[[Discrete subgroup]]
# Lattice subgroup

A **lattice** $L$ of a [[Locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] [[Abelian group|abelian]] [[topological group]] $G$ is a [[discrete subgroup]] $L \leq G$ such that the quotient $G / L$ is [[Compact space|compact]]. #m/def/group
The above definition generalizes and is motivated by the case where $G = \mathbb{Q}^n$, where we define a [[Rational lattice]].

## Classical lattice

A **classical lattice** $L$ is a lattice in the [[topological vector space]] $\mathbb{K}^n$ where $\mathbb{K} = \mathbb{R}$ or $\mathbb{K} = \mathbb{Q}$,
and is called **complete** iff $\Span_{\mathbb{K}}L = \mathbb{K}^n$.

Let $V$ be an $n$-dimensional space vector space over $\mathbb{K}$.
Let $L \leq V$ be a $\mathbb{Z}$-[[submodule]] spanning $V$. The following are equivalent[^1999] #m/thm/topology 

1. $L$ is a complete lattice subgroup of $V$;
2. $L$ is generated by $n$ elements;
3. $L \cong \mathbb{Z}^n$ in $\lMod{\mathbb{Z}}$.

> [!check]- Proof
> Suppose $L \leq_{\mathbb{Z}} V$ is discrete.
> 
> Then $L$ is closed.
> For if $U$ is an isolating neighbourhood of 0, 
> then
> $$
> \begin{align*}
> U' = \bigcap _{x \in U} (x- (-))^{-1} U
> \end{align*}
> $$
> is an open neighbourhood of $0$ such that $U' \sube U$ and the difference of any elements of $U'$ lies in $U$.
> If there were an $x \notin L$ such that $x \in \Cl L$, then there would be a two distinct elements $l_{1},l_{2} \in x + U'$ such that $0 \neq l_{1}-l_{2} \in U' - U' \sube U$,
> so $0$ is not isolated in $U$, a contradiction.
> 
> Now let $\mathcal{B}= \{ u_{i} \}_{i=1}^n \sube L$ be a $\mathbb{K}$-basis of $V$, and let $L_{0} = \Span_{\mathbb{Z}} \mathcal{B} \leq_{\mathbb{Z}} L$.
> We will show that the [[Lagrange's theorem|Lagrange index]] $\abs{L / L_{0}}$ is finite.
> Let $[l_{i}] \in L / L_{0}$ for $i \in I$ be a complete system of representatives for each coset.
> Letting
> $$
> \begin{align*}
> \Phi_{0} = \Span_{[0,1)} \mathcal{B}
> \end{align*}
> $$
> (this is an abuse of notation but the meaning is clear)
> we have
> $$
> \begin{align*}
> l_{i} &= \mu_{i} + l_{0i}, & \mu_{i} &\in \Phi_{0}, & l_{0i} \in L_{0} \leq_{\mathbb{Z}} V
> \end{align*} 
> $$
> where
> $$
> \begin{align*}
> \mu_{i} = l_{i} - l_{0i} \in L
> \end{align*}
> $$
> lie discretely in the bounded set $\Phi_{0}$.
> Since $L \cap \Cl(\Phi_{0})$ is compact and discrete, and thus finite, it follows $L \cap \Phi_{0}$ is finite,
> so the $\mu_{i}$ are finite and thus $q = \abs{L / L_{0}}$ is finite.
> 
> It follows $q L \leq_{\mathbb{Z}} L_{0}$, whence
> $$
> \begin{align*}
> L \leq_{\mathbb{Z}} \frac{1}{q} L_{0} = \Span_{\mathbb{Z}} \left( \frac{1}{q} \mathcal{B} \right) 
> \end{align*}
> $$
> implying $L$ possesses a $\mathbb{Z}$-basis of length less than $n$.

  [^1999]: 1999\. [[Sources/@neukirchAlgebraicNumberTheory1999|Algebraic number theory]], ¶I.4.2, p. 25

## See also

- Not to be confused with [[Lattice order]]

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