The Stacks project

Remark 13.12.4. Let $\mathcal{A}$ be an abelian category. Let $K^\bullet $ be a complex of $\mathcal{A}$. Let $a \in \mathbf{Z}$. We claim there is a canonical distinguished triangle

\[ \tau _{\leq a}K^\bullet \to K^\bullet \to \tau _{\geq a + 1}K^\bullet \to (\tau _{\leq a}K^\bullet )[1] \]

in $D(\mathcal{A})$. Here we have used the canonical truncation functors $\tau $ from Homology, Section 12.14. Namely, we first take the distinguished triangle associated by our $\delta $-functor (Lemma 13.12.1) to the short exact sequence of complexes

\[ 0 \to \tau _{\leq a}K^\bullet \to K^\bullet \to K^\bullet /\tau _{\leq a}K^\bullet \to 0 \]

Next, we use that the map $K^\bullet \to \tau _{\geq a + 1}K^\bullet $ factors through a quasi-isomorphism $K^\bullet /\tau _{\leq a}K^\bullet \to \tau _{\geq a + 1}K^\bullet $ by the description of cohomology groups in Homology, Section 12.14. In a similar way we obtain canonical distinguished triangles

\[ \tau _{\leq a}K^\bullet \to \tau _{\leq a + 1}K^\bullet \to H^{a + 1}(K^\bullet )[-a-1] \to (\tau _{\leq a}K^\bullet )[1] \]

and

\[ H^ a(K^\bullet )[-a] \to \tau _{\geq a}K^\bullet \to \tau _{\geq a + 1}K^\bullet \to H^ a(K^\bullet )[-a + 1] \]

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