---- Assume $(X, \mathcal{O}_{X})$ is a [[locally Noetherian scheme|Noetherian]], [[integral scheme|integral]] [[scheme]] [[scheme over a field|over]] a [[field]] $k$ which is [[regular scheme|regular]] in [[codimension of a closed subspace|codimension]] $1$. (This is the default for this chapter of the course.) Recall that $K(X)$ denotes the [[generic point of an integral scheme|function field]] of $X$, $K(X)^{*}$ its [[unit|nonzero elements]], and that any [[prime divisor in a scheme|prime divisor]] $Y \subset X$ gives rise to a [[discrete valuation]] $\nu_{Y}:K(X)^{*} \to \mathbb{Z}$. Write $\text{Div}(X)$ for [[group]] of Weil divisors on $X$, i.e., the [[free abelian group]] generated by [[prime divisor in a scheme|prime divisors]] of $X$. > [!definition] Definition. ([[divisor (of zeros and poles)]]) > Let $f \in K(X)^{*}$. The **divisor (of zeros and poles)** of $f$, denoted $(f)$, is the [[prime divisor in a scheme|Weil divisor]] specified by the $\mathbb{Z}$-[[linear combination]]$(f):= \sum_{Y \subset X \text{ a prime divisor}}\nu_{Y}(f) \ Y \ \ \in \text{Div}(X).$ > ([[almost all prime divisors vanish on a given function|This result]] ensures such a summation is indeed finite.) > > Two Weil divisors $D,D' \in \text{Div}(X)$ are **linearly equivalent**[^3] if $D-D'=(f)$ for some $f \in K(X)^{*}$. Write $D \sim D'$. If $D \sim 0$, we say $D$ is a **principal divisor**. > > There is a [[group homomorphism]][^1] $\begin{align} > K(X)^{*} &\to \text{Div}(X) \\ > f & \mapsto (f) > \end{align}$ called the **divisor map** of $X$. Its image consists of all principal divisors of $X$. The [[quotient group|quotient]] of $\text{Div}(X)$ by this [[subgroup]] is known as the **class group** of $X$: $\text{Cl }X:= \frac{\text{Div} \ X}{ \{ (f): f \in K(X)^{*} \}}.$ > It identifies divisors (elements of $\text{Div}(X)$ ) up to linear equivalence[^2], and measures the failure of divisors to be principal. > [!note] Note. > $\text{Cl }X$ in some sense only depends on certain parts of $X$: often we may delete a [[closed set|closed subspace]] of $X$ without losing much (or any) information about $\text{Cl }X$. This can be very useful for computations. See [[computing class group by deleting a subspace]]. ^note > [!basicexample] > >- For which $X$ is $\text{Cl }X=0$ (i.e., when is every [[prime divisor in a scheme|Weil divisor]] principal)? One example is $\text{Spec}$ of a [[UFD]]. In fact, this is the only example among [[normal scheme|normal]] [[affine scheme|affine schemes]]. See [[class group quantifies failure of unique factorization in a ring]].[^4] >- Using [[computing class group by deleting a subspace]], can compute $\text{Cl }\mathbb{P}^{n}_{k}=\text{Cl }\text{Proj }k[X_{0},\dots,X_{n}]\cong \mathbb{Z} \langle H \rangle$, where $H$ is the class of a hyperplane $H=V(\langle X_{0} \rangle)$. See [[class group of projective space]]. ^basic-example ---- #### [^1]: This map is indeed a [[group homomorphism]], since $\nu_{Y}$ is: $\begin{align} (fg) &= \sum_{Y \subset X \text{ p.d.}} \nu_{Y}(fg) \ Y \\ &= \sum_{Y \subset X \text{ p.d.}} (\nu_{Y}(f) + \nu_{Y}(g)) \ Y \\ &= \sum_{Y \subset X \text{ p.d.}} \nu_{Y}(f ) \ Y + \sum_{Y \subset X \text{ p.d.}} \nu_{Y}(g) \ Y \\ &= (f) + (g). \end{align}$ [^2]: Usual group quotienting stuff: $[D]=[D']$ iff $D-D' \in \{ (f): f \in K(X)^{*} \}$ iff $D \sim D'$. [^3]:This is indeed an [[equivalence relation]]: $D-D=0=(1)$. If $D-D'=(f)$ then $D'-D=(f ^{-1})$. If $D-D'=(f)$ and $D'-D''=(g)$, then $D-D''=\underbrace{ (D-D') }_{ (f) }+\underbrace{ (D'' -D') }_{ (g)})=(f)+(g)=(fg).$ [^4]: That proof uses [[Algebraic Hartog's Lemma]], as well as [[UFD iff height-1 primes are principal]] and [[every UFD is integrally closed]]. ---- #### 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 > ```