# Tuesday, March 22

## Classification of finitely-generated modules over a Dedekind domain

:::{.definition title="Dedekind domain"}
A ring $A$ is a **Dedekind domain** iff

- $A\in\Noeth\Domain, \krulldim A = 1$,
- The local rings $A_p$ are DVRs for all $p\in \mspec A$.

:::

:::{.theorem title="Structure theorem for Dedekind domains"}
If $M\in \amod, A\in \Dedekind\Domain$,
then 
\[
M\isoas{\amod} T \oplus F \da \qty{ \bigoplus (A/\mfp_i)^{n_i} } \oplus \qty{ \bigoplus \mcl_i }
\]
where $\mcl_i \in \amod^{\locfree, \rank=1}$ (and are in particular torsionfree) and the $A/\mfp_i$ are torsion.
:::

:::{.exercise title="?"}
Show that if $p\in \spec A$ then $M_p \cong A_p^m$ where $m = \rank M$.

> Hint: use that $M_p \oplus \ff(A_p) = \ff(A)^n = \ff(A)^n$

:::

:::{.exercise title="?"}
Show that for $A$ a Dedekind domain and $M\in \amod^\fg$, TFAE:

- $M\in \amod^\flat$
- $M\in \amod(\proj)$
- $M\in \amod^{\locfree}$
- $M$ is torsionfree

Conclude that the following SES splits:
\[
0\to M_\tors \to M \to M_{\tors\free} \to 0
.\]


> Hint: the first three already hold for Noetherian domains.
> To show flat $\implies$ torsionfree, that $0\to A \mapsvia{\cdot a} A \to A/\gens{a} \to 0$ for $a\in \Tors(M)$ and tensor with $M$.
> For the converse, show $M_p$ is flat for all $p$ and that $A_p$ is a DVR -- if ${a\over s'}{m\over s} = 0 \implies \tilde s' s am = 0$ for some $\tilde s$, then $m$ is torsion.

:::

:::{.remark}
Note that torsionfree $\implies$ flat fails for most rings!
:::

:::{.exercise title="?"}
Show that if $M\in \amod^\fg$ is torsion over a Dedekind domain $A$ then $M \cong \bigoplus _{i} (A/p_i^{n_i})^{m_i}$.

> Hint: use that $\supp M$ is finite to produce a map $M\to \bigoplus _i M_{p_i}$ and show it is locally an isomorphism since $(M_{p})_q = M_q$ when $q\neq p$.
> Also use that $M_p = \bigoplus _j A_p/p^{i_j} \cong \oplus_j A/p^{i_j}$.

:::

:::{.exercise title="?"}
Show that for $M\in \amod^\fg$ torsionfree over $A$ a Dedekind domain, there is a (not necessarily unique) decomposition $M \cong \bigoplus _i \mcl_i$ with $\mcl_i$ locally free of rank 1.

> Induct on rank, where it's ETS there exists an $\mcl$ where $M/\mcl_i$ is torsionfree since $0\to \mcl \to M\to M/\mcl_i\to 0$ splits (tensor to fraction field).
> To find such an $\mcl$, take any $m\neq 0$ and take $\mcl$ to be the preimage of $(M/\gens{m})_\tors$ under $M \to M/\gens{m}$; then $\mcm/\mcl$ will be torsionfree and is rank 1 since $\mcl \tensor \ff(A) \cong \gens{m}\tensor \ff(A)$.

:::

## Classification of locally free rank 1 modules over a Dedekind domain

:::{.exercise title="?"}
Show that if $I\normal A$ is nonzero for $A$ a Dedekind domain, then $I$ is locally free of rank 1.

> Hint: show that $I_p\leq A_p$ is nonzero, so $I_p = \gens{\pi^n}$ for $\pi$ a uniformizer of $A_p$.

:::

:::{.definition title="Invertible modules"}
A module $M\in \amod$ is **invertible** iff $M$ is locally free of rank 1.
Equivalently, defining $M\dual \da \amod(M, A)$ there is an evaluation isomorphism $M\tensor_A M\dual \iso A$.
Define \[
\Pic(A) \da (\amod, \tensor_A)/\cong
.\]
Note that for $A = \OO_K, K\in \NF$, 
\[
\Pic(A) \cong \Cl(K)
.\]
:::

:::{.definition title="(Weil) Divisors"}
For $A$ a Dedekind domain, a **divisor** is a formal linear combination \[
D = \sum_{\mfp\neq 0 \in \spec A} n_p \mfp \in \Free_{\zmod}(\spec A)
.\]such that $n_p = 0$ for almost all $p$.
These form a group $\Div(A)$.
A divisor $D$ is **effective** iff $n_p\geq 0$ for all $p$, yielding a submonoid $\Div^+(A)$ of effective divisors.
:::

:::{.remark}
More generally, this will be a sum over height 1 primes.
:::

:::{.exercise title="?"}
Show that there is a natural bijection
\[
\Div^+(A) &\mapstofrom \Id(A)\smz \\
\sum n_p p &\mapstofrom \prod p^{n_p}
.\]

> Hint: if $\krulldim A = 1$, use that $\Id(A)\cong \ts{I_p \normal A_p}_{p\in P}$ where $I_p = A_p$ for almost all $p$, and each $I_p \cong \gens{\pi_p^{n_p}}$.

:::

:::{.exercise title="?"}
Show that if $\mcl$ is invertible then $\mcl\dual$ is invertible.

> Hint: show $\mcl\dual$ is locally free of rank 1, using $\mcl_p\dual \iso \amod(\mcl, A)_p \iso \mods{A_p}(\mcl_p, A_p)$ by post-composing with localization.
> Then check $\mcl\tensor \amod(\mcl, A)\to A$ where $(n, f) \mapsto f(n)$ is an isomorphism by checking locally where everything is free.

:::

:::{.proposition title="Picard group SES"}
Write $D\in \Div(A)$ as $D = D^+ - D^-$ where $D^{\pm} \in \Div^+(A)$ are effective.
There is a SES

\begin{tikzcd}
	0 && {\ker f} && {\Div(A)} && {\Pic(A)} && 0 \\
	&&&& {D^+ - D^-} && {\qty{\prod_{p} p^{n_p^+} } \tensor \qty{ \prod_p p^{n_p^-}}\dual}
	\arrow["f", two heads, from=1-5, to=1-7]
	\arrow[maps to, from=2-5, to=2-7]
	\arrow[from=1-7, to=1-9]
	\arrow[from=1-3, to=1-5]
	\arrow[from=1-1, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbNCwwLCJcXERpdihBKSJdLFs2LDAsIlxcUGljKEEpIl0sWzQsMSwiRF4rIC0gRF4tIl0sWzYsMSwiXFxxdHl7XFxwcm9kX3twfSBwXntuX3BeK30gfSBcXHRlbnNvciBcXHF0eXsgXFxwcm9kX3AgcF57bl9wXi19fVxcZHVhbCJdLFs4LDAsIjAiXSxbMiwwLCJcXGtlciBmIl0sWzAsMCwiMCJdLFswLDEsImYiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMiwzLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzEsNF0sWzUsMF0sWzYsNV1d)

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:::{.remark}
This is generally far from injective, and will instead biject with *fractional ideals*:
:::

:::{.definition title="Fractional ideals"}
For $A \in \Dedekind$, a **fractional ideal** is a nonzero $A\dash$submodule $\mcl\leq A$ where $\mcl\in \amod^\fg$. 
:::

:::{.example title="?"}
\envlist

- $I\normal A$ or $I\normal \ff(A)$ any ideal
- ${1\over a}I\normal \ff(A)$.

:::

:::{.exercise title="?"}
Show that any fractional ideal is invertible.

> Hint: use that any such $I$ is torsionfree since it's a subset of $\ff(A)$, and $\mcl \tensor_A \ff(A) \cong \ff(A)$ implies rank 1.

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