# Tuesday, March 15

## Artin Rings

:::{.remark}
Last time: $A/\jacobsonrad{A}$ is a product of Artin local rings.
:::

:::{.exercise title="?"}
Show that if $A$ is Artin local, then $\mfm_A^n = 0$ for some $n$.

> Hint: use that $\ts{\mfm^k}_{k\geq 0}$ stabilizes, so $\mfm^n/\mfm^{n+1} \cong \mfm^n/\mfm \mfm^{n} = 0$ so $\mfm^n=0$ since $A$ is Noetherian.

:::

:::{.exercise title="?"}
Show that if $A$ is an Artin local ring that is finitely generated over an algebraically closed field $k$.
Then $A$ is a quotient of $\kxn/\gens{x_1,\cdots,x_n}^N$ for some $n$ and $N$.
:::

:::{.solution}
Hint: use that $\mfm$ is finitely generated, construct a surjection $\kxn\to A$, and show $I_{N} \da \gens{x_1,\cdots, x_n}^N$ is in the kernel.
Also use the corollary of the Nullstellensatz (EEKS) that finite extensions of $k$ which are fields are necessarily $k$ itself.
Apply the snake lemma:

\begin{tikzcd}
	&&& 0 \\
	0 & {\gens{x_1,\cdots, x_n}} & \kxn & k & 0 \\
	\\
	0 & \mfm & A & {A/\mfm \cong k} & 0 \\
	& 0 & \textcolor{rgb,255:red,92;green,92;blue,214}{\therefore 0} & 0
	\arrow[from=2-1, to=2-2]
	\arrow[from=2-2, to=2-3]
	\arrow[from=2-3, to=2-4]
	\arrow[from=2-4, to=2-5]
	\arrow[from=4-1, to=4-2]
	\arrow[from=4-2, to=4-3]
	\arrow[from=4-3, to=4-4]
	\arrow[from=4-4, to=4-5]
	\arrow["\sim", from=2-2, to=4-2]
	\arrow[two heads, from=2-3, to=4-3]
	\arrow["{?}", color={rgb,255:red,92;green,92;blue,214}, from=4-3, to=5-3]
	\arrow["\sim", from=2-4, to=4-4]
	\arrow[from=5-2, to=5-3]
	\arrow[from=5-3, to=5-4]
	\arrow[from=1-4, to=5-2]
\end{tikzcd}

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

This shows surjectivity, and $\gens{x_1,\cdots, x_n}^N$ being in the kernel follows from the previous proposition.
:::

## DVRs

:::{.exercise title="?"}
Recall that a Noetherian local domain $A$ is a DVR if $\mfm\neq 0$ is principal.
Show that $\dim_k \mfm/\mfm^2 = 1$ for $k=A/\mfm$.

> Hint: use Nakayama to bound the dimension by 1.

:::

:::{.example title="?"}
Examples of DVRs:

- $k\fps{t}$
- $\ZZpadic$
- $\ZZ_{(p)} = \ZZ\adjoin{\ts{ q\inv \st q\neq p }}$.

A non-example: $k\fps{x,y}$.
:::

:::{.exercise title="?"}
Show that if $A$ is a DVR with uniformizer $\pi$ and $a\in A\smz$, then there is a unique $n\in \ZZ_{\geq 0}$ such that $a = \pi^n a_u$ with $a_u$ a unit.

> Hint: for existence, set $N\da \max \ts{N\st a\in \gens{\pi^N}}$ which exists because $\intersect_N \gens{\pi^N} = 0$.
  Use that $\pi I = I \implies I=0$ by Nakayama, to write $a = \pi^n a_0$, and if $a_0$ is not a unit then $a_0\in\gens{\pi}$ contradicting maximality.

:::

:::{.corollary title="?"}
If $A$ is a DVR, $\Id(A) = \ts{\gens 0, \gens{\pi^n} }_{n\geq 0}$ and $\spec A = \ts{\gens\pi, \gens 0}$.
This follows from writing 
\[
I = \gens{a_1,\cdots, a_N} = \gens{\pi^{n_1} b_1,\cdots, \pi^{n_N} b_N } = \gens{\pi^m},\qquad m\da \min\ts{n_j}_{j\leq N}
.\]
:::

:::{.exercise title="?"}
Show that DVRs $A$ biject with fields $K$ equipped with a valuation $v: K\to \ZZ\union\ts{\infty}$ satisfying:

- $v(a+b) \geq \min( v(a), v(b) )$,
- $v(ab) = v(a) + v(b)$,
- $v(a) =\infty \iff a=0$,
- $v(K\units) \neq \ZZ$.

> Hint: for $A\in\DVR$, set $K = \ff(A)$ and $v(a/b)\da v(a)-v(b)$ where $v(\pi^n a_0) = n$.
  Given $(K, v)$, set $A = \ts{x\in K \st v(x)\geq 0}$ with $\mfm = \ts{v(x) > 0}$, showing $\mfm^c \subseteq A\units$ and $\mfm$ is generated by any $x$ with $v(x) = 1$?

:::

## Classifying finitely-generated modules over a DVR

:::{.remark}
Recall that $M\in\mods{A}$ is torsionfree iff $\Ann_A(M) \da \ts{a\in A\st am=0}$ contains only zero divisors iff $M \mapsvia{\times a} M$ is injective for all nonzero $a\in A$.
:::

:::{.exercise title="?"}
Show that if $A\in \CRing^{\Noeth}$ and $M\in \mods{A}^\fg$ then $0\to M_\tors\to M\to M'\to 0$ is a SES where $M'$ is torsionfree, and moreover there exists some $a\in A$ such that $a M_\tors = 0$.

> Hint: for the latter statement, use that $M_\tors$ is finitely-generated and take a product of annihilators of generators.
  For the former, take $a\in \Ann_A(\tilde m)$ for some $\tilde m\in M'$, lift to $m\in M$ and show $am\in M_\tors$ for some $a$.

:::

:::{.exercise title="?"}
Show that if $A$ is a PID, then $M\in\mods{A}$ is flat iff $M$ is torsionfree.

> Hint: use $A \mapsvia{\times a} A$ and apply $(\wait) \tensor_A M$.
  For the reverse, show $\Tor^1(M, A/I) = 0$ for any ideal in $A$ and compute using the projective resolution $0\to A \mapsvia{\times a} A \to A/\gens{a} \to 0$.
  Note that $H_1(M \mapsvia{\times a} M ) = \ker(M \mapsvia{\times a} M) = 0$ since $M$ is torsionfree.

:::

:::{.exercise title="?"}
Show that for $A$ a DVR and $M\in \mods{A}^\fg$, then $M$ torsionfree implies $M$ is free.

> Hint: torsionfree $\implies$ flat $\implies$ free for finitely-generated Noetherian local rings.

:::

:::{.exercise title="?"}
Show that if $M \in \mods{A}^\fg$ for $A$ a DVR then $M \cong M_\tors \oplus A^n$ for some $n$.

> Hint: use that the SES involving $M_\tors \to M$ splits.

:::

:::{.exercise title="?"}
Show that any finitely-generated torsion module over $A$ is isomorphic to $\bigoplus_i A/\pi^{n_i} A$.
Use this to classify all finitely-generated modules over $k\fps{t}$.
:::