## 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 theassociated total complexis 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 5411–5427 (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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