Locally principal ideal of a quaternion order

Let $K$ be a number field with $R=\mathcal O_K$ , and let $\mathcal O$ be an order in a quaternion algebra $B/K$ . A right fractional $\mathcal O$ -ideal $I$ is locally principal if, for every nonzero prime $\mathfrak p$ of $R$ ,

I_{\mathfrak p}=a_{\mathfrak p}\mathcal O_{\mathfrak p} \quad\text{for some }a_{\mathfrak p}\in B^\times.

Here $I_{\mathfrak p}=R_{\mathfrak p}\otimes_RI$ and $\mathcal O_{\mathfrak p}=R_{\mathfrak p}\otimes_R\mathcal O$ use localization at the prime. For left ideals the generator is placed on the right.

Local versus global

A principal right ideal $a\mathcal O$ is locally principal. The converse can fail: the local generators need not arise from one global element of $B^\times$ . The ideal class set records that obstruction.

For a maximal quaternion order over a number ring, every full fractional one-sided ideal is locally principal. The fullness condition is necessary in split algebras.

References

John Voight, Quaternion Algebras, author's text, Definition 16.2.4 and Corollary 17.2.3.

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Locally principal ideal of a quaternion order

Basic properties