nLab ramification of ideals




Ramification of ideals through ring/algebra homomorphisms is the dual incarnation of branch points of branched covering spaces.


Given a ring injection i:BAi \colon B \to A (for instance the inclusion of 𝒪 K\mathcal{O}_K in 𝒪 L\mathcal{O}_L, where KLK \to L is a field extension and 𝒪 K\mathcal{O}_K (resp. 𝒪 L\mathcal{O}_L) is the ring of integers of KK (resp. LL)), then a prime ideal JBJ \subset B is said to be ramified in AA if ι(J)AA\iota(J) A\subset A is not a prime ideal anymore.

Here ι(J)A\iota(J)A will be a product of powers of prime ideals of AA, and the ramification index of JJ at a prime ideal of AA is the power with which this appears.



  • Noah Snyder, section 1.5.2 of Artin L-Functions: A Historical Approach, 2002 (pdf)

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