Maps of sheaves glue.

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. $\square$

Comment #3034 by Brian Lawrence on

Suggested slogan: Maps of sheaves glue.

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