Exposé XIV. Depth and Lefschetz theorems in étale cohomology

by Mme M. Raynaud, after unpublished notes of A. Grothendieck¹

Translator note. Per Raynaud's opening footnote, this Exposé adopts the modern terminology in which scheme denotes what Exposés I–XIII of SGA 2 called prescheme, and separated scheme denotes what they called scheme. This Exposé therefore breaks with the prescheme/scheme convention used elsewhere in this translation; the shift is deliberate and matches Raynaud's own.

Typographic note. Throughout this Exposé we write $R{\Gamma }_Y$ for the sheafified derived functor (underlined in source) and $R{\Gamma }_Y$ for the global one; context disambiguates. Likewise, ${\mathcal{H}}_Y^p(F)$ denotes the sheafified local cohomology and $H_Y^p(X,F)$ the global one.

In §1 we define a notion of "étale depth" which is, in étale cohomology, the analogue of the notion of depth studied in III in the cohomology of coherent sheaves. After a technical part, we prove in §4 some "Lefschetz theorems", the central theorem being 4.2. Let $X$ be a scheme, $Y$ a closed part of $X$, $U$ the complementary open set $X-Y$, and $F$ an abelian sheaf on $X$ for the étale topology; in a general manner, the aim of the Lefschetz theorems is to show that, if $F$ satisfies certain local conditions on $U$, expressible in terms of étale depth at the points of $U$, then under certain supplementary global conditions on $U$ (for example $U$ affine), the natural map of étale cohomology groups

H^i(X, F) → H^i(Y, F|Y)

is an isomorphism for values $i<n$, where $n$ is a certain explicit integer. By taking for $F$ a constant sheaf, one obtains in this way conditions for ${\pi }_0(X)$ to equal ${\pi }_0(Y)$ and conditions for the abelianized fundamental groups of $X$ and $Y$ to be the same. In §5, the introduction of a notion of "geometrical depth" enables us to give useful particular cases of the Lefschetz theorems (5.7). Finally in §6 we mention some conjectures, concerning in particular "non-commutative" variants of the theorems obtained.

1. Cohomological and homotopical depth

1.0. Fix the following notations. Let $X$ be a scheme², $Y$ a closed part of $X$, $U$ the complementary open set, and $i:Y=X-U\to X$ the canonical immersion. Let $\Gamma Y$ be the functor that, to an abelian sheaf on $X$, associates the "sheaf of sections with support in $Y$", that is $\Gamma Y=i_{\ast }{i}^{!}$ (cf. SGA 4 IV 3.8 and VIII 6.6), and let ${\Gamma }_Y$ be the functor $\Gamma \circ \Gamma Y$ (where $\Gamma$ is the "global sections" functor). Consider the derived category $D^{+}(X)$ and the derived functor $R{\Gamma }_Y$ (resp. $R{\Gamma }_Y$) of ${\Gamma }_Y$ (resp. of $\Gamma Y$) (cf. [3]). Given a complex $F$ of abelian sheaves on $X$ bounded below, we may consider it as an element of $D^{+}(X)$; we then denote by ${\mathcal{H}}_Y^p(F)$ the $p$-th cohomology sheaf of $R{\Gamma }_Y(F)$ (sheafified) and by $H_Y^p(X,F)$ the $p$-th cohomology group of $R{\Gamma }_Y(F)$. The results of (SGA 4 V 4.3 and 4.4) extend trivially to ${\mathcal{H}}_Y^p(F)$ and $H_Y^p(X,F)$.

Proposition 1.1.

Let $X$ be a scheme, $Y$ a closed part of $X$, $U$ the complementary open set, and $i:U\to X$ the canonical immersion. Denote by $F$ either a sheaf of sets on $X$, a sheaf of groups on $X$, or a complex of abelian sheaves on $X$ bounded below. Fix the following notations: if $X^{\prime }\to X$ is a morphism, $U^{\prime }$ and $F^{\prime }$ denote the inverse images of $U$ and $F$ on $X^{\prime }$; furthermore, if $y$ is a geometric point of $X$, $\bar{X}$ denotes the strict localization of $X$ at $y$ and Ū, $\bar{F}$ the inverse images of $U$ and $F$ on $\bar{X}$.

1°) Let $F$ be a sheaf of sets on $X$ and $n$ an integer $⩽2$; then the following conditions are equivalent:

(i) The canonical morphism

F → i_* i^* F

is injective if $n⩾1$, bijective if $n⩾2$.

(ii) For every scheme $X^{\prime }$ étale over $X$, the canonical morphism

H⁰(X′, F′) → H⁰(U′, F′)

is injective if $n⩾1$, bijective if $n⩾2$.

Suppose moreover that $U$ is retrocompact in $X$; then the preceding conditions are equivalent to the following:

(iii) For every geometric point $y$ of $Y$, the canonical morphism

H⁰(X̄, F̄) → H⁰(Ū, F̄)

is injective if $n⩾1$, and bijective if $n⩾2$.

2°) Let $F$ be a sheaf of groups on $X$ and $n$ an integer $⩽3$; then the following conditions are equivalent:

(i) The canonical morphism

F → i_* i^* F

is injective if $n⩾1$, bijective if $n⩾2$, and if $n⩾3$, in addition to the preceding conditions, the pointed sheaf of sets $R^1{i}_{\ast }(i^{\ast }F)$ is null.

(ii) For every scheme $X^{\prime }$ étale over $X$, the canonical morphism

H⁰(X′, F′) → H⁰(U′, F′)

is injective if $n⩾1$, bijective if $n⩾2$, and moreover the canonical morphism

H¹(X′, F′) → H¹(U′, F′)

is injective if $n⩾2$, bijective if $n⩾3$.

(ii bis) Identical to (ii), except in the case $n=2$ where one only supposes $H^0(X^{\prime },F^{\prime })\to H^0(U^{\prime },F^{\prime })$ bijective.

Suppose moreover that $U$ is retrocompact in $X$; then the preceding conditions are also equivalent to the following:

(iii) For every geometric point $y$ of $Y$, the canonical morphism

H⁰(X̄, F̄) → H⁰(Ū, F̄)

is injective if $n⩾1$, bijective if $n⩾2$; finally if $n⩾3$, in addition to the preceding conditions, $H^1(\bar{U},\bar{F})$ is null.

3°) Let $F$ be a complex of abelian sheaves bounded below and $n$ an integer; then the following conditions are equivalent:

(i) One has ${\mathcal{H}}_Y^p(F)=0$ for $p<n$ (cf. 1.0).

(ii) For every scheme $X^{\prime }$ étale over $X$, the canonical morphism

H^p(X′, F′) → H^p(U′, F′)

is bijective for $p<n-1$, injective for $p=n-1$.

Suppose $U$ retrocompact in $X$, then the preceding conditions are also equivalent to the following:

(iii) For every geometric point $y$ of $Y$, the canonical morphism

H^p(X̄, F̄) → H^p(Ū, F̄)

is bijective for $p<n-1$, injective for $p=n-1$.

In the case where $F$ is an abelian sheaf and $n⩾2$, conditions (i) and (ii) are also equivalent to the following:

(ii bis) For every scheme $X^{\prime }$ étale over $X$, the canonical morphism

H^p(X′, F′) → H^p(U′, F′)

is bijective for $p<n-1$.

Proof.

1°) It is clear that (i) ⇔ (ii). Let us show that, if $U$ is retrocompact in $X$, (i) ⇔ (iii). Indeed, (i) amounts to saying that for every geometric point $y$ of $X$, the morphism $F_y\to (i_{\ast }{i}^{\ast }F{)}_y$ is injective if $n⩽1$ and bijective if $n⩽2$ (SGA 4 VIII 3.6). Since this morphism is bijective in any case when $y$ is a geometric point of $U$, one can restrict to geometric points $y$ of $Y$. Now it follows from the fact that $i$ is quasi-compact and from (SGA 4 VIII 5.3) that the morphism

F_y → (i_* i^* F)_y

is canonically identified with the morphism

H⁰(X̄, F̄) → H⁰(Ū, F̄),

whence the equivalence of (i) and (iii).

2°) (i) ⇒ (ii). The assertions about $H^0$ follow from 1°). Let $i^{\prime }$ be the canonical immersion of $U^{\prime }$ into $X^{\prime }$; the assertions about $H^1$ follow from the exact sequence (SGA 4 XII 3.2)

0 → H¹(X′, i′_* i′^* F′) → H¹(U′, F′) → H⁰(X′, R¹ i′_* (i′^* F′)).

(ii bis) ⇒ (i). By 1°), it suffices to show that, for $n⩾3$, one has $R^1{i}_{\ast }(i^{\ast }F)=0$. Now $R^1{i}_{\ast }(i^{\ast }F)$ is the sheaf associated to the presheaf $X^{\prime }\mapsto H^1(U^{\prime },F^{\prime })$, that is, by hypothesis, the sheaf associated to the presheaf $X^{\prime }\mapsto H^1(X^{\prime },F^{\prime })$, which is null.

(i) ⇔ (iii). Taking 1°) into account, the only thing that remains to see is that the relation $R^1{i}_{\ast }(i^{\ast }F)=0$ is equivalent to the fact that $H^1(\bar{U},\bar{F})=0$ for every geometric point $y$ of $Y$. Since $R^1{i}_{\ast }(i^{\ast }F)$ is null outside $Y$, it amounts to the same to say that $R^1{i}_{\ast }(i^{\ast }F)=0$ or that $(R^1{i}_{\ast }(i^{\ast }F){)}_y=0$ for every geometric point $y$ of $Y$. It then suffices to note that, $i$ being quasi-compact, one has $(R^1{i}_{\ast }(i^{\ast }F){)}_y=H^1(\bar{U},\bar{F})$ (SGA 4 VIII 5.3).

3°) (i) ⇒ (ii). Let $X^{\prime }$ be a scheme étale over $X$; one has the exact sequence (SGA 4 V 4.5)

(∗)    → H^p_{Y′}(X′, F′) → H^p(X′, F′) → H^p(U′, F′) →;

so (ii) is equivalent to $H_{Y^{\prime }}^p(X^{\prime },F^{\prime })=0$ for $p<n$ and for every scheme $X^{\prime }$ étale over $X$. Consider then the spectral sequence

E_2^{pq} = H^p(X′, ℋ^q_{Y′}(F)) ⟹ H^{p+q}_{Y′}(X′, F′);

by hypothesis, ${\mathcal{H}}_Y^q(F)=0$ for $q<n$, whence $E_2^{pq}=0$ for $p+q<n$ and consequently $H_{Y^{\prime }}^p(X^{\prime },F^{\prime })=0$ for $p<n$.

(ii) ⇒ (i). The sheaf ${\mathcal{H}}_Y^p(F)$ is associated to the presheaf $X^{\prime }\mapsto H_{Y^{\prime }}^p(X^{\prime },F^{\prime })$; since we have already remarked that (ii) is equivalent to the relation $H_{Y^{\prime }}^p(X^{\prime },F^{\prime })=0$ for $p<n$ and for every scheme $X^{\prime }$ étale over $X$, one indeed has ${\mathcal{H}}_Y^p(F)=0$ for $p<n$.

(i) ⇔ (iii). The sheaves ${\mathcal{H}}_Y^p(F)$ are concentrated on $Y$; consequently it amounts to the same

to say that ${\mathcal{H}}_Y^p(F)=0$ or that, for every geometric point $y$ of $Y$, the fiber $({\mathcal{H}}_Y^p(F){)}_y$ is null. Now, $i$ being quasi-compact, one deduces from (SGA 4 VIII 5.2) that one has $({\mathcal{H}}_Y^p(F){)}_y=H_{\bar{Y}}^p(\bar{X},\bar{F})$. The equivalence of (i) and (iii) follows, taking into account the analogue on $\bar{X}$ of the exact sequence (∗).

(ii bis) ⇒ (ii) in the case where $F$ is an abelian sheaf. The only thing that remains to show is that ${\mathcal{H}}_Y^{n-1}(F)=0$. Now, for $n>2$, the sheaf ${\mathcal{H}}_Y^{n-1}(F)$ is associated to the presheaf $X^{\prime }\mapsto H^{n-2}(U^{\prime },F^{\prime })=H^{n-2}(X^{\prime },F^{\prime })$, hence is null. The case $n=2$ follows from the fact that ${\mathcal{H}}_Y^1(F)$ is the cokernel of the morphism $F\to i_{\ast }{i}^{\ast }F$.

Definition 1.2.

The notations are those of 1.1. One says that $F$ is of $Y$-étale depth $⩾n$ and writes

$$pro{f}_Y(F)⩾n$$

if $F$ satisfies the equivalent conditions (i) and (ii) of 1.1. If $F$ is a complex of abelian sheaves, one calls $Y$-étale depth of $F$ the supremum of the $n$

for which $pro{f}_Y(F)⩾n$; one will use the same notation if $F$ is a sheaf of sets, resp. of not-necessarily-commutative groups (so that one then has $0⩽pro{f}_Y(F)⩽2$, resp. $0⩽pro{f}_Y(F)⩽3$, when context does not allow confusion as to which of the three variants envisaged here one is using).

If $L$ is a set of prime numbers, one says that the $Y$-étale depth for $L$ of $X$ is $⩾n$ and writes

$$pro{f}_Y^L(X)⩾n$$

if, for every constant sheaf of the form $\mathbb{Z}/\ell \mathbb{Z}$ with $\ell \in L$, one has $pro{f}_Y(\mathbb{Z}/\ell \mathbb{Z})⩾n$. One defines in the obvious way the $Y$-étale depth for $L$ of $X$. If $L=P$, the set of all prime numbers, and if there is no risk of confusion with the notation of (EGA IV 5.7.1) (relative to the case $F=O_X$), one omits $L$ in the notation; otherwise one writes $pro{f}_Y^{\acute{e}}t(X)$.

Finally one says that $X$ is of $Y$-homotopical depth $⩾3$ for $L$ and writes

$$pro{f}_Y^{hopL}(X)⩾3$$

if, for every constant finite sheaf of $L$-groups $F$ on $X$, one has $pro{f}_Y(F)⩾3$. If $L=P$, one omits $L$ in the notation.

Corollary 1.3.

Under the conditions of 1.1, if $pro{f}_Y(F)⩾n$, then for every closed subset $Z$ of $Y$, one has

$$pro{f}_Z(F)⩾n.$$

Let us, for example, carry out the reasoning in the case where $F$ is a complex of abelian sheaves bounded below. We use 1.1 3°) (ii). Let $V=X-Z$ and consider, for every integer $p$, the sequence of morphisms

H^p(X, F) --f--> H^p(V, F) --g--> H^p(U, F).

By hypothesis $g$ and $f\circ g$ are bijective for $p<n-1$ and injective for $p=n-1$; the same therefore holds for $f$. Since the reasoning is valid when one replaces $X$ by a scheme $X^{\prime }$ étale over $X$, this proves 1.3.

Corollary 1.4.

The notations are those of 1.1 2°). If $X^{\prime }$ is a scheme over $X$, denote by ${\Phi }^{\prime }$ the functor that associates to an étale covering of $X^{\prime }$ its restriction to $U^{\prime }$, and by ${\Phi }_{F^{\prime }}^{\prime }$ the functor that associates to a torsor³ under $F^{\prime }$ its restriction to $U^{\prime }$. Then the following conditions are equivalent:

(i) One has $pro{f}_Y(F)⩾1$ (resp. $pro{f}_Y(F)⩾2$, resp. $pro{f}_Y(F)⩾3$).

(ii) For every scheme $X^{\prime }$ étale over $X$, the functor ${\Phi }_{F^{\prime }}^{\prime }$ is faithful (resp. fully faithful, resp. an equivalence of categories).

In particular, in order that $pro{f}_Y(X)⩾1$ (resp. $pro{f}_Y(X)⩾2$, resp. $pro{f}_Y^{hop}(X)⩾3$), it is necessary and sufficient that the functor ${\Phi }^{\prime }$ be faithful (resp. fully faithful, resp. an equivalence of categories).

This indeed follows from 1.1 2°) (ii), taking into account the interpretation of $H^1(X^{\prime },F^{\prime })$ as the set of classes (mod isomorphism) of torsors under $F^{\prime }$ (SGA 4 VII 2), and of étale coverings $Z$ of degree $n$ of a scheme as associated to Galois principal coverings with group the symmetric group $S_n$, where to $Z$ is associated the covering $Iso{m}_X(Z_0,Z)$, where $Z_0$ is the trivial covering of $X$ of degree $n$.

Corollary 1.5.

Under the conditions of 1.1 3°), suppose that $pro{f}_Y(F)⩾n$; then one has

H^n_Y(X, F) ≃ H⁰(X, ℋ^n_Y(F)).

The corollary follows from the spectral sequence

E_2^{pq} = H^p(X, ℋ^q_Y(F)) ⟹ H^{p+q}_Y(X, F).

Indeed, one has by hypothesis $E_2^{pq}=0$ for $q<n$, whence

H^n_Y(X, F) = E_2^{0,n} = H⁰(X, ℋ^n_Y(F)).

Remarks 1.6.

a) The notion of $Y$-depth, in the form of the equivalent conditions (i) and (ii) of 1.1, makes sense for any site. In the particular case where $X$ is a locally noetherian scheme equipped with the Zariski topology, and $F$ a sheaf of coherent O_X-modules, one finds the usual notion of $Y$-depth as the infimum of the depths at the points of $Y$ (III).

b) For $n⩽2$, the notion of $Y$-étale depth of $X$ is independent of $L$. For $n=1$, it simply means that $U$ is dense in $X$. Indeed this condition is necessary in order that $pro{f}_Y(F)⩾1$, and it is also sufficient since one may suppose $X$ reduced, the case in which the condition $U$ dense in $X$ is preserved by étale base change (EGA IV 11.10.5 (ii) b)). If $U$ is retrocompact in $X$, the relation $pro{f}_Y(X)⩾1$ is also equivalent to saying that $Y$ contains no maximal point of $X$ (EGA I 6.6.5). For $n=2$ and $U$ retrocompact in $X$, the condition $pro{f}_Y(X)⩾2$ is equivalent to the fact that, for every geometric point $y$ of $Y$, Ū is connected non-empty, that is, "$Y$ does not disconnect $X$ locally for the étale topology".

c) If $X$ is of $Y$-depth $⩾n$ for $L$ and $U$ retrocompact in $X$, then for every locally constant abelian sheaf of $L$-torsion $F$ on $X$, one has $pro{f}_Y(F)⩾n$. Indeed, since the property $pro{f}_Y(F)⩾n$ is local for the étale topology, one may suppose $F$ constant; then $F$ is a filtered direct limit of sheaves that are finite sums of sheaves of the form $\mathbb{Z}/p^m\mathbb{Z}$, where $m$ is a positive integer and $p\in L$. Using 1.1 (iii)

and (SGA 4 VII 3.3), one sees that one may reduce to the case $F=\mathbb{Z}/p^m\mathbb{Z}$, then, by induction on $m$, to the case $F=\mathbb{Z}/p\mathbb{Z}$ for which the assertion follows from the definition.

d) By 1.4, if $pro{f}_Y(X)⩾3$, the pair $(X,Y)$ is pure in the sense of X 3.1. In fact the pure pairs that one encounters in practice (cf. X 3.4) satisfy the stronger condition of homotopical depth $⩾3$, and this notion may therefore advantageously be substituted for that of pure pair.

e) Let $F$ be a complex of abelian sheaves and $T(F)$ the complex obtained by applying to $F$ the translation functor ([3]); then one evidently has:

$$pro{f}_Y(T(F))=pro{f}_Y(F)-1.$$

f) Let us note that the recent works of Artin-Mazur ([1]) allow one to define the notion of homotopical depth $⩾n$ for every integer $n$ (not only for $n⩾3$).

g) Under the conditions of 1.1 3°), in order that $pro{f}_Y(F)=\infty$, it is necessary and sufficient that $F\stackrel{\sim }{\to }R{i}_{\ast }(i^{\ast }F)$ in $D^{+}(X)$. Indeed, the ${\mathcal{H}}_Y^p(F)$ are the cohomology sheaves of the cone (= mapping cylinder) of the canonical morphism $F\to R{i}_{\ast }{i}^{\ast }(F)$.

Definition 1.7.

Let $X$ be a scheme, $x$ a point of $X$, $\bar{x}$ a geometric point above $x$, and $\bar{X}$ the strict localization of $X$ at $\bar{x}$. As before, $F$ denotes either a sheaf of sets on $X$, a sheaf of groups on $X$, or a complex of abelian sheaves on $X$ bounded below; $\bar{F}$ its inverse image on $\bar{X}$, and $L$ a set of prime numbers. One says that $F$ is of étale depth $⩾n$ at the point $x$ (resp. that the étale depth for $L$ of $X$ at $x$ is

$⩾n$, resp. that the homotopical depth for $L$ of $X$ at $x$ is $⩾3$) and one writes $pro{f}_x(F)⩾n$ (resp. $pro{f}_x^L(X)⩾n$, resp. $pro{f}_x^{hopL}(X)⩾3$) if one has $pro{f}_{\bar{x}}(\bar{F})⩾n$ (resp. $pro{f}_{\bar{x}}^L(\bar{X})⩾n$, resp. $pro{f}_{\bar{x}}^{hopL}(\bar{X})⩾3$). One defines in the obvious way the integer $pro{f}_x^L(X)$ and, if $F$ is a complex of abelian sheaves, the integer $pro{f}_x(F)$. If $L$ is the set of all prime numbers, one omits $L$ in the notation $pro{f}_x^L(X)$, unless there is a risk of confusion with the notation of (EGA IV 5.7.1), in which case one writes $pro{f}_x^{\acute{e}}t(X)$.

One then has the following pointwise characterization of depth:

Theorem 1.8.

Let $X$ be a scheme, $Y$ a closed part of $X$ such that the open set $U=X-Y$ is retrocompact in $X$. If $F$ is either a sheaf of sets on $X$, a sheaf of groups on $X$, or a complex of abelian sheaves on $X$ bounded below, then one has

prof_Y(F) = inf_{y ∈ Y} prof_y(F).

1.8.1. Let us first show that, for every point $y$ of $Y$, one has the inequality $pro{f}_y(F)⩾pro{f}_Y(F)$. Indeed, let ȳ be a geometric point above $y$, $\bar{X}$ the strict localization of $X$ at ȳ, $\bar{F}$ and Ȳ the inverse images of $F$ and $Y$ on $\bar{X}$. By 1.7 and 1.3,

prof_y(F) = prof_{ȳ}(F̄) ⩾ prof_{Ȳ}(F̄) ⩾ prof_Y(F),

the last inequality using the hypothesis "$U$ retrocompact", via the conditions (iii) in 1.1 and the transitivity in the formation of strict localizations.

1.8.2. Conversely, suppose that, for every point $y$ of $Y$, one has $pro{f}_y(F)⩾n$ ($n$ an integer) and let us show that one then has $pro{f}_Y(F)⩾n$.

Let us first recall the following well-known results (SGA 4 VIII):

Lemma 1.8.2.1.

Let $X$ be a scheme, $F$ a sheaf of sets on $X$ (resp. $G\to F$ a monomorphism of sheaves of sets on $X$). Then, in order that two sections $s$ and $s^{\prime }$ of $F$ coincide (resp. that a section $s$ of $F$ come from a section of $G$), it is necessary and sufficient that this hold locally. In particular, if $s$ and $s^{\prime }$ are two sections of $F$, there exists a largest open set $V$ of $X$ on which they coincide (resp. if $s$ is a section of $F$ over $X$, there exists a largest open set $V$ of $X$ such that $s|V$ comes from a section of $G$ over $V$). This open set is also the set of points $x$ of $X$ such that, denoting by $\bar{x}$ a geometric point above $x$, the sections $s$ and $s^{\prime }$ have the same image in the fiber $F_{\bar{x}}$ (resp. that the image of $s$ in $F_{\bar{x}}$ comes from an element of $G_{\bar{x}}$).

Let us return to the proof of 1.8.

1°) Case where $F$ is a sheaf of sets. If $n=1$, it suffices to show that the canonical morphism

H⁰(X, F) → H⁰(U, F)

is injective, the result still applying when one replaces $X$ by a scheme étale over $X$. Let $s$ and $s^{\prime }$ be two sections of $F$ over $X$ having the same image in $H^0(U,F)$ and let $V$ be the largest open set over which they are equal; one evidently has $V\supset U$. Suppose $V\ne X$ and let $y$ be a maximal point of $X-V$, ȳ a

geometric point above $y$, $\bar{X}$ the strict localization of $X$ at ȳ, and $\bar{V}$ and $\bar{F}$ the inverse images of $V$ and $F$ on $\bar{X}$. By the choice of $y$, one has $\bar{X}-\bar{y}=\bar{V}$, and by hypothesis the morphism

H⁰(X̄, F̄) → H⁰(X̄ − ȳ, F̄) = H⁰(V̄, F̄)

is injective. It follows that $s$ and $s^{\prime }$ coincide at the point $y$, which is absurd. If $n=2$, it suffices to show, taking the preceding into account, that the morphism

H⁰(X, F) → H⁰(U, F) = H⁰(X, i_* i^* F)

is surjective (where $i$ is the canonical immersion of $U$ in $X$). Let $s$ be a section of $i_{\ast }{i}^{\ast }F$ over $X$ and $V$ the largest open set over which it comes from a section of $F$. Suppose $V\ne X$ and let $y$ be a maximal point of $X-V$; with the preceding notations, it follows from the hypothesis that the canonical morphism

H⁰(X̄, F̄) → H⁰(X̄ − ȳ, F̄) = H⁰(V̄, F̄)

is bijective; consequently $s|V$ extends to the point $y$, which is absurd and completes the proof in case 1°).

2°) Case where $F$ is a sheaf of groups. Taking 1°) into account, the only thing that remains to show is that, in the case $n=3$, the morphism

H¹(X, F) = H¹(X, i_* i^* F) → H¹(U, F)

is bijective. One already knows that it is injective by 1°) and 1.1 2°) (ii bis). For surjectivity, one uses the exact sequence (SGA 4 XII 3.2)

0 → H¹(X, i_* i^* F) → H¹(U, F) --d--> H⁰(X, R¹ i_* (i^* F)).

Let $s\in H^1(U,F)$ and $V\supset U$ the largest open set over which $d(s)=0$; it is also the largest open set such that $s$ comes from an element of $H^1(V,F)$. Suppose $V\ne X$ and let $y$ be a maximal point of $X-V$; if $\bar{X}$ is the strict localization of $X$ at a geometric point ȳ above $y$, one has, with obvious notations, the exact sequence

0 → H¹(X̄, i_* (i^* F̄)) → H¹(Ū, F̄) --d--> H⁰(X̄, R¹ i_* (i^* F̄)).

Since $i:U\to X$ is quasi-compact, $R^1{i}_{\ast }(i^{\ast }\bar{F})$ is the inverse image of $R^1{i}_{\ast }(i^{\ast }F)$ by the morphism $\bar{X}\to X$, whence $H^0(\bar{X},R^1{i}_{\ast }(i^{\ast }\bar{F}))=(R^1{i}_{\ast }(i^{\ast }F){)}_{\bar{y}}$. By hypothesis and given that $y\in Y$, the morphism

H¹(X̄, F̄) → H¹(V̄, F̄)

is bijective. The image $\bar{s}$ of $s$ in $H^1(\bar{U},\bar{F})$, which extends to $\bar{V}$ by definition of $V$, therefore also extends to $\bar{X}$; it follows that $d(\bar{s})=0$, hence the image of $d(s)$ in the geometric fiber $(R^1{i}_{\ast }(i^{\ast }F){)}_{\bar{y}}$ is null; but this contradicts the definition of $V$, whence case 2°).

3°) Case where $F$ is a complex of abelian sheaves bounded below. One reasons by induction on $n$. The conclusion is satisfied for $n$ sufficiently small, since $F$ is bounded below. So suppose that $pro{f}_Y(F)⩾n-1$ and let us show that $pro{f}_Y(F)⩾n$, knowing that, for every point $y$ of $Y$, one has $pro{f}_y(F)⩾n$. It suffices to see that the canonical morphism

(∗)    H^{n−2}(X, F) → H^{n−2}(U, F)

is surjective and that

(∗∗)   H^{n−1}(X, F) → H^{n−1}(U, F)

is injective (the result applying when one replaces $X$ by a scheme étale over $X$).

a) Surjectivity of (∗). The proof is analogous to that of 2°). Taking 1.5 and (SGA 4 V 4.5) into account, one has the exact sequence

H^{n−2}(X, F) → H^{n−2}(U, F) --d--> H^{n−1}_Y(X, F) = H⁰(X, ℋ^{n−1}_Y(F)).

Let $s\in H^{n-2}(U,F)$ and $V\supset U$ the largest open set over which $d(s)=0$, which is also the largest open set such that $s$ extends to $H^{n-2}(V,F)$. Suppose $V\ne X$ and let $y$ be a maximal point of $X-V$ and $\bar{X}$ the strict localization of $X$ at a geometric point ȳ above $y$. Since $i:U\to X$ is quasi-compact, the formation of ${\mathcal{H}}_Y^{n-1}(F)$ commutes with the base change $\bar{X}\to X$ and one therefore has (with obvious notations) the exact sequence

H^{n−2}(X̄, F̄) → H^{n−2}(Ū, F̄) --d--> H^{n−1}_Ȳ(X̄, F̄) = (ℋ^{n−1}_Y(F))_ȳ,

the last equality resulting from the retrocompactness hypothesis on $U$.

Now one has by hypothesis the isomorphism

H^{n−2}(X̄, F̄) ⥲ H^{n−2}(X̄ − ȳ, F̄) = H^{n−2}(V̄, F̄);

consequently the image $\bar{s}$ of $s$ in $H^{n-2}(\bar{U},\bar{F})$, which extends (by definition of $V$) to $H^{n-2}(\bar{V},\bar{F})$, also extends to $H^{n-2}(\bar{X},\bar{F})$; but this shows that $d(\bar{s})=0$, that is, that $d(s)$ is null at $y$, which is absurd.

b) Injectivity of (∗∗). Using the surjectivity of (∗), one obtains the exact sequence

0 → H⁰(X, ℋ^{n−1}_Y(F)) → H^{n−1}(X, F) → H^{n−1}(U, F)

and one must show that every element $s\in H^0(X,{\mathcal{H}}_Y^{n-1}(F))$ is null. Let $V$ be the largest open set over which $s=0$. Suppose $V\ne X$ and let $y$ be a maximal point of $X-V$, $\bar{X}$ a strict localization of $X$ at a geometric point ȳ above $y$. By the inductive hypothesis and by 1.8.1, one has the relation $pro{f}_{\bar{Y}}(\bar{F})⩾pro{f}_Y(F)⩾n-1$, whence the fact that the map $e$ in the diagram below is injective:

H⁰(X̄, ℋ^{n−1}_Ȳ(F̄)) = (ℋ^{n−1}_Y(F))_ȳ --e--> H^{n−1}(X̄, F̄) --f--> H^{n−1}(V̄, F̄).

The same holds for $f$ by virtue of the hypothesis; the left equality follows from the retrocompactness hypothesis on $U$. Let $\bar{s}$ be the image of $s$ in $({\mathcal{H}}_Y^{n-1}(F){)}_{\bar{y}}$; since $s$ vanishes over $V$, one has $f\cdot e(\bar{s})=0$, whence $\bar{s}=0$, which contradicts the choice of $y$ and completes the proof.

Remark 1.9.

A result analogous to 1.8 is doubtless valid in the case where one replaces the étale topos of a scheme $X$ by a "topos locally of finite type", that is, definable by a site locally of finite type (SGA 4 VI 1.1). To see this, one must use a result of P. Deligne (SGA 4 VI.9), asserting that there are "sufficiently many fiber functors" in such a topos.

We are going to deduce from 1.8 important cases where one can determine the étale depth.

Theorem 1.10 (Cohomological semi-purity theorem)⁴.

Denote by $X$ either a smooth scheme over a field $k$, or a regular excellent scheme (EGA IV 7.8.2) of characteristic zero (N.B. if one admits resolution of singularities in the sense of (SGA 4 XIX), it suffices to suppose, more generally, that $X$ is a regular excellent scheme of equal characteristic). Let $Y$ be a closed part of $X$ and $L$ the set

of prime numbers distinct from the characteristic of $X$. Then one has

$$pro{f}_Y^L(X)=2codim(Y,X).$$

Proof. It follows from 1.8 that one has

prof_Y^L(X) = inf_{y ∈ Y} prof_y^L(X).

Since on the other hand codim(Y, X) = inf_{y ∈ Y} dim O_{X,y}, one is reduced to showing that

$$pro{f}_y^L(X)=2\dim O_{X,y},$$

which follows from (SGA 4 XVI 3.7 and XIX 3.2).

Theorem 1.11 (Homotopical purity theorem)⁵.

If $X$ is a locally noetherian scheme that is regular (resp. whose local rings are complete intersections), $Y$ a closed part of $X$ such that $codim(Y,X)⩾2$ (resp. $codim(Y,X)⩾3$), then one has

$$pro{f}_Y^{hop}(X)⩾3.$$

It indeed follows from 1.8 that one has prof_Y^{hop}(X) = inf_{y ∈ Y} prof_y^{hop}(X). Now the strictly local rings of $X$ at the various points of $Y$ are regular rings of dimension $⩾2$ (resp. complete intersections of dimension $⩾3$). It then follows from the purity theorem X 3.4 that $pro{f}_y^{hop}(X)⩾3$, which proves the theorem.

Example 1.12.

Let $X$ be a locally noetherian scheme, $Y$ a closed part of $X$, and $n=1$ or 2. Then, if $pro{f}_Y(O_X)⩾n$ ($pro{f}_Y(O_X)$ denoting the $Y$-depth in the sense of coherent sheaves, cf. 1.6 a)), one also has $pro{f}_Y(X)⩾n$; this is evident for $n=1$ and, for $n=2$, it is none other than Hartshorne's theorem

(III 1). On the other hand, the analogous assertion is false for $n⩾3$. Take for example an

affine space of dimension $⩾3$ over a field of characteristic $\ne 2$ and let $\mathbb{Z}/2\mathbb{Z}$ act by symmetry with respect to the origin. Let $X$ be the quotient and $Y=x$ the image of the origin in $X$. Then $O_{X,x}$ is a Cohen-Macaulay ring, hence one has $pro{f}_x(O_X)⩾3$; but the affine space minus the origin is an étale covering of $X-x$ that does not extend to an étale covering of $X$; hence one has by 1.4 $pro{f}_Y(X)=2$.

The following theorem is the analogue of (EGA IV 6.3.1):

Theorem 1.13.

Let $f:X\to S$ be a morphism of schemes, $Y$ a closed part of $X$, $Z$ a closed part of $S$ such that $f(Y)\subset Z$. Suppose that the local rings of $X$ at the various points of $Y$ are noetherian and that the open sets $X-Y$ and $S-Z$ are retrocompact in $X$ and $S$ respectively. Let p, q, r be integers such that $p⩾-r$, $q⩾0$, $L$ a set of prime numbers and $F$ a complex of abelian sheaves of $L$-torsion on $S$ such that the cohomology sheaves $H^i(F)$ are null for $i<-r$. Suppose that

a) The morphism $f$ is locally $(p+q+r-2)$-acyclic for $L$ (SGA 4 XV 1.11).

b) One has

$$pro{f}_Z(F)⩾p.$$

c) For every point $s$ of $Z$, one has

$$pro{f}_{Y_s}^L(X_s)⩾q.$$

Then one has

prof_Y(f^* F) ⩾ p + q.

We shall need the following lemma:

Lemma 1.13.1.

Let $L$ be a set of prime numbers, $n$ and $r$ integers, $f:X\to S$ a morphism locally $n$-acyclic for $L$. Let $F$ be a complex of abelian sheaves with cohomology sheaves of $L$-torsion such that $H^i(F)=0$ for $i<-r$, $Z$ a closed part of $S$ such that $S-Z$ is retrocompact in $S$, and $T=f^{-1}(Z)$. Then the canonical morphism

f^* (ℋ^i_Z(F)) → ℋ^i_T(f^* F)

is bijective for $i<n-r+2$ and injective for $i=n-r+2$.

Set $U=S-Z$ and $V=X-T$, so that one has the cartesian square

V --g--> U
|        |
k        j
|        |
v        v
X --f--> S

Consider the commutative diagram below whose rows are exact

→ f^*(ℋ^i_Z(F))    → f^*(H^i(F))    → f^*(H^i(R j_*(j^* F)))    →
                       ≀
→ ℋ^i_T(f^* F)     → H^i(f^* F)     → H^i(R k_*(k^* f^* F))     → ;

it follows that one is reduced to showing that the morphism

f^*(H^i(R j_*(j^* F))) → H^i(R k_*(k^* f^* F))

is bijective for $i<n-r+1$ and injective for $i=n-r+1$. Now such a morphism comes from the following morphism between hypercohomology spectral sequences

f^* E_2^{p,q} = f^* (R^p j_*(H^q(j^* F)))     ⟹ f^*(H^*(R j_*(j^* F)))
       ↓                                         ↓
E′_2^{p,q} = R^p k_*(H^q(k^* f^* F))           ⟹ H^*(R k_*(k^* f^* F)).

Since $j$ is quasi-compact, it follows from (SGA 4 XV 1.10) that the morphism $f^{\ast }(E_2^{p,q})\to E_2^{\prime p,q}$ is bijective for $p⩽n$ and injective for $p=n+1$; in particular it is bijective for $p+q⩽n-r$ and injective for $p+q=n-r+1$. The conclusion follows immediately.

Let us return to the proof of 1.13. Let $T=f^{-1}(Z)$. By 1.13.1 and condition a), the canonical morphism $f^{\ast }({\mathcal{H}}_Z^i(F))\to {\mathcal{H}}_T^i(f^{\ast }F)$ is an isomorphism for $i⩽p+q$. It therefore follows from b) that ${\mathcal{H}}_T^i(f^{\ast }F)=0$ for $i<p$ and, for $i<p+q$, ${\mathcal{H}}_T^i(f^{\ast }F)$ restricted to $T$ is the inverse image of a sheaf $G_i$ on $Z$. Let

$$f_T:T\to Z$$

be the restriction of $f$ to $T$. It then follows from c) and from the corollary that follows that ${\mathcal{H}}_Y^j(f_T^{\ast }(G_i))=0$ for $j<q$. One concludes that

ℋ^j_Y(ℋ^i_T(f^* F)) = 0 for i + j < p + q,

since the inequality $i+j<p+q$ entails either $i<p$ and then ${\mathcal{H}}_T^i(f^{\ast }F)=0$, or $j<q$ and then ${\mathcal{H}}_Y^j({\mathcal{H}}_T^i(f^{\ast }F))=0$. Given that one has, with the notations of 1.0, ${\Gamma }_Y={\Gamma }_Y\cdot {\Gamma }_T$, one has the spectral sequence

(1.13.2)    E_2^{i,j} = ℋ^j_Y(ℋ^i_T(f^* F)) ⟹ ℋ^{i+j}_Y(f^* F);

since $E_2^{i,j}=0$ for $i+j<p+q$, one sees that ${\mathcal{H}}_Y^k(f^{\ast }F)=0$ for $k<p+q$.

The theorem will therefore be proved if one proves the following corollary (which is the particular case of 1.13 obtained by taking $Z=S$, $r=p=0$ and $F$ reduced to degree 0).

Corollary 1.14.

Let $f:X\to S$ be a morphism, $Y$ a closed part such that the complementary open set $X-Y$ is retrocompact in $X$ and that the local rings of $X$ at the various points of $Y$ are noetherian. Let $L$ be a set of prime numbers, $q$ an integer, and $F$ an abelian sheaf of $L$-torsion on $S$. Suppose

that $f$ is locally $(q-2)$-acyclic for $L$ and that, for every point $s$ of $S$, one has $pro{f}_{Y_s}^L(X_s)⩾q$. Then one has $pro{f}_Y(f^{\ast }F)⩾q$.

1°) Reduction to the case where $X$ and $S$ are strictly local schemes, $f$ a local morphism, and $Y$ reduced to a closed point of $X$.

By 1.8, to establish 1.14, one must show that one has for every point $y$ of $Y$:

$$pro{f}_y(f^{\ast }F)⩾q.$$

Let $s=f(y)$, $\bar{s}$ a geometric point above $s$, ȳ a geometric point above $y$ and over $\bar{s}$, $\bar{X}$ and $\bar{S}$ the strict localizations of $X$ and $S$ at ȳ and $\bar{s}$ respectively, $\bar{f}:\bar{X}\to \bar{S}$ the canonical morphism, and $\bar{F}$ the inverse image of $F$ on $\bar{S}$. Since one has the relation $pro{f}_y(f^{\ast }F)=pro{f}_{\bar{y}}({\bar{f}}^{\ast }\bar{F})$, it suffices to show that the hypotheses of 1.14 are preserved when one replaces $f$ (resp. $Y$, resp. $F$) by $\bar{f}$ (resp. $\bar{y}$, resp. $\bar{F}$). The retrocompactness condition

follows from the noetherian hypothesis on $O_{X,x}$, which implies that $\bar{X}$ is noetherian. By (SGA 4 XV 1.10 (i)), $\bar{f}$ is still locally $(q-2)$-acyclic for $L$. Moreover the fiber $(\bar{X}{)}_{\bar{s}}$ of $\bar{X}$ over $\bar{s}$ is identified with the strict localization of $X_s$ at ȳ, hence satisfies the relation $pro{f}_{\bar{y}}((\bar{X}{)}_{\bar{s}})⩾q$. Since an analogous relation is trivially verified for the fibers of the $\bar{S}$-scheme $\bar{X}$ other than the closed fiber, this completes the reduction.

2°) Case where $X$ and $S$ are strictly local, $f$ a local homomorphism, and $Y$ reduced to the closed point of $X$. Let

g : U = X − {y} → S

be the structural morphism of $U$. One must show that the canonical morphism

u_i : H^i(X, f^* F) → H^i(U, f^* F)

is bijective for $i⩽q-2$ and injective for $i=q-1$. Consider the commutative diagram

H^i(X, f^* F) --u_i--> H^i(U, f^* F)
       ↖ v_i           ↗ w_i
          H^i(S, F)

The morphism $v_i$ is evidently bijective for every $i$. On the other hand $g$ is locally $(q-2)$-acyclic for $L$; moreover its fibers are $(q-2)$-acyclic for $L$, as follows from the fact that $pro{f}_y(X_s)⩾q$ and that the fibers of $f$ are $(q-2)$-acyclic for $L$; since $g$ is quasi-compact (as $X$ is noetherian), it follows from (SGA 4 XV 1.16) that $g$ is $(q-2)$-acyclic for $L$. Consequently $w_i$, hence also $u_i$, is bijective

for $i⩽q-2$ and injective for $i=q-1$, which completes the proof of 1.14.

Corollary 1.15.

Let $f:X\to S$ be a morphism of schemes, $L$ a set of prime numbers, $m$ and $r$ integers, and $F$ a complex of abelian sheaves of $L$-torsion on $S$ such that $H^i(F)=0$ for $i<-r$. Let $x$ be a point of $X$, $s=f(x)$

and suppose that the local ring $O_{X,x}$ is noetherian. Then, if $f$ is locally $m$-acyclic for $L$, one has the relation

(∗)    prof_x(f^* F) ⩾ inf(prof_s(F) + prof_x^L(X_s), n)   where n = m − r + 2.

In particular, if $n⩾pro{f}_s(F)+pro{f}_x^L(X_s)$, for example if $f$ is locally acyclic for $L$, one has

(∗∗)   prof_x(f^* F) ⩾ prof_s(F) + prof_x^L(X_s).

If $L$ is reduced to one element $\ell$ and if one has $n⩾pro{f}_s(F)+pro{f}_x^L(X_s)$, the preceding inequality is an equality.

One reduces to the case where $s$ and $x$ are closed points by taking the strict localizations of $S$ and $x$ at geometric points $\bar{s}$ above $s$ and $\bar{x}$ above $x$ and $\bar{s}$. If one has the inequality $n⩾pro{f}_s(F)+pro{f}_x^L(X_s)$, then (∗) is obtained from 1.13 by taking $p=pro{f}_s(F)$ and $q=pro{f}_x^L(X_s)$ (the hypothesis that $S-s$ is retrocompact in $S$ follows from the fact that $X-X_s$ is retrocompact in $X$ and that $f$ is surjective since it is $(-1)$-acyclic (except perhaps if the conclusion of 1.15 is empty)). If $n<pro{f}_s(F)+pro{f}_x^L(X_s)$, the inequality (∗) is again obtained from 1.13 by taking for example $p=pro{f}_s(F)$ and $q=n-p$. It remains to prove the last assertion. Let $p=pro{f}_s(F)$ and $q=pro{f}_x(X_s)$; it follows from (1.13.2) that one has

$$H_x^{p+q}(f^{\ast }(F))\simeq H_x^q(f^{\ast }(H_s^p(F))).$$

Since $pro{f}_s(F)=p$, the sheaf $H_s^p(F)$ is a sheaf of $\ell$-torsion, constant on $s$, non-zero. Consequently the sheaf $G=f^{\ast }(H_s^p(F))$ is a sheaf of $\ell$-torsion, constant on $X_s$, non-zero, hence contains a subsheaf isomorphic to $\mathbb{Z}/\ell \mathbb{Z}$; since $H_x^q(\mathbb{Z}/\ell \mathbb{Z})$ is non-zero, one indeed has $H_x^q(G)\ne 0$.

Corollary 1.16.

Let $f:X\to S$ be a regular morphism of excellent schemes (EGA IV 7.8.2) of characteristic zero, $\ell$ a prime number, and $F$ a complex of sheaves of $\ell$-torsion on $S$. Let $x\in X$, $s=f(x)$; then one has

prof_x(f^* F) = prof_s(F) + 2 dim(O_{X_s,x}).

Indeed $f$ is locally acyclic (SGA 4 XIX 4.1). It then follows from 1.15 that one has

prof_x(f^* F) = prof_s(F) + prof_x(X_s).

Now one has by 1.10

$$pro{f}_x(X_s)=2\dim O_{X_s,x},$$

whence the result.

Remark 1.17.

It follows from 1.15 that 1.13 remains valid when one replaces b) and c) by the conditions:

b′) For every point $s\in f(Y)$, one has $pro{f}_s(F)⩾p$.

c′) For every point $x\in Y$, if $s=f(x)$, one has $pro{f}_x^L(X_s)⩾q$.

In the case of a sheaf of sets or of groups, one has the following theorem analogous to 1.13.

Theorem 1.18.

Let $f:X\to S$ be a morphism of schemes, $Y$ a closed part of $X$ such that $X-Y$ is retrocompact in $X$ and that, for every point $x$ of $Y$, the local ring $O_{X,x}$ is noetherian.

1°) Let $F$ be a sheaf of sets on $S$ and $n$ an integer equal to 1 or 2. Suppose that $f$ is locally $(n-2)$-acyclic and that, for every point $s$ of $f(Y)$, one has:

$$pro{f}_{Y_s}(X_s)+pro{f}_s(F)⩾n.$$

Then one has:

$$pro{f}_Y(f^{\ast }F)⩾n.$$

2°) Let $L$ be a set of prime numbers and $F$ a sheaf of ind-$L$-groups. Suppose that $f$ is locally 1-aspherical for $L$ (SGA 4 XV 1.11) and that, for every point $s$ of $f(Y)$, one has:

prof_{Y_s}^{hopL}(X_s) + prof_s(F) ⩾ 3.

Then one has:

$$pro{f}_Y(f^{\ast }F)⩾3.$$

Proof. One reduces, as in 1.14 and 1.15, to the case where $X$ and $S$ are strictly local schemes, $f$ a local homomorphism, and $Y$ the closed point $x$ of $X$. Let $s=f(x)$ be the closed point of $S$; one has the commutative diagram:

X − X_s --i--> X − {x} --j--> X
        \         |          /
         \        g         /
          \       |        / f
           \      v       /
            \   S − {s}  /
             \    |k    /
              \   v    /
                  S

1°) a) Case $n=1$.

If $pro{f}_s(F)⩾1$, then the morphism $F\to k_{\ast }{k}^{\ast }F$ is injective, hence the morphism $f^{\ast }F\to f^{\ast }(k_{\ast }{k}^{\ast }F)$ is also injective. On the other hand it follows from the fact that $f$ is locally $(-1)$-acyclic that the morphism $f^{\ast }(k_{\ast }{k}^{\ast }F)\to (j\cdot i{)}_{\ast }(f^{\ast }F{|}_{X-X_s})$ is injective. Finally, the composite morphism $f^{\ast }F\to (j\cdot i{)}_{\ast }(f^{\ast }F{|}_{X-X_s})$ is injective, which shows that one has $pro{f}_{X_s}(f^{\ast }F)⩾1$, hence also $pro{f}_x(f^{\ast }F)⩾1$.

If $pro{f}_x(X_s)⩾1$, one considers the commutative diagram

                          v
H⁰(X, f^* F)          ----→    H⁰(X − {x}, f^* F)
    | ≀                            ↑
    v                              |
H⁰(X_s, f^* F)        --v′--→ H⁰(X_s − {x}, f^* F);
(∗)

By hypothesis, $v^{\prime }$ is injective, hence the same holds for $v$.

b) Case $n=2$. One considers the commutative diagram

            u
H⁰(S, F)  ----→   H⁰(S − {s}, F)
   m ≀                  n ≀
   |                      |
   v                      v
            v                     w
H⁰(X, f^* F) --→ H⁰(X − {x}, f^* F) --→ H⁰(X − X_s, f^* F);
(∗∗)

one must show that $v$ is bijective. The morphism $m$ is evidently bijective, and, since $f$ is 0-acyclic, $n$ is also bijective.

If $pro{f}_s(F)⩾2$, $u$ is bijective. As one has seen in a), the single hypothesis $pro{f}_s(F)⩾1$ entails the relation $pro{f}_{X_s}(f^{\ast }F)⩾1$; consequently $v$ and $w$ are injective; it then follows from (∗∗) that $v$ is bijective.

If $pro{f}_x(X_s)⩾2$, then $g$ is 0-acyclic (since it is locally 0-acyclic and its fibers are 0-acyclic). It follows that $v\cdot m$ is bijective, hence $v$ is bijective.

If $pro{f}_s(F)⩾1$ and $pro{f}_x(X_s)⩾1$, then one already knows that $v$ and $w$ are injective. Let $z$ be a maximal point of $X_s-x$ (such a point exists by the hypothesis $pro{f}_x(X_s)⩾1$), $\bar{Z}$ the strict localization of $X$ at a geometric point above $z$, and $f^{\ast }\bar{F}$ the inverse image of $f^{\ast }F$ on $\bar{Z}$. Consider the commutative diagram

H⁰(S, F) ----→ H⁰(S − {s}, F)
   |m ≀                |n ≀
   v                    v
H⁰(X, f^* F) --v--→ H⁰(X − {x}, f^* F) --w--→ H⁰(X − X_s, f^* F)
   |m′ ≀                |r                       |
   v                    v                        v
H⁰(Z̄, f^* F̄) ----→  H⁰(Z̄ − {z̄}, f^* F̄)

The morphism $m^{\prime }\cdot m$ is evidently bijective, and it follows from the fact that $f$ is locally 0-acyclic that $n^{\prime }\cdot n$ is bijective; consequently $m^{\prime }$ and $n^{\prime }$ are also bijective. Since $w$ is injective, $r$ is also injective, and consequently $v$ is bijective.

2°) Taking b) into account, one already knows that $pro{f}_x(f^{\ast }F)⩾2$.

If $pro{f}_s(F)⩾3$, then $R^1{k}_{\ast }(k^{\ast }F)=1$⁶. Since $f$ is locally 1-aspherical,

one has $R^1(j\cdot i{)}_{\ast }(f^{\ast }F{|}_{X-X_s})=f^{\ast }(R^1{k}_{\ast }(k^{\ast }F))=1$. One therefore has $pro{f}_{X_s}(f^{\ast }F)⩾3$ and consequently $pro{f}_x(f^{\ast }F)⩾3$.

If $pro{f}_x(X_s)⩾3$, then $g$ is 1-aspherical (since $g$ is locally 1-aspherical and its fibers are 1-aspherical). One therefore has $H^1(X-x,f^{\ast }F)=H^1(S,F)=1$ and consequently $pro{f}_x(f^{\ast }F)⩾3$.

If $pro{f}_s(F)⩾2$ and $pro{f}_x(X_s)⩾1$, one uses the exact sequence (SGA 4 XII 3.2):

1 → R¹ j_*(i_*(f^* F|_{X − X_s})) → R¹(j · i)_*(f^* F|_{X − X_s}) → j_*(R¹ i_*(f^* F|_{X − X_s})).

Since $f$ and $g$ are locally 1-aspherical, one has

R¹(j · i)_*(f^* F|_{X − X_s}) ≃ f^*(R¹ k_*(k^* F))
R¹ i_*(f^* F|_{X − X_s}) ≃ g^*(R¹ k_*(k^* F));

the preceding exact sequence then writes in the form

(∗∗∗)  1 → R¹ j_*(i_*(f^* F|_{X − X_s})) → f^*(R¹ k_*(k^* F)) --a--> j_*(j^*(f^*(R¹ k_*(k^* F)))).

The hypothesis $pro{f}_s(F)⩾2$ shows that the morphism $F\to k_{\ast }{k}^{\ast }F$ is bijective; applying $g^{\ast }$, one finds, taking into account the fact that $g$ is locally 0-acyclic, $f^{\ast }F{|}_{X-x}=i_{\ast }(f^{\ast }F{|}_{X-X_s})$. The hypothesis $pro{f}_x(X_s)⩾1$ shows that the morphism $a$ is injective (note that $f^{\ast }(R^1{k}_{\ast }(k^{\ast }F))$ is a sheaf equal to 1 outside $X_s$ and constant on $X_s$). It then follows from (∗∗∗) that one has $R^1{j}_{\ast }(f^{\ast }F{|}_{X-x})=1$, hence

$pro{f}_x(f^{\ast }F)⩾3$.

If $pro{f}_s(F)⩾1$ and $pro{f}_x(f^{\ast }F)⩾2$, one considers the sheaf of homogeneous spaces $G$ defined by the exact sequence

1 → F → k_* k^* F → G → 1.

Applying to this exact sequence the exact functor $g^{\ast }$ and using (SGA 4 XII 3.1), one obtains the following commutative diagram whose rows are exact:

f^*(k_* k^* F)       →     f^* G                   → 1
        |                    |b
        v                    v u
j_*(g^*(k_* k^* F)) --u-->  j_*(g^* G)      → R¹ j_*(g^* F) → R¹ j_*(g^*(k_* k^* F)).

Since $pro{f}_x(X_s)⩾2$, the morphism $b$ is bijective hence $u$ is surjective, and one thus has a map with kernel reduced to the neutral element:

1 → R¹ j_*(g^* F) → R¹ j_*(g^*(k_* k^* F)) = R.

Since $g^{\ast }(k_{\ast }{k}^{\ast }F)\simeq i_{\ast }(f^{\ast }F{|}_{X-X_s})$ (because $g$ is locally 0-acyclic), $R$ is identified with the first term of the exact sequence (∗∗∗); now one saw in the preceding case that $R=1$ as soon as one has $pro{f}_x(X_s)⩾1$, which shows that $pro{f}_x(f^{\ast }F)⩾3$ and completes the proof of 1.18.

The following corollaries are generalizations of (SGA 4 XVI 3.2 and 3.3).

Corollary 1.19.

Let $f:X\to S$ be a flat morphism with separable fibers of locally noetherian schemes,

and $Y$ a closed part of $X$. Suppose that for every point $s\in f(Y)$, the fiber $Y_s$ is rare⁷ in $X_s$ and that one of the two following conditions is verified:

a) the closure of $f(Y)$ is rare in $S$.

b) $X_s$ is geometrically unibranch at the points of $Y_s$.

Then one has

$$pro{f}_Y(X)⩾2.$$

It indeed follows from the hypothesis on $f$ that $f$ is locally 0-acyclic (SGA 4 XV 4.1). One then applies 1.13. The hypothesis $Y_s$ rare in $X_s$ (resp. $f(Y)$ rare in $S$) is by 1.6 b) equivalent to the relation $pro{f}_{Y_s}(X_s)⩾1$ (resp. $pro{f}_{f(Y)}(S)⩾1$). The hypothesis $X_s$ geometrically unibranch at each point of $Y_s$ is equivalent to saying that the strict localization of $X_s$ at a geometric point of $Y_s$ is irreducible; knowing that $Y_s$ is rare in $X_s$, this evidently entails $pro{f}_{Y_s}(X_s)⩾2$, by 1.8. In either case 1.13 indeed gives $pro{f}_Y(X)⩾2$.

Corollary 1.20.

Let $f:X\to S$ be a regular morphism (EGA IV 6.8.1) of locally noetherian schemes, $Y$ a closed part of $X$. Suppose that, for every point $s\in f(Y)$, one of the following conditions is realized:

a) One has $codim(Y_s,X_s)⩾2$.

b) One has $codim(Y_s,X_s)⩾1$ and $pro{f}_s(S)⩾1$.

c) One has $pro{f}_s^{hop}(S)⩾3$.

Then one has

$$pro{f}_Y^{hop}(X)⩾3.$$

This indeed follows from 1.18, given that hypothesis a) implies $pro{f}_{Y_s}^{hop}(X_s)⩾3$ (cf. 1.11), and that the condition $codim(Y_s,X_s)⩾1$ evidently implies $pro{f}_Y(X)⩾2$.

2. Technical lemmas

2.1. Let $S$ be a locally noetherian scheme, $f:X\to S$ a morphism locally of finite type, $t$ a point of $S$. If $x\in X$ is such that $s=f(x)\in \mathrm{Spec}{O}_{S,t}$, one sets

δ_t(x) = deg.tr. k(x)/k(s) + dim({s}),

where $s$ denotes the closure of $s$ in $\mathrm{Spec}{O}_{S,t}$, $k(x)$ and $k(s)$ the residue fields of $x$ and $s$ respectively. If $S$ is a local ring with closed point $t$, one writes also $\delta (x)$ instead of ${\delta }_t(x)$ (cf. SGA 4 XIV 2.2).

Lemma 2.1.1.

Let

X′ --h--> X
|         |
f′        f
|         |
v         v
S′ --g--> S

be a cartesian square, where $S$ and $S^{\prime }$ are noetherian local rings with closed points $t$ and $t^{\prime }$ respectively, $g$ a faithfully flat morphism such that $g^{-1}(t)=t^{\prime }$, $f$ a morphism locally

of finite type. Let $x^{\prime }\in X^{\prime }$, $x=h(x^{\prime })$, $s=f(x)$, $s^{\prime }=f^{\prime }(x^{\prime })$; then one has

$$\delta (x^{\prime })⩽\delta (x).$$

Moreover the preceding inequality is an equality if and only if one has:

deg.tr. k(x)/k(s) = deg.tr. k(x′)/k(s′)   and   dim({s}) = dim({s′}).

In particular, given $x\in X$, one can find $x^{\prime }$ such that $\delta (x)=\delta (x^{\prime })$.

One has indeed (EGA IV 6.11)

$$\dim (s)=\dim g^{-1}(s).$$

It follows that, for every point $s^{\prime }$ of $g^{-1}(s)$, one has the relation $\dim (s^{\prime })⩽\dim (s)$, and that, $s$ being given, one can find $s^{\prime }\in g^{-1}(s)$ such that one has the equality. Denote then by $Z$ the schematic closure of $x$ in the fiber $X_s$ of $X$ at $s$, and let $Z^{\prime }=Z{\times }_{\mathrm{Spec}k(s)}\mathrm{Spec}k(s^{\prime })$. Then $Z^{\prime }$ is equidimensional of dimension $deg.tr.k(x)/k(s)$; one therefore has, for every point $x^{\prime }\in Z_x^{\prime }$,

deg.tr. k(x′)/k(s′) ⩽ deg.tr. k(x)/k(s),  with equality

when $x^{\prime }$ is a maximal point of $Z_x^{\prime }$. Whence immediately the announced conclusion.

2.2. Let $f:X\to S$ be a morphism locally of finite type and $T$ a closed part of $S$. Let $x\in X$, $s=f(x)$; we shall set

δ_T(x) = deg.tr. k(x)/k(s) + codim({s} ∩ T, {s}) = inf_{t ∈ T ∩ {s}} δ_t(x).

Lemma 2.2.1.

Let

X′ --h--> X
|         |
f′        f
|         |
v         v
S′ --g--> S

be a cartesian square, where the schemes $S$ and $S^{\prime }$ are locally noetherian, catenary, the morphism $f$ locally of finite type, and $g$ faithfully flat. Let $T$ be a closed part of $S$, $T^{\prime }$ a closed part of $S^{\prime }$ such that $g(T^{\prime })\subset T$, $x^{\prime }$ an element of $X^{\prime }$, $x=h(x^{\prime })$ and

h_{x′} : Spec O_{X′,x′} → Spec O_{X,x}

the morphism induced by $h$. Then one has:

δ_T(x) − δ_{T′}(x′) ⩽ dim h_{x′}^{−1}(x).

Let $s^{\prime }=f^{\prime }(x^{\prime })$, $s=f(x)$. By definition:

δ_T(x) − δ_{T′}(x′) = deg.tr. k(x)/k(s) − deg.tr. k(x′)/k(s′)
                     + codim({s} ∩ T, {s}) − codim({s′} ∩ T′, {s′}).

Since $g$ is faithfully flat, it follows from (EGA IV 6.1.4) that one has

(∗) codim({s} ∩ T, {s}) = codim(g^{−1}({s}) ∩ g^{−1}(T), g^{−1}({s}))
                       ⩽ codim(g^{−1}({s}) ∩ T′, g^{−1}({s}));

since $S^{\prime }$ is catenary, one has, by (EGA 0_IV 14.3.2 b)):

codim({s′} ∩ T′, g^{−1}({s})) = codim({s′} ∩ T′, {s′}) + codim({s′}, g^{−1}({s}))
   = codim({s′} ∩ T′, g^{−1}({s}) ∩ T′) + codim(g^{−1}({s}) ∩ T′, g^{−1}({s})).

One deduces from this relation and (∗)

δ_T(x) − δ_{T′}(x′) ⩽ deg.tr. k(x)/k(s) − deg.tr. k(x′)/k(s′) + codim({s′}, g^{−1}({s})).

Let us compute $codim(s^{\prime },g^{-1}(s))=\dim O_{S_s^{\prime },s^{\prime }}$ (where $S_s^{\prime }$ is the fiber of $S^{\prime }$ at $s$). Let $Z$ be the closed image of $x$ in $X_s$ and $Z^{\prime }\subset X_s^{\prime }$ the scheme defined by the cartesian square