---- > [!definition] Definition. ([[Picard group]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]] and let $\mathcal{L}$ be a [[locally free sheaf|line bundle]] on $X$. Observe that $\mathcal{L}^{\vee} \otimes_{\mathcal{O}_{X}} \mathcal{L} \cong \mathcal{O}_{X}$, since its [[lines bundles and transition functions|transition functions]] are [[identity map|identities]].[^1] > > Now let $X$ be a [[scheme]]. The **Picard group** of $X$, $\text{Pic }X$, is the set of [[morphism of sheaves of modules|isomorphism classes]] of [[locally free sheaf|line bundles]] on $X$. It is made into a [[group]] by defining multiplication via the [[tensor product sheaf of modules|tensor product of line bundles]] construction, with inverses coming from the above observation. The identity is the trivial bundle $\mathcal{O}_{X}$. That is: > - $\mathcal{L}_{1} \cdot \mathcal{L}_{2}:= \mathcal{L}_{1} \otimes_{\mathcal{O}_{X}}\mathcal{L}_{2}$ ; > - $\mathcal{L}^{-1}=\mathcal{L}^{\vee}$; > - $1_{\text{Pic }X}=\mathcal{O}_{X}$. > > ---- #### [^1]: Indeed, if $\mathcal{L}$ has transition functions $g_{ij}$, then the [[dual line bundle]] $\mathcal{L}^{\vee}$ has transition functions $g_{ij}^{-1}$, hence the [[tensor product sheaf of modules#^basic-example|tensor product line bundle]] has transition functions $g_{ij}g_{ij}^{-1}=1$. ---- #### 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 > ```