§10. Formal Schemes

10.1. Formal affine schemes

(10.1.1) Let $A$ be an admissible topological ring (0, 7.1.2); for every ideal of definition $\mathfrak{J}$ of $A$, $\mathrm{Spec}(A/\mathfrak{J})$ is identified with the closed subspace $V(\mathfrak{J})$ of $\mathrm{Spec}(A)$ (1.1.11), the set of open prime ideals of $A$; this topological space does not depend on the ideal of definition $\mathfrak{J}$ considered; let us denote it by $\mathfrak{X}$. Let $({\mathfrak{J}}_{\lambda })$ be a fundamental system of neighborhoods of $0$ in $A$, formed of ideals of definition, and for each $\lambda$, let ${\mathcal{O}}_{\lambda }$ be the structure sheaf of $\mathrm{Spec}(A/{\mathfrak{J}}_{\lambda })$; this sheaf is induced on $\mathfrak{X}$ by $\widetilde{A/{\mathfrak{J}}_{\lambda }}$ (which is zero off $\mathfrak{X}$). For ${\mathfrak{J}}_{\mu }\subset {\mathfrak{J}}_{\lambda }$, the canonical homomorphism $A/{\mathfrak{J}}_{\mu }\to A/{\mathfrak{J}}_{\lambda }$ therefore defines a homomorphism $u_{\lambda \mu }:{\mathcal{O}}_{\mu }\to {\mathcal{O}}_{\lambda }$ of sheaves of rings (1.6.1), and $({\mathcal{O}}_{\lambda })$ is a projective system of sheaves of rings for these homomorphisms. Since the topology of $\mathfrak{X}$ admits a base formed of quasi-compact opens, one may associate to each ${\mathcal{O}}_{\lambda }$ a sheaf of pseudo-discrete topological rings (0, 3.8.1) which has ${\mathcal{O}}_{\lambda }$ as its underlying sheaf of rings (without topology), and which we shall again denote by ${\mathcal{O}}_{\lambda }$; and the ${\mathcal{O}}_{\lambda }$ again form a projective system of sheaves of topological rings (0, 3.8.2). We shall designate by ${\mathcal{O}}_{\mathfrak{X}}$ the sheaf of topological rings on $\mathfrak{X}$, the projective limit of the system $({\mathcal{O}}_{\lambda })$; for every quasi-compact open $U$ of $\mathfrak{X}$, $\Gamma (U,{\mathcal{O}}_{\mathfrak{X}})$ is therefore the topological ring projective limit of the system of discrete rings $\Gamma (U,{\mathcal{O}}_{\lambda })$ (0, 3.2.6).

Definition (10.1.2). Given an admissible topological ring $A$, one calls formal spectrum of $A$, and denotes by $\mathrm{Spf}(A)$, the closed subspace $\mathfrak{X}$ of $\mathrm{Spec}(A)$ formed of the open prime ideals of $A$. One says that a topologically ringed space is a formal affine scheme if it is isomorphic to a formal spectrum $\mathrm{Spf}(A)=\mathfrak{X}$ equipped with the sheaf of topological rings ${\mathcal{O}}_{\mathfrak{X}}$, projective limit of the sheaves of pseudo-discrete rings $(\widetilde{A/{\mathfrak{J}}_{\lambda }})|\mathfrak{X}$, where ${\mathfrak{J}}_{\lambda }$ runs through the filtered set of ideals of definition of $A$.

When we speak of a formal spectrum $\mathfrak{X}=\mathrm{Spf}(A)$ as a formal affine scheme, it will always be a question of the topologically ringed space $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ where ${\mathcal{O}}_{\mathfrak{X}}$ is defined as above.

One will note that every affine scheme $X=\mathrm{Spec}(A)$ may be considered as a formal affine scheme in one and only one way, by considering $A$ as a discrete topological ring: the topological rings $\Gamma (U,{\mathcal{O}}_X)$ are then discrete when $U$ is quasi-compact (but not in general when $U$ is an arbitrary open of $X$).

Proposition (10.1.3). If $\mathfrak{X}=\mathrm{Spf}(A)$, where $A$ is an admissible ring, then $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ is topologically isomorphic to $A$.

Proof. Indeed, since $\mathfrak{X}$ is closed in $\mathrm{Spec}(A)$, it is quasi-compact, and consequently $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ is topologically isomorphic to the projective limit of the discrete rings $\Gamma (\mathfrak{X},{\mathcal{O}}_{\lambda })$; but $\Gamma (\mathfrak{X},{\mathcal{O}}_{\lambda })$ is isomorphic to $A/{\mathfrak{J}}_{\lambda }$ (1.3.7); since $A$ is separated and complete, it is topologically isomorphic to $\underleftarrow{\lim }A/{\mathfrak{J}}_{\lambda }$ (0, 7.2.1), whence the proposition.

Proposition (10.1.4). Let $A$ be an admissible ring, $\mathfrak{X}=\mathrm{Spf}(A)$, and for every $f\in A$, let $\mathfrak{D}(f)=D(f)\cap \mathfrak{X}$; the topologically ringed space $(\mathfrak{D}(f),{\mathcal{O}}_{\mathfrak{X}}|\mathfrak{D}(f))$ is isomorphic to the formal affine spectrum $\mathrm{Spf}(A_{\{f\}})$ (0, 7.6.15).

Proof. For every ideal of definition $\mathfrak{J}$ of $A$, the discrete ring $\widetilde{A_{\{f\}}/{\mathfrak{J}}_{\{f\}}}$ is identified canonically with $A_f/{\mathfrak{J}}_f$ (0, 7.6.9); hence (1.2.5 and 1.2.6) the topological space $\mathrm{Spf}(A_{\{f\}})$ is identified canonically with $\mathfrak{D}(f)$. Moreover, for every quasi-compact open $U$ of $\mathfrak{X}$ contained in $\mathfrak{D}(f)$, $\Gamma (U,{\mathcal{O}}_{\mathfrak{X}})$ is identified with the module of sections of the structure sheaf of $\mathrm{Spec}(A_f/{\mathfrak{J}}_f)$ over $U$ (1.3.6); hence, if one sets $\mathfrak{Y}=\mathrm{Spf}(A_{\{f\}})$, then $\Gamma (U,{\mathcal{O}}_{\mathfrak{X}})$ is identified with the module of sections $\Gamma (U,{\mathcal{O}}_{\mathfrak{Y}})$, which proves the proposition.

(10.1.5) As a sheaf of rings without topology, the structure sheaf ${\mathcal{O}}_{\mathfrak{X}}$ of $\mathrm{Spf}(A)$ admits, for every $x\in \mathfrak{X}$, a stalk which, by virtue of (10.1.4), is identified with the inductive limit $\underrightarrow{\lim }{A}_{\{f\}}$ for $f\notin x$. Consequently (0, 7.6.17 and 7.6.18):

Proposition (10.1.6). For every $x\in \mathfrak{X}=\mathrm{Spf}(A)$, the stalk ${\mathcal{O}}_x$ is a local ring whose residue field is isomorphic to $\kappa (x)=A_x/{\mathfrak{j}}_x{A}_x$. If in addition $A$ is adic and Noetherian, ${\mathcal{O}}_x$ is a Noetherian ring.

Since $\kappa (x)$ is not reduced to $0$, one concludes from this result that the support of the sheaf of rings ${\mathcal{O}}_{\mathfrak{X}}$ is equal to $\mathfrak{X}$.

10.2. Morphisms of formal affine schemes

(10.2.1) Let $A$, $B$ be two admissible rings, and let $\phi :B\to A$ be a continuous homomorphism. The continuous map ${}^a\phi :\mathrm{Spec}(A)\to \mathrm{Spec}(B)$ (1.2.1) then sends $\mathfrak{X}=\mathrm{Spf}(A)$ into $\mathfrak{Y}=\mathrm{Spf}(B)$, for the inverse image under $\phi$ of an open prime ideal of $A$ is an open prime ideal of $B$. On the other hand, for every $g\in B$, $\phi$ defines a continuous homomorphism $\Gamma (\mathfrak{D}(g),{\mathcal{O}}_{\mathfrak{Y}})\to \Gamma (\mathfrak{D}(\phi (g)),{\mathcal{O}}_{\mathfrak{X}})$ by virtue of (10.1.4), (10.1.3), and (0, 7.6.7); since these homomorphisms satisfy the conditions of compatibility for the restrictions corresponding to the passage from $g$ to a multiple of $g$, and since $\mathfrak{D}(\phi (g))={}^a{\phi }^{-1}(\mathfrak{D}(g))$, they define a continuous homomorphism of sheaves of topological rings $\theta :{\mathcal{O}}_{\mathfrak{Y}}\to {\psi }_{\ast }({\mathcal{O}}_{\mathfrak{X}})$ (0, 3.2.5), which we shall again denote by $\tilde{\phi }$; one has thus obtained a morphism $({}^a\phi ,\tilde{\phi })$ of topologically ringed spaces $\mathfrak{X}\to \mathfrak{Y}$. One will note that, as a homomorphism of sheaves of rings without topology, $\tilde{\phi }$ defines a homomorphism ${\tilde{\phi }}_x^{♯}:{\mathcal{O}}_{{}^a\phi (x)}\to {\mathcal{O}}_x$ on the stalks, for every $x\in \mathfrak{X}$.

Proposition (10.2.2). Let $A$, $B$ be two admissible topological rings, and let $\mathfrak{X}=\mathrm{Spf}(A)$, $\mathfrak{Y}=\mathrm{Spf}(B)$. For a morphism $u=(\psi ,\theta ):\mathfrak{X}\to \mathfrak{Y}$ of topologically ringed spaces to be of the form $({}^a\phi ,\tilde{\phi })$, where $\phi$ is a continuous ring homomorphism $B\to A$, it is necessary and sufficient that for every $x\in \mathfrak{X}$, ${\theta }_x^{♯}$ be a local homomorphism ${\mathcal{O}}_{\psi (x)}\to {\mathcal{O}}_x$.

Proof. The condition is necessary: indeed, let $\mathfrak{p}={\mathfrak{j}}_x\in \mathrm{Spf}(A)$, and let $\mathfrak{q}={\phi }^{-1}(\mathfrak{p})$; if $g\notin \mathfrak{q}$, one has $\phi (g)\notin \mathfrak{p}$, and it is immediate that the homomorphism $B_{\mathfrak{q}}\to A_{\mathfrak{p}}$ deduced from $\phi$ (0, 7.6.7) transforms $\mathfrak{q}{B}_{\mathfrak{q}}$ into a part of $\mathfrak{p}{A}_{\mathfrak{p}}$; passing to the inductive limit, one sees therefore (taking account of (10.1.5) and of (0, 7.6.17)) that ${\tilde{\phi }}_x^{♯}$ is a local homomorphism.

Conversely, let $(\psi ,\theta )$ be a morphism verifying the condition of the statement; by virtue of (10.1.3), $\theta$ defines a continuous ring homomorphism $$\phi =\Gamma (\theta ):B=\Gamma (\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}})\to \Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})=A.$$ By virtue of the hypothesis on $\theta$, for the section $\phi (g)$ of ${\mathcal{O}}_{\mathfrak{X}}$ over $\mathfrak{X}$ to have an invertible germ at the point $x$, it is necessary and sufficient that $g$ have an invertible germ at the point $\psi (x)$. But by virtue of (0, 7.6.17), the sections of ${\mathcal{O}}_{\mathfrak{X}}$ (resp. ${\mathcal{O}}_{\mathfrak{Y}}$) over $\mathfrak{X}$ (resp. $\mathfrak{Y}$) whose germ is not invertible at the point $x$ (resp. $\psi (x)$) are exactly the elements of ${\mathfrak{j}}_x$ (resp. ${\mathfrak{j}}_{\psi (x)}$); the preceding remark thus shows that $\psi ={}^a\phi$. Finally, for every $g\in B$, the diagram $$\begin{array}{ccc}B=\Gamma (\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}})& ⟶& \Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})=A\\ \downarrow & & \downarrow \\ B_{\{g\}}=\Gamma (\mathfrak{D}(g),{\mathcal{O}}_{\mathfrak{Y}})& ⟶& \Gamma (\mathfrak{D}(\phi (g)),{\mathcal{O}}_{\mathfrak{X}})=A_{\{\phi (g)\}}\end{array}$$ is commutative; by the universal property of the complete rings of fractions (0, 7.6.6), ${\theta }_g$ is equal to ${\tilde{\phi }}_g$ for every $g\in B$, hence (0, 3.2.5) one has $\theta =\tilde{\phi }$.

We shall say that a morphism $(\psi ,\theta )$ of topologically ringed spaces verifying the condition of (10.2.2) is a morphism of formal affine schemes. One may say that the functors $\mathrm{Spf}(A)$ in $A$ and $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$ in $\mathfrak{X}$ define an equivalence between the category of admissible rings and the dual category of the category of formal affine schemes (T, I, 1.2).

(10.2.3) As a particular case of (10.2.2), let us note that, for $f\in A$, the canonical injection of the formal affine scheme induced by $\mathfrak{X}$ on $\mathfrak{D}(f)$ corresponds to the canonical continuous homomorphism $A\to A_{\{f\}}$. Under the hypotheses of (10.2.2), let $h$ be an element of $B$ and $g$ an element of $A$, a multiple of $\phi (h)$; one then has $\psi (\mathfrak{D}(g))\subset \mathfrak{D}(h)$; the restriction of $u$ to $\mathfrak{D}(g)$, considered as a morphism of $\mathfrak{D}(g)$ into $\mathfrak{D}(h)$, is the unique morphism $v$ making commutative the diagram $$\begin{array}{ccc}\mathfrak{D}(g)& \stackrel{v}{\to }& \mathfrak{D}(h)\\ \downarrow & & \downarrow \\ \mathfrak{X}& \stackrel{u}{\to }& \mathfrak{Y}\end{array}$$ This morphism corresponds to the unique continuous homomorphism ${\phi }^{\prime }:B_{\{h\}}\to A_{\{g\}}$ (0, 7.6.7) making commutative the diagram $$\begin{array}{ccc}A& \stackrel{\phi }{\leftarrow }& B\\ \downarrow & & \downarrow \\ A_{\{g\}}& \stackrel{{\phi }^{\prime }}{\leftarrow }& B_{\{h\}}\end{array}$$

10.3. Ideals of definition of a formal affine scheme

(10.3.1) Let $A$ be an admissible ring, $\mathfrak{J}$ an open ideal of $A$, $\mathfrak{X}$ the formal affine scheme $\mathrm{Spf}(A)$. Let $({\mathfrak{J}}_{\lambda })$ be the set of ideals of definition of $A$ contained in $\mathfrak{J}$; then $\widetilde{\mathfrak{J}/{\mathfrak{J}}_{\lambda }}$ is a sheaf of ideals of $\widetilde{A/{\mathfrak{J}}_{\lambda }}$. Let us designate by ${\mathfrak{J}}^{\Delta }$ the projective limit of the sheaves induced on $\mathfrak{X}$ by $\widetilde{\mathfrak{J}/{\mathfrak{J}}_{\lambda }}$, which is identified with a sheaf of ideals of ${\mathcal{O}}_{\mathfrak{X}}$ (0, 3.2.6). For every $f\in A$, $\Gamma (\mathfrak{D}(f),{\mathfrak{J}}^{\Delta })$ is the projective limit of ${\mathfrak{J}}_f/({\mathfrak{J}}_{\lambda }{)}_f$, in other words is identified with the open ideal ${\mathfrak{J}}_{\{f\}}$ of the ring $A_{\{f\}}$ (0, 7.6.9), and in particular $\Gamma (\mathfrak{X},{\mathfrak{J}}^{\Delta })=\mathfrak{J}$; one concludes (the $\mathfrak{D}(f)$ forming a base for the topology of $\mathfrak{X}$) that one has

$${\mathfrak{J}}_{\{f\}}=({\mathfrak{J}}^{\Delta }{)}_{\{f\}}\text{\hspace{1em}(10.3.1.1)}$$

(10.3.2) With the notations of (10.3.1), for every $f\in A$, the canonical map of $A_{\{f\}}=\Gamma (\mathfrak{D}(f),{\mathcal{O}}_{\mathfrak{X}})$ into $\Gamma (\mathfrak{D}(f),\widetilde{(A/\mathfrak{J}{)}_f})=A_f/{\mathfrak{J}}_f$ is surjective and has for kernel $\Gamma (\mathfrak{D}(f),{\mathfrak{J}}^{\Delta })={\mathfrak{J}}_{\{f\}}$ (0, 7.6.9); these maps therefore define a continuous surjective homomorphism, said to be canonical, of the sheaf of topological rings ${\mathcal{O}}_{\mathfrak{X}}$ onto the sheaf of discrete rings $(\widetilde{A/\mathfrak{J}})|\mathfrak{X}$ whose kernel is ${\mathfrak{J}}^{\Delta }$; this homomorphism is moreover none other than $\tilde{\phi }$ (10.2.1), where $\phi$ is the continuous homomorphism $A\to A/\mathfrak{J}$; the morphism $(\psi ,\theta ):\mathrm{Spec}(A/\mathfrak{J})\to \mathfrak{X}$ of formal affine schemes (where $\psi$ is moreover the identity homeomorphism of $\mathfrak{X}$ onto itself) is again said to be canonical. One therefore has, by what precedes, a canonical isomorphism

$${\mathcal{O}}_{\mathfrak{X}}/{\mathfrak{J}}^{\Delta }\cong (\widetilde{A/\mathfrak{J}})|\mathfrak{X}\text{\hspace{1em}(10.3.2.1)}$$

It is clear (by virtue of $\Gamma (\mathfrak{X},{\mathfrak{J}}^{\Delta })=\mathfrak{J}$) that the map $\mathfrak{J}\to {\mathfrak{J}}^{\Delta }$ is strictly increasing; by what precedes, for $\mathfrak{J}\subset {\mathfrak{J}}^{\prime }$, the sheaf ${\mathfrak{J}}^{\Delta }/{\mathfrak{J}}^{\prime \Delta }$ is canonically isomorphic to $\widetilde{\mathfrak{J}/{\mathfrak{J}}^{\prime }}=(\mathfrak{J}/{\mathfrak{J}}^{\prime }{)}^{\sim }$.

(10.3.3) The hypotheses and notations still being those of (10.3.1), we shall say that a sheaf of ideals $\mathcal{I}$ of ${\mathcal{O}}_{\mathfrak{X}}$ is a sheaf of ideals of definition of $\mathfrak{X}$ (or an Ideal of definition of $\mathfrak{X}$) if, for every $x\in \mathfrak{X}$, there exists an open neighborhood of $x$ of the form $\mathfrak{D}(f)$, where $f\in A$, such that $\mathcal{I}|\mathfrak{D}(f)$ is of the form ${\mathfrak{g}}^{\Delta }$, where $\mathfrak{g}$ is an ideal of definition of $A_{\{f\}}$.

Proposition (10.3.4). For every $f\in A$, every Ideal of definition of $\mathfrak{X}$ induces an Ideal of definition of $\mathfrak{D}(f)$.

Proof. This results from (10.3.1.1).

Proposition (10.3.5). If $A$ is an admissible ring, every Ideal of definition of $\mathfrak{X}=\mathrm{Spf}(A)$ is of the form ${\mathfrak{J}}^{\Delta }$, where $\mathfrak{J}$ is an ideal of definition of $A$, uniquely determined.

Proof. Indeed, let $\mathcal{I}$ be an Ideal of definition of $\mathfrak{X}$; by hypothesis, and since $\mathfrak{X}$ is quasi-compact, there is a finite number of elements $f_i\in A$ such that the $\mathfrak{D}(f_i)$ cover $\mathfrak{X}$ and that $\mathcal{I}|\mathfrak{D}(f_i)={\mathfrak{g}}_i^{\Delta }$, where ${\mathfrak{g}}_i$ is an ideal of definition of $A_{\{f_i\}}$. For each $i$, there exists therefore an open ideal ${\mathfrak{K}}_i$ of $A$ such that $({\mathfrak{K}}_i{)}_{\{f_i\}}={\mathfrak{g}}_i$ (0, 7.6.9); let ${\mathfrak{J}}_1$ be an ideal of definition of $A$ contained in all the ${\mathfrak{K}}_i$. The canonical image of $\mathcal{I}/{\mathfrak{J}}_1^{\Delta }$ in the structure sheaf $(\widetilde{A/{\mathfrak{J}}_1})$ of $\mathrm{Spec}(A/{\mathfrak{J}}_1)$ (10.3.2) is then such that its restriction to $\mathfrak{D}(f_i)$ is equal to that of $(\widetilde{{\mathfrak{K}}_i/{\mathfrak{J}}_1})$; one concludes that this canonical image is a quasi-coherent sheaf on $\mathrm{Spec}(A/{\mathfrak{J}}_1)$, hence of the form $(\widetilde{\mathfrak{J}/{\mathfrak{J}}_1})$, where $\mathfrak{J}$ is an ideal of $A$ containing ${\mathfrak{J}}_1$ (1.4.1), whence $\mathcal{I}={\mathfrak{J}}^{\Delta }$ (10.3.2); moreover, since for each $i$ there exists an integer $n_i$ such that ${\mathfrak{g}}_i^{n_i}\subset ({\mathfrak{J}}_1{)}_{\{f_i\}}$, one will have, denoting by $n$ the largest of the $n_i$, $(\mathfrak{J}/{\mathfrak{J}}_1{)}^n|\mathfrak{D}(f_i)=0$, and consequently (10.3.2) $((\mathfrak{J}/{\mathfrak{J}}_1{)}^n{)}^{\Delta }=0$, whence finally $(\mathfrak{J}/{\mathfrak{J}}_1{)}^n=0$ (1.3.13), which proves that $\mathfrak{J}$ is an ideal of definition of $A$ (0, 7.1.4).

Proposition (10.3.6). Let $A$ be an adic ring, $\mathfrak{J}$ an ideal of definition of $A$ such that $\mathfrak{J}/{\mathfrak{J}}^2$ is an $(A/\mathfrak{J})$-module of finite type. For every integer $n>0$, one then has $({\mathfrak{J}}^{\Delta }{)}^n=({\mathfrak{J}}^n{)}^{\Delta }$.

Proof. Indeed, for every $f\in A$, one has (since ${\mathfrak{J}}^n$ is an open ideal) $$(({\mathfrak{J}}^{\Delta }{)}^n{)}_{\{f\}}=(({\mathfrak{J}}^{\Delta }{)}_{\{f\}}{)}^n=({\mathfrak{J}}_{\{f\}}{)}^n=({\mathfrak{J}}^n{)}_{\{f\}}=(({\mathfrak{J}}^n{)}^{\Delta }{)}_{\{f\}}$$ by virtue of (10.3.1.1) and of (0, 7.6.12). Since $({\mathfrak{J}}^{\Delta }{)}^n$ is associated to the presheaf $U\to (\Gamma (U,{\mathfrak{J}}^{\Delta }){)}^n$ (0, 4.1.6), the corollary results from this, since the $\mathfrak{D}(f)$ form a base for the topology of $\mathfrak{X}$.

(10.3.7) One says that a family $({\mathcal{I}}_{\lambda })$ of Ideals of definition of $\mathfrak{X}$ is a fundamental system of Ideals of definition if every Ideal of definition of $\mathfrak{X}$ contains one of the ${\mathcal{I}}_{\lambda }$; since ${\mathcal{I}}_{\lambda }={\mathfrak{J}}_{\lambda }^{\Delta }$, it amounts to the same to say that the ${\mathfrak{J}}_{\lambda }$ form a fundamental system of neighborhoods of $0$ in $A$. Let $(f_{\alpha })$ be a family of elements of $A$ such that the $\mathfrak{D}(f_{\alpha })$ cover $\mathfrak{X}$. If $({\mathcal{I}}_{\lambda })$ is a filtered decreasing family of ideals of ${\mathcal{O}}_{\mathfrak{X}}$ such that for every $\alpha$, the family $({\mathcal{I}}_{\lambda }|\mathfrak{D}(f_{\alpha }))$ is a fundamental system of Ideals of definition of $\mathfrak{D}(f_{\alpha })$, then $({\mathcal{I}}_{\lambda })$ is a fundamental system of Ideals of definition of $\mathfrak{X}$.

Proof. Indeed, for every Ideal of definition $\mathcal{I}$ of $\mathfrak{X}$, there is a finite covering of $\mathfrak{X}$ by sets $\mathfrak{D}(f_i)$ such that, for each $i$, $\mathcal{I}|\mathfrak{D}(f_i)$ is an Ideal of definition of $\mathfrak{D}(f_i)$ contained in ${\mathcal{I}}_{\lambda }|\mathfrak{D}(f_i)$. If $\mu$ is an index such that ${\mathcal{I}}_{\mu }\subset {\mathcal{I}}_{{\lambda }_i}$ for all $i$, it results from (10.3.3) that ${\mathcal{I}}_{\mu }$ is an Ideal of definition of $\mathfrak{X}$, evidently contained in $\mathcal{I}$, whence our assertion.

10.4. Formal preschemes and morphisms of formal preschemes

(10.4.1) Given a topologically ringed space $X$, one says that an open $U\subset X$ is a formal affine open (resp. an adic formal affine open, resp. a Noetherian formal affine open) if the topologically ringed space induced by $X$ on $U$ is a formal affine scheme (resp. such a scheme whose ring is adic, resp. adic and Noetherian).

Definition (10.4.2). One calls formal prescheme a topologically ringed space $\mathfrak{X}$ every point of which admits a formal affine open neighborhood. One says that the formal prescheme $\mathfrak{X}$ is adic (resp. locally Noetherian) if every point of $\mathfrak{X}$ admits an adic (resp. Noetherian) formal affine open neighborhood. One says that $\mathfrak{X}$ is Noetherian if it is locally Noetherian and if its underlying space is quasi-compact (hence Noetherian).

Proposition (10.4.3). If $\mathfrak{X}$ is a formal prescheme (resp. locally Noetherian), the formal affine opens (resp. Noetherian affine opens) form a base for the topology of $\mathfrak{X}$.

Proof. This results from (10.4.2) and (10.1.4), taking account of the fact that if $A$ is an adic Noetherian ring, so is $A_{\{f\}}$ for every $f\in A$ (0, 7.6.11).

Corollary (10.4.4). If $\mathfrak{X}$ is a formal prescheme (resp. a locally Noetherian formal prescheme, resp. Noetherian), the topologically ringed space induced on every open of $\mathfrak{X}$ is again a formal prescheme (resp. a locally Noetherian formal prescheme, resp. Noetherian).

Definition (10.4.5). Given two formal preschemes $\mathfrak{X}$, $\mathfrak{Y}$, one calls morphism (of formal preschemes) of $\mathfrak{X}$ into $\mathfrak{Y}$ every morphism $(\psi ,\theta )$ of topologically ringed spaces such that, for every $x\in \mathfrak{X}$, ${\theta }_x^{♯}$ is a local homomorphism ${\mathcal{O}}_{\psi (x)}\to {\mathcal{O}}_x$.

It is immediate that the composite of two morphisms of formal preschemes is again such a morphism; the formal preschemes therefore form a category, and one will denote by $\mathrm{Hom}(\mathfrak{X},\mathfrak{Y})$ the set of morphisms of a formal prescheme $\mathfrak{X}$ into a formal prescheme $\mathfrak{Y}$.

If $U$ is an open part of $\mathfrak{X}$, the canonical injection into $\mathfrak{X}$ of the formal prescheme induced by $\mathfrak{X}$ on $U$ is a morphism of formal preschemes (and even a monomorphism of topologically ringed spaces (0, 4.1.1)).

Proposition (10.4.6). Let $\mathfrak{X}$ be a formal prescheme, $\mathfrak{G}=\mathrm{Spf}(A)$ a formal affine scheme. There exists a canonical one-to-one correspondence between the morphisms of the formal prescheme $\mathfrak{X}$ into the formal prescheme $\mathfrak{G}$ and the continuous homomorphisms of the ring $A$ into the topological ring $\Gamma (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$.

Proof. The demonstration is the same as that of (2.2.4), replacing “homomorphism” by “continuous homomorphism”, “affine open” by “formal affine open”, and using (10.2.2) instead of (1.7.3); we leave the details to the reader.

(10.4.7) Given a formal prescheme $\mathfrak{G}$, one says that the datum of a formal prescheme $\mathfrak{X}$ and of a morphism $\phi :\mathfrak{X}\to \mathfrak{G}$ defines a formal prescheme $\mathfrak{X}$ over $\mathfrak{G}$, or a $\mathfrak{G}$-formal prescheme, $\phi$ being called the structure morphism of the $\mathfrak{G}$-prescheme $\mathfrak{X}$. If $\mathfrak{G}=\mathrm{Spf}(A)$, where $A$ is an admissible ring, one also says that the $\mathfrak{G}$-formal prescheme $\mathfrak{X}$ is an $A$-formal prescheme or a formal prescheme over $A$. An arbitrary formal prescheme may always be considered as a formal prescheme over $\mathbf{Z}$ (equipped with the discrete topology).

If $\mathfrak{X}$, $\mathfrak{Y}$ are two $\mathfrak{G}$-formal preschemes, one says that a morphism $u:\mathfrak{X}\to \mathfrak{Y}$ is a $\mathfrak{G}$-morphism if the diagram $$\begin{array}{ccc}\mathfrak{X}& \stackrel{u}{\to }& \mathfrak{Y}\\ & \searrow \swarrow & \\ & \mathfrak{G}& \end{array}$$ (where the oblique arrows are the structure morphisms) is commutative. With this definition, the $\mathfrak{G}$-formal preschemes (for fixed $\mathfrak{G}$) form a category. One designates by ${\mathrm{Hom}}_{\mathfrak{G}}(\mathfrak{X},\mathfrak{Y})$ the set of $\mathfrak{G}$-morphisms of the $\mathfrak{G}$-formal prescheme $\mathfrak{X}$ into the $\mathfrak{G}$-formal prescheme $\mathfrak{Y}$. When $\mathfrak{G}=\mathrm{Spf}(A)$, one also says $A$-morphism instead of $\mathfrak{G}$-morphism.

(10.4.8) Since every affine scheme may be considered as a formal affine scheme (10.1.2), every (ordinary) prescheme may be considered as a formal prescheme. It moreover results from (10.4.5) that for ordinary preschemes, the morphisms (resp. $\mathfrak{G}$-morphisms) of formal preschemes coincide with the morphisms (resp. $S$-morphisms) defined in §2.

10.5. Ideals of definition of formal preschemes

(10.5.1) Let $\mathfrak{X}$ be a formal prescheme; one says that an ${\mathcal{O}}_{\mathfrak{X}}$-Ideal $\mathcal{I}$ is a sheaf of ideals of definition (or an Ideal of definition) of $\mathfrak{X}$ if every $x\in \mathfrak{X}$ possesses a formal affine open neighborhood $U$ such that $\mathcal{I}|U$ is an Ideal of definition of the formal affine scheme induced by $\mathfrak{X}$ on $U$ (10.3.3); by virtue of (10.3.1.1) and (10.4.3), for every open $V\subset \mathfrak{X}$, $\mathcal{I}|V$ is then an Ideal of definition of the formal prescheme induced by $\mathfrak{X}$ on $V$.

One says that a family $({\mathcal{I}}_{\lambda })$ of Ideals of definition of $\mathfrak{X}$ is a fundamental system of Ideals of definition if there exists a covering $(U_{\alpha })$ of $\mathfrak{X}$ by formal affine opens such that, for every $\alpha$, the family of ${\mathcal{I}}_{\lambda }|U_{\alpha }$ is a fundamental system of Ideals of definition (10.3.6) of the formal affine scheme induced by $\mathfrak{X}$ on $U_{\alpha }$. It results from the final remark of (10.3.7) that when $\mathfrak{X}$ is a formal affine scheme, this definition coincides with the definition given in (10.3.7). For every open $V$ of $\mathfrak{X}$, the restrictions ${\mathcal{I}}_{\lambda }|V$ then form a fundamental system of Ideals of definition of the formal prescheme induced on $V$, by virtue of (10.3.1.1). If $\mathfrak{X}$ is a locally Noetherian formal prescheme, and $\mathcal{I}$ an Ideal of definition of $\mathfrak{X}$, it results from (10.3.6) that the powers ${\mathcal{I}}^n$ form a fundamental system of Ideals of definition of $\mathfrak{X}$.

(10.5.2) Let $\mathfrak{X}$ be a formal prescheme, $\mathcal{I}$ an Ideal of definition of $\mathfrak{X}$. Then the ringed space $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{I})$ is an (ordinary) prescheme, which is affine (resp. locally Noetherian, resp. Noetherian) when $\mathfrak{X}$ is a formal affine scheme (resp. a locally Noetherian formal prescheme, resp. Noetherian); one is indeed at once reduced to the affine case, and then the proposition has already been demonstrated in (10.3.2). Moreover, if $\theta :{\mathcal{O}}_{\mathfrak{X}}\to {\mathcal{O}}_{\mathfrak{X}}/\mathcal{I}$ is the canonical homomorphism, $u=(1_{\mathfrak{X}},\theta )$ is a morphism (said to be canonical) of formal preschemes $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{I})\to (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})$, for here again, this has been seen in the affine case (10.3.2), to which one reduces at once.

Proposition (10.5.3). Let $\mathfrak{X}$ be a formal prescheme, $({\mathcal{I}}_{\lambda })$ a fundamental system of Ideals of definition of $\mathfrak{X}$. Then the sheaf of topological rings ${\mathcal{O}}_{\mathfrak{X}}$ is the projective limit of the sheaves of pseudo-discrete rings (0, 3.8.1) ${\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda }$.

Proof. Since the topology of $\mathfrak{X}$ admits a base of quasi-compact formal affine opens (10.4.3), one is reduced to the affine case, where the proposition is a consequence of (10.3.5), (10.3.2), and of the definition (10.1.1).

It is not certain that every formal prescheme admits Ideals of definition. However:

Proposition (10.5.4). Let $\mathfrak{X}$ be a locally Noetherian formal prescheme. There exists a largest Ideal of definition $\mathcal{T}$ of $\mathfrak{X}$; it is the only Ideal of definition $\mathcal{I}$ such that the prescheme $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{I})$ is reduced. If $\mathcal{I}$ is an Ideal of definition of $\mathfrak{X}$, $\mathcal{T}$ is the inverse image under ${\mathcal{O}}_{\mathfrak{X}}\to {\mathcal{O}}_{\mathfrak{X}}/\mathcal{I}$ of the Nilradical of ${\mathcal{O}}_{\mathfrak{X}}/\mathcal{I}$.

Proof. Suppose first that $\mathfrak{X}=\mathrm{Spf}(A)$, where $A$ is an adic Noetherian ring. The existence and properties of $\mathcal{T}$ result immediately from (10.3.4) and (5.1.1), taking account of the existence and properties of the largest ideal of definition of $A$ (0, 7.1.6 and 7.1.7).

To prove the existence and properties of $\mathcal{T}$ in the general case, it suffices to show that if $U\supset V$ are two Noetherian formal affine opens of $\mathfrak{X}$, the largest Ideal of definition ${\mathcal{T}}_U$ of $U$ induces the largest Ideal of definition ${\mathcal{T}}_V$ of $V$; but since $(V,({\mathcal{O}}_{\mathfrak{X}}|V)/({\mathcal{T}}_U|V))$ is reduced, this results from what precedes.

One designates by ${\mathfrak{X}}_{\mathrm{red}}$ the (ordinary) reduced prescheme $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{T})$.

Corollary (10.5.5). Let $\mathfrak{X}$ be a locally Noetherian formal prescheme, $\mathcal{T}$ the largest Ideal of definition of $\mathfrak{X}$; for every open $V$ of $\mathfrak{X}$, $\mathcal{T}|V$ is the largest Ideal of definition of the formal prescheme induced by $\mathfrak{X}$ on $V$.

Proposition (10.5.6). Let $\mathfrak{X}$, $\mathfrak{Y}$ be two formal preschemes, $\mathcal{I}$ (resp. $\mathcal{J}$) an Ideal of definition of $\mathfrak{X}$ (resp. $\mathfrak{Y}$), $f:\mathfrak{X}\to \mathfrak{Y}$ a morphism of formal preschemes.

(i) If $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$, there exists a unique morphism $f^{\prime }:(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{I})\to (\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/\mathcal{J})$ of ordinary preschemes making commutative the diagram

$$\begin{array}{ccc}(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}})& \stackrel{f}{\to }& (\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}})\\ \uparrow & & \uparrow \\ (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/\mathcal{I})& \stackrel{f^{\prime }}{\to }& (\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/\mathcal{J})\end{array}\text{\hspace{1em}(10.5.6.1)}$$

where the vertical arrows are the canonical morphisms.

(ii) Suppose that $\mathfrak{X}=\mathrm{Spf}(A)$, $\mathfrak{Y}=\mathrm{Spf}(B)$ are formal affine schemes, $\mathcal{I}={\mathfrak{J}}^{\Delta }$, $\mathcal{J}={\mathfrak{K}}^{\Delta }$, where $\mathfrak{J}$ (resp. $\mathfrak{K}$) is an ideal of definition of $A$ (resp. $B$), and $f=({}^a\phi ,\tilde{\phi })$, where $\phi :B\to A$ is a continuous homomorphism; for $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$, it is necessary and sufficient that $\phi (\mathfrak{K})\subset \mathfrak{J}$, and $f^{\prime }$ is then the morphism $({}^a{\phi }^{\prime },\widetilde{{\phi }^{\prime }})$, where ${\phi }^{\prime }:B/\mathfrak{K}\to A/\mathfrak{J}$ is the homomorphism deduced from $\phi$ by passage to the quotients.

Proof. (i) If $f=(\psi ,\theta )$, the hypothesis entails that the image under ${\theta }^{\ast }:{\psi }^{\ast }(\mathcal{J})\to {\mathcal{O}}_{\mathfrak{X}}$ of the sheaf of ideals ${\psi }^{\ast }(\mathcal{J})$ of ${\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}})$ is contained in $\mathcal{I}$ (0, 4.3.5). By passage to the quotients, one therefore deduces from ${\theta }^{♯}$ a homomorphism of sheaves of rings $$\omega :{\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}}/\mathcal{J})={\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}})/{\psi }^{\ast }(\mathcal{J})\to {\mathcal{O}}_{\mathfrak{X}}/\mathcal{I};$$ moreover, since for every $x\in \mathfrak{X}$, ${\theta }_x^{♯}$ is a local homomorphism, so is ${\omega }_x$. The morphism of ringed spaces $(\psi ,\omega )$ is therefore (2.2.1) the unique morphism $f^{\prime }$ of ringed spaces answering the question.

(ii) The canonical functorial correspondence between morphisms of formal affine preschemes and continuous homomorphisms of rings (10.2.2) shows that in the case considered, the relation $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$ entails that one has $f^{\prime }=({}^a{\phi }^{\prime },\widetilde{{\phi }^{\prime }})$, where ${\phi }^{\prime }:B/\mathfrak{K}\to A/\mathfrak{J}$ is the unique homomorphism making commutative the diagram

$$\begin{array}{ccc}B& \stackrel{\phi }{\to }& A\\ \downarrow & & \downarrow \\ B/\mathfrak{K}& \stackrel{{\phi }^{\prime }}{\to }& A/\mathfrak{J}\end{array}\text{\hspace{1em}(10.5.6.2)}$$

The existence of ${\phi }^{\prime }$ therefore implies that $\phi (\mathfrak{K})\subset \mathfrak{J}$. Conversely, if this condition is verified, then, denoting by ${\phi }^{\prime }$ the unique homomorphism making commutative the diagram (10.5.6.2) and setting $f^{\prime }=({}^a{\phi }^{\prime },\widetilde{{\phi }^{\prime }})$, it is clear that the diagram (10.5.6.1) is commutative; the consideration of the homomorphisms ${\psi }^{\ast }(\mathcal{J})\to \mathcal{I}$ and ${\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}}/\mathcal{J})\to {\mathcal{O}}_{\mathfrak{X}}/\mathcal{I}$ corresponding to $f$ and $f^{\prime }$ respectively then shows that this entails the relation $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$.

It is clear that the correspondence $f\to f^{\prime }$ defined above is functorial.

10.6. Formal preschemes as inductive limits of preschemes

(10.6.1) Let $\mathfrak{X}$ be a formal prescheme, $({\mathcal{I}}_{\lambda })$ a fundamental system of Ideals of definition of $\mathfrak{X}$; for each $\lambda$, let $f_{\lambda }$ be the canonical morphism $(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda })\to \mathfrak{X}$ (10.5.2); for ${\mathcal{I}}_{\mu }\subset {\mathcal{I}}_{\lambda }$, the canonical homomorphism ${\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\mu }\to {\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda }$ defines a canonical morphism $f_{\lambda \mu }:(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda })\to (\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\mu })$ of (ordinary) preschemes such that one has $f_{\lambda }=f_{\mu }\circ f_{\lambda \mu }$. The preschemes $X_{\lambda }=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda })$ and the morphisms $f_{\lambda \mu }$ therefore constitute (by virtue of (10.4.8)) an inductive system in the category of formal preschemes.

Proposition (10.6.2). With the notations of (10.6.1), the formal prescheme $\mathfrak{X}$ and the morphisms $f_{\lambda }$ constitute an inductive limit (T, I, 1.8) of the system $(X_{\lambda },f_{\lambda \mu })$ in the category of formal preschemes.

Proof. Let $\mathfrak{Y}$ be a formal prescheme, and for each index $\lambda$, let $$g_{\lambda }=({\psi }_{\lambda },{\theta }_{\lambda }):X_{\lambda }\to \mathfrak{Y}$$ be a morphism such that one has $g_{\lambda }=g_{\mu }\circ f_{\lambda \mu }$ for ${\mathcal{I}}_{\mu }\subset {\mathcal{I}}_{\lambda }$. This last condition and the definition of the $X_{\lambda }$ entail first that all the ${\psi }_{\lambda }$ are identical to one and the same continuous map $\psi :\mathfrak{X}\to \mathfrak{Y}$ of the underlying spaces; moreover, the homomorphisms ${\theta }_{\lambda }^{♯}:{\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}})\to {\mathcal{O}}_{X_{\lambda }}={\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda }$ form a projective system of homomorphisms of sheaves of rings. By passage to the projective limit, one therefore deduces a homomorphism $\omega :{\psi }^{\ast }({\mathcal{O}}_{\mathfrak{Y}})\to \underleftarrow{\lim }{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda }={\mathcal{O}}_{\mathfrak{X}}$, and it is clear that the morphism $g=(\psi ,\omega )$ of ringed spaces is the only one making commutative the diagrams

$$\begin{array}{ccc}{X}_{\lambda }& \stackrel{g_{\lambda }}{\to }& \mathfrak{Y}\\ \downarrow & \nearrow & \\ \mathfrak{X}& & \end{array}\text{\hspace{1em}(10.6.2.1)}$$

It remains therefore to prove that $g$ is a morphism of formal preschemes; the question being local on $\mathfrak{X}$ and $\mathfrak{Y}$, one may suppose $\mathfrak{X}=\mathrm{Spf}(A)$, $\mathfrak{Y}=\mathrm{Spf}(B)$, $A$ and $B$ being admissible rings, with ${\mathcal{I}}_{\lambda }={\mathfrak{J}}_{\lambda }^{\Delta }$, where $({\mathfrak{J}}_{\lambda })$ is a fundamental system of ideals of definition of $A$ (10.3.5); since $A=\underleftarrow{\lim }A/{\mathfrak{J}}_{\lambda }$, the existence of a morphism of formal affine schemes $g$ making commutative the diagrams (10.6.2.1) results then from the one-to-one correspondence (10.2.2) between morphisms of formal affine schemes and continuous homomorphisms of rings, and from the definition of the projective limit. But the uniqueness of $g$ as a morphism of ringed spaces shows that it coincides with the morphism denoted the same way at the beginning of the demonstration.

The following proposition establishes, under certain supplementary conditions, the existence of the inductive limit of a given inductive system of (ordinary) preschemes in the category of formal preschemes:

Proposition (10.6.3). Let $X$ be a topological space, $({\mathcal{O}}_i,u_{ij})$ a projective system of sheaves of rings on $X$, having $\mathbf{N}$ for index set. Let ${\mathcal{J}}_i$ be the kernel of $u_{i0}:{\mathcal{O}}_i\to {\mathcal{O}}_0$. Suppose that:

a) the ringed space $(X,{\mathcal{O}}_0)$ is a prescheme $X_0$.

b) for every $x\in X$ and every $i$, there exists an open neighborhood $U_i$ of $x$ in $X$ such that the restriction ${\mathcal{J}}_i|U_i$ is nilpotent.

c) the homomorphisms $u_{ij}$ are surjective.

Let ${\mathcal{O}}_{\infty }$ be the sheaf of topological rings projective limit of the sheaves of pseudo-discrete rings ${\mathcal{O}}_i$, and let $u_{\infty i}:{\mathcal{O}}_{\infty }\to {\mathcal{O}}_i$ be the canonical homomorphism. Then the topologically ringed space $(X,{\mathcal{O}}_{\infty })$ is a formal prescheme; the homomorphisms $u_{\infty i}$ are surjective; their kernels ${\mathcal{J}}_{\infty i}$ form a fundamental system of Ideals of definition of $\mathfrak{X}$, and ${\mathcal{J}}_{\infty i}$ is the projective limit of the sheaves of ideals ${\mathcal{J}}_{ji}$.

Proof. Let us first note that on each stalk, $u_{ij}$ is a surjective homomorphism and a fortiori a local homomorphism; hence $v_{ij}=(1_X,u_{ij})$ is a morphism of preschemes $X_i\to X_j$ ($i\ge j$) (2.2.1). Suppose first that each $X_i$ is an affine scheme with ring $A_i$. There exists a ring homomorphism ${\phi }_{ij}:A_j\to A_i$ such that $u_{ij}=\widetilde{{\phi }_{ij}}$ (1.7.3); consequently (1.6.3), the sheaf ${\mathcal{O}}_i$ is a quasi-coherent ${\mathcal{O}}_j$-Module on $X_j$ (for the external law defined by $u_{ij}$), associated to $A_i$ considered as an $A_j$-module by means of ${\phi }_{ij}$. For every $f\in A_j$, let $f^{\prime }={\phi }_{ij}(f)$; by hypothesis, the opens $D(f)$ and $D(f^{\prime })$ are identical in $X$, and the homomorphism of $\Gamma (D(f),{\mathcal{O}}_j)=(A_j{)}_f$ into $\Gamma (D(f),{\mathcal{O}}_i)=(A_i{)}_{f^{\prime }}$ corresponding to $u_{ij}$ is none other than $({\phi }_{ij}{)}_f$ (1.6.1). But when one considers $A_i$ as an $A_j$-module, $(A_i{)}_{f^{\prime }}$ is the $(A_j{)}_f$-module $(A_i{)}_f$, hence one also has $u_{ij}=\widetilde{{\phi }_{ij}}$ when ${\phi }_{ij}$ is this time considered as a homomorphism of $A_j$-modules. Then, since $u_{ij}$ is surjective, one concludes that ${\phi }_{ij}$ is also (1.3.9), and if ${\mathfrak{J}}_{ij}$ is the kernel of ${\phi }_{ij}$, the kernel of $u_{ij}$ is a quasi-coherent ${\mathcal{O}}_j$-Module equal to $\widetilde{{\mathfrak{J}}_{ij}}$. In particular, one has ${\mathcal{J}}_i=\widetilde{{\mathfrak{J}}_i}$, where ${\mathfrak{J}}_i$ is the kernel of ${\phi }_{i0}:A_i\to A_0$. Hypothesis b) entails that ${\mathcal{J}}_i$ is nilpotent: indeed, since $X$ is quasi-compact, one may cover $X$ by a finite number of opens $V_k$ such that $({\mathcal{J}}_i|V_k{)}^{n_k}=0$, and taking for $n$ the largest of the $n_k$, one has ${\mathcal{J}}_i^n=0$. One concludes that ${\mathfrak{J}}_i$ is nilpotent (1.3.13). Then the ring $A=\underleftarrow{\lim }{A}_i$ is admissible (0, 7.2.2), the canonical homomorphism ${\phi }_i:A\to A_i$ is surjective and its kernel ${\mathfrak{J}}_{\infty i}$ is equal to the projective limit of the ${\mathfrak{J}}_{ki}$ for $k\ge i$; the ${\mathfrak{J}}_{\infty i}$ form a fundamental system of neighborhoods of $0$ in $A$. The assertions of (10.6.3) result in this case from (10.1.1) and (10.3.2), $(X,{\mathcal{O}}_{\infty })$ being none other than $\mathrm{Spf}(A)$.

Still in this same particular case, let us note that if $f=(f_i)$ is an element of the projective limit $A=\underleftarrow{\lim }{A}_i$, all the opens $D(f_i)$ (affine open in $X_i$) are identified with the open $\mathfrak{D}(f)$ of $\mathfrak{X}$, the prescheme induced by $X_i$ on $\mathfrak{D}(f)$ being thus identified with the affine scheme $\mathrm{Spec}((A_i{)}_f)$.

In the general case, let us remark first that for every quasi-compact open $U$ of $X$, each of the ${\mathcal{J}}_i|U$ is nilpotent, as the reasoning made above shows. We shall see that for every $x\in X$, there is an open neighborhood $U$ of $x$ in $X$ which is an affine open for all the $X_i$. Indeed, take $U$ affine open for $X_0$, and observe that ${\mathcal{O}}_{x_i}={\mathcal{O}}_{x_0}/{\mathcal{J}}_i$. Since ${\mathcal{J}}_i|U$ is nilpotent, by virtue of what precedes, $U$ is affine open also for each $X_i$ by virtue of (5.1.9). This being so, for every $U$ satisfying the preceding conditions, the study of the affine case made above shows that $(U,{\mathcal{O}}_{\infty }|U)$ is a formal prescheme of which the ${\mathcal{J}}_{\infty i}|U$ form a fundamental system of ideals of definition and ${\mathcal{O}}_{\infty }|U$ is the projective limit of the ${\mathcal{O}}_i|U$; whence the conclusion.

Corollary (10.6.4). Suppose that for $i\ge j$, the kernel of $u_{ij}$ is ${\mathcal{J}}_i^{j+1}$ and that ${\mathcal{J}}_1/{\mathcal{J}}_1^2$ is of finite type over ${\mathcal{O}}_0={\mathcal{O}}_1/{\mathcal{J}}_1$. Then $\mathfrak{X}$ is an adic formal prescheme, and if ${\mathcal{J}}_{\infty }^{(n)}$ is the kernel of ${\mathcal{O}}_{\infty }\to {\mathcal{O}}_n$, one has ${\mathcal{J}}_{\infty }^{(n)}={\mathcal{J}}_{\infty }^{n+1}$ and ${\mathcal{J}}_{\infty }/{\mathcal{J}}_{\infty }^2$ is isomorphic to ${\mathcal{J}}_1$. If in addition $X_0$ is locally Noetherian (resp. Noetherian), $\mathfrak{X}$ is locally Noetherian (resp. Noetherian).

Proof. Since the spaces underlying $\mathfrak{X}$ and $X_0$ are the same, the question is local and one may suppose all the $X_i$ affine; taking account of the relations ${\mathcal{J}}_i=\widetilde{{\mathfrak{J}}_i}$ (with the notations of (10.6.3)), one is at once reduced to the corresponding assertions of (0, 7.2.7 and 7.2.8), noting that ${\mathfrak{J}}_1/{\mathfrak{J}}_1^2$ is then an $A_0$-module of finite type (1.3.9).

In particular, every locally Noetherian formal prescheme $\mathfrak{X}$ is the inductive limit of a sequence $(X_n)$ of (ordinary) locally Noetherian preschemes verifying the conditions of (10.6.3) and (10.6.4): it suffices to consider an Ideal of definition $\mathcal{I}$ of $\mathfrak{X}$ (10.5.4) and to take $X_n=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}^{n+1})$ ((10.5.1) and (10.6.2)).

Corollary (10.6.5). Let $A$ be an admissible ring. For the formal affine scheme $\mathfrak{X}=\mathrm{Spf}(A)$ to be Noetherian, it is necessary and sufficient that $A$ be adic and Noetherian.

Proof. The condition is evidently sufficient. Conversely, suppose that $\mathfrak{X}$ is Noetherian, and let $\mathfrak{J}$ be an ideal of definition of $A$, $\mathcal{I}={\mathfrak{J}}^{\Delta }$ the corresponding Ideal of definition of $\mathfrak{X}$. The (ordinary) preschemes $X_n=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}^{n+1})$ are then affine and Noetherian, hence the rings $A_n=A/{\mathfrak{J}}^{n+1}$ are Noetherian (6.1.3), whence one concludes that $\mathfrak{J}/{\mathfrak{J}}^2$ is an $(A/\mathfrak{J})$-module of finite type. Since the ${\mathcal{I}}^n$ form a fundamental system of Ideals of definition of $\mathfrak{X}$ (10.5.1), one has ${\mathcal{O}}_{\mathfrak{X}}=\underleftarrow{\lim }({\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}^n)$ (10.5.3); one concludes (10.1.3) that $A$ is topologically isomorphic to $\underleftarrow{\lim }A/{\mathfrak{J}}^n$, hence is adic and Noetherian (0, 7.2.8).

Remark (10.6.6). With the notations of (10.6.3), let ${\mathcal{F}}_i$ be an ${\mathcal{O}}_i$-Module, and suppose given, for $i\ge j$, a $v_{ij}$-morphism ${\theta }_{ij}:{\mathcal{F}}_i\to {\mathcal{F}}_j$, so that ${\theta }_{jk}\circ {\theta }_{ij}={\theta }_{ik}$ for $i\ge j\ge k$. Since the continuous map underlying $v_{ij}$ is the identity, ${\theta }_{ij}$ is a homomorphism of sheaves of abelian groups on the space $X$; moreover, if $\mathcal{F}$ is the projective limit of the projective system $({\mathcal{F}}_i)$ of sheaves of abelian groups, the fact that the ${\theta }_{ij}$ are $v_{ij}$-morphisms permits one to define on $\mathcal{F}$ a structure of ${\mathcal{O}}_{\infty }$-Module by passage to the projective limit; equipped with this structure, we shall say that $\mathcal{F}$ is the projective limit (for the ${\theta }_{ij}$) of the system of ${\mathcal{O}}_i$-Modules $({\mathcal{F}}_i)$. In the particular case where $v_{ij}^{\ast }({\mathcal{F}}_j)={\mathcal{F}}_i$ and where ${\theta }_{ij}$ is the identity, we shall say for brevity that $\mathcal{F}$ is the projective limit of a system $({\mathcal{F}}_i)$ such that $v_{ij}^{\ast }({\mathcal{F}}_j)={\mathcal{F}}_i$ for $j\le i$ (without mentioning the ${\theta }_{ij}$).

(10.6.7) Let $\mathfrak{X}$, $\mathfrak{Y}$ be two formal preschemes, $\mathcal{I}$ (resp. $\mathcal{J}$) an Ideal of definition of $\mathfrak{X}$ (resp. $\mathfrak{Y}$), $f:\mathfrak{X}\to \mathfrak{Y}$ a morphism such that $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$. One then has, for every integer $n>0$, $f^{\ast }({\mathcal{J}}^{n+1}){\mathcal{O}}_{\mathfrak{X}}=(f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}{)}^{n+1}\subset {\mathcal{I}}^{n+1}$; one may therefore (10.5.6) deduce from $f$ a morphism of (ordinary) preschemes $f_n:X_n\to Y_n$, on setting $X_n=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}^{n+1})$, $Y_n=(\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/{\mathcal{J}}^{n+1})$, and it results at once from the definitions that the diagrams

$$\begin{array}{ccc}{X}_n& \stackrel{f_n}{\to }& Y_n\\ \uparrow & & \uparrow \\ X_m& \stackrel{f_m}{\to }& Y_m\end{array}\text{\hspace{1em}(10.6.7.1)}$$

are commutative for $m\le n$; in other words, $(f_n)$ is an inductive system of morphisms.

(10.6.8) Conversely, let $(X_n)$ (resp. $(Y_n)$) be an inductive system of (ordinary) preschemes satisfying the conditions b) and c) of (10.6.3), and let $\mathfrak{X}$ (resp. $\mathfrak{Y}$) be its inductive limit. By definition of inductive limits, every sequence $(f_n)$ of morphisms $X_n\to Y_n$ forming an inductive system admits an inductive limit $f:\mathfrak{X}\to \mathfrak{Y}$, which is the unique morphism of formal preschemes making commutative the diagrams

$$\begin{array}{ccc}{X}_n& \stackrel{f_n}{\to }& Y_n\\ \downarrow & & \downarrow \\ \mathfrak{X}& \stackrel{f}{\to }& \mathfrak{Y}\end{array}$$

Proposition (10.6.9). Let $\mathfrak{X}$, $\mathfrak{Y}$ be two locally Noetherian formal preschemes, $\mathcal{I}$ (resp. $\mathcal{J}$) an Ideal of definition of $\mathfrak{X}$ (resp. $\mathfrak{Y}$); the map $f\to (f_n)$ defined in (10.6.7) is a bijection of the set of morphisms $f:\mathfrak{X}\to \mathfrak{Y}$ such that $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$ onto the set of sequences $(f_n)$ of morphisms making commutative the diagrams (10.6.7.1).

Proof. If $f$ is the inductive limit of such a sequence, it must be shown that $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$. The question being local on $\mathfrak{X}$ and $\mathfrak{Y}$, one may restrict oneself to the case where $\mathfrak{X}=\mathrm{Spf}(A)$, $\mathfrak{Y}=\mathrm{Spf}(B)$ are affine, $A$ and $B$ being adic Noetherian, $\mathcal{I}={\mathfrak{J}}^{\Delta }$, $\mathcal{J}={\mathfrak{K}}^{\Delta }$, where $\mathfrak{J}$ (resp. $\mathfrak{K}$) is an ideal of definition of $A$ (resp. $B$). One then has $X_n=\mathrm{Spec}(A_n)$, $Y_n=\mathrm{Spec}(B_n)$, with $A_n=A/{\mathfrak{J}}^{n+1}$ and $B_n=B/{\mathfrak{K}}^{n+1}$, by virtue of (10.3.6) and (10.3.2); $f_n=({}^a{\phi }_n,\widetilde{{\phi }_n})$, where the homomorphisms ${\phi }_n:B_n\to A_n$ form a projective system, hence $f=({}^a\phi ,\tilde{\phi })$, where $\phi =\underleftarrow{\lim }{\phi }_n$. The commutativity of the diagram (10.6.7.1) for $m=0$ then gives the condition ${\phi }_n(\mathfrak{K}/{\mathfrak{K}}^{n+1})\subset \mathfrak{J}/{\mathfrak{J}}^{n+1}$ for every $n$, hence, passing to the projective limit, $\phi (\mathfrak{K})\subset \mathfrak{J}$, and this entails $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$ (10.5.6, (ii)).

Corollary (10.6.10). Let $\mathfrak{X}$, $\mathfrak{Y}$ be two locally Noetherian formal preschemes, $\mathcal{T}$ the largest Ideal of definition of $\mathfrak{X}$ (10.5.4).

(i) For every Ideal of definition $\mathcal{J}$ of $\mathfrak{Y}$ and every morphism $f:\mathfrak{X}\to \mathfrak{Y}$, one has $f^{\ast }(\mathcal{J}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{T}$.

(ii) There is a canonical one-to-one correspondence between $\mathrm{Hom}(\mathfrak{X},\mathfrak{Y})$ and the set of sequences $(f_n)$ of morphisms making commutative the diagrams (10.6.7.1), where $X_n=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{T}}^{n+1})$, $Y_n=(\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/{\mathcal{J}}^{n+1})$.

Proof. (ii) results at once from (i) and (10.6.9). To demonstrate (i), one may restrict oneself to the case where $\mathfrak{X}=\mathrm{Spf}(A)$, $\mathfrak{Y}=\mathrm{Spf}(B)$, $A$ and $B$ being Noetherian, $\mathcal{T}={\mathfrak{T}}^{\Delta }$, $\mathcal{J}={\mathfrak{K}}^{\Delta }$, where $\mathfrak{T}$ is the largest ideal of definition of $A$ and $\mathfrak{K}$ an ideal of definition of $B$. Let $f=({}^a\phi ,\tilde{\phi })$, where $\phi :B\to A$ is a continuous homomorphism; since the elements of $\mathfrak{K}$ are topologically nilpotent (0, 7.1.4, (ii)), so are those of $\phi (\mathfrak{K})$, hence $\phi (\mathfrak{K})\subset \mathfrak{T}$ since $\mathfrak{T}$ is the set of topologically nilpotent elements of $A$ (0, 7.1.6); whence the conclusion by virtue of (10.5.6, (ii)).

Corollary (10.6.11). Let $\mathfrak{S}$, $\mathfrak{X}$, $\mathfrak{Y}$ be three locally Noetherian formal preschemes, $f:\mathfrak{X}\to \mathfrak{S}$, $g:\mathfrak{Y}\to \mathfrak{S}$ morphisms making $\mathfrak{X}$ and $\mathfrak{Y}$ into $\mathfrak{S}$-formal preschemes. Let $\mathcal{L}$ (resp. $\mathcal{I}$, $\mathcal{J}$) be an Ideal of definition of $\mathfrak{S}$ (resp. $\mathfrak{X}$, $\mathfrak{Y}$), and suppose that $f^{\ast }(\mathcal{L}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$, $g^{\ast }(\mathcal{L}){\mathcal{O}}_{\mathfrak{Y}}\subset \mathcal{J}$; set $S_n=(\mathfrak{S},{\mathcal{O}}_{\mathfrak{S}}/{\mathcal{L}}^{n+1})$, $X_n=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}^{n+1})$, $Y_n=(\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/{\mathcal{J}}^{n+1})$. There is then a canonical one-to-one correspondence between ${\mathrm{Hom}}_{\mathfrak{S}}(\mathfrak{X},\mathfrak{Y})$ and the set of sequences $(u_n)$ of $S_n$-morphisms $u_n:X_n\to Y_n$ making commutative the diagrams (10.6.7.1).

Proof. For every $\mathfrak{S}$-morphism $u:\mathfrak{X}\to \mathfrak{Y}$, one has by definition $f=g\circ u$, hence $$u^{\ast }(g^{\ast }(\mathcal{L})){\mathcal{O}}_{\mathfrak{X}}=u^{\ast }(g^{\ast }(\mathcal{L}){\mathcal{O}}_{\mathfrak{Y}}){\mathcal{O}}_{\mathfrak{X}}=f^{\ast }(\mathcal{L}){\mathcal{O}}_{\mathfrak{X}}\subset \mathcal{I}$$ the corollary thus follows from (10.6.9).

One will note that, for $m\le n$, the datum of a morphism $f_m:X_m\to Y_m$ determines one and only one morphism $f_n:X_n\to Y_n$ making commutative the diagram (10.6.7.1), as one sees at once by reducing to the affine case; one has thus defined a map ${\rho }_{nm}:{\mathrm{Hom}}_{S_m}(X_m,Y_m)\to {\mathrm{Hom}}_{S_n}(X_n,Y_n)$ and the ${\mathrm{Hom}}_{S_n}(X_n,Y_n)$ form for the ${\rho }_{nm}$ a projective system of sets; (10.6.11) may be stated again by saying that there exists a canonical bijection $${\mathrm{Hom}}_{\mathfrak{S}}(\mathfrak{X},\mathfrak{Y})\cong \underset{n}{\underleftarrow{\lim }}{\mathrm{Hom}}_{S_n}(X_n,Y_n).$$

10.7. Product of formal preschemes

(10.7.1) Let $\mathfrak{S}$ be a formal prescheme; the $\mathfrak{S}$-formal preschemes forming a category, one may define the notion of product of $\mathfrak{S}$-formal preschemes.

Proposition (10.7.2). Let $\mathfrak{X}=\mathrm{Spf}(B)$, $\mathfrak{Y}=\mathrm{Spf}(C)$ be two formal affine schemes over a formal affine scheme $\mathfrak{S}=\mathrm{Spf}(A)$. Let $\mathfrak{Z}=\mathrm{Spf}(B{\hat{\otimes }}_{A}C)$, $p_1$, $p_2$ the $\mathfrak{S}$-morphisms corresponding (10.2.2) to the canonical (continuous) $A$-homomorphisms $b$ and $c$ of $B$ and $C$ into $B{\hat{\otimes }}_{A}C$; then $(\mathfrak{Z},p_1,p_2)$ is a product of the $\mathfrak{S}$-formal affine schemes $\mathfrak{X}$ and $\mathfrak{Y}$.

Proof. By virtue of (10.4.6), everything reduces to verifying that if, to every continuous $A$-homomorphism $\phi :B{\hat{\otimes }}_{A}C\to D$, where $D$ is an admissible ring which is a topological $A$-algebra, one associates the pair $(\phi \circ b,\phi \circ c)$, one defines a bijection $${\mathrm{Hom}}_A(B{\hat{\otimes }}_{A}C,D)\cong {\mathrm{Hom}}_A(B,D)\times {\mathrm{Hom}}_A(C,D)$$ which is none other than the universal property of the completed tensor product (0, 7.7.6).

Proposition (10.7.3). Given two $\mathfrak{S}$-formal preschemes $\mathfrak{X}$, $\mathfrak{Y}$, the product $\mathfrak{X}{\times }_{\mathfrak{S}}\mathfrak{Y}$ exists.

Proof. The demonstration is identical to that of (3.2.6), replacing therein the affine schemes (resp. the affine opens) by the formal affine schemes (resp. the formal affine opens), and the prop. (3.2.2) by (10.7.2).

All the formal properties of the product of preschemes (3.2.7 and 3.2.8, 3.3.1 to 3.3.12) are valid without any modification for the product of formal preschemes.

(10.7.4) Let $\mathfrak{S}$, $\mathfrak{X}$, $\mathfrak{Y}$ be three formal preschemes and let $f:\mathfrak{X}\to \mathfrak{S}$, $g:\mathfrak{Y}\to \mathfrak{S}$ be two morphisms. Suppose that there exist in $\mathfrak{S}$, $\mathfrak{X}$, $\mathfrak{Y}$ respectively three fundamental systems of Ideals of definition $({\mathcal{L}}_{\lambda })$, $({\mathcal{I}}_{\lambda })$, $({\mathcal{J}}_{\lambda })$ respectively, having the same index set $I$, such that $f^{\ast }({\mathcal{L}}_{\lambda }){\mathcal{O}}_{\mathfrak{X}}\subset {\mathcal{I}}_{\lambda }$ and $g^{\ast }({\mathcal{L}}_{\lambda }){\mathcal{O}}_{\mathfrak{Y}}\subset {\mathcal{J}}_{\lambda }$ for every $\lambda$. Set $S_{\lambda }=(\mathfrak{S},{\mathcal{O}}_{\mathfrak{S}}/{\mathcal{L}}_{\lambda })$, $X_{\lambda }=(\mathfrak{X},{\mathcal{O}}_{\mathfrak{X}}/{\mathcal{I}}_{\lambda })$, $Y_{\lambda }=(\mathfrak{Y},{\mathcal{O}}_{\mathfrak{Y}}/{\mathcal{J}}_{\lambda })$; for ${\mathcal{L}}_{\mu }\subset {\mathcal{L}}_{\lambda }$, ${\mathcal{I}}_{\mu }\subset {\mathcal{I}}_{\lambda }$, ${\mathcal{J}}_{\mu }\subset {\mathcal{J}}_{\lambda }$, note that $S_{\mu }$ (resp. $X_{\mu }$, $Y_{\mu }$) is a closed subprescheme of $S_{\lambda }$ (resp. $X_{\lambda }$, $Y_{\lambda }$) having the same underlying space (10.6.1). Since $S_{\mu }\to S_{\lambda }$ is a monomorphism of preschemes, one sees first that the products $X_{\mu }{\times }_{S_{\mu }}{Y}_{\mu }$ and $X_{\mu }{\times }_{S_{\lambda }}{Y}_{\mu }$ are identical (3.2.4), then that $X_{\mu }{\times }_{S_{\mu }}{Y}_{\mu }$ is identified with a closed subprescheme of $X_{\lambda }{\times }_{S_{\lambda }}{Y}_{\lambda }$ having the same underlying space (4.3.1). This being so, the product $\mathfrak{X}{\times }_{\mathfrak{S}}\mathfrak{Y}$ is the inductive limit of the ordinary preschemes $X_{\lambda }{\times }_{S_{\lambda }}{Y}_{\lambda }$: indeed, one sees as in (10.6.2) that one may reduce to the case where $\mathfrak{S}$, $\mathfrak{X}$, and $\mathfrak{Y}$ are formal affine schemes. Taking account of (10.5.6, (ii)) and of the hypothesis on the fundamental systems of Ideals of definition of $\mathfrak{S}$, $\mathfrak{X}$, and $\mathfrak{Y}$, one sees at once that our assertion results from the definition of the completed tensor product of two algebras (0, 7.7.1).

Moreover, let $\mathfrak{Z}$ be an $\mathfrak{S}$-formal prescheme, $({\mathcal{K}}_{\lambda })$ a fundamental system of Ideals of definition of $\mathfrak{Z}$ having $I$ as index set, $u:\mathfrak{Z}\to \mathfrak{X}$, $v:\mathfrak{Z}\to \mathfrak{Y}$ two $\mathfrak{S}$-morphisms such that $u^{\ast }({\mathcal{I}}_{\lambda }){\mathcal{O}}_{\mathfrak{Z}}\subset {\mathcal{K}}_{\lambda }$ and $v^{\ast }({\mathcal{J}}_{\lambda }){\mathcal{O}}_{\mathfrak{Z}}\subset {\mathcal{K}}_{\lambda }$. If one sets $Z_{\lambda }=(\mathfrak{Z},{\mathcal{O}}_{\mathfrak{Z}}/{\mathcal{K}}_{\lambda })$, and if $u_{\lambda }:Z_{\lambda }\to X_{\lambda }$ and $v_{\lambda }:Z_{\lambda }\to Y_{\lambda }$ are the $S_{\lambda }$-morphisms corresponding to $u$ and $v$ (10.5.6), one verifies at once that $(u,v{)}_{\mathfrak{S}}$ is the inductive limit of the $S_{\lambda }$-morphisms $(u_{\lambda },v_{\lambda }{)}_{S_{\lambda }}$.

The considerations of this number apply in particular when $\mathfrak{S}$, $\mathfrak{X}$, and $\mathfrak{Y}$ are locally Noetherian, taking as fundamental systems of Ideals of definition the systems formed of the powers of an Ideal of definition (10.5.1). But one will note that $\mathfrak{X}{\times }_{\mathfrak{S}}\mathfrak{Y}$ is not necessarily locally Noetherian (see however (10.13.5)).

10.8. Formal completion of a prescheme along a closed part

(10.8.1) Let $X$ be a locally Noetherian (ordinary) prescheme, $X^{\prime }$ a closed part of the space underlying $X$; designate by $\Phi$ the set of coherent sheaves of ideals $\mathcal{I}$ in ${\mathcal{O}}_X$ such that the support of ${\mathcal{O}}_X/\mathcal{I}$ is $X^{\prime }$. The set $\Phi$ is not empty (5.2.1, 4.1.4, and 6.1.1); we shall order it by the relation $\supset$.

Lemma (10.8.2). The ordered set $\Phi$ is filtered; if $X$ is Noetherian, then for every $\mathcal{I}\in \Phi$, the set of powers ${\mathcal{I}}^n$ ($n>0$) is cofinal with $\Phi$.

Proof. Indeed, if ${\mathcal{I}}_1$ and ${\mathcal{I}}_2$ belong to $\Phi$, and if one sets $\mathcal{I}={\mathcal{I}}_1{\mathcal{I}}_2$, $\mathcal{I}$ is coherent since ${\mathcal{O}}_X$ is coherent (6.1.1 and 0, 5.3.4), and one has ${\mathcal{I}}_x=({\mathcal{I}}_1{)}_x({\mathcal{I}}_2{)}_x$ for every $x\in X$, hence ${\mathcal{I}}_x={\mathcal{O}}_x$ for $x\notin X^{\prime }$ and ${\mathcal{I}}_x\ne {\mathcal{O}}_x$ for $x\in X^{\prime }$, which proves that $\mathcal{I}\in \Phi$. On the other hand, if $X$ is Noetherian, and if ${\mathcal{I}}_1$ and $\mathcal{I}$ belong to $\Phi$, there exists an integer $n>0$ such that ${\mathcal{I}}^n\subset {\mathcal{I}}_1$ (9.3.4), which signifies that ${\mathcal{I}}^n\subset {\mathcal{I}}_1$.