---- Let $(X, \mathcal{O}_{X})$ be a [[scheme]]. > [!theorem] Theorem. ([[relating Cartier divisors and line bundles]]) > There exists a [[group homomorphism]] between the [[group]] of [[Cartier divisor|Cartier divisors]] on $X$ and the [[Picard group]] on $X$ $\begin{align} \Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right) & \to \text{Pic }X \ (*) \\ D &\mapsto \mathcal{O}_{X}(D) \end{align}$ and this morphism descends to an [[group embedding|embedding]] $\text{CaCl }X \hookrightarrow \text{Pic }X \ (**)$ If $X$ is an [[integral scheme]], then the map $(*)$ is [[surjection|surjective]] and so the embedding $(* *)$ is in fact an [[isomorphism]]. ^theorem > [!note] Note. > Being [[integral scheme|integral]] is generally considered a light condition to place on a [[scheme]]. So it is generally not offensive to image [[locally free sheaf|line bundles]] as [[Cartier divisor|Cartier divisors]] modulo linear [[equivalence relation|equivalence]]. ^note > [!proof]- Proof. ([[relating Cartier divisors and line bundles]]) > What we will do: > 1. Show the map $(*)$ is indeed a [[group homomorphism]] . > 2. Show that $D$ is [[Cartier divisor|principal]] if and only if $\mathcal{O}_{X}(D)$ is the trivial element $\mathcal{O}_{X}=:1 \in \text{Pic }X$. [[first isomorphism theorem|This gives rise to the desired embedding.]] $(* *)$. > 3. Show that adding in the assumption that $X$ is [[integral scheme|integral]] makes $(* )$ into a [[surjection]]. Since its [[kernel]] consists precisely of the principal Cartier divisors, the map $(*)$ is therefore an [[isomorphism]]. > > - [ ] bring over from handwritten notes ---- #### ----- #### 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 > ```