---- > [!definition] Definition. ([[sheaf cokernel]]) > Let $\mathcal{F} \xrightarrow{f} \mathcal{G}$ be a [[morphism of (pre)sheaves|morphism of sheaves]]. The **sheaf cokernel** of $f$ is the [[sheafification]] of the [[presheaf cokernel]], (still) denoted by $\text{coker }f$. ^definition > [!note] Remark. > There is an asymmetry between [[sheaf cokernel]] and [[(pre)sheaf kernel|sheaf kernel]]: the former requires [[sheafification]] of the [[presheaf cokernel]] while the latter does not require [[sheafification]] of the [[(pre)sheaf kernel|presheaf kernel]]. This asymmetry is the same asymmetry between injectivity and surjectivity at play in the proof of [[sheaf isomorphism iff isomorphism on stalks]]. ^note > [!warning] Why we need to sheafify. > Let $X=\mathbb{P}^{1}$. Let $p,q \in X$ be distinct points. Let $\mathcal{G}$ be the [[sheaf]] of [[projective variety|regular functions]] on $X$, and $\mathcal{F}$ be the [[sheaf]] of regular functions on $X$ which vanish at $p$ and $q$. Let $f:\mathcal{F} \hookrightarrow \mathcal{G}$ be the obvious inclusion.[^1] Note $\mathcal{F}(U)=\mathcal{G}(U)$ if $U \cap \{ p,q \}=\emptyset$. > > *Claim: the [[presheaf cokernel]] of $f$ is not a [[sheaf]].* > > To see this, first note $\mathcal{G}(X)=k$ and $\mathcal{F}(X)=0$, because regular functions on $X$ are constants. Then [[cover]] $X$ with [[affine variety|affines]] $U:=X-\{ p \} \cong \mathbb{A}^{1}$ and $V:=X-\{ q \} \cong \mathbb{A}^{1}$. > > > We have at the [[presheaf]] level: > > - $(\operatorname{coker }f)(X)=\cancel{ \mathcal{G}(X) }^{k} /\cancel{ \mathcal{F}(X) }^{0}=k$. > - $(\operatorname{coker }f)(U)=\mathcal{G}(U) / \mathcal{F}(U)=k[T] / \langle T \rangle=k$ if $T$ is the coordinate on $\mathbb{A}^{1}=X-\{ p \}$ with $q$ at $T=0$. > - Similarly, $(\text{coker }f)(V)=\mathcal{G}(V) / \mathcal{F}(V)=k$. > - $(\operatorname{coker }f)(U \cap V)=\mathcal{G}(U \cap V) / \mathcal{F}(U \cap V)=0$, since $p,q\notin U \cap V$. > > Now choose any $s_{U} \in (\operatorname{coker }f)(U)$, $s_{V} \in (\operatorname{coker }f)(V)$. Certainly $s_{U}$ and $s_V$ agree on overlap, if $\operatorname{coker }f$ were a sheaf, then $(\operatorname{coker }f)(X) \subset k \oplus k$. (?) > > > > ---- #### [^1]: Recall that there is no need to [[sheafification|sheafify]] the [[presheaf image]] when the [[morphism of (pre)sheaves|morphism]] is [[injective sheaf morphism|injective]]. So this is an earnest inclusion. ---- #### 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 > ```