The Stacks project

79.10 Final bootstrap

The following result goes quite a bit beyond the earlier results.

Theorem 79.10.1. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Any one of the following conditions implies that $F$ is an algebraic space:

  1. $F = U/R$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation,

  2. $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation,

  3. $F$ is a sheaf and there exists an algebraic space $U$ and a morphism $U \to F$ which is representable by algebraic spaces, surjective, flat and locally of finite presentation,

  4. $F$ is a sheaf and there exists a scheme $U$ and a morphism $U \to F$ which is representable by algebraic spaces or schemes, surjective, flat and locally of finite presentation,

  5. $F$ is a sheaf, $\Delta _ F$ is representable by algebraic spaces, and there exists an algebraic space $U$ and a morphism $U \to F$ which is surjective, flat, and locally of finite presentation, or

  6. $F$ is a sheaf, $\Delta _ F$ is representable, and there exists a scheme $U$ and a morphism $U \to F$ which is surjective, flat, and locally of finite presentation.

Proof. Trivial observations: (6) is a special case of (5) and (4) is a special case of (3). We first prove that cases (5) and (3) reduce to case (1). Namely, by bootstrapping the diagonal Lemma 79.5.3 we see that (3) implies (5). In case (5) we set $R = U \times _ F U$ which is an algebraic space by assumption. Moreover, by assumption both projections $s, t : R \to U$ are surjective, flat and locally of finite presentation. The map $j : R \to U \times _ S U$ is clearly an equivalence relation. By Lemma 79.4.6 the map $U \to F$ is a surjection of sheaves. Thus $F = U/R$ which reduces us to case (1).

Next, we show that (1) reduces to (2). Namely, let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation. Choose a scheme $U'$ and a surjective étale morphism $U' \to U$. Let $R' = R|_{U'}$ be the restriction of $R$ to $U'$. By Groupoids in Spaces, Lemma 77.19.6 we see that $U/R = U'/R'$. Since $s', t' : R' \to U'$ are also flat and locally of finite presentation (see More on Groupoids in Spaces, Lemma 78.8.1) this reduces us to the case where $U$ is a scheme. As $j$ is an equivalence relation we see that $j$ is a monomorphism. As $s : R \to U$ is locally of finite presentation we see that $j : R \to U \times _ S U$ is locally of finite type, see Morphisms of Spaces, Lemma 66.23.6. By Morphisms of Spaces, Lemma 66.27.10 we see that $j$ is locally quasi-finite and separated. Hence if $U$ is a scheme, then $R$ is a scheme by Morphisms of Spaces, Proposition 66.50.2. Thus we reduce to proving the theorem in case (2).

Assume $F = U/R$ where $(U, R, s, t, c)$ is a groupoid scheme over $S$ such that $s, t$ are flat and locally of finite presentation, and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation. By Lemma 79.8.1 we reduce to that case where $s, t$ are flat, locally of finite presentation, and locally quasi-finite. Let $U = \bigcup _{i \in I} U_ i$ be an affine open covering (with index set $I$ of cardinality $\leq $ than the size of $U$ to avoid set theoretic problems later – most readers can safely ignore this remark). Let $(U_ i, R_ i, s_ i, t_ i, c_ i)$ be the restriction of $R$ to $U_ i$. It is clear that $s_ i, t_ i$ are still flat, locally of finite presentation, and locally quasi-finite as $R_ i$ is the open subscheme $s^{-1}(U_ i) \cap t^{-1}(U_ i)$ of $R$ and $s_ i, t_ i$ are the restrictions of $s, t$ to this open. By Lemma 79.7.1 (or the simpler Spaces, Lemma 64.10.2) the map $U_ i/R_ i \to U/R$ is representable by open immersions. Hence if we can show that $F_ i = U_ i/R_ i$ is an algebraic space, then $\coprod _{i \in I} F_ i$ is an algebraic space by Spaces, Lemma 64.8.4. As $U = \bigcup U_ i$ is an open covering it is clear that $\coprod F_ i \to F$ is surjective. Thus it follows that $U/R$ is an algebraic space, by Spaces, Lemma 64.8.5. In this way we reduce to the case where $U$ is affine and $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j$ is an equivalence.

Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite, and $j$ is an equivalence relation. Choose $u \in U$. We apply More on Groupoids in Spaces, Lemma 78.15.13 to $u \in U, R, s, t, c$. We obtain an affine scheme $U'$, an étale morphism $g : U' \to U$, a point $u' \in U'$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ is quasi-split over $u'$. Note that the image $g(U')$ is open as $g$ is étale and contains $u$. Hence, repeatedly applying the lemma, we can find finitely many points $u_ i \in U$, $i = 1, \ldots , n$, affine schemes $U'_ i$, étale morphisms $g_ i : U_ i' \to U$, points $u'_ i \in U'_ i$ with $g(u'_ i) = u_ i$ such that (a) each restriction $R'_ i$ is quasi-split over some point in $U'_ i$ and (b) $U = \bigcup _{i = 1, \ldots , n} g_ i(U'_ i)$. Now we rerun the last part of the argument in the preceding paragraph: Using Lemma 79.7.1 (or the simpler Spaces, Lemma 64.10.2) the map $U'_ i/R'_ i \to U/R$ is representable by open immersions. If we can show that $F_ i = U'_ i/R'_ i$ is an algebraic space, then $\coprod _{i \in I} F_ i$ is an algebraic space by Spaces, Lemma 64.8.4. As $\{ g_ i : U'_ i \to U\} $ is an étale covering it is clear that $\coprod F_ i \to F$ is surjective. Thus it follows that $U/R$ is an algebraic space, by Spaces, Lemma 64.8.5. In this way we reduce to the case where $U$ is affine and $s, t$ are flat, locally of finite presentation, and locally quasi-finite, $j$ is an equivalence, and $R$ is quasi-split over $u$ for some $u \in U$.

Assume $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, $u \in U$ such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation and $R$ is quasi-split over $u$. Let $P \subset R$ be a quasi-splitting of $R$ over $u$. By Lemma 79.9.1 we see that $(U, R, s, t, c)$ is the restriction of a groupoid $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ by a surjective finite locally free morphism $U \to \overline{U}$ such that $P = U \times _{\overline{U}} U$. Note that $s$ admits a factorization

\[ R = U \times _{\overline{U}, \overline{t}} \overline{R} \times _{\overline{s}, \overline{U}} U \xrightarrow {\text{pr}_{23}} \overline{R} \times _{\overline{s}, \overline{U}} U \xrightarrow {\text{pr}_2} U \]

The map $\text{pr}_2$ is the base change of $\overline{s}$, and the map $\text{pr}_{23}$ is a base change of the surjective finite locally free map $U \to \overline{U}$. Since $s$ is flat, locally of finite presentation, and locally quasi-finite and since $\text{pr}_{23}$ is surjective finite locally free (as a base change of such), we conclude that $\text{pr}_2$ is flat, locally of finite presentation, and locally quasi-finite by Descent, Lemmas 35.27.1 and 35.28.1 and Morphisms, Lemma 29.20.18. Since $\text{pr}_2$ is the base change of the morphism $\overline{s}$ by $U \to \overline{U}$ and $\{ U \to \overline{U}\} $ is an fppf covering we conclude $\overline{s}$ is flat, locally of finite presentation, and locally quasi-finite, see Descent, Lemmas 35.23.15, 35.23.11, and 35.23.24. The same goes for $\overline{t}$. Consider the commutative diagram

\[ \xymatrix{ U \times _{\overline{U}} U \ar@{=}[r] \ar[rd] & P \ar[r] \ar[d] & R \ar[d] \\ & \overline{U} \ar[r]^{\overline{e}} & \overline{R} } \]

It is a general fact about restrictions that the outer four corners form a cartesian diagram. By the equality we see the inner square is cartesian. Since $P$ is open in $R$ (by definition of a quasi-splitting) we conclude that $\overline{e}$ is an open immersion by Descent, Lemma 35.23.16. An application of Groupoids, Lemma 39.20.5 shows that $U/R = \overline{U}/\overline{R}$. Hence we have reduced to the case where $(U, R, s, t, c)$ is a groupoid scheme over $S$, with $U$ affine, $u \in U$ such that $s, t$ are flat, locally of finite presentation, and locally quasi-finite and $j = (t, s) : R \to U \times _ S U$ is an equivalence relation and $e : U \to R$ is an open immersion!

But of course, if $e$ is an open immersion and $s, t$ are flat and locally of finite presentation then the morphisms $t, s$ are étale. For example you can see this by applying More on Groupoids, Lemma 40.4.1 which shows that $\Omega _{R/U} = 0$ which in turn implies that $s, t : R \to U$ is G-unramified (see Morphisms, Lemma 29.35.2), which in turn implies that $s, t$ are étale (see Morphisms, Lemma 29.36.16). And if $s, t$ are étale then finally $U/R$ is an algebraic space by Spaces, Theorem 64.10.5. $\square$


Comments (2)

Comment #7111 by F. Liu on

1) A typos in the 4th paragraph of the proof: in the statement "... the image is open as is étale and contains . ", should be . 2) In the 5th paragraph it was claimed that are the base changes of the morphisms by . This seems incorrect. Anyway, the desired property of can be deduced as follows. Note that admits a factorization . The second map is the mentioned base change of , and the first map is a base change of the surjective finite locally free map . By 'descent' we get the desired property of from the same property of .


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