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66.27 Étale localization

Reading this section should be avoided at all cost.

Let $X \to Y$ be an étale morphism of algebraic spaces. Then $X$ is an object of $Y_{spaces, {\acute{e}tale}}$ and it is immediate from the definitions, see also the proof of Lemma 66.18.11, that
\begin{equation} \label{spaces-properties-equation-localize} X_{spaces, {\acute{e}tale}} = Y_{spaces, {\acute{e}tale}}/X \end{equation}

where the right hand side is the localization of the site $Y_{spaces, {\acute{e}tale}}$ at the object $X$, see Sites, Definition 7.25.1. Moreover, this identification is compatible with the structure sheaves by Lemma 66.26.1. Hence the ringed site $(X_{spaces, {\acute{e}tale}}, \mathcal{O}_ X)$ is identified with the localization of the ringed site $(Y_{spaces, {\acute{e}tale}}, \mathcal{O}_ Y)$ at the object $X$:
\begin{equation} \label{spaces-properties-equation-localize-ringed} (X_{spaces, {\acute{e}tale}}, \mathcal{O}_ X) = (Y_{spaces, {\acute{e}tale}}/X, \mathcal{O}_ Y|_{Y_{spaces, {\acute{e}tale}}/X}) \end{equation}

The localization of a ringed site used on the right hand side is defined in Modules on Sites, Definition 18.19.1.

Assume now $X \to Y$ is an étale morphism of algebraic spaces and $X$ is a scheme. Then $X$ is an object of $Y_{\acute{e}tale}$ and it follows that
\begin{equation} \label{spaces-properties-equation-localize-at-scheme} X_{\acute{e}tale}= Y_{\acute{e}tale}/X \end{equation}

\begin{equation} \label{spaces-properties-equation-localize-at-scheme-ringed} (X_{\acute{e}tale}, \mathcal{O}_ X) = (Y_{\acute{e}tale}/X, \mathcal{O}_ Y|_{Y_{\acute{e}tale}/X}) \end{equation}

as above.

Finally, if $X \to Y$ is an étale morphism of algebraic spaces and $X$ is an affine scheme, then $X$ is an object of $Y_{affine, {\acute{e}tale}}$ and
\begin{equation} \label{spaces-properties-equation-localize-at-affine} X_{affine, {\acute{e}tale}} = Y_{affine, {\acute{e}tale}}/X \end{equation}

\begin{equation} \label{spaces-properties-equation-localize-at-affine-ringed} (X_{affine, {\acute{e}tale}}, \mathcal{O}_ X) = (Y_{affine, {\acute{e}tale}}/X, \mathcal{O}_ Y|_{Y_{affine, {\acute{e}tale}}/X}) \end{equation}

as above.

Next, we show that these localizations are compatible with morphisms.

Lemma 66.27.1. Let $S$ be a scheme. Let

\[ \xymatrix{ U \ar[d]_ p \ar[r]_ g & V \ar[d]^ q \\ X \ar[r]^ f & Y } \]

be a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ étale. Via the identifications ( for $U \to X$ and $V \to Y$ the morphism of ringed topoi

\[ (g_{spaces, {\acute{e}tale}}, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{spaces, {\acute{e}tale}}), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (V_{spaces, {\acute{e}tale}}), \mathcal{O}_ V) \]

is $2$-isomorphic to the morphism $(f_{spaces, {\acute{e}tale}, c}, f_ c^\sharp )$ constructed in Modules on Sites, Lemma 18.20.2 starting with the morphism of ringed sites $(f_{spaces, {\acute{e}tale}}, f^\sharp )$ and the map $c : U \to V \times _ Y X$ corresponding to $g$.

Proof. The morphism $(f_{spaces, {\acute{e}tale}, c}, f_ c^\sharp )$ is defined as a composition $f' \circ j$ of a localization and a base change map. Similarly $g$ is a composition $U \to V \times _ Y X \to V$. Hence it suffices to prove the lemma in the following two cases: (1) $f = \text{id}$, and (2) $U = X \times _ Y V$. In case (1) the morphism $g : U \to V$ is étale, see Lemma 66.16.6. Hence $(g_{spaces, {\acute{e}tale}}, g^\sharp )$ is a localization morphism by the discussion surrounding Equations ( and ( which is exactly the content of the lemma in this case. In case (2) the morphism $g_{spaces, {\acute{e}tale}}$ comes from the morphism of ringed sites given by the functor $V_{spaces, {\acute{e}tale}} \to U_{spaces, {\acute{e}tale}}$, $V'/V \mapsto V' \times _ V U/U$ which is also what the morphism $f'$ is defined by, see Sites, Lemma 7.28.1. We omit the verification that $(f')^\sharp = g^\sharp $ in this case (both are the restriction of $f^\sharp $ to $U_{spaces, {\acute{e}tale}}$). $\square$

Lemma 66.27.2. Same notation and assumptions as in Lemma 66.27.1 except that we also assume $U$ and $V$ are schemes. Via the identifications ( for $U \to X$ and $V \to Y$ the morphism of ringed topoi

\[ (g_{small}, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (V_{\acute{e}tale}), \mathcal{O}_ V) \]

is $2$-isomorphic to the morphism $(f_{small, s}, f_ s^\sharp )$ constructed in Modules on Sites, Lemma 18.22.3 starting with $(f_{small}, f^\sharp )$ and the map $s : h_ U \to f_{small}^{-1}h_ V$ corresponding to $g$.

Proof. Note that $(g_{small}, g^\sharp )$ is $2$-isomorphic as a morphism of ringed topoi to the morphism of ringed topoi associated to the morphism of ringed sites $(g_{spaces, {\acute{e}tale}}, g^\sharp )$. Hence we conclude by Lemma 66.27.1 and Modules on Sites, Lemma 18.22.4. $\square$

Finally, we discuss the relationship between sheaves of sets on the small étale site $Y_{\acute{e}tale}$ of an algebraic space $Y$ and algebraic spaces étale over $Y$. Let $S$ be a scheme and let $Y$ be an algebraic space over $S$. Let $\mathcal{F}$ be an object of $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. Consider the functor

\[ X : (\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets} \]

defined by the rule

\[ X(T) = \{ (y, s) \mid y : T \to Y\text{ is a morphism over }S\text{ and } s \in \Gamma (T, y_{small}^{-1}\mathcal{F})\} \]

Given a morphism $g : T' \to T$ the restriction map sends $(y, s)$ to $(y \circ g, g_{small}^{-1}s)$. This makes sense as $y_{small} \circ g_{small} = (y \circ g)_{small}$ by Lemma 66.18.8.

Lemma 66.27.3. Let $S$ be a scheme and let $Y$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of sets on $Y_{\acute{e}tale}$. Provided a set theoretic condition is satisfied (see proof) the functor $X$ associated to $\mathcal{F}$ above is an algebraic space and there is an étale morphism $f : X \to Y$ of algebraic spaces such that $\mathcal{F} = f_{small, *}*$ where $*$ is the final object of the category $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ (constant sheaf with value a singleton).

Proof. Let us prove that $X$ is a sheaf for the fppf topology. Namely, suppose that $\{ g_ i : T_ i \to T\} $ is a covering of $(\mathit{Sch}/S)_{fppf}$ and $(y_ i, s_ i) \in X(T_ i)$ satisfy the glueing condition, i.e., the restriction of $(y_ i, s_ i)$ and $(y_ j, s_ j)$ to $T_ i \times _ T T_ j$ agree. Then since $Y$ is a sheaf for the fppf topology, we see that the $y_ i$ give rise to a unique morphism $y : T \to Y$ such that $y_ i = y \circ g_ i$. Then we see that $y_{i, small}^{-1}\mathcal{F} = g_{i, small}^{-1}y_{small}^{-1}\mathcal{F}$. Hence the sections $s_ i$ glue uniquely to a section of $y_{small}^{-1}\mathcal{F}$ by Étale Cohomology, Lemma 59.39.2.

The construction that sends $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}))$ to $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ preserves finite limits and all colimits since each of the functors $y_{small}^{-1}$ have this property. Of course, if $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$, then the construction sends the representable sheaf $h_ V$ on $Y_{\acute{e}tale}$ to the representable functor represented by $V$.

By Sites, Lemma 7.12.5 we can find a set $I$, for each $i \in I$ an object $V_ i$ of $Y_{\acute{e}tale}$ and a surjective map of sheaves

\[ \coprod h_{V_ i} \longrightarrow \mathcal{F} \]

on $Y_{\acute{e}tale}$. The set theoretic condition we need is that the index set $I$ is not too large1. Then $V = \coprod V_ i$ is an object of $(\mathit{Sch}/S)_{fppf}$ and therefore an object of $Y_{\acute{e}tale}$ and we have a surjective map $h_ V \to \mathcal{F}$.

Observe that the product of $h_ V$ with itself in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ is $h_{V \times _ Y V}$. Consider the fibre product

\[ h_ V \times _\mathcal {F} h_ V \subset h_{V \times _ Y V} \]

There is an open subscheme $R$ of $V \times _ Y V$ such that $h_ V \times _\mathcal {F} h_ V = h_ R$, see Lemma 66.20.1 (small detail omitted). By the Yoneda lemma we obtain two morphisms $s, t : R \to V$ in $Y_{\acute{e}tale}$ and we find a coequalizer diagram

\[ \xymatrix{ h_ R \ar@<1ex>[r] \ar@<-1ex>[r] & h_ V \ar[r] & \mathcal{F} } \]

in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. Of course the morphisms $s, t$ are étale and define an étale equivalence relation $(t, s) : R \to V \times _ S V$.

By the discussion in the preceding two paragraphs we find a coequalizer diagram

\[ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & V \ar[r] & X } \]

in $(\mathit{Sch}/S)_{fppf}$. Thus $X = V/R$ is an algebraic space by Spaces, Theorem 65.10.5. The other statements follow readily from this; details omitted. $\square$

[1] It suffices if the supremum of the cardinalities of the stalks of $\mathcal{F}$ at geometric points of $Y$ is bounded by the size of some object of $(\mathit{Sch}/S)_{fppf}$.

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