# The Stacks Project

## Tag 02P6

Definition 41.2.1. Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $\kappa$. Let $M$ be a finite length $R$-module. Say $l = \text{length}_R(M)$.

1. Given elements $x_1, \ldots, x_r \in M$ we denote $\langle x_1, \ldots, x_r \rangle = Rx_1 + \ldots + Rx_r$ the $R$-submodule of $M$ generated by $x_1, \ldots, x_r$.
2. We will say an $l$-tuple of elements $(e_1, \ldots, e_l)$ of $M$ is admissible if $\mathfrak m e_i \subset \langle e_1, \ldots, e_{i - 1} \rangle$ for $i = 1, \ldots, l$.
3. A symbol $[e_1, \ldots, e_l]$ will mean $(e_1, \ldots, e_l)$ is an admissible $l$-tuple.
4. An admissible relation between symbols is one of the following:
1. if $(e_1, \ldots, e_l)$ is an admissible sequence and for some $1 \leq a \leq l$ we have $e_a \in \langle e_1, \ldots, e_{a - 1}\rangle$, then $[e_1, \ldots, e_l] = 0$,
2. if $(e_1, \ldots, e_l)$ is an admissible sequence and for some $1 \leq a \leq l$ we have $e_a = \lambda e'_a + x$ with $\lambda \in R^*$, and $x \in \langle e_1, \ldots, e_{a - 1}\rangle$, then $$[e_1, \ldots, e_l] = \overline{\lambda} [e_1, \ldots, e_{a - 1}, e'_a, e_{a + 1}, \ldots, e_l]$$ where $\overline{\lambda} \in \kappa^*$ is the image of $\lambda$ in the residue field, and
3. if $(e_1, \ldots, e_l)$ is an admissible sequence and $\mathfrak m e_a \subset \langle e_1, \ldots, e_{a - 2}\rangle$ then $$[e_1, \ldots, e_l] = - [e_1, \ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \ldots, e_l].$$
5. We define the determinant of the finite length $R$-module $M$ to be $$\det\nolimits_\kappa(M) = \left\{ \frac{\kappa\text{-vector space generated by symbols}} {\kappa\text{-linear combinations of admissible relations}} \right\}$$

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\begin{definition}
\label{definition-determinant}
Let $R$ be a local ring with maximal ideal $\mathfrak m$ and
residue field $\kappa$. Let $M$ be a finite length $R$-module.
Say $l = \text{length}_R(M)$.
\begin{enumerate}
\item Given elements $x_1, \ldots, x_r \in M$ we denote
$\langle x_1, \ldots, x_r \rangle = Rx_1 + \ldots + Rx_r$ the
$R$-submodule of $M$ generated by $x_1, \ldots, x_r$.
\item We will say an $l$-tuple of elements
$(e_1, \ldots, e_l)$ of $M$ is {\it admissible} if
$\mathfrak m e_i \subset \langle e_1, \ldots, e_{i - 1} \rangle$
for $i = 1, \ldots, l$.
\item A {\it symbol} $[e_1, \ldots, e_l]$ will mean
$(e_1, \ldots, e_l)$ is an admissible $l$-tuple.
\item An {\it admissible relation} between symbols is one of the following:
\begin{enumerate}
\item if $(e_1, \ldots, e_l)$ is an admissible sequence and
for some $1 \leq a \leq l$ we have
$e_a \in \langle e_1, \ldots, e_{a - 1}\rangle$, then
$[e_1, \ldots, e_l] = 0$,
\item if $(e_1, \ldots, e_l)$ is an admissible sequence and
for some $1 \leq a \leq l$ we have $e_a = \lambda e'_a + x$
with $\lambda \in R^*$, and
$x \in \langle e_1, \ldots, e_{a - 1}\rangle$, then
$$[e_1, \ldots, e_l] = \overline{\lambda} [e_1, \ldots, e_{a - 1}, e'_a, e_{a + 1}, \ldots, e_l]$$
where $\overline{\lambda} \in \kappa^*$ is the image of $\lambda$ in
the residue field, and
\item if $(e_1, \ldots, e_l)$ is an admissible sequence and
$\mathfrak m e_a \subset \langle e_1, \ldots, e_{a - 2}\rangle$ then
$$[e_1, \ldots, e_l] = - [e_1, \ldots, e_{a - 2}, e_a, e_{a - 1}, e_{a + 1}, \ldots, e_l].$$
\end{enumerate}
\item
We define the {\it determinant of the finite length $R$-module $M$} to be
$$\det\nolimits_\kappa(M) = \left\{ \frac{\kappa\text{-vector space generated by symbols}} {\kappa\text{-linear combinations of admissible relations}} \right\}$$
\end{enumerate}
\end{definition}

Comment #2609 by Ko Aoki on June 24, 2017 a 6:09 am UTC

Typo in the definition of an admissible tuple: "$\mathfrak m e_i \in \langle e_1, \ldots, e_{i - 1} \rangle$" should be replaced by "$\mathfrak m e_i \subset \langle e_1, \ldots, e_{i - 1} \rangle$".

Comment #2632 by Johan (site) on July 7, 2017 a 12:50 pm UTC

Thanks, fixed here.

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