The Stacks project

Lemma 78.9.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $P \to R$ be monomorphism of schemes. Assume that

  1. $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t}P})$ is a groupoid scheme,

  2. $s|_ P, t|_ P : P \to U$ are finite locally free,

  3. $j|_ P : P \to U \times _ S U$ is a monomorphism.

  4. $U$ is affine, and

  5. $j : R \to U \times _ S U$ is separated and locally quasi-finite,

Then $U/P$ is representable by an affine scheme $\overline{U}$, the quotient morphism $U \to \overline{U}$ is finite locally free, and $P = U \times _{\overline{U}} U$. Moreover, $R$ is the restriction of a groupoid scheme $(\overline{U}, \overline{R}, \overline{s}, \overline{t}, \overline{c})$ on $\overline{U}$ via the quotient morphism $U \to \overline{U}$.

Proof. Conditions (1), (2), (3), and (4) and Groupoids, Proposition 39.23.9 imply the affine scheme $\overline{U}$ representing $U/P$ exists, the morphism $U \to \overline{U}$ is finite locally free, and $P = U \times _{\overline{U}} U$. The identification $P = U \times _{\overline{U}} U$ is such that $t|_ P = \text{pr}_0$ and $s|_ P = \text{pr}_1$, and such that composition is equal to $\text{pr}_{02} : U \times _{\overline{U}} U \times _{\overline{U}} U \to U \times _{\overline{U}} U$. A product of finite locally free morphisms is finite locally free (see Spaces, Lemma 63.5.7 and Morphisms, Lemmas 29.48.4 and 29.48.3). To get $\overline{R}$ we are going to descend the scheme $R$ via the finite locally free morphism $U \times _ S U \to \overline{U} \times _ S \overline{U}$. Namely, note that

\[ (U \times _ S U) \times _{(\overline{U} \times _ S \overline{U})} (U \times _ S U) = P \times _ S P \]

by the above. Thus giving a descent datum (see Descent, Definition 35.31.1) for $R / U \times _ S U / \overline{U} \times _ S \overline{U}$ consists of an isomorphism

\[ \varphi : R \times _{(U \times _ S U), t \times t} (P \times _ S P) \longrightarrow (P \times _ S P) \times _{s \times s, (U \times _ S U)} R \]

over $P \times _ S P$ satisfying a cocycle condition. We define $\varphi $ on $T$-valued points by the rule

\[ \varphi : (r, (p, p')) \longmapsto ((p, p'), p^{-1} \circ r \circ p') \]

where the composition is taken in the groupoid category $(U(T), R(T), s, t, c)$. This makes sense because for $(r, (p, p'))$ to be a $T$-valued point of the source of $\varphi $ it needs to be the case that $t(r) = t(p)$ and $s(r) = t(p')$. Note that this map is an isomorphism with inverse given by $((p, p'), r') \mapsto (p \circ r' \circ (p')^{-1}, (p, p'))$. To check the cocycle condition we have to verify that $\varphi _{02} = \varphi _{12} \circ \varphi _{01}$ as maps over

\[ (U \times _ S U) \times _{(\overline{U} \times _ S \overline{U})} (U \times _ S U) \times _{(\overline{U} \times _ S \overline{U})} (U \times _ S U) = (P \times _ S P) \times _{s \times s, (U \times _ S U), t \times t} (P \times _ S P) \]

By explicit calculation we see that

\[ \begin{matrix} \varphi _{02} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), (p_1 \circ p_2)^{-1} \circ r \circ (p_1' \circ p_2')) \\ \varphi _{01} & (r, (p_1, p_1'), (p_2, p_2')) & \mapsto & ((p_1, p_1'), p_1^{-1} \circ r \circ p_1', (p_2, p_2')) \\ \varphi _{12} & ((p_1, p_1'), r, (p_2, p_2')) & \mapsto & ((p_1, p_1'), (p_2, p_2'), p_2^{-1} \circ r \circ p_2') \end{matrix} \]

(with obvious notation) which implies what we want. As $j$ is separated and locally quasi-finite by (5) we may apply More on Morphisms, Lemma 37.50.1 to get a scheme $\overline{R} \to \overline{U} \times _ S \overline{U}$ and an isomorphism

\[ R \to \overline{R} \times _{(\overline{U} \times _ S \overline{U})} (U \times _ S U) \]

which identifies the descent datum $\varphi $ with the canonical descent datum on $\overline{R} \times _{(\overline{U} \times _ S \overline{U})} (U \times _ S U)$, see Descent, Definition 35.31.10.

Since $U \times _ S U \to \overline{U} \times _ S \overline{U}$ is finite locally free we conclude that $R \to \overline{R}$ is finite locally free as a base change. Hence $R \to \overline{R}$ is surjective as a map of sheaves on $(\mathit{Sch}/S)_{fppf}$. Our choice of $\varphi $ implies that given $T$-valued points $r, r' \in R(T)$ these have the same image in $\overline{R}$ if and only if $p^{-1} \circ r \circ p'$ for some $p, p' \in P(T)$. Thus $\overline{R}$ represents the sheaf

\[ T \longmapsto \overline{R(T)} = P(T)\backslash R(T)/P(T) \]

with notation as in the discussion preceding the lemma. Hence we can define the groupoid structure on $(\overline{U} = U/P, \overline{R} = P\backslash R/P)$ exactly as in the discussion of the “plain” groupoid case. It follows from this that $(U, R, s, t, c)$ is the pullback of this groupoid structure via the morphism $U \to \overline{U}$. This concludes the proof. $\square$


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