Exposé V. The Fundamental Group: Generalities

Introduction

The present Seminar is the continuation of the 1960 Seminar. We refer to the latter by sigla such as I.9.7, meaning: Séminaire de Géométrie Algébrique, Exposé I, no. 9.7. The numbers of the 1961 exposés will follow those of 1960. We refer to the Éléments de Géométrie Algébrique of Dieudonné-Grothendieck by sigla such as EGA I 8.7.3.

The present exposé summarizes, with slight additions, the last exposés of 1960, which had not been written up.

As in 1961, we shall generally restrict ourselves to locally noetherian preschemes, although this restriction is often inessential. In Exposé VI we shall admit the theory of faithfully flat descent, summarized in Bourbaki Seminar no. 190. If need be, we shall give a more detailed exposition in a later exposé,¹ once the reader has had occasion to convince himself of the usefulness of this technique for the theory of the fundamental group.

1. Prescheme with a Finite Group of Operators; Quotient Prescheme

Let $X$ be a prescheme, and let $G$ be a finite group operating on $X$ by automorphisms, on the right to fix ideas. If $X$ is affine with ring $A$, then $G$ operates by automorphisms on the left on $A$.

For every prescheme $Z$, $G$ operates on the left on the set $\mathrm{Hom}(X,Z)$, so one may consider the set

$$\mathrm{Hom}(X,Z{)}^G$$

of morphisms invariant under $G$. This depends functorially on $Z$; one may ask whether this functor is “representable”, i.e. isomorphic to a functor $Z\mapsto \mathrm{Hom}(Y,Z)$. This means that one can find a prescheme $Y$ and a morphism invariant under $G$,

$$p:X\to Y,$$

such that for every $Z$, the corresponding map $g\mapsto gp$,

$$\mathrm{Hom}(Y,Z)\to \mathrm{Hom}(X,Z{)}^G$$

is bijective. One then says that $(Y,p)$ is a quotient prescheme of $X$ by $G$; it is determined up to unique isomorphism.

Proposition.

Let $A$ be a ring on which the finite group $G$ operates on the left, let $B=A^G$ be the subring of invariants of $A$, let $X=\mathrm{Spec}(A)$ and $Y=\mathrm{Spec}(B)$, and let $p:X\to Y$ be the canonical morphism, evidently invariant under $G$. Then:

1. $A$ is integral over $B$, i.e. $p$ is an integral morphism.
2. The morphism $p$ is surjective, its fibers are the orbits of $G$, and the topology of $Y$ is the quotient of that of $X$.
3. Let $x\in X$, $y=p(x)$, and let $G_x$ be the stabilizer of $x$. Then $\kappa (x)$ is a quasi-Galois algebraic extension of $\kappa (y)$, and the canonical map from $G_x$ to the group $\mathrm{Gal}(\kappa (x)/\kappa (y))$ of $\kappa (y)$-automorphisms of $\kappa (x)$ is surjective.
4. $(Y,p)$ is a quotient prescheme of $X$ by $G$.

The statements (i), (ii), (iii) are well known in commutative algebra² and are included only for reference, except for the assertion on the topology, which comes from the following general fact, an easy consequence of the Cohen-Seidenberg theorem: an integral morphism is closed, i.e. transforms closed sets into closed sets. Let us note at once:

Corollary.

Under the preceding conditions, the natural homomorphism

$${\mathcal{O}}_Y\to p_{\ast }({\mathcal{O}}_X{)}^G$$

is an isomorphism.

This follows at once from the formula

$$(S^{-1}A{)}^G=S^{-1}(A^G)$$

valid for every multiplicatively stable subset $S$ of $B=A^G$. This formula is modular and is stated more generally for a base change $A\to A^{\prime }$ that is flat; apply it to the case where $S$ is generated by an element $f$ of $B$.

Assertion (ii) and Corollary V.1.2 easily imply (iv). More generally, we shall have the following.

Proposition.

Let $X$ be a prescheme with a finite group $G$ of automorphisms, and let $p:X\to Y$ be an invariant affine morphism such that

$${\mathcal{O}}_Y\to p_{\ast }({\mathcal{O}}_X{)}^G$$

is an isomorphism. Then the conclusions (i), (ii), (iii), (iv) of V.1.1 are still valid.

Indeed, for (i), (ii), (iii), we may suppose $Y$ and hence $X$ affine; if $B$ and $A$ are their rings, the hypothesis implies $B=A^G$, and it suffices to apply V.1.1. For (iv), use (ii) and ${\mathcal{O}}_Y=p_{\ast }({\mathcal{O}}_X{)}^G$.

Corollary.

Under the conditions of V.1.3, for every open $U$ of $Y$, $U$ is a quotient of $X|U=p^{-1}(U)$ by $G$.

Indeed, $p^{-1}(U)\to U$ induced by $p$ satisfies the same hypotheses as $p$.

If now $X$ is a $Z$-prescheme and the operations of $G$ are $Z$-automorphisms, then by (iv) $Y$ is a $Z$-prescheme. With this understood:

Corollary.

In order that $X$ be affine, respectively separated, over $Z$, it is necessary and sufficient that $Y$ be so. If $X$ is of finite type over $Z$, then it is finite over $Y$; if moreover $Z$ is locally noetherian, then $Y$ is of finite type over $Z$.

Since $X$ is affine, and a fortiori separated, over $Y$, if $Y$ is affine, respectively separated, over $Z$, then so is $X$. Conversely, suppose $X$ affine over $Z$; we prove that $Y$ is so. By V.1.4 we may suppose $Z$ affine, and we are reduced to proving that if $X$ is affine, then $Y$ is affine. This follows from the explicit determination of $Y$ as $\mathrm{Spec}(A^G)$ made in V.1.1. Similarly, since $p:X\to Y$ is integral, hence universally closed, and surjective, it follows that if $X$ is separated over $Z$, then so is $Y$; this is a lemma to be isolated. Indeed, in the diagram

X ×_Z X  --p×_Zp-->  Y ×_Z Y
↑                     ↑
X   --------p------>  Y,

where the vertical arrows are the diagonals, the morphism $X{\times }_{Z}X\to Y{\times }_{Z}Y$ is closed, hence transforms the diagonal, closed in $X{\times }_{Z}X$, into a closed subset of $Y{\times }_{Z}Y$, which is moreover just the diagonal of the latter since $p$ is surjective.

If $X$ is of finite type over $Z$, it is a fortiori of finite type over $Y$; hence it is finite over $Y$, since it is already integral over $Y$. Suppose moreover $Z$ locally noetherian; we prove that $Y$ is of finite type over $Z$. By V.1.4 we may suppose $Z$ affine. Since the topological space $X$ is quasi-compact and $p:X\to Y$ is surjective, $Y$ is also quasi-compact, hence a finite union of affine opens; by V.1.4 we are reduced to the case where $Y$ is affine, and hence $X$ affine. But then the ring $A$ of $X$ is an algebra of finite type over the ring $C$ of $Z$, which is noetherian, and it is known that $B=A^G$ is then also an algebra of finite type over $C$: indeed, $A$ is integral, hence finite, over a subalgebra $B^{\prime }$ of $B$ of finite type over $C$; since $B^{\prime }$ is noetherian, $B$ is also finite over $B^{\prime }$, hence of finite type over $C$.

Corollary.

In order that $X$ be affine, respectively a scheme, it is necessary and sufficient that $Y$ be so.

Definition.

Let $X$ be a prescheme on which a finite group $G$ operates on the right. We say that $G$ operates admissibly if there exists a morphism $p:X\to Y$ having the properties of V.1.3, which implies that $X/G$ exists and is isomorphic to $Y$.

Proposition.

Let $X$ be a prescheme on which the finite group $G$ operates on the right. In order that $G$ operate admissibly, it is necessary and sufficient that $X$ be a union of affine opens invariant under $G$, or again that every orbit of $G$ in $X$ be contained in an affine open.

The latter condition is evidently implied by the first, and in turn it implies the first. Indeed, let $T$ be an orbit of $G$ and $U$ an affine open containing it. The intersection of the transforms of $U$ by the $g$ in $G$ is then an open $U^{\prime }$ stable under $G$, containing $T$ and contained in the affine open $U$. Since in $U$ every finite subset has a fundamental system of affine open neighborhoods, there exists an affine open neighborhood $V$ of $T$ contained in $U^{\prime }$. Its transforms by the $g$ in $G$ are affine and contained in $U^{\prime }$, which is separated; hence their intersection $U^{\prime \prime }$ is an affine open, invariant under $G$ and containing $T$.

With this established, the condition considered in V.1.8 is necessary, for one takes the inverse images $X_i$ of affine opens $Y_i$ covering $Y$. It is sufficient, because then by V.1.1 one can construct the quotients $Y_i=X_i/G$. In each $Y_i$, the image of $X_i\cap X_j$ is an open $Y_{ij}$ identifying with $X_{ij}/G$ by V.1.4; in particular, one deduces isomorphisms $Y_{ij}\to Y_{ji}$ allowing the $Y_i$ to be glued to construct $Y$. Serre prefers to construct directly the topological quotient space $Y$ of $X$ by $G$, put on it the sheaf $p_{\ast }({\mathcal{O}}_X{)}^G$, and verify that $Y$ thereby becomes a prescheme and that one is then under the conditions of V.1.3.

Corollary.

If $G$ operating on $X$ is admissible, the same is true for every subgroup $H$ of $G$; hence $X/H$ exists.

This may also be verified directly in the situation V.1.3, noting that one may always suppose $X$ affine over some $Z$ and the $s\in G$ operate by $Z$-automorphisms, for instance by taking $Z=Y$. Indeed:

Corollary.

Suppose $X$ is affine over $Z$, and the operations of $G$ are $Z$-automorphisms. Then $G$ operates on $X$ admissibly. If $X$ is defined by a quasi-coherent sheaf of algebras $\mathcal{A}$, $Y$ is defined by the sheaf ${\mathcal{A}}^G$ of invariants of $\mathcal{A}$ under $G$.

Proposition.

Suppose $G$ operates admissibly on $X$, and $X/G=Y$ is a prescheme over $Z$. Consider a base-change morphism $Z^{\prime }\to Z$, and put $X^{\prime }=X{\times }_Z{Z}^{\prime }$ and $Y^{\prime }=Y{\times }_Z{Z}^{\prime }$, so that $G$ still operates on $X^{\prime }$ by transport of structure, the morphism $p^{\prime }:X^{\prime }\to Y^{\prime }$ being invariant. If $Z^{\prime }$ is flat over $Z$, then $p^{\prime }$ still satisfies the hypotheses of V.1.3, i.e.

$${\mathcal{O}}_{Y^{\prime }}\to p_{\ast }^{\prime }({\mathcal{O}}_{X^{\prime }}{)}^G$$

is an isomorphism, $p^{\prime }$ being affine in any case. Thus $G$ operates admissibly on $X^{\prime }$, and

(X/G) ×_Z Z′ ≃ (X ×_Z Z′)/G.

We may evidently suppose $Z=Y$, reducing to the case where moreover $Y$ and $Y^{\prime }$ are affine. One must show that if $B$ is the subring of invariants of $G$ operating in $A$, and if $B^{\prime }$ is an algebra over $B$ flat over $B$, then $B^{\prime }$ is the subalgebra of invariants of $A^{\prime }=A{\otimes }_B{B}^{\prime }$. This is immediate, because the exact sequence

0 → B --i→ A --j→ A^(G)

where the last term means a power of $A$, and where $j(x)$ is the system of $s\cdot x-x$, $s\in G$, remains exact after tensoring by $B^{\prime }$.

Care must be taken that the flatness hypothesis is essential for the validity of the result. In particular, if $Y^{\prime }$ is a closed sub-prescheme of $Y$, for instance even a closed point of $Y$, and $X^{\prime }$ is its inverse image in $X$, then $Y^{\prime }$ does not identify in general with $X^{\prime }/G$. We shall see that it does if $X$ is étale over $Y$.

Finally, let us give a formalism that is as convenient as it is trivial. Let $Y$ be a prescheme. Since direct sums exist in the category of preschemes, for every set $E$ one may consider the prescheme that is the sum of a family $(Y_i{)}_{i\in E}$ of preschemes all identical to $Y$; this prescheme will be denoted $Y\times E$. It is characterized by the formula

Hom(Y × E,Z) = Hom(E,Hom(Y,Z)),

where the second Hom evidently denotes the set of maps from the set $E$ to the set $\mathrm{Hom}(Y,Z)$. There is a canonical morphism

$$Y\times E\to Y$$

making $Y\times E$ a prescheme over $Y$. Since fiber products commute with direct sums in the category of preschemes, if $Y$ is a prescheme over another $Z$, then for a base change $Z^{\prime }\to Z$ one has

(Y × E) ×_Z Z′ = (Y ×_Z Z′) × E,

a formula useful especially if $Z=Y$. On the other hand, one concludes trivially from the definition that

(Y × E) × F = Y × (E × F) = (Y × E) ×_Y (Y × F),

the last formula, however, following from the commutativity noted above.

For fixed $Y$, one may regard $Y\times E$ as a functor in $E$, with values in the preschemes over $Y$. By the preceding formula this functor commutes with finite products, allowing for example every ordinary group $G$ to correspond to a group scheme $Y\times G$ over $Y$, which is finite over $Y$ if $G$ is, etc. More generally, this functor is “left exact”, but we shall not need that here. This functor also trivially commutes with direct sums, and it is also “right exact”, as one sees at once from the defining formula $V.1.\ast$. In particular, if the finite group $G$ operates on the right on the set $E$, then it operates on the right on $Y\times E$, and one has

(Y × E)/G = Y × (E/G),

where in fact the quotient on the left satisfies the conditions of V.1.3; this is immediate.

2. Decomposition and Inertia Groups. The Étale Case

Let $G$ be a finite group operating on the right on the prescheme $X$. If $x\in X$, the decomposition group of $x$ is the stabilizer $G_d(x)$ of $x$. This group operates canonically, on the left, on the residue field $\kappa (x)$, and the set of elements of $G_d(x)$ that operate trivially is called the inertia group of $x$, denoted $G_i(x)$.

Suppose $G$ operates on $X$ admissibly and $Y$ is a prescheme over a prescheme $Z$. Fix $z\in Z$, and an algebraically closed extension $\Omega$ of $\kappa (z)$ having transcendence degree greater than that of the $\kappa (x)/\kappa (z)$, where $x$ is a point of $X$ over $z$. We may regard $\mathrm{Spec}(\Omega )$ as a $Z$-scheme, and the points of $X$ with values in $\Omega$ correspond to homomorphisms of $\kappa (z)$-algebras $\kappa (x)\to \Omega$, where $x$ is a point of $X$ over $z$. Since $\Omega$ was taken large enough, every point $x$ of $X$ over $z$ is the locality of a point of $X$ with values in $\Omega$. If $X(\Omega )$ and $Y(\Omega )$ denote respectively the sets of points of $X$ and $Y$ with values in $\Omega$, there is a natural map

$$X(\Omega )\to Y(\Omega ).$$

On the other hand, $G$ operates on $X(\Omega )$, and the preceding map is invariant under $G$. With this understood, conclusions (ii) and (iii) of V.1.3 are also interpreted as follows: the preceding map is surjective and identifies $Y(\Omega )$ with the quotient $X(\Omega )/G$. Moreover, if $x$ is the locality of $a\in X(\Omega )$, then the stabilizer of $a$ in $G$ is exactly the inertia group $G_i(x)$. All this is in fact true without supposing $\Omega$ “large enough”; this last hypothesis serves only to ensure that the inertia group of every element of $X$ over $z$ can be characterized as a “geometric” stabilizer. One concludes at once, for example:

Proposition.

Make a base extension $Z^{\prime }\to Z$, whence $X^{\prime }=X{\times }_Z{Z}^{\prime }$. Let $x^{\prime }$ be a point of $X^{\prime }$ and $x$ its image in $X$. Then $G_i(x)=G_i(x^{\prime })$.

It suffices, in the considerations above, to take $\Omega$ to be a sufficiently large extension of $\kappa (z^{\prime })$, where $z$ and $z^{\prime }$ are the images of $x$ and $x^{\prime }$ in $Z$ and $Z^{\prime }$.

Proposition.

Under the conditions of V.1.3, suppose $Y$ locally noetherian and $X$ finite over $Y$. Let $H$ be a subgroup of $G$, consider $X^{\prime }=X/H$, cf. V.1.7, and let $x\in X$, $x^{\prime }$ its image in $X^{\prime }$, and $y$ its image in $Y$.

1. If $H\supset G_d(x)$, then the homomorphism ${\mathcal{O}}_y\to {\mathcal{O}}_{x^{\prime }}$ induces an isomorphism on completions.
2. If $H\supset G_i(x)$, then the homomorphism ${\mathcal{O}}_y\to {\mathcal{O}}_{x^{\prime }}$ is étale, i.e. $X^{\prime }$ is étale over $Y$ at $x^{\prime }$.

Let $Y_1=\mathrm{Spec}({\hat{\mathcal{O}}}_y)$. Make the base change $Y_1\to Y$. We obtain $X_1=X{\times }_Y{Y}_1$ finite over $Y_1$, on which $G$ operates, with quotient $Y_1$ by V.1.9. Let $y_1$ be the unique point of $Y_1$ over $y$. Since $\kappa (y)=\kappa (y_1)$, it follows that the fiber of $X$ at $y$ is isomorphic to that of $X_1$ at $y_1$, whence a unique point $x_1$ of $X_1$ over $x$. Moreover, by V.1.9 we have $X_1/H=X_1^{\prime }=X^{\prime }{\times }_Y{Y}_1$. Let $x_1^{\prime }$ be the image of $x_1$ in $X_1^{\prime }$. It lies over $x^{\prime }$, and one verifies easily, since $X^{\prime }$ is of finite type over $Y$, that the homomorphism ${\mathcal{O}}_{x^{\prime }}\to {\mathcal{O}}_{x_1^{\prime }}$ induces an isomorphism on completions. Thus we are reduced to the case where $Y$ is the spectrum of a complete local ring $B$; hence $X$ is the spectrum of a finite ring $A$ over $B$, a product of finitely many local rings $A_x$ corresponding to the points $x_i$ of $X$ over $Y$. If $A_0$ corresponds to $x=x_0$, then $A$ identifies with the ring ${\mathrm{Hom}}_{G_d}(G,A_0)$ of functions $f:G\to A_0$ such that $f(st)=sf(t)$ for $s\in G_d$, the operations of $G$ on these functions being defined by $(uf)(t)=f(tu)$. Thus, if $H$ is any subgroup of $G$, $A^H$ is the ring of functions $f:G\to A_0$ such that

f(stu) = s f(t),     s ∈ G_d, u ∈ H.

It is therefore a semi-local ring whose local components correspond to the double classes $G_{d}aH$ in $G$; to the double class defined by $a\in G$ corresponds, by the map $f\mapsto f(a)$, the subring $A_0^{H(a)}$ of $A_0$, where $H(a)=G_d\cap aH{a}^{-1}$. Moreover, the local component of $A^H$ corresponding to the image $x^{\prime }$ of $x$ is also the one corresponding to the double class $G_{d}H$ of the identity element; its local component is therefore $A_0^{G_d\cap H}$. If $G_d\subset H$, one finds $A_0^{G_d}=A^G=B$, which proves (i). To prove (ii), by passing to a suitable finite flat extension of $A$ and using V.2.1, one may reduce to the case where the residual extension $\kappa (x)/\kappa (y)$ is trivial. But then $G_i(x)=G_d(x)$, and one is reduced to the preceding case.

Corollary.

Under the conditions of V.2.2, suppose $G_i(x)=(e)$. Then $X$ is étale over $Y$ at $x$. Hence if $G_i(x)=(e)$ for every $x\in X$, then $X\to Y$ is an étale morphism.

There is a partial converse:

Corollary.

Suppose $X$ connected and the group $G$ faithful on $X$. In order that $p:X\to Y=X/G$ be étale, it is necessary and sufficient that the inertia groups of the points of $X$ be reduced to the identity element. If this is so, $G$ identifies with the group of all $Y$-automorphisms of the $Y$-scheme $X$.

Taking V.2.3 into account, we may suppose $X$ étale over $Y$. But if $s\in G$ lies in some $G_i(x)$, it follows from I.5.4 that $s$ operates trivially on $X$, hence is the identity element since $G$ is faithful. This proves the first assertion. Let $u$ be a $Y$-automorphism of $X$, and let $x\in X$. By Proposition V.1.3, there exists $s\in G$ such that $s(x)=u(x)$, inducing the same residual homomorphism $\kappa (x)\to \kappa (x^{\prime })$ as $u$. By the cited place, one has $s=u$, completing the proof.

Remark.

The hypothesis that $G$ operates faithfully is obviously not superfluous in Corollary V.2.4. The same is true of the hypothesis that $X$ is connected, as one sees for example by taking $X=Y\times E$, with $E$ a finite set, and $G$ the group of permutations of $E$: $G$ operates with plenty of inertia, nevertheless $(Y\times E)/G=Y\times (E/G)=Y$, and $X$ is étale over $Y$. Taking for $G$ a group strictly smaller than the symmetric group of $E$, but operating transitively on $E$, one sees that there will also be $Y$-automorphisms of $X$ not coming from $G$.

The typical example of a group $G$ operating without inertia is that of $Y\times G$, on which $G$ operates through its operations on the factor $G$ by right translations: a $Y$-prescheme $X$ with a right group of operators $G$ is said to be trivial if it is isomorphic to $Y\times G$.

To make the link between preschemes with finite groups of operators and the notion of principal bundle in a category, a link we shall not need in the sequel of the seminar but which is important in other contexts, the following considerations are useful. We fix a base prescheme $Y$ and place ourselves in the category of $Y$-preschemes. If $G$ is a finite group, write for short $G_Y=Y\times G$. This is a finite group scheme over $Y$, cf. no. 1; and if $X$ is a $Y$-prescheme, then

X ×_Y G_Y = X × G.

Giving a $Y$-morphism $X{\times }_Y{G}_Y\to X$ is therefore equivalent to giving a $Y$-morphism $X\times G\to X$, i.e. to giving, for every $g\in G$, a $Y$-morphism $T_g:X\to X$. One verifies at once that in order for the data of the $T_g$ to define on $X$ a structure of prescheme with a right group of operators $G$, i.e. $T_{g{g}^{\prime }}=T_{g^{\prime }}{T}_g$ and $T_e=i{d}_X$, it is necessary and sufficient that the corresponding $Y$-morphism $X{\times }_Y{G}_Y\to X$ define on $X$ a structure of $Y$-prescheme with $Y$-group scheme of operators, in the general sense of objects with $\mathcal{C}$-group of operators in a category $\mathcal{C}$. Suppose this is so. Recall that $X$ is said to be formally principal homogeneous under G_Y³ if the canonical morphism

X ×_Y G_Y → X ×_Y X,

whose components are respectively $p{r}_1$ and the multiplication morphism $\pi :X{\times }_Y{G}_Y\to X$, is an isomorphism. In the present case, identifying the first member with $X\times G$, the morphism considered is the one that associates to every $g\in G$ the morphism

(id_X,T_g) = (id_X ×_Y T_g) Δ_{X/Y}: X → X ×_Y X.

Thus to say that $X$ is formally principal homogeneous under G_Y also means that $X{\times }_{Y}X$ is isomorphic to the direct sum of the transforms of the diagonal by the elements $(e,g)$ of $G\times G$, operating on $X{\times }_{Y}X$ in the evident way, where $e$ denotes the identity element of $G$. If one does not want to distinguish left and right, and wants to give a formula that remains applicable to a product of more than two factors identical to $X$, one may formulate the condition by saying that the canonical morphism

X ×_G (G × G) → X ×_Y X

obtained by attaching to the pair $(g,g^{\prime })$ the morphism

(T_g,T_{g′}) = (T_g ×_Y T_{g′}) Δ_{X/Y}: X → X ×_Y X

and making $G$ operate on the left on $G\times G$ by the diagonal homomorphism,

$$s(g,g^{\prime })=(sg,s{g}^{\prime }),$$

is an isomorphism.

The notion of principal homogeneous space is deduced from that of formally principal homogeneous space by adding an additional axiom, ensuring that the “quotient” of $X$ by G_Y exists and is precisely the right unit object of the category, here $Y$. This axiom may vary with the context, and is often most conveniently made explicit, in the yoga of “descent”, by requiring that the object with operators become “trivial”, i.e. isomorphic to the product $X{\times }_Y{G}_Y$, here $X\times G$, after a suitable base change of specified type, so as in practice to allow descent techniques; cf. Grothendieck, Technique de descente et théorèmes d'existence en Géométrie Algébrique, Séminaire Bourbaki no. 190, pp. 26-28.⁴ In this spirit, let us note here the characterization of principal homogeneous bundles with group $G$, in the sense of the cited place:

Proposition.

Let $Y$ be a locally noetherian prescheme, and let $X$ be a $Y$-prescheme with a finite group $G$ of operators operating on the right. The following conditions are equivalent:

1. $X$ is finite over $Y$, $Y=X/G$, and the inertia groups of the points of $X$ are reduced to the identity.
2. There exists a faithfully flat and quasi-compact base change $Y_1\to Y$ such that $X_1=X{\times }_Y{Y}_1$ is a trivial $Y_1$-prescheme with operators, i.e. isomorphic to $Y_1\times G$.
3. As in (ii), but with $Y_1\to Y$ finite, étale, and surjective.
4. $X$ is formally principal homogeneous under G_Y, and faithfully flat and quasi-compact over $Y$.

Proof. (i) ⇒ (ii bis). Take $Y_1=X$, noting that $X\to Y$ is indeed finite, étale by V.2.3, and surjective. We show that $X_1$ is then trivial over $Y_1$; this will follow from:

Corollary.

If (i) holds and $X$ has a section over $Y$, then $X$ is a trivial space with operators.

Indeed, this section allows one to define a $G$-morphism $X\times G\to X$, surjective because $G$ is transitive on the fibers of $X$, injective because $G$ operates without inertia; finally, it is a local isomorphism by I.5.3 since $X$ is étale over $Y$. Hence it is an isomorphism.

(ii bis) trivially implies (ii), which implies (i), because the ingredients of (i) are “invariant” under faithfully flat quasi-compact extension of the base: for “finite”, cf. the Bourbaki seminar cited above; for inertia groups, apply V.2.1; and for $Y=X/G$, use a converse to V.1.9 in the case of a faithfully flat base change, which we forgot to spell out.

We proved (i) ⇒ (iii) in passing, by proving (i) ⇒ (ii bis). Finally, (iii) ⇒ (ii), because the first hypothesis in (iii) means precisely that $X$ becomes trivial after the base change $Y_1=X$; hence (ii), since $X$ is faithfully flat and quasi-compact over $Y$.

Definition.

A $Y$-prescheme $X$ with right group of operators $G$ satisfying the equivalent conditions of V.2.6 is called a principal covering of $Y$, with Galois group $G$.

3. Automorphisms and Morphisms of Étale Coverings

Proposition.

Let $X$ be étale, separated, and of finite type over locally noetherian $Y$, and let $G$ be a finite group operating on $X$ by $Y$-automorphisms. Then $G$ operates admissibly, and the quotient prescheme $X/G$ is étale over $Y$.

We do not suppose $X$ finite over $Y$; however, $X$ is quasi-projective over $Y$, whence the existence of $X/G$ by V.1.8. We first prove:

Corollary.

The morphism $X\to X/G$ is étale.

We may evidently suppose $G$ transitive on the set of connected components of $X$; then, by considering the stabilizer of a connected component, we may suppose $X$ itself connected. Finally, we may suppose $G$ operates faithfully. But then, as in V.2.4, $G$ operates without inertia, so by V.2.3 it follows that $X\to X/G$ is étale. We conclude using:

Lemma (remorse about Exposé I).

Let $X\to X^{\prime }\to Y$ be morphisms of finite type, and let $x$ be a point of $X$, with images $x^{\prime }$ and $y$. Suppose $Y$ locally noetherian. If two of the morphisms under consideration are étale at the marked points, then so is the third.

It remains only to consider the case where $X\to X^{\prime }$ and $X\to Y$ are étale at $x$ and prove that $X^{\prime }\to Y$ is étale at $x^{\prime }$, which is the case needed for V.3.1. Making a suitable flat extension of the base $Y$, one is reduced to the case where the residual extension $\kappa (x)/\kappa (y)$ is trivial. Consider the homomorphisms ${\mathcal{O}}_y\to {\mathcal{O}}_{x^{\prime }}\to {\mathcal{O}}_x$ and the homomorphisms deduced by passage to completions. The hypothesis means that ${\hat{\mathcal{O}}}_y\to {\hat{\mathcal{O}}}_x$ and ${\hat{\mathcal{O}}}_{x^{\prime }}\to {\hat{\mathcal{O}}}_x$ are isomorphisms; hence at once ${\hat{\mathcal{O}}}_y\to {\hat{\mathcal{O}}}_{x^{\prime }}$ is one, proving the lemma.

Corollary.

If $X$ is finite and étale over $Y$, then $X/G$ is finite and étale over $Y$.

Proposition.

Let $X$ and $X^{\prime }$ be two étale coverings of $Y$. Then every $Y$-morphism $f:X\to X^{\prime }$ factors as the product of a surjective étale morphism $X\to X^{\prime \prime }$ and the canonical immersion $X^{\prime \prime }\to X^{\prime }$ of a subset $X^{\prime \prime }$ of $X^{\prime }$ that is both open and closed.

We know by I.4.8 that $f$ is étale, hence an open morphism. On the other hand, since $X$ is finite over $Y$, $f$ is closed, so $f(X)=X^{\prime \prime }$ is both open and closed in $X^{\prime }$. This finishes the proof. It would have been enough for $X^{\prime }$, instead of being an étale covering, to be unramified over $Y$.

Corollary.

With the preceding notation, $X\to X^{\prime \prime }$ is a strict epimorphism in the category of preschemes, and $X^{\prime \prime }\to X^{\prime }$ is a monomorphism, indeed a strict monomorphism, in the category of preschemes.

The first assertion means by definition that the sequence of morphisms

X ×_{X″} X ⇉ X → X″

is exact, and this follows from the fact that $X\to X^{\prime \prime }$ is finite and faithfully flat, as is easily seen; cf. Grothendieck, loc. cit. The dual assertion for $X^{\prime \prime }\to X^{\prime }$ is even more trivial.

Corollary V.3.6 will be useful for the theory of the fundamental group in the next number. For those who do not like the notion of strict epimorphism, it is possible to replace Corollary V.3.6 by whatever variant the reader arranges to his personal taste. Let us only take the occasion to point out that a factorization $f=f^{\prime }{f}^{\prime \prime }$, with $f^{\prime \prime }$ a strict epimorphism and $f^{\prime }$ a monomorphism, is necessarily unique up to unique isomorphism in any category. However, there may simultaneously exist a factorization $f=f_1{f}_2$ having the dual properties: $f_2$ is an epimorphism and $f_1$ a strict monomorphism, also unique up to unique isomorphism, which is not isomorphic to the preceding one. It is enough to take, for example, the category of topological vector spaces, separated if desired, and for $u:X\to X^{\prime }$ a morphism such that $u(X)$ is not closed.

Proposition.

Let $Y$ be a connected locally noetherian prescheme, let $y$ be a point of $Y$, and let $\Omega$ be an algebraically closed extension of $\kappa (y)$. For every $X$ over $Y$, denote by $X(\Omega )$ the set of points of $X$ with values in $\Omega$. Let $X$ and $X^{\prime }$ be étale coverings of $Y$, and let $u:X\to X^{\prime }$ be a $Y$-morphism such that the corresponding map $X(\Omega )\to X^{\prime }(\Omega )$ is an isomorphism. Then $u$ is an isomorphism.

We are immediately reduced to the case where $X^{\prime }$ is connected. Since $X\to X^{\prime }$ is finite and étale, we know that the geometric number of points in a fiber of $X\to X^{\prime }$ is constant, and is equal to 1 if and only if the morphism under consideration is an isomorphism. But this number is also the number of elements in a fiber of $X(\Omega )\to X^{\prime }(\Omega )$, whence the conclusion.

4. Axiomatic Conditions for a Galois Theory

Let $\mathcal{C}$ be a category, and let $F$ be a covariant functor from $\mathcal{C}$ to the category of finite sets. Suppose the following conditions are satisfied:

(G 1) $\mathcal{C}$ has a final object,⁵ and the fiber product of two objects over a third exists in $\mathcal{C}$. This axiom may also be stated by saying that finite projective limits exist in $\mathcal{C}$.

(G 2) Finite sums in $\mathcal{C}$ exist, hence also an initial object ${\varnothing }_{\mathcal{C}}$ playing the role of the empty set, as does the quotient of an object of $\mathcal{C}$ by a finite group of automorphisms.

(G 3) Let $u:X\to Y$ be a morphism in $\mathcal{C}$. Then $u$ factors as a product $X--u^{\prime }\to Y^{\prime }--u^{\prime \prime }\to Y$, with $u^{\prime }$ a strict epimorphism and $u^{\prime \prime }$ a monomorphism, which is an isomorphism onto a direct summand of $Y$.

(G 4) The functor $F$ is left exact, i.e. transforms the right unit into the right unit and commutes with fiber products.

(G 5) $F$ commutes with finite direct sums, transforms strict epimorphisms into epimorphisms, and commutes with passage to the quotient by a finite group of automorphisms.

(G 6) Let $u:X\to Y$ be a morphism in $\mathcal{C}$ such that $F(u)$ is an isomorphism. Then $u$ is an isomorphism.

Our aim is to construct a topological group $\pi$, a projective limit of finite groups, and an equivalence of the category $\mathcal{C}$ with the category $\mathcal{C}(\pi )$ of finite sets on which $\pi$ operates continuously, i.e. so that the stabilizer of a point is an open subgroup, or equivalently so that there exists a discrete quotient group already operating on the set in question. The equivalence constructed will transform the given functor $F$ into the evident inclusion functor from $\mathcal{C}(\pi )$ into the category of finite sets. Note at once that the category $\mathcal{C}(\pi )$ constructed from a topological group $\pi$, and the preceding inclusion functor, do satisfy conditions (G 1) to (G 6).

We proceed in several steps.

1. Let $u:X\to Y$ be in $\mathcal{C}$. In order that $u$ be a monomorphism, it is necessary and sufficient that $F(u)$ be one. This uses (G 1), (G 4), (G 6).

Indeed, to say that $u$ is a monomorphism means that the projection $p{r}_1:X{\times }_{Y}X\to X$ is an isomorphism.

1. Every object $X$ of $\mathcal{C}$ is artinian.

Indeed, if $X^{\prime }\to X^{\prime \prime }\to X$ are monomorphisms such that $F(X^{\prime })$ and $F(X^{\prime \prime })$ have the same image in $F(X)$, then by (a) $F(X^{\prime })\to F(X^{\prime \prime })$ is an isomorphism, hence $X^{\prime }\to X^{\prime \prime }$ is an isomorphism by (G 6).

1. The functor $F$ is strictly pro-representable; cf. Grothendieck, Technique de descente et théorèmes d'existence en Géométrie Algébrique, II, Séminaire Bourbaki 195, February 1960.

Indeed, by the cited place, Proposition V.3.1, this follows from (b) and (G 4). We may therefore find a projective system over a filtered ordered set $I$,

$$P=(P_i{)}_{i\in I},$$

in $\mathcal{C}$, regarded as a pro-object of $\mathcal{C}$, and a functorial isomorphism

F(X) = Hom_{Pro(𝒞)}(P,X) = colim_i Hom_𝒞(P_i,X).

This isomorphism is realized by an element

φ ∈ lim_i F(P_i) = F(P).

One may moreover suppose that the transition homomorphisms ${\varphi }_{ji}:P_i\to P_j$, for $i\ge j$, are epimorphisms, and that every epimorphism $P_i\to P^{\prime }$ is equivalent to an epimorphism $P_i\to P_j$ for suitable $j\le i$. This determines the projective system $P$ in an essentially unique way.

An object $X\in \mathcal{C}$ is called connected if it is not isomorphic to the sum of two objects of $\mathcal{C}$ not isomorphic to the initial object ${\varnothing }_{\mathcal{C}}$.

1. The $P_i$ are connected and not isomorphic to ${\varnothing }_{\mathcal{C}}$.

If $X$ is a left unit, then $F(X)=\varnothing$ by (G 5), applied to the direct sum of an empty family, and conversely by (G 6). Thus if $X^{\prime }$ is an object of $\mathcal{C}$ that is not a left unit, i.e. such that $F(X^{\prime })\ne \varnothing$, there is no morphism from $X^{\prime }$ to $X$. Hence if some $P_i$ is a left unit, then $i$ is a greatest element of the filtered ordered index set $I$, and formula $V.4.\ast$ would mean $F(X)=\mathrm{Hom}(P_i,X)$, a one-element set for every $X$; this is absurd since $F({\varnothing }_{\mathcal{C}})=\varnothing$. Thus the $P_i$ are not isomorphic to ${\varnothing }_{\mathcal{C}}$.

Suppose $P_i=A⨿B$. By (G 5), $F(P_i)=F(A)⨿F(B)$. In particular the element $a_i$ of $F(P_i)$, corresponding by $V.4.\ast$ to the identity homomorphism $P_i\to P_i$, lies in $F(A)⨿F(B)$, for instance in $F(A)$. This means that there exists $j\ge i$ such that ${\varphi }_{ij}:P_j\to P_i$ factors as $P_j\to A\to P_i=A⨿B$, where the second arrow is canonical. Thus $F(P_j)\to F(P_i)$ factors as $F(P_j)\to F(A)\to F(P_i)=F(A)⨿F(B)$; since $F(P_j)\to F(P_i)$ is surjective by (G 5), it follows that $F(B)=\varnothing$, hence $B$ is isomorphic to ${\varnothing }_{\mathcal{C}}$.

1. Every morphism $u:X\to Y$ in $\mathcal{C}$, with $X$ not isomorphic to ${\varnothing }_{\mathcal{C}}$ and $Y$ connected, is a strict epimorphism. Every endomorphism of a connected object is an automorphism.

Consider the factorization (G 3) of $u$. Since $X\ne {\varnothing }_{\mathcal{C}}$, by (G 6) $F(X)\ne \varnothing$, hence $F(Y^{\prime })\ne \varnothing$, and therefore $Y^{\prime }\ne {\varnothing }_{\mathcal{C}}$. Since $Y$ is connected, $Y^{\prime }$ identifies with $Y$, so $u$ is a strict epimorphism. Suppose $u$ is an endomorphism of the connected object $X$; we prove it is an automorphism. We may suppose $X$ not isomorphic to ${\varnothing }_{\mathcal{C}}$, hence $u$ is a strict epimorphism by what precedes. Thus $F(u)$ is an epimorphism by (G 5), and since $F(X)$ is a finite set, $F(u)$ is bijective. Therefore $u$ is an automorphism by (G 6).

In particular, every endomorphism of a $P_i$ is an automorphism.

1. The following conditions on a $P_i$ are equivalent:

2. The natural injective map $\mathrm{Hom}(P_i,P_i)\to \mathrm{Hom}(P,P_i)\simeq F(P_i)$ is also surjective; i.e. for every $u:P\to P_i$ there exists $v:P_i\to P_i$ such that $u=v{\varphi }_i$, where ${\varphi }_i$ is the canonical homomorphism $P\to P_i$.

3. The group $\mathrm{Aut}(P_i)$ operates transitively on $F(P_i)$.

4. The group $\mathrm{Aut}(P_i)$ operates simply transitively on $F(P_i)$.

Indeed, identifying $\mathrm{Hom}(P,P_i)$ with $F(P_i)$, the map considered in (i) is just $v\mapsto F(v)({\varphi }_i)$. The equivalence of the three conditions then comes from the fact that $\mathrm{Hom}(P_i,P_i)=\mathrm{Aut}(P_i)$ and that the preceding map is already injective.

A $P_i$ satisfying the equivalent conditions (i), (ii), (iii) of (f) is called Galois.

1. For every $X$ in $\mathcal{C}$, there exists a Galois $P_i$ such that every $u\in \mathrm{Hom}(P,X)$ factors as $P--{\varphi }_i\to P_i\to X$.

Let $J=\mathrm{Hom}(P,X)=F(X)$. This is a finite set, so there exists $P_j$ such that every $u:P\to X$ factors as $P\to P_j\to X$, or equivalently such that the natural morphism

P → X^J,     J = Hom(P,X),

factors as

P --φ_j→ P_j → X^J.

By (G 3), the morphism $P_j\to X^J$ factors as a product of a monomorphism and a strict epimorphism, which may be taken in the form ${\varphi }_{ij}:P_j\to P_i$. We are therefore reduced to proving that $P_i$ is Galois. Let $k$ be an index $\ge j$ such that every morphism $P\to P_i$ factors through $P--{\varphi }_k\to P_k\to P_i$. Note that the natural morphism $P_k\to X^J$ still factors as the composite

P_k --φ_{ik}→ P_i --U→ X^J,

where the first arrow is a strict epimorphism by (e), and the second a monomorphism. We want to prove that for a given morphism $\psi :P_k\to P_i$, there exists an endomorphism $v$ of $P_i$ such that $\psi =v{\varphi }_{ik}$. For every $u\in \mathrm{Hom}(P_i,X)$, consider $u\psi \in \mathrm{Hom}(P_k,X)$. It is therefore of the form $u^{\prime }{\varphi }_{ik}$, with $u^{\prime }\in \mathrm{Hom}(P_i,X)$ uniquely determined. The map $u\mapsto u^{\prime }$ from $J$ to $J$ thus defined by $\psi$ is injective, since $\psi$ is an epimorphism by (e); it is therefore bijective, since $J$ is finite. The bijective map $u\mapsto u^{\prime }$ from $J$ to $J$ therefore defines an isomorphism $\alpha :X^J\to X^J$ making the diagram

P_k --φ_{ik}→ P_i --U→ X^J
 |                         | α ≃
 =                         v
P_k ---ψ--→ P_i --U→ X^J

commutative. By the uniqueness properties of factoring a morphism as a product of a monomorphism and a strict epimorphism, it follows, since $\psi$ is also a strict epimorphism by (e), that one can find a morphism $v:P_i\to P_i$ making the diagram commute, as required.

It follows in particular that the Galois $P_i$ form a cofinal system in the system of the $P_j$. Therefore, since for a Galois object $P_i$ one has

Hom(P,P_i) = Hom(P_i,P_i) = Aut(P_i),

passing to the limit gives

Hom(P,P) = lim_i Hom(P,P_i) = lim_i Hom(P_i,P_i) = lim_i Aut(P_i),

where the projective limit is taken over the Galois $P_i$. Moreover, under the identification $\mathrm{Hom}(P,P_i)=F(P_i)$, and taking into account that $F$ transforms epimorphisms into epimorphisms, one sees that the transition homomorphisms in the preceding projective system are surjective. We conclude from all this:

1. One has

Hom(P,P) = Aut(P) = lim_i F(P_i) = lim_i Aut(P_i),

where the projective limit is taken over the Galois $P_i$.

In particular, $\mathrm{Aut}(P)$ appears as the projective limit of a projective system of finite groups, with surjective transition homomorphisms; we equip it with the projective-limit topology from the discrete topologies. We denote by $\pi$ and call the fundamental group of $\mathcal{C}$ equipped with $F$ the group opposite to $\mathrm{Aut}(P)$. This group therefore operates on the right on $P$; it is the projective limit of finite groups ${\pi }_i$ operating on the right on the Galois $P_i$, where ${\pi }_i$ is the group opposite to $\mathrm{Aut}(P_i)$.

Taking the functorial isomorphism

$$F(X)=\mathrm{Hom}(P,X)$$

and the definition of $\pi$ into account, one sees that $\pi$ operates on the left on $F(X)$, and moreover continuously by (g), since with the notation of (g), it is in fact ${\pi }_i$ that operates on $F(X)$. It is trivial that for every morphism $u:X\to Y$ in $\mathcal{C}$, the morphism $F(u):F(X)\to F(Y)$ is compatible with the operations of $\pi$. Thus from now on one may regard $F$ as a covariant functor

$$F:\mathcal{C}\to \mathcal{C}(\pi ),$$

where $\mathcal{C}(\pi )$ is the category of finite sets on which $\pi$ operates on the left continuously.

We now define a functor in the opposite direction:

$$G:\mathcal{C}\leftarrow \mathcal{C}(\pi )$$

by the formula

G(E) = P ×_π E,

where $P{\times }_{\pi }E$ is defined as the solution of the universal problem summarized by

Hom_𝒞(P ×_π E, X) ≃ Hom_π(E, Hom(P,X)).

In the second member $\mathrm{Hom}(P,X)=F(X)$ is regarded as a set on which $\pi$ operates on the left. One must prove the existence of the object $P{\times }_{\pi }E$.

1. Let $Q$ be an object of $\mathcal{C}$ on which a finite group $G$ operates on the right, and let $E$ be a finite set on which $G$ operates on the left. Then $Q{\times }_{G}E$ exists, and the canonical map

F(Q) ×_G E → F(Q ×_G E)

is an isomorphism.

Since finite direct sums exist in $\mathcal{C}$ by (G 2), and $F$ commutes with them by (G 5), one is immediately reduced to the case where $G$ operates transitively on $E$; if the $E_j$ are the orbits of $G$ in $E$, one will have

Q ×_G E = ⨿_j Q ×_G E_j.

Let $a\in E$, and let $H$ be its stabilizer. One sees at once from the definition that $Q{\times }_{G}E$ identifies with $Q/H$. Hence existence follows from (G 2), and the commutation property for $F$ from (G 5).

1. Let $E$ be an object of $\mathcal{C}(\pi )$, and let $P_i$ be Galois such that ${\pi }_i$ already operates on $E$. Then $P_i{\times }_{{\pi }_i}E$ exists and there is a canonical isomorphism

E → F(P_i ×_{π_i} E).

If $j\ge i$ is such that $P_j$ is Galois, then the canonical homomorphism $P_j{\times }_{{\pi }_j}E\to P_i{\times }_{{\pi }_i}E$ is an isomorphism.

The first assertion is a special case of (i), taking into account that ${\pi }_i$ operates simply transitively on $F(P_i)$, which is equipped with a marked point ${\varphi }_i$, whence an isomorphism $F(P_i){\times }_{{\pi }_i}E\simeq E$. For the second assertion, use for example (G 6).

For every $j$, let ${\mathcal{C}}_j$ be the full subcategory of $\mathcal{C}$ formed by the $X$ such that $\mathrm{Hom}(P_j,X)\to \mathrm{Hom}(P,X)\simeq F(X)$ is bijective. We know by (g) that $\mathcal{C}$ is the filtered union of the ${\mathcal{C}}_j$. Thus for $X\in {\mathcal{C}}_j$ one has

Hom_π(E,Hom(P,X))
  ≃ Hom_π(E,Hom(P_j,X))
  ≃ Hom_{π_j}(E,Hom(P_j,X))
  ≃ Hom(P_j ×_{π_j} E, X).

Taking the last assertion in (j) into account, one finds an isomorphism, functorial in the object $X$ of ${\mathcal{C}}_j$,

Hom_π(E,Hom(P,X)) ≃ Hom(P_i ×_{π_i} E, X).

Since this is true for every $j$, and since these functorial isomorphisms for varying $j$ induce one another, we conclude:

1. Under the conditions of (j), the composite of the canonical morphisms

E → Hom(P_i, P_i ×_{π_i} E) → Hom(P, P_i ×_{π_i} E)

makes $P_i{\times }_{{\pi }_i}E$ a solution of the universal problem defining $P{\times }_{\pi }E$; i.e. the latter exists and there is an isomorphism

P ×_π E → P_i ×_{π_i} E.

This completes the construction of the functor $G(E)$. On the other hand, there is a functorial homomorphism

$$\alpha :i{d}_{\mathcal{C}(\pi )}\to FG,$$

i.e. a homomorphism functorial in the object $E$ of $\mathcal{C}(\pi )$,

α(E): E → FG(E) = F(P ×_π E),

namely the composite of the canonical morphisms

E → F(P) ×_π E → F(P ×_π E),

where the first comes from the marked point $\varphi \in F(P)$. Combining (j) and (k), one finds:

1. The homomorphism $\alpha$ is an isomorphism.

One similarly defines a functorial homomorphism

$$\beta :GF\to i{d}_{\mathcal{C}},$$

i.e. a homomorphism functorial in the object $X$ of $\mathcal{C}$,

β(X): P ×_π F(X) → X,

as associated with the $\pi$-homomorphism

$$F(X)\to \mathrm{Hom}(P,X)$$

inverse to the canonical isomorphism $\mathrm{Hom}(P,X)\to F(X)$.

1. The composites

F(X) --α(F(X))→ FGF(X) --F(β(X))→ F(X),
G(E) --G(α(E))→ GFG(E) --β(G(E))→ G(E)

are the identity isomorphisms.

The donkey trots.

Taking (l) into account, it follows:

1. The homomorphism $\beta$ is an isomorphism.

We have thus obtained the promised result:

Theorem.

Let $\mathcal{C}$ be a category satisfying conditions (G 1), (G 2), (G 3) from the beginning of this number, and let $F$ be a covariant functor from $\mathcal{C}$ to the category of finite sets satisfying (G 4), (G 5), and (G 6). Then the preceding canonical constructions define quasi-inverse equivalences of categories $F:\mathcal{C}\to \mathcal{C}(\pi )$ and $G:\mathcal{C}(\pi )\to \mathcal{C}$. More precisely, there exists a pro-object $P$ of $\mathcal{C}$ and a functorial isomorphism $F(X)\leftarrow \mathrm{Hom}(P,X)$; $\pi$ is the group opposite to the automorphism group of $P$, topologized suitably, so that $\pi$ operates continuously on the sets $\mathrm{Hom}(P,X)\simeq F(X)$. Finally, $G$ is given by $G(E)\simeq P{\times }_{\pi }E$.

Remarks.

The statement of conditions (G 1) to (G 6) becomes simpler and more agreeable if one replaces (G 2) and (G 5), respectively, by:

(G′ 2) Finite inductive limits exist in $\mathcal{C}$.

(G′ 5) The functor $F$ is right exact, i.e. commutes with finite inductive limits.

These conditions appear stronger than (G 2) and (G 5), but it follows at once from the structure theorem V.4.1 that they are implied by (G 1) to (G 6). Note, however, that in the cases that will interest us, verifying (G 2) and (G 5) seems effectively simpler than verifying (G′ 2) and (G′ 5). I do not know whether, in condition (G 3), the fact that $u^{\prime \prime }$ is an isomorphism onto a direct summand of $Y$ could be omitted.

5. Galois Categories

Definition.

A Galois category is a category $\mathcal{C}$ equivalent to a category $\mathcal{C}(\pi )$, where $\pi$ is a compact group, a projective limit of finite groups, i.e. totally disconnected.

For the definition of $\mathcal{C}(\pi )$, cf. the beginning of V.4. By Theorem V.4.1, $\mathcal{C}$ is Galois if and only if it satisfies conditions (G 1) to (G 3), and there exists a functor $F$ from $\mathcal{C}$ to the category of finite sets satisfying conditions (G 4) to (G 6), i.e. which is exact and conservative, in general terminology. Such a functor will be called a fundamental functor of the Galois category $\mathcal{C}$;⁶ it is pro-representable by a pro-object that we denote P_F. A pro-object $P$ such that the associated functor $F$ is fundamental is called a fundamental pro-object.

In this way, the category of fundamental functors on $\mathcal{C}$ is anti-equivalent to the category of fundamental pro-objects. If $F$ and $P$ correspond, the group $\mathrm{Aut}F$ is therefore isomorphic to the opposite of the group $\mathrm{Aut}P$; hence the group denoted $\pi$ in the preceding number is none other than $\mathrm{Aut}P$. Recall that in the preceding number, starting from a given fundamental functor $F$, we constructed an equivalence of $\mathcal{C}$ with $\mathcal{C}(\pi )$, where $\pi =\mathrm{Aut}(F)$, that transforms $F$ into the canonical functor from $\mathcal{C}(\pi )$ to the category of finite sets. In the typical case $\mathcal{C}=\mathcal{C}(\pi )$, with $F$ the canonical functor, the fundamental pro-object associated with $F$ is nothing other than the projective system of the discrete quotients ${\pi }_i$ of $\pi$.

It may be useful to spell out the category of pro-objects of $\mathcal{C}(\pi )$. One finds:

Proposition.

The category $Pro-\mathcal{C}(\pi )$ is canonically equivalent to the category ${\mathcal{C}}^{\prime }(\pi )$ of spaces, with topological group $\pi$ of operators, which are compact and totally disconnected.

Since the latter contains $\mathcal{C}(\pi )$ as a full subcategory, corresponding to compact discrete spaces with operators, and since projective limits exist in it, we have in any case a canonical functor

$$g:Pro-\mathcal{C}(\pi )\to {\mathcal{C}}^{\prime }(\pi ),$$

which sends the projective system $Q=(Q_i)$ to the object $X={\lim }_i{Q}_i$ of ${\mathcal{C}}^{\prime }(\pi )$. To define a functor in the opposite direction, it is enough to define a contravariant functor from ${\mathcal{C}}^{\prime }(\pi )$ to the category of left-exact functors $\mathcal{C}\to Set$; for $X\in {\mathcal{C}}^{\prime }(\pi )$, take the functor

$$h(X)(E)=\mathrm{Hom}(X,E),$$

where Hom is taken in ${\mathcal{C}}^{\prime }(\pi )$. It is immediate from the definitions that $h$ and $g$ are adjoint to one another, and that hg is canonically isomorphic to the identity functor of $Pro-\mathcal{C}(\pi )$. It remains, in order to prove that $g$ and $h$ are quasi-inverse to one another, to show that every object of ${\mathcal{C}}^{\prime }(\pi )$ is isomorphic to an object of the form $g(Q)$, with $Q\in Pro-\mathcal{C}(\pi )$; in other words: every space $X$ with topological group $\pi$ of operators, compact and totally disconnected, is isomorphic to a projective limit of finite discrete spaces with operators.

Since $X$ is the projective limit of its finite discrete quotients, as a topological space without operators, we are reduced to showing that, among these quotients, there is a cofinal system invariant under $\pi$. For this it is enough to show that, for such a quotient $X^{\prime }$, the set of transforms of this quotient by the operations of $\pi$ is finite; one then takes the supremum of these transforms, which will be an invariant quotient dominating $X^{\prime }$. Equivalently, there is an open invariant subgroup ${\pi }^{\prime }$ of $\pi$ whose elements leave $X^{\prime }$ fixed. Now $X^{\prime }$ corresponds to a finite partition of $X$ into open sets Xᵢ. By continuity and compactness of $\pi$, there exists a neighborhood $V$ of the identity element of $\pi$ such that $s\in V$ implies $s\cdot X_i\subset X_i$ for every $i$, and hence $s$ leaves $X^{\prime }$ fixed. But the open invariant subgroups of $\pi$ are known to form a fundamental system of neighborhoods of the identity element. This finishes the proof.

Let us note that one sees still more simply that the category $Ind-\mathcal{C}(\pi )$ is canonically equivalent to the category of sets on which $\pi$ operates continuously. We shall not need this here.

Proposition.

Let $\mathcal{C}$ be a Galois category, $F$ a fundamental functor on $\mathcal{C}$, and $P=(P_i)$ the associated pro-object, normalized in the usual way. Let $X\in \mathcal{C}$. Then $X$ is connected if and only if $\pi$ operates transitively on $E=F(X)$.

This reduces to the typical case $\mathcal{C}=\mathcal{C}(\pi )$, with $F$ the canonical functor, where it is trivial.

Corollary.

For $X$, the following conditions are equivalent:

1. $X$ is connected and X ≄ ∅_𝒞.
2. The group $\pi$ is transitive on $E=F(X)$, and $F(X)\ne \varnothing$.
3. $X$ is isomorphic to some Pᵢ.

The equivalence of (1) and (3) also follows already easily from V.4, e).

Proposition.

Let $Q=(Q_i{)}_i\in I$ be a pro-object of $\mathcal{C}$, normalized in the usual way, and let $G$ be the corresponding functor $G(X)=\mathrm{Hom}(Q,X)$ from $\mathcal{C}$ to Set. The following conditions are equivalent:

1. $G$ commutes with finite direct sums.
2. $G$ commutes with the sum of two objects.
3. The Qᵢ are connected and Qᵢ ≄ ∅_𝒞.
4. $Q$ is isomorphic to $\pi /H$, where $H$ is a closed subgroup of $\pi$.
5. The functor $G$ is isomorphic to the functor $E\mapsto E^H$, the set of $H$-invariants, defined by a closed subgroup $H$ of $\pi$.

N.B. In the statement of (4) and (5), one assumes that a fundamental functor has been chosen, allowing $\mathcal{C}$ to be identified with the category $\mathcal{C}(\pi )$.

Proof. We may suppose $\mathcal{C}=\mathcal{C}(\pi )$. The implication (1) ⇒ (2) is trivial, and (2) ⇒ (3) is proved as property d) of V.4. Let us prove (3) ⇒ (4). Indeed, ${\lim }_i{Q}_i$ is nonempty as a projective limit of nonempty finite sets. Let $a$ be a point of ${\lim }_i{Q}_i$; it defines a homomorphism of spaces with operators

$$\pi \to Q$$

which is surjective, since for every $i$ the composite $\pi \to Q\to Q_i$ is surjective, because $\pi$ is transitive on Qᵢ by V.5.3. If $H$ is the stabilizer subgroup of $a$, one obtains an isomorphism $\pi /H\simeq Q$. The implications (4) ⇒ (5) and (5) ⇒ (1) are again trivial.

Proposition.

Let $\mathcal{C}$ be a Galois category, $P$ a fundamental pro-object of $\mathcal{C}$, and $F$ the associated fundamental functor. Let $P^{\prime }=(P_i^{\prime }{)}_i\in I$ be a pro-object of $\mathcal{C}$, put in normal form, and let $F^{\prime }$ be the associated functor $F^{\prime }(X)=\mathrm{Hom}(P^{\prime },X)$ from $\mathcal{C}$ to Set. The following conditions are equivalent:

1. $P^{\prime }\simeq P$, or equivalently $F^{\prime }\simeq F$.
2. $P^{\prime }$ is fundamental, or equivalently $F^{\prime }$ is fundamental.
3. $F^{\prime }$ transforms a sum of two objects into a sum, and X ≄ ∅_𝒞 implies $F(X)\ne \varnothing$.
4. The objects of $\mathcal{C}$ that are connected and $\ne {\varnothing }_{\mathcal{C}}$ are exactly the objects isomorphic to some $P_i^{\prime }$.

We have trivially (1) ⇒ (3) and (1) ⇒ (2); furthermore (2) ⇒ (4) by V.5.4, applied to $P^{\prime }$ instead of $P$. Moreover, (3) or (4) implies, by V.5.5, that $P^{\prime }$ is of the form $\pi /H$, where $H$ is a closed subgroup of $\pi$. In case (3), for every open invariant subgroup ${\pi }^{\prime }$ of $\pi$ there exists a $\pi$-homomorphism $P^{\prime }=\pi /H\to \pi /{\pi }^{\prime }$, hence $H\subset {\pi }^{\prime }$; thus $H=(0)$, and consequently (1), as required.

Corollary.

Let $\mathcal{C}$ be a Galois category. The fundamental pro-objects are isomorphic; the fundamental functors are isomorphic.

In other words, the category of fundamental functors is a connected groupoid $\Gamma$, which one may call the fundamental groupoid of the Galois category $\mathcal{C}$. If $\mathcal{C}=\mathcal{C}(\pi )$, the automorphism group of an object of the fundamental groupoid is isomorphic to $\pi$, this isomorphism being well determined up to inner automorphism. Here a groupoid means a category in which all morphisms are isomorphisms, and a connected groupoid means a groupoid all of whose objects are isomorphic. The fundamental pro-objects of $\mathcal{C}$ form a connected groupoid equivalent to the opposite of the fundamental groupoid.

If $F,F^{\prime }$ are two fundamental functors, associated with fundamental pro-objects $P,P^{\prime }$, then $\mathrm{Hom}(F,F^{\prime })=Isom(F,F^{\prime })$ is sometimes denoted ${\pi }_{F^{\prime },F}$ and plays the role of a “set of path classes from $F$ to $F^{\prime }$”. In particular, ${\pi }_{F,F}={\pi }_F$ is nothing other than the fundamental group of $\mathcal{C}$ at $F$ constructed in the preceding number. As for the pro-object $P$ associated with $F$, it plays the role of a universal covering at $F$ of the final object $e_{\mathcal{C}}$ of $\mathcal{C}$.

It can be convenient to have a description of $\mathcal{C}$, up to equivalence, in terms of its fundamental groupoid $\Gamma$, without going through the choice of one particular object $F$ of $\Gamma$. To every object $X$ of $\mathcal{C}$ there is associated the functor E_X on the fundamental groupoid, defined by

$$E_X(F)=F(X),$$

with values in Set. Such a functor is known in topology under the name “local system” on the groupoid. $F(X)=E_X(F)$ may be called the fiber of $X$ at $F$, and the functor E_X the fiber-functor associated with $X$. The functor E_X has the following property:

for every $F$, $E_X(F)$ is a finite set on which the topological group ${\pi }_F=\mathrm{Aut}(F)$ operates continuously.

For a given covariant functor $\xi$ from the fundamental groupoid to Set, the preceding condition is moreover equivalent to the same condition for one arbitrary fixed $F$. This being so:

Proposition.

The functor $X\mapsto E_X$ is an equivalence of the category $\mathcal{C}$ with the category of covariant functors from the fundamental groupoid $\Gamma$ of $\mathcal{C}$ to Set that satisfy the condition displayed above.

Indeed, let $F_0$ be an object of the fundamental groupoid, and let ${\pi }_0=\mathrm{Aut}(F_0)$. Then the functor $\xi \mapsto \xi (F_0)$ is an equivalence from the second category considered in V.5.8 to the category $\mathcal{C}({\pi }_0)$, as one sees at once. On the other hand, the composite of this functor with $X\mapsto E_X$ is the natural equivalence $\mathcal{C}\to \mathcal{C}({\pi }_0)$. It follows that the functor $X\mapsto E_X$ itself is an equivalence.

Corollary.

The category $Pro-\mathcal{C}$ is canonically equivalent to the category of covariant functors $\xi$ from the fundamental groupoid $\Gamma$ to the category of topological spaces satisfying the following condition: for every object $F$ of $\Gamma$, $\xi (F)$ is a compact totally disconnected space with topological group ${\pi }_F$ of operators.

Here again, to check this condition on $\xi$, it is enough to check it for one $F$. The proof is the same as for V.5.8.

Remark.

Let $(F_s{)}_{s\in S}$ be a family of objects of the fundamental groupoid $\Gamma$. Put, for $s,s^{\prime }\in S$,

$$\mathrm{Hom}(s,s^{\prime })=\mathrm{Hom}(F_s,F_s^{\prime }),$$

so that $S$ itself becomes a connected groupoid, and the map $s\mapsto F_s$ becomes a fully faithful functor $f$ from $S$ to $\Gamma$. Considering then the functor $X\mapsto E_X\circ f$ from $\mathcal{C}$ to the category of functors $\mathrm{Hom}(S,Set)$, one obtains a variant of V.5.8, and V.5.9, with $\Gamma$ replaced by $S$. The statement so obtained reduces to Theorem V.4.1 when $S$ is reduced to a point, and is none other than V.5.8 itself if $S$ is the set of objects of $\Gamma$.

We are going to use V.5.9 to define a canonical pro-object of $\mathcal{C}$. For this, we consider the functor from $\Gamma$ to the category of topological spaces, indeed of topological groups,

f: F ↦ Aut(F) = π_F.

This functor satisfies the condition considered in V.5.8: the space with operators $f(F)$, under ${\pi }_F$, is none other than ${\pi }_F$ considered as a space with operators under itself by inner automorphisms. Thus the functor $f$ corresponds to a pro-object of $\mathcal{C}$, determined up to unique isomorphism, which is even a pro-group of $\mathcal{C}$ and is called the fundamental pro-group of $\mathcal{C}$, playing the role of a local system of fundamental groups. It is therefore a pro-group $\Pi$ of $\mathcal{C}$ defined by the condition that one have an isomorphism functorial in $F$:

$$F(\Pi )\simeq {\pi }_F.$$

If $X$ is any pro-object of $\mathcal{C}$, one has a canonical morphism

$$\Pi \times X\to X$$

which makes $X$ an object with a left group of operators $\Pi$ in $Pro-\mathcal{C}$. For this it is enough to note that, for variable $F$, one has a canonical map

$$\Pi (F)\times X(F)\to X(F),$$

i.e.

Aut(F) × E_X(F) → E_X(F),    or    π_F × F(X) → F(X),

which is functorial in $F$. It is also functorial in $X$, so for every morphism $X\to Y$ of pro-objects, the corresponding diagram

$$\Pi \times X\to X\downarrow \downarrow \Pi \times Y\to Y$$

is commutative.

Remark.

One should be careful not to confuse a fundamental pro-object $P$, which is not endowed with a group structure and is connected, with the fundamental pro-group, which is a pro-group and in general is not connected. More precisely, $\Pi$ is connected if and only if ${\pi }_F$, operating on itself by inner automorphisms, is transitive, i.e. if $\pi$ is reduced to the identity element, or again if $\mathcal{C}$ is equivalent to the category of finite sets. Another essential difference is that $\Pi$ is determined up to unique isomorphism, while $P$ is determined only up to non-unique isomorphism.

Let $E$ be a finite set, and consider the constant functor on the groupoid $\Gamma$ with value $E$. By V.5.8, it defines an object of $\mathcal{C}$, denoted $E_{\mathcal{C}}$, which can also be interpreted as the sum of $E$ copies of the final object $e_{\mathcal{C}}$ of $\mathcal{C}$. One may regard $E_{\mathcal{C}}$ as a functor in $E$, from the category of finite sets to the category $\mathcal{C}$, and this functor is exact; hence it transforms finite groups into $\mathcal{C}$-groups, etc. Thus if $X$ is an object of $\mathcal{C}$ on which the finite group $G$ operates on the right, one sees that $X$ may be regarded as an object of $\mathcal{C}$ having a right $\mathcal{C}$-group of operators $G_{\mathcal{C}}$.

By extension of the general terminology concerning objects with $\mathcal{C}$-groups of operators, we shall therefore say that $X$ is formally principal homogeneous under $G$ if $X$ is formally principal homogeneous under $G_{\mathcal{C}}$, i.e. if the canonical morphism

X × G_𝒞 → X × X

deduced from the right operation of $G_{\mathcal{C}}$ on $X$ is an isomorphism. We say that $X$ is principal homogeneous under $G$ if it is so under $G_{\mathcal{C}}$, i.e. if it is formal in the preceding sense and if, moreover, the quotient $X/G=X/G_{\mathcal{C}}$ is $e_{\mathcal{C}}$.

If a fundamental functor is fixed, hence an equivalence of $\mathcal{C}$ with a category $\mathcal{C}(\pi )$, then $X$ corresponds to a set $E=F(X)$ on which $\pi$ operates continuously on the left. Making $G$ operate on $X$ on the right then amounts to making $G$ operate on the set $E$ on the right, in such a way that the operations of $G$ commute with those of $\pi$. One checks at once that $X$ is principal homogeneous under $G$ if and only if the set $E$ is a principal homogeneous space under $G$, i.e. if and only if $G$ operates on it simply transitively. Moreover, $X$ is formally principal homogeneous if and only if $E$ is principal homogeneous or empty.

Comparing with V.5.3, one sees that if $X$ is principal homogeneous under $G$ and connected, then the given homomorphism from $G$ to the group opposite to $\mathrm{Aut}(X)$ is an isomorphism; and moreover, for an object $X$ of $\mathcal{C}$ to be connected and principal homogeneous under the group opposite to $\mathrm{Aut}(X)$, it is necessary and sufficient, with the notation of V.4, that it be isomorphic to a Galois Pᵢ. In the typical case $\mathcal{C}=\mathcal{C}(\pi )$, this means that $X$ is isomorphic to a quotient of $\pi$ by an invariant subgroup.

Suppose still that a fundamental functor $F$ is given. Then the data of an $X$ principal homogeneous under a finite group $G$ operating on the right, together with a point $a\in F(X)$, is equivalent to the data of a homomorphism from $\pi$ to the group $G$.

Indeed, to such a homomorphism one associates the set $E=G$, making $\pi$ operate on it on the left by means of the given homomorphism $\pi \to G$ and the left translations of $G$, and making $G$ operate on it on the right by right translation; the marked point $a$ of $E$ is the identity element of $G$. By what precedes, one thus obtains, in an essentially unique way, every triple $(X,G,a)$ having the properties considered above, since a pointed set that is principal homogeneous under a group $G$ is identified with that group. In this way, one has a direct geometric interpretation of the functor $G\mapsto \mathrm{Hom}(\pi ,G)$ from the category of finite groups to Set, a functor which is pro-representable by $\pi$ and whose consideration would therefore give another construction of the group $\pi$ associated with $F$.

6. Exact Functors from One Galois Category to Another

Proposition.

Let $\mathcal{C},{\mathcal{C}}^{\prime }$ be two Galois categories, $H:\mathcal{C}\to {\mathcal{C}}^{\prime }$ a covariant functor, $F^{\prime }$ a fundamental functor on ${\mathcal{C}}^{\prime }$, and $F=F^{\prime }\circ H$. The following conditions are equivalent:

1. $H$ is exact, i.e. left exact and right exact.
2. $H$ is left exact, transforms finite sums into finite sums, and transforms epimorphisms into epimorphisms; equivalently, it transforms objects ≉ ∅_𝒞 into objects ≉ ∅_𝒞′.
3. $F$ is a fundamental functor on $\mathcal{C}$.

The implication (1) ⇒ (2) is a general fact about categories. Moreover, the first form given for (2) implies the second: if $X$ is an object of $\mathcal{C}$, then $X$ is ≉ ∅_𝒞 if and only if the morphism $X\to e_{\mathcal{C}}$ is an epimorphism; one notes that $F$, being assumed left exact, transforms $e_{\mathcal{C}}$ into $e_{\mathcal{C}}^{\prime }$. The second form of (2) implies (3), because $F$, being left exact and hence pro-representable, falls under the criterion V.5.6, (3). Finally, (3) implies (1), as follows from the fact that $F$ is exact and “conservative”, i.e. satisfies axiom (G 6) of V.4.

Let $\Gamma$ be the fundamental groupoid of $\mathcal{C}$ and ${\Gamma }^{\prime }$ that of ${\mathcal{C}}^{\prime }$. Thus, if $H$ is exact, then

$$F^{\prime }\mapsto F^{\prime }\circ H$$

is a functor from the groupoid ${\Gamma }^{\prime }$ to the groupoid $\Gamma$, which we shall denote by ᵗH:

$${}^{t}H(F^{\prime })(X)=F^{\prime }(H(X)).$$

This may also be written, with the notation $F(X)=E_X(F)$ introduced in V.6, as

$$E_{H(X)}(F^{\prime })=E_X({}^{t}H(F^{\prime })).$$

This last formula shows, taking V.5.8 or V.4.1 into account, that the exact functor $H$ is determined, up to unique isomorphism, once the corresponding functor ᵗH is known. Fix an $F^{\prime }$, and put $F={}^{t}H(F^{\prime })$. Then ᵗH defines a homomorphism

ᵗH: π_{F′} → π_F,    where F = ᵗH(F′) = F′ ∘ H.

Moreover, the formula above shows, taking V.5.8 into account, that this homomorphism has the following property: for every finite set $E$ on which ${\pi }_F$ operates continuously, the group ${\pi }_{F^{\prime }}$ also operates continuously by means of the preceding homomorphism ${\pi }_{F^{\prime }}\to {\pi }_F$. Applying this to the quotients of ${\pi }_F$ by its open invariant subgroups, one sees that the preceding condition also says that the homomorphism under consideration is continuous.

Conversely, suppose we are given an object $F$ of $\Gamma$, an object $F^{\prime }$ of ${\Gamma }^{\prime }$, and a continuous homomorphism

$$u:{\pi }_{F^{\prime }}\to {\pi }_F.$$

To it there corresponds a functor from $\mathcal{C}(\pi )$ to $\mathcal{C}({\pi }^{\prime })$, manifestly exact; hence, by V.4.1, there corresponds to it a functor $H$ from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ which is exact and such that ${}^{t}H:{\pi }_{F^{\prime }}\to {\pi }_F$ is precisely $u$. One may also, instead of starting from a group homomorphism, start from a functor

$$U:{\Gamma }^{\prime }\to \Gamma$$

such that, for every $F^{\prime }\in {\Gamma }^{\prime }$, or for one $F^{\prime }\in {\Gamma }^{\prime }$, which comes to the same thing, the corresponding homomorphism ${\pi }_{F^{\prime }}\to {\pi }_F$ is continuous. Such a functor is isomorphic to a functor of the form ᵗH, where $H:\mathcal{C}\to {\mathcal{C}}^{\prime }$ is an exact functor determined up to unique isomorphism. Thus:

Corollary.

For a functor $H:\mathcal{C}\to {\mathcal{C}}^{\prime }$ of Galois categories to be exact, it is necessary and sufficient that there exist equivalences $\mathcal{C}(\pi )\to \mathcal{C}$ and ${\mathcal{C}}^{\prime }\to \mathcal{C}({\pi }^{\prime })$ that transform the functor $H$ into the functor $\mathcal{C}(\pi )\to \mathcal{C}({\pi }^{\prime })$ associated with a homomorphism of topological groups ${\pi }^{\prime }\to \pi$.

Corollary.

Let $\mathcal{C},{\mathcal{C}}^{\prime }$ be two Galois categories, and let $\Gamma ,{\Gamma }^{\prime }$ be their fundamental groupoids. Then the category of exact functors from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ is equivalent to the category of functors $U:{\Gamma }^{\prime }\to \Gamma$ having the following property: for every $F^{\prime }$ in ${\Gamma }^{\prime }$, or for one $F^{\prime }$ in ${\Gamma }^{\prime }$, which comes to the same thing, if $F=U(F^{\prime })$, the homomorphism

π_{F′} = Aut(F′) → π_F = Aut(F)

defined by $U$ is continuous.

Consider the fundamental pro-group $\Pi$ of $\mathcal{C}$. An exact functor $H$ transforms it into a pro-group $H(\Pi )$ of ${\mathcal{C}}^{\prime }$. We are going to define a homomorphism

$${\Pi }^{\prime }\to H(\Pi ),$$

where ${\Pi }^{\prime }$ is the fundamental pro-group of ${\mathcal{C}}^{\prime }$, by requiring that, for every object $F^{\prime }$ of ${\Gamma }^{\prime }$, the corresponding homomorphism

F′(Π′) = π_{F′} → F′(H(Π)) = π_F    where F = F′ ∘ H = ᵗH(F′)

be the natural homomorphism

$$\mathrm{Aut}(F^{\prime })\to \mathrm{Aut}(F^{\prime }\circ H).$$

Since the latter is functorial in $F^{\prime }$, it indeed defines, by V.5.8, a homomorphism of pro-objects, and in fact of pro-groups, of ${\mathcal{C}}^{\prime }$. This homomorphism is said to be associated with the functor $H$.

Let now $H^{\prime }$ be a second exact functor, from the Galois category ${\mathcal{C}}^{\prime }$ to a Galois category ${\mathcal{C}}^{\prime \prime }$. It is trivial that

$${}^t(H^{\prime }H)={}^{t}H{}^t{H}^{\prime }.$$

N.B. this is an identity of functors, and not merely a canonical isomorphism. There is an analogous transitivity property for the associated homomorphisms of fundamental pro-groups.

We shall now interpret the properties of the exact functor $H$ in terms of the corresponding homomorphism

u: π_{F′} → π_F,    where F = F′ ∘ H.

It is convenient to introduce the notion of a pointed object of the Galois category $\mathcal{C}$, endowed with its fundamental functor $F$. By definition, this is an object $X$ of $\mathcal{C}$ together with an element $a$ of $F(X)$. It is therefore interpreted as a finite set on which ${\pi }_F$ operates continuously on the left, together with a point $a$. Thus the connected pointed objects of $\mathcal{C}$ are identified, by V.5.3, with the open subgroups of ${\pi }_F$. If $U$ and $V$ are two such subgroups, corresponding to connected pointed objects $X$ and $Y$ of $\mathcal{C}$, then there exists a pointed homomorphism from $X$ to $Y$ if and only if $U\subset V$, and that homomorphism is then unique.

Of course the functor $H$ transforms pointed objects into pointed objects, since $F=F^{\prime }\circ H$. On the other hand, note that a closed subgroup of a group such as ${\pi }_F$ is the intersection of the open subgroups containing it; consequently, if $M$ and $N$ are two closed subgroups, then $M\subset N$ if and only if every open subgroup that contains $N$ also contains $M$. With these remarks, one easily proves the following results:

Proposition.

Let $X$ be a connected pointed object of $\mathcal{C}$, associated with an open subgroup $U$ of ${\pi }_F$. In order that $U$ contain $u({\pi }_{F^{\prime }})$, respectively the closed invariant subgroup generated by $u({\pi }_{F^{\prime }})$, it is necessary and sufficient that $H(X)$ admit a pointed section, respectively be completely decomposed.