## 12.18 Double complexes and associated total complexes

We discuss double complexes and associated total complexes.

Definition 12.18.1. Let $\mathcal{A}$ be an additive category. A *double complex* in $\mathcal{A}$ is given by a system $(\{ A^{p, q}, d_1^{p, q}, d_2^{p, q}\} _{p, q\in \mathbf{Z}})$, where each $A^{p, q}$ is an object of $\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$ such that the following rules hold:

$d_1^{p + 1, q} \circ d_1^{p, q} = 0$

$d_2^{p, q + 1} \circ d_2^{p, q} = 0$

$d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$

for all $p, q \in \mathbf{Z}$.

This is just the cochain version of the definition. It says that each $A^{p, \bullet }$ is a cochain complex and that each $d_1^{p, \bullet }$ is a morphism of complexes $A^{p, \bullet } \to A^{p + 1, \bullet }$ such that $d_1^{p + 1, \bullet } \circ d_1^{p, \bullet } = 0$ as morphisms of complexes. In other words a double complex can be seen as a complex of complexes. So in the diagram

\[ \xymatrix{ \ldots & \ldots & \ldots & \ldots \\ \ldots \ar[r] & A^{p, q + 1} \ar[r]^{d_1^{p, q + 1}} \ar[u] & A^{p + 1, q + 1} \ar[r] \ar[u] & \ldots \\ \ldots \ar[r] & A^{p, q} \ar[r]^{d_1^{p, q}} \ar[u]^{d_2^{p, q}} & A^{p + 1, q} \ar[r] \ar[u]_{d_2^{p + 1, q}} & \ldots \\ \ldots & \ldots \ar[u] & \ldots \ar[u] & \ldots } \]

any square commutes. Warning: In the literature one encounters a different definition where a “bicomplex” or a “double complex” has the property that the squares in the diagram anti-commute.

Example 12.18.2. Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be additive categories. Suppose that

\[ \otimes : \mathcal{A} \times \mathcal{B} \longrightarrow \mathcal{C}, \quad (X, Y) \longmapsto X \otimes Y \]

is a functor which is bilinear on morphisms, see Categories, Definition 4.2.20 for the definition of $\mathcal{A} \times \mathcal{B}$. Given complexes $X^\bullet $ of $\mathcal{A}$ and $Y^\bullet $ of $\mathcal{B}$ we obtain a double complex

\[ K^{\bullet , \bullet } = X^\bullet \otimes Y^\bullet \]

in $\mathcal{C}$. Here the first differential $K^{p, q} \to K^{p + 1, q}$ is the morphism $X^ p \otimes Y^ q \to X^{p + 1} \otimes Y^ q$ induced by the morphism $X^ p \to X^{p + 1}$ and the identity on $Y^ q$. Similarly for the second differential.

Definition 12.18.3. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet }$ be a double complex. The *associated simple complex*, denoted $sA^\bullet $, also often called the *associated total complex*, denoted $\text{Tot}(A^{\bullet , \bullet })$, is given by

\[ sA^ n = \text{Tot}^ n(A^{\bullet , \bullet }) = \bigoplus \nolimits _{n = p + q} A^{p, q} \]

(if it exists) with differential

\[ d_{sA^\bullet }^ n = d_{\text{Tot}(A^{\bullet , \bullet })}^ n = \sum \nolimits _{n = p + q} (d_1^{p, q} + (-1)^ p d_2^{p, q}) \]

If countable direct sums exist in $\mathcal{A}$ or if for each $n$ at most finitely many $A^{p, n - p}$ are nonzero, then $\text{Tot}(A^{\bullet , \bullet })$ exists. Note that the definition is *not* symmetric in the indices $(p, q)$.

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