The Stacks project

Definition 42.68.2. 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

   \[ \det \nolimits _\kappa (M) = \left\{ \frac{\kappa \text{-vector space generated by symbols}}{\kappa \text{-linear combinations of admissible relations}} \right\} \]