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Tag 02PI

Chapter 41: Chow Homology and Chern Classes > Section 41.3: Periodic complexes and Herbrand quotients

Lemma 41.3.3. Let $R$ be a local ring.

  1. If $(M, N, \varphi, \psi)$ is a $2$-periodic complex such that $M$, $N$ have finite length. Then $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
  2. If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex such that $M$ has finite length. Then $e_R(M, \varphi, \psi) = 0$.
  3. Suppose that we have a short exact sequence of $2$-periodic complexes $$ 0 \to (M_1, N_1, \varphi_1, \psi_1) \to (M_2, N_2, \varphi_2, \psi_2) \to (M_3, N_3, \varphi_3, \psi_3) \to 0 $$ If two out of three have cohomology modules of finite length so does the third and we have $$ e_R(M_2, N_2, \varphi_2, \psi_2) = e_R(M_1, N_1, \varphi_1, \psi_1) + e_R(M_3, N_3, \varphi_3, \psi_3). $$

Proof. Proof of (3). Abbreviate $A = (M_1, N_1, \varphi_1, \psi_1)$, $B = (M_2, N_2, \varphi_2, \psi_2)$ and $C = (M_3, N_3, \varphi_3, \psi_3)$. We have a long exact cohomology sequence $$ \ldots \to H^1(C) \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to H^1(C) \to \ldots $$ This gives a finite exact sequence $$ 0 \to I \to H^0(A) \to H^0(B) \to H^0(C) \to H^1(A) \to H^1(B) \to K \to 0 $$ with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of the length function (Algebra, Lemma 10.51.3) we see the result. The proofs of (1) and (2) are omitted. $\square$

    The code snippet corresponding to this tag is a part of the file chow.tex and is located in lines 1048–1074 (see updates for more information).

    \begin{lemma}
    \label{lemma-periodic-length}
    Let $R$ be a local ring.
    \begin{enumerate}
    \item If $(M, N, \varphi, \psi)$ is a $2$-periodic complex
    such that $M$, $N$ have finite length. Then
    $e_R(M, N, \varphi, \psi) = \text{length}_R(M) - \text{length}_R(N)$.
    \item If $(M, \varphi, \psi)$ is a $(2, 1)$-periodic complex
    such that $M$ has finite length. Then
    $e_R(M, \varphi, \psi) = 0$.
    \item Suppose that we have a short exact sequence of
    $2$-periodic complexes
    $$
    0 \to (M_1, N_1, \varphi_1, \psi_1)
    \to (M_2, N_2, \varphi_2, \psi_2)
    \to (M_3, N_3, \varphi_3, \psi_3)
    \to 0
    $$
    If two out of three have cohomology modules of finite length so does
    the third and we have
    $$
    e_R(M_2, N_2, \varphi_2, \psi_2) =
    e_R(M_1, N_1, \varphi_1, \psi_1) +
    e_R(M_3, N_3, \varphi_3, \psi_3).
    $$
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Proof of (3). Abbreviate $A = (M_1, N_1, \varphi_1, \psi_1)$,
    $B = (M_2, N_2, \varphi_2, \psi_2)$ and $C = (M_3, N_3, \varphi_3, \psi_3)$.
    We have a long exact cohomology sequence
    $$
    \ldots
    \to H^1(C)
    \to H^0(A)
    \to H^0(B)
    \to H^0(C)
    \to H^1(A)
    \to H^1(B)
    \to H^1(C)
    \to \ldots
    $$
    This gives a finite exact sequence
    $$
    0 \to I
    \to H^0(A)
    \to H^0(B)
    \to H^0(C)
    \to H^1(A)
    \to H^1(B)
    \to K \to 0
    $$
    with $0 \to K \to H^1(C) \to I \to 0$ a filtration. By additivity of
    the length function (Algebra, Lemma \ref{algebra-lemma-length-additive})
    we see the result.
    The proofs of (1) and (2) are omitted.
    \end{proof}

    Comments (2)

    Comment #2608 by Ko Aoki on June 24, 2017 a 6:02 am UTC

    Typo in the statement of (3): "$(2,1)$-periodic complexes" should be replaced by "$2$-periodic complexes."

    Comment #2631 by Johan (site) on July 7, 2017 a 12:48 pm UTC

    Thanks, fixed here.

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