Exposé XII. Applications to projective algebraic schemes

1. Projective duality theorem and finiteness theorem

¹

The following theorem, essentially contained in FAC² (except that at the time Serre did not yet have at his disposal the language of Ext of sheaves of modules³), is the global analogue of the local duality theorem (Exp. IV), which was modelled on it.

Theorem.

Let $k$ be a field, $X=P_k^r$ projective space of dimension $r$ over $k$, and $F$ a variable coherent module on $X$. Then one has an isomorphism of $\partial$-functors in $F$:

Hⁱ(X, F)′ ⥲ Ext^{r−i}(X; F, Ωʳ_{X/k}),

where one sets

$${\Omega }_{X/k}^r=O_{P_k^r}(-r-1).$$

Remark.

Of course, this module is also the module of relative differentials of degree $r$ of $X$ over $k$. In this form, the theorem remains true if $X$ is a proper and smooth scheme over $k$ (for the projective case, see A. Grothendieck, "Théorèmes de dualité pour les faisceaux algébriques cohérents", Séminaire Bourbaki, May 1957).⁴ When $F$ is locally free, one recovers Serre's duality theorem

Hⁱ(X, F)′ ⥲ H^{r−i}(Hom_{O_X}(F, Ωʳ_{X/k})).

Theorem 1.1 (which moreover recovers the case of $X$ projective over $k$, as in loc. cit.) will suffice for our purposes.

The homomorphism (1) is deduced from the Yoneda pairing

Hⁱ(X, F) × Ext^{r−i}(X; F, Ωʳ_{X/k}) → Hʳ(X, Ωʳ_{X/k}),

and from a well-known isomorphism (cf. FAC, or EGA III 2.1.12):

Hʳ(X, Ωʳ_{X/k}) = Hʳ(P^r_k, O_{P^r_k}(−r − 1)) ⥲ k.

To show that (1) is an isomorphism, one proceeds as in the case of the local duality theorem, noting that $H^r(X,F)$, as a functor in $F$, is right exact (since $H^n(X,F)=0$ for $n>r$), and that every coherent module is isomorphic to a quotient of a direct sum of modules of the form $O(-m)$ with $m$ large. This reduces us, by descending induction on $i$, to making the verification for a sheaf of the form $O(-m)$, where it is contained in the well-known explicit computations (FAC, or EGA III 2.1.12). One may moreover assume $-m⩽-r-1$, in which case $H^i(X,O(-m))=0$ for $i\ne r$.

Corollary.

For $F$ coherent and given, and $m$ large enough, one has a canonical isomorphism

Hⁱ(X, F(−m))′ ⥲ H⁰(X, ℰxt^{r−i}_{O_X}(F, Ωʳ_{X/k})(m))

(where ${}^{\prime }$ denotes the vector-space dual).

Indeed, on projective space $X$, for any pair of coherent sheaves $F$, $G$ and for $n$ large enough one has a canonical isomorphism:

Extⁿ(X; F(−m), G) ≅ Extⁿ(X; F, G(m)) ⥲ H⁰(X, ℰxtⁿ_{O_X}(F, G)(m)),

(the isomorphism of the first two terms being trivially true for any $m$), as follows from the spectral sequence of global Ext

Hᵖ(X, ℰxt^q_{O_X}(F, G(m))) ⇒ Ext^•(X; F, G(m)),

which degenerates for $m$ large thanks to the fact that

ℰxt^q_{O_X}(F, G(m)) ≅ ℰxt^q_{O_X}(F, G)(m),

and that the $\mathcal{E}x{t}_{O_X}^q(F,G)$ are coherent sheaves. Hence (5) follows from (6) and (1).

Corollary.

For given i, F, the following conditions are equivalent:

1. $H^i(X,F(-m))=0$ for $m$ large.
2. (i bis) $H^i(X,F(\cdot ))={⨁}_{m\in \mathbb{Z}}{H}^i(X,F(-m))$ is a finitely generated $S$-module, where $S=k[t_0,\cdots ,t_r]$.
3. $\mathcal{E}x{t}_{O_X}^{r-i}(F,{\Omega }_{X/k}^r)=0$.
4. (ii bis) $\mathcal{E}x{t}_{O_X}^{r-i}(F,O_X)=0$.
5. $H_x^i(F_x)=0$ for every closed point $x$ of $X$.
6. $H_x^{i+1}(F_x)=0$ for every closed point $x$ of the punctured projecting cone $X=\mathrm{Spec}(S)-\mathrm{Spec}(k)$ of $X$, where $F$ denotes the inverse image of $F$ under the canonical morphism $X\to X$.

Proof.

(i) ⇔ (i bis) since the submodule of $H^i(X,F(\cdot ))$ formed by the sum of the homogeneous components of degree $⩾\nu$ is finitely generated over $S$ (in fact, for $i\ne 0$, it is even finitely generated over $k$), cf. FAC or EGA III 2.2.1 and 2.3.2.

(i) ⇔ (ii) by virtue of Corollary 1.2.

(ii) ⇔ (ii bis) since ${\Omega }_{X/k}^r$ is locally isomorphic to O_X.

(ii bis) ⇔ (iii) by virtue of the local duality theorem for $O_{X,x}$ (which is a regular local ring of dimension $r$), according to which the "dual" of $\mathcal{E}x{t}_{O_X}^{r-i}(F,O_X{)}_x$ is identified with $H_x^i(F_x)$ (V 2.1).

(ii bis) is equivalent to the analogous relation

$$\mathcal{E}x{t}_{O_X}^{r-i}(F,O_{\tilde{X}})=0$$

(thanks to the fact that $X\to X$ is faithfully flat, so the inverse image of $\mathcal{E}x{t}_{O_X}^{r-i}(F,O_X)$ is isomorphic to $\mathcal{E}x{t}_{O_X}^{r-i}(F,O_X)$), and this last relation is equivalent to (iv) by the local duality theorem for the local ring $O_{\tilde{X},x}$, which is regular of dimension $r+1$.

In particular, applying this to all $i⩽n$, one finds:

Corollary.

Equivalent conditions for given n, F:

1. $H^i(X,F(-m))=0$ for $i⩽n$ and $m$ large.
2. (i bis) $H^i(X,F(\cdot ))$ is a finitely generated $S$-module for $i⩽n$.
3. $prof(F_x)>n$ for every closed point $x$ of $X$.
4. $prof(F_x)>n+1$ for every closed point $x$ of $X$.

The interest of Corollaries 1.3 and 1.4 is to express a global condition (i) or (i bis) in terms of local conditions, namely the vanishing of local invariants such as $H_x^i(X,F_x)$ or $H_x^i(X,F_x)$, or an inequality on depth. In this form, these results remain trivially valid for an arbitrary projective scheme $X$ and a very ample invertible sheaf $O_X(1)$ on $X$, as one sees by inducing the latter using a suitable projective immersion $X\to P_k^r$. (Of course, conditions 1.3 (ii) and 1.3 (ii bis) are no longer equivalent to the others in this general case, except if one assumes for example that $X$ is regular.) One may moreover generalize to the case of a projective morphism $X\to S$ as follows:

Proposition.

Let $f:X\to S$ be a projective morphism with $S$ noetherian, $O_X(1)$ an invertible module on $X$ very ample relatively to $S$, $F$ a coherent module on $X$, flat with respect to $S$, $s$ an element of $S$, $X_s$ the fiber of $X$ at $s$ (considered as a projective scheme over $k(s)$), $F_s$ the sheaf induced on $X_s$ by $F$, finally $i$ an integer. Suppose that for every closed point $x$ of $X_s$, one has $H_x^i(F_{s,x})=0$ (for example $prof(F_{s,x})>i$). Then there exists an open neighborhood $U$ of $s$ such that the same condition is verified for $s^{\prime }\in U$. Moreover, for such a $U$, one has

Rⁱf_∗(F(−m)) = 0 for m large,

and if $\mathcal{S}$ is a graded quasi-coherent algebra on $S$, generated by ${\mathcal{S}}^1$, that defines $X$ together with $O_X(1)$ as $X=\mathrm{Proj}(\mathcal{S})$, $O_X(1)=\mathrm{Proj}(\mathcal{S}(1))$, then the $\mathcal{S}$-module

Rⁱf_∗(F(·)) = ⨁_{m ∈ ℤ} Rⁱf_∗(F(m))

is finitely generated on $U$.

Embed $X$ in some $X^{\prime }=P_S^r$ so that $O_X(1)$ is induced by $O_{X^{\prime }}(1)$ (which is possible, possibly by replacing $S$ by an affine neighborhood of $s$). Set⁵ for every integer $j$ and every $t\in S$:

$$E^j(t)=Ex{t}_{O_{X_t^{\prime }}}^j(F_t^{\prime },O_{X_t^{\prime }}(-r-1)).$$

Thus $E^j(t)$ is a coherent module on $X_t$. I claim that, for variable $t$, the family of these modules is "constructible" in the following sense: for every $t\in S$ there exists a non-empty open subset $V$ of the closure of $t$, which one endows with the induced reduced structure, and a coherent module $E^j(V)$ on $X_V=X{\times }_{S}V$, flat relatively to $V$, such that for every $t^{\prime }\in V$, $E^j(t^{\prime })$ is isomorphic to the module induced by $E^j(V)$ on $X_{t^{\prime }}$. To verify this assertion, setting $Z=t$ with its induced structure, one considers the coherent modules

$$E^j(Z)=Ex{t}_{O_{X_Z^{\prime }}}^j(F_Z,O_{X_Z^{\prime }}(-r-1))$$

(where the subscript $Z$ means again that one induces over $Z$), and one takes for $V$ a non-empty open subset of $Z$ such that the modules $E^j(Z)$ are flat over $V$: this is possible since one checks immediately that $E^j(Z)=0$ for $j$ not lying in the interval [0, r], and one may apply SGA 1 IV 6.11. One then takes $E^j(V)=E^j(Z)|X_V$, and one verifies easily that it answers the question.

From the preceding remark it follows that there exists a finite partition of $S$ formed by sets $V_{\alpha }$ of the form $V=V(t)$ as above (noetherian induction), and applying Serre's theorem EGA III 2.2.1 to the $E^j(V_{\alpha })$, one sees that there exists an integer $m_0$ such that

Rⁱf_{V_α∗}(Eʲ(V_α)) = 0 for i ≠ 0, m ⩾ m₀, for all j,

whence it follows, using the flatness of $E^j(V_{\alpha })$ with respect to $V_{\alpha }$ and easy Künneth-type relations (cf. EGA III, §7), that

Hⁱ(X_t, Eʲ(t)(m)) = 0 for i ≠ 0, m ⩾ m₀, for all j,

for every $t\in V_{\alpha }$, hence for every $t$ since the $V_{\alpha }$ cover $S$. From this and the spectral sequence of global Ext follows, thanks to 1.1 and as in the proof of 1.2, an isomorphism

Hⁱ(X_t, F_t(−m))′ ⥲ H⁰(X_t, E^{r−i}(t)(m)) for m ⩾ m₀,

every integer $i$, and every $t\in S$.

Let us now use the hypothesis on $F_s$, which is written

$$E^{r-i}(s)=0,$$

and which, thanks to (8), is equivalent to

Hⁱ(X_s, F_s(−m)) = 0 for m ⩾ m₀.

Since $F$, hence $F(-m)$, is flat with respect to $S$, it follows by the Künneth-type relations already invoked that (for $m$ given) the same relation (10) holds when $s$ is replaced by a $t$ near $s$, in particular for any generization $t$ of $s$. By virtue of (8), one will therefore have, for such a generization,

$$E^{r-i}(t)=0,$$

now the set of $t\in S$ for which this relation holds is plainly a constructible set (since it induces an open set on each $V_{\alpha }$); since it contains the generizations of $s$, it contains an open neighborhood $U$ of $s$. This proves the first assertion of 1.5. Moreover, for $t\in U$, one concludes from (11) and (8) that

Hⁱ(X_t, F_t(−m)) = 0 for m ⩾ m₀, t ∈ U,

which, by virtue of the Künneth-type relations, implies (in fact, is much stronger than)

Rⁱf_∗(F(−m)) = 0 on U, for m ⩾ m₀.

This proves the second assertion of 1.5. Finally the last assertion follows at once, by proceeding as at the start of the proof of 1.3.

Remark.

⁶

The proof simplifies notably (by eliminating any consideration of constructibility) when one assumes already that the hypothesis made for $s\in S$ is verified at every $s^{\prime }\in S$. In fact, when one makes the hypothesis that $F_s$ is of depth $>i$ at the closed points of $X_s$, one has at one's disposal a general statement, local in nature on $X$, which says that the same condition is verified for all $X_t$, on condition of replacing $X$ by a suitable open neighborhood of the fiber $X_s$ (in other words, a certain part of $X$, defined by conditions on the modules induced by $F$ on the fibers, is open, cf. EGA IV). Since $f$ is proper here, one may therefore take this neighborhood of the form $f^{-1}(U)$, which recovers the first assertion of 1.5 without any tedious dévissage. In this general case, one may still prove by the method of loc. cit. that the first assertion of 1.5 (proved here by global means, using that $X$ is projective over $S$) follows from a purely local statement on $X$ (which the reader will spell out if he thinks it useful).

2. Lefschetz theory for a projective morphism: Grauert's comparison theorem

It is the following theorem:

Theorem.

Let $f:X\to S$ be a projective morphism with $S$ noetherian, $O_X(1)$ an invertible module on $X$, ample relatively to $S$, $Y$ the prescheme of zeros of a section $t$ of $O_X(1)$, $\mathcal{J}$ the ideal defining $Y$, $X_n$ the subprescheme of $X$ defined by ${\mathcal{J}}^{n+1}$, $\hat{X}$ the formal completion of $X$ along $Y$, $\hat{f}:\hat{X}\to S$ the composite morphism $\hat{X}\to X\to S$, $F$ a coherent module on $X$, flat relatively to $S$. Suppose moreover that for every $s\in S$, the module $F_s$ induced on the fiber $X_s$ is of depth $>n$ at the points of that fiber, and that $t$ is $F$-regular. Under these conditions:

1. The canonical homomorphism

   Rⁱf_∗(F) → Rⁱf̂_∗(F̂)
   

   is an isomorphism for $i<n$, a monomorphism for $i=n$.

2. The canonical homomorphism

   Rⁱf̂_∗(F̂) → lim_m Rⁱf_∗(F_m)
   

   is an isomorphism for $i⩽n$.

Proof.

One reduces at once to the case where $S$ is affine, and to proving in this case the following:

Corollary.

Under the conditions of 2.1, suppose moreover that $S$ is affine. Then:

1. The canonical homomorphism

   Hⁱ(X, F) → Hⁱ(X̂, F̂)
   

   is an isomorphism for $i<n$, a monomorphism for $i=n$.

2. The canonical homomorphism

   Hⁱ(X̂, F̂) → lim_m Hⁱ(X_m, F_m)
   

   is an isomorphism for $i⩽n$.

Replacing $O_X(1)$ by a tensor power, and $t$ by a power of $t$ if necessary, one may assume $O_X(1)$ very ample relatively to $S$. On the other hand, $t$, hence tᵐ, being $F$-regular, multiplication by tᵐ, considered as a homomorphism from $F(-m)$ to $F$, is injective; so one has for every $m⩾0$ an exact sequence:

0 → F(−m) ──tᵐ──→ F → F_m → 0,

whence a cohomology exact sequence

Hⁱ(X, F(−m)) → Hⁱ(X, F) → Hⁱ(X, F_m) → H^{i+1}(X, F(−m)).

Now by virtue of 1.5 one has $H^i(X,F(-m))=0$ for $i⩽n$ and $m$ large enough, which proves the following:

Lemma.

For $m$ large, the canonical homomorphism

Hⁱ(X, F) → Hⁱ(X, F_m)

is bijective if $i<n$, injective if $i=n$.

This shows that for $i<n$, the projective system $(H^i(X_m,F_m){)}_{m⩾0}$ is essentially constant, a fortiori satisfies the Mittag-Leffler condition; therefore (taking into account $\hat{F}=\lim F_m$) one concludes (ii) by EGA 0_III 13.3. On the other hand, (i) follows trivially, taking into account 2.3.

Corollary.

⁷

Let $f:X\to S$ be a flat projective morphism with $S$ locally noetherian, $O_X(1)$ an invertible module on $X$, ample relatively to $S$, $t$ a section of this module that is O_X-regular, $Y$ the subprescheme of zeros of $t$, $\hat{X}$ the formal completion of $X$ along $Y$. Suppose that for every $s\in S$, $X_s$ is of depth $⩾1$ (resp. of depth $⩾2$) at its closed points. Then for every open neighborhood $U$ of $Y$, the functor

$$F\mapsto \hat{F}$$

from the category of locally free coherent modules on $U$ to the category of locally free coherent modules on $\hat{X}$ is faithful (resp. fully faithful, i.e. the Lefschetz condition (Lef) of X 2 is verified).

For two locally free modules $F$ and $G$ on $U$ introduce the module

$$H={\mathrm{Hom}}_{O_U}(F,G);$$

one is reduced to proving that the canonical homomorphism

H⁰(U, H) → H⁰(Û, Ĥ)

is injective (resp. bijective). Now the modules $H_t$ are of depth $⩾1$ (resp. $⩾2$) at the closed points of $X_t$; one may therefore apply 2.1, which implies the conclusion of 2.4 in the case where $U=X$. In the case of an arbitrary $U$, one notes that the question is local on $S$, so one may assume $S$ affine. Then every coherent module on $X$ is a quotient of a locally free coherent module (since $O_X(1)$ is a relatively ample invertible module on $X$). Since the dual module $H^{\prime }=\mathrm{Hom}(H,O_U)$ extends to a coherent module on $X$, which is therefore isomorphic to a cokernel of a homomorphism of locally free modules on $X$, it follows by transposition that one may find a homomorphism

$$u^{\prime }:L^{\prime 0}\to L^{\prime 1}$$

of locally free modules on $X$, inducing a homomorphism

$$u:L^0\to L^1$$

of locally free modules on $U$, such that one has an exact sequence

0 → H → L⁰ ──u──→ L¹.

Using the five lemma (which becomes the three lemma), and the left exactness of the functor $H^0$, one is reduced to proving that (15) is injective (resp. bijective) when $H$ is replaced by $L^0$, $L^1$, which reduces us to the case where $H$ is induced by a locally free module $H^{\prime }$ on $X$. Moreover, in the non-respective case this reduction is even unnecessary, since the kernel of (15) is in any case formed of the sections of $H$ on $U$ that vanish in a suitable open neighborhood $V$ of $Y$; now the restriction homomorphism $H^0(U,H)\to H^0(V,H)$ is injective, since $H$ is of depth $⩾1$ at the points of any closed subset $Z$ of $X$ not meeting $Y$ (cf. the lemma below). In the respective case, one is reduced to proving that

H⁰(X, H′) → H⁰(U, H′)

is bijective, which follows from the fact that $H^{\prime }$ is of depth $⩾2$ at every point of a closed subset $Z=X-U$ of $X$ not meeting $Y$. One therefore needs only to prove the following:

Lemma.

Let $F$ be a coherent module on $X$, flat with respect to $S$, such that for every $s\in S$, $F_s$ is of depth $⩾n$ at every closed point of $X_s$. Then for any closed subset $Z$ of $X$ not meeting $Y$, $F$ is of depth $⩾n$ at every point of $Z$.

Indeed, for every $x\in X$, setting $s=f(x)$, one has

$$prof(F_x)⩾prof(F_{s,x}),$$

as one sees by lifting in any way a maximal $F_{s,x}$-regular sequence of elements of $\mathfrak{r}(O_{X_s,x})$, which yields an $F_x$-regular sequence by virtue of SGA 1 IV 5.7. Now if $x$ belongs to a $Z$ as in Lemma 2.5, then $x$ is necessarily closed in $X_s$; in other words, $Z$ is finite over $S$. Indeed $Z$ (endowed with a structure induced by $X$) is projective over $S$ as a closed subprescheme of $X$ which is so, and $Z$ is affine over $S$ as a closed subprescheme of $X-Y$, which is so.

Remark.

Suppose that for every $s\in S$ the section $t_s$ of $O_{X_s}(1)$ induced by $t$ is $O_{X_s}$-regular (which implies, by SGA 1 IV 5.7, that $t$ is O_X-regular). Then the hypotheses made are stable under base extension $S^{\prime }\to S$ ($S^{\prime }$ locally noetherian). Hence the conclusion remains valid after any base change.

3. Lefschetz theory for a projective morphism: existence theorem

Theorem.

⁸

Let $f:X\to S$ be a projective morphism, with $S$ noetherian, $O_X(1)$ an invertible module on $X$ ample relatively to $S$, $X_0$ the subprescheme of zeros of a section $t$ of $O_X(1)$, $\hat{X}$ the formal completion of $X$ along $X_0$, $\mathcal{F}$ a coherent module on $\hat{X}$, ${\mathcal{F}}_0$ the module that it induces on $X_0$. Suppose moreover:

• a) $\mathcal{F}$ is flat with respect to $S$.
• b) For every $s\in S$, the section $t_s$ induced by $t$ on $X_s$ is ${\mathcal{F}}_s$-regular (which implies that ${\mathcal{F}}_0$ is also flat with respect to $S$, cf. SGA 1 IV 5.7).
• c) For every $s\in S$, ${\mathcal{F}}_{0,s}$ is of depth $⩾2$ at the closed points of $X_{0,s}$.

Suppose moreover that $S$ admits an ample invertible sheaf. Under these conditions, there exists a coherent module $F$ on $X$ and an isomorphism of its formal completion $\hat{F}$ with $\mathcal{F}$.

This statement will follow from the following:

Corollary.

Under conditions a), b), c) above, one has the following:

1. The module ${\hat{f}}_{\ast }(\mathcal{F})$ on $S$ is coherent; hence for every $n$, the module ${\hat{f}}_{\ast }(\mathcal{F}(n))$ on $S$ is coherent.

2. For $n$ large, the canonical homomorphism $\hat{f}\ast {\hat{f}}_{\ast }(\mathcal{F}(n))\to \mathcal{F}(n)$ is surjective.

Let us admit the corollary for the moment, and prove 3.1. Thanks to the last hypothesis made in 3.1, one may reduce to the case where $X=P_S^r$, by replacing $O_X(1)$, $t$ by a suitable power. I claim that one may moreover assume that for every $s$, $t_s\ne 0$. Otherwise, indeed, one has ${\mathcal{F}}_s=0$ by b), or what amounts to the same by Nakayama, ${\mathcal{F}}_{0,s}=0$ i.e. $s$ does not belong to the image of $Supp{\mathcal{F}}_0$ by the morphism $f_0:X_0\to S$ induced by $f$. Now this image $S^{\prime }$ is open by virtue of a), b), since ${\mathcal{F}}_0$ is flat with respect to $S$; and it is obvious that it suffices to prove the conclusion of 3.1 in the situation obtained by restricting above $S^{\prime }$, since the coherent module $F^{\prime }$ on $X|S^{\prime }$ obtained will be the restriction of a coherent module $F$ on $X$, which will answer the question. One may therefore assume that, in addition to hypotheses a), b), c), the following hypotheses are also verified:

• a′) O_X is flat with respect to $S$.
• b′) For every $s\in S$, the section $t_s$ is $O_{X_s}$-regular.
• c′) For every $s\in S$, $O_{X_{0,s}}$ is of depth $⩾2$ at the closed points of $X_{0,s}$.

(It suffices to choose $X=P_S^r$ with $r⩾3$, which is permissible.)

Now 3.2 implies that one may find an epimorphism

$$\hat{L}\to \mathcal{F}\to 0,$$

where $L$ is a module on $X$ of the form $f\ast (G)(-n)$, $G$ being a locally free coherent module on $S$: for $n$ large, it suffices indeed to represent the coherent module ${\hat{f}}_{\ast }(\mathcal{F})$ on $S$ as a quotient of such a $G$. On the other hand, the hypotheses a), b), c) on $f$, $t$ imply that $\hat{L}$ satisfies the same conditions a), b), c) as $\mathcal{F}$. One concludes easily that the same holds for the kernel of (17), to which one may therefore apply the same argument; so that $\mathcal{F}$ is represented as a cokernel of a homomorphism

$${\hat{L}}^{\prime }\to \hat{L},$$

where $L$, $L^{\prime }$ are locally free modules on $X$. Now by virtue of a′) and the second part of c′), and of 2.1 or 2.4 as preferred, the homomorphism (18) comes from a homomorphism $L^{\prime }\to L$ of modules on $X$. It suffices now to take for $F$ the cokernel of $L^{\prime }\to L$, and one wins.

It remains to prove 3.2. This had been done in the seminar by a somewhat tedious expedient, consisting in interpreting everything in terms of cohomology on the punctured projecting cone of $X$ relative to $S$, in order to reduce to Theorem 2.1. A more direct and more satisfactory way (although substantially the same) seems to me now the following. It consists in noting that in IX, no. 2 (and with the notation of that exposé), the hypothesis that the morphism $f:\mathcal{X}\to {\mathcal{X}}^{\prime }$ be adic does not intervene anywhere in the proof of 2.1, via EGA 0_III 13.7.7; it suffices to assume in its place that $\mathcal{X}$ is also adic, and to choose two ideals of definition $\mathcal{J}$ for ${\mathcal{X}}^{\prime }$, $\mathcal{I}$ for $\mathcal{X}$, such that $\mathcal{J}{O}_{\mathcal{X}}\subset \mathcal{I}$, and to define $\mathcal{S}=g{r}_{\mathcal{J}}(O_{{\mathcal{X}}^{\prime }})$, and to consider $g{r}_{\mathcal{I}}(\mathcal{F})$. In any case, 2.1 may be applied directly to the morphism $\hat{f}:\hat{X}\to S$ considered in the present section, where one simply takes $\mathcal{J}=0$. Thus, to verify that ${\hat{f}}_{\ast }(\mathcal{F})$ is coherent, it suffices, by virtue of loc. cit., to verify that Rⁱf₀_∗(gr_ℐ(ℱ)) is coherent on $S$ for $i=0,1$; for this one notes that by virtue of a) and b), the module considered is none other than ⨁_{m⩾0} Rⁱf₀_∗(ℱ₀(−m)), which is indeed coherent by virtue of hypothesis c) and of 1.5.

This proves 3.2 (i). For 3.2 (ii), we shall need the following:

Lemma.

Under conditions a), b), c) of 3.1, set

G_m = f̂_∗(ℱ(·)_m) = ⨁_n f_∗(ℱ_m(n)).

Then the projective system $(G_m)$ satisfies the Mittag-Leffler condition.

One may assume $S$ affine, with ring $A$. Let $\mathcal{S}$ then be a finitely generated graded $A$-algebra with positive degrees, and $t^{\prime }\in {\mathcal{S}}_1$, such that $X$ immerses into $\mathrm{Proj}(\mathcal{S})$, $O_X(1)$ being induced by $O(1)$ and the section $t$ being the image of $t^{\prime }$. Equip $\mathcal{S}$ with the $\mathcal{J}$-adic filtration, where $\mathcal{J}=t^{\prime }\mathcal{S}$, and consider the projective system of the $\mathcal{F}(\cdot {)}_m$ in the category of abelian sheaves on $X_0$. One is again under the preliminary conditions of EGA 0_III 13.7.7⁹ and moreover $H^i(X_0,gr(\mathcal{F}(\cdot )))$ is a finitely generated module on $g{r}_{\mathcal{J}}(\mathcal{S})$ for $i=0,1$. Indeed, since $t^{\prime }$ is $\mathcal{F}$-regular, one sees at once that as a module on $(\mathcal{S}/t^{\prime }\mathcal{S})[T]$ (of which $g{r}_{\mathcal{J}}(\mathcal{S})$ is a quotient), the module under consideration is identified with $H^i(X_0,{\mathcal{F}}_0(\cdot )){\otimes }_{\mathcal{S}/t^{\prime }\mathcal{S}}(\mathcal{S}/t^{\prime }\mathcal{S})[T]$; now by virtue of 1.5, $H^i(X_0,{\mathcal{F}}_0(\cdot ))$ is finitely generated on $\mathcal{S}$, hence on $\mathcal{S}/t^{\prime }\mathcal{S}$, for $i=0,1$, which proves our assertion. Consequently one is under the conditions for applying 0_III 13.7.7 with $n=1$, which proves 3.3.

This point being acquired (and assuming $S$ still affine, which is permissible for proving 3.2 (ii)) let $m_0$ be such that $m⩾m_0$ implies $Im(G_m\to G_0)=Im(G_{m_0}\to G_0)$, so that both sides are also equal to $Im(\lim G_m\to G_0)=Im({\hat{f}}_{\ast }(\mathcal{F}(\cdot ))\to f_{\ast }{\mathcal{F}}_0(\cdot ))$. Note now that for $n$ large, ${\mathcal{F}}_{m_0}(n)$ is generated by its sections; hence ${\mathcal{F}}_0(n)$ is generated by sections that lift to ${\mathcal{F}}_{m_0}$, and so (thanks to the choice of $m_0$) that lift to $\mathcal{F}$. So the sections of $\mathcal{F}$ generate ${\mathcal{F}}_0$, hence also $\mathcal{F}$ thanks to Nakayama. This proves 3.2 (ii), hence 3.1.

Corollary.

Let $f:X\to S$ be a flat projective morphism with $S$ locally noetherian, $O_X(1)$ an invertible module on $X$, ample relatively to $S$, $t$ a section of this module such that for every $s\in S$ the section $t_s$ induced on the fiber $X_s$ is $O_{X_s}$-regular, $X_0$ the subprescheme of zeros of $t$, $\hat{X}$ the formal completion of $X$ along $X_0$. Suppose that for every $s\in S$, $X_{0,s}$ is of depth $⩾2$ at its closed points (i.e. $X_s$ is of depth $⩾3$ at the closed points of $X_{0,s}$), and $X_s$ is of depth $⩾2$ at its closed points. Under these conditions, the pair $(X,X_0)$ satisfies the effective Lefschetz condition (Leff) of paragraph 2 of Exposé X, i.e.:

1. For every open neighborhood $U$ of $X_0$ in $X$, the functor

   F ↦ F̂
   

   from the category of locally free coherent modules on $U$ to the category of locally free coherent modules on $\hat{X}$ is fully faithful.

2. For every locally free coherent module $\mathcal{F}$ on $\hat{X}$, there exists an open neighborhood $U$ of $X_0$, and a locally free coherent module $F$ on $U$ such that $\hat{F}$ is isomorphic to $\mathcal{F}$.

Indeed, a) has already been noted in 2.4 under weaker conditions. For b), one applies 3.1, which gives the conclusion, at least if $S$ is noetherian and admits an absolutely ample invertible module, in particular if $S$ is affine. Indeed, if $F$ is a coherent module on $X$ such that $\hat{F}$ is isomorphic to $\mathcal{F}$ and hence locally free, it follows that $F$ is locally free on a neighborhood $U$ of $X_0$, and $F|U$ will satisfy the required condition. But let us now note that by virtue of 2.5, for such an $F$, its image under the immersion $U\to X$ is coherent, and moreover is independent of the chosen solution $(U,F)$ (taking into account the fact that two solutions coincide in a neighborhood of $X_0$, by virtue of a)). Precisely, one may find a coherent module $F$ on $X$ and an isomorphism $\hat{F}\stackrel{\sim }{\to }\mathcal{F}$ such that $F$ is of depth $⩾2$ at every point of $X$ that is closed in its fiber, and this determines $F$ up to a unique isomorphism. Thanks to this uniqueness property, the solutions of the problem found by inducing above the affine open subsets of $S$ glue together, yielding a coherent $F$ on all of $X$ and an isomorphism $\hat{F}\cong \mathcal{F}$. Restricting $F$ to the open subset $U$ of points where it is free, one finds what one was looking for.

Thanks to 2.4 and 3.4, one may exploit, in the situation of a projective algebraic scheme and a "hyperplane section" thereof, the general facts established in Exposés X and XI concerning the conditions (Lef) and (Leff). Thus:

Corollary.

Let $X$ be a projective algebraic scheme equipped with an ample invertible module $O_X(1)$, let $t$ be a section of this module that is O_X-regular, and let $X_0$ be the subscheme of zeros of $t$. Suppose that $X$ is of depth $⩾2$ at its closed points (resp. and of depth $⩾3$ at the closed points of $X_0$). Then ${\pi }_0(X_0)\to {\pi }_0(X)$ is bijective, in particular $X$ is connected if and only if $X_0$ is, and choosing a geometric base point in $X_0$, ${\pi }_1(X_0)\to {\pi }_1(X)$ is surjective, and more generally for every open subset $U\supset X_0$, the homomorphism ${\pi }_1(X_0)\to {\pi }_1(U)$ is surjective (resp. the homomorphism ${\pi }_1(X_0)\to {\lim }_U{\pi }_1(U)$ is bijective). In the respective case, if one assumes moreover that the local ring of every closed point of $X$ not in $X_0$ is pure (3.2) (for example is regular, or only a complete intersection), then ${\pi }_1(X_0)\to {\pi }_1(X)$ is an isomorphism.

One applies 2.4 and 3.4. One will note that in the respective case the hypothesis that $X$ be of depth $⩾3$ at the closed points of $X_0$ implies that all the irreducible components of dimension $\ne 0$ of $X$ are of dimension $⩾3$ (as one sees by noting that any such component necessarily meets $X_0$, and looking at a closed point of the intersection).

Remark.

When $X$ is normal, of dimension $⩾2$ at all its points, it is of depth $⩾2$ at its closed points and one is under the non-respective conditions of 3.5. In this case, one has a more elementary proof of the surjectivity of ${\pi }_1(X_0)\to {\pi }_1(X)$ using Bertini's theorem (cf. SGA 1 X.2.10). If one assumes moreover $X_0$ normal, and $X$ of dimension $⩾3$ at all its points, then one is under the respective conditions of 3.5. In this case, 3.5 was established by Grauert (indeed, thanks to the normality hypothesis, one then succeeds in dispensing with the existence theorem 3.1 by certain expedients). It is this proof of Grauert that was the starting point of the "Lefschetz theory" that is the subject of the present seminar.

Corollary.

Let $X,O_X(1),t,X_0$ be as in 3.5. Suppose that $X$ is of depth $⩾2$ at its closed points, and that

$$H^i(X_0,O_{X_0}(-n))=0$$

for $n>0$ and for $i=1$ (resp. for $i=1$ and for $i=2$), which implies by virtue of 1.4 that $X_0$ is of depth $⩾2$ (resp. $⩾3$) at its closed points, i.e. that $X$ is of depth $⩾3$ (resp. $⩾4$) at the closed points of $X_0$. Under these conditions, for every open neighborhood $U$ of $X_0$, $\mathrm{Pic}(U)\to \mathrm{Pic}(X_0)$ is injective, in particular $\mathrm{Pic}(X)\to \mathrm{Pic}(X_0)$ is injective (resp. lim_U Pic(U) → Pic(X₀) is bijective). In the respective case, if one assumes moreover that the local ring of $X$ at every closed point not in $X_0$ is parafactorial (3.1) (for example is regular, or more generally a complete intersection), then $\mathrm{Pic}(X)\to \mathrm{Pic}(X_0)$ is bijective.

One applies XI 3.12 and 3.13, noting that the respective hypothesis implies that the irreducible components of dimension $\ne 0$ of $X$ are of dimension $⩾4$. One finds in particular, by applying this to the case where $X$ is a global complete intersection of dimension $⩾4$ in projective space:

Corollary.

Let $X$ be an algebraic scheme of dimension $⩾3$, which is a complete intersection in a scheme $P_k^r$. Then $\mathrm{Pic}(X)$ is the free group generated by the class of the sheaf $O_X(1)$.

One reasons by induction on the number of hypersurfaces of which $X$ is the intersection, applying 3.6 and noting that for a complete intersection $X$ of dimension $⩾3$, one has $H^i(X,O_X(n))=0$ for $i=1,2$ and every $n$.

Remark.

In the case where $X$ is a non-singular hypersurface, 3.7 is due to Andreotti. The result 3.7 may also be expressed (when $X$ is non-singular) by saying that the homogeneous coordinate ring of $X$ is factorial, and in this form is contained in XI 3.13 (ii). Let us also point out that Serre had given a proof of 3.7 in the non-singular case, by transcendental means, using a specialization argument to reduce to the case of characteristic 0, where one has the Lefschetz theorem in its classical form. Of course, the fact that the purely algebraic proof given here makes it possible to dispense with non-singularity hypotheses in the statement of Lefschetz's theorem invites one to reconsider the latter also in the classical case. Cf. the following exposé, which proposes conjectures in this direction.

In Corollaries 3.5 and 3.7 we have placed ourselves over a base field, whereas the key theorems 2.4 and 3.4 are valid over an arbitrary base. To generalize Corollaries 3.5 and 3.6 to a general $S$, we must give serviceable criteria for a point of $X$ (flat over $S$) to have a "pure" or, respectively, parafactorial local ring. This will be the object of the following section.

4. Formal completion and normal flatness

Theorem.

Let $X$ be a locally noetherian prescheme, locally immersible in a regular scheme, $Y$ a closed part of $X$, $U=X-Y$, $X_0$ a closed subprescheme of $X$ defined by an ideal $\mathcal{J}$, $\hat{X}$ the formal completion of $X$ along $X_0$, $U_0$ the trace of $X_0$ on $U$, Û the formal completion of $U$ along $U_0$, $i:U\to X$ and $\hat{i}:\hat{U}\to \hat{X}$ the canonical immersions, $n$ an integer. Suppose:

• a) $X$ is normally flat along $X_0$ at the points of $Y\cap X_0$, i.e. at these points the modules ${\mathcal{J}}^n/{\mathcal{J}}^{n+1}$ on $X_0$ are flat, i.e. locally free.
• b) For every $x\in Y\cap X_0$, one has $prof{O}_{X_0,x}⩾n+2$.

Under these conditions, one has the following:

1. Let $F$ be a coherent module on $U$; suppose that one has:

   • c) For every $x\in Y-Y\cap X_0$, one has $prof{O}_{X,x}⩾n+2$.
   • d) $F$ is free at the points of $U_0$, and of depth $⩾n+1$ at every point of $U$ where it is not free.

   Then the graded module

   ⨁_{m⩾0} Rᵖi_∗(𝒥ᵐF)
   

   on ${⨁}_{m⩾0}{\mathcal{J}}^m$ is finitely generated for $p⩽n$.

2. Let $F$ be a coherent module on Û. Then the graded module

   ⨁_{m⩾0} Rᵖi_∗(𝒥ᵐF/𝒥^{m+1}F)
   

   on ${⨁}_{m⩾0}{\mathcal{J}}^m/{\mathcal{J}}^{m+1}=g{r}_{\mathcal{J}}(O_X)$ is finitely generated for $p⩽n$.

Proof.

(1) Let $X^{\prime }=\mathrm{Spec}({⨁}_{m⩾0}{\mathcal{J}}^m)$; the base change $f:X^{\prime }\to X$ then defines $U^{\prime }=X^{\prime }-Y^{\prime }$, $X_0^{\prime }$, $U_0^{\prime }=X_0^{\prime }\cap U^{\prime }$, and immersions $i^{\prime }:U^{\prime }\to X^{\prime }$, $i_0^{\prime }:U_0^{\prime }\to X_0^{\prime }$. One has therefore a cartesian square

       i′
X′ ←──── U′
│         │
f│         │g
↓    i    ↓
X ←────── U

and one has

⨁_{m⩾0} Rᵖi_∗(𝒥ᵐF) = Rᵖi_∗(⨁_{m⩾0} 𝒥ᵐF) = Rᵖi_∗(g_∗(F′)),

where $F^{\prime }=g\ast F$, so that one has indeed a canonical isomorphism

$$g_{\ast }(F^{\prime })\stackrel{\sim }{\to }\underset{m⩾0}{⨁}{\mathcal{J}}^{m}F,$$

since this is true at the points of $U_0$, due to the fact that $F$ is free there by virtue of d), and also at the points outside $U_0$, due to the fact that there one has ${\mathcal{J}}^m=O_U$ (so that in both cases, ${\mathcal{J}}^m{\otimes }_{O_U}F\to {\mathcal{J}}^{m}F$ is an isomorphism).

On the other hand, since $f$ and consequently $g$ are affine, one has

Rᵖi_∗(g_∗(F′)) = Rᵖ(ig)_∗(F′) = Rᵖ(fi′)_∗(F′) = f_∗(Rᵖi′_∗(F′)),

so comparing (19) and (20), one sees that assertion (1) is equivalent to the following: $R^p{i}_{\ast }^{\prime }(F^{\prime })$ is a finitely generated module, i.e. coherent on $X^{\prime }$, for every $p⩽n$. Now since $X$ is locally immersible in a regular scheme, the same holds of $X^{\prime }$, which is of finite type over $X$, and one may apply the coherence criterion VIII 2.3 to a coherent extension $F^{\prime \prime }$ of $F^{\prime }$: one wants to express that $H_{Y^{\prime }}^p(F^{\prime \prime })$ is coherent for $p⩽n+1$, and this is also equivalent to saying that for every $x^{\prime }\in U^{\prime }$ such that

$$codim(x^{\prime }\cap Y^{\prime },x^{\prime })=1,$$

one has

$$prof{F}_{x^{\prime }}^{\prime }⩾n+1.$$

Now this condition is verified at the points $x^{\prime }$ where $F^{\prime }$ is not free, since for such an $x^{\prime }$ one has $x^{\prime }\notin U_0^{\prime }$ by virtue of d), so $g$ is an isomorphism there, and by virtue of d) again, $F$ is of depth $⩾n+1$ at $g(x^{\prime })$, so $F^{\prime }$ is of depth $⩾n+1$ at $x^{\prime }$. It therefore suffices to verify condition (21) at the $x^{\prime }\in U^{\prime }$ satisfying (20 bis) and at which $F^{\prime }$ is free. For this, it suffices to prove that one has

$$prof{O}_{X^{\prime },x^{\prime }}⩾n+1$$

at these points, a fortiori it suffices to establish that one has this relation at all points $x^{\prime }$ of $U^{\prime }$ satisfying (20 bis). Now, again by virtue of criterion 2.3 of Exposé VIII, this is equivalent to the assertion that the modules

Hᵖ_{Y′}(O_{X′}) for p ⩽ n + 1

are coherent. In fact, we are going to prove that they are even zero, or what amounts to the same by virtue of Exposé III, that one has

prof O_{X′,x′} ⩾ n + 2 for every x′ ∈ Y′.

For this, we distinguish two cases. If $x^{\prime }\notin X_0^{\prime }$, then $f$ is an immersion at $x^{\prime }$, and it is necessary to verify that $F$ is of depth $⩾n+2$ at the image $x=f(x^{\prime })$, which is none other than condition c). If on the contrary $x^{\prime }\in X_0^{\prime }$, i.e. $x=f(x^{\prime })\in X_0$ so $x\in Y\cap X_0$, one applies conditions a) and b) thanks to the following:

Lemma.

Let $X$ be a locally noetherian prescheme, $X_0$ a closed subprescheme of $X$ defined by an ideal $\mathcal{J}$, $X^{\prime }=\mathrm{Spec}({⨁}_{m⩾0}{\mathcal{J}}^m)$, $X_0^{\prime }=\mathrm{Spec}({⨁}_{m⩾0}{\mathcal{J}}^m/{\mathcal{J}}^{m+1})=X^{\prime }{\times }_X{X}_0$, $x$ a point of $X_0$ at which $X$ is normally flat along $X_0$, i.e. such that $g{r}_{\mathcal{J}}(O_X)={⨁}_{m⩾0}{\mathcal{J}}^m/{\mathcal{J}}^{m+1}$ is flat there as a module on $X_0$. Then for any sequence of elements $f_i(1⩽i⩽m)$ of $O_{X,x}$ whose images in $O_{X_0,x}$ form an $O_{X_0,x}$-regular sequence, and for every $x^{\prime }\in X^{\prime }$ above $x$, the images of the $f_i$ in $O_{X^{\prime },x^{\prime }}$ (resp. in $O_{X_0^{\prime },x^{\prime }}$) form respectively an $O_{X^{\prime },x^{\prime }}$-regular sequence (resp. an $O_{X_0^{\prime },x^{\prime }}$-regular sequence); in particular one has

prof O_{X′,x′} ⩾ prof O_{X₀,x},  prof O_{X′₀,x′} ⩾ prof O_{X₀,x}.

To prove this, one may assume that $X$ is local with closed point $x$, hence affine of ring $A=O_{X,x}$, $\mathcal{J}$ being defined by an ideal $J$; and it suffices to prove that for any sequence $f_i(1⩽i⩽m)$ of elements of $A$ whose images in $A/J$ form an $A/J$-regular sequence, the $f_i$ form an $({⨁}_{m⩾0}{J}^m)$-regular sequence and an $({⨁}_{m⩾0}{J}^m/J^{m+1})$-regular sequence, i.e. for every $m$, they form a Jᵐ-regular sequence and a $J^m/J^{m+1}$-regular sequence. The second assertion is trivial, since $J^m/J^{m+1}$ is a free module on $A/J$. The first follows by looking at the $J$-adic filtration of Jᵐ and noting that, for the graded module associated with Jᵐ for this filtration, the sequence of the $f_i$ is regular.

This proves 4.2 and consequently 4.1, (1).

Let us prove 4.1, (2). For this, let us use the cartesian square

        i′₀
X′₀ ←────── U′₀
│            │
f₀│            │g₀
↓     i₀     ↓
X₀ ←──────── U₀

and proceeding as at the beginning of the proof of (1), one finds that

⨁_{m⩾0} Rᵖi_∗(𝒥ᵐF/𝒥^{m+1}F) ≅ f₀_∗(Rᵖi′₀_∗(F′₀)),

where $F_0=F/\mathcal{J}F$ and $F_0^{\prime }=g_0\ast (F_0)$ (using the fact that $F$ is locally free). Hence the conclusion of (2) amounts to saying that for $p⩽n$, Rᵖi′₀_∗(F′₀) is a coherent module.

Here again, taking into account that $F_0^{\prime }$ is locally free, criterion VIII 2.3 lets us reduce to proving that this is so when one replaces $F_0^{\prime }$ by $O_{U_0^{\prime }}$, i.e. to proving that the modules

Hᵖ_{Y′₀}(O_{X′₀}) for p ⩽ n + 1  (where Y′₀ = Y′ ∩ X′₀ = X′₀ − U′₀)

are coherent. One proves again that they are in fact zero, i.e. that one has

prof O_{X′₀,x′} ⩾ n + 2 for every x′ ∈ Y′₀.

Now this indeed follows from conditions a) and b), taking into account 4.2. This completes the proof of 4.1.

Remark.

One sees at once, by descent, that the hypothesis: $X$ locally immersible in a regular scheme, may be replaced by the following weaker one: there exists a morphism $\bar{X}\to X$, faithfully flat and quasi-compact, such that $\bar{X}$ is locally immersible in a regular scheme.

Theorem 4.1 puts us in a position to apply the results of Exposé IX (comparison and existence theorems). We shall be particularly interested in the following:

Corollary.

Suppose conditions a), b), c) of Theorem 4.1 verified, with $n=1$, and $X=\mathrm{Spec}(A)$, $A$ being separated and complete for the $\mathcal{J}$-adic topology. Then:

1. The functor $F\mapsto \hat{F}$ from the category of locally free coherent modules on $U$ to the category of locally free coherent modules on Û is fully faithful.
2. For every locally free coherent module $\mathcal{F}$ on Û, there exists a coherent module $F$ on $U$ and an isomorphism $\hat{F}\stackrel{\sim }{\to }\mathcal{F}$.

In particular, if for every $x\in U$ whose closure in $U$ does not meet $U_0$, i.e. such that $x\cap X_0\subset Y$, one has $prof{O}_{U,x}⩾2$, then the pair $(U,U_0)$ satisfies the effective Lefschetz condition (Leff) of Exposé X.

(For the last assertion, one proceeds as in X 2.1.)

A particular case of 4.4:

Corollary.

Let $A$ be a noetherian ring, $J$ an ideal of $A$ contained in the radical, $A_0=A/J$. Suppose

1. $prof{A}_0⩾3$.
2. $g{r}_J(A)$ is a free $A_0$-module.
3. $A$ is complete for the $J$-adic topology.

Let $X=\mathrm{Spec}(A)$, $X_0=\mathrm{Spec}(A_0)=V(J)$, $a$ the closed point of $X$, $U=X-a$, $U_0=X_0-a$, Û the formal completion of $U$ along $U_0$. Then the functor $F\mapsto \hat{F}$ from the category of locally free coherent modules on $U$ to the category of locally free coherent modules on Û is fully faithful. Moreover, for every locally free coherent module $\mathcal{F}$ on Û, there exists a coherent module (not necessarily locally free!) $F$ on $U$, and an isomorphism $\hat{F}\cong \mathcal{F}$.

One will note that thanks to 4.3, we did not have to suppose that $A$ is a quotient of a regular ring, since the completion of $A$ for the $\mathfrak{r}(A)$-adic topology satisfies this condition in any case.

Proceeding as in Exposés X and XI, one concludes from 4.5:

Corollary.

Under the conditions of 4.5, one has the following:

• a) $U$ and $U_0$ are connected (III 3.1).

  Choosing a geometric base point in $U_0$, the homomorphism

  π₁(U₀) → π₁(U)
  

  is surjective.

• b) The homomorphism

  Pic(U) → Pic(U₀)
  

  is injective.

To prove b), taking 4.5 into account, this amounts to verifying that any isomorphism $L_0^{\prime }\stackrel{\sim }{\to }{L}_0$ lifts to an isomorphism ${\hat{L}}^{\prime }\stackrel{\sim }{\to }\hat{L}$. Now for this one lifts step by step to isomorphisms $L_n^{\prime }\stackrel{\sim }{\to }{L}_n$; the obstructions lie in $H^1(U_0,{\mathcal{J}}^n/{\mathcal{J}}^{n+1})$, and these modules are zero because $J^n/J^{n+1}$ is free and $prof{A}_0⩾3$.

We are now in a position to prove the following:

Theorem.

Let $A$ be a noetherian local ring, $J$ an ideal of $A$ contained in its radical, $A_0=A/J$. Suppose

1. $prof{A}_0⩾3$.
2. $g{r}_J(A)$ is a free module on $A_0$.

Then, if $A_0$ is "pure" (X 3.1) (resp. parafactorial (XI 3.1)), so is $A$.

Proof.

By descent, one may assume that one also has

1. $A$ is complete for the $J$-adic topology.

Indeed, by virtue of (i) and (ii), one has $prof(A)⩾3$, hence $prof(\hat{A})⩾3$, where Â is the completion of $A$ for the $J$-adic topology, and one applies X 3.6 and XI 3.6. One is therefore under the conditions of 4.5. Since $prof(A)⩾3⩾2$, to say that $A$ is parafactorial means simply that $\mathrm{Pic}(U)=0$, and by virtue of 4.6 b) it suffices for this that $\mathrm{Pic}(U_0)=0$, i.e. that $A_0$ be parafactorial. To prove that $A$ is "pure" if $A_0$ is, one needs to prove that if $V$ is an étale cover of $U$, defined by an algebra $B$ on $U$, then $H^0(U,B)$ is a finite étale algebra over $A$. Now $A_0$ being pure, the same holds of the $A_n$ (which differ from it only by nilpotent elements), so for every $n$, $B_n=H^0(U,B_n)$ is an étale algebra over $A/J^{n+1}$, and these algebras of course glue, so that $\lim B_n$ is an étale algebra over $A$. Now by virtue of 4.5, this algebra is none other than $H^0(U,B)$, which establishes our assertion.

Corollary.

Let $f:X\to Y$ be a flat morphism of locally noetherian preschemes, $x\in X$, $y=f(x)$; suppose that $O_{X_y,x}$ is a "pure" (resp. parafactorial) local ring of depth $⩾3$. Then the same holds for $O_{X,x}$.

This is the result of the type promised at the end of the preceding section, in order to generalize Corollaries 3.5 and following. One thus finds, using 3.4, the following:

Corollary.

Let $f:X\to S$ be a flat projective morphism with $S$ locally noetherian, $O_X(1)$ an invertible module on $X$ ample with respect to $S$, $t$ a section of $O_X(1)$ such that for every $s\in S$ the section $t_s$ induced on $X_s$ is $O_{X_s}$-regular, $X_0$ the subscheme of zeros of $t$, $X_m$ the subscheme of zeros of $t^{m+1}$. Suppose that for every $s\in S$, $X_s$ is of depth $⩾3$ at all its closed points. Then:

• a) If the local rings of the closed points of $X_s-X_{0,s}$ ($s\in S$) are "pure", for example are complete intersections, then the functor $X^{\prime }\mapsto X_0^{\prime }=X^{\prime }{\times }_X{X}_0$ from the category of étale covers of $X$ to the category of étale covers of $X_0$ is an equivalence of categories; in particular, choosing a geometric base point in $X_0$, the homomorphism

  π₁(X₀) → π₁(X)
  

  is an isomorphism.¹⁰

• b) If the local rings of the closed points of $X_s-X_{0,s}$ ($s\in S$) are "parafactorial", for example regular, or complete intersections of dimension $⩾4$, then for every integer $m$ such that Rⁱf₀_∗(O_{X₀}(−n)) = 0 for $n>m$ and $i=1,2$, the map $\mathrm{Pic}(X)\to \mathrm{Pic}(X_m)$ is bijective.

Moreover, if $S$ is noetherian and the $X_{0,s}$ are of depth $⩾3$ at their closed points, there exist such $m$ (cf. 1.5).

Remark.

Under the conditions of the last assertion of 4.9 b), one has seen in 1.5 that there exists an $m$ such that $n>m$ implies even $H^i(X_0,O_{X_0}(-n))=0$ for $i=1,2$ (and even for $i⩽2$). This condition is stronger than $R^i{f}_{\ast }(O_{X_0}(-n))=0$ for $i=1,2$, and it has moreover the advantage of being stable under base change. The same holds of the depth hypotheses made in 4.9, and also of a hypothesis of the type "the $X_s$ are locally complete intersections". It then follows, under these conditions, that 4.9 b) also implies that the functor morphism

$${\mathrm{Pic}}_{X/S}\to {\mathrm{Pic}}_{X_n/S}$$

in $Sch/S$ is an isomorphism, hence also the morphism for the relative Picard schemes, when these exist:

$${\mathrm{Pic}}_{X/S}\to {\mathrm{Pic}}_{X_n/S}.$$

Even in the case where $S$ is the spectrum of an algebraically closed field, this statement is markedly more precise than the statement saying merely that $\mathrm{Pic}(X)\to \mathrm{Pic}(X_n)$ is bijective.

One may ask whether one can always take $n=0$ in the preceding conclusions (assuming therefore the $X_{0,s}$ of depth $⩾3$ at their closed points). When $X_0$ is smooth over $S$ and the residue characteristics of $S$ are zero, this is indeed so, by virtue of Kodaira's "vanishing theorem" (proved by transcendental means, using a Kählerian metric) which implies that for every smooth connected projective scheme of dimension $n$ over a field $k$ of characteristic zero, and every ample invertible module $L$ on $X$, one has $H^i(X,L^{-1})=0$ for $i\ne n$. It is not known¹¹ at present whether this theorem may be replaced by a generalization in characteristic $p>0$, and whether the smoothness hypothesis may be replaced by a hypothesis of a more general nature (bearing on depth, or of "complete intersection" type ...).

5. Universal finiteness conditions for a non-proper morphism

Let us recall for the record the following:

Proposition.

Let $f:X\to S$ be a proper morphism of preschemes with $S$ locally noetherian, $U$ an open part of $X$, $g:U\to X$ the canonical immersion, $h=fg:U\to S$, $F$ a module on $U$. Suppose that the modules $R^i{g}_{\ast }(F)$ are coherent for $i⩽n$ (a hypothesis of local nature on $X$, which is verified in practice using criterion VIII 2.3). Then $R^i{h}_{\ast }(F)$ is coherent for $i⩽n$.

This follows at once from the Leray spectral sequence

$$E_2^{p,q}=R^p{f}_{\ast }(R_{\ast }^{qg}(F))\Rightarrow R^{\ast }{h}_{\ast }(F),$$

and from the fact that the higher direct images by $f$ of a coherent module on $X$ are coherent (EGA III 3.2.1).

Proposition.

Let $S$ be a locally noetherian prescheme, $\mathcal{S}$ a quasi-coherent graded algebra of finite type on $S$, generated by ${\mathcal{S}}_1$, $X$ a subprescheme of $\mathrm{Proj}(\mathcal{S})$, $O_X(1)$ the invertible module on $X$ very ample relatively to $S$ induced by $\mathrm{Proj}(\mathcal{S}(1))$, $U$ an open part of $X$, $g:U\to X$ the canonical immersion, $h=fg:U\to S$, $F$ a quasi-coherent module on $U$, whence twisted modules $F(m)=F\otimes O_X(m)$ ($m\in \mathbb{Z}$), $n$ an integer, $m_0$ an integer. The following conditions are equivalent:

1. $R^i{g}_{\ast }(F)$ is coherent for $i⩽n$.
2. ${⨁}_{m>m_0}{R}^i{h}_{\ast }(F(m))$ is a finitely generated $\mathcal{S}$-module for $i⩽n$.

Proof.

Replacing $F$ by $F(m)$ in the spectral sequence above one finds a spectral sequence of graded $\mathcal{S}$-modules

E₂^{p,q} = ⨁_{m⩾m₀} Rᵖf_∗(R^qg_∗(F(m))) ⇒ ⨁_{m⩾m₀} R^∗h_∗(F(m)).

Since one has

$$R_{\ast }^{qg}(F(m))\cong R_{\ast }^{qg}(F)(m),$$

one sees that if the $R^i{g}_{\ast }(F)$ are coherent, $E_2^{p,q}$ is finitely generated on $\mathcal{S}$ for $q⩽n$, thanks to part a) of Lemma 5.3 below, which implies that the abutment is finitely generated on $\mathcal{S}$ in degree $i⩽n$. This proves (i) ⇒ (ii). Moreover, reasoning in the abelian category of graded $\mathcal{S}$-modules modulo the thick subcategory $C$ of those that are quasi-coherent of finite type, one finds by the preceding spectral sequence

⨁_{m⩾m₀} R^{n+1}h_∗(F(m)) ≅ ⨁_{m⩾m₀} f_∗(R^{n+1}g_∗(F)(m)) mod C,

which proves that if the left-hand side is a finitely generated $\mathcal{S}$-module, then $R^{n+1}{g}_{\ast }(F)$ is coherent, by virtue of part b) of Lemma 5.3. This proves the implication (ii) ⇒ (i) by induction on $n$. It remains to prove:

Lemma.

Let $S$, $\mathcal{S}$, $X$, $f$ be as in 5.2, and $G$ a quasi-coherent module on $X$, $m_0$ an integer. Then:

• a) If $G$ is coherent, then for every integer $i$, the graded module

  ⨁_{m⩾m₀} Rⁱf_∗(G(m))
  

  on $\mathcal{S}$ is finitely generated.

• b) Conversely, suppose that the module ${⨁}_{m⩾m_0}{R}^i{f}_{\ast }(G(m))$ on $\mathcal{S}$ is finitely generated; then $G$ is coherent.

Proof of 5.3.

For a), the case $i=0$ is given in EGA III 2.3.2; the case $i>0$ follows from EGA III 2.2.1 (i)(ii), which says that the $R^i{f}_{\ast }G(m)$ are coherent, and zero for $m$ large (if one assumes $S$ noetherian, which is permissible).

For b), one notes that $G$ is isomorphic to $\mathrm{Proj}({⨁}_{m⩾m_0}{f}_{\ast }(G(m)))$ (EGA II 3.4.4 and 3.4.2), which proves that $G$ is coherent if ${⨁}_{m⩾m_0}{f}_{\ast }(G(m))$ is finitely generated on $\mathcal{S}$, by virtue of loc. cit. 3.4.4.

Corollary.

($S$ noetherian.) Suppose that $R^i{g}_{\ast }(F)$ is coherent for $i⩽n$; then for $i⩽n+1$ and $m$ large, one has a canonical isomorphism:

$$R^i{h}_{\ast }(F(m))\cong f_{\ast }(R^i{g}_{\ast }(F)(m)).$$

Indeed the spectral sequence (26) for $F(m)$ then degenerates in degree $⩽n$, by EGA III 2.2.1 (ii), whence at once the result (which moreover recovers the implication (ii) ⇒ (i) of 5.2).

Corollary.

Under the preliminary conditions of 5.2, $S$ noetherian, the following conditions are equivalent:

1. ${⨁}_{m⩾m_0}{h}_{\ast }(F(m))$ is finitely generated on $\mathcal{S}$, and $R^i{h}_{\ast }(F(m))=0$ for $0<i⩽n$ and $m$ large.
2. $g_{\ast }(F)$ is coherent, and $R^i{g}_{\ast }(F)=0$ for $0<i⩽n$.
3. (ii bis) $g_{\ast }(F)$ is coherent, and $pro{f}_Y{g}_{\ast }(F)>n+1$.

The equivalence of (ii) and (ii bis) is contained in III 3.3. Moreover, by virtue of 5.2 conditions (i) and (ii) both imply that the $R^i{g}_{\ast }(F)$ ($i⩽n$) are coherent. The equivalence of (i) and (ii) then follows from 5.4, taking into account the fact that for a coherent module $G$ on $X$, one has $G=0$ if and only if $f_{\ast }(G(m))=0$ for $m$ large, for instance by virtue of EGA III 2.2.1 (iii).

Remark.

One may interpret criteria 5.2 and 5.5 by saying that the "simultaneous finiteness condition" 5.2 (ii) is expressed by properties of local regularity (in terms of depth, thanks to VIII 2.1) of $F$ at the points of $U$ neighboring $Y=X-U$, whereas the "asymptotic vanishing condition" 5.5 (i) is of a markedly stronger nature, and is expressed by conditions of local regularity of $g_{\ast }(F)$ at the points of $Y$ itself. It would be interesting, in order to generalize the Lefschetz-type theorems for projective morphisms to quasi-projective morphisms, to find local criteria on $X$ necessary and sufficient for the $\mathcal{S}$-modules ${⨁}_{m⩾0}{R}^i{h}_{\ast }(F(m))$ for $i⩽n$ to be finitely generated. When $S$ is the spectrum of a field (and doubtless more generally, when it is the spectrum of an artinian ring) and $Y=X-U$ is finite, one can show that it is necessary and sufficient that the following conditions be verified:

1. $prof{F}_x>n$ for every closed point $x$ of $U$ (compare 1.4).
2. $R^i{g}_{\ast }(F)$ is coherent for $i⩽n$, or what amounts to the same, there exists an open neighborhood $V$ of $Y$ such that for every closed point $x$ of $U\cap V$, one has $prof{F}_x>n+1$.

Proposition.

Let $S$ be a locally noetherian prescheme, $g:U\to X$ a morphism of preschemes of finite type over $S$,¹² with structural morphisms $h$ and $f$, $F$ a quasi-coherent module on $U$, $n$ an integer. The following conditions are equivalent:

1. For every base change $S^{\prime }\to S$ with $S^{\prime }$ noetherian, the module $R^n{g}_{\ast }^{\prime }(F^{\prime })$ on $X^{\prime }$ is coherent.

2. For every base change as above, and every coherent ideal $J$ on $S^{\prime }$, denoting by $\mathcal{I}$ the ideal $J{O}_{X^{\prime }}$ on $X^{\prime }$, the graded module

   ⨁_{m⩾0} Rⁿg′_∗(ℐᵐF′)
   

   on ${⨁}_{m⩾0}{\mathcal{I}}^m$ is finitely generated.

3. For every base change $S^{\prime }\to S$, and $J$ as above, the graded module

   ⨁_{m⩾0} Rⁿg′_∗(ℐᵐF′/ℐ^{m+1}F′)
   

   on $g{r}_{\mathcal{I}}(O_{X^{\prime }})={⨁}_{m⩾0}{\mathcal{I}}^m/{\mathcal{I}}^{m+1}$ is finitely generated.

Plainly (ii) ⇒ (i) and (iii) ⇒ (i), as one sees by setting $\mathcal{I}=0$ in conditions (ii) and (iii). The reverse implications are obtained by applying (i) to the composite base change $S^{\prime \prime }\to S^{\prime }\to S$, where $S^{\prime \prime }$ is equal to $\mathrm{Spec}({⨁}_{m⩾0}{J}^m)$ resp. $\mathrm{Spec}({⨁}_{m⩾0}{J}^m/J^{m+1})$.

The interest of this proposition is that conditions of form (ii) are those that intervene in the "algebraic-formal comparison theorems", whereas conditions of form (iii) intervene in the "existence theorems" that complement them, cf. Exposé IX. A first interesting case is the one where $f:X\to S$ is the identity, and where it is therefore a question of conditions on a morphism $h:U\to S$ locally of finite type and a quasi-coherent module $F$ on $U$ flat with respect to $S$. To obtain sufficient conditions, we are going to assume that $U$ embeds, via $g:U\to X$, as an open subprescheme of an $X$ proper over $S$. Applying 5.1, one sees therefore:

Corollary.

Let $f:X\to S$ be a proper morphism with $S$ locally noetherian, $U$ an open subset of $X$, $g:U\to X$ the canonical immersion, $h=fg:U\to S$, $F$ a quasi-coherent module on $X$, flat with respect to $S$. Suppose that for every base change $S^{\prime }\to S$ with $S^{\prime }$ locally noetherian, one has $R^i{g}_{\ast }^{\prime }(F^{\prime })$ coherent on $X^{\prime }$ for $i⩽n$. Then one has the following:

1. For every base change $S^{\prime }\to S$ with $S^{\prime }$ locally noetherian, $R^i{h}_{\ast }^{\prime }(F^{\prime })$ is coherent on $S^{\prime }$ for $i⩽n$.

2. For every $S^{\prime }\to S$ as above, and every coherent ideal $J$ on $S^{\prime }$, the graded modules

   ⨁_{m⩾0} Rⁱh′_∗(JᵐF′)
   

   on ${⨁}_{m⩾0}{J}^m$ are finitely generated for $i⩽n$.

3. For every $S^{\prime }\to S$ and $J$ as above, the graded modules

   ⨁_{m⩾0} Rⁱh′_∗(JᵐF′/J^{m+1}F′)
   

   on $g{r}_J(O_{S^{\prime }})={⨁}_{m⩾0}{J}^m/J^{m+1}$ are finitely generated for $i⩽n$.