---- > [!definition] Definition. ([[Cartier divisor]]) > ~ > Let $(X, \mathcal{O}_{X})$ be a [[scheme]]. Denote by $\mathcal{K}_{X}$ the [[sheaf]] [[sheaf of rational functions|of rational functions]] on $X$, and by $\mathcal{K}_{X}^{*}$ its sheaf (of [[abelian group|abelian groups]]) of [[unit|invertible]] elements. > > We have a natural inclusion $\mathcal{O}_{X}^{*} \hookrightarrow \mathcal{K}_{X}^{*}$.[^1] A **Cartier divisor** on $X$ is a global section of $\frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}}$, i.e., an element of $\Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right)$. We call $\Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right)$ the **group of Cartier divisors on $X$**. > > Cf. the discussion in [[quotient sheaf|quotient sheaf]], any Cartier divisor is (nonuniquely) represented by a collection of pairs ("local equations") $\{ (U_{i}, g_{i} )\}$, where the $U_{i}$ [[cover|openly cover]] $X$, $g_{i} \in \Gamma(U_{i}, \mathcal{K}_{X}^{*})$, and[^2] $\frac{g_{i}}{g_{j}} \in \Gamma(U_{i} \cap U_{j}, \mathcal{O}_{X}^{*})$ on overlaps $U_{i} \cap U_{j}$. > > A Cartier divisor $D$ is **principal** if it lies in the image of $\Gamma(X, \mathcal{K}_{X}^{*}) \xrightarrow{\pi_{X}} \Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right)$.[^3] Concretely, $\pi_{X}$ takes a global rational function $f \in \Gamma(X, \mathcal{K}_{X}^{*})$ and sends it to the Cartier divisor represented by $\{ (X, f) \}$.[^4] We write $D=(f)$. > > > Two Cartier divisors are **linearly equivalent** if their difference is principal.[^5] > > The **Cartier class group** $\text{CaCl }X$ is the [[categorical cokernel|cokernel]] of the map $\Gamma(X, \mathcal{K}_{X}^{*}) \to \Gamma\left( X, \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}} \right)$, i.e., the group of Cartier divisors on $X$ modulo linear equivalence. ---- #### [^1]: This is certainly [[injective sheaf morphism|injective]] at the [[presheaf]] level, because each [[localization|localization map]] $\mathcal{O}_{X}(U) \to S(U) ^{-1} \mathcal{O}_{X}(U)$ has trivial [[kernel of a ring homomorphism|kernel]] (Recall: in general $\operatorname{ker}(R \to S ^{-1} R)=\{ 0 \} \iff \text{zero-divisors}(R) \cap S=\emptyset$. By construction, $S(U)$ has no [[zero-divisor|zero-divisors]] inside it — since those would induce zero-divisor [[(pre)sheaf stalk|germs]]), and the localization map restricts to an injective map on [[unit|units]] ([[ring homomorphisms preserve structure|ring morphisms preserve units]]) at the presheaf level. Then recall that sheafifying preserves injectivity. [^2]: Note the multiplicative notation. In additive notation, this would be $g_{i}-g_{j} \in \Gamma(U_{i} \cap U_{j}, \mathcal{O}_{X}^{*})$, i.e., $[g_{i}]=[g_{j}]$. [^3]: Recall from example sheet 2 that such a map exists and may not be [[surjection|surjective]]. Specifically, it is the global sections [[covariant functor|functor]] applied to the natural [[quotient sheaf|quotient map]] $\mathcal{K}_{X}^{*} \xrightarrow{\pi} \frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}}$, i.e., the $X$-[[natural transformation|component]] $\pi_{X}:\mathcal{K}_{X}(X) \to (\frac{\mathcal{K}_{X}^{*}}{\mathcal{O}_{X}^{*}})(X)$. [^4]: This is unpacked in a footnote in [[quotient sheaf]]. [^5]: Note the verbage 'difference' used here even though the groups are multiplicative. ---- #### 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 > ```