Lemma 7.26.1. Let \mathcal{C} be a site. Let \{ U_ i \to U\} be a covering of \mathcal{C}. Let \mathcal{F}, \mathcal{G} be sheaves on \mathcal{C}. Given a collection
\varphi _ i : \mathcal{F}|_{\mathcal{C}/U_ i} \longrightarrow \mathcal{G}|_{\mathcal{C}/U_ i}
of maps of sheaves such that for all i, j \in I the maps \varphi _ i, \varphi _ j restrict to the same map \varphi _{ij} : \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j} \to \mathcal{G}|_{\mathcal{C}/U_ i \times _ U U_ j} then there exists a unique map of sheaves
\varphi : \mathcal{F}|_{\mathcal{C}/U} \longrightarrow \mathcal{G}|_{\mathcal{C}/U}
whose restriction to each \mathcal{C}/U_ i agrees with \varphi _ i.
Proof.
The restrictions used in the lemma are those of Lemma 7.25.8. Let V/U be an object of \mathcal{C}/U. Set V_ i = U_ i \times _ U V and denote \mathcal{V} = \{ V_ i \to V\} . Observe that (U_ i \times _ U U_ j) \times _ U V = V_ i \times _ V V_ j. Then we have \mathcal{F}|_{\mathcal{C}/U_ i}(V_ i/U_ i) = \mathcal{F}(V_ i) and \mathcal{F}|_{\mathcal{C}/U_ i \times _ U U_ j}(V_ i \times _ V V_ j/U_ i \times _ U U_ j) = \mathcal{F}(V_ i \times _ V V_ j) and similarly for \mathcal{G}. Thus we can define \varphi on sections over V as the dotted arrows in the diagram
\xymatrix{ \mathcal{F}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{F}) \ar@{..>}[d] \ar[r] & \prod \mathcal{F}(V_ i) \ar[d]_{\prod \varphi _ i} \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{F}(V_ i \times _ V V_ j) \ar[d]_{\prod \varphi _{ij}} \\ \mathcal{G}(V) \ar@{=}[r] & H^0(\mathcal{V}, \mathcal{G}) \ar[r] & \prod \mathcal{G}(V_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod \mathcal{G}(V_ i \times _ V V_ j) }
The equality signs come from the sheaf condition; see Section 7.10 for the notation H^0(\mathcal{V}, -). We omit the verification that these maps are compatible with the restriction maps.
\square
Comments (1)
Comment #3034 by Brian Lawrence on