Lemma 40.9.4 (04MX)—The Stacks project

Lemma 40.9.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ and $g' : U'' \to U'$ be morphisms of schemes. Set $g'' = g \circ g'$. Let $(U', R', s', t', c')$ be the restriction of $R$ to $U'$. Let $h = s \circ \text{pr}_1 : U' \times _{g, U, t} R \to U$, let $h' = s' \circ \text{pr}_1 : U'' \times _{g', U', t} R \to U'$, and let $h'' = s \circ \text{pr}_1 : U'' \times _{g'', U, t} R \to U$. Let $\tau \in \{ Zariski, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic, \linebreak[0] fppf, \linebreak[0] fpqc\} $. Let $\mathcal{P}$ be a property of morphisms of schemes which is preserved under base change, and which is local on the target for the $\tau $-topology. If

$h(U' \times _ U R)$ is open in $U$,
$\{ h : U' \times _ U R \to h(U' \times _ U R)\} $ is a $\tau $-covering,
$h'$ has property $\mathcal{P}$,

then $h''$ has property $\mathcal{P}$. Conversely, if

$\{ t : R \to U\} $ is a $\tau $-covering,
$h''$ has property $\mathcal{P}$,

then $h'$ has property $\mathcal{P}$.

Proof. This follows formally from the properties of the diagram of Lemma 40.9.3. In the first case, note that the image of the morphism $h''$ is contained in the image of $h$, as $g'' = g \circ g'$. Hence we may replace the $U$ in the lower right corner of the diagram by $h(U' \times _ U R)$. This explains the significance of conditions (1) and (2) in the lemma. In the second case, note that $\{ \text{pr}_0 : U' \times _{g, U, t} R \to U'\} $ is a $\tau $-covering as a base change of $\tau $ and condition (a). $\square$

The Stacks project

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