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
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
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. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)