Last modified date: <%+ tp.file.last_modified_date() %>

- Tags: - #todo/untagged - Refs: - Summary of adelic class groups: https://www2.math.upenn.edu/~yeya/class_group.pdf#page=1 #resources/summaries - Links: - #todo/create-links

Picard group

Adelic class groups

The algebraic class group \(\operatorname{Cl} (\operatorname{GL}_n(K))\) is a moduli of lattices in \(K^n\).

ideal class group

For Noetherian domains

Write \({ \ddot{\operatorname{Id}} }(R)\) for the monoid of fractional ideals.

• For a Noetherian domain \(R\): the ideal group \({ \ddot{\operatorname{Id}} }^{\times}(R)\) is the abelian group of invertible fractional ideals. The elements are not necessarily ideals. - Uses that \({ \ddot{\operatorname{Id}} }(R)\) is commutative and associative under multiplication with unit \(R = \left\langle{1}\right\rangle\), so the subset of invertible elements forms a commutative group.
• Every principal fractional ideal is invertible using \(\left\langle{x}\right\rangle^{-1}= \left\langle{x^{-1}}\right\rangle\), and they are closed under multiplication since \(\left\langle{x}\right\rangle\left\langle{y}\right\rangle = \left\langle{xy}\right\rangle\), so the principal fractional ideals form a subgroup \(\mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }(R) \leq { \ddot{\operatorname{Id}} }^{\times}(R)\).
  • The ideal class group or Picard group of \(R\) is the quotient \({ \operatorname{cl}} (R) \coloneqq\operatorname{Pic}(R) \coloneqq{ \ddot{\operatorname{Id}} }^{\times}(R)/\mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }(R)\).

For orders in number fields

• For \({\mathcal{O}}\in{ \mathrm{Ord} }(K)\), let \({ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})\) be the invertible fractional ideals and \(\mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}}) \subseteq { \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})\) be the principal fractional ideals of \({\mathcal{O}}\). The quotient \(\operatorname{Cl} ({\mathcal{O}}) \coloneqq{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}}) / \mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})\) is the ideal class group of the order \({\mathcal{O}}\).
• There is a SES in ${}_{G_K}{\mathsf{Mod}} $: \begin{align*}1 \to \mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}}) \to { \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}}) \to \operatorname{Cl} ({\mathcal{O}}) \to 1,\end{align*} and the corresponding LES starts \begin{align*} 1 \to \mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})^G \to { \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})^G\to \operatorname{Cl} ({\mathcal{O}})^G \to H^1(\mathop{\mathrm{Prin}}{ \ddot{\operatorname{Id}} }^{\times}({\mathcal{O}})) \to 1 .\end{align*}

Todo: not sure if this is actually for orders, or just for class groups of rings as in the previous section.

Exercises

• Show that a DVR \(R\) with a uniformizer has \({ \operatorname{cl}} (R) \cong {\mathbf{Z}}\).
• Show that a Dedekind domain \(R\) is a UFD iff \({ \operatorname{cl}} (R) = 1\).
  • Show that if \(R\) is an integrally closed Noetherian domain then \({ \operatorname{cl}} (R) = 1\) when \(R\) is a UFD.
  • Show that the converse holds if \({ \operatorname{cl}} (R)\) is replaced with the divisor class group.
  • Show that the ideal class group and divisor class group coincide for DVRs