Lattice

Last updated January 27, 2022

Definition

Suppose $\mathcal{O}$ is an integral domain, $k$ its field of fractions, and $K/k$ a finite separable extension.

Definition. An $\mathcal{O}$-lattice in $K$ is a finitely generated $\mathcal{O}$-submodule of $K$ which contains a basis of the $k$-vector space $K$.

Properties

1. Suppose $N\subset K$ is an $\mathcal{O}$-lattice, $\mathfrak{p}\subset\mathcal{O}$ a prime ideal.
   1. If $\mathcal{O}$ is a PID then $N$ is a free $\mathcal{O}$-module of rank $n$.
   2. If $N$ is an $\mathcal{O}$-lattice, then $N_\mathfrak{p}\coloneqq N\mathcal{O}_p$ is an $\mathcal{O}_\mathfrak{p}$-lattice.