The behavior of the base ring
R of a lattice
M strongly influences the behavior of
M. If
R is a
Dedekind domain,
M is completely decomposable (with respect to a suitable basis) as a direct sum of fractional ideals. Every lattice over a Dedekind domain is
projective. Lattices are well-behaved under
localization and
completion: A lattice
M is equal to the intersection of all the localizations M_{(\mathfrak{p})} of
M at \mathfrak{p}. Further, two lattices are equal if and only if their localizations are equal at all primes. Over a
Dedekind domain, the local-global-dictionary is even more robust: any two full
R-lattices are equal all all but finitely many localizations, and for any choice of R_{(\mathfrak{p})}-lattices N_{(\mathfrak{p})} there exists an
R-lattice
M satisfying M_{(\mathfrak{p})} = N_{(\mathfrak{p})}. Over Dedekind domains a similar correspondence exists between
R-lattices and collections of lattices N_\mathfrak{p} over the completions of
R with respect at primes \mathfrak{p}. A pair of lattices
M and
N over
R admit a notion of
relative index analogous to that of integer lattices in \mathbb{R}^n. If
M and
N are projective (e.g. if
R is a Dedekind domain), then
M and
N have trivial relative index if and only if
M = N. == Pure sublattices ==