# 5 Quotient preschemes

[FGA 3.III]
Grothendieck, A.
“Technique de descente et théorèmes d’existence en géométrie algébrique, III: Préschemas quotients”.
*Séminaire Bourbaki* **13** (1960–61), Talk no. 212.

## Introduction

*[Comp.]*
We note that the application (of the theory developed here) in FGA 3.V (“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 §8, the most important gap in the theory presented here is the lack of existence criteria 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 [15].
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 FGA 3.I and FGA 3.II.
Indeed, the question of *effectiveness of a descent data* on a *Picard scheme* (for the definition, see FGA 3.II, C.3) of an

The problem of passing to the quotient in preschemes again offers unresolved questions.
The most important is mentioned in §8.
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.

## 5.1 Equivalence relations, effective equivalence relations

Let *equivalence pair*” in *target* *source* *equivalence {\mathcal{C}}-relation* on

If

Every time that we have a pair of morphisms *cokernel* of the pair as an object

We now start with a morphism
*associated* with the morphism *associated* with the morphism

We say that a pair of morphisms *effective equivalence pair* if

- the cokernel
Y=X/(p_1,p_2) exists ; - the fibre product
X\times_Y X exists ; and - the morphism
R\to X\times_Y X with componentsp_1 andp_2 is an isomorphism.

Then the pair *effective equivalence relation*.

We say that a morphism *effective epimorphism* if

- the fibre product
R=X\times_Y X exists ; - the quotient
X/(p_1,p_2) exists, wherep_1 andp_2 are the projections fromR toX ; and - the morphism
X/(p_1,p_2)\to Y induced byf is an isomorphism.

Then *effective quotient* of

The above definitions imply the following “*Galois correspondence*”:

There is a bijective correspondence, respecting the natural orders, between the set of effective equivalence relations

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

## 5.2 Example: finite preschemes over S

Let *all* preschemes.

As we mentioned in FGA 3.I, *there are non-effective epimorphisms in {\mathcal{C}}* (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

We can also consider the case of a prescheme *finite equivalence relation*.
Supposing, for simplicity, that *we do not know, even in this case, if there exists a quotient X/R=Y, and if the canonical morphism X\to Y is finite*.
(The most simple case is that where we suppose that

*existence*of a quotient

*finiteness*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

The only general positive fact known to the author is the following:

Let *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 *radicial and surjective* (and, in fact, a surjective closed immersion, since it is a monomorphism).

## 5.3 The case of a group with operators

We now suppose that

Suppose that *freely* on the set *acts freely* on *principal {\mathcal{C}}-space under G*).
The equivalence relation associated to the pair

*equivalence relation defined by the group*G acting freely on

*monomorphism*.

Of course, even if

The cokernel in question will often be denoted by

## 5.4 Equivalence pre-relations

Recall 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, *objects* and the latter the set of *arrows*, endowed with the following structures:

- a pair of maps
p_1,p_2\colon R\rightrightarrows X called the*source map*and the*target map*; - a map
\pi\colon(R,p_2)\times_X(R,p_1) \to R 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 *symmetry* of

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 * {\mathcal{C}}-category*” (resp.

*) on a pair of objects*{\mathcal{C}} -groupoid

*source morphism*and the

*target morphism*, and, if the fibre product in question exists, a morphism

*composition morphism*; these three morphisms then suffice to determine the structure of a category (resp. groupoid) on

It is important, in practice, to know how to understand the morphisms *simplicial operations* in a suitable semi-simplicial or simplicial objects of *simplex types* as the category whose objects are finite sets of the form
*arbitrary maps* between these finite sets.
We note that the category *non-empty* sets, where we take the morphisms to be maps between finite sets.
In *non-empty* 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 *increasing maps* between the *does* exist;
consider

A *simplicial object* (resp. *semi-simplicial object*) in a category *symmetry operations* in the

With the above, for all *simplicial set* (resp. *semi-simplicial set*), which is said to be *associated to the category Z*, and denoted by

The functor *categories* and the category of semi-simplicial sets, i.e. contravariant functors *that send amalgamated sums A\coprod_C B (of the type described above) to fibre products of sets*.

Similarly, the functor *groupoids* and the category of simplicial sets, i.e. contravariant functors *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:

The above claim remains true when we replace categories, groupoids, and simplicial sets with *provided that* fibre products exist in

The semi-simplicial object

We now define an *equivalence pre-relation* on an object

An equivalence relation on an object

A

We can avoid the logical difficulties that arise in a statement such as Proposition 4.1 by implicitly assuming that all the objects in question can be found in a fixed “universe” (that is itself a set).

## 5.5 Quotient by a finite and flat equivalence relation

Let

B is integral overA , i.e.f is an integral morphism.- The morphism
f is surjective, and its fibres are the set-theoretic equivalence classesp_2(p_1^{-1}(x)) inX modulo{\mathcal{R}} , and the topology ofY is the quotient of that ofX . Y is the quotient ofX by{\mathcal{R}} in the category of preschemes.- If
{\mathcal{R}} comes from an equivalence*relation*, then the morphismf\colon X\to Y is finite and locally free (i.e.B is a projectiveA -module of finite type), and the equivalence relation is effective, i.e.R_1\to X\times_Y X is an isomorphism.

This theorem generalises the well-known theorem concerning the case of a finite group

The canonical morphism *surjective*.

Let *admissible*.
With this definition:

Let

We can in fact easily show that every equivalence class modulo *stable* under

Suppose that this condition is satisfied, and, further, that

We then immediately conclude, by descent, the following:

Under the conditions of Corollary 5.4, for

In summary:

The data of a finite, locally free, and surjective morphism

—

We 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

f\colon X\to Y such thatf_*({\mathcal{O}}_X) admits ap -basis with respect to{\mathcal{O}}_Y (whereX is a given prescheme whose sheaf{\mathcal{O}}_X is annihilated by the prime numberp>0 ). We note that this result can also be easily expressed without any regularity hypothesis on the local rings, and without supposing thatX is an algebraic scheme over a field. The theory of Jacobson–Bourbaki is obtained by takingX to be the spectrum of a field of characteristicp .Gabriel had already obtained a particular case of Theorem 5.3 in the theory of passing to the quotient for finite commutative groups over a field

k . (Compare with Corollary 7.4).

## 5.6 Quotient by a proper and flat equivalence relation

Let *locally Noetherian* prescheme, *quasi-projective*

p_1\colon R_1\to X is proper and flat; andR_1\to X\times_S X is a finite morphism (or, equivalently, by (a), a morphism with finite fibres, which is a condition that is automatically satisfied if{\mathcal{R}} comes from an equivalence relation).

Under these conditions:

Y=X/{\mathcal{R}} exists, and (ifS is Noetherian) is quasi-projective^{10}overS .The canonical morphism

f\colon X\to Y is surjective, proper, and open, and its fibres are the equivalence classesp_2(p_1^{-1}(x)) inX modulo{\mathcal{R}} , and soY can be identified with the topological quotient space ofX by the set-theoretic equivalence relation defined by{\mathcal{R}} . Finally,R_1\to X\times_Y X is surjective.If

{\mathcal{R}} comes from an equivalence relation, then the equivalence relation in question is effective, i.e.R_1\to X\times_Y X is an isomorphism, and, further,f\colon X\to Y is flat (and thus faithfully flat).

For the proof, we can reduce to Theorem 5.1 by considering suitable quasi-sections of

In summary:

Let

The same method gives the following result:

Let

p_1\colon R_1\to X is flat and of finite type; and- the morphism
R_1\to X\times_S X is quasi-finite (i.e. has finite fibres).

Then there exists a *dense* open subset *saturated* for

If

{\mathcal{R}}_U is the equivalence pre-relation induced by{\mathcal{R}} onU , thenU/{\mathcal{R}}_U exists, and is of finite type overS .The canonical morphism

U\to U/{\mathcal{R}}_U is surjective and open, and its fibres are the set-theoretic equivalence classes for{\mathcal{R}}_U (and thusU/{\mathcal{R}}_U is a topological quotient space ofU by the set-theoretic equivalence relation defined by{\mathcal{R}}_U ). Finally, the morphism({\mathcal{R}}_U)_1\to U\times_{U/{\mathcal{R}}_U}U is surjective.If

{\mathcal{R}} comes from an equivalence relation, then we can suppose thatU\to U/{\mathcal{R}}_U is faithfully flat and that{\mathcal{R}}_U is effective.

This is a result of an essentially “birational” nature.

—

I do not know if, in Theorem 6.1 and Theorem 6.2, 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 Theorem 6.1 and Theorem 6.2 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

p_1\colon R\to X is flat and quasi-finite (but not finite), and withX affine, it can still be the case thatR 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).

## 5.7 Applications

As we said in the introduction, the most important application of Theorem 6.1 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:

Let

*Proof*. We can suppose that

Finally, essentially known arguments allow us to extract from Theorem 6.2 the following result:

Let

F is flat overS ; and- the kernel of
u is finite.

Under these conditions, the quotient scheme

Under these conditions, for *quotient group* of

The situation is particularly simple if

Let

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

## 5.8 A conjecture

*[Comp.]*
It now appears that the conjectures in this section are false, even for non-singular varieties over a field of characteristic

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 *multiplicative group*, and often denoted by

Let *freely* on

Under the above conditions:

The equivalence relation defined by

G is effective, the quotientY=X/G is of finite type overS , and the canonical morphismf\colon X\to Y is flat and surjective (and thusX becomes a homogeneous principal bundle onY , with groupG\times_S Y=\operatorname{GP}(n)_Y ).Let

{\mathscr{L}}' be the invertible sheaf onY induced from{\mathscr{L}} by “faithfully flat descent” underf (cf. FGA 3.I, B, Theorem 1). Then{\mathscr{L}}' is “pre-ample” onY with respect toS , i.e. there exists an integerm and a quasi-finite morphism fromY to some suitable projective-type scheme\mathbb{P}_S^N such that({\mathscr{L}}')^{\otimes m} is isomorphic to the inverse image of{\mathcal{O}}_{\mathbb{P}_S^N}(1) .

We note that, even if

::: {.rmenv #fga-3-iii-remarks-8.2 title=“Remarks 8.2” latex=“{Remarks 8.2}”}f
We have assumed that *Let X be an affine scheme over a field k of characteristic 0, on which the group \operatorname{GL}(n)_k or \operatorname{GP}(n-1)_k acts freely. Then the equivalence relation defined by G is effective, the quotient X/G is affine, and the morphism X\to X/G is flat and surjective.*
The proof uses the following fact (that, for now, has only been proven in characteristic

*[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, …).

### References

*Bull. Amer. Math. Soc.*

**67**(1961), 483–486.

*[Comp.]*The fact thatY=X/{\mathcal{R}} is quasi-projective overS has only been proven, for now, in the case where{\mathcal{R}} comes from an equivalence relation.↩︎