----- > [!proposition] Proposition. ([[the section functor is left-exact]]) > Let $X$ be a [[topological space]]. For any open set $U \subset X$, the [[section functor|section]] [[covariant functor|functor]] $\Gamma(U, \cdot)$ is [[exact functor|left-exact]], i.e., given an [[exact sequence]] of [[sheaf|sheaves]] $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}''$ we obtain an [[exact sequence]] $0 \to \Gamma(U, \mathcal{F}') \to \Gamma(U, \mathcal{F} ) \to \Gamma(U, \mathcal{F}'')$ of (say) [[abelian group|abelian groups]]. ^proposition > [!proof]- Proof. ([[the section functor is left-exact]]) > Label the maps involved as > > $0 \to \mathcal{F}' \xrightarrow{f} \mathcal{F} \xrightarrow{g} \mathcal{F}''$ > and $0 \to \mathcal{F}'(U) \xrightarrow{f_{U}}\mathcal{F}(U) \xrightarrow{g_{U}} \mathcal{F}''(U).$ > We need to show: > 1. $\ker f_{U}=\{ 0 \}$ > 2. $\im f_{U}=\ker g_{U}$ > > $(1)$ is immediate: since $f$ is an [[injective sheaf morphism]], $\ker f_{U}=\{ 0 \}$. > > Initially, $(2)$ does not appear immediate: while we know that sections in the [[sheafification]] of the [[presheaf image]] of $f$ at $U$ agree exactly with sections in $\ker g_{U}$, this is *a priori* not the same as having $\im f_{U}=\ker g_{U}$ because of the involved [[sheafification]]. However, since $f$ is [[injective sheaf morphism|injective]], the [[sheafification]] does nothing — the presheaf image is already a sheaf, and so actually $(2)$ *is* immediate. (See [[sheaf image#^properties]].) ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```