Lemma 40.7.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $r, r' \in R$ with $t(r) = t(r')$ in $U$. Set $u = s(r)$, $u' = s(r')$. Denote $F_ u = s^{-1}(u)$ and $F_{u'} = s^{-1}(u')$ the scheme theoretic fibres.
There exists a common field extension $\kappa (u) \subset k$, $\kappa (u') \subset k$ and an isomorphism $(F_ u)_ k \cong (F_{u'})_ k$.
We may choose the isomorphism of (1) such that a point lying over $r$ maps to a point lying over $r'$.
If the morphisms $s$, $t$ are flat then the morphisms of germs $s : (R, r) \to (U, u)$ and $s : (R, r') \to (U, u')$ are flat locally on the base isomorphic.
If the morphisms $s$, $t$ are étale (resp. smooth, syntomic, or flat and locally of finite presentation) then the morphisms of germs $s : (R, r) \to (U, u)$ and $s : (R, r') \to (U, u')$ are locally on the base isomorphic in the étale (resp. smooth, syntomic, or fppf) topology.
Proof.
We repeatedly use the properties and the existence of diagram (40.3.0.1). By the properties of the diagram (and Schemes, Lemma 26.17.5) there exists a point $\xi $ of $R \times _{s, U, t} R$ with $\text{pr}_0(\xi ) = r$ and $c(\xi ) = r'$. Let $\tilde r = \text{pr}_1(\xi ) \in R$.
Proof of (1). Set $k = \kappa (\tilde r)$. Since $t(\tilde r) = u$ and $s(\tilde r) = u'$ we see that $k$ is a common extension of both $\kappa (u)$ and $\kappa (u')$ and in fact that both $(F_ u)_ k$ and $(F_{u'})_ k$ are isomorphic to the fibre of $\text{pr}_1 : R \times _{s, U, t} R \to R$ over $\tilde r$. Hence (1) is proved.
Part (2) follows since the point $\xi $ maps to $r$, resp. $r'$.
Part (3) is clear from the above (using the point $\xi $ for $\tilde u$ and $\tilde u'$) and the definitions.
If $s$ and $t$ are flat and of finite presentation, then they are open morphisms (Morphisms, Lemma 29.25.10). Hence the image of some affine open neighbourhood $V''$ of $\tilde r$ will cover an open neighbourhood $V$ of $u$, resp. $V'$ of $u'$. These can be used to show that properties (1) and (2) of the definition of “locally on the base isomorphic in the $\tau $-topology”.
$\square$
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