§5. An existence theorem for coherent algebraic sheaves

5.1. Statement of the theorem

(5.1.1)

Let $A$ be an adic Noetherian ring, $\mathfrak{J}$ an ideal of definition of $A$, so that $A$ is separated and complete for the $\mathfrak{J}$-adic topology. If $Y=\mathrm{Spec}(A)$, the affine formal scheme $Spf(A)$ is identified with the completion Ŷ of $Y$ along the closed subset $Y^{\prime }=V(\mathfrak{J})$ (I, 10.10.1). Let $X$ be a (usual) $Y$-prescheme of finite type, $f:X\to Y$ the structure morphism; we shall denote by $\mathfrak{X}$ the completion of $X$ along the closed subset $X^{\prime }=f^{-1}(Y^{\prime })$, or equivalently the Ŷ-formal prescheme $X{\times }_Y\hat{Y}$; by $\hat{f}:\mathfrak{X}\to \hat{Y}$ the extension of $f$ to the completions; finally, for every coherent ${\mathcal{O}}_X$-module $\mathcal{F}$, we shall denote by $\hat{\mathcal{F}}$ its completion ${\mathcal{F}}_{/X^{\prime }}$, which is a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module.

Proposition (5.1.2).

The hypotheses and notation being those of (5.1.1), let $\mathcal{F}$ be a coherent ${\mathcal{O}}_X$-module whose support is proper over $Y$ (II, 5.4.10). The canonical homomorphisms (4.1.4)

  ρ_i : H^i(X, ℱ) → H^i(𝔛, 𝓕̂)

are then isomorphisms.

Proof. Since $H^i(X,\mathcal{F})$ is an $A$-module of finite type (3.2.4), hence identical to its Hausdorff completion for the $\mathfrak{J}$-preadic topology $(0_I,\mathrm{7.3.6})$, the proposition is only a particular case of (4.1.10).

Recall that the canonical isomorphisms ${\rho }_i$ commute with the coboundaries for every exact sequence $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ of coherent ${\mathcal{O}}_X$-modules ((0, 12.1.6) and (I, 10.8.9)).

Corollary (5.1.3).

Let $\mathcal{F}$, $\mathcal{G}$ be two coherent ${\mathcal{O}}_X$-modules such that the intersection of their supports is proper over $Y$. Then the canonical homomorphism

  Hom_{𝒪_X}(ℱ, 𝒢) → Hom_{𝒪_𝔛}(𝓕̂, 𝓖̂)                                          (5.1.3.1)

which associates to every homomorphism $u:\mathcal{F}\to \mathcal{G}$ its completion $\hat{u}:\hat{\mathcal{F}}\to \hat{\mathcal{G}}$, is an isomorphism. Moreover, when the morphism $f$ is closed, in order that û be injective (resp. surjective) it is necessary and sufficient that $u$ be so.

Proof. The first assertion is a particular case of (4.5.3), again due to the fact that the first member of (5.1.3.1) is an $A$-module of finite type, hence identical to its Hausdorff completion. To prove the second, note by virtue of (I, 10.8.14) that û is injective (resp. surjective) if and only if there exists a neighbourhood of $X^{\prime }$ in which $u$ is injective (resp. surjective).

The conclusion therefore follows from the following lemma:

Lemma (5.1.3.1).

Under the hypotheses of (5.1.1), if one assumes in addition that the morphism $f$ is closed, then every neighbourhood of $X^{\prime }$ in $X$ is identical to $X$.

Proof. First, we may reduce to the case where $f(X)=Y$. Indeed, by hypothesis, $f(X)$ is a closed subset $Z$ of $Y$; we may in addition replace $f$ by $f_{red}$ (I, 6.3.4), and suppose consequently $X$ and $Y$ reduced; we may then replace $Y$ by the reduced closed subprescheme of $Y$ with $Z$ as underlying space (I, 5.2.2), since every ideal of $A$ is closed, and every quotient ring of $A$ is therefore adic and Noetherian. One then has $f(X^{\prime })=Y^{\prime }$; if $V$ is an open neighbourhood of $X^{\prime }$ in $X$, $f(X-V)$ is closed in $Y$ by hypothesis, and does not meet $Y^{\prime }$; but this is impossible unless $X-V$ is empty, since $\mathfrak{J}$ is contained in the radical of $A$ (I, 1.1.15 and 0_I, 7.1.10), whence the conclusion.

When one restricts to coherent ${\mathcal{O}}_X$-modules whose support is proper over $Y$, (5.1.3) can be stated, in the language of categories, by saying that the functor $\mathcal{F}\mapsto \hat{\mathcal{F}}$ is fully faithful from the category of ${\mathcal{O}}_X$-modules of the preceding type into the category of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules, and consequently establishes an equivalence of the first of these categories with a full subcategory of the second (0, 8.1.6). The existence theorem will prove that when $X$ is proper over $Y$, this subcategory is in fact the category of all coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules. More precisely:

Theorem (5.1.4).

Let $A$ be an adic Noetherian ring, $Y=\mathrm{Spec}(A)$, $\mathfrak{J}$ an ideal of definition of $A$, $Y^{\prime }=V(\mathfrak{J})$, $f:X\to Y$ a separated morphism of finite type, $X^{\prime }=f^{-1}(Y^{\prime })$. Let $\hat{Y}=Y_{/Y^{\prime }}=Spf(A)$, $\mathfrak{X}=X_{/X^{\prime }}$ be the completions of $Y$ and $X$ along $Y^{\prime }$ and $X^{\prime }$, $\hat{f}:\mathfrak{X}\to \hat{Y}$ the extension of $f$ to the completions; then the functor $\mathcal{F}\mapsto \hat{\mathcal{F}}={\mathcal{F}}_{/X^{\prime }}$ is an equivalence of the category of coherent ${\mathcal{O}}_X$-modules of support proper over $\mathrm{Spec}(A)$ with the category of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules of support proper over $Spf(A)$.

In other words, taking (5.1.3) into account:

Corollary (5.1.5).

For an ${\mathcal{O}}_{\mathfrak{X}}$-module to be isomorphic to the completion of an ${\mathcal{O}}_X$-module which is coherent and of support proper over $\mathrm{Spec}(A)$, it is necessary and sufficient that it be coherent and of support proper over $Spf(A)$.

The most important case is the:

Corollary (5.1.6).

Suppose $X$ proper over $Y=\mathrm{Spec}(A)$. Then the functor $\mathcal{F}\mapsto \hat{\mathcal{F}}$ is an equivalence of the category of coherent ${\mathcal{O}}_X$-modules and of the category of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules.

Scholium (5.1.7).

If one takes into account the characterization of coherent sheaves on formal preschemes (I, 10.11.3), one sees that under the conditions of (5.1.1), the data of a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module of support proper over $\mathrm{Spec}(A)$ is equivalent (on setting $Y_n=\mathrm{Spec}(A/{\mathfrak{J}}^{n+1})$ and $X_n=X{\times }_Y{Y}_n$) to the data of a projective system of coherent ${\mathcal{O}}_{X_n}$-modules $({\mathcal{F}}_n)$ such that for $m\le n$ one has ${\mathcal{F}}_m={\mathcal{F}}_n{\otimes }_{{\mathcal{O}}_{Y_n}}{\mathcal{O}}_{Y_m}$ (or equivalently ${\mathcal{F}}_m={\mathcal{F}}_n/{\mathfrak{J}}^{m+1}{\mathcal{F}}_n$) and that the support of ${\mathcal{F}}_0$ is a part of X_0 proper over Y_0. By means of (I, 10.11.4), one likewise interprets homomorphisms of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules as homomorphisms of projective systems of coherent ${\mathcal{O}}_{X_n}$-modules.

In all known cases of application, $A$ is in fact an adic local Noetherian ring, so the $Y_n$ are spectra of artinian local rings, and the results of this section and the preceding ones reduce in large measure algebraic geometry over an adic local Noetherian ring to algebraic geometry over artinian local rings.

Corollary (5.1.8).

Under the conditions of (5.1.4), the map $Z\mapsto \hat{Z}=Z_{/(Z\cap X^{\prime })}$ is a bijection of the set of closed subpreschemes $Z$ of $X$, proper over $Y$, onto the set of closed formal subpreschemes of $\mathfrak{X}$, proper over Ŷ.

Proof. Indeed, a closed formal subprescheme of $\mathfrak{X}$ is of the form $(\mathfrak{T},({\mathcal{O}}_{\mathfrak{X}}/\mathcal{A})|\mathfrak{T})$, where $\mathcal{A}$ is a coherent Ideal of ${\mathcal{O}}_{\mathfrak{X}}$ (I, 10.14.2); if $\mathfrak{T}$ is proper over Ŷ, it follows from (5.1.4) that ${\mathcal{O}}_{\mathfrak{X}}/\mathcal{A}$ is isomorphic to an ${\mathcal{O}}_{\mathfrak{X}}$-module of the form $\hat{\mathcal{F}}$, where $\mathcal{F}$ is a coherent ${\mathcal{O}}_X$-module of support proper over $Y$; in addition, it follows from (5.1.3) that the canonical homomorphism ${\mathcal{O}}_{\mathfrak{X}}\to {\mathcal{O}}_{\mathfrak{X}}/\mathcal{A}$ is of the form û, where $u:{\mathcal{O}}_X\to \mathcal{F}$ is a surjective homomorphism of ${\mathcal{O}}_X$-modules. Hence $\mathcal{F}$ is of the form ${\mathcal{O}}_X/\mathcal{N}$, where $\mathcal{N}$ is a coherent Ideal of ${\mathcal{O}}_X$, and $\mathcal{A}=\hat{\mathcal{N}}$ (I, 10.8.8), whence the conclusion (I, 10.14.7).

5.2. Proof of the existence theorem: projective and quasi-projective cases

(5.2.1)

Under the conditions of (5.1.4), we shall say provisionally that a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module is algebraizable if it is isomorphic to a completion $\hat{\mathcal{F}}$ of a coherent ${\mathcal{O}}_X$-module $\mathcal{F}$ of support proper over $Y$.

Lemma (5.2.2).

Let ${\mathcal{F}}^{\prime }$, ${\mathcal{G}}^{\prime }$ be two algebraizable ${\mathcal{O}}_{\mathfrak{X}}$-modules. For every homomorphism $u:{\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }$, $Ker(u)$, $Im(u)$ and $Coker(u)$ are algebraizable.

Proof. Indeed, if ${\mathcal{F}}^{\prime }=\hat{\mathcal{F}}$, ${\mathcal{G}}^{\prime }=\hat{\mathcal{G}}$, where $\mathcal{F}$ and $\mathcal{G}$ are coherent ${\mathcal{O}}_X$-modules of supports proper over $Y$, one has $u=\hat{v}$, where $v:\mathcal{F}\to \mathcal{G}$ is a homomorphism (5.1.3). By virtue of the exactness of the functor $\mathcal{F}\mapsto \hat{\mathcal{F}}$, $Ker(u)$ is isomorphic to $\widehat{Ker(v)}$, and since the support of $Ker(v)$ is contained in that of $\mathcal{F}$, one sees that $Ker(u)$ is algebraizable; analogous proof for $Im(u)$ and $Coker(u)$.

Proposition (5.2.3).

Let $A$ be an adic Noetherian ring, $\mathfrak{J}$ an ideal of definition of $A$, $\mathfrak{Y}=Spf(A)$, $f:\mathfrak{X}\to \mathfrak{Y}$ a proper morphism of formal preschemes. Set $Y_k=\mathrm{Spec}(A/{\mathfrak{J}}^{k+1})$, $X_k=\mathfrak{X}{\times }_{\mathfrak{Y}}{Y}_k$, and for every ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$, ${\mathcal{F}}_k=\mathcal{F}{\otimes }_{{\mathcal{O}}_{\mathfrak{X}}}{\mathcal{O}}_{X_k}=\mathcal{F}/{\mathfrak{J}}^{k+1}\mathcal{F}$. Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_{\mathfrak{X}}$-module, and suppose that ${\mathcal{L}}_0=\mathcal{L}/\mathfrak{J}\mathcal{L}$ is an ample ${\mathcal{O}}_{X_0}$-module; for every ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$ and every integer $n$, set $\mathcal{F}(n)=\mathcal{F}\otimes {\mathcal{L}}^{\otimes n}$. Then, for every coherent ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$, there exists an integer $n_0$ such that, for every $n\ge n_0$, the following properties hold:

(i) The canonical homomorphism $H^0(\mathfrak{X},\mathcal{F}(n))\to H^0(\mathfrak{X},{\mathcal{F}}_k(n))$ is surjective for every $k\ge 0$.

(ii) One has $H^q(\mathfrak{X},\mathcal{F}(n))=0$ for every $q>0$.

Proof. We know that the underlying spaces of $\mathfrak{X}$ and X_0 are the same; the sheaves ${\mathcal{M}}_k={\mathfrak{J}}^k\mathcal{F}/{\mathfrak{J}}^{k+1}\mathcal{F}$, being annihilated by $\mathfrak{J}$, may be considered as coherent ${\mathcal{O}}_{X_0}$-modules $(0_I,\mathrm{5.3.10})$; in addition, if one sets ${\mathcal{M}}_k(n)={\mathcal{M}}_k{\otimes }_{{\mathcal{O}}_{X_0}}{\mathcal{L}}_0^{\otimes n}$, one sees at once that ${\mathcal{M}}_k(n)={\mathfrak{J}}^k\mathcal{F}(n)/{\mathfrak{J}}^{k+1}\mathcal{F}(n)$. Note that, since ${\mathcal{L}}_0$ is ample for $f_0:X_0\to Y_0$, and

$f_0$ is proper, we conclude that $f_0$ is projective (II, 5.5.4). Let $\mathcal{S}={\oplus }_{k\ge 0}{\mathfrak{J}}^k/{\mathfrak{J}}^{k+1}$ be the graded $A$-algebra associated to the $\mathfrak{J}$-adic filtration of $A$, which is of finite type since $A$ is Noetherian; if one sets ${\mathcal{S}}^{\prime }=f_0^{\ast }(\tilde{\mathcal{S}})$, ${\mathcal{S}}^{\prime }$ is a quasi-coherent ${\mathcal{O}}_{X_0}$-algebra of finite type, and $\mathcal{M}={\oplus }_{k\ge 0}{\mathcal{M}}_k$ a graded quasi-coherent ${\mathcal{S}}^{\prime }$-module of finite type (since ${\mathcal{F}}_0$ is coherent and generates the ${\mathcal{S}}^{\prime }$-module $\mathcal{M}$). We are therefore in the conditions of application of theorem (2.4.1, (ii)), and we conclude that there exists $n_0$ such that, for $n\ge n_0$ and for every $k$, one has

  H^q(X_0, ℳ_k(n)) = 0           for every q > 0.                              (5.2.3.1)

One therefore also has $H^q(\mathfrak{X},{\mathcal{M}}_k(n))=0$ for $q>0$ and $n\ge n_0$, ${\mathcal{M}}_k(n)$ being this time considered as ${\mathcal{O}}_{\mathfrak{X}}$-module. Applying the exact cohomology sequence to

  0 → 𝔍^h 𝓕(n) / 𝔍^{k+1} 𝓕(n) → 𝔍^h 𝓕(n) / 𝔍^k 𝓕(n) →
                                              𝔍^k 𝓕(n) / 𝔍^{k+1} 𝓕(n) → 0,

one deduces first that for $0\le h<k$, $n\ge n_0$ and $q>0$, one has, by induction on $k-h$,

  H^q(𝔛, 𝔍^h 𝓕(n) / 𝔍^k 𝓕(n)) = 0                                              (5.2.3.2)

and in particular for $h=0$,

  H^q(𝔛, 𝓕_k(n)) = 0           for n ≥ n_0, k ≥ 0 and q > 0.                   (5.2.3.3)

Another portion of the exact cohomology sequence, for $h=0$, gives the exact sequence

  H^0(𝔛, 𝓕_{k+1}(n)) → H^0(𝔛, 𝓕_k(n)) → H^1(𝔛, 𝔍^k 𝓕(n) / 𝔍^{k+1} 𝓕(n)) = 0,   (5.2.3.4)

whence one deduces that for $h\le k$, the canonical map

  H^0(𝔛, 𝓕_{k+1}(n)) → H^0(𝔛, 𝓕_h(n))                                          (5.2.3.5)

is surjective. For every $q$, the projective system $(H^q(\mathfrak{X},{\mathcal{F}}_k(n)){)}_{k\ge 0}$ therefore satisfies condition (ML) for $n\ge n_0$. Moreover, every formal affine open $U$ of $\mathfrak{X}$ is also an affine open in each of the $X_k$ (I, 10.5.2), hence one has $H^q(U,{\mathcal{F}}_k(n))=0$ for every $q>0$ (1.3.1), and $H^0(U,{\mathcal{F}}_{k+1}(n))\to H^0(U,{\mathcal{F}}_k(n))$ is surjective for $h\le k$ (I, 1.3.9). The conditions of application of $(0_{III},\mathrm{13.3.1})$ are consequently fulfilled, and we conclude that, for $n\ge n_0$:

1° For every $q>0$, $H^q(\mathfrak{X},\mathcal{F}(n))\to \underleftarrow{\lim }{H}^q(\mathfrak{X},{\mathcal{F}}_k(n))$ is bijective, hence, by virtue of (5.2.3.3), $H^q(\mathfrak{X},\mathcal{F}(n))=0$.

2° The homomorphism $H^0(\mathfrak{X},\mathcal{F}(n))\to \underleftarrow{\lim }{H}^0(\mathfrak{X},{\mathcal{F}}_k(n))$ is bijective; moreover, since the homomorphisms (5.2.3.5) are surjective, so is each of the homomorphisms

  lim_← H^0(𝔛, 𝓕_k(n)) → H^0(𝔛, 𝓕_h(n)),

which completes the proof.

Corollary (5.2.4).

The hypotheses being those of (5.2.3), for every coherent ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$, there exists an integer $N$ such that for $n\ge N$, $\mathcal{F}(n)$ is generated by its sections above

$\mathfrak{X}$; in other words, $\mathcal{F}$ is isomorphic to the quotient of an ${\mathcal{O}}_{\mathfrak{X}}$-module of the form $({\mathcal{L}}^{\otimes (-n)}{)}^h$.

Proof. Since X_0 is Noetherian, it follows from the hypothesis on ${\mathcal{L}}_0$ and from (II, 4.5.5) that there exists $n_0$ such that, for $n\ge n_0$, ${\mathcal{F}}_0(n)$ is generated by its sections above X_0; moreover, one may suppose $n_0$ chosen large enough that the homomorphism $\Gamma (\mathfrak{X},\mathcal{F}(n))\to \Gamma (X_0,{\mathcal{F}}_0(n))$ is surjective for $n\ge n_0$ (5.2.3). There thus exists a finite number of sections $s_i\in \Gamma (\mathfrak{X},\mathcal{F}(n))$ whose images in $\Gamma (X_0,{\mathcal{F}}_0(n))$ generate ${\mathcal{F}}_0(n)$ $(0_I,\mathrm{5.2.3})$. Since $\mathfrak{J}$ is contained in the maximal ideal of the local ring at every point of $\mathfrak{X}$, it follows from Nakayama's lemma, applied to these local rings, that the $s_i$ generate $\mathcal{F}(n)$ $(0_I,\mathrm{5.1.1})$.

(5.2.5) Proof of the existence theorem: projective case.

The notation being that of (5.1.4), suppose $f$ projective, so that there exists an ample ${\mathcal{O}}_X$-module $\mathcal{L}$ (I, 5.5.4). By definition, $X_n=X{\times }_Y{Y}_n$ is equal to the closed subprescheme $X_n{\times }_Y{Y}_n=f^{-1}(Y_n)$ of $X$; if ${\mathcal{L}}^{\prime }$ is the completion $\mathcal{L}{\otimes }_{{\mathcal{O}}_X}{\mathcal{O}}_{\mathfrak{X}}$ of $\mathcal{L}$, one has ${\mathcal{L}}_0^{\prime }=\mathcal{L}/\mathfrak{J}\mathcal{L}$, considered as ${\mathcal{O}}_{X_0}$-module; one knows that ${\mathcal{L}}_0^{\prime }$ is ample (II, 4.6.13, (i bis)). One may therefore apply to ${\mathcal{L}}^{\prime }$ and to every coherent ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$ Corollary (5.2.4); one sees thus that $\mathcal{F}$ is isomorphic to a quotient of $\mathcal{G}=({\mathcal{L}}^{\prime \otimes (-n)}{)}^k$ for suitable integers $n>0$ and $k>0$. Now, it is clear that $\mathcal{G}$ is the completion of $({\mathcal{L}}^{\otimes (-n)}{)}^k$ (I, 10.8.10), and is therefore algebraizable. Next consider the canonical surjective homomorphism $u:\mathcal{G}\to \mathcal{F}$, and let $\mathcal{H}=Ker(u)$, which is a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module $(0_I,\mathrm{5.3.4})$. One sees in the same way that there exists an algebraizable ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{K}$ and a homomorphism $v:\mathcal{K}\to \mathcal{G}$ such that $\mathcal{H}=Im(v)$. One then has $\mathcal{F}=Coker(v)$, and $\mathcal{F}$ is algebraizable by virtue of (5.2.2).

(5.2.6) Proof of the existence theorem: quasi-projective case.

The notation being still that of (5.1.4), suppose now that $f$ is quasi-projective. Then there exists a projective morphism $g:Z\to Y$ such that $X$ is identified with the $Y$-prescheme induced on an open subset of $Z$ (II, 5.3.2); if one sets $Z^{\prime }=g^{-1}(Y^{\prime })$, one has $X^{\prime }=X\cap Z^{\prime }$. Consequently, the completion $\mathfrak{X}=X_{/X^{\prime }}$ is identified with the formal prescheme induced by the completion $\mathfrak{J}=Z_{/Z^{\prime }}$ on the open subset $X\cap Z^{\prime }$ of $\mathfrak{J}$ (I, 10.8.5). Let $\mathcal{F}$ be a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module whose support $T^{\prime }$ is proper over Ŷ; this means by definition that there exists a closed subprescheme of $X^{\prime }$, having $T^{\prime }\subset X^{\prime }$ as underlying space, such that the restriction $T^{\prime }\to Y^{\prime }$ of $f$ is proper; it follows that $T^{\prime }$ is proper over $Y$, hence closed in $Z^{\prime }$ (II, 5.4.10). It follows that $\mathcal{F}$ is the sheaf induced on $\mathfrak{X}$ by the ${\mathcal{O}}_{\mathfrak{J}}$-module ${\mathcal{F}}^{\prime }$ obtained by glueing of $\mathcal{F}$ (defined on the open subset $\mathfrak{X}$ of $\mathfrak{J}$) and the sheaf 0 on the open subset $\mathfrak{J}-T^{\prime }$ of $\mathfrak{J}$, these two sheaves coinciding on the intersection open subset $\mathfrak{X}-T^{\prime }$. It is clear that ${\mathcal{F}}^{\prime }$ is coherent; by virtue of (5.2.5), there exists a coherent ${\mathcal{O}}_Z$-module $\mathcal{G}$ such that ${\mathcal{F}}^{\prime }=\hat{\mathcal{G}}$; let $T$ be the support of $\mathcal{G}$, so that $T^{\prime }=T\cap Z^{\prime }$ (I, 10.8.12). If $h$ is the restriction of $g$ to the reduced closed subprescheme of $Z$ having $T$ as underlying space, one then has $T^{\prime }=h^{-1}(Y^{\prime })=T\cap g^{-1}(Y^{\prime })$, and consequently $X\cap T$ is an open subset of $T$ containing $T^{\prime }$.

Since $h$ is proper (II, 5.4.2), hence closed, it follows from (5.1.3.1) that $X\cap T=T$; in other words $T\subset X$, and since $T$ is closed in $Z$, $T$ is proper over $Y$. If $\mathcal{F}$ is the sheaf induced on $X$ by $\mathcal{G}$, its completion $\hat{\mathcal{F}}$ is induced on $\mathfrak{X}$ by $\hat{\mathcal{G}}$ (I, 10.8.4), hence is equal to ${\mathcal{F}}^{\prime }$, which completes the proof.

5.3. Proof of the existence theorem: general case

Lemma (5.3.1).

Under the conditions of (5.1.4), if $0\to \mathcal{H}\to \mathcal{F}\to \mathcal{G}\to 0$ is an exact sequence of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules such that $\mathcal{G}$ and $\mathcal{H}$ are algebraizable, then $\mathcal{F}$ is algebraizable.

Proof. Indeed, suppose $\mathcal{G}=\hat{\mathcal{B}}$, $\mathcal{H}=\hat{\mathcal{C}}$, where $\mathcal{B}$ and $\mathcal{C}$ are coherent ${\mathcal{O}}_X$-modules with supports proper over $Y$; $\mathcal{F}$ canonically defines an element of the $A$-module $Ex{t}_{{\mathcal{O}}_{\mathfrak{X}}}^1(\mathfrak{X};\hat{\mathcal{B}},\hat{\mathcal{C}})$ (0, 12.3.2), and the hypotheses imply that this $A$-module is canonically isomorphic to $Ex{t}_{{\mathcal{O}}_X}^1(X;\mathcal{B},\mathcal{C})$ (4.5.2); there thus exists an exact sequence $0\to \mathcal{C}\to \mathcal{A}\to \mathcal{B}\to 0$ of coherent ${\mathcal{O}}_X$-modules such that the canonical image of the element of $Ex{t}_{{\mathcal{O}}_X}^1(X;\mathcal{B},\mathcal{C})$ corresponding to $\mathcal{A}$ is the element of $Ex{t}_{{\mathcal{O}}_{\mathfrak{X}}}^1(\mathfrak{X};\hat{\mathcal{B}},\hat{\mathcal{C}})$ corresponding to $\mathcal{F}$. But by definition (taking (I, 10.8.8, (ii)) into account), this means that $\mathcal{F}$ is isomorphic to $\hat{\mathcal{A}}$, whence the lemma, since $Supp(\mathcal{A})$ is contained in the union of $Supp(\mathcal{B})$ and $Supp(\mathcal{C})$, hence is proper over $Y$.

Corollary (5.3.2).

Under the conditions of (5.1.1), let $u:\mathcal{F}\to \mathcal{G}$ be a homomorphism of coherent ${\mathcal{O}}_{\mathfrak{X}}$-modules; if $\mathcal{G}$, $Ker(u)$ and $Coker(u)$ are algebraizable, then so is $\mathcal{F}$.

Proof. The lemma (5.2.2) applied to the homomorphism $\mathcal{G}\to Coker(u)$ shows indeed that $Im(u)$ is algebraizable, and it then suffices to apply lemma (5.3.1) to the exact sequence $0\to Ker(u)\to \mathcal{F}\to Im(u)\to 0$.

Lemma (5.3.3).

Under the conditions of (5.1.1), let $h:Z\to Y$ be a morphism of finite type, $\mathfrak{z}=Z_{/Z^{\prime }}$ the completion of $Z$ along $Z^{\prime }=h^{-1}(Y^{\prime })$, $g:Z\to X$ a proper $Y$-morphism, $\hat{g}:\mathfrak{z}\to \mathfrak{X}$ its extension to the completions. For every algebraizable ${\mathcal{O}}_{\mathfrak{z}}$-module ${\mathcal{F}}^{\prime }$, ${\hat{g}}_{\ast }({\mathcal{F}}^{\prime })$ is an algebraizable ${\mathcal{O}}_{\mathfrak{X}}$-module.

Proof. Indeed, if $\mathcal{F}$ is a coherent ${\mathcal{O}}_Z$-module such that ${\mathcal{F}}^{\prime }=\hat{\mathcal{F}}$, it follows from the first comparison theorem (4.1.5) that ${\hat{g}}_{\ast }({\mathcal{F}}^{\prime })$ is isomorphic to the completion of $g_{\ast }(\mathcal{F})$.

Lemma (5.3.4).

Let $X$ be a (usual) Noetherian scheme, $X^{\prime }$ a closed subset of $X$, $f:Z\to X$ a proper morphism, $Z^{\prime }=f^{-1}(X^{\prime })$, $\mathfrak{X}=X_{/X^{\prime }}$, $\mathfrak{z}=Z_{/Z^{\prime }}$, $\hat{f}:\mathfrak{z}\to \mathfrak{X}$ the extension of $f$ to the completions. Let $\mathcal{M}$ be a coherent Ideal of ${\mathcal{O}}_X$ such that, if $U=X-Supp({\mathcal{O}}_X/\mathcal{M})$, the restriction $f^{-1}(U)\to U$ of $f$ is an isomorphism. Then, for every coherent ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$, there exists an integer $n>0$ such that the kernel and cokernel of the canonical homomorphism ${\rho }_{\mathcal{F}}:\mathcal{F}\to {\hat{f}}_{\ast }({\hat{f}}^{\ast }(\mathcal{F}))$ $(0_I,\mathrm{4.4.3})$ are annihilated by ${\hat{\mathcal{M}}}^n$.

Proof. We may restrict to the case where $X=\mathrm{Spec}(B)$, $B$ a Noetherian ring, hence $X^{\prime }=V(\mathfrak{K})$, where $\mathfrak{K}$ is an ideal of $B$. We are going to see that one may reduce to the case where $B$ is an adic Noetherian ring and $\mathfrak{K}$ an ideal of definition of $B$. Indeed, let B_1 be the Hausdorff completion of $B$ for the $\mathfrak{K}$-preadic topology; if ${\mathfrak{K}}_1=\mathfrak{K}{B}_1$, B_1 is therefore an

adic Noetherian ring of which ${\mathfrak{K}}_1$ is an ideal of definition. Set $X_1=\mathrm{Spec}(B_1)$ and let $h:X_1\to X$ be the morphism corresponding to the canonical homomorphism $B\to B_1$; if $X_1^{\prime }=h^{-1}(X^{\prime })$, one then has $X_1^{\prime }=V({\mathfrak{K}}_1)$. Set finally $Z_1=Z{\times }_X{X}_1=Z_{(X_1)}$, $f_1=f_{(X_1)}:Z_1\to X_1$, which is a proper morphism (II, 5.4.2), and denote by ${\mathfrak{X}}_1$ the completion of X_1 along $X_1^{\prime }$, by ${\mathfrak{z}}_1=Z_1{\times }_X{\mathfrak{X}}_1$ the completion of Z_1 along $Z_1^{\prime }=f_1^{-1}(X_1^{\prime })$, by ${\hat{f}}_1$ the extension of $f_1$ to the completions. It is immediate that the extension $\hat{h}:{\mathfrak{X}}_1\to \mathfrak{X}$ of $h$ to the completions is an isomorphism, corresponding to the identity map of B_1 (I, 10.9.1); one concludes that the corresponding homomorphism ${\mathfrak{z}}_1\to \mathfrak{z}$ is also an isomorphism, these isomorphisms identifying ${\hat{f}}_1$ and $\hat{f}$. Finally, ${\mathcal{M}}_1=h^{\ast }(\mathcal{M})$ is a coherent Ideal of ${\mathcal{O}}_{X_1}$ and $Supp({\mathcal{O}}_{X_1}/{\mathcal{M}}_1)=h^{-1}(Supp({\mathcal{O}}_X/\mathcal{M}))$ (I, 9.1.13), hence, if $U_1=X_1-Supp({\mathcal{O}}_{X_1}/{\mathcal{M}}_1)$, one has $U_1=h^{-1}(U)$, whence it follows at once that the restriction $f_1^{-1}(U_1)\to U_1$ of $f_1$ is an isomorphism (I, 3.2.7); in addition, the completions $\hat{\mathcal{M}}$ and ${\hat{\mathcal{M}}}_1$ are identified by ĥ (I, 10.9.5). All hypotheses of (5.3.4) are therefore fulfilled by X_1, $X_1^{\prime }$, $f_1$ and ${\mathcal{M}}_1$, and one may therefore from now on suppose $B$ adic Noetherian and $\mathfrak{K}$ an ideal of definition of $B$. One then has $\mathfrak{X}=Spf(B)$, and $\mathcal{F}=N^{\Delta }$, where $N$ is a $B$-module of finite type, whence $\mathcal{F}=\hat{\mathcal{G}}$, where $\mathcal{G}$ is the coherent ${\mathcal{O}}_X$-module Ñ (I, 10.10.5), and consequently ${\hat{f}}^{\ast }(\mathcal{F})=\widehat{f^{\ast }(\mathcal{G})}$ (I, 10.9.5). In addition, by virtue of the first comparison theorem (4.1.5), ${\hat{f}}_{\ast }(\widehat{f^{\ast }(\mathcal{G})})$ is canonically identified with $\widehat{f_{\ast }(f^{\ast }(\mathcal{G}))}$, and the canonical homomorphism ${\rho }_{\mathcal{F}}$ is none other than ${\hat{\rho }}_{\mathcal{G}}$ by virtue of (5.1.3). Now, the kernel $\mathcal{P}$ and the cokernel $\mathcal{Q}$ of ${\rho }_{\mathcal{G}}:\mathcal{G}\to f_{\ast }(f^{\ast }(\mathcal{G}))$ are coherent ${\mathcal{O}}_X$-modules, and by hypothesis their restrictions to $U$ are obviously zero. There thus exists an integer $n>0$ such that ${\mathcal{M}}^n\mathcal{P}={\mathcal{M}}^n\mathcal{Q}=0$ (I, 9.3.4); one concludes that ${\hat{\mathcal{M}}}^n\hat{\mathcal{P}}={\hat{\mathcal{M}}}^n\hat{\mathcal{Q}}=0$ (I, 10.8.8 and 10.8.10).

5.3.5. End of the proof of the existence theorem.

The hypotheses being those of (5.1.4), we shall use the principle of Noetherian induction $(0_I,\mathrm{2.2.2})$, supposing therefore the theorem true for every closed subprescheme $T$ of $X$ whose underlying space is distinct from $X$ (the completion $\mathfrak{T}$ being of course the completion of $T$ along $T\cap X^{\prime }$). We may suppose $X$ non-empty. Since $f$ is separated and of finite type, we may apply Chow's lemma (II, 5.6.1): there thus exists a $Y$-scheme $Z$ and a $Y$-morphism $g:Z\to X$ such that the structure morphism $h:Z\to Y$ is quasi-projective, the morphism $g$ projective and surjective, and in addition a non-empty open subset $U$ of $X$ such that the restriction $g^{-1}(U)\to U$ is an isomorphism. Let $\mathcal{M}$ be a coherent Ideal of ${\mathcal{O}}_X$ defining a closed subprescheme of underlying space $X-U$ (I, 5.2.2), and let $\mathcal{F}$ be a coherent ${\mathcal{O}}_{\mathfrak{X}}$-module whose support $E$ is proper over $Y$; denote by $\mathfrak{z}$ the completion of $Z$ along $h^{-1}(Y^{\prime })$, by $\hat{g}:\mathfrak{z}\to \mathfrak{X}$ the extension of $g$ to the completions. Then ${\hat{g}}^{\ast }(\mathcal{F})$ is a coherent ${\mathcal{O}}_{\mathfrak{z}}$-module whose support is contained in $g^{-1}(E)$ and is consequently proper over $Y$, since $g$ is projective, hence proper (II, 5.4.6). Since $h$ is quasi-projective, ${\hat{g}}^{\ast }(\mathcal{F})$ is algebraizable by virtue of (5.2.6). We conclude

that ${\hat{g}}_{\ast }({\hat{g}}^{\ast }(\mathcal{F}))$ is an algebraizable ${\mathcal{O}}_{\mathfrak{X}}$-module (5.3.3) since $g$ is proper. We may now apply to $\mathcal{M}$ and to $g$ the result of (5.3.4): the kernel $\mathcal{P}$ and the cokernel $\mathcal{Q}$ of the homomorphism ${\rho }_{\mathcal{F}}:\mathcal{F}\to {\hat{g}}_{\ast }({\hat{g}}^{\ast }(\mathcal{F}))$ are annihilated by a power ${\hat{\mathcal{M}}}^n$; let $T$ be the closed subprescheme of $X$ defined by ${\mathcal{M}}^n$, having $X-U$ as underlying space, $j:T\to X$ the canonical injection, so that the extension to the completions $\hat{j}:\mathfrak{T}\to \mathfrak{X}$ is the canonical injection (I, 10.14.7). One may therefore write $\mathcal{P}={\hat{j}}_{\ast }({\hat{j}}^{\ast }(\mathcal{P}))$ and $\mathcal{Q}={\hat{j}}_{\ast }({\hat{j}}^{\ast }(\mathcal{Q}))$, and since $U$ is non-empty, it follows from the induction hypothesis that ${\hat{j}}^{\ast }(\mathcal{P})$ and ${\hat{j}}^{\ast }(\mathcal{Q})$ are algebraizable ${\mathcal{O}}_{\mathfrak{T}}$-modules; by virtue of (5.3.3), $\mathcal{P}$ and $\mathcal{Q}$ are algebraizable, and one may then apply (5.3.2), which finally proves that $\mathcal{F}$ is algebraizable. Q.E.D.

5.4. Application: comparison of morphisms of usual schemes and of morphisms of formal schemes. Algebraizable formal schemes

Theorem (5.4.1).

Let $A$ be an adic Noetherian ring, $\mathfrak{J}$ an ideal of definition of $A$, $S=\mathrm{Spec}(A)$, $S^{\prime }=V(\mathfrak{J})$. Let $u:X\to S$ be a proper morphism, $v:Y\to S$ a separated morphism of finite type, and let Ŝ, $\mathfrak{X}$, $\mathfrak{Y}$ be the completions of $S$, $X$, $Y$ along $S^{\prime }$, $u^{-1}(S^{\prime })$, $v^{-1}(S^{\prime })$ respectively. If, for every $S$-morphism $f:X\to Y$, $\hat{f}:\mathfrak{X}\to \mathfrak{Y}$ is the extension of $f$ to the completions, the map $f\mapsto \hat{f}$ is a bijection

  Hom_S(X, Y) ⥲ Hom_Ŝ(𝔛, 𝔜).

Proof. Let us first show that $f\mapsto \hat{f}$ is injective. Suppose indeed that two $S$-morphisms $f$, $g$ from $X$ to $Y$ are such that $\hat{f}=\hat{g}$. One then knows (I, 10.9.4) that there exists an open neighbourhood $V$ of $X^{\prime }=u^{-1}(S^{\prime })$ in which $f$ and $g$ coincide. Now, since $u$ is a closed map, one has $V=X$ (5.1.3.1), whence $f=g$.

Let us now prove that $f\mapsto \hat{f}$ is surjective, and let $h$ therefore be an Ŝ-morphism $\mathfrak{X}\to \mathfrak{Y}$. Let $Z=X{\times }_{S}Y$, and denote by $p:Z\to X$ and $q:Z\to Y$ the canonical projections; $Z$ is of finite type over $S$ (I, 6.3.4), hence Noetherian; denote by $\mathfrak{z}$ its completion along $Z^{\prime }=p^{-1}(u^{-1}(S^{\prime }))$; one knows that $\mathfrak{z}$ is canonically identified with $\mathfrak{X}{\times }_{\hat{S}}\mathfrak{Y}$, the projections $\mathfrak{z}\to \mathfrak{X}$ and $\mathfrak{z}\to \mathfrak{Y}$ being identified with the extensions $\hat{p}$ and $\hat{q}$ (I, 10.9.7). Since $Y$ is separated over $S$, $\mathfrak{Y}$ is separated over Ŝ (I, 10.15.7), hence the graph morphism ${\Gamma }_h=(1_{\mathfrak{X}},h):\mathfrak{X}\to \mathfrak{z}$ is a closed immersion (I, 10.15.4). Let $\mathfrak{T}$ be the closed formal subprescheme of $\mathfrak{z}$ associated to this immersion, and $j:\mathfrak{T}\to \mathfrak{z}$ the canonical injection, so that ${\Gamma }_h=j\circ w$, where $w:\mathfrak{X}\to \mathfrak{T}$ is an isomorphism (I, 10.14.3) whose inverse isomorphism is $\hat{p}\circ j$; in addition, $\mathfrak{T}$ is obviously proper over Ŝ, since $\mathfrak{X}$ is; one concludes (5.1.8) that there exists a closed subprescheme $T$ of $Z$ such that $\mathfrak{T}=\hat{T}=T_{/(T\cap Z^{\prime })}$, and that $j=\hat{i}$, where $i$ is the canonical injection $T\to Z$ (I, 10.14.7). Then $p\circ i:T\to X$ is an isomorphism, since it is so for $\widehat{p\circ i}=\hat{p}\circ \hat{i}$ by hypothesis, and it suffices to apply

(4.6.8), noting as above that $S$ is the only neighbourhood of $S^{\prime }$ in $S$. Let $g:X\to T$ be the inverse isomorphism of $p\circ i$, and set $f=q\circ i\circ g$, which is a morphism $X\to Y$ whose graph is by definition ${\Gamma }_f=i\circ g$. Since ĝ is the inverse isomorphism of $\widehat{p\circ i}=w$, one has $\widehat{{\Gamma }_f}=\hat{i}\circ \hat{g}=j\circ w={\Gamma }_h$. But one knows that $\widehat{{\Gamma }_f}={\Gamma }_{\hat{f}}$ (I, 10.9.8), whence finally $h=\hat{f}$, which completes the proof.

One may therefore say, in the language of categories, that the functor $X\mapsto \mathfrak{X}$ is fully faithful (0, 8.1.6) from the category of proper schemes over $\mathrm{Spec}(A)$ into the category of proper formal schemes over $Spf(A)$, for every adic Noetherian ring $A$; it consequently establishes an equivalence between the first of these categories and a subcategory of the second; the objects of the latter will be called algebraizable formal schemes. For such a scheme $\mathfrak{X}$, there exists a usual scheme $X$, proper over $\mathrm{Spec}(A)$, determined up to unique isomorphism, such that $\mathfrak{X}$ is isomorphic to $\hat{X}$.

Scholium (5.4.2).

With the notation of (5.4.1), set $S_n=\mathrm{Spec}(A/{\mathfrak{J}}^{n+1})$, $X_n=X{\times }_S{S}_n$, $Y_n=Y{\times }_S{S}_n$. It follows from (5.4.1) and from (I, 10.12.3) that to give an $S$-morphism $f:X\to Y$ is equivalent to giving an $(S_n)$-adic inductive system (I, 10.12.2) of $S_n$-morphisms $f_n:X_n\to Y_n$.

Remark (5.4.3).

Contrary to what the existence theorem (5.1.6) might suggest, there exist formal schemes proper over $Spf(A)$ that are not algebraizable (just as there exist compact analytic spaces that do not come from complex algebraic varieties). We shall later encounter such schemes in "moduli theory", which deals precisely (when the base field is $\mathbb{C}$) with the infinitesimal variations of the complex structure of a complete algebraic variety, and it is known that such variations may give rise to analytic varieties that are not algebraic.

Proposition (5.4.4).

Let $A$ be an adic Noetherian ring, $\mathfrak{S}=Spf(A)$, $g:\mathfrak{X}\to \mathfrak{S}$, $h:\mathfrak{Y}\to \mathfrak{S}$ two proper morphisms of formal schemes, $f:\mathfrak{X}\to \mathfrak{Y}$ an $\mathfrak{S}$-morphism. If $f$ is finite and if $\mathfrak{Y}$ is algebraizable, then $\mathfrak{X}$ is algebraizable.

Proof. Note that the hypotheses on $g$ and $h$ already entail that $f$ is proper (3.4.1), and for $f$ to be finite, it suffices that for every $y\in \mathfrak{Y}$, the fibre $f^{-1}(y)$ is finite (4.8.11). The hypothesis entails that $\mathcal{B}=f_{\ast }({\mathcal{O}}_{\mathfrak{X}})$ is a coherent ${\mathcal{O}}_{\mathfrak{Y}}$-Algebra (4.8.6), hence it follows from the existence theorem that, if $\mathfrak{Y}=\hat{Y}$ and $h=\hat{w}$, where $w:Y\to \mathrm{Spec}(A)$ is a proper morphism of usual schemes, there exists a coherent ${\mathcal{O}}_Y$-Algebra $\mathcal{C}$ such that $\mathcal{B}=\hat{\mathcal{C}}$. Let $X=\mathrm{Spec}(\mathcal{C})$, and $u:X\to Y$ the structure morphism; it then follows at once from the definition of $\mathfrak{X}$ from $\mathcal{B}$ (4.8.7) that $\mathfrak{X}$ is canonically isomorphic to $\hat{X}$ and that $f$ is identified with û (it suffices to see this for the case where $Y$ is affine).

Note that (5.1.8) is a particular case of (5.4.4).

Theorem (5.4.5).

Let $A$ be an adic Noetherian ring, $\mathfrak{J}$ an ideal of definition of $A$, $S=\mathrm{Spec}(A)$, $\mathfrak{S}=\hat{S}=Spf(A)$, $f:\mathfrak{X}\to \mathfrak{S}$ a proper morphism of formal schemes. Set $S_k=\mathrm{Spec}(A/{\mathfrak{J}}^{k+1})$, $X_k=\mathfrak{X}{\times }_{\mathfrak{S}}{S}_k$, and for every ${\mathcal{O}}_{\mathfrak{X}}$-module $\mathcal{F}$, ${\mathcal{F}}_k=\mathcal{F}{\otimes }_{{\mathcal{O}}_{\mathfrak{X}}}{\mathcal{O}}_{X_k}=\mathcal{F}/{\mathfrak{J}}^{k+1}\mathcal{F}$.

Let $\mathcal{L}$ be an invertible ${\mathcal{O}}_{\mathfrak{X}}$-module, and suppose that ${\mathcal{L}}_0=\mathcal{L}/\mathfrak{J}\mathcal{L}$ is an ample ${\mathcal{O}}_{X_0}$-module. Then $\mathfrak{X}$ is algebraizable, and if $X$ is a proper $S$-scheme such that $\mathfrak{X}$ is isomorphic to $\hat{X}$, there exists an ample ${\mathcal{O}}_X$-module $\mathcal{M}$ such that $\mathcal{L}$ is isomorphic to $\hat{\mathcal{M}}$ (which implies that $X$ is projective over $S$).

Proof. Let us apply (5.2.3) to $\mathcal{F}={\mathcal{O}}_{\mathfrak{X}}$: there thus exists an integer $n_0$ such that for $n\ge n_0$, the canonical homomorphism $\Gamma (\mathfrak{X},{\mathcal{L}}^{\otimes n})\to \Gamma (X_0,{\mathcal{L}}_0^{\otimes n})$ is surjective. One may suppose $n\ge n_0$ chosen large enough that ${\mathcal{L}}_0^{\otimes n}$ is very ample for S_0 (II, 4.5.10). Since the morphism $f_0:X_0\to S_0$ is proper, $\Gamma (X_0,{\mathcal{L}}_0^{\otimes n})$ is an $A$-module of finite type (3.2.1), hence there exists a sub-$A$-module of finite type $E$ of $\Gamma (\mathfrak{X},{\mathcal{L}}^{\otimes n})$ whose image in $\Gamma (X_0,{\mathcal{L}}_0^{\otimes n})$ is this latter module in its entirety. This being so, for every $k\ge 0$, consider the homomorphism $u_k:E/{\mathfrak{J}}^{k+1}E\to \Gamma (X_k,{\mathcal{L}}_k^{\otimes n})$ deduced from the canonical injection $E\to \Gamma (\mathfrak{X},{\mathcal{L}}^{\otimes n})$. Note that $(f_k{)}_{\ast }({\mathcal{L}}_k^{\otimes n})$ is quasi-coherent, and since $\Gamma (S_k,(f_k{)}_{\ast }({\mathcal{L}}_k^{\otimes n}))=\Gamma (X_k,{\mathcal{L}}_k^{\otimes n})$, $u_k$ defines a homomorphism $u_k:(E/{\mathfrak{J}}^{k+1}E{)}^{\sim }\to (f_k{)}_{\ast }({\mathcal{L}}_k^{\otimes n})$, and consequently also a homomorphism $u_k^{♯}:f_k^{\ast }((E/{\mathfrak{J}}^{k+1}E{)}^{\sim })\to {\mathcal{L}}_k^{\otimes n}$. Moreover, if one sets ${\mathcal{G}}_k=f_k^{\ast }((E/{\mathfrak{J}}^{k+1}E{)}^{\sim })$, one has ${\mathcal{G}}_k={\mathcal{G}}_0/{\mathfrak{J}}^k{\mathcal{G}}_0$ (I, 9.1.5), hence $u_k^{♯}:{\mathcal{G}}_k/{\mathfrak{J}}^k{\mathcal{G}}_k\to {\mathcal{L}}_k^{\otimes n}/{\mathfrak{J}}^k{\mathcal{L}}_k^{\otimes n}$ is deduced from $u_0^{♯}$ by passage to quotients. Now, by definition of $E$, $u_0^{♯}$ is none other than the canonical homomorphism $\sigma :f_0^{\ast }((f_0{)}_{\ast }({\mathcal{L}}_0^{\otimes n}))\to {\mathcal{L}}_0^{\otimes n}$, and the hypothesis that ${\mathcal{L}}_0^{\otimes n}$ is very ample entails that $u_0^{♯}$ is surjective (II, 4.4.3); one then deduces from Nakayama's lemma that each of the $u_k^{♯}$ is also surjective. Each of the $u_k^{♯}$ therefore defines (II, 4.2.2) an $S_k$-morphism $g_k:X_k\to P_k=\mathbf{P}(E/{\mathfrak{J}}^{k+1}E)$, and since $P_h=P_k{\times }_{S_k}{S}_h$ for $h\le k$ by virtue of (II, 4.1.3), $(g_k)$ is an $(S_k)$-adic inductive system (I, 10.12.2) by virtue of the relations between the $u_k^{♯}$ and of (II, 4.2.10). The $g_k$ therefore define an $\mathfrak{S}$-morphism of formal schemes $g:\mathfrak{X}\to \mathfrak{P}$, where $\mathfrak{P}$ is the inductive limit of the system $(P_k)$, or equivalently the completion $\hat{P}$, where $P=\mathbf{P}(E)$. In addition, the hypothesis that ${\mathcal{L}}_0^{\otimes n}$ is very ample entails that $g_0$ is a closed immersion (II, 4.4.3); one concludes that $g$ is a closed immersion of formal schemes (4.8.10), hence $\mathfrak{X}$ is algebraizable (5.1.8). The fact that $\mathcal{L}$ is isomorphic to the completion $\hat{\mathcal{M}}$ of an invertible ${\mathcal{O}}_X$-module then follows from the existence theorem (5.1.6). In addition, ${\mathcal{L}}^{\otimes n}$ is then the completion of ${\mathcal{M}}^{\otimes n}$ (I, 10.8.10), and the homomorphisms $u_k^{♯}$ define a well-determined homomorphism $\hat{v}:f^{\ast }(\tilde{E})\to {\hat{\mathcal{M}}}^{\otimes n}$ (5.1.7); moreover, since $\hat{v}$ is surjective, so is $v$ (I, 10.11.5), hence so is $u$ (5.1.3); in addition, the morphism $r:X\to P$ defined by $v$ (II, 4.2.2) has as extension to the completions $g$, and since $g$ is a closed immersion, so is $r$, by (5.1.8) and (5.4.1); one concludes that ${\mathcal{M}}^{\otimes n}$ is very ample (II, 4.4.6) and $\mathcal{M}$ is ample (II, 4.5.10).

Remark (5.4.6).

Let $A$ be an adic Noetherian ring, $\mathfrak{S}=Spf(A)$, $f:\mathfrak{X}\to \mathfrak{S}$ a proper morphism of formal schemes. Let $\mathcal{N}$ be the coherent Ideal of ${\mathcal{O}}_{\mathfrak{X}}$ such that for every formal affine open $U$ of $\mathfrak{X}$, $\Gamma (U,\mathcal{N})$ is the nilradical of $\Gamma (U,{\mathcal{O}}_{\mathfrak{X}})$; the existence of this Ideal follows easily from (I, 10.10.2) and from the fact that every ring homomorphism $B\to C$ sends the nilradical of $B$ into that of $C$. Let ${\mathfrak{X}}^{\prime }$ be the closed formal subscheme of $\mathfrak{X}$ defined by $\mathcal{N}$ (I, 10.14.2); it would be interesting to know whether, when ${\mathfrak{X}}^{\prime }$ is algebraizable, $\mathfrak{X}$ itself is algebraizable. One would no doubt arrive at a solution to this problem if one knew how to classify (for example by means of invariants of

cohomological nature) the extensions of a structure sheaf ${\mathcal{O}}_{\mathfrak{X}}$ (for a usual prescheme or a formal prescheme) by an Ideal of square zero, in other words the ${\mathcal{O}}_{\mathfrak{X}}$-Algebras $\mathcal{A}$ such that ${\mathcal{O}}_{\mathfrak{X}}$ is isomorphic to $\mathcal{A}/\mathcal{J}$, where $\mathcal{J}$ is an Ideal of square zero of $\mathcal{A}$.

5.5. A decomposition of certain schemes

Proposition (5.5.1).

Let $A$ be an adic Noetherian ring, $\mathfrak{J}$ an ideal of definition of $A$, $Y=\mathrm{Spec}(A)$. Let $f:X\to Y$ be a separated morphism of finite type; set $Y_0=\mathrm{Spec}(A/\mathfrak{J})$, $X_0=X{\times }_Y{Y}_0=f^{-1}(Y_0)$. Let Z_0 be an open part of X_0, proper over Y_0; then there exists in $X$ an open and closed part $Z$, proper over $Y$ and such that $Z\cap X_0=Z_0$.

Proof. By hypothesis, there is an open subset $T$ of $X$ such that $T\cap X_0=Z_0$; let $\mathfrak{T}$ be the completion along Z_0 of the scheme induced by $X$ on the open subset $T$; the support of ${\mathcal{O}}_{\mathfrak{T}}$ being Z_0, which is proper over Y_0, $\mathfrak{T}$ is proper over $\hat{Y}=Spf(A)$ (3.4.1). It follows from (5.1.8) that there exists a closed subscheme $Z$ of $T$ proper over $Y$ such that, if $i:Z\to T$ is the canonical injection, $\hat{i}:\hat{\mathcal{Z}}\to \mathfrak{T}$ is an isomorphism ($\hat{\mathcal{Z}}$ being the completion of $Z$ along Z_0). One concludes (4.6.8) that there exists in $T$ an open neighbourhood $V$ of Z_0 such that the restriction $i^{-1}(V)\to V$ of $i$ is an isomorphism. But $i^{-1}(V)$ is a neighbourhood of Z_0 in $Z$, hence is necessarily identical to $Z$ (5.1.3.1). One concludes that $Z$ is open in $T$, hence in $X$, which completes the proof.

Corollary (5.5.2).

If X_0 is proper over Y_0, $X$ is the union of two disjoint open parts $Z$ and $Z^{\prime }$ such that $Z$ is proper over $Y$ and contains X_0; in addition, every closed part $P$ of $X$, proper over $Y$, is contained in $Z$.

Proof. The last assertion follows from the fact that $P\cap Z^{\prime }$, being closed in $P$, is proper over $Y$; if $P\cap Z^{\prime }$ were not empty, $f(P\cap Z^{\prime })$ would be closed non-empty in $Y$, hence would meet Y_0 (5.1.3.1), which contradicts the definition of $Z$.

(To be continued.)