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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