---- > [!definition] Definition. ([[(pre)sheaf kernel]]) > Let $f:\mathcal{F} \to \mathcal{G}$ be a [[morphism of (pre)sheaves|morphism of presheaves]] valued in a [[category]] where [[categorical kernel|kernels]] make sense. The **presheaf kernel** of $f$, $\ker f$, is the [[presheaf]] specified by $(\ker f)(U):=\ker \big(f_{U}:\mathcal{F}(U) \to \mathcal{G}(U) \big)$. > If $f:\mathcal{F} \to \mathcal{G}$ is a [[morphism of (pre)sheaves|morphism of]] [[sheaf|sheaves]], then $\ker \mathcal{F}$ is in fact a [[sheaf]]. ^definition > [!justification] > Quick check that $\ker \mathcal{F}$ is in fact a [[sheaf]] when $\mathcal{F}$, $\mathcal{G}$ are. ^justification > [!basicproperties] > Taking kernels and [[(pre)sheaf stalk|stalks]] are compatible: $(\ker f)_{p}=\ker (f_{p}: \mathcal{F}_{p} \to \mathcal{G}_{p})$ for all $p \in X$. ^properties > [!proof] > > There is really not much to show. > > Have a map $\begin{align} > (\ker f)_{p} &\to \ker f_{p} > \end{align}$ > taking a germ $[U, s] \in (\ker f)_{p}$ to itself viewed as an element of $\mathcal{F}_{p}$ sent to zero by $f_{p}$. Well-defined because if $[U,s] \in (\ker f)_{p}$, then $s \in \ker f_{U}$, so $f_{p}([U,s])=[U, f_{p}(s)]=[U,0]$ and thus $[U,s] \in \ker f_{p}$. Now we want to show injective and surjective. > > **Injective.** If $[U,s] =0 \text{ in } \mathcal{F}_{p}$, then $s$ restricts to zero on some neighborhood $V \ni p$, hence $[U,s]=0$ in $(\ker f)_{p}$. > > **Surjective.** If $[U,s] \in \ker f_{p}$, then $0=f_{p}([U,s])=[U, f_{U}(s)]$ in $\mathcal{G}_{p}$. Thus there exists $p \in V \subset U$ such that $f_{U}(s) |_{V}=0$. So $s |_{V} \in (\ker f)(V)$. Thus $(V, s |_{V}) \in (\ker f)_{p}$. ---- #### ---- #### 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 > ```