Exposé XIII. Cohomological Properness of Sheaves of Sets and of Sheaves of Noncommutative Groups

By Mme M. Raynaud. [Translator note: according to unpublished notes of A. Grothendieck.]

This exposé proposes to use étale cohomology to generalize certain results of Exposés IX and X. It also shows how one can extend to sheaves of not necessarily commutative groups those results of SGA 5 II that still make sense for such sheaves. The notions of étale cohomology set out in SGA 4 are assumed known.

The main result (XIII.2.4) gives an important example of a nonproper morphism $f:U\to S$ that is "cohomologically proper in dimension ≤ 1," that is, such that, for certain sheaves of groups $F$ on $U$, in the sense of the étale topology, the formation of $f_{\ast }F$ and $R^1{f}_{\ast }F$ commutes with every base change $S^{\prime }\to S$. This property is indeed satisfied by the open $U$ of a scheme $X$ proper over $S$, the complement of a divisor $D$ with normal crossings relative to $S$, at least if one requires $F$ to be finite constant of order prime to the residual characteristics of $S$. If $F$ is no longer assumed of order prime to the residual characteristics of $S$, one has an analogous result by replacing $R^1{f}_{\ast }F$ by the subsheaf $R_{tame}^1{f}_{\ast }F$ obtained by restricting to torsors under $F$ "tamely ramified on $X$ relative to $S$." In particular, this makes it possible to show that the tamely ramified fundamental group of a proper smooth algebraic curve over a separably closed field, with finitely many closed points removed, is topologically of finite type (XIII.2.12).

No. XIII.4 is devoted to the homotopy exact sequence and to the Künneth formula.

Finally, an appendix gives useful variants of Abhyankar’s lemma proved in X.3.6.

0. Reminders on the Theory of Stacks

We shall use in what follows the theory of stacks set out in [XIII.1] and [XIII.2]. We restrict ourselves to the case of the étale site of a scheme. Given a scheme X, write X_et for the étale site of X. Recall that a stack F on X is a fibered category over X_et such that, for every scheme X′ étale over X and every pair of objects x, y of the fiber F_X′, the presheaf SheafHom_X′(x,y) is a sheaf, and such that, for every surjective étale morphism X″ → X′, every object of F_X″ endowed with a descent datum relative to X″ → X′ is the inverse image of an object of F_X′. [Translator note: the corrected source writes “stack F” and corrects “pair of object” to “pair of objects.”]

We write F(X′) for the category of cartesian sections of F/X′. More generally, if Sch_X is the category of schemes over X endowed with the étale topology, the stack F may be extended to a stack 𝓕 on Sch_X, and for every morphism f: X′ → X, one again writes F(X′) for the category of cartesian sections of this stack 𝓕 over X′.

A gerbe is a stack such that, for every scheme X′ étale over X and every pair of objects x, y of F_X′, every morphism from x to y is an isomorphism, x and y are locally isomorphic, and the set of objects X′ of X_et such that F_X′ is nonempty is a refinement of X_et. For example, the stack of torsors under a sheaf of groups is a gerbe which, moreover, has a cartesian section. Conversely, a gerbe which has a section, that is, such that there exists an object x of F_X, is equivalent to the stack of torsors under the sheaf

of groups SheafAut_X(x).

There is an evident notion of subgerbe and maximal subgerbe of a stack F. Given a cartesian section x of F(X), there exists a unique maximal subgerbe G_x of F such that x factors through G_x. One calls G_x the subgerbe generated by x; it is by definition a trivial gerbe. The presheaf SF

defined by

SF(X′) = { maximal subgerbes of F|X′ }

is a sheaf, called the sheaf of maximal subgerbes of F. Let O be the presheaf defined by

O(X′) = { classes of objects of F_X′ modulo isomorphism }.

By associating to every object x of F_X′ the maximal subgerbe of F|X′ generated by x, one obtains a morphism

$$O\to SF;$$

by [XIII.2, III 2.1.4], this morphism makes SF a sheaf associated with O.

A stack F is said to be constructible, respectively ind-ℒ-finite, where ℒ is a set of prime numbers, if for every scheme X′ étale over X and every object x of F_X′, the same is true of the sheaf SheafAut_X′(x) [XIII.2, VII 2.2.1]. A stack is said to be 1-constructible if it is constructible and if its sheaf of maximal subgerbes is constructible.

1. Cohomological Properness

1.0.

Let S be a scheme, and let f: X → Y be a morphism of S-schemes. If S′ is an S-scheme, consider the following diagram, all of whose squares are cartesian:

X′ → X
 |    |
f′   f
 |    |
Y′ → Y
 |    |
S′ → S.

If Y₁ is a scheme étale over Y, put X₁ = X ×_Y Y₁ and Y₁′ = Y′ ×_Y Y₁, and consider the cartesian square

X₁′ → X₁
 |      |
f₁′    f₁
 |      |
Y₁′ → Y₁.

Definition.

Let F be a stack on X. One says that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively in dimension ≤ 0, respectively in dimension ≤ 1, if for every S-scheme S′, the canonical functor, defined in the evident way by the universal property of inverse image of stacks,

$$g\ast f_{\ast }F\to f_{\ast }^{\prime }h\ast F(cf.1.0.1)$$

is faithful, respectively fully faithful, respectively an equivalence of categories.

If there is no possible confusion about S, in particular if S = Y, we say cohomologically proper instead of cohomologically proper relative to S.

1.2.

Let F be a sheaf of sets on X, and let Φ be the stack in discrete categories associated with F, that is, the stack whose fiber above every scheme X₁ étale over X is the discrete category whose set of objects is F(X₁). One says that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively in dimension ≤ 0, if (Φ,f) is cohomologically proper relative to S in dimension ≤ 0, respectively in dimension ≤ 1.

The

canonical morphism

$$g\ast f_{\ast }F\to f_{\ast }^{\prime }h\ast F$$

gives, after passage to the associated stacks in discrete categories, the canonical morphism

$$g\ast f_{\ast }\Phi \to f_{\ast }^{\prime }h\ast \Phi .$$

Consequently, saying that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively in dimension ≤ 0, is equivalent to saying that, for every S-scheme S′, the morphism above is injective, respectively bijective.

1.3.

Let F be a sheaf of groups on X, and let Φ be the stack of torsors on X with group F [XIII.1, II 2.3.2]. One says that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1, if (Φ,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1. The condition of cohomological properness can be made explicit as follows.

Subproposition.

The notation is that of (XIII.1.0.1) and (XIII.1.0.2). Let F be a sheaf of groups on X. Write F′, respectively F₁, F₁′, etc., for the inverse image of F on X′, respectively on X₁, X₁′, etc. Then the following conditions are equivalent:

(i) (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1.

(ii) For every morphism S′ → S, every scheme Y₁ étale over Y, and every torsor P on X₁ with group F₁, if ^P F₁ denotes the group obtained by twisting F₁ by P [XIII.1, II 4.1.2.3], the canonical morphism

a₀: g₁*(f₁*(^P F₁)) → f₁′*(^P′ F₁′)

is injective, respectively a₀ is bijective and the canonical morphism

$$a_1:g\ast (R^1{f}_{\ast }F)\to R^1{f}_{\ast }^{\prime }{F}^{\prime }$$

is

injective, respectively a₀ and a₁ are bijective.

(ii bis) For every morphism S′ → S, every scheme Y₁ étale over Y, every torsor P on X₁ with group F₁, and every torsor R under ^P F₁, the canonical morphism

$${\alpha }_0:g_1\ast (f_1\ast R)\to f_1^{\prime }\ast R^{\prime }$$

is injective, respectively α₀ is bijective, respectively the morphisms α₀ and

α₁: g₁*(R¹f₁*(^P F₁)) → R¹f₁′*(^P′ F₁′)

are bijective.

Proof.

(i) ⇒ (ii bis). By [XIII.1, II 4.2.5], every torsor R with group ^P F₁ is of the form

R = Q ∧^F₁ P°,

where $Q$ is a torsor with group $F_1$ and $P^{\circ }$ is the opposite of $P$. One then has $R^{\prime }\simeq Q^{\prime }{{\wedge }_1^F}^{\prime }{P}^{\prime \circ }$. Let $\Phi$ be the stack of torsors under $F$, and let $x$, $y$, respectively $x^{\prime }$, $y^{\prime }$, be the objects of the fiber category (g*f_*Φ)_Y₁′, respectively (f′_*Φ′)_Y₁′, associated with $P$, $Q$, respectively $P^{\prime }$, $Q^{\prime }$. One has the relation

Q ∧^F₁ P° ≃ SheafHom_F₁(P,Q),

and hence canonical isomorphisms

SheafHom_Y₁′(x,y) ≃ g₁*f₁*(Q ∧^F₁ P°),
SheafHom_Y₁′(x′,y′) ≃ f₁′*(Q′ ∧^F₁′ P′°).

Consequently the morphism α₀ identifies with the morphism

SheafHom_Y₁′(x,y) → SheafHom_Y₁′(x′,y′).

It follows that, if (F,f) is cohomologically proper relative to S in dimension ≤ −1, α₀ is injective, and if (F,f) is cohomologically proper in dimension ≤ 0, α₀ is bijective.

Suppose now that (F,f) is cohomologically proper relative to S in dimension ≤ 1, that is, that the canonical morphism

$$\varphi :g\ast f_{\ast }\Phi \to f_{\ast }^{\prime }{\Phi }^{\prime }$$

is an equivalence. Let $G$ be the sheaf of maximal subgerbes of the stack $f_{\ast }\Phi$ [XIII.1, III 2.1.8]; one then has an isomorphism $G\simeq R^1{f}_{\ast }F$. Since $g\ast G$ is the sheaf of maximal subgerbes of $g\ast f_{\ast }\Phi$ [XIII.2, III 2.1.5.5], the morphism ${\alpha }_1$ is obtained from $\varphi |Y_1^{\prime }$ by taking sheaves of maximal subgerbes, and hence is an isomorphism.

(ii bis) ⇒ (ii). It is enough to show that, if the morphisms ${\alpha }_0$ are bijective, then the morphisms $a_1$ are injective. Let $Y_1^{\prime }$ be a scheme étale over $Y^{\prime }$, and let $s$ and $t$ be two elements of $g\ast (R^1{f}_{\ast }F)(Y_1^{\prime })$ having the same image in $R^1{f}_{\ast }^{\prime }{F}^{\prime }(Y_1^{\prime })$; let us show that $s=t$. The assertion is local for the étale topology of $Y_1^{\prime }$, and, taking into account the definition of the inverse image $g\ast (R^1{f}_{\ast }F)$, one may suppose that $Y_1^{\prime }$ is the inverse image of a scheme $Y_1$ étale over $Y$ and that $s$ and $t$ come from torsors $P$ and $Q$ on $X_1$. The hypothesis on $s$ and $t$ then means that the inverse images $P^{\prime }$ and $Q^{\prime }$ of $P$ and $Q$ on $X_1^{\prime }$ are locally isomorphic for the étale topology of $Y_1^{\prime }$. If one puts R = SheafHom_F₁(P,Q), the fact that the morphism

$$g_1\ast f_1\ast R\to f_1^{\prime }\ast R^{\prime }$$

is bijective proves that P and Q are locally isomorphic for the étale topology of Y₁, hence that s = t.

(ii) ⇒ (i). To prove that $\varphi$ is faithful, respectively fully faithful, it is enough to show that, if $Y_1$ is a scheme étale over $Y$, if $P$, $Q$ are two torsors on $X_1$ with group $F_1$, and if $x$, $y$, respectively $x^{\prime }$, $y^{\prime }$, are the objects of (g*f_*Φ)_Y₁′, respectively (f′_*Φ′)_Y₁′, associated with $P$, $Q$, respectively $P^{\prime }$, $Q^{\prime }$, then the morphism

$$a:\mathrm{Hom}(x,y)\to \mathrm{Hom}(x^{\prime },y^{\prime })$$

is injective, respectively bijective. But a identifies with the canonical morphism

H⁰(Y₁′, g₁*f₁*(Q ∧^F₁ P°))
  → H⁰(Y₁′, f₁′*(Q′ ∧^F₁′ P′°)).

If

Hom(x,y) ≠ ∅, then Q ∧^F₁ P° is a torsor under ^P F₁ locally trivial on Y₁; hence f₁*(Q ∧^F₁ P°) is a torsor under f₁*(^P F₁), and g₁f₁(Q ∧^F₁ P°) is a trivial torsor. The morphism a then identifies with the canonical morphism

H⁰(Y₁, g₁*f₁*(^P F₁)) → H⁰(Y₁′, f₁′*(^P′ F₁′)).

The same is true if Hom(x′,y′) ≠ ∅ and if a₁ is injective; for then Q′ ∧^F₁′ P′° is trivial, and it follows from the injectivity of a₁ that P and Q are locally isomorphic on Y₁. We conclude that, if a₀ is injective, respectively if a₀ is bijective and a₁ injective, then φ is faithful, respectively fully faithful.

It remains to show that, if $a_0$ and $a_1$ are bijective, the functor $\varphi$ is essentially surjective. Let $Y^{\prime \prime }$ be a scheme étale over $Y^{\prime }$, put $X^{\prime \prime }=X^{\prime }{\times }_Y^{\prime }{Y}^{\prime \prime }$, and let $P^{\prime \prime }$ be a torsor on $X^{\prime \prime }$ with group $F^{\prime \prime }=F^{\prime }|X^{\prime \prime }$. We shall show that there exists an element $x$ of $(g\ast f_{\ast }\Phi {)}_Y^{\prime \prime }$ whose image in $(f_{\ast }^{\prime }{\Phi }^{\prime }{)}_Y^{\prime \prime }$ is isomorphic to $P^{\prime \prime }$. Let $p^{\prime \prime }$ be the class of $P^{\prime \prime }$. Since $a_1$ is surjective, one can find a surjective étale morphism $Y_1^{\prime \prime }\to Y^{\prime \prime }$, an étale morphism $Y_1\to Y$, a morphism $Y_1^{\prime \prime }\to Y_1^{\prime }$, and a torsor $P_1$ on $X_1$ with group $F_1$ whose inverse image $P_1^{\prime \prime }$ on $X_1^{\prime \prime }$ is isomorphic to the inverse image of $P^{\prime \prime }$. Using the fact that $\varphi$ is fully faithful, one sees that the object $x_1$ of ${(g\ast f_{\ast }\Phi {)}_Y^{\prime \prime }}_1$ corresponding to $P_1^{\prime \prime }$ is endowed with a descent datum relative to $Y_1^{\prime \prime }\to Y^{\prime \prime }$, and hence comes from an element $x$ of $(g\ast f_{\ast }\Phi {)}_Y^{\prime \prime }$. Since the image of $x$ in $(f_{\ast }^{\prime }{\Phi }^{\prime }{)}_Y^{\prime \prime }$ is $P^{\prime \prime }$, this proves that $\varphi$ is essentially surjective and completes the proof.

Example.

Let f: X → Y be a proper morphism. It follows from [XIII.2, VII 2.2.2] that, for every ind-finite stack F on X, the pair (F,f) is cohomologically proper, relative to Y, in dimension ≤ 1. In particular, for every sheaf of sets, respectively every sheaf of groups, respectively every sheaf of ind-finite groups, F on X, the pair (F,f) is cohomologically

proper in dimension ≤ 0, respectively in dimension ≤ 0, respectively in dimension ≤ 1.

Remarks.

a. Let F be a sheaf of groups on X such that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0. If F is regarded as a sheaf of sets, then (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, but the converse is false.

For example, let Y be the spectrum of a strictly local discrete valuation ring, with closed point t and generic point s; let f: X → Y be a nonempty scheme over Y whose closed fiber is empty; let F be a nontrivial constant sheaf of groups on X; and let P be a torsor under F such that H⁰(X_s, ^P F|X_s) = 1. Then (^P F,f) is cohomologically proper relative to Y in dimension ≤ −1 when ^P F is regarded as a sheaf of sets. If ^P F is regarded as a sheaf of groups, one has an isomorphism ^P°(^P F) ≃ F; since the canonical morphism

$$H^0(X,F)\to H^0(X_t,F|X_t)=1$$

is not injective, this proves that (^P F,f) is not cohomologically proper relative to Y in dimension ≤ −1.

b. Suppose f is coherent, that is, quasi-compact and quasi-separated. Let F be a stack on X. For every geometric point ȳ of Y′, write Ȳ, respectively Ȳ′, for the strict localization of Y, respectively Y′, at ȳ, and put X̄ = X ×_Y Ȳ, X̄′ = X′ ×_Y′ Ȳ′, etc. For (F,f) to be cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1, it is necessary and sufficient that, for every S-scheme S′ and every geometric point ȳ of Y′, the canonical functor

$$\bar{F}(\bar{X})\to {\bar{F}}^{\prime }({\bar{X}}^{\prime })$$

be faithful, respectively fully faithful, respectively an equivalence.

Indeed,

if S′ is an S-scheme, then for the functor

$$g\ast f_{\ast }F\to f_{\ast }^{\prime }{F}^{\prime }$$

to be faithful, respectively fully faithful, respectively an equivalence, it is necessary and sufficient that the same be true of the functor induced on the fibers at the various geometric points ȳ′ of Ȳ′ [XIII.2, III 2.1.5.9]. The assertion therefore follows from the calculation of geometric fibers of the direct image of a stack by a coherent morphism [XIII.2, VII 2.1.5].

c. Let F be a stack on X. The fact that (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1, is local on Y for the étale topology.

Let S′ be an S-scheme, and let F′ be the inverse image of F on X′; cf. (XIII.1.0.1). If (F,f) is cohomologically proper relative to S in dimension ≤ 1, then the same is true of (F′,f′). But if (F,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, this need not remain true for (F′,f′).

For example, let S′ be a discrete valuation ring, let f′: E_S′ → S′ be affine space over S′, let x be a closed point of E_S′ above the generic point of S′, and let F′ be the sheaf of sets on E_S′ whose restriction to E_S′ − {x} is the constant sheaf with one element and whose fiber at a geometric point above x has two elements. Then (F′,f′) is not cohomologically proper relative to S′ in dimension ≤ −1. Let S = S′[Z], let f: E_S → S be affine space over S, and let T be a closed subset of X = E_S which does not meet the closed subset Z = 0 and such that f(T) contains the generic point of S. Let G be the inverse image of F′ on X, and let F be the sheaf on X obtained by extending G|X−T by the empty sheaf. Then (F,f) is cohomologically proper relative to S in dimension ≤ −1, but this is no longer true after the base change S′ → S defined by Z = 0.

d.

Let F be a stack on X such that (F,f) is cohomologically proper relative to Y in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1. Then, for every geometric point ȳ of Y, the canonical functor

$$(f_{\ast }F{)}_{\bar{y}}\to F(X_{\bar{y}})$$

is faithful, respectively fully faithful, respectively an equivalence of categories.

Proposition.

Let f: X → Y and g: Y → Z be two S-morphisms, and let Φ be a stack on X.

1. Suppose that $(\Phi ,f)$ and $(f_{\ast }\Phi ,g)$ are cohomologically proper relative to $S$ in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1. Then the same is true of $(\Phi ,gf)$.

2. Suppose that $(\Phi ,gf)$ is cohomologically proper relative to $S$ in dimension ≤ −1, respectively that $(\Phi ,gf)$ is cohomologically proper relative to $S$ in dimension ≤ 0 and $(\Phi ,f)$ cohomologically proper relative to $S$ in dimension ≤ −1, respectively that $(\Phi ,gf)$ is cohomologically proper relative to $S$ in dimension ≤ 1 and $(\Phi ,f)$ cohomologically proper relative to $S$ in dimension ≤ 0. Then $(f_{\ast }\Phi ,g)$ is cohomologically proper relative to $S$ in dimension ≤ −1, respectively in dimension ≤ 0, respectively in dimension ≤ 1.

For every S-scheme S′, consider the following diagram, all of whose squares are cartesian:

X′ --f′→ Y′ --g′→ Z′ → S′
 |        |        |     |
h        k        m     |
 |        |        |     |
X  --f→  Y  --g→  Z  → S.

[Translator note: the corrected source capitalizes the sentence after the diagram.] The canonical morphism

$$m\ast (g_{\ast }{f}_{\ast }\Phi )\to g_{\ast }^{\prime }{f}_{\ast }^{\prime }(h\ast \Phi )$$

identifies

with the composite of the canonical morphisms

m*(g_*f_*Φ) --i→ g′_*(k*f_*Φ) --j→ g′_*f′_*(h*Φ).

1. The hypothesis implies that i and j are faithful, respectively fully faithful, respectively equivalences; hence the same is true of ji.

2. The hypothesis implies that ji is faithful, respectively that ji is fully faithful and j is faithful, respectively that ji is an equivalence and j is fully faithful. [Translator note: the corrected source fixes a typo in “fully.”] It follows that i is faithful, respectively fully faithful, respectively an equivalence.

Corollary.

Let $f:X\to Y$ and $g:Y\to Z$ be two $S$-morphisms, and let $F$ be a sheaf of groups on $X$. Suppose that (F,gf) is cohomologically proper relative to $S$ in dimension ≤ −1, respectively that (F,gf) is cohomologically proper relative to $S$ in dimension ≤ 0 and that $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ −1. Then $(f_{\ast }F,g)$ is cohomologically proper relative to $S$ in dimension ≤ −1, respectively in dimension ≤ 0.

Return to the notation of (XIII.1.6.1), and for every scheme $Y_1$ étale over $Y$, write $f_1$, $F_1$ for the respective inverse images of $f$, $F$ by the morphism $Y_1\to Y$. Let $\Phi$ be the stack of torsors under $F$, and let $\Psi$ be the stack of torsors under $f_{\ast }F$. There is a canonical functor

$$\varphi :\Psi \to f_{\ast }\Phi ,$$

obtained by associating to every scheme Y₁ étale over Y and every torsor P on Y₁ with group f₁F₁ the torsor P̃ on X₁ deduced from f₁P by extension of the structural group f₁f₁F₁ → F₁. The functor φ is fully faithful. Indeed, if P and Q are two torsors on Y₁ with group f₁*F₁, one has a canonical morphism

SheafIsom_f₁*F₁(P,Q) → f₁*(SheafIsom_F₁(P̃,Q̃))

which

is an isomorphism because it is so locally. It follows that the canonical morphism

Isom_f₁*F₁(P,Q) → Isom_F₁(P̃,Q̃)

is an isomorphism, hence that φ is fully faithful.

There is a commutative diagram

$$g_{\ast }^{\prime }k\ast \Psi \to m\ast (g_{\ast }\Psi )\downarrow \downarrow g_{\ast }^{\prime }k\ast (f_{\ast }\Phi )\to m\ast (g_{\ast }{f}_{\ast }\Phi ),$$

where the horizontal arrows are the base-change morphisms. It follows from XIII.1.6 2 that the lower horizontal arrow is faithful, respectively fully faithful. Since g′k(φ) and m*g*(φ) are fully faithful, the diagram above implies that the upper horizontal arrow is faithful, respectively fully faithful.

Corollary.

Let f: X → Y be a coherent S-morphism, let g: Y → Z be a proper S-morphism, and let Φ be an ind-finite stack on X [XIII.2, VII 2.2.1]. Suppose that (Φ,f) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1. Then the same is true of (Φ,gf).

Since $f$ is coherent, $f_{\ast }\Phi$ is an ind-finite stack (SGA 4 IX 1.6 (ii)). The corollary therefore follows from XIII.1.6 1 and XIII.1.4.

Corollary.

Let f: X → Y be an integral S-morphism, and let g: Y → Z be an S-morphism. If F is a sheaf of sets on X, then (f**F,g) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, if and only if the same is true of (F,gf). If F is a sheaf of groups on X, then (f**F,g) is cohomologically proper relative to S in dimension ≤ −1, respectively ≤ 0, respectively ≤ 1, if and only if the same is true of (F,gf).

The assertion

concerning the case of a sheaf of sets follows from XIII.1.6 and from the fact that $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ 0. Let $F$ be a sheaf of groups on $X$, and let $\Phi$ be the stack of torsors under $F$. By SGA 4 VIII 5.8, every torsor under $F$ is locally trivial on $Y$. It follows that the stack $f_{\ast }\Phi$ is equivalent to the stack of torsors under $f_{\ast }F$, the equivalence being obtained by associating to every scheme $Y_1$ étale over $Y$ and every torsor $P$ on $X_1=X{\times }_Y{Y}_1$ with group $F|X_1$ the torsor $f_{\ast }P$ with group $f_{\ast }F|Y_1$. Since $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ 1, the corollary follows from XIII.1.6.

Definitions.

1.10.1.

Let E be a category and consider a diagram

Φ --p→ Φ₁ ⇉ Φ₂,

where Φ, Φ₁, Φ₂ are fibered categories over E and the arrows are morphisms of fibered categories, together with an isomorphism of functors

$$a:p_{1}p\simeq p_{2}p.$$

One says that the diagram above is exact if the following condition is satisfied:

a. For every pair of objects x, y of Φ and every morphism f: p(x) → p(y) such that p₁(f) = p₂(f), with p₁p and p₂p identified by means of a, there exists a unique morphism g: x → y such that p(g) = f.

1.10.2.

Consider the diagram

Φ --p→ Φ₁ ⇉ Φ₂ ⇉⇉ Φ₃,

where

Φ and Φ_i, 1 ≤ i ≤ 3, are fibered categories over E and the arrows are morphisms of fibered categories. Suppose given isomorphisms of functors

a: p₁p ≃ p₂p,
a₁: p₃₁p₂ ≃ p₁₂p₁,     a₂: p₁₂p₂ ≃ p₂₃p₁,     a₃: p₂₃p₂ ≃ p₃₁p₁,

such that the evident hexagonal compatibility diagram commutes. We identify p₁p with p₂p, p₃₁p₂ with p₁₂p₁, and so on.

One says that the diagram above is exact if the following conditions are satisfied:

a. The analogue of condition a of XIII.1.10.1.

b. For every object x₁ of Φ₁ and every isomorphism

$$u:p_1(x_1)\simeq p_2(x_1)$$

such that

$$(\mathrm{1.10.2.1})p_{23}(u)p_{31}(u)=p_{12}(u{)}^{-1},$$

there exists an object x of Φ and an isomorphism i: p(x) ≃ x₁ making the diagram

p₁p(x) = p₂p(x)
  |        |
p₁(i)    p₂(i)
  |        |
p₁(x₁) --u→ p₂(x₁)

commute.

1.10.3.

One defines in the evident way the notion of a morphism of exact diagrams of fibered categories over a category E.

1.10.4.

We shall use

in particular the notion of exact diagram in the case where E is a site and Φ, Φ_i, 1 ≤ i ≤ 3, are stacks on E.

Let f: E → E′ be a morphism of sites and let

Φ → Φ₁ ⇉ Φ₂ ⇉⇉ Φ₃

be an exact diagram of stacks on E. Taking direct images gives a diagram

f_*Φ → f_*Φ₁ ⇉ f_*Φ₂ ⇉⇉ f_*Φ₃

which is evidently exact.

If h: E″ → E is a morphism of sites, one likewise has an exact diagram

h*Φ → h*Φ₁ ⇉ h*Φ₂ ⇉⇉ h*Φ₃.

Let us first verify condition a of XIII.1.10.2. Let $F^{\prime \prime }$ be an object of $E^{\prime \prime }$, let $x^{\prime \prime }$ and $y^{\prime \prime }$ be two objects of $(h\ast \Phi {)}_F^{\prime \prime }$, let $x_1^{\prime \prime }$ and $y_1^{\prime \prime }$ be their respective images in $h\ast {\Phi }_1$, and let $x_2^{\prime \prime }$ and $y_2^{\prime \prime }$ be their images in $h\ast {\Phi }_2$. Let $u_1^{\prime \prime }:x_1^{\prime \prime }\to y_1^{\prime \prime }$ be a morphism such that $p_1^{\prime \prime }(u_1^{\prime \prime })=p_2^{\prime \prime }(u_1^{\prime \prime })$, and let us prove that $u_1^{\prime \prime }$ comes from a unique morphism $u^{\prime \prime }:x^{\prime \prime }\to y^{\prime \prime }$. Since the question is local on $F^{\prime \prime }$, one may suppose that one has an object $F_1$ of $E$, a morphism from $F^{\prime \prime }$ to the inverse image $F_1^{\prime \prime }$ of $F_1$ by $h$, and objects $x$, $y$ of Φ_F₁ whose inverse images on $F^{\prime \prime }$ are $x^{\prime \prime }$ and $y^{\prime \prime }$. Let $x_1$, $y_1$, respectively $x_2$, $y_2$, be the images of $x$, $y$ in ${\Phi }_1$, respectively ${\Phi }_2$. We may suppose that $u_1^{\prime \prime }$ comes from a morphism $u_1:x_1\to y_1$ such that $p_1(u_1)=p_2(u_1)$. By exactness of (XIII.1.10.4.1), one obtains a unique morphism $u:x\to y$ whose inverse image by $h$ is the desired morphism $u^{\prime \prime }$.

The

condition b of XIII.1.10.2 is verified analogously. Let $x_1^{\prime \prime }$ be an object of $(h\ast {\Phi }_1{)}_F^{\prime \prime }$, and let $u^{\prime \prime }:p_1^{\prime \prime }(x_1^{\prime \prime })\to p_2^{\prime \prime }(x_1^{\prime \prime })$ be a morphism satisfying

$$p_{23}^{\prime \prime }(u^{\prime \prime })p_{31}^{\prime \prime }(u^{\prime \prime })=p_{12}^{\prime \prime }(u^{\prime \prime }{)}^{-1}.$$

We must prove that there exists an object $x^{\prime \prime }$ of $(h\ast \Phi {)}_F^{\prime \prime }$ and an isomorphism $i^{\prime \prime }:p^{\prime \prime }(x^{\prime \prime })\simeq x_1^{\prime \prime }$ making a diagram analogous to (XIII.1.10.2.2) commute. Since the question is local on $F^{\prime \prime }$, one may suppose that one has an object $F_1$, a morphism $F^{\prime \prime }\to F_1^{\prime \prime }$ as above, and an object $x_1$ of (Φ₁)_F₁ whose inverse image in $(h\ast {\Phi }_1{)}_F^{\prime \prime }$ is $x_1^{\prime \prime }$. Likewise one may suppose that $u^{\prime \prime }$ comes from a morphism $u:p_1(x_1)\to p_2(x_1)$ satisfying (XIII.1.10.2.1). The existence of an object $x$ of Φ_F₁ whose inverse image by $h$ is an element $x^{\prime \prime }$ answering the question follows from the exactness of (XIII.1.10.4.1).

Examples.

1. Let f: X₁ → X be a morphism of descent for the category of étale sheaves on variable schemes, for example a universally submersive morphism (SGA 4 VIII 9.3). Let X₂ = X₁ ×_X X₁, let g: X₂ → X be the canonical projection, and let F be a sheaf of sets on X. It then follows from loc. cit. that one has an exact sequence of sheaves of sets

F → f_*f*F ⇉ g_*g*F.

If Φ is the stack in discrete categories associated with F and Φ₃ is the final stack on X, that is, the stack all of whose fibers are reduced to a single object with the identity as its only morphism, saying that the sequence (XIII.1.11.1) is exact is equivalent to saying that the following diagram of stacks is exact:

Φ → f_*f*Φ ⇉ g_*g*Φ ⇉⇉ Φ₃.

Let f: X₁ → X be a morphism of effective descent for the category of étale sheaves on variable schemes, for example a proper surjective morphism, an integral surjective morphism, or a faithfully flat morphism locally of finite presentation (SGA 4 VIII 9.4). Let X₂ = X₁ ×X X₁, let g: X₂ → X be the canonical projection, let X₃ = X₁ ×_X X₁ ×_X X₁, and let h: X₃ → X be the canonical morphism. Let Φ be a stack on X, Φ₁ = f**fΦ, Φ₂ = gg*Φ, and Φ₃ = hh*Φ. Then one has an exact diagram

Φ → Φ₁ ⇉ Φ₂ ⇉⇉ Φ₃,

where the arrows are the canonical morphisms associated with the projections.

Indeed, regard Φ as a stack on the category Sch_X of schemes over X, endowed with the étale topology. Then, by [XIII.2, VII 2.2.8], Φ is also a stack for the finest topology on Sch_X for which the covering morphisms are the morphisms of effective descent for the category of étale sheaves. The exactness of the diagram above follows immediately.

Proposition.

Let S be a scheme, and let f: X → Y be an S-morphism.

1. Let

Φ → Φ₁ ⇉ Φ₂

be an exact diagram of stacks on X. Suppose that (Φ₁,f) is cohomologically proper relative to S in dimension ≤ 0 and that (Φ₂,f) is cohomologically proper relative to S in dimension ≤ −1. Then (Φ,f) is cohomologically proper relative to S in dimension ≤ 0.

1. Let

Φ → Φ₁ ⇉ Φ₂ ⇉⇉ Φ₃

be an exact diagram of stacks on X. Suppose that (Φ₁,f) is cohomologically proper relative to S in dimension ≤ 1, that (Φ₂,f) is cohomologically proper

relative to S in dimension ≤ 0, and that (Φ₃,f) is cohomologically proper relative to S in dimension ≤ −1. Then (Φ,f) is cohomologically proper relative to S in dimension ≤ 1. [Translator note: the corrected source inserts a missing “relative.”]

For every S-scheme S′, consider the following commutative diagram, all of whose squares are cartesian:

X′ --f′→ Y′ → S′
 |        |     |
h        g     |
 |        |     |
X  --f→  Y  → S.

Let us prove 2; the proof of 1 is analogous. Since direct-image and inverse-image functors send exact diagrams of stacks to exact diagrams (XIII.1.10.4), one has the following morphism of exact diagrams of stacks:

g*f_*Φ  → g*f_*Φ₁ ⇉ g*f_*Φ₂ ⇉⇉ g*f_*Φ₃
  ↓          ↓           ↓           ↓
f′_*h*Φ → f′_*h*Φ₁ ⇉ f′_*h*Φ₂ ⇉⇉ f′_*h*Φ₃.

By hypothesis, the second vertical functor is an equivalence of categories, the third is fully faithful, and the fourth is faithful. It follows from the preceding diagram that the first vertical functor is an equivalence.

Proposition.

Let f: X → Y be an S-morphism.

1. Let

F → G ⇉ H

be an exact diagram of sheaves of sets on X. Suppose that (G,f) is cohomologically proper relative to S in dimension ≤ 0 and that (H,f) is cohomologically proper relative to S in dimension ≤ −1. Then (F,f) is cohomologically proper relative to S in dimension ≤ 0.

Let F → G be a monomorphism of sheaves of groups on X. If Y₁ is a scheme étale over Y, put X₁ = Y₁ ×_Y X, and write f₁, respectively F₁, respectively G₁, for the inverse image of f, respectively F, respectively G, on Y₁; cf. (XIII.1.0.2). Suppose that (G,f) is cohomologically proper relative to S in dimension ≤ 0, respectively in dimension ≤ 1, and that for every scheme Y₁ étale over Y and every torsor Q under G₁, the pair (Q/F₁,f₁) is cohomologically proper relative to S in dimension ≤ −1, respectively in dimension ≤ 0. Then (F,f) is cohomologically proper relative to S in dimension ≤ 0, respectively in dimension ≤ 1.

1. Let F → G be a monomorphism of sheaves of groups on X. Suppose that (F,f) is cohomologically proper relative to S in dimension ≤ 1 and that (G,f) is cohomologically proper relative to S in dimension ≤ 0. Then, for every torsor Q under G, the pair (Q/F,f) is cohomologically proper relative to S in dimension ≤ 0.

Proof.

1. Let Φ, respectively Φ₁, respectively Φ₂, be the stack in discrete categories associated with F, respectively G, respectively H, and let Φ₃ be the final stack on X. One then has an exact diagram

Φ → Φ₁ ⇉ Φ₂ ⇉⇉ Φ₃.

By hypothesis, (Φ₁,f) is cohomologically proper relative to S in dimension ≤ 1 and (Φ₂,f) is cohomologically proper relative to S in dimension ≤ 0 (XIII.1.2). Since (Φ₃,f) is evidently cohomologically proper relative to S in dimension ≤ −1, it follows from XIII.1.12 that (Φ,f) is cohomologically proper relative to S in dimension ≤ 1, that is, that (F,f) is cohomologically proper relative to S in dimension ≤ 0.

1. Let us first show that, if (G,f) is cohomologically proper relative to S in dimension ≤ 0 and if the (Q/F₁,f₁) are cohomologically

proper relative to S in dimension ≤ −1, then (F,f) is cohomologically proper relative to S in dimension ≤ 0. By XIII.1.3.1, it is enough to prove that, for every scheme Y₁ étale over Y and every torsor P on X₁ with group F₁, the pair (^P F₁,f₁) is cohomologically proper relative to S in dimension ≤ 0 when ^P F₁ is regarded as a sheaf of sets, and that the canonical morphism

$$d:g\ast (R^1{f}_{\ast }F)\to R^1{f}_{\ast }^{\prime }{F}^{\prime }$$

is injective. The first assertion follows at once from 1: if Q denotes the torsor deduced from P by extension of the structural group F₁ → G₁, one has an isomorphism

^Q G₁ / ^P F₁ ≃ Q/F₁.

Let us show that d is injective. It is enough to prove that, if Y₁ is a scheme étale over Y and P, P̃ are two torsors under F₁ whose inverse images P′ and P̃′ on X₁′ are isomorphic, then, after possibly making a surjective étale extension of Y₁, P and P̃ become isomorphic. Choose an isomorphism p′: P′ ≃ P̃′. Let Q, respectively Q̃, be the torsor deduced from P, respectively P̃, by extension of the structural group F₁ → G₁. The inverse images Q′, respectively Q̃′, of Q, respectively Q̃, on X₁′ are deduced from P′, respectively P̃′, by extension of the structural group F₁′ → G₁′; [Translator note: the corrected source inserts the missing article before “extension.”] let q′: Q′ ≃ Q̃′ be the isomorphism obtained in the same way from p′. Since (G,f) is cohomologically proper relative to S in dimension ≤ 0, after possibly making a surjective étale extension of Y₁, one may suppose that q′ is the image of an isomorphism q: Q ≃ Q̃. To the torsor P, respectively P̃, is associated a section x of Q/F₁, respectively a section x̃ of Q̃/F₁, and for P and P̃ to be isomorphic, it is necessary and sufficient that there be an isomorphism Q ≃ Q̃ such that the induced isomorphism

$$e:H^0(X_1,Q/F_1)\to H^0(X_1,\tilde{Q}/F_1)$$

sends x to x̃. We take the isomorphism q. The sections e(x) and x̃ of H⁰(X₁,Q̃/F₁) have the same image in H⁰(X₁′,Q̃′/F₁′). Since (Q̃/F₁,f₁) is cohomologically proper relative to S in dimension ≤ −1,

after possibly making a surjective étale extension of Y₁, one indeed has e(x) = x̃. This proves the injectivity of d.

To finish the proof, it remains to prove that if (G,f) is cohomologically proper relative to S in dimension ≤ 1, and if for every scheme Y₁ étale over Y and every torsor Q on X₁ with group G₁, the pair (Q/F₁,f₁) is cohomologically proper relative to S in dimension ≤ 0, then the morphism d is surjective. Let P′ be a torsor on X₁′ with group F₁′, and let Q′ be the torsor under G₁′ obtained from P′ by extension of the structural group. Giving P′ is equivalent to giving Q′ and a section x′ of H⁰(X₁′,Q′/F₁′). It then follows from the surjectivity of the morphism

$$g\ast (R^1{f}_{\ast }G)\to R^1{f}_{\ast }^{\prime }{G}^{\prime }$$

that, after possibly making a surjective étale extension of Y₁, there exists a torsor Q under G₁ whose inverse image on X₁′ is isomorphic to Q′. Using the fact that (Q/F₁,f₁) is cohomologically proper relative to S in dimension ≤ 0, one may similarly suppose that there exists an element x of H⁰(X₁,Q/F₁) whose image in H⁰(X₁′,Q′/F₁′) is x′. The data of Q and x determine a torsor P under F₁ whose inverse image on X₁′ is isomorphic to P′, which proves the surjectivity of d.

1. Let us show that (Q/F,f) is cohomologically proper relative to S in dimension ≤ −1, that is, that for every S-scheme S′ and every scheme Y₁ étale over Y, if x and x̃ are two elements of H⁰(X₁,Q₁/F₁) whose images x′ and x̃′ in H⁰(X₁′,Q₁′/F₁′) are equal, then, after a surjective extension of Y₁, one has x = x̃. To x, respectively x̃, is associated a torsor P, respectively P̃, under F₁, such that Q₁ is deduced from P, respectively P̃, by extension of the structural group F₁ → G₁. From the relation x′ = x̃′ it follows that one has an isomorphism u′: P′ ≃ P̃′ such that the isomorphism induced on Q₁′ by u′ is the identity. Since (F,f) is cohomologically proper relative to S in dimension ≤ 0, it follows that,

after a surjective étale extension of Y₁, one has an isomorphism u: P → P̃ lifting u′. The fact that (G,f) is cohomologically proper relative to S in dimension ≤ −1 then implies that x = x̃.

Let us show that (Q/F,f) is cohomologically proper relative to S in dimension ≤ 0. Let Y″ be a scheme étale over Y′, and let x″ be an element of H⁰(X″,Q″/F″). To x″ is associated a torsor P″ on X″ with group F″. Since (F,f) is cohomologically proper relative to S in dimension ≤ 1, one can find surjective étale morphisms Y″₁ → Y″ and Y₁ → Y, with a morphism Y″₁ → Y₁′, and a torsor P on X₁ with group F₁ whose inverse image on X″₁ is isomorphic to the inverse image of P″. It then follows from the fact that (G,f) is cohomologically proper relative to S in dimension ≤ 0 that one may even choose Y″₁ and Y₁ so that the torsor deduced from P by extension of the structural group F₁ → G₁ is isomorphic to Q₁. To P there corresponds an element x of H⁰(X₁,Q₁/F₁), whose image in H⁰(X″₁,Q″₁/F″₁) is isomorphic to the inverse image of x″. This completes the proof.

Proposition.

Let f: X → S be an S-scheme, and let F be a sheaf of sets or groups on X, respectively a sheaf of ind-ℒ-groups, where ℒ is a set of prime numbers. Suppose F is locally constant, (F,f) is cohomologically proper in dimension ≤ 0, respectively in dimension ≤ 1, and f is locally 0-acyclic, respectively locally 1-aspherical for ℒ (SGA 4 XV 1.11). Then, for every specialization s̄₁ → s̄₂ of geometric points of S, the specialization morphism (SGA 4 VIII 7.1)

a₀: (f_*F)_s̄₂ → (f_*F)_s̄₁

is an isomorphism; and, if F is a sheaf of groups, the morphism

a₁: (R¹f_*F)_s̄₂ → (R¹f_*F)_s̄₁

is injective, respectively the morphisms a₀ and a₁ are isomorphisms.

The

proof is obtained by copying word for word that of SGA 4 XVI 2.3, but replacing the expression “proper” by the expression “cohomologically proper.”

Corollary.

Let f: X → S be a morphism, let Φ be a stack on X, and let ℒ be a set of prime numbers. Suppose that, for every scheme X₁ étale over X and every pair of objects x, y of Φ_X₁, the sheaf SheafHom_X₁(x,y) is locally constant, that the sheaf SheafAut_X₁(x) is a locally constant ind-ℒ-group, and that the sheaf SΦ of maximal subgerbes of Φ [XIII.1, III 2.1.7] is locally constant. Suppose that (Φ,f) is cohomologically proper in dimension ≤ 1 and that f is locally 1-aspherical for ℒ. Then, for every specialization s̄₁ → s̄₂ of geometric points of S, the specialization morphism

a: (f_*Φ)_s̄₂ → (f_*Φ)_s̄₁

is an equivalence of categories.

Let S̄₁, respectively S̄₂, be the strict localization of S at s̄₁, respectively at s̄₂, let X̄₂, Φ̄₂, respectively X̄₁, Φ̄₁, be the inverse images of X₂, Φ₂ on S̄₂, respectively of X₁, Φ₁ on S̄₁, and consider the cartesian square

X̄₁ --h→ X̄₂
 |        |
f̄₁      f̄₂
 |        |
S̄₁ --g→ S̄₂.

We must show that the functor

$$\varphi :{\bar{\Phi }}_2({\bar{X}}_2)\to {\bar{\Phi }}_1({\bar{X}}_1)$$

is an equivalence. The functor φ is fully faithful. Indeed, let

$x$ and $y$ be two objects of (Φ̄₂)_X̄₂; the canonical morphism

Hom_X̄₂(x,y) → Hom_X̄₁(φ(x),φ(y))

identifies with the canonical morphism

H⁰(X̄₂,SheafHom_X̄₂(x,y)) → H⁰(X̄₁,h*(SheafHom_X̄₂(x,y))).

This morphism is an isomorphism by XIII.1.14.

Let us show that φ is an equivalence. Let x₁ be an object of Φ̄₁(X̄₁), and let G₁ be the maximal subgerbe of Φ̄₁ generated by x₁. The morphism

$$H^0({\bar{X}}_2,S{\bar{\Phi }}_2)\to H^0({\bar{X}}_1,h\ast (S{\bar{\Phi }}_2))=H^0({\bar{X}}_1,S{\bar{\Phi }}_1)$$

is bijective; hence there exists a maximal subgerbe G₂ of Φ̄₂ such that h*G₂ is isomorphic to G₁. It remains only to prove that the functor

$$G_2\to h_{\ast }h\ast G_2$$

is an equivalence. But in this form the question is local for the étale topology on X̄₂. We may therefore suppose that G₂ is a gerbe of torsors under the automorphism group of an object of G₂, a case in which the assertion follows from XIII.1.14.

Corollary.

The hypotheses are those of XIII.1.14. If in addition $F$ is a sheaf of sets, respectively of ind-$\mathcal{L}$-groups, and $f_{\ast }F$, respectively $R^1{f}_{\ast }F$, is constructible, then $f_{\ast }F$, respectively $R^1{f}_{\ast }F$, is locally constant.

The corollary follows from XIII.1.14 by SGA 4 IX 2.11.

Remark.

Recall that the condition that f be locally 0-acyclic is satisfied if f is flat with separable fibers and X and Y are locally noetherian (SGA 4 XV 4.1), and that the condition that f be locally 1-aspherical

for ℒ is satisfied if f is smooth, ℒ being the set of prime numbers distinct from the residual characteristics of S (SGA 4 XV 2.1).

2. A Special Case of Cohomological Properness: Relative Normal-Crossings Divisors

2.0.

Let $R$ be a discrete valuation ring with field of fractions $K$, and let $L$ be an étale $K$-algebra. Then $L$ is a direct product of finitely many fields $L_i$, where $L_i$ is an étale extension of $K$. If $L_i^{\prime }$ denotes the Galois extension generated by $L_i$ in an algebraic closure of $L_i$, one says that $L$ is tamely ramified over $R$ if the $L_i^{\prime }$ are tamely ramified extensions in the sense of X.3, that is, if an inertia group $I_i$ of $L_i^{\prime }|K$ has order prime to the residual characteristic $p$ of $R$.

One knows that $I_i$ is in any case an extension of a cyclic group of order prime to $p$ by a $p$-group. This follows from [XIII.5, ch. IV, prop. 7, cor. 4] when the residual extension of $R$ is separable. The proof given there extends to the general case as follows. Resume the hypotheses and notation of loc. cit., but without assuming the residual extension separable. Let $H_i$ be the subgroup of the inertia group $G_0$ consisting of the elements $s$ of $G_0$ such that $s\pi /\pi \in U^i$ for every uniformizer $\pi$ of A_L. One then verifies that $G_0/H_1$ is a group of order prime to $p$ and that, for $i\ge 1$, the $H^i/H^{i+1}$ are $p$-groups, from which the announced result follows.

2.0.1.

In the case where $R$ is strictly local, one has the following simple characterization: the $K$-algebra $L$ is tamely ramified over $R$ if and only if the $[L_i:K]$ are prime to $p$. Moreover, if $L$ is tamely ramified over $R$, the $L_i$ are cyclic extensions of $K$. Indeed, when $R$ is strictly local, $I_i$ is equal to the Galois group

of $L_i^{\prime }$ over $K$. As just recalled, $I_i$ is an extension of a cyclic group of order prime to $p$ by a $p$-group. If $L_i^{\prime }$ is assumed tamely ramified over $R$, $I_i$ is then a cyclic group of order prime to $p$. It follows that $[L_i:K]$ is prime to $p$ and that $L_i=L_i^{\prime }$. Conversely, if $[L_i:K]$ is prime to $p$, $I_i$ cannot contain a nontrivial normal $p$-subgroup; hence $I_i$ is a cyclic group of order prime to $p$, which proves that $L_i$ is tamely ramified over $R$.

2.0.2.

Let R be a discrete valuation ring with field of fractions K, let L be an étale K-algebra, and let R̄ be a strict localization of R, K̄ its field of fractions, and L̄ = L ⊗_K K̄. Then L is tamely ramified over R if and only if L̄ is tamely ramified over R̄. Indeed, one reduces to the case where L is a field. Let L̄ = ∏_i L̄_i, where the L̄_i are fields extending K̄. If L′ is the Galois extension generated by L, and if L̄′ = L′ ⊗_K K̄, [Translator note: the corrected source fixes L' to \bar{L}' here and in the following displayed product.] then one likewise has a decomposition of L̄′ as a product of fields,

$${\bar{L}}^{\prime }=\prod _j{\bar{L}}_j^{\prime },$$

and each ${\bar{L}}_i$ is a subextension of at least one of the ${\bar{L}}_j^{\prime }$. Since $L^{\prime }$ is a Galois extension of $K$, the ${\bar{L}}_j^{\prime }$ are Galois extensions of $\bar{K}$. Suppose $L$ is tamely ramified over $R$. Since the Galois group of ${\bar{L}}_j^{\prime }|\bar{K}$ is isomorphic to the inertia group of $L^{\prime }|K$, the ${\bar{L}}_j^{\prime }$ are also tamely ramified over $\bar{R}$, and hence so are the ${\bar{L}}_i$. Conversely, suppose $\bar{L}$ is tamely ramified over $\bar{R}$. For each $j$, let $v_j$ be the discrete valuation of ${\bar{L}}_j^{\prime }$ extending the valuation of $\bar{K}$, and again write $v_j$ for the valuation induced on $L^{\prime }$. As $j$ varies, $v_j$ runs through the set of valuations of $L^{\prime }$ extending the valuation of $K$. Let $G=\mathrm{Gal}(L^{\prime }|K)$, $H=\mathrm{Gal}(L^{\prime }|L)$, $I_j$ be the inertia group of $L^{\prime }|K$ at $v_j$, and $J_j$ the inertia group of $L^{\prime }|L$ at $v_j$. The group $I_j$ is an extension of a cyclic group of order prime to $p$ by a $p$-group $P_j$. Since the ${\bar{L}}_i$ are tamely ramified over $\bar{R}$, $I_j/J_j$ has order prime to $p$, hence $P_j\subset J_j$. Consequently $H$ contains all the $P_j$, and therefore also the group $P$ generated by

the P_j as j varies. But P is invariant in G, because an inner automorphism of G transforms the I_j among themselves and hence also the P_j among themselves. It follows that P is a subgroup of H normal in G; hence, since L′ is the Galois extension generated by L, one has P = 1. This proves that L is tamely ramified over R.

More generally, let R → R′ be a morphism of discrete valuation rings such that the image of a uniformizer π of R is a uniformizer π′ of R′ and such that the residual extension κ(R′) is a separable extension of κ(R). Let K be the field of fractions of R, K′ the field of fractions of R′, L an étale K-algebra, and L′ = L ⊗*K K′. Then L is tamely ramified over R if and only if L′ is tamely ramified over R′. Indeed, one may suppose R and R′ strictly local. By XIII.2.0.1 it is enough to prove that, when L is a field, L′ is also a field. Let R̃ be the normalization of R in L, let π̃ be a uniformizer of R̃, and let R̃′ = R̃ ⊗_R R′. Since the extension κ(R̃)|κ(R) is radicial and the extension κ(R′)|κ(R) is étale, κ(R̃) ⊗*κ(R) κ(R′) is a field [EGA IV 4.3.2 and 4.3.5]. This proves that R̃′ is a local ring; and since π maps to π′ in R′, one has κ(R̃′) = R′/(π̃). Consequently R̃′ is a discrete valuation ring [XIII.5, ch. I, §2, prop. 2], and hence L′ is a field.

2.0.3.

By reduction to the strictly local case, one sees that a subalgebra of a tamely ramified algebra is tamely ramified, that the tensor product of two tamely ramified algebras is tamely ramified, that a tamely ramified algebra remains tamely ramified after extension of the discrete valuation ring, and that an algebra which becomes tamely ramified after a tamely ramified extension is tamely ramified.

2.1.

Let

$X$ be an $S$-scheme, and let $D$ be a divisor $\ge 0$ on $X$. Recall (SGA 5 II 4.2) that $D$ is said to be strictly with normal crossings relative to $S$ if there exists a finite family $(f_i{)}_i\in I$ of elements of $\Gamma (X,{\mathcal{O}}_X)$, such that $D={\sum }_{i\in I}divisor(f_i)$ and the following condition is satisfied:

2.1.0.

For every point $x$ of Supp D, $X$ is smooth over $S$ at $x$, and, if $I(x)$ denotes the set of $i\in I$ such that $f_i(x)=0$, the subscheme $V((f_i{)}_i\in I(x))$ is smooth over $S$ of codimension card I(x) in $X$.

The divisor D is said to be with normal crossings relative to S if, locally on X for the étale topology, it is strictly with normal crossings.

Let D be a divisor with normal crossings relative to S. Put Y = Supp D and U = X − Y, and write i: U → X for the canonical immersion. For every geometric point s̄ of S and every maximal point y of the geometric fiber Y_s̄, the ring R = 𝒪_X_s̄,y is a discrete valuation ring.

In the sequel of this number, we shall use the following technical definition:

Subdefinition.

Let F be a sheaf of sets on U. One says that F is tamely ramified on X, along D, relative to S, if for every geometric point s̄ of S, the following condition is satisfied:

For every maximal point y of Y_s̄, the restriction of F to the field of fractions K of 𝒪_X_s̄,y is representable by the spectrum of an étale K-algebra L, tamely ramified over 𝒪_X_s̄,y.

Most often, when no confusion can result, we shall omit mention of D in the terminology.

Subdefinition.

If

F is a sheaf of groups on U, tamely ramified on X relative to S, we denote by

$$H_{tame}^1(U,F)$$

the subset of H¹(U,F) formed by the classes of torsors under F that are tamely ramified on X relative to S.

Let

U --i→ X
 \    |
  \   f
   g  |
    \ |
      T

be a commutative diagram of S-schemes, with i as in XIII.2.1. We denote by

$$R_{tame}^1{g}_{\ast }F$$

the sheaf on T associated with the presheaf

$$T^{\prime }\mapsto H_{tame}^1(U^{\prime },F),$$

where $T^{\prime }$ ranges over the schemes étale over $T$ and $U^{\prime }=U{\times }_T{T}^{\prime }$. The sheaf $R_{tame}^1{g}_{\ast }F$ is a subsheaf of $R^1{g}_{\ast }F$.

Notice that, if g is coherent, if t̄ is a geometric point of T, T̄ is the strict localization of T at t̄, and Ū = U ×_T T̄, one has an isomorphism

$$(R_{tame}^1{g}_{\ast }F{)}_{\bar{t}}\simeq H_{tame}^1(\bar{U},\bar{F}).$$

2.1.3.

Let C_t((U,X)/S), or simply C_t,

be the category of étale coverings of U that are tamely ramified on X relative to S. Suppose U is connected and let a be a geometric point of U. Let Γ_t be the functor which associates to an étale covering U′ of U, tamely ramified on X relative to S, the set of geometric points of U′ above a. It follows from XIII.2.0 that the pair (C_t,Γ_t) satisfies axioms (G₁) to (G₆) of V.4. Consequently Γ_t is representable by a pro-object called the tamely ramified universal covering

of (U,X) relative to S pointed at a. The group opposite to the group of U-automorphisms of the tamely ramified universal covering is called the tamely ramified fundamental group and is denoted

π₁^tame((U,X)/S,a), or simply π₁^tame(U,a), or even π₁^tame(U).

It is evidently a quotient of the fundamental group π₁(U,a) (V.6.9).

2.1.4.

Let F be a sheaf of groups on U, let P be a right torsor with group F, and let Q be a left torsor with group F. Suppose P and Q are tamely ramified on X relative to S. Then the same is true of P ∧^F Q. Indeed, one reduces to showing that, if R is a discrete valuation ring with field of fractions K, if F is a finite étale group scheme over K, and if P and Q are two torsors under F tamely ramified over R, then P ∧^F Q is also tamely ramified over R. But T = P ∧^F Q is a quotient of P ×_K Q. If L, M, N denote the K-algebras representing T, P, Q respectively, then L is a subalgebra of M ⊗_K N, and it follows from XIII.2.0.3 that L is tamely ramified over R.

From the preceding one deduces that, if F is a sheaf of groups on U and if there exists a torsor with group F tamely ramified on X relative to S, then F is tamely ramified on X relative to S. Indeed, the opposite torsor P° of P is tamely ramified on X relative to S, since it is isomorphic to P as a sheaf of sets. If ^P F is the group obtained by twisting F by P, one has an isomorphism

F ≃ P° ∧^(^P F) P,

and consequently F is tamely ramified on X relative to S.

As before, one sees that, if F → F′ is a morphism of sheaves of groups on U, tamely ramified on X relative to S, and if P is a torsor under F tamely ramified on X relative

to S, then the torsor P′ deduced from P by extension of the structural group F → F′ is tamely ramified on X relative to S.

In particular, the canonical morphism

$$H^1(U,F)\to H^1(U,F^{\prime })$$

restricts to a canonical morphism

$$H_{tame}^1(U,F)\to H_{tame}^1(U,F^{\prime }).$$

2.1.5.

Let S′ → S be a morphism, and write U′, respectively X′, etc., for the inverse image of U, respectively X, etc., on S′. If F is a sheaf of sets on U tamely ramified on X relative to S, it follows from Definition XIII.2.1.1 and from XIII.2.0.3 that F′ is tamely ramified on X′ relative to S′.

If now F is a sheaf of groups on U, the inverse image on S′ of a torsor under F tamely ramified on X relative to S is a torsor under F′ tamely ramified on X′ relative to S′. In particular, one has a canonical functor

$$C_t((U,X)/S)\to C_t((U^{\prime },X^{\prime })/S^{\prime }).$$

Suppose U and U′ are connected, and let a be a geometric point of U and a′ a geometric point of U′ above a. From the preceding one deduces a canonical morphism

$${\pi }_1^{tame}(U^{\prime },a^{\prime })\to {\pi }_1^{tame}(U,a).$$

If S′ → S is a morphism and h: T′ → T is the canonical projection, the morphism

$$h\ast (R^1{g}_{\ast }F)\to R^1{g}_{\ast }^{\prime }{F}^{\prime }$$

restricts to a canonical morphism

h*(R¹_tame g_*F) → R¹_tame g′_*F′.

2.1.6.

Let

F be a sheaf of groups on U, tamely ramified on X relative to S. With the notation of XIII.2.1.2, one has canonical exact sequences:

1 → H¹(X,i_*F) → H¹_tame(U,F) → H⁰(X,R¹_tame i_*F),

1 → R¹f_*(i_*F) → R¹_tame g_*F → f_*(R¹_tame i_*F).

The first is obtained from the exact sequence (SGA 4 III 3.2)

$$1\to H^1(X,i_{\ast }F)\to H^1(U,F)\to H^0(X,R^1{i}_{\ast }F).$$

Indeed, it is enough to show that the image of H¹(X,i**F) in H¹(U,F) is in fact contained in H¹_tame(U,F), and that the image of H¹_tame(U,F) in H⁰(X,R¹i**F) is in fact contained in H⁰(X,R¹tame iF). But the inverse image on U of a torsor under iF is a torsor under ii**F which is evidently tamely ramified on X relative to S; hence the same is true after extension of the structural group ii*F → F. This proves the existence of the arrow H¹(X,i**F) → H¹*tame(U,F). The fact that the image of H¹_tame(U,F) in H⁰(X,R¹iF) is contained in H⁰(X,R¹*tame iF) follows at once from the definition of R¹*tame i**F. This proves the existence of the first exact sequence, and the second is deduced from it by localization.

2.2.

We keep the notation of XIII.2.1. We shall define a notion of tamely ramified object of a stack Φ on U when this stack is given, locally on X and on S for the étale topology, as the inverse image of a stack Ψ on S.

First let G be a gerbe on U, and suppose given a surjective étale morphism S₁ → S, a surjective étale morphism X₂ → X ×_S S₁, a trivial gerbe H on S₁, and an isomorphism

$$G|U_2\to H|U_2,$$

where

U₁ = U ×_X X₁ and U₂ = U ×_X X₂. When one chooses a trivialization of H|X₂, the isomorphism above identifies G|U₂ with the stack of torsors under a sheaf of groups F. One says that an element x of G_U is tamely ramified on X relative to S if the restriction of x to U₂ is a torsor tamely ramified on X relative to S. By XIII.2.1.4, this notion does not depend on the way in which H|X₂ has been trivialized.

Now let Φ be a stack on U, and suppose given a surjective étale morphism S₁ → S, a surjective étale morphism X₂ → X ×_S S₁, a stack Ψ on S₁, and an isomorphism

$$i:\Phi |U_2\to \Psi |U_2.$$

Let x be an element of Φ_U, let G_x be the maximal subgerbe of Φ generated by x [XIII.1, III 2.1.7], and let SΦ be the sheaf of maximal subgerbes of Φ. The isomorphism i induces an isomorphism

$$S\Phi |U_2\to S\Psi |U_2.$$

It follows from XIII.5.7 that, after replacing S₁ by a surjective étale extension if necessary, one has a unique maximal subgerbe H of Ψ, which may be supposed trivial, such that i defines an isomorphism

$$G_x|U_2\to H|U_2.$$

One says that the element x is tamely ramified on X relative to S if it is so as an element of G_x endowed with the isomorphism above.

2.2.1.

Let Φ be a stack on U given, locally on X and S, as the inverse image of a stack on S, and let

U --i→ X
 \    |
  \   f
   g  |
    \ |
      T

be a diagram as in XIII.2.1.2. For every scheme $T^{\prime }$ étale over $T$, if $U^{\prime }=U{\times }_T{T}^{\prime }$, consider the subset $(g_{\ast }^{tame}\Phi {)}_T^{\prime }$

of $(g_{\ast }\Phi {)}_T^{\prime }={\Phi }_U^{\prime }$ formed

by the elements of Φ_U′ which are tamely ramified on X relative to S. The tamely ramified direct image of Φ by g, denoted

$$g_{\ast }^{tame}\Phi ,$$

is the full subcategory of $g_{\ast }\Phi$ whose objects over a scheme $T^{\prime }$ étale over $T$ are the elements of $(g_{\ast }^{tame}\Phi {)}_T^{\prime }$. It is clear that $g_{\ast }^{tame}\Phi$ is a substack of $g_{\ast }\Phi$.

2.2.2.

By reduction to the case of a stack of torsors, one sees that, if Φ is a stack on U which is locally for the étale topology of S and X the inverse image of a stack on S, the canonical morphism

$$h\ast (g_{\ast }\Phi )\to g_{\ast }^{\prime }{\Phi }^{\prime }$$

restricts to a canonical morphism

h*(g_*^tame Φ) → g′_*^tame Φ′.

Remarks.

a. If F is a locally constant constructible sheaf of sets on U, then for F to be tamely ramified on X relative to S, it is enough that the condition of XIII.2.1.1 be satisfied for the geometric points of S above the maximal points of S. To see this, one may suppose the divisor D strictly with normal crossings. The sheaf F is representable by an étale covering V of U. If s̄ is a geometric point of S and y is a maximal point of Y_s̄, write S̄ for the strict localization of S at s̄, X̄ for the strict localization of X at ȳ, Ū = U ×_X X̄, and V̄ = V ×_X X̄. If the condition of XIII.2.1.1 is satisfied at the geometric points above the maximal points of S, it follows from XIII.5.5 below that V̄ is a covering of Ū tamely ramified on X̄ relative to S̄. Consequently V is an étale covering of U tamely ramified on X relative to S.

b.

Let F be a sheaf of groups on U tamely ramified on X relative to S. If s̄ is a geometric point of S and y is a maximal point of Y_s̄, write K for the field of fractions of 𝒪_X_s̄,y. Suppose that, for every s̄ and every y, the K-algebra L whose spectrum represents F|K has rank prime to the residual characteristic p of 𝒪_X_s̄. We shall sometimes say, by abuse of language, that F is prime to the residual characteristics of S. When this holds, every torsor P under F is tamely ramified on X relative to S.

Indeed, let R̄ be the strict localization of 𝒪_X_s̄,y at ȳ, let K̄ be its field of fractions, and let F̄ be the inverse image of F on K̄. Let us show that one may suppose F constant. Since F is tamely ramified on X relative to S, F̄ is representable by the spectrum of a K̄-algebra L = ∏ L_i, where the L_i are extensions of K̄ of degree prime to p. One can therefore find an extension K′ of K̄ of degree prime to p such that F̄|K′ is a constant sheaf. By XIII.2.0.3, to prove that P|K̄ is tamely ramified over R̄, it is enough to see that P|K′ is tamely ramified over the integral closure of R̄ in K′; this gives the reduction to the case where F̄ is constant. Suppose from now on that F̄ is constant. The K̄-algebra H representing P|K̄ is then a product of mutually isomorphic extensions H_i of K̄. Since the rank of H is prime to p, so is [H₁:K̄], which proves that H is tamely ramified on X relative to S.

c. Let X be a regular scheme, let D be a divisor with normal crossings on X (SGA 5 I 3.1.5), let U = X − Supp D, and let F be a sheaf of sets on U. If y is a maximal point of Supp D, write K for the field of fractions of 𝒪_X,y. One says that F is tamely ramified relative to D if, for every maximal point y of Supp D, F|K is representable by a K-algebra tamely ramified over 𝒪_X,y.

Theorem.

Let

f: X → S be an S-scheme, let D be a divisor on X with normal crossings relative to S (XIII.2.1), let Y = Supp D, let U = X − Y, and let i: U → X be the canonical immersion. Let F be a sheaf of sets, respectively of groups, on U, satisfying one of the following conditions:

a. F is locally for the étale topology on X and on S the inverse image of a sheaf of sets, respectively a constructible sheaf of groups, on S.

b. F is locally constant constructible on U and tamely ramified on X relative to S.

Then the following conclusions hold:

1. $(F,i)$ is cohomologically proper relative to $S$ in dimension ≤ 0; respectively, for every morphism $h:S^{\prime }\to S$, if $i^{\prime }:U^{\prime }\to X^{\prime }$ is the inverse image of $i$ on $S^{\prime }$, if $F^{\prime }=F|U^{\prime }$, and if $k=h_X$, the canonical morphism

Ψ: k*(R¹_tame i_*F) → R¹_tame i′_*F′

is an isomorphism.

If F is a sheaf of groups prime to the residual characteristics of S (XIII.2.3 b), tamely ramified on X relative to S, then (F,i) is cohomologically proper relative to S in dimension ≤ 1.

1. If F is a constructible sheaf of sets, respectively of groups, then i**F, respectively R¹_tame i**F, is constructible.

Proof. For every S-scheme S′, consider the following diagram, all of whose squares are cartesian:

U′ → U
|     |
i′    i
|     |
X′ → X
|     |
S′ → S.

Since

the question is local on X for the étale topology, one may suppose that D is a divisor strictly with normal crossings relative to S (XIII.2.1). Moreover, after restricting X to a neighborhood of Y if necessary, one may suppose X smooth over S.

Proof of XIII.2.4 1.

2.4.1.

Case of a sheaf of sets satisfying a. We may suppose that F = g*G, where G is a sheaf on S. It then follows from SGA 4 XVI 3.2 that the canonical morphism

$$f\ast G\to i_{\ast }F$$

is an isomorphism. For every $S$-scheme $h:S^{\prime }\to S$, one likewise has an isomorphism $f^{\prime }\ast G^{\prime }\to i_{\ast }^{\prime }{F}^{\prime }$; consequently the canonical morphism

$$\varphi :k\ast (i_{\ast }F)\to i_{\ast }^{\prime }{F}^{\prime }$$

identifies with the natural isomorphism

$$k\ast f\ast G\simeq f^{\prime }\ast G^{\prime }.$$

2.4.2.

Case of a sheaf of sets satisfying b. We must show that φ is an isomorphism, and it is enough to check this at every geometric point x̄′ of X′. Let S̄, respectively X̄, S̄′, X̄′, be the strict localization of S, respectively X, S′, X′, at x̄′, and put Ū = U*(X̄), Ū′ = U*(X̄′), etc. The morphism φ_x̄ identifies with the canonical morphism

$$\bar{\varphi }:H^0(\bar{U},\bar{F})\to H^0({\bar{U}}^{\prime },{\bar{F}}^{\prime }).$$

One can find a principal covering V of Ū of the type occurring in XIII.5.4 such that the inverse image of F̄ on V is a constant sheaf with value C. If Π is the Galois group of V over Ū, Π acts on F̄|V, and one has

$$H^0(\bar{U},\bar{F})\simeq H^0(V,C_V{)}^{\Pi },$$

where the second member denotes the set of elements of H⁰(V,C_V) invariant

under Π. Since V′ = V ×_Ū Ū′ is a principal covering of Ū′ with Galois group Π′ ≃ Π, one sees that the morphism φ̄ is obtained, by taking invariants under Π, from the canonical morphism

$$H^0(V,C_V)\to H^0(V^{\prime },C_V^{\prime }).$$

Since V and V′ are connected (XIII.5.4), this morphism, and hence also φ̄, is an isomorphism.

Notice that if moreover F is a sheaf of groups and if P is a torsor on Ū with group F̄, tamely ramified relative to D, it follows from the preceding proof and from XIII.2.2 that (^P F̄,ī) is cohomologically proper relative to S in dimension ≤ 0.

2.4.3.

Case of a sheaf of groups. To show that Ψ is an isomorphism, it is enough to prove that, for every geometric point ȳ′ of Y′, the morphism

Ψ_ȳ′: (k*(R¹_tame i_*F))_ȳ′ → (R¹_tame i′_*F′)_ȳ′

is an isomorphism. But, by XIII.2.1.2, Ψ_ȳ′ identifies with the canonical morphism

$$\bar{\Psi }:H_{tame}^1(\bar{U},\bar{F})\to H_{tame}^1({\bar{U}}^{\prime },{\bar{F}}^{\prime }).$$

Let Ũ be the tamely ramified universal covering of Ū (XIII.2.1.3), and let F̃ be the inverse image of F̄ on Ũ. It follows from XIII.5.7 in case a and from XIII.5.5 in case b that H¹_tame(Ū,F̄) identifies with the subset of H¹(Ū,F̄) formed by the classes of F̄-torsors whose inverse image on Ũ is trivial. On the other hand, a classical argument (cf. IX.5, p. 300) shows that the set of elements of H¹(Ū,F̄) whose inverse image on Ũ is trivial identifies with

$$H^1({\pi }_1^{tame}(\bar{U}),H^0(U,F)).$$

Thus one obtains a canonical isomorphism

$$H_{tame}^1(\bar{U},\bar{F})\simeq H^1({\pi }_1^{tame}(\bar{U}),H^0(U,F)).$$

Consequently the morphism Ψ̄ identifies with the canonical morphism

$$H^1({\pi }_1^{tame}(\bar{U}),H^0(U,F))\to H^1({\pi }_1^{tame}({\bar{U}}^{\prime }),H^0(U^{\prime },F^{\prime })).$$

Let us show

that this morphism is an isomorphism. The morphism π₁^tame(Ū′) → π₁^tame(Ū) is an isomorphism by XIII.5.6, and the same is true of the morphism H⁰(Ũ,F̃) → H⁰(Ũ′,F̃′). Indeed, this is evident in case b, since F̃ is constant and Ũ and Ũ′ are connected. In case a, let Ḡ be a constructible sheaf of groups on S̄ such that F̄ = ḡ*Ḡ. Since the morphisms Ũ → S̄ and Ũ′ → S̄′ are 0-acyclic (XIII.5.7), one has

H⁰(Ũ,F̃) ≃ H⁰(S̄,Ḡ) ≃ H⁰(S̄′,Ḡ′) ≃ H⁰(Ũ′,F̃′),

which implies that Ψ is an isomorphism. The last assertion of XIII.2.4 1 follows from the preceding, taking XIII.2.3 b into account.

Proof of XIII.2.4 2.

The case of a constructible sheaf of sets satisfying a follows at once from (XIII.2.4.1.1). Let $F=g\ast G$ be a sheaf of groups satisfying a, where $G$ is a constructible sheaf. We may suppose $S$ affine. Let $(S_j{)}_j\in J$ be a finite family of reduced closed subschemes of $S$ whose union covers $S$, such that the inverse image of $G$ on $S_j$ is locally constant. Taking XIII.2.4 1 into account, to establish that $R_{tame}^1{i}_{\ast }F$ is constructible, it is enough to see that it becomes so after the base change $S_j\to S$ for each $j\in J$. We are therefore reduced to case b, where $F$ is locally constant.

From now on suppose that F is a sheaf of sets or groups satisfying b. Since the question is local for the étale topology on X, one may suppose X of finite presentation over S, and, by passage to the limit, one may suppose X and S noetherian.

Let $D={\sum }_{1\le i\le r}divisor{f}_i$, where, for each point $x$ of Supp D, if $I(x)$ is the set of $i$ such that $f_i(x)=0$, the subscheme $V((f_i{)}_i\in I(x))$ is smooth over $S$ of codimension card I(x) in $X$. Let $\mathcal{P}$ be the set of subsets of [1,r], and for each $I\in \mathcal{P}$ put

X_I = (⋂_{i∈I} V(f_i)) ∩ (⋂_{i∉I} X_{f_i}).

Let

z be a point of X_I. After first restricting to an étale neighborhood of z, one can find an open W of X containing z and a principal covering V of U ∩ W, tamely ramified on W relative to S, of the type considered in XIII.5.6.1, such that the inverse image of F on V is a constant sheaf with value C. Let π be the Galois group of the covering V. For every geometric point x̄ of X_I, one then has, by (XIII.2.4.2.1),

$$H^0(\bar{U},\bar{F})\simeq H^0(V,C_V{)}^{\pi }.$$

It follows that i**F|X_I∩W is locally constant (SGA 4 IX 2.13), and hence that i**F is constructible.

Finally let us show that if F is a locally constant sheaf of groups, R¹tame i*F|X_I is constructible. If x̄ is a geometric point of X, we obtained in (XIII.2.4.3.1) the expression

$$H_{tame}^1(\bar{U},\bar{F})\simeq H^1({\pi }_1^{tame}(\bar{U}),H^0(U,F)).$$

If p is the residual characteristic of X̄, then, by XIII.5.6,

π₁^tame(Ū) = ∏_{ℓ≠p} ℤ_ℓ(1)^card I.

Let ℒ be the set of prime numbers dividing the order of the finite group H⁰(Ũ,F̃), and let

K = ∏_{ℓ∈ℒ, ℓ≠p} ℤ_ℓ(1)^card I.

It follows from [XIII.4, I §5 ex. 2] that

$$H_{tame}^1(\bar{U},\bar{F})\simeq H^1(K,H^0(U,F)).$$

Since K is topologically of finite type and H⁰(Ũ,F̃) is finite, it follows first that the fibers of the sheaf R¹tame i**F|XI are finite. On the other hand, the set ℒ does not depend on the point x̄. For every q ∈ ℒ, let X_I,q be the closed subset of X_I with equation q = 0, and let X_I′ be the open subset of X_I complementary to the union of the X_I,q. Then R¹_tame i*F|XI,q and R¹_tame i**F|X_I′ are locally constant: indeed, a specialization arrow of geometric points of X_I,q, respectively of X_I′, induces an isomorphism on the groups K (XIII.5.6.1), hence also on the sets H¹_tame(Ū,F̄), and one may apply SGA 4 IX 2.13.

Corollary.

Let

f: X → S be a morphism, let D be a divisor on X with normal crossings relative to S (XIII.2.1), let Y = Supp D, let U = X − Y, and let i: U → X be the canonical immersion. Let Φ be a stack on U and suppose given surjective étale morphisms S₁ → S and X₂ → X ×_S S₁, a stack Ψ on S₁, and an isomorphism Φ|U₂ ≃ Ψ|U₂; cf. XIII.2.2.

Then, for every morphism $h:S^{\prime }\to S$, if $k=h_X$, the canonical functor

$$\varphi :k\ast i_{\ast }\Phi \to i_{\ast }^{\prime }{\Phi }^{\prime }$$

is fully faithful. If Ψ is constructible, the canonical functor

ψ: k*i_*^tame Φ → i′_*^tame Φ′

is an equivalence of categories.

Moreover, if the stack Ψ is constructible, respectively if Ψ is 1-constructible (XIII.0) and S is locally noetherian, then i**Φ is constructible, respectively i**^tame Φ is 1-constructible.

Let us show that φ is fully faithful. It is enough to see that, for every geometric point x̄′ of X′, the same is true of the functor

$$\bar{\varphi }:\Phi (\bar{U})\to {\Phi }^{\prime }({\bar{U}}^{\prime }),$$

where we have resumed the notation of XIII.2.4.2. Let a, b be two elements of Φ(Ū), and let a′, b′ be their images by φ̄. Since the morphism Ū → S̄ is locally 0-acyclic (XIII.5.7), one has an isomorphism

$$H^0(\bar{U},S\bar{\Phi })\simeq H^0(\bar{S},S\bar{\Psi }).$$

Consequently a and b come by inverse image from elements of Ψ, and the same is therefore true of F = SheafHom_Ū(a,b). Since the sheaf F′ = SheafHom_Ū′(a′,b′) is the inverse image of F on Ū′, it follows from XIII.2.4.1 that the canonical morphism

$$H^0(\bar{U},F)\to H^0({\bar{U}}^{\prime },F^{\prime })$$

is an isomorphism, which proves that φ̄ is fully faithful.

Let us show

that, for every geometric point x̄′ of X′, the functor

ψ̄: i_*^tame Φ(X̄′) → i′_*^tame Φ′(X̄′)

is an equivalence. By what precedes, ψ̄ is fully faithful. Let us show that ψ̄ is essentially surjective. Let a′ be an element of Φ′(Ū′) tamely ramified on X′ relative to S′ (XIII.2.2), and let us show that it is the image of a tamely ramified element of Φ(Ū). It follows from XIII.2.4 1 that the canonical morphism

$$H^0(\bar{U},S\bar{\Phi })\to H^0({\bar{U}}^{\prime },S{\bar{\Phi }}^{\prime })$$

is an isomorphism. Let G′ be the maximal subgerbe of Φ′ generated by a′. There exists a maximal subgerbe G of Φ̄, inverse image of a gerbe on S̄, such that

$$\bar{m}\ast G\simeq G^{\prime },$$

where m is the morphism Ū′ → Ū. The canonical functor

(*)   k̄*ī_*^tame G → ī′_*^tame G′

is an equivalence, because G identifies with a gerbe of torsors under a constructible sheaf of groups coming from S̄, and one can apply XIII.2.4 1. It then follows from (*) that there exists an element a of G(Ū), tamely ramified on X relative to S, whose inverse image on Ū′ is a′. This proves that ψ is an equivalence.

If $\Psi$ is constructible, then so is $i_{\ast }\Phi$. Indeed, an object $x$ of $i_{\ast }\Phi$ is, locally for the étale topology of $S$, the inverse image of an object $y$ of $\Psi$; it therefore follows from (XIII.2.4.1.1) that $SheafAut(x)$ is the inverse image of $SheafAut(y)$, and hence is constructible. Finally, if $\Psi$ is 1-constructible, then $i_{\ast }^{tame}\Phi$ is also 1-constructible by XIII.6.3 below.

Corollary.

The

notation is that of XIII.2.4. Suppose that S has characteristic zero at every point s such that Y_s ≠ ∅. Then, if 𝓕 is a locally constant constructible sheaf of groups on U, respectively a constructible stack on U which is locally on X and S the inverse image of a constructible stack on S, the pair (𝓕,i) is cohomologically proper relative to S in dimension ≤ 1.

Since every constructible sheaf of sets on U is tamely ramified on X relative to S, the corollary follows from XIII.2.4, respectively XIII.2.5.

Corollary.

The notation is that of XIII.2.4, but in addition one is given an S-scheme T and a proper morphism p: X → T, and X and T are assumed of finite presentation over S. Let q = pi. Let 𝓕 be a constructible sheaf of sets on U satisfying one of conditions a or b of XIII.2.4, respectively a sheaf of groups satisfying one of conditions a, b of XIII.2.4, respectively a stack on U which is locally on X and S the inverse image of a constructible stack G on S. Then the following conclusions hold:

1. $(\mathcal{F},q)$ is cohomologically proper relative to $S$ in dimension ≤ 0; respectively, for every morphism $h:S^{\prime }\to S$, if $m=h_T$, the canonical morphism

Θ: m*(R¹_tame q_*𝓕) → R¹_tame q′_*𝓕′

is an isomorphism; respectively, for every morphism S′ → S, the canonical morphism

ξ: m*(q_*^tame 𝓕) → q′_*^tame 𝓕′

is an equivalence.

1. The sheaf $q_{\ast }\mathcal{F}$, respectively the sheaf $R_{tame}^1{q}_{\ast }\mathcal{F}$, respectively the stack $q_{\ast }^{tame}\mathcal{F}$, is constructible. In the last case, if $S$ is assumed locally noetherian and $G$ is 1-constructible, then $q_{\ast }^{tame}\mathcal{F}$ is also 1-constructible.

The first part follows at once from XIII.2.4, XIII.2.5, and the proof of XIII.1.8. Let us prove 2. If $\mathcal{F}$ is a constructible sheaf of sets on $U$ satisfying XIII.2.4 a or XIII.2.4 b, it follows from XIII.2.4 2 that $i_{\ast }\mathcal{F}$ is constructible; hence the same is true of $q_{\ast }\mathcal{F}=p_{\ast }(i_{\ast }\mathcal{F})$ (SGA 4 XIV 1.1).

Let 𝓕 be a constructible sheaf of groups on U satisfying XIII.2.4 a or XIII.2.4 b, and let us prove that R¹tame q*𝓕 is constructible. By passage to the limit (EGA IV 8.10.5 and 17.7.8) and using 1, one may suppose S noetherian. Let Φ be the stack on X whose fiber over every scheme X′ étale over X is formed by the torsors on U′ = U ×_X X′, with group 𝓕|U, that are tamely ramified on X relative to S. Thus one has

S(i_*^tame Φ) ≃ R¹_tame i_*𝓕,

and this sheaf is constructible by XIII.2.4 2. [Translator note: the source correction says the reference here was wrong in the older text.] It follows from XIII.6.3 below that S(p**Φ) is constructible, that is, that R¹_tame q**𝓕 is constructible.

Finally, if $\mathcal{F}$ is a stack on $U$ which is locally on $X$ and $S$ the inverse image of a constructible stack on $S$, then $i_{\ast }^{tame}\mathcal{F}$ is constructible, and hence so is $q_{\ast }^{tame}\mathcal{F}=p_{\ast }(i_{\ast }^{tame}\mathcal{F})$. If in addition $S$ is locally noetherian and SG is constructible, then $S(i_{\ast }^{tame}\mathcal{F})$ is constructible by XIII.6.3; hence the same is true of $S(q_{\ast }{i}_{\ast }^{tame}\mathcal{F})$, that is, of $S(q_{\ast }^{tame}\mathcal{F})$, by XIII.6.2.

Corollary.

Let

U --i→ X
 \    |
  \   f
   g  |
    \ |
      S

be a commutative diagram of schemes in which U is the open complement

in X of a divisor with normal crossings relative to S, and f is a proper morphism of finite presentation. Let ℒ be a set of prime numbers. Suppose g is locally 0-acyclic, respectively locally 1-aspherical for ℒ. Then, if 𝓕 is a sheaf of sets on U, respectively a sheaf of ℒ-groups on U, locally constant constructible and tamely ramified on X relative to S, f**𝓕, respectively R¹_tame f**𝓕, is locally constant constructible and (𝓕,f) is cohomologically proper relative to S in dimension ≤ 0; respectively, the formation of R¹*tame f**𝓕 commutes with every base change S′ → S. In the non-respective case, if 𝓕 is a sheaf of groups, then for every specialization s̄₁ → s̄₂ of geometric points of S, the specialization morphism

(R¹_tame f_*𝓕)_s̄₂ → (R¹_tame f_*𝓕)_s̄₁

is injective.

The corollary follows at once from XIII.2.6 and XIII.1.14, respectively from the analogue of XIII.1.14 for R¹tame f*𝓕, which is proved as in loc. cit.

Corollary.

Let

U --i→ X
 \    |
  \   f
   g  |
    \ |
      S

be a commutative diagram of schemes in which $U$ is the open complement in $X$ of a divisor with normal crossings relative to $S$, and $f$ is a proper smooth morphism of finite presentation. Let $\mathcal{L}$ be the set of prime numbers distinct from the residual characteristics of $S$. Let $\mathcal{F}$ be a locally constant constructible sheaf of $\mathcal{L}$-groups on $U$, tamely ramified on $X$ relative to $S$. Then $R^1{f}_{\ast }\mathcal{F}$ is locally constant constructible and $(\mathcal{F},f)$ is cohomologically proper relative

to S in dimension ≤ 1.

The corollary follows from XIII.2.8 and from the fact that $R_{tame}^1{f}_{\ast }\mathcal{F}=R^1{f}_{\ast }\mathcal{F}$ (XIII.2.3 b).

2.10.

If U is a connected scheme, a is a geometric point of U, and ℒ is a set of prime numbers, write

$${\pi }_1^{\mathcal{L}}(U,a)$$

for the projective limit of the finite quotients of π₁(U,a) whose orders have all their prime factors in ℒ.

We shall define specialization morphisms for the fundamental group, generalizing X.2.

Let g: U → S be a coherent morphism with geometrically connected fibers; respectively, let g be of the form g = fi, where f: X → S is a proper morphism of finite presentation and i: U → X is an open immersion such that U is the complement in X of a divisor with normal crossings relative to S (cf. XIII.2.8). Let ℒ be a set of prime numbers and suppose, in the non-respective case, that for every finite constant ℒ-group C, the pair (C_U,g) is cohomologically proper relative to S in dimension ≤ 1. Let s̄₁ → s̄₂ be a specialization morphism of geometric points of S, let S̄ be the strict localization of S at s̄₂, and let Ū = U ×_S S̄. One has a commutative diagram

U_s̄₁ → Ū ← U_s̄₂
  |      |     |
 s̄₁  → S̄ ← s̄₂.

If a₁ is a geometric point of U_s̄₁ and a₂ a geometric point of U_s̄₂, the two morphisms define canonical morphisms

π₁: π₁^ℒ(U_s̄₁,a₁) → π₁^ℒ(Ū,a₁),
π₂: π₁^ℒ(U_s̄₂,a₂) → π₁^ℒ(Ū,a₂),

respectively

π₁: π₁^tame(U_s̄₁,a₁) → π₁^tame(Ū,a₁),
π₂: π₁^tame(U_s̄₂,a₂) → π₁^tame(Ū,a₂).

See V.7 and (XIII.2.1.5.2). The hypotheses of cohomological properness, respectively XIII.2.8, prove that π₂ is an isomorphism. If one chooses a path class from a₁ to a₂, one obtains an isomorphism

$${\pi }_{12}:{\pi }_1^{\mathcal{L}}(\bar{U},a_1)\simeq {\pi }_1^{\mathcal{L}}(\bar{U},a_2),$$

respectively

$${\pi }_{12}:{\pi }_1^{tame}(\bar{U},a_1)\simeq {\pi }_1^{tame}(\bar{U},a_2),$$

[Translator note: the corrected source notes that the last isomorphism had been written only as a morphism.] hence a morphism π = π₂⁻¹π₁₂π₁:

π: π₁^ℒ(U_s̄₁,a₁) → π₁^ℒ(U_s̄₂,a₂),

respectively

π: π₁^tame(U_s̄₁,a₁) → π₁^tame(U_s̄₂,a₂).

Changing the path class from a₁ to a₂ modifies π by an inner automorphism of π₁^ℒ(X_s̄₂,a₂), respectively of π₁^tame(X_s̄₂,a₂). One calls one of the morphisms defined above the specialization morphism for the fundamental group associated with the morphism s̄₁ → s̄₂, and writes simply

π: π₁^ℒ(X_s̄₁) → π₁^ℒ(X_s̄₂),

respectively

π: π₁^tame(X_s̄₁) → π₁^tame(X_s̄₂).

Lemma.

Let f: X → S be a proper morphism of finite presentation, let D be a divisor on X with normal crossings relative to S, let Y = Supp D, let U = X − Y, let i: U → X be the canonical morphism, let s̄₁ → s̄₂ be a specialization morphism of geometric points of S, and let y₁ be a geometric point of Y_s̄₁ and y₂ a geometric point of Y_s̄₂ such that the projection z₁ of y₁ on X is a generization of the projection z₂ of y₂. Let I_y₁ be an inertia subgroup of π₁^tame(U_s̄₁) at y₁. Then the image of I_y₁ by the specialization morphism

π: π₁^tame(U_s̄₁) → π₁^tame(U_s̄₂)

is

an inertia subgroup of π₁^tame(U_s̄₂) at y₂.

Indeed, let X̄, respectively X̃, be the strict localization of X at y₂, respectively at y₁, and let Ū = U ×_X X̄, respectively Ũ = U ×_X X̃. There is a canonical morphism Ũ → Ū, and it follows from XIII.1.10 that one has a commutative diagram

π₁^tame(Ũ_s̄₁) → π₁^tame(Ū_s̄₂)
      ↓                 ↓
π₁^tame(U_s̄₁) → π₁^tame(U_s̄₂),

where the upper horizontal morphism π′ is the composite of the canonical morphism π₁^tame(Ũ_s̄₁) → π₁^tame(Ū_s̄₁) and the specialization morphism. Since π₁^tame(Ũ_s̄₁), respectively π₁^tame(Ū_s̄₁), is an inertia group of π₁^tame(U_s̄₁) at y₁, respectively of π₁^tame(U_s̄₂) at y₂, it is enough to prove that π′ is surjective. This follows from the expression obtained in XIII.5.6.

Corollary.

Let X be a connected proper smooth curve of genus g over a separably closed field k of characteristic p ≥ 0. Let U be the open subset obtained by removing from X n distinct closed points a₁, …, a_n. Then the tamely ramified fundamental group π₁^tame(U) (XIII.2.1.3) can be generated by 2g + n elements x_i, y_i, σ_j, with 1 ≤ i ≤ g and 1 ≤ j ≤ n, such that σ_j is a generator of an inertia group corresponding to a_j and one has the relation

(*)   ∏_{1≤i≤g}(x_i y_i x_i⁻¹ y_i⁻¹) · ∏_{1≤j≤n} σ_j = 1.

For every finite group G of order prime to p, generated by elements x̄_i, ȳ_i, σ̄_j satisfying relation (*), there exists an étale covering of U with group G, corresponding to a homomorphism π₁^tame(U) → G which

sends x_i, y_i, σ_j to x̄_i, ȳ_i, σ̄_j respectively. In other words, if p′ denotes the set of prime numbers distinct from p, π₁^p′(U) is the pro-p′ group generated by the generators x_i, y_i, σ_j subject to the single relation (*).

Proof. We may suppose $k$ algebraically closed. First suppose $k$ has characteristic zero. There then exists an algebraically closed subextension $k^{\prime }$ of $k$, of finite transcendence degree over $\mathbb{Q}$, such that $X$ comes by extension of scalars from a proper smooth curve $X^{\prime }$ defined over $k^{\prime }$, and one may suppose that the points $a_1,\cdots ,a_n$ come from rational points $a_1^{\prime },\cdots ,a_n^{\prime }$ of $X^{\prime }$. Since $k^{\prime }$ has finite transcendence degree over $\mathbb{Q}$, one can find an embedding of $k^{\prime }$ into the field of complex numbers $\mathbb{C}$. Let $\tilde{U}=U^{\prime }{\times }_k^{\prime }\mathbb{C}$. Let $k^{\prime \prime }$ be an algebraically closed extension of $k^{\prime }$ such that there are $k^{\prime }$-morphisms from $k$ and from $\mathbb{C}$ to $k^{\prime \prime }$. If $g^{\prime }:U^{\prime }\to k^{\prime }$ is the structural morphism, and if $\mathcal{F}$ is a finite constant sheaf of groups on $U^{\prime \prime }$, it follows from XIII.2.9 that the specialization morphisms

$$(R^1{g}_{\ast }^{\prime }{\mathcal{F}}^{\prime }{)}_{\mathbb{C}}\to (R^1{g}_{\ast }^{\prime }{\mathcal{F}}^{\prime }{)}_k^{\prime \prime }\leftarrow (R^1{g}_{\ast }^{\prime }{\mathcal{F}}^{\prime }{)}_k$$

are isomorphisms. In terms of fundamental groups, this shows that one has an isomorphism, defined up to inner automorphism,

$${\pi }_1(U)\to {\pi }_1(\tilde{U}),$$

and it is clear that this isomorphism transforms an inertia group relative to a point of X′ − U′ into an inertia group relative to the same point. We may therefore suppose k = ℂ. In this last case, it follows from the Riemann existence theorem (XII.5.2) that the fundamental group π₁(U) is nothing other than the completion, for the topology of subgroups of finite index, of the fundamental group of the analytic space associated with U. But the latter can be computed transcendently [XIII.3, ch. 7, §47]:

it can be generated by 2g + n elements x_i, y_i, σ_j such that σ_j is the image of a generator of the local fundamental group π₁(D_j) of a small disk centered at a_j, that is, a generator of an inertia group corresponding to the point a_j, these elements satisfying the single relation (*).

Now suppose k has characteristic p > 0. One can find a complete discrete valuation ring A, with residue field k and field of fractions K of characteristic zero, and a connected scheme X₁, proper and smooth over S = Spec A, such that X₁ ×_S Spec k ≃ X (III.7.4). The points a_j then lift to sections s_j of X₁ over S. Let Y₁j be the reduced closed subscheme with underlying space s_j(S), let Y₁ be the union of the Y₁j, let U₁ = X₁ − Y₁, and let g₁: U₁ → S be the structural morphism. Let K̄ be an algebraically closed extension of K and let Ū = U₁ ×_S K̄. If C is a finite constant group, it follows from XIII.2.8 that the specialization morphism

(R¹g₁*C_U₁)_k → (R¹g₁*C_U₁)_K̄

is injective, and even bijective if C has order prime to p. In terms of fundamental groups, this means that the specialization morphism (XIII.1.10)

$$\pi :{\pi }_1(\bar{U})\to {\pi }_1^{tame}(U)$$

is surjective, and that the specialization morphism

$${{\pi }_1^p}^{\prime }(\bar{U})\to {{\pi }_1^p}^{\prime }(U)$$

is bijective. Finally, if x_i, y_i, σ_j are generators of π₁(Ū) such that σ_j is a generator of an inertia group corresponding to the point b_j = Y₁j(K̄) of X̄, then by XIII.1.11, π(σ_j) is a generator of an inertia group corresponding to a_j. This completes the proof.

Remark (M. Raynaud, added in 2003).

Let k be an algebraically closed field of characteristic p > 0, let X be a connected proper smooth algebraic curve over k of genus g, and let U be an affine open of X, the complement of r ≥ 1 rational points of X. One has the fundamental group π₁(U), its quotient π₁^tame(U), which classifies finite étale coverings of U tamely ramified at the points of X − U, and the quotient π₁^p′(U) of π₁^tame(U), which classifies Galois étale coverings of U with Galois group of order prime to p.

In “Coverings of algebraic curves,” Amer. J. Math. 79 (1957), pp. 825-856, S. Abhyankar formulated a number of conjectures on the structure of the finite groups that occur as Galois groups of finite étale coverings of U.

The conjectures concerning the finite groups which are Galois groups of connected étale coverings of U of order prime to p are proved and made precise in Corollary XIII.2.12. Before turning to the finite quotients of π₁(U), let us begin by giving some indications about the “size” of this group. Let X = Spec(A). One knows that Hom_cont(π₁(U),ℤ/pℤ) is described by Artin-Schreier theory (Corollary XI.6.9). This group is isomorphic to A/wp(A), where wp is the map from A to A sending a to aᵖ − a. First suppose that U is the affine line 𝔸¹, with ring k[T]. Let E be the set of elements of k[T] of the form ∑_i a_iT^i, with a_i = 0 when p divides i. It is immediate that the composite map E → k[T] → k[T]/wp k[T] is bijective. Hence the coefficients a_i, with (i,p) = 1, behave like coordinates of a space parametrizing cyclic degree-p coverings of the affine line. In particular, one deduces that π₁(𝔸¹) is not topologically of finite type and that, if one makes an extension k → k′ of algebraically closed fields, the natural morphism π₁(𝔸¹ ×_k k′) → π₁(𝔸¹), which is surjective, is not bijective, unlike the proper case. In the general case, the curve U can be realized as a finite scheme over the affine line, and one concludes that the same phenomena occur for π₁(U), and indeed more generally for π₁(V) for every connected affine k-scheme V of finite type and dimension ≥ 1.

With this said, if G is a finite group, write G^(p′) for the largest quotient group of G of order prime to p. For a finite group G to be a topological quotient of π₁(U), it is necessary that G^(p′) be a topological quotient of π₁^p′(U), a condition one knows in principle how to answer by Corollary XIII.2.12. In the article cited above, Abhyankar conjectures that this necessary condition is also sufficient. This conjecture was proved by M. Raynaud in the case of the affine line and by D. Harbater in the general case (Invent. Math. 116 (1994), pp. 425-462, and 117, pp. 1-25). For example, in the case of the affine line, π₁^p′(𝔸¹) = 1, and one concludes that a finite group G is the Galois group of a connected étale covering of 𝔸¹ if and only if G^(p′) = 1, that is, if and only if G is generated by its Sylow p-subgroups. Thus every finite simple group whose order is a multiple of p is suitable.

3. Cohomological Properness and Generic Local Acyclicity

Theorem.

Let S be an irreducible scheme with generic point s, let X and Y be two S-schemes of finite presentation, and let f: X → Y be an S-morphism. For every S-scheme S′, write Y′, X′, etc. for the inverse images of Y, X, etc. by the morphism S′ → S. The following properties hold:

1. a. One can find a nonempty open subset $S^{\prime }$ of $S$ such that, for every finite constant sheaf of sets $F^{\prime }$ on $X^{\prime }$, $f_{\ast }^{\prime }{F}^{\prime }$ is constructible and $(F^{\prime },f^{\prime })$ is cohomologically proper relative to $S^{\prime }$ in dimension ≤ 0.

2. b. Let $F$ be a constructible sheaf of sets on $X$. Then one can find a nonempty open subset $S^{\prime }$ of $S$, depending on $F$, such that $f_{\ast }^{\prime }{F}^{\prime }$ is constructible and $(F^{\prime },f^{\prime })$ is cohomologically proper relative to $S^{\prime }$ in dimension ≤ 0.

3. Suppose that schemes of finite type of dimension ≤ dim X_s over an algebraic closure k̄ of κ(s) are strongly desingularizable (SGA 5 I 3.1.5). Then one has in addition the following properties:

4. a. One can find a nonempty open subset $S^{\prime }$ of $S$ such that, for every finite constant sheaf of groups $F^{\prime }$ on $X^{\prime }$ of order prime to the residual characteristics of $S$, if ${\Phi }^{\prime }$ is the stack of torsors under $F^{\prime }$, then $f_{\ast }^{\prime }{\Phi }^{\prime }$ is 1-constructible and $(F^{\prime },f^{\prime })$ is cohomologically proper relative to $S^{\prime }$ in dimension ≤ 1.

5. b. Let $\mathcal{L}$ be the set of prime numbers distinct from the residual characteristics of $S$, and let $\Phi$ be a 1-constructible ind-$\mathcal{L}$-stack on $X$ (XIII.0) such that, for every scheme $X_1$ étale over $X$ and every pair of objects $x$, $x_1$ of Φ_X₁, the sheaf SheafHom_X₁(x,x₁) is constructible. Assume moreover that $S$ is locally noetherian. Then one can find a nonempty open subset $S^{\prime }$ of $S$ such that $f_{\ast }^{\prime }{\Phi }^{\prime }$ is 1-constructible, such that, for every pair of objects $y$, $y_1$ of a fiber (f′_*Φ′)_Y₁, the sheaf SheafHom_Y₁(y,y₁) is constructible,

and such that (Φ′,f′) is cohomologically proper relative to S′ in dimension ≤ 1.

Proof. One may suppose S affine. By SGA 4 VIII 1.1, one may suppose S integral. Finally, by passage to the limit, one may suppose that S is the spectrum of an algebra of finite type over ℤ; in particular S is then noetherian. Since the question is local on Y, one may suppose Y affine. Moreover, to prove the theorem it is enough to do so after a finite extension S′ → S, where S′ is an integral scheme and S′ → S is a composite of étale morphisms and finite surjective radicial morphisms.

1. The Case of Constant Sheaves of Sets

1.1. Reduction to the case where $X$ is normal over $S$. Let X₁_s̄ be the normalization of $(X_{\bar{s}}{)}_{red}$. After replacing $S$ by a nonempty open subset and making a radicial extension of $S$, one may suppose that X₁_s̄ comes from a scheme $X_1$ normal over $S$, and that the morphism X₁_s̄ → X_s̄ comes from a finite surjective morphism $p:X_1\to X$ (EGA IV 8.8.2 and 9.6.1). Suppose the theorem proved for fp. After restricting $S$ to an open subset, one may suppose that, for every constant sheaf of sets $F$ on $X$, $(p\ast F,fp)$ is cohomologically proper relative to $S$ in dimension ≤ 0 and $f_{\ast }{p}_{\ast }(p\ast F)$ is constructible. By XIII.1.9, $(p_{\ast }p\ast F,f)$ is then cohomologically proper relative to $S$ in dimension ≤ 0. The morphism

$$F\to p_{\ast }p\ast F=G$$

is a monomorphism. It already follows, since $f_{\ast }F$ is a subsheaf of $f_{\ast }G$, that $f_{\ast }F$ is constructible (SGA 4 IX 2.9 (ii)) and that $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ −1.

Let X₂ = X₁ ×_X X₁, and let q: X₂ → X be the canonical morphism. By XIII.1.11 1,

one has an exact sequence

F → G ⇉ q_*q*F.

By what we have just proved, applied to fq instead of $f$, one may suppose, after restricting $S$ to a nonempty open subset, that for every constant sheaf of sets $F$ on $X$, $(q\ast F,fq)$ is cohomologically proper relative to $S$ in dimension ≤ −1, hence that $(q_{\ast }q\ast F,f)$ is cohomologically proper relative to $S$ in dimension ≤ −1. It then follows from XIII.1.13 1 that $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ 0.

1.2. Reduction to the case where X is normal affine over S.

Let U_s be an affine open subset of X_s dense in X_s. After restricting S to a nonempty open subset, one may suppose that U_s → X_s lifts to an open immersion i: U → X, schematically dominant relative to S (EGA IV 8.9.1). Since the morphism X → S is normal, one has by SGA 2 XIV 1.18:

$$pro{f}_{et},S-U(X)\ge 2;$$

hence, for every constant sheaf F on X, the canonical morphism

$$F\to i_{\ast }i\ast F$$

is an isomorphism. It follows that, if the theorem is supposed proved for fi and i, then, after restricting S to a nonempty open subset, (iF,i) and (iF,fi) are cohomologically proper relative to S in dimension ≤ 0. The same is therefore true of (F,f) (XIII.1.6 2). Since moreover f**F = (fi)**(i*F) is constructible, this completes the reduction.

1.3. End of the proof.

One may suppose S normal (EGA IV 7.8.3). One can find a compactification of X_s:

X_s → P_s
 \    |
  \   g_s
   f_s|
     Y_s,

where j_s is a dominant open immersion and g_s is proper. After making a radicial extension of κ(s) and replacing P_s by its normalization, which does not change X_s, one may suppose P_s geometrically normal. After restricting S to a nonempty open subset and making a surjective radicial extension, one may suppose that the diagram above comes from a diagram

X → P
 \  |
  \ g
   f|
    Y,

where $P$ is a scheme normal over $S$, $j$ is an open immersion schematically dominant relative to $S$, and $g$ is proper (EGA IV 6.9.1, 9.9.4, and 9.6.1). For every finite constant sheaf of sets $F$ on $X$ with value $C$, $j_{\ast }F$ is the constant sheaf with value $C$ (SGA 4 2.14.1), and the same remains true after every base change $S^{\prime }\to S$. It follows that $(F,j)$ is cohomologically proper relative to $S$ in dimension ≤ 0. Hence the same is true of $(F,f)$, since $g$ is proper (XIII.1.8). Since $g$ is proper, $f_{\ast }F=g_{\ast }{C}_P$ is constructible, which completes the proof of 1 a.

2. The Case of a Constructible Sheaf of Sets

Let F be a constructible sheaf of sets on X. By SGA 4 IX 2.14 (ii), one can find a finite family of morphisms p_i: Z_i → X and, on each Z_i, a finite constant sheaf of sets C_i, such that one has a monomorphism

j: F → ∏_i (p_i)_*C_i = G.

By 1 a, after restricting S to a nonempty open subset, one may suppose that the (Ci,fp_i) are cohomologically proper relative to S in dimension ≤ 0, and that the f**(pi)*Ci are constructible. One already concludes that (G,f) is cohomologically proper relative to S in dimension ≤ 0 (XIII.1.9), hence that (F,f) is cohomologically proper relative to S in dimension ≤ −1, and that f**F is constructible. Let K be the amalgamated sum K = G ⨿_F G. Since F and G are constructible, so is K. We therefore conclude from the preceding that, after restricting S to a nonempty open subset, one may suppose that (K,f) is cohomologically proper relative to S in dimension ≤ −1. It then follows from XIII.1.13 1 that (F,f) is cohomologically proper relative to S in dimension ≤ 0.

3. The Case of Constant Sheaves of Groups

If F is a constant sheaf of groups on X, write Φ for the stack of torsors under F.

3.1. Let us first show that, after restricting $S$ to a nonempty open subset, for every constant sheaf of groups $F$ on $X$ whose order is prime to the residual characteristics of $S$, the pair $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ 0 and $f_{\ast }\Phi$ is constructible.

For this, one reduces to the case where X is smooth over S. After making a finite extension of κ(s), which is allowed since it may be regarded as the composite of an étale extension and a radicial extension, one can find a proper surjective morphism p_s: X₁s → X_s, where X₁s is a scheme smooth over S of the same dimension as X_s. After restricting S to a nonempty open subset, one may suppose that p_s comes from a proper surjective morphism p: X₁ → X, where X₁ is a scheme smooth over S

(EGA IV 9.6.1 and 12.1.6). Let X₂ = X₁ ×_X X₁, and let q: X₂ → X be the canonical morphism. There is an exact diagram of stacks on X

Φ → p_*p*Φ ⇉ q_*q*Φ

(XIII.1.11 2). Supposing the theorem proved in the smooth case, one sees first that one may suppose $f_{\ast }{p}_{\ast }p\ast \Phi$ constructible; hence the same is true of $f_{\ast }\Phi$ (XIII.3.1.1 below). Moreover, by XIII.1.6 2, one may suppose $(p_{\ast }p\ast \Phi ,f)$ cohomologically proper relative to $S$ in dimension ≤ 0. It follows that $(\Phi ,f)$ is cohomologically proper relative to $S$ in dimension ≤ −1. One may therefore suppose that $(q_{\ast }q\ast \Phi ,f)$ is cohomologically proper relative to $S$ in dimension ≤ −1, and this implies that $(\Phi ,f)$ is cohomologically proper relative to $S$ in dimension ≤ 0 (XIII.1.12 1).

One then reduces, as in 1.2, to the case where X is smooth and affine over S. Let

X → P
 \  |
  \ q
   f|
    Y

be a compactification of X, where i is a dominant open immersion and q is proper. Since dim P_s = dim X_s, the hypothesis of resolution of singularities can be applied to P_s̄. After making an étale extension and a radicial extension of S, one can find a proper morphism r: Z → P, where Z is smooth over S, r⁻¹(X) ≃ X, and r⁻¹(X) is the complement in Z of a divisor with normal crossings relative to S. Every torsor under F is then tamely ramified on Z relative to S (XIII.2.3 b). It follows from XIII.2.7 that (F,f) is cohomologically proper relative to S in dimension ≤ 0, proving our assertion.

3.2. Reduction to the case where X is smooth over S.

After making a finite extension of κ(s), one can find a proper surjective morphism p_s: X₁s → X, where X₁s is a scheme smooth over s, and one may suppose that p_s comes from a proper surjective morphism p: X₁ → X, where X₁ is smooth over S. Suppose the theorem proved for fp and let us prove it for f. Let F be a finite constant sheaf of groups on X, of order prime to the residual characteristics of S, and let Φ be the stack of torsors under F. Let X₂ = X₁ ×_X X₁, X₃ = X₁ ×_X X₁ ×_X X₁, and let q: X₂ → X, r: X₃ → X be the canonical morphisms. By XIII.1.11 2, one has an exact diagram of stacks

Φ → p_*p*Φ ⇉ q_*q*Φ ⇉⇉ r_*r*Φ.

To prove that (Φ,f) is cohomologically proper relative to S in dimension ≤ 1, it is enough to show that the same is true of (pp*Φ,f), that (qqΦ,f) is cohomologically proper relative to S in dimension ≤ 0, and that (r**rΦ,f) is cohomologically proper relative to S in dimension ≤ −1 (XIII.1.12 2). By 3.1 above, one may suppose that, for every finite constant sheaf of groups F, the pairs (qΦ,fq), (qΦ,q), (rΦ,fr), and (rΦ,r) are cohomologically proper relative to S in dimension ≤ 0. It then follows from XIII.1.6 2 that (qq*Φ,f) and (rrΦ,f) are cohomologically proper relative to S in dimension ≤ 0. Since the theorem is assumed proved in the smooth case, (pΦ,fp) and (pΦ,p), and hence also (p**pΦ,f) (XIII.1.6 2), are cohomologically proper relative to S in dimension ≤ 1. This shows that (Φ,f) is cohomologically proper relative to S in dimension ≤ 1.

Moreover $f_{\ast }{p}_{\ast }p\ast \Phi$ is 1-constructible by hypothesis. By XIII.3.1 one may suppose $f_{\ast }{q}_{\ast }q\ast \Phi$ constructible; it therefore follows from XIII.3.1.1 below that $f_{\ast }\Phi$ is 1-constructible.

3.3. Reduction to the case where X is smooth affine over S.

By 3.2, one may suppose $X$ smooth over $S$. Let $(U_i{)}_i\in I$ be a finite covering of $X$ by affine opens, let $X_1$ be the direct sum of the $U_i$, and let $p:X_1\to X$ be the canonical morphism. Since $p$ is a morphism of effective descent for the category of étale sheaves of finite type on variable schemes, one sees as in 3.2 that, if the theorem is assumed proved for the $U_i$, that is, for $X_1$, it is also true for $X$.

3.4. The case where X is smooth affine over S.

As in 3.1, after restricting S to a nonempty open subset and making an étale extension and a surjective radicial extension of S, one can find a commutative diagram

X → P
 \  |
  \ q
   f|
    Y

where $P$ is smooth over $S$, $X$ is the complement in $P$ of a divisor with normal crossings relative to $S$, and $q$ is proper. If $F$ is a constant sheaf of order prime to the residual characteristics of $S$, every torsor under $F$ is tamely ramified on $P$ relative to $S$ (XIII.2.3 b). The fact that $(F,f)$ is cohomologically proper relative to $S$ in dimension ≤ 1 and that $f_{\ast }\Phi$ is constructible then follows from XIII.2.7.

4. Proof of 2 b

4.1. The case where Φ is a gerbe.

One can find a surjective étale morphism of finite type p: X₁ → X such that p*Φ is a trivial gerbe. By descent, as in 3.2, it is enough to prove the theorem for X₁, X₁ ×_X X₁, and

X₁ ×_X X₁ ×_X X₁. We are therefore reduced to the case where Φ is the gerbe of torsors under a constructible sheaf of groups F whose fibers have order prime to the residual characteristics of S.

By SGA 4 IX 2.14, one can find a finite family of finite morphisms p_i: Z_i → X and, for each i, a finite constant sheaf of groups C_i, of order prime to the residual characteristics of S, such that one has a monomorphism [Translator note: the corrected source fixes “morphism” to “monomorphism.”]

j: F → ∏_i p_i*C_i = G.

Let Φi be the stack of torsors under C_i, and let Ψ be the stack of torsors under G. It follows from 2 a that, after restricting S to a nonempty open subset, one may suppose that the (C_i,fp_i) are cohomologically proper relative to S in dimension ≤ 1, and that the stacks f**p_iΦi are 1-constructible. It then follows from XIII.1.9 that the (p_iC_i,f) are cohomologically proper relative to S in dimension ≤ 1; hence the same is true of (G,f). Moreover, since p_iΦ_i is equivalent to the stack of torsors under the group p_i*C_i (SGA 4 VIII 5.8), one sees that f*Ψ is 1-constructible.

Since $R^1{f}_{\ast }G$ is constructible, one can find a sheaf representable by an étale $Y$-scheme of finite type $T$ and an epimorphism

$$a:T\to R^1{f}_{\ast }G$$

(SGA 4 IX 2.7). Moreover, one may suppose that the image of the identity section of T(T) is defined by a torsor Q on X ×Y T = X_T with group G|X_T. Let f_T: X_T → T be the canonical morphism, and put F_T = F|X_T, etc. By 1 b, after restricting S to a nonempty open subset, one may suppose that (Q/F_T,f_T) is cohomologically proper relative to S in dimension ≤ 0 and that f_T*(Q/F_T) is constructible. It then follows from XIII.3.1.2 that f*Ψ is constructible.

Let us show that (F,f) is cohomologically proper relative to

S in dimension ≤ 1. By XIII.1.13 2, it is enough to prove that, for every scheme Y₁ étale over Y and every torsor Q₁ on X₁ = X ×_Y Y₁, if f₁: X₁ → Y₁ is the canonical morphism, then (Q₁/F₁,f₁) is cohomologically proper relative to S in dimension ≤ 0. But by definition of T, Q₁ is, locally for the étale topology of Y₁, the inverse image of Q. This proves our reduction.

4.2. The general case.

Using Lemma XIII.6.1.1, 4.1, and 1 a, one sees that, after restricting $S$ to a nonempty open subset, one may suppose $S(f_{\ast }\Phi )$ constructible and $(S\Phi ,f)$ cohomologically proper relative to $S$ in dimension ≤ 0. One can then find a sheaf representable by an étale $Y$-scheme of finite type $T$ and an epimorphism

$$a:T\to S(f_{\ast }\Phi ).$$

We resume the notation of 4.1 and put moreover $Z=T{\times }_{Y}T$, $X_Z=X{\times }_{Y}Z$, and write $f_Z:X_Z\to Z$ for the canonical morphism. One may suppose $T$ chosen so that the image $q$ of the identity section of $T(T)$ by $a$ is defined by an object $p$ of (f_*Φ)_T = Φ_X_T. Let $p_1$ and $p_2$, respectively $q_1$ and $q_2$, be the inverse images of $p$, respectively $q$, by the two projections from $Z$ to $T$. After restricting $S$ to a nonempty open subset, one may suppose that (SheafAut_X_T(p),f_T) is cohomologically proper relative to $S$ in dimension ≤ 1, that (SheafHom_X_Z(p₁,p₂),f_Z) is cohomologically proper relative to $S$ in dimension ≤ 0, and that f_T_*(SheafAut_X_T(p)) and f_Z_*(SheafHom_X_Z(p₁,p₂)) are constructible (1 a and 4.1).

One first deduces that, for every scheme $Y_1$ étale over $Y$ and every pair of objects $y$, $y_1$ of (f_*Φ)_Y₁, the sheaf SheafAut_Y₁(y), respectively SheafHom_Y₁(y,y₁), is constructible: such a sheaf is, locally for the étale topology of $Y_1$, the inverse image of f_T_*(SheafAut_X_T(p)), respectively of f_Z_*(SheafHom_X_Z(p₁,p₂)).

It remains to prove that (Φ,f) is cohomologically proper relative

to S in dimension ≤ 1. For this it is enough to show that, for every S-scheme S′ and every geometric point ȳ′ of Y, if Ȳ denotes the strict localization of Y at y′, X̄ = X ×_Y Ȳ, etc., then the canonical functor

$$\bar{\varphi }:\Phi (\bar{X})\to {\Phi }^{\prime }({\bar{X}}^{\prime })$$

is an equivalence of categories.

Let us show that φ̄ is fully faithful. Let x, y ∈ Φ(X̄), and let x′, y′ be their images in Φ′(X̄′). We must show that the canonical morphism

$$(\ast ){\mathrm{Hom}}_{\bar{X}}(x,y)\to {\mathrm{Hom}}_{\bar{X}}^{\prime }(x^{\prime },y^{\prime })$$

is bijective. By definition of T, there exist two morphisms from Ȳ to T such that x and y are the inverse images of p by these two morphisms. This amounts to saying that there is a morphism Ȳ → Z, hence a morphism h: X̄ → X_Z, such that

h*(p₁) = x,     h*(p₂) = y.

Consequently one has a canonical isomorphism

SheafHom_X̄(x,y) = h*(SheafHom_X_Z(p₁,p₂)).

But, taking into account the fact that (SheafHom_X_Z(p₁,p₂),f_Z) is cohomologically proper relative to S in dimension ≤ 0, one sees that the same is true of (SheafHom_X̄(x,y),f̄), which proves that the morphism (*) is bijective.

Let us show that φ̄ is essentially surjective. Let x′ ∈ Φ′(X̄′). Since (SΦ,f) is cohomologically proper relative to S in dimension ≤ 0, the canonical morphism

$$H^0(\bar{X},S\bar{\Phi })\to H^0({\bar{X}}^{\prime },S{\bar{\Phi }}^{\prime })$$

is bijective. Let G′ be the maximal subgerbe of Φ′ generated by x′. There then exists a maximal subgerbe G of Φ̄ whose inverse image on X̄′ is G′. By 4.1, (G,f̄) is cohomologically proper relative to S in dimension ≤ 1; consequently the canonical functor

$$G(\bar{X})\to G^{\prime }({\bar{X}}^{\prime })$$

is an equivalence of categories. This proves the existence of an element x of Φ(X̄) whose image in Φ′(X′) is isomorphic to x′, and completes the proof of the theorem.

Sublemma.

Let S be a locally noetherian scheme, and let

Φ → Φ₁ ⇉ Φ₂

be an exact diagram of stacks on $S$ (XIII.1.10.1). If ${\Phi }_1$ is constructible, then so is $\Phi$. If, for every scheme $S^{\prime }$ étale over $S$ and every pair of objects $x_1$, $y_1$ of $({\Phi }_1{)}_S^{\prime }$, the sheaf $SheafHo{m}_S^{\prime }(x_1,y_1)$ is constructible, then for every pair of objects $x$, $y$ of ${\Phi }_S^{\prime }$, the same is true of $SheafHo{m}_S^{\prime }(x,y)$. [Translator note: the corrected source fixes "object" to "objects."] Suppose ${\Phi }_1$ is 1-constructible (XIII.0) and ${\Phi }_2$ is constructible; then $\Phi$ is 1-constructible.

For every scheme S′ étale over S and every object x of Φ_S′, one has a monomorphism

$$SheafAu{t}_S^{\prime }(x)\to SheafAu{t}_S^{\prime }(p(x)).$$

It follows from SGA 4 IX 2.9 that, if Φ₁ is constructible, then so is Φ; the second assertion of the lemma is proved in the same way.

Now suppose Φ₁ is 1-constructible and Φ₂ constructible. The morphism p induces on the sheaves of maximal subgerbes a morphism

$$\varphi :S\Phi \to S{\Phi }_1.$$

Let G be the image of SΦ by φ. By SGA 4 IX 2.9, G is a constructible sheaf. Thus one can find a sheaf representable by an étale S-scheme of finite type T and an epimorphism

$$a:T\to G$$

(SGA 4 IX 2.7).

Moreover, $T$ may be chosen so that the image $y$ of the identity section of $T(T)$ by $a$ is defined by an object $x_1$ of $({\Phi }_1{)}_T$ of the form $x_1=p(x)$, with $x\in {\Phi }_T$.

It is enough to show that, for every point $s$ of $S$, there exists a nonempty open subset $U$ of $closure(s)$ such that $S\Phi |U$ is locally constant constructible. Let $s\in S$, let $\bar{s}$ be a geometric point above $s$, and let ${\bar{y}}_1,\cdots ,{\bar{y}}_n$ be the elements of $G_{\bar{s}}$. By definition of $T$, there exist morphisms $h_i:\bar{s}\to T$ such that $h_i\ast (y)={\bar{y}}_i$. Let $S^{\prime }$ be the fiber product over $S$ of $n$ schemes isomorphic to $T$, let $h:\bar{s}\to S^{\prime }$ be the fiber product of the $h_i$, and let $y_i$, respectively $x_i$, be the inverse image of $y$, respectively $x$, by the $i$-th projection from $S^{\prime }$ to $T$. Let $F_i$ be the subsheaf of $S\Phi |S^{\prime }$ inverse image of $y_i$, and let us show that the $F_i$ are constructible. The sheaf $F_i$ is a quotient of the sheaf $F_i^{\prime }$ such that, for every scheme $S^{\prime \prime }$ étale over $S^{\prime }$,