The Stacks project

85.27 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 64.12.5). Then we have

\[ X_{spaces, {\acute{e}tale}} = X_{red, spaces, {\acute{e}tale}},\quad X_{\acute{e}tale}= X_{red, {\acute{e}tale}},\quad X_{affine, {\acute{e}tale}} = X_{red, affine, {\acute{e}tale}} \]

by More on Morphisms of Spaces, Theorem 74.8.1 and Lemma 74.8.2. 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 85.27.1. Let $S$ be a scheme. Let $X$ be a formal algebraic space with reduction $X_{red}$ (Lemma 85.7.2).

  1. The small étale site $X_{\acute{e}tale}$ of $X$ is the site $X_{red, {\acute{e}tale}}$ of Properties of Spaces, Definition 64.18.1.

  2. The site $X_{spaces, {\acute{e}tale}}$ is the site $X_{red, spaces, {\acute{e}tale}}$ of Properties of Spaces, Definition 64.18.2.

  3. The site $X_{affine, {\acute{e}tale}}$ is the site $X_{red, affine, {\acute{e}tale}}$ of Properties of Spaces, Lemma 64.18.5.

In Lemma 85.27.6 we will see that $X_{spaces, {\acute{e}tale}}$ 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 64.18.3 and 64.18.5 we have identifications

85.27.1.1
\begin{equation} \label{formal-spaces-equation-etale-topos} \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) = \mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}) \end{equation}

We will call this the (small) étale topos of $X$.

Lemma 85.27.2. 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 $Y_{spaces, {\acute{e}tale}} \to X_{spaces, {\acute{e}tale}}$ which induces a morphism of sites

    \[ f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}. \]
  2. The rule $f \mapsto f_{spaces, {\acute{e}tale}}$ is compatible with compositions, in other words $(f \circ g)_{spaces, {\acute{e}tale}} = f_{spaces, {\acute{e}tale}} \circ g_{spaces, {\acute{e}tale}}$ (see Sites, Definition 7.14.5).

  3. The morphism of topoi associated to $f_{spaces, {\acute{e}tale}}$ induces, via (85.27.1.1), a morphism of topoi $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ 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 85.7.2. Hence this lemma is immediate from the corresponding lemma for morphisms of algebraic spaces (Properties of Spaces, Lemma 64.18.7). $\square$

If the morphism of formal algebraic spaces $X \to Y$ is étale, then the morphism of topoi $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ is a localization. Here is a statement.

Lemma 85.27.3. 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 $j : X_{\acute{e}tale}\to Y_{\acute{e}tale}$. The morphism of topoi $f_{small}$ is the morphism of topoi associated to $j$, see Sites, Lemma 7.21.1. Moreover, $j$ is continuous as well, hence Sites, Lemma 7.21.5 applies.

Proof. This will follow immediately from the case of algebraic spaces (Properties of Spaces, Lemma 64.18.10) 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 64.7. Hence $X_{red} = X \times _ Y Y_{red}$ as desired. $\square$

Lemma 85.27.4. Let $S$ be a scheme. Let $X$ be an affine formal algebraic space over $S$. Then $X_{affine, {\acute{e}tale}}$ 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 85.14.7. Recall that $X_{affine, {\acute{e}tale}} = X_{red, affine, {\acute{e}tale}}$. Hence we can define a functor

\[ \mathcal{C} \longrightarrow X_{affine, {\acute{e}tale}},\quad (U \to X) \longmapsto U \times _ X X_{red} \]

because $U \times _ X X_{red}$ is an affine scheme.

To finish the proof we will construct a quasi-inverse. Namely, write $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ as in Definition 85.5.1. For each $\lambda $ we have $X_{red} \subset X_\lambda $ is a thickening. Thus for every $\lambda $ we have an equivalence

\[ X_{red, affine, {\acute{e}tale}} = X_{\lambda , affine, {\acute{e}tale}} \]

for example by More on Algebra, Lemma 15.11.2. 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

\[ U = \mathop{\mathrm{colim}}\nolimits U_\lambda \]

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. $\square$

Lemma 85.27.5. 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 $(X_{affine, {\acute{e}tale}})^{opp}$ is equivalent to the category whose

  1. objects are $A$-algebras of the form $B^\wedge = \mathop{\mathrm{lim}}\nolimits B/JB$ 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.

Lemma 85.27.6. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{spaces, {\acute{e}tale}}$ 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 $X_{spaces, {\acute{e}tale}} = X_{red, spaces, {\acute{e}tale}}$. Hence we can define a functor

\[ \mathcal{C} \longrightarrow X_{spaces, {\acute{e}tale}},\quad (U \to X) \longmapsto U \times _ X X_{red} \]

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 $X_{red, spaces, {\acute{e}tale}}$. Consider the functor $U_{V, \psi } : (\mathit{Sch}/S)_{fppf} \to \textit{Sets}$ 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 85.7.1. 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 85.7.1. On the other hand, we have seen in the proof of Lemma 85.27.3 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. $\square$

Lemma 85.27.7. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{affine, {\acute{e}tale}}$ 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 85.27.6 and 85.13.3. $\square$


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