Lemma 7.38.2. Let \mathcal{C} be a site and let \{ p_ i\} _{i\in I} be a conservative family of points. Then
Given any map of sheaves \varphi : \mathcal{F} \to \mathcal{G} we have \forall i, \varphi _{p_ i} injective implies \varphi injective.
Given any map of sheaves \varphi : \mathcal{F} \to \mathcal{G} we have \forall i, \varphi _{p_ i} surjective implies \varphi surjective.
Given any pair of maps of sheaves \varphi _1, \varphi _2 : \mathcal{F} \to \mathcal{G} we have \forall i, \varphi _{1, p_ i} = \varphi _{2, p_ i} implies \varphi _1 = \varphi _2.
Given a finite diagram \mathcal{G} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), a sheaf \mathcal{F} and morphisms q_ j : \mathcal{F} \to \mathcal{G}_ j then (\mathcal{F}, q_ j) is a limit of the diagram if and only if for each i the stalk (\mathcal{F}_{p_ i}, (q_ j)_{p_ i}) is one.
Given a finite diagram \mathcal{F} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}), a sheaf \mathcal{G} and morphisms e_ j : \mathcal{F}_ j \to \mathcal{G} then (\mathcal{G}, e_ j) is a colimit of the diagram if and only if for each i the stalk (\mathcal{G}_{p_ i}, (e_ j)_{p_ i}) is one.
Comments (2)
Comment #8356 by ZL on
Comment #8962 by Stacks project on