The Stacks project

Remark 57.3.3. Let $X^0 \to X^1 \to X^2 \to X^0[1]$ and $Y^0 \to Y^1 \to Y^2 \to Y^0[1]$ be distinguished triangles in a triangulated category. For $p \in \mathbf{Z}$ write $p = 3n + i$ with $i \in \{ 0, 1, 2\} $ and set we set $X^ p = X^ i[n]$. Simlarly for $Y^ q$. Consider the double complex with terms

The differential $d_1 : K^{p, q} \to K^{p + 1, q}$ is given by the map $X^{-p - 1} \to X^{-p}$ (equal to the corresponding map in the first distinguished triangle up the a shift) and the differential $d_2 : K^{p, q} \to K^{p, q + 1}$ likewise by the map $Y^ q \to Y^{q + 1}$. From Derived Categories, Lemma 13.4.2 we see that the rows and columns of this double complex are exact. Furthermore, we see that $K^{p, q} = K^{p + 3, q - 3}$ and these equalities are compatible with the differentials. Finally, axiom TR3 implies one additional property: given $\alpha \in K^{p, q}$ and $\beta \in K^{p - 1, q + 1}$ such that $d_2 \alpha = d_1 \beta $, there exists a $\gamma \in K^{p - 2, q + 2}$ such that $d_1 \gamma = d_2 \beta $ in $K^{p - 1, q + 2}$ and $d_2 \gamma = d_1 \alpha $ in $K^{p - 2, q + 3} = K^{p + 1, q}$. (Hint: for $p = q = 0$ this is exactly the statement of TR3 and for other indices prove it by shifting.) A double complex with these properties is called a matress (see [Bondal-Kapranov]).