Lemma 13.35.7. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{D} = D(\mathcal{A})$. Let $\mathcal{E} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be a subset which we view as a subset of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ also. Let $K$ be an object of $\mathcal{D}$.

Let $b \geq a$ and assume $H^ i(K)$ is zero for $i \not\in [a, b]$ and $H^ i(K) \in \mathcal{E}$ if $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$.

Let $b \geq a$ and assume $H^ i(K)$ is zero for $i \not\in [a, b]$ and $H^ i(K) \in smd(add(\mathcal{E}))$ if $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$.

Let $b \geq a$ and assume $K$ can be represented by a complex $K^\bullet $ with $K^ i = 0$ for $i \not\in [a, b]$ and $K^ i \in \mathcal{E}$ for $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$.

Let $b \geq a$ and assume $K$ can be represented by a complex $K^\bullet $ with $K^ i = 0$ for $i \not\in [a, b]$ and $K^ i \in smd(add(\mathcal{E}))$ for $i \in [a, b]$. Then $K$ is in $smd(add(\mathcal{E}[a, b])^{\star (b - a + 1)})$.

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