How to compute $\chi (M)$?

Question

$A$ is a Dedekind domain.

How to compute $\chi_A (M)$ in the above example? What's the series using?

Answer 1

Since $A$ is a Dedekind domain the fractional ideals are a group and remembering that $A=(1)$ we have that

$$\frac{\mathfrak{b}}{\mathfrak{a}}=\frac{A}{\mathfrak{a}\mathfrak{b}^{-1}}$$

This we take a maximal chain $$\mathfrak{a}=\mathfrak{a}_0\subseteq \mathfrak{a}_1 \subseteq \cdots \mathfrak{a}_n=\mathfrak{b}$$ and by definition

$$\chi(\mathfrak{b}/\mathfrak{a})=\prod \mathfrak{a}_i\mathfrak{a}_{i+1}^{-1}=\mathfrak{a}\mathfrak{b}^{-1}$$