of modules* for *formal scheme of modules* for

Here, if we wish to apply the technique of descent, the “finite” automorphisms of

Suppose that

—

If

X_0 is not assumed to be simple overk , thenF(I_A,\xi) can be identified with a sub-A -module of\operatorname{Ext}_{{\mathscr{O}}_{P_A}}^1(P_A;{\mathscr{I}}_{X_A},{\mathscr{O}}_{X_A}) where we setP_A=X_A\times_A X_A , where{\mathscr{O}}_{X_A} is considered as a coherent sheaf onP_A via the diagonal morphismX_A\to P_A , and where{\mathscr{I}}_{X_A} denotes the coherent sheaf of ideals onP_A defined by the diagonal morphism. More precisely, an easy globalisation of Hochschild theory shows that the\operatorname{Ext}^1 above can be identified with the set of classes, up to isomorphism, of sheaves ofI_A -algebras{\mathscr{O}} that are flat overX_A , and endowed with an augmentation isomorphism{\mathscr{O}}\otimes_{I_A}A\to{\mathscr{O}}_{X_A} (recall that we setI_A=At/(t^2) ). The submoduleF(I_A,\xi) is that which corresponds to the sheaves of*commutative*algebras. The simplicity hypotheses are thus not essential in the theory of modules, as [11] implies.Recall (loc. cit.) that, in particular, every

*simple*and*proper algebraic curve*X_0 overk admits a formal scheme of modules that is simple over\Lambda , and of relative dimension equal to3g-3 if the genusg is\geqslant 2 , and tog ifg=0,1 . These two latter cases no longer figure directly in Proposition 4.1. We can, however, recover them in the case of elliptic curves (g=1 ) thanks to the remarks that will follow.

We can, of course, vary Proposition 4.1 *ad libitum* by considering systems of schemes over *abelian scheme* over *groups* over *abelian* schemes over *one* fibre, and provided that *simple* over *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 *Greenberg functor*” to prove its worth…

## 4.10 Extension of coverings

Let *formal* Noetherian prescheme [11], *[Comp.]* It is also necessary to assume that the section defining *flat coverings* of *unramified over U*, the evident map

*injective*; in particular,

*an automorphism of*{\mathfrak{X}}' that induces the identity on X'_0 is the identity. This allows us to apply the technique of descent to the situation. We start, in particular, with a flat covering

If

V varies amongst open subset of{\mathfrak{X}} , then theG(V) form a*sheaf*on{\mathfrak{X}} , say{\mathscr{G}}_{\mathfrak{X}}={\mathscr{G}} . The restriction of this sheaf toU is the*constant*sheaf whose fibres consist of a single element.More generally, describing the fibres of

{\mathscr{G}}_{\mathfrak{X}} is a question of complete local rings, in a precise way:For all

x\in{\mathfrak{X}} , we have{\mathscr{G}}_x = G(\operatorname{Spec}({\mathscr{O}}_{{\mathfrak{X}},x})) \subset G(\operatorname{Spec}(\widehat{{\mathscr{O}}}_{{\mathfrak{X}},x})) (i.e. isomorphism classes of finite free algebrasB over\widehat{{\mathscr{O}}}_{{\mathfrak{X}},x} endowed with an isomorphism fromB\otimes_{\widehat{{\mathscr{O}}}_{{\mathfrak{X}},x}}{\mathscr{O}}_{X_0,x} to(\widehat{{\mathscr{O}}}'_0)_x , where{\mathscr{O}}'_0 is the sheaf of algebras onX_0 that definesX'_0 ).We have a canonical isomorphism

{\mathscr{G}}_{\mathfrak{X}}=\varprojlim{\mathscr{G}}_{{\mathfrak{X}}_n} ; in other words, for every open subsetV of{\mathfrak{X}} , we haveG(V)=\varprojlim G(V_n) .Suppose that

{\mathfrak{X}} comes from an ordinary*proper*schemeX over a complete local Noetherian ring\Lambda that has ideal of definition{\mathfrak{m}} by taking the{\mathscr{J}} -adic completion of{\mathscr{O}}_X , where{\mathscr{J}}={\mathfrak{m}}\cdot{\mathscr{O}}_X . ThenG({\mathfrak{X}}) is canonically isomorphic to the set of classes of flat coverings of the*ordinary*schemeX that are “reducible alongX'_0 ”.

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

The above functor is pro-representable.

Of course, by (a), if *is not Noetherian*.
Its existence, however, shows, in a striking manner, the *“continuous” nature* of the set *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 finish, we note that the situation simplifies if *we can take arbitrary “local” extensions at the ramification points*.
Further, if *simple* over

### References

[10] A. Grothendieck. “Sur quelques points d’algèbre homologique.” *Tohoku Math. J.* **9** (1957), 119–221.

[11] A. Grothendieck. “Géométrie formelle et géométrie algébrique.” *Séminaire Bourbaki*. **11** (n.d.), Talk no. 182.

[12] A. Grothendieck. “Technique de descente et théorèmes d’existence en géométrie algébrique, I: Généralités, descente pas morphismes fidèlement plats.” *Séminaire Bourbaki*. **12** (n.d.), Talk no. 190.

[26] J.-P. Serre. “Corps locaux et isogénies.” *Séminaire Bourbaki*. **11** (n.d.), Talk no. 185.

*[Trans.]*Here the word “Hausdorff” is implicit.↩︎