Full Euclidean lattice and covolume

Fix $n\ge 1$. A full lattice in ${\mathbb{R}}^n$ is a subgroup $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ of the form

$\mathrm{\Lambda }=\mathbb{Z}{b}_1\oplus \cdots \oplus \mathbb{Z}{b}_n=\{{\sum }_{i=1}^n{m}_i{b}_i:m_i\in \mathbb{Z}\},$

where $b_1,\dots ,b_n\in {\mathbb{R}}^n$ are linearly independent over $\mathbb{R}$, that is, they form a real basis of ${\mathbb{R}}^n$. Write $B=(b_1\cdots b_n)$ for the $n\times n$ matrix whose $i$-th column is $b_i$; determinants of square matrices are as in For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix. The covolume of $\mathrm{\Lambda }$ is

$\mathrm{covol}(\mathrm{\Lambda }):=\mathrm{\mid }\det B\mathrm{\mid }\mathrm{.}$

The value is positive: linear independence of the columns makes $B$ invertible, so $\det B\mathrm{\ne }0$.

Basis independence. Suppose $c_1,\dots ,c_n$ is a second integer basis of $\mathrm{\Lambda }$, with matrix $C=(c_1\cdots c_n)$. Each $c_j$ is an integer combination of the $b_i$ and each $b_i$ is an integer combination of the $c_j$, so there are matrices $A,A^{\mathrm{\prime }}\in M_n(\mathbb{Z})$ with $C=BA$ and $B=C{A}^{\mathrm{\prime }}$. Substituting gives $B=BA{A}^{\mathrm{\prime }}$, hence $A{A}^{\mathrm{\prime }}=I_n$ because $B$ is invertible, and symmetrically $A^{\mathrm{\prime }}A=I_n$. Taking determinants, $(\det A)(\det A^{\mathrm{\prime }})=1$ with both factors integers, so $\det A=\pm 1$ and therefore $\mathrm{\mid }\det C\mathrm{\mid }=\mathrm{\mid }\det B\mathrm{\mid }\mathrm{\mid }\det A\mathrm{\mid }=\mathrm{\mid }\det B\mathrm{\mid }$. Thus the covolume does not depend on the chosen basis, and the definition above is unambiguous.

Discrete subgroups. A subgroup $\mathrm{\Lambda }\subseteq {\mathbb{R}}^n$ is a full lattice in the sense above exactly when it is discrete in the Euclidean topology and spans ${\mathbb{R}}^n$; this equivalence is Milne's Lemma 4.14 together with the identification of full lattices with discrete spanning subgroups (Milne, Ch. 4, pp.73-75). The half-open fundamental parallelotope $P=\{{\sum }_i{t}_i{b}_i:0\le t_i<1\}$ of a full lattice tiles ${\mathbb{R}}^n$ by $\mathrm{\Lambda }$-translates with volume $\mathrm{covol}(\mathrm{\Lambda })$, and bounded sets meet $\mathrm{\Lambda }$ in finitely many points; both facts are proved in this batch and used below.