## 40.7 Comparing fibres

Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Diagram (40.3.0.1) gives us a way to compare the fibres of the map $s : R \to U$ in a groupoid. For a point $u \in U$ we will denote $F_ u = s^{-1}(u)$ the scheme theoretic fibre of $s : R \to U$ over $u$. For example the diagram implies that if $u, u' \in U$ are points such that $s(r) = u$ and $t(r) = u'$, then $(F_ u)_{\kappa (r)} \cong (F_{u'})_{\kappa (r)}$. This is a special case of the more general and more precise Lemma 40.7.1 below. To see this take $r' = i(r)$.

A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is sometimes called the germ of $X$ at $x$. A morphism of germs $f : (X, x) \to (S, s)$ is a morphism $f : U \to S$ defined on an open neighbourhood of $x$ with $f(x) = s$. Two such $f$, $f'$ are said to give the same morphism of germs if and only if $f$ and $f'$ agree in some open neighbourhood of $x$. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. We temporarily introduce the following concept: We say that two morphisms of germs $f : (X, x) \to (S, s)$ and $f' : (X', x') \to (S', s')$ are isomorphic locally on the base in the $\tau$-topology, if there exists a pointed scheme $(S'', s'')$ and morphisms of germs $g : (S'', s'') \to (S, s)$, and $g' : (S'', s'') \to (S', s')$ such that

1. $g$ and $g'$ are an open immersion (resp. étale, smooth, syntomic, flat and locally of finite presentation) at $s''$,

2. there exists an isomorphism

$(S'' \times _{g, S, f} X, \tilde x) \cong (S'' \times _{g', S', f'} X', \tilde x')$

of germs over the germ $(S'', s'')$ for some choice of points $\tilde x$ and $\tilde x'$ lying over $(s'', x)$ and $(s'', x')$.

Finally, we simply say that the maps of germs $f : (X, x) \to (S, s)$ and $f' : (X', x') \to (S', s')$ are flat locally on the base isomorphic if there exist $S'', s'', g, g'$ as above but with (1) replaced by the condition that $g$ and $g'$ are flat at $s''$ (this is much weaker than any of the $\tau$ conditions above as a flat morphism need not be open).

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.

1. There exists a common field extension $\kappa (u) \subset k$, $\kappa (u') \subset k$ and an isomorphism $(F_ u)_ k \cong (F_{u'})_ k$.

2. We may choose the isomorphism of (1) such that a point lying over $r$ maps to a point lying over $r'$.

3. 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.

4. 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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