The Stacks project

85.7 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 78.3 and 78.4.

Definition 85.7.1. Let $S$ be a scheme. We say a sheaf $X$ on $(\mathit{Sch}/S)_{fppf}$ 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 85.7.6). 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 85.5.11. By Bootstrap, Lemma 78.4.6 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. However, they do have an underlying reduced algebraic space as the following lemma demonstrates.

Lemma 85.7.2. 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 = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 85.5.1. 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 85.5.3 and the fact that compositions of thickenings are thickenings. We omit the verification of the universal property (use Schemes, Definition 26.12.5, Schemes, Lemma 26.12.7, Properties of Spaces, Definition 64.12.5, and Properties of Spaces, Lemma 64.12.4).

Let $X$ and $\{ X_ i \to X\} _{i \in I}$ be as in Definition 85.7.1. 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 85.5.11). Hence $X_{i, red} \times _ X X_ j$ is reduced (see Descent, Lemma 35.15.1). 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 63.10.5. 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 78.5.2 (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 63.5.6 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 64.7). 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. $\square$

Lemma 85.7.3. 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 85.7.1. 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 85.5.2 and 85.5.11). These maps are jointly surjective hence $U \times _ X V$ is an algebraic space by Bootstrap, Theorem 78.10.1. 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 85.5.2) and an étale morphism, see Descent on Spaces, Lemma 72.18.2. Hence we conclude that $U \times _ X V$ is a scheme by Morphisms of Spaces, Proposition 65.50.2. Thus $\Delta $ is representable, see Spaces, Lemma 63.5.10.

In fact, since we've shown above that the morphisms of schemes $U \times _ X V \to U \times _ S V$ are aways 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 63.5.11. Then we can use the principle of Spaces, Lemma 63.5.8 to see that $\Delta $ is separated and locally of finite type. Namely, a monomorphism of schemes is separated (Schemes, Lemma 26.23.3) and a locally quasi-finite morphism of schemes is locally of finite type (follows from the definition in Morphisms, Section 29.20). $\square$

Lemma 85.7.4. 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 63.5.5) and $U \times _ Y Z$ is a scheme by Lemma 85.7.3. Hence the result by Bootstrap, Theorem 78.10.1. $\square$

Remark 85.7.5. Modulo set theoretic issues the category of formal schemes à la EGA (see Section 85.2) is equivalent to a full subcategory of the category of formal algebraic spaces. To explain this we assume our base scheme is $\mathop{\mathrm{Spec}}(\mathbf{Z})$. By Lemma 85.2.2 the functor of points $h_\mathfrak X$ associated to a formal scheme $\mathfrak X$ is a sheaf in the fppf topology. By Lemma 85.2.1 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 85.5.8. 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 85.7.1 and we see that $h_\mathfrak X$ is a formal algebraic space.

Remark 85.7.6. Let $S$ be a scheme and let $(\mathit{Sch}/S)_{fppf}$ be a big fppf site as in Topologies, Definition 34.7.8. As our set theoretic condition on $X$ in Definitions 85.5.1 and 85.7.1 we take: there exist objects $U, R$ of $(\mathit{Sch}/S)_{fppf}$, 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 = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } X_\lambda $ and the system satisfies conditions (1) and (2) of Definition 85.5.1. 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 85.5.2. Hence if $U$ is representable by an object of $(\mathit{Sch}/S)_{fppf}$ then $U \times _ S U$ is too (see Sets, Lemma 3.9.9) and the set theoretic condition is satisfied. This is always the case if $\Lambda $ is countable, see Sets, Lemma 3.9.9.

  2. Sanity check. Let $\{ X_ i \to X\} _{i \in I}$ be as in Definition 85.7.1 (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 $(\mathit{Sch}'/S)_{fppf}$ 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 63.15.2 shows that $X$ is an algebraic space for $(\mathit{Sch}/S)_{fppf}$.

  3. Let $\{ X_ i \to X\} _{i \in I}$ be a family of maps of sheaves satisfying conditions (1), (2), (3) of Definition 85.7.1. For each $i$ we can pick $U_ i \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ 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 $(\mathit{Sch}/S)_{fppf}$. To get $R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ 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.

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