different

Last updated January 27, 2022

Definition

Let $\mathcal{O}$ be a dedekind domain, $k$ its field of fractions and $K/k$ a finite separable extension. Let $\mathcal{O}K$ be the integral closure, and $D\mathcal{O}(\mathcal{O}_K)$ for the dual module.

The different, written $\mathfrak{D}$ or $\mathfrak{D}_{K/k}$, is an ideal of $\mathcal{O}K$ defined by the formula $$\mathfrak{D}\coloneqq (D\mathcal{O}(\mathcal{O}_K))^{-1}$$