Infinite places

Conductor of a ring

Idea: for \(A\leq_{\mathsf{Ring}} B\) with \(B\) integral over \(A\), the conductor \({\mathfrak{c}}(B_{/ {A}} )\) is the largest ideal \(I \in \operatorname{Id}(A) \cap\operatorname{Id}(B)\). Equivalently, \({\mathfrak{c}}(B_{/ {A}} ) = \operatorname{Ann}_{ {}_{A}{\mathsf{Mod}} }(B/A)\).

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Conductor of an order

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Conductor of an extension

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Conductor of an abelian extension

attachments/Pasted%20image%2020220126165905.png # Conductor of an elliptic curve

#todo

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order

An order \({\mathcal{O}}\) is a Noetherian integral domain of dimension one with nonzero conductor. Equivalently, \begin{align*} \operatorname{cl}^{\mathrm{int}} {\mathcal{O}}\in {}_{{\mathcal{O}}}{\mathsf{Mod}} ^{\mathrm{fg}}.\end{align*}