Lemma 39.20.6. Let \tau \in \{ Zariski, {\acute{e}tale}, fppf, smooth, syntomic\} . Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U' \to U a morphism of schemes over S. Let (U', R', s', t', c') be the restriction of (U, R, s, t, c) to U'. Assume U, U', R, S are objects of a \tau -site \mathit{Sch}_\tau . The map of quotient sheaves
U'/R' \longrightarrow U/R
is injective. If the composition
\xymatrix{ U' \times _{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R \ar[r]_ s & U }
defines a surjection of sheaves in the \tau -topology then the map is bijective. This holds for example if \{ h : U' \times _{g, U, t} R \to U\} is a \tau -covering, or if U' \to U defines a surjection of sheaves in the \tau -topology, or if \{ g : U' \to U\} is a covering in the \tau -topology.
Proof.
Injectivity follows on combining Lemmas 39.13.2 and 39.20.5. To see surjectivity (see Sites, Section 7.11 for a characterization of surjective maps of sheaves) we argue as follows. Suppose that T is a scheme and \sigma \in U/R(T). There exists a covering \{ T_ i \to T\} such that \sigma |_{T_ i} is the image of some element f_ i \in U(T_ i). Hence we may assume that \sigma is the image of f \in U(T). By the assumption that h is a surjection of sheaves, we can find a \tau -covering \{ \varphi _ i : T_ i \to T\} and morphisms f_ i : T_ i \to U' \times _{g, U, t} R such that f \circ \varphi _ i = h \circ f_ i. Denote f'_ i = \text{pr}_0 \circ f_ i : T_ i \to U'. Then we see that f'_ i \in U'(T_ i) maps to g \circ f'_ i \in U(T_ i) and that g \circ f'_ i \sim _{T_ i} h \circ f_ i = f \circ \varphi _ i notation as in (39.20.0.1). Namely, the element of R(T_ i) giving the relation is \text{pr}_1 \circ f_ i. This means that the restriction of \sigma to T_ i is in the image of U'/R'(T_ i) \to U/R(T_ i) as desired.
If \{ h\} is a \tau -covering, then it induces a surjection of sheaves, see Sites, Lemma 7.12.4. If U' \to U is surjective, then also h is surjective as s has a section (namely the neutral element e of the groupoid scheme).
\square
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