The Stacks project

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.

  1. 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

    1. $M'$ is a lattice,

    2. $\text{length}_ R(M'/M)$ is finite, and

    3. $M'$ is finitely generated.

  2. 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.

  3. If $M$, $M'$ are lattices in $V$, then so are $M \cap M'$ and $M + M'$.

  4. If $M \subset M' \subset M'' \subset V$ are lattices in $V$ then

    \[ \text{length}_ R(M''/M) = \text{length}_ R(M'/M) + \text{length}_ R(M''/M'). \]
  5. 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*}

Proof. Proof of (1). Assume (1)(a). Say $y_1, \ldots , y_ m$ generate $M'$. Then each $y_ i = x_ i/f_ i$ for some $x_ i \in M$ and nonzero $f_ i \in R$. Hence we see that $f_1 \ldots f_ m M' \subset M$. Since $R$ is Noetherian local of dimension $1$ we see that $\mathfrak m^ n \subset (f_1 \ldots f_ m)$ for some $n$ (for example combine Lemmas 10.60.13 and Proposition 10.60.7 or combine Lemmas 10.119.9 and 10.52.4). In other words $\mathfrak m^ nM' \subset M$ for some $n$ Hence $\text{length}(M'/M) < \infty $ by Lemma 10.52.8, in other words (1)(b) holds. Assume (1)(b). Then $M'/M$ is a finite $R$-module (see Lemma 10.52.2). Hence $M'$ is a finite $R$-module as an extension of finite $R$-modules. Hence (1)(c). The implication (1)(c) $\Rightarrow $ (1)(a) follows from the remark following Definition 10.121.3.

Proof of (2). Suppose $M$ is a lattice in $V$ and $M' \subset M$ is an $R$-submodule. We have seen in (1) that if $M'$ is a lattice, then $\text{length}_ R(M/M') < \infty $. Conversely, assume that $\text{length}_ R(M/M') < \infty $. Then $M'$ is finitely generated as $R$ is Noetherian and for some $n$ we have $\mathfrak m^ n M \subset M'$ (Lemma 10.52.4). Hence it follows that $M'$ contains a basis for $V$, and $M'$ is a lattice.

Proof of (3). Assume $M$, $M'$ are lattices in $V$. Since $R$ is Noetherian the submodule $M \cap M'$ of $M$ is finite. As $M$ is a lattice we can find $x_1, \ldots , x_ n \in M$ which form a $K$-basis for $V$. Because $M'$ is a lattice we can write $x_ i = y_ i/f_ i$ with $y_ i \in M'$ and $f_ i \in R$. Hence $f_ ix_ i \in M \cap M'$. Hence $M \cap M'$ is a lattice also. The fact that $M + M'$ is a lattice follows from part (1).

Part (4) follows from additivity of lengths (Lemma 10.52.3) and the exact sequence

\[ 0 \to M'/M \to M''/M \to M''/M' \to 0 \]

Part (5) follows from repeatedly applying part (4). $\square$

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