Lemma 36.11.6. Let $X$ be a Noetherian scheme. Let $E$ in $D(\mathcal{O}_ X)$ be perfect. Then

$E$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

if $L$ is in $D_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D_{\textit{Coh}}(\mathcal{O}_ X)$,

if $L$ is in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$,

if $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$,

if $L$ is in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ then $E \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ and $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(E, L)$ are in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$.

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