---- > [!definition] Definition. ([[morphism of sheaves of modules]]) > Let $(X ,\mathcal{O}_X)$ be a [[ringed space]]. A **morphism of sheaves of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]** (or simply a **morphism of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]**) $\mathcal{F} \xrightarrow{f} \mathcal{G}$ is [[morphism of (pre)sheaves|morphism]] $f$ of the underlying [[sheaf|sheaves]] such that for all open $U \subset X$, the $U$-[[natural transformation|component]] $f_{U}:\mathcal{F}(U) \to \mathcal{G}(U)$ is $\mathcal{O}_{X}(U)$-[[linear map|linear]]. > $\text{Hom}_{\mathsf{Sh}(X)}(\mathcal{F}, \mathcal{G})$ denotes the [[abelian group]] of ([[sheaf]] of) $\mathcal{O}_{X}$-module morphisms $\mathcal{F} \to \mathcal{G}$. Analogous to [[homsets in R-mod are R-modules]], it is an $\mathcal{O}_{X}(X)=\Gamma(X, \mathcal{O}_{X})$-[[module]] via the evident [[module|action]] $(sf)_{U}(\cdot) \mapsto s |_{U} \cdot f_{U}(\cdot).$ Indeed, the rule $U \mapsto \text{Hom}_{\mathsf{Sh}(U)}(\mathcal{F} |_{U}, \mathcal{G} |_{U})$ prescribes a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]], denoted $\mathcal{Hom}_{X}(\mathcal{F} , \mathcal{G})$ and called the **internal hom sheaf** in direct analogue to the [[homsets in R-mod are R-modules|internal hom module]]. > [!definition] Kernels, cokernels, and images. > Just as the [[kernel of a module homomorphism|kernel]], [[cokernel of a module homomorphism|cokernel]], and [[image|image]] of a [[module|module homomorphism]] are defined to be those of the underlying [[abelian group|abelian group]] [[group homomorphism|homomorphism]], the **kernel**, **cokernel**, and **image** of a morphism $\mathcal{F} \xrightarrow{f} \mathcal{G}$ of [[sheaf|sheaves]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] are respectively defined to be the [[(pre)sheaf kernel|sheaf kernel]], [[sheaf cokernel]], and [[sheaf image]] of the underlying [[morphism of (pre)sheaves|morphism of sheaves of]] [[abelian group|abelian groups]]. It is evident that the $\mathcal{O}_{X}$-action restricts to each of $\operatorname{ker }f$, $\operatorname{coker }f$, $\operatorname{im }f$, making each into a [[sheaf of modules|sheaf of]] $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. ^definition For $\operatorname{ker }f$ this really is immediate. For $\operatorname{coker }f$ and $\operatorname{im }f$, somewhat more thinking is required because a [[sheafification]] is involved. ---- #### - [ ] check that internal hom sheaf is indeed a sheaf ---- #### 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 > ```