# Monday, November 22

:::{.remark}
Last time: a local ring $R$ is **regular** iff the number of generators of $\mfm_R$ is equal to the Krull dimension of $R$.
There is a slightly weaker notion: a scheme is regular in codimension 1 iff every local ring $\OO_{X, x}$ of dimension 1 is regular.
Note that this is locally the generic point associated to a height 1 prime ideal.
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:::{.remark}
Note that not every local ring is a domain, e.g. $\OO_{V(xy), \vector 0}$.
Algebra fact: a 1-dimensional regular local ring is a DVR, since this forces $\mfm$ to be generated by 1 element and thus principal.
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:::{.remark}
A new standing assumption: $X\in \Sch$ is 

- Noetherian,
- Integral (covered by spec of integral domains, equivalently reduced and irreducible)
- Separated,
- Regular in codimension 1.

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:::{.example title="?"}
Examples are smooth projective varieties, but may include singular varieties, e.g. $V(xy - z^2) \subseteq \AA^3\slice k$.
Note that the partials vanish at $\vector 0$, an singularity is picked up by the fact that $k[x,y]/\gens{f} \localize{\gens{ x, y} }$ and this is conical hyperboloid:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/Schemes/sections/figures}{2021-11-22_11-43.pdf_tex} };
\end{tikzpicture}

Note that some curves may be singular, namely those passing through $0$, but generically they are nonsingular.
An equation of a line on $X$ might be $V(x, z)$, so consider $\mfp = \gens{x, y} \in k[x,y,z]/\gens{xy-z^2}$.

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:::{.exercise title="A good one"}
Check that $R\localize{\mfp}$ is regular, and $\mfp R\localize{\mfp}$ is principal.
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:::{.definition title="Prime divisors"}
A **prime divisor** on $X$ is an integrable subscheme of codimension 1.
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:::{.example title="?"}
Take $V(y^2-x^3) \subseteq \AA^2\slice k$, or $V(x, z) \subseteq \spec R$.
Generally, if $f\in R$ is irreducible then $V(f) \subseteq \spec R$ is a prime divisor.
Note that the second example is codimension 1 in $\AA^2\slice k$.
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:::{.remark}
If $X=\spec R$, then the prime divisors are in 1-to-1 correspondence with height 1 prime ideals in $R$.
Check that $\gens{0} \subsetneq \mfp$ since $R$ is a domain, and no prime ideal can fit between these.
Note that in the above example, $\gens{0} \subsetneq \gens{x, z} \subseteq k[x,y,z]/\gens{xy-z^2}$, where e.g. $\gens{x}$ isn't prime because $xy\in \gens{x} \implies z^2\in \gens{x} \implies z\in \gens{x}$.
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:::{.example title="?"}
Some examples of prime divisors:

- For $X$ a nice variety: the irreducible subvarieties of codimension 1.
- For $X = \spec \ZZ$: closed points, i.e. any maximal ideal.
- For $X=\spec \ZZ\adjoin{\sqrt{-5}}$: an example might be $\gens{2, 1 + \sqrt{-5}}$.
- For $X= \FF_3[t]$, consider $\gens{t-a_i}$ for $a_i=0,1,2$ and $\gens{t^2-2}$.
  Note that being a prime ideal is not preserved under base change, e.g.

\begin{tikzcd}
	{X' = \spec \FF_3[t]\tensor \FF_q = \spec \FF_q[t]} && {X = \spec \FF_3[t]} \\
	\\
	{\spec \FF_q} && {\spec \FF_3}
	\arrow[from=3-1, to=3-3]
	\arrow[from=1-3, to=3-3]
	\arrow[from=1-1, to=1-3]
	\arrow[from=1-1, to=3-1]
	\arrow["\lrcorner"{anchor=center, pos=0.125, rotate=45}, draw=none, from=1-1, to=3-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJYJyA9IFxcc3BlYyBcXEZGXzNbdF1cXHRlbnNvciBcXEZGX3EgPSBcXHNwZWMgXFxGRl9xW3RdIl0sWzIsMCwiWCA9IFxcc3BlYyBcXEZGXzNbdF0iXSxbMiwyLCJcXHNwZWMgXFxGRl8zIl0sWzAsMiwiXFxzcGVjIFxcRkZfcSJdLFszLDJdLFsxLDJdLFswLDFdLFswLDNdLFswLDIsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==)

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:::{.definition title="Weil Divisor"}
The **Weil divisors** on $X$ is the free $\ZZ\dash$module on the prime divisors, and is denoted $\Div(X)$.
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:::{.example title="?"}

- $1[\gens{x, y}] \in \Div(\spec k[x,y,z] / \gens{xy-z^2})$.
- $2[\gens{z}] - [\gens{3}] + 8[\gens 7] \in \Div(\spec \ZZ)$.
- $[V(y^2-x^3)] + 2[V(y)] \in \Div(\AA^2\slice k)$
- $[0] - [\infty] \in \Div(\PP^1\slice k)$.
- For $C$ an irreducible reduced curve, any linear combination of closed points.

Note that *Cartier* divisors are those locally cut out by a single equation.
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:::{.remark}
Since $X$ is integral, it has a generic point $\eta$, so define a **rational function** as a nonzero element of $\OO_{X, \eta}$.
Equivalently, if $\spec R \subseteq X$ is an affine chart, an element of $k\units$ where $k = \ff(R)$.
Note that this is independent of further localizing $R$!
Any rational function $\phi$ on $X$ gives an element $\phi \in \ff \OO_{X, x} \cong k$ for any point $x\in X$.
In particular, the standing assumptions (specifically being regular in codimension 1) implies that $\OO_{X, x}$ is a DVR when $x$ is the generic point of a prime divisor.
Let $Y \subseteq X$ be a prime divisor, then define $v_Y(\phi)$ to be the valuation $v(\phi)$ in $\OO_{X, Y}$.
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:::{.example title="?"}
The element $4/7$ is a rational function on $\spec \ZZ$, which is exactly $\QQ\units$.
Moreover $4/7\in \OO_{\spec \ZZ, 2}$ and $\val_\mfm(4/7) = 2$ for $\mfm = \gens{2}$.
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:::{.definition title="Divisors of functions, principal divisors, class groups"}
\[
\Div(\phi) \da \sum_{Y \subseteq X \text{prime}} v_Y(\phi) [Y]
.\]
These are called **principal divisors**, and form a group $\cl(X)$ the **class group**.
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:::{.example title="?"}
$\Div(4/7) = 2[2] - 1[7]$.
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