The Stacks project

Lemma 13.41.3. Let $\mathcal{D}$ be a triangulated category. Consider Postnikov systems for complexes of length $n$.

  1. For $n = 0$ Postnikov systems always exist and any morphism (13.41.1.1) of complexes extends to a unique morphism of Postnikov systems.

  2. For $n = 1$ Postnikov systems always exist and any morphism (13.41.1.1) of complexes extends to a (nonunique) morphism of Postnikov systems.

  3. For $n = 2$ Postnikov systems always exist but morphisms (13.41.1.1) of complexes in general do not extend to morphisms of Postnikov systems.

  4. For $n > 2$ Postnikov systems do not always exist.

Proof. The case $n = 0$ is immediate as isomorphisms are invertible. The case $n = 1$ follows immediately from TR1 (existence of triangles) and TR3 (extending morphisms to triangles). For the case $n = 2$ we argue as follows. Set $Y_0 = X_0$. By the case $n = 1$ we can choose a Postnikov system

\[ Y_1 \to X_1 \to Y_0 \to Y_1[1] \]

Since the composition $X_2 \to X_1 \to X_0$ is zero, we can factor $X_2 \to X_1$ (nonuniquely) as $X_2 \to Y_1 \to X_1$ by Lemma 13.4.2. Then we simply fit the morphism $X_2 \to Y_1$ into a distinguished triangle

\[ Y_2 \to X_2 \to Y_1 \to Y_2[1] \]

to get the Postnikov system for $n = 2$. For $n > 2$ we cannot argue similarly, as we do not know whether the composition $X_ n \to X_{n - 1} \to Y_{n - 1}$ is zero in $\mathcal{D}$. $\square$

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