---- > [!definition] Definition. ([[quasicoherent sheaf]]) > Let $(X, \mathcal{O}_{X})$ be a [[scheme]], $\mathcal{F}$ a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. We say $\mathcal{F}$ is **quasi-coherent** if there exists an [[cover|open cover]] $\{ U_{i} \}$ of $X$ by [[affine scheme|affines]] $U_{i}=\text{Spec }A_{i}$ such that, for all $i$, $\mathcal{F} |_{U_{i}} \cong \widetilde{M}_{i}$ for some $A_{i}$-[[module]] $M_{i}$.[^1] > > We furthermore say $\mathcal{F}$ is **coherent** if each $M_{i}$ can be taken to be [[submodule generated by a subset|finitely generated]] as an $A_{i}$-[[module]]. > > [[Quasicoherent sheaves over a scheme form an abelian category]]. > ^definition > [!note] Slogan. > [[ring|Rings]] are to [[module|modules]] as [[scheme|schemes]] are to (quasi)coherent sheaves. ^note > [!basicexample] > A [[locally free sheaf]] $\mathcal{F}$ of $\mathcal{O}_{X}$-[[sheaf of modules|modules]], $X$ a [[scheme]], is always quasi-coherent. Indeed, with $\{ U_{i} \}=\{ \text{Spec }A_{i} \}$ an open affine cover of $X$, $\begin{align} \mathcal{F} |_{U_{i}} &\cong (\mathcal{O}_{X} |_{U_{i}})^{\oplus J} \text{ for some index set }J \text{ (def. of locally free sheaf)} \\ & \cong \mathcal{O}_{\text{Spec }A_{i}}^{ \ \ \oplus J} \\ & \cong \widetilde{M}_{i}, \text{ where } \widetilde{M}_{i}= A_{i}^{\oplus J}. \end{align}$ (See also the example in [[sheaf associated to a module]].) If $\mathcal{F}$ is of finite rank, then it is in fact coherent. ^basic-example [^1]: Here, $\widetilde{M}$ denotes the [[sheaf associated to a module]]. ---- #### ---- #### 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 > ```