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Tag 02QT

Chapter 41: Chow Homology and Chern Classes > Section 41.10: Cycle associated to a closed subscheme

Lemma 41.10.1. Let $(S, \delta)$ be as in Situation 41.8.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme.

  1. Let $Z' \subset Z$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then $$ \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{O}_{Z, \xi} < \infty $$
  2. If $\dim_\delta(Z) \leq k$ and $\xi \in Z$ with $\delta(\xi) = k$, then $\xi$ is a generic point of an irreducible component of $Z$.

Proof. Let $Z' \subset Z$, $\xi \in Z'$ be as in (1). Then $\dim(\mathcal{O}_{Z, \xi}) = 0$ (for example by Properties, Lemma 27.10.3). Hence $\mathcal{O}_{Z, \xi}$ is Noetherian local ring of dimension zero, and hence has finite length over itself (see Algebra, Proposition 10.59.6). Hence, it also has finite length over $\mathcal{O}_{X, \xi}$, see Algebra, Lemma 10.51.12.

Assume $\xi \in Z$ and $\delta(\xi) = k$. Consider the closure $Z' = \overline{\{\xi\}}$. It is an irreducible closed subscheme with $\dim_\delta(Z') = k$ by definition. Since $\dim_\delta(Z) = k$ it must be an irreducible component of $Z$. Hence we see (2) holds. $\square$

    The code snippet corresponding to this tag is a part of the file chow.tex and is located in lines 3212–3228 (see updates for more information).

    \begin{lemma}
    \label{lemma-multiplicity-finite}
    Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
    Let $X$ be locally of finite type over $S$.
    Let $Z \subset X$ be a closed subscheme.
    \begin{enumerate}
    \item Let $Z' \subset Z$ be an irreducible component and
    let $\xi \in Z'$ be its generic point.
    Then
    $$
    \text{length}_{\mathcal{O}_{X, \xi}} \mathcal{O}_{Z, \xi} < \infty
    $$
    \item If $\dim_\delta(Z) \leq k$ and $\xi \in Z$ with
    $\delta(\xi) = k$, then $\xi$ is a generic point of an
    irreducible component of $Z$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Let $Z' \subset Z$, $\xi \in Z'$ be as in (1).
    Then $\dim(\mathcal{O}_{Z, \xi}) = 0$ (for example by
    Properties, Lemma \ref{properties-lemma-codimension-local-ring}).
    Hence $\mathcal{O}_{Z, \xi}$ is Noetherian
    local ring of dimension zero, and hence has finite length over
    itself (see
    Algebra, Proposition \ref{algebra-proposition-dimension-zero-ring}).
    Hence, it also has finite length over $\mathcal{O}_{X, \xi}$, see
    Algebra, Lemma \ref{algebra-lemma-pushdown-module}.
    
    \medskip\noindent
    Assume $\xi \in Z$ and $\delta(\xi) = k$.
    Consider the closure $Z' = \overline{\{\xi\}}$. It is an irreducible
    closed subscheme with $\dim_\delta(Z') = k$ by definition.
    Since $\dim_\delta(Z) = k$ it must be an irreducible component
    of $Z$. Hence we see (2) holds.
    \end{proof}

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