nLab ramification of ideals

Idea

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

Definition

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

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

Examples

• Under the function field analogy ramifications of algebraic number fields as field extensions of the rational numbers are analogous to branched covers of the Riemann sphere in complex geometry.

Related concepts

• Kummer's theorem?

• unramified morphism

• formally unramified morphism

References

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