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 (Étale 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. $\square$


Comments (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07S5. Beware of the difference between the letter 'O' and the digit '0'.