The Stacks project

66.14 Obtaining a scheme

We have used in the previous section that the quotient $U/R$ of an affine scheme $U$ by an equivalence relation $R$ is a scheme if the morphisms $s, t : R \to U$ are finite étale. This is a special case of the following result.

Proposition 66.14.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume

  1. $s, t : R \to U$ finite locally free,

  2. $j = (t, s)$ is an equivalence, and

  3. for a dense set of points $u \in U$ the $R$-equivalence class $t(s^{-1}(\{ u\} ))$ is contained in an affine open of $U$.

Then there exists a finite locally free morphism $U \to M$ of schemes over $S$ such that $R = U \times _ M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology.

Proof. By assumption (3) and Groupoids, Lemma 39.24.1 we can find an open covering $U = \bigcup U_ i$ such that each $U_ i$ is an $R$-invariant affine open of $U$. Set $R_ i = R|_{U_ i}$. Consider the fppf sheaves $F = U/R$ and $F_ i = U_ i/R_ i$. By Spaces, Lemma 65.10.2 the morphisms $F_ i \to F$ are representable and open immersions. By Groupoids, Proposition 39.23.9 the sheaves $F_ i$ are representable by affine schemes. If $T$ is a scheme and $T \to F$ is a morphism, then $V_ i = F_ i \times _ F T$ is open in $T$ and we claim that $T = \bigcup V_ i$. Namely, fppf locally on $T$ we can lift $T \to F$ to a morphism $f : T \to U$ and in that case $f^{-1}(U_ i) \subset V_ i$. Hence we conclude that $F$ is representable by a scheme, see Schemes, Lemma 26.15.4. $\square$

For example, if $U$ is isomorphic to a locally closed subscheme of an affine scheme or isomorphic to a locally closed subscheme of $\text{Proj}(A)$ for some graded ring $A$, then the third assumption holds by Properties, Lemma 28.29.5. In particular we can apply this to free actions of finite groups and finite group schemes on quasi-affine or quasi-projective schemes. For example, the quotient $X/G$ of a quasi-projective variety $X$ by a free action of a finite group $G$ is a scheme. Here is a detailed statement.

Lemma 66.14.2. Let $S$ be a scheme. Let $G \to S$ be a group scheme. Let $X \to S$ be a morphism of schemes. Let $a : G \times _ S X \to X$ be an action. Assume that

  1. $G \to S$ is finite locally free,

  2. the action $a$ is free,

  3. $X \to S$ is affine, or quasi-affine, or projective, or quasi-projective, or $X$ is isomorphic to an open subscheme of an affine scheme, or $X$ is isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$, or $G \to S$ is radicial.

Then the fppf quotient sheaf $X/G$ is a scheme and $X \to X/G$ is an fppf $G$-torsor.

Proof. We first show that $X/G$ is a scheme. Since the action is free the morphism $j = (a, \text{pr}) : G \times _ S X \to X \times _ S X$ is a monomorphism and hence an equivalence relation, see Groupoids, Lemma 39.10.3. The maps $s, t : G \times _ S X \to X$ are finite locally free as we've assumed that $G \to S$ is finite locally free. To conclude it now suffices to prove the last assumption of Proposition 66.14.1 holds. Since the action of $G$ is over $S$ it suffices to prove that any finite set of points in a fibre of $X \to S$ is contained in an affine open of $X$. If $X$ is isomorphic to an open subscheme of an affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$ for some graded ring $A$ this follows from Properties, Lemma 28.29.5. If $X \to S$ is affine, or quasi-affine, or projective, or quasi-projective, we may replace $S$ by an affine open and we get back to the case we just dealt with. If $G \to S$ is radicial, then the orbits of points on $X$ under the action of $G$ are singletons and the condition trivially holds. Some details omitted.

To see that $X \to X/G$ is an fppf $G$-torsor (Groupoids, Definition 39.11.3) we have to show that $G \times _ S X \to X \times _{X/G} X$ is an isomorphism and that $X \to X/G$ fppf locally has sections. The second part is clear from the fact that $X \to X/G$ is surjective as a map of fppf sheaves (by construction). The first part follows from the isomorphism $R = U \times _ M U$ in the conclusion of Proposition 66.14.1 (note that $R = G \times _ S X$ in our case). $\square$

Lemma 66.14.3. Notation and assumptions as in Proposition 66.14.1. Then

  1. if $U$ is quasi-separated over $S$, then $U/R$ is quasi-separated over $S$,

  2. if $U$ is quasi-separated, then $U/R$ is quasi-separated,

  3. if $U$ is separated over $S$, then $U/R$ is separated over $S$,

  4. if $U$ is separated, then $U/R$ is separated, and

  5. add more here.

Similar results hold in the setting of Lemma 66.14.2.

Proof. Since $M$ represents the quotient sheaf we have a cartesian diagram

\[ \xymatrix{ R \ar[r]_-j \ar[d] & U \times _ S U \ar[d] \\ M \ar[r] & M \times _ S M } \]

of schemes. Since $U \times _ S U \to M \times _ S M$ is surjective finite locally free, to show that $M \to M \times _ S M$ is quasi-compact, resp. a closed immersion, it suffices to show that $j : R \to U \times _ S U$ is quasi-compact, resp. a closed immersion, see Descent, Lemmas 35.23.1 and 35.23.19. Since $j : R \to U \times _ S U$ is a morphism over $U$ and since $R$ is finite over $U$, we see that $j$ is quasi-compact as soon as the projection $U \times _ S U \to U$ is quasi-separated (Schemes, Lemma 26.21.14). Since $j$ is a monomorphism and locally of finite type, we see that $j$ is a closed immersion as soon as it is proper (√Čtale Morphisms, Lemma 41.7.2) which will be the case as soon as the projection $U \times _ S U \to U$ is separated (Morphisms, Lemma 29.41.7). This proves (1) and (3). To prove (2) and (4) we replace $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Definition 66.3.1. Since Lemma 66.14.2 is proved through an application of Proposition 66.14.1 the final statement is clear too. $\square$

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