Lemma 7.41.2. Let f : \mathcal{D} \to \mathcal{C} be a morphism of sites associated to the continuous functor u : \mathcal{C} \to \mathcal{D}. Assume that for any object U of \mathcal{C} and any covering \{ V_ j \to u(U)\} in \mathcal{D} there exists a covering \{ U_ i \to U\} in \mathcal{C} such that the map of sheaves
\coprod h_{u(U_ i)}^\# \to h_{u(U)}^\#
factors through the map of sheaves
\coprod h_{V_ j}^\# \to h_{u(U)}^\# .
Then f_* transforms surjective maps of sheaves into surjective maps of sheaves.
Proof.
Let a : \mathcal{F} \to \mathcal{G} be a surjective map of sheaves on \mathcal{D}. Let U be an object of \mathcal{C} and let s \in f_*\mathcal{G}(U) = \mathcal{G}(u(U)). By assumption there exists a covering \{ V_ j \to u(U)\} and sections s_ j \in \mathcal{F}(V_ j) with a(s_ j) = s|_{V_ j}. Now we may think of the sections s, s_ j and a as giving a commutative diagram of maps of sheaves
\xymatrix{ \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ h_{u(U)}^\# \ar[r]^ s & \mathcal{G} }
By assumption there exists a covering \{ U_ i \to U\} such that we can enlarge the commutative diagram above as follows
\xymatrix{ & \coprod h_{V_ j}^\# \ar[r]_-{\coprod s_ j} \ar[d] & \mathcal{F} \ar[d]^ a \\ \coprod h_{u(U_ i)}^\# \ar[r] \ar[ur] & h_{u(U)}^\# \ar[r]^ s & \mathcal{G} }
Because \mathcal{F} is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections s_ i \in \mathcal{F}(u(U_ i)), i.e., sections s_ i \in f_*\mathcal{F}(U_ i). The commutativity of the diagram implies that a(s_ i) is equal to the restriction of s to U_ i. In other words we have shown that f_*a is a surjective map of sheaves.
\square
Comments (0)