---- > [!definition] Definition. ([[dual sheaf of modules]]) > Let $(X, \mathcal{O}_{X})$ be a [[scheme]] and $\mathcal{F}$ a [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. The [[morphism of sheaves of modules|internal hom sheaf]] $\mathcal{Hom}_{\mathcal{O}_{X}}\big( \mathcal{F}, \mathcal{O}_{X} \big)$ is called the **dual sheaf** of $\mathcal{F}$. it is a clean analogue to the notion of a [[dual vector space|dual]] $R$-[[module]] for $R$ a [[ring]]. ^definition > [!basicexample] Example. (Dual Line Bundle) > Let $\mathcal{L}$ be a [[locally free sheaf|line bundle]] on $(X, \mathcal{O}_{X})$. Its **dual line bundle** is $\mathcal{L}^{\vee}=\mathcal{Hom}\big( \mathcal{L}, \mathcal{O}_{X} \big).$ This is indeed also a line bundle: letting $\{ U_{i} \}$ be a [[locally free sheaf|trivializing cover]], $\mathcal{L} |_{U_{i}} \cong \mathcal{O}_{U_{i}}$; we have[^1] $\big(\mathcal{Hom}_{\mathcal{O}_{X}}(\mathcal{L} , \mathcal{O}_{X}) \big)|_{U_{i}} \cong\mathcal{Hom}_{\mathcal{O}_{U_{i}}}(\mathcal{O}_{U_{i}}, \mathcal{O}_{U_{i}}) \cong \mathcal{O}_{U_{i}}.$ If $\mathcal{L}$ has [[lines bundles and transition functions|transition functions]] $g_{ij}$, then $\mathcal{L}^{\vee}$ has transition functions given by $g_{ij} ^{-1}$. ^basic-example ---- #### [^1]: That $\mathcal{Hom}_{\mathcal{O}_{U_{i}}}(\mathcal{O}_{U_{i}}, \mathcal{O}_{U_{i}}) \cong \mathcal{O}_{U_{i}}$ is not surprising following from the fact that for any [[ring]] $R$, $\text{Hom}_{R}(R, R) \cong R$: one [[ring homomorphism]] per choice of destination for $1_{R}$. ---- #### 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 > ```