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Tag 012Z

Chapter 12: Homological Algebra > Section 12.22: Spectral sequences: double complexes

Definition 12.22.3. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet}$ be a double complex. The associated simple complex $sA^\bullet$, also sometimes called the associated total complex is given by $$ sA^n = \bigoplus\nolimits_{n = p + q} A^{p, q} $$ (if it exists) with differential $$ d_{sA}^n = \sum\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q}) $$ Alternatively, we sometimes write $\text{Tot}(A^{\bullet, \bullet})$ to denote this complex.

    The code snippet corresponding to this tag is a part of the file homology.tex and is located in lines 5543–5559 (see updates for more information).

    \begin{definition}
    \label{definition-associated-simple-complex}
    Let $\mathcal{A}$ be an additive category.
    Let $A^{\bullet, \bullet}$ be a double complex.
    The {\it associated simple complex $sA^\bullet$}, also
    sometimes called the {\it associated total complex} is
    given by
    $$
    sA^n = \bigoplus\nolimits_{n = p + q} A^{p, q}
    $$
    (if it exists) with differential
    $$
    d_{sA}^n = \sum\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q})
    $$
    Alternatively, we sometimes write $\text{Tot}(A^{\bullet, \bullet})$
    to denote this complex.
    \end{definition}

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