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Tag 02M0

Chapter 10: Commutative Algebra > Section 10.51: Length

Lemma 10.51.12. Let $A$ be a local ring with maximal ideal $\mathfrak m$. Let $B$ be a semi-local ring with maximal ideals $\mathfrak m_i$, $i = 1, \ldots, n$. Suppose that $A \to B$ is a homomorphism such that each $\mathfrak m_i$ lies over $\mathfrak m$ and such that $$ [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)] < \infty. $$ Let $M$ be a $B$-module of finite length. Then $$ \text{length}_A(M) = \sum\nolimits_{i = 1, \ldots, n} [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)] \text{length}_{B_{\mathfrak m_i}}(M_{\mathfrak m_i}), $$ in particular $\text{length}_A(M) < \infty$.

Proof. Choose a maximal chain $$ 0 = M_0 \subset M_1 \subset M_2 \subset \ldots \subset M_n = M $$ by $B$-submodules as in Lemma 10.51.11. Then each quotient $M_i/M_{i - 1}$ is isomorphic to $\kappa(\mathfrak m_{j(i)})$ for some $j(i) \in \{1, \ldots, n\}$. Moreover $\text{length}_A(\kappa(\mathfrak m_i)) = [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)]$ by Lemma 10.51.6. The lemma follows by additivity of lengths (Lemma 10.51.3). $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 11967–11985 (see updates for more information).

    \begin{lemma}
    \label{lemma-pushdown-module}
    Let $A$ be a local ring with maximal ideal $\mathfrak m$.
    Let $B$ be a semi-local ring with maximal ideals $\mathfrak m_i$,
    $i = 1, \ldots, n$.
    Suppose that $A \to B$ is a homomorphism such that each $\mathfrak m_i$
    lies over $\mathfrak m$ and such that
    $$
    [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)] < \infty.
    $$
    Let $M$ be a $B$-module of finite length.
    Then
    $$
    \text{length}_A(M) = \sum\nolimits_{i = 1, \ldots, n}
    [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)]
    \text{length}_{B_{\mathfrak m_i}}(M_{\mathfrak m_i}),
    $$
    in particular $\text{length}_A(M) < \infty$.
    \end{lemma}
    
    \begin{proof}
    Choose a maximal chain
    $$
    0 = M_0
    \subset M_1
    \subset M_2
    \subset \ldots
    \subset M_n = M
    $$
    by $B$-submodules as in Lemma \ref{lemma-simple-pieces}.
    Then each quotient $M_i/M_{i - 1}$ is isomorphic to
    $\kappa(\mathfrak m_{j(i)})$ for some $j(i) \in \{1, \ldots, n\}$.
    Moreover
    $\text{length}_A(\kappa(\mathfrak m_i)) =
    [\kappa(\mathfrak m_i) : \kappa(\mathfrak m)]$ by
    Lemma \ref{lemma-dimension-is-length}. The lemma follows
    by additivity of lengths (Lemma \ref{lemma-length-additive}).
    \end{proof}

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