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*}

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