localization

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Localization

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attachments/Pasted%20image%2020220123193015.png

At a prime ideal:

attachments/Pasted%20image%2020220123193450.png attachments/Pasted%20image%2020220123193551.png

Localization for modules as base change: attachments/Pasted%20image%2020220123193747.png

When is \(M\) a submodule of one of its localizations? attachments/Pasted%20image%2020220123193853.png # Examples

p-adic integers attachments/Pasted%20image%2020220123185345.png

Valuations for Laurent series attachments/Pasted%20image%2020220123185457.png

Localizations in \(k[x]\), and function evaluation as a morphism onto the residue field: attachments/Pasted%20image%2020220123193648.png

🗓️ Timeline
  • 2021-04-25

    The most interesting commutative algebras are \(S/I\) where \(S = k[x_1, \cdots, x_n]\) for \(k\) a field, \({\mathbb{Z}}\), or the localization of a ring at a prime ideal.

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