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

on $Y_{\acute{e}tale}$. The set theoretic condition we need is that the index set $I$ is not too large^{1}. 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

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

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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #8459 by ZL on