The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.120.4. 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 10.59.12 and Proposition 10.59.6 or combine Lemmas 10.118.9 and 10.51.4). In other words $\mathfrak m^ nM' \subset M$ for some $n$ Hence $\text{length}(M'/M) < \infty $ by Lemma 10.51.8, in other words (1)(b) holds. Assume (1)(b). Then $M'/M$ is a finite $R$-module (see Lemma 10.51.2). 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 10.120.3.

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 10.51.4). 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 10.51.3) and the exact sequence

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

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


Comments (0)

There are also:

  • 2 comment(s) on Section 10.120: Orders of vanishing

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02MF. Beware of the difference between the letter 'O' and the digit '0'.