Formal Algebraic Spaces

Introduction

Formal schemes were introduced in [EGA]. A more general version of formal schemes was introduced in [McQuillan] and another in [Yasuda]. Formal algebraic spaces were introduced in [Kn]. Related material and much besides can be found in [Abbes] and [Fujiwara-Kato]. This chapter introduces the notion of formal algebraic spaces we will work with. Our definition is general enough to allow most classes of formal schemes/spaces in the literature as full subcategories.

Although we do discuss the comparison of some of these alternative theories with ours, we do not always give full details when it is not necessary for the logical development of the theory.

Besides introducing formal algebraic spaces, we also prove a few very basic properties and we discuss a few types of morphisms.

Formal schemes à la EGA

In this section we review the construction of formal schemes in [EGA]. This notion, although very useful in algebraic geometry, may not always be the correct one to consider. Perhaps it is better to say that in the setup of the theory a number of choices are made, where for different purposes others might work better. And indeed in the literature one can find many different closely related theories adapted to the problem the authors may want to consider. Still, one of the major advantages of the theory as sketched here is that one gets to work with definite geometric objects.

Before we start we should point out an issue with the sheaf condition for sheaves of topological rings or more generally sheaves of topological spaces. Namely, the big categories

1. category of topological spaces,

2. category of topological groups,

3. category of topological rings,

4. category of topological modules over a given topological ring,

endowed with their natural forgetful functors to \(\textit{Sets}\) are not examples of types of algebraic structures as defined in Sheaves, Section 007L. Thus we cannot blithely apply to them the machinery developed in that chapter. On the other hand, each of the categories listed above has limits and equalizers and the forgetful functor to sets, groups, rings, modules commutes with them (see Topology, Lemmas 08ZT, 0B20, 0B23, and 0B26). Thus we can define the notion of a sheaf as in Sheaves, Definition 0072 and the underlying presheaf of sets, groups, rings, or modules is a sheaf. The key difference is that for an open covering \(U = \bigcup_{i \in I} U_i\) the diagram \[\xymatrix{ \mathcal{F}(U) \ar[r] & \prod\nolimits_{i\in I} \mathcal{F}(U_i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod\nolimits_{(i_0, i_1) \in I \times I} \mathcal{F}(U_{i_0} \cap U_{i_1}) }\] has to be an equalizer diagram in the category of topological spaces, topological groups, topological rings, topological modules, i.e., that the first map identifies \(\mathcal{F}(U)\) with a subspace of \(\prod_{i \in I} \mathcal{F}(U_i)\) which is endowed with the product topology.

The stalk \(\mathcal{F}_x\) of a sheaf \(\mathcal{F}\) of topological spaces, topological groups, topological rings, or topological modules at a point \(x \in X\) is defined as the colimit over open neighbourhoods \[\mathcal{F}_x = \colim_{x\in U} \mathcal{F}(U)\] in the corresponding category. This is the same as taking the colimit on the level of sets, groups, rings, or modules (see Topology, Lemmas 0B1X, 0B21, 0B24, and 0B27) but comes equipped with a topology. Warning: the topology one gets depends on which category one is working with, see Examples, Section 0B2Y. One can sheafify presheaves of topological spaces, topological groups, topological rings, or topological modules and taking stalks commutes with this operation, see Remark 0AI4.

Let \(f : X \to Y\) be a continuous map of topological spaces. There is a functor \(f_*\) from the category of sheaves of topological spaces, topological groups, topological rings, topological modules, to the corresponding category of sheaves on \(Y\) which is defined by setting \(f_*\mathcal{F}(V) = \mathcal{F}(f^{-1}V)\) as usual. (We delay discussing the pullback in this setting till later.) We define the notion of an \(f\)-map \(\xi : \mathcal{G} \to \mathcal{F}\) between a sheaf of topological spaces \(\mathcal{G}\) on \(Y\) and a sheaf of topological spaces \(\mathcal{F}\) on \(X\) in exactly the same manner as in Sheaves, Definition 008J with the additional constraint that \(\xi_V : \mathcal{G}(V) \to \mathcal{F}(f^{-1}V)\) be continuous for every open \(V \subset Y\). We have \[\{f\text{-maps from }\mathcal{G}\text{ to }\mathcal{F}\} = \Mor_{\Sh(Y, \textit{Top})}(\mathcal{G}, f_*\mathcal{F})\] as in Sheaves, Lemma 008K. Similarly for sheaves of topological groups, topological rings, topological modules. Finally, let \(\xi : \mathcal{G} \to \mathcal{F}\) be an \(f\)-map as above. Then given \(x \in X\) with image \(y = f(x)\) there is a continuous map \[\xi_x : \mathcal{G}_y \longrightarrow \mathcal{F}_x\] of stalks defined in exactly the same manner as in the discussion following Sheaves, Definition 008L.

Using the discussion above, we can define a category \(LTRS\) of “locally topologically ringed spaces”. An object is a pair \((X, \mathcal{O}_X)\) consisting of a topological space \(X\) and a sheaf of topological rings \(\mathcal{O}_X\) whose stalks \(\mathcal{O}_{X, x}\) are local rings (if one forgets about the topology). A morphism \((X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)\) of \(LTRS\) is a pair \((f, f^\sharp)\) where \(f : X \to Y\) is a continuous map of topological spaces and \(f^\sharp : \mathcal{O}_Y \to \mathcal{O}_X\) is an \(f\)-map such that for every \(x \in X\) the induced map \[f^\sharp_x : \mathcal{O}_{Y, f(x)} \longrightarrow \mathcal{O}_{X, x}\] is a local homomorphism of local rings (forgetting about the topologies). The composition works in exactly the same manner as composition of morphisms of locally ringed spaces.

Assume now that the topological space \(X\) has a basis consisting of quasi-compact opens. Given a sheaf \(\mathcal{F}\) of sets, groups, rings, modules over a ring, one can endow \(\mathcal{F}\) with the structure of a sheaf of topological spaces, topological groups, topological rings, topological modules. Namely, if \(U \subset X\) is quasi-compact open, we endow \(\mathcal{F}(U)\) with the discrete topology. If \(U \subset X\) is arbitrary, then we choose an open covering \(U = \bigcup_{i \in I} U_i\) by quasi-compact opens and we endow \(\mathcal{F}(U)\) with the induced topology from \(\prod_{i \in I} \mathcal{F}(U_i)\) (as we should do according to our discussion above). The reader may verify (omitted) that we obtain a sheaf of topological spaces, topological groups, topological rings, topological modules in this fashion. Let us say that a sheaf of topological spaces, topological groups, topological rings, topological modules is pseudo-discrete if the topology on \(\mathcal{F}(U)\) is discrete for every quasi-compact open \(U \subset X\). Then the construction given above is an adjoint to the forgetful functor and induces an equivalence between the category of sheaves of sets and the category of pseudo-discrete sheaves of topological spaces (similarly for groups, rings, modules).

Grothendieck and Dieudonné first define formal affine schemes. These correspond to admissible topological rings \(A\), see More on Algebra, Definition 07E8. Namely, given \(A\) one considers a fundamental system \(I_\lambda\) of ideals of definition for the ring \(A\). (In any admissible topological ring the family of all ideals of definition forms a fundamental system.) For each \(\lambda\) we can consider the scheme \(\Spec(A/I_\lambda)\). For \(I_\lambda \subset I_\mu\) the induced morphism \[\Spec(A/I_\mu) \to \Spec(A/I_\lambda)\] is a thickening because \(I_\mu^n \subset I_\lambda\) for some \(n\). Another way to see this, is to notice that the image of each of the maps \[\Spec(A/I_\lambda) \to \Spec(A)\] is a homeomorphism onto the set of open prime ideals of \(A\). This motivates the definition \[\text{Spf}(A) = \{\text{open prime ideals }\mathfrak p \subset A\}\] endowed with the topology coming from \(\Spec(A)\). For each \(\lambda\) we can consider the structure sheaf \(\mathcal{O}_{\Spec(A/I_\lambda)}\) as a sheaf on \(\text{Spf}(A)\). Let \(\mathcal{O}_\lambda\) be the corresponding pseudo-discrete sheaf of topological rings, see above. Then we set \[\mathcal{O}_{\text{Spf}(A)} = \lim \mathcal{O}_\lambda\] where the limit is taken in the category of sheaves of topological rings. The pair \((\text{Spf}(A), \mathcal{O}_{\text{Spf}(A)})\) is called the formal spectrum of \(A\).

At this point one should check several things. The first is that the stalks \(\mathcal{O}_{\text{Spf}(A), x}\) are local rings (forgetting about the topology). The second is that given \(f \in A\), for the corresponding open \(D(f) \cap \text{Spf}(A)\) we have \[\Gamma(D(f) \cap \text{Spf}(A), \mathcal{O}_{\text{Spf}(A)}) = A_{\{f\}} = \lim (A/I_\lambda)_f\] as topological rings where \(I_\lambda\) is a fundamental system of ideals of definition as above. Moreover, the ring \(A_{\{f\}}\) is admissible too and \((\text{Spf}(A_f), \mathcal{O}_{\text{Spf}(A_{\{f\}})})\) is isomorphic to \((D(f) \cap \text{Spf}(A), \mathcal{O}_{\text{Spf}(A)}|_{D(f) \cap \text{Spf}(A)})\). Finally, given a pair of admissible topological rings \(A, B\) we have [0AHZ]\[\begin{equation} \Mor_{LTRS}((\text{Spf}(B), \mathcal{O}_{\text{Spf}(B)}), (\text{Spf}(A), \mathcal{O}_{\text{Spf}(A)})) = \Hom_{cont}(A, B) \end{equation}\] where \(LTRS\) is the category of “locally topologically ringed spaces” as defined above.

Having said this, in [EGA] a formal scheme is defined as a pair \((\mathfrak X, \mathcal{O}_\mathfrak X)\) where \(\mathfrak X\) is a topological space and \(\mathcal{O}_\mathfrak X\) is a sheaf of topological rings such that every point has an open neighbourhood isomorphic (in \(LTRS\)) to an affine formal scheme. A morphism of formal schemes \(f : (\mathfrak X, \mathcal{O}_\mathfrak X) \to (\mathfrak Y, \mathcal{O}_\mathfrak Y)\) is a morphism in the category \(LTRS\).

Let \(A\) be a ring endowed with the discrete topology. Then \(A\) is admissible and the formal scheme \(\text{Spf}(A)\) is equal to \(\Spec(A)\). The structure sheaf \(\mathcal{O}_{\text{Spf}(A)}\) is the pseudo-discrete sheaf of topological rings associated to \(\mathcal{O}_{\Spec(A)}\), in other words, its underlying sheaf of rings is equal to \(\mathcal{O}_{\Spec(A)}\) and the ring \(\mathcal{O}_{\text{Spf}(A)}(U) = \mathcal{O}_{\Spec(A)}(U)\) over a quasi-compact open \(U\) has the discrete topology, but not in general. Thus we can associate to every affine scheme a formal affine scheme. In exactly the same manner we can start with a general scheme \((X, \mathcal{O}_X)\) and associate to it \((X, \mathcal{O}'_X)\) where \(\mathcal{O}'_X\) is the pseudo-discrete sheaf of topological rings whose underlying sheaf of rings is \(\mathcal{O}_X\). This construction is compatible with morphisms and defines a functor [0AI0]\[\begin{equation} \textit{Schemes} \longrightarrow \textit{Formal Schemes} \end{equation}\] It follows in a straightforward manner from (0AHZ) that this functor is fully faithful.

Let \(\mathfrak X\) be a formal scheme. Let us define the size of the formal scheme by the formula \(\text{size}(\mathfrak X) = \max(\aleph_0, \kappa_1, \kappa_2)\) where \(\kappa_1\) is the cardinality of the formal affine opens of \(\mathfrak X\) and \(\kappa_2\) is the supremum of the cardinalities of \(\mathcal{O}_\mathfrak X(\mathfrak U)\) where \(\mathfrak U \subset \mathfrak X\) is such a formal affine open.

Lemma

Choose a category of schemes \(\Sch_\alpha\) as in Sets, Lemma 000J. Given a formal scheme \(\mathfrak X\) let \[h_\mathfrak X : (\Sch_\alpha)^{opp} \longrightarrow \textit{Sets},\quad h_\mathfrak X(S) = \Mor_{\textit{Formal Schemes}}(S, \mathfrak X)\] be its functor of points. Then we have \[\Mor_{\textit{Formal Schemes}}(\mathfrak X, \mathfrak Y) = \Mor_{\textit{PSh}(\Sch_\alpha)}(h_\mathfrak X, h_\mathfrak Y)\] provided the size of \(\mathfrak X\) is not too large.

Proof

First we observe that \(h_\mathfrak X\) satisfies the sheaf property for the Zariski topology for any formal scheme \(\mathfrak X\) (see Schemes, Definition 01JI). This follows from the local nature of morphisms in the category of formal schemes. Also, for an open immersion \(\mathfrak V \to \mathfrak W\) of formal schemes, the corresponding transformation of functors \(h_\mathfrak V \to h_\mathfrak W\) is injective and representable by open immersions (see Schemes, Definition 01JI). Choose an open covering \(\mathfrak X = \bigcup \mathfrak U_i\) of a formal scheme by affine formal schemes \(\mathfrak U_i\). Then the collection of functors \(h_{\mathfrak U_i}\) covers \(h_\mathfrak X\) (see Schemes, Definition 01JI). Finally, note that \[h_{\mathfrak U_i} \times_{h_\mathfrak X} h_{\mathfrak U_j} = h_{\mathfrak U_i \cap \mathfrak U_j}\] Hence in order to give a map \(h_\mathfrak X \to h_\mathfrak Y\) is equivalent to giving a family of maps \(h_{\mathfrak U_i} \to h_\mathfrak Y\) which agree on overlaps. Thus we can reduce the bijectivity (resp. injectivity) of the map of the lemma to bijectivity (resp. injectivity) for the pairs \((\mathfrak U_i, \mathfrak Y)\) and injectivity (resp. nothing) for \((\mathfrak U_i \cap \mathfrak U_j, \mathfrak Y)\). In this way we reduce to the case where \(\mathfrak X\) is an affine formal scheme. Say \(\mathfrak X = \text{Spf}(A)\) for some admissible topological ring \(A\). Also, choose a fundamental system of ideals of definition \(I_\lambda \subset A\).

We can also localize on \(\mathfrak Y\). Namely, suppose that \(\mathfrak V \subset \mathfrak Y\) is an open formal subscheme and \(\varphi : h_\mathfrak X \to h_\mathfrak Y\). Then \[h_\mathfrak V \times_{h_\mathfrak Y, \varphi} h_\mathfrak X \to h_\mathfrak X\] is representable by open immersions. Pulling back to \(\Spec(A/I_\lambda)\) for all \(\lambda\) we find an open subscheme \(U_\lambda \subset \Spec(A/I_\lambda)\). However, for \(I_\lambda \subset I_\mu\) the morphism \(\Spec(A/I_\lambda) \to \Spec(A/I_\mu)\) pulls back \(U_\mu\) to \(U_\lambda\). Thus these glue to give an open formal subscheme \(\mathfrak U \subset \mathfrak X\). A straightforward argument (omitted) shows that \[h_\mathfrak U = h_\mathfrak V \times_{h_\mathfrak Y} h_\mathfrak X\] In this way we see that given an open covering \(\mathfrak Y = \bigcup \mathfrak V_j\) and a transformation of functors \(\varphi : h_\mathfrak X \to h_\mathfrak Y\) we obtain a corresponding open covering of \(\mathfrak X\). Since \(\mathfrak X\) is affine, we can refine this covering by a finite open covering \(\mathfrak X = \mathfrak U_1 \cup \ldots \cup \mathfrak U_n\) by affine formal subschemes. In other words, for each \(i\) there is a \(j\) and a map \(\varphi_i : h_{\mathfrak U_i} \to h_{\mathfrak V_j}\) such that \[\xymatrix{ h_{\mathfrak U_i} \ar[r]_{\varphi_i} \ar[d] & h_{\mathfrak V_j} \ar[d] \\ h_{\mathfrak X} \ar[r]^\varphi & h_\mathfrak Y }\] commutes. With a few additional arguments (which we omit) this implies that it suffices to prove the bijectivity of the lemma in case both \(\mathfrak X\) and \(\mathfrak Y\) are affine formal schemes.

Assume \(\mathfrak X\) and \(\mathfrak Y\) are affine formal schemes. Say \(\mathfrak X = \text{Spf}(A)\) and \(\mathfrak Y = \text{Spf}(B)\). Let \(\varphi : h_\mathfrak X \to h_\mathfrak Y\) be a transformation of functors. Let \(I_\lambda \subset A\) be a fundamental system of ideals of definition. The canonical inclusion morphism \(i_\lambda : \Spec(A/I_\lambda) \to \mathfrak X\) maps to a morphism \(\varphi(i_\lambda) : \Spec(A/I_\lambda) \to \mathfrak Y\). By (0AHZ) this corresponds to a continuous map \(\chi_\lambda : B \to A/I_\lambda\). Since \(\varphi\) is a transformation of functors it follows that for \(I_\lambda \subset I_\mu\) the composition \(B \to A/I_\lambda \to A/I_\mu\) is equal to \(\chi_\mu\). In other words we obtain a ring map \[\chi = \lim \chi_\lambda : B \longrightarrow \lim A/I_\lambda = A\] This is a continuous homomorphism because the inverse image of \(I_\lambda\) is open for all \(\lambda\) (as \(A/I_\lambda\) has the discrete topology and \(\chi_\lambda\) is continuous). Thus we obtain a morphism \(\text{Spf}(\chi) : \mathfrak X \to \mathfrak Y\) by (0AHZ). We omit the verification that this construction is the inverse to the map of the lemma in this case.

Set theoretic remarks. To make this work on the given category of schemes \(\Sch_\alpha\) we just have to make sure all the schemes used in the proof above are isomorphic to objects of \(\Sch_\alpha\). In fact, a careful analysis shows that it suffices if the schemes \(\Spec(A/I_\lambda)\) occurring above are isomorphic to objects of \(\Sch_\alpha\). For this it certainly suffices to assume the size of \(\mathfrak X\) is at most the size of a scheme contained in \(\Sch_\alpha\).

Lemma

Let \(\mathfrak X\) be a formal scheme. The functor of points \(h_\mathfrak X\) (see Lemma 0AI1) satisfies the sheaf condition for fpqc coverings.

Proof

Topologies, Lemma 022H reduces us to the case of a Zariski covering and a covering \(\{\Spec(S) \to \Spec(R)\}\) with \(R \to S\) faithfully flat. We observed in the proof of Lemma 0AI1 that \(h_\mathfrak X\) satisfies the sheaf condition for Zariski coverings.

Suppose that \(R \to S\) is a faithfully flat ring map. Denote \(\pi : \Spec(S) \to \Spec(R)\) the corresponding morphism of schemes. It is surjective and flat. Let \(f : \Spec(S) \to \mathfrak X\) be a morphism such that \(f \circ \text{pr}_1 = f \circ \text{pr}_2\) as maps \(\Spec(S \otimes_R S) \to \mathfrak X\). By Descent, Lemma 02KI we see that as a map on the underlying sets \(f\) is of the form \(f = g \circ \pi\) for some (set theoretic) map \(g : \Spec(R) \to \mathfrak X\). By Morphisms, Lemma 02JY and the fact that \(f\) is continuous we see that \(g\) is continuous.

Pick \(y \in \Spec(R)\). Choose \(\mathfrak U \subset \mathfrak X\) an affine formal open subscheme containing \(g(y)\). Say \(\mathfrak U = \text{Spf}(A)\) for some admissible topological ring \(A\). By the above we may choose an \(r \in R\) such that \(y \in D(r) \subset g^{-1}(\mathfrak U)\). The restriction of \(f\) to \(\pi^{-1}(D(r))\) into \(\mathfrak U\) corresponds to a continuous ring map \(A \to S_r\) by (0AHZ). The two induced ring maps \(A \to S_r \otimes_{R_r} S_r = (S \otimes_R S)_r\) are equal by assumption on \(f\). Note that \(R_r \to S_r\) is faithfully flat. By Descent, Lemma 023M the equalizer of the two arrows \(S_r \to S_r \otimes_{R_r} S_r\) is \(R_r\). We conclude that \(A \to S_r\) factors uniquely through a map \(A \to R_r\) which is also continuous as it has the same (open) kernel as the map \(A \to S_r\). This map in turn gives a morphism \(D(r) \to \mathfrak U\) by (0AHZ).

What have we proved so far? We have shown that for any \(y \in \Spec(R)\) there exists a standard affine open \(y \in D(r) \subset \Spec(R)\) such that the morphism \(f|_{\pi^{-1}(D(r))} : \pi^{-1}(D(r)) \to \mathfrak X\) factors uniquely though some morphism \(D(r) \to \mathfrak X\). We omit the verification that these morphisms glue to the desired morphism \(\Spec(R) \to \mathfrak X\).

Remark

There is a variant of the construction of formal schemes due to McQuillan, see [McQuillan]. He suggests a slight weakening of the condition of admissibility. Namely, recall that an admissible topological ring is a complete (and separated by our conventions) topological ring \(A\) which is linearly topologized such that there exists an ideal of definition: an open ideal \(I\) such that any neighbourhood of \(0\) contains \(I^n\) for some \(n \geq 1\). McQuillan works with what we will call weakly admissible topological rings. A weakly admissible topological ring \(A\) is a complete (and separated by our conventions) topological ring which is linearly topologized such that there exists an weak ideal of definition: an open ideal \(I\) such that for all \(f \in I\) we have \(f^n \to 0\) for \(n \to \infty\). Similarly to the admissible case, if \(I\) is a weak ideal of definition and \(J \subset A\) is an open ideal, then \(I \cap J\) is a weak ideal of definition. Thus the weak ideals of definition form a fundamental system of open neighbourhoods of \(0\) and one can proceed along much the same route as above to define a larger category of formal schemes based on this notion. The analogues of Lemmas 0AI1 and 0AI2 still hold in this setting (with the same proof).

Remark

In this remark we briefly discuss sheafification of presheaves of topological spaces. The exact same arguments work for presheaves of topological abelian groups, topological rings, and topological modules (over a given topological ring). In order to do this in the correct generality let us work over a site \(\mathcal{C}\). The reader who is interested in the case of (pre)sheaves over a topological space \(X\) should think of objects of \(\mathcal{C}\) as the opens of \(X\), of morphisms of \(\mathcal{C}\) as inclusions of opens, and of coverings in \(\mathcal{C}\) as coverings in \(X\), see Sites, Example 00VJ. Denote \(\Sh(\mathcal{C}, \textit{Top})\) the category of sheaves of topological spaces on \(\mathcal{C}\) and denote \(\textit{PSh}(\mathcal{C}, \textit{Top})\) the category of presheaves of topological spaces on \(\mathcal{C}\). Let \(\mathcal{F}\) be a presheaf of topological spaces on \(\mathcal{C}\). The sheafification \(\mathcal{F}^\#\) should satisfy the formula \[\Mor_{\textit{PSh}(\mathcal{C}, \textit{Top})}(\mathcal{F}, \mathcal{G}) = \Mor_{\Sh(\mathcal{C}, \textit{Top})}(\mathcal{F}^\#, \mathcal{G})\] functorially in \(\mathcal{G}\) from \(\Sh(\mathcal{C}, \textit{Top})\). In other words, we are trying to construct the left adjoint to the inclusion functor \(\Sh(\mathcal{C}, \textit{Top}) \to \textit{PSh}(\mathcal{C}, \textit{Top})\). We first claim that \(\Sh(\mathcal{C}, \textit{Top})\) has limits and that the inclusion functor commutes with them. Namely, given a category \(\mathcal{I}\) and a functor \(i \mapsto \mathcal{G}_i\) into \(\Sh(\mathcal{C}, \textit{Top})\) we simply define \[(\lim \mathcal{G}_i)(U) = \lim \mathcal{G}_i(U)\] where we take the limit in the category of topological spaces (Topology, Lemma 08ZT). This defines a sheaf because limits commute with limits (Categories, Lemma 002M) and in particular products and equalizers (which are the operations used in the sheaf axiom). Finally, a morphism of presheaves from \(\mathcal{F} \to \lim \mathcal{G}_i\) is clearly the same thing as a compatible system of morphisms \(\mathcal{F} \to \mathcal{G}_i\). In other words, the object \(\lim \mathcal{G}_i\) is the limit in the category of presheaves of topological spaces and a fortiori in the category of sheaves of topological spaces. Our second claim is that any morphism of presheaves \(\mathcal{F} \to \mathcal{G}\) with \(\mathcal{G}\) an object of \(\Sh(\mathcal{C}, \textit{Top})\) factors through a subsheaf \(\mathcal{G}' \subset \mathcal{G}\) whose size is bounded. Here we define the size \(|\mathcal{H}|\) of a sheaf of topological spaces \(\mathcal{H}\) to be the cardinal \(\sup_{U \in \Ob(\mathcal{C})} |\mathcal{H}(U)|\). To prove our claim we let \[\mathcal{G}'(U) = \left\{ \quad s \in \mathcal{G}(U) \quad \middle| \quad \begin{matrix} \text{there exists a covering }\{U_i \to U\}_{i \in I} \\ \text{such that } s|_{U_i} \in \Im(\mathcal{F}(U_i) \to \mathcal{G}(U_i)) \end{matrix} \quad \right\}\] We endow \(\mathcal{G}'(U)\) with the induced topology. Then \(\mathcal{G}'\) is a sheaf of topological spaces (details omitted) and \(\mathcal{G}' \to \mathcal{G}\) is a morphism through which the given map \(\mathcal{F} \to \mathcal{G}\) factors. Moreover, the size of \(\mathcal{G}'\) is bounded by some cardinal \(\kappa\) depending only on \(\mathcal{C}\) and the presheaf \(\mathcal{F}\) (hint: use that coverings in \(\mathcal{C}\) form a set by our conventions). Putting everything together we see that the assumptions of Categories, Theorem 0AHQ are satisfied and we obtain sheafification as the left adjoint of the inclusion functor from sheaves to presheaves. Finally, let \(p\) be a point of the site \(\mathcal{C}\) given by a functor \(u : \mathcal{C} \to \textit{Sets}\), see Sites, Definition 00Y5. For a topological space \(M\) the presheaf defined by the rule \[U \mapsto \text{Map}(u(U), M) = \prod\nolimits_{x \in u(U)} M\] endowed with the product topology is a sheaf of topological spaces. Hence the exact same argument as given in the proof of Sites, Lemma 00Y8 shows that \(\mathcal{F}_p = \mathcal{F}^\#_p\), in other words, sheafification commutes with taking stalks at a point.

Conventions and notation

The conventions from now on will be similar to the conventions in Properties of Spaces, Section 03BQ. Thus from now on the standing assumption is that all schemes are contained in a big fppf site \(\Sch_{fppf}\). And all rings \(A\) considered have the property that \(\Spec(A)\) is (isomorphic) to an object of this big site. For topological rings \(A\) we assume only that all discrete quotients have this property (but usually we assume more, compare with Remark 0AIS).

Let \(S\) be a scheme and let \(X\) be a “space” over \(S\), i.e., a sheaf on \((\Sch/S)_{fppf}\). In this chapter we will write \(X \times_S X\) for the product of \(X\) with itself in the category of sheaves on \((\Sch/S)_{fppf}\) instead of \(X \times X\). Moreover, if \(X\) and \(Y\) are “spaces” then we say "let \(f : X \to Y\) be a morphism" to indicate that \(f\) is a natural transformation of functors, i.e., a map of sheaves on \((\Sch/S)_{fppf}\). Similarly, if \(U\) is a scheme over \(S\) and \(X\) is a “space” over \(S\), then we say "let \(f : U \to X\) be a morphism" or "let \(g : X \to U\) be a morphism" to indicate that \(f\) or \(g\) is a map of sheaves \(h_U \to X\) or \(X \to h_U\) where \(h_U\) is as in Categories, Example 001O.

Topological rings and modules

This section is a continuation of More on Algebra, Section 07E7. Let \(R\) be a topological ring and let \(M\) be a linearly topologized \(R\)-module. When we say “let \(M_\lambda\) be a fundamental system of open submodules” we will mean that each \(M_\lambda\) is an open submodule and that any neighbourhood of \(0\) contains one of the \(M_\lambda\). In other words, this means that \(M_\lambda\) is a fundamental system of neighbourhoods of \(0\) in \(M\) consisting of submodules. Similarly, if \(R\) is a linearly topologized ring, then we say “let \(I_\lambda\) be a fundamental system of open ideals” to mean that \(I_\lambda\) is a fundamental system of neighbourhoods of \(0\) in \(R\) consisting of ideals.

Example

Let \(R\) be a linearly topologized ring and let \(M\) be a linearly topologized \(R\)-module. Let \(I_\lambda\) be a fundamental system of open ideals in \(R\) and let \(M_\mu\) be a fundamental system of open submodules of \(M\). The continuity of \(+ : M \times M \to M\) is automatic and the continuity of \(R \times M \to M\) signifies \[\forall f, x, \mu\ \exists \lambda, \nu,\ (f + I_\lambda)(x + M_\nu) \subset fx + M_\mu\] Since \(fM_\nu + I_\lambda M_\nu \subset M_\mu\) if \(M_\nu \subset M_\mu\) we see that the condition is equivalent to \[\forall x, \mu\ \exists \lambda\ I_\lambda x \subset M_\mu\] However, it need not be the case that given \(\mu\) there is a \(\lambda\) such that \(I_\lambda M \subset M_\mu\). For example, consider \(R = k[[t]]\) with the \(t\)-adic topology and \(M = \bigoplus_{n \in \mathbf{N}} R\) with fundamental system of open submodules given by \[M_m = \bigoplus\nolimits_{n \in \mathbf{N}} t^{nm}R\] Since every \(x \in M\) has finitely many nonzero coordinates we see that, given \(m\) and \(x\) there exists a \(k\) such that \(t^k x \in M_m\). Thus \(M\) is a linearly topologized \(R\)-module, but it isn’t true that given \(m\) there is a \(k\) such that \(t^kM \subset M_m\). On the other hand, if \(R \to S\) is a continuous map of linearly topologized rings, then the corresponding statement does hold, i.e., for every open ideal \(J \subset S\) there exists an open ideal \(I \subset R\) such that \(IS \subset J\) (as the reader can easily deduce from continuity of the map \(R \to S\)).

Lemma

Let \(R\) be a topological ring. Let \(M\) be a linearly topologized \(R\)-module and let \(M_\lambda\), \(\lambda \in \Lambda\) be a fundamental system of open submodules. Let \(N \subset M\) be a submodule. The closure of \(N\) is \(\bigcap_{\lambda \in \Lambda} (N + M_\lambda)\).

Proof

Since each \(N + M_\lambda\) is open, it is also closed. Hence the intersection is closed. If \(x \in M\) is not in the closure of \(N\), then \((x + M_\lambda) \cap N = 0\) for some \(\lambda\). Hence \(x \not \in N + M_\lambda\). This proves the lemma.

Unless otherwise mentioned we endow submodules and quotient modules with the induced topology. Let \(M\) be a linearly topologized module over a topological ring \(R\), and let \(0 \to N \to M \to Q \to 0\) be a short exact sequence of \(R\)-modules. If \(M_\lambda\) is a fundamental system of open submodules of \(M\), then \(N \cap M_\lambda\) is a fundamental system of open submodules of \(N\). If \(\pi : M \to Q\) is the quotient map, then \(\pi(M_\lambda)\) is a fundamental system of open submodules of \(Q\). In particular these induced topologies are linear topologies.

Lemma

Let \(R\) be a topological ring. Let \(M\) be a linearly topologized \(R\)-module. Let \(N \subset M\) be a submodule. Then

1. \(0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge\) is exact, and

2. \(N^\wedge\) is the closure of the image of \(N \to M^\wedge\).

Proof

Let \(M_\lambda\), \(\lambda \in \Lambda\) be a fundamental system of open submodules. Then \(N \cap M_\lambda\) is a fundamental system of open submodules of \(N\) and \(M_\lambda + N/N\) is a fundamental system of open submodules of \(M/N\). Thus we see that (1) follows from the exactness of the sequences \[0 \to N/N \cap M_\lambda \to M/M_\lambda \to M/(M_\lambda + N) \to 0\] and the fact that taking limits commutes with limits. The second statement follows from this and the fact that \(N \to N^\wedge\) has dense image and that the kernel of \(M^\wedge \to (M/N)^\wedge\) is closed.

Lemma

Let \(R\) be a topological ring. Let \(M\) be a complete, linearly topologized \(R\)-module. Let \(N \subset M\) be a closed submodule. If \(M\) has a countable fundamental system of neighbourhoods of \(0\), then \(M/N\) is complete and the map \(M \to M/N\) is open.

Proof

Let \(M_n\), \(n \in \mathbf{N}\) be a fundamental system of open submodules of \(M\). We may assume \(M_{n + 1} \subset M_n\) for all \(n\). The system \((M_n + N)/N\) is a fundamental system in \(M/N\). Hence we have to show that \(M/N = \lim M/(M_n + N)\). Consider the short exact sequences \[0 \to N/N \cap M_n \to M/M_n \to M/(M_n + N) \to 0\] Since the transition maps of the system \(\{N/N\cap M_n\}\) are surjective we see that \(M = \lim M/M_n\) (by completeness of \(M\)) surjects onto \(\lim M/(M_n + N)\) by Algebra, Lemma 0598. As \(N\) is closed we see that the kernel of \(M \to \lim M/(M_n + N)\) is \(N\) (see Lemma 0AMS). Finally, \(M \to M/N\) is open by definition of the quotient topology.

Lemma

Let \(R\) be a topological ring. Let \(M\) be a linearly topologized \(R\)-module. Let \(N \subset M\) be a submodule. Assume \(M\) has a countable fundamental system of neighbourhoods of \(0\). Then

1. \(0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge \to 0\) is exact,

2. \(N^\wedge\) is the closure of the image of \(N \to M^\wedge\),

3. \(M^\wedge \to (M/N)^\wedge\) is open.

Proof

We have \(0 \to N^\wedge \to M^\wedge \to (M/N)^\wedge\) is exact and statement (2) by Lemma 0ARZ. This produces a canonical map \(c : M^\wedge/N^\wedge \to (M/N)^\wedge\). The module \(M^\wedge/N^\wedge\) is complete and \(M^\wedge \to M^\wedge/N^\wedge\) is open by Lemma 0AMT. By the universal property of completion we obtain a canonical map \(b : (M/N)^\wedge \to M^\wedge/N^\wedge\). Then \(b\) and \(c\) are mutually inverse as they are on a dense subset.

Lemma

Let \(R\) be a topological ring. Let \(M\) be a topological \(R\)-module. Let \(I \subset R\) be a finitely generated ideal. Assume \(M\) has an open submodule whose topology is \(I\)-adic. Then \(M^\wedge\) has an open submodule whose topology is \(I\)-adic and we have \(M^\wedge/I^n M^\wedge = M/I^nM\) for all \(n \geq 1\).

Proof

Let \(M' \subset M\) be an open submodule whose topology is \(I\)-adic. Then \(\{I^nM'\}_{n \geq 1}\) is a fundamental system of open submodules of \(M\). Thus \(M^\wedge = \lim M/I^nM'\) contains \((M')^\wedge = \lim M'/I^nM'\) as an open submodule and the topology on \((M')^\wedge\) is \(I\)-adic by Algebra, Lemma 05GG. Since \(I\) is finitely generated, \(I^n\) is finitely generated, say by \(f_1, \ldots, f_r\). Observe that the surjection \((f_1, \ldots, f_r) : M^{\oplus r} \to I^n M\) is continuous and open by our description of the topology on \(M\) above. By Lemma 0AS0 applied to this surjection and to the short exact sequence \(0 \to I^nM \to M \to M/I^nM \to 0\) we conclude that \[(f_1, \ldots, f_r) : (M^\wedge)^{\oplus r} \longrightarrow M^\wedge\] surjects onto the kernel of the surjection \(M^\wedge \to M/I^nM\). Since \(f_1, \ldots, f_r\) generate \(I^n\) we conclude.

Definition

Let \(R\) be a topological ring. Let \(M\) and \(N\) be linearly topologized \(R\)-modules. The tensor product of \(M\) and \(N\) is the (usual) tensor product \(M \otimes_R N\) endowed with the linear topology defined by declaring \[\Im(M_\mu \otimes_R N + M \otimes_R N_\nu \longrightarrow M \otimes_R N)\] to be a fundamental system of open submodules, where \(M_\mu \subset M\) and \(N_\nu \subset N\) run through fundamental systems of open submodules in \(M\) and \(N\). The completed tensor product \[M \widehat{\otimes}_R N = \lim M \otimes_R N/(M_\mu \otimes_R N + M \otimes_R N_\nu) = \lim M/M_\mu \otimes_R N/N_\nu\] is the completion of the tensor product.

Observe that the topology on \(R\) is immaterial for the construction of the tensor product or the completed tensor product. If \(R \to A\) and \(R \to B\) are continuous maps of linearly topologized rings, then the construction above gives a tensor product \(A \otimes_R B\) and a completed tensor product \(A \widehat{\otimes}_R B\).

We record here the notions introduced in Remark 0AI3.

Definition

Let \(A\) be a linearly topologized ring.

1. An element \(f \in A\) is called topologically nilpotent if \(f^n \to 0\) as \(n \to \infty\).

2. A weak ideal of definition for \(A\) is an open ideal \(I \subset A\) consisting entirely of topologically nilpotent elements.

3. We say \(A\) is weakly pre-admissible if \(A\) has a weak ideal of definition.

4. We say \(A\) is weakly admissible if \(A\) is weakly pre-admissible and complete¹.

Given a weak ideal of definition \(I\) in a linearly topologized ring \(A\) and an open ideal \(J\) the intersection \(I \cap J\) is a weak ideal of definition. Hence if there is one weak ideal of definition, then there is a fundamental system of open ideals consisting of weak ideals of definition. In particular, given a weakly admissible topological ring \(A\) then \(A = \lim A/I_\lambda\) where \(\{I_\lambda\}\) is a fundamental system of weak ideals of definition.

Lemma

Let \(A\) be a weakly admissible topological ring. Let \(I \subset A\) be a weak ideal of definition. Then \((A, I)\) is a henselian pair.

Proof

Let \(A \to A'\) be an étale ring map and let \(\sigma : A' \to A/I\) be an \(A\)-algebra map. By More on Algebra, Lemma 09XI it suffices to lift \(\sigma\) to an \(A\)-algebra map \(A' \to A\). To do this, as \(A\) is complete, it suffices to find, for every open ideal \(J \subset I\), a unique \(A\)-algebra map \(A' \to A/J\) lifting \(\sigma\). Since \(I\) is a weak ideal of definition, the ideal \(I/J\) is locally nilpotent. We conclude by More on Algebra, Lemma 0ALI.

Lemma

Let \(B\) be a linearly topologized ring. The set of topologically nilpotent elements of \(B\) is a closed, radical ideal of \(B\). Let \(\varphi : A \to B\) be a continuous map of linearly topologized rings.

1. If \(f \in A\) is topologically nilpotent, then \(\varphi(f)\) is topologically nilpotent.

2. If \(I \subset A\) consists of topologically nilpotent elements, then the closure of \(\varphi(I)B\) consists of topologically nilpotent elements.

Proof

Let \(\mathfrak b \subset B\) be the set of topologically nilpotent elements. We omit the proof of the fact that \(\mathfrak b\) is a radical ideal (good exercise in the definitions). Let \(g\) be an element of the closure of \(\mathfrak b\). Our goal is to show that \(g\) is topologically nilpotent. Let \(J \subset B\) be an open ideal. We have to show \(g^e \in J\) for some \(e \geq 1\). We have \(g \in \mathfrak b + J\) by Lemma 0AMS. Hence \(g = f + h\) for some \(f \in \mathfrak b\) and \(h \in J\). Pick \(m \geq 1\) such that \(f^m \in J\). Then \(g^{m + 1} \in J\) as desired.

Let \(\varphi : A \to B\) be as in the statement of the lemma. Assertion (1) is clear and assertion (2) follows from this and the fact that \(\mathfrak b\) is a closed ideal.

Lemma

Let \(A \to B\) be a continuous map of linearly topologized rings. Let \(I \subset A\) be an ideal. The closure of \(IB\) is the kernel of \(B \to B \widehat{\otimes}_A A/I\).

Proof

Let \(J_\mu\) be a fundamental system of open ideals of \(B\). The closure of \(IB\) is \(\bigcap (IB + J_\lambda)\) by Lemma 0AMS. Let \(I_\mu\) be a fundamental system of open ideals in \(A\). Then \[B \widehat{\otimes}_A A/I = \lim (B/J_\lambda \otimes_A A/(I_\mu + I)) = \lim B/(J_\lambda + I_\mu B + I B)\] Since \(A \to B\) is continuous, for every \(\lambda\) there is a \(\mu\) such that \(I_\mu B \subset J_\lambda\), see discussion in Example 0AMR. Hence the limit can be written as \(\lim B/(J_\lambda + IB)\) and the result is clear.

Lemma

Let \(B \to A\) and \(B \to C\) be continuous homomorphisms of linearly topologized rings.

1. If \(A\) and \(C\) are weakly pre-admissible, then \(A \widehat{\otimes}_B C\) is weakly admissible.

2. If \(A\) and \(C\) are pre-admissible, then \(A \widehat{\otimes}_B C\) is admissible.

3. If \(A\) and \(C\) have a countable fundamental system of open ideals, then \(A \widehat{\otimes}_B C\) has a countable fundamental system of open ideals.

4. If \(A\) and \(C\) are pre-adic and have finitely generated ideals of definition, then \(A \widehat{\otimes}_B C\) is adic and has a finitely generated ideal of definition.

5. If \(A\) and \(C\) are pre-adic Noetherian rings and \(B/\mathfrak b \to A/\mathfrak a\) is of finite type where \(\mathfrak a \subset A\) and \(\mathfrak b \subset B\) are the ideals of topologically nilpotent elements, then \(A \widehat{\otimes}_B C\) is adic Noetherian.

Proof

Let \(I_\lambda \subset A\), \(\lambda \in \Lambda\) and \(J_\mu \subset C\), \(\mu \in M\) be fundamental systems of open ideals, then by definition \[A \widehat{\otimes}_B C = \lim_{\lambda, \mu} A/I_\lambda \otimes_B C/J_\mu\] with the limit topology. Thus a fundamental system of open ideals is given by the kernels \(K_{\lambda, \mu}\) of the maps \(A \widehat{\otimes}_B C \to A/I_\lambda \otimes_B C/J_\mu\). Note that \(K_{\lambda, \mu}\) is the closure of the ideal \(I_\lambda(A \widehat{\otimes}_B C) + J_\mu(A \widehat{\otimes}_B C)\). Finally, we have a ring homomorphism \(\tau : A \otimes_B C \to A \widehat{\otimes}_B C\) with dense image.

Proof of (1). If \(I_\lambda\) and \(J_\mu\) consist of topologically nilpotent elements, then so does \(K_{\lambda, \mu}\) by Lemma 0AMW. Hence \(A \widehat{\otimes}_B C\) is weakly admissible by definition.

Proof of (2). Assume for some \(\lambda_0\) and \(\mu_0\) the ideals \(I = I_{\lambda_0} \subset A\) and \(J_{\mu_0} \subset C\) are ideals of definition. Thus for every \(\lambda\) there exists an \(n\) such that \(I^n \subset I_\lambda\). For every \(\mu\) there exists an \(m\) such that \(J^m \subset J_\mu\). Then \[\left(I(A \widehat{\otimes}_B C) + J(A \widehat{\otimes}_B C)\right)^{n + m} \subset I_\lambda(A \widehat{\otimes}_B C) + J_\mu(A \widehat{\otimes}_B C)\] It follows that the open ideal \(K = K_{\lambda_0, \mu_0}\) satisfies \(K^{n + m} \subset K_{\lambda, \mu}\). Hence \(K\) is an ideal of definition of \(A \widehat{\otimes}_B C\) and \(A \widehat{\otimes}_B C\) is admissible by definition.

Proof of (3). If \(\Lambda\) and \(M\) are countable, so is \(\Lambda \times M\).

Proof of (4). Assume \(\Lambda = \mathbf{N}\) and \(M = \mathbf{N}\) and we have finitely generated ideals \(I \subset A\) and \(J \subset C\) such that \(I_n = I^n\) and \(J_n = J^n\). Then \[I(A \widehat{\otimes}_B C) + J(A \widehat{\otimes}_B C)\] is a finitely generated ideal and it is easily seen that \(A \widehat{\otimes}_B C\) is the completion of \(A \otimes_B C\) with respect to this ideal. Hence (4) follows from Algebra, Lemma 05GG.

Proof of (5). Let \(\mathfrak c \subset C\) be the ideal of topologically nilpotent elements. Since \(A\) and \(C\) are adic Noetherian, we see that \(\mathfrak a\) and \(\mathfrak c\) are ideals of definition (details omitted). From part (4) we already know that \(A \widehat{\otimes}_B C\) is adic and that \(\mathfrak a(A \widehat{\otimes}_B C) + \mathfrak c(A \widehat{\otimes}_B C)\) is a finitely generated ideal of definition. Since \[A \widehat{\otimes}_B C / \left(\mathfrak a(A \widehat{\otimes}_B C) + \mathfrak c(A \widehat{\otimes}_B C)\right) = A/\mathfrak a \otimes_{B/\mathfrak b} C/\mathfrak c\] is Noetherian as a finite type algebra over the Noetherian ring \(C/\mathfrak c\) we conclude by Algebra, Lemma 05GH.

Taut ring maps

It turns out to be convenient to have a name for the following property of continuous maps between linearly topologized rings.

Definition

Let \(\varphi : A \to B\) be a continuous map of linearly topologized rings. We say \(\varphi\) is taut² if for every open ideal \(I \subset A\) the closure of the ideal \(\varphi(I)B\) is open and these closures form a fundamental system of open ideals.

If \(\varphi : A \to B\) is a continuous map of linearly topologized rings and \(I_\lambda\) a fundamental system of open ideals of \(A\), then \(\varphi\) is taut if and only if the closures of \(I_\lambda B\) are open and form a fundamental system of open ideals in \(A\).

Lemma

Let \(\varphi : A \to B\) be a continuous map of weakly admissible topological rings. The following are equivalent

1. \(\varphi\) is taut,

2. for every weak ideal of definition \(I \subset A\) the closure of \(\varphi(I)B\) is a weak ideal of definition of \(B\) and these form a fundamental system of weak ideals of definition of \(B\).

Proof

The remarks following Definition 0AMX show that (2) implies (1). Conversely, assume \(\varphi\) is taut. If \(I \subset A\) is a weak ideal of definition, then the closure of \(\varphi(I)B\) is open by definition of tautness and consists of topologically nilpotent elements by Lemma 0AMW. Hence the closure of \(\varphi(I)B\) is a weak ideal of definition. Furthermore, by definition of tautness these ideals form a fundamental system of open ideals and we see that (2) is true.

Lemma

Let \(A\) be a linearly topologized ring. The map \(A \to A^\wedge\) from \(A\) to its completion is taut.

Proof

Let \(I_\lambda\) be a fundamental system of open ideals of \(A\). Recall that \(A^\wedge = \lim A/I_\lambda\) with the limit topology, which means that the kernels \(J_\lambda = \Ker(A^\wedge \to A/I_\lambda)\) form a fundamental system of open ideals of \(A^\wedge\). Since \(J_\lambda\) is the closure of \(I_\lambda A^\wedge\) (compare with Lemma 0AMZ) we conclude.

Lemma

Let \(A \to B\) and \(B \to C\) be continuous homomorphisms of linearly topologized rings. If \(A \to B\) and \(B \to C\) are taut, then \(A \to C\) is taut.

Proof

Omitted. Hint: if \(I \subset A\) is an ideal and \(J\) is the closure of \(IB\), then the closure of \(JC\) is equal to the closure of \(IC\).

Lemma

Let \(A \to B\) and \(B \to C\) be continuous homomorphisms of linearly topologized rings. If \(A \to C\) is taut, then \(B \to C\) is taut.

Proof

Let \(J \subset B\) be an open ideal with inverse image \(I \subset A\). Then the closure of \(JC\) contains the closure of \(IC\). Hence this closure is open as \(A \to C\) is taut. Let \(I_\lambda\) be a fundamental system of open ideals of \(A\). Let \(K_\lambda\) be the closure of \(I_\lambda C\). Since \(A \to C\) is taut, these form a fundamental system of open ideals of \(C\). Denote \(J_\lambda \subset B\) the inverse image of \(K_\lambda\). Then the closure of \(J_\lambda C\) is \(K_\lambda\). Hence we see that the closures of the ideals \(JC\), where \(J\) runs over the open ideals of \(B\) form a fundamental system of open ideals of \(C\).

Lemma

Let \(A \to B\) and \(A \to C\) be continuous homomorphisms of linearly topologized rings. If \(A \to B\) is taut, then \(C \to B \widehat{\otimes}_A C\) is taut.

Proof

Let \(K \subset C\) be an open ideal. Choose any open ideal \(I \subset A\) whose image in \(C\) is contained in \(J\). By assumption the closure \(J\) of \(IB\) is open. Since \(A \to B\) is taut we see that \(B \widehat{\otimes}_A C\) is the limit of the rings \(B/J \otimes_{A/I} C/K\) over all choices of \(K\) and \(I\), i.e, the ideals \(J(B \widehat{\otimes}_A C) + K(B \widehat{\otimes}_A C)\) form a fundamental system of open ideals. Now, since \(B \to B \widehat{\otimes}_A C\) is continuous we see that \(J\) maps into the closure of \(K(B \widehat{\otimes}_A C)\) (as \(I\) maps into \(K\)). Hence this closure is equal to \(J(B \widehat{\otimes}_A C) + K(B \widehat{\otimes}_A C)\) and the proof is complete.

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism of linearly topologized rings. If \(\varphi\) is taut and \(A\) has a countable fundamental system of open ideals, then \(B\) has a countable fundamental system of open ideals.

Proof

Immediate from the definitions.

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism of linearly topologized rings. If \(\varphi\) is taut and \(A\) is weakly pre-admissible, then \(B\) is weakly pre-admissible.

Proof

Let \(I \subset A\) be a weak ideal of definition. Then the closure \(J\) of \(IB\) is open and consists of topologically nilpotent elements by Lemma 0AMW. Hence \(J\) is a weak ideal of definition of \(B\).

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism of linearly topologized rings. If \(\varphi\) is taut and \(A\) is pre-admissible, then \(B\) is pre-admissible.

Proof

Let \(I \subset A\) be an ideal of definition. Let \(I_\lambda \subset A\) be a fundamental system of open ideals. Then the closure \(J\) of \(IB\) is open and the closures \(J_\lambda\) of \(I_\lambda B\) are open and form a fundamental system of open ideals of \(B\). For every \(\lambda\) there is an \(n\) such that \(I^n \subset I_\lambda\). Observe that \(J^n\) is contained in the closure of \(I^nB\). Thus \(J^n \subset J_\lambda\) and we conclude \(J\) is an ideal of definition.

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism of linearly topologized rings. Assume

1. \(\varphi\) is taut and has dense image,

2. \(A\) is complete and has a countable fundamental system of open ideals, and

3. \(B\) is separated.

Then \(\varphi\) is surjective and open, \(B\) is complete, and \(B = A/K\) for some closed ideal \(K \subset A\).

Proof

By the open mapping lemma (More on Algebra, Lemma 0CQW) combined with tautness of \(\varphi\), we see the map \(\varphi\) is open. Since the image of \(\varphi\) is dense, we see that \(\varphi\) is surjective. The kernel \(K\) of \(\varphi\) is closed as \(\varphi\) is continuous. It follows that \(B = A/K\) is complete, see for example Lemma 0AMT.

Adic ring maps

Let us make the following definition.

Definition

Let \(A\) and \(B\) be pre-adic topological rings. A ring homomorphism \(\varphi : A \to B\) is adic³ if there exists an ideal of definition \(I \subset A\) such that the topology on \(B\) is the \(I\)-adic topology.

If \(\varphi : A \to B\) is an adic homomorphism of pre-adic rings, then \(\varphi\) is continuous and the topology on \(B\) is the \(I\)-adic topology for every ideal of definition \(I\) of \(A\).

Lemma

Let \(A \to B\) and \(B \to C\) be continuous homomorphisms of pre-adic rings. If \(A \to B\) and \(B \to C\) are adic, then \(A \to C\) is adic.

Proof

Omitted.

Lemma

Let \(A \to B\) and \(B \to C\) be continuous homomorphisms of pre-adic rings. If \(A \to C\) is adic, then \(B \to C\) is adic.

Proof

Choose an ideal of definition \(I\) of \(A\). As \(A \to C\) is adic, we see that \(IC\) is an ideal of definition of \(C\). As \(B \to C\) is continuous, we can find an ideal of definition \(J \subset B\) mapping into \(IC\). As \(A \to B\) is continuous the inverse image \(I' \subset I\) of \(J\) in \(I\) is an ideal of definition of \(A\) too. Hence \(I'C \subset JC \subset IC\) is sandwiched between two ideals of definition, hence is an ideal of definition itself.

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism between pre-adic topological rings. If \(\varphi\) is adic, then \(\varphi\) is taut.

Proof

Immediate from the definitions.

The next lemma says two things

1. the property of being adic ascents along taut maps of complete linearly topologized rings, and

2. the properties “\(\varphi\) is taut” and “\(\varphi\) is adic” are equivalent for continuous maps \(\varphi : A \to B\) between adic rings if \(A\) has a finitely generated ideal of definition.

Because of (2) we can say that “tautness” generalizes “adicness” to continuous ring maps between arbitrary linearly topologized rings. See also Section 0AQ2.

Lemma

Let \(\varphi : A \to B\) be a continuous map of linearly topologized rings. If \(\varphi\) is taut, \(A\) is pre-adic and has a finitely generated ideal of definition, and \(B\) is complete, then \(B\) is adic and has a finitely generated ideal of definition and the ring map \(\varphi\) is adic.

Proof

Choose a finitely generated ideal of definition \(I\) of \(A\). Let \(J_n\) be the closure of \(\varphi(I^n)B\) in \(B\). Since \(B\) is complete we have \(B = \lim B/J_n\). Let \(B' = \lim B/I^nB\) be the \(I\)-adic completion of \(B\). By Algebra, Lemma 05GG, the \(I\)-adic topology on \(B'\) is complete and \(B'/I^nB' = B/I^nB\). Thus the ring map \(B' \to B\) is continuous and has dense image as \(B' \to B/I^nB \to B/J_n\) is surjective for all \(n\). Finally, the map \(B' \to B\) is taut because \((I^nB')B = I^nB\) and \(A \to B\) is taut. By Lemma 0APT we see that \(B' \to B\) is open and surjective. Thus the topology on \(B\) is the \(I\)-adic topology and the proof is complete.

Weakly adic rings

We suggest the reader skip this section. The following is a natural generalization of adic rings.

Definition

Let \(A\) be a linearly topologized ring.

1. We say \(A\) is weakly pre-adic⁴ if there exists an ideal \(I \subset A\) such that the closure of \(I^n\) is open for all \(n \geq 0\) and these closures form a fundamental system of open ideals.

2. We say \(A\) is weakly adic if \(A\) is weakly pre-adic and complete⁵.

For complete linearly topologized rings we have the following implications \[\xymatrix{ \text{adic + Noetherian} \ar@{=>}[d] \\ \text{adic + finitely generated ideal of definition} \ar@{=>}[d] \\ \text{adic} \ar@{=>}[d] \\ \text{weakly adic} \ar@{=>}[d] \\ \text{admissible + first countable} \ar@{=>}[d] \ar@{=>}[r] & \text{admissible} \ar@{=>}[d] \\ \text{weakly admissible + first countable} \ar@{=>}[r] & \text{weakly admissible} }\] where “first countable” means that our topological ring has a countable fundamental system of open ideals. There is a similar diagram of implications for noncomplete linearly topologized rings (i.e., using the notions of pre-adic, weakly pre-adic, pre-admissible, and weakly pre-admissible). Contrary to what happens with pre-adic rings the completion of a weakly pre-adic ring is weakly adic as the following lemma characterizing weakly pre-adic rings shows.

Lemma

Let \(A\) be a linearly topologized ring. The following are equivalent

1. \(A\) is weakly pre-adic,

2. there exists a taut continuous ring map \(A' \to A\) where \(A'\) is a pre-adic topological ring, and

3. \(A\) is pre-admissible and there exists an ideal of definition \(I\) such that the closure of \(I^n\) is open for all \(n \geq 1\), and

4. \(A\) is pre-admissible and for every ideal of definition \(I\) the closure of \(I^n\) is open for all \(n \geq 1\).

The completion of a weakly pre-adic ring is weakly adic. If \(A\) is weakly adic, then \(A\) is admissible and has a countable fundamental system of open ideals.

Proof

Assume (1). Choose an ideal \(I\) such that the closure of \(I^n\) is open for all \(n\) and such that these closures form a fundamental system of open ideals. Denote \(A' = A\) endowed with the \(I\)-adic topology. Then \(A' \to A\) is taut by definition and we see that (2) holds.

Assume (2). Let \(I' \subset A'\) be an ideal of definition. Denote \(I\) the closure of \(I'A\). Tautness of \(A' \to A\) means that the closures \(I_n\) of \((I')^nA\) are open and form a fundamental system of open ideals. Thus \(I = I_1\) is open and the closures of \(I^n\) are equal to \(I_n\) and hence open and form a fundamental system of open ideals. Thus certainly \(I\) is an ideal of definition such that the closure of \(I^n\) is open for all \(n\). Hence (3) holds.

If \(I \subset A\) is as in (3), then \(I\) is an ideal as in Definition 0GXE and we see that (1) holds. Also, if \(I' \subset A\) is any other ideal of definition, then \(I'\) is open (see More on Algebra, Definition 07E8) and hence contains \(I^n\) for some \(n \geq 1\). Thus \((I')^m\) contains \(I^{nm}\) for all \(m \geq 1\) and we conclude that the closures of \((I')^m\) are open for all \(m\). In this way we see that (3) implies (4). The implication (4) \(\Rightarrow\) (3) is trivial.

Let \(A\) be weakly pre-adic. Choose \(A' \to A\) as in (2). By Lemmas 0GX2 and 0GX3 the composition \(A' \to A^\wedge\) is taut. Hence \(A^\wedge\) is weakly pre-adic by the equivalence of (2) and (1). Since the completion of a linearly topologized ring \(A\) is complete (More on Algebra, Section 07E7) we see that \(A^\wedge\) is weakly adic.

Let \(A\) be weakly adic. Then \(A\) is complete and and pre-admissible by (1) \(\Rightarrow\) (3) and hence \(A\) is admissible. Of course by definition \(A\) has a countable fundamental system of open ideals.

We give two criteria that guarantee that a weakly adic ring is adic and has a finitely generated ideal of definition.

Lemma

Let \(A\) be a complete linearly topologized ring. Let \(I \subset A\) be a finitely generated ideal such that the closure of \(I^n\) is open for all \(n \geq 0\) and these closures form a fundamental system of open ideals. Then \(A\) is adic and has a finitely generated ideal of definition.

Proof

Denote \(A'\) the ring \(A\) endowed with the \(I\)-adic topology. The assumptions tells us that \(A' \to A\) is taut. We conclude by Lemma 0APU (to be sure, this lemma also tells us that \(I\) is an ideal of definition).

Lemma

Let \(A\) be a weakly adic topological ring. Let \(I\) be an ideal of definition such that \(I/I_2\) is a finitely generated module where \(I_2\) is the closure of \(I^2\). Then \(A\) is adic and has a finitely generated ideal of definition.

Proof

We use the characterization of Lemma 0GXF without further mention. Choose \(f_1, \ldots, f_r \in I\) which map to generators of \(I/I_2\). Set \(I' = (f_1, \ldots, f_r)\). We have \(I' + I_2 = I\). Then \(I_2\) is the closure of \(I^2 = (I' + I_2)^2 \subset I' + I_3\) where \(I_3\) is the closure of \(I^3\). Hence \(I' + I_3 = I\). Continuing in this fashion we see that \(I' + I_n = I\) for all \(n \geq 2\) where \(I_n\) is the closure of \(I^n\). In other words, the closure of \(I'\) in \(A\) is \(I\). Hence the closure of \((I')^n\) is \(I_n\). Thus the closures of \((I')^n\) are a fundamental system of open ideals of \(A\). We conclude by Lemma 0GXG.

A key feature of the property “weakly pre-adic” is that it ascents along taut ring homomorphisms of linearly topologized rings.

Lemma

Let \(\varphi : A \to B\) be a continuous homomorphism of linearly topologized rings. If \(\varphi\) is taut and \(A\) is weakly pre-adic, then \(B\) is weakly pre-adic.

Proof

Let \(I \subset A\) be an ideal such that the closure \(I_n\) of \(I^n\) is open and these closures define a fundamental system of open ideals. Then the closure of \(I^nB\) is equal to the closure of \(I_nB\). Since \(\varphi\) is taut, these closures are open and form a fundamental system of open ideals of \(B\). Hence \(B\) is weakly pre-adic.

Lemma

Let \(B \to A\) and \(B \to C\) be continuous homomorphisms of linearly topologized rings. If \(A\) and \(C\) are weakly pre-adic, then \(A \widehat{\otimes}_B C\) is weakly adic.

Proof

We will use the characterization of Lemma 0GXF without further mention. By Lemma 0GB4 we know that \(A \widehat{\otimes}_B C\) is admissible. Moreover, the proof of that lemma shows that the closure \(K \subset A \widehat{\otimes}_B C\) is an ideal of definition, when \(I \subset A\) and \(J \subset C\) of \(I(A \widehat{\otimes}_B C) + J(A \widehat{\otimes}_B C)\) are ideals of definition. Then it suffices to show that the closure of \(K^n\) is open for all \(n \geq 1\). Since the ideal \(K^n\) contains \(I^n(A \widehat{\otimes}_B C) + J^n(A \widehat{\otimes}_B C)\), since the closure of \(I^n\) in \(A\) is open, and since the closure of \(J^n\) in \(C\) is open, we see that the closure of \(K^n\) is open in \(A \widehat{\otimes}_B C\).

Descending properties

In this section we consider the following situation

1. \(\varphi : A \to B\) is a continuous map of linearly topologized topological rings,

2. \(\varphi\) is taut, and

3. for every open ideal \(I \subset A\) if \(J \subset B\) denotes the closure of \(IB\), then the map \(A/I \to B/J\) is faithfully flat.

We are going to show that properties of \(B\) are inherited by \(A\) in this situation.

Lemma

In the situation above, if \(B\) has a countable fundamental system of open ideals, then \(A\) has a countable fundamental system of open ideals.

Proof

Choose a fundamental system \(B \supset J_1 \supset J_2 \supset \ldots\) of open ideals. By tautness of \(\varphi\), for every \(n\) we can find an open ideal \(I_n\) such that \(J_n \supset I_nB\). We claim that \(I_n\) is a fundamental system of open ideals of \(A\). Namely, suppose that \(I \subset A\) is open. As \(\varphi\) is taut, the closure of \(IB\) is open and hence contains \(J_n\) for some \(n\) large enough. Hence \(I_nB \subset IB\). Let \(J\) be the closure of \(IB\) in \(B\). Since \(A/I \to B/J\) is faithfully flat, it is injective. Hence, since \(I_n \to A/I \to B/J\) is zero as \(I_nB \subset IB \subset J\), we conclude that \(I_n \to A/I\) is zero. Hence \(I_n \subset I\) and we win.

Lemma

In the situation above, if \(B\) is weakly pre-admissible, then \(A\) is weakly pre-admissible.

Proof

Let \(J \subset B\) be a weak ideal of definition. Let \(I \subset A\) be an open ideal such that \(IB \subset J\). To show that \(I\) is a weak ideal of definition we have to show that any \(f \in I\) is topologically nilpotent. Let \(I' \subset A\) be an open ideal. Denote \(J' \subset B\) the closure of \(I'B\). Then \(A/I' \to B/J'\) is faithfully flat, hence injective. Thus in order to show that \(f^n \in I'\) it suffices to show that \(\varphi(f)^n \in J'\). This holds for \(n \gg 0\) since \(\varphi(f) \in J\), the ideal \(J\) is a weak ideal of definition of \(B\), and \(J'\) is open in \(B\).

Lemma

In the situation above, if \(B\) is pre-admissible, then \(A\) is pre-admissible.

Proof

Let \(J \subset B\) be a weak ideal of definition. Let \(I \subset A\) be an open ideal such that \(IB \subset J\). Let \(I' \subset A\) be an open ideal. To show that \(I\) is an ideal of definition we have to show that \(I^n \subset I'\) for \(n \gg 0\). Denote \(J' \subset B\) the closure of \(I'B\). Then \(A/I' \to B/J'\) is faithfully flat, hence injective. Thus in order to show that \(I^n \subset I'\) it suffices to show that \(\varphi(I)^n \subset J'\). This holds for \(n \gg 0\) since \(\varphi(I) \subset J\), the ideal \(J\) is an ideal of definition of \(B\), and \(J'\) is open in \(B\).

Lemma

In the situation above, if \(B\) is weakly pre-adic, then \(A\) is weakly pre-adic.

Proof

We will use the characterization of weakly pre-adic rings given in Lemma 0GXF without further mention. By Lemma 0GXN the topological ring \(A\) is pre-admissible. Let \(I \subset A\) be an ideal of definition. Fix \(n \geq 1\). To prove the lemma we have to show that the closure of \(I^n\) is open. Let \(I_\lambda \subset A\) be a fundamental system of open ideals. Denote \(J \subset B\), resp. \(J_\lambda \subset B\) the closure of \(IB\), resp. \(I_\lambda B\). Since \(B\) is weakly pre-adic, the closure of \(J^n\) is open. Hence there exists a \(\lambda\) such that \[J_\lambda \subset \bigcap\nolimits_\mu (J^n + J_\mu)\] because the right hand side is the closure of \(J^n\) by Lemma 0AMS. This means that the image of \(J_\lambda\) in \(B/J_\mu\) is contained in the image of \(J^n\) in \(B/J_\mu\). Observe that the image of \(J^n\) in \(B/J_\mu\) is equal to the image of \(I^nB\) in \(B/J_\mu\) (since every element of \(J\) is congruent to an element of \(IB\) modulo \(J_\mu\)). Since \(A/I_\mu \to B/J_\mu\) is faithfully flat and since \(I_\lambda B \subset J_\lambda\), we conclude that the image of \(I_\lambda\) in \(A/I_\mu\) is contained in the image of \(I^n\). We conclude that \(I_\lambda\) is contained in the closure of \(I^n\) and the proof is complete.

Lemma

In the situation above, if \(B\) is adic and has a finitely generated ideal of definition and \(A\) is complete, then \(A\) is adic and has a finitely generated ideal of definition.

Proof

We already know that \(A\) is weakly adic and a fortiori admissible by Lemma 0GXP (and Lemma 0GXF to see that adic rings are weakly adic). Let \(I \subset A\) be an ideal of definition. Let \(J \subset B\) be a finitely generated ideal of definition. Since the closure of \(IB\) is open, we can find an \(n > 0\) such that \(J^n\) is contained in the closure of \(IB\). Thus after replacing \(J\) by \(J^n\) we may assume \(J\) is a finitely generated ideal of definition contained in the closure of \(IB\). By Lemma 0AMS this certainly implies that \[J \subset IB + J^2\] Consider the finitely generated \(A\)-module \(M = (J + IB)/IB\). The displayed equation shows that \(JM = M\). By Lemma 0DCZ (for example) we see that \(J\) is contained in the Jacobson radical of \(B\). Hence by Nakayama’s lemma, more precisely part (2) of Algebra, Lemma 00DV, we conclude \(M = 0\). Thus \(J \subset IB\).

Since \(J\) is finitely generated, we can find a finitely generated ideal \(I' \subset I\) such that \(J \subset I'B\). Since \(A \to B\) is continuous, \(J \subset B\) is open, and \(I\) is an ideal of definition, we can find an \(n > 0\) such that \(I^nB \subset J\). Let \(J_{n + 1} \subset B\) be the closure of \(I^{n + 1}B\). We have \[I^n \cdot (B/J_{n + 1}) \subset J \cdot (B/J_{n + 1}) \subset I' \cdot (B/J_{n + 1})\] Since \(A/I^{n + 1} \to B/J_{n + 1}\) is faithfully flat, this implies \(I^n \cdot (A/I^{n + 1}) \subset I' \cdot (A/I^{n + 1})\) which in turn means \[I^n \subset I' + I^{n + 1}\] This implies \(I^n \subset I' + I^{n + k}\) for all \(k \geq 1\) which in turn implies that \(I^{nm} \subset (I')^m + I^{nm + k}\) for all \(k, m \geq 1\). This implies that the closure of \((I')^m\) contains \(I^{nm}\). Since the closure of \(I^{nm}\) is open as \(A\) is weakly adic, we conclude that the closure \((I')^m\) is open for all \(m\). Since these closures form a fundamental system of open ideals of \(A\) (as the same thing is true for the closures of \(I^n\)) we conclude by Lemma 0GXG.

Affine formal algebraic spaces

In this section we introduce affine formal algebraic spaces. These will in fact be the same as what are called affine formal schemes in [BVGD]. However, we will call them affine formal algebraic spaces, in order to prevent confusion with the notion of an affine formal scheme as defined in [EGA].

Recall that a thickening of schemes is a closed immersion which induces a surjection on underlying topological spaces, see More on Morphisms, Definition 04EX.

Definition

Let \(S\) be a scheme. We say a sheaf \(X\) on \((\Sch/S)_{fppf}\) is an affine formal algebraic space if there exist

1. a directed set \(\Lambda\),

2. a system \((X_\lambda, f_{\lambda \mu})\) over \(\Lambda\) in \((\Sch/S)_{fppf}\) where

   1. each \(X_\lambda\) is affine,

   2. each \(f_{\lambda \mu} : X_\lambda \to X_\mu\) is a thickening,

such that \[X \cong \colim_{\lambda \in \Lambda} X_\lambda\] as fppf sheaves and \(X\) satisfies a set theoretic condition (see Remark 0AIS). A morphism of affine formal algebraic spaces over \(S\) is a map of sheaves.

Observe that the system \((X_\lambda, f_{\lambda \mu})\) is not part of the data. Suppose that \(U\) is a quasi-compact scheme over \(S\). Since the transition maps are monomorphisms, we see that \[X(U) = \colim X_\lambda(U)\] by Sites, Lemma 0738. Thus the fppf sheafification inherent in the colimit of the definition is a Zariski sheafification which does not do anything for quasi-compact schemes.

Lemma

Let \(S\) be a scheme. If \(X\) is an affine formal algebraic space over \(S\), then the diagonal morphism \(\Delta : X \to X \times_S X\) is representable and a closed immersion.

Proof

Suppose given \(U \to X\) and \(V \to X\) where \(U, V\) are schemes over \(S\). Let us show that \(U \times_X V\) is representable. Write \(X = \colim X_\lambda\) as in Definition 0AI7. The discussion above shows that Zariski locally on \(U\) and \(V\) the morphisms factors through some \(X_\lambda\). In this case \(U \times_X V = U \times_{X_\lambda} V\) which is a scheme. Thus the diagonal is representable, see Spaces, Lemma 025W. Given \((a, b) : W \to X \times_S X\) where \(W\) is a scheme over \(S\) consider the map \(X \times_{\Delta, X \times_S X, (a, b)} W \to W\). As before locally on \(W\) the morphisms \(a\) and \(b\) map into the affine scheme \(X_\lambda\) for some \(\lambda\) and then we get the morphism \(X_\lambda \times_{\Delta_\lambda, X_\lambda \times_S X_\lambda, (a, b)} W \to W\). This is the base change of \(\Delta_\lambda : X_\lambda \to X_\lambda \times_S X_\lambda\) which is a closed immersion as \(X_\lambda \to S\) is separated (because \(X_\lambda\) is affine). Thus \(X \to X \times_S X\) is a closed immersion.

A morphism of schemes \(X \to X'\) is a thickening if it is a closed immersion and induces a surjection on underlying sets of points, see (More on Morphisms, Definition 04EX). Hence the property of being a thickening is preserved under arbitrary base change and fpqc local on the target, see Spaces, Section 02WE. Thus Spaces, Definition 025V applies to “thickening” and we know what it means for a representable transformation \(F \to G\) of presheaves on \((\Sch/S)_{fppf}\) to be a thickening. We observe that this does not clash with our definition (More on Morphisms of Spaces, Definition 05ZK) of thickenings in case \(F\) and \(G\) are algebraic spaces.

Lemma

Let \(X_\lambda, \lambda \in \Lambda\) and \(X = \colim X_\lambda\) be as in Definition 0AI7. Then \(X_\lambda \to X\) is representable and a thickening.

Proof

The statement makes sense by the discussion in Spaces, Section 025U and 02WI. By Lemma 0AI8 the morphisms \(X_\lambda \to X\) are representable. Given \(U \to X\) where \(U\) is a scheme, then the discussion following Definition 0AI7 shows that Zariski locally on \(U\) the morphism factors through some \(X_\mu\) with \(\lambda \leq \mu\). In this case \(U \times_X X_\lambda = U \times_{X_\mu} X_\lambda\) so that \(U \times_X X_\lambda \to U\) is a base change of the thickening \(X_\lambda \to X_\mu\).

Lemma

Let \(X_\lambda, \lambda \in \Lambda\) and \(X = \colim X_\lambda\) be as in Definition 0AI7. If \(Y\) is a quasi-compact algebraic space over \(S\), then any morphism \(Y \to X\) factors through an \(X_\lambda\).

Proof

Choose an affine scheme \(V\) and a surjective étale morphism \(V \to Y\). The composition \(V \to Y \to X\) factors through \(X_\lambda\) for some \(\lambda\) by the discussion following Definition 0AI7. Since \(V \to Y\) is a surjection of sheaves, we conclude.

Lemma

Let \(S\) be a scheme. Let \(X\) be a sheaf on \((\Sch/S)_{fppf}\). Then \(X\) is an affine formal algebraic space if and only if the following hold

1. any morphism \(U \to X\) where \(U\) is an affine scheme over \(S\) factors through a morphism \(T \to X\) which is representable and a thickening with \(T\) an affine scheme over \(S\), and

2. a set theoretic condition as in Remark 0AIS.

Proof

It follows from Lemmas 0AI9 and 0AIA that an affine formal algebraic space satisfies (1) and (2). In order to prove the converse we may assume \(X\) is not empty. Let \(\Lambda\) be the category of representable morphisms \(T \to X\) which are thickenings where \(T\) is an affine scheme over \(S\). This category is directed. Since \(X\) is not empty, \(\Lambda\) contains at least one object. If \(T \to X\) and \(T' \to X\) are in \(\Lambda\), then we can factor \(T \amalg T' \to X\) through \(T'' \to X\) in \(\Lambda\). Between any two objects of \(\Lambda\) there is a unique arrow or none. Thus \(\Lambda\) is a directed set and by assumption \(X = \colim_{T \to X\text{ in }\Lambda} T\). To finish the proof we need to show that any arrow \(T \to T'\) in \(\Lambda\) is a thickening. This is true because \(T' \to X\) is a monomorphism of sheaves, so that \(T = T \times_{T'} T' = T \times_X T'\) and hence the morphism \(T \to T'\) equals the projection \(T \times_X T' \to T'\) which is a thickening because \(T \to X\) is a thickening.

For a general affine formal algebraic space \(X\) there is no guarantee that \(X\) has enough functions to separate points (for example). See Examples, Section 0ANY. To characterize those that do we offer the following lemma.

Lemma

Let \(S\) be a scheme. Let \(X\) be an fppf sheaf on \((\Sch/S)_{fppf}\) which satisfies the set theoretic condition of Remark 0AIS. The following are equivalent:

1. there exists a weakly admissible topological ring \(A\) over \(S\) (see Remark 0AI3) such that \(X = \colim_{I \subset A\text{ weak ideal of definition}} \Spec(A/I)\),

2. \(X\) is an affine formal algebraic space and there exists an \(S\)-algebra \(A\) and a map \(X \to \Spec(A)\) such that for a closed immersion \(T \to X\) with \(T\) an affine scheme the composition \(T \to \Spec(A)\) is a closed immersion,

3. \(X\) is an affine formal algebraic space and there exists an \(S\)-algebra \(A\) and a map \(X \to \Spec(A)\) such that for a closed immersion \(T \to X\) with \(T\) a scheme the composition \(T \to \Spec(A)\) is a closed immersion,

4. \(X\) is an affine formal algebraic space and for some choice of \(X = \colim X_\lambda\) as in Definition 0AI7 the projections \(\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})\) are surjective,

5. \(X\) is an affine formal algebraic space and for any choice of \(X = \colim X_\lambda\) as in Definition 0AI7 the projections \(\lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda}) \to \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})\) are surjective.

Moreover, the weakly admissible topological ring is \(A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})\) endowed with its limit topology and the weak ideals of definition classify exactly the morphisms \(T \to X\) which are representable and thickenings.

Proof

It is clear that (5) implies (4).

Assume (4) for \(X = \colim X_\lambda\) as in Definition 0AI7. Set \(A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})\). Let \(T \to X\) be a closed immersion with \(T\) a scheme (note that \(T \to X\) is representable by Lemma 0AI8). Since \(X_\lambda \to X\) is a thickening, so is \(X_\lambda \times_X T \to T\). On the other hand, \(X_\lambda \times_X T \to X_\lambda\) is a closed immersion, hence \(X_\lambda \times_X T\) is affine. Hence \(T\) is affine by Limits, Proposition 05YU. Then \(T \to X\) factors through \(X_\lambda\) for some \(\lambda\) by Lemma 0AIA. Thus \(A \to \Gamma(X_\lambda, \mathcal{O}) \to \Gamma(T, \mathcal{O})\) is surjective. In this way we see that (3) holds.

It is clear that (3) implies (2).

Assume (2) for \(A\) and \(X \to \Spec(A)\). Write \(X = \colim X_\lambda\) as in Definition 0AI7. Then \(A_\lambda = \Gamma(X_\lambda, \mathcal{O})\) is a quotient of \(A\) by assumption (2). Hence \(A^\wedge = \lim A_\lambda\) is a complete topological ring, see discussion in More on Algebra, Section 07E7. The maps \(A^\wedge \to A_\lambda\) are surjective as \(A \to A_\lambda\) is. We claim that for any \(\lambda\) the kernel \(I_\lambda \subset A^\wedge\) of \(A^\wedge \to A_\lambda\) is a weak ideal of definition. Namely, it is open by definition of the limit topology. If \(f \in I_\lambda\), then for any \(\mu \in \Lambda\) the image of \(f\) in \(A_\mu\) is zero in all the residue fields of the points of \(X_\mu\). Hence it is a nilpotent element of \(A_\mu\). Hence some power \(f^n \in I_\mu\). Thus \(f^n \to 0\) as \(n \to 0\). Thus \(A^\wedge\) is weakly admissible. Finally, suppose that \(I \subset A^\wedge\) is a weak ideal of definition. Then \(I \subset A^\wedge\) is open and hence there exists some \(\lambda\) such that \(I \supset I_\lambda\). Thus we obtain a morphism \(\Spec(A^\wedge/I) \to \Spec(A_\lambda) \to X\). Then it follows that \(X = \colim \Spec(A^\wedge/I)\) where now the colimit is over all weak ideals of definition. Thus (1) holds.

Assume (1). In this case it is clear that \(X\) is an affine formal algebraic space. Let \(X = \colim X_\lambda\) be any presentation as in Definition 0AI7. For each \(\lambda\) we can find a weak ideal of definition \(I \subset A\) such that \(X_\lambda \to X\) factors through \(\Spec(A/I) \to X\), see Lemma 0AIA. Then \(X_\lambda = \Spec(A/I_\lambda)\) with \(I \subset I_\lambda\). Conversely, for any weak ideal of definition \(I \subset A\) the morphism \(\Spec(A/I) \to X\) factors through \(X_\lambda\) for some \(\lambda\), i.e., \(I_\lambda \subset I\). It follows that each \(I_\lambda\) is a weak ideal of definition and that they form a cofinal subset of the set of weak ideals of definition. Hence \(A = \lim A/I = \lim A/I_\lambda\) and we see that (5) is true and moreover that \(A = \lim \Gamma(X_\lambda, \mathcal{O}_{X_\lambda})\).

With this lemma in hand we can make the following definition.

Definition

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). We say \(X\) is McQuillan if \(X\) satisfies the equivalent conditions of Lemma 0AIC. Let \(A\) be the weakly admissible topological ring associated to \(X\). We say

1. \(X\) is classical if \(X\) is McQuillan and \(A\) is admissible (More on Algebra, Definition 07E8),

2. \(X\) is weakly adic if \(X\) is McQuillan and \(A\) is weakly adic (Definition 0GXE),

3. \(X\) is adic if \(X\) is McQuillan and \(A\) is adic (More on Algebra, Definition 07E8),

4. \(X\) is adic* if \(X\) is McQuillan, \(A\) is adic, and \(A\) has a finitely generated ideal of definition, and

5. \(X\) is Noetherian if \(X\) is McQuillan and \(A\) is both Noetherian and adic.

In [Fujiwara-Kato] they use the terminology “of finite ideal type” for the property that an adic topological ring \(A\) contains a finitely generated ideal of definition. Given an affine formal algebraic space \(X\) here are the implications among the notions introduced in the definition: \[\xymatrix{ X\text{ Noetherian} \ar@{=>}[r] & X\text{ adic*} \ar@{=>}[r] & X\text{ adic} \ar@{=>}[lld] \\ X\text{ weakly adic} \ar@{=>}[r] & X\text{ classical} \ar@{=>}[r] & X\text{ McQuillan} }\] See discussion in Section 0GXD and for a precise statement see Lemma 0AIK.

Remark

The classical affine formal algebraic spaces correspond to the affine formal schemes considered in EGA ([EGA]). To explain this we assume our base scheme is \(\Spec(\mathbf{Z})\). Let \(\mathfrak X = \text{Spf}(A)\) be an affine formal scheme. Let \(h_\mathfrak X\) be its functor of points as in Lemma 0AI1. Then \(h_\mathfrak X = \colim h_{\Spec(A/I)}\) where the colimit is over the collection of ideals of definition of the admissible topological ring \(A\). This follows from (0AHZ) when evaluating on affine schemes and it suffices to check on affine schemes as both sides are fppf sheaves, see Lemma 0AI2. Thus \(h_\mathfrak X\) is an affine formal algebraic space. In fact, it is a classical affine formal algebraic space by Definition 0AID. Thus Lemma 0AI1 tells us the category of affine formal schemes is equivalent to the category of classical affine formal algebraic spaces.

Having made the connection with affine formal schemes above, it seems natural to make the following definition.

Definition

Let \(S\) be a scheme. Let \(A\) be a weakly admissible topological ring over \(S\), see Definition 0AMV⁶. The formal spectrum of \(A\) is the affine formal algebraic space \[\text{Spf}(A) = \colim \Spec(A/I)\] where the colimit is over the set of weak ideals of definition of \(A\) and taken in the category \(\Sh((\Sch/S)_{fppf})\).

Such a formal spectrum is McQuillan by construction and conversely every McQuillan affine formal algebraic space is isomorphic to a formal spectrum. To be sure, in our theory there exist affine formal algebraic spaces which are not the formal spectrum of any weakly admissible topological ring. Following [Yasuda] we could introduce \(S\)-pro-rings to be pro-objects in the category of \(S\)-algebras, see Categories, Remark 05PX. Then every affine formal algebraic space over \(S\) would be the formal spectrum of such an \(S\)-pro-ring. We will not do this and instead we will work directly with the corresponding affine formal algebraic spaces.

The construction of the formal spectrum is functorial. To explain this let \(\varphi : B \to A\) be a continuous map of weakly admissible topological rings over \(S\). Then \[\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)\] is the unique morphism of affine formal algebraic spaces such that the diagrams \[\xymatrix{ \Spec(B/J) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }\] commute for all weak ideals of definition \(I \subset A\) and \(J \subset B\) with \(\varphi(I) \subset J\). Since continuity of \(\varphi\) implies that for every weak ideal of definition \(J \subset B\) there is a weak ideal of definition \(I \subset A\) with the required property, we see that the required commutativities uniquely determine and define \(\text{Spf}(\varphi)\).

Lemma

Let \(S\) be a scheme. Let \(A\), \(B\) be weakly admissible topological rings over \(S\). Any morphism \(f : \text{Spf}(B) \to \text{Spf}(A)\) of affine formal algebraic spaces over \(S\) is equal to \(\text{Spf}(f^\sharp)\) for a unique continuous \(S\)-algebra map \(f^\sharp : A \to B\).

Proof

Let \(f : \text{Spf}(B) \to \text{Spf}(A)\) be as in the lemma. Let \(J \subset B\) be a weak ideal of definition. By Lemma 0AIA there exists a weak ideal of definition \(I \subset A\) such that \(\Spec(B/J) \to \text{Spf}(B) \to \text{Spf}(A)\) factors through \(\Spec(A/I)\). By Schemes, Lemma 01I1 we obtain an \(S\)-algebra map \(A/I \to B/J\). These maps are compatible for varying \(J\) and define the map \(f^\sharp : A \to B\). This map is continuous because for every weak ideal of definition \(J \subset B\) there is a weak ideal of definition \(I \subset A\) such that \(f^\sharp(I) \subset J\). The equality \(f = \text{Spf}(f^\sharp)\) holds by our choice of the ring maps \(A/I \to B/J\) which make up \(f^\sharp\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a map of presheaves on \((\Sch/S)_{fppf}\). If \(X\) is an affine formal algebraic space and \(f\) is representable by algebraic spaces and locally quasi-finite, then \(f\) is representable (by schemes).

Proof

Let \(T\) be a scheme over \(S\) and \(T \to Y\) a map. We have to show that the algebraic space \(X \times_Y T\) is a scheme. Write \(X = \colim X_\lambda\) as in Definition 0AI7. Let \(W \subset X \times_Y T\) be a quasi-compact open subspace. The restriction of the projection \(X \times_Y T \to X\) to \(W\) factors through \(X_\lambda\) for some \(\lambda\). Then \[W \to X_\lambda \times_S T\] is a monomorphism (hence separated) and locally quasi-finite (because \(W \to X \times_Y T \to T\) is locally quasi-finite by our assumption on \(X \to Y\), see Morphisms of Spaces, Lemma 03XN). Hence \(W\) is a scheme by Morphisms of Spaces, Proposition 03XX. Thus \(X \times_Y T\) is a scheme by Properties of Spaces, Lemma 03JH.

Countably indexed affine formal algebraic spaces

These are the affine formal algebraic spaces as in the following lemma.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). The following are equivalent

1. there exists a system \(X_1 \to X_2 \to X_3 \to \ldots\) of thickenings of affine schemes over \(S\) such that \(X = \colim X_n\),

2. there exists a choice \(X = \colim X_\lambda\) as in Definition 0AI7 such that \(\Lambda\) is countable.

Proof

This follows from the observation that a countable directed set has a cofinal subset isomorphic to \((\mathbf{N}, \geq)\). See proof of Algebra, Lemma 0597.

Definition

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). We say \(X\) is countably indexed if the equivalent conditions of Lemma 0AII are satisfied.

In the language of [BVGD] this is expressed by saying that \(X\) is an \(\aleph_0\)-ind scheme.

Lemma

Let \(X\) be an affine formal algebraic space over a scheme \(S\).

1. If \(X\) is Noetherian, then \(X\) is adic*.

2. If \(X\) is adic*, then \(X\) is adic.

3. If \(X\) is adic, then \(X\) is weakly adic.

4. If \(X\) is weakly adic, then \(X\) is classical.

5. If \(X\) is weakly adic, then \(X\) is countably indexed.

6. If \(X\) is countably indexed, then \(X\) is McQuillan.

Proof

Statements (1), (2), (3), and (4) follow by writing \(X = \text{Spf}(A)\) and where \(A\) is a weakly admissible (hence complete) linearly topologized ring and using the implications between the various types of such rings discussed in Section 0GXD.

Proof of (5). By definition there exists a weakly adic topological ring \(A\) such that \(X = \colim \Spec(A/I)\) where the colimit is over the ideals of definition of \(A\). As \(A\) is weakly adic, there exits in particular a countable fundamental system \(I_\lambda\) of open ideals, see Definition 0GXE. Then \(X = \colim \Spec(A/I_n)\) by definition of \(\text{Spf}(A)\). Thus \(X\) is countably indexed.

Proof of (6). Write \(X = \colim X_n\) for some system \(X_1 \to X_2 \to X_3 \to \ldots\) of thickenings of affine schemes over \(S\). Then \[A = \lim \Gamma(X_n, \mathcal{O}_{X_n})\] surjects onto each \(\Gamma(X_n, \mathcal{O}_{X_n})\) because the transition maps are surjections as the morphisms \(X_n \to X_{n + 1}\) are closed immersions. Hence \(X\) is McQuillan.

Lemma

Let \(S\) be a scheme. Let \(X\) be a presheaf on \((\Sch/S)_{fppf}\). The following are equivalent

1. \(X\) is a countably indexed affine formal algebraic space,

2. \(X = \text{Spf}(A)\) where \(A\) is a weakly admissible topological \(S\)-algebra which has a countable fundamental system of neighbourhoods of \(0\),

3. \(X = \text{Spf}(A)\) where \(A\) is a weakly admissible topological \(S\)-algebra which has a fundamental system \(A \supset I_1 \supset I_2 \supset I_3 \supset \ldots\) of weak ideals of definition,

4. \(X = \text{Spf}(A)\) where \(A\) is a complete topological \(S\)-algebra with a fundamental system of open neighbourhoods of \(0\) given by a countable sequence \(A \supset I_1 \supset I_2 \supset I_3 \supset \ldots\) of ideals such that \(I_n/I_{n + 1}\) is locally nilpotent, and

5. \(X = \text{Spf}(A)\) where \(A = \lim B/J_n\) with the limit topology where \(B \supset J_1 \supset J_2 \supset J_3 \supset \ldots\) is a sequence of ideals in an \(S\)-algebra \(B\) with \(J_n/J_{n + 1}\) locally nilpotent.

Proof

Assume (1). By Lemma 0AIK we can write \(X = \text{Spf}(A)\) where \(A\) is a weakly admissible topological \(S\)-algebra. For any presentation \(X = \colim X_n\) as in Lemma 0AII part (1) we see that \(A = \lim A_n\) with \(X_n = \Spec(A_n)\) and \(A_n = A/I_n\) for some weak ideal of definition \(I_n \subset A\). This follows from the final statement of Lemma 0AIC which moreover implies that \(\{I_n\}\) is a fundamental system of open neighbourhoods of \(0\). Thus we have a sequence \[A \supset I_1 \supset I_2 \supset I_3 \supset \ldots\] of weak ideals of definition with \(A = \lim A/I_n\). In this way we see that condition (1) implies each of the conditions (2) – (5).

Assume (5). First note that the limit topology on \(A = \lim B/J_n\) is a linearly topologized, complete topology, see More on Algebra, Section 07E7. If \(f \in A\) maps to zero in \(B/J_1\), then some power maps to zero in \(B/J_2\) as its image in \(J_1/J_2\) is nilpotent, then a further power maps to zero in \(J_2/J_3\), etc, etc. In this way we see the open ideal \(\Ker(A \to B/J_1)\) is a weak ideal of definition. Thus \(A\) is weakly admissible. In this way we see that (5) implies (2).

It is clear that (4) is a special case of (5) by taking \(B = A\). It is clear that (3) is a special case of (2).

Assume \(A\) is as in (2). Let \(E_n\) be a countable fundamental system of neighbourhoods of \(0\) in \(A\). Since \(A\) is a weakly admissible topological ring we can find open ideals \(I_n \subset E_n\). We can also choose a weak ideal of definition \(J \subset A\). Then \(J \cap I_n\) is a fundamental system of weak ideals of definition of \(A\) and we get \(X = \text{Spf}(A) = \colim \Spec(A/(J \cap I_n))\) which shows that \(X\) is a countably indexed affine formal algebraic space.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space. The following are equivalent

1. \(X\) is Noetherian,

2. \(X\) is adic* and for every closed immersion \(T \to X\) with \(T\) a scheme, \(T\) is Noetherian,

3. \(X\) is adic* and for some choice of \(X = \colim X_\lambda\) as in Definition 0AI7 the schemes \(X_\lambda\) are Noetherian, and

4. \(X\) is weakly adic and for some choice \(X = \colim X_\lambda\) as in Definition 0AI7 the schemes \(X_\lambda\) are Noetherian.

Proof

Assume \(X\) is Noetherian. Then \(X = \text{Spf}(A)\) where \(A\) is a Noetherian adic ring. Let \(T \to X\) be a closed immersion where \(T\) is a scheme. By Lemma 0AIC we see that \(T\) is affine and that \(T \to \Spec(A)\) is a closed immersion. Since \(A\) is Noetherian, we see that \(T\) is Noetherian. In this way we see that (1) \(\Rightarrow\) (2).

The implications (2) \(\Rightarrow\) (3) and (2) \(\Rightarrow\) (4) are immediate (see Lemma 0AIK).

To prove (3) \(\Rightarrow\) (1) write \(X = \text{Spf}(A)\) for some adic ring \(A\) with finitely generated ideal of definition \(I\). We are also given that the rings \(A/I_\lambda\) are Noetherian for some fundamental system of open ideals \(I_\lambda\). Since \(I\) is open, we can find a \(\lambda\) such that \(I_\lambda \subset I\). Then \(A/I\) is Noetherian and we conclude that \(A\) is Noetherian by Algebra, Lemma 05GH.

To prove (4) \(\Rightarrow\) (3) write \(X = \text{Spf}(A)\) for some weakly adic ring \(A\). Then \(A\) is admissible and has an ideal of definition \(I\) and the closure \(I_2\) of \(I^2\) is open, see Lemma 0GXF. We are also given that the rings \(A/I_\lambda\) are Noetherian for some fundamental system of open ideals \(I_\lambda\). Choose a \(\lambda\) such that \(I_\lambda \subset I_2\). Then \(A/I_2\) is Noetherian as a quotient of \(A/I_\lambda\). Hence \(I/I_2\) is a finite \(A\)-module. Hence \(A\) is an adic ring with a finitely generated ideal of definition by Lemma 0GXH. Thus \(X\) is adic* and (3) holds.

Formal algebraic spaces

We take a break from our habit of introducing new concepts first for rings, then for schemes, and then for algebraic spaces, by introducing formal algebraic spaces without first introducing formal schemes. The general idea will be that a formal algebraic space is a sheaf in the fppf topology which étale locally is an affine formal scheme in the sense of [BVGD]. Related material can be found in [Yasuda].

In the definition of a formal algebraic space we are going to borrow some terminology from Bootstrap, Sections 02YP and 046E.

Definition

Let \(S\) be a scheme. We say a sheaf \(X\) on \((\Sch/S)_{fppf}\) is a formal algebraic space if there exist a family of maps \(\{X_i \to X\}_{i \in I}\) of sheaves such that

1. \(X_i\) is an affine formal algebraic space,

2. \(X_i \to X\) is representable by algebraic spaces and étale,

3. \(\coprod X_i \to X\) is surjective as a map of sheaves

and \(X\) satisfies a set theoretic condition (see Remark 0AIS). A morphism of formal algebraic spaces over \(S\) is a map of sheaves.

Discussion. Sanity check: an affine formal algebraic space is a formal algebraic space. In the situation of the definition the morphisms \(X_i \to X\) are representable (by schemes), see Lemma 0AIG. By Bootstrap, Lemma 04S1 we could instead of asking \(\coprod X_i \to X\) to be surjective as a map of sheaves, require that it be surjective (which makes sense because it is representable).

Our notion of a formal algebraic space is very general. In fact, even affine formal algebraic spaces as defined above are very nasty objects.

Lemma

Let \(S\) be a scheme. If \(X\) is a formal algebraic space over \(S\), then the diagonal morphism \(\Delta : X \to X \times_S X\) is representable, a monomorphism, locally quasi-finite, locally of finite type, and separated.

Proof

Suppose given \(U \to X\) and \(V \to X\) with \(U, V\) schemes over \(S\). Then \(U \times_X V\) is a sheaf. Choose \(\{X_i \to X\}\) as in Definition 0AIM. For every \(i\) the morphism \[(U \times_X X_i) \times_{X_i} (V \times_X X_i) = (U \times_X V) \times_X X_i \to U \times_X V\] is representable and étale as a base change of \(X_i \to X\) and its source is a scheme (use Lemmas 0AI8 and 0AIG). These maps are jointly surjective hence \(U \times_X V\) is an algebraic space by Bootstrap, Theorem 04S6. The morphism \(U \times_X V \to U \times_S V\) is a monomorphism. It is also locally quasi-finite, because on precomposing with the morphism displayed above we obtain the composition \[(U \times_X X_i) \times_{X_i} (V \times_X X_i) \to (U \times_X X_i) \times_S (V \times_X X_i) \to U \times_S V\] which is locally quasi-finite as a composition of a closed immersion (Lemma 0AI8) and an étale morphism, see Descent on Spaces, Lemma 06F5. Hence we conclude that \(U \times_X V\) is a scheme by Morphisms of Spaces, Proposition 03XX. Thus \(\Delta\) is representable, see Spaces, Lemma 025W.

In fact, since we’ve shown above that the morphisms of schemes \(U \times_X V \to U \times_S V\) are always monomorphisms and locally quasi-finite we conclude that \(\Delta : X \to X \times_S X\) is a monomorphism and locally quasi-finite, see Spaces, Lemma 0CB7. Then we can use the principle of Spaces, Lemma 02YO to see that \(\Delta\) is separated and locally of finite type. Namely, a monomorphism of schemes is separated (Schemes, Lemma 01L4) and a locally quasi-finite morphism of schemes is locally of finite type (follows from the definition in Morphisms, Section 01TC).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism from an algebraic space over \(S\) to a formal algebraic space over \(S\). Then \(f\) is representable by algebraic spaces.

Proof

Let \(Z \to Y\) be a morphism where \(Z\) is a scheme over \(S\). We have to show that \(X \times_Y Z\) is an algebraic space. Choose a scheme \(U\) and a surjective étale morphism \(U \to X\). Then \(U \times_Y Z \to X \times_Y Z\) is representable surjective étale (Spaces, Lemma 02WL) and \(U \times_Y Z\) is a scheme by Lemma 0AIP. Hence the result by Bootstrap, Theorem 04S6.

Remark

Modulo set theoretic issues the category of formal schemes à la EGA (see Section 0AHY) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is \(\Spec(\mathbf{Z})\). By Lemma 0AI2 the functor of points \(h_\mathfrak X\) associated to a formal scheme \(\mathfrak X\) is a sheaf in the fppf topology. By Lemma 0AI1 the assignment \(\mathfrak X \mapsto h_\mathfrak X\) is a fully faithful embedding of the category of formal schemes into the category of fppf sheaves. Given a formal scheme \(\mathfrak X\) we choose an open covering \(\mathfrak X = \bigcup \mathfrak X_i\) with \(\mathfrak X_i\) affine formal schemes. Then \(h_{\mathfrak X_i}\) is an affine formal algebraic space by Remark 0AIE. The morphisms \(h_{\mathfrak X_i} \to h_\mathfrak X\) are representable and open immersions. Thus \(\{h_{\mathfrak X_i} \to h_\mathfrak X\}\) is a family as in Definition 0AIM and we see that \(h_\mathfrak X\) is a formal algebraic space.

Remark

Let \(S\) be a scheme and let \((\Sch/S)_{fppf}\) be a big fppf site as in Topologies, Definition 021S. As our set theoretic condition on \(X\) in Definitions 0AI7 and 0AIM we take: there exist objects \(U, R\) of \((\Sch/S)_{fppf}\), a morphism \(U \to X\) which is a surjection of fppf sheaves, and a morphism \(R \to U \times_X U\) which is a surjection of fppf sheaves. In other words, we require our sheaf to be a coequalizer of two maps between representable sheaves. Here are some observations which imply this notion behaves reasonably well:

1. Suppose \(X = \colim_{\lambda \in \Lambda} X_\lambda\) and the system satisfies conditions (1) and (2) of Definition 0AI7. Then \(U = \coprod_{\lambda \in \Lambda} X_\lambda \to X\) is a surjection of fppf sheaves. Moreover, \(U \times_X U\) is a closed subscheme of \(U \times_S U\) by Lemma 0AI8. Hence if \(U\) is representable by an object of \((\Sch/S)_{fppf}\) then \(U \times_S U\) is too (see Sets, Lemma 000R) and the set theoretic condition is satisfied. This is always the case if \(\Lambda\) is countable, see Sets, Lemma 000R.

2. Sanity check. Let \(\{X_i \to X\}_{i \in I}\) be as in Definition 0AIM (with the set theoretic condition as formulated above) and assume that each \(X_i\) is actually an affine scheme. Then \(X\) is an algebraic space. Namely, if we choose a larger big fppf site \((\Sch'/S)_{fppf}\) such that \(U' = \coprod X_i\) and \(R' = \coprod X_i \times_X X_j\) are representable by objects in it, then \(X' = U'/R'\) will be an object of the category of algebraic spaces for this choice. Then an application of Spaces, Lemma 04W1 shows that \(X\) is an algebraic space for \((\Sch/S)_{fppf}\).

3. Let \(\{X_i \to X\}_{i \in I}\) be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition 0AIM. For each \(i\) we can pick \(U_i \in \Ob((\Sch/S)_{fppf})\) and \(U_i \to X_i\) which is a surjection of sheaves. Thus if \(I\) is not too large (for example countable) then \(U = \coprod U_i \to X\) is a surjection of sheaves and \(U\) is representable by an object of \((\Sch/S)_{fppf}\). To get \(R \in \Ob((\Sch/S)_{fppf})\) surjecting onto \(U \times_X U\) it suffices to assume the diagonal \(\Delta : X \to X \times_S X\) is not too wild, for example this always works if the diagonal of \(X\) is quasi-compact, i.e., \(X\) is quasi-separated.

The reduction

All formal algebraic spaces have an underlying reduced algebraic space as the following lemma demonstrates.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). There exists a reduced algebraic space \(X_{red}\) and a representable morphism \(X_{red} \to X\) which is a thickening. A morphism \(U \to X\) with \(U\) a reduced algebraic space factors uniquely through \(X_{red}\).

Proof

First assume that \(X\) is an affine formal algebraic space. Say \(X = \colim X_\lambda\) as in Definition 0AI7. Since the transition morphisms are thickenings, the affine schemes \(X_\lambda\) all have isomorphic reductions \(X_{red}\). The morphism \(X_{red} \to X\) is representable and a thickening by Lemma 0AI9 and the fact that compositions of thickenings are thickenings. We omit the verification of the universal property (use Schemes, Definition 01J4, Schemes, Lemma 0356, Properties of Spaces, Definition 047X, and Properties of Spaces, Lemma 03JJ).

Let \(X\) and \(\{X_i \to X\}_{i \in I}\) be as in Definition 0AIM. For each \(i\) let \(X_{i, red} \to X_i\) be the reduction as constructed above. For \(i, j \in I\) the projection \(X_{i, red} \times_X X_j \to X_{i, red}\) is an étale (by assumption) morphism of schemes (by Lemma 0AIG). Hence \(X_{i, red} \times_X X_j\) is reduced (see Descent, Lemma 034E). Thus the projection \(X_{i, red} \times_X X_j \to X_j\) factors through \(X_{j, red}\) by the universal property. We conclude that \[R_{ij} = X_{i, red} \times_X X_j = X_{i, red} \times_X X_{j, red} = X_i \times_X X_{j, red}\] because the morphisms \(X_{i, red} \to X_i\) are injections of sheaves. Set \(U = \coprod X_{i, red}\), set \(R = \coprod R_{ij}\), and denote \(s, t : R \to U\) the two projections. As a sheaf \(R = U \times_X U\) and \(s\) and \(t\) are étale. Then \((t, s) : R \to U\) defines an étale equivalence relation by our observations above. Thus \(X_{red} = U/R\) is an algebraic space by Spaces, Theorem 02WW. By construction the diagram \[\xymatrix{ \coprod X_{i, red} \ar[r] \ar[d] & \coprod X_i \ar[d] \\ X_{red} \ar[r] & X }\] is cartesian. Since the right vertical arrow is étale surjective and the top horizontal arrow is representable and a thickening we conclude that \(X_{red} \to X\) is representable by Bootstrap, Lemma 046J (to verify the assumptions of the lemma use that a surjective étale morphism is surjective, flat, and locally of finite presentation and use that thickenings are separated and locally quasi-finite). Then we can use Spaces, Lemma 03KD to conclude that \(X_{red} \to X\) is a thickening (use that being a thickening is equivalent to being a surjective closed immersion).

Finally, suppose that \(U \to X\) is a morphism with \(U\) a reduced algebraic space over \(S\). Then each \(X_i \times_X U\) is étale over \(U\) and therefore reduced (by our definition of reduced algebraic spaces in Properties of Spaces, Section 03E5). Then \(X_i \times_X U \to X_i\) factors through \(X_{i, red}\). Hence \(U \to X\) factors through \(X_{red}\) because \(\{X_i \times_X U \to U\}\) is an étale covering.

Example

Let \(A\) be a weakly admissible topological ring. In this case we have \[\text{Spf}(A)_{red} = \Spec(A/\mathfrak a)\] where \(\mathfrak a \subset A\) is the ideal of topologically nilpotent elements. Namely, \(\mathfrak a\) is a radical ideal (Lemma 0AMW) which is open because \(A\) is weakly admissible.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\) which is representable by algebraic spaces and smooth (for example étale). Then \(X_{red} = X \times_Y Y_{red}\).

Proof

(The étale case follows directly from the construction of the underlying reduced algebraic space in the proof of Lemma 0AIN.) Assume \(f\) is smooth. Observe that \(X \times_Y Y_{red} \to Y_{red}\) is a smooth morphism of algebraic spaces. Hence \(X \times_Y Y_{red}\) is a reduced algebraic space by Descent on Spaces, Lemma 0GB3. Then the universal property of reduction shows that the canonical morphism \(X_{red} \to X \times_Y Y_{red}\) is an isomorphism.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\) which is representable by algebraic spaces. Then \(f\) is surjective in the sense of Bootstrap, Definition 03XZ if and only if \(f_{red} : X_{red} \to Y_{red}\) is a surjective morphism of algebraic spaces.

Proof

Omitted.

Colimits of algebraic spaces along thickenings

A special type of formal algebraic space is one which can globally be written as a cofiltered colimit of algebraic spaces along thickenings as in the following lemma. We will see later (in Section 0AJD) that any quasi-compact and quasi-separated formal algebraic space is such a global colimit.

Lemma

Let \(S\) be a scheme. Suppose given a directed set \(\Lambda\) and a system of algebraic spaces \((X_\lambda, f_{\lambda \mu})\) over \(\Lambda\) where each \(f_{\lambda \mu} : X_\lambda \to X_\mu\) is a thickening. Then \(X = \colim_{\lambda \in \Lambda} X_\lambda\) is a formal algebraic space over \(S\).

Proof

Since we take the colimit in the category of fppf sheaves, we see that \(X\) is a sheaf. Choose and fix \(\lambda \in \Lambda\). Choose an étale covering \(\{X_{i, \lambda} \to X_\lambda\}\) where \(X_i\) is an affine scheme over \(S\), see Properties of Spaces, Lemma 03FX. For each \(\mu \geq \lambda\) there exists a cartesian diagram \[\xymatrix{ X_{i, \lambda} \ar[r] \ar[d] & X_{i, \mu} \ar[d] \\ X_\lambda \ar[r] & X_\mu }\] with étale vertical arrows, see More on Morphisms of Spaces, Theorem 05ZH (this also uses that a thickening is a surjective closed immersion which satisfies the conditions of the theorem). Moreover, these diagrams are unique up to unique isomorphism and hence \(X_{i, \mu} = X_\mu \times_{X_{\mu'}} X_{i, \mu'}\) for \(\mu' \geq \mu\). The morphisms \(X_{i, \mu} \to X_{i, \mu'}\) is a thickening as a base change of a thickening. Each \(X_{i, \mu}\) is an affine scheme by Limits of Spaces, Proposition 07VT and the fact that \(X_{i, \lambda}\) is affine. Set \(X_i = \colim_{\mu \geq \lambda} X_{i, \mu}\). Then \(X_i\) is an affine formal algebraic space. The morphism \(X_i \to X\) is étale because given an affine scheme \(U\) any \(U \to X\) factors through \(X_\mu\) for some \(\mu \geq \lambda\) (details omitted). In this way we see that \(X\) is a formal algebraic space.

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). How does one prove or check that \(X\) is a global colimit as in Lemma 0AIU? To do this we look for maps \(i : Z \to X\) where \(Z\) is an algebraic space over \(S\) and \(i\) is surjective and a closed immersion, in other words, \(i\) is a thickening. This makes sense as \(i\) is representable by algebraic spaces (Lemma 0AIQ) and we can use Bootstrap, Definition 03XZ as before.

Example

Let \((A, \mathfrak m, \kappa)\) be a valuation ring, which is \((\pi)\)-adically complete for some nonzero \(\pi \in \mathfrak m\). Assume also that \(\mathfrak m\) is not finitely generated. An example is \(A = \mathcal{O}_{\mathbf{C}_p}\) and \(\pi = p\) where \(\mathcal{O}_{\mathbf{C}_p}\) is the ring of integers of the field of \(p\)-adic complex numbers \(\mathbf{C}_p\) (this is the completion of the algebraic closure of \(\mathbf{Q}_p\)). Another example is \[A = \left\{ \sum\nolimits_{\alpha \in \mathbf{Q},\ \alpha \geq 0} a_\alpha t^\alpha \middle| \begin{matrix} a_\alpha \in \kappa \text{ and for all }n\text{ there are only a} \\ \text{finite number of nonzero }a_\alpha \text{ with }\alpha \leq n \end{matrix} \right\}\] and \(\pi = t\). Then \(X = \text{Spf}(A)\) is an affine formal algebraic space and \(\Spec(\kappa) \to X\) is a thickening which corresponds to the weak ideal of definition \(\mathfrak m \subset A\) which is however not an ideal of definition.

Remark

Let \(\mathfrak X\) be a formal scheme in the sense of McQuillan, see Remark 0AI3. An weak ideal of definition for \(\mathfrak X\) is an ideal sheaf \(\mathcal{I} \subset \mathcal{O}_\mathfrak X\) such that for all \(\mathfrak U \subset \mathfrak X\) affine formal open subscheme the ideal \(\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)\) is a weak ideal of definition of the weakly admissible topological ring \(\mathcal{O}_\mathfrak X(\mathfrak U)\). It suffices to check the condition on the members of an affine open covering. There is a one-to-one correspondence \[\{\text{weak ideals of definition for }\mathfrak X\} \leftrightarrow \{\text{thickenings }i : Z \to h_\mathfrak X\text{ as above}\}\] This correspondence associates to \(\mathcal{I}\) the scheme \(Z = (\mathfrak X, \mathcal{O}_\mathfrak X/\mathcal{I})\) together with the obvious morphism to \(\mathfrak X\). A fundamental system of weak ideals of definition is a collection of weak ideals of definition \(\mathcal{I}_\lambda\) such that on every affine open formal subscheme \(\mathfrak U \subset \mathfrak X\) the ideals \[I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X)\] form a fundamental system of weak ideals of definition of the weakly admissible topological ring \(A\). It suffices to check on the members of an affine open covering. We conclude that the formal algebraic space \(h_\mathfrak X\) associated to the McQuillan formal scheme \(\mathfrak X\) is a colimit of schemes as in Lemma 0AIU if and only if there exists a fundamental system of weak ideals of definition for \(\mathfrak X\).

Remark

Let \(\mathfrak X\) be a formal scheme à la EGA. An ideal of definition for \(\mathfrak X\) is an ideal sheaf \(\mathcal{I} \subset \mathcal{O}_\mathfrak X\) such that for all \(\mathfrak U \subset \mathfrak X\) affine formal open subscheme the ideal \(\mathcal{I}(\mathfrak U) \subset \mathcal{O}_\mathfrak X(\mathfrak U)\) is an ideal of definition of the admissible topological ring \(\mathcal{O}_\mathfrak X(\mathfrak U)\). It suffices to check the condition on the members of an affine open covering. We do not get the same correspondence between ideals of definition and thickenings \(Z \to h_\mathfrak X\) as in Remark 0AIV; an example is given in Example 0CB8. A fundamental system of ideals of definition is a collection of ideals of definition \(\mathcal{I}_\lambda\) such that on every affine open formal subscheme \(\mathfrak U \subset \mathfrak X\) the ideals \[I_\lambda = \mathcal{I}_\lambda(\mathfrak U) \subset A = \Gamma(\mathfrak U, \mathcal{O}_\mathfrak X)\] form a fundamental system of ideals of definition of the admissible topological ring \(A\). It suffices to check on the members of an affine open covering. Suppose that \(\mathfrak X\) is quasi-compact and that \(\{\mathcal{I}_\lambda\}_{\lambda \in \Lambda}\) is a fundamental system of weak ideals of definition. If \(A\) is an admissible topological ring then all sufficiently small open ideals are ideals of definition (namely any open ideal contained in an ideal of definition is an ideal of definition). Thus since we only need to check on the finitely many members of an affine open covering we see that \(\mathcal{I}_\lambda\) is an ideal of definition for \(\lambda\) sufficiently large. Using the discussion in Remark 0AIV we conclude that the formal algebraic space \(h_\mathfrak X\) associated to the quasi-compact formal scheme \(\mathfrak X\) à la EGA is a colimit of schemes as in Lemma 0AIU if and only if there exists a fundamental system of ideals of definition for \(\mathfrak X\).

Completion along a closed subset

Our notion of a formal algebraic space is well adapted to taking the completion along a closed subset.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine scheme over \(S\). Let \(T \subset |X|\) be a closed subset. Then the functor \[(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\}\] is a McQuillan affine formal algebraic space.

Proof

Say \(X = \Spec(A)\) and \(T\) corresponds to the radical ideal \(I \subset A\). Let \(U = \Spec(B)\) be an affine scheme over \(S\) and let \(f : U \to X\) be an element of \(F(U)\). Then \(f\) corresponds to a ring map \(\varphi : A \to B\) such that every prime of \(B\) contains \(\varphi(I) B\). Thus every element of \(\varphi(I)\) is nilpotent in \(B\), see Algebra, Lemma 00E0. Setting \(J = \Ker(\varphi)\) we conclude that \(I/J\) is a locally nilpotent ideal in \(A/J\). Equivalently, \(V(J) = V(I) = T\). In other words, the functor of the lemma equals \(\colim \Spec(A/J)\) where the colimit is over the collection of ideals \(J\) with \(V(J) = T\). Thus our functor is an affine formal algebraic space. It is McQuillan (Definition 0AID) because the maps \(A \to A/J\) are surjective and hence \(A^\wedge = \lim A/J \to A/J\) is surjective, see Lemma 0AIC.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset. Then the functor \[(\Sch/S)_{fppf} \longrightarrow \textit{Sets},\quad U \longmapsto \{f : U \to X \mid f(|U|) \subset T\}\] is a formal algebraic space.

Proof

Denote \(F\) the functor. Let \(\{U_i \to U\}\) be an fppf covering. Then \(\coprod |U_i| \to |U|\) is surjective. Since \(X\) is an fppf sheaf, it follows that \(F\) is an fppf sheaf.

Let \(\{g_i : X_i \to X\}\) be an étale covering such that \(X_i\) is affine for all \(i\), see Properties of Spaces, Lemma 03FX. The morphisms \(F \times_X X_i \to F\) are étale (see Spaces, Lemma 02WL) and the map \(\coprod F \times_X X_i \to F\) is a surjection of sheaves. Thus it suffices to prove that \(F \times_X X_i\) is an affine formal algebraic space. A \(U\)-valued point of \(F \times_X X_i\) is a morphism \(U \to X_i\) whose image is contained in the closed subset \(g_i^{-1}(T) \subset |X_i|\). Thus this follows from Lemma 0AIY.

Definition

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset. The formal algebraic space of Lemma 0AIZ is called the completion of \(X\) along \(T\).

In [EGA, Chapter I, Section 10.8] the notation \(X_{/T}\) is used to denote the completion and we will occasionally use this notation as well. Let \(f : X \to X'\) be a morphism of algebraic spaces over a scheme \(S\). Suppose that \(T \subset |X|\) and \(T' \subset |X'|\) are closed subsets such that \(|f|(T) \subset T'\). Then it is clear that \(f\) defines a morphism of formal algebraic spaces \[X_{/T} \longrightarrow X'_{/T'}\] between the completions.

Lemma

Let \(S\) be a scheme. Let \(f : X' \to X\) be a morphism of algebraic spaces over \(S\). Let \(T \subset |X|\) be a closed subset and let \(T' = |f|^{-1}(T) \subset |X'|\). Then \[\xymatrix{ X'_{/T'} \ar[r] \ar[d] & X' \ar[d]^f \\ X_{/T} \ar[r] & X }\] is a cartesian diagram of sheaves. In particular, the morphism \(X'_{/T'} \to X_{/T}\) is representable by algebraic spaces.

Proof

Namely, suppose that \(Y \to X\) is a morphism from a scheme into \(X\) such that \(|Y|\) maps into \(T\). Then \(Y \times_X X' \to X\) is a morphism of algebraic spaces such that \(|Y \times_X X'|\) maps into \(T'\). Hence the functor \(Y \times_{X_{/T}} X'_{/T'}\) is represented by \(Y \times_X X'\) and we see that the lemma holds.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset. The reduction \((X_{/T})_{red}\) of the completion \(X_{/T}\) of \(X\) along \(T\) is the reduced induced closed subspace \(Z\) of \(X\) corresponding to \(T\).

Proof

It follows from Lemma 0AIN, Properties of Spaces, Definition 047X (which uses Properties of Spaces, Lemma 03IQ to construct \(Z\)), and the definition of \(X_{/T}\) that \(Z\) and \((X_{/T})_{red}\) are reduced algebraic spaces characterized the same mapping property: a morphism \(g : Y \to X\) whose source is a reduced algebraic space factors through them if and only if \(|Y|\) maps into \(T \subset |X|\).

Lemma

Let \(S\) be a scheme. Let \(X = \Spec(A)\) be an affine scheme over \(S\). Let \(T \subset X\) be a closed subset. Let \(X_{/T}\) be the formal completion of \(X\) along \(T\).

1. If \(X \setminus T\) is quasi-compact, i.e., \(T\) is constructible, then \(X_{/T}\) is adic*.

2. If \(T = V(I)\) for some finitely generated ideal \(I \subset A\), then \(X_{/T} = \text{Spf}(A^\wedge)\) where \(A^\wedge\) is the \(I\)-adic completion of \(A\).

3. If \(X\) is Noetherian, then \(X_{/T}\) is Noetherian.

Proof

By Algebra, Lemma 00F6 if (1) holds, then we can find an ideal \(I \subset A\) as in (2). If (3) holds then we can find an ideal \(I \subset A\) as in (2). Moreover, completions of Noetherian rings are Noetherian by Algebra, Lemma 0316. All in all we see that it suffices to prove (2).

Proof of (2). Let \(I = (f_1, \ldots, f_r) \subset A\) cut out \(T\). If \(Z = \Spec(B)\) is an affine scheme and \(g : Z \to X\) is a morphism with \(g(Z) \subset T\) (set theoretically), then \(g^\sharp(f_i)\) is nilpotent in \(B\) for each \(i\). Thus \(I^n\) maps to zero in \(B\) for some \(n\). Hence we see that \(X_{/T} = \colim \Spec(A/I^n) = \text{Spf}(A^\wedge)\).

The following lemma is due to Ofer Gabber.

Lemma

Let \(S\) be a scheme. Let \(X = \Spec(A)\) be an affine scheme over \(S\). Let \(T \subset X\) be a closed subscheme.

1. If the formal completion \(X_{/T}\) is countably indexed and there exist countably many \(f_1, f_2, f_3, \ldots \in A\) such that \(T = V(f_1, f_2, f_3, \ldots)\), then \(X_{/T}\) is adic*.

2. The conclusion of (1) is wrong if we omit the assumption that \(T\) can be cut out by countably many functions in \(X\).

Proof

The assumption that \(X_{/T}\) is countably indexed means that there exists a sequence of ideals \[A \supset J_1 \supset J_2 \supset J_3 \supset \ldots\] with \(V(J_n) = T\) such that every ideal \(J \subset A\) with \(V(J) = T\) there exists an \(n\) such that \(J \supset J_n\).

To construct an example for (2) let \(\omega_1\) be the first uncountable ordinal. Let \(k\) be a field and let \(A\) be the \(k\)-algebra generated by \(x_\alpha\), \(\alpha \in \omega_1\) and \(y_{\alpha \beta}\) with \(\alpha \in \beta \in \omega_1\) subject to the relations \(x_\alpha = y_{\alpha \beta} x_\beta\). Let \(T = V(x_\alpha)\). Let \(J_n = (x_\alpha^n)\). If \(J \subset A\) is an ideal such that \(V(J) = T\), then \(x_\alpha^{n_\alpha} \in J\) for some \(n_\alpha \geq 1\). One of the sets \(\{\alpha \mid n_\alpha = n\}\) must be unbounded in \(\omega_1\). Then the relations imply that \(J_n \subset J\).

To see that (2) holds it now suffices to show that \(A^\wedge = \lim A/J_n\) is not a ring complete with respect to a finitely generated ideal. For \(\gamma \in \omega_1\) let \(A_\gamma\) be the quotient of \(A\) by the ideal generated by \(x_\alpha\), \(\alpha \in \gamma\) and \(y_{\alpha \beta}\), \(\alpha \in \gamma\). As \(A/J_1\) is reduced, every topologically nilpotent element \(f\) of \(\lim A/J_n\) is in \(J_1^\wedge = \lim J_1/J_n\). This means \(f\) is an infinite series involving only a countable number of generators. Hence \(f\) dies in \(A_\gamma^\wedge = \lim A_\gamma/J_nA_\gamma\) for some \(\gamma\). Note that \(A^\wedge \to A_\gamma^\wedge\) is continuous and open by Lemma 0AS0. If the topology on \(A^\wedge\) was \(I\)-adic for some finitely generated ideal \(I \subset A^\wedge\), then \(I\) would go to zero in some \(A_\gamma^\wedge\). This would mean that \(A_\gamma^\wedge\) is discrete, which is not the case as there is a surjective continuous and open (by Lemma 0AS0) map \(A_\gamma^\wedge \to k[[t]]\) given by \(x_\alpha \mapsto t\), \(y_{\alpha \beta} \mapsto 1\) for \(\gamma = \alpha\) or \(\gamma \in \alpha\).

Before we prove (1) we first prove the following: If \(I \subset A^\wedge\) is a finitely generated ideal whose closure \(\bar I\) is open, then \(I = \bar I\). Since \(V(J_n^2) = T\) there exists an \(m\) such that \(J_n^2 \supset J_m\). Thus, we may assume that \(J_n^2 \supset J_{n + 1}\) for all \(n\) by passing to a subsequence. Set \(J_n^\wedge = \lim_{k \geq n} J_n/J_k \subset A^\wedge\). Since the closure \(\bar I = \bigcap (I + J_n^\wedge)\) (Lemma 0AMS) is open we see that there exists an \(m\) such that \(I + J_n^\wedge \supset J_m^\wedge\) for all \(n \geq m\). Fix such an \(m\). We have \[J_{n - 1}^\wedge I + J_{n + 1}^\wedge \supset J_{n - 1}^\wedge (I + J_{n + 1}^\wedge) \supset J_{n - 1}^\wedge J_m^\wedge\] for all \(n \geq m + 1\). Namely, the first inclusion is trivial and the second was shown above. Because \(J_{n - 1}J_m \supset J_{n - 1}^2 \supset J_n\) these inclusions show that the image of \(J_n\) in \(A^\wedge\) is contained in the ideal \(J_{n - 1}^\wedge I + J_{n + 1}^\wedge\). Because this ideal is open we conclude that \[J_{n - 1}^\wedge I + J_{n + 1}^\wedge \supset J_n^\wedge.\] Say \(I = (g_1, \ldots, g_t)\). Pick \(f \in J_{m + 1}^\wedge\). Using the last displayed inclusion, valid for all \(n \geq m + 1\), we can write by induction on \(c \geq 0\) \[f = \sum f_{i, c} g_i \mod J_{m + 1+ c}^\wedge\] with \(f_{i, c} \in J_m^\wedge\) and \(f_{i, c} \equiv f_{i, c - 1} \bmod J_{m + c}^\wedge\). It follows that \(IJ_m^\wedge \supset J_{m + 1}^\wedge\). Combined with \(I + J_{m + 1}^\wedge \supset J_m^\wedge\) we conclude that \(I\) is open.

Proof of (1). Assume \(T = V(f_1, f_2, f_3, \ldots)\). Let \(I_m \subset A^\wedge\) be the ideal generated by \(f_1, \ldots, f_m\). We distinguish two cases.

Case I: For some \(m\) the closure of \(I_m\) is open. Then \(I_m\) is open by the result of the previous paragraph. For any \(n\) we have \((J_n)^2 \supset J_{n+1}\) by design, so the closure of \((J_n^\wedge)^2\) contains \(J_{n+1}^\wedge\) and thus is open. Taking \(n\) large, it follows that the closure of the product of any two open ideals in \(A^\wedge\) is open. Let us prove \(I_m^k\) is open for \(k \ge 1\) by induction on \(k\). The case \(k = 1\) is our hypothesis on \(m\) in Case I. For \(k > 1\), suppose \(I_m^{k - 1}\) is open. Then \(I_m^k = I_m^{k - 1} \cdot I_m\) is the product of two open ideals and hence has open closure. But then since \(I_m^k\) is finitely generated it follows that \(I_m^k\) is open by the previous paragraph (applied to \(I = I_m^k\)), so we can continue the induction on \(k\). As each element of \(I_m\) is topologically nilpotent, we conclude that \(I_m\) is an ideal of definition which proves that \(A^\wedge\) is adic with a finitely generated ideal of definition, i.e., \(X_{/T}\) is adic*.

Case II. For all \(m\) the closure \(\bar I_m\) of \(I_m\) is not open. Then the topology on \(A^\wedge/\bar I_m\) is not discrete. This means we can pick \(\phi(m) \geq m\) such that \[\Im(J_{\phi(m)} \to A/(f_1, \ldots, f_m)) \not = \Im(J_{\phi(m) + 1} \to A/(f_1, \ldots, f_m))\] To see this we have used that \(A^\wedge/(\bar I_m + J_n^\wedge) = A/((f_1, \ldots, f_m) + J_n)\). Choose exponents \(e_i > 0\) such that \(f_i^{e_i} \in J_{\phi(m) + 1}\) for \(0 < m < i\). Let \(J = (f_1^{e_1}, f_2^{e_2}, f_3^{e_3}, \ldots)\). Then \(V(J) = T\). We claim that \(J \not \supset J_n\) for all \(n\) which is a contradiction proving Case II does not occur. Namely, the image of \(J\) in \(A/(f_1, \ldots, f_m)\) is contained in the image of \(J_{\phi(m) + 1}\) which is properly contained in the image of \(J_m\).

Fibre products

Obligatory section about fibre products of formal algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(\{X_i \to X\}_{i \in I}\) be a family of maps of sheaves on \((\Sch/S)_{fppf}\). Assume (a) \(X_i\) is a formal algebraic space over \(S\), (b) \(X_i \to X\) is representable by algebraic spaces and étale, and (c) \(\coprod X_i \to X\) is a surjection of sheaves. Then \(X\) is a formal algebraic space over \(S\).

Proof

For each \(i\) pick \(\{X_{ij} \to X_i\}_{j \in J_i}\) as in Definition 0AIM. Then \(\{X_{ij} \to X\}_{i \in I, j \in J_i}\) is a family as in Definition 0AIM for \(X\).

Lemma

Let \(S\) be a scheme. Let \(X, Y\) be formal algebraic spaces over \(S\) and let \(Z\) be a sheaf whose diagonal is representable by algebraic spaces. Let \(X \to Z\) and \(Y \to Z\) be maps of sheaves. Then \(X \times_Z Y\) is a formal algebraic space.

Proof

Choose \(\{X_i \to X\}\) and \(\{Y_j \to Y\}\) as in Definition 0AIM. Then \(\{X_i \times_Z Y_j \to X \times_Z Y\}\) is a family of maps which are representable by algebraic spaces and étale. Thus Lemma 0AJ1 tells us it suffices to show that \(X \times_Z Y\) is a formal algebraic space when \(X\) and \(Y\) are affine formal algebraic spaces.

Assume \(X\) and \(Y\) are affine formal algebraic spaces. Write \(X = \colim X_\lambda\) and \(Y = \colim Y_\mu\) as in Definition 0AI7. Then \(X \times_Z Y = \colim X_\lambda \times_Z Y_\mu\). Each \(X_\lambda \times_Z Y_\mu\) is an algebraic space. For \(\lambda \leq \lambda'\) and \(\mu \leq \mu'\) the morphism \[X_\lambda \times_Z Y_\mu \to X_\lambda \times_Z Y_{\mu'} \to X_{\lambda'} \times_Z Y_{\mu'}\] is a thickening as a composition of base changes of thickenings. Thus we conclude by applying Lemma 0AIU.

Lemma

Let \(S\) be a scheme. The category of formal algebraic spaces over \(S\) has fibre products.

Proof

Special case of Lemma 0AJ2 because formal algebraic spaces have representable diagonals, see Lemma 0AIP.

Lemma

Let \(S\) be a scheme. Let \(X \to Z\) and \(Y \to Z\) be morphisms of formal algebraic spaces over \(S\). Then \((X \times_Z Y)_{red} = (X_{red} \times_{Z_{red}} Y_{red})_{red}\).

Proof

This follows from the universal property of the reduction in Lemma 0AIN.

We have already proved the following lemma (without knowing that fibre products exist).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The diagonal morphism \(\Delta : X \to X \times_Y X\) is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated.

Proof

Let \(T\) be a scheme and let \(T \to X \times_Y X\) be a morphism. Then \[T \times_{(X \times_Y X)} X = T \times_{(X \times_S X)} X\] Hence the result follows immediately from Lemma 0AIP.

Separation axioms for formal algebraic spaces

This section is about “absolute” separation conditions on formal algebraic spaces. We will discuss separation conditions for morphisms of formal algebraic spaces later.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). The following are equivalent

1. the reduction of \(X\) (Lemma 0AIN) is a quasi-separated algebraic space,

2. for \(U \to X\), \(V \to X\) with \(U\), \(V\) quasi-compact schemes the fibre product \(U \times_X V\) is quasi-compact,

3. for \(U \to X\), \(V \to X\) with \(U\), \(V\) affine the fibre product \(U \times_X V\) is quasi-compact.

Proof

Observe that \(U \times_X V\) is a scheme by Lemma 0AIP. Let \(U_{red}, V_{red}, X_{red}\) be the reduction of \(U, V, X\). Then \[U_{red} \times_{X_{red}} V_{red} = U_{red} \times_X V_{red} \to U \times_X V\] is a thickening of schemes. From this the equivalence of (1) and (2) is clear, keeping in mind the analogous lemma for algebraic spaces, see Properties of Spaces, Lemma 0AHR. We omit the proof of the equivalence of (2) and (3).

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). The following are equivalent

1. the reduction of \(X\) (Lemma 0AIN) is a separated algebraic space,

2. for \(U \to X\), \(V \to X\) with \(U\), \(V\) affine the fibre product \(U \times_X V\) is affine and \[\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V)\] is surjective.

Proof

If (2) holds, then \(X_{red}\) is a separated algebraic space by applying Properties of Spaces, Lemma 0AHR to morphisms \(U \to X_{red}\) and \(V \to X_{red}\) with \(U, V\) affine and using that \(U \times_{X_{red}} V = U \times_X V\).

Assume (1). Let \(U \to X\) and \(V \to X\) be as in (2). Observe that \(U \times_X V\) is a scheme by Lemma 0AIP. Let \(U_{red}, V_{red}, X_{red}\) be the reduction of \(U, V, X\). Then \[U_{red} \times_{X_{red}} V_{red} = U_{red} \times_X V_{red} \to U \times_X V\] is a thickening of schemes. It follows that \((U \times_X V)_{red} = (U_{red} \times_{X_{red}} V_{red})_{red}\). In particular, we see that \((U \times_X V)_{red}\) is an affine scheme and that \[\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}((U \times_X V)_{red})\] is surjective, see Properties of Spaces, Lemma 0AHR. Then \(U \times_X V\) is affine by Limits of Spaces, Proposition 07VT. On the other hand, the morphism \(U \times_X V \to U \times V\) of affine schemes is the composition \[U \times_X V = X \times_{(X \times_S X)} (U \times_S V) \to U \times_S V \to U \times V\] The first morphism is a monomorphism and locally of finite type (Lemma 0AIP). The second morphism is an immersion (Schemes, Lemma 01KR). Hence the composition is a monomorphism which is locally of finite type. On the other hand, the composition is integral as the map on underlying reduced affine schemes is a closed immersion by the above and hence universally closed (use Morphisms, Lemma 01WM). Thus the ring map \[\mathcal{O}(U) \otimes_\mathbf{Z} \mathcal{O}(V) \longrightarrow \mathcal{O}(U \times_X V)\] is an epimorphism which is integral of finite type hence finite hence surjective (use Morphisms, Lemma 01WJ and Algebra, Lemma 04VT).

Definition

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). We say

1. \(X\) is quasi-separated if the equivalent conditions of Lemma 0AJ5 are satisfied.

2. \(X\) is separated if the equivalent conditions of Lemma 0AJ6 are satisfied.

The following lemma implies in particular that the completed tensor product of weakly admissible topological rings is a weakly admissible topological ring.

Lemma

Let \(S\) be a scheme. Let \(X \to Z\) and \(Y \to Z\) be morphisms of formal algebraic spaces over \(S\). Assume \(Z\) separated.

1. If \(X\) and \(Y\) are affine formal algebraic spaces, then so is \(X \times_Z Y\).

2. If \(X\) and \(Y\) are McQuillan affine formal algebraic spaces, then so is \(X \times_Z Y\).

3. If \(X\), \(Y\), and \(Z\) are McQuillan affine formal algebraic spaces corresponding to the weakly admissible topological \(S\)-algebras \(A\), \(B\), and \(C\), then \(X \times_Z Y\) corresponds to \(A \widehat{\otimes}_C B\).

Proof

Write \(X = \colim X_\lambda\) and \(Y = \colim Y_\mu\) as in Definition 0AI7. Then \(X \times_Z Y = \colim X_\lambda \times_Z Y_\mu\). Since \(Z\) is separated the fibre products are affine, hence we see that (1) holds. Assume \(X\) and \(Y\) corresponds to the weakly admissible topological \(S\)-algebras \(A\) and \(B\) and \(X_\lambda = \Spec(A/I_\lambda)\) and \(Y_\mu = \Spec(B/J_\mu)\). Then \[X_\lambda \times_Z Y_\mu \to X_\lambda \times Y_\mu \to \Spec(A \otimes B)\] is a closed immersion. Thus one of the conditions of Lemma 0AIC holds and we conclude that \(X \times_Z Y\) is McQuillan. If also \(Z\) is McQuillan corresponding to \(C\), then \[X_\lambda \times_Z Y_\mu = \Spec(A/I_\lambda \otimes_C B/J_\mu)\] hence we see that the weakly admissible topological ring corresponding to \(X \times_Z Y\) is the completed tensor product (see Definition 0AMU).

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). Let \(U \to X\) be a morphism where \(U\) is a separated algebraic space over \(S\). Then \(U \to X\) is separated.

Proof

The statement makes sense because \(U \to X\) is representable by algebraic spaces (Lemma 0AIQ). Let \(T\) be a scheme and \(T \to X\) a morphism. We have to show that \(U \times_X T \to T\) is separated. Since \(U \times_X T \to U \times_S T\) is a monomorphism, it suffices to show that \(U \times_S T \to T\) is separated. As this is the base change of \(U \to S\) this follows. We used in the argument above: Morphisms of Spaces, Lemmas 03KL, 03KQ, 042N, and 04ZI.

Quasi-compact formal algebraic spaces

Here is the characterization of quasi-compact formal algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). The following are equivalent

1. the reduction of \(X\) (Lemma 0AIN) is a quasi-compact algebraic space,

2. we can find \(\{X_i \to X\}_{i \in I}\) as in Definition 0AIM with \(I\) finite,

3. there exists a morphism \(Y \to X\) representable by algebraic spaces which is étale and surjective and where \(Y\) is an affine formal algebraic space.

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). We say \(X\) is quasi-compact if the equivalent conditions of Lemma 0AJ9 are satisfied.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. the induced map \(f_{red} : X_{red} \to Y_{red}\) between reductions (Lemma 0AIN) is a quasi-compact morphism of algebraic spaces,

2. for every quasi-compact scheme \(T\) and morphism \(T \to Y\) the fibre product \(X \times_Y T\) is a quasi-compact formal algebraic space,

3. for every affine scheme \(T\) and morphism \(T \to Y\) the fibre product \(X \times_Y T\) is a quasi-compact formal algebraic space, and

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that each \(X \times_Y Y_j\) is a quasi-compact formal algebraic space.

Proof

Omitted.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). We say \(f\) is quasi-compact if the equivalent conditions of Lemma 0AJB are satisfied.

This agrees with the already existing notion when the morphism is representable by algebraic spaces (and in particular when it is representable).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\) which is representable by algebraic spaces. Then \(f\) is quasi-compact in the sense of Definition 0AJC if and only if \(f\) is quasi-compact in the sense of Bootstrap, Definition 03XZ.

Proof

This is immediate from the definitions and Lemma 0AJB.

Quasi-compact and quasi-separated formal algebraic spaces

The following result is due to Yasuda, see [Yasuda, Proposition 3.32].

Lemma

Let \(S\) be a scheme. Let \(X\) be a quasi-compact and quasi-separated formal algebraic space over \(S\). Then \(X = \colim X_\lambda\) for a system of algebraic spaces \((X_\lambda, f_{\lambda \mu})\) over a directed set \(\Lambda\) where each \(f_{\lambda \mu} : X_\lambda \to X_\mu\) is a thickening.

Proof

By Lemma 0AJ9 we may choose an affine formal algebraic space \(Y\) and a representable surjective étale morphism \(Y \to X\). Write \(Y = \colim Y_\lambda\) as in Definition 0AI7.

Pick \(\lambda \in \Lambda\). Then \(Y_\lambda \times_X Y\) is a scheme by Lemma 0AIG. The reduction (Lemma 0AIN) of \(Y_\lambda \times_X Y\) is equal to the reduction of \(Y_{red} \times_{X_{red}} Y_{red}\) which is quasi-compact as \(X\) is quasi-separated and \(Y_{red}\) is affine. Therefore \(Y_\lambda \times_X Y\) is a quasi-compact scheme. Hence there exists a \(\mu \geq \lambda\) such that \(\text{pr}_2 : Y_\lambda \times_X Y \to Y\) factors through \(Y_\mu\), see Lemma 0AIA. Let \(Z_\lambda\) be the scheme theoretic image of the morphism \(\text{pr}_2 : Y_\lambda \times_X Y \to Y_\mu\). This is independent of the choice of \(\mu\) and we can and will think of \(Z_\lambda \subset Y\) as the scheme theoretic image of the morphism \(\text{pr}_2 : Y_\lambda \times_X Y \to Y\). Observe that \(Z_\lambda\) is also equal to the scheme theoretic image of the morphism \(\text{pr}_1 : Y \times_X Y_\lambda \to Y\) since this is isomorphic to the morphism used to define \(Z_\lambda\). We claim that \(Z_\lambda \times_X Y = Y \times_X Z_\lambda\) as subfunctors of \(Y \times_X Y\). Namely, since \(Y \to X\) is étale we see that \(Z_\lambda \times_X Y\) is the scheme theoretic image of the morphism \[\text{pr}_{13} = \text{pr}_1 \times \text{id}_Y : Y \times_X Y_\lambda \times_X Y \longrightarrow Y \times_X Y\] by Morphisms of Spaces, Lemma 082Z. By the same token, \(Y \times_X Z_\lambda\) is the scheme theoretic image of the morphism \[\text{pr}_{13} = \text{id}_Y \times \text{pr}_2 : Y \times_X Y_\lambda \times_X Y \longrightarrow Y \times_X Y\] The claim follows. Then \(R_\lambda = Z_\lambda \times_X Y = Y \times_X Z_\lambda\) together with the morphism \(R_\lambda \to Z_\lambda \times_S Z_\lambda\) defines an étale equivalence relation. In this way we obtain an algebraic space \(X_\lambda = Z_\lambda/R_\lambda\). By construction the diagram \[\xymatrix{ Z_\lambda \ar[r] \ar[d] & Y \ar[d] \\ X_\lambda \ar[r] & X }\] is cartesian (because \(X\) is the coequalizer of the two projections \(R = Y \times_X Y \to Y\), because \(Z_\lambda \subset Y\) is \(R\)-invariant, and because \(R_\lambda\) is the restriction of \(R\) to \(Z_\lambda\)). Hence \(X_\lambda \to X\) is representable and a closed immersion, see Spaces, Lemma 03I2. On the other hand, since \(Y_\lambda \subset Z_\lambda\) we see that \((X_\lambda)_{red} = X_{red}\), in other words, \(X_\lambda \to X\) is a thickening. Finally, we claim that \[X = \colim X_\lambda\] We have \(Y \times_X X_\lambda = Z_\lambda \supset Y_\lambda\). Every morphism \(T \to X\) where \(T\) is a scheme over \(S\) lifts étale locally to a morphism into \(Y\) which lifts étale locally into a morphism into some \(Y_\lambda\). Hence \(T \to X\) lifts étale locally on \(T\) to a morphism into \(X_\lambda\). This finishes the proof.

Remark

In this remark we translate the statement and proof of Lemma 0AJE into the language of formal schemes à la EGA. Looking at Remark 0AIW we see that the lemma can be translated as follows

• Every quasi-compact and quasi-separated formal scheme has a fundamental system of ideals of definition.

To prove this we first use the induction principle (reformulated for quasi-compact and quasi-separated formal schemes) of Cohomology of Schemes, Lemma 08DR to reduce to the following situation: \(\mathfrak X = \mathfrak U \cup \mathfrak V\) with \(\mathfrak U\), \(\mathfrak V\) open formal subschemes, with \(\mathfrak V\) affine, and the result is true for \(\mathfrak U\), \(\mathfrak V\), and \(\mathfrak U \cap \mathfrak V\). Pick any ideals of definition \(\mathcal{I} \subset \mathcal{O}_\mathfrak U\) and \(\mathcal{J} \subset \mathcal{O}_\mathfrak V\). By our assumption that we have a fundamental system of ideals of definition on \(\mathfrak U\) and \(\mathfrak V\) and because \(\mathfrak U \cap \mathfrak V\) is quasi-compact, we can find ideals of definition \(\mathcal{I}' \subset \mathcal{I}\) and \(\mathcal{J}' \subset \mathcal{J}\) such that \[\mathcal{I}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{J}|_{\mathfrak U \cap \mathfrak V} \quad\text{and}\quad \mathcal{J}'|_{\mathfrak U \cap \mathfrak V} \subset \mathcal{I}|_{\mathfrak U \cap \mathfrak V}\] Let \(U \to U' \to \mathfrak U\) and \(V \to V' \to \mathfrak V\) be the closed immersions determined by the ideals of definition \(\mathcal{I}' \subset \mathcal{I} \subset \mathcal{O}_\mathfrak U\) and \(\mathcal{J}' \subset \mathcal{J} \subset \mathcal{O}_\mathfrak V\). Let \(\mathfrak U \cap V\) denote the open subscheme of \(V\) whose underlying topological space is that of \(\mathfrak U \cap \mathfrak V\). By our choice of \(\mathcal{I}'\) there is a factorization \(\mathfrak U \cap V \to U'\). We define similarly \(U \cap \mathfrak V\) which factors through \(V'\). Then we consider \[Z_U = \text{scheme theoretic image of } U \amalg (\mathfrak U \cap V) \longrightarrow U'\] and \[Z_V = \text{scheme theoretic image of } (U \cap \mathfrak V) \amalg V \longrightarrow V'\] Since taking scheme theoretic images of quasi-compact morphisms commutes with restriction to opens (Morphisms, Lemma 01R8) we see that \(Z_U \cap \mathfrak V = \mathfrak U \cap Z_V\). Thus \(Z_U\) and \(Z_V\) glue to a scheme \(Z\) which comes equipped with a morphism \(Z \to \mathfrak X\). Analogous to the discussion in Remark 0AIV we see that \(Z\) corresponds to a weak ideal of definition \(\mathcal{I}_Z \subset \mathcal{O}_\mathfrak X\). Note that \(Z_U \subset U'\) and that \(Z_V \subset V'\). Thus the collection of all \(\mathcal{I}_Z\) constructed in this manner forms a fundamental system of weak ideals of definition. Hence a subfamily gives a fundamental system of ideals of definition, see Remark 0AIW.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). Then \(X\) is an affine formal algebraic space if and only if its reduction \(X_{red}\) (Lemma 0AIN) is affine.

Proof

By Lemmas 0AJ5 and 0AJ9 and Definitions 0AJ7 and 0AJA we see that \(X\) is quasi-compact and quasi-separated. By Yasuda’s lemma (Lemma 0AJE) we can write \(X = \colim X_\lambda\) as a filtered colimit of thickenings of algebraic spaces. However, each \(X_\lambda\) is affine by Limits of Spaces, Lemma 07VU because \((X_\lambda)_{red} = X_{red}\). Hence \(X\) is an affine formal algebraic space by definition.

Morphisms representable by algebraic spaces

Let \(f : X \to Y\) be a morphism of formal algebraic spaces which is representable by algebraic spaces. For these types of morphisms we have a lot of theory at our disposal, thanks to the work done in the chapters on algebraic spaces.

Lemma

The composition of morphisms representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).

Proof

See Bootstrap, Lemma 03Y1.

Lemma

A base change of a morphism representable by algebraic spaces is representable by algebraic spaces. The same holds for representable (by schemes).

Proof

See Bootstrap, Lemma 03Y0.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of formal algebraic spaces over \(S\). If \(g \circ f : X \to Z\) is representable by algebraic spaces, then \(f : X \to Y\) is representable by algebraic spaces.

Proof

Note that the diagonal of \(Y \to Z\) is representable by Lemma 0AN2. Thus \(X \to Y\) is representable by algebraic spaces by Bootstrap, Lemma 0AMN.

The property of being representable by algebraic spaces is local on the source and the target.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent:

1. the morphism \(f\) is representable by algebraic spaces,

2. there exists a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are formal algebraic spaces, the vertical arrows are representable by algebraic spaces, \(U \to X\) is surjective étale, and \(U \to V\) is representable by algebraic spaces,

3. for any commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are formal algebraic spaces and the vertical arrows are representable by algebraic spaces, the morphism \(U \to V\) is representable by algebraic spaces,

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM such that \(X_{ji} \to Y_j\) is representable by algebraic spaces for each \(j\) and \(i\),

5. there exist a covering \(\{X_i \to X\}\) as in Definition 0AIM and for each \(i\) a factorization \(X_i \to Y_i \to Y\) where \(Y_i\) is an affine formal algebraic space, \(Y_i \to Y\) is representable by algebraic spaces, such that \(X_i \to Y_i\) is representable by algebraic spaces, and

6. add more here.

Proof

It is clear that (1) implies (2) because we can take \(U = X\) and \(V = Y\). Conversely, (2) implies (1) by Bootstrap, Lemma 0AMP applied to \(U \to X \to Y\).

Assume (1) is true and consider a diagram as in (3). Then \(U \to Y\) is representable by algebraic spaces (as the composition \(U \to X \to Y\), see Bootstrap, Lemma 03Y1) and factors through \(V\). Thus \(U \to V\) is representable by algebraic spaces by Lemma 0AQ0.

It is clear that (3) implies (2). Thus now (1) – (3) are equivalent.

Observe that the condition in (4) makes sense as the fibre product \(Y_j \times_Y X\) is a formal algebraic space by Lemma 0AJ3. It is clear that (4) implies (5).

Assume \(X_i \to Y_i \to Y\) as in (5). Then we set \(V = \coprod Y_i\) and \(U = \coprod X_i\) to see that (5) implies (2).

Finally, assume (1) – (3) are true. Thus we can choose any covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) any covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM. Then \(X_{ij} \to Y_j\) is representable by algebraic spaces by (3) and we see that (4) is true. This concludes the proof.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an affine formal algebraic space over \(S\). Let \(f : X \to Y\) be a map of sheaves on \((\Sch/S)_{fppf}\) which is representable by algebraic spaces. Then \(X\) is a formal algebraic space.

Proof

Write \(Y = \colim Y_\lambda\) as in Definition 0AI7. For each \(\lambda\) the fibre product \(X \times_Y Y_\lambda\) is an algebraic space. Hence \(X = \colim X \times_Y Y_\lambda\) is a formal algebraic space by Lemma 0AIU.

Lemma

Let \(S\) be a scheme. Let \(Y\) be a formal algebraic space over \(S\). Let \(f : X \to Y\) be a map of sheaves on \((\Sch/S)_{fppf}\) which is representable by algebraic spaces. Then \(X\) is a formal algebraic space.

Proof

Let \(\{Y_i \to Y\}\) be as in Definition 0AIM. Then \(X \times_Y Y_i \to X\) is a family of morphisms representable by algebraic spaces, étale, and jointly surjective. Thus it suffices to show that \(X \times_Y Y_i\) is a formal algebraic space, see Lemma 0AJ1. This follows from Lemma 0AJH.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then \(f\) is representable (by schemes) and affine.

Proof

We will show that \(f\) is affine; it will then follow that \(f\) is representable and affine by Morphisms of Spaces, Lemma 03WG. Write \(Y = \colim Y_\mu\) and \(X = \colim X_\lambda\) as in Definition 0AI7. Let \(T \to Y\) be a morphism where \(T\) is a scheme over \(S\). We have to show that \(X \times_Y T \to T\) is affine, see Bootstrap, Definition 03XZ. To do this we may assume that \(T\) is affine and we have to prove that \(X \times_Y T\) is affine. In this case \(T \to Y\) factors through \(Y_\mu \to Y\) for some \(\mu\), see Lemma 0AIA. Since \(f\) is quasi-compact we see that \(X \times_Y T\) is quasi-compact (Lemma 0AJB). Hence \(X \times_Y T \to X\) factors through \(X_\lambda\) for some \(\lambda\). Similarly \(X_\lambda \to Y\) factors through \(Y_\mu\) after increasing \(\mu\). Then \(X \times_Y T = X_\lambda \times_{Y_\mu} T\). We conclude as fibre products of affine schemes are affine.

Lemma

Let \(S\) be a scheme. Let \(\varphi : A \to B\) be a continuous map of weakly admissible topological rings over \(S\). The following are equivalent

1. \(\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)\) is representable by algebraic spaces,

2. \(\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)\) is representable (by schemes),

3. \(\varphi\) is taut, see Definition 0AMX.

Proof

Parts (1) and (2) are equivalent by Lemma 0AKN.

Assume the equivalent conditions (1) and (2) hold. If \(I \subset A\) is a weak ideal of definition, then \(\Spec(A/I) \to \text{Spf}(A)\) is representable and a thickening (this is clear from the construction of the formal spectrum but it also follows from Lemma 0AIC). Then \(\Spec(A/I) \times_{\text{Spf}(A)} \text{Spf}(B) \to \text{Spf}(B)\) is representable and a thickening as a base change. Hence by Lemma 0AIC there is a weak ideal of definition \(J(I) \subset B\) such that \(\Spec(A/I) \times_{\text{Spf}(A)} \text{Spf}(B) = \Spec(B/J(I))\) as subfunctors of \(\text{Spf}(B)\). We obtain a cartesian diagram \[\xymatrix{ \Spec(B/J(I)) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }\] By Lemma 0AN3 we see that \(B/J(I) = B \widehat{\otimes}_A A/I\). It follows that \(J(I)\) is the closure of the ideal \(\varphi(I)B\), see Lemma 0AMZ. Since \(\text{Spf}(A) = \colim \Spec(A/I)\) with \(I\) as above, we find that \(\text{Spf}(B) = \colim \Spec(B/J(I))\). Thus the ideals \(J(I)\) form a fundamental system of weak ideals of definition (see Lemma 0AIC). Hence (3) holds.

Assume (3) holds. We are essentially just going to reverse the arguments given in the previous paragraph. Let \(I \subset A\) be a weak ideal of definition. By Lemma 0AN3 we get a cartesian diagram \[\xymatrix{ \text{Spf}(B \widehat{\otimes}_A A/I) \ar[d] \ar[r] & \Spec(A/I) \ar[d] \\ \text{Spf}(B) \ar[r] & \text{Spf}(A) }\] If \(J(I)\) is the closure of \(IB\), then \(J(I)\) is open in \(B\) by tautness of \(\varphi\). Hence if \(J\) is open in \(B\) and \(J \subset J(B)\), then \(B/J \otimes_A A/I = B/(IB + J) = B/J(I)\) because \(J(I) = \bigcap_{J \subset B\text{ open}} (IB + J)\) by Lemma 0AMS. Hence the limit defining the completed tensor product collapses to give \(B \widehat{\otimes}_A A/I = B/J(I)\). Thus \(\text{Spf}(B \widehat{\otimes}_A A/I) = \Spec(B/J(I))\). This proves that \(\text{Spf}(B) \times_{\text{Spf}(A)} \Spec(A/I)\) is representable for every weak ideal of definition \(I \subset A\). Since every morphism \(T \to \text{Spf}(A)\) with \(T\) quasi-compact factors through \(\Spec(A/I)\) for some weak ideal of definition \(I\) (Lemma 0AIA) we conclude that \(\text{Spf}(\varphi)\) is representable, i.e., (2) holds. This finishes the proof.

Lemma

Let \(S\) be a scheme. Let \(Y\) be an affine formal algebraic space. Let \(f : X \to Y\) be a map of sheaves on \((\Sch/S)_{fppf}\) which is representable and affine. Then

1. \(X\) is an affine formal algebraic space,

2. if \(Y\) is countably indexed, then \(X\) is countably indexed,

3. if \(Y\) is countably indexed and classical, then \(X\) is countably indexed and classical,

4. if \(Y\) is weakly adic, then \(X\) is weakly adic,

5. if \(Y\) is adic*, then \(X\) is adic*, and

6. if \(Y\) is Noetherian and \(f\) is (locally) of finite type, then \(X\) is Noetherian.

Proof

Proof of (1). Write \(Y = \colim_{\lambda \in \Lambda} Y_\lambda\) as in Definition 0AI7. Since \(f\) is representable and affine, the fibre products \(X_\lambda = Y_\lambda \times_Y X\) are affine. And \(X = \colim Y_\lambda \times_Y X\). Thus \(X\) is an affine formal algebraic space.

Proof of (2). If \(Y\) is countably indexed, then in the argument above we may assume \(\Lambda\) is countable. Then we immediately see that \(X\) is countably indexed too.

Proof of (3), (4), and (5). In each of these cases the assumptions imply that \(Y\) is a countably indexed affine formal algebraic space (Lemma 0AIK) and hence \(X\) is too by (2). Thus we may write \(X = \text{Spf}(A)\) and \(Y = \text{Spf}(B)\) for some weakly admissible topological \(S\)-algebras \(A\) and \(B\), see Lemma 0AN1. By Lemma 0AN0 the morphism \(f\) corresponds to a continuous \(S\)-algebra homomorphism \(\varphi : B \to A\). We see from Lemma 0AN5 that \(\varphi\) is taut. We conclude that (3) follows from Lemma 0GX8, (4) follows from Lemma 0GXI, and (5) follows from Lemma 0APU.

Proof of (6). Combining (3) with Lemma 0AIK we see that \(X\) is adic*. Thus we can use the criterion of Lemma 0AKM. First, it tells us the affine schemes \(Y_\lambda\) are Noetherian. Then \(X_\lambda \to Y_\lambda\) is of finite type, hence \(X_\lambda\) is Noetherian too (Morphisms, Lemma 01T6). Then the criterion tells us \(X\) is Noetherian and the proof is complete.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces. Then

1. if \(Y\) is countably indexed, then \(X\) is countably indexed,

2. if \(Y\) is countably indexed and classical, then \(X\) is countably indexed and classical,

3. if \(Y\) is weakly adic, then \(X\) is weakly adic,

4. if \(Y\) is adic*, then \(X\) is adic*, and

5. if \(Y\) is Noetherian and \(f\) is (locally) of finite type, then \(X\) is Noetherian.

Proof

Combine Lemmas 0AKN and 0AKP.

Example

Let \(B\) be a weakly admissible topological ring. Let \(B \to A\) be a ring map (no topology). Then we can consider \[A^\wedge = \lim A/JA\] where the limit is over all weak ideals of definition \(J\) of \(B\). Then \(A^\wedge\) (endowed with the limit topology) is a complete linearly topologized ring. The (open) kernel \(I\) of the surjection \(A^\wedge \to A/JA\) is the closure of \(JA^\wedge\), see Lemma 0AMS. By Lemma 0AMW we see that \(I\) consists of topologically nilpotent elements. Thus \(I\) is a weak ideal of definition of \(A^\wedge\) and we conclude \(A^\wedge\) is a weakly admissible topological ring. Thus \(\varphi : B \to A^\wedge\) is taut map of weakly admissible topological rings and \[\text{Spf}(A^\wedge) \longrightarrow \text{Spf}(B)\] is a special case of the phenomenon studied in Lemma 0AN5.

Remark

The discussion in Lemmas 0AN5, 0AKP, and 0AKQ is sharp in the following two senses:

1. If \(A\) and \(B\) are weakly admissible rings and \(\varphi : A \to B\) is a continuous map, then \(\text{Spf}(\varphi) : \text{Spf}(B) \to \text{Spf}(A)\) is in general not representable.

2. If \(f : Y \to X\) is a representable morphism of affine formal algebraic spaces and \(X = \text{Spf}(A)\) is McQuillan, then it does not follow that \(Y\) is McQuillan.

An example for (1) is to take \(A = k\) a field (with discrete topology) and \(B = k[[t]]\) with the \(t\)-adic topology. An example for (2) is given in Examples, Section 0ANY.

The warning above notwithstanding, we do have the following result.

Lemma

Let \(S\) be a scheme. Let \(Y\) be a McQuillan affine formal algebraic space over \(S\), i.e., \(Y = \text{Spf}(B)\) for some weakly admissible topological \(S\)-algebra \(B\). Then there is an equivalence of categories between

1. the category of morphisms \(f : X \to Y\) of affine formal algebraic spaces which are representable by algebraic spaces and étale, and

2. the category of topological \(B\)-algebras of the form \(A^\wedge\) where \(A\) is an étale \(B\)-algebra and \(A^\wedge = \lim A/JA\) with \(J \subset B\) running over the weak ideals of definition of \(B\).

The equivalence is given by sending \(A^\wedge\) to \(X = \text{Spf}(A^\wedge)\). In particular, any \(X\) as in (1) is McQuillan.

Proof

Let \(A\) be an étale \(B\)-algebra. Then \(B/J \to A/JA\) is étale for every open ideal \(J \subset B\). Hence the morphism \(\text{Spf}(A^\wedge) \to Y\) is representable and étale. The functor \(\text{Spf}\) is fully faithful by Lemma 0AN0. To finish the proof we will show in the next paragraph that any \(X \to Y\) as in (1) is in the essential image.

Choose a weak ideal of definition \(J_0 \subset B\). Set \(Y_0 = \Spec(B/J_0)\) and \(X_0 = Y_0 \times_Y X\). Then \(X_0 \to Y_0\) is an étale morphism of affine schemes (see Lemma 0AKN). Say \(X_0 = \Spec(A_0)\). By Algebra, Lemma 04D1 we can find an étale algebra map \(B \to A\) such that \(A_0 \cong A/J_0A\). Consider an ideal of definition \(J \subset J_0\). As above we may write \(\Spec(B/J) \times_Y X = \Spec(\bar A)\) for some étale ring map \(B/J \to \bar A\). Then both \(B/J \to \bar A\) and \(B/J \to A/JA\) are étale ring maps lifting the étale ring map \(B/J_0 \to A_0\). By More on Algebra, Lemma 0ALI there is a unique \(B/J\)-algebra isomorphism \(\varphi_J : A/JA \to \bar A\) lifting the identification modulo \(J_0\). Since the maps \(\varphi_J\) are unique they are compatible for varying \(J\). Thus \[X = \colim \Spec(B/J) \times_Y X = \colim \Spec(A/JA) = \text{Spf}(A)\] and we see that the lemma holds.

Lemma

With notation and assumptions as in Lemma 0AN8 let \(f : X \to Y\) correspond to \(B \to A^\wedge\). The following are equivalent

1. \(f : X \to Y\) is surjective,

2. \(B \to A\) is faithfully flat,

3. for every weak ideal of definition \(J \subset B\) the ring map \(B/J \to A/JA\) is faithfully flat, and

4. for some weak ideal of definition \(J \subset B\) the ring map \(B/J \to A/JA\) is faithfully flat.

Proof

Let \(J \subset B\) be a weak ideal of definition. As every element of \(J\) is topologically nilpotent, we see that every element of \(1 + J\) is a unit. It follows that \(J\) is contained in the Jacobson radical of \(B\) (Algebra, Lemma 0AME). Hence a flat ring map \(B \to A\) is faithfully flat if and only if \(B/J \to A/JA\) is faithfully flat (Algebra, Lemma 00HQ). In this way we see that (2) – (4) are equivalent. If (1) holds, then for every weak ideal of definition \(J \subset B\) the morphism \(\Spec(A/JA) = \Spec(B/J) \times_Y X \to \Spec(B/J)\) is surjective which implies (3). Conversely, assume (3). A morphism \(T \to Y\) with \(T\) quasi-compact factors through \(\Spec(B/J)\) for some ideal of definition \(J\) of \(B\) (Lemma 0AIA). Hence \(X \times_Y T = \Spec(A/JA) \times_{\Spec(B/J)} T \to T\) is surjective as a base change of the surjective morphism \(\Spec(A/JA) \to \Spec(B/J)\). Thus (1) holds.

Types of formal algebraic spaces

In this section we define “locally Noetherian”, “locally adic*”, “locally weakly adic”, “locally countably indexed and classical”, and “locally countably indexed” formal algebraic spaces. The types “locally adic”, “locally classical”, and “locally McQuillan” are missing as we do not know how to prove the analogue of the following lemmas for those cases (it would suffice to prove the analogue of these lemmas for étale coverings between affine formal algebraic spaces).

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then \(X\) is countably indexed if and only if \(Y\) is countably indexed.

Proof

Assume \(X\) is countably indexed. We write \(X = \colim X_n\) as in Lemma 0AII. Write \(Y = \colim Y_\lambda\) as in Definition 0AI7. For every \(n\) we can pick a \(\lambda_n\) such that \(X_n \to Y\) factors through \(Y_{\lambda_n}\), see Lemma 0AIA. On the other hand, for every \(\lambda\) the scheme \(Y_\lambda \times_Y X\) is affine (Lemma 0AKN) and hence \(Y_\lambda \times_Y X \to X\) factors through \(X_n\) for some \(n\) (Lemma 0AIA). Picture \[\xymatrix{ Y_\lambda \times_Y X \ar[r] \ar[d] & X_n \ar[r] \ar[d] & X \ar[d] \\ Y_\lambda \ar@{..>}[r] \ar@/_1pc/[rr] & Y_{\lambda_n} \ar[r] & Y }\] If we can show the dotted arrow exists, then we conclude that \(Y = \colim Y_{\lambda_n}\) and \(Y\) is countably indexed. To do this we pick a \(\mu\) with \(\mu \geq \lambda\) and \(\mu \geq \lambda_n\). Thus both \(Y_\lambda \to Y\) and \(Y_{\lambda_n} \to Y\) factor through \(Y_\mu \to Y\). Say \(Y_\mu = \Spec(B_\mu)\), the closed subscheme \(Y_\lambda\) corresponds to \(J \subset B_\mu\), and the closed subscheme \(Y_{\lambda_n}\) corresponds to \(J' \subset B_\mu\). We are trying to show that \(J' \subset J\). By the diagram above we know \(J'A_\mu \subset JA_\mu\) where \(Y_\mu \times_Y X = \Spec(A_\mu)\). Since \(X \to Y\) is surjective and flat the morphism \(Y_\lambda \times_Y X \to Y_\lambda\) is a faithfully flat morphism of affine schemes, hence \(B_\mu \to A_\mu\) is faithfully flat. Thus \(J' \subset J\) as desired.

Assume \(Y\) is countably indexed. Then \(X\) is countably indexed by Lemma 0AKQ.

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then \(X\) is countably indexed and classical if and only if \(Y\) is countably indexed and classical.

Proof

We have already seen the implication in one direction in Lemma 0AKQ. For the other direction, note that by Lemma 0AKS we may assume both \(X\) and \(Y\) are countably indexed. Thus \(X = \text{Spf}(A)\) and \(Y = \text{Spf}(B)\) for some weakly admissible topological \(S\)-algebras \(A\) and \(B\), see Lemma 0AN1. By Lemma 0AN0 the morphism \(X \to Y\) corresponds to a continuous \(S\)-algebra homomorphism \(\varphi : B \to A\). We see from Lemma 0AN5 that \(\varphi\) is taut. Let \(J \subset B\) be an open ideal and let \(I \subset A\) be the closure of \(JA\). By Lemmas 0AN3 and 0AMZ we see that \(\Spec(B/J) \times_Y X = \Spec(A/I)\). Hence \(B/J \to A/I\) is faithfully flat (since \(X \to Y\) is surjective and flat). This means that \(\varphi : B \to A\) is as in Section 0GXK (with the roles of \(A\) and \(B\) swapped). We conclude that the lemma holds by Lemma 0GXM.

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then \(X\) is weakly adic if and only if \(Y\) is weakly adic.

Proof

The proof is exactly the same as the proof of Lemma 0GXR except that at the end we use Lemma 0GXP.

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, and flat. Then \(X\) is adic* if and only if \(Y\) is adic*.

Proof

The proof is exactly the same as the proof of Lemma 0GXR except that at the end we use Lemma 0GXQ.

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) be a morphism of affine formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and (locally) of finite type. Then \(X\) is Noetherian if and only if \(Y\) is Noetherian.

Proof

Observe that a Noetherian affine formal algebraic space is adic*, see Lemma 0AIK. Thus by Lemma 0AKT we may assume that both \(X\) and \(Y\) are adic*. We will use the criterion of Lemma 0AKM to see that the lemma holds. Namely, write \(Y = \colim Y_n\) as in Lemma 0AII. For each \(n\) set \(X_n = Y_n \times_Y X\). Then \(X_n\) is an affine scheme (Lemma 0AKN) and \(X = \colim X_n\). Each of the morphisms \(X_n \to Y_n\) is faithfully flat and of finite type. Thus the lemma follows from the fact that in this situation \(X_n\) is Noetherian if and only if \(Y_n\) is Noetherian, see Algebra, Lemma 033E (to go down) and Algebra, Lemma 00FN (to go up).

Lemma

Let \(S\) be a scheme. Let \[P \in \left\{ \begin{matrix} countably\ indexed,\\ countably\ indexed\ and\ classical,\\ weakly\ adic,\ adic*,\ Noetherian \end{matrix} \right\}\] Let \(X\) be a formal algebraic space over \(S\). The following are equivalent

1. if \(Y\) is an affine formal algebraic space and \(f : Y \to X\) is representable by algebraic spaces and étale, then \(Y\) has property \(P\),

2. for some \(\{X_i \to X\}_{i \in I}\) as in Definition 0AIM each \(X_i\) has property \(P\).

Proof

It is clear that (1) implies (2). Assume (2) and let \(Y \to X\) be as in (1). Since the fibre products \(X_i \times_X Y\) are formal algebraic spaces (Lemma 0AJ2) we can pick coverings \(\{X_{ij} \to X_i \times_X Y\}\) as in Definition 0AIM. Since \(Y\) is quasi-compact, there exist \((i_1, j_1), \ldots, (i_n, j_n)\) such that \[X_{i_1 j_1} \amalg \ldots \amalg X_{i_n j_n} \longrightarrow Y\] is surjective and étale. Then \(X_{i_kj_k} \to X_{i_k}\) is representable by algebraic spaces and étale hence \(X_{i_kj_k}\) has property \(P\) by Lemma 0AKQ. Then \(X_{i_1 j_1} \amalg \ldots \amalg X_{i_n j_n}\) is an affine formal algebraic space with property \(P\) (small detail omitted on finite disjoint unions of affine formal algebraic spaces). Hence we conclude by applying one of Lemmas 0AKS, 0GXR, 0GXS, 0AKT, and 0AKW.

The previous lemma clears the way for the following definition.

Definition

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). We say \(X\) is locally countably indexed, locally countably indexed and classical, locally weakly adic, locally adic*, or locally Noetherian if the equivalent conditions of Lemma 0AKX hold for the corresponding property.

The formal completion of a locally Noetherian algebraic space along a closed subset is a locally Noetherian formal algebraic space.

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset. Let \(X_{/T}\) be the formal completion of \(X\) along \(T\).

1. If \(X \setminus T \to X\) is quasi-compact, then \(X_{/T}\) is locally adic*.

2. If \(X\) is locally Noetherian, then \(X_{/T}\) is locally Noetherian.

Proof

Choose a surjective étale morphism \(U \to X\) with \(U = \coprod U_i\) a disjoint union of affine schemes, see Properties of Spaces, Lemma 03FX. Let \(T_i \subset U_i\) be the inverse image of \(T\). We have \(X_{/T} \times_X U_i = (U_i)_{/T_i}\) (Lemma 0APV). Hence \(\{(U_i)_{/T_i} \to X_{/T}\}\) is a covering as in Definition 0AIM. Moreover, if \(X \setminus T \to X\) is quasi-compact, so is \(U_i \setminus T_i \to U_i\) and if \(X\) is locally Noetherian, so is \(U_i\). Thus the lemma follows from the affine case which is Lemma 0GBA.

Remark

Suppose \(X = \Spec(A)\) and \(T \subset X\) is the zero locus of a finitely generated ideal \(I \subset A\). Let \(J = \sqrt{I}\) be the radical of \(I\). Then from the definitions we see that \(X_{/T} = \text{Spf}(A^\wedge)\) where \(A^\wedge = \lim A/I^n\) is the \(I\)-adic completion of \(A\). On the other hand, the map \(A^\wedge \to \lim A/J^n\) from the \(I\)-adic completion to the \(J\)-adic completion can fail to be a ring isomorphism. As an example let \[A = \bigcup\nolimits_{n \geq 1} \mathbf{C}[t^{1/n}]\] and \(I = (t)\). Then \(J = \mathfrak m\) is the maximal ideal of the valuation ring \(A\) and \(J^2 = J\). Thus the \(J\)-adic completion of \(A\) is \(\mathbf{C}\) whereas the \(I\)-adic completion is the valuation ring described in Example 0CB8 (but in particular it is easy to see that \(A \subset A^\wedge\)).

Lemma

Let \(S\) be a scheme. Let \(X \to Y\) and \(Z \to Y\) be morphisms of formal algebraic space over \(S\). Then

1. If \(X\) and \(Z\) are locally countably indexed, then \(X \times_Y Z\) is locally countably indexed.

2. If \(X\) and \(Z\) are locally countably indexed and classical, then \(X \times_Y Z\) is locally countably indexed and classical.

3. If \(X\) and \(Z\) are weakly adic, then \(X \times_Y Z\) is weakly adic.

4. If \(X\) and \(Z\) are locally adic*, then \(X \times_Y Z\) is locally adic*.

5. If \(X\) and \(Z\) are locally Noetherian and \(X_{red} \to Y_{red}\) is locally of finite type, then \(X \times_Y Z\) is locally Noetherian.

Proof

Choose a covering \(\{Y_j \to Y\}\) as in Definition 0AIM. For each \(j\) choose a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM. For each \(j\) choose a covering \(\{Z_{jk} \to Y_j \times_Y Z\}\) as in Definition 0AIM. Observe that \(X_{ji} \times_{Y_j} Z_{jk}\) is an affine formal algebraic space by Lemma 0AN3. Hence \[\{X_{ji} \times_{Y_j} Z_{jk} \to X \times_Y Z\}\] is a covering as in Definition 0AIM. Thus it suffices to prove (1), (2), (3), and (4) in case \(X\), \(Y\), and \(Z\) are affine formal algebraic spaces.

Assume \(X\) and \(Z\) are countably indexed. Say \(X = \colim X_n\) and \(Z = \colim Z_m\) as in Lemma 0AII. Write \(Y = \colim_{\lambda \in \Lambda} Y_\lambda\) as in Definition 0AI7. For each \(n\) and \(m\) we can find \(\lambda_{n, m} \in \Lambda\) such that \(X_n \to Y\) and \(Z_m \to Y\) factor through \(Y_{\lambda_{n, m}}\) (for example see Lemma 0AIA). Pick \(\lambda_0 \in \Lambda\). By induction for \(t \geq 1\) pick an element \(\lambda_t \in \Lambda\) such that \(\lambda_t \geq \lambda_{n, m}\) for all \(1 \leq n, m \leq t\) and \(\lambda_t \geq \lambda_{t - 1}\). Set \(Y' = \colim Y_{\lambda_t}\). Then \(Y' \to Y\) is a monomorphism such that \(X \to Y\) and \(Z \to Y\) factor through \(Y'\). Hence we may replace \(Y\) by \(Y'\), i.e., we may assume that \(Y\) is countably indexed.

Assume \(X\), \(Y\), and \(Z\) are countably indexed. By Lemma 0AN1 we can write \(X = \text{Spf}(A)\), \(Y = \text{Spf}(B)\), \(Z = \text{Spf}(C)\) for some weakly admissible topological rings \(A\), \(B\), and \(C\). The morphsms \(X \to Y\) and \(Z \to Y\) are given by continuous ring maps \(B \to A\) and \(B \to C\), see Lemma 0AN0. By Lemma 0AN3 we see that \(X \times_Y Z = \text{Spf}(A \widehat{\otimes}_B C)\) and that \(A \widehat{\otimes}_B C\) is a weakly admissible topological ring. In particular, we see that \(X \times_Y Z\) is countably indexed by Lemma 0GB4 part (3). This proves (1).

Proof of (2). In this case \(X\) and \(Z\) are countably indexed and hence the arguments above show that \(X \times_Y Z\) is the formal spectrum of \(A \widehat{\otimes}_B C\) where \(A\) and \(C\) are admissible. Then \(A \widehat{\otimes}_B C\) is admissible by Lemma 0GB4 part (2).

Proof of (3). As before we conclude that \(X \times_Y Z\) is the formal spectrum of \(A \widehat{\otimes}_B C\) where \(A\) and \(C\) are weakly adic. Then \(A \widehat{\otimes}_B C\) is weakly adic by Lemma 0GXJ.

Proof of (4). Arguing as above, this follows from Lemma 0GB4 part (4).

Proof of (5). To deduce case (5) from Lemma 0GB4 part (5) we need to show the hypotheses match. Namely, with notation as in the first parapgrah of the proof, if \(X_{red} \to Y_{red}\) is locally of finite type, then \((X_{ji})_{red} \to (Y_j)_{red}\) is locally of finite type. This follows from Morphisms of Spaces, Lemma 040Y and the fact that in the commutative diagram \[\xymatrix{ (X_{ji})_{red} \ar[d] \ar[r] & (Y_j)_{red} \ar[d] \\ X_{red} \ar[r] & Y_{red} }\] the vertical morphisms are étale. Namely, we have \((X_{ji})_{red} = X_{ij} \times_X X_{red}\) and \((Y_j)_{red} = Y_j \times_Y Y_{red}\) by Lemma 0GB7. Thus as above we reduce to the case where \(X\), \(Y\), \(Z\) are affine formal algebraic spaces, \(X\), \(Z\) are Noetherian, and \(X_{red} \to Y_{red}\) is of finite type. Next, in the second paragraph of the proof we replaced \(Y\) by \(Y'\) but by construction \(Y_{red} = Y'_{red}\), hence the finite type assumption is preserved by this replacement. Then we see that \(X, Y, Z\) correspond to \(A, B, C\) and \(X \times_Y Z\) to \(A \widehat{\otimes}_B C\) with \(A\), \(C\) Noetherian adic. Finally, taking the reduction corresponds to dividing by the ideal of topologically nilpotent elements (Example 0GB6) hence the fact that \(X_{red} \to Y_{red}\) is of finite type does indeed mean that \(B/\mathfrak b \to A/\mathfrak a\) is of finite type and the proof is complete.

Lemma

Let \(S\) be a scheme. Let \(X\) be a locally Noetherian formal algebraic space over \(S\). Then \(X = \colim X_n\) for a system \(X_1 \to X_2 \to X_3 \to \ldots\) of finite order thickenings of locally Noetherian algebraic spaces over \(S\) where \(X_1 = X_{red}\) and \(X_n\) is the \(n\)th infinitesimal neighbourhood of \(X_1\) in \(X_m\) for all \(m \geq n\).

Proof

We only sketch the proof and omit some of the details. Set \(X_1 = X_{red}\). Define \(X_n \subset X\) as the subfunctor defined by the rule: a morphism \(f : T \to X\) where \(T\) is a scheme factors through \(X_n\) if and only if the \(n\)th power of the ideal sheaf of the closed immersion \(X_1 \times_X T \to T\) is zero. Then \(X_n \subset X\) is a subsheaf as vanishing of quasi-coherent modules can be checked fppf locally. We claim that \(X_n \to X\) is representable by schemes, a closed immersion, and that \(X = \colim X_n\) (as fppf sheaves). To check this we may work étale locally on \(X\). Hence we may assume \(X = \text{Spf}(A)\) is a Noetherian affine formal algebraic space. Then \(X_1 = \Spec(A/\mathfrak a)\) where \(\mathfrak a \subset A\) is the ideal of topologically nilpotent elements of the Noetherian adic topological ring \(A\). Then \(X_n = \Spec(A/\mathfrak a^n)\) and we obtain what we want.

Morphisms and continuous ring maps

In this section we denote \(\textit{WAdm}\) the category of weakly admissible topological rings and continuous ring homomorphisms. We define full subcategories \[\textit{WAdm} \supset \textit{WAdm}^{count} \supset \textit{WAdm}^{cic} \supset \textit{WAdm}^{weakly\ adic} \supset \textit{WAdm}^{adic*} \supset \textit{WAdm}^{Noeth}\] whose objects are

1. \(\textit{WAdm}^{count}\): those weakly admissible topological rings \(A\) which have a countable fundamental system of open ideals,

2. \(\textit{WAdm}^{cic}\): the admissible topological rings \(A\) which have a countable fundamental system of open ideals,

3. \(\textit{WAdm}^{weakly\ adic}\): the weakly adic topological rings (Section 0GXD),

4. \(\textit{WAdm}^{adic*}\): the adic topological rings which have a finitely generated ideal of definition, and

5. \(\textit{WAdm}^{Noeth}\): the adic topological rings which are Noetherian.

Clearly, the formal spectra of these types of rings are the basic building blocks of locally countably indexed, locally countably indexed and classical, locally weakly adic, locally adic*, and locally Noetherian formal algebraic spaces.

We briefly review the relationship between morphisms of countably indexed, affine formal algebraic spaces and morphisms of \(\textit{WAdm}^{count}\). Let \(S\) be a scheme. Let \(X\) and \(Y\) be countably indexed, affine formal algebraic spaces. Write \(X = \text{Spf}(A)\) and \(Y = \text{Spf}(B)\) topological \(S\)-algebras \(A\) and \(B\) in \(\textit{WAdm}^{count}\), see Lemma 0AN1. By Lemma 0AN0 there is a 1-to-1 correspondence between morphisms \(f : X \to Y\) and continuous maps \[\varphi : B \longrightarrow A\] of topological \(S\)-algebras. The relationship is given by \(f \mapsto f^\sharp\) and \(\varphi \mapsto \text{Spf}(\varphi)\).

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of locally countably indexed formal algebraic spaces. Consider a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with \(U\) and \(V\) affine formal algebraic spaces and \(U \to X\) and \(V \to Y\) representable by algebraic spaces and étale. By Definition 0AKY (and hence via Lemma 0AKX) we see that \(U\) and \(V\) are countably indexed affine formal algebraic spaces. By the discussion in the previous paragraph we see that \(U \to V\) is isomorphic to \(\text{Spf}(\varphi)\) for some continuous map \[\varphi : B \longrightarrow A\] of topological \(S\)-algebras in \(\textit{WAdm}^{count}\).

Lemma

Let \(A \in \Ob(\textit{WAdm})\). Let \(A \to A'\) be a ring map (no topology). Let \((A')^\wedge = \lim_{I \subset A\text{ w.i.d}} A'/IA'\) be the object of \(\textit{WAdm}\) constructed in Example 0AN6.

1. If \(A\) is in \(\textit{WAdm}^{count}\), so is \((A')^\wedge\).

2. If \(A\) is in \(\textit{WAdm}^{cic}\), so is \((A')^\wedge\).

3. If \(A\) is in \(\textit{WAdm}^{weakly\ adic}\), so is \((A')^\wedge\).

4. If \(A\) is in \(\textit{WAdm}^{adic*}\), so is \((A')^\wedge\).

5. If \(A\) is in \(\textit{WAdm}^{Noeth}\) and \(A'\) is Noetherian, then \((A')^\wedge\) is in \(\textit{WAdm}^{Noeth}\).

Proof

Recall that \(A \to (A')^\wedge\) is taut, see discussion in Example 0AN6. Hence statements (1), (2), (3), and (4) follow from Lemmas 0GX6, 0GX8, 0GXI, and 0APU. Finally, assume that \(A\) is Noetherian and adic. By (4) we know that \((A')^\wedge\) is adic. By Algebra, Lemma 0316 we see that \((A')^\wedge\) is Noetherian. Hence (5) holds.

Situation

Let \(P\) be a property of morphisms of \(\textit{WAdm}^{count}\). Consider commutative diagrams [0ANC]\[\begin{equation} \vcenter{ \xymatrix{ A \ar[r] & (A')^\wedge \\ B \ar[r] \ar[u]^\varphi & (B')^\wedge \ar[u]_{\varphi'} } } \end{equation}\] satisfying the following conditions

1. \(A\) and \(B\) are objects of \(\textit{WAdm}^{count}\),

2. \(A \to A'\) and \(B \to B'\) are étale ring maps,

3. \((A')^\wedge = \lim A'/IA'\), resp. \((B')^\wedge = \lim B'/JB'\) where \(I \subset A\), resp. \(J \subset B\) runs through the weakly admissible ideals of definition of \(A\), resp. \(B\),

4. \(\varphi : B \to A\) and \(\varphi' : (B')^\wedge \to (A')^\wedge\) are continuous.

By Lemma 0ANB the topological rings \((A')^\wedge\) and \((B')^\wedge\) are objects of \(\textit{WAdm}^{count}\). We say \(P\) is a local property if the following axioms hold:

1. for any diagram (0ANC) we have \(P(\varphi) \Rightarrow P(\varphi')\),

2. for any diagram (0ANC) with \(A \to A'\) faithfully flat we have \(P(\varphi') \Rightarrow P(\varphi)\),

3. if \(P(B \to A_i)\) for \(i = 1, \ldots, n\), then \(P(B \to \prod_{i = 1, \ldots, n} A_i)\).

Axiom (0ANF) makes sense as \(\textit{WAdm}^{count}\) has finite products.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of locally countably indexed formal algebraic spaces over \(S\). Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\). The following are equivalent

1. for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with \(U\) and \(V\) affine formal algebraic spaces, \(U \to X\) and \(V \to Y\) representable by algebraic spaces and étale, the morphism \(U \to V\) corresponds to a morphism of \(\textit{WAdm}^{count}\) with property \(P\),

2. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM such that each \(X_{ji} \to Y_j\) corresponds to a morphism of \(\textit{WAdm}^{count}\) with property \(P\), and

3. there exist a covering \(\{X_i \to X\}\) as in Definition 0AIM and for each \(i\) a factorization \(X_i \to Y_i \to Y\) where \(Y_i\) is an affine formal algebraic space, \(Y_i \to Y\) is representable by algebraic spaces and étale, and \(X_i \to Y_i\) corresponds to a morphism of \(\textit{WAdm}^{count}\) with property \(P\).

Proof

It is clear that (1) implies (2) and that (2) implies (3). Assume \(\{X_i \to X\}\) and \(X_i \to Y_i \to Y\) as in (3) and let a diagram as in (1) be given. Since \(Y_i \times_Y V\) is a formal algebraic space (Lemma 0AJ2) we may pick coverings \(\{Y_{ij} \to Y_i \times_Y V\}\) as in Definition 0AIM. For each \((i, j)\) we may similarly choose coverings \(\{X_{ijk} \to Y_{ij} \times_{Y_i} X_i \times_X U\}\) as in Definition 0AIM. Since \(U\) is quasi-compact we can choose \((i_1, j_1, k_1), \ldots, (i_n, j_n, k_n)\) such that \[X_{i_1 j_1 k_1} \amalg \ldots \amalg X_{i_n j_n k_n} \longrightarrow U\] is surjective. For \(s = 1, \ldots, n\) consider the commutative diagram \[\xymatrix{ & & & X_{i_s j_s k_s} \ar[ld] \ar[d] \ar[rd] \\ X \ar[d] & X_{i_s} \ar[l] \ar[d] & X_{i_s} \times_X U \ar[l] \ar[d] & Y_{i_s j_s} \ar[ld] \ar[rd] & X_{i_s} \times_X U \ar[d] \ar[r] & U \ar[d] \ar[r] & X \ar[d] \\ Y & Y_{i_s} \ar[l] & Y_{i_s} \times_Y V \ar[l] & & Y_{i_s} \times_Y V \ar[r] & V \ar[r] & Y }\] Let us say that \(P\) holds for a morphism of countably indexed affine formal algebraic spaces if it holds for the corresponding morphism of \(\textit{WAdm}^{count}\). Observe that the maps \(X_{i_s j_s k_s} \to X_{i_s}\), \(Y_{i_s j_s} \to Y_{i_s}\) are given by completions of étale ring maps, see Lemma 0AN8. Hence we see that \(P(X_{i_s} \to Y_{i_s})\) implies \(P(X_{i_s j_s k_s} \to Y_{i_s j_s})\) by axiom (0AND). Observe that the maps \(Y_{i_s j_s} \to V\) are given by completions of étale rings maps (same lemma as before). By axiom (0ANE) applied to the diagram \[\xymatrix{ X_{i_s j_s k_s} \ar@{=}[r] \ar[d] & X_{i_s j_s k_s} \ar[d] \\ Y_{i_s j_s} \ar[r] & V }\] (this is permissible as identities are faithfully flat ring maps) we conclude that \(P(X_{i_s j_s k_s} \to V)\) holds. By axiom (0ANF) we find that \(P(\coprod_{s = 1, \ldots, n} X_{i_s j_s k_s} \to V)\) holds. Since the morphism \(\coprod X_{i_s j_s k_s} \to U\) is surjective by construction, the corresponding morphism of \(\textit{WAdm}^{count}\) is the completion of a faithfully flat étale ring map, see Lemma 0AN9. One more application of axiom (0ANE) (with \(B' = B\)) implies that \(P(U \to V)\) is true as desired.

Remark

Let \(P\) be a property of morphisms of \(\textit{WAdm}^{adic*}\). We say \(P\) is a local property if axioms (0AND), (0ANE), (0ANF) of Situation 0CBA hold for morphisms of \(\textit{WAdm}^{adic*}\). In exactly the same way we obtain a variant of Lemma 0ANG for morphisms between locally adic* formal algebraic spaces over \(S\).

Remark

Let \(P\) be a property of morphisms of \(\textit{WAdm}^{Noeth}\). We say \(P\) is a local property if axioms (0AND), (0ANE), (0ANF), of Situation 0CBA hold for morphisms of \(\textit{WAdm}^{Noeth}\). In exactly the same way we obtain a variant of Lemma 0ANG for morphisms between locally Noetherian formal algebraic spaces over \(S\).

Situation

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\), see Situation 0CBA. We say \(P\) is stable under base change if given \(B \to A\) and \(B \to C\) in \(\textit{WAdm}^{count}\) we have \(P(B \to A) \Rightarrow P(C \to A \widehat{\otimes}_B C)\). This makes sense as \(A \widehat{\otimes}_B C\) is an object of \(\textit{WAdm}^{count}\) by Lemma 0GB4.

Lemma

Let \(S\) be a scheme. Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\) which is stable under base change. Let \(f : X \to Y\) and \(g : Z \to Y\) be morphisms of locally countably indexed formal algebraic spaces over \(S\). If \(f\) satisfies the equivalent conditions of Lemma 0ANG then so does \(\text{pr}_2 : X \times_Y Z \to Z\).

Proof

Choose a covering \(\{Y_j \to Y\}\) as in Definition 0AIM. For each \(j\) choose a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM. For each \(j\) choose a covering \(\{Z_{jk} \to Y_j \times_Y Z\}\) as in Definition 0AIM. Observe that \(X_{ji} \times_{Y_j} Z_{jk}\) is an affine formal algebraic space which is countably indexed, see Lemma 0GBB. Then we see that \[\{X_{ji} \times_{Y_j} Z_{jk} \to X \times_Y Z\}\] is a covering as in Definition 0AIM. Moreover, the morphisms \(X_{ji} \times_{Y_j} Z_{jk} \to Z\) factor through \(Z_{jk}\). By assumption we know that \(X_{ji} \to Y_j\) corresponds to a morphism \(B_j \to A_{ji}\) of \(\text{WAdm}^{count}\) having property \(P\). The morphisms \(Z_{jk} \to Y_j\) correspond to morphisms \(B_j \to C_{jk}\) in \(\text{WAdm}^{count}\). Since \(X_{ji} \times_{Y_j} Z_{jk} = \text{Spf}(A_{ji} \widehat{\otimes}_{B_j} C_{jk})\) by Lemma 0AN3 we see that it suffices to show that \(C_{jk} \to A_{ji} \widehat{\otimes}_{B_j} C_{jk}\) has property \(P\) which is exactly what the condition that \(P\) is stable under base change guarantees.

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{adic*}\), see Remark 0ANH. We say \(P\) is stable under base change if given \(B \to A\) and \(B \to C\) in \(\textit{WAdm}^{adic*}\) we have \(P(B \to A) \Rightarrow P(C \to A \widehat{\otimes}_B C)\). This makes sense as \(A \widehat{\otimes}_B C\) is an object of \(\textit{WAdm}^{adic*}\) by Lemma 0GB4. In exactly the same way we obtain a variant of Lemma 0GBD for morphisms between locally adic* formal algebraic spaces over \(S\).

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{Noeth}\), see Remark 0ANI. We say \(P\) is stable under base change if given \(B \to A\) and \(B \to C\) in \(\textit{WAdm}^{Noeth}\) the property \(P(B \to A)\) implies both that \(A \widehat{\otimes}_B C\) is adic Noetherian⁷ and that \(P(C \to A \widehat{\otimes}_B C)\). In exactly the same way we obtain a variant of Lemma 0GBD for morphisms between locally Noetherian formal algebraic spaces over \(S\).

Remark

Let \(P\) and \(Q\) be local properties of morphisms of \(\textit{WAdm}^{Noeth}\), see Remark 0ANI. We say \(P\) is stable under base change by \(Q\) if given \(B \to A\) and \(B \to C\) in \(\textit{WAdm}^{Noeth}\) satisfying \(P(B \to A)\) and \(Q(B \to C)\), then \(A \widehat{\otimes}_B C\) is adic Noetherian and \(P(C \to A \widehat{\otimes}_B C)\) holds. Arguing exactly as in the proof of Lemma 0GBD we obtain the following statement: given morphisms \(f : X \to Y\) and \(g : Y \to Z\) of locally Noetherian formal algebraic spaces over \(S\) such that

1. the equivalent conditions of Lemma 0ANG hold for \(f\) and \(P\),

2. the equivalent conditions of Lemma 0ANG hold for \(g\) and \(Q\),

then the equivalent conditions of Lemma 0ANG hold for \(\text{pr}_2 : X \times_Y Z \to Z\) and \(P\).

Situation

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\), see Situation 0CBA. We say \(P\) is stable under composition if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{count}\) we have \(P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)\).

Lemma

Let \(S\) be a scheme. Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\) which is stable under composition. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of locally countably indexed formal algebraic spaces over \(S\). If \(f\) and \(g\) satisfies the equivalent conditions of Lemma 0ANG then so does \(g \circ f : X \to Z\).

Proof

Choose a covering \(\{Z_k \to Z\}\) as in Definition 0AIM. For each \(k\) choose a covering \(\{Y_{kj} \to Z_k \times_Z Y\}\) as in Definition 0AIM. For each \(k\) and \(j\) choose a covering \(\{X_{kji} \to Y_{kj} \times_Y X\}\) as in Definition 0AIM. If \(f\) and \(g\) satisfies the equivalent conditions of Lemma 0ANG then \(X_{kji} \to Y_{jk}\) and \(Y_{jk} \to Z_k\) correspond to arrows \(B_{kj} \to A_{kji}\) and \(C_k \to B_{kj}\) of \(\text{WAdm}^{count}\) having property \(P\). Hence the compositions do too and we conclude.

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{adic*}\), see Remark 0ANH. We say \(P\) is stable under composition if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{adic*}\) we have \(P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)\). In exactly the same way we obtain a variant of Lemma 0GBI for morphisms between locally adic* formal algebraic spaces over \(S\).

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{Noeth}\), see Remark 0ANI. We say \(P\) is stable under composition if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{Noeth}\) we have \(P(B \to A) \wedge P(C \to B) \Rightarrow P(C \to A)\). In exactly the same way we obtain a variant of Lemma 0GBI for morphisms between locally Noetherian formal algebraic spaces over \(S\).

Situation

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\), see Situation 0CBA. We say \(P\) has the cancellation property if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{count}\) we have \(P(C \to B) \wedge P(C \to A) \Rightarrow P(B \to A)\).

Lemma

Let \(S\) be a scheme. Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{count}\) which has the cancellation property. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of locally countably indexed formal algebraic spaces over \(S\). If \(g \circ f\) and \(g\) satisfies the equivalent conditions of Lemma 0ANG then so does \(f : X \to Y\).

Proof

Choose a covering \(\{Z_k \to Z\}\) as in Definition 0AIM. For each \(k\) choose a covering \(\{Y_{kj} \to Z_k \times_Z Y\}\) as in Definition 0AIM. For each \(k\) and \(j\) choose a covering \(\{X_{kji} \to Y_{kj} \times_Y X\}\) as in Definition 0AIM. Let \(X_{kji} \to Y_{jk}\) and \(Y_{jk} \to Z_k\) correspond to arrows \(B_{kj} \to A_{kji}\) and \(C_k \to B_{kj}\) of \(\text{WAdm}^{count}\). If \(g \circ f\) and \(g\) satisfies the equivalent conditions of Lemma 0ANG then \(C_k \to B_{kj}\) and \(C_k \to A_{kji}\) satisfy \(P\). Hence \(B_{kj} \to A_{kji}\) does too and we conclude.

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{adic*}\), see Remark 0ANH. We say \(P\) has the cancellation property if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{adic*}\) we have \(P(C \to A) \wedge P(C \to B) \Rightarrow P(B \to A)\). In exactly the same way we obtain a variant of Lemma 0GBI for morphisms between locally adic* formal algebraic spaces over \(S\).

Remark

Let \(P\) be a local property of morphisms of \(\textit{WAdm}^{Noeth}\), see Remark 0ANI. We say \(P\) has the cancellation property if given \(B \to A\) and \(C \to B\) in \(\textit{WAdm}^{Noeth}\) we have \(P(C \to B) \wedge P(C \to A) \Rightarrow P(C \to B)\). In exactly the same way we obtain a variant of Lemma 0GBI for morphisms between locally Noetherian formal algebraic spaces over \(S\).

Taut ring maps and representability by algebraic spaces

In this section we briefly show that morphisms between locally countably index formal algebraic spaces correspond étale locally to taut continuous ring homomorphisms between weakly admissible topological rings having countable fundamental systems of open ideals. In fact, this is rather clear from Lemma 0AN5 and we encourage the reader to skip this section.

Lemma

Let \(B \to A\) be an arrow of \(\textit{WAdm}^{count}\). The following are equivalent

1. \(B \to A\) is taut (Definition 0AMX),

2. for \(B \supset J_1 \supset J_2 \supset J_3 \supset \ldots\) a fundamental system of weak ideals of definitions there exist a commutative diagram \[\xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] }\] such that \(A_{n + 1}/J_nA_{n + 1} = A_n\) and \(A = \lim A_n\) as topological ring.

Moreover, these equivalent conditions define a local property, i.e., they satisfy axioms (0AND), (0ANE), (0ANF).

Proof

The equivalence of (a) and (b) is immediate. Below we will give an algebraic proof of the axioms, but it turns out we’ve already proven them. Namely, using Lemma 0AN5 the equivalent conditions (a) and (b) translate to saying the corresponding morphism of affine formal algebraic spaces is representable by algebraic spaces. Since this condition is “étale local on the source and target” by Lemma 0AN4 we immediately get axioms (0AND), (0ANE), and (0ANF).

Direct algebraic proof of (0AND), (0ANE), (0ANF). Let a diagram (0ANC) as in Situation 0CBA be given. By Example 0AN6 the maps \(A \to (A')^\wedge\) and \(B \to (B')^\wedge\) satisfy (a) and (b).

Assume (a) and (b) hold for \(\varphi\). Let \(J \subset B\) be a weak ideal of definition. Then the closure of \(JA\), resp. \(J(B')^\wedge\) is a weak ideal of definition \(I \subset A\), resp. \(J' \subset (B')^\wedge\). Then the closure of \(I(A')^\wedge\) is a weak ideal of definition \(I' \subset (A')^\wedge\). A topological argument shows that \(I'\) is also the closure of \(J(A')^\wedge\) and of \(J'(A')^\wedge\). Finally, as \(J\) runs over a fundamental system of weak ideals of definition of \(B\) so do the ideals \(I\) and \(I'\) in \(A\) and \((A')^\wedge\). It follows that (a) holds for \(\varphi'\). This proves (0AND).

Assume \(A \to A'\) is faithfully flat and that (a) and (b) hold for \(\varphi'\). Let \(J \subset B\) be a weak ideal of definition. Using (a) and (b) for the maps \(B \to (B')^\wedge \to (A')^\wedge\) we find that the closure \(I'\) of \(J(A')^\wedge\) is a weak ideal of definition. In particular, \(I'\) is open and hence the inverse image of \(I'\) in \(A\) is open. Now we have (explanation below) \[\begin{align*} A \cap I' & = A \cap \bigcap (J(A')^\wedge + \Ker((A')^\wedge \to A'/I_0A')) \\ & = A \cap \bigcap \Ker((A')^\wedge \to A'/JA' + I_0 A') \\ & = \bigcap (JA + I_0) \end{align*}\] which is the closure of \(JA\) by Lemma 0AMS. The intersections are over weak ideals of definition \(I_0 \subset A\). The first equality because a fundamental system of neighbourhoods of \(0\) in \((A')^\wedge\) are the kernels of the maps \((A')^\wedge \to A'/I_0A'\). The second equality is trivial. The third equality because \(A \to A'\) is faithfully flat, see Algebra, Lemma 05CK. Thus the closure of \(JA\) is open. By Lemma 0AMW the closure of \(JA\) is a weak ideal of definition of \(A\). Finally, given a weak ideal of definition \(I \subset A\) we can find \(J\) such that \(J(A')^\wedge\) is contained in the closure of \(I(A')^\wedge\) by property (a) for \(B \to (B')^\wedge\) and \(\varphi'\). Thus we see that (a) holds for \(\varphi\). This proves (0ANE).

We omit the proof of (0ANF).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of locally countably indexed formal algebraic spaces over \(S\). The following are equivalent

1. for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with \(U\) and \(V\) affine formal algebraic spaces, \(U \to X\) and \(V \to Y\) representable by algebraic spaces and étale, the morphism \(U \to V\) corresponds to a taut map \(B \to A\) of \(\textit{WAdm}^{count}\),

2. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM such that each \(X_{ji} \to Y_j\) corresponds to a taut ring map in \(\textit{WAdm}^{count}\),

3. there exist a covering \(\{X_i \to X\}\) as in Definition 0AIM and for each \(i\) a factorization \(X_i \to Y_i \to Y\) where \(Y_i\) is an affine formal algebraic space, \(Y_i \to Y\) is representable by algebraic spaces and étale, and \(X_i \to Y_i\) corresponds to a taut ring map in \(\textit{WAdm}^{count}\), and

4. \(f\) is representable by algebraic spaces.

Proof

The property of a map in \(\textit{WAdm}^{count}\) being “taut” is a local property by Lemma 0ANJ. Thus Lemma 0ANG exactly tells us that (1), (2), and (3) are equivalent. On the other hand, by Lemma 0AN5 being “taut” on maps in \(\textit{WAdm}^{count}\) corresponds exactly to being “representable by algebraic spaces” for the corresponding morphisms of countably indexed affine formal algebraic spaces. Thus the implication (1) \(\Rightarrow\) (2) of Lemma 0AN4 shows that (4) implies (1) of the current lemma. Similarly, the implication (4) \(\Rightarrow\) (1) of Lemma 0AN4 shows that (2) implies (4) of the current lemma.

Adic morphisms

This section matches the occasionally used notion of an “adic morphism” \(f : X \to Y\) of locally adic* formal algebraic spaces \(X\) and \(Y\) on the one hand with representability of \(f\) by algebraic spaces and on the other hand with our notion of taut continuous ring homomorphisms. First we recall that tautness is equivalent to adicness for adic rings with finitely generated ideal of definition.

Lemma

Let \(A\) and \(B\) be pre-adic topological rings. Let \(\varphi : A \to B\) be a continuous ring homomorphism.

1. If \(\varphi\) is adic, then \(\varphi\) is taut.

2. If \(B\) is complete, \(A\) has a finitely generated ideal of definition, and \(\varphi\) is taut, then \(\varphi\) is adic.

In particular the conditions “\(\varphi\) is adic” and “\(\varphi\) is taut” are equivalent on the category \(\textit{WAdm}^{adic*}\).

Proof

Part (1) is Lemma 0GXC. Part (2) is Lemma 0APU. The final statement is a consequence of (1) and (2).

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of locally adic* formal algebraic spaces over \(S\). By Lemma 0ANK the following are equivalent

1. \(f\) is representable by algebraic spaces (in other words, the equivalent conditions of Lemma 0AN4 hold),

2. for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with \(U\) and \(V\) affine formal algebraic spaces, \(U \to X\) and \(V \to Y\) representable by algebraic spaces and étale, the morphism \(U \to V\) corresponds to an adic⁸ map in \(\textit{WAdm}^{adic*}\).

In this situation we will say that \(f\) is an adic morphism (the formal definition is below). This notion/terminology will only be defined/used for morphisms between formal algebraic spaces which are locally adic* since otherwise we don’t have the equivalence between (1) and (2) above.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). Assume \(X\) and \(Y\) are locally adic*. We say \(f\) is an adic morphism if \(f\) is representable by algebraic spaces. See discussion above.

Morphisms of finite type

Due to how things are setup in the Stacks project, the following is really the correct thing to do and stronger notions should have a different name.

Definition

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of formal algebraic spaces over \(S\).

1. We say \(f\) is locally of finite type if \(f\) is representable by algebraic spaces and is locally of finite type in the sense of Bootstrap, Definition 03XZ.

2. We say \(f\) is of finite type if \(f\) is locally of finite type and quasi-compact (Definition 0AJC).

We will discuss the relationship between finite type morphisms of certain formal algebraic spaces and continuous ring maps \(A \to B\) which are topologically of finite type in Section 0ALL.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. \(f\) is of finite type,

2. \(f\) is representable by algebraic spaces and is of finite type in the sense of Bootstrap, Definition 03XZ.

Proof

This follows from Bootstrap, Lemma 0AM1, the implication “quasi-compact \(+\) locally of finite type \(\Rightarrow\) finite type” for morphisms of algebraic spaces, and Lemma 0AM2.

Lemma

The composition of finite type morphisms is of finite type. The same holds for locally of finite type.

Proof

See Bootstrap, Lemma 046G and use Morphisms of Spaces, Lemma 03XG.

Lemma

A base change of a finite type morphism is finite type. The same holds for locally of finite type.

Proof

See Bootstrap, Lemma 046F and use Morphisms of Spaces, Lemma 03XH.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of formal algebraic spaces over \(S\). If \(g \circ f : X \to Z\) is locally of finite type, then \(f : X \to Y\) is locally of finite type.

Proof

By Lemma 0AQ0 we see that \(f\) is representable by algebraic spaces. Let \(T\) be a scheme and let \(T \to Z\) be a morphism. Then we can apply Morphisms of Spaces, Lemma 0462 to the morphisms \(T \times_Z X \to T \times_Z Y \to T\) of algebraic spaces to conclude.

Being locally of finite type is local on the source and the target.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent:

1. the morphism \(f\) is locally of finite type,

2. there exists a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are formal algebraic spaces, the vertical arrows are representable by algebraic spaces and étale, \(U \to X\) is surjective, and \(U \to V\) is locally of finite type,

3. for any commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U\), \(V\) are formal algebraic spaces and vertical arrows representable by algebraic spaces and étale, the morphism \(U \to V\) is locally of finite type,

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM such that \(X_{ji} \to Y_j\) is locally of finite type for each \(j\) and \(i\),

5. there exist a covering \(\{X_i \to X\}\) as in Definition 0AIM and for each \(i\) a factorization \(X_i \to Y_i \to Y\) where \(Y_i\) is an affine formal algebraic space, \(Y_i \to Y\) is representable by algebraic spaces and étale, such that \(X_i \to Y_i\) is locally of finite type, and

6. add more here.

Proof

In each of the 5 cases the morphism \(f : X \to Y\) is representable by algebraic spaces, see Lemma 0AN4. We will use this below without further mention.

It is clear that (1) implies (2) because we can take \(U = X\) and \(V = Y\). Conversely, assume given a diagram as in (2). Let \(T\) be a scheme and let \(T \to Y\) be a morphism. Then we can consider \[\xymatrix{ U \times_Y T \ar[d] \ar[r] & V \times_Y T \ar[d] \\ X \times_Y T \ar[r] & T }\] The vertical arrows are étale and the top horizontal arrow is locally of finite type as base changes of such morphisms. Hence by Morphisms of Spaces, Lemma 040Y we conclude that \(X \times_Y T \to T\) is locally of finite type. In other words (1) holds.

Assume (1) is true and consider a diagram as in (3). Then \(U \to Y\) is locally of finite type (as the composition \(U \to X \to Y\), see Bootstrap, Lemma 046G). Let \(T\) be a scheme and let \(T \to V\) be a morphism. Then the projection \(T \times_V U \to T\) factors as \[T \times_V U = (T \times_Y U) \times_{(V \times_Y V)} V \to T \times_Y U \to T\] The second arrow is locally of finite type (as a base change of the composition \(U \to X \to Y\)) and the first is the base change of the diagonal \(V \to V \times_Y V\) which is locally of finite type by Lemma 0AN2.

It is clear that (3) implies (2). Thus now (1) – (3) are equivalent.

Observe that the condition in (4) makes sense as the fibre product \(Y_j \times_Y X\) is a formal algebraic space by Lemma 0AJ3. It is clear that (4) implies (5).

Assume \(X_i \to Y_i \to Y\) as in (5). Then we set \(V = \coprod Y_i\) and \(U = \coprod X_i\) to see that (5) implies (2).

Finally, assume (1) – (3) are true. Thus we can choose any covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) any covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM. Then \(X_{ij} \to Y_j\) is locally of finite type by (3) and we see that (4) is true. This concludes the proof.

Example

Let \(S\) be a scheme. Let \(A\) be a weakly admissible topological ring over \(S\). Let \(A \to A'\) be a finite type ring map. Then \[(A')^\wedge = \lim_{I \subset A\ w.i.d.} A'/IA'\] is a weakly admissible ring and the corresponding morphism \(\text{Spf}((A')^\wedge) \to \text{Spf}(A)\) is representable, see Example 0AN6. If \(T \to \text{Spf}(A)\) is a morphism where \(T\) is a quasi-compact scheme, then this factors through \(\Spec(A/I)\) for some weak ideal of definition \(I \subset A\) (Lemma 0AIA). Then \(T \times_{\text{Spf}(A)} \text{Spf}((A')^\wedge)\) is equal to \(T \times_{\Spec(A/I)} \Spec(A'/IA')\) and we see that \(\text{Spf}((A')^\wedge) \to \text{Spf}(A)\) is of finite type.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). If \(Y\) is locally Noetherian and \(f\) locally of finite type, then \(X\) is locally Noetherian.

Proof

Pick \(\{Y_j \to Y\}\) and \(\{X_{ij} \to Y_j \times_Y X\}\) as in Lemma 0ANL. Then it follows from Lemma 0AKP that each \(X_{ij}\) is Noetherian. This proves the lemma.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) and \(Z \to Y\) be morphisms of formal algebraic spaces over \(S\). If \(Z\) is locally Noetherian and \(f\) locally of finite type, then \(Z \times_Y X\) is locally Noetherian.

Proof

The morphism \(Z \times_Y X \to Z\) is locally of finite type by Lemma 0AQ5. Hence this follows from Lemma 0AQ7.

Surjective morphisms

By Lemma 0GB8 the following definition does not clash with the already existing definitions for morphisms of algebraic spaces or morphisms of formal algebraic spaces which are representable by algebraic spaces.

Definition

Let \(S\) be a scheme. A morphism \(f : X \to Y\) of formal algebraic spaces over \(S\) is said to be surjective if it induces a surjective morphism \(X_{red} \to Y_{red}\) on underlying reduced algebraic spaces.

Lemma

The composition of two surjective morphisms is a surjective morphism.

Proof

Omitted.

Lemma

A base change of a surjective morphism is a surjective morphism.

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. \(f\) is surjective,

2. for every scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a surjective morphism of formal algebraic spaces,

3. for every affine scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a surjective morphism of formal algebraic spaces,

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that each \(X \times_Y Y_j \to Y_j\) is a surjective morphism of formal algebraic spaces,

5. there exists a surjective morphism \(Z \to Y\) of formal algebraic spaces such that \(X \times_Y Z \to Z\) is surjective, and

6. add more here.

Proof

Omitted.

Monomorphisms

Here is the definition.

Definition

Let \(S\) be a scheme. A morphism of formal algebraic spaces over \(S\) is called a monomorphism if it is an injective map of sheaves.

An example is the following. Let \(X\) be an algebraic space and let \(T \subset |X|\) be a closed subset. Then the morphism \(X_{/T} \to X\) from the formal completion of \(X\) along \(T\) to \(X\) is a monomorphism. In particular, monomorphisms of formal algebraic spaces are in general not representable.

Lemma

The composition of two monomorphisms is a monomorphism.

Proof

Omitted.

Lemma

A base change of a monomorphism is a monomorphism.

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. \(f\) is a monomorphism,

2. for every scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a monomorphism of formal algebraic spaces,

3. for every affine scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a monomorphism of formal algebraic spaces,

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that each \(X \times_Y Y_j \to Y_j\) is a monomorphism of formal algebraic spaces, and

5. there exists a family of morphisms \(\{Y_j \to Y\}\) such that \(\coprod Y_j \to Y\) is a surjection of sheaves on \((\Sch/S)_{fppf}\) such that each \(X \times_Y Y_j \to Y_j\) is a monomorphism for all \(j\),

6. there exists a morphism \(Z \to Y\) of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that \(X \times_Y Z \to X\) is a monomorphism, and

7. add more here.

Proof

Omitted.

Closed immersions

Here is the definition.

Definition

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of formal algebraic spaces over \(S\). We say \(f\) is a closed immersion if \(f\) is representable by algebraic spaces and a closed immersion in the sense of Bootstrap, Definition 03XZ.

Please skip the initial the obligatory lemmas when reading this section.

Lemma

The composition of two closed immersions is a closed immersion.

Proof

Omitted.

Lemma

A base change of a closed immersion is a closed immersion.

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. \(f\) is a closed immersion,

2. for every scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a closed immersion,

3. for every affine scheme \(T\) and morphism \(T \to Y\) the projection \(X \times_Y T \to T\) is a closed immersion,

4. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that each \(X \times_Y Y_j \to Y_j\) is a closed immersion, and

5. there exists a morphism \(Z \to Y\) of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that \(X \times_Y Z \to X\) is a closed immersion, and

6. add more here.

Proof

Omitted.

Lemma

Let \(S\) be a scheme. Let \(X\) be a McQuillan affine formal algebraic space over \(S\). Let \(f : Y \to X\) be a closed immersion of formal algebraic spaces over \(S\). Then \(Y\) is a McQuillan affine formal algebraic space and \(f\) corresponds to a continuous homomorphism \(A \to B\) of weakly admissible topological \(S\)-algebras which is taut, has closed kernel, and has dense image.

Proof

Write \(X = \text{Spf}(A)\) where \(A\) is a weakly admissible topological ring. Let \(I_\lambda\) be a fundamental system of weakly admissible ideals of definition in \(A\). Then \(Y \times_X \Spec(A/I_\lambda)\) is a closed subscheme of \(\Spec(A/I_\lambda)\) and hence affine (Definition 0ANP). Say \(Y \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)\). The ring map \(A/I_\lambda \to B_\lambda\) is surjective. Hence the projections \[B = \lim B_\lambda \longrightarrow B_\lambda\] are surjective as the compositions \(A \to B \to B_\lambda\) are surjective. It follows that \(Y\) is McQuillan by Lemma 0AIC. The ring map \(A \to B\) is taut by Lemma 0AN5. The kernel is closed because \(B\) is complete and \(A \to B\) is continuous. Finally, as \(A \to B_\lambda\) is surjective for all \(\lambda\) we see that the image of \(A\) in \(B\) is dense.

Even though we have the result above, in general we do not know how closed immersions behave when the target is a McQuillan affine formal algebraic space, see Remark 0AJK.

Example

Let \(S\) be a scheme. Let \(A\) be a weakly admissible topological ring over \(S\). Let \(K \subset A\) be a closed ideal. Setting \[B = (A/K)^\wedge = \lim_{I \subset A\ w.i.d.} A/(I + K)\] the morphism \(\text{Spf}(B) \to \text{Spf}(A)\) is representable, see Example 0AN6. If \(T \to \text{Spf}(A)\) is a morphism where \(T\) is a quasi-compact scheme, then this factors through \(\Spec(A/I)\) for some weak ideal of definition \(I \subset A\) (Lemma 0AIA). Then \(T \times_{\text{Spf}(A)} \text{Spf}(B)\) is equal to \(T \times_{\Spec(A/I)} \Spec(A/(K + I))\) and we see that \(\text{Spf}(B) \to \text{Spf}(A)\) is a closed immersion. The kernel of \(A \to B\) is \(K\) as \(K\) is closed, but beware that in general the ring map \(A \to B = (A/K)^\wedge\) need not be surjective.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces. Assume

1. \(f\) is representable by algebraic spaces,

2. \(f\) is a monomorphism,

3. the inclusion \(Y_{red} \to Y\) factors through \(f\), and

4. \(f\) is locally of finite type or \(Y\) is locally Noetherian.

Then \(f\) is a closed immersion.

Proof

Assumptions (2) and (3) imply that \(X_{red} = X \times_Y Y_{red} = Y_{red}\). We will use this without further mention.

If \(Y' \to Y\) is an étale morphism of formal algebraic spaces over \(S\), then the base change \(f' : X \times_Y Y' \to Y'\) satisfies conditions (1) – (4). Hence by Lemma 0GHY we may assume \(Y\) is an affine formal algebraic space.

Say \(Y = \colim_{\lambda \in \Lambda} Y_\lambda\) as in Definition 0AI7. Then \(X_\lambda = X \times_Y Y_\lambda\) is an algebraic space endowed with a monomorphism \(f_\lambda : X_\lambda \to Y_\lambda\) which induces an isomorphism \(X_{\lambda, red} \to Y_{\lambda, red}\). Thus \(X_\lambda\) is an affine scheme by Limits of Spaces, Proposition 07VT (as \(X_{\lambda, red} \to X_\lambda\) is surjective and integral). To finish the proof it suffices to show that \(X_\lambda \to Y_\lambda\) is a closed immersion which we will do in the next paragraph.

Let \(X \to Y\) be a monomorphism of affine schemes such that \(X_{red} = X \times_Y Y_{red} = Y_{red}\). In general, this does not imply that \(X \to Y\) is a closed immersion, see Examples, Section 06RH. However, under our assumption (4) we know that in the previous parapgrah either \(X_\lambda \to Y_\lambda\) is of finite type or \(Y_\lambda\) is Noetherian. This means that \(X \to Y\) corresponds to a ring map \(R \to A\) such that \(R/I \to A/IA\) is an isomorphism where \(I \subset R\) is the nil radical (ie., the maximal locally nilpotent ideal of \(R\)) and either \(R \to A\) is of finite type or \(R\) is Noetherian. In the first case \(R \to A\) is surjective by Algebra, Lemma 07RD and in the second case \(I\) is finitely generated, hence nilpotent, hence \(R \to A\) is surjective by Nakayama’s lemma, see Algebra, Lemma 00DV part (11).

Restricted power series

Let \(A\) be a topological ring complete with respect to a linear topology (More on Algebra, Definition 07E8). Let \(I_\lambda\) be a fundamental system of open ideals. Let \(r \geq 0\) be an integer. In this setting one often denotes \[A\{x_1, \ldots, x_r\} = \lim_\lambda A/I_\lambda[x_1, \ldots, x_r] = \lim_\lambda (A[x_1, \ldots, x_r]/I_\lambda A[x_1, \ldots, x_r])\] endowed with the limit topology. In other words, this is the completion of the polynomial ring with respect to the ideals \(I_\lambda\). We can think of elements of \(A\{x_1, \ldots, x_r\}\) as power series \[f = \sum\nolimits_{E = (e_1, \ldots, e_r)} a_E x_1^{e_1} \ldots x_r^{e_r}\] in \(x_1, \ldots, x_r\) with coefficients \(a_E \in A\) which tend to zero in the topology of \(A\). In other words, for any \(\lambda\) all but a finite number of \(a_E\) are in \(I_\lambda\). For this reason elements of \(A\{x_1, \ldots, x_r\}\) are sometimes called restricted power series. Sometimes this ring is denoted \(A\langle x_1, \ldots, x_r\rangle\); we will refrain from using this notation.

Remark

Let \(A \to C\) be a continuous map of complete linearly topologized rings. Then any \(A\)-algebra map \(A[x_1, \ldots x_r] \to C\) extends uniquely to a continuous map \(A\{x_1, \ldots, x_r\} \to C\) on restricted power series.

Remark

Let \(A\) be a ring and let \(I \subset A\) be an ideal. If \(A\) is \(I\)-adically complete, then the \(I\)-adic completion \(A[x_1, \ldots, x_r]^\wedge\) of \(A[x_1, \ldots, x_r]\) is the restricted power series ring over \(A\) as a ring. However, it is not clear that \(A[x_1, \ldots, x_r]^\wedge\) is \(I\)-adically complete. We think of the topology on \(A\{x_1, \ldots, x_r\}\) as the limit topology (which is always complete) whereas we often think of the topology on \(A[x_1, \ldots, x_r]^\wedge\) as the \(I\)-adic topology (not always complete). If \(I\) is finitely generated, then \(A\{x_1, \ldots, x_r\} = A[x_1, \ldots, x_r]^\wedge\) as topological rings, see Algebra, Lemma 05GG.

Algebras topologically of finite type

Here is our definition. This definition is not generally agreed upon. Many authors impose further conditions, often because they are only interested in specific types of rings and not the most general case.

Definition

Let \(A \to B\) be a continuous map of topological rings (More on Algebra, Definition 07E8). We say \(B\) is topologically of finite type over \(A\) if there exists an \(A\)-algebra map \(A[x_1, \ldots, x_n] \to B\) whose image is dense in \(B\).

If \(A\) is a complete, linearly topologized ring, then the restricted power series ring \(A\{x_1, \ldots, x_r\}\) is topologically of finite type over \(A\). If \(k\) is a field, then the power series ring \(k[[x_1, \ldots, x_r]]\) is topologically of finite type over \(k\).

For continuous taut maps of weakly admissible topological rings, being topologically of finite type corresponds exactly to morphisms of finite type between the associated affine formal algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \(\varphi : A \to B\) be a continuous map of weakly admissible topological rings over \(S\). The following are equivalent

1. \(\text{Spf}(\varphi) : Y = \text{Spf}(B) \to \text{Spf}(A) = X\) is of finite type,

2. \(\varphi\) is taut and \(B\) is topologically of finite type over \(A\).

Proof

We can use Lemma 0AN5 to relate tautness of \(\varphi\) to representability of \(\text{Spf}(\varphi)\). We will use this without further mention below. It follows that \(X = \colim \Spec(A/I)\) and \(Y = \colim \Spec(B/J(I))\) where \(I \subset A\) runs over the weak ideals of definition of \(A\) and \(J(I)\) is the closure of \(IB\) in \(B\).

Assume (2). Choose a ring map \(A[x_1, \ldots, x_r] \to B\) whose image is dense. Then \(A[x_1, \ldots, x_r] \to B \to B/J(I)\) has dense image too which means that it is surjective. Therefore \(B/J(I)\) is of finite type over \(A/I\). Let \(T \to X\) be a morphism with \(T\) a quasi-compact scheme. Then \(T \to X\) factors through \(\Spec(A/I)\) for some \(I\) (Lemma 0AIA). Then \(T \times_X Y = T \times_{\Spec(A/I)} \Spec(B/J(I))\), see proof of Lemma 0AN5. Hence \(T \times_Y X \to T\) is of finite type as the base change of the morphism \(\Spec(B/J(I)) \to \Spec(A/I)\) which is of finite type. Thus (1) is true.

Assume (1). Pick any \(I \subset A\) as above. Since \(\Spec(A/I) \times_X Y = \Spec(B/J(I))\) we see that \(A/I \to B/J(I)\) is of finite type. Choose \(b_1, \ldots, b_r \in B\) mapping to generators of \(B/J(I)\) over \(A/I\). We claim that the image of the ring map \(A[x_1, \ldots, x_r] \to B\) sending \(x_i\) to \(b_i\) is dense. To prove this, let \(I' \subset I\) be a second weak ideal of definition. Then we have \[B/(J(I') + IB) = B/J(I)\] because \(J(I)\) is the closure of \(IB\) and because \(J(I')\) is open. Hence we may apply Algebra, Lemma 07RD to see that \((A/I')[x_1, \ldots, x_r] \to B/J(I')\) is surjective. Thus (2) is true, concluding the proof.

Let \(A\) be a topological ring complete with respect to a linear topology. Let \((I_\lambda)\) be a fundamental system of open ideals. Let \(\mathcal{C}\) be the category of inverse systems \((B_\lambda)\) where

1. \(B_\lambda\) is a finite type \(A/I_\lambda\)-algebra, and

2. \(B_\mu \to B_\lambda\) is an \(A/I_\mu\)-algebra homomorphism which induces an isomorphism \(B_\mu/I_\lambda B_\mu \to B_\lambda\).

Morphisms in \(\mathcal{C}\) are given by compatible systems of homomorphisms.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). Assume \(X\) is McQuillan and let \(A\) be the weakly admissible topological ring associated to \(X\). Then there is an anti-equivalence of categories between

1. the category \(\mathcal{C}\) introduced above, and

2. the category of maps \(Y \to X\) of finite type of affine formal algebraic spaces.

Proof

Let \((I_\lambda)\) be a fundamental system of weakly admissible ideals of definition in \(A\). Consider \(Y\) as in (2). Then \(Y \times_X \Spec(A/I_\lambda)\) is affine (Definition 0AM4 and Lemma 0AKN). Say \(Y \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)\). The ring map \(A/I_\lambda \to B_\lambda\) is of finite type because \(\Spec(B_\lambda) \to \Spec(A/I_\lambda)\) is of finite type (by Definition 0AM4). Then \((B_\lambda)\) is an object of \(\mathcal{C}\).

Conversely, given an object \((B_\lambda)\) of \(\mathcal{C}\) we can set \(Y = \colim \Spec(B_\lambda)\). This is an affine formal algebraic space. We claim that \[Y \times_X \Spec(A/I_\lambda) = \left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)\] To show this it suffices we get the same values if we evaluate on a quasi-compact scheme \(U\). A morphism \(U \to \left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda)\) comes from a morphism \(U \to \Spec(B_\mu) \times_{\Spec(A/I_\mu)} \Spec(A/I_\lambda)\) for some \(\mu \geq \lambda\) (use Lemma 0AIA two times). Since \(\Spec(B_\mu) \times_{\Spec(A/I_\mu)} \Spec(A/I_\lambda) = \Spec(B_\lambda)\) by our second assumption on objects of \(\mathcal{C}\) this proves what we want. Using this we can show the morphism \(Y \to X\) is of finite type. Namely, we note that for any morphism \(U \to X\) with \(U\) a quasi-compact scheme, we get a factorization \(U \to \Spec(A/I_\lambda) \to X\) for some \(\lambda\) (see lemma cited above). Hence \[Y \times_X U = Y \times_X \Spec(A/I_\lambda)) \times_{\Spec(A/I_\lambda)} U = \Spec(B_\lambda) \times_{\Spec(A/I_\lambda)} U\] is a scheme of finite type over \(U\) as desired. Thus the construction \((B_\lambda) \mapsto \colim \Spec(B_\lambda)\) does give a functor from category (1) to category (2).

To finish the proof we show that the above constructions define quasi-inverse functors between the categories (1) and (2). In one direction you have to show that \[\left(\colim_\mu \Spec(B_\mu)\right) \times_X \Spec(A/I_\lambda) = \Spec(B_\lambda)\] for any object \((B_\lambda)\) in the category \(\mathcal{C}\). This we proved above. For the other direction you have to show that \[Y = \colim (Y \times_X \Spec(A/I_\lambda))\] given \(Y\) in the category (2). Again this is true by evaluating on quasi-compact test objects and because \(X = \colim \Spec(A/I_\lambda)\).

Remark

Let \(A\) be a weakly admissible topological ring and let \((I_\lambda)\) be a fundamental system of weak ideals of definition. Let \(X = \text{Spf}(A)\), in other words, \(X\) is a McQuillan affine formal algebraic space. Let \(f : Y \to X\) be a morphism of affine formal algebraic spaces. In general it will not be true that \(Y\) is McQuillan. More specifically, we can ask the following questions:

1. Assume that \(f : Y \to X\) is a closed immersion. Then \(Y\) is McQuillan and \(f\) corresponds to a continuous map \(\varphi : A \to B\) of weakly admissible topological rings which is taut, whose kernel \(K \subset A\) is a closed ideal, and whose image \(\varphi(A)\) is dense in \(B\), see Lemma 0ANQ. What conditions on \(A\) guarantee that \(B = (A/K)^\wedge\) as in Example 0ANR?

2. What conditions on \(A\) guarantee that closed immersions \(f : Y \to X\) correspond to quotients \(A/K\) of \(A\) by closed ideals, in other words, the corresponding continuous map \(\varphi\) is surjective and open?

3. Suppose that \(f : Y \to X\) is of finite type. Then we get \(Y = \colim \Spec(B_\lambda)\) where \((B_\lambda)\) is an object of \(\mathcal{C}\) by Lemma 0AL1. In this case it is true that there exists a fixed integer \(r\) such that \(B_\lambda\) is generated by \(r\) elements over \(A/I_\lambda\) for all \(\lambda\) (the argument is essentially already given in the proof of (1) \(\Rightarrow\) (2) in Lemma 0ANT). However, it is not clear that the projections \(\lim B_\lambda \to B_\lambda\) are surjective, i.e., it is not clear that \(Y\) is McQuillan. Is there an example where \(Y\) is not McQuillan?

4. Suppose that \(f : Y \to X\) is of finite type and \(Y\) is McQuillan. Then \(f\) corresponds to a continuous map \(\varphi : A \to B\) of weakly admissible topological rings. In fact \(\varphi\) is taut and \(B\) is topologically of finite type over \(A\), see Lemma 0ANT. In other words, \(f\) factors as \[Y \longrightarrow \mathbf{A}^r_X \longrightarrow X\] where the first arrow is a closed immersion of McQuillan affine formal algebraic spaces. However, then questions (1) and (2) are in force for \(Y \to \mathbf{A}^r_X\).

Below we will answer these questions when \(X\) is countably indexed, i.e., when \(A\) has a countable fundamental system of open ideals. If you have answers to these questions in greater generality, or if you have counter examples, please email stacks.project@gmail.com.

Lemma

Let \(S\) be a scheme. Let \(X\) be a countably indexed affine formal algebraic space over \(S\). Let \(f : Y \to X\) be a closed immersion of formal algebraic spaces over \(S\). Then \(Y\) is a countably indexed affine formal algebraic space and \(f\) corresponds to \(A \to A/K\) where \(A\) is an object of \(\textit{WAdm}^{count}\) (Section 0ANA) and \(K \subset A\) is a closed ideal.

Proof

By Lemma 0AN1 we see that \(X = \text{Spf}(A)\) where \(A\) is an object of \(\textit{WAdm}^{count}\). Since a closed immersion is representable and affine, we conclude by Lemma 0AKP that \(Y\) is an affine formal algebraic space and countably index. Thus applying Lemma 0AN1 again we see that \(Y = \text{Spf}(B)\) with \(B\) an object of \(\textit{WAdm}^{count}\). By Lemma 0ANQ we conclude that \(f\) is given by a morphism \(A \to B\) of \(\textit{WAdm}^{count}\) which is taut and has dense image. To finish the proof we apply Lemma 0APT.

Lemma

Let \(B \to A\) be an arrow of \(\textit{WAdm}^{count}\), see Section 0ANA. The following are equivalent

1. \(B \to A\) is taut and \(B/J \to A/I\) is of finite type for every weak ideal of definition \(J \subset B\) where \(I \subset A\) is the closure of \(JA\),

2. \(B \to A\) is taut and \(B/J_\lambda \to A/I_\lambda\) is of finite type for a cofinal system \((J_\lambda)\) of weak ideals of definition of \(B\) where \(I_\lambda \subset A\) is the closure of \(J_\lambda A\),

3. \(B \to A\) is taut and \(A\) is topologically of finite type over \(B\),

4. \(A\) is isomorphic as a topological \(B\)-algebra to a quotient of \(B\{x_1, \ldots, x_n\}\) by a closed ideal.

Moreover, these equivalent conditions define a local property, i.e., they satisfy Axioms (0AND), (0ANE), (0ANF).

Proof

The implications (a) \(\Rightarrow\) (b), (c) \(\Rightarrow\) (a), (d) \(\Rightarrow\) (c) are straightforward from the definitions. Assume (b) holds and let \(J \subset B\) and \(I \subset A\) be as in (a). Choose a commutative diagram \[\xymatrix{ A \ar[r] & \ldots \ar[r] & A_3 \ar[r] & A_2 \ar[r] & A_1 \\ B \ar[r] \ar[u] & \ldots \ar[r] & B/J_3 \ar[r] \ar[u] & B/J_2 \ar[r] \ar[u] & B/J_1 \ar[u] }\] such that \(A_{n + 1}/J_nA_{n + 1} = A_n\) and such that \(A = \lim A_n\) as in Lemma 0ANJ. For every \(m\) there exists a \(\lambda\) such that \(J_\lambda \subset J_m\). Since \(B/J_\lambda \to A/I_\lambda\) is of finite type, this implies that \(B/J_m \to A/I_m\) is of finite type. Let \(\alpha_1, \ldots, \alpha_n \in A_1\) be generators of \(A_1\) over \(B/J_1\). Since \(A\) is a countable limit of a system with surjective transition maps, we can find \(a_1, \ldots, a_n \in A\) mapping to \(\alpha_1, \ldots, \alpha_n\) in \(A_1\). By Remark 0AJM we find a continuous map \(B\{x_1, \ldots, x_n\} \to A\) mapping \(x_i\) to \(a_i\). This map induces surjections \((B/J_m)[x_1, \ldots, x_n] \to A_m\) by Algebra, Lemma 07RD. For \(m \geq 1\) we obtain a short exact sequence \[0 \to K_m \to (B/J_m)[x_1, \ldots, x_n] \to A_m \to 0\] The induced transition maps \(K_{m + 1} \to K_m\) are surjective because \(A_{m + 1}/J_mA_{m + 1} = A_m\). Hence the inverse limit of these short exact sequences is exact, see Algebra, Lemma 0598. Since \(B\{x_1, \ldots, x_n\} = \lim (B/J_m)[x_1, \ldots, x_n]\) and \(A = \lim A_m\) we conclude that \(B\{x_1, \ldots, x_n\} \to A\) is surjective and open. As \(A\) is complete the kernel is a closed ideal. In this way we see that (a), (b), (c), and (d) are equivalent.

Let a diagram (0ANC) as in Situation 0CBA be given. By Example 0ANM the maps \(A \to (A')^\wedge\) and \(B \to (B')^\wedge\) satisfy (a), (b), (c), and (d). Moreover, by Lemma 0ANJ in order to prove Axioms (0AND) and (0ANE) we may assume both \(B \to A\) and \((B')^\wedge \to (A')^\wedge\) are taut. Now pick a weak ideal of definition \(J \subset B\). Let \(J' \subset (B')^\wedge\), \(I \subset A\), \(I' \subset (A')^\wedge\) be the closure of \(J(B')^\wedge\), \(JA\), \(J(A')^\wedge\). By what was said above, it suffices to consider the commutative diagram \[\xymatrix{ A/I \ar[r] & (A')^\wedge/I' \\ B/J \ar[r] \ar[u]^{\overline{\varphi}} & (B')^\wedge/J' \ar[u]_{\overline{\varphi}'} }\] and to show (1) \(\overline{\varphi}\) finite type \(\Rightarrow \overline{\varphi}'\) finite type, and (2) if \(A \to A'\) is faithfully flat, then \(\overline{\varphi}'\) finite type \(\Rightarrow \overline{\varphi}\) finite type. Note that \((B')^\wedge/J' = B'/JB'\) and \((A')^\wedge/I' = A'/IA'\) by the construction of the topologies on \((B')^\wedge\) and \((A')^\wedge\). In particular the horizontal maps in the diagram are étale. Part (1) now follows from Algebra, Lemma 00F4 and part (2) from Descent, Lemma 0367 as the ring map \(A/I \to (A')^\wedge/I' = A'/IA'\) is faithfully flat and étale.

We omit the proof of Axiom (0ANF).

Lemma

In Lemma 0ANU if \(B\) is admissible (for example adic), then the equivalent conditions (a) – (d) are also equivalent to

1. \(B \to A\) is taut and \(B/J \to A/I\) is of finite type for some ideal of definition \(J \subset B\) where \(I \subset A\) is the closure of \(JA\).

Proof

It is enough to show that (e) implies (a). Let \(J' \subset B\) be a weak ideal of definition and let \(I' \subset A\) be the closure of \(J'A\). We have to show that \(B/J' \to A/I'\) is of finite type. If the corresponding statement holds for the smaller weak ideal of definition \(J'' = J' \cap J\), then it holds for \(J'\). Thus we may assume \(J' \subset J\). As \(J\) is an ideal of definition (and not just a weak ideal of definition), we get \(J^n \subset J'\) for some \(n \geq 1\). Thus we can consider the diagram \[\xymatrix{ 0 \ar[r] & I/I' \ar[r] & A/I' \ar[r] & A/I \ar[r] & 0 \\ 0 \ar[r] & J/J' \ar[r] \ar[u] & B/J' \ar[r] \ar[u] & B/J \ar[r] \ar[u] & 0 }\] with exact rows. Since \(I' \subset A\) is open and since \(I\) is the closure of \(J A\) we see that \(I/I' = (J/J') \cdot A/I'\). Because \(J/J'\) is a nilpotent ideal and as \(B/J \to A/I\) is of finite type, we conclude from Algebra, Lemma 0G8U that \(A/I'\) is of finite type over \(B/J'\) as desired.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of affine formal algebraic spaces. Assume \(Y\) countably indexed. The following are equivalent

1. \(f\) is locally of finite type,

2. \(f\) is of finite type,

3. \(f\) corresponds to a morphism \(B \to A\) of \(\textit{WAdm}^{count}\) (Section 0ANA) satisfying the equivalent conditions of Lemma 0ANU.

Proof

Since \(X\) and \(Y\) are affine it is clear that conditions (1) and (2) are equivalent. In cases (1) and (2) the morphism \(f\) is representable by algebraic spaces by definition, hence affine by Lemma 0AKN. Thus if (1) or (2) holds we see that \(X\) is countably indexed by Lemma 0AKP. Write \(X = \text{Spf}(A)\) and \(Y = \text{Spf}(B)\) for topological \(S\)-algebras \(A\) and \(B\) in \(\textit{WAdm}^{count}\), see Lemma 0AN1. By Lemma 0AN0 we see that \(f\) corresponds to a continuous map \(B \to A\). Hence now the result follows from Lemma 0ANT.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of locally countably indexed formal algebraic spaces over \(S\). The following are equivalent

1. for every commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] with \(U\) and \(V\) affine formal algebraic spaces, \(U \to X\) and \(V \to Y\) representable by algebraic spaces and étale, the morphism \(U \to V\) corresponds to a morphism of \(\textit{WAdm}^{count}\) which is taut and topologically of finite type,

2. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM and for each \(j\) a covering \(\{X_{ji} \to Y_j \times_Y X\}\) as in Definition 0AIM such that each \(X_{ji} \to Y_j\) corresponds to a morphism of \(\textit{WAdm}^{count}\) which is taut and topologically of finite type,

3. there exist a covering \(\{X_i \to X\}\) as in Definition 0AIM and for each \(i\) a factorization \(X_i \to Y_i \to Y\) where \(Y_i\) is an affine formal algebraic space, \(Y_i \to Y\) is representable by algebraic spaces and étale, and \(X_i \to Y_i\) corresponds to a morphism of \(\textit{WAdm}^{count}\) which is, taut and topologically of finite type, and

4. \(f\) is locally of finite type.

Proof

By Lemma 0ANU the property \(P(\varphi)=\)“\(\varphi\) is taut and topologically of finite type” is local on \(\text{WAdm}^{count}\). Hence by Lemma 0ANG we see that conditions (1), (2), and (3) are equivalent. On the other hand, by Lemma 0ANV the condition \(P\) on morphisms of \(\textit{WAdm}^{count}\) corresponds exactly to morphisms of countably indexed, affine formal algebraic spaces being locally of finite type. Thus the implication (1) \(\Rightarrow\) (3) of Lemma 0ANL shows that (4) implies (1) of the current lemma. Similarly, the implication (4) \(\Rightarrow\) (1) of Lemma 0ANL shows that (2) implies (4) of the current lemma.

Separation axioms for morphisms

This section is the analogue of Morphisms of Spaces, Section 03HJ for morphisms of formal algebraic spaces.

Definition

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). Let \(\Delta_{X/Y} : X \to X \times_Y X\) be the diagonal morphism.

1. We say \(f\) is separated if \(\Delta_{X/Y}\) is a closed immersion.

2. We say \(f\) is quasi-separated if \(\Delta_{X/Y}\) is quasi-compact.

Since \(\Delta_{X/Y}\) is representable (by schemes) by Lemma 0AN2 we can test this by considering morphisms \(T \to X \times_Y X\) from affine schemes \(T\) and checking whether \[E = T \times_{X \times_Y X} X \longrightarrow T\] is quasi-compact or a closed immersion, see Lemma 0AM2 or Definition 0ANP. Note that the scheme \(E\) is the equalizer of two morphisms \(a, b : T \to X\) which agree as morphisms into \(Y\) and that \(E \to T\) is a monomorphism and locally of finite type.

Lemma

All of the separation axioms listed in Definition 0ARN are stable under base change.

Proof

Let \(f : X \to Y\) and \(Y' \to Y\) be morphisms of formal algebraic spaces. Let \(f' : X' \to Y'\) be the base change of \(f\) by \(Y' \to Y\). Then \(\Delta_{X'/Y'}\) is the base change of \(\Delta_{X/Y}\) by the morphism \(X' \times_{Y'} X' \to X \times_Y X\). Each of the properties of the diagonal used in Definition 0ARN is stable under base change. Hence the lemma is true.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Z\), \(g : Y \to Z\) and \(Z \to T\) be morphisms of formal algebraic spaces over \(S\). Consider the induced morphism \(i : X \times_Z Y \to X \times_T Y\). Then

1. \(i\) is representable (by schemes), locally of finite type, locally quasi-finite, separated, and a monomorphism,

2. if \(Z \to T\) is separated, then \(i\) is a closed immersion, and

3. if \(Z \to T\) is quasi-separated, then \(i\) is quasi-compact.

Proof

By general category theory the following diagram \[\xymatrix{ X \times_Z Y \ar[r]_i \ar[d] & X \times_T Y \ar[d] \\ Z \ar[r]^-{\Delta_{Z/T}} \ar[r] & Z \times_T Z }\] is a fibre product diagram. Hence \(i\) is the base change of the diagonal morphism \(\Delta_{Z/T}\). Thus the lemma follows from Lemma 0AN2.

Lemma

All of the separation axioms listed in Definition 0ARN are stable under composition of morphisms.

Proof

Let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of formal algebraic spaces to which the axiom in question applies. The diagonal \(\Delta_{X/Z}\) is the composition \[X \longrightarrow X \times_Y X \longrightarrow X \times_Z X.\] Our separation axiom is defined by requiring the diagonal to have some property \(\mathcal{P}\). By Lemma 0ARQ above we see that the second arrow also has this property. Hence the lemma follows since the composition of (representable) morphisms with property \(\mathcal{P}\) also is a morphism with property \(\mathcal{P}\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). Let \(\mathcal{P}\) be any of the separation axioms of Definition 0ARN. The following are equivalent

1. \(f\) is \(\mathcal{P}\),

2. for every scheme \(Z\) and morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) of \(f\) is \(\mathcal{P}\),

3. for every affine scheme \(Z\) and every morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) of \(f\) is \(\mathcal{P}\),

4. for every affine scheme \(Z\) and every morphism \(Z \to Y\) the formal algebraic space \(Z \times_Y X\) is \(\mathcal{P}\) (see Definition 0AJ7),

5. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that the base change \(Y_j \times_Y X \to Y_j\) has \(\mathcal{P}\) for all \(j\).

Proof

We will repeatedly use Lemma 0ARP without further mention. In particular, it is clear that (1) implies (2) and (2) implies (3).

Assume (3) and let \(Z \to Y\) be a morphism where \(Z\) is an affine scheme. Let \(U\), \(V\) be affine schemes and let \(a : U \to Z \times_Y X\) and \(b : V \to Z \times_Y X\) be morphisms. Then \[U \times_{Z \times_Y X} V = (Z \times_Y X) \times_{\Delta, (Z \times_Y X) \times_Z (Z \times_Y X)} (U \times_Z V)\] and we see that this is quasi-compact if \(\mathcal{P} =\)“quasi-separated” or an affine scheme equipped with a closed immersion into \(U \times_Z V\) if \(\mathcal{P} =\)“separated”. Thus (4) holds.

Assume (4) and let \(Z \to Y\) be a morphism where \(Z\) is an affine scheme. Let \(U\), \(V\) be affine schemes and let \(a : U \to Z \times_Y X\) and \(b : V \to Z \times_Y X\) be morphisms. Reading the argument above backwards, we see that \(U \times_{Z \times_Y X} V \to U \times_Z V\) is quasi-compact if \(\mathcal{P} =\)“quasi-separated” or a closed immersion if \(\mathcal{P} =\)“separated”. Since we can choose \(U\) and \(V\) as above such that \(U\) varies through an étale covering of \(Z \times_Y X\), we find that the corresponding morphisms \[U \times_Z V \to (Z \times_Y X) \times_Z (Z \times_Y X)\] form an étale covering by affines. Hence we conclude that \(\Delta : (Z \times_Y X) \to (Z \times_Y X) \times_Z (Z \times_Y X)\) is quasi-compact, resp. a closed immersion. Thus (3) holds.

Let us prove that (3) implies (5). Assume (3) and let \(\{Y_j \to Y\}\) be as in Definition 0AIM. We have to show that the morphisms \[\Delta_j : Y_j \times_Y X \longrightarrow (Y_j \times_Y X) \times_{Y_j} (Y_j \times_Y X) = Y_j \times_Y X \times_Y X\] has the corresponding property (i.e., is quasi-compact or a closed immersion). Write \(Y_j = \colim Y_{j, \lambda}\) as in Definition 0AI7. Replacing \(Y_j\) by \(Y_{j, \lambda}\) in the formula above, we have the property by our assumption that (3) holds. Since the displayed arrow is the colimit of the arrows \(\Delta_{j, \lambda}\) and since we can test whether \(\Delta_j\) has the corresponding property by testing after base change by affine schemes mapping into \(Y_j \times_Y X \times_Y X\), we conclude by Lemma 0AIA.

Let us prove that (5) implies (1). Let \(\{Y_j \to Y\}\) be as in (5). Then we have the fibre product diagram \[\xymatrix{ \coprod Y_j \times_Y X \ar[r] \ar[d] & X \ar[d] \\ \coprod Y_j \times_Y X \times_Y X \ar[r] & X \times_Y X }\] By assumption the left vertical arrow is quasi-compact or a closed immersion. It follows from Spaces, Lemma 03KD that also the right vertical arrow is quasi-compact or a closed immersion.

Proper morphisms

Here is the definition we will use.

Definition

Let \(S\) be a scheme. Let \(f : Y \to X\) be a morphism of formal algebraic spaces over \(S\). We say \(f\) is proper if \(f\) is representable by algebraic spaces and is proper in the sense of Bootstrap, Definition 03XZ.

It follows from the definitions that a proper morphism is of finite type.

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). The following are equivalent

1. \(f\) is proper,

2. for every scheme \(Z\) and morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) of \(f\) is proper,

3. for every affine scheme \(Z\) and every morphism \(Z \to Y\) the base change \(Z \times_Y X \to Z\) of \(f\) is proper,

4. for every affine scheme \(Z\) and every morphism \(Z \to Y\) the formal algebraic space \(Z \times_Y X\) is an algebraic space proper over \(Z\),

5. there exists a covering \(\{Y_j \to Y\}\) as in Definition 0AIM such that the base change \(Y_j \times_Y X \to Y_j\) is proper for all \(j\).

Proof

Omitted.

Lemma

Proper morphisms of formal algebraic spaces are preserved by base change.

Proof

This is an immediate consequence of Lemma 0ART and transitivity of base change.

Formal algebraic spaces and fpqc coverings

This section is the analogue of Properties of Spaces, Section 03W8. Please read that section first.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). Then \(X\) satisfies the sheaf property for the fpqc topology.

Proof

The proof is identical to the proof of Properties of Spaces, Proposition 0APL. Since \(X\) is a sheaf for the Zariski topology it suffices to show the following. Given a surjective flat morphism of affines \(f : T' \to T\) we have: \(X(T)\) is the equalizer of the two maps \(X(T') \to X(T' \times_T T')\). See Topologies, Lemma 022H.

Let \(a, b : T \to X\) be two morphisms such that \(a \circ f = b \circ f\). We have to show \(a = b\). Consider the fibre product \[E = X \times_{\Delta_{X/S}, X \times_S X, (a, b)} T.\] By Lemma 0AIP the morphism \(\Delta_{X/S}\) is a representable monomorphism. Hence \(E \to T\) is a monomorphism of schemes. Our assumption that \(a \circ f = b \circ f\) implies that \(T' \to T\) factors (uniquely) through \(E\). Consider the commutative diagram \[\xymatrix{ T' \times_T E \ar[r] \ar[d] & E \ar[d] \\ T' \ar[r] \ar@/^5ex/[u] \ar[ru] & T }\] Since the projection \(T' \times_T E \to T'\) is a monomorphism with a section we conclude it is an isomorphism. Hence we conclude that \(E \to T\) is an isomorphism by Descent, Lemma 02L4. This means \(a = b\) as desired.

Next, let \(c : T' \to X\) be a morphism such that the two compositions \(T' \times_T T' \to T' \to X\) are the same. We have to find a morphism \(a : T \to X\) whose composition with \(T' \to T\) is \(c\). Choose a formal affine scheme \(U\) and an étale morphism \(U \to X\) such that the image of \(|U| \to |X_{red}|\) contains the image of \(|c| : |T'| \to |X_{red}|\). This is possible by Definition 0AIM, Properties of Spaces, Lemma 03BX, the fact that a finite union of formal affine algebraic spaces is a formal affine algebraic space, and the fact that \(|T'|\) is quasi-compact (small argument omitted). The morphism \(U \to X\) is representable by schemes (Lemma 0AIG) and separated (Lemma 0APX). Thus \[V = U \times_{X, c} T' \longrightarrow T'\] is an étale and separated morphism of schemes. It is also surjective by our choice of \(U \to X\) (if you do not want to argue this you can replace \(U\) by a disjoint union of formal affine algebraic spaces so that \(U \to X\) is surjective everything else still works as well). The fact that \(c \circ \text{pr}_0 = c \circ \text{pr}_1\) means that we obtain a descent datum on \(V/T'/T\) (Descent, Definition 023V) because \[\begin{align*} V \times_{T'} (T' \times_T T') & = U \times_{X, c \circ \text{pr}_0} (T' \times_T T') \\ & = (T' \times_T T') \times_{c \circ \text{pr}_1, X} U \\ & = (T' \times_T T') \times_{T'} V \end{align*}\] The morphism \(V \to T'\) is ind-quasi-affine by More on Morphisms, Lemma 0AP9 (because étale morphisms are locally quasi-finite, see Morphisms, Lemma 03WS). By More on Groupoids, Lemma 0APK the descent datum is effective. Say \(W \to T\) is a morphism such that there is an isomorphism \(\alpha : T' \times_T W \to V\) compatible with the given descent datum on \(V\) and the canonical descent datum on \(T' \times_T W\). Then \(W \to T\) is surjective and étale (Descent, Lemmas 02KV and 02VN). Consider the composition \[b' : T' \times_T W \longrightarrow V = U \times_{X, c} T' \longrightarrow U\] The two compositions \(b' \circ (\text{pr}_0, 1), b' \circ (\text{pr}_1, 1) : (T' \times_T T') \times_T W \to T' \times_T W \to U\) agree by our choice of \(\alpha\) and the corresponding property of \(c\) (computation omitted). Hence \(b'\) descends to a morphism \(b : W \to U\) by Descent, Lemma 023Q. The diagram \[\xymatrix{ T' \times_T W \ar[r] \ar[d] & W \ar[r]_b & U \ar[d] \\ T' \ar[rr]^c & & X }\] is commutative. What this means is that we have proved the existence of \(a\) étale locally on \(T\), i.e., we have an \(a' : W \to X\). However, since we have proved uniqueness in the first paragraph, we find that this étale local solution satisfies the glueing condition, i.e., we have \(\text{pr}_0^*a' = \text{pr}_1^*a'\) as elements of \(X(W \times_T W)\). Since \(X\) is an étale sheaf we find a unique \(a \in X(T)\) restricting to \(a'\) on \(W\).

Maps out of affine formal schemes

We prove a few results that will be useful later. In the paper [Bhatt-Algebraize] the reader can find very general results of a similar nature.

Lemma

Let \(S\) be a scheme. Let \(A\) be a weakly admissible topological \(S\)-algebra. Let \(X\) be an affine scheme over \(S\). Then the natural map \[\Mor_S(\Spec(A), X) \longrightarrow \Mor_S(\text{Spf}(A), X)\] is bijective.

Proof

If \(X\) is affine, say \(X = \Spec(B)\), then we see from Lemma 0AN0 that morphisms \(\text{Spf}(A) \to \Spec(B)\) correspond to continuous \(S\)-algebra maps \(B \to A\) where \(B\) has the discrete topology. These are just \(S\)-algebra maps, which correspond to morphisms \(\Spec(A) \to \Spec(B)\).

Lemma

Let \(S\) be a scheme. Let \(A\) be a weakly admissible topological \(S\)-algebra such that \(A/I\) is a local ring for some weak ideal of definition \(I \subset A\). Let \(X\) be a scheme over \(S\). Then the natural map \[\Mor_S(\Spec(A), X) \longrightarrow \Mor_S(\text{Spf}(A), X)\] is bijective.

Proof

Let \(\varphi : \text{Spf}(A) \to X\) be a morphism. Since \(\Spec(A/I)\) is local we see that \(\varphi\) maps \(\Spec(A/I)\) into an affine open \(U \subset X\). However, this then implies that \(\Spec(A/J)\) maps into \(U\) for every ideal of definition \(J\). Hence we may apply Lemma 0AQF to see that \(\varphi\) comes from a morphism \(\Spec(A) \to X\). This proves surjectivity of the map. We omit the proof of injectivity.

Lemma

Let \(S\) be a scheme. Let \(R\) be a complete local Noetherian \(S\)-algebra. Let \(X\) be an algebraic space over \(S\). Then the natural map \[\Mor_S(\Spec(R), X) \longrightarrow \Mor_S(\text{Spf}(R), X)\] is bijective.

Proof

Let \(\mathfrak m\) be the maximal ideal of \(R\). We have to show that \[\Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X)\] is bijective for \(R\) as above.

Injectivity: Let \(x, x' : \Spec(R) \to X\) be two morphisms mapping to the same element in the right hand side. Consider the fibre product \[T = \Spec(R) \times_{(x, x'), X \times_S X, \Delta} X\] Then \(T\) is a scheme and \(T \to \Spec(R)\) is locally of finite type, monomorphism, separated, and locally quasi-finite, see Morphisms of Spaces, Lemma 03HK. In particular \(T\) is locally Noetherian, see Morphisms, Lemma 01T6. Let \(t \in T\) be the unique point mapping to the closed point of \(\Spec(R)\) which exists as \(x\) and \(x'\) agree over \(R/\mathfrak m\). Then \(R \to \mathcal{O}_{T, t}\) is a local ring map of Noetherian rings such that \(R/\mathfrak m^n \to \mathcal{O}_{T, t}/\mathfrak m^n\mathcal{O}_{T, t}\) is an isomorphism for all \(n\) (because \(x\) and \(x'\) agree over \(\Spec(R/\mathfrak m^n)\) for all \(n\)). Since \(\mathcal{O}_{T, t}\) maps injectively into its completion (see Algebra, Lemma 00IP) we conclude that \(R = \mathcal{O}_{T, t}\). Hence \(x\) and \(x'\) agree over \(R\).

Surjectivity: Let \((x_n)\) be an element of the right hand side. Choose a scheme \(U\) and a surjective étale morphism \(U \to X\). Denote \(x_0 : \Spec(k) \to X\) the morphism induced on the residue field \(k = R/\mathfrak m\). The morphism of schemes \(U \times_{X, x_0} \Spec(k) \to \Spec(k)\) is surjective étale. Thus \(U \times_{X, x_0} \Spec(k)\) is a nonempty disjoint union of spectra of finite separable field extensions of \(k\), see Morphisms, Lemma 02GL. Hence we can find a finite separable field extension \(k'/k\) and a \(k'\)-point \(u_0 : \Spec(k') \to U\) such that \[\xymatrix{ \Spec(k') \ar[d] \ar[r]_-{u_0} & U \ar[d] \\ \Spec(k) \ar[r]^-{x_0} & X }\] commutes. Let \(R \subset R'\) be the finite étale extension of Noetherian complete local rings which induces \(k'/k\) on residue fields (see Algebra, Lemmas 04GK and 04GM). Denote \(x'_n\) the restriction of \(x_n\) to \(\Spec(R'/\mathfrak m^nR')\). By More on Morphisms of Spaces, Lemma 0616 we can find an element \((u'_n) \in \lim \Mor_S(\Spec(R'/\mathfrak m^nR'), U)\) mapping to \((x'_n)\). By Lemma 0AQG the family \((u'_n)\) comes from a unique morphism \(u' : \Spec(R') \to U\). Denote \(x' : \Spec(R') \to X\) the composition. Note that \(R' \otimes_R R'\) is a finite product of spectra of Noetherian complete local rings to which our current discussion applies. Hence the diagram \[\xymatrix{ \Spec(R' \otimes_R R') \ar[r] \ar[d] & \Spec(R') \ar[d]^{x'} \\ \Spec(R') \ar[r]^{x'} & X }\] is commutative by the injectivity shown above and the fact that \(x'_n\) is the restriction of \(x_n\) which is defined over \(R/\mathfrak m^n\). Since \(\{\Spec(R') \to \Spec(R)\}\) is an fppf covering we conclude that \(x'\) descends to a morphism \(x : \Spec(R) \to X\). We omit the proof that \(x_n\) is the restriction of \(x\) to \(\Spec(R/\mathfrak m^n)\).

Lemma

Let \(S\) be a scheme. Let \(X\) be an algebraic space over \(S\). Let \(T \subset |X|\) be a closed subset such that \(X \setminus T \to X\) is quasi-compact. Let \(R\) be a complete local Noetherian \(S\)-algebra. Then an adic morphism \(p : \text{Spf}(R) \to X_{/T}\) corresponds to a unique morphism \(g : \Spec(R) \to X\) such that \(g^{-1}(T) = \{\mathfrak m_R\}\).

Proof

The statement makes sense because \(X_{/T}\) is adic* by Lemma 0AQ1 (and hence we’re allowed to use the terminology adic for morphisms, see Definition 0AQ3). Let \(p\) be given. By Lemma 0AQH we get a unique morphism \(g : \Spec(R) \to X\) corresponding to the composition \(\text{Spf}(R) \to X_{/T} \to X\). Let \(Z \subset X\) be the reduced induced closed subspace structure on \(T\). The incusion morphism \(Z \to X\) corresponds to a morphism \(Z \to X_{/T}\). Since \(p\) is adic it is representable by algebraic spaces and we find \[\text{Spf}(R) \times_{X_{/T}} Z = \text{Spf}(R) \times_X Z\] is an algebraic space endowed with a closed immersion to \(\text{Spf}(R)\). (Equality holds because \(X_{/T} \to X\) is a monomorphism.) Thus this fibre product is equal to \(\Spec(R/J)\) for some ideal \(J \subset R\) which contains \(\mathfrak m_R^{n_0}\) for some \(n_0 \geq 1\). This implies that \(\Spec(R) \times_X Z\) is a closed subscheme of \(\Spec(R)\), say \(\Spec(R) \times_X Z = \Spec(R/I)\), whose intersection with \(\Spec(R/\mathfrak m_R^n)\) for \(n \geq n_0\) is equal to \(\Spec(R/J)\). In algebraic terms this says \(I + \mathfrak m_R^n = J + \mathfrak m_R^n = J\) for all \(n \geq n_0\). By Krull’s intersection theorem this implies \(I = J\) and we conclude.

The small étale site of a formal algebraic space

The motivation for the following definition comes from classical formal schemes: the underlying topological space of a formal scheme \((\mathfrak X, \mathcal{O}_\mathfrak X)\) is the underlying topological space of the reduction \(\mathfrak X_{red}\).

An important remark is the following. Suppose that \(X\) is an algebraic space with reduction \(X_{red}\) (Properties of Spaces, Definition 047X). Then we have \[X_{spaces, \etale} = X_{red, spaces, \etale},\quad X_\etale = X_{red, \etale},\quad X_{affine, \etale} = X_{red, affine, \etale}\] by More on Morphisms of Spaces, Theorem 05ZH and Lemma 07VW. Therefore the following definition does not conflict with the already existing notion in case our formal algebraic space happens to be an algebraic space.

Definition

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space with reduction \(X_{red}\) (Lemma 0AIN).

1. The small étale site \(X_\etale\) of \(X\) is the site \(X_{red, \etale}\) of Properties of Spaces, Definition 03ED.

2. The site \(X_{spaces, \etale}\) is the site \(X_{red, spaces, \etale}\) of Properties of Spaces, Definition 03G0.

3. The site \(X_{affine, \etale}\) is the site \(X_{red, affine, \etale}\) of Properties of Spaces, Lemma 04JS.

In Lemma 0DEG we will see that \(X_{spaces, \etale}\) can be described by in terms of morphisms of formal algebraic spaces which are representable by algebraic spaces and étale. By Properties of Spaces, Lemmas 03G1 and 04JS we have identifications [0DEB]\[\begin{equation} \Sh(X_\etale) = \Sh(X_{spaces, \etale}) = \Sh(X_{affine, \etale}) \end{equation}\] We will call this the (small) étale topos of \(X\).

Lemma

Let \(S\) be a scheme. Let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\).

1. There is a continuous functor \(Y_{spaces, \etale} \to X_{spaces, \etale}\) which induces a morphism of sites \[f_{spaces, \etale} : X_{spaces, \etale} \to Y_{spaces, \etale}.\]

2. The rule \(f \mapsto f_{spaces, \etale}\) is compatible with compositions, in other words \((f \circ g)_{spaces, \etale} = f_{spaces, \etale} \circ g_{spaces, \etale}\) (see Sites, Definition 03CC).

3. The morphism of topoi associated to \(f_{spaces, \etale}\) induces, via (0DEB), a morphism of topoi \(f_{small} : \Sh(X_\etale) \to \Sh(Y_\etale)\) whose construction is compatible with compositions.

Proof

The only point here is that \(f\) induces a morphism of reductions \(X_{red} \to Y_{red}\) by Lemma 0AIN. Hence this lemma is immediate from the corresponding lemma for morphisms of algebraic spaces (Properties of Spaces, Lemma 03G2).

If the morphism of formal algebraic spaces \(X \to Y\) is étale, then the morphism of topoi \(\Sh(X_\etale) \to \Sh(Y_\etale)\) is a localization. Here is a statement.

Lemma

Let \(S\) be a scheme, and let \(f : X \to Y\) be a morphism of formal algebraic spaces over \(S\). Assume \(f\) is representable by algebraic spaces and étale. In this case there is a cocontinuous functor \(j : X_\etale \to Y_\etale\). The morphism of topoi \(f_{small}\) is the morphism of topoi associated to \(j\), see Sites, Lemma 00XO. Moreover, \(j\) is continuous as well, hence Sites, Lemma 00XR applies.

Proof

This will follow immediately from the case of algebraic spaces (Properties of Spaces, Lemma 03LP) if we can show that the induced morphism \(X_{red} \to Y_{red}\) is étale. Observe that \(X \times_Y Y_{red}\) is an algebraic space, étale over the reduced algebraic space \(Y_{red}\), and hence reduced itself (by our definition of reduced algebraic spaces in Properties of Spaces, Section 03E5. Hence \(X_{red} = X \times_Y Y_{red}\) as desired.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). Then \(X_{affine, \etale}\) is equivalent to the category whose objects are morphisms \(\varphi : U \to X\) of formal algebraic spaces such that

1. \(U\) is an affine formal algebraic space,

2. \(\varphi\) is representable by algebraic spaces and étale.

Proof

Denote \(\mathcal{C}\) the category introduced in the lemma. Observe that for \(\varphi : U \to X\) in \(\mathcal{C}\) the morphism \(\varphi\) is representable (by schemes) and affine, see Lemma 0AKN. Recall that \(X_{affine, \etale} = X_{red, affine, \etale}\). Hence we can define a functor \[\mathcal{C} \longrightarrow X_{affine, \etale},\quad (U \to X) \longmapsto U \times_X X_{red}\] because \(U \times_X X_{red}\) is an affine scheme.

To finish the proof we will construct a quasi-inverse. Namely, write \(X = \colim X_\lambda\) as in Definition 0AI7. For each \(\lambda\) we have \(X_{red} \subset X_\lambda\) is a thickening. Thus for every \(\lambda\) we have an equivalence \[X_{red, affine, \etale} = X_{\lambda, affine, \etale}\] for example by More on Algebra, Lemma 0ALI. Hence if \(U_{red} \to X_{red}\) is an étale morphism with \(U_{red}\) affine, then we obtain a system of étale morphisms \(U_\lambda \to X_\lambda\) of affine schemes compatible with the transition morphisms in the system defining \(X\). Hence we can take \[U = \colim U_\lambda\] as our affine formal algebraic space over \(X\). The construction gives that \(U \times_X X_\lambda = U_\lambda\). This shows that \(U \to X\) is representable and étale. We omit the verification that the constructions are mutually inverse to each other.

Lemma

Let \(S\) be a scheme. Let \(X\) be an affine formal algebraic space over \(S\). Assume \(X\) is McQuillan, i.e., equal to \(\text{Spf}(A)\) for some weakly admissible topological \(S\)-algebra \(A\). Then \((X_{affine, \etale})^{opp}\) is equivalent to the category whose

1. objects are \(A\)-algebras of the form \(B^\wedge = \lim B/JB\) where \(A \to B\) is an étale ring map and \(J\) runs over the weak ideals of definition of \(A\), and

2. morphisms are continuous \(A\)-algebra homomorphisms.

Proof

Combine Lemmas 0DEE and 0AN8.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). Then \(X_{spaces, \etale}\) is equivalent to the category whose objects are morphisms \(\varphi : U \to X\) of formal algebraic spaces such that \(\varphi\) is representable by algebraic spaces and étale.

Proof

Denote \(\mathcal{C}\) the category introduced in the lemma. Recall that \(X_{spaces, \etale} = X_{red, spaces, \etale}\). Hence we can define a functor \[\mathcal{C} \longrightarrow X_{spaces, \etale},\quad (U \to X) \longmapsto U \times_X X_{red}\] because \(U \times_X X_{red}\) is an algebraic space étale over \(X_{red}\).

To finish the proof we will construct a quasi-inverse. Choose an object \(\psi : V \to X_{red}\) of \(X_{red, spaces, \etale}\). Consider the functor \(U_{V, \psi} : (\Sch/S)_{fppf} \to \textit{Sets}\) given by \[U_{V, \psi}(T) = \{(a, b) \mid a : T \to X, \ b : T \times_{a, X} X_{red} \to V, \ \psi \circ b = a|_{T \times_{a, X} X_{red}}\}\] We claim that the transformation \(U_{V, \psi} \to X\), \((a, b) \mapsto a\) defines an object of the category \(\mathcal{C}\). First, let’s prove that \(U_{V, \psi}\) is a formal algebraic space. Observe that \(U_{V, \psi}\) is a sheaf for the fppf topology (some details omitted). Next, suppose that \(X_i \to X\) is an étale covering by affine formal algebraic spaces as in Definition 0AIM. Set \(V_i = V \times_{X_{red}} X_{i, red}\) and denote \(\psi_i : V_i \to X_{i, red}\) the projection. Then we have \[U_{V, \psi} \times_X X_i = U_{V_i, \psi_i}\] by a formal argument because \(X_{i, red} = X_i \times_X X_{red}\) (as \(X_i \to X\) is representable by algebraic spaces and étale). Hence it suffices to show that \(U_{V_i, \psi_i}\) is an affine formal algebraic space, because then we will have a covering \(U_{V_i, \psi_i} \to U_{V, \psi}\) as in Definition 0AIM. On the other hand, we have seen in the proof of Lemma 0DED that \(\psi_i : V_i \to X_i\) is the base change of a representable and étale morphism \(U_i \to X_i\) of affine formal algebraic spaces. Then it is not hard to see that \(U_i = U_{V_i, \psi_i}\) as desired.

We omit the verification that \(U_{V, \psi} \to X\) is representable by algebraic spaces and étale. Thus we obtain our functor \((V, \psi) \mapsto (U_{V, \psi} \to X)\) in the other direction. We omit the verification that the constructions are mutually inverse to each other.

Lemma

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). Then \(X_{affine, \etale}\) is equivalent to the category whose objects are morphisms \(\varphi : U \to X\) of formal algebraic spaces such that

1. \(U\) is an affine formal algebraic space,

2. \(\varphi\) is representable by algebraic spaces and étale.

Proof

This follows by combining Lemmas 0DEG and 0DE8.

The structure sheaf

As in Section 0DE9 the motivation for the discussion in this section comes from classical formal schemes. These are defined as pairs \((\mathfrak X, \mathcal{O}_\mathfrak X)\) where \(\mathcal{O}_\mathfrak X\) is a sheaf of topological rings.

Let \(X\) be a formal algebraic space. A structure sheaf for \(X\) is a sheaf of topological rings \(\mathcal{O}_X\) on the étale site \(X_\etale\) (which we defined in Section 0DE9) such that \[\mathcal{O}_X(U_{red}) = \lim \Gamma(U_\lambda, \mathcal{O}_{U_\lambda})\] as topological rings whenever

1. \(\varphi : U \to X\) is a morphism of formal algebraic spaces,

2. \(U\) is an affine formal algebraic space,

3. \(\varphi\) is representable by algebraic spaces and étale,

4. \(U_{red} \to X_{red}\) is the corresponding affine object of \(X_\etale\), see Lemma 0DEH,

5. \(U = \colim U_\lambda\) is a colimit representation for \(U\) as in Definition 0AI7.

Structure sheaves exist but may behave in unexpected manner.

Lemma

Every formal algebraic space has a structure sheaf.

Proof

Let \(S\) be a scheme. Let \(X\) be a formal algebraic space over \(S\). By (0DEB) it suffices to construct \(\mathcal{O}_X\) as a sheaf of topological rings on \(X_{affine, \etale}\). Denote \(\mathcal{C}\) the category whose objects are morphisms \(\varphi : U \to X\) of formal algebraic spaces such that \(U\) is an affine formal algebraic space and \(\varphi\) is representable by algebraic spaces and étale. By Lemma 0DEH the functor \(U \mapsto U_{red}\) is an equivalence of categories \(\mathcal{C} \to X_{affine, \etale}\). Hence by the rule given above the lemma, we already have \(\mathcal{O}_X\) as a presheaf of topological rings on \(X_{affine, \etale}\). Thus it suffices to check the sheaf condition.

By definition of \(X_{affine, \etale}\) a covering corresponds to a finite family \(\{g_i : U_i \to U\}_{i = 1, \ldots, n}\) of morphisms of \(\mathcal{C}\) such that \(\{U_{i, red} \to U_{red}\}\) is an étale covering. The morphisms \(g_i\) are representably by algebraic spaces (Lemma 0AQ0) hence affine (Lemma 0AKN). Then \(g_i\) is étale (follows formally from Properties of Spaces, Lemma 03FV as \(U_i\) and \(U\) are étale over \(X\) in the sense of Bootstrap, Section 046E). Finally, write \(U = \colim U_\lambda\) as in Definition 0AI7.

With these preparations out of the way, we can prove the sheaf property as follows. For each \(\lambda\) we set \(U_{i, \lambda} = U_i \times_U U_\lambda\) and \(U_{ij, \lambda} = (U_i \times_U U_j) \times_U U_\lambda\). By the above, these are affine schemes, \(\{U_{i, \lambda} \to U_\lambda\}\) is an étale covering, and \(U_{ij, \lambda} = U_{i, \lambda} \times_{U_\lambda} U_{j, \lambda}\). Also we have \(U_i = \colim U_{i, \lambda}\) and \(U_i \times_U U_j = \colim U_{ij, \lambda}\). For each \(\lambda\) we have an exact sequence \[0 \to \Gamma(U_\lambda, \mathcal{O}_{U_\lambda}) \to \prod\nolimits_i \Gamma(U_{i, \lambda}, \mathcal{O}_{U_{i, \lambda}}) \to \prod\nolimits_{i, j} \Gamma(U_{ij, \lambda}, \mathcal{O}_{U_{ij, \lambda}})\] as we have the sheaf condition for the structure sheaf on \(U_\lambda\) and the étale topology (see Étale Cohomology, Proposition 03OG). Since limits commute with limits, the inverse limit of these exact sequences is an exact sequence \[0 \to \lim \Gamma(U_\lambda, \mathcal{O}_{U_\lambda}) \to \prod\nolimits_i \lim \Gamma(U_{i, \lambda}, \mathcal{O}_{U_{i, \lambda}}) \to \prod\nolimits_{i, j} \lim \Gamma(U_{ij, \lambda}, \mathcal{O}_{U_{ij, \lambda}})\] This means that \[0 \to \mathcal{O}_X(U_{red}) \to \prod\nolimits_i \mathcal{O}_X(U_{i, red}) \to \prod\nolimits_{i, j} \mathcal{O}_X((U_i \times_U U_j)_{red})\] is exact as a sequence of abelian groups. We still have to show that the limit topology on \(\mathcal{O}_X(U_{red})\) is induced from the limit topology on \(\prod_{i = 1, \ldots, n} \mathcal{O}_X(U_{i, red})\). The kernels of the maps \(\mathcal{O}_X(U_{red}) \to \mathcal{O}_{U_\lambda}(U_\lambda)\) form a fundamental system of open submodules. Since these are the inverse images of the kernels of the maps \(\prod_i \mathcal{O}_X(U_{i, red}) \to \prod_i \mathcal{O}_{U_{i, \lambda}}(U_{i, \lambda})\) and these kernels form a fundamental system of open submodules of \(\mathcal{O}_X(U_{red}) \to \mathcal{O}_{U_\lambda}(U_\lambda)\) we win.

Remark

The structure sheaf does not always have “enough sections”. In Examples, Section 0ANY we have seen that there exist affine formal algebraic spaces which aren’t McQuillan and there are even examples whose points are not separated by regular functions.

In the next lemma we prove that the structure sheaf on a countably indexed affine formal scheme has vanishing higher cohomology. For non-countably indexed ones, presumably this generally doesn’t hold.

Lemma

If \(X\) is a countably indexed affine formal algebraic space, then we have \(H^n(X_\etale, \mathcal{O}_X) = 0\) for \(n > 0\).

Proof

We may work with \(X_{affine, \etale}\) as this gives the same topos. We will apply Cohomology on Sites, Lemma 03F9 to show we have vanishing. Since \(X_{affine, \etale}\) has finite disjoint unions, this reduces us to the Čech complex of a covering given by a single arrow \(\{U_{red} \to V_{red}\}\) in \(X_{affine, \etale} = X_{red, affine, \etale}\) (see Étale Cohomology, Lemma 03OZ). Thus we have to show that \[0 \to \mathcal{O}_X(V_{red}) \to \mathcal{O}_X(U_{red}) \to \mathcal{O}_X(U_{red} \times_{V_{red}} U_{red}) \to \ldots\] is exact. We will do this below in the case \(V_{red} = X_{red}\). The general case is proven in exactly the same way.

Recall that \(X = \text{Spf}(A)\) where \(A\) is a weakly admissible topological ring having a countable fundamental system of weak ideals of definition. We have seen in Lemmas 0DEE and 0DEF that the object \(U_{red}\) in \(X_{affine, \etale}\) corresponds to a morphism \(U \to X\) of affine formal algebraic spaces which is representable by algebraic space and étale and \(U = \text{Spf}(B^\wedge)\) where \(B\) is an étale \(A\)-algebra. By our rule for the structure sheaf we see \[\mathcal{O}_X(U_{red}) = B^\wedge\] We recall that \(B^\wedge = \lim B/JB\) where the limit is over weak ideals of definition \(J \subset A\). Working through the definitions we obtain \[\mathcal{O}_X(U_{red} \times_{X_{red}} U_{red}) = (B \otimes_A B)^\wedge\] and so on. Since \(U \to X\) is a covering the map \(A \to B\) is faithfully flat, see Lemma 0AN9. Hence the complex \[0 \to A \to B \to B \otimes_A B \to B \otimes_A B \otimes_A B \to \ldots\] is universally exact, see Descent, Lemma 023M. Our goal is to show that \[H^n(0 \to A^\wedge \to B^\wedge \to (B \otimes_A B)^\wedge \to (B \otimes_A B \otimes_A B)^\wedge \to \ldots)\] is zero for \(n > 0\). To see what is going on, let’s split our exact complex (before completion) into short exact sequences \[0 \to A \to B \to M_1 \to 0,\quad 0 \to M_i \to B^{\otimes_A i + 1} \to M_{i + 1} \to 0\] By what we said above, these are universally exact short exact sequences. Hence \(JM_i = M_i \cap J(B^{\otimes_A i + 1})\) for every ideal \(J\) of \(A\). In particular, the topology on \(M_i\) as a submodule of \(B^{\otimes_A i + 1}\) is the same as the topology on \(M_i\) as a quotient module of \(B^{\otimes_A i}\). Therefore, since there exists a countable fundamental system of weak ideals of definition in \(A\), the sequences \[0 \to A^\wedge \to B^\wedge \to M_1^\wedge \to 0,\quad 0 \to M_i^\wedge \to (B^{\otimes_A i + 1})^\wedge \to M_{i + 1}^\wedge \to 0\] remain exact by Lemma 0AS0. This proves the lemma.

Remark

Even if the structure sheaf has good properties, this does not mean there is a good theory of quasi-coherent modules. For example, in Examples, Section 0ALF we have seen that for almost any Noetherian affine formal algebraic spaces the most natural notion of a quasi-coherent module leads to a category of modules which is not abelian.

Colimits of formal algebraic spaces

In this section we generalize the result of Section 0AIT to the case of systems of morphisms of formal algebraic spaces. We remark that in the lemmas below the condition “\(f_{\lambda \mu} : X_\lambda \to X_\mu\) is a closed immersion inducing an isomorphism \(X_{\lambda, red} \to X_{\mu, red}\)” can be reformulated as “\(f_{\lambda \mu}\) is representable and a thickening”.