The Stacks project

40.7 Comparing fibres

Let $(U, R, s, t, c, e, i)$ be a groupoid scheme over $S$. Diagram ( 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 ( 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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