The Stacks project

Remark 12.18.4. Let $\mathcal{A}$ be an additive category. Let $A^{\bullet , \bullet , \bullet }$ be a triple complex. The associated total complex is the complex with terms

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

and differential

\[ d^ n_{\text{Tot}(A^{\bullet , \bullet , \bullet })} = \sum \nolimits _{p + q + r = n} d_1^{p, q, r} + (-1)^ pd_2^{p, q, r} + (-1)^{p + q}d_3^{p, q, r} \]

With this definition a simple calculation shows that the associated total complex is equal to

\[ \text{Tot}(A^{\bullet , \bullet , \bullet }) = \text{Tot}(\text{Tot}_{12}(A^{\bullet , \bullet , \bullet })) = \text{Tot}(\text{Tot}_{23}(A^{\bullet , \bullet , \bullet })) \]

In other words, we can either first combine the first two of the variables and then combine sum of those with the last, or we can first combine the last two variables and then combine the first with the sum of the last two.

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