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Tag 011K

Chapter 12: Homological Algebra > Section 12.13: Homotopy and the shift functor

Lemma 12.13.11. Notation and assumptions as in Lemma 12.13.10 above. Assume in addition that $\mathcal{A}$ is abelian. The morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$ induces the maps $$ H^i(\delta) : H^i(C^\bullet) \longrightarrow H^i(A[1]^\bullet) = H^{i + 1}(A^\bullet) $$ which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma 12.12.12.

Proof. Omitted. $\square$

    The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 3127–3141 (see updates for more information).

    \begin{lemma}
    \label{lemma-ses-termwise-split-long-cochain}
    Notation and assumptions as in
    Lemma \ref{lemma-ses-termwise-split-cochain} above.
    Assume in addition that $\mathcal{A}$ is abelian.
    The morphism of complexes $\delta : C^\bullet \to A[1]^\bullet$
    induces the maps
    $$
    H^i(\delta) :
    H^i(C^\bullet) \longrightarrow H^i(A[1]^\bullet) = H^{i + 1}(A^\bullet)
    $$
    which occur in the long exact homology sequence associated
    to the short exact sequence of cochain complexes by
    Lemma \ref{lemma-long-exact-sequence-cochain}.
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

    Comments (1)

    Comment #2795 by Tim on September 3, 2017 a 5:13 pm UTC

    Is there a reference, where the proof is carried out in detail?

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