Lemma 10.121.4. Let R be a Noetherian local domain of dimension 1 with fraction field K. Let V be a finite dimensional K-vector space.
If M is a lattice in V and M \subset M' \subset V is an R-submodule of V containing M then the following are equivalent
M' is a lattice,
\text{length}_ R(M'/M) is finite, and
M' is finitely generated.
If M is a lattice in V and M' \subset M is an R-submodule of M then M' is a lattice if and only if \text{length}_ R(M/M') is finite.
If M, M' are lattices in V, then so are M \cap M' and M + M'.
If M \subset M' \subset M'' \subset V are lattices in V then
\text{length}_ R(M''/M) = \text{length}_ R(M'/M) + \text{length}_ R(M''/M').If M, M', N, N' are lattices in V and N \subset M \cap M', M + M' \subset N', then we have
\begin{eqnarray*} & & \text{length}_ R(M/M \cap M') - \text{length}_ R(M'/M \cap M')\\ & = & \text{length}_ R(M/N) - \text{length}_ R(M'/N) \\ & = & \text{length}_ R(M + M' / M') - \text{length}_ R(M + M'/M) \\ & = & \text{length}_ R(N' / M') - \text{length}_ R(N'/M) \end{eqnarray*}
Comments (0)
There are also: