index of lattice

Last updated January 27, 2022

Definition

Let $\mathcal{O}$ be a dedekind domain. Suppose $M$ and $N$ are two $\mathcal{O}$- lattice. Then the index $$(M:N)_\mathcal{O}$$ is a fractional ideal of $\mathcal{O}$ in $k$ (the field of fractions) defined as follows:

1. If both $M$ and $N$ are free of rank $n$, then $$(M:N)_\mathcal{O}=(\text{det }A),$$ (the ideal generated by the determinant of $A$) where $A$ is the matrix of coordinates of elements of the basis of $N$ with respect to the basis of $M$.
2. Generally, $$(M:N)\mathcal{O}\coloneqq \prod\mathfrak{p}\mathfrak{p}^{v_\mathfrak{p}((M_\mathfrak{p}:N_\mathfrak{p})_{\mathcal{O}_\mathfrak{p}})}$$

Properties

1. $(M:N)\mathcal{O}(N:T)\mathcal{O}=(M:T)_\mathcal{O}$
2. $(M:M)_\mathcal{O}=\mathcal{O}$
3. Suppose $N\subset M$. Then $(M:N)\mathcal{O}\subset\mathcal{O}$, with equality iff $M=N$. If $\mathcal{O}=\mathbb{Z}$, then $(M:N)\mathbb{Z}$ is generated by the usual index $(M:N)$.