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Tag 02P6

Chapter 41: Chow Homology and Chern Classes > Section 41.2: Determinants of finite length modules

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}

    Comments (2)

    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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