40.11 Morphisms of groupoids on fields
This section studies morphisms between groupoids on fields. This is slightly more general, but very akin to, studying morphisms of groupschemes over a field.
Situation 40.11.1. Let $S$ be a scheme. Let $U = \mathop{\mathrm{Spec}}(k)$ be a scheme over $S$ with $k$ a field. Let $(U, R_1, s_1, t_1, c_1)$, $(U, R_2, s_2, t_2, c_2)$ be groupoid schemes over $S$ with identical first component. Let $a : R_1 \to R_2$ be a morphism such that $(\text{id}_ U, a)$ defines a morphism of groupoid schemes over $S$, see Groupoids, Definition 39.13.1. In particular, the following diagrams commute
\[ \vcenter { \xymatrix{ R_1 \ar[rrd]^{t_1} \ar[rdd]_{s_1} \ar[rd]_ a \\ & R_2 \ar[d]^{t_2} \ar[r]_{s_2} & U \\ & U } } \quad \quad \vcenter { \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r]_-{c_1} \ar[d]_{a \times a} & R_1 \ar[d]^ a \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r]^-{c_2} & R_2 } } \]
The following lemma is a generalization of Groupoids, Lemma 39.7.7.
Lemma 40.11.2. Notation and assumptions as in Situation 40.11.1. If $a(R_1)$ is open in $R_2$, then $a(R_1)$ is closed in $R_2$.
Proof.
Let $r_2 \in R_2$ be a point in the closure of $a(R_1)$. We want to show $r_2 \in a(R_1)$. Pick $k \subset k'$ and $r_2' \in R'_2$ adapted to $(U, R_2, s_2, t_2, c_2)$ and $r_2$ as in Lemma 40.10.5. Let $R_ i'$ be the restriction of $R_ i$ via the morphism $U' = \mathop{\mathrm{Spec}}(k') \to U = \mathop{\mathrm{Spec}}(k)$. Let $a' : R'_1 \to R_2'$ be the base change of $a$. The diagram
\[ \xymatrix{ R'_1 \ar[r]_{a'} \ar[d]_{p_1} & R'_2 \ar[d]^{p_2} \\ R_1 \ar[r]^ a & R_2 } \]
is a fibre square. Hence the image of $a'$ is the inverse image of the image of $a$ via the morphism $p_2 : R'_2 \to R_2$. By Lemma 40.10.4 the map $p_2$ is surjective and open. Hence by Topology, Lemma 5.6.4 we see that $r_2'$ is in the closure of $a'(R'_1)$. This means that we may assume that $r_2 \in R_2$ has the property that the maps $k \to \kappa (r_2)$ induced by $s_2$ and $t_2$ are isomorphisms.
In this case we can use Lemma 40.10.6. This lemma implies $c(r_2, a(R_1))$ is an open neighbourhood of $r_2$. Hence $a(R_1) \cap c(r_2, a(R_1)) \not= \emptyset $ as we assumed that $r_2$ was a point of the closure of $a(R_1)$. Using the inverse of $R_2$ and $R_1$ we see this means $c_2(a(R_1), a(R_1))$ contains $r_2$. As $c_2(a(R_1), a(R_1)) \subset a(c_1(R_1, R_1)) = a(R_1)$ we conclude $r_2 \in a(R_1)$ as desired.
$\square$
Lemma 40.11.3. Notation and assumptions as in Situation 40.11.1. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then $c_2(Z \times _{s_2, U, t_2} Z) \subset Z$ set theoretically.
Proof.
Consider the commutative diagram
\[ \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r] \ar[d] & R_1 \ar[d] \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r] & R_2 } \]
By Varieties, Lemma 33.24.2 the closure of the image of the left vertical arrow is (set theoretically) $Z \times _{s_2, U, t_2} Z$. Hence the result follows.
$\square$
Lemma 40.11.4. Notation and assumptions as in Situation 40.11.1. Assume that $k$ is perfect. Let $Z \subset R_2$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $a : R_1 \to R_2$. Then
\[ (U, Z, s_2|_ Z, t_2|_ Z, c_2|_ Z) \]
is a groupoid scheme over $S$.
Proof.
We first explain why the statement makes sense. Since $U$ is the spectrum of a perfect field $k$, the scheme $Z$ is geometrically reduced over $k$ (via either projection), see Varieties, Lemma 33.6.3. Hence the scheme $Z \times _{s_2, U, t_2} Z \subset Z$ is reduced, see Varieties, Lemma 33.6.7. Hence by Lemma 40.11.3 we see that $c$ induces a morphism $Z \times _{s_2, U, t_2} Z \to Z$. Finally, it is clear that $e_2$ factors through $Z$ and that the map $i_2 : R_2 \to R_2$ preserves $Z$. Since the morphisms of the septuple $(U, R_2, s_2, t_2, c_2, e_2, i_2)$ satisfies the axioms of a groupoid, it follows that after restricting to $Z$ they satisfy the axioms.
$\square$
Lemma 40.11.5. Notation and assumptions as in Situation 40.11.1. If the image $a(R_1)$ is a locally closed subset of $R_2$ then it is a closed subset.
Proof.
Let $k \subset k'$ be a perfect closure of the field $k$. Let $R_ i'$ be the restriction of $R_ i$ via the morphism $U' = \mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k)$. Note that the morphisms $R_ i' \to R_ i$ are universal homeomorphisms as compositions of base changes of the universal homeomorphism $U' \to U$ (see diagram in statement of Lemma 40.10.4). Hence it suffices to prove that $a'(R_1')$ is closed in $R_2'$. In other words, we may assume that $k$ is perfect.
If $k$ is perfect, then the closure of the image is a groupoid scheme $Z \subset R_2$, by Lemma 40.11.4. By the same lemma applied to $\text{id}_{R_1} : R_1 \to R_1$ we see that $(R_2)_{red}$ is a groupoid scheme. Thus we may apply Lemma 40.11.2 to the morphism $a|_{(R_2)_{red}} : (R_2)_{red} \to Z$ to conclude that $Z$ equals the image of $a$.
$\square$
Lemma 40.11.6. Notation and assumptions as in Situation 40.11.1. Assume that $a : R_1 \to R_2$ is a quasi-compact morphism. Let $Z \subset R_2$ be the scheme theoretic image (see Morphisms, Definition 29.6.2) of $a : R_1 \to R_2$. Then
\[ (U, Z, s_2|_ Z, t_2|_ Z, c_2|_ Z) \]
is a groupoid scheme over $S$.
Proof.
The main difficulty is to show that $c_2|_{Z \times _{s_2, U, t_2} Z}$ maps into $Z$. Consider the commutative diagram
\[ \xymatrix{ R_1 \times _{s_1, U, t_1} R_1 \ar[r] \ar[d]^{a \times a} & R_1 \ar[d] \\ R_2 \times _{s_2, U, t_2} R_2 \ar[r] & R_2 } \]
By Varieties, Lemma 33.24.3 we see that the scheme theoretic image of $a \times a$ is $Z \times _{s_2, U, t_2} Z$. By the commutativity of the diagram we conclude that $Z \times _{s_2, U, t_2} Z$ maps into $Z$ by the bottom horizontal arrow. As in the proof of Lemma 40.11.4 it is also true that $i_2(Z) \subset Z$ and that $e_2$ factors through $Z$. Hence we conclude as in the proof of that lemma.
$\square$
Lemma 40.11.7. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field. Let $Z \subset U \times _ S U$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $j = (t, s) : R \to U \times _ S U$. Then $\text{pr}_{02}(Z \times _{\text{pr}_1, U, \text{pr}_0} Z) \subset Z$ set theoretically.
Proof.
As $(U, U \times _ S U, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$ is a groupoid scheme over $S$ this is a special case of Lemma 40.11.3. But we can also prove it directly as follows.
Write $U = \mathop{\mathrm{Spec}}(k)$. Denote $R_ s$ (resp. $Z_ s$, resp. $U^2_ s$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $s$ (resp. $\text{pr}_1|_ Z$, resp. $\text{pr}_1$). Similarly, denote ${}_ tR$ (resp. ${}_ tZ$, resp. ${}_ tU^2$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $t$ (resp. $\text{pr}_0|_ Z$, resp. $\text{pr}_0$). The morphism $j$ induces morphisms of schemes $j_ s : R_ s \to U^2_ s$ and ${}_ tj : {}_ tR \to {}_ tU^2$ over $k$. Consider the commutative diagram
\[ \xymatrix{ R_ s \times _ k {}_ tR \ar[r]^ c \ar[d]_{j_ s \times {}_ tj} & R \ar[d]^ j \\ U^2_ s \times _ k {}_ tU^2 \ar[r] & U \times _ S U } \]
By Varieties, Lemma 33.24.2 we see that the closure of the image of $j_ s \times {}_ tj$ is $Z_ s \times _ k {}_ tZ$. By the commutativity of the diagram we conclude that $Z_ s \times _ k {}_ tZ$ maps into $Z$ by the bottom horizontal arrow.
$\square$
Lemma 40.11.8. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a perfect field. Let $Z \subset U \times _ S U$ be the reduced closed subscheme (see Schemes, Definition 26.12.5) whose underlying topological space is the closure of the image of $j = (t, s) : R \to U \times _ S U$. Then
\[ (U, Z, \text{pr}_0|_ Z, \text{pr}_1|_ Z, \text{pr}_{02}|_{Z \times _{\text{pr}_1, U, \text{pr}_0} Z}) \]
is a groupoid scheme over $S$.
Proof.
As $(U, U \times _ S U, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$ is a groupoid scheme over $S$ this is a special case of Lemma 40.11.4. But we can also prove it directly as follows.
We first explain why the statement makes sense. Since $U$ is the spectrum of a perfect field $k$, the scheme $Z$ is geometrically reduced over $k$ (via either projection), see Varieties, Lemma 33.6.3. Hence the scheme $Z \times _{\text{pr}_1, U, \text{pr}_0} Z \subset Z$ is reduced, see Varieties, Lemma 33.6.7. Hence by Lemma 40.11.7 we see that $\text{pr}_{02}$ induces a morphism $Z \times _{\text{pr}_1, U, \text{pr}_0} Z \to Z$. Finally, it is clear that $\Delta _{U/S}$ factors through $Z$ and that the map $\sigma : U \times _ S U \to U \times _ S U$, $(x, y) \mapsto (y, x)$ preserves $Z$. Since $(U, U \times _ S U, \text{pr}_0, \text{pr}_1, \text{pr}_{02}, \Delta _{U/S}, \sigma )$ satisfies the axioms of a groupoid, it follows that after restricting to $Z$ they satisfy the axioms.
$\square$
Lemma 40.11.9. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U$ is the spectrum of a field and assume $R$ is quasi-compact (equivalently $s, t$ are quasi-compact). Let $Z \subset U \times _ S U$ be the scheme theoretic image (see Morphisms, Definition 29.6.2) of $j = (t, s) : R \to U \times _ S U$. Then
\[ (U, Z, \text{pr}_0|_ Z, \text{pr}_1|_ Z, \text{pr}_{02}|_{Z \times _{\text{pr}_1, U, \text{pr}_0} Z}) \]
is a groupoid scheme over $S$.
Proof.
As $(U, U \times _ S U, \text{pr}_1, \text{pr}_0, \text{pr}_{02})$ is a groupoid scheme over $S$ this is a special case of Lemma 40.11.6. But we can also prove it directly as follows.
The main difficulty is to show that $\text{pr}_{02}|_{Z \times _{\text{pr}_1, U, \text{pr}_0} Z}$ maps into $Z$. Write $U = \mathop{\mathrm{Spec}}(k)$. Denote $R_ s$ (resp. $Z_ s$, resp. $U^2_ s$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $s$ (resp. $\text{pr}_1|_ Z$, resp. $\text{pr}_1$). Similarly, denote ${}_ tR$ (resp. ${}_ tZ$, resp. ${}_ tU^2$) the scheme $R$ (resp. $Z$, resp. $U \times _ S U$) viewed as a scheme over $k$ via $t$ (resp. $\text{pr}_0|_ Z$, resp. $\text{pr}_0$). The morphism $j$ induces morphisms of schemes $j_ s : R_ s \to U^2_ s$ and ${}_ tj : {}_ tR \to {}_ tU^2$ over $k$. Consider the commutative diagram
\[ \xymatrix{ R_ s \times _ k {}_ tR \ar[r]^ c \ar[d]_{j_ s \times {}_ tj} & R \ar[d]^ j \\ U^2_ s \times _ k {}_ tU^2 \ar[r] & U \times _ S U } \]
By Varieties, Lemma 33.24.3 we see that the scheme theoretic image of $j_ s \times {}_ tj$ is $Z_ s \times _ k {}_ tZ$. By the commutativity of the diagram we conclude that $Z_ s \times _ k {}_ tZ$ maps into $Z$ by the bottom horizontal arrow. As in the proof of Lemma 40.11.8 it is also true that $\sigma (Z) \subset Z$ and that $\Delta _{U/S}$ factors through $Z$. Hence we conclude as in the proof of that lemma.
$\square$
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