Lemma 13.6.3. Let H : \mathcal{D} \to \mathcal{A} be a homological functor of a pre-triangulated category into an abelian category. Let \mathcal{D}' be the full subcategory of \mathcal{D} with objects
\mathop{\mathrm{Ob}}\nolimits (\mathcal{D}') = \{ X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) \mid H(X[n]) = 0\text{ for all }n \in \mathbf{Z}\}
Then \mathcal{D}' is a strictly full saturated pre-triangulated subcategory of \mathcal{D}. If \mathcal{D} is a triangulated category, then \mathcal{D}' is a triangulated subcategory.
Proof.
It is clear that \mathcal{D}' is preserved under [1] and [-1]. If (X, Y, Z, f, g, h) is a distinguished triangle of \mathcal{D} and H(X[n]) = H(Y[n]) = 0 for all n, then also H(Z[n]) = 0 for all n by the long exact sequence (13.3.5.1). Hence we may apply Lemma 13.4.16 to see that \mathcal{D}' is a pre-triangulated subcategory (respectively a triangulated subcategory if \mathcal{D} is a triangulated category). The assertion of being saturated follows from
\begin{align*} H((X \oplus Y)[n]) = 0 & \Rightarrow H(X[n] \oplus Y[n]) = 0 \\ & \Rightarrow H(X[n]) \oplus H(Y[n]) = 0 \\ & \Rightarrow H(X[n]) = H(Y[n]) = 0 \end{align*}
for all n \in \mathbf{Z}.
\square
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