# 3 Generalities, and descent by faithfully flat morphisms

Grothendieck, A.
“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.

(*[Trans.]*
Sections 3.1 to 3.4 were numbered A.1 to A.4 in the original;
sections 3.5 to 3.9 were numbered B.1 to B.5.)

*[Comp.]*
For various details concerning the theory of descent, see also SGA 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

Currently, it seems that the general technique of descent that will be explained (combined with, when necessary, the fundamental theorems of “formal geometry”, cf. [9]) is at the base of the majority of existence theorems in algebraic geometry.^{7}
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, the specialists will know how to prove the “analytic” analogues of the existence theorems in formal geometry that will be given in article 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, show reasonably clearly the necessity of methods that are closer to the theory of schemes (which should naturally complement transcendental techniques).

In the present article (namely article I) we will discuss the most elementary case of descent (the one indicated in the title). The applications of Theorem 1, Theorem 2, and Theorem 3 below (in §B.1) 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 [6].
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

**A.** Preliminaries on categories

## 3.1 Fibred categories, descent data, {\mathcal{F}} -descent morphisms

### 3.1.1 (a)

A *fibred category {\mathcal{F}} with base {\mathcal{C}}* (or

*over*{\mathcal{C}} ) consists of

a category

{\mathcal{C}} for every

X\in{\mathcal{C}} , a category{\mathcal{F}}_X for every

{\mathcal{C}} -morphismf\colon X\to Y , a functorf^*\colon{\mathcal{F}}_Y\to{\mathcal{F}}_X , which we also write asf^*(\xi) = \xi \times_Y X for\xi\in{\mathcal{F}}_Y (withX being thought of as an “object of{\mathcal{C}} overY ”, i.e. as being endowed with the morphismf )for any two composible morphisms

X\xrightarrow{f}Y\xrightarrow{g}Z , an isomorphism of functorsc_{f,g}\colon (gf)^* \to f^*g^*

with the above data being subject to the conditions that

\operatorname{id}^*=\operatorname{id} c_{f,g} is the identity isomorphism iff org is an identity isomorphismfor any three composible morphisms

X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}T , the following diagram, given by using the isomorphisms of the formc_{u,v} , commutes:\begin{CD} (h(gf))^* @= ((hg)f)^* \\@VVV @VVV \\(gf)^*h^* @. f^*(hg)^* \\@VVV @VVV \\(f^*g^*)h^* @= f^*(g^*h^*) \end{CD}

Let *fibre product*

Let *inverse image of sheaves of modules* functor.
We thus obtain a category fibred over

### 3.1.2 (b)

A diagram of maps of sets
*exact* if

Let * {\mathcal{F}}-exact* if, for every pair

(where

In this above diagram, for simplicity, we have identified

Let *gluing data* on *compatible with the gluing data* if the following diagram commutes:

With this definition, the objects of *category*.
If *canonical functor* from the category * {\mathcal{F}}-exact* if the above functor is

*fully faithful*, i.e. if the above functor defines an equivalence between the category

We say that a gluing data on *effective* (with respect to

In the case where the diagram (+) is *the category {\mathcal{F}}_S is equivalent to the category of objects of {\mathcal{F}}_{S'} endowed with effective gluing data*.

### 3.1.3 (c)

The most important case is that where

\Delta^*(\varphi) = \operatorname{id}_\xi , where\Delta\colon S'\to S'\times_S S' denotes the diagonal morphism, and where we identify\Delta^* p_i^*(\xi') with(p_i\Delta)^*(\xi')=\xi' .p_{31}^*(\varphi) = p_{32}^*(\varphi)p_{21}^*(\varphi) , wherep_{ij} denotes the canonical projection fromS'\times_S S'\times_S S' to the partial product of itsi th andj th factors.

We define *descent data* on

A morphism * {\mathcal{F}}-descent morphism* if the diagram of morphisms

*strict*{\mathcal{F}} -descent morphismif, further, every descent data (Definition 1.6) on any object of

This latter condition (of strictness) can also be stated in a more evocative way:
“giving an object of

Note that, if an ^{8} *section*

### 3.1.4 (d)

We can present the above notions in a more intuitive manner, by introducing, for an object

i bis.

ii bis.

We can show that (ii bis) implies that

## 3.2 Exact diagrams and strict epimorphisms, descent morphisms, and examples

### 3.2.1 (a)

Let *exact* if, for all *kernel* of the pair

This kernel is evidently determined up to unique isomorphism.
If *cokernel* of the pair

A morphism *strict epimorphism* if it is an epimorphism and, for every morphism

If the fibre product *strict monomorphism*.

To make the relation between the ideas of

A morphism *universal epimorphism* (resp. a *strict universal epimorphism*) if, for every

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

A morphism *descent morphism* (resp. a *strict descent morphism*) if it is an

If

### 3.2.2 (b)

Let *finite* over

is exact if and only if the diagram of sheaves on *if S is a Noetherian prescheme, then every finite morphism S'\to S 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.

### 3.2.3 (c)

If (+) is an exact diagram of finite morphisms, then, for every *flat* morphism *flat* over *flat* *flat* quasi-coherent sheaves on the prescheme *effectiveness* of a gluing data (and, more specifically, of a descent data, when *The effectiveness of \varphi then implies the following: M is a flat A-module, and the above homomorphism is an isomorphism.*

In 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 *fields*, over which everything, in fact, is flat!).

## 3.3 Application to étalements

Let *étalé* over

B is flat overA ; andB/{\mathfrak{m}}B is a separable finite extension ofA/{\mathfrak{m}}=k (where{\mathfrak{m}} denotes the maximal ideal ofA ).

If *étale at x\in T*, or

*étalé over*S at x , if

*étale*, or an

*étalement*, or

*étalé over*S , if

If *étale covering* *étale coverings* of *finite*) étale coverings of *the Grauert–Remmert theorem remains true without any normality hypotheses on S*.
First let

*completions*of the modules that are involved). From this, we obtain a coherent sheaf of algebras

*reduced*scheme, i.e. such that

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 *universal homeomorphism*).
For every *Then the functor T\mapsto T' is an equivalence between the category of preschemes T that are étalé over S and the category of preschemes T' that are étalé over S'.*
(We use the bijectivity of

*topological invariance of the fundamental group of a prescheme*”).

## 3.4 Relations to 1 -cohomology

### 3.4.1 (a)

Let *simplicial object* of *if u and v are morphisms T\to T', then the corresponding morphisms K_T\to K_{T} are homotopic*.
We say that

*dominates*

*abelian*category

Suppose that

Note that, as per usual, we can define *augmentation*)

### 3.4.2 (b)

For example, let *saying that the augmentation morphism*
*is an isomorphism for every pair of elements \xi,\eta\in{\mathcal{F}}_S implies that \alpha\colon S'\to S is an {\mathcal{F}}-descent morphism* (Definition 1.7).

### 3.4.3 (c)

Similarly, for * Z^1(S'/S,G) is canonically identified with the set of descent data on \xi'=\xi\times_S S' with respect to S'\to S* (Definition 1.6), and that

*. Then,*\operatorname{H}^1(S'/S,G) can be identified with the set of isomorphism classes of objects of {\mathcal{F}}_{S'} endowed with a descent data relative to \alpha\colon S'\to S that are isomorphic, as objects of {\mathcal{F}}_{S'} , to \xi'=\xi\times_S S'

*if*\alpha\colon S'\to S is an {\mathcal{F}} -descent morphism(cf. §A.4.b),

*then*\operatorname{H}^1(S'/S,G) contains as a subset the set of isomorphism classes of objects \eta of {\mathcal{F}}_S such that \eta\times_S S' is isomorphic (in {\mathcal{F}}_{S'} ) to \xi\times_S S' ; further,

*this inclusion is the identity if and only if every descent data on*\xi'=\xi\times_S S' with respect to \alpha\colon S'\to S is effective. (This will be the case, in particular, if

The cochain complexes of the form

### 3.4.4 (d)

Let * C^\bullet(S'/S,F) is canonically isomorphic to the simplicial group C^\bullet(\Gamma,F(S')) of standard homogeneous cochains, and so \operatorname{H}^\bullet(S'/S,F) is canonically isomorphic to \operatorname{H}^\bullet(\Gamma,F(S')*.

### 3.4.5 (e)

Let

We note that the following conditions concerning a morphism

- The augmentation homomorphism
\operatorname{H}^0(S,{\mathscr{O}}_S) = \operatorname{G_a}(S)\to\operatorname{H}^0(S'/S,\operatorname{G_a}) is an isomorphism. \alpha\colon S'\to S is an{\mathcal{F}} -descent morphism (where{\mathcal{F}} is the fibred category over{\mathcal{C}} described above).\alpha\colon S'\to S is a strict epimorphism (cf. §A.2.c).

Now suppose that *In all the cases known to the speaker, \operatorname{H}^i(A'/A,\operatorname{G_a})=0 for i>0, and, if A is local, then \operatorname{H}^1(A'/A,\operatorname{G_m})=0, and, more generally, \operatorname{H}^1(A'/A,\operatorname{GL}(n))=(e)* (if

*Hilbert’s “Theorem 90” is exactly the fact that*\operatorname{H}^1(S'/S,\operatorname{G_m})=0 if A is a field and A' is a finite Galois extension of A (cf. Example 1),

*and we can also express it by saying that, in the case in question,*S'\to S is a strict descent morphisms for the fibred category of locally free sheaves of rank 1 .This latter statement is the one that is most suitable to generalise Hilbert’s theorem, by varying the hypotheses both on the morphism

Finally, we note that, for a local *Artinian*

\operatorname{H}^1(A'_k/A_k,\operatorname{G_a})=0 for allk .\operatorname{H}^1(A'_k/A_k,\operatorname{G_m})=0 for allk .\operatorname{H}^1(A'_k/A_k,\operatorname{GL}(n))=(e) for allk and alln .

If *strict* descent morphism for free modules (not necessarily of finite type) over

The definition of the groups

**B.** Descent by faithfully flat morphisms

## 3.5 Statement of the descent theorems

A morphism *flat* if *faithfully flat* if it is flat and surjective.

For example, if *exact* and *faithful* functor in the

A morphism *quasi-compact* if the inverse image of every quasi-compact open subset *finite* union of affine open subsets).

It evidently suffices to verify this property for the *affine* open subsets of

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.

Let *faithfully flat* and *quasi-compact*.
Then *strict descent morphism* (cf. Definition 1.7 for the fibred category

This statement implies two things:

If

{\mathcal{F}} and{\mathscr{G}} are quasi-coherent sheaves onS , and{\mathcal{F}}' and{\mathscr{G}}' their inverse images onS' , then the natural homomorphism\operatorname{Hom}({\mathcal{F}},{\mathscr{G}}) \to \operatorname{Hom}({\mathcal{F}}',{\mathscr{G}}') is a bijection from the left-hand side to the subgroup of the right-hand side consisting of homomorphisms{\mathcal{F}}'\to{\mathscr{G}}' that are compatible with the canonical descent data on these sheaves, i.e. whose inverse images under the two projections ofS''=S'\times_S S' toS' give the same homomorphism{\mathcal{F}}''\to{\mathscr{G}}'' .Every quasi-coherent sheaf

{\mathcal{F}}' onS' endowed with a descent data with respect to the morphism\alpha\colon S'\to S (cf. Definition 1.6 is isomorphic (endowed with this data) to the inverse image of a quasi-coherent sheaf{\mathcal{F}} onS .

Setting

Let *exact* (cf. Definition 1.1.

Also, the combination of (i) and (ii) with Definition 1.1 gives:

Let

Of course, we have an equivalent statement in terms of quotient sheaves.
As we have already seen in §A.4.e, Theorem 1 should be thought of as a generalisation of Hilbert’s “Theorem 90”, and implies, as particular cases, various formulations in terms of

Let *resolution* of

*Proof*. It suffices to prove that the augmented complex induced from the above by extension of the base

We note, in passing, the following corollary, which generalises a well-known statement in Galois cohomology (compare with §A.4.e):

If

*Proof*. *Part (ii).*
To prove part (ii) of Theorem 1, we proceed, as for (i), by restricting to the case where

We can evidently vary Theorem 1 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

Let *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 *exact*, i.e.

Let *topological quotient space of S'*, i.e. every subset

To complete Theorem 2, we must give effectiveness criteria for a descent data on an *such a descent data is not necessarily effective*, even if *For a descent data on X'/S' with respect to \alpha\colon S'\to S (assumed to be faithfully flat and quasi-compact) to be effective, it is necessary and sufficient that X' be a union of open subsets X'_i that are affine over S' and “stable” under the descent data on X'.*
This is certainly the case (for any

*radicial*(i.e. injective, and with radicial residual extensions). We can also show that this is the case if

*finite*, and every finite subset of

*Weil criterion*). It is, in particular, the case if

Let *radicial*, then it is a *strict descent morphism*.
If

I do not know if, in the second claim above, the hypothesis that *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 S there exists an open neighbourhood U, a finite and faithfully flat U' over U, and an S-morphism from U' to S'*.
A type of case that is not covered by the above is that where

*[Comp.]*
A morphism *non-isotrivial* under an abelian scheme.

## 3.6 Application to the descent of certain properties of morphisms

Let *faithfully flat* and *quasi-compact* (this latter hypothesis being sometimes overly strong).
We can see this directly without difficulty if

## 3.7 Decent by finite faithfully flat morphisms

Let *finite* morphism, corresponding to a sheaf of algebras *locally free* and of finite type as a sheaf of modules, and everywhere non-zero.
Then

where

In other words, *a descent data on {\mathcal{F}}' is equivalent to a representation of the sheaf {\mathscr{U}}=\underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathscr{A}}',{\mathscr{A}}') of {\mathscr{O}}_S-algebras in the sheaf \underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathcal{F}}',{\mathcal{F}}') of {\mathscr{O}}_S-algebras that satisfies the two linearity conditions (3.1)*.
If we have a pairing of quasi-coherent sheaves on

*equivalent to the given pairing*, in the evident meaning of the phrase, if and only if the following condition is satisfied:

For every section

for every pair *compatible with the diagonal map of {\mathscr{U}}*, with respect to the given pair).
In particular, equations (3.1) to (3.4) allow us to understand, in terms of representations of algebras via diagonal maps, the descent data on a quasi-coherent sheaf of

*algebras*on

From here, we obtain an analogous interpretation of descent data on an arbitrary *over S* endowed with a homomorphism of

(*Weil*).
Suppose that *Galois étale covering* with Galois group * {\mathscr{U}} admits, as a left A'-module, a basis given by the sections of {\mathscr{U}} that correspond to elements of \Gamma*.

(*Cartier*).
Let *characteristic p*), that

*radicial of height*1 ), and that the sheaf of algebras

*locally admits a*p -basis(i.e. a family

It follows from the above lemma that a descent data on the quasi-coherent sheaf

(This is what we may call a *linear connection on {\mathcal{F}}*, which is further

*flat*and

*compatible with the*p -th powers). We can similarly write down the notion of a descent data on an

*derivations*of

Note that we have not needed to impose any flatness, non-singular, or finiteness hypotheses on

## 3.8 Application to rationality criteria

Let *invertible sheaf* (i.e. locally free of rank *section of {\mathcal{F}} over s* to be a section of the invertible sheaf

*if*{\mathcal{F}}_1 and {\mathcal{F}}_2 are isomorphic, then there exists exactly one isomorphism from {\mathcal{F}}_1 to {\mathcal{F}}_2 that is compatible with the sections in question(i.e. sending the first to the second). We also, independently of the section

*equivalent*if every point of

*every invertible sheaf*{\mathcal{F}} on X is equivalent to an invertible sheaf {\mathcal{F}}_1 endowed with a marked section over s (we take

*and*{\mathcal{F}}_1 is determined up to isomorphism. In other words, the classification

*up to equivalence*of invertible sheaves on

*up to isomorphism*of invertible sheaves endowed with a marked section.

Since these properties remain true under flat extensions

*With the prescheme X/S being as above, and admitting a section s, let \alpha\colon S'\to S be a faithfully flat and quasi-compact morphism; let {\mathcal{F}}' be an invertible sheaf on X'=X\times_S S'.*

*For*{\mathcal{F}}' to be equivalent to the inverse image on X' of an invertible sheaf {\mathcal{F}}' on X , it is necessary and sufficient for its inverse images p_1^*({\mathcal{F}}') and p_2^*({\mathcal{F}}') on X'\times_X X'=X\times_S(S'\times_S S') to be equivalent.

*If this is the case, then*{\mathcal{F}} is determined up to equivalence.(We then say that

*rational*on

Considering this principle in the case where *rationality criteria of Weil and of Cartier*.
(We note that the authors restrict to the case where *sheaf of 1-differentials of X' with respect to X*).
Since the restrictions of the

*if*\Omega_{S'/S}^1 is locally free on S (as in the Cartier case),

*then*\xi defines a section of \operatorname{R}^1f'({\mathscr{O}}_{X'})\otimes\Omega_{S'/S}^1 on S' (called the

*Atiyah–Cartier class of the invertible sheaf*{\mathcal{F}} on X'/S )

*whose vanishing is necessary and sufficient for the inverse images of*{\mathcal{F}}' under the two projections of

*to*X' to be equivalent(where

*necessary*for the inverse images of

*connection*of

## 3.9 Application to the restriction of the base scheme to an abelian scheme

Let *abelian scheme* over *regular* (i.e. that its local rings are regular), then we can show, using the *connection theorem* of Murre [20]
(at least in the case “of equal characteristics”, where the cited theorem is currently proven) that *every rational section of X over S is everywhere defined* (i.e. is a section) (which generalises a classical theorem of Weil).
It then follows, more generally, that, if

*with*S Noetherian and regular, and K denoting its ring of rational functions(a direct sum of fields),

*let*X be an abelian scheme over K ; if X is isomorphic to a K -scheme of the form X_0\times_S\operatorname{Spec}(K) , where X_0 is an abelian scheme over S , then X_0 is determined up to unique isomorphism.

Using the above uniqueness result, we see that the question of restriction of the base to *simple* morphism of finite type, if *then X'_0 is endowed with a canonical descent data with respect to \alpha*.
Taking Theorem 3 into account, we thus conclude:

Let *unramified over S*, let

The speaker does not know if we can replace the hypothesis that *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

## 3.10 Application to local triviality and isotriviality criteria

Let *prescheme of groups*” over * G acts*” (on the right).
We say that

*formally principal homogeneous*under

*isomorphism*. From now on, we assume

*flat*over

*principal homogeneous bundle*under

*faithfully flat*and

*quasi-compact*over

*faithfully flat*and

*quasi-compact extension*

*trivial*, i.e. isomorphic to

*closed*points of

*finite and flat*over

*simple*over

*étale*over

*geometric case*”), then

We can consider other, stronger, types of conditions on *isotrivial* (resp. *strictly isotrivial*) if, for all *finite and faithfully flat* morphism (resp. a *surjective étale covering*)

If *affine* over

that every principal homogeneous bundle under

G (resp. every locally free sheaf of rankn onS ) becomes isomorphic to the “trivial” objectG (resp.{\mathscr{O}}_S^n ) under a suitable faithfully flat and quasi-compact extension ofS ;that we can descend the type of objects in question (principal homogeneous bundles under

G , resp. locally free sheaves of rankn ) by such morphisms; and, finallythat the automorphism group of the trivial bundle on any

S'/S is functorially isomorphic to the automorphism group of the trivial locally free sheaf of rankn onS' ,

we “formally” conclude that it is “equivalent” to give, on *equivalence of fibred categories*).
We thus conclude, in particular:

Every principal homogeneous bundle under the group

By known arguments, we thus conclude the same result for others structure groups such as *every* principal homogeneous bundle of structure group

We also point out that, for most groups (linear or not) that are simple over *à la* Cartier (cf.Example 2) is obviously isotrivial.

One 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.

### References

[4] J. Dieudonné, A. Grothendieck. “Eléments de géométrie algébrique.” *Publications Mathématiques de L’Institut Des Hautes Etudes Scientifiques*. (n.d.).

[6] H. Grauert, R. Remmert. “Komplexe Räume.” *Math. Annalen*. **136** (1958), 245–318.

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

[20] J.P. Murre. “On a connectedness theorem for a birational transformation at a simple point.” *Amer. J. Math.* **80** (1958), 3–15.

[24] J.-P. Serre. “Géométrie algébrique et géométrie analytique.” *Ann. Institut Fourier Grenoble*. **6** (1956), 1–42.

[25] J.-P. Serre. “Espaces fibrés algébriques.” *Séminaire Chevalley*. **3** (1958), Talk no. 1.

*[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 articles of this current series (i.e. “Techniques of descent and existence theorems in algebraic geometry”), but not for the projective techniques (articles IV, V, and VI).↩︎*[Comp.]*It is useless to assume here that\alpha is an{\mathcal{F}} -descent morphism.↩︎