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39.14. Finite groupoids

A groupoid scheme $(U, R, s, t, c)$ is sometimes called finite if the morphisms $s$ and $t$ are finite. This is potentially confusing as it doesn't imply that $U$ or $R$ or the quotient sheaf $U/R$ are finite over anything.

Lemma 39.14.1. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists a sequence of $R$-invariant closed subschemes $$ U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots $$ such that $\bigcap Z_r = \emptyset$ and such that $s^{-1}(Z_{r - 1}) \setminus s^{-1}(Z_r) \to Z_{r - 1} \setminus Z_r$ is finite locally free of rank $r$.

Proof. Let $\{Z_r\}$ be the stratification of $U$ given by the Fitting ideals of the finite type quasi-coherent modules $s_*\mathcal{O}_R$. See Divisors, Lemma 30.9.6. Since the identity $e : U \to R$ is a section to $s$ we see that $s_*\mathcal{O}_R$ contains $\mathcal{O}_S$ as a direct summand. Hence $U = Z_{-1} = Z_0$ (details omitted). Since formation of Fitting ideals commutes with base change (More on Algebra, Lemma 15.8.4) we find that $s^{-1}(Z_r)$ corresponds to the $r$th Fitting ideal of $\text{pr}_{1, *}\mathcal{O}_{R \times_{s, U, t} R}$ because the lower right square of diagram (39.3.0.1) is cartesian. Using the fact that the lower left square is also cartesian we conclude that $s^{-1}(Z_r) = t^{-1}(Z_r)$, in other words $Z_r$ is $R$-invariant. The morphism $s^{-1}(Z_{r - 1}) \setminus s^{-1}(Z_r) \to Z_{r - 1} \setminus Z_r$ is finite locally free of rank $r$ because the module $s_*\mathcal{O}_R$ pulls back to a finite locally free module of rank $r$ on $Z_{r - 1} \setminus Z_r$ by Divisors, Lemma 30.9.6. $\square$

Lemma 39.14.2. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite. There exists an open subscheme $W \subset U$ and a closed subscheme $W' \subset W$ such that

  1. $W$ and $W'$ are $R$-invariant,
  2. $U = t(s^{-1}(\overline{W}))$ set theoretically,
  3. $W$ is a thickening of $W'$, and
  4. the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$ are finite locally free.

Proof. Consider the stratification $U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots$ of Lemma 39.14.1.

We will construct disjoint unions $W = \coprod_{r \geq 1} W_r$ and $W' = \coprod_{r \geq 1} W'_r$ with each $W'_r \to W_r$ a thickening of $R$-invariant subschemes of $U$ such that the morphisms $s_r', t_r'$ of the restrictions $(W_r', R_r', s_r', t_r', c_r')$ are finite locally free of rank $r$. To begin we set $W_1 = W'_1 = U \setminus Z_1$. This is an $R$-invariant open subscheme of $U$, it is true that $W_0$ is a thickening of $W'_0$, and the maps $s_1'$, $t_1'$ of the restriction $(W_1', R_1', s_1', t_1', c_1')$ are isomorphisms, i.e., finite locally free of rank $1$. Moreover, every point of $U \setminus Z_1$ is in $t(s^{-1}(\overline{W_1}))$.

Assume we have found subschemes $W'_r \subset W_r \subset U$ for $r \leq n$ such that

  1. $W_1, \ldots, W_n$ are disjoint,
  2. $W_r$ and $W_r'$ are $R$-invariant,
  3. $U \setminus Z_n \subset \bigcup_{r \leq n} t(s^{-1}(\overline{W_r}))$ set theoretically,
  4. $W_r$ is a thickening of $W'_r$,
  5. the maps $s_r'$, $t_r'$ of the restriction $(W_r', R_r', s_r', t_r', c_r')$ are finite locally free of rank $r$.

Then we set $$ W_{n + 1} = Z_n \setminus \left( Z_{n + 1} \cup \bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r})) \right) $$ set theoretically and $$ W'_{n + 1} = Z_n \setminus \left( Z_{n + 1} \cup \bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r})) \right) $$ scheme theoretically. Then $W_{n + 1}$ is an $R$-invariant open subscheme of $U$ because $Z_{n + 1} \setminus \overline{U \setminus Z_{n + 1}}$ is open in $U$ and $\overline{U \setminus Z_{n + 1}}$ is contained in the closed subset $\bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r}))$ we are removing by property (3) and the fact that $t$ is a closed morphism. It is clear that $W'_{n + 1}$ is a closed subscheme of $W_{n + 1}$ with the same underlying topological space. Finally, properties (1), (2) and (3) are clear and property (5) follows from Lemma 39.14.1.

By Lemma 39.14.1 we have $\bigcap Z_r = \emptyset$. Hence every point of $U$ is contained in $U \setminus Z_n$ for some $n$. Thus we see that $U = \bigcup_{r \geq 1} t(s^{-1}(\overline{W_r}))$ set theoretically and we see that (2) holds. Thus $W' \subset W$ satisfy (1), (2), (3), and (4). $\square$

Let $(U, R, s, t, c)$ be a groupoid scheme. Given a point $u \in U$ the $R$-orbit of $u$ is the subset $t(s^{-1}(\{u\}))$ of $U$.

Lemma 39.14.3. In Lemma 39.14.2 assume in addition that $s$ and $t$ are of finite presentation. Then

  1. the morphism $W' \to W$ is of finite presentation, and
  2. if $u \in U$ is a point whose $R$-orbit consists of generic points of irreducible components of $U$, then $u \in W$.

Proof. In this case the stratification $U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots$ of Lemma 39.14.1 is given by closed immersions $Z_k \to U$ of finite presentation, see Divisors, Lemma 30.9.6. Part (1) follows immediately from this as $W' \to W$ is locally given by intersecting the open $W$ by $Z_r$. To see part (2) let $\{u_1, \ldots, u_n\}$ be the orbit of $u$. Since the closed subschemes $Z_k$ are $R$-invariant and $\bigcap Z_k = \emptyset$, we find an $k$ such that $u_i \in Z_k$ and $u_i \not \in Z_{k + 1}$ for all $i$. The image of $Z_k \to U$ and $Z_{k + 1} \to U$ is locally constructible (Morphisms, Theorem 28.21.3). Since $u_i \in U$ is a generic point of an irreducible component of $U$, there exists an open neighbourhood $U_i$ of $u_i$ which is contained in $Z_k \setminus Z_{k + 1}$ set theoretically (Properties, Lemma 27.2.2). In the proof of Lemma 39.14.2 we have constructed $W$ as a disjoint union $\coprod W_r$ with $W_r \subset Z_{r - 1} \setminus Z_r$ such that $U = \bigcup t(s^{-1}(\overline{W_r}))$. As $\{u_1, \ldots, u_n\}$ is an $R$-orbit we see that $u \in t(s^{-1}(\overline{W_r}))$ implies $u_i \in \overline{W_r}$ for some $i$ which implies $U_i \cap W_r \not = \emptyset$ which implies $r = k$. Thus we conclude that $u$ is in $$ W_{k + 1} = Z_k \setminus \left( Z_{k + 1} \cup \bigcup\nolimits_{r \leq k} t(s^{-1}(\overline{W_r})) \right) $$ as desired. $\square$

Lemma 39.14.4. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite and of finite presentation and $U$ quasi-separated. Let $u_1, \ldots, u_m \in U$ be points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \subset V \subset U$ such that

  1. $u_1, \ldots, u_m \in V'$,
  2. $V$ is open in $U$,
  3. $V'$ and $V$ are affine,
  4. $V' \subset V$ is a thickening of finite presentation,
  5. the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free.

Proof. Let $W' \subset W \subset U$ be as in Lemma 39.14.2. By Lemma 39.14.3 we get $u_j \in W$ and that $W' \to W$ is a thickening of finite presentation. By Limits, Lemma 31.11.3 it suffices to find an $R$-invariant affine open subscheme $V'$ of $W'$ containing $u_j$ (because then we can let $V \subset W$ be the corresponding open subscheme which will be affine). Thus we may replace $(U, R, s, t, c)$ by the restriction $(W', R', s', t', c')$ to $W'$. In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$ whose morphisms $s$ and $t$ are finite locally free. By Properties, Lemma 27.29.1 we can find an affine open containing the union of the orbits of $u_1, \ldots, u_m$. Finally, we can apply Groupoids, Lemma 38.24.1 to conclude. $\square$

The following lemma is a special case of Lemma 39.14.4 but we redo the argument as it is slightly easier in this case (it avoids using Lemma 39.14.3).

Lemma 39.14.5. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \ldots, u_m \in U$ points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \subset V \subset U$ such that

  1. $u_1, \ldots, u_m \in V'$,
  2. $V$ is open in $U$,
  3. $V'$ and $V$ are affine,
  4. $V' \subset V$ is a thickening,
  5. the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free.

Proof. Let $\{u_{j1}, \ldots, u_{jn_j}\}$ be the orbit of $u_j$. Let $W' \subset W \subset U$ be as in Lemma 39.14.2. Since $U = t(s^{-1}(\overline{W}))$ we see that at least one $u_{ji} \in \overline{W}$. Since $u_{ji}$ is a generic point of an irreducible component and $U$ locally Noetherian, this implies that $u_{ji} \in W$. Since $W$ is $R$-invariant, we conclude that $u_j \in W$ and in fact the whole orbit is contained in $W$. By Cohomology of Schemes, Lemma 29.13.3 it suffices to find an $R$-invariant affine open subscheme $V'$ of $W'$ containing $u_1, \ldots, u_m$ (because then we can let $V \subset W$ be the corresponding open subscheme which will be affine). Thus we may replace $(U, R, s, t, c)$ by the restriction $(W', R', s', t', c')$ to $W'$. In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$ whose morphisms $s$ and $t$ are finite locally free. By Properties, Lemma 27.29.1 we can find an affine open containing $\{u_{ij}\}$ (a locally Noetherian scheme is quasi-separated by Properties, Lemma 27.5.4). Finally, we can apply Groupoids, Lemma 38.24.1 to conclude. $\square$

Lemma 39.14.6. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$ with $s, t$ integral. Let $g : U' \to U$ be an integral morphism such that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. If $u' \in U'$ is contained in an $R'$-invariant affine open, then the image $u \in U$ is contained in an $R$-invariant affine open of $U$.

Proof. Let $W' \subset U'$ be an $R'$-invariant affine open. Set $\tilde R = U' \times_{g, U, t} R$ with maps $\text{pr}_0 : \tilde R \to U'$ and $h = s \circ \text{pr}_1 : \tilde R \to U$. Observe that $\text{pr}_0$ and $h$ are integral. It follows that $\tilde W = \text{pr}_0^{-1}(W')$ is affine. Since $W'$ is $R'$-invariant, the image $W = h(\tilde W)$ is set theoretically $R$-invariant and $\tilde W = h^{-1}(W)$ set theoretically (details omitted). Thus, if we can show that $W$ is open, then $W$ is a scheme and the morphism $\tilde W \to W$ is integral surjective which implies that $W$ is affine by Limits, Proposition 31.11.2. However, our assumption on orbits meeting $U'$ implies that $h : \tilde R \to U$ is surjective. Since an integral surjective morphism is submersive (Topology, Lemma 5.6.5 and Morphisms, Lemma 28.42.7) it follows that $W$ is open. $\square$

The following technical lemma produces ''almost'' invariant functions in the situation of a finite groupoid on a quasi-affine scheme.

Lemma 39.14.7. Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of finite presentation. Let $u_1, \ldots, u_m \in U$ be points whose $R$-orbits consist of generic points of irreducible components of $U$. Let $j : U \to \mathop{\rm Spec}(A)$ be an immersion. Let $I \subset A$ be an ideal such that $j(U) \cap V(I) = \emptyset$ and $V(I) \cup j(U)$ is closed in $\mathop{\rm Spec}(A)$. Then there exists an $h \in I$ such that $j^{-1}D(h)$ is an $R$-invariant affine open subscheme of $U$ containing $u_1, \ldots, u_m$.

Proof. Let $u_1, \ldots, u_m \in V' \subset V \subset U$ be as in Lemma 39.14.4. Since $U \setminus V$ is closed in $U$, $j$ an immersion, and $V(I) \cup j(U)$ is closed in $\mathop{\rm Spec}(A)$, we can find an ideal $J \subset I$ such that $V(J) = V(I) \cup j(U \setminus V)$. For example we can take the ideal of elements of $I$ which vanish on $j(U \setminus V)$. Thus we can replace $(U, R, s, t, c)$, $j : U \to \mathop{\rm Spec}(A)$, and $I$ by $(V', R', s', t', c')$, $j|_{V'} : V' \to \mathop{\rm Spec}(A)$, and $J$. In other words, we may assume that $U$ is affine and that $s$ and $t$ are finite locally free. Take any $f \in I$ which does not vanish at all the points in the $R$-orbits of $u_1, \ldots, u_m$ (Algebra, Lemma 10.14.2). Consider $$ g = \text{Norm}_s(t^\sharp(j^\sharp(f))) \in \Gamma(U, \mathcal{O}_U) $$ Since $f \in I$ and since $V(I) \cup j(U)$ is closed we see that $U \cap D(f) \to D(f)$ is a closed immersion. Hence $f^ng$ is the image of an element $h \in I$ for some $n > 0$. We claim that $h$ works. Namely, we have seen in Groupoids, Lemma 38.23.2 that $g$ is an $R$-invariant function, hence $D(g) \subset U$ is $R$-invariant. Since $f$ does not vanish on the orbit of $u_j$, the function $g$ does not vanish at $u_j$. Moreover, we have $V(g) \supset V(j^\sharp(f))$ and hence $j^{-1}D(h) = D(g)$. $\square$

Lemma 39.14.8. Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite, and $u, u' \in R$ are distinct points in the same orbit, then $u'$ is not a specialization of $u$.

Proof. Let $r \in R$ with $s(r) = u$ and $t(r) = u'$. If $u \leadsto u'$ then we can find a nontrivial specialization $r \leadsto r'$ with $s(r') = u'$, see Schemes, Lemma 25.19.8. Set $u'' = t(r')$. Note that $u'' \not = u'$ as there are no specializations in the fibres of a finite morphism. Hence we can continue and find a nontrivial specialization $r' \leadsto r''$ with $s(r'') = u''$, etc. This shows that the orbit of $u$ contains an infinite sequence $u \leadsto u' \leadsto u'' \leadsto \ldots$ of specializations which is nonsense as the orbit $t(s^{-1}(\{u\}))$ is finite. $\square$

Lemma 39.14.9. Let $j : V \to \mathop{\rm Spec}(A)$ be a quasi-compact immersion of schemes. Let $f \in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \cap V(f)$ is closed. Then $V$ is affine.

Proof. This follows from Morphisms, Lemma 28.11.14 but we will also give a direct proof. Let $A' = \Gamma(V, \mathcal{O}_V)$. Then $j' : V \to \mathop{\rm Spec}(A')$ is a quasi-compact open immersion, see Properties, Lemma 27.18.3. Let $f' \in A'$ be the image of $f$. Then $(j')^{-1}D(f') = j^{-1}D(f)$ is affine. On the other hand, $j'(V) \cap V(f')$ is a subscheme of $\mathop{\rm Spec}(A')$ which maps isomorphically to the closed subscheme $j(V) \cap V(f)$ of $\mathop{\rm Spec}(A)$. Hence it is closed in $\mathop{\rm Spec}(A')$ for example by Schemes, Lemma 25.21.12. Thus we may replace $A$ by $A'$ and assume that $j$ is an open immersion and $A = \Gamma(V, \mathcal{O}_V)$.

In this case we claim that $j(V) = \mathop{\rm Spec}(A)$ which finishes the proof. If not, then we can find a principal affine open $D(g) \subset \mathop{\rm Spec}(A)$ which meets the complement and avoids the closed subset $j(V) \cap V(f)$. Note that $j$ maps $j^{-1}D(f)$ isomorphically onto $D(f)$, see Properties, Lemma 27.18.2. Hence $D(g)$ meets $V(f)$. On the other hand, $j^{-1}D(g)$ is a principal open of the affine open $j^{-1}D(f)$ hence affine. Hence by Properties, Lemma 27.18.2 again we see that $D(g)$ is isomorphic to $j^{-1}D(g) \subset j^{-1}D(f)$ which implies that $D(g) \subset D(f)$. This contradiction finishes the proof. $\square$

Lemma 39.14.10. Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \in U$. Assume

  1. $s, t$ are finite morphisms,
  2. $U$ is separated and locally Noetherian,
  3. $\dim(\mathcal{O}_{U, u'}) \leq 1$ for every point $u'$ in the orbit of $u$.

Then $u$ is contained in an $R$-invariant affine open of $U$.

Proof. The $R$-orbit of $u$ is finite. By conditions (2) and (3) it is contained in an affine open $U'$ of $U$, see Varieties, Proposition 32.41.7. Then $t(s^{-1}(U \setminus U'))$ is an $R$-invariant closed subset of $U$ which does not contain $u$. Thus $U \setminus t(s^{-1}(U \setminus U'))$ is an $R$-invariant open of $U'$ containing $u$. Replacing $U$ by this open we may assume $U$ is quasi-affine.

By Lemma 39.14.6 we may replace $U$ by its reduction and assume $U$ is reduced. This means $R$-invariant subschemes $W' \subset W \subset U$ of Lemma 39.14.2 are equal $W' = W$. As $U = t(s^{-1}(\overline{W}))$ some point $u'$ of the $R$-orbit of $u$ is contained in $\overline{W}$ and by Lemma 39.14.6 we may replace $U$ by $\overline{W}$ and $u$ by $u'$. Hence we may assume there is a dense open $R$-invariant subscheme $W \subset U$ such that the morphisms $s_W, t_W$ of the restriction $(W, R_W, s_W, t_W, c_W)$ are finite locally free.

If $u \in W$ then we are done by Groupoids, Lemma 38.24.1 (because $W$ is quasi-affine so any finite set of points of $W$ is contained in an affine open, see Properties, Lemma 27.29.5). Thus we assume $u \not \in W$ and hence none of the points of the orbit of $u$ is in $W$. Let $\xi \in U$ be a point with a nontrivial specialization to a point $u'$ in the orbit of $u$. Since there are no specializations among the points in the orbit of $u$ (Lemma 39.14.8) we see that $\xi$ is not in the orbit. By assumption (3) we see that $\xi$ is a generic point of $U$ and hence $\xi \in W$. As $U$ is Noetherian there are finitely many of these points $\xi_1, \ldots, \xi_m \in W$. Because $s_W, t_W$ are flat the orbit of each $\xi_j$ consists of generic points of irreducible components of $W$ (and hence $U$).

Let $j : U \to \mathop{\rm Spec}(A)$ be an immersion of $U$ into an affine scheme (this is possible as $U$ is quasi-affine). Let $J \subset A$ be an ideal such that $V(J) \cap j(W) = \emptyset$ and $V(J) \cup j(W)$ is closed. Apply Lemma 39.14.7 to the groupoid scheme $(W, R_W, s_W, t_W, c_W)$, the morphism $j|_W : W \to \mathop{\rm Spec}(A)$, the points $\xi_j$, and the ideal $J$ to find an $f \in J$ such that $(j|_W)^{-1}D(f)$ is an $R_W$-invariant affine open containing $\xi_j$ for all $j$. Since $f \in J$ we see that $j^{-1}D(f) \subset W$, i.e., $j^{-1}D(f)$ is an $R$-invariant affine open of $U$ contained in $W$ containing all $\xi_j$.

Let $Z$ be the reduced induced closed subscheme structure on $$ U \setminus j^{-1}D(f) = j^{-1}V(f). $$ Then $Z$ is set theoretically $R$-invariant (but it may not be scheme theoretically $R$-invariant). Let $(Z, R_Z, s_Z, t_Z, c_Z)$ be the restriction of $R$ to $Z$. Since $Z \to U$ is finite, it follows that $s_Z$ and $t_Z$ are finite. Since $u \in Z$ the orbit of $u$ is in $Z$ and agrees with the $R_Z$-orbit of $u$ viewed as a point of $Z$. Since $\dim(\mathcal{O}_{U, u'}) \leq 1$ and since $\xi_j \not \in Z$ for all $j$, we see that $\dim(\mathcal{O}_{Z, u'}) \leq 0$ for all $u'$ in the orbit of $u$. In other words, the $R_Z$-orbit of $u$ consists of generic points of irreducible components of $Z$.

Let $I \subset A$ be an ideal such that $V(I) \cap j(U) =\emptyset$ and $V(I) \cup j(U)$ is closed. Apply Lemma 39.14.7 to the groupoid scheme $(Z, R_Z, s_Z, t_Z, c_Z)$, the restriction $j|_Z$, the ideal $I$, and the point $u \in Z$ to obtain $h \in I$ such that $j^{-1}D(h) \cap Z$ is an $R_Z$-invariant open affine containing $u$.

Consider the $R_W$-invariant (Groupoids, Lemma 38.23.2) function $$ g = \text{Norm}_{s_W}(t_W^\sharp(j^\sharp(h)|_W)) \in \Gamma(W, \mathcal{O}_W) $$ (In the following we only need the restriction of $g$ to $j^{-1}D(f)$ and in this case the norm is along a finite locally free morphism of affines.) We claim that $$ V = (W_g \cap j^{-1}D(f)) \cup (j^{-1}D(h) \cap Z) $$ is an $R$-invariant affine open of $U$ which finishes the proof of the lemma. It is set theoretically $R$-invariant by construction. As $V$ is a constuctible set, to see that it is open it suffices to show it is closed under generalization in $U$ (Topology, Lemma 5.19.9 or the more general Topology, Lemma 5.23.5). Since $W_g \cap j^{-1}D(f)$ is open in $U$, it suffices to consider a specialization $u_1 \leadsto u_2$ of $U$ with $u_2 \in j^{-1}D(h) \cap Z$. This means that $h$ is nonzero in $j(u_2)$ and $u_2 \in Z$. If $u_1 \in Z$, then $j(u_1) \leadsto j(u_2)$ and since $h$ is nonzero in $j(u_2)$ it is nonzero in $j(u_1)$ which implies $u_1 \in V$. If $u_1 \not \in Z$ and also not in $W_g \cap j^{-1}D(f)$, then $u_1 \in W$, $u_1 \not \in W_g$ because the complement of $Z = j^{-1}V(f)$ is contained in $W \cap j^{-1}D(f)$. Hence there exists a point $r_1 \in R$ with $s(r_1) = u_1$ such that $h$ is zero in $t(r_1)$. Since $s$ is finite we can find a specialization $r_1 \leadsto r_2$ with $s(r_2) = u_2$. However, then we conclude that $f$ is zero in $u'_2 = t(r_2)$ which contradicts the fact that $j^{-1}D(h) \cap Z$ is $R$-invariant and $u_2$ is in it. Thus $V$ is open.

Observe that $V \subset j^{-1}D(h)$ for our function $h \in I$. Thus we obtain an immersion $$ j' : V \longrightarrow \mathop{\rm Spec}(A_h) $$ Let $f' \in A_h$ be the image of $f$. Then $(j')^{-1}D(f')$ is the principal open determined by $g$ in the affine open $j^{-1}D(f)$ of $U$. Hence $(j')^{-1}D(f)$ is affine. Finally, $j'(V) \cap V(f') = j'(j^{-1}D(h) \cap Z)$ is closed in $\mathop{\rm Spec}(A_h/(f')) = \mathop{\rm Spec}((A/f)_h) = D(h) \cap V(f)$ by our choice of $h \in I$ and the ideal $I$. Hence we can apply Lemma 39.14.9 to conclude that $V$ is affine as claimed above. $\square$

    The code snippet corresponding to this tag is a part of the file more-groupoids.tex and is located in lines 2412–2957 (see updates for more information).

    \section{Finite groupoids}
    \label{section-finite-groupoids}
    
    \noindent
    A groupoid scheme $(U, R, s, t, c)$ is sometimes called {\it finite} if the
    morphisms $s$ and $t$ are finite. This is potentially confusing as it doesn't
    imply that $U$ or $R$ or the quotient sheaf $U/R$ are finite over anything.
    
    \begin{lemma}
    \label{lemma-finite-stratify}
    Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$
    are finite. There exists a sequence of $R$-invariant closed subschemes
    $$
    U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots
    $$
    such that $\bigcap Z_r = \emptyset$ and such that
    $s^{-1}(Z_{r - 1}) \setminus s^{-1}(Z_r) \to Z_{r - 1} \setminus Z_r$
    is finite locally free of rank $r$.
    \end{lemma}
    
    \begin{proof}
    Let $\{Z_r\}$ be the stratification of $U$ given by the Fitting ideals
    of the finite type quasi-coherent modules $s_*\mathcal{O}_R$. See
    Divisors, Lemma \ref{divisors-lemma-locally-free-rank-r-pullback}.
    Since the identity $e : U \to R$ is a section to $s$ we see that
    $s_*\mathcal{O}_R$ contains $\mathcal{O}_S$ as a direct summand.
    Hence $U = Z_{-1} = Z_0$ (details omitted).
    Since formation of Fitting ideals commutes with base change
    (More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-basics})
    we find that $s^{-1}(Z_r)$ corresponds to the $r$th Fitting ideal
    of $\text{pr}_{1, *}\mathcal{O}_{R \times_{s, U, t} R}$ because
    the lower right square of diagram (\ref{equation-diagram}) is cartesian.
    Using the fact that the lower left square is also cartesian we conclude
    that $s^{-1}(Z_r) = t^{-1}(Z_r)$, in other words $Z_r$ is $R$-invariant.
    The morphism
    $s^{-1}(Z_{r - 1}) \setminus s^{-1}(Z_r) \to Z_{r - 1} \setminus Z_r$
    is finite locally free of rank $r$ because the module
    $s_*\mathcal{O}_R$ pulls back to a finite locally free module of rank $r$
    on $Z_{r - 1} \setminus Z_r$ by
    Divisors, Lemma \ref{divisors-lemma-locally-free-rank-r-pullback}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-finite-flat-over-almost-dense-subscheme}
    Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$
    are finite. There exists an open subscheme $W \subset U$ and a closed
    subscheme $W' \subset W$ such that
    \begin{enumerate}
    \item $W$ and $W'$ are $R$-invariant,
    \item $U = t(s^{-1}(\overline{W}))$ set theoretically,
    \item $W$ is a thickening of $W'$, and
    \item the maps $s'$, $t'$ of the restriction $(W', R', s', t', c')$
    are finite locally free.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Consider the stratification $U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots$
    of Lemma \ref{lemma-finite-stratify}.
    
    \medskip\noindent
    We will construct disjoint unions $W = \coprod_{r \geq 1} W_r$ and
    $W' = \coprod_{r \geq 1} W'_r$ with each $W'_r \to W_r$ a thickening
    of $R$-invariant subschemes of $U$ such that the morphisms
    $s_r', t_r'$ of the restrictions $(W_r', R_r', s_r', t_r', c_r')$
    are finite locally free of rank $r$. To begin we set
    $W_1 = W'_1 = U \setminus Z_1$. This is an $R$-invariant open
    subscheme of $U$, it is true that $W_0$ is a thickening of $W'_0$,
    and the maps $s_1'$, $t_1'$ of the
    restriction $(W_1', R_1', s_1', t_1', c_1')$ are isomorphisms, i.e.,
    finite locally free of rank $1$.
    Moreover, every point of $U \setminus Z_1$ is in $t(s^{-1}(\overline{W_1}))$.
    
    \medskip\noindent
    Assume we have found subschemes $W'_r \subset W_r \subset U$ for $r \leq n$
    such that
    \begin{enumerate}
    \item $W_1, \ldots, W_n$ are disjoint,
    \item $W_r$ and $W_r'$ are $R$-invariant,
    \item $U \setminus Z_n \subset \bigcup_{r \leq n} t(s^{-1}(\overline{W_r}))$
    set theoretically,
    \item $W_r$ is a thickening of $W'_r$,
    \item the maps $s_r'$, $t_r'$ of the restriction
    $(W_r', R_r', s_r', t_r', c_r')$ are finite locally free of rank $r$.
    \end{enumerate}
    Then we set
    $$
    W_{n + 1} = Z_n \setminus
    \left(
    Z_{n + 1} \cup \bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r}))
    \right)
    $$
    set theoretically and
    $$
    W'_{n + 1} = Z_n \setminus
    \left(
    Z_{n + 1} \cup \bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r}))
    \right)
    $$
    scheme theoretically. Then $W_{n + 1}$ is an $R$-invariant open subscheme
    of $U$ because $Z_{n + 1} \setminus \overline{U \setminus Z_{n + 1}}$
    is open in $U$ and $\overline{U \setminus Z_{n + 1}}$ is contained
    in the closed subset $\bigcup\nolimits_{r \leq n} t(s^{-1}(\overline{W_r}))$
    we are removing by property (3) and the fact that $t$ is a closed morphism.
    It is clear that $W'_{n + 1}$ is a closed subscheme
    of $W_{n + 1}$ with the same underlying topological space.
    Finally, properties (1), (2) and (3) are clear and property (5) follows from
    Lemma \ref{lemma-finite-stratify}.
    
    \medskip\noindent
    By Lemma \ref{lemma-finite-stratify} we have $\bigcap Z_r = \emptyset$.
    Hence every point of $U$ is contained in $U \setminus Z_n$
    for some $n$. Thus we see that
    $U = \bigcup_{r \geq 1} t(s^{-1}(\overline{W_r}))$
    set theoretically and we see that (2) holds.
    Thus $W' \subset W$ satisfy (1), (2), (3), and (4).
    \end{proof}
    
    \noindent
    Let $(U, R, s, t, c)$ be a groupoid scheme. Given a point $u \in U$
    the {\it $R$-orbit} of $u$ is the subset $t(s^{-1}(\{u\}))$ of $U$.
    
    \begin{lemma}
    \label{lemma-finite-flat-over-almost-dense-subscheme-addendum}
    In Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}
    assume in addition that $s$ and $t$ are of finite presentation.
    Then
    \begin{enumerate}
    \item the morphism $W' \to W$ is of finite presentation, and
    \item if $u \in U$ is a point whose $R$-orbit consists of
    generic points of irreducible components of $U$, then $u \in W$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    In this case the stratification
    $U = Z_0 \supset Z_1 \supset Z_2 \supset \ldots$ of
    Lemma \ref{lemma-finite-stratify} is given by closed immersions $Z_k \to U$
    of finite presentation, see
    Divisors, Lemma \ref{divisors-lemma-locally-free-rank-r-pullback}.
    Part (1) follows immediately from this as $W' \to W$ is locally given
    by intersecting the open $W$ by $Z_r$. To see part (2)
    let $\{u_1, \ldots, u_n\}$ be the orbit of $u$.
    Since the closed subschemes $Z_k$ are $R$-invariant and
    $\bigcap Z_k = \emptyset$, we find an $k$ such that $u_i \in Z_k$
    and $u_i \not \in Z_{k + 1}$ for all $i$.
    The image of $Z_k \to U$ and $Z_{k + 1} \to U$ is locally constructible
    (Morphisms, Theorem \ref{morphisms-theorem-chevalley}).
    Since $u_i \in U$ is a generic point of an irreducible component
    of $U$, there exists an open neighbourhood $U_i$ of $u_i$ which
    is contained in $Z_k \setminus Z_{k + 1}$ set theoretically
    (Properties, Lemma \ref{properties-lemma-generic-point-in-constructible}).
    In the proof of Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}
    we have constructed $W$ as a disjoint union $\coprod W_r$
    with $W_r \subset Z_{r - 1} \setminus Z_r$ such that
    $U = \bigcup t(s^{-1}(\overline{W_r}))$. As $\{u_1, \ldots, u_n\}$
    is an $R$-orbit we see that $u \in t(s^{-1}(\overline{W_r}))$
    implies $u_i \in \overline{W_r}$ for some $i$ which implies
    $U_i \cap W_r \not = \emptyset$ which implies $r = k$.
    Thus we conclude that $u$ is in
    $$
    W_{k + 1} = Z_k \setminus
    \left(
    Z_{k + 1} \cup \bigcup\nolimits_{r \leq k} t(s^{-1}(\overline{W_r}))
    \right)
    $$
    as desired.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-invariant-affine-open-around-generic-point}
    Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$
    are finite and of finite presentation and $U$ quasi-separated. Let
    $u_1, \ldots, u_m \in U$ be points whose orbits consist of generic points
    of irreducible components of $U$. Then there exist $R$-invariant subschemes
    $V' \subset V \subset U$ such that
    \begin{enumerate}
    \item $u_1, \ldots, u_m \in V'$,
    \item $V$ is open in $U$,
    \item $V'$ and $V$ are affine,
    \item $V' \subset V$ is a thickening of finite presentation,
    \item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$
    are finite locally free.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let $W' \subset W \subset U$ be as in
    Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}.
    By Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme-addendum}
    we get $u_j \in W$ and that $W' \to W$ is a thickening of finite presentation.
    By Limits, Lemma \ref{limits-lemma-affines-glued-in-closed-affine}
    it suffices to find an $R$-invariant affine open subscheme
    $V'$ of $W'$ containing $u_j$ (because then we can let $V \subset W$
    be the corresponding open subscheme which will be affine).
    Thus we may replace $(U, R, s, t, c)$ by the restriction
    $(W', R', s', t', c')$ to $W'$.
    In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$
    whose morphisms $s$ and $t$ are finite locally free.
    By Properties, Lemma \ref{properties-lemma-maximal-points-affine}
    we can find an affine open containing the union of the orbits of
    $u_1, \ldots, u_m$. Finally, we can apply
    Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine}
    to conclude.
    \end{proof}
    
    \noindent
    The following lemma is a special case of
    Lemma \ref{lemma-invariant-affine-open-around-generic-point}
    but we redo the argument as it is slightly easier in this case
    (it avoids using
    Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme-addendum}).
    
    \begin{lemma}
    \label{lemma-invariant-affine-open-around-generic-point-Noetherian}
    Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$.
    Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \ldots, u_m \in U$
    points whose orbits consist of generic points of irreducible
    components of $U$. Then there exist $R$-invariant subschemes
    $V' \subset V \subset U$ such that
    \begin{enumerate}
    \item $u_1, \ldots, u_m \in V'$,
    \item $V$ is open in $U$,
    \item $V'$ and $V$ are affine,
    \item $V' \subset V$ is a thickening,
    \item the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$
    are finite locally free.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let $\{u_{j1}, \ldots, u_{jn_j}\}$ be the orbit of $u_j$.
    Let $W' \subset W \subset U$ be as in
    Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}.
    Since $U = t(s^{-1}(\overline{W}))$ we see that at least
    one $u_{ji} \in \overline{W}$. Since $u_{ji}$ is a generic point
    of an irreducible component and $U$ locally Noetherian,
    this implies that $u_{ji} \in W$. Since $W$ is $R$-invariant, we
    conclude that $u_j \in W$ and in fact the whole orbit is contained in $W$.
    By Cohomology of Schemes, Lemma
    \ref{coherent-lemma-image-affine-finite-morphism-affine-Noetherian}
    it suffices to find an $R$-invariant affine open subscheme $V'$
    of $W'$ containing $u_1, \ldots, u_m$ (because then we can let $V \subset W$
    be the corresponding open subscheme which will be affine).
    Thus we may replace $(U, R, s, t, c)$
    by the restriction $(W', R', s', t', c')$ to $W'$.
    In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$
    whose morphisms $s$ and $t$ are finite locally free.
    By Properties, Lemma \ref{properties-lemma-maximal-points-affine}
    we can find an affine open containing $\{u_{ij}\}$
    (a locally Noetherian scheme is quasi-separated by
    Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated}).
    Finally, we can apply
    Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine}
    to conclude.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-find-affine-integral}
    Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$
    with $s, t$ integral. Let $g : U' \to U$ be an integral morphism
    such that every $R$-orbit in $U$ meets $g(U')$. Let $(U', R', s', t', c')$
    be the restriction of $R$ to $U'$. If $u' \in U'$ is contained in an
    $R'$-invariant affine open, then the image $u \in U$ is contained
    in an $R$-invariant affine open of $U$.
    \end{lemma}
    
    \begin{proof}
    Let $W' \subset U'$ be an $R'$-invariant affine open.
    Set $\tilde R = U' \times_{g, U, t} R$ with maps
    $\text{pr}_0 : \tilde R \to U'$ and $h = s \circ \text{pr}_1 : \tilde R \to U$.
    Observe that $\text{pr}_0$ and $h$ are integral.
    It follows that $\tilde W = \text{pr}_0^{-1}(W')$ is affine.
    Since $W'$ is $R'$-invariant, the image
    $W = h(\tilde W)$ is set theoretically $R$-invariant and
    $\tilde W = h^{-1}(W)$ set theoretically (details omitted).
    Thus, if we can show that $W$ is open, then $W$ is a scheme
    and the morphism $\tilde W \to W$ is integral surjective
    which implies that $W$ is affine by
    Limits, Proposition \ref{limits-proposition-affine}.
    However, our assumption on orbits meeting $U'$ implies
    that $h : \tilde R \to U$ is surjective. Since an
    integral surjective morphism is submersive
    (Topology, Lemma \ref{topology-lemma-closed-morphism-quotient-topology}
    and Morphisms, Lemma \ref{morphisms-lemma-integral-universally-closed})
    it follows that $W$ is open.
    \end{proof}
    
    \noindent
    The following technical lemma produces ``almost'' invariant
    functions in the situation of a finite groupoid on a quasi-affine
    scheme.
    
    \begin{lemma}
    \label{lemma-find-almost-invariant-function}
    Let $(U, R, s, t, c)$ be a groupoid scheme with $s, t$ finite and of
    finite presentation. Let $u_1, \ldots, u_m \in U$ be points whose $R$-orbits
    consist of generic points of irreducible components of $U$.
    Let $j : U \to \Spec(A)$ be an immersion.
    Let $I \subset A$ be an ideal such that $j(U) \cap V(I) = \emptyset$
    and $V(I) \cup j(U)$ is closed in $\Spec(A)$.
    Then there exists an $h \in I$ such that $j^{-1}D(h)$
    is an $R$-invariant affine open subscheme of $U$ containing
    $u_1, \ldots, u_m$.
    \end{lemma}
    
    \begin{proof}
    Let $u_1, \ldots, u_m \in V' \subset V \subset U$ be as in
    Lemma \ref{lemma-invariant-affine-open-around-generic-point}.
    Since $U \setminus V$ is closed in $U$, $j$ an immersion, and $V(I) \cup j(U)$
    is closed in $\Spec(A)$, we can find an ideal
    $J \subset I$ such that $V(J) = V(I) \cup j(U \setminus V)$.
    For example we can take the ideal of elements of $I$ which
    vanish on $j(U \setminus V)$. Thus we can replace
    $(U, R, s, t, c)$, $j : U \to \Spec(A)$, and $I$ by
    $(V', R', s', t', c')$, $j|_{V'} : V' \to \Spec(A)$, and $J$.
    In other words, we may assume that $U$ is affine and that
    $s$ and $t$ are finite locally free.
    Take any $f \in I$ which does not vanish at all the
    points in the $R$-orbits of $u_1, \ldots, u_m$
    (Algebra, Lemma \ref{algebra-lemma-silly}). Consider
    $$
    g = \text{Norm}_s(t^\sharp(j^\sharp(f))) \in \Gamma(U, \mathcal{O}_U)
    $$
    Since $f \in I$ and since $V(I) \cup j(U)$ is closed we see that
    $U \cap D(f) \to D(f)$ is a closed immersion. Hence $f^ng$ is the
    image of an element $h \in I$ for some $n > 0$. We claim that $h$ works.
    Namely, we have seen in
    Groupoids, Lemma \ref{groupoids-lemma-determinant-trick}
    that $g$ is an $R$-invariant function, hence $D(g) \subset U$
    is $R$-invariant. Since $f$ does not vanish on the orbit of $u_j$,
    the function $g$ does not vanish at $u_j$. Moreover, we have
    $V(g) \supset V(j^\sharp(f))$ and hence $j^{-1}D(h) = D(g)$.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-no-specializations-map-to-same-point}
    Let $(U, R, s, t, c)$ be a groupoid scheme. If $s, t$ are finite,
    and $u, u' \in R$ are distinct points in the same orbit,
    then $u'$ is not a specialization of $u$.
    \end{lemma}
    
    \begin{proof}
    Let $r \in R$ with $s(r) = u$ and $t(r) = u'$. If $u \leadsto u'$
    then we can find a nontrivial specialization $r \leadsto r'$ with
    $s(r') = u'$, see 
    Schemes, Lemma \ref{schemes-lemma-quasi-compact-closed}.
    Set $u'' = t(r')$. Note that $u'' \not = u'$ as there are no
    specializations in the fibres of a finite morphism.
    Hence we can continue and find a nontrivial specialization
    $r' \leadsto r''$ with $s(r'') = u''$, etc. This shows that the
    orbit of $u$ contains an infinite sequence
    $u \leadsto u' \leadsto u'' \leadsto \ldots$
    of specializations which is nonsense as the orbit
    $t(s^{-1}(\{u\}))$ is finite.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-get-affine}
    Let $j : V \to \Spec(A)$ be a quasi-compact immersion of schemes.
    Let $f \in A$ be such that $j^{-1}D(f)$ is affine and $j(V) \cap V(f)$
    is closed. Then $V$ is affine.
    \end{lemma}
    
    \begin{proof}
    This follows from Morphisms, Lemma \ref{morphisms-lemma-get-affine}
    but we will also give a direct proof.
    Let $A' = \Gamma(V, \mathcal{O}_V)$. Then $j' : V \to \Spec(A')$ is a
    quasi-compact open immersion, see
    Properties, Lemma \ref{properties-lemma-quasi-affine}.
    Let $f' \in A'$ be the image of $f$. Then $(j')^{-1}D(f') = j^{-1}D(f)$
    is affine. On the other hand, $j'(V) \cap V(f')$ is a subscheme of
    $\Spec(A')$ which maps isomorphically to the closed subscheme
    $j(V) \cap V(f)$ of $\Spec(A)$. Hence it is closed in $\Spec(A')$
    for example by Schemes, Lemma \ref{schemes-lemma-section-immersion}.
    Thus we may replace $A$ by $A'$ and assume that $j$ is an open immersion
    and $A = \Gamma(V, \mathcal{O}_V)$.
    
    \medskip\noindent
    In this case we claim that $j(V) = \Spec(A)$ which finishes the proof.
    If not, then we can find a principal affine open $D(g) \subset \Spec(A)$
    which meets the complement and avoids the closed subset $j(V) \cap V(f)$.
    Note that $j$ maps $j^{-1}D(f)$ isomorphically onto $D(f)$, see
    Properties, Lemma \ref{properties-lemma-invert-f-affine}.
    Hence $D(g)$ meets $V(f)$. On the other hand, $j^{-1}D(g)$
    is a principal open of the affine open $j^{-1}D(f)$ hence affine.
    Hence by
    Properties, Lemma \ref{properties-lemma-invert-f-affine}
    again we see that $D(g)$ is isomorphic to $j^{-1}D(g) \subset j^{-1}D(f)$
    which implies that $D(g) \subset D(f)$. This contradiction finishes
    the proof.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-find-affine-codimension-1}
    Let $(U, R, s, t, c)$ be a groupoid scheme. Let $u \in U$. Assume
    \begin{enumerate}
    \item $s, t$ are finite morphisms,
    \item $U$ is separated and locally Noetherian,
    \item $\dim(\mathcal{O}_{U, u'}) \leq 1$ for every point $u'$
    in the orbit of $u$.
    \end{enumerate}
    Then $u$ is contained in an $R$-invariant affine open of $U$.
    \end{lemma}
    
    \begin{proof}
    The $R$-orbit of $u$ is finite. By conditions (2) and (3) it is contained
    in an affine open $U'$ of $U$, see
    Varieties, Proposition
    \ref{varieties-proposition-finite-set-of-points-of-codim-1-in-affine}.
    Then $t(s^{-1}(U \setminus U'))$ is an $R$-invariant
    closed subset of $U$ which does not contain $u$. Thus
    $U \setminus t(s^{-1}(U \setminus U'))$ is an $R$-invariant open
    of $U'$ containing $u$.
    Replacing $U$ by this open we may assume $U$ is quasi-affine.
    
    \medskip\noindent
    By Lemma \ref{lemma-find-affine-integral} we may replace $U$ by its reduction
    and assume $U$ is reduced. This means $R$-invariant subschemes
    $W' \subset W \subset U$ of
    Lemma \ref{lemma-finite-flat-over-almost-dense-subscheme}
    are equal $W' = W$. As $U = t(s^{-1}(\overline{W}))$ some point
    $u'$ of the $R$-orbit of $u$ is contained in $\overline{W}$
    and by Lemma \ref{lemma-find-affine-integral}
    we may replace $U$ by $\overline{W}$ and $u$ by $u'$.
    Hence we may assume there is
    a dense open $R$-invariant subscheme $W \subset U$ such that
    the morphisms $s_W, t_W$ of the restriction $(W, R_W, s_W, t_W, c_W)$ are
    finite locally free.
    
    \medskip\noindent
    If $u \in W$ then we are done by
    Groupoids, Lemma \ref{groupoids-lemma-find-invariant-affine}
    (because $W$ is quasi-affine so any finite set of points
    of $W$ is contained in an affine open, see
    Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}).
    Thus we assume $u \not \in W$ and hence none of the points of the
    orbit of $u$ is in $W$. Let $\xi \in U$
    be a point with a nontrivial specialization to a point $u'$ in the orbit
    of $u$. Since there are no specializations among the points in the
    orbit of $u$ (Lemma \ref{lemma-no-specializations-map-to-same-point})
    we see that $\xi$ is not in the orbit.
    By assumption (3) we see that $\xi$ is a generic point of $U$
    and hence $\xi \in W$.
    As $U$ is Noetherian there are finitely many of these
    points $\xi_1, \ldots, \xi_m \in W$. Because $s_W, t_W$ are flat the orbit
    of each $\xi_j$ consists of generic points of irreducible components
    of $W$ (and hence $U$).
    
    \medskip\noindent
    Let $j : U \to \Spec(A)$ be an immersion of $U$ into an affine scheme
    (this is possible as $U$ is quasi-affine). Let $J \subset A$
    be an ideal such that $V(J) \cap j(W) = \emptyset$ and $V(J) \cup j(W)$
    is closed. Apply Lemma \ref{lemma-find-almost-invariant-function}
    to the groupoid scheme $(W, R_W, s_W, t_W, c_W)$, the morphism
    $j|_W : W \to \Spec(A)$, the points $\xi_j$, and the ideal $J$
    to find an $f \in J$ such that $(j|_W)^{-1}D(f)$ is an $R_W$-invariant
    affine open containing $\xi_j$ for all $j$. Since $f \in J$
    we see that $j^{-1}D(f) \subset W$, i.e., $j^{-1}D(f)$ is
    an $R$-invariant affine open of $U$ contained in $W$
    containing all $\xi_j$.
    
    \medskip\noindent
    Let $Z$ be the reduced induced closed subscheme structure on
    $$
    U \setminus j^{-1}D(f) = j^{-1}V(f).
    $$
    Then $Z$ is set theoretically
    $R$-invariant (but it may not be scheme theoretically $R$-invariant).
    Let $(Z, R_Z, s_Z, t_Z, c_Z)$ be the restriction of $R$ to $Z$.
    Since $Z \to U$ is finite, it follows that $s_Z$ and $t_Z$ are finite.
    Since $u \in Z$ the orbit of $u$ is in $Z$ and agrees with the
    $R_Z$-orbit of $u$ viewed as a point of $Z$. Since
    $\dim(\mathcal{O}_{U, u'}) \leq 1$ and since $\xi_j \not \in Z$
    for all $j$, we see that $\dim(\mathcal{O}_{Z, u'}) \leq 0$ for
    all $u'$ in the orbit of $u$. In other words, the $R_Z$-orbit of $u$
    consists of generic points of irreducible components of $Z$.
    
    \medskip\noindent
    Let $I \subset A$ be an ideal such that $V(I) \cap j(U) =\emptyset$
    and $V(I) \cup j(U)$ is closed. Apply
    Lemma \ref{lemma-find-almost-invariant-function} to
    the groupoid scheme $(Z, R_Z, s_Z, t_Z, c_Z)$, the restriction $j|_Z$,
    the ideal $I$, and the point $u \in Z$ to obtain $h \in I$ such that
    $j^{-1}D(h) \cap Z$ is an $R_Z$-invariant open affine containing $u$.
    
    \medskip\noindent
    Consider the $R_W$-invariant (Groupoids, Lemma
    \ref{groupoids-lemma-determinant-trick}) function
    $$
    g = 
    \text{Norm}_{s_W}(t_W^\sharp(j^\sharp(h)|_W)) \in \Gamma(W, \mathcal{O}_W)
    $$
    (In the following we only need the restriction of $g$ to $j^{-1}D(f)$ and
    in this case the norm is along a finite locally free morphism of affines.)
    We claim that
    $$
    V = (W_g \cap j^{-1}D(f)) \cup (j^{-1}D(h) \cap Z)
    $$
    is an $R$-invariant affine open of $U$ which finishes the proof of the lemma.
    It is set theoretically $R$-invariant by construction. As $V$ is a
    constuctible set, to see that it is open it suffices to show it is
    closed under generalization in $U$ (Topology, Lemma
    \ref{topology-lemma-characterize-closed-Noetherian}
    or the more general
    Topology, Lemma
    \ref{topology-lemma-constructible-stable-specialization-closed}).
    Since $W_g \cap j^{-1}D(f)$ is open in $U$, it suffices to consider
    a specialization $u_1 \leadsto u_2$ of $U$ with
    $u_2 \in j^{-1}D(h) \cap Z$.
    This means that $h$ is nonzero in $j(u_2)$ and $u_2 \in Z$.
    If $u_1 \in Z$, then $j(u_1) \leadsto j(u_2)$ and since
    $h$ is nonzero in $j(u_2)$ it is nonzero in $j(u_1)$ which
    implies $u_1 \in V$. If $u_1 \not \in Z$ and
    also not in $W_g \cap j^{-1}D(f)$, then $u_1 \in W$, $u_1 \not \in W_g$
    because the complement of $Z = j^{-1}V(f)$ is contained in $W \cap j^{-1}D(f)$.
    Hence there exists a point $r_1 \in R$ with $s(r_1) = u_1$
    such that $h$ is zero in $t(r_1)$. Since $s$ is finite we
    can find a specialization $r_1 \leadsto r_2$ with $s(r_2) = u_2$.
    However, then we conclude that $f$ is zero in $u'_2 = t(r_2)$
    which contradicts the fact that $j^{-1}D(h) \cap Z$
    is $R$-invariant and $u_2$ is in it. Thus $V$ is open.
    
    \medskip\noindent
    Observe that $V \subset j^{-1}D(h)$ for our function $h \in I$.
    Thus we obtain an immersion
    $$
    j' : V \longrightarrow \Spec(A_h)
    $$
    Let $f' \in A_h$ be the image of $f$. Then $(j')^{-1}D(f')$
    is the principal open determined by $g$ in the affine
    open $j^{-1}D(f)$ of $U$.
    Hence $(j')^{-1}D(f)$ is affine. Finally,
    $j'(V) \cap V(f') = j'(j^{-1}D(h) \cap Z)$
    is closed in $\Spec(A_h/(f')) = \Spec((A/f)_h) = D(h) \cap V(f)$
    by our choice of $h \in I$ and the ideal $I$. Hence we can apply
    Lemma \ref{lemma-get-affine}
    to conclude that $V$ is affine as claimed above.
    \end{proof}

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