Discrete subgroup

Lattice subgroup

A lattice of a locally compact Hausdorff abelian topological group is a discrete subgroup such that the quotient is compact. #m/def/group The above definition generalizes and is motivated by the case where , where we define a Rational lattice.

Classical lattice

A classical lattice is a lattice in the topological vector space where or , and is called complete iff .

Let be an -dimensional space vector space over . Let be a -submodule spanning . The following are equivalent1 #m/thm/topology

  1. is a complete lattice subgroup of ;
  2. is generated by elements;
  3. in .
Proof

Suppose is discrete.

Then is closed. For if is an isolating neighbourhood of 0, then

is an open neighbourhood of such that and the difference of any elements of lies in . If there were an such that , then there would be a two distinct elements such that , so is not isolated in , a contradiction.

Now let be a -basis of , and let . We will show that the Lagrange index is finite. Let for be a complete system of representatives for each coset. Letting

(this is an abuse of notation but the meaning is clear) we have

where

lie discretely in the bounded set . Since is compact and discrete, and thus finite, it follows is finite, so the are finite and thus is finite.

It follows , whence

implying possesses a -basis of length less than .

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 1999. Algebraic number theory, ¶I.4.2, p. 25 ↩